D28: Internal seiche mixing study - Hydromod
D28: Internal seiche mixing study - Hydromod
D28: Internal seiche mixing study - Hydromod
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Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
Work package No. 7: Mixing by internal <strong>seiche</strong>s<br />
Lead contractor: SOG<br />
Main objective: Key Processes<br />
Strategic leader: Malgorzata Loga-Karpinska (WuT)<br />
Responsible task leader: Claude GUILBAUD (SOG)<br />
Main contributor involved: Organisation E-Mail<br />
Claude GUILBAUD SOG claude.guilbaud@sogreah.fr<br />
Eckard HOLLAN ISF isf.eurolakes@lfula.lfu.bwl.de<br />
Bernd WAHL ISF isf.eurolakes@lfula.lfu.bwl.de<br />
Kurt DUWE HYD duwe@hydromod.de<br />
Ulrich LEMMIN EPF ulrich.lemmin@epfl.ch<br />
Lars UMLAUF EPF lars.umlauf@epfl.ch<br />
Maciej FILOCHA WUT maciej.Filocha@is.pw.edu.pl<br />
Dissemination level: Public<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 1 of 92
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
Table of contents<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 2 of 92<br />
1 Introduction 4<br />
1.1 GENERAL CONSIDERATIONS 4<br />
1.2 OBJECTIVES 5<br />
2 Lac du Bourget 6<br />
2.1 INTRODUCTION 6<br />
2.2 THE ENVIRONMENT OF LAC DU BOURGET 6<br />
2.3 MEASUREMENT ON THE LAKE 7<br />
2.3.1 The data 7<br />
2.4 APPLICATIONS 11<br />
2.4.1 TELEMAC System 11<br />
2.4.2 SIMULATED SEICHES – ANALYSIS 17<br />
2.5 CONCLUSIONS 20<br />
2.6 REFERENCES 20<br />
3 <strong>Internal</strong> <strong>seiche</strong>s and <strong>mixing</strong> in Lac Léman 21<br />
3.1 THE LAC LÉMAN ENVIRONMENT 21<br />
3.2 FIELD STUDIES 22<br />
3.2.1 Modes detected 22<br />
3.2.2 Wave propagation patterns 24<br />
3.2.3 Evidence of Poincaré waves 31<br />
3.2.4 Progressive vector analysis 31<br />
3.2.5 Mixing by internal <strong>seiche</strong>s 32<br />
3.3 NUMERICAL SIMULATIONS 35<br />
3.3.1 Model Description 35<br />
3.3.2 Mixing by Long <strong>Internal</strong> Waves - Model Results 36<br />
3.4 CONCLUSIONS 42<br />
3.5 REFERENCES 44<br />
4 Maps and tables of free internal <strong>seiche</strong>s in Upper Lake Constance for practical use 46<br />
4.1 OBJECTIVE 46<br />
4.2 OUTLINE OF THE CALCULATION 47<br />
4.3 AUXILIARY FORMULATIONS FOR EVALUATION OF EIGEN-PERIODS WITH RESPECT TO DIFFERENT<br />
STRATIFICATIONS 52<br />
4.4 THE EIGEN-PERIODS INCLUDING THE CORIOLIS EFFECT 54<br />
4.5 THE HORIZONTAL STRUCTURES OF THE INTERFACE AMPLITUDES 58<br />
4.6 THE HORIZONTAL STRUCTURES OF THE CURRENTS 74<br />
4.7 REFERENCES 81
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 3 of 92<br />
5 Loch Lomond 82<br />
5.1 INTRODUCTION 82<br />
5.2 MEASUREMENT INFORMATION ON INTERNAL WAVES 84<br />
5.3 ANALYSIS AND INTERPRETATION 87<br />
5.4 REFERENCES 87<br />
6 Conclusions 91<br />
6.1 LAC DU BOURGET 91<br />
6.2 LAC LÉMAN 91<br />
6.3 LAKE CONSTANCE (BODENSEE) 92<br />
6.4 LOCH LOMOND 92
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
1 INTRODUCTION<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 4 of 92<br />
1.1 GENERAL CONSIDERATIONS<br />
During summer, heating of lake surface water leads to the constitution of a stable interface<br />
with strong thermal and density gradients at intermediate depth, the mesolimnion.<br />
Wind stress at the lake surface induces a tilting of this density interface. After the<br />
wind event, the interface oscillates in a standing wave movement (<strong>seiche</strong>s) to converge<br />
slowly to the new equilibrium position. The characteristics of these internal waves depend<br />
on the wind intensity and period between successive wind events. <strong>Internal</strong> wave<br />
amplitude may by amplified depending on the lake geometry and bathymetry. The dynamics<br />
of these phenomena are important for vertical <strong>mixing</strong> and therefore for the reaeration<br />
of mesolimnion waters<br />
Long internal waves in lakes commonly take the form of standing waves (<strong>seiche</strong>s) of frequency<br />
and form determined by basin shape and density (temperature) stratification in the<br />
water column. Recognised by Thoulet (1894) and Richter (1897), the "temperature <strong>seiche</strong>"<br />
phenomenon was first systematically explored and interpreted by Wedderburn (1912 and<br />
earlier papers). It was later shown (Mortimer 1953, 1963, 1993) to be a widespread response<br />
to thermocline tilt initiated by wind action on stratified lakes of all sizes and<br />
shapes, modified by Earth's rotation as the lake size increases. Various modes of oscillation<br />
can be excited, commonly modes No 1 and 2 (depending upon the vertical density<br />
profile) and several horizontal modes with No 1 usually dominant. The history of research<br />
on such waves has been reviewed by Mortimer (1993).<br />
In small lakes, internal <strong>seiche</strong>s are end-to-end motions along the lake axis (Lemmin<br />
1987). In large lakes, the Coriolis force transforms the end-to-end motion into rotating<br />
(amphidromic) wave patterns, if the characteristic length scale of the lake (typically the<br />
lake's width), exceeds the Rossby radius of deformation a = ci/f. Here f is the latitudedependent<br />
Coriolis parameter and ci is the speed of long internal waves in the absence of<br />
rotation. As described below, the first horizontal mode resembles a shore-hugging Kelvin<br />
wave traveling counter-clockwise (cyclonically) around the lake basin in northern hemisphere<br />
lakes, as was first demonstrated in the Lake of Geneva (Mortimer 1963) and later<br />
in Lake Biwa, Japan (Kanari 1975, 1984).<br />
For higher horizontal modes, the resulting rotating wave patterns in large lakes depend on<br />
the basin shape and size and become more complicated. Observations are often lacking.<br />
The spatial structure of higher modes can best be visualized through comparisons of numerical<br />
models with observations. In thermally stratified lakes, those observations usually<br />
take the form of vertical oscillations of near thermocline isotherms and/or current measurements.<br />
Four large and deep lakes in Europe are investigated in relation with the internal<br />
<strong>seiche</strong>s phenomenon: Lac du Bourget, Lake Constance, Lac Léman and Loch Lomond.
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 5 of 92<br />
Different approach were used:<br />
• Lac du Bourget : Analysis of measurements available and numerical simulation<br />
with a three-dimensional model of the internals waves<br />
• Lake Constance : Estimation of the internal waves through linear theory model<br />
• Lac Léman: Analysis of measurements and numerical simulation with a threedimensional<br />
model of the internals waves<br />
• Loch Lomond: Measurements analysis<br />
1.2 OBJECTIVES<br />
To quantified the influence of internal <strong>seiche</strong>s it is crucial to know the properties of possible<br />
internal gravity waves (periods, amplitude, located depth, Coriolis influence, associated<br />
current characteristics).Then the influence of internal <strong>seiche</strong>s on <strong>mixing</strong> processes<br />
and there impact concerning the water management, constructions or executives<br />
measures could be investigated.<br />
During the period of summer stratification, <strong>mixing</strong> in lakes can be cause by vertical<br />
<strong>mixing</strong>, boundary <strong>mixing</strong>, interbasin density current and river inflows. In Lac Léman,<br />
river inflows and interbasin density currents are of little importance during summer<br />
stratification. Vertical <strong>mixing</strong> can be related to wind induced shear or to internal <strong>seiche</strong><br />
and internal wave activity.<br />
The assessment of the local intensity of internal <strong>seiche</strong>s due to their variable structure<br />
is of considerable concern in applications of water management, certain water constructions<br />
and other executive measures, for which essential information on transient<br />
currents and corresponding water displacements are required.
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
2 LAC DU BOURGET<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 6 of 92<br />
2.1 INTRODUCTION<br />
The Lac du Bourget is the smallest and the less deep of the studied lakes of the project.<br />
There is very few data available in the Lac du Bourget in relation to the internal gravity<br />
waves (P.E. Bournet 1996). These data were analysed in order to identify and quantify<br />
the <strong>seiche</strong>s in the lake. Three episodes located principally in winter have been selected.<br />
The numerical model (TELEMAC-3D) was used to reproduce the ”real world”. We tried<br />
to reproduce the hydraulic respond of the stratified lake under the wind forcing. The<br />
comparison between the model results and the measurement is shown general qualitative<br />
agreement.<br />
2.2 THE ENVIRONMENT OF LAC DU BOURGET<br />
The Lac du Bourget is the largest natural<br />
lake in France located in the French Alps<br />
(area: 44.5 km 2 , length: 18 km, average<br />
depth: 85 m; maximum depth: 145 m,<br />
lake surface altitude: 231 m). Lac du<br />
Bourget is set in a depression that geologists<br />
call a syncline, resulting from the<br />
folding of the Alpine chain during the Tertiary<br />
era. The depression was further<br />
B point<br />
deepened by almost 145 m.<br />
Figure 2 : Lac du Bourget It is oriented in<br />
the North-south direction and it is composed<br />
of two main basins with similar<br />
size: the northern basin with the maximum<br />
depth (145 m); and the Southern<br />
one with a maximum depth of 70 m.<br />
The two basins are separate by a con-<br />
River Sierroz traction of the coastline. The maximum<br />
width is about 3.2 km at Grésine and the<br />
South basin<br />
minimum width at Saint-Innocent is 1.6<br />
T point<br />
km. The rivers Leysse and Sierroz are<br />
the two main tributaries recharging the<br />
lake. At the northern end, near the<br />
Chautagne marshes, Savière canal links<br />
the lake with the river Rhone, is the main<br />
outflow.<br />
Canal de Savière<br />
North basin<br />
Figure 1 :<br />
Lac du Bourget<br />
River Leysse
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 7 of 92<br />
2.3 MEASUREMENT ON THE LAKE<br />
P.E. Bournet (1996) made the major measurements project on the Lac du Bourget between<br />
1994 and 1996. During this period 18 campaigns of vertical temperature profiles<br />
at the T point were conducted. The thermistor chain recorded temperature every 10 min<br />
over 9 spaced depths between –10 m to –51m. By a spectral analysis of the measurements<br />
P.E Bournet extracted the energy density as a function of the depth.<br />
From this analysis we have extracted three periods to define scenarios that are interesting<br />
to be reproduced by the numerical model. In the following table, we have compiled<br />
for each scenario the first horizontal mode period and the more energetic depth.<br />
Scenario Dates 1er Mode<br />
Period<br />
1 12-19 December<br />
1995<br />
2 03-09 April 1994<br />
09-15 April 1994<br />
3 27/Nov.<br />
1994<br />
to 7/Dec.<br />
Depth<br />
71 h –50 m<br />
50 h<br />
85 h<br />
43 h -25 m<br />
-20 m to –30 m<br />
-40 m<br />
2.3.1 The data<br />
In the Figure 3, Figure 5 and Figure 7 we present the temperature time series measured<br />
at the T point (see Figure 2) for the chosen scenarios at different depth. The<br />
curves are shifted for comprehension.<br />
For the scenario 1 (Figure 3), The decrease in time of the temperature at the depth –<br />
23.0m (similar for the other depth) is due to refreshment of the air above the lac. The existence<br />
internal waves at depth –35.0 and –41.0m are directly identified. This <strong>seiche</strong> is<br />
related to the high wind speed at the surface of the lac on the 13 and 14 September<br />
(see Figure 4).<br />
For the scenario 2 (Figure 5) the internal waves are not well identified, particularly due<br />
to the meteorological variability (see Figure 6).<br />
The measurements for the scenario 3 (Figure 7), shows a well-developed internal wave<br />
located around –23.0 m but without correlation with the wind speed!
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
8.81°C<br />
7.93°C<br />
7.34°C<br />
12.12.95 14.12.95 16.12.95 18.12.95 20.12.95<br />
4<br />
22.12.95<br />
Figure 3 : Temperature series at the T point; Scenario 1. Curves are shifted by 1°C.<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
Figure 4 : Wind speed series at Voglans meteorological station; Scenario 1.<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Wind Speed<br />
-23.0 m<br />
-35.0 m<br />
-41.0 m<br />
12/12/95 13/12/95 14/12/95 15/12/95 16/12/95 17/12/95 18/12/95 19/12/95 20/12/95<br />
9<br />
8,5<br />
8<br />
7,5<br />
7<br />
6,5<br />
6<br />
5,5<br />
5<br />
4,5<br />
Page 8 of 92
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
8.00°C<br />
6.39°C<br />
6.09°C<br />
3<br />
03/04/94 04/04/94 05/04/94 06/04/94 07/04/94 08/04/94 09/04/94 10/04/94 11/04/94 12/04/94<br />
Figure 5 : Temperature series at the T point; Scenario 2. Curves are shifted by 1°C.<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
Figure 6 : Wind speed series at Voglans meteorological station; Scenario 2.<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Wind Speed<br />
-10.0 m<br />
-23.0 m<br />
-29.0 m<br />
0<br />
03/04/94 05/04/94 07/04/94 09/04/94 11/04/94 13/04/94 15/04/94<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
Page 9 of 92
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
8.19°C<br />
7.48°C<br />
7.21°C<br />
30/11/94 01/12/94 02/12/94 03/12/94 04/12/94 05/12/94 06/12/94 07/12/94<br />
Figure 7 : Temperature series at the T point; Scenario 2. Curves are shifted by 1°C.<br />
Figure 8 : Wind speed series at Voglans meteorological station; Scenario 3.<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
-23 m<br />
-26 m<br />
-29 m<br />
Wind Speed<br />
0<br />
28/11/94 30/11/94 02/12/94 04/12/94 06/12/94 08/12/94<br />
11<br />
10.5<br />
10<br />
9.5<br />
9<br />
8.5<br />
8<br />
7.5<br />
7<br />
6.5<br />
Page 10 of 92
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
2.4 APPLICATIONS<br />
FP5_Contract No.: EVK1-CT1999-00004<br />
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File: <strong>D28</strong>.doc<br />
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2.4.1 TELEMAC System<br />
The TELEMAC system is a powerful integrated modelling tool for use in the field of<br />
free-surface flows (J.M. JANIN 1992).<br />
The TELEMAC-3D software solves 3D hydraulic equations (with the assumption of hydrostatic<br />
pressure conditions and time-dependent surface) and transport-diffusion<br />
equations for intrinsic values (temperature, salinity, concentration). The main results<br />
obtained at each point of the computational mesh are velocity in three directions and<br />
the concentration of transported quantities. The main result for the surface mesh is the<br />
water depth.<br />
The space is discretised in the form of an unstructured grid of triangular elements<br />
(Figure 9 on the left), which means that it can be refined particularly in areas of special<br />
interest.<br />
Bathymé-<br />
Figure 9 : Mesh and bathymetry used by TELEMAC-3D
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
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File: <strong>D28</strong>.doc<br />
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2.4.1.1 TELEMAC-3D Equations<br />
The code solves the three-dimensional hydrodynamic equations under the following assumptions:<br />
• Navier-Stokes 3D equations with free surface changing in time,<br />
• Negligible density variation in the mass conservation equation,<br />
• Hydrostatic pressure assumed,<br />
• Boussinesq approximation for momentum.<br />
Given these assumptions, the following 3D equations are solved:<br />
∂u<br />
∂u<br />
∂u<br />
∂u<br />
1 ∂p<br />
∂ � ∂u<br />
� ∂ � ∂u<br />
� ∂ � ∂u<br />
�<br />
+ u + v + w = − + �νH<br />
� +<br />
F<br />
t x y z x x x y<br />
�<br />
�νH<br />
y<br />
�<br />
� + �νH<br />
� +<br />
∂ ∂ ∂ ∂ ρ0<br />
∂ ∂ � ∂ � ∂ � ∂ � ∂z<br />
� ∂z<br />
�<br />
∂v<br />
∂v<br />
∂v<br />
∂v<br />
1 ∂p<br />
∂ � ∂v<br />
� ∂ � ∂v<br />
� ∂ � ∂v<br />
�<br />
+ u + v + w = − + �νH<br />
� + H<br />
H F<br />
t x y z 0 y x x y �<br />
�ν<br />
y �<br />
� + �ν<br />
� +<br />
∂ ∂ ∂ ∂ ρ ∂ ∂ � ∂ � ∂ � ∂ � ∂z<br />
� ∂z<br />
�<br />
S ∆ρ<br />
p = ρ0g(<br />
S − z)<br />
+ ρ0g�<br />
dz<br />
z ρ0<br />
∂u<br />
∂v<br />
∂w<br />
+ + = 0<br />
∂x<br />
∂y<br />
∂z<br />
∂T<br />
∂T<br />
∂T<br />
∂T<br />
∂ � ∂T<br />
� ∂ � ∂T<br />
� ∂ � ∂T<br />
�<br />
+ u + v + w = �νHT<br />
� + HT + �νHT<br />
� + Q<br />
t x y z x x y �<br />
�ν<br />
y �<br />
�<br />
∂ ∂ ∂ ∂ ∂ � ∂ � ∂ � ∂ � ∂z<br />
� ∂z<br />
�<br />
with:<br />
h (m) water depth Zf (m) bottom elevation<br />
S (m) free surface elevation<br />
u, v, w<br />
(m/s)<br />
ρ 0 (X) reference density<br />
velocity components ∆ρ (X) variation in density<br />
T (°C) active or passive<br />
tracer<br />
t (s) time<br />
P (X) pressure x, y (m) horizontal space<br />
components<br />
g (m/s2)<br />
νH,νz<br />
(m2/s)<br />
νHT,νzT<br />
(m2/s)<br />
acceleration due to<br />
gravity<br />
velocity diffusion<br />
coefficients<br />
tracer diffusion coefficients<br />
Fx, Fy<br />
(m/s2)<br />
Q (tracer<br />
unit)<br />
source terms<br />
tracer source or<br />
sink<br />
In order to reproduce properly the internal waves with TELEMAC-3D, we need horizontal<br />
level meshes between the surface down to the thermocline. So we cut the<br />
bathymetry above the depth –70.0m bellow the surface. Only the bathymetry below this<br />
depth is the real one. The resulted bathymetry is presented on Figure 9 (right).<br />
x<br />
y
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
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File: <strong>D28</strong>.doc<br />
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2.4.1.2 Lac du Bourget application<br />
To compare the results from the model with the measurement at the T point, we simulate<br />
and analyse the scenario 1.<br />
The simulation characteristics for the model run are:<br />
• initial lac status without velocity,<br />
• horizontal homogeneous stratification, initial vertical profile of temperature from<br />
the measurements (Figure 10),<br />
• outflow and inflow neglected.<br />
0<br />
6<br />
-20<br />
6.5 7 7.5 8 8.5 9 9.5<br />
-40<br />
-60<br />
-80<br />
-100<br />
-120<br />
-140<br />
Initial temperature<br />
Figure 10 : Initial temperature of the lac.<br />
The measured wind and temperature at the Voglans station (between 12 of december<br />
to 22 of december 1995), are impose all over the lac. The time series of the lac surface<br />
boundary condition are reported on the Figure 11.<br />
14<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
Int ensit é du vent<br />
Températ ure<br />
0<br />
-2<br />
12/12/95 14/12/95 16/12/95 18/12/95 20/12/95 22/12/95<br />
Figure 11 : Vent et Température de l’air à Voglans.<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0
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On the Figure 12, we have reported the time evolution of the lac temperature at the<br />
depth –23.0m for the measurement and the simulation. The decreasing of the water<br />
temperature during the simulated period is connected the a lower air temperature compare<br />
to the lac temperature associated with high wind intensity (Figure 11) which mixed<br />
the upper water level of the lac.<br />
8.9<br />
8.7<br />
8.5<br />
8.3<br />
8.1<br />
7.9<br />
7.7<br />
7.5<br />
Profondeur : -23.0 m<br />
Simulation pt T<br />
Mesures<br />
Simulation pt B<br />
12/12/95 13/12/95 14/12/95 15/12/95 16/12/95 17/12/95 18/12/95 19/12/95 20/12/95 21/12/95 22/12/95<br />
Figure 12 : Temporal evolution of the temperature at –23.0 m below the surface. Comparison between<br />
the measurement and the results from the simulation at the T point. Results from the simulation<br />
at the B point<br />
The comparison at the T point (south part of the lac see Fehler! Verweisquelle konnte<br />
nicht gefunden werden.) is showing a good agreement, until the 18 of December,<br />
between the measurement and the simulated results (purple curves and dark blue<br />
curve). After this date the shift (about 0.3°C) is constant between the two curves. The<br />
rapid variation of the temperature measured (12/12, 16/12 and 18/12) is not reproduce<br />
by the model. The rapid decreasing of temperature the 16/12 is reflecting the <strong>seiche</strong>s<br />
existed below.<br />
The comparison between the simulation results at the T point and the B point demonstrate<br />
that at the considered depth the wind action is homogeneous all over the lake<br />
between the surface and –21.0 m (see Figure 10).<br />
At the depth –35.0 m (Figure 13), the measurement at the T point (curve in purple)<br />
show clearly the existence of the internal gravity wave at this depth. The amplitude of<br />
the temperature is bigger than 1°C.<br />
At this depth, the results of the simulation are in a good agreement until mid of the<br />
15/12. After this date, the first oscillation of the temperature is well reproducing in time<br />
but the amplitude is two times smaller than the measurements. At the end of the simulated<br />
time the shift in time and amplitude is bigger.<br />
The amplitude of the <strong>seiche</strong>s at the B point is lower than at the T point, and there is a<br />
shift in phase.
