D28: Internal seiche mixing study - Hydromod

hydromod

D28: Internal seiche mixing study - Hydromod

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D28: Internal seiche mixing study

D28: Internal seiche mixing study

Work package No. 7: Mixing by internal seiches

Lead contractor: SOG

Main objective: Key Processes

Strategic leader: Malgorzata Loga-Karpinska (WuT)

Responsible task leader: Claude GUILBAUD (SOG)

Main contributor involved: Organisation E-Mail

Claude GUILBAUD SOG claude.guilbaud@sogreah.fr

Eckard HOLLAN ISF isf.eurolakes@lfula.lfu.bwl.de

Bernd WAHL ISF isf.eurolakes@lfula.lfu.bwl.de

Kurt DUWE HYD duwe@hydromod.de

Ulrich LEMMIN EPF ulrich.lemmin@epfl.ch

Lars UMLAUF EPF lars.umlauf@epfl.ch

Maciej FILOCHA WUT maciej.Filocha@is.pw.edu.pl

Dissemination level: Public

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D28: Internal seiche mixing study

Table of contents

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1 Introduction 4

1.1 GENERAL CONSIDERATIONS 4

1.2 OBJECTIVES 5

2 Lac du Bourget 6

2.1 INTRODUCTION 6

2.2 THE ENVIRONMENT OF LAC DU BOURGET 6

2.3 MEASUREMENT ON THE LAKE 7

2.3.1 The data 7

2.4 APPLICATIONS 11

2.4.1 TELEMAC System 11

2.4.2 SIMULATED SEICHES – ANALYSIS 17

2.5 CONCLUSIONS 20

2.6 REFERENCES 20

3 Internal seiches and mixing in Lac Léman 21

3.1 THE LAC LÉMAN ENVIRONMENT 21

3.2 FIELD STUDIES 22

3.2.1 Modes detected 22

3.2.2 Wave propagation patterns 24

3.2.3 Evidence of Poincaré waves 31

3.2.4 Progressive vector analysis 31

3.2.5 Mixing by internal seiches 32

3.3 NUMERICAL SIMULATIONS 35

3.3.1 Model Description 35

3.3.2 Mixing by Long Internal Waves - Model Results 36

3.4 CONCLUSIONS 42

3.5 REFERENCES 44

4 Maps and tables of free internal seiches in Upper Lake Constance for practical use 46

4.1 OBJECTIVE 46

4.2 OUTLINE OF THE CALCULATION 47

4.3 AUXILIARY FORMULATIONS FOR EVALUATION OF EIGEN-PERIODS WITH RESPECT TO DIFFERENT

STRATIFICATIONS 52

4.4 THE EIGEN-PERIODS INCLUDING THE CORIOLIS EFFECT 54

4.5 THE HORIZONTAL STRUCTURES OF THE INTERFACE AMPLITUDES 58

4.6 THE HORIZONTAL STRUCTURES OF THE CURRENTS 74

4.7 REFERENCES 81


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5 Loch Lomond 82

5.1 INTRODUCTION 82

5.2 MEASUREMENT INFORMATION ON INTERNAL WAVES 84

5.3 ANALYSIS AND INTERPRETATION 87

5.4 REFERENCES 87

6 Conclusions 91

6.1 LAC DU BOURGET 91

6.2 LAC LÉMAN 91

6.3 LAKE CONSTANCE (BODENSEE) 92

6.4 LOCH LOMOND 92


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D28: Internal seiche mixing study

1 INTRODUCTION

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1.1 GENERAL CONSIDERATIONS

During summer, heating of lake surface water leads to the constitution of a stable interface

with strong thermal and density gradients at intermediate depth, the mesolimnion.

Wind stress at the lake surface induces a tilting of this density interface. After the

wind event, the interface oscillates in a standing wave movement (seiches) to converge

slowly to the new equilibrium position. The characteristics of these internal waves depend

on the wind intensity and period between successive wind events. Internal wave

amplitude may by amplified depending on the lake geometry and bathymetry. The dynamics

of these phenomena are important for vertical mixing and therefore for the reaeration

of mesolimnion waters

Long internal waves in lakes commonly take the form of standing waves (seiches) of frequency

and form determined by basin shape and density (temperature) stratification in the

water column. Recognised by Thoulet (1894) and Richter (1897), the "temperature seiche"

phenomenon was first systematically explored and interpreted by Wedderburn (1912 and

earlier papers). It was later shown (Mortimer 1953, 1963, 1993) to be a widespread response

to thermocline tilt initiated by wind action on stratified lakes of all sizes and

shapes, modified by Earth's rotation as the lake size increases. Various modes of oscillation

can be excited, commonly modes No 1 and 2 (depending upon the vertical density

profile) and several horizontal modes with No 1 usually dominant. The history of research

on such waves has been reviewed by Mortimer (1993).

In small lakes, internal seiches are end-to-end motions along the lake axis (Lemmin

1987). In large lakes, the Coriolis force transforms the end-to-end motion into rotating

(amphidromic) wave patterns, if the characteristic length scale of the lake (typically the

lake's width), exceeds the Rossby radius of deformation a = ci/f. Here f is the latitudedependent

Coriolis parameter and ci is the speed of long internal waves in the absence of

rotation. As described below, the first horizontal mode resembles a shore-hugging Kelvin

wave traveling counter-clockwise (cyclonically) around the lake basin in northern hemisphere

lakes, as was first demonstrated in the Lake of Geneva (Mortimer 1963) and later

in Lake Biwa, Japan (Kanari 1975, 1984).

For higher horizontal modes, the resulting rotating wave patterns in large lakes depend on

the basin shape and size and become more complicated. Observations are often lacking.

The spatial structure of higher modes can best be visualized through comparisons of numerical

models with observations. In thermally stratified lakes, those observations usually

take the form of vertical oscillations of near thermocline isotherms and/or current measurements.

Four large and deep lakes in Europe are investigated in relation with the internal

seiches phenomenon: Lac du Bourget, Lake Constance, Lac Léman and Loch Lomond.


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Different approach were used:

• Lac du Bourget : Analysis of measurements available and numerical simulation

with a three-dimensional model of the internals waves

• Lake Constance : Estimation of the internal waves through linear theory model

• Lac Léman: Analysis of measurements and numerical simulation with a threedimensional

model of the internals waves

• Loch Lomond: Measurements analysis

1.2 OBJECTIVES

To quantified the influence of internal seiches it is crucial to know the properties of possible

internal gravity waves (periods, amplitude, located depth, Coriolis influence, associated

current characteristics).Then the influence of internal seiches on mixing processes

and there impact concerning the water management, constructions or executives

measures could be investigated.

During the period of summer stratification, mixing in lakes can be cause by vertical

mixing, boundary mixing, interbasin density current and river inflows. In Lac Léman,

river inflows and interbasin density currents are of little importance during summer

stratification. Vertical mixing can be related to wind induced shear or to internal seiche

and internal wave activity.

The assessment of the local intensity of internal seiches due to their variable structure

is of considerable concern in applications of water management, certain water constructions

and other executive measures, for which essential information on transient

currents and corresponding water displacements are required.


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2 LAC DU BOURGET

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2.1 INTRODUCTION

The Lac du Bourget is the smallest and the less deep of the studied lakes of the project.

There is very few data available in the Lac du Bourget in relation to the internal gravity

waves (P.E. Bournet 1996). These data were analysed in order to identify and quantify

the seiches in the lake. Three episodes located principally in winter have been selected.

The numerical model (TELEMAC-3D) was used to reproduce the ”real world”. We tried

to reproduce the hydraulic respond of the stratified lake under the wind forcing. The

comparison between the model results and the measurement is shown general qualitative

agreement.

2.2 THE ENVIRONMENT OF LAC DU BOURGET

The Lac du Bourget is the largest natural

lake in France located in the French Alps

(area: 44.5 km 2 , length: 18 km, average

depth: 85 m; maximum depth: 145 m,

lake surface altitude: 231 m). Lac du

Bourget is set in a depression that geologists

call a syncline, resulting from the

folding of the Alpine chain during the Tertiary

era. The depression was further

B point

deepened by almost 145 m.

Figure 2 : Lac du Bourget It is oriented in

the North-south direction and it is composed

of two main basins with similar

size: the northern basin with the maximum

depth (145 m); and the Southern

one with a maximum depth of 70 m.

The two basins are separate by a con-

River Sierroz traction of the coastline. The maximum

width is about 3.2 km at Grésine and the

South basin

minimum width at Saint-Innocent is 1.6

T point

km. The rivers Leysse and Sierroz are

the two main tributaries recharging the

lake. At the northern end, near the

Chautagne marshes, Savière canal links

the lake with the river Rhone, is the main

outflow.

Canal de Savière

North basin

Figure 1 :

Lac du Bourget

River Leysse


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2.3 MEASUREMENT ON THE LAKE

P.E. Bournet (1996) made the major measurements project on the Lac du Bourget between

1994 and 1996. During this period 18 campaigns of vertical temperature profiles

at the T point were conducted. The thermistor chain recorded temperature every 10 min

over 9 spaced depths between –10 m to –51m. By a spectral analysis of the measurements

P.E Bournet extracted the energy density as a function of the depth.

From this analysis we have extracted three periods to define scenarios that are interesting

to be reproduced by the numerical model. In the following table, we have compiled

for each scenario the first horizontal mode period and the more energetic depth.

Scenario Dates 1er Mode

Period

1 12-19 December

1995

2 03-09 April 1994

09-15 April 1994

3 27/Nov.

1994

to 7/Dec.

Depth

71 h –50 m

50 h

85 h

43 h -25 m

-20 m to –30 m

-40 m

2.3.1 The data

In the Figure 3, Figure 5 and Figure 7 we present the temperature time series measured

at the T point (see Figure 2) for the chosen scenarios at different depth. The

curves are shifted for comprehension.

For the scenario 1 (Figure 3), The decrease in time of the temperature at the depth –

23.0m (similar for the other depth) is due to refreshment of the air above the lac. The existence

internal waves at depth –35.0 and –41.0m are directly identified. This seiche is

related to the high wind speed at the surface of the lac on the 13 and 14 September

(see Figure 4).

For the scenario 2 (Figure 5) the internal waves are not well identified, particularly due

to the meteorological variability (see Figure 6).

The measurements for the scenario 3 (Figure 7), shows a well-developed internal wave

located around –23.0 m but without correlation with the wind speed!


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D28: Internal seiche mixing study

8.81°C

7.93°C

7.34°C

12.12.95 14.12.95 16.12.95 18.12.95 20.12.95

4

22.12.95

Figure 3 : Temperature series at the T point; Scenario 1. Curves are shifted by 1°C.

12

10

8

6

4

2

0

Figure 4 : Wind speed series at Voglans meteorological station; Scenario 1.

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Wind Speed

-23.0 m

-35.0 m

-41.0 m

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

9

8,5

8

7,5

7

6,5

6

5,5

5

4,5

Page 8 of 92


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8.00°C

6.39°C

6.09°C

3

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

Figure 5 : Temperature series at the T point; Scenario 2. Curves are shifted by 1°C.

10

9

8

7

6

5

4

3

2

1

Figure 6 : Wind speed series at Voglans meteorological station; Scenario 2.

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Wind Speed

-10.0 m

-23.0 m

-29.0 m

0

03/04/94 05/04/94 07/04/94 09/04/94 11/04/94 13/04/94 15/04/94

9

8

7

6

5

4

Page 9 of 92


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8.19°C

7.48°C

7.21°C

30/11/94 01/12/94 02/12/94 03/12/94 04/12/94 05/12/94 06/12/94 07/12/94

Figure 7 : Temperature series at the T point; Scenario 2. Curves are shifted by 1°C.

Figure 8 : Wind speed series at Voglans meteorological station; Scenario 3.

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8

7

6

5

4

3

2

1

-23 m

-26 m

-29 m

Wind Speed

0

28/11/94 30/11/94 02/12/94 04/12/94 06/12/94 08/12/94

11

10.5

10

9.5

9

8.5

8

7.5

7

6.5

Page 10 of 92


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2.4 APPLICATIONS

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2.4.1 TELEMAC System

The TELEMAC system is a powerful integrated modelling tool for use in the field of

free-surface flows (J.M. JANIN 1992).

The TELEMAC-3D software solves 3D hydraulic equations (with the assumption of hydrostatic

pressure conditions and time-dependent surface) and transport-diffusion

equations for intrinsic values (temperature, salinity, concentration). The main results

obtained at each point of the computational mesh are velocity in three directions and

the concentration of transported quantities. The main result for the surface mesh is the

water depth.

The space is discretised in the form of an unstructured grid of triangular elements

(Figure 9 on the left), which means that it can be refined particularly in areas of special

interest.

Bathymé-

Figure 9 : Mesh and bathymetry used by TELEMAC-3D


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2.4.1.1 TELEMAC-3D Equations

The code solves the three-dimensional hydrodynamic equations under the following assumptions:

• Navier-Stokes 3D equations with free surface changing in time,

• Negligible density variation in the mass conservation equation,

• Hydrostatic pressure assumed,

• Boussinesq approximation for momentum.

Given these assumptions, the following 3D equations are solved:

∂u

∂u

∂u

∂u

1 ∂p

∂ � ∂u

� ∂ � ∂u

� ∂ � ∂u


+ u + v + w = − + �νH

� +

F

t x y z x x x y


�νH

y


� + �νH

� +

∂ ∂ ∂ ∂ ρ0

∂ ∂ � ∂ � ∂ � ∂ � ∂z

� ∂z


∂v

∂v

∂v

∂v

1 ∂p

∂ � ∂v

� ∂ � ∂v

� ∂ � ∂v


+ u + v + w = − + �νH

� + H

H F

t x y z 0 y x x y �

�ν

y �

� + �ν

� +

∂ ∂ ∂ ∂ ρ ∂ ∂ � ∂ � ∂ � ∂ � ∂z

� ∂z


S ∆ρ

p = ρ0g(

S − z)

+ ρ0g�

dz

z ρ0

∂u

∂v

∂w

+ + = 0

∂x

∂y

∂z

∂T

∂T

∂T

∂T

∂ � ∂T

� ∂ � ∂T

� ∂ � ∂T


+ u + v + w = �νHT

� + HT + �νHT

� + Q

t x y z x x y �

�ν

y �


∂ ∂ ∂ ∂ ∂ � ∂ � ∂ � ∂ � ∂z

� ∂z


with:

h (m) water depth Zf (m) bottom elevation

S (m) free surface elevation

u, v, w

(m/s)

ρ 0 (X) reference density

velocity components ∆ρ (X) variation in density

T (°C) active or passive

tracer

t (s) time

P (X) pressure x, y (m) horizontal space

components

g (m/s2)

νH,νz

(m2/s)

νHT,νzT

(m2/s)

acceleration due to

gravity

velocity diffusion

coefficients

tracer diffusion coefficients

Fx, Fy

(m/s2)

Q (tracer

unit)

source terms

tracer source or

sink

In order to reproduce properly the internal waves with TELEMAC-3D, we need horizontal

level meshes between the surface down to the thermocline. So we cut the

bathymetry above the depth –70.0m bellow the surface. Only the bathymetry below this

depth is the real one. The resulted bathymetry is presented on Figure 9 (right).

x

y


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2.4.1.2 Lac du Bourget application

To compare the results from the model with the measurement at the T point, we simulate

and analyse the scenario 1.

