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Motivation

Modal analysis is a common method used to analyse vibration problems in physics and engineering. For two-layer-stratified lakes, the method of calculating basin-scale modes has been known for many years^1),7). However, a lack of theoretical understanding prevented physical limnologists from taking full advantage of the method. My idea was to enable a wide range of applications of this method as in other fields, by developing the theory of modal analysis for stratified rotating lakes. We made a first step in theoretical development, and then applied it to Lake Biwa to investigate horizontal structure and excitation of the basin-scale motions.

Study Site

Lake Biwa is the largest lake in Japan, with ~60-km length, ~20-km width, and ~100-m maximum depth. During the stratified period, Lake Biwa supports many basin-scale internal wave modes, as well as strong and persistent gyres (typical current speed ~0.1 m/s).

Basin-Scale Modes

Basin-scale modes in Lake Biwa were computed using two-layer approximation. The internal wave modes are shown in Fig. 1, and the Rossby wave modes in Fig. 2. (The Rossby wave modes have very long periods, and effectively represent quasi-geostropic flows, such as gyres.)

Theory and Applications of Modal Analysis

Following previous theories for homogeneous rotating basins^9)-12), basin-scale modes can be used as basis functions for generalized Fourier series. For example, averaged velocity in $k$th layer $\vec{v}_k (\vec{x}, t)$ ($k=1$ and $2$ for upper and lower layer) can be decomposed into modal components as

\begin{align} \vec{v}_k=\sum_n{\vec{v}_k^{(n)} a^{(n)}}, \end{align}

where $\vec{v}_k^{(n)}(\vec{x})$ is the complex-valued layer-average velocity due to $n$th mode, and $a^{(n)}(t)$ is its amplitude. Then, neglecting nonlinearity and damping, $a^{(n)}$ follows an decoupled modal amplitude equation:

\begin{align} \frac{d a^{(n)}}{dt} =& i \omega^{(n)} a^{(n)} + \frac{f^{(n)}}{e^{(n)}}, \\ f^{(n)}=&\int{}\vec{v}_1^{(n)*} \cdot \vec{\tau} dS, \end{align}

where $\vec{\tau}$ is the wind stress vector, $e^{(n)}$ is an arbitrary normalization factor of $n$th mode, and the integral is taken over the lake surface.

Based on this theory, we proposed the following new applications of modal analysis:

• To understand the effectiveness of wind forcing in exciting different modes using the modal force $f^{(n)}$.
• To extract the amplitudes and energetics of individual modes from the results of numerical simulations based on the modal decomposition (Fig. 3).

Results and Implications for Lake Biwa

• The structure and periods of the computed internal waves agree well with previous observations in Lake Biwa^2)-6),8),13).
• The internal wave modes can be preferentially excited by winds depending on wind direction (Fig. 4).
• Excitation of the gyres (Rossby wave modes) requires by wind stress curl.
• The internal waves are effectively excited by winds but dampened within few days, whereas gyres are excited ineffectively but dampened slowly (Fig. 5). This implies the importance of internal waves as energy sources for mixing, and gyres for long-term horizontal transport.

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Related Publications

Details are available in

Shimizu, K., J. Imberger, and M. Kumagai. 2007. Horizontal structure and excitation of primary motions in a strongly stratified lake. Limnology and Oceanography 52: 2641-2655. doi:10.4319/lo.2007.52.6.2641.

The modal analysis has also been applied to Lake Biwa in winter:

Homma, H., T. Nagai, K. Shimizu, and H. Yamazaki. 2016. The early-winter mixing event associated with baroclinic motions in weakly stratified Lake Biwa. Inland Waters, 6: 364-378, doi:10.5268/IW-6.3.898.

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