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The development of the <strong>seiche</strong> is complete at the depth –41.0 (Figure 14), the measurements<br />
at the T point (curve in purple) illustrate this point. The amplitude of the temperature<br />
is bigger than 2°C and decrease with time.<br />
At this depth, the results of the simulation are in a very good agreement until mid of the<br />
15/12. After this date, the next oscillation of the temperature is relatively well reproducing<br />
in time but the amplitude is underestimated compared to the measurements. At the<br />
end of the simulated time the shift in time and amplitude is bigger.<br />
9<br />
8.5<br />
8<br />
7.5<br />
7<br />
6.5<br />
Profondeur: -35.0 m<br />
Simulation pt T<br />
Mesures<br />
Simulation pt B<br />
12/12/95 13/12/95 14/12/95 15/12/95 16/12/95 17/12/95 18/12/95 19/12/95 20/12/95 21/12/95 22/12/95<br />
Figure 13 : Temporal evolution of the temperature at –35.0 m below the surface. Comparison between<br />
the measurement and the results from the simulation at the T point. Results from the simulation<br />
at the B point<br />
8.5<br />
8.3<br />
8.1<br />
7.9<br />
7.7<br />
7.5<br />
7.3<br />
7.1<br />
6.9<br />
6.7<br />
6.5<br />
Profondeur : -41.0 m<br />
Simulation pt T<br />
Mesures<br />
Simulation pt B<br />
12/12/95 13/12/95 14/12/95 15/12/95 16/12/95 17/12/95 18/12/95 19/12/95 20/12/95 21/12/95 22/12/95<br />
Figure 14 : Temporal evolution of the temperature at –41.0 m below the surface. Comparison between<br />
the measurement and the results from the simulation at the T point. Results from the simulation<br />
at the B point
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2.4.1.3 Frequency analysis<br />
A simple decomposition of the <strong>seiche</strong>s is used through a trigonometric function decomposition<br />
of the calculated temperature of the lac at the T point at –41.0 min based on<br />
the following formula:<br />
� i<br />
( ω t + )<br />
−αi<br />
S = C + A e cos ϕ<br />
i<br />
Where S is the simulated temperature, C a constant and i is the order of the decomposition.<br />
The first level of decomposition gave the following parameters:<br />
C = 7.<br />
6<br />
A1 = −0.<br />
9<br />
1 10800 = ϕ<br />
267600 − t<br />
α1<br />
=<br />
500000<br />
2π<br />
ω 1 =<br />
300000<br />
This gave a period close to 83 hours for the first mode. On Figure 15 is shown the<br />
graphical comparison between the simulated temperature at T point at –41.0m and the<br />
first level of trigonometric decomposition. For the selected period (between 200000s<br />
and 700000s) the both curve are very close.<br />
The second level of the decomposition gave the following parameters:<br />
2 0.<br />
12 = A<br />
2π<br />
ω 1 =<br />
94000<br />
i<br />
1 = ϕ<br />
i<br />
7200<br />
With this method the second mode is estimated to 26 hours.<br />
8.5<br />
8.3<br />
8.1<br />
7.9<br />
7.7<br />
7.5<br />
7.3<br />
7.1<br />
6.9<br />
200000 300000 400000 500000 600000 700000<br />
Figure 15: Simulated temperature at T point at –41.0m (S in dark Blue), first level of trigonometric<br />
decomposition (in red), first plus second level in light blue.<br />
S<br />
First level<br />
1+2
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2.4.2 SIMULATED SEICHES – ANALYSIS<br />
2.4.2.1 Rotation<br />
The Figure 16 shows the temperature<br />
iso-contour level 7.6°C at six different<br />
times at –41.0 m depth. These times<br />
are chosen in order to illustrate the rotation<br />
of the internal wave around the<br />
amphidromic point due to the coriolis<br />
forcing. The coloured stars are associated<br />
to the position of the temperature<br />
iso-contour that has the same colour.<br />
The black arrow that follow the stars<br />
explicit the wave rotation direction.<br />
The both lines Red and Purple, which<br />
are superimposed, are related to the<br />
wave period of the <strong>seiche</strong>s (80 hours).<br />
The intersection of the six curve is located<br />
at the middle of the lac, determine<br />
the amphidromic point of the internal<br />
wave.<br />
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Figure 16 : Coriolis effect on the internal wave propagation.<br />
°C<br />
°C<br />
°C<br />
°C<br />
°C<br />
°C
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2.4.2.2 Vertical displacement<br />
In order to evaluate the amplitude of the wave and the affected area we show on Figure<br />
17 (depth –35.0 m at left and –41.0 m on the right), the difference between the temporal<br />
(during all the simulation) maximum of the temperature and the initial temperature at<br />
the considered depth.<br />
The maximum difference is obtain on the depth –41.0 m, as the analysis made by P.E.<br />
Bournet shown. At the middle of the lac, the difference is zero in both cases, that confirm<br />
the amphidromic point (Figure 16). The maximum value are located on the west<br />
side of the lac. The North basin is more exposed to high amplitude of internal wave.<br />
Figure 17 :Difference between the maximum temperature and the initial one (-35.0 m on the right, -<br />
41.0 m on the left)
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2.4.2.3 Vertical velocity<br />
The maximum of the vertical velocity, during all the simulation time, for the depth –<br />
35.0m and –41.0m are shown on Figure 18.<br />
We observe a very similar value for both levels. The maximum of the vertical velocity is<br />
concentrated along the coast line with some spot located at particular points.<br />
Figure 18 : Maximum vertical velocity during the simulation (-35.0 m on the right, -41.0 m on the<br />
left)
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2.5 CONCLUSIONS<br />
The analysed measurements are shown the existence of internal gravity wave in the<br />
Lac du Bourget with a period between 40hours to 80 hours depending on the vertical<br />
stratification structure.<br />
The simulation using the TELEMAC-3D model is used to reproduce one <strong>seiche</strong> event<br />
on the Lac du Bourget. The numerical results are in a quite good qualitative agreement<br />
with the measurements.<br />
The numerical results are given some interesting new information about the hydraulic<br />
respond of the lake under wind forcing. After the generation of the internal <strong>seiche</strong>s, the<br />
wave propagation is showing a rotating structure, due to the Coriolis influence, with an<br />
amphidromic point at the middle of the lake (Figure 16).<br />
In spite of the bad representation of the bathymetry along the coast, the maximum dissipation<br />
of the internal wave illustrated by the two figures (Figure 17 and Figure 18) is<br />
located all around the coastline of the lake. And the maximum vertical velocity is less<br />
than 1. 10 -3 m/s.<br />
2.6 REFERENCES<br />
Bournet PE. 1996. Contribution à l’étude hydrodynamique et thermique du Lac du<br />
Bourget. PhD Thesis.<br />
Janin JM, Lepeintre F, and Pechon P. 1992. TELEMAC-3D: a finite element code to<br />
solve 3D free surface flows problems. Proceedings of computer modelling of seas and<br />
Coastal regions. Southampton, UK.
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3 INTERNAL SEICHES AND MIXING IN LAC LÉMAN<br />
3.1 THE LAC LÉMAN ENVIRONMENT<br />
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The Lac Léman (Figure 19) is a warm monomictic lake situated between Switzerland<br />
and France. It is curved in shape and composed of two main basins: a deep central<br />
eastern basin (310 m maximum depth, 157 m mean depth, mean width 10 km) called<br />
Grand Lac ("big lake") and a relatively small and narrow section in the west called Petit<br />
Lac ("small lake," maximum depth 70 m; mean width around 4 km). The lake has a<br />
total length of 70.2 km and a width of 13.8 km in the central part which corresponds to<br />
2.4 Rossby radii under typical summer stratification conditions. The eastern part of the<br />
lake is surrounded by high mountains sheltering it from most strong winds. The central<br />
and western part of the lake form part of the Swiss central plateau. The windfield over<br />
the lake is affected by the mountains and the plateau. It is consequently dominated by<br />
events of strong winds from the northeast and southwest which may last from several<br />
hours to several days. The winds from the northeast have been observed to cause<br />
thermocline depressions of more than 20 m in the Petit Lac (Lemmin and D'Adamo<br />
1996). These wind events can be considered as the principal forcing to initiate internal<br />
<strong>seiche</strong>s.<br />
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Figure 19 : Map of the Lac Léman (Lake Geneva) with surrounding topography and positions of<br />
recording stations discussed in this text. Depth contours are given with reference to the water<br />
surface level which is at an elevation of 371 m above sea level.<br />
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Field measurements concerning temperature and currents have been carried out in the<br />
Lac Léman over the last twenty years using moored instruments. The recording interval<br />
was 30 min. or 60 min. and instruments were deployed in campaigns lasting for several<br />
months. Most of these data were collected along the northern shore of the lake at<br />
mooring stations covering the slope down to about 175 m during fall stratification. However,<br />
some measurements were also made during summer periods. The data were<br />
analyzed for the presence of internal <strong>seiche</strong> modes using spectral analysis. For spectral<br />
analysis of the records, a standard Fast Fourier Transform with segment overlap was<br />
used.<br />
3.2.1 Modes detected<br />
Modes were identified by comparing the periods P corresponding to prominent spectral<br />
peaks with those predicted for a two-layered approximation fitted to the lake dimensions<br />
and density distribution using the Merian formula<br />
P n =2L b<br />
(h1 +h2) nh1h2 g(ρ2 − ρ1) ρ2 where Lb is the basin length, h1 and h2 are the depth of the epilimnon and the hypolimnion<br />
and ρ1 and ρ2 are the respective densities. n is the mode number. In the calculations<br />
we assumed, for the summer (values for the fall in parenthesis) a top layer h1 =<br />
15 (25) m; a bottom layer h2= 175 (165) m; a top layer temperature T1 = 19 (8) °C; and<br />
a bottom layer temperature T2= 5.5 °C. The analysis of all the data indicates that, independent<br />
of season and station location, only certain modes are excited. Differences<br />
were found between the number and type of modes observed at different locations in<br />
the lake in particular with distance from the shore.
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Figure 20 : Cumulative energy spectra of alongshore current component at stations P1 and P2 (for<br />
positions see Figure 19). The maximum at the nearshore station P1 is at the Kelvin wave period<br />
while that for the offshore station P2 is at the Poincaré period. (cpd = cycles per day; modified<br />
from Bohle-Carbonell, 1986)<br />
At the stations closest to shore, the most prominent <strong>seiche</strong> is the first mode (n = 1;<br />
called L1 hereinafter; Figure 20). Its period is near 81.5 h in summer, increasing to 131<br />
h in fall, as the density structure of the water column changes and ci decreases. In the<br />
spectra from the narrow western end of the lake (not shown here) the L1 mode response<br />
is always seen most clearly. Recently, timeseries measurements were carried<br />
out during summer stratification with a current meter placed about 5 m above the lake<br />
bottom on the southern side of the deep central plateau at station S (Figure 19). Spectra<br />
from this station (Figure 21) show again a broad peak at the Kelvin <strong>seiche</strong> period.<br />
Seiche modes n = 2 to 9 were not observed in any spectra.<br />
The first cross basin or transverse mode (called T1 hereinafter), with summer and fall<br />
periods of 10.7 and 13.5 hours respectively. It is seen most clearly in the spectra at offshore<br />
stations in the central part of the lake basin (Figure 20) and is often found in the<br />
eastern part. Modes higher than 10 cannot be detected with certainty because of the<br />
cut-off imposed by the time step of the data records. The T1 period is also clearly seen<br />
in spectra from the recent deep measurements at S (Figure 21).
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10 3<br />
Lake Geneva, midlake station, depth 310m, from 6 July 14:00 to 12 October 24:00, 2001<br />
10 2<br />
10 1<br />
10 0<br />
10 -7 10 -6 10 -5 10 -4 10 -3<br />
10 -1<br />
north component at 304m<br />
east component at 304m<br />
frequency (cph)<br />
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Figure 21 : Spectra of north and east component of the currents at 304m depth recorded during<br />
summer 2001 at station S. For station location see Figure 19.<br />
3.2.2 Wave propagation patterns<br />
3.2.2.1 Mode L1<br />
It should be noted that the thermocline oscillations are accompanied by oscillations in<br />
lake surface level which are out-of-phase with and about 1/1000 smaller in amplitude<br />
than the oscillations of thermocline isotherms. It is therefore possible (e.g. Caloi et al.<br />
1961; Sirkes 1987) to use numerically filtered deviations of surface level from equilibrium<br />
to follow the progress of internal motions. This possibility arose for the Lake of<br />
Geneva, because the Swiss Service Fédérale des Eaux, (SFE, 1954) published tables<br />
of water levels measured in 1950 by 13 high-precision water level recorders spaced<br />
around the lake shore (see Fig. 1). We have analyzed these limnigraph records for the<br />
wave propagation pattern. Cross spectral analysis reveals the wave propagation pattern.<br />
Coherence and phase were determined between pairs of stations.<br />
Coherence is always high at the mode L1 period and falls off rapidly below the confidence<br />
limit for surrounding periods. A calculation was made for all possible combinations.<br />
The progression for the L1 mode during the summer period is presented in Figure<br />
22. In that case, station 2 was taken as the reference station and pairs were formed<br />
with all other stations around the lake basin. Indicated for each station are coherence
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and phase angle. Included, for comparison, is the L1 mode amphidromic pattern obtained<br />
from the numerical model under the conditions specified above.<br />
During summer, coherence was high for most station pairs. For the stations along the<br />
northern shore, coherence increases again from E to W and became high for those<br />
stations in the narrow Petit Lac. The phase angles calculated from the data can be<br />
compared to those predicted by the numerical model. Along the southern shore, satisfactory<br />
agreement in phase angle was found; and cyclonic progression was clearly established.<br />
Agreement was less satisfactory on the northern shore, particularly, at the<br />
entrance to the Petit Lac at stations 10 and 11, even though the coherence was high.<br />
At the western end of the lake, agreement was again good, indicating that the L1 <strong>seiche</strong><br />
mode completed a full cycle around the lake. During fall, interstation coherence decreased<br />
from W to E along the southern shore.