The simulation characteristics for the model run are:

• initial lac status without velocity,

• horizontal homogeneous stratification, initial vertical profile of temperature from

the measurements (Figure 10),

• outflow and inflow neglected.

0

6

-20

6.5 7 7.5 8 8.5 9 9.5

-40

-60

-80

-100

-120

-140

Initial temperature

Figure 10 : Initial temperature of the lac.

The measured wind and temperature at the Voglans station (between 12 of december

to 22 of december 1995), are impose all over the lac. The time series of the lac surface

boundary condition are reported on the Figure 11.

14

12

10

8

6

4

2

Int ensit é du vent

Températ ure

0

-2

12/12/95 14/12/95 16/12/95 18/12/95 20/12/95 22/12/95

Figure 11 : Vent et Température de l’air à Voglans.

12

10

8

6

4

2

0


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On the Figure 12, we have reported the time evolution of the lac temperature at the

depth –23.0m for the measurement and the simulation. The decreasing of the water

temperature during the simulated period is connected the a lower air temperature compare

to the lac temperature associated with high wind intensity (Figure 11) which mixed

the upper water level of the lac.

8.9

8.7

8.5

8.3

8.1

7.9

7.7

7.5

Profondeur : -23.0 m

Simulation pt T

Mesures

Simulation pt B

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

Figure 12 : Temporal evolution of the temperature at –23.0 m below the surface. Comparison between

the measurement and the results from the simulation at the T point. Results from the simulation

at the B point

The comparison at the T point (south part of the lac see Fehler! Verweisquelle konnte

nicht gefunden werden.) is showing a good agreement, until the 18 of December,

between the measurement and the simulated results (purple curves and dark blue

curve). After this date the shift (about 0.3°C) is constant between the two curves. The

rapid variation of the temperature measured (12/12, 16/12 and 18/12) is not reproduce

by the model. The rapid decreasing of temperature the 16/12 is reflecting the seiches

existed below.

The comparison between the simulation results at the T point and the B point demonstrate

that at the considered depth the wind action is homogeneous all over the lake

between the surface and –21.0 m (see Figure 10).

At the depth –35.0 m (Figure 13), the measurement at the T point (curve in purple)

show clearly the existence of the internal gravity wave at this depth. The amplitude of

the temperature is bigger than 1°C.

At this depth, the results of the simulation are in a good agreement until mid of the

15/12. After this date, the first oscillation of the temperature is well reproducing in time

but the amplitude is two times smaller than the measurements. At the end of the simulated

time the shift in time and amplitude is bigger.

The amplitude of the seiches at the B point is lower than at the T point, and there is a

shift in phase.


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The development of the seiche is complete at the depth –41.0 (Figure 14), the measurements

at the T point (curve in purple) illustrate this point. The amplitude of the temperature

is bigger than 2°C and decrease with time.

At this depth, the results of the simulation are in a very good agreement until mid of the

15/12. After this date, the next oscillation of the temperature is relatively well reproducing

in time but the amplitude is underestimated compared to the measurements. At the

end of the simulated time the shift in time and amplitude is bigger.

9

8.5

8

7.5

7

6.5

Profondeur: -35.0 m

Simulation pt T

Mesures

Simulation pt B

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

Figure 13 : Temporal evolution of the temperature at –35.0 m below the surface. Comparison between

the measurement and the results from the simulation at the T point. Results from the simulation

at the B point

8.5

8.3

8.1

7.9

7.7

7.5

7.3

7.1

6.9

6.7

6.5

Profondeur : -41.0 m

Simulation pt T

Mesures

Simulation pt B

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

Figure 14 : Temporal evolution of the temperature at –41.0 m below the surface. Comparison between

the measurement and the results from the simulation at the T point. Results from the simulation

at the B point


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2.4.1.3 Frequency analysis

A simple decomposition of the seiches is used through a trigonometric function decomposition

of the calculated temperature of the lac at the T point at –41.0 min based on

the following formula:

� i

( ω t + )

−αi

S = C + A e cos ϕ

i

Where S is the simulated temperature, C a constant and i is the order of the decomposition.

The first level of decomposition gave the following parameters:

C = 7.

6

A1 = −0.

9

1 10800 = ϕ

267600 − t

α1

=

500000


ω 1 =

300000

This gave a period close to 83 hours for the first mode. On Figure 15 is shown the

graphical comparison between the simulated temperature at T point at –41.0m and the

first level of trigonometric decomposition. For the selected period (between 200000s

and 700000s) the both curve are very close.

The second level of the decomposition gave the following parameters:

2 0.

12 = A


ω 1 =

94000

i

1 = ϕ

i

7200

With this method the second mode is estimated to 26 hours.

8.5

8.3

8.1

7.9

7.7

7.5

7.3

7.1

6.9

200000 300000 400000 500000 600000 700000

Figure 15: Simulated temperature at T point at –41.0m (S in dark Blue), first level of trigonometric

decomposition (in red), first plus second level in light blue.

S

First level

1+2


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2.4.2 SIMULATED SEICHES – ANALYSIS

2.4.2.1 Rotation

The Figure 16 shows the temperature

iso-contour level 7.6°C at six different

times at –41.0 m depth. These times

are chosen in order to illustrate the rotation

of the internal wave around the

amphidromic point due to the coriolis

forcing. The coloured stars are associated

to the position of the temperature

iso-contour that has the same colour.

The black arrow that follow the stars

explicit the wave rotation direction.

The both lines Red and Purple, which

are superimposed, are related to the

wave period of the seiches (80 hours).

The intersection of the six curve is located

at the middle of the lac, determine

the amphidromic point of the internal

wave.

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Figure 16 : Coriolis effect on the internal wave propagation.

°C

°C

°C

°C

°C

°C


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2.4.2.2 Vertical displacement

In order to evaluate the amplitude of the wave and the affected area we show on Figure

17 (depth –35.0 m at left and –41.0 m on the right), the difference between the temporal

(during all the simulation) maximum of the temperature and the initial temperature at

the considered depth.

The maximum difference is obtain on the depth –41.0 m, as the analysis made by P.E.

Bournet shown. At the middle of the lac, the difference is zero in both cases, that confirm

the amphidromic point (Figure 16). The maximum value are located on the west

side of the lac. The North basin is more exposed to high amplitude of internal wave.

Figure 17 :Difference between the maximum temperature and the initial one (-35.0 m on the right, -

41.0 m on the left)


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2.4.2.3 Vertical velocity

The maximum of the vertical velocity, during all the simulation time, for the depth –

35.0m and –41.0m are shown on Figure 18.

We observe a very similar value for both levels. The maximum of the vertical velocity is

concentrated along the coast line with some spot located at particular points.

Figure 18 : Maximum vertical velocity during the simulation (-35.0 m on the right, -41.0 m on the

left)


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2.5 CONCLUSIONS

The analysed measurements are shown the existence of internal gravity wave in the

Lac du Bourget with a period between 40hours to 80 hours depending on the vertical

stratification structure.

The simulation using the TELEMAC-3D model is used to reproduce one seiche event

on the Lac du Bourget. The numerical results are in a quite good qualitative agreement

with the measurements.

The numerical results are given some interesting new information about the hydraulic

respond of the lake under wind forcing. After the generation of the internal seiches, the

wave propagation is showing a rotating structure, due to the Coriolis influence, with an

amphidromic point at the middle of the lake (Figure 16).

In spite of the bad representation of the bathymetry along the coast, the maximum dissipation

of the internal wave illustrated by the two figures (Figure 17 and Figure 18) is

located all around the coastline of the lake. And the maximum vertical velocity is less

than 1. 10 -3 m/s.

2.6 REFERENCES

Bournet PE. 1996. Contribution à l’étude hydrodynamique et thermique du Lac du

Bourget. PhD Thesis.

Janin JM, Lepeintre F, and Pechon P. 1992. TELEMAC-3D: a finite element code to

solve 3D free surface flows problems. Proceedings of computer modelling of seas and

Coastal regions. Southampton, UK.


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3 INTERNAL SEICHES AND MIXING IN LAC LÉMAN

3.1 THE LAC LÉMAN ENVIRONMENT

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The Lac Léman (Figure 19) is a warm monomictic lake situated between Switzerland

and France. It is curved in shape and composed of two main basins: a deep central

eastern basin (310 m maximum depth, 157 m mean depth, mean width 10 km) called

Grand Lac ("big lake") and a relatively small and narrow section in the west called Petit

Lac ("small lake," maximum depth 70 m; mean width around 4 km). The lake has a

total length of 70.2 km and a width of 13.8 km in the central part which corresponds to

2.4 Rossby radii under typical summer stratification conditions. The eastern part of the

lake is surrounded by high mountains sheltering it from most strong winds. The central

and western part of the lake form part of the Swiss central plateau. The windfield over

the lake is affected by the mountains and the plateau. It is consequently dominated by

events of strong winds from the northeast and southwest which may last from several

hours to several days. The winds from the northeast have been observed to cause

thermocline depressions of more than 20 m in the Petit Lac (Lemmin and D'Adamo

1996). These wind events can be considered as the principal forcing to initiate internal

seiches.


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Figure 19 : Map of the Lac Léman (Lake Geneva) with surrounding topography and positions of

recording stations discussed in this text. Depth contours are given with reference to the water

surface level which is at an elevation of 371 m above sea level.

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3.2 FIELD STUDIES

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Field measurements concerning temperature and currents have been carried out in the

Lac Léman over the last twenty years using moored instruments. The recording interval

was 30 min. or 60 min. and instruments were deployed in campaigns lasting for several

months. Most of these data were collected along the northern shore of the lake at

mooring stations covering the slope down to about 175 m during fall stratification. However,

some measurements were also made during summer periods. The data were

analyzed for the presence of internal seiche modes using spectral analysis. For spectral

analysis of the records, a standard Fast Fourier Transform with segment overlap was

used.

3.2.1 Modes detected

Modes were identified by comparing the periods P corresponding to prominent spectral

peaks with those predicted for a two-layered approximation fitted to the lake dimensions

and density distribution using the Merian formula

P n =2L b

(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

and ρ1 and ρ2 are the respective densities. n is the mode number. In the calculations

we assumed, for the summer (values for the fall in parenthesis) a top layer h1 =

15 (25) m; a bottom layer h2= 175 (165) m; a top layer temperature T1 = 19 (8) °C; and

a bottom layer temperature T2= 5.5 °C. The analysis of all the data indicates that, independent

of season and station location, only certain modes are excited. Differences

were found between the number and type of modes observed at different locations in

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

positions see Figure 19). The maximum at the nearshore station P1 is at the Kelvin wave period

while that for the offshore station P2 is at the Poincaré period. (cpd = cycles per day; modified

from Bohle-Carbonell, 1986)

At the stations closest to shore, the most prominent seiche is the first mode (n = 1;

called L1 hereinafter; Figure 20). Its period is near 81.5 h in summer, increasing to 131

h in fall, as the density structure of the water column changes and ci decreases. In the

spectra from the narrow western end of the lake (not shown here) the L1 mode response

is always seen most clearly. Recently, timeseries measurements were carried

out during summer stratification with a current meter placed about 5 m above the lake

bottom on the southern side of the deep central plateau at station S (Figure 19). Spectra

from this station (Figure 21) show again a broad peak at the Kelvin seiche period.

Seiche modes n = 2 to 9 were not observed in any spectra.

The first cross basin or transverse mode (called T1 hereinafter), with summer and fall

periods of 10.7 and 13.5 hours respectively. It is seen most clearly in the spectra at offshore

stations in the central part of the lake basin (Figure 20) and is often found in the

eastern part. Modes higher than 10 cannot be detected with certainty because of the

cut-off imposed by the time step of the data records. The T1 period is also clearly seen

in spectra from the recent deep measurements at S (Figure 21).


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10 3

Lake Geneva, midlake station, depth 310m, from 6 July 14:00 to 12 October 24:00, 2001

10 2

10 1

10 0

10 -7 10 -6 10 -5 10 -4 10 -3

10 -1

north component at 304m

east component at 304m

frequency (cph)

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Figure 21 : Spectra of north and east component of the currents at 304m depth recorded during

summer 2001 at station S. For station location see Figure 19.

3.2.2 Wave propagation patterns

3.2.2.1 Mode L1

It should be noted that the thermocline oscillations are accompanied by oscillations in

lake surface level which are out-of-phase with and about 1/1000 smaller in amplitude

than the oscillations of thermocline isotherms. It is therefore possible (e.g. Caloi et al.

1961; Sirkes 1987) to use numerically filtered deviations of surface level from equilibrium

to follow the progress of internal motions. This possibility arose for the Lake of

Geneva, because the Swiss Service Fédérale des Eaux, (SFE, 1954) published tables

of water levels measured in 1950 by 13 high-precision water level recorders spaced

around the lake shore (see Fig. 1). We have analyzed these limnigraph records for the

wave propagation pattern. Cross spectral analysis reveals the wave propagation pattern.

Coherence and phase were determined between pairs of stations.

Coherence is always high at the mode L1 period and falls off rapidly below the confidence

limit for surrounding periods. A calculation was made for all possible combinations.

The progression for the L1 mode during the summer period is presented in Figure

22. In that case, station 2 was taken as the reference station and pairs were formed

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

from the numerical model under the conditions specified above.

During summer, coherence was high for most station pairs. For the stations along the

northern shore, coherence increases again from E to W and became high for those

stations in the narrow Petit Lac. The phase angles calculated from the data can be

compared to those predicted by the numerical model. Along the southern shore, satisfactory

agreement in phase angle was found; and cyclonic progression was clearly established.

Agreement was less satisfactory on the northern shore, particularly, at the

entrance to the Petit Lac at stations 10 and 11, even though the coherence was high.

