Motivation

An analytical model of bottom boundary layers (BBLs) provides not only an conceptual understanding of the dynamics, but also bottom boundary conditions for the far-field flow. In oceans and lakes, bottom-generated turbulence tends to create a well-mixed BBL capped by a sharp density gradient at the top. To include the effects of this capping, the use of quadratic eddy viscosity was proposed for turbulent bottom Ekman layers^2),3). By applying this suggestion to rotating oscillatory BBLs, I proposed a new analytical model of capped turbulent oscillatory BBLs. Then, I used the model to develop a simpler, layer-averaged linear BBL model, which can be used in modal analysis for semi-enclosed basins (see examples for Lake Biwa and Lake Kinneret).

A Quadratic-Eddy-Viscosity Model

Fig. 1 shows the idealized capped BBL used for this study and the definition of relevant variables. The governing equations for the Ekman part of the velocity $\vec{v}_E=(u_E, v_E)$ are⁷⁾

\begin{equation} \frac{\partial \vec{v}_E}{\partial t} = - f \vec{k} \times \vec{v}_E + \frac{\partial}{\partial z} \left( \kappa z \left( 1-\frac{z}{d} \right) u_\tau \frac{\partial \vec{v}_E }{\partial z} \right), \end{equation}

where $f$ is the Coriolis parameter, $\vec{k}$ is the vertical unit vector, and $u_\tau$ is the friction velocity. We assume an oscillatory interior flow

\begin{equation} \vec{v}_I=\frac{1}{2}\tilde{\vec{v}}_I e^{\mathrm{i} \omega t} + c.c., \end{equation}

where $\omega$ is the angular frequency, and $c.c.$ represents the complex conjugate of the preceding term. The solution has the form:

\begin{equation} \begin{bmatrix} u_E \\ v_E \end{bmatrix} = -\frac{1}{2} \begin{bmatrix} H_{//} & -H_{\perp} \\ H_{\perp} & H_{//} \end{bmatrix} \begin{bmatrix} \tilde{u}_I \\ \tilde{v}_I \end{bmatrix} e^{\mathrm{i} \omega t} + c.c., \end{equation}

where $H_{//}$ and $H_{\perp}$ are vertical structure functions. Although this model is conceptually simple, the solution is unfortunately a bit difficult to handle because $H_{//}$ and $H_{\perp}$ are written in terms of the hypergeometric functions $_2F_1$ with complex-valued arguments. One of the important parameters coming out of the BBL model is the bottom shear stress

\begin{equation} \begin{bmatrix} \tau_x \\ \tau_y \end{bmatrix} = \frac{\rho_0}{2} \begin{bmatrix} c_{//} & -c_{\perp} \\ c_{\perp} & c_{//} \end{bmatrix} \begin{bmatrix} \tilde{u}_I \\ \tilde{v}_I \end{bmatrix} e^{\mathrm{i} \omega t} + c.c., \end{equation}

where $\rho_0$ is the reference density, and $c_{//}$ and $c_{\perp}$ are two complex-valued friction coefficients calculated from $H_{//}$ and $H_{\perp}$.

Choosing the friction velocity $u_\tau$ provides a 'closure' to this nonlinear problem. Since I was interested in applying this model to damping of internal waves, I proposed to choose $u_\tau$ such that the model yields the correct average dissipation rate over a period. The common choice of $u_\tau$ based on maximum shear stress was found unsatisfactory for rotating flows from an energetics point of view.

Comparisons with Field Data and Simulation results

• Comparison with the observation of M₂ tidal currents in a narrow open channel¹⁾ (Fig. 2). The magnitude is predicted well by the theory; however, the phase varies only half as much as the measurements, which is a known limitation of time-independent eddy viscosity models⁸⁾.
• Comparison with Large Eddy Simulation (LES) of capped Ekman layers⁶⁾ (Fig. 3). The velocity profiles are not predicted well; however, the parameters important for setting bottom boundary conditions, e.g., the bottom shear stress, veering angle at the bottom, and Ekman transport, are predicted well by the theory.

Reduction to A Layer-Averaged BBL Model

The quadratic-eddy-viscosity model was used to develop a simpler layer-averaged BBL model that is compatible with modal analysis for semi-enclosed basins. This layer-averaged BBL model is based on

\begin{align} \frac{\partial \vec{v}_E}{\partial t} = f \vec{k} \times \vec{v}_E - \frac{1}{\rho h} \vec{\tau}, \\ \vec{\tau} = \chi C_b v_0 \left( \vec{v}_I + \vec{v_E} \right), \end{align}

where $C_b$ is the usual drag coefficient, and $v_0$ is the typical interior velocity. Note that the flow is allowed to slip at the top of the BBL. The model constant $\chi$ is introduced to minimize the difference in energy dissipation rate between this layer-averaged model and the quadratic-eddy-viscosity model. The layer-averaged model show frequency characteristics different from capped BBLs with constant eddy viscosity, but similar to those with quadratic eddy viscosity (Fig. 4).

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Related Publications

Details of the quadratic-eddy-viscosity model are available in

Shimizu, K. 2010. An analytical model of capped turbulent oscillatory bottom boundary layers. Journal of Geophysical Research – Oceans, 115: C03011. doi:10.1029/2009JC005548.

The layer-averaged BBL model for modal analysis is described in

Shimizu, K., and J. Imberger. 2010. Seasonal differences in the evolution of damped internal waves in a stratified lake. Limnology and Oceanography, 57: 1449-1462. doi:10.4319/lo.2010.55.3.1449.

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