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Figure 22: The L1 longitudinal mode during the summer interval 4 June to 24 Aug. 1950: top panel,<br />
structure as predicted by the model. At each limnigraph station marked by an encircled number in<br />
the top panel, phase and coherence with reference to station 2 obtained from the analysis of the<br />
waterlevel data are indicated. Middle panels, spectra of SFE water level fluctuations at station 2<br />
and 7. The small peak at the L3 mode period is not significant; T1 is the first transversal mode<br />
wave to be discussed below. Bottom panels, coherence and phase angle between 2 and 7 (out of<br />
phase at -180°). Coherence and phase at other stations relative to 2 are shown in the top panel.
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a.1 Evidence for "Kelvin-<strong>seiche</strong>" response<br />
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Initial analysis of part of the water level data had already shown (Mortimer 1963) that<br />
periodicity in the surface elevation signals corresponded to the L1 mode "Kelvin<br />
<strong>seiche</strong>." That signal was visually correlated with the thermocline oscillation at Geneva.<br />
The surface elevation signal progressed around the lake perimeter at a nearly constant<br />
speed corresponding to a L1 mode internal <strong>seiche</strong>.<br />
The results of the present statistical analysis indicate that, for the L1 mode, coherence<br />
is high in the narrow western basin falling off toward the eastern end of the lake where<br />
the amplitude is greatly reduced and the wave form is perturbed by either T1 mode<br />
waves or non-wave effects.<br />
Analysis of lake temperature and current data has previously shown (Bohle-Carbonell<br />
1986) that typically one or two whole-basin L1 circuits are completed when post-storm<br />
calm persisted for long enough. Evidence for the Kelvin wave character of the L1 mode<br />
wave also came from current and temperature recordings, carried out in the western<br />
part of the Grand Lac in 1982/1983 (Mortimer 1993) from December to January when<br />
the lake was still weakly stratified with a thermocline at about 90 m depth. Parts of this<br />
analysis are reproduced in Figure 23. Station A1 is close to shore while station A3 is<br />
further offshore (for station location see Figure 19). Three wind events occurred between<br />
9 and 20 December.
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Figure 23: Current and temperature records of fall/winter 1982/1982. Winds were measured at<br />
Cointrin Airport near Geneva at the west end of the lake basin. The wind speed squared is multiplied<br />
with the direction of the wind. Positive winds are coming from the NE; negative winds are<br />
coming from the SW (adapted from Mortimer, 1993). Measurement depths are indicated on each<br />
panel. For station locations, see Figure 19.<br />
At roughly 100 h after the start of each wind event, the currents at 15 m at mooring A1<br />
reversed suddenly from eastgoing to westgoing at the mooring nearest shore. Those<br />
reversals, R1, R2, R3, and R7, were each accompanied by a sudden depression of the<br />
thermocline (i.e. a sudden temperature rise) at mooring A1, marking the passage of a<br />
surge. Similar saw-toothed "waves" were also seen at moorings further offshore at A3<br />
but with reduced amplitude. The reversals in current direction, however, were confined<br />
to the nearshore instrument, A1.Figure 23 indicates a pattern, which starting on the<br />
south shore, had traveled around the lake (the Petit Lac was destratified and took no<br />
part in the <strong>seiche</strong>) at internal <strong>seiche</strong> speed.
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After the wind event R3, as would be expected for a Kelvin wave, a free oscillation persisted<br />
for three cycles with amplitudes of the current reversal R3 to R5 and amplitudes<br />
of the associated temperature waves decreasing steadily in time. It can be seen that<br />
each time a strong current reversal in the upper layer occurred at station A1, but not at<br />
station A3, in accordance with the Kelvin wave exponential decay of wave amplitude<br />
with distance from shore. The temperature records again supported the Kelvin wave<br />
pattern: pronounced signals were observed at A1; at A3 (not shown) the amplitude was<br />
rather small.<br />
At that time of the year, the thermocline was below the depth of the Petit Lac basin.<br />
The path of a Kelvin wave should, therefore, be confined to the contour of the Grand<br />
Lac basin. Data from station G (see Figure 19) can be used to investigate this point.<br />
While the signal from the L1 mode is seen in the alongshore (east-west) component at<br />
A1, it appears as south-going currents at G (Figure 23). Thus the Kelvin wave has<br />
turned, following the deep basin contour of the Grand Lac instead of continuing to move<br />
along shore into the Petit Lac.<br />
Evidence for a Kelvin <strong>seiche</strong> response can also be seen in the spectra in Figure 21<br />
where the energy level of the along shore component is clearly dominant in the Kelvin<br />
<strong>seiche</strong> range.
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3.2.2.2 Mode T1<br />
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For the only other strongly expressed mode, the first transverse mode T1 (n=10), the<br />
results of the cross spectral analysis are given in Figure 24, in that case for cross-basin<br />
stations 3 and 9. For the mode T1, the amphidromic pattern (Figure 23) displayed coherence<br />
between stations, which are part of the same or neighbouring amphidromic<br />
cell, but no coherence between stations in distant cells.<br />
Figure 24: The first transverse mode T1 during the summer interval 4 June to 24 Aug. 1950: top<br />
panel, structure and period predicted by the model. Bottom panels, coherence and phase between<br />
stations 3 and 9 from spectral analysis of SFE records of surface level fluctuations. Phase angle of<br />
+180° indicates a cross-basin oscillation. No coherence is found for station pairs which are not in<br />
the same amphidromic cell.
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3.2.3 Evidence of Poincaré waves<br />
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Is this T1 mode a cross basin <strong>seiche</strong> or is it a Poincaré wave? The fact that a wave of<br />
near 11 h period has been detected in the shore-based water level records would favour<br />
interpretation as a standing cross-basin wave in the central part of the lake. However<br />
since a constant periodicity appears in the central part of the basin, independent of<br />
local topography, a Poincaré wave interpretation may be more likely. This is supported<br />
by results from the numerical model (Bäuerle, 1985), which predicts clockwise rotation<br />
in the amphidromic cells in that part of the lake.<br />
Poincaré waves are clearly seen in Figure 23 superimposed on the Kelwin wave pattern,<br />
particularly at the offshore station A3 and appear to be excited together with the<br />
Kelwin waves. To investigate this point further, the current records at station S in the<br />
central basin were examined in detail.<br />
3.2.4 Progressive vector analysis<br />
Progressive vector analysis was carried out for current components measured at station<br />
S in order to better understand the pattern described above. Figure 25 shows the vector<br />
diagram for the full record. This is a predominantly eastward oriented transport and the<br />
total running length is about twice that of the east-west length of the central bottom<br />
plateau. From the time markers it is obvious that there are periods of relatively slow<br />
transport as would be expected during summer stratification.<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
Lake Geneva, midlake station, depth 310m, from 6 June 14:00 to 12 October 24:00, 2001<br />
instrument depth 304 m<br />
marker every 24 h<br />
-3 0 3 6 9 12 15 18 21<br />
east west displacement (km)<br />
Figure 25 : Progressive vector diagram of currents measured at station S. For station location see<br />
Figure 19<br />
Frequently there are undulating sections in the progressive vector curve in Figure 25<br />
with two waves occurring between two 24 h markers. This obviously reflects a periodic<br />
oscillation of about 12 h clearly seen in the analysis above. The cusped motion pattern<br />
is evidently a superposition of a circular motion with a linear motion with a mean speed
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which is greater than the rotating vector. To learn more about the rotating motion, an<br />
excerpt of the progressive vector diagram has been produced with Figure 26. From this<br />
figure it is obvious that circular motions with periods close to 12 h are always executed<br />
in a clockwise sense of rotation.<br />
2.40<br />
2.15<br />
1.90<br />
1.65<br />
1.40<br />
Lake Geneva; midlake station, depth 310m, from 1 Sept. 14:00 to 5 Sept 14:00, 2001<br />
325<br />
343<br />
307<br />
379 397<br />
361<br />
415<br />
289<br />
253<br />
271<br />
433<br />
469<br />
451 487<br />
235<br />
505512<br />
12.75 13.00 13.25 13.50 13.75 14.00 14.25<br />
217<br />
199<br />
181<br />
163<br />
145<br />
91109<br />
127<br />
instrument depth 304m<br />
marker every 3 h<br />
73<br />
55<br />
1 37<br />
east west displacement (km)<br />
Figure 26 : Progressive vector plot for selected period of the trajectory shown in Figure 25.<br />
Further support for the Poincaré wave concept comes from spectra calculated by<br />
Mortimer et al. (1983) for current and temperature data at different stations in the central<br />
part of the lake. A statistically significant peak at the period of the T1 mode is always<br />
clearly seen in the temperature data at stations offshore (but rarely at the nearshore<br />
stations). However, it has to be realized that the T1 mode has Poincaré wave<br />
characteristics only in the central part of the Grand Lac basin.<br />
3.2.5 Mixing by internal <strong>seiche</strong>s<br />
During the period of summer stratification, <strong>mixing</strong> in lakes can be caused by vertical<br />
<strong>mixing</strong>, boundary <strong>mixing</strong>, interbasin density currents and river inflows. In Lac Léman,<br />
river inflows and interbasin density currents are of little importance during summer<br />
stratification. Vertical <strong>mixing</strong> can be related to wind induced shear or to internal <strong>seiche</strong><br />
and internal wave activity.<br />
Turbulent <strong>mixing</strong> can be expressed in terms of turbulent <strong>mixing</strong> coefficients, Kz, in<br />
analogy to molecular diffusion coefficients. This coefficient can be determined by the<br />
flux gradient method from integral changes of a tracer, such as temperature, over a<br />
certain time and a certain depth in the following way (Jassby and Powell, 1975):<br />
Kz =−1/ ( [ A(z)∆T<br />
∆z]<br />
) A(z') ∆T<br />
∆t dz'<br />
zmax �z<br />
19
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A(z’) is the lake surface at depth z’, T is temperature, t is time, z and zmax is the bottom<br />
of the lake. The flux gradient method is valid during periods when lakes warm up but is<br />
not valid when convective cooling overwhelms wind-induced <strong>mixing</strong>. In the present<br />
<strong>study</strong> we will apply this equation to temperature profiles taken in the central part of the<br />
Grand Lac basin in order to investigate the origin of vertical <strong>mixing</strong> in Lac Léman. This<br />
selection will minimize the effect of thermocline tilt due to internal <strong>seiche</strong>s and wind setup.<br />
The calculation of the vertical <strong>mixing</strong> coefficient Kz were carried out based on a set of<br />
monthly temperature profiles taken between 1987 and 1991 for the warming period<br />
from May to September. For each month, average profiles were established from the<br />
data by interpolating to the same depth intervals and eliminating the linear trend due to<br />
progressive longterm warming of the lake. Details are given in Michalski and Lemmin<br />
(1995). Results of this calculation are shown in Figure 27.
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Figure 27 : Logarithm of the vertical turbulent <strong>mixing</strong> coef. Kz vs. logarithm of the Brunt-Väisälä<br />
frequency N 2 . The data are plotted as depth profiles with the surface at right and the bottom at left.<br />
The corresponding depths are indicated on each curve. A straight line is drawn for the section of<br />
each profile where the relation Kz ∝a(N 2 ) b can be validated. (a) for May (b = -0.5). (b) for the<br />
whole warming season from May to September (b = –0.4). (c) for a multiannual trend (b = –0.6).<br />
The hypothesis that the <strong>mixing</strong> coefficients result essentially from vertical <strong>mixing</strong> has<br />
been examined by looking at the correlation between Kz and the Brunt Vaisala frequency<br />
N. The Brunt Vaisala frequency is defined as<br />
N 2 =−(dρ/ dz)(g /ρ)<br />
where g is the gravitational acceleration. As in most lakes, the vertical density gradient<br />
is controlled by the temperature gradient dT/dz. The conversion from temperature into<br />
density is carried out using standard formulae. Different theoretical models exist for the<br />
relation between Kz and N2 of which all have the form<br />
Kz ∝a(N 2 ) b
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where a and b are constants. Using dimensional analysis, Welander (1968) suggested<br />
that, depending on the origins of turbulence, two limiting cases given by b = -0.5 for<br />
shear induced turbulence and b = -1 for cascading 2D turbulence. Heinz et al. (1990)<br />
summarized results from different lakes where b was found to range between – 0.4 and<br />
–0.7 indicating <strong>mixing</strong> in lakes is predominantly generated by local shear and internal<br />
waves. In Lac Léman, values of b are found between –0.4 and –0.6 (Figure 27) indicating<br />
that energy cascading is not important. A value of –0.4 falls outside the range<br />
predicted by Welander. However, Jassby and Powell (1975) who found the same value<br />
already noted that the assumptions of steady state and horizontal homogeneity made<br />
by Welander are most likely not fulfilled over periods of several months.<br />
Our results indicate that during summer stratification, turbulent <strong>mixing</strong> in the upper water<br />
column is predominantly caused by processes related to internal <strong>seiche</strong>s and progressive<br />
internal waves (Lemmin et al., 1998; Thorpe et al. 1996; Thorpe and Jiang,<br />
1998) quantified by the order of magnitude of the <strong>mixing</strong> coefficient. The increase of Kz<br />
with N 2b (and depth) ends at a depth of ≈ 90 m (see Figure 27). Below that depth an<br />
exponential correlation between Kz and N 2b cannot be established indicating that processes<br />
other than those considered by Welander dominate the turbulent <strong>mixing</strong>.<br />
3.3 NUMERICAL SIMULATIONS<br />
3.3.1 Model Description<br />
We use a slightly modified version of the three-dimensional numerical code ‘GETM’,<br />
developed by Burchard and Bolding (2002). A detailed mathematical description of this<br />
model is given in their report. Here we only mention its main features.<br />
The model solves the three-dimensional shallow-water equations for momentum and<br />
heat with a free surface as the upper boundary condition. The Boussinesq assumption<br />
has been adopted, implying that the balance of mass simplifies to a statement of zero<br />
divergence of the velocity field. As in all models of this type, pressure is computed from<br />
the hydrostatic balance, and certain types of barotropic and baroclinic waves cannot be<br />
reproduced. Among them are waves short in comparison with the local water depth and<br />
non-linear solitary waves. However, in the context of this section, <strong>mixing</strong> caused by long<br />
internal waves is emphasized, and these restrictions are hardly relevant.<br />
Quite to the contrary, the capability of the model to predict the evolution and propagation<br />
of steep non-linear waves or ‘bores’ is relevant in Lac Léman as illustrated below.<br />
To capture the essential physics of these waves, it is necessary to retain the non-linear<br />
advection terms in the horizontal momentum balance. Since it is well-known that the<br />
classical first-order upstream schemes for the numerical discretisation of these terms<br />
have great difficulties in reproducing steep waves because of their excessive numerical
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diffusion, we discretise the advection terms with so-called Total Variation Diminishing<br />
(TVD) schemes, as described in Burchard and Bolding (2002).<br />
The size and stratification of all lakes considered in the EUROLAKES projects suggests<br />
that the effects of the rotation of the earth cannot be neglected, and thus the Coriolis<br />
force is included in the model.<br />
The model equations are discretised on a staggered Arakawa-C grid in the horizontal<br />
coordinate, using generalised sigma-coordinates in the vertical. Due to the extremely<br />
steep topography of Lac Léman and the use of sigma-coordinates, great care has to be<br />
taken in computing the internal pressure-gradient, the driving force for the internal<br />
waves discussed in this section. We alleviate these problems using high horizontal<br />
resolution (250 m) and a special discretisation technique for the internal pressuregradient<br />
(Burchard and Petersen, 1997). The vertical resolution is 40 sigma levels,<br />
which is about the minimum to resolve the turbulent structure in the boundary layers<br />
and around the thermocline.<br />
Evidently, because <strong>mixing</strong> by internal waves is to be investigated here, a good turbulence<br />
model is required. We use a so-called Algebraic Reynolds Stress model (Canuto<br />
et al., 2001), solved in connection with two differential transport equations for the turbulent<br />
kinetic energy, k, and the specific dissipation rate, ω (see Umlauf et al., 2002).<br />
3.3.2 Mixing by Long <strong>Internal</strong> Waves - Model Results<br />
We investigated the activity of long internal waves at two positions in the lake which are<br />
dynamically very different and serve as two extreme cases spanning the range of <strong>mixing</strong><br />
activity in Lac Léman. The two positions are marked by yellow dots in Figure 28.<br />
Position A is located near the bottom of the deep plateau of the main basin. Measurements<br />
at this point have been discussed above. Position B is located at the entrance of<br />
the ‘Petit Lac’, the elongated south western appendix of Lac Léman. Measurements<br />
and model results at these two locations will be considered below.
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Figure 28: Smoothed topography of Lac Léman used as the input for the numerical model. The<br />
measuring positions are marked by yellow dots. The geographical orientation of the lake has been<br />
turned 17 degrees in the clockwise direction on this and the following plots.<br />
The computation of the possibly complicated structure of the wind field over Lac Léman<br />
with a meteorological model was not part of the EUROLAKES project. Therefore, we<br />
had to force our model with highly idealized winds. We used homogeneous wind fields<br />
from SW (along the basin) and from NE (across the basin). These fields can be thought<br />
of as a first-order approximation of the well-known winds ‘Vent’ and ‘Bise’, respectively,<br />
which typically last only for a few days. Evidently, the current pattern close to the surface<br />
will exhibit large errors due to these simplistic wind fields. However, the excitation<br />
of basin-scale internal waves and their decay due to <strong>mixing</strong>, the main topic of this section,<br />
should be reproduced with much better accuracy. All runs have been initiated with<br />
zero velocities and an idealised typical late summer stratification with a well mixed<br />
epilimnion of 18 °C, a homogeneous hypolimnion of 5 °C, and a transitional thermocline<br />
at about 20 m depth.<br />
3.3.2.1 Mixing at Deepest Part of Lac Léman<br />
The measurements at point A (marked as S in Figure 19) near the bottom of the deepest<br />
part of the main basin have been discussed above. The position of the spectral<br />
peaks and the clockwise rotation of the velocity vector have been interpreted as strong<br />
indicators for the dominance of Poincaré waves (in winter possibly degenerated to simple<br />
inertial oscillations) at this location. In summer, considerable energy was also found<br />
in the along-basin component of the velocity at much lower frequency. Current speeds<br />
were demonstrated to be weak at all times and of the order of 1 to 3 cm/s. Additional<br />
measurements showed that the temperature profile in the lowest 50 meters of the water<br />
column is weakly unstable, most likely due to the geothermal heat flux. This thermal instability,<br />
however, is always compensated by the much stronger stabilizing effect of salinity<br />
in this region. These observations point towards very weak or negligible <strong>mixing</strong><br />
near the bottom, even though a definite conclusion has to await direct measurements of<br />
turbulence.