At the western end of the lake, agreement was again good, indicating that the L1 seiche

mode completed a full cycle around the lake. During fall, interstation coherence decreased

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,

structure as predicted by the model. At each limnigraph station marked by an encircled number in

the top panel, phase and coherence with reference to station 2 obtained from the analysis of the

waterlevel data are indicated. Middle panels, spectra of SFE water level fluctuations at station 2

and 7. The small peak at the L3 mode period is not significant; T1 is the first transversal mode

wave to be discussed below. Bottom panels, coherence and phase angle between 2 and 7 (out of

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-seiche" response

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Initial analysis of part of the water level data had already shown (Mortimer 1963) that

periodicity in the surface elevation signals corresponded to the L1 mode "Kelvin

seiche." That signal was visually correlated with the thermocline oscillation at Geneva.

The surface elevation signal progressed around the lake perimeter at a nearly constant

speed corresponding to a L1 mode internal seiche.

The results of the present statistical analysis indicate that, for the L1 mode, coherence

is high in the narrow western basin falling off toward the eastern end of the lake where

the amplitude is greatly reduced and the wave form is perturbed by either T1 mode

waves or non-wave effects.

Analysis of lake temperature and current data has previously shown (Bohle-Carbonell

1986) that typically one or two whole-basin L1 circuits are completed when post-storm

calm persisted for long enough. Evidence for the Kelvin wave character of the L1 mode

wave also came from current and temperature recordings, carried out in the western

part of the Grand Lac in 1982/1983 (Mortimer 1993) from December to January when

the lake was still weakly stratified with a thermocline at about 90 m depth. Parts of this

analysis are reproduced in Figure 23. Station A1 is close to shore while station A3 is

further offshore (for station location see Figure 19). Three wind events occurred between

9 and 20 December.


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Figure 23: Current and temperature records of fall/winter 1982/1982. Winds were measured at

Cointrin Airport near Geneva at the west end of the lake basin. The wind speed squared is multiplied

with the direction of the wind. Positive winds are coming from the NE; negative winds are

coming from the SW (adapted from Mortimer, 1993). Measurement depths are indicated on each

panel. For station locations, see Figure 19.

At roughly 100 h after the start of each wind event, the currents at 15 m at mooring A1

reversed suddenly from eastgoing to westgoing at the mooring nearest shore. Those

reversals, R1, R2, R3, and R7, were each accompanied by a sudden depression of the

thermocline (i.e. a sudden temperature rise) at mooring A1, marking the passage of a

surge. Similar saw-toothed "waves" were also seen at moorings further offshore at A3

but with reduced amplitude. The reversals in current direction, however, were confined

to the nearshore instrument, A1.Figure 23 indicates a pattern, which starting on the

south shore, had traveled around the lake (the Petit Lac was destratified and took no

part in the seiche) at internal seiche speed.


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After the wind event R3, as would be expected for a Kelvin wave, a free oscillation persisted

for three cycles with amplitudes of the current reversal R3 to R5 and amplitudes

of the associated temperature waves decreasing steadily in time. It can be seen that

each time a strong current reversal in the upper layer occurred at station A1, but not at

station A3, in accordance with the Kelvin wave exponential decay of wave amplitude

with distance from shore. The temperature records again supported the Kelvin wave

pattern: pronounced signals were observed at A1; at A3 (not shown) the amplitude was

rather small.

At that time of the year, the thermocline was below the depth of the Petit Lac basin.

The path of a Kelvin wave should, therefore, be confined to the contour of the Grand

Lac basin. Data from station G (see Figure 19) can be used to investigate this point.

While the signal from the L1 mode is seen in the alongshore (east-west) component at

A1, it appears as south-going currents at G (Figure 23). Thus the Kelvin wave has

turned, following the deep basin contour of the Grand Lac instead of continuing to move

along shore into the Petit Lac.

Evidence for a Kelvin seiche response can also be seen in the spectra in Figure 21

where the energy level of the along shore component is clearly dominant in the Kelvin

seiche range.


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3.2.2.2 Mode T1

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For the only other strongly expressed mode, the first transverse mode T1 (n=10), the

results of the cross spectral analysis are given in Figure 24, in that case for cross-basin

stations 3 and 9. For the mode T1, the amphidromic pattern (Figure 23) displayed coherence

between stations, which are part of the same or neighbouring amphidromic

cell, but no coherence between stations in distant cells.

Figure 24: The first transverse mode T1 during the summer interval 4 June to 24 Aug. 1950: top

panel, structure and period predicted by the model. Bottom panels, coherence and phase between

stations 3 and 9 from spectral analysis of SFE records of surface level fluctuations. Phase angle of

+180° indicates a cross-basin oscillation. No coherence is found for station pairs which are not in

the same amphidromic cell.


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3.2.3 Evidence of Poincaré waves

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Is this T1 mode a cross basin seiche or is it a Poincaré wave? The fact that a wave of

near 11 h period has been detected in the shore-based water level records would favour

interpretation as a standing cross-basin wave in the central part of the lake. However

since a constant periodicity appears in the central part of the basin, independent of

local topography, a Poincaré wave interpretation may be more likely. This is supported

by results from the numerical model (Bäuerle, 1985), which predicts clockwise rotation

in the amphidromic cells in that part of the lake.

Poincaré waves are clearly seen in Figure 23 superimposed on the Kelwin wave pattern,

particularly at the offshore station A3 and appear to be excited together with the

Kelwin waves. To investigate this point further, the current records at station S in the

central basin were examined in detail.

3.2.4 Progressive vector analysis

Progressive vector analysis was carried out for current components measured at station

S in order to better understand the pattern described above. Figure 25 shows the vector

diagram for the full record. This is a predominantly eastward oriented transport and the

total running length is about twice that of the east-west length of the central bottom

plateau. From the time markers it is obvious that there are periods of relatively slow

transport as would be expected during summer stratification.

5

4

3

2

1

0

-1

Lake Geneva, midlake station, depth 310m, from 6 June 14:00 to 12 October 24:00, 2001

instrument depth 304 m

marker every 24 h

-3 0 3 6 9 12 15 18 21

east west displacement (km)

Figure 25 : Progressive vector diagram of currents measured at station S. For station location see

Figure 19

Frequently there are undulating sections in the progressive vector curve in Figure 25

with two waves occurring between two 24 h markers. This obviously reflects a periodic

oscillation of about 12 h clearly seen in the analysis above. The cusped motion pattern

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

excerpt of the progressive vector diagram has been produced with Figure 26. From this

figure it is obvious that circular motions with periods close to 12 h are always executed

in a clockwise sense of rotation.

2.40

2.15

1.90

1.65

1.40

Lake Geneva; midlake station, depth 310m, from 1 Sept. 14:00 to 5 Sept 14:00, 2001

325

343

307

379 397

361

415

289

253

271

433

469

451 487

235

505512

12.75 13.00 13.25 13.50 13.75 14.00 14.25

217

199

181

163

145

91109

127

instrument depth 304m

marker every 3 h

73

55

1 37

east west displacement (km)

Figure 26 : Progressive vector plot for selected period of the trajectory shown in Figure 25.

Further support for the Poincaré wave concept comes from spectra calculated by

Mortimer et al. (1983) for current and temperature data at different stations in the central

part of the lake. A statistically significant peak at the period of the T1 mode is always

clearly seen in the temperature data at stations offshore (but rarely at the nearshore

stations). However, it has to be realized that the T1 mode has Poincaré wave

characteristics only in the central part of the Grand Lac basin.

3.2.5 Mixing by internal seiches

During the period of summer stratification, mixing in lakes can be caused by vertical

mixing, boundary mixing, interbasin density currents and river inflows. In Lac Léman,

river inflows and interbasin density currents are of little importance during summer

stratification. Vertical mixing can be related to wind induced shear or to internal seiche

and internal wave activity.

Turbulent mixing can be expressed in terms of turbulent mixing coefficients, Kz, in

analogy to molecular diffusion coefficients. This coefficient can be determined by the

flux gradient method from integral changes of a tracer, such as temperature, over a

certain time and a certain depth in the following way (Jassby and Powell, 1975):

Kz =−1/ ( [ A(z)∆T

∆z]

) A(z') ∆T

∆t dz'

zmax �z

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

of the lake. The flux gradient method is valid during periods when lakes warm up but is

not valid when convective cooling overwhelms wind-induced mixing. In the present

study we will apply this equation to temperature profiles taken in the central part of the

Grand Lac basin in order to investigate the origin of vertical mixing in Lac Léman. This

selection will minimize the effect of thermocline tilt due to internal seiches and wind setup.

The calculation of the vertical mixing coefficient Kz were carried out based on a set of

monthly temperature profiles taken between 1987 and 1991 for the warming period

from May to September. For each month, average profiles were established from the

data by interpolating to the same depth intervals and eliminating the linear trend due to

progressive longterm warming of the lake. Details are given in Michalski and Lemmin

(1995). Results of this calculation are shown in Figure 27.


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Figure 27 : Logarithm of the vertical turbulent mixing coef. Kz vs. logarithm of the Brunt-Väisälä

frequency N 2 . The data are plotted as depth profiles with the surface at right and the bottom at left.

The corresponding depths are indicated on each curve. A straight line is drawn for the section of

each profile where the relation Kz ∝a(N 2 ) b can be validated. (a) for May (b = -0.5). (b) for the

whole warming season from May to September (b = –0.4). (c) for a multiannual trend (b = –0.6).

The hypothesis that the mixing coefficients result essentially from vertical mixing has

been examined by looking at the correlation between Kz and the Brunt Vaisala frequency

N. The Brunt Vaisala frequency is defined as

N 2 =−(dρ/ dz)(g /ρ)

where g is the gravitational acceleration. As in most lakes, the vertical density gradient

is controlled by the temperature gradient dT/dz. The conversion from temperature into

density is carried out using standard formulae. Different theoretical models exist for the

relation between Kz and N2 of which all have the form

Kz ∝a(N 2 ) b


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where a and b are constants. Using dimensional analysis, Welander (1968) suggested

that, depending on the origins of turbulence, two limiting cases given by b = -0.5 for

shear induced turbulence and b = -1 for cascading 2D turbulence. Heinz et al. (1990)

summarized results from different lakes where b was found to range between – 0.4 and

–0.7 indicating mixing in lakes is predominantly generated by local shear and internal

waves. In Lac Léman, values of b are found between –0.4 and –0.6 (Figure 27) indicating

that energy cascading is not important. A value of –0.4 falls outside the range

predicted by Welander. However, Jassby and Powell (1975) who found the same value

already noted that the assumptions of steady state and horizontal homogeneity made

by Welander are most likely not fulfilled over periods of several months.

Our results indicate that during summer stratification, turbulent mixing in the upper water

column is predominantly caused by processes related to internal seiches and progressive

internal waves (Lemmin et al., 1998; Thorpe et al. 1996; Thorpe and Jiang,

1998) quantified by the order of magnitude of the mixing coefficient. The increase of Kz

with N 2b (and depth) ends at a depth of ≈ 90 m (see Figure 27). Below that depth an

exponential correlation between Kz and N 2b cannot be established indicating that processes

other than those considered by Welander dominate the turbulent mixing.

3.3 NUMERICAL SIMULATIONS

3.3.1 Model Description

We use a slightly modified version of the three-dimensional numerical code ‘GETM’,

developed by Burchard and Bolding (2002). A detailed mathematical description of this

model is given in their report. Here we only mention its main features.

The model solves the three-dimensional shallow-water equations for momentum and

heat with a free surface as the upper boundary condition. The Boussinesq assumption

has been adopted, implying that the balance of mass simplifies to a statement of zero

divergence of the velocity field. As in all models of this type, pressure is computed from

the hydrostatic balance, and certain types of barotropic and baroclinic waves cannot be

reproduced. Among them are waves short in comparison with the local water depth and

non-linear solitary waves. However, in the context of this section, mixing caused by long

internal waves is emphasized, and these restrictions are hardly relevant.

Quite to the contrary, the capability of the model to predict the evolution and propagation

of steep non-linear waves or ‘bores’ is relevant in Lac Léman as illustrated below.

To capture the essential physics of these waves, it is necessary to retain the non-linear

advection terms in the horizontal momentum balance. Since it is well-known that the

classical first-order upstream schemes for the numerical discretisation of these terms

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

(TVD) schemes, as described in Burchard and Bolding (2002).

The size and stratification of all lakes considered in the EUROLAKES projects suggests

that the effects of the rotation of the earth cannot be neglected, and thus the Coriolis

force is included in the model.

The model equations are discretised on a staggered Arakawa-C grid in the horizontal

coordinate, using generalised sigma-coordinates in the vertical. Due to the extremely

steep topography of Lac Léman and the use of sigma-coordinates, great care has to be

taken in computing the internal pressure-gradient, the driving force for the internal

waves discussed in this section. We alleviate these problems using high horizontal

resolution (250 m) and a special discretisation technique for the internal pressuregradient

(Burchard and Petersen, 1997). The vertical resolution is 40 sigma levels,

which is about the minimum to resolve the turbulent structure in the boundary layers

and around the thermocline.

Evidently, because mixing by internal waves is to be investigated here, a good turbulence

model is required. We use a so-called Algebraic Reynolds Stress model (Canuto

et al., 2001), solved in connection with two differential transport equations for the turbulent

kinetic energy, k, and the specific dissipation rate, ω (see Umlauf et al., 2002).

3.3.2 Mixing by Long Internal Waves - Model Results

We investigated the activity of long internal waves at two positions in the lake which are

dynamically very different and serve as two extreme cases spanning the range of mixing

activity in Lac Léman. The two positions are marked by yellow dots in Figure 28.

Position A is located near the bottom of the deep plateau of the main basin. Measurements

at this point have been discussed above. Position B is located at the entrance of

the ‘Petit Lac’, the elongated south western appendix of Lac Léman. Measurements

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

measuring positions are marked by yellow dots. The geographical orientation of the lake has been

turned 17 degrees in the clockwise direction on this and the following plots.

The computation of the possibly complicated structure of the wind field over Lac Léman

with a meteorological model was not part of the EUROLAKES project. Therefore, we

had to force our model with highly idealized winds. We used homogeneous wind fields

from SW (along the basin) and from NE (across the basin). These fields can be thought

of as a first-order approximation of the well-known winds ‘Vent’ and ‘Bise’, respectively,

which typically last only for a few days. Evidently, the current pattern close to the surface

will exhibit large errors due to these simplistic wind fields. However, the excitation

of basin-scale internal waves and their decay due to mixing, the main topic of this section,

should be reproduced with much better accuracy. All runs have been initiated with

zero velocities and an idealised typical late summer stratification with a well mixed

epilimnion of 18 °C, a homogeneous hypolimnion of 5 °C, and a transitional thermocline

at about 20 m depth.

