# Linear Stability Analysis¶

## One-Layer Shallow Water¶

In this subsection we consider the one-layer reduced gravity RSW model with topography below. We define the following:

• $$H$$: mean depth of the layer
• $$z=\eta$$: height of the free surface
• $$z=-H + \eta_B$$: height of the topography.
• $$h = H + \eta - \eta_B$$: total depth of layer
• $$(u,v)$$: horizontal velocity
• $$g'$$: reduced gravity
• $$\rho_0$$: reference density

The governing nonlinear equations are,

\begin{split}\begin{aligned} \frac{\partial u}{\partial t} + {\vec u} \cdot \vec \nabla u - f v & = - g \frac{\partial}{\partial x} \left( h + \eta_B \right) , \\ \frac{\partial v}{\partial t} + {\vec u} \cdot \vec \nabla v + f u & = - g \frac{\partial}{\partial y} \left( h + \eta_B \right) , \\ \frac{\partial h}{\partial t} + \vec\nabla \cdot \left( h \vec u_1 \right) & = 0.\end{aligned}\end{split}

## Basic State¶

To study shear flows in a meridional channel we consider solutions of the form,

\begin{split}\begin{aligned} u & = U_B(y), \\ v & = 0,\\ h & = H_B(y).\end{aligned}\end{split}

For this to be an exact solution we require that the flow is in geostrophic balance,

$f U_B = - g \frac{d}{dy}\left( H_B + \eta_B \right).$

## Perturbation¶

We perturb the basic state with infinitesimal quantities,

\begin{split}\begin{aligned} u & = U_B(y) + u', \\ v & = 0 + v',\\ h & = H_B(y) + h'.\end{aligned}\end{split}

We substitute our perturbation into the governing equations and drop the primes (for brevity) and cancelling out the geostrophic terms

\begin{split}\begin{aligned} \frac{\partial u}{\partial t} + (u + U_B) \frac{\partial u}{\partial x} + v \frac{\partial}{\partial y} \left(u + U_B \right) - f v & = - g \frac{\partial h}{\partial x}, \\ \frac{\partial v}{\partial t} + (u + U_B) \frac{\partial v}{\partial x} + v \frac{\partial v }{\partial y} + f u & = - g \frac{\partial h}{\partial y}, \\ \frac{\partial h}{\partial t} + (u + U_B) \frac{\partial h}{\partial x} + v \frac{\partial}{\partial y}(H_B + h) + (H_B + h) & \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0.\end{aligned}\end{split}

Now we neglect the quadratic terms to obtain the linearized equations,

\begin{split}\begin{aligned} \frac{\partial u}{\partial t} & = - U_B \frac{\partial u}{\partial x} + \left( f - \frac{d U_B}{d y} \right) v - g \frac{\partial h_1}{\partial x}, \\ \frac{\partial v}{\partial t} & = - f u - U_B \frac{\partial v}{\partial x} - g \frac{\partial h_1}{\partial y}, \\ \frac{\partial h}{\partial t} & = - H_B \frac{\partial u}{\partial x} - v \frac{d H_B}{d y} - H_B \frac{\partial v}{\partial y} - U_B \frac{\partial h}{\partial x} .\end{aligned}\end{split}

Finally, we assume a normal mode decomposition in the zonal direction and time,

\begin{aligned} [u, v, h] = \mbox{Re}\left\{ e^{ik(x - c t)} [\hat u, ik \hat v, \hat h] \right\},\end{aligned}

which we can substitute into the above equations to yield in the inviscid limit

\begin{split}\begin{aligned} c \hat u & = U_B \hat u - (f - \frac{dU_B}{dy}) \hat v + g \hat h, \\ c \hat v & =- \frac{f}{k^2} \hat u + U_B \hat v -\frac{ g}{k^2} \frac{d h}{d y}, \\ c \hat h &= H_B \hat u + \frac{d}{d y}\left(H_B \hat v\right) + U_B \hat h.\end{aligned}\end{split}