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To investigate the ability of the model to reproduce these results, we started our runs<br />
with a typical late summer stratification for Lac Léman. The response to both idealized<br />
wind fields mentioned above, each with a duration of 1 day and a wind speed roughly<br />
corresponding to 7 m/s, was analysed at point A.<br />
v ( m / s )<br />
0 . 0 3<br />
0 . 0 2 5<br />
0 . 0 2<br />
0 . 0 1 5<br />
0 . 0 1<br />
0 . 0 0 5<br />
- 0 . 0 0 5<br />
- 0 . 0 1<br />
- 0 . 0 1 5<br />
0<br />
u<br />
S i m u l a t e d b o t t o m s p e e d f o r " V e n t "<br />
v<br />
- 0 . 0 2<br />
0 1 2 3 4 5<br />
t ( d a y s )<br />
v ( m / s )<br />
- 3<br />
x 1 0<br />
S i m u l a t e d b o t t o m s p e e d f o r " B i s e "<br />
1 0<br />
- 2<br />
- 4<br />
- 6<br />
8<br />
6<br />
4<br />
2<br />
0<br />
v<br />
u<br />
- 8<br />
0 1 2 3 4 5<br />
t ( d a y s )<br />
Figure 29: Time series of the x- and y-components u, v of the velocity at point A near the bottom.<br />
Left panel: with along-basin wind ‘Vent’. Right panel: with cross-basin wind ‘Bise’. Note the different<br />
velocity scales.<br />
The velocity components in x- and y-direction (see Figure 28) near the bottom are displayed<br />
in Figure 29 for both wind fields. It is evident that the cross-basin v-component is<br />
dominated by oscillations of roughly 10 hours in both cases. In particular for the ‘Bise’<br />
situation (right panel), it is clearly visible that these oscillations are correlated to similar<br />
oscillations in the along basin u-component with a respective phase shift of 90 degrees,<br />
a clear indication for Poincaré wave activity. This correlation can also be seen, though<br />
less clearly, for the ‘Vent’ situation, where the u-component is dominated by a low frequency<br />
contribution with a ‘period’ of approximately 3 days. Spectral analysis (not<br />
shown) confirms these results, in particular the peak around 10 hours and the higher<br />
energy at low frequencies in the u-component for the ‘Vent’ forcing. Current speeds associated<br />
with the higher frequency motions are at most 1 cm/s, and only the low frequency<br />
contribution of the along basin current can reach 2-3 cm/s for the relatively<br />
strong winds applied here.<br />
We conclude that the numerical model is able to correctly predict the most important<br />
components of the currents at this location. The somewhat shorter period of the Poincaré-waves<br />
is due to the fact that the model stratification in this example was slightly<br />
stronger than the measured one, leading to higher wave speeds and shorter periods. In<br />
addition, the measured spectra (Figure 21) represent an average over many weeks with<br />
periods of varying stratification, and perfect agreement cannot be expected.<br />
The turbulence model predicts zero turbulence at the bottom. Even though this is in apparent<br />
agreement with the absence of a measured well-mixed layer near the bottom<br />
(see above), this model result should not be overemphasized: It is well known that the<br />
class of turbulence models we adopted is not suited for the prediction of laminar-
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turbulent transition and strongly intermittent turbulence at very low Reynolds numbers.<br />
Both effects, however, have to be expected at the location investigated. At present,<br />
there exists no satisfactory theory for this regime of turbulent flows, and only when the<br />
boundary layer is fully turbulent, Reynolds stress models as that use here can yield<br />
reasonable results even when turbulence is rather weak. This has been shown recently<br />
by Lorke et al. (2002) for the bottom boundary layer of a small lake.<br />
3.3.2.2 Mixing at the Entrance of the Petit Lac<br />
The dynamics of mean currents and turbulence at point B, the entrance of the S-W appendix<br />
(‘Petit Lac’, see Figure 28), is quite different. Temperature profiles measured at<br />
this point for late summer and early winter 1987 are displayed in Figure 30. For the<br />
summer period (August/September), the temperature profiles exhibit a typical structure,<br />
which is also found in other years:<br />
• a well-mixed upper layer of about 10 m thickness<br />
• a strongly stratified upper thermocline from about 10 m to 30 m<br />
• a weakly stratified lower thermocline from about 30 m to 50 – 60 m<br />
• a well-mixed bottom boundary layer of 10 to 15 m thickness<br />
Later in the year, the thermocline is slowly mixed downward and eroded by penetrative<br />
convection, as is particularly visible from the slightly unstable temperature profile in December<br />
(see Figure 30). Finally, starting from January, the whole water column at this<br />
point is well mixed until re-stratification starts in spring (not shown). Current records at<br />
the entrance of the Petit Lac for a period of the same year 1987 are plotted in the right<br />
panel of Figure 30.<br />
d e p t h ( m )<br />
- 1 0<br />
- 2 0<br />
- 3 0<br />
- 4 0<br />
- 5 0<br />
- 6 0<br />
- 7 0<br />
0<br />
1 6 D e c<br />
T e m p e r a t u r e a t t h e e n t r a n c e o f t h e P e t i t L a c<br />
1 7 N o v<br />
2 8 O k t<br />
1 0 S e p<br />
1 8 A u g<br />
- 8 0<br />
5 1 0 1 5 2 0 2 5<br />
t ( d e g C )<br />
v ( m / s )<br />
0 . 4<br />
0 . 3<br />
0 . 2<br />
0 . 1<br />
- 0 . 1<br />
- 0 . 2<br />
- 0 . 3<br />
- 0 . 4<br />
0<br />
S p e e d a t t h e e n t r a n c e o f t h e P e t i t L a c<br />
6 0 m<br />
- 0 . 5<br />
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5<br />
d a y s f r o m 2 1 . O k t . 8 7<br />
Figure 30 : Left panel: profiles of temperature at the entrance of the Petit Lac for the second half of<br />
1987. Right panel: measured in- and outflow velocities at the entrance of the Petit Lac at 10 m and<br />
60 m depth. Outflow is positive.<br />
1 0 m
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It is clearly visible, that the speeds at this point are at least one order of magnitude<br />
higher than at the bottom of the main basin. In addition, currents measured in the<br />
epilimnion and the hypolimnion close to the bottom are of comparable magnitude, but<br />
almost always of opposite sign causing a strong shear across the thermocline. It is to<br />
be noted, that waves of fluid entering and leaving the Petit Lac can be non-linear. An<br />
extreme example is the current reversal in the lower and upper layer at day 25 (see<br />
Figure 30): Currents in both layers change sign within less than 30 minutes (the resolution<br />
of our measurements), indicating a strongly non-linear internal ‘bore’ entering the<br />
Petit Lac. It is very likely that turbulence caused by these processes is crucial for shaping<br />
the vertical profiles of passive and active scalars in the water column.<br />
We tried to model the basic features at position B by forcing our model with a strong<br />
‘Vent’-type along-basin wind of about 7 m/s, which lasts for one day and was then<br />
switched off. Numerical studies showed that a ‘Bise’ wind event also causes high velocities<br />
at the entrance of the Petit Lac. However, turbulence characteristics and vertical<br />
shear are quite similar in both cases, and we considered it sufficient to look only at one.<br />
The left panel of Figure 31 illustrates the structure of the velocity field 38 hours after the<br />
wind has started (and terminated after 24 hours). This is precisely the time at which the<br />
return current, driven by the pressure gradient caused by the interface set-up, reaches<br />
the entrance of the Petit Lac.<br />
As can be seen from this figure, the return current occurs in the form of a strongly nonlinear,<br />
coastally trapped ‘bore’ in the Northern part of the entrance, and in form of a<br />
more gradual adjustment in the Southern part. After the shock wave has passed, the<br />
inflow currents of warm surface water into the Petit Lac are distributed fairly homogeneous<br />
across the entrance (not shown). As discussed above, the existence of such<br />
non-linear waves is also indicated by our current measurements, if the initial wind forcing<br />
is strong. The correct prediction of these waves depends to a large degree on the<br />
ability of the model to reproduce strong horizontal gradients in scalar and vector fields,<br />
i.e. on the quality of the advection scheme.
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Figure 31: Left panel: surface velocities at the entrance of the Petit Lac (cf. Figure 28 for the geometry)<br />
38 hours after the start of an along-basin wind lasting for 24 hours. Dark red arrows indicate<br />
‘warm’ water, light red/orange arrows ‘cold’ water. Right panel: vertical velocity profile at<br />
point B (entrance of the Petit Lac) 50 hours after the start of the wind.<br />
The modelled vertical velocity profile 50 hours after the start of the wind, long after the<br />
‘bore’ of the return current has passed, is illustrated in the right panel of Figure 32.<br />
Since no measured velocity profiles are available at point B, we remark only on the<br />
most evident features of this profile, namely a strong shear near the bottom, a sharp<br />
maximum at about 47 m depth, a strong shear across the thermocline, and a more or<br />
less well-mixed upper layer.<br />
Figure 32: Left panel: vertical profile of the modelled temperature at point B (entrance of the Petit<br />
Lac) 50 hours after the start of the wind. Right panel: same as left panel, but now the turbulent diffusivity<br />
of momentum is displayed.<br />
The temperature profile at the same time is shown on the left panel of Figure 30. Remarkably,<br />
the vertical structure of this profile is very similar to that measured at the<br />
same location in late summer (see above): One easily identifies a well-mixed layer near
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the surface and the bottom, a strongly stratified upper thermocline, and a weakly stratified<br />
lower thermocline. Also the vertical extent and position of these different zones<br />
closely correspond to the measurements.<br />
The physical processes that shaped the thermocline structure become evident from the<br />
right panel of the same figure, which displays the turbulent diffusivity computed by the<br />
Reynolds stress model at the same time. As can be seen from this figure, the wellmixed<br />
bottom boundary layer is caused by strong turbulent diffusivities in the lowest 15<br />
meters, driven by the strong velocity shear near in this region. At the region of the velocity<br />
maximum (at about 47 m, see above), this shear is zero and diffusivities drop to<br />
very small values. This happens exactly at the lower edge of the stratified region.<br />
Above, the shear becomes strong again, turbulent diffusivities increase and cause an<br />
erosion of the thermocline from below. The result can be found in both, the measured<br />
and the modelled temperature profiles. In this region, the gradient Richardson number<br />
predicted by the turbulence model is approximately 0.25 (not shown), and thus <strong>mixing</strong><br />
occurs at high efficiency. From 30 to 10 meters depth, stratification becomes too strong,<br />
and turbulence is completely suppressed by local buoyancy effects. Only in the upper<br />
10 meters, wind <strong>mixing</strong> is strong enough to create a well-mixed region. Note, that at 50<br />
hours as in Figure 30 the wind has already been switched off, and diffusivities are only<br />
weak. This figure also gives a nice impression of the vertical numerical resolution required<br />
to resolve the basic features of the profile.<br />
3.4 CONCLUSIONS<br />
An analysis of internal <strong>seiche</strong>s dynamics was carried for Lac Léman combining field<br />
measurements and numerical modeling. Using field data of temperature, currents and<br />
surface elevation, it has been shown that only two modes of internals <strong>seiche</strong>s are sufficiently<br />
excited in Lac Léman to be considered significant. The first one is a Kelvin wave<br />
and the second one is a Poincaré wave. Model calculations have indicated that other<br />
<strong>seiche</strong> modes can only be excited by winds from certain directions. However, due to the<br />
topographic constraints particularly in the eastern part of the lake basin the wind field<br />
over the lake is strongly canalized and these winds do not exist in nature.<br />
The Kelvin wave progresses around the perimeter of the lake basin and its effects are<br />
most prominent in the nearshore zone. During the passage of the wave the thermocline<br />
descends by several meters and this provokes a thermocline displacement over the<br />
weakly sloping lateral zone which may easily reach 100 m. The combined alongshoredownslope<br />
velocity vector can reach speeds high enough to cause sediment erosion<br />
with potentially negative effects for the drinking water intake structures which are placed<br />
in the same zone. Since Kelvin waves are rather frequent in the Lake of Geneva, it can<br />
be expected that the dynamics of the nearshore zone vary on a periodic level during<br />
stratification.
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As shown above, Poincaré waves are most prominent in the offshore regions of the<br />
central basin. However, water surface recorder records near the shore also indicated<br />
their presence. Their mayor axis is oriented in the cross-lake direction. The transport<br />
resulting from these waves will link the water masses in the center of the lake with the<br />
near shore zones. This will affect the water mass residence times, in particular shortening<br />
those in the central part of the basin. We also observed that oxygen concentration<br />
in the near bottom layers in the center of the lake fluctuates with the period of<br />
Poincaré waves. This is further indication of the link between the water masses in the<br />
center of the lake and the lateral zones. Due to their period of about 12 h during summer,<br />
a certain correlation in the forcing with the diurnal windfield over the lake (Lemmin<br />
and D’Adamo, 1996) can be envisioned.<br />
From our analysis of field studies of the longterm mean conditions of <strong>mixing</strong> it is indicated<br />
that internal <strong>seiche</strong>s are important in providing vertical <strong>mixing</strong>. Recently, it has<br />
been shown though that most of this <strong>mixing</strong> is actually generated in the near shore<br />
zone and then propagates into the open waters (Wuest et al., 2000). Thus, the interaction<br />
between near shore zones and the open water is also important for <strong>mixing</strong>. Furthermore,<br />
we have pointed to the importance of the interaction with the sloping sides of<br />
the lake and short progressive internal waves (Thorpe and Lemmin, 1999a, Lemmin et<br />
al., 1998). These waves and their breaking play a role in the production and redistribution<br />
of currents and stratification as well as <strong>mixing</strong> (Thorpe and Jiang, 1998). From our<br />
studies it appears that short progressive internal waves are often produced in the passage<br />
of non-linear internal <strong>seiche</strong>s (Thorpe et al., 1996).<br />
A state-of-the-art numerical model for the three-dimensional shallow-water equations<br />
has been compared to measured currents at two dynamically very different locations in<br />
Lac Léman: A low-energy point close to the bottom in the deepest part of the lake, and<br />
a very active region at the entrance of the appendix ‘Petit-Lac’, both for a typical latesummer<br />
stratification. Even though the wind field was highly idealized, the basic components<br />
of the currents at both locations could be reproduced: The nearly linear Poincaré-wave<br />
signal with very low current speeds at the deepest point of the lake, and the<br />
pattern of fast inflow and outflow currents including effects of the non-linear return-wave<br />
at the entrance of the Petit Lac after the wind had stopped.<br />
Clearly, the basic theory of linear shallow water waves in stratified basins is known<br />
since many decades, and the existence of these waves in Lac Léman does not come<br />
as a great surprise. However, the ability of our model to predict these waves, in particular<br />
their non-linear steepening at the entrance of the Petit Lac, can be taken as an<br />
indication for the reliability of the numerical scheme.<br />
Much more important as the mere prediction of internal waves is the question of their<br />
contribution to the overall <strong>mixing</strong> in Lac Léman. This was the major topic of the work<br />
package, and new insight into the physics of this process has been obtained. It must,<br />
however, be noted that due to the low current speeds in the deepest part of the lake, no<br />
useful information about the turbulent characteristics in the deep hypolimnion can be<br />
expected from the turbulence model. Since our turbulence model is state-of-the-art in
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geophysical modelling, this deficiency points into the direction of more fundamental investigations<br />
on this topic. As long as no measurements of microstructure in the water<br />
column are available, it can not even be definitely concluded at what level (if at all) turbulence<br />
exists in the lowest quarter of the hypolimnion.<br />
Quite encouraging results have, however, been obtained from the comparison of<br />
measured and computed quantities at the entrance of the Petit Lac. The measured<br />
characteristics of the velocity time series (namely the non-linear wave) in the case of a<br />
strong wind event could be reproduced. In addition, the good agreement of the measured<br />
and computed structure of the temperature profile suggest that the turbulence<br />
model yields reasonable turbulent diffusivities. All these results indicate that the Petit<br />
Lac could serve as the main ‘mixer’ of hypolimnetic water in the whole basin, even<br />
though enhanced turbulence at the lateral boundaries of the lake may also play a role.<br />
The precise mechanisms of how <strong>mixing</strong> in the Petit Lac affects the main basin are not<br />
yet known. The following possibilities appear to be, however, the most reasonable:<br />
• mixed water from the Petit Lac enters the hypolimnion of the main basin by advection<br />
through the strong mean currents in this region<br />
• mixed water intrudes horizontally into the main basin, driven by the density differences<br />
due to differential <strong>mixing</strong><br />
• heavy water is ‘strained’ over lighter water and causes additional local <strong>mixing</strong><br />
due to static instability of the water column<br />
All processes could be identified in the model results. Experimental confirmation, however,<br />
would be required to confirm if these processes in fact occur in the lake, and to<br />
what extent each of them contributes to the overall <strong>mixing</strong> in Lac Léman.<br />
3.5 REFERENCES<br />
Bäuerle, E. (1985) <strong>Internal</strong> free oscillation in the Lake of Geneva. Ann. Geophysicae.<br />
2/3: 199-206.<br />
Bohle-Carbonell, M. (1986) Currents in Lake Leman. Limnol. Oceanogr. 31: 1255-1266.<br />
Bohle-Carbonell, M., and D. v. Senden (1990) On internal <strong>seiche</strong>s and noisy current<br />
fields- theoretical concepts versus observations. Large Lakes Ed. M. T. a. C. Seruya.<br />
Springer. 81-105.<br />
Burchard, H., and K. Bolding(2002) GETM – A General Estuarine Transport Model,<br />
EUR 20253 EN, European Commission Joint Research Center, 21020 Ispra, Italy<br />
Burchard, H., and O. Petersen (1997) Hybridisation between sigma and z coordinates<br />
for improving the internal pressure gradient calculation in marine models with steep<br />
bottom slope, International Journal for Numerical Methods in Fluids, 25, 1003-1023,<br />
Caloi, P., M. Migani, and G. Pannocchia (1961) Ancora sulle onde interne del lago di<br />
Bracciano e sui fenomeni ad esse collegati. Ann. Geofisica, Roma 14: 345-355.