3.3.2.1 Mixing at Deepest Part of Lac Léman

The measurements at point A (marked as S in Figure 19) near the bottom of the deepest

part of the main basin have been discussed above. The position of the spectral

peaks and the clockwise rotation of the velocity vector have been interpreted as strong

indicators for the dominance of Poincaré waves (in winter possibly degenerated to simple

inertial oscillations) at this location. In summer, considerable energy was also found

in the along-basin component of the velocity at much lower frequency. Current speeds

were demonstrated to be weak at all times and of the order of 1 to 3 cm/s. Additional

measurements showed that the temperature profile in the lowest 50 meters of the water

column is weakly unstable, most likely due to the geothermal heat flux. This thermal instability,

however, is always compensated by the much stronger stabilizing effect of salinity

in this region. These observations point towards very weak or negligible mixing

near the bottom, even though a definite conclusion has to await direct measurements of

turbulence.


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To investigate the ability of the model to reproduce these results, we started our runs

with a typical late summer stratification for Lac Léman. The response to both idealized

wind fields mentioned above, each with a duration of 1 day and a wind speed roughly

corresponding to 7 m/s, was analysed at point A.

v ( m / s )

0 . 0 3

0 . 0 2 5

0 . 0 2

0 . 0 1 5

0 . 0 1

0 . 0 0 5

- 0 . 0 0 5

- 0 . 0 1

- 0 . 0 1 5

0

u

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 "

v

- 0 . 0 2

0 1 2 3 4 5

t ( d a y s )

v ( m / s )

- 3

x 1 0

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 "

1 0

- 2

- 4

- 6

8

6

4

2

0

v

u

- 8

0 1 2 3 4 5

t ( d a y s )

Figure 29: Time series of the x- and y-components u, v of the velocity at point A near the bottom.

Left panel: with along-basin wind ‘Vent’. Right panel: with cross-basin wind ‘Bise’. Note the different

velocity scales.

The velocity components in x- and y-direction (see Figure 28) near the bottom are displayed

in Figure 29 for both wind fields. It is evident that the cross-basin v-component is

dominated by oscillations of roughly 10 hours in both cases. In particular for the ‘Bise’

situation (right panel), it is clearly visible that these oscillations are correlated to similar

oscillations in the along basin u-component with a respective phase shift of 90 degrees,

a clear indication for Poincaré wave activity. This correlation can also be seen, though

less clearly, for the ‘Vent’ situation, where the u-component is dominated by a low frequency

contribution with a ‘period’ of approximately 3 days. Spectral analysis (not

shown) confirms these results, in particular the peak around 10 hours and the higher

energy at low frequencies in the u-component for the ‘Vent’ forcing. Current speeds associated

with the higher frequency motions are at most 1 cm/s, and only the low frequency

contribution of the along basin current can reach 2-3 cm/s for the relatively

strong winds applied here.

We conclude that the numerical model is able to correctly predict the most important

components of the currents at this location. The somewhat shorter period of the Poincaré-waves

is due to the fact that the model stratification in this example was slightly

stronger than the measured one, leading to higher wave speeds and shorter periods. In

addition, the measured spectra (Figure 21) represent an average over many weeks with

periods of varying stratification, and perfect agreement cannot be expected.

The turbulence model predicts zero turbulence at the bottom. Even though this is in apparent

agreement with the absence of a measured well-mixed layer near the bottom

(see above), this model result should not be overemphasized: It is well known that the

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.

Both effects, however, have to be expected at the location investigated. At present,

there exists no satisfactory theory for this regime of turbulent flows, and only when the

boundary layer is fully turbulent, Reynolds stress models as that use here can yield

reasonable results even when turbulence is rather weak. This has been shown recently

by Lorke et al. (2002) for the bottom boundary layer of a small lake.

3.3.2.2 Mixing at the Entrance of the Petit Lac

The dynamics of mean currents and turbulence at point B, the entrance of the S-W appendix

(‘Petit Lac’, see Figure 28), is quite different. Temperature profiles measured at

this point for late summer and early winter 1987 are displayed in Figure 30. For the

summer period (August/September), the temperature profiles exhibit a typical structure,

which is also found in other years:

• a well-mixed upper layer of about 10 m thickness

• a strongly stratified upper thermocline from about 10 m to 30 m

• a weakly stratified lower thermocline from about 30 m to 50 – 60 m

• a well-mixed bottom boundary layer of 10 to 15 m thickness

Later in the year, the thermocline is slowly mixed downward and eroded by penetrative

convection, as is particularly visible from the slightly unstable temperature profile in December

(see Figure 30). Finally, starting from January, the whole water column at this

point is well mixed until re-stratification starts in spring (not shown). Current records at

the entrance of the Petit Lac for a period of the same year 1987 are plotted in the right

panel of Figure 30.

d e p t h ( m )

- 1 0

- 2 0

- 3 0

- 4 0

- 5 0

- 6 0

- 7 0

0

1 6 D e c

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

1 7 N o v

2 8 O k t

1 0 S e p

1 8 A u g

- 8 0

5 1 0 1 5 2 0 2 5

t ( d e g C )

v ( m / s )

0 . 4

0 . 3

0 . 2

0 . 1

- 0 . 1

- 0 . 2

- 0 . 3

- 0 . 4

0

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

6 0 m

- 0 . 5

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5

d a y s f r o m 2 1 . O k t . 8 7

Figure 30 : Left panel: profiles of temperature at the entrance of the Petit Lac for the second half of

1987. Right panel: measured in- and outflow velocities at the entrance of the Petit Lac at 10 m and

60 m depth. Outflow is positive.

1 0 m


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It is clearly visible, that the speeds at this point are at least one order of magnitude

higher than at the bottom of the main basin. In addition, currents measured in the

epilimnion and the hypolimnion close to the bottom are of comparable magnitude, but

almost always of opposite sign causing a strong shear across the thermocline. It is to

be noted, that waves of fluid entering and leaving the Petit Lac can be non-linear. An

extreme example is the current reversal in the lower and upper layer at day 25 (see

Figure 30): Currents in both layers change sign within less than 30 minutes (the resolution

of our measurements), indicating a strongly non-linear internal ‘bore’ entering the

Petit Lac. It is very likely that turbulence caused by these processes is crucial for shaping

the vertical profiles of passive and active scalars in the water column.

We tried to model the basic features at position B by forcing our model with a strong

‘Vent’-type along-basin wind of about 7 m/s, which lasts for one day and was then

switched off. Numerical studies showed that a ‘Bise’ wind event also causes high velocities

at the entrance of the Petit Lac. However, turbulence characteristics and vertical

shear are quite similar in both cases, and we considered it sufficient to look only at one.

The left panel of Figure 31 illustrates the structure of the velocity field 38 hours after the

wind has started (and terminated after 24 hours). This is precisely the time at which the

return current, driven by the pressure gradient caused by the interface set-up, reaches

the entrance of the Petit Lac.

As can be seen from this figure, the return current occurs in the form of a strongly nonlinear,

coastally trapped ‘bore’ in the Northern part of the entrance, and in form of a

more gradual adjustment in the Southern part. After the shock wave has passed, the

inflow currents of warm surface water into the Petit Lac are distributed fairly homogeneous

across the entrance (not shown). As discussed above, the existence of such

non-linear waves is also indicated by our current measurements, if the initial wind forcing

is strong. The correct prediction of these waves depends to a large degree on the

ability of the model to reproduce strong horizontal gradients in scalar and vector fields,

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)

38 hours after the start of an along-basin wind lasting for 24 hours. Dark red arrows indicate

‘warm’ water, light red/orange arrows ‘cold’ water. Right panel: vertical velocity profile at

point B (entrance of the Petit Lac) 50 hours after the start of the wind.

The modelled vertical velocity profile 50 hours after the start of the wind, long after the

‘bore’ of the return current has passed, is illustrated in the right panel of Figure 32.

Since no measured velocity profiles are available at point B, we remark only on the

most evident features of this profile, namely a strong shear near the bottom, a sharp

maximum at about 47 m depth, a strong shear across the thermocline, and a more or

less well-mixed upper layer.

Figure 32: Left panel: vertical profile of the modelled temperature at point B (entrance of the Petit

Lac) 50 hours after the start of the wind. Right panel: same as left panel, but now the turbulent diffusivity

of momentum is displayed.

The temperature profile at the same time is shown on the left panel of Figure 30. Remarkably,

the vertical structure of this profile is very similar to that measured at the

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

lower thermocline. Also the vertical extent and position of these different zones

closely correspond to the measurements.

The physical processes that shaped the thermocline structure become evident from the

right panel of the same figure, which displays the turbulent diffusivity computed by the

Reynolds stress model at the same time. As can be seen from this figure, the wellmixed

bottom boundary layer is caused by strong turbulent diffusivities in the lowest 15

meters, driven by the strong velocity shear near in this region. At the region of the velocity

maximum (at about 47 m, see above), this shear is zero and diffusivities drop to

very small values. This happens exactly at the lower edge of the stratified region.

Above, the shear becomes strong again, turbulent diffusivities increase and cause an

erosion of the thermocline from below. The result can be found in both, the measured

and the modelled temperature profiles. In this region, the gradient Richardson number

predicted by the turbulence model is approximately 0.25 (not shown), and thus mixing

occurs at high efficiency. From 30 to 10 meters depth, stratification becomes too strong,

and turbulence is completely suppressed by local buoyancy effects. Only in the upper

10 meters, wind mixing is strong enough to create a well-mixed region. Note, that at 50

hours as in Figure 30 the wind has already been switched off, and diffusivities are only

weak. This figure also gives a nice impression of the vertical numerical resolution required

to resolve the basic features of the profile.

3.4 CONCLUSIONS

An analysis of internal seiches dynamics was carried for Lac Léman combining field

measurements and numerical modeling. Using field data of temperature, currents and

surface elevation, it has been shown that only two modes of internals seiches are sufficiently

excited in Lac Léman to be considered significant. The first one is a Kelvin wave

and the second one is a Poincaré wave. Model calculations have indicated that other

seiche modes can only be excited by winds from certain directions. However, due to the

topographic constraints particularly in the eastern part of the lake basin the wind field

over the lake is strongly canalized and these winds do not exist in nature.

The Kelvin wave progresses around the perimeter of the lake basin and its effects are

most prominent in the nearshore zone. During the passage of the wave the thermocline

descends by several meters and this provokes a thermocline displacement over the

weakly sloping lateral zone which may easily reach 100 m. The combined alongshoredownslope

velocity vector can reach speeds high enough to cause sediment erosion

with potentially negative effects for the drinking water intake structures which are placed

in the same zone. Since Kelvin waves are rather frequent in the Lake of Geneva, it can

be expected that the dynamics of the nearshore zone vary on a periodic level during

stratification.


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As shown above, Poincaré waves are most prominent in the offshore regions of the

central basin. However, water surface recorder records near the shore also indicated

their presence. Their mayor axis is oriented in the cross-lake direction. The transport

resulting from these waves will link the water masses in the center of the lake with the

near shore zones. This will affect the water mass residence times, in particular shortening

those in the central part of the basin. We also observed that oxygen concentration

in the near bottom layers in the center of the lake fluctuates with the period of

Poincaré waves. This is further indication of the link between the water masses in the

center of the lake and the lateral zones. Due to their period of about 12 h during summer,

a certain correlation in the forcing with the diurnal windfield over the lake (Lemmin

and D’Adamo, 1996) can be envisioned.

From our analysis of field studies of the longterm mean conditions of mixing it is indicated

that internal seiches are important in providing vertical mixing. Recently, it has

been shown though that most of this mixing is actually generated in the near shore

zone and then propagates into the open waters (Wuest et al., 2000). Thus, the interaction

between near shore zones and the open water is also important for mixing. Furthermore,

we have pointed to the importance of the interaction with the sloping sides of

the lake and short progressive internal waves (Thorpe and Lemmin, 1999a, Lemmin et

al., 1998). These waves and their breaking play a role in the production and redistribution

of currents and stratification as well as mixing (Thorpe and Jiang, 1998). From our

studies it appears that short progressive internal waves are often produced in the passage

of non-linear internal seiches (Thorpe et al., 1996).

A state-of-the-art numerical model for the three-dimensional shallow-water equations

has been compared to measured currents at two dynamically very different locations in

Lac Léman: A low-energy point close to the bottom in the deepest part of the lake, and

a very active region at the entrance of the appendix ‘Petit-Lac’, both for a typical latesummer

stratification. Even though the wind field was highly idealized, the basic components

of the currents at both locations could be reproduced: The nearly linear Poincaré-wave

signal with very low current speeds at the deepest point of the lake, and the

pattern of fast inflow and outflow currents including effects of the non-linear return-wave

at the entrance of the Petit Lac after the wind had stopped.

Clearly, the basic theory of linear shallow water waves in stratified basins is known

since many decades, and the existence of these waves in Lac Léman does not come

as a great surprise. However, the ability of our model to predict these waves, in particular

their non-linear steepening at the entrance of the Petit Lac, can be taken as an

indication for the reliability of the numerical scheme.

Much more important as the mere prediction of internal waves is the question of their

contribution to the overall mixing in Lac Léman. This was the major topic of the work

package, and new insight into the physics of this process has been obtained. It must,

however, be noted that due to the low current speeds in the deepest part of the lake, no

useful information about the turbulent characteristics in the deep hypolimnion can be

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

on this topic. As long as no measurements of microstructure in the water

column are available, it can not even be definitely concluded at what level (if at all) turbulence

exists in the lowest quarter of the hypolimnion.

Quite encouraging results have, however, been obtained from the comparison of

measured and computed quantities at the entrance of the Petit Lac. The measured

characteristics of the velocity time series (namely the non-linear wave) in the case of a

strong wind event could be reproduced. In addition, the good agreement of the measured

and computed structure of the temperature profile suggest that the turbulence

model yields reasonable turbulent diffusivities. All these results indicate that the Petit

Lac could serve as the main ‘mixer’ of hypolimnetic water in the whole basin, even

though enhanced turbulence at the lateral boundaries of the lake may also play a role.

The precise mechanisms of how mixing in the Petit Lac affects the main basin are not

yet known. The following possibilities appear to be, however, the most reasonable:

• mixed water from the Petit Lac enters the hypolimnion of the main basin by advection

through the strong mean currents in this region

• mixed water intrudes horizontally into the main basin, driven by the density differences

due to differential mixing

• heavy water is ‘strained’ over lighter water and causes additional local mixing

due to static instability of the water column

All processes could be identified in the model results. Experimental confirmation, however,

would be required to confirm if these processes in fact occur in the lake, and to

what extent each of them contributes to the overall mixing in Lac Léman.

