The Sawyer-Eliassen equation

The Sawyer-Eliassen equation (Sawyer and Sutton, 1956; Eliassen, 1962) describes the ageostrophic overturning circulations at fronts. The derivation here largely follows Mooers (1975) although we allow for arbitary momentum and buoyancy forcing. Consider a 2D set-up with a buoyancy field $B(x,z)$ in thermal wind balance $\partial B/\partial x = f\partial V/\partial z$ with a geostrophic velocity $V(x,z)$. Assuming that perturbations from this state are also 2D the Boussinesq equations can be written.

\[\begin{align*} \frac{\partial u}{\partial t} - fv + \frac{1}{\rho_0}\frac{\partial p}{\partial x} & = \mathcal{F}^{(x)} \\ \frac{\partial v}{\partial t} + u\frac{\partial V}{\partial x} + w\frac{\partial V}{\partial z} + fu & = \mathcal{F}^{(y)} \\ \frac{\partial w}{\partial t} + \frac{1}{\rho_0}\frac{\partial p}{\partial z} - b & = \mathcal{F}^{(z)} \\ \frac{\partial b}{\partial t} + u\frac{\partial B}{\partial x} + w\frac{\partial B}{\partial z} & = \mathcal{B} \\ \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} & = 0 \end{align*}\]

where the non-linear terms have been absorbed into the arbitary RHS forcing. We form the evolution equation for the component of vorticity into the page, $\zeta = \partial u/\partial z - \partial w/\partial x$,

\[\begin{equation} \label{eq:vorticity-evolution} \frac{\partial\zeta}{\partial t} = f\frac{\partial v}{\partial z} - \frac{\partial b}{\partial x} + \frac{\partial \mathcal{F}^{(x)}}{\partial z} - \frac{\partial \mathcal{F}^{(z)}}{\partial x}. \end{equation}\]

Introducing a streamfunction $\psi$, such that $u = -\partial\psi/\partial z$ and $w = \partial\psi/\partial x$, the $v$ and $b$ perturbations are determined by

\[\begin{align*} \frac{\partial