D28: Internal seiche mixing study - Hydromod

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Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas EUROLAKES D28: Internal seiche mixing study FP5_Contract No.: EVK1-CT1999-00004 Version: 1.2 Date: 24.08.2004 File: D28.doc 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.
Integrated Water Resource Management for Important Deep European Lakes and their Catchment Areas EUROLAKES D28: Internal seiche mixing study momentary surface h 1 h 2 FP5_Contract No.: EVK1-CT1999-00004 Version: 1.2 Date: 24.08.2004 File: D28.doc 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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D28: Internal seiche mixing study - Hydromod

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