3.5 REFERENCES

Bäuerle, E. (1985) Internal free oscillation in the Lake of Geneva. Ann. Geophysicae.

2/3: 199-206.

Bohle-Carbonell, M. (1986) Currents in Lake Leman. Limnol. Oceanogr. 31: 1255-1266.

Bohle-Carbonell, M., and D. v. Senden (1990) On internal seiches and noisy current

fields- theoretical concepts versus observations. Large Lakes Ed. M. T. a. C. Seruya.

Springer. 81-105.

Burchard, H., and K. Bolding(2002) GETM – A General Estuarine Transport Model,

EUR 20253 EN, European Commission Joint Research Center, 21020 Ispra, Italy

Burchard, H., and O. Petersen (1997) Hybridisation between sigma and z coordinates

for improving the internal pressure gradient calculation in marine models with steep

bottom slope, International Journal for Numerical Methods in Fluids, 25, 1003-1023,

Caloi, P., M. Migani, and G. Pannocchia (1961) Ancora sulle onde interne del lago di

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:

One-point closure model. Momentum and heat diffusivities. Journal of Physical Oceanography,

31, 1413-1426,

Heinz, G., et al (1990) Vertical mixing in Ueberlinger See, western part of Lake Constance.

Aquat. Sci. 52:256-268.

Jassby A. and T. Powell (1975) Vertical patterns of eddy diffusion during stratification in

Castle Lake, California. Limnol. Oceanogr. 20: 530-543.

Kanari, S. (1984) Internal waves and seiches, Lake Biwa. S. Hone, ed. , Junk. Dordrecht,

pp. 185- 235.

Kanari, S. (1975) The long-period internal waves in Lake Biwa. Limnol. Oceanogr. 20:

544-553.

Lemmin, U. (1987) The structure and dynamics of internal waves in Baldeggersee.

Limnol. Oceanogr. 32: 43-61.

Lemmin, U., and N. D'Adamo (1996) Summertime winds and direct cyclonic circulation:

observations from Lake Leman. Ann. Geophysicae 14: 1207-1220.

Lemmin, U., et al (1998) Finescale dynamics of stratified waters near a sloping boundary

of a lake. Physical processes in lakes and oceans Ed. J. Imberger. Amer. Geophys.

Un., Washington, DC, Coastal and Estuarine Studies, 54:461-474.

Lorke, A., L. Umlauf, T. Jonas, and A. Wüst, Dynamics of turbulence in low-speed oscillating

bottom-boundary layers of stratified basins, Environmental Fluid Mechanics,

accepted 2002

Michalski, J. and U. Lemmin (1995) Dynamics of vertical mixing in the hypolimnion of a

deep lake: Lake Leman. Limnol. Oceanogr. 40: 809-816.

Mortimer, C.H. (1955) Some effects of earth rotation on water movements in stratified

lakes. Verh. Int. Ver. Limnol. 12: 66-77.

Mortimer, C..H. (1963) Frontiers in physical limnology with particular reference to long

waves in rotating basins. Proc. 5th Conf. Great Lakes Res. Div., Univ. Michigan, 9-42.

Mortimer, C.H. (1993) Long internal waves in lakes: review of a century of research.,

Special report, No. 42, Univ. Wisconsin-Milwaukee, Center for Great Lakes Studies,

117 pp.

Mortimer, C.H., et al (1984) Internal oscillatory responses of Lake Geneva to wind impulses

during 1977/78 compared with waves in rotating channel models. Commun. Lab.

Hydraul., Ecole Polytech. Fed. Lausanne, No. 50, 89 pp.

Richter, E. (1897) Seenstudien. Pencks Geogr. Abh., Wien 6: 121-191.

Saggio, A., and J. Imberger (1998) Internal wave weather in a stratified lake. Limnol.

Oceanogr. 43: 1780-1795.

Service fédéral des eaux; SFE (1954). Les dénivellations du lac Léman. Département

fédéral des postes et des chemins de fer. Report, maps, figures.

Sirkes, Z. (1987) Surface manifestations of internal oscillations in a highly saline lake

(the Dead Sea). Limnol. & Oceanogr. 32: 76-82.

Thorpe, S. A. and U. Lemmin (1999a). Internal waves and temperature fronts on

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

using a submarine. Limnol. Oceanogr. 44(6): 1575-1582.

Thorpe, S. A. and R. Jiang (1998). “Estimating internal waves and diapycnical mixing

from conventional mooring data in a lake.” Limnol. Oceanogr. 43: 936-945.

Thorpe, S. A., et al (1996) High frequency internal waves in Lake Leman. Phil. Trans.

Roy. Soc. London A 354: 237-257.

Thoulet, M. J. (1894) Contribution à l'étude des lacs des Vosges. Bull. Soc. Geographie

15: 557-604.

Umlauf, L., H. Burchard, and K. Hutter, Extending the k-omega turbulence model towards

oceanic applications, Ocean Modelling, accepted 2002.

Wuest, A., G. Piepke, et al. (2000). “Turbulent kinetic energy balance as a tool for estimating

vertical eddy diffusivity in wind forced stratified waters.” Limnol. Oceanogr. 45:

1388-1400.

4 MAPS AND TABLES OF FREE INTERNAL SEICHES IN UPPER LAKE

CONSTANCE FOR PRACTICAL USE

4.1 OBJECTIVE

The assessment of the local intensity of internal seiches due to their variable structure

is of considerable concern in applications of water management, certain water constructions

and other executive measures, for which essential information on transient

currents and corresponding water displacements is required. For instance, the spill of

harmful substances, drift of lost bodies, mixing and dispersion in various limnological

and hydrological problems may depend during the period of stratification occasionally

strongly on internal seiches. Their local effect in such cases can be estimated on the

basis of adequately resolved graphical representations of their variable spatial intensity.

As rough information it is often sufficient to get an idea on the local variation of potential

activity by superposed internal seiches of different order.

To enable the expert community without resort to detailed knowledge of the physics of

internal waves, the oscillations have to be displayed in a form easy to grasp by inspection.

This was done in a different delineation than the structures usually shown in terms

of wave parameters such as lines of equal range and phase, which give a condensed

overview. Instead, the resolution into momentary wave stages and local amplitude

variations with time has been presented providing better imagination and at the same

time more refined quantitative information on the structure. The corresponding horizontal

motion is given in field representations of ellipses of the rotating current vectors

with indication of the sense of rotation and the zero phase position for a defined moment

of the vertical elevation. Such a description consists of several diagrams for one

mode and sums up to a collection of numerous illustrations as the number of modes

represented increases. Despite this inconvenience the use is nevertheless facilitated,

as the corresponding diagrams of each mode have the same scale and the thematic

content has been drawn in the same graphically proper forms. This description has

been compiled for the first 15 modes of Upper Lake Constance. A few examples are

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

Hollan in a project of the ISF. Later refined calculations by Bäuerle resulted in more

details but confirmed the main structures of the oscillations already calculated in this

early approach. Thus the compilation from 1991 is still worthwhile in the present context

and may also serve as an example for the other large lakes under consideration. In

particular reference to Lake Leman, the calculation with the same model has been carried

out for the first 12 modes also by Bäuerle (1985). The extension to a similar presentation

was included for merely two modes, thus not allowing for scanning the local

potential contribution of all the 12 modes. The usefulness of the chosen graphic description

of internal seiches has been demonstrated very early by Bäuerle and Hollan

(1983) for the case of the fundamental and a transverse mode of Lake Tanganyika enclosed

in the monography on the lakes of the warm belt by Serruya and Pollingher

(1983). This reference is quoted within the preceding one. C. Serruya recommended in

this context to carry out such work on other large stratified lakes as accomplished later

in the advanced version for Lake Constance here.

4.2 OUTLINE OF THE CALCULATION

The strength of internal seiches is most adequately described in terms of forced oscillations.

However, this approach implies precise knowledge of the driving agent, which is

mostly the wind field over the lake. The horizontal variation of this quantity is generally

not sufficiently known for that purpose, in particular, for large lakes. Moreover, the superposed

different internal modes excited during one event have to be decomposed for

identification of the associated single contributions. Desisting from such a difficult description,

which is practically beyond reach, the relative structures and natural periods

in terms of free oscillations may already serve for essential information to quantify the

phenomenon. If certain observations exist on the mean amplitude of internal seiches

with respect to typical wind fields and the stratification in the lake, the determination

relative to an arbitrary factor may be converted to absolute values, which often suffices

for assessment.


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Page 48 of 92

Figure 33 : Typical temperature stratification in the western part of Lake Constance (Lake Überlingen)

mid October 1972 and the corresponding density stratification (relation ρ = ρ(

T ( z))

see

text, from Bäuerle (1981)

The problem is most conveniently solved by an eigenvalue-method, as the structures

and periods of the oscillations are calculated as distinct constituents of each solution

and result therefore very precisely. The question is, how detailed the hydrodynamic

model is formulated to simulate nature. In the present context a two-layer model has

been adopted including the Coriolis force. By the same reason, as the forcing has been

kept out of concern, friction is not considered, since it represents generally a lakespecific

process. It may be introduced by empirical information and rough linear assumption,

if required for estimation. The approach under these conditions yields still a

very useful description, when the calculations cover also characteristical stratifications

during the warm season, what has been carried out for Upper Lake Constance. The

simplification by a two-layer model restricts the solutions to the fundamental vertical order.

This limitation is to a certain extent serious, as internal seiches of second vertical

order exist in lakes and should be included into the consideration. Since they are not so

frequent and appear less pronounced generally than the fundamental vertical modes,

their omittance may be tolerable for the time being.


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momentary surface

h 1

h 2

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mean position

of surface

mean depth position

of interface momentary depth

position of interface

density ρ 2

density ρ 1

Page 49 of 92

Figure 34 : General sketch and definition of quantities of a two-layer model with respect to the

stratification in a deep temperate lake in summer

Lake Constance is a large deep lake in the moderate zone and develops therefore

during the warm season a stratification, which consists of a relatively shallow surface

layer (epilimnion) over a deep lower layer (hypolimnion). A typical example of the vertical

density variation is given in Figure 33 for early autumn (October 1972). The relation

between density and temperature applied here is valid for the waters of Lake Con-

2

stance (Hollan and Simons, 1978) and reads: ρ = ρ0

−α

( T − T0

) with ρ0 = 1.000145

g/cm³, T in °C, T0 = 4°C, α = 7.3 ·10 -6 g/(cm² °C²). While the upper depth range of 20 m

is covered by the epilimnion, the hypolimnion extends from about 30 m to the maximum

depth of 254 m, or on the average between 30 m and the mean depth of 100 m. The

difference of thicknesses is even increased during summer, since the surface layer is

generally less deep till October, when cooling becomes stronger. Such conditions are

appropriate for a two-layer model of constant equivalent depth he . This quantity appears

in the fundamental equations of internal seiches in a two-layer system and allows for a

description similar to surface (barotropic) seiches, if it can be assumed as constant.

With the definition of the two-layer model (see Figure 34) by an idealized step-like density

stratification:

�ρ1

0 ≤ z < h1

ρ0

= �

�ρ2

h1

≤ z < h(

x,

y)

where x,y,z represent the coordinates of a Cartesian system with z directed vertically

downward (z=0: surface), and ρ1, ρ2 constant densities with ρ2 > ρ1. The equivalent depth

h e

h1

h

=

h + h

1

represents a quantity of low variation as to the deep and steep depth configuration of

Lake Constance.

It is therefore reasonable to approximate he by a constant value,


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(1)

h1

h2

=

h + h

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h e

1

2

Page 50 of 92

where h2 gives now a corresponding constant thickness of the lower layer. In contrast to

the gravity g determining the hydrodynamic pressure forces of surface seiches, the low

density difference between both layers at the interface implies that the corresponding

forces as to internal seiches are governed by reduced gravity, i.e.:


(2) g = g ⋅ε

with

ρ2

− ρ1

ε =

ρ

From this reason result relatively high internal amplitudes of several meters and considerably

long periods of several hours up to several days compared to those of surface

seiches in the same basin. As example , the period T1 of the fundamental mode and the

corresponding frequency ω1 in a rectangular lake of length L, constant equivalent depth

he, without influence of the earth’s rotation, read according to Merian’s formula:

(3)

L

T =


g h

2

1

e

and

= π

ω1 2


g he

In (3) the term g he


represents the phase velocity ci of long internal waves in this system,

which is considerably lower than that of long surface waves cs = g H , with H =

h1+h2. Since the periods of surface seiches obey the corresponding relation as in (3)

with respect to cs , the great difference is evident.

The third physical parameter in the fundamental equations accounts for the influence of

the earth’s rotation. The effective component of the rotational vector of the earth, the

Coriolis parameter f, reads:

(4) f = 2Ωsinϕ

with Ω = 7.29 ·10 -5 s -1 the angular velocity of the earth and ϕ the geographical latitude.

For the mean geographical latitude of Lake Constance, ϕ = 48°N, f amounts to

1.07 ·10 -4 s -1 .

The derivation of the governing equations and their numerical solution is treated by

Bäuerle (1981). After separation of the sinusoidal time dependency, the system of

equations is solved for distinct eigen-frequencies ωn (n = 1,2,3 ...). It describes the dynamical

relations for the dependent variables of the lower layer, which are the amplitudes

of the volume transport 2 V� (x,y) = (U2(x,y), V2(x,y)) and the amplitude of the vertical

displacement of the interface ζ2(x,y) at the top of this layer. It reads:

∗ ∂ζ

2

iω U 2 + fV2

+ g he

= 0

∂x

∗ ∂ζ

2

(5a) iω V2

− fU 2 + g he

= 0

∂y

∂V2

iω ζ 2 + fU 2 + = 0

∂y

with the boundary condition:

L


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(5b) V ⋅ n = 0

� �

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The condition (5b) means that there is no volume transport through to the rigid boundary

expressed by the vanishing scalar product of 2 V� with the unit vector n � normal to the

boundary. (5a) together with the condition (5b) forms an eigenvalue problem with an infinite

number of distinct eigen-frequencies and corresponding eigen-solutions. The

imaginary unit i = −1

in the equations (5a) indicates that the solutions are given in


complex notation. Thus the eigen-solutions of the volume transports V 2,

n (x,y) and the

amplitudes ζ 2,

n (x,y) represent complex functions and the corresponding real functions

are evaluated for presentation in the course of the complex mathematical formulation.

This treatment is also inherent to the nature of the oscillations as horizontally rotating

long waves, which is caused by the influence of the Coriolis force. Due to this pattern a

peculiar task of sufficiently resolved delineation for practical use has to be solved by a

proper design, which is demonstrated in this report.