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Canuto, V. M., A. Howard, Y. Cheng, and M. S. Dubovikov ( 2001) Ocean turbulence I:<br />
One-point closure model. Momentum and heat diffusivities. Journal of Physical Oceanography,<br />
31, 1413-1426,<br />
Heinz, G., et al (1990) Vertical <strong>mixing</strong> in Ueberlinger See, western part of Lake Constance.<br />
Aquat. Sci. 52:256-268.<br />
Jassby A. and T. Powell (1975) Vertical patterns of eddy diffusion during stratification in<br />
Castle Lake, California. Limnol. Oceanogr. 20: 530-543.<br />
Kanari, S. (1984) <strong>Internal</strong> waves and <strong>seiche</strong>s, Lake Biwa. S. Hone, ed. , Junk. Dordrecht,<br />
pp. 185- 235.<br />
Kanari, S. (1975) The long-period internal waves in Lake Biwa. Limnol. Oceanogr. 20:<br />
544-553.<br />
Lemmin, U. (1987) The structure and dynamics of internal waves in Baldeggersee.<br />
Limnol. Oceanogr. 32: 43-61.<br />
Lemmin, U., and N. D'Adamo (1996) Summertime winds and direct cyclonic circulation:<br />
observations from Lake Leman. Ann. Geophysicae 14: 1207-1220.<br />
Lemmin, U., et al (1998) Finescale dynamics of stratified waters near a sloping boundary<br />
of a lake. Physical processes in lakes and oceans Ed. J. Imberger. Amer. Geophys.<br />
Un., Washington, DC, Coastal and Estuarine Studies, 54:461-474.<br />
Lorke, A., L. Umlauf, T. Jonas, and A. Wüst, Dynamics of turbulence in low-speed oscillating<br />
bottom-boundary layers of stratified basins, Environmental Fluid Mechanics,<br />
accepted 2002<br />
Michalski, J. and U. Lemmin (1995) Dynamics of vertical <strong>mixing</strong> in the hypolimnion of a<br />
deep lake: Lake Leman. Limnol. Oceanogr. 40: 809-816.<br />
Mortimer, C.H. (1955) Some effects of earth rotation on water movements in stratified<br />
lakes. Verh. Int. Ver. Limnol. 12: 66-77.<br />
Mortimer, C..H. (1963) Frontiers in physical limnology with particular reference to long<br />
waves in rotating basins. Proc. 5th Conf. Great Lakes Res. Div., Univ. Michigan, 9-42.<br />
Mortimer, C.H. (1993) Long internal waves in lakes: review of a century of research.,<br />
Special report, No. 42, Univ. Wisconsin-Milwaukee, Center for Great Lakes Studies,<br />
117 pp.<br />
Mortimer, C.H., et al (1984) <strong>Internal</strong> oscillatory responses of Lake Geneva to wind impulses<br />
during 1977/78 compared with waves in rotating channel models. Commun. Lab.<br />
Hydraul., Ecole Polytech. Fed. Lausanne, No. 50, 89 pp.<br />
Richter, E. (1897) Seenstudien. Pencks Geogr. Abh., Wien 6: 121-191.<br />
Saggio, A., and J. Imberger (1998) <strong>Internal</strong> wave weather in a stratified lake. Limnol.<br />
Oceanogr. 43: 1780-1795.<br />
Service fédéral des eaux; SFE (1954). Les dénivellations du lac Léman. Département<br />
fédéral des postes et des chemins de fer. Report, maps, figures.<br />
Sirkes, Z. (1987) Surface manifestations of internal oscillations in a highly saline lake<br />
(the Dead Sea). Limnol. & Oceanogr. 32: 76-82.<br />
Thorpe, S. A. and U. Lemmin (1999a). <strong>Internal</strong> waves and temperature fronts on<br />
slopes. Ann. Geophysicae 17: 1227-1234.
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Thorpe, S. T., U. Lemmin, et al. (1999b). Observations of the thermal structure of a lake<br />
using a submarine. Limnol. Oceanogr. 44(6): 1575-1582.<br />
Thorpe, S. A. and R. Jiang (1998). “Estimating internal waves and diapycnical <strong>mixing</strong><br />
from conventional mooring data in a lake.” Limnol. Oceanogr. 43: 936-945.<br />
Thorpe, S. A., et al (1996) High frequency internal waves in Lake Leman. Phil. Trans.<br />
Roy. Soc. London A 354: 237-257.<br />
Thoulet, M. J. (1894) Contribution à l'étude des lacs des Vosges. Bull. Soc. Geographie<br />
15: 557-604.<br />
Umlauf, L., H. Burchard, and K. Hutter, Extending the k-omega turbulence model towards<br />
oceanic applications, Ocean Modelling, accepted 2002.<br />
Wuest, A., G. Piepke, et al. (2000). “Turbulent kinetic energy balance as a tool for estimating<br />
vertical eddy diffusivity in wind forced stratified waters.” Limnol. Oceanogr. 45:<br />
1388-1400.<br />
4 MAPS AND TABLES OF FREE INTERNAL SEICHES IN UPPER LAKE<br />
CONSTANCE FOR PRACTICAL USE<br />
4.1 OBJECTIVE<br />
The assessment of the local intensity of internal <strong>seiche</strong>s due to their variable structure<br />
is of considerable concern in applications of water management, certain water constructions<br />
and other executive measures, for which essential information on transient<br />
currents and corresponding water displacements is required. For instance, the spill of<br />
harmful substances, drift of lost bodies, <strong>mixing</strong> and dispersion in various limnological<br />
and hydrological problems may depend during the period of stratification occasionally<br />
strongly on internal <strong>seiche</strong>s. Their local effect in such cases can be estimated on the<br />
basis of adequately resolved graphical representations of their variable spatial intensity.<br />
As rough information it is often sufficient to get an idea on the local variation of potential<br />
activity by superposed internal <strong>seiche</strong>s of different order.<br />
To enable the expert community without resort to detailed knowledge of the physics of<br />
internal waves, the oscillations have to be displayed in a form easy to grasp by inspection.<br />
This was done in a different delineation than the structures usually shown in terms<br />
of wave parameters such as lines of equal range and phase, which give a condensed<br />
overview. Instead, the resolution into momentary wave stages and local amplitude<br />
variations with time has been presented providing better imagination and at the same<br />
time more refined quantitative information on the structure. The corresponding horizontal<br />
motion is given in field representations of ellipses of the rotating current vectors<br />
with indication of the sense of rotation and the zero phase position for a defined moment<br />
of the vertical elevation. Such a description consists of several diagrams for one<br />
mode and sums up to a collection of numerous illustrations as the number of modes<br />
represented increases. Despite this inconvenience the use is nevertheless facilitated,<br />
as the corresponding diagrams of each mode have the same scale and the thematic<br />
content has been drawn in the same graphically proper forms. This description has<br />
been compiled for the first 15 modes of Upper Lake Constance. A few examples are<br />
selected in the following to give an idea for practising.
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The work had been completed by Bäuerle and Ollinger (1991) in collaboration with<br />
Hollan in a project of the ISF. Later refined calculations by Bäuerle resulted in more<br />
details but confirmed the main structures of the oscillations already calculated in this<br />
early approach. Thus the compilation from 1991 is still worthwhile in the present context<br />
and may also serve as an example for the other large lakes under consideration. In<br />
particular reference to Lake Leman, the calculation with the same model has been carried<br />
out for the first 12 modes also by Bäuerle (1985). The extension to a similar presentation<br />
was included for merely two modes, thus not allowing for scanning the local<br />
potential contribution of all the 12 modes. The usefulness of the chosen graphic description<br />
of internal <strong>seiche</strong>s has been demonstrated very early by Bäuerle and Hollan<br />
(1983) for the case of the fundamental and a transverse mode of Lake Tanganyika enclosed<br />
in the monography on the lakes of the warm belt by Serruya and Pollingher<br />
(1983). This reference is quoted within the preceding one. C. Serruya recommended in<br />
this context to carry out such work on other large stratified lakes as accomplished later<br />
in the advanced version for Lake Constance here.<br />
4.2 OUTLINE OF THE CALCULATION<br />
The strength of internal <strong>seiche</strong>s is most adequately described in terms of forced oscillations.<br />
However, this approach implies precise knowledge of the driving agent, which is<br />
mostly the wind field over the lake. The horizontal variation of this quantity is generally<br />
not sufficiently known for that purpose, in particular, for large lakes. Moreover, the superposed<br />
different internal modes excited during one event have to be decomposed for<br />
identification of the associated single contributions. Desisting from such a difficult description,<br />
which is practically beyond reach, the relative structures and natural periods<br />
in terms of free oscillations may already serve for essential information to quantify the<br />
phenomenon. If certain observations exist on the mean amplitude of internal <strong>seiche</strong>s<br />
with respect to typical wind fields and the stratification in the lake, the determination<br />
relative to an arbitrary factor may be converted to absolute values, which often suffices<br />
for assessment.
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Figure 33 : Typical temperature stratification in the western part of Lake Constance (Lake Überlingen)<br />
mid October 1972 and the corresponding density stratification (relation ρ = ρ(<br />
T ( z))<br />
see<br />
text, from Bäuerle (1981)<br />
The problem is most conveniently solved by an eigenvalue-method, as the structures<br />
and periods of the oscillations are calculated as distinct constituents of each solution<br />
and result therefore very precisely. The question is, how detailed the hydrodynamic<br />
model is formulated to simulate nature. In the present context a two-layer model has<br />
been adopted including the Coriolis force. By the same reason, as the forcing has been<br />
kept out of concern, friction is not considered, since it represents generally a lakespecific<br />
process. It may be introduced by empirical information and rough linear assumption,<br />
if required for estimation. The approach under these conditions yields still a<br />
very useful description, when the calculations cover also characteristical stratifications<br />
during the warm season, what has been carried out for Upper Lake Constance. The<br />
simplification by a two-layer model restricts the solutions to the fundamental vertical order.<br />
This limitation is to a certain extent serious, as internal <strong>seiche</strong>s of second vertical<br />
order exist in lakes and should be included into the consideration. Since they are not so<br />
frequent and appear less pronounced generally than the fundamental vertical modes,<br />
their omittance may be tolerable for the time being.
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momentary surface<br />
h 1<br />
h 2<br />
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mean position<br />
of surface<br />
mean depth position<br />
of interface momentary depth<br />
position of interface<br />
density ρ 2<br />
density ρ 1<br />
Page 49 of 92<br />
Figure 34 : General sketch and definition of quantities of a two-layer model with respect to the<br />
stratification in a deep temperate lake in summer<br />
Lake Constance is a large deep lake in the moderate zone and develops therefore<br />
during the warm season a stratification, which consists of a relatively shallow surface<br />
layer (epilimnion) over a deep lower layer (hypolimnion). A typical example of the vertical<br />
density variation is given in Figure 33 for early autumn (October 1972). The relation<br />
between density and temperature applied here is valid for the waters of Lake Con-<br />
2<br />
stance (Hollan and Simons, 1978) and reads: ρ = ρ0<br />
−α<br />
( T − T0<br />
) with ρ0 = 1.000145<br />
g/cm³, T in °C, T0 = 4°C, α = 7.3 ·10 -6 g/(cm² °C²). While the upper depth range of 20 m<br />
is covered by the epilimnion, the hypolimnion extends from about 30 m to the maximum<br />
depth of 254 m, or on the average between 30 m and the mean depth of 100 m. The<br />
difference of thicknesses is even increased during summer, since the surface layer is<br />
generally less deep till October, when cooling becomes stronger. Such conditions are<br />
appropriate for a two-layer model of constant equivalent depth he . This quantity appears<br />
in the fundamental equations of internal <strong>seiche</strong>s in a two-layer system and allows for a<br />
description similar to surface (barotropic) <strong>seiche</strong>s, if it can be assumed as constant.<br />
With the definition of the two-layer model (see Figure 34) by an idealized step-like density<br />
stratification:<br />
�ρ1<br />
0 ≤ z < h1<br />
ρ0<br />
= �<br />
�ρ2<br />
h1<br />
≤ z < h(<br />
x,<br />
y)<br />
where x,y,z represent the coordinates of a Cartesian system with z directed vertically<br />
downward (z=0: surface), and ρ1, ρ2 constant densities with ρ2 > ρ1. The equivalent depth<br />
h e<br />
h1<br />
h<br />
=<br />
h + h<br />
1<br />
represents a quantity of low variation as to the deep and steep depth configuration of<br />
Lake Constance.<br />
It is therefore reasonable to approximate he by a constant value,
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(1)<br />
h1<br />
h2<br />
=<br />
h + h<br />
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h e<br />
1<br />
2<br />
Page 50 of 92<br />
where h2 gives now a corresponding constant thickness of the lower layer. In contrast to<br />
the gravity g determining the hydrodynamic pressure forces of surface <strong>seiche</strong>s, the low<br />
density difference between both layers at the interface implies that the corresponding<br />
forces as to internal <strong>seiche</strong>s are governed by reduced gravity, i.e.:<br />
∗<br />
(2) g = g ⋅ε<br />
with<br />
ρ2<br />
− ρ1<br />
ε =<br />
ρ<br />
From this reason result relatively high internal amplitudes of several meters and considerably<br />
long periods of several hours up to several days compared to those of surface<br />
<strong>seiche</strong>s in the same basin. As example , the period T1 of the fundamental mode and the<br />
corresponding frequency ω1 in a rectangular lake of length L, constant equivalent depth<br />
he, without influence of the earth’s rotation, read according to Merian’s formula:<br />
(3)<br />
L<br />
T =<br />
∗<br />
g h<br />
2<br />
1<br />
e<br />
and<br />
= π<br />
ω1 2<br />
∗<br />
g he<br />
In (3) the term g he<br />
∗<br />
represents the phase velocity ci of long internal waves in this system,<br />
which is considerably lower than that of long surface waves cs = g H , with H =<br />
h1+h2. Since the periods of surface <strong>seiche</strong>s obey the corresponding relation as in (3)<br />
with respect to cs , the great difference is evident.<br />
The third physical parameter in the fundamental equations accounts for the influence of<br />
the earth’s rotation. The effective component of the rotational vector of the earth, the<br />
Coriolis parameter f, reads:<br />
(4) f = 2Ωsinϕ<br />
with Ω = 7.29 ·10 -5 s -1 the angular velocity of the earth and ϕ the geographical latitude.<br />
For the mean geographical latitude of Lake Constance, ϕ = 48°N, f amounts to<br />
1.07 ·10 -4 s -1 .<br />
The derivation of the governing equations and their numerical solution is treated by<br />
Bäuerle (1981). After separation of the sinusoidal time dependency, the system of<br />
equations is solved for distinct eigen-frequencies ωn (n = 1,2,3 ...). It describes the dynamical<br />
relations for the dependent variables of the lower layer, which are the amplitudes<br />
of the volume transport 2 V� (x,y) = (U2(x,y), V2(x,y)) and the amplitude of the vertical<br />
displacement of the interface ζ2(x,y) at the top of this layer. It reads:<br />
∗ ∂ζ<br />
2<br />
iω U 2 + fV2<br />
+ g he<br />
= 0<br />
∂x<br />
∗ ∂ζ<br />
2<br />
(5a) iω V2<br />
− fU 2 + g he<br />
= 0<br />
∂y<br />
∂V2<br />
iω ζ 2 + fU 2 + = 0<br />
∂y<br />
with the boundary condition:<br />
L
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(5b) V ⋅ n = 0<br />
� �<br />
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2<br />
Page 51 of 92<br />
The condition (5b) means that there is no volume transport through to the rigid boundary<br />
expressed by the vanishing scalar product of 2 V� with the unit vector n � normal to the<br />
boundary. (5a) together with the condition (5b) forms an eigenvalue problem with an infinite<br />
number of distinct eigen-frequencies and corresponding eigen-solutions. The<br />
imaginary unit i = −1<br />
in the equations (5a) indicates that the solutions are given in<br />
�<br />
complex notation. Thus the eigen-solutions of the volume transports V 2,<br />
n (x,y) and the<br />
amplitudes ζ 2,<br />
n (x,y) represent complex functions and the corresponding real functions<br />
are evaluated for presentation in the course of the complex mathematical formulation.<br />
This treatment is also inherent to the nature of the oscillations as horizontally rotating<br />
long waves, which is caused by the influence of the Coriolis force. Due to this pattern a<br />
peculiar task of sufficiently resolved delineation for practical use has to be solved by a<br />
proper design, which is demonstrated in this report.<br />
A general property of the mathematical solution of (5a,b) is that the eigen-functions<br />
�<br />
V 2,<br />
n and ζ 2,<br />
n are determined except for a free factor. Thus the calculation yields the<br />
structure of the modes relative to 100% either of the maximum elevation of ζ 2,<br />
n or of the<br />
�<br />
maximum volume transport V 2,<br />
n encountered in the lake. Since the most interesting<br />
�<br />
functions for application with respect to the structure are ζ 2,<br />
n and V 1,<br />
n , which represents<br />
the amplitude of the volume transport in the upper layer, the relation of the latter quantity<br />
has to be supplemented here:<br />
� �<br />
(6) V 1,<br />
n = −V2,<br />
n<br />
From (6) the vertically averaged current in the epilimnion is inferred by<br />
�<br />
� V1,<br />
n<br />
(7a)<br />
v1,<br />
n =<br />
h<br />
while that in the lower layer results from<br />
�<br />
� V2,<br />
n<br />
(7b)<br />
v2,<br />
n =<br />
h<br />
with h2 given according to (1), if he and h1 had been prescribed.<br />
1<br />
2<br />
The equations (5a,b) are not solvable analytically even with constant he neither for very<br />
simple geometrical approximations of the basin nor for the irregular shape assumed<br />
here. Therefore, the solutions are determined by horizontal discretisation of the dependent<br />
variables and the rigid boundary. Their derivations in the fundamental equations<br />
(5a) are replaced by central differences of the variables discretised in a square<br />
grid of 1.4 km mesh size, as used by Hollan et al. (1980) for calculation of the surface<br />
<strong>seiche</strong>s of Lake Constance. The grid is shown Figure 35.
