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Motivation

Linear modal analysis for a stratified rotating lake was proven to be a useful tool to understand basin-scale internal waves in medium-sized lakes (see examples from Lake Biwa and Lake Kinneret). Since basin-scale internal waves in lakes can steepen and degenerate into solitary waves, it is natural to extend the method to include nonlinearity. I developed the nonlinear theory during my PhD³⁾, but it was my collaborator, Alberto de la Fuente (Uni. of Chile, Santiago), who applied the theory to Lake Constance.

Theory of Fully-nonlinear Modal Analysis

I had to come up with a mathematical trick to extend the theory of the modal analysis to include full nonlinearity. This is because it is much preferred to have the kinetic energy in a quadratic form from a mathematical view point, but nonlinearity introduces cubic terms. One solution was to expand the layer-averaged velocity $\vec{v}_k$ and the layer transport $\vec{q}_k=h_k \vec{v}_k$ ($h_k$ is the layer thickness and $k$ is a layer index) into modal components individually, following the idea by Imai²⁾ who proposed a similar approach to define energy in electromagnetism. Omitting further details, the amplitude of individual modes $a^{(n)}$ follow evolutionary equations that look like, neglecting forcing and damping terms, \begin{align} \frac{d a^{(n)}}{dt} = i \omega^{(n)} a^{(n)} + \sum_{l,m} n^{(n,l,m)} a^{(l)} a^{(m)}, \end{align} where $\omega^{(n)}$ is the angular frequency of $n$^th mode, and $n^{(n,l,m)}$ is the nonlinear interaction coefficient among $n$^th, $l$^th, and $m$^th modes. There is also equations corresponding to the modal energy equations \begin{align} \frac{d E^{(n)}}{dt} = \sum_{l,m} \dot{F}^{(n,m)}, \end{align} where $\dot{F}^{(n,m)}$ is the nonlinear "energy" transfer rate. Although the sum of $E^{(n)}$ over all the modes is equal to the total energy, $E^{(n)}$ is not a positive definite quantity under strong nonlinearity. So $E^{(n)}$ should be regarded as a pseudo-energy.

Study Site and Observations

Lake Constance is a long, deep lake located between Germany and Switzerland (Fig. 1A,B). It consists of the main basin (Upper Lake Constance) and a sub-basin (Lake Überlingen) separated by Mainau Sill. Previous studies showed that the fundamental mode Kelvin wave with ~90 h period transforms into an internal surge as it travels along the northern shore¹⁾, and then degenerates into solitary and other high-frequency internal waves⁴⁾.

Fig. 1C shows an example of the steepened basin-scale internal Kelvin wave in the lake, observed near the northern shore by a thermistor chain in 2003.

Model Set-up

Assuming two-layer stratification, we calculated 360 basin-scale modes with periods between 5.4 and 105 d. The modes include Kelvin and Poincaré (gravity) waves, and topographic waves.

The nonlinear model is run for 3 days from Day 294 in 2003. To excite lower-mode internal Kelvin waves, the model is forced with a measured northwesterly wind event with ~10 m/s wind speed and ~12 h duration. Then the model is left free for 2.5 days to investigate the nonlinear evolution of the Kelvin waves.

Results and Discussion

The model reproduced the steepened Kelvin wave at S5 with reasonable amplitude and shape (Fig. 2). In modal analysis, the steepening is represented by superposition of higher azimuthal-mode Kelvin waves that are excited by nonlinear interaction among different modes. The nonlinear transfer towards higher modes (or cascade) can also be seen in the "energy spectra" before and after the wave steepening (Fig. 3).

The above results are familiar from the spectral approach in fluid mechanics, for example, that used in the turbulence theory and Direct Numerical Simulation (DNS). This is not surprising because the two becomes equivalent if we simplify the Kelvin waves in Lake Constance as internal seiche in a non-rotating rectangular basin. The theory proposed in this study extends the nonlinear spectral method to a stratified rotating basin with realistic bathymetry.

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Related Publications

Details are available in

de la Fuente, A., K. Shimizu, Y. Niňo, and J. Imberger. 2010. Nonlinear and weakly nonhydrostatic evolution of basin-scale waves in a large deep lake. Journal of Geophysical Research – Oceans, 115: C12045, doi:10.1029/2009JC005839.

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