A general property of the mathematical solution of (5a,b) is that the eigen-functions


V 2,

n and ζ 2,

n are determined except for a free factor. Thus the calculation yields the

structure of the modes relative to 100% either of the maximum elevation of ζ 2,

n or of the


maximum volume transport V 2,

n encountered in the lake. Since the most interesting


functions for application with respect to the structure are ζ 2,

n and V 1,

n , which represents

the amplitude of the volume transport in the upper layer, the relation of the latter quantity

has to be supplemented here:

� �

(6) V 1,

n = −V2,

n

From (6) the vertically averaged current in the epilimnion is inferred by


� V1,

n

(7a)

v1,

n =

h

while that in the lower layer results from


� V2,

n

(7b)

v2,

n =

h

with h2 given according to (1), if he and h1 had been prescribed.

1

2

The equations (5a,b) are not solvable analytically even with constant he neither for very

simple geometrical approximations of the basin nor for the irregular shape assumed

here. Therefore, the solutions are determined by horizontal discretisation of the dependent

variables and the rigid boundary. Their derivations in the fundamental equations

(5a) are replaced by central differences of the variables discretised in a square

grid of 1.4 km mesh size, as used by Hollan et al. (1980) for calculation of the surface

seiches of Lake Constance. The grid is shown Figure 35.


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Figure 35 : Outline of the numerical grid of 1.4 km mesh size adopted for Upper Lake Constance

(Rao - grid in Hollan et al. (1980))

The variables U2,n, V2,n and ζ2,n are defined on this grid, each staggered by 700 m. The

details of the numerical solution are treated by Bäuerle (1981). The minimum depth at

the numerical grid points is assumed to be somewhat larger than the thickness h1 of the

epilimnion . This problem is discussed in Bäuerle’s (1981) treatise in more detail.

4.3 AUXILIARY FORMULATIONS FOR EVALUATION OF EIGEN-PERIODS WITH

RESPECT TO DIFFERENT STRATIFICATIONS

The advantage of the two-layer model with constant equivalent depth consists in the

property that the eigen-periods determined for selected different stratifications may be

transformed to the cases of other stratifications by simple auxiliary relations. In order to

arrive at a concise representation of the eigen-periods, as they vary with the modal order,

with stratification and with respect to the Coriolis effect, this convenience of the

model is utilized as follows. The transformation of the eigen-periods to different stratifications,

if they are computed for one case, is non-ambiguously given, when there is no

influence of the earth’s rotation, f = 0. Such internal seiches represent standing oscillations.

Their period spectrum is determined exclusively by the configuration of the basin

and differs for each peculiar stratification by a constant factor. This factor q is inferred

from the phase velocities ci of long internal waves


(8) ci

= g he

which is characteristic for each case of stratification, as mentioned in the context of

equation (3).

When the eigen-frequencies have been calculated for a selected stratification which is

considered formally as reference stratification and are designated by ωn(ref), the conversion

factor q for the corresponding frequencies ωn(novel) of another stratification is

defined by the ratio:

ci

( novel)

(9)

q =

c ( ref )

and the relation of the eigen-frequencies reads:

(10) ω ( ) = q ⋅ω

( ref ) n = 1, 2, 3...

n

i

novel n

Since the graphical representation of the dependency on the stratification and the influence

of the earth’s rotation is very lucid and simple at the same time, if the frequencies

are normalised to the fundamental frequency ω1 for f = 0, the conversion relation (10)

may be put alternatively into the form


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D28: Internal seiche mixing study

(11) ω ( ) = p ⋅ω1(

novel)

n = 1, 2 ,3...

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n

novel n

Page 53 of 92

where the factors pn represent the normalised eigen-frequencies determined for one

reference case of stratification by the relation (11), i.e. by pn = ωn(ref) / ω1(ref) for f = 0.

The corresponding relations to (10) and (11) in terms of eigen-periods Tn read:

(10)* ( novel)

= T ( ref ) / q

Tn n

(11)* T n ( novel)

= T1

( novel)

/ pn

with pn = T1

( ref ) / Tn

( ref ) , resp.

In order to make use of (11), merely ω1(novel) of the new stratification has to be determined

from equations (9) and (10) for n = 1. It is this notation which had been adopted

due to its feasibility for evaluation and delineation.

The definition and evaluation for a reference stratification with f = 0 has been compiled

in Table 1.


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mode ωn in 10 -5 sec Tn in hours

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σ n = ωn

1 1.910 91.38 1.00

2 3.349 52.11 1.75

3 4.699 37.14 2.46

4 6.255 27.90 3.27

5 7.587 23.00 3.97

6 8.504 20.52 4.45

7 9.096 19.19 4.76

8 10.538 16.56 5.52

9 11.223 15.55 5.88

10 11.849 14.73 6.20

11 12.836 13.60 6.72

12 13.087 13.34 6.85

13 14.016 12.45 7.34

14 14.762 11.82 7.73

15 15.430 11.31 8.08

ω1

Page 54 of 92

Table 1: Eigen-frequencies, eigen-periods and normalised eigen-frequencies of the first 15 modes

for an exemplary reference stratification with f = 0, ε = 5.3 ·10 -4 , he = 21m, h1 = 30m and

h2 = 70m. This case has been observed in the western part of Lake Constance in October 1972 and

had been assumed for the explanation of transverse internal oscillations in this region (Hollan

1974, Bäuerle 1981)

4.4 THE EIGEN-PERIODS INCLUDING THE CORIOLIS EFFECT

As to the eigen-frequencies, the generally strong and complicated influence of the

Coriolis force can be condensed into a very instructive and handy diagram for evaluation

with the preceding consideration in mind. The effect of the earth’s rotation increases

the larger the lake is and the smaller the phase velocity of the internal wave is.

The latter may be due to weakening of the stratification or lowering the values of the

equivalent depth by diminishing the thickness of the upper layer. The dependency on

rotation is at best expressed with respect to the rotation number

( 0)

(12)

F = f /ω

1

( 0)

where F is the ratio of the local Coriolis frequency f and the eigenfrequency ω 1 of the

fundamental mode without earth rotation.


σσ

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σ

10

9

8

7

6

5

4

3

2

1

0

0 5 10 15 20

Figure 36 : Dependency of the dimensionless eigen-frequencies

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F

σn ωn

ω

Page 55 of 92

= on the rotation

( 0)

1

( 0)

F = f /ω for 15 modes of internal seiches in Upper Lake Constance on the basis of a

number

1

two-layer constant equivalent depth model (Figure 35) The vertical dashed lines indicate the dates

for which the calculations of the 1991 report were done. The circles refer to Figures 38, 41, 44, 47

and Figure 50 through Figure 55.

( 0)

Since the stratification enters into ω 1 proportionally to the phase velocity according to

(8), the rotation number F increases, when either the stratification as given in (2) or the

equivalent depth he diminish. The latter is effectuated by reduction of h1, the thickness of

the surface layer. As mentioned above, the influence of the size of the lake is, by the

way, detected with the help of (3) in terms of the length L in a rectangular basin. In this

case the rotation number is proportional to L.

With constant Coriolis frequency the rotation number F depends (inversely) on the

( 0)

( 0)

value of ω 1 . As discussed above, ω 1 decreases with the length of the basin and with

decreasing phase velocity. In other words, the larger and the less stratified the lake is,

the larger is the influence of the earth rotation.

At low F the internal seiches resemble standing oscillations. With increasing F the effect

of rotation becomes dominant and the structure of the oscillations takes the form of a

rotating wave propagating around amphidromic points, where the amplitude of vertical

displacements is zero. The number of amphidromic systems for the lower order modes

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

form in the diagram shown in Figure 36. Some explanations are necessary to

make use of the information compiled in this figure. Increasing influence of rotation is

represented by increasing rotation number F on the abscissa. On the other side, the or-

( 0)

dinate gives the normalised eigen-frequencies σ n = ωn

ω1

, discerned by increasing

modal order up to 15. Thus, a family of 15 curves is displayed, each curve starting on

the ordinate at F = 0, with the fixed normalised distribution for the case of Upper Lake

Constance. These numbers σ n for f = 0 are listed in the fourth column of Table 1, as

their general values result from the adopted reference stratification. For f = 0, they are

the same for any different stratification as outlined in the previous section. These points

on the ordinate mean the onset of the corresponding function showing the increasing

influence of the earth’s rotation on the modal relative frequency in question.

The effect of the stratification and varying equivalent depth is incorporated in the nor-

( 0)

malising fundamental frequency ω 1 . As to the other coordinate, the rotational number

F gives the form, which reflects the dependency on the earth’s rotation with respect to

the size and stratification of the lake. Since both variables are normalised to the same

( 0)

absolute quantity ω 1 , there is a simple method of evaluation with the help of this diagram.

Given a particular stratification and equivalent depth, the corresponding rotation number,

say Fe, is determined according to (12) and (3). The vertical line at this value on the

abscissa crosses the curve family at 15 respective ordinate values σ n(

F1

) ,

n = 1, 2, .., 15. From these relative eigen-frequencies the absolute values ωn(Fe) are

( 0)

obtained by multiplication with ω 1 . The corresponding eigen-periods result from the

formula Tn ( F1

) = 2π

/ ωn

( Fe

) . This evaluation is presented for two examples selected from

33 cases of stratification taken from observations in the lake during the period from

1985 through 1989. The 15 eigen-periods for each case are tabulated by Bäuerle &

Ollinger (1991) in their original report. The data of both examples shown here concern a

case in spring on 13 April 1989 and another one in late summer on 30 August 1989.

They represent a situation, which is strongly influenced by rotation (spring) and predominantly

by stratification (late summer), respectively. The extracted stratification pa-

( 0)

( 0)

rameters and resulting fundamental frequencies ω1 and periods T 1 , as well as the

rotation numbers F are listed (Table 2).


13.4.

1989

1/q=

3.3276

30.8.

1989

1/q=

0.7419

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date h1 he

13.04.1989 20 16.0

30.08.1989 15 12.7

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T1

T2

ρ1

ρ2

7.10 1.000075

5.00 1.000138

18.80 0.998546

5.30 1.000133

ε ci

ω

( 0)

1

( 0)

1

Page 57 of 92

T F

0.63 9.93 0.57 304.2 18.65

15.86 44.54 2.57 67.8 4.16

Table 2 : Approximation of temperature profiles from the central as well as deepest (254 m) position

Fischbach-Uttwil of Upper Lake Constance in spring and late summer 1989 by two layers of

constant density and resulting parameters of a two-layer equivalent depth model of internal

seiches. The total depth of the constant equivalent depth model is :

H = h1+h2 = 100 m.

date date of observation

h1 depth of the upper layer [m]

he equivalent depth [m]

T1, T2 constant temperature of the upper and lower layer [°C]

constant density of the upper and lower layer [g/cm³]

ρ1, ρ2

ε relative density difference ε = (ρ2 - ρ1)/ρ2 , [x10 -4 ]

phase velocity of long internal waves [cm/s]

ω eigen-frequency of the fundamental mode resulting from the two-

ci

( 0)

1

layer equivalent depth model for f = 0 [x10 -5 /s]

( 0)

T 1 corresponding eigen-period [h]

( 0)

F rotation number F = f / ω

1

F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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

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

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

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

Table 3 : Eigen-periods (in h) of the first 15 modes of internal seiches in Lake Constance for stratification

in spring (13 April 1989) and late summer (30 August 1989) displayed with and without

(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

from the diagram in Figure 36. The absolute eigen-periods T ) resulting from

n F

T F ) = T / σ ( F ) are compiled in Table 3.

n(

i 1 n i

In the spring case (13.4.1989) the periods T1 through T15, are rather long varying from

443 h through 31 h, resp. Consequently the rotation number is relatively high,

F1 = 18.65, and the influence of the Coriolis force is dominant. The effect is evident by

the associated periods for the same case without rotation (f = 0) given in the second

row for this date. The latter periods have been calculated from the internal phase velocity

in this stratification quoted in Table 2, by using the relations (9) and (10)* with respect

to the periods of the reference case listed in Table 1. There are considerable differences

in the modal pairs of periods for f ≠ 0 and f = 0 in this case, which exhibit the

strong rotational effects. It is remarkable that the eigen-periods of the lowest modes are

considerably reduced for f = 0 compared to those influenced by rotation. For higher

modal order from seven on the periods for f = 0 are higher. It is worthwhile to notice,

that all the eigen-periods are greater than the inertial period. The consequences will be

discussed in the following section.

In the late summer case (30.8.1989) the rotation number (F2 = 4.16) is much smaller.

While the periods vary from 72.9 h through 8.0 h for f ≠ 0 with the modal order from 1

through 15, the corresponding periods for f = 0 deviate relatively less compared to the

lowest orders of the previous case. The associated periods for f = 0 range from 67.8 to

8.4. Despite that seemingly small effect on the eigen-periods, the effect of the earth’s

rotation is of crucial importance for the higher modes. From general reasons, which are

not delineated here, this is the case for the modes higher than order 5 which is indicated

in Figure 36 by the straight line from the origin, σ = F, and the modes above it at

F2 = 4.16. This transition is also reflected in the shift of relative amount of the periods

between modal orders 5 and 6 given in Table 3. for f = 0 and f ≠ 0 in the last two lines,

respectively.

A final remark is in order for the application in case of an observed stratification which is

not represented in the tabulated cases for f ≠ 0. Such a situation is generally to be expected

and the large amount of 33 calculated examples has been achieved to meet this

problem as to the periods. As approximation it is considered to be sufficient, if the most

similar calculated case is found out as to the interesting stratification. If more precise

assessment is required, an interpolation between two adjacent calculated cases may

serve in the way that ci of one of them is greater and of the other is lower than that of

the stratification in question.

4.5 THE HORIZONTAL STRUCTURES OF THE INTERFACE AMPLITUDES

Progressive rotating waves are usually illustrated in a diagram, which shows corange

and so-called cotidal lines representing lines of equal amplitudes and of equal phases,

resp. This delineation is inherent to the mathematical expression of such eigenoscillations.

For a given eigen-frequency ωn the corresponding amplitude function of the

solution as to the interface elevations reads:

(13) ζ ( , y,

t)

= A ( x,

y)

cos( ω t + ϕ ( x,

y))

n = 1, 2, ...

n x n

n n

n( Fi


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Since free eigen-oscillations are determined except for an arbitrary constant factor, the

amplitudes An(x,y) are merely known to their relative distribution. Their absolute maximum

may be ascribed the value of 100%, which occurs for every modal order generally

at a different position in the interface. The corange lines show the horizontal structure

and are displayed by equal percentage increments. The evaluation of the solution (13)

is in terms of the formula

(13a) A n ( x,

y)

= Pk

with Pk in steps of 10% from -100% through +100%, correspondingly identified by the

index k in the range {-10 (1) 10}.