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Page 52 of 92<br />
Figure 35 : Outline of the numerical grid of 1.4 km mesh size adopted for Upper Lake Constance<br />
(Rao - grid in Hollan et al. (1980))<br />
The variables U2,n, V2,n and ζ2,n are defined on this grid, each staggered by 700 m. The<br />
details of the numerical solution are treated by Bäuerle (1981). The minimum depth at<br />
the numerical grid points is assumed to be somewhat larger than the thickness h1 of the<br />
epilimnion . This problem is discussed in Bäuerle’s (1981) treatise in more detail.<br />
4.3 AUXILIARY FORMULATIONS FOR EVALUATION OF EIGEN-PERIODS WITH<br />
RESPECT TO DIFFERENT STRATIFICATIONS<br />
The advantage of the two-layer model with constant equivalent depth consists in the<br />
property that the eigen-periods determined for selected different stratifications may be<br />
transformed to the cases of other stratifications by simple auxiliary relations. In order to<br />
arrive at a concise representation of the eigen-periods, as they vary with the modal order,<br />
with stratification and with respect to the Coriolis effect, this convenience of the<br />
model is utilized as follows. The transformation of the eigen-periods to different stratifications,<br />
if they are computed for one case, is non-ambiguously given, when there is no<br />
influence of the earth’s rotation, f = 0. Such internal <strong>seiche</strong>s represent standing oscillations.<br />
Their period spectrum is determined exclusively by the configuration of the basin<br />
and differs for each peculiar stratification by a constant factor. This factor q is inferred<br />
from the phase velocities ci of long internal waves<br />
∗<br />
(8) ci<br />
= g he<br />
which is characteristic for each case of stratification, as mentioned in the context of<br />
equation (3).<br />
When the eigen-frequencies have been calculated for a selected stratification which is<br />
considered formally as reference stratification and are designated by ωn(ref), the conversion<br />
factor q for the corresponding frequencies ωn(novel) of another stratification is<br />
defined by the ratio:<br />
ci<br />
( novel)<br />
(9)<br />
q =<br />
c ( ref )<br />
and the relation of the eigen-frequencies reads:<br />
(10) ω ( ) = q ⋅ω<br />
( ref ) n = 1, 2, 3...<br />
n<br />
i<br />
novel n<br />
Since the graphical representation of the dependency on the stratification and the influence<br />
of the earth’s rotation is very lucid and simple at the same time, if the frequencies<br />
are normalised to the fundamental frequency ω1 for f = 0, the conversion relation (10)<br />
may be put alternatively into the form
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(11) ω ( ) = p ⋅ω1(<br />
novel)<br />
n = 1, 2 ,3...<br />
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n<br />
novel n<br />
Page 53 of 92<br />
where the factors pn represent the normalised eigen-frequencies determined for one<br />
reference case of stratification by the relation (11), i.e. by pn = ωn(ref) / ω1(ref) for f = 0.<br />
The corresponding relations to (10) and (11) in terms of eigen-periods Tn read:<br />
(10)* ( novel)<br />
= T ( ref ) / q<br />
Tn n<br />
(11)* T n ( novel)<br />
= T1<br />
( novel)<br />
/ pn<br />
with pn = T1<br />
( ref ) / Tn<br />
( ref ) , resp.<br />
In order to make use of (11), merely ω1(novel) of the new stratification has to be determined<br />
from equations (9) and (10) for n = 1. It is this notation which had been adopted<br />
due to its feasibility for evaluation and delineation.<br />
The definition and evaluation for a reference stratification with f = 0 has been compiled<br />
in Table 1.
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mode ωn in 10 -5 sec Tn in hours<br />
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σ n = ωn<br />
1 1.910 91.38 1.00<br />
2 3.349 52.11 1.75<br />
3 4.699 37.14 2.46<br />
4 6.255 27.90 3.27<br />
5 7.587 23.00 3.97<br />
6 8.504 20.52 4.45<br />
7 9.096 19.19 4.76<br />
8 10.538 16.56 5.52<br />
9 11.223 15.55 5.88<br />
10 11.849 14.73 6.20<br />
11 12.836 13.60 6.72<br />
12 13.087 13.34 6.85<br />
13 14.016 12.45 7.34<br />
14 14.762 11.82 7.73<br />
15 15.430 11.31 8.08<br />
ω1<br />
Page 54 of 92<br />
Table 1: Eigen-frequencies, eigen-periods and normalised eigen-frequencies of the first 15 modes<br />
for an exemplary reference stratification with f = 0, ε = 5.3 ·10 -4 , he = 21m, h1 = 30m and<br />
h2 = 70m. This case has been observed in the western part of Lake Constance in October 1972 and<br />
had been assumed for the explanation of transverse internal oscillations in this region (Hollan<br />
1974, Bäuerle 1981)<br />
4.4 THE EIGEN-PERIODS INCLUDING THE CORIOLIS EFFECT<br />
As to the eigen-frequencies, the generally strong and complicated influence of the<br />
Coriolis force can be condensed into a very instructive and handy diagram for evaluation<br />
with the preceding consideration in mind. The effect of the earth’s rotation increases<br />
the larger the lake is and the smaller the phase velocity of the internal wave is.<br />
The latter may be due to weakening of the stratification or lowering the values of the<br />
equivalent depth by diminishing the thickness of the upper layer. The dependency on<br />
rotation is at best expressed with respect to the rotation number<br />
( 0)<br />
(12)<br />
F = f /ω<br />
1<br />
( 0)<br />
where F is the ratio of the local Coriolis frequency f and the eigenfrequency ω 1 of the<br />
fundamental mode without earth rotation.
σσ<br />
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σ<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 5 10 15 20<br />
Figure 36 : Dependency of the dimensionless eigen-frequencies<br />
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F<br />
σn ωn<br />
ω<br />
Page 55 of 92<br />
= on the rotation<br />
( 0)<br />
1<br />
( 0)<br />
F = f /ω for 15 modes of internal <strong>seiche</strong>s in Upper Lake Constance on the basis of a<br />
number<br />
1<br />
two-layer constant equivalent depth model (Figure 35) The vertical dashed lines indicate the dates<br />
for which the calculations of the 1991 report were done. The circles refer to Figures 38, 41, 44, 47<br />
and Figure 50 through Figure 55.<br />
( 0)<br />
Since the stratification enters into ω 1 proportionally to the phase velocity according to<br />
(8), the rotation number F increases, when either the stratification as given in (2) or the<br />
equivalent depth he diminish. The latter is effectuated by reduction of h1, the thickness of<br />
the surface layer. As mentioned above, the influence of the size of the lake is, by the<br />
way, detected with the help of (3) in terms of the length L in a rectangular basin. In this<br />
case the rotation number is proportional to L.<br />
With constant Coriolis frequency the rotation number F depends (inversely) on the<br />
( 0)<br />
( 0)<br />
value of ω 1 . As discussed above, ω 1 decreases with the length of the basin and with<br />
decreasing phase velocity. In other words, the larger and the less stratified the lake is,<br />
the larger is the influence of the earth rotation.<br />
At low F the internal <strong>seiche</strong>s resemble standing oscillations. With increasing F the effect<br />
of rotation becomes dominant and the structure of the oscillations takes the form of a<br />
rotating wave propagating around amphidromic points, where the amplitude of vertical<br />
displacements is zero. The number of amphidromic systems for the lower order modes<br />
is generally identical with the modal number.
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The influence of the rotation number on the eigen-frequencies is presented in condensed<br />
form in the diagram shown in Figure 36. Some explanations are necessary to<br />
make use of the information compiled in this figure. Increasing influence of rotation is<br />
represented by increasing rotation number F on the abscissa. On the other side, the or-<br />
( 0)<br />
dinate gives the normalised eigen-frequencies σ n = ωn<br />
ω1<br />
, discerned by increasing<br />
modal order up to 15. Thus, a family of 15 curves is displayed, each curve starting on<br />
the ordinate at F = 0, with the fixed normalised distribution for the case of Upper Lake<br />
Constance. These numbers σ n for f = 0 are listed in the fourth column of Table 1, as<br />
their general values result from the adopted reference stratification. For f = 0, they are<br />
the same for any different stratification as outlined in the previous section. These points<br />
on the ordinate mean the onset of the corresponding function showing the increasing<br />
influence of the earth’s rotation on the modal relative frequency in question.<br />
The effect of the stratification and varying equivalent depth is incorporated in the nor-<br />
( 0)<br />
malising fundamental frequency ω 1 . As to the other coordinate, the rotational number<br />
F gives the form, which reflects the dependency on the earth’s rotation with respect to<br />
the size and stratification of the lake. Since both variables are normalised to the same<br />
( 0)<br />
absolute quantity ω 1 , there is a simple method of evaluation with the help of this diagram.<br />
Given a particular stratification and equivalent depth, the corresponding rotation number,<br />
say Fe, is determined according to (12) and (3). The vertical line at this value on the<br />
abscissa crosses the curve family at 15 respective ordinate values σ n(<br />
F1<br />
) ,<br />
n = 1, 2, .., 15. From these relative eigen-frequencies the absolute values ωn(Fe) are<br />
( 0)<br />
obtained by multiplication with ω 1 . The corresponding eigen-periods result from the<br />
formula Tn ( F1<br />
) = 2π<br />
/ ωn<br />
( Fe<br />
) . This evaluation is presented for two examples selected from<br />
33 cases of stratification taken from observations in the lake during the period from<br />
1985 through 1989. The 15 eigen-periods for each case are tabulated by Bäuerle &<br />
Ollinger (1991) in their original report. The data of both examples shown here concern a<br />
case in spring on 13 April 1989 and another one in late summer on 30 August 1989.<br />
They represent a situation, which is strongly influenced by rotation (spring) and predominantly<br />
by stratification (late summer), respectively. The extracted stratification pa-<br />
( 0)<br />
( 0)<br />
rameters and resulting fundamental frequencies ω1 and periods T 1 , as well as the<br />
rotation numbers F are listed (Table 2).
13.4.<br />
1989<br />
1/q=<br />
3.3276<br />
30.8.<br />
1989<br />
1/q=<br />
0.7419<br />
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date h1 he<br />
13.04.1989 20 16.0<br />
30.08.1989 15 12.7<br />
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T1<br />
T2<br />
ρ1<br />
ρ2<br />
7.10 1.000075<br />
5.00 1.000138<br />
18.80 0.998546<br />
5.30 1.000133<br />
ε ci<br />
ω<br />
( 0)<br />
1<br />
( 0)<br />
1<br />
Page 57 of 92<br />
T F<br />
0.63 9.93 0.57 304.2 18.65<br />
15.86 44.54 2.57 67.8 4.16<br />
Table 2 : Approximation of temperature profiles from the central as well as deepest (254 m) position<br />
Fischbach-Uttwil of Upper Lake Constance in spring and late summer 1989 by two layers of<br />
constant density and resulting parameters of a two-layer equivalent depth model of internal<br />
<strong>seiche</strong>s. The total depth of the constant equivalent depth model is :<br />
H = h1+h2 = 100 m.<br />
date date of observation<br />
h1 depth of the upper layer [m]<br />
he equivalent depth [m]<br />
T1, T2 constant temperature of the upper and lower layer [°C]<br />
constant density of the upper and lower layer [g/cm³]<br />
ρ1, ρ2<br />
ε relative density difference ε = (ρ2 - ρ1)/ρ2 , [x10 -4 ]<br />
phase velocity of long internal waves [cm/s]<br />
ω eigen-frequency of the fundamental mode resulting from the two-<br />
ci<br />
( 0)<br />
1<br />
layer equivalent depth model for f = 0 [x10 -5 /s]<br />
( 0)<br />
T 1 corresponding eigen-period [h]<br />
( 0)<br />
F rotation number F = f / ω<br />
1<br />
F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15<br />
18.65 443.2 206.7 140.6 104.8 83.6 70.7 60.6 53.7 48.3 43.4 39.5 37.1 34.7 32.7 30.5<br />
0 304.1 173.4 123.6 92.8 76.5 68.3 63.9 55.1 51.7 49.0 45.3 44.4 41.4 39.3 37.6<br />
4.16 72.9 41.7 28.5 21.3 17.5 15.1 12.9 11.2 10.7 10.5 9.6 9.3 8.7 8.3 8.0<br />
0 67.8 38.7 27.6 20.7 17.1 15.2 14.2 12.3 11.5 10.9 10.1 9.9 9.2 8.8 8.4<br />
Table 3 : Eigen-periods (in h) of the first 15 modes of internal <strong>seiche</strong>s in Lake Constance for stratification<br />
in spring (13 April 1989) and late summer (30 August 1989) displayed with and without<br />
(F = 0) the effect of the earth’s rotation.
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Finally, the relative frequencies σ ( i ) with i = 1, 2 designating both cases, are read<br />
from the diagram in Figure 36. The absolute eigen-periods T ) resulting from<br />
n F<br />
T F ) = T / σ ( F ) are compiled in Table 3.<br />
n(<br />
i 1 n i<br />
In the spring case (13.4.1989) the periods T1 through T15, are rather long varying from<br />
443 h through 31 h, resp. Consequently the rotation number is relatively high,<br />
F1 = 18.65, and the influence of the Coriolis force is dominant. The effect is evident by<br />
the associated periods for the same case without rotation (f = 0) given in the second<br />
row for this date. The latter periods have been calculated from the internal phase velocity<br />
in this stratification quoted in Table 2, by using the relations (9) and (10)* with respect<br />
to the periods of the reference case listed in Table 1. There are considerable differences<br />
in the modal pairs of periods for f ≠ 0 and f = 0 in this case, which exhibit the<br />
strong rotational effects. It is remarkable that the eigen-periods of the lowest modes are<br />
considerably reduced for f = 0 compared to those influenced by rotation. For higher<br />
modal order from seven on the periods for f = 0 are higher. It is worthwhile to notice,<br />
that all the eigen-periods are greater than the inertial period. The consequences will be<br />
discussed in the following section.<br />
In the late summer case (30.8.1989) the rotation number (F2 = 4.16) is much smaller.<br />
While the periods vary from 72.9 h through 8.0 h for f ≠ 0 with the modal order from 1<br />
through 15, the corresponding periods for f = 0 deviate relatively less compared to the<br />
lowest orders of the previous case. The associated periods for f = 0 range from 67.8 to<br />
8.4. Despite that seemingly small effect on the eigen-periods, the effect of the earth’s<br />
rotation is of crucial importance for the higher modes. From general reasons, which are<br />
not delineated here, this is the case for the modes higher than order 5 which is indicated<br />
in Figure 36 by the straight line from the origin, σ = F, and the modes above it at<br />
F2 = 4.16. This transition is also reflected in the shift of relative amount of the periods<br />
between modal orders 5 and 6 given in Table 3. for f = 0 and f ≠ 0 in the last two lines,<br />
respectively.<br />
A final remark is in order for the application in case of an observed stratification which is<br />
not represented in the tabulated cases for f ≠ 0. Such a situation is generally to be expected<br />
and the large amount of 33 calculated examples has been achieved to meet this<br />
problem as to the periods. As approximation it is considered to be sufficient, if the most<br />
similar calculated case is found out as to the interesting stratification. If more precise<br />
assessment is required, an interpolation between two adjacent calculated cases may<br />
serve in the way that ci of one of them is greater and of the other is lower than that of<br />
the stratification in question.<br />
4.5 THE HORIZONTAL STRUCTURES OF THE INTERFACE AMPLITUDES<br />
Progressive rotating waves are usually illustrated in a diagram, which shows corange<br />
and so-called cotidal lines representing lines of equal amplitudes and of equal phases,<br />
resp. This delineation is inherent to the mathematical expression of such eigenoscillations.<br />
For a given eigen-frequency ωn the corresponding amplitude function of the<br />
solution as to the interface elevations reads:<br />
(13) ζ ( , y,<br />
t)<br />
= A ( x,<br />
y)<br />
cos( ω t + ϕ ( x,<br />
y))<br />
n = 1, 2, ...<br />
n x n<br />
n n<br />
n( Fi
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Since free eigen-oscillations are determined except for an arbitrary constant factor, the<br />
amplitudes An(x,y) are merely known to their relative distribution. Their absolute maximum<br />
may be ascribed the value of 100%, which occurs for every modal order generally<br />
at a different position in the interface. The corange lines show the horizontal structure<br />
and are displayed by equal percentage increments. The evaluation of the solution (13)<br />
is in terms of the formula<br />
(13a) A n ( x,<br />
y)<br />
= Pk<br />
with Pk in steps of 10% from -100% through +100%, correspondingly identified by the<br />
index k in the range {-10 (1) 10}.<br />
A<br />
B D<br />
C<br />
G<br />
F<br />
E<br />
Figure 37 : Contour of the numerical grid of Upper Lake Constance shown in Figure 35with the 19<br />
sites, from where the time dependent amplitude variation is illustrated in Figure 40, Figure 43,<br />
Figure 46 and Figure 49.<br />
Superposed in the same diagram are usually the cotidal lines, which represent the process<br />
of wave propagation. They are evaluated in equal steps of phase increase in the<br />
argument of the cosine function in (13). If the maximum amplitude is considered as<br />
phase stage, the cotidal lines are described by the formula<br />
2π<br />
k<br />
(13b) + ϕ n ( x,<br />
y)<br />
= 0 with k = 0, 1, 2, ..., K,<br />
K<br />
where K is the integer which divides the wave cycle into equal phase steps of 2π / K .<br />
The general pattern of cotidal lines in horizontally rotating waves is the radial arrangement<br />
of the curves meeting another in amphidromic points. When the wave progresses<br />
in the mathematical positive sense, i.e. is turning around the amphidromic point in anticlockwise<br />
direction, this point and system is called cyclonic, as the sense of the earth’s<br />
rotation is the same. The opposite sense of wave propagation designates the amphidromic<br />
system as anticyclonic.<br />
In the following, four sets of diagrams are presented as a selection from a total number<br />
of 30 which have been compiled for consultation of structural details of the first 15<br />
modes. The doubling results from the elaboration for the two aforementioned contrasting<br />
stratifications, namely, the case dominated by rotation in spring time (13 April 1989)<br />
and the case of governing gravity effects in late summer (30 August 1989). Particularly,<br />
the first and ninth modes are presented here, because the pattern of the structures and<br />
their variation with increasing modal order are disclosed well with this choice. Moreover,<br />
J<br />
I<br />
H<br />
L<br />
K<br />
M<br />
O<br />
N<br />
P<br />
R<br />
Q<br />
S
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they give a good idea of the instructive and feasible survey possible by the structurally<br />
more resolving illustrations.<br />
Before entering into the graphical inspection a reference map of 19 adjacent places on<br />
the boundary and in the interior of the lake has to be considered. These selected<br />
points, designated by the letters from A through S, are shown in Figure 37 on and inside<br />
the contour of the numerical grid. They serve for a proper display of local time histories<br />
of the interface displacements and are referred to in this peculiar thematic diagram<br />
in the sequence of illustrations for each selected mode. Besides these examples,<br />
there is another subset of diagrams, which shows transverse and longitudinal sections<br />
of the structure of interface displacements throughout the lake or in bays. Although<br />
these illustrations provide another comprehensive information, they are only mentioned<br />
here for the sake of brevity.<br />
After this preliminary note, we resort to the standard representations by corange and<br />
cotidal lines as outlined above. The corresponding diagrams are given in the first place<br />
of each series of representations in Figure 38 and Figure 41 as well as Figure 44 and<br />
Figure 47. The corange diagrams are resolved to 10% steps of the maximum, while the<br />
associated cotidal behaviour is displayed in steps of 1 /12 of the period. It is obvious from<br />
these delineations that a comparison of several consecutive modes at particular places<br />
in the lake is toilsome and will not help sufficiently for a comparative detailed description<br />
of the different relative local contributions. Therefore, the solutions (13) are represented<br />
additionally by realistic stages of the motion, particularly in time steps of the interface<br />
elevations, as they form in the process of wave propagation. This graphical<br />
resolution has been calculated for the same phase increments as in the cotidal diagrams<br />
and is illustrated by six stages for the first half of the corresponding periods in<br />
Figure 39 and Figure 42 and correspondingly in Figure 45 and Figure 48. The corange<br />
lines in these diagrams are resolved again in 10% steps of the maximum. They are correspondingly<br />
the same in the second half of the oscillation except for the change of the<br />
sign of elevations and need not be repeated.
