D28: Internal seiche mixing study - Hydromod

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 35 of 92
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