A

B D

C

G

F

E

Figure 37 : Contour of the numerical grid of Upper Lake Constance shown in Figure 35with the 19

sites, from where the time dependent amplitude variation is illustrated in Figure 40, Figure 43,

Figure 46 and Figure 49.

Superposed in the same diagram are usually the cotidal lines, which represent the process

of wave propagation. They are evaluated in equal steps of phase increase in the

argument of the cosine function in (13). If the maximum amplitude is considered as

phase stage, the cotidal lines are described by the formula


k

(13b) + ϕ n ( x,

y)

= 0 with k = 0, 1, 2, ..., K,

K

where K is the integer which divides the wave cycle into equal phase steps of 2π / K .

The general pattern of cotidal lines in horizontally rotating waves is the radial arrangement

of the curves meeting another in amphidromic points. When the wave progresses

in the mathematical positive sense, i.e. is turning around the amphidromic point in anticlockwise

direction, this point and system is called cyclonic, as the sense of the earth’s

rotation is the same. The opposite sense of wave propagation designates the amphidromic

system as anticyclonic.

In the following, four sets of diagrams are presented as a selection from a total number

of 30 which have been compiled for consultation of structural details of the first 15

modes. The doubling results from the elaboration for the two aforementioned contrasting

stratifications, namely, the case dominated by rotation in spring time (13 April 1989)

and the case of governing gravity effects in late summer (30 August 1989). Particularly,

the first and ninth modes are presented here, because the pattern of the structures and

their variation with increasing modal order are disclosed well with this choice. Moreover,

J

I

H

L

K

M

O

N

P

R

Q

S


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they give a good idea of the instructive and feasible survey possible by the structurally

more resolving illustrations.

Before entering into the graphical inspection a reference map of 19 adjacent places on

the boundary and in the interior of the lake has to be considered. These selected

points, designated by the letters from A through S, are shown in Figure 37 on and inside

the contour of the numerical grid. They serve for a proper display of local time histories

of the interface displacements and are referred to in this peculiar thematic diagram

in the sequence of illustrations for each selected mode. Besides these examples,

there is another subset of diagrams, which shows transverse and longitudinal sections

of the structure of interface displacements throughout the lake or in bays. Although

these illustrations provide another comprehensive information, they are only mentioned

here for the sake of brevity.

After this preliminary note, we resort to the standard representations by corange and

cotidal lines as outlined above. The corresponding diagrams are given in the first place

of each series of representations in Figure 38 and Figure 41 as well as Figure 44 and

Figure 47. The corange diagrams are resolved to 10% steps of the maximum, while the

associated cotidal behaviour is displayed in steps of 1 /12 of the period. It is obvious from

these delineations that a comparison of several consecutive modes at particular places

in the lake is toilsome and will not help sufficiently for a comparative detailed description

of the different relative local contributions. Therefore, the solutions (13) are represented

additionally by realistic stages of the motion, particularly in time steps of the interface

elevations, as they form in the process of wave propagation. This graphical

resolution has been calculated for the same phase increments as in the cotidal diagrams

and is illustrated by six stages for the first half of the corresponding periods in

Figure 39 and Figure 42 and correspondingly in Figure 45 and Figure 48. The corange

lines in these diagrams are resolved again in 10% steps of the maximum. They are correspondingly

the same in the second half of the oscillation except for the change of the

sign of elevations and need not be repeated.


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Figure 38 : Corange and cotidal diagram of the fundamental internal seiche for F = 18.65 (13 April

1989) with the eigen-period of T1 = 407.3 h. The corange lines are given in 10% steps of the maximum,

while the cotidal lines resolve the wave propagation in phase steps of 1 /12 T1 . The oscillation

is cyclonic (anti-clockwise).

If the continuous horizontal representation at different moments is changed to continuous

display of the time history at different sites, a very instructive information results.

This has been calculated for 16 sites on the shore and three in the middle of the lake as

indicated in Figure 37. The resulting 19 diagrams are assembled in Figure 40 and

Figure 43 as complement to the preceding figure and corresponding to the summer

stratification in Figure 46 and Figure 49. The amplitude scale differs in this kind of diagrams

from the previous relative representations, as it is showing the variation normalised

to a maximum of 5 m. Such an assumption meets a realistic size of interface elevations

associated often with internal seiches in Lake Constance. How instructive this

subset of local time histories is, may be perceived from comparable inspection of different

modes for different stratification at a fixed place or at neighbouring places in relation

with the discrete stages, in time, but represented continuously in horizontal dimension

in the preceding figures. These complementary delineations of the amplitude structures

serve a great deal for practical use in the context with the maps of the associated current

field in the surface layer shown in the next section.

A remark on the general structure and its variation with growing influence of the earth’s

rotation on one hand and with gravity on the other hand has to be accomplished, as this

pattern is manifested in the complete series of diagrams in the unpublished German

report and is also apparent in the examples presented here. In the case of high rotation

number, i.e. the case in spring time with extraordinary long eigen-periods, higher amplitudes

are confined to the near-shore region throughout all calculated modes. Moreover,

all amphidromic points in structures up to the 15 th mode are cyclonic and distributed

very regularly, as if suspended on a mid-lake line from one end to the other. This

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

the Bay of Rorschach at the greatest width in the eastern half of the basin.

The general regular structure is contrasted by the modes in the late summer stratification.

Here, we realize a reduced confinement of higher amplitudes near the shores in

the wider part of the basin. By contrast, the high amplitudes in the less wide parts as in

the western half and at the eastern end are distributed similar to those of standing

waves at vanishing rotation number. Thus there is a hybrid form of the general structure

which resembles to a certain extent that of standing oscillations.

As to the amphidromic points with increasing modal number, a characteristic difference

appears with the ninth mode, which is represented for spring and summer stratification

in Figure 41 and Figure 47, resp.. Regarding the latter case it has to be premised that

up to the eighth mode all amphidromic systems are cyclonic, their centres aligned along

mid-lake from the western to the eastern end, except for one situated in the Bay of Rorschach.

In the ninth mode shown in Figure 47 a dominant anticyclonic amphidromic

system appears just east of the centre of the lake and a second lateral cyclonic system

appears off the Bay of Constance. For higher modes the structure is seemingly more

complicated, since a few amphidromic systems exist near each other. However, this

pattern develops with increasing mode number more into forms which resemble nodal

lines of standing waves. This is disclosed by crowded cotidal lines connecting certain

adjacent amphidromic points. Such structures represent the change of nodal lines in

standing waves by weak rotational influences in so far, as the jump of the phase by π

across the node is resolved into a steady variation within a narrow band along the node.

Desisting from further considerations of the details an open question has still to be

mentioned. The limitation of the calculations to two typical stratifications with respect to

the structures leaves the user with the uncertainty, whether there are essential variations

of the structures for intermediate cases of stratifications. Certainly, the late summer

situation represents conditions of stratification, which resemble more each other

throughout most of the warm season. Therefore, this part has more bearing on application,

while the displayed case of strong rotational influence in spring may merely provide

rough insight also into weaker autumnal stratification with deepened surface layer.

To the pending completion of the calculations as to other interesting stratifications, the

late summer example allows nevertheless for considerable insight and assessments, if

the assumption of gradual variation of the structures with moderate change of stratification

is correct. As this behaviour can be recognised in the structures with increasing

modal order for both selected stratifications, it is to be expected also for modifications

of these cases.


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Figure 39 : Momentary interface topographies of the first mode shown in Figure 38 at the first six

phase steps of 1 /12 T1 . The lines of zero elevation are congruent with the cotidal lines in Figure 38.

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

sites on the shores and in the lake, however normalised to 5 m maximum amplitude. Map of sites

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

13 April 1989 with the eigen-period of T9 = 48.2 h (further explanation see Figure 38). The oscillation

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

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

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 seiche for F = 4.16 (30 August

1989) with the eigen-period of T1 = 72.5 h (further explanation see Figure 38). The oscillation is cyclonic

(anti-clockwise).


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Figure 45 : Momentary interface topographies of the first mode shown in Figure 44 (30 August

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

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 seiche for the stratification on 30

August 1989 with F = 4.16 and the eigen-period of T9 = 10.7 h (further explanation see Figure 38).

All the amphidromic points are cyclonic (anti-clockwise) except for the central one in the mid of

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

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

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

The information about the transports associated with the different modes of internal

seiches has important practical reasons as well. It allows for assessment of horizontal

oscillatory displacements and dispersion of either harmful substances or other interesting

compounds in the water, when the site considered is subjected to enhanced activity

of internal seiches. For this purpose, the diagrams have been extended to the horizontal

dependency of the currents. To achieve this, another instructive graphical description

had been elaborated, which meets the vectorial character of this quantity, and delivers

the variation of speed and direction during one wave cycle in synoptic charts. Borrowing

from the illustration of tidal currents, a similar graphical method had been

adopted, namely by showing the current ellipses and, separately, the sense of rotation

indicated at a peculiar moment of the current distribution.

Figure 50 : Current field of the first internal mode in the surface-layer on 13 April 1989 with

T1 = 407.3 h. Upper diagram: current ellipses. The normalised maximum transport at the southern

shore of the entrance of Lake Überlingen is about 600 cm²/s). Lower diagram: momentary current

distribution and sense of current vector rotation.

Before treatment of details, the dimension of transport used here needs a comment.

The numbers of transport are given in cm²/s which results from the vertical integration

of the velocity throughout the upper or lower layer. This quantity is meant as transport

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

unique. However, a clear (and reasonable) graphical representation is only possible for

the relative horizontal distribution of the transports in the upper or lower layer, according

to

V 1 = - V2

→ →

respectively. As the isotherm displacements due to internal oscillations are

much easier to measure than the currents, normally it will be the task to refer the transports

to the amplitudes of the vertical displacements of the interface between the upper

and lower layer, which we have presented in the preceding section. Knowing the phase

velocity and the more easily observable and assessable amplitude of the vertical interface

displacement of a specific mode at any location in the basin it is possible to determine

definite transports by evaluating the relative results of the numerical calculations.

Figure 51 : Current field of the ninth internal mode in the surface-layer on 13 April 1989 with

T9 = 48.2 h. The normalised maximum transport is situated at the mouth of the Bay of Constance

and amounts to about 1800 cm²/s. (further explanation inFigure 50)

To give an example: The vertical displacement of the interface by the fundamental

mode at 30 August 1989 (Figure 44) has its maximum amplitude (100 %) at the very

end of Lake Überlingen. The maximum transport of that mode occurs near the Sill of

Mainau (Figure 53) indicating strong exchange flow between Lake Überlingen and the

main basin of Upper Lake Constance. If we assume the value of 100% to be equivalent

to 100 cm of real interface displacement and normalise the vertically integrated transport

in the upper layer to this quantity, the amplitude of the horizontal transport at the

central position of the Sill of Mainau would be about 4300 cm 2 /s with the definition of

the dimension explained above. Taking the actual depth of the upper and lower layer,

respectively, the horizontal velocities result for the present case to 4300/1500 cm/s and

4300/8500 cm/s as to the lower and upper layer, respectively. These numbers are given

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

central position of Lake Überlingen, where amplitudes of about 12 m are of common

occurrence (Bäuerle et al., 1998): From Figure 44 it is inferred that at central Lake

Überlingen the amplitude of the interface displacement is about 90 % of the maximum.

This in turn yields that 13.3 m is the corresponding amplitude at the very end of Lake

Überlingen. Finally, in analogy to the above case of reference, we get

13.3 x 4300/1500 = 38 cm/s and 13.3 x 4300/8500 = 6.7 cm/s as vertically averaged

velocities in the upper and lower layer in the Straits of Mainau, respectively, which correspond

to an amplitude of 12 m measured at a central position of Lake Überlingen.

Figure 52 : Current field of the eleventh internal mode in the surface-layer on 13 April 1989 with

T11 = 39.8 h. The normalised maximum transport occurs at the mouth of the Bay of Constance and

amounts to about 5600 cm²/s. The cross mark near the southern shore in the eastern half of the

lake designates the site of the waste water intake discussed in the text. (further explanation in

Figure 50)

It should be mentioned that the same value of the phase velocity of long internal waves

ci results from different combinations of he and ε according to the relations (1), (2), (8).

Since the eigen-solutions of the problem (5) are uniquely determined for a very phase

speed, ci, the respective definite transport field of the mode in question has to be

evaluated with regard to he, i.e. the associated combination of h1 and h2, which fit together

with ε in the relation (8). In this sense, there is a certain variety of two-layer

manifestations equivalently related to a unique set of eigen-solutions. Thus, for the

same maximum amplitude of a mode of them different definite transports result at a

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,

ε, on the other hand. This peculiar variety of two-layer cases covered by one

set of eigenfunctions means a useful advantage, as the same calculations can be exploited

for certain different two-layer approaches.

The current fields of the first and ninth modes are depicted in Figure 50 and Figure 51

for the case in spring and in Figure 53 and Figure 54 for that in late summer. Additionally,

the currents of the eleventh mode are enclosed in Figure 52 and Figure 55 in order

to emphasize the practical use by a peculiar application, which Hollan (1995) carried

out and is outlined below. Except for the reference to the amplitude normalisation the

relative variation of the structures in the current fields can be detected well from the

diagrams. They consist each of a pair, showing the current ellipses in the upper diagram

which indicate the position of the head of the current vector when turning during

one cycle of oscillation. The main axis of the ellipses shows the orientation of the predominant

current during a wave cycle, while the small axis gives the maximum transverse

currents a quarter of the period later than the main currents. In the lower diagram

the information is compiled, which concerns the sense of rotation at a given moment.

Included into this stick representation is the sense of rotation of the current vector. The

illustration of the phase relation and the sense of rotation throughout the lake is secondary

for a rough evaluation, as this information is more important for inspection of

closely adjacent conditions. The assessment of the local intensity of the currents is very

comprehensive in the upper diagram. There is remarkable variation of the intensity and

relative difference between the main and the transverse currents. This is easily perceived

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

T1 = 72.5 h. The transport normalised to 1 m maximum vertical displacement of the interface is

4300 cm²/s at the Sill of Mainau, resulting in a current velocity of about 3 cm/s, in the upper layer

(h1 = 15 m) and about 0.5 cm/s in the lower layer (h2 = 85 m), respectively. The evaluation is given

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

h. The normalised maximum transport (at the mouth of the Bay of Constance) is about 1800 cm 2 /s.

(further explanation in Figure 50)

Not shown here is an example of another additional set of diagrams on current ellipses.