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Figure 38 : Corange and cotidal diagram of the fundamental internal <strong>seiche</strong> for F = 18.65 (13 April<br />
1989) with the eigen-period of T1 = 407.3 h. The corange lines are given in 10% steps of the maximum,<br />
while the cotidal lines resolve the wave propagation in phase steps of 1 /12 T1 . The oscillation<br />
is cyclonic (anti-clockwise).<br />
If the continuous horizontal representation at different moments is changed to continuous<br />
display of the time history at different sites, a very instructive information results.<br />
This has been calculated for 16 sites on the shore and three in the middle of the lake as<br />
indicated in Figure 37. The resulting 19 diagrams are assembled in Figure 40 and<br />
Figure 43 as complement to the preceding figure and corresponding to the summer<br />
stratification in Figure 46 and Figure 49. The amplitude scale differs in this kind of diagrams<br />
from the previous relative representations, as it is showing the variation normalised<br />
to a maximum of 5 m. Such an assumption meets a realistic size of interface elevations<br />
associated often with internal <strong>seiche</strong>s in Lake Constance. How instructive this<br />
subset of local time histories is, may be perceived from comparable inspection of different<br />
modes for different stratification at a fixed place or at neighbouring places in relation<br />
with the discrete stages, in time, but represented continuously in horizontal dimension<br />
in the preceding figures. These complementary delineations of the amplitude structures<br />
serve a great deal for practical use in the context with the maps of the associated current<br />
field in the surface layer shown in the next section.<br />
A remark on the general structure and its variation with growing influence of the earth’s<br />
rotation on one hand and with gravity on the other hand has to be accomplished, as this<br />
pattern is manifested in the complete series of diagrams in the unpublished German<br />
report and is also apparent in the examples presented here. In the case of high rotation<br />
number, i.e. the case in spring time with extraordinary long eigen-periods, higher amplitudes<br />
are confined to the near-shore region throughout all calculated modes. Moreover,<br />
all amphidromic points in structures up to the 15 th mode are cyclonic and distributed<br />
very regularly, as if suspended on a mid-lake line from one end to the other. This<br />
is true except for two amphidromic points which evade gradually to the southern shore
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from the tenth mode on, namely towards the mouth of the Bay of Constance and into<br />
the Bay of Rorschach at the greatest width in the eastern half of the basin.<br />
The general regular structure is contrasted by the modes in the late summer stratification.<br />
Here, we realize a reduced confinement of higher amplitudes near the shores in<br />
the wider part of the basin. By contrast, the high amplitudes in the less wide parts as in<br />
the western half and at the eastern end are distributed similar to those of standing<br />
waves at vanishing rotation number. Thus there is a hybrid form of the general structure<br />
which resembles to a certain extent that of standing oscillations.<br />
As to the amphidromic points with increasing modal number, a characteristic difference<br />
appears with the ninth mode, which is represented for spring and summer stratification<br />
in Figure 41 and Figure 47, resp.. Regarding the latter case it has to be premised that<br />
up to the eighth mode all amphidromic systems are cyclonic, their centres aligned along<br />
mid-lake from the western to the eastern end, except for one situated in the Bay of Rorschach.<br />
In the ninth mode shown in Figure 47 a dominant anticyclonic amphidromic<br />
system appears just east of the centre of the lake and a second lateral cyclonic system<br />
appears off the Bay of Constance. For higher modes the structure is seemingly more<br />
complicated, since a few amphidromic systems exist near each other. However, this<br />
pattern develops with increasing mode number more into forms which resemble nodal<br />
lines of standing waves. This is disclosed by crowded cotidal lines connecting certain<br />
adjacent amphidromic points. Such structures represent the change of nodal lines in<br />
standing waves by weak rotational influences in so far, as the jump of the phase by π<br />
across the node is resolved into a steady variation within a narrow band along the node.<br />
Desisting from further considerations of the details an open question has still to be<br />
mentioned. The limitation of the calculations to two typical stratifications with respect to<br />
the structures leaves the user with the uncertainty, whether there are essential variations<br />
of the structures for intermediate cases of stratifications. Certainly, the late summer<br />
situation represents conditions of stratification, which resemble more each other<br />
throughout most of the warm season. Therefore, this part has more bearing on application,<br />
while the displayed case of strong rotational influence in spring may merely provide<br />
rough insight also into weaker autumnal stratification with deepened surface layer.<br />
To the pending completion of the calculations as to other interesting stratifications, the<br />
late summer example allows nevertheless for considerable insight and assessments, if<br />
the assumption of gradual variation of the structures with moderate change of stratification<br />
is correct. As this behaviour can be recognised in the structures with increasing<br />
modal order for both selected stratifications, it is to be expected also for modifications<br />
of these cases.
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Figure 39 : Momentary interface topographies of the first mode shown in Figure 38 at the first six<br />
phase steps of 1 /12 T1 . The lines of zero elevation are congruent with the cotidal lines in Figure 38.<br />
The increment of the corange lines is 10% of the maximum. (13 April 1989, T1 = 407.3 h)
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Figure 40 : Time histories of interface elevations of the first internal mode as in Figure 38 at 19<br />
sites on the shores and in the lake, however normalised to 5 m maximum amplitude. Map of sites<br />
in Figure 37 (13 April 1989, T1 = 407.3 h)
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Figure 41 : Corange and cotidal diagram of the ninth internal mode for the stratification on<br />
13 April 1989 with the eigen-period of T9 = 48.2 h (further explanation see Figure 38). The oscillation<br />
around all the amphidromic points is cyclonic (anti-clockwise).
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Figure 42 : Momentary structures of the amplitude distribution of the ninth mode shown in Figure<br />
41, 13 April 1989, T9 = 48.2 h (further explanation see Figure 39).
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Figure 43 : Time histories of interface elevations of the ninth internal mode at 19 sites on the<br />
shore and in the lake (13 April 1989, T9 = 48.2 h and further explanation in Figure 40)
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Figure 44 : Corange and cotidal diagram of the fundamental internal <strong>seiche</strong> for F = 4.16 (30 August<br />
1989) with the eigen-period of T1 = 72.5 h (further explanation see Figure 38). The oscillation is cyclonic<br />
(anti-clockwise).
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Figure 45 : Momentary interface topographies of the first mode shown in Figure 44 (30 August<br />
1989, T1 = 72.5 h, further explanation see Figure 39)
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Figure 46 : Time histories of interface elevations of the first mode at 19 sites on the shore and in<br />
the lake (30 August 1989, T1 = 72.5 h, further explanation see Figure 40)
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Figure 47 : Corange and cotidal diagram of the ninth internal <strong>seiche</strong> for the stratification on 30<br />
August 1989 with F = 4.16 and the eigen-period of T9 = 10.7 h (further explanation see Figure 38).<br />
All the amphidromic points are cyclonic (anti-clockwise) except for the central one in the mid of<br />
the main basin, where the oscillation turns clockwise.
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Figure 48 : Momentary interface topographies of the ninth mode shown in Figure 47 (30 August<br />
1989, T9 = 10.7 h, further explanation in Figure 39
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Figure 49 : Time histories of interface elevations of the ninth mode at 19 sites on the shore and in<br />
the lake (30 August 1989, T9 = 10.7 h, further explanation in Figure 39)
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4.6 THE HORIZONTAL STRUCTURES OF THE CURRENTS<br />
The information about the transports associated with the different modes of internal<br />
<strong>seiche</strong>s has important practical reasons as well. It allows for assessment of horizontal<br />
oscillatory displacements and dispersion of either harmful substances or other interesting<br />
compounds in the water, when the site considered is subjected to enhanced activity<br />
of internal <strong>seiche</strong>s. For this purpose, the diagrams have been extended to the horizontal<br />
dependency of the currents. To achieve this, another instructive graphical description<br />
had been elaborated, which meets the vectorial character of this quantity, and delivers<br />
the variation of speed and direction during one wave cycle in synoptic charts. Borrowing<br />
from the illustration of tidal currents, a similar graphical method had been<br />
adopted, namely by showing the current ellipses and, separately, the sense of rotation<br />
indicated at a peculiar moment of the current distribution.<br />
Figure 50 : Current field of the first internal mode in the surface-layer on 13 April 1989 with<br />
T1 = 407.3 h. Upper diagram: current ellipses. The normalised maximum transport at the southern<br />
shore of the entrance of Lake Überlingen is about 600 cm²/s). Lower diagram: momentary current<br />
distribution and sense of current vector rotation.<br />
Before treatment of details, the dimension of transport used here needs a comment.<br />
The numbers of transport are given in cm²/s which results from the vertical integration<br />
of the velocity throughout the upper or lower layer. This quantity is meant as transport<br />
per cm width transverse to the current and thus yields the usual dimension cm³/s.
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The evaluation of the current fields with respect to their part of the eigen-solutions is<br />
unique. However, a clear (and reasonable) graphical representation is only possible for<br />
the relative horizontal distribution of the transports in the upper or lower layer, according<br />
to<br />
V 1 = - V2<br />
→ →<br />
respectively. As the isotherm displacements due to internal oscillations are<br />
much easier to measure than the currents, normally it will be the task to refer the transports<br />
to the amplitudes of the vertical displacements of the interface between the upper<br />
and lower layer, which we have presented in the preceding section. Knowing the phase<br />
velocity and the more easily observable and assessable amplitude of the vertical interface<br />
displacement of a specific mode at any location in the basin it is possible to determine<br />
definite transports by evaluating the relative results of the numerical calculations.<br />
Figure 51 : Current field of the ninth internal mode in the surface-layer on 13 April 1989 with<br />
T9 = 48.2 h. The normalised maximum transport is situated at the mouth of the Bay of Constance<br />
and amounts to about 1800 cm²/s. (further explanation inFigure 50)<br />
To give an example: The vertical displacement of the interface by the fundamental<br />
mode at 30 August 1989 (Figure 44) has its maximum amplitude (100 %) at the very<br />
end of Lake Überlingen. The maximum transport of that mode occurs near the Sill of<br />
Mainau (Figure 53) indicating strong exchange flow between Lake Überlingen and the<br />
main basin of Upper Lake Constance. If we assume the value of 100% to be equivalent<br />
to 100 cm of real interface displacement and normalise the vertically integrated transport<br />
in the upper layer to this quantity, the amplitude of the horizontal transport at the<br />
central position of the Sill of Mainau would be about 4300 cm 2 /s with the definition of<br />
the dimension explained above. Taking the actual depth of the upper and lower layer,<br />
respectively, the horizontal velocities result for the present case to 4300/1500 cm/s and<br />
4300/8500 cm/s as to the lower and upper layer, respectively. These numbers are given<br />
in the legend of Figure 53. How the procedure would work, if the observations came
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from another location, tells the following example. Take the initial information at the<br />
central position of Lake Überlingen, where amplitudes of about 12 m are of common<br />
occurrence (Bäuerle et al., 1998): From Figure 44 it is inferred that at central Lake<br />
Überlingen the amplitude of the interface displacement is about 90 % of the maximum.<br />
This in turn yields that 13.3 m is the corresponding amplitude at the very end of Lake<br />
Überlingen. Finally, in analogy to the above case of reference, we get<br />
13.3 x 4300/1500 = 38 cm/s and 13.3 x 4300/8500 = 6.7 cm/s as vertically averaged<br />
velocities in the upper and lower layer in the Straits of Mainau, respectively, which correspond<br />
to an amplitude of 12 m measured at a central position of Lake Überlingen.<br />
Figure 52 : Current field of the eleventh internal mode in the surface-layer on 13 April 1989 with<br />
T11 = 39.8 h. The normalised maximum transport occurs at the mouth of the Bay of Constance and<br />
amounts to about 5600 cm²/s. The cross mark near the southern shore in the eastern half of the<br />
lake designates the site of the waste water intake discussed in the text. (further explanation in<br />
Figure 50)<br />
It should be mentioned that the same value of the phase velocity of long internal waves<br />
ci results from different combinations of he and ε according to the relations (1), (2), (8).<br />
Since the eigen-solutions of the problem (5) are uniquely determined for a very phase<br />
speed, ci, the respective definite transport field of the mode in question has to be<br />
evaluated with regard to he, i.e. the associated combination of h1 and h2, which fit together<br />
with ε in the relation (8). In this sense, there is a certain variety of two-layer<br />
manifestations equivalently related to a unique set of eigen-solutions. Thus, for the<br />
same maximum amplitude of a mode of them different definite transports result at a<br />
selected place, just depending on the differences allowed for by both the associated
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variations of the pair of upper and lower layer depths on one hand and the density difference,<br />
ε, on the other hand. This peculiar variety of two-layer cases covered by one<br />
set of eigenfunctions means a useful advantage, as the same calculations can be exploited<br />
for certain different two-layer approaches.<br />
The current fields of the first and ninth modes are depicted in Figure 50 and Figure 51<br />
for the case in spring and in Figure 53 and Figure 54 for that in late summer. Additionally,<br />
the currents of the eleventh mode are enclosed in Figure 52 and Figure 55 in order<br />
to emphasize the practical use by a peculiar application, which Hollan (1995) carried<br />
out and is outlined below. Except for the reference to the amplitude normalisation the<br />
relative variation of the structures in the current fields can be detected well from the<br />
diagrams. They consist each of a pair, showing the current ellipses in the upper diagram<br />
which indicate the position of the head of the current vector when turning during<br />
one cycle of oscillation. The main axis of the ellipses shows the orientation of the predominant<br />
current during a wave cycle, while the small axis gives the maximum transverse<br />
currents a quarter of the period later than the main currents. In the lower diagram<br />
the information is compiled, which concerns the sense of rotation at a given moment.<br />
Included into this stick representation is the sense of rotation of the current vector. The<br />
illustration of the phase relation and the sense of rotation throughout the lake is secondary<br />
for a rough evaluation, as this information is more important for inspection of<br />
closely adjacent conditions. The assessment of the local intensity of the currents is very<br />
comprehensive in the upper diagram. There is remarkable variation of the intensity and<br />
relative difference between the main and the transverse currents. This is easily perceived<br />
in the different examples and left to the reader for comparison.
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Figure 53 : Current field of the first internal mode in the surface layer on 30 August 1989 with<br />
T1 = 72.5 h. The transport normalised to 1 m maximum vertical displacement of the interface is<br />
4300 cm²/s at the Sill of Mainau, resulting in a current velocity of about 3 cm/s, in the upper layer<br />
(h1 = 15 m) and about 0.5 cm/s in the lower layer (h2 = 85 m), respectively. The evaluation is given<br />
on the previous pages (further explanation in Figure 50).
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Figure 54: Current field of the ninth internal mode in the surface layer on 30 August 1989, T9 = 10.7<br />
h. The normalised maximum transport (at the mouth of the Bay of Constance) is about 1800 cm 2 /s.<br />
(further explanation in Figure 50)<br />
Not shown here is an example of another additional set of diagrams on current ellipses.<br />
For more detailed local insight, the currents have been depicted in this sort of diagrams<br />
on a larger scale in selected sub-regions. The amount and phase variation with time<br />
along with the sense of rotation has been graphically well resolved. This very instructive<br />
information has been compiled for a few modes in sub-regions covering the complete<br />
Upper Lake Constance. As to this survey report, the quotation of this elaboration may<br />
suffice.
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Figure 55 : Current field of the eleventh internal mode in the surface layer on 30 August 1989,<br />
T11 = 9.6 h. The cross mark near the southern shore in the eastern half of the lake designates the<br />
site of the waste water intake discussed in the text. The normalised maximum transport (at the<br />
mouth of the Bay of Constance) is about 3400 cm 2 /s. The normalised transport at the waste water<br />
intake is about 3200 cm 2 /s. (further explanation in Figure 50)<br />
The above mentioned application by Hollan (1995) elucidates this way of consideration.<br />
For information, the site of a great waste water inlet, which was considered for alternative<br />
construction, is marked by a cross in the lower diagrams of Figure 52 and Figure<br />
55. These figures show the current fields of the eleventh mode in both cases. Compared<br />
to those of the first and ninth mode they differ in the region of this site remarkably,<br />
in particular for the late summer stratification, which is rather representative for the<br />
summer season. The investigation of dispersion conditions in front of this shore section,<br />
which is just aside of the mouth of the Old Rhine, resorted also to the potential activity<br />
of internal <strong>seiche</strong>s in this region. From the comparison of the relative current contributions<br />
by different modes of internal <strong>seiche</strong>s it was deduced, that considerable variability<br />
had to be expected from this process. These conditions can be recognised even from<br />
the examples shown here, as mentioned above. The result of the calculations with this<br />
application is supported by long-time experiences of local fishermen, who reported to<br />
the author (Hollan) about the difficulties with the retrieval of their drift nets in this region<br />
due to transient high vertical current shear in the thermocline. Such phenomenon is<br />
very probably caused by internal <strong>seiche</strong>s, since otherwise vertical shear would be associated<br />
with drift and compensating gradient current during stronger wind attack over the<br />
lake. However, during such conditions fishermen interrupt their work on the lake, what<br />
underscores the first interpretation.
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4.7 REFERENCES<br />
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Bäuerle E. (1981) : Die Eigenschwingungen abgeschlossener, zweigeschichteter Wasserbecken<br />
bei variabler Bodentopographie. Berichte aus dem Institut für Meereskunde<br />
an der Univ. Kiel, Nr. 85, Kiel, 79pp.<br />
Bäuerle E. and E. Hollan (1983): Calculation of the fundamental and a transverse mode<br />
of internal <strong>seiche</strong>s in Lake Tanganyika. In: Serruya C. and U. Pollingher: Lakes of the<br />
Warm Belt. Appendix, p.499-503, Cambridge, pp.569, Cambridge University Press.<br />
Bäuerle E. (1985): <strong>Internal</strong> free oscillations in the Lake of Geneva. Annales Geophysicae,<br />
Vol. 3, p.199-206.<br />
Bäuerle E. and D. Ollinger (1991): Karten-Dokumentation der internen Seiches des Bodensee-Obersees<br />
für den Gebrauch in der wasserwirtschaftlichen und limnologischen<br />
Anwendung. Unpublished report of the Institut zur Erforschung und zum Schutz der<br />
Gewässer Ottendorf, by contract with the Landesanstalt für Umweltschutz Baden-<br />
Württemberg, Institut für Seenforschung, Langenargen. 10p. with numerous tables and<br />
figures.<br />
Bäuerle E., D. Ollinger and J. Ilmberger (1998): Some meteorological, hydrological, and<br />
hydrodynamical aspects of Upper Lake Constance. In: Bäuerle, E. and Gaedke, U.<br />
(eds.): Lake Constance, Characterization of an ecosystem in transition. Arch. Hydrobiol.<br />
Spec. Issues Advanc. Limnol. 53, p. 31-83.<br />
Hollan E. (1974): Strömungsmessungen im Bodensee. Arbeitsgemeinschaft Wasserwerke<br />
Bodensee-Rhein (AWBR), Sechster Bericht, p.111-187.<br />
Hollan E. and T.J. Simons (1978): Wind-induced Changes of Temperature and Currents<br />
in Lake Constance. Arch. f. Meteorologie, Geophysik u. Bioklimatologie, Ser. A,<br />
Vol 27, p333-373.<br />
Hollan E., D.B. Rao and E. Bäuerle (1980): Free Surface Oscillations in Lake Constance<br />
with an Interpretation of the “Wonder of the Rising Water “ at Konstanz in 1549.<br />
Arch. f. Meteorologie, Geophysik u. Bioklimatologie, Ser. A, Vol. 29, No. 3, p.301-325.<br />
Hollan E. (1995): Ausbreitungsverhalten des Abwassers aus der Kläranlage Altenrhein<br />
im Fernbereich zweier alternativ projektierter Auslauföffnungen an der Halde südwestlich<br />
des Rheinspitz-Canyons im Bodensee-Obersee. <strong>Internal</strong> report of the Institut für<br />
Seenforschung (Landesanstalt für Umweltschutz Baden-Württemberg), by contract with<br />
the Amt für Umweltschutz of the Kanton St. Gallen, Langenargen, pp.17, 1map, 12 figures.
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5 LOCH LOMOND<br />
Although there were no specific modelling activities foreseen within EUROLAKES investigating<br />
<strong>mixing</strong> by internal waves in Loch Lomond it was nevertheless found advantageous<br />
to record recent experiences from field campaigns on the importance of long<br />
internal waves in this lake of irregular shape.<br />
5.1 INTRODUCTION<br />
Loch Lomond is a warm monomictic lake with three very distinct basins. The northern<br />
and central basins are narrow fjord-like (with up to 200 m water depth) and are separated<br />
by a chain of islands from the shallow south basin with a maximum depth of 30 m.<br />
Geologically the loch is a long deep trough of glacial origin and is separated from the<br />
sea by a moraine dam in the south. The mean lake level of Loch Lomond is not higher<br />
than 8 metres above mean sea level. Stratified conditions progressively occur from May<br />
through the summer months particularly in the north basin where a thermocline develops<br />
at a depth of about 15-25 m, separating the warmer epilimnion (ca. 14°C) from the<br />
cooler hypolimnion (ca. 6° C) which was already noted by Slack (1957).<br />
There are, however, only a few past recordings of temperature profiles, the only consistent<br />
approach has been done between 1969 and 1972 by Tippett (1994) with roughly<br />
one measurement profile of temperature and oxygen per month in the centre of every<br />
basin. The result of these experiments are depicted in the following figures for the years<br />
1970 and 1971. Whereas in the south basin thermal stratification occurred only for a<br />
few weeks the deeper portions of the loch show considerable stratification during the<br />
summer months.<br />
Figure 56 : Temperature profiles in the southern basin of Loch Lomond during 1970 and 1971<br />
(data according to Tippett, 1994) with short periods of thermal stratification
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Figure 57 : Temperature profiles in the northern (above) and central basins (below) of Loch Lomond<br />
during 1970 and 1971 (data according to Tippett, 1994) with pronounced stratification and<br />
high probability of long internal waves
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5.2 MEASUREMENT INFORMATION ON INTERNAL WAVES<br />
As there is an apparent lack of information concerning the temporal development of the<br />
seasonal thermo- and pycnoclines in Loch Lomond it was decided within the EURO-<br />
LAKES project to carry out a long-term monitoring survey with thermistor-chains<br />
moored at three locations along the main north-south axis of the loch. These thermistor<br />
chains were planned to provide a complete one year data set on the seasonal changes<br />
in water temperature on three vertical profiles from the surface down to a maximum<br />
depth of 50 meters. In conjunction with a number of quasi-synoptic CTD-surveys and<br />
further meteorological monitoring data on solar radiation, air temperature, wind, etc. the<br />
thermistor chain data will provide more comprehensive information to obtain a better<br />
understanding of Loch Lomond’s physics and even more to get a better understanding<br />
of its complete ecosystem.<br />
North Basin<br />
Mid Basin<br />
South Basin<br />
Figure 58 : Location of long-term deployments of thermistor chains in 2002
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The monitoring survey has been carried out with thermistor chains (AANDERAA Instruments,<br />
Norway) which were provided by the “Institut für Seenforschung” in Langenargen,<br />
Germany within the frame of the scientific co-operation in the research project<br />
EUROLAKES. Management and operations during the survey were carried out by the<br />
project partners from University of Glasgow. The thermistor chain measurements were<br />
planned to cover a time period of at least one year. To gain a high resolution data set<br />
regarding the thermodynamic processes during the development phase of the summer<br />
thermocline a sampling rate of 10 minutes was chosen for the first deployment in the<br />
mid basin in May 2002. Realising that a 10 minutes measuring interval would need data<br />
collection at least every month, it was decided to switch over to a sampling rate of 30<br />
minutes for all deployments from June 2002 onwards. In this report data from the first<br />
measurement period in summer 2002 are used to look at the relative importance of<br />
long internal waves.<br />
Figure 59 : Temperature measured by individual thermistors from May 3 rd to June 8 th 2002 by the<br />
upper chain (in 3 m to 23 m water depth every two metres) in the mid basin (Post, 2002)
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Figure 60 : Temperature profiles from 3 rd May to 9 th June 2002 in the mid basin position (Post,<br />
2002)<br />
The recordings from all three locations show considerable short-term fluctuations and at<br />
certain times sudden vertical homogeneity in the thermal conditions. Interpretation of<br />
these (horizontally seen) point measurements is not as straightforward as it might seem<br />
because Loch Lomond has a very irregular shape. This means that a generalisation of<br />
these profiles as “basin characteristic” is not possible. Investigations with 3D models<br />
(described in other reports of EUROLAKES) did show strong wind-driven currents with<br />
pronounced upwelling/downwelling processes near steep lake shores and the possible<br />
occurrence of longer period internal standing waves in the fjord-like section of the lake.<br />
Generally, however, it can be stated that long internal waves in the south basin are very<br />
intermittent and associated <strong>mixing</strong> processes will be much smaller than the turbulent<br />
<strong>mixing</strong> associated with the wind stress at the surface. Therefore in this report analysis<br />
of data is confined to the mid and north basin locations where stratification is strong in<br />
the upper 40 metres of the water column. Measurements are scheduled to proceed until<br />
spring 2003 but we are concentrating here on the summer situation.<br />
12.7<br />
12.2<br />
11.7<br />
11.2<br />
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10.2<br />
9.7<br />
9.2<br />
8.7<br />
8.2<br />
7.7<br />
7.2<br />
6.7
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5.3 ANALYSIS AND INTERPRETATION<br />
In order to obtain a clearer picture of the kind of temperature fluctuations occurring in<br />
the upper water column in Loch Lomond the thermistor recordings from both north and<br />
mid basin locations were analysed statistically for the period of 8 th August and 18 th<br />
September 2002.<br />
In the following temperature histograms are shown for two vertical levels (6 metres and<br />
45 metres below surface) for north and mid basin deployments depicting the existence<br />
of pronounced long internal waves in Loch Lomond with a stronger variability of temperature<br />
in the central basin.<br />
Frequency analysis provides a few conspicuous periods ranging between about 5 hours<br />
and 24 hours (a frequency of 0.25 in the figures is connected with a period of 2 hours =<br />
time step of recording divided by frequency). The periods coincide with theoretical values<br />
for long standing waves in a narrow channel of 20 km length (north + mid basin) for<br />
strong vertical density stratification near the surface and less strong ones below 20 metres.<br />
The results for the mid basin prove quite clearly that it reacts as an appendix to the<br />
north basin because the long periods cannot be explained by its own basin length of<br />
roughly five kilometres. The differences in periods are due to different thermal stratification<br />
conditions.<br />
5.4 REFERENCES<br />
Post, J. (2002): Thermistor measurements in Loch Lomond. HYDROMOD Scientific<br />
Consulting, Wedel – unpublished report.<br />
Slack H. D. (ed.) (1957) Studies on Loch Lomond 1. Blackie and Son Ltd., Glasgow.<br />
Tippett R. (1994): An Introduction to Loch Lomond, Hydrobiologia, 290, 11-15. Kluwer<br />
Academic Publishers, Dordrecht / Boston / London.
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Figure 61 : Histogram of temperature at 6 metres water depth for north basin (above) and mid basin<br />
(below) during 8 th August to 18 th 0.0<br />
15.0 15.4 15.8 16.2 16.6 17.0 17.4 17.8 18.2 18.6 19.0<br />
September 2002
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Figure 62 : Histogram of temperature at 45 metres water depth for north basin (above) and mid basin<br />
(below) between 8 th August and 18 th 0<br />
7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2<br />
September 2002
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Figure 63 : Frequency analysis of temperature fluctuations in mid basin (at 16 to 41 metres, picture<br />
above) and north basin (20 to 51 metres, picture below) for low pass-filtered data.
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6 CONCLUSIONS<br />
The better understanding and knowledge of the internal gravity waves (<strong>seiche</strong>s) in deep<br />
large lakes were the main objectives of this <strong>study</strong>. We have focused the work principally<br />
on the determination of the waves properties through measurements analysis and numerical<br />
simulation. The <strong>seiche</strong>s qualification has been done all over the four studied<br />
lakes. The influence of the <strong>seiche</strong>s on <strong>mixing</strong> processes was also studied.<br />
Several approaches of the internal gravity wave in large deep lakes have been presented:<br />
• Analysis of existing measurement on Loch Lomond, Lac du Bourget, Lac<br />
Léman,<br />
• Linear analysis (eigen-value method): Lake Constance,<br />
• Numerical 3D model analysis: Lac du Bourget, Lac Léman<br />
6.1 LAC DU BOURGET<br />
The analysed measurements are shown the existence of internal gravity wave in the<br />
Lac du Bourget with a period between 40hours to 80 hours depending on the vertical<br />
stratification structure.<br />
The simulation using the TELEMAC-3D model is used to reproduce one <strong>seiche</strong> event<br />
on the Lac du Bourget. The numerical results are in a quite good qualitative agreement<br />
with the measurements.<br />
The numerical results are given some interesting new information about the hydraulic<br />
respond of the lake under wind forcing. After the generation of the internal <strong>seiche</strong>s, the<br />
wave propagation is showing a rotating structure, due to the Coriolis influence, with an<br />
amphidromic point at the middle of the lake (Figure 16).<br />
6.2 LAC LÉMAN<br />
An analysis of internal <strong>seiche</strong>s dynamics was carried out for Lac Léman combining field<br />
measurements and numerical modeling. Using field data of temperature, currents and<br />
surface elevation, it has been shown that only two modes of internals <strong>seiche</strong>s are sufficiently<br />
excited in Lac Léman to be considered significant. The first one is a Kelvin wave<br />
and the second one is a Poincaré wave. Model calculations have indicated that other<br />
<strong>seiche</strong> modes can only be excited by winds from certain directions. However, due to the<br />
topographic constraints particularly in the eastern part of the lake basin the wind field<br />
over the lake is strongly canalized and these winds do not exist in nature.<br />
From our analysis of field studies of the longterm mean conditions of <strong>mixing</strong> it is indicated<br />
that internal <strong>seiche</strong>s are important in providing vertical <strong>mixing</strong>. Recently, it has<br />
been shown though that most of this <strong>mixing</strong> is actually generated in the near shore<br />
zone and then propagates into the open waters (Wuest et al., 2000). Thus, the interaction<br />
between near shore zones and the open water is also important for <strong>mixing</strong>. Furthermore,<br />
we have pointed to the importance of the interaction with the sloping sides of<br />
the lake and short progressive internal waves (Thorpe and Lemmin, 1999a, Lemmin et
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas<br />
EUROLAKES<br />
<strong>D28</strong>: <strong>Internal</strong> <strong>seiche</strong> <strong>mixing</strong> <strong>study</strong><br />
FP5_Contract No.: EVK1-CT1999-00004<br />
Version: 1.2<br />
Date: 24.08.2004<br />
File: <strong>D28</strong>.doc<br />
Page 92 of 92<br />
al., 1998). These waves and their breaking play a role in the production and redistribution<br />
of currents and stratification as well as <strong>mixing</strong> (Thorpe and Jiang, 1998). From our<br />
studies it appears that short progressive internal waves are often produced in the passage<br />
of non-linear internal <strong>seiche</strong>s (Thorpe et al., 1996).<br />
The measured characteristics of the velocity time series (namely the non-linear wave) in<br />
the case of a strong wind event could be reproduced. In addition, the good agreement<br />
of the measured and computed structure of the temperature profile suggest that the<br />
turbulence model yields reasonable turbulent diffusivities.<br />
6.3 LAKE CONSTANCE (BODENSEE)<br />
The method of calculating the eigen-periods of free internal oscillations in a two-layermodel<br />
also permits a relatively simple evaluation of the influence of the stratification on<br />
the eigen-periods. But also the variation of the structure with respect to the stratification<br />
is calculated at the same time and provides the characteristical differences, which result<br />
from strong or diminishing influence of the earth´s rotation via varying stratification.<br />
The application of the eigen-value method should be calculated for other lakes in order<br />
permit the comparison and evaluation of the lake-specific characteristics<br />
6.4 LOCH LOMOND<br />
In order to obtain a clearer picture of the kind of temperature fluctuations occurring in<br />
the upper water column in Loch Lomond the thermistor recordings from both north and<br />
mid basin locations were analysed statistically for the period of 8 th August and 18 th<br />
September 2002. Frequency analysis provides a few conspicuous periods ranging between<br />
about 5 hours and 24 hours.