For more detailed local insight, the currents have been depicted in this sort of diagrams

on a larger scale in selected sub-regions. The amount and phase variation with time

along with the sense of rotation has been graphically well resolved. This very instructive

information has been compiled for a few modes in sub-regions covering the complete

Upper Lake Constance. As to this survey report, the quotation of this elaboration may

suffice.


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Figure 55 : Current field of the eleventh internal mode in the surface layer on 30 August 1989,

T11 = 9.6 h. The cross mark near the southern shore in the eastern half of the lake designates the

site of the waste water intake discussed in the text. The normalised maximum transport (at the

mouth of the Bay of Constance) is about 3400 cm 2 /s. The normalised transport at the waste water

intake is about 3200 cm 2 /s. (further explanation in Figure 50)

The above mentioned application by Hollan (1995) elucidates this way of consideration.

For information, the site of a great waste water inlet, which was considered for alternative

construction, is marked by a cross in the lower diagrams of Figure 52 and Figure

55. These figures show the current fields of the eleventh mode in both cases. Compared

to those of the first and ninth mode they differ in the region of this site remarkably,

in particular for the late summer stratification, which is rather representative for the

summer season. The investigation of dispersion conditions in front of this shore section,

which is just aside of the mouth of the Old Rhine, resorted also to the potential activity

of internal seiches in this region. From the comparison of the relative current contributions

by different modes of internal seiches it was deduced, that considerable variability

had to be expected from this process. These conditions can be recognised even from

the examples shown here, as mentioned above. The result of the calculations with this

application is supported by long-time experiences of local fishermen, who reported to

the author (Hollan) about the difficulties with the retrieval of their drift nets in this region

due to transient high vertical current shear in the thermocline. Such phenomenon is

very probably caused by internal seiches, since otherwise vertical shear would be associated

with drift and compensating gradient current during stronger wind attack over the

lake. However, during such conditions fishermen interrupt their work on the lake, what

underscores the first interpretation.


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Bäuerle E. (1981) : Die Eigenschwingungen abgeschlossener, zweigeschichteter Wasserbecken

bei variabler Bodentopographie. Berichte aus dem Institut für Meereskunde

an der Univ. Kiel, Nr. 85, Kiel, 79pp.

Bäuerle E. and E. Hollan (1983): Calculation of the fundamental and a transverse mode

of internal seiches in Lake Tanganyika. In: Serruya C. and U. Pollingher: Lakes of the

Warm Belt. Appendix, p.499-503, Cambridge, pp.569, Cambridge University Press.

Bäuerle E. (1985): Internal free oscillations in the Lake of Geneva. Annales Geophysicae,

Vol. 3, p.199-206.

Bäuerle E. and D. Ollinger (1991): Karten-Dokumentation der internen Seiches des Bodensee-Obersees

für den Gebrauch in der wasserwirtschaftlichen und limnologischen

Anwendung. Unpublished report of the Institut zur Erforschung und zum Schutz der

Gewässer Ottendorf, by contract with the Landesanstalt für Umweltschutz Baden-

Württemberg, Institut für Seenforschung, Langenargen. 10p. with numerous tables and

figures.

Bäuerle E., D. Ollinger and J. Ilmberger (1998): Some meteorological, hydrological, and

hydrodynamical aspects of Upper Lake Constance. In: Bäuerle, E. and Gaedke, U.

(eds.): Lake Constance, Characterization of an ecosystem in transition. Arch. Hydrobiol.

Spec. Issues Advanc. Limnol. 53, p. 31-83.

Hollan E. (1974): Strömungsmessungen im Bodensee. Arbeitsgemeinschaft Wasserwerke

Bodensee-Rhein (AWBR), Sechster Bericht, p.111-187.

Hollan E. and T.J. Simons (1978): Wind-induced Changes of Temperature and Currents

in Lake Constance. Arch. f. Meteorologie, Geophysik u. Bioklimatologie, Ser. A,

Vol 27, p333-373.

Hollan E., D.B. Rao and E. Bäuerle (1980): Free Surface Oscillations in Lake Constance

with an Interpretation of the “Wonder of the Rising Water “ at Konstanz in 1549.

Arch. f. Meteorologie, Geophysik u. Bioklimatologie, Ser. A, Vol. 29, No. 3, p.301-325.

Hollan E. (1995): Ausbreitungsverhalten des Abwassers aus der Kläranlage Altenrhein

im Fernbereich zweier alternativ projektierter Auslauföffnungen an der Halde südwestlich

des Rheinspitz-Canyons im Bodensee-Obersee. Internal report of the Institut für

Seenforschung (Landesanstalt für Umweltschutz Baden-Württemberg), by contract with

the Amt für Umweltschutz of the Kanton St. Gallen, Langenargen, pp.17, 1map, 12 figures.


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5 LOCH LOMOND

Although there were no specific modelling activities foreseen within EUROLAKES investigating

mixing by internal waves in Loch Lomond it was nevertheless found advantageous

to record recent experiences from field campaigns on the importance of long

internal waves in this lake of irregular shape.

5.1 INTRODUCTION

Loch Lomond is a warm monomictic lake with three very distinct basins. The northern

and central basins are narrow fjord-like (with up to 200 m water depth) and are separated

by a chain of islands from the shallow south basin with a maximum depth of 30 m.

Geologically the loch is a long deep trough of glacial origin and is separated from the

sea by a moraine dam in the south. The mean lake level of Loch Lomond is not higher

than 8 metres above mean sea level. Stratified conditions progressively occur from May

through the summer months particularly in the north basin where a thermocline develops

at a depth of about 15-25 m, separating the warmer epilimnion (ca. 14°C) from the

cooler hypolimnion (ca. 6° C) which was already noted by Slack (1957).

There are, however, only a few past recordings of temperature profiles, the only consistent

approach has been done between 1969 and 1972 by Tippett (1994) with roughly

one measurement profile of temperature and oxygen per month in the centre of every

basin. The result of these experiments are depicted in the following figures for the years

1970 and 1971. Whereas in the south basin thermal stratification occurred only for a

few weeks the deeper portions of the loch show considerable stratification during the

summer months.

Figure 56 : Temperature profiles in the southern basin of Loch Lomond during 1970 and 1971

(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

during 1970 and 1971 (data according to Tippett, 1994) with pronounced stratification and

high probability of long internal waves


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5.2 MEASUREMENT INFORMATION ON INTERNAL WAVES

As there is an apparent lack of information concerning the temporal development of the

seasonal thermo- and pycnoclines in Loch Lomond it was decided within the EURO-

LAKES project to carry out a long-term monitoring survey with thermistor-chains

moored at three locations along the main north-south axis of the loch. These thermistor

chains were planned to provide a complete one year data set on the seasonal changes

in water temperature on three vertical profiles from the surface down to a maximum

depth of 50 meters. In conjunction with a number of quasi-synoptic CTD-surveys and

further meteorological monitoring data on solar radiation, air temperature, wind, etc. the

thermistor chain data will provide more comprehensive information to obtain a better

understanding of Loch Lomond’s physics and even more to get a better understanding

of its complete ecosystem.

North Basin

Mid Basin

South Basin

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,

Norway) which were provided by the “Institut für Seenforschung” in Langenargen,

Germany within the frame of the scientific co-operation in the research project

EUROLAKES. Management and operations during the survey were carried out by the

project partners from University of Glasgow. The thermistor chain measurements were

planned to cover a time period of at least one year. To gain a high resolution data set

regarding the thermodynamic processes during the development phase of the summer

thermocline a sampling rate of 10 minutes was chosen for the first deployment in the

mid basin in May 2002. Realising that a 10 minutes measuring interval would need data

collection at least every month, it was decided to switch over to a sampling rate of 30

minutes for all deployments from June 2002 onwards. In this report data from the first

measurement period in summer 2002 are used to look at the relative importance of

long internal waves.

Figure 59 : Temperature measured by individual thermistors from May 3 rd to June 8 th 2002 by the

upper chain (in 3 m to 23 m water depth every two metres) in the mid basin (Post, 2002)


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Depth [m]

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Water Temperature in Loch Lomond near Ross Point

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Figure 60 : Temperature profiles from 3 rd May to 9 th June 2002 in the mid basin position (Post,

2002)

The recordings from all three locations show considerable short-term fluctuations and at

certain times sudden vertical homogeneity in the thermal conditions. Interpretation of

these (horizontally seen) point measurements is not as straightforward as it might seem

because Loch Lomond has a very irregular shape. This means that a generalisation of

these profiles as “basin characteristic” is not possible. Investigations with 3D models

(described in other reports of EUROLAKES) did show strong wind-driven currents with

pronounced upwelling/downwelling processes near steep lake shores and the possible

occurrence of longer period internal standing waves in the fjord-like section of the lake.

Generally, however, it can be stated that long internal waves in the south basin are very

intermittent and associated mixing processes will be much smaller than the turbulent

mixing associated with the wind stress at the surface. Therefore in this report analysis

of data is confined to the mid and north basin locations where stratification is strong in

the upper 40 metres of the water column. Measurements are scheduled to proceed until

spring 2003 but we are concentrating here on the summer situation.

12.7

12.2

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10.2

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6.7


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5.3 ANALYSIS AND INTERPRETATION

In order to obtain a clearer picture of the kind of temperature fluctuations occurring in

the upper water column in Loch Lomond the thermistor recordings from both north and

mid basin locations were analysed statistically for the period of 8 th August and 18 th

September 2002.

In the following temperature histograms are shown for two vertical levels (6 metres and

45 metres below surface) for north and mid basin deployments depicting the existence

of pronounced long internal waves in Loch Lomond with a stronger variability of temperature

in the central basin.

Frequency analysis provides a few conspicuous periods ranging between about 5 hours

and 24 hours (a frequency of 0.25 in the figures is connected with a period of 2 hours =

time step of recording divided by frequency). The periods coincide with theoretical values

for long standing waves in a narrow channel of 20 km length (north + mid basin) for

strong vertical density stratification near the surface and less strong ones below 20 metres.

The results for the mid basin prove quite clearly that it reacts as an appendix to the

north basin because the long periods cannot be explained by its own basin length of

roughly five kilometres. The differences in periods are due to different thermal stratification

conditions.

5.4 REFERENCES

Post, J. (2002): Thermistor measurements in Loch Lomond. HYDROMOD Scientific

Consulting, Wedel – unpublished report.

Slack H. D. (ed.) (1957) Studies on Loch Lomond 1. Blackie and Son Ltd., Glasgow.

Tippett R. (1994): An Introduction to Loch Lomond, Hydrobiologia, 290, 11-15. Kluwer

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

(below) during 8 th August to 18 th 0.0

15.0 15.4 15.8 16.2 16.6 17.0 17.4 17.8 18.2 18.6 19.0

September 2002


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Figure 62 : Histogram of temperature at 45 metres water depth for north basin (above) and mid basin

(below) between 8 th August and 18 th 0

7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2

September 2002


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Figure 63 : Frequency analysis of temperature fluctuations in mid basin (at 16 to 41 metres, picture

above) and north basin (20 to 51 metres, picture below) for low pass-filtered data.


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6 CONCLUSIONS

The better understanding and knowledge of the internal gravity waves (seiches) in deep

large lakes were the main objectives of this study. We have focused the work principally

on the determination of the waves properties through measurements analysis and numerical

simulation. The seiches qualification has been done all over the four studied

lakes. The influence of the seiches on mixing processes was also studied.

Several approaches of the internal gravity wave in large deep lakes have been presented:

• Analysis of existing measurement on Loch Lomond, Lac du Bourget, Lac

Léman,

• Linear analysis (eigen-value method): Lake Constance,

• Numerical 3D model analysis: Lac du Bourget, Lac Léman

6.1 LAC DU BOURGET

The analysed measurements are shown the existence of internal gravity wave in the

Lac du Bourget with a period between 40hours to 80 hours depending on the vertical

stratification structure.

The simulation using the TELEMAC-3D model is used to reproduce one seiche event

on the Lac du Bourget. The numerical results are in a quite good qualitative agreement

with the measurements.

The numerical results are given some interesting new information about the hydraulic

respond of the lake under wind forcing. After the generation of the internal seiches, the

wave propagation is showing a rotating structure, due to the Coriolis influence, with an

amphidromic point at the middle of the lake (Figure 16).

6.2 LAC LÉMAN

An analysis of internal seiches dynamics was carried out for Lac Léman combining field

measurements and numerical modeling. Using field data of temperature, currents and

surface elevation, it has been shown that only two modes of internals seiches are sufficiently

excited in Lac Léman to be considered significant. The first one is a Kelvin wave

and the second one is a Poincaré wave. Model calculations have indicated that other

seiche modes can only be excited by winds from certain directions. However, due to the

topographic constraints particularly in the eastern part of the lake basin the wind field

over the lake is strongly canalized and these winds do not exist in nature.

From our analysis of field studies of the longterm mean conditions of mixing it is indicated

that internal seiches are important in providing vertical mixing. Recently, it has

been shown though that most of this mixing is actually generated in the near shore

zone and then propagates into the open waters (Wuest et al., 2000). Thus, the interaction

between near shore zones and the open water is also important for mixing. Furthermore,

we have pointed to the importance of the interaction with the sloping sides of

the lake and short progressive internal waves (Thorpe and Lemmin, 1999a, Lemmin et


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al., 1998). These waves and their breaking play a role in the production and redistribution

of currents and stratification as well as mixing (Thorpe and Jiang, 1998). From our

studies it appears that short progressive internal waves are often produced in the passage

of non-linear internal seiches (Thorpe et al., 1996).

The measured characteristics of the velocity time series (namely the non-linear wave) in

the case of a strong wind event could be reproduced. In addition, the good agreement

of the measured and computed structure of the temperature profile suggest that the

turbulence model yields reasonable turbulent diffusivities.

6.3 LAKE CONSTANCE (BODENSEE)

The method of calculating the eigen-periods of free internal oscillations in a two-layermodel

also permits a relatively simple evaluation of the influence of the stratification on

the eigen-periods. But also the variation of the structure with respect to the stratification

is calculated at the same time and provides the characteristical differences, which result

from strong or diminishing influence of the earth´s rotation via varying stratification.

The application of the eigen-value method should be calculated for other lakes in order

permit the comparison and evaluation of the lake-specific characteristics

6.4 LOCH LOMOND

In order to obtain a clearer picture of the kind of temperature fluctuations occurring in

the upper water column in Loch Lomond the thermistor recordings from both north and

mid basin locations were analysed statistically for the period of 8 th August and 18 th

September 2002. Frequency analysis provides a few conspicuous periods ranging between

about 5 hours and 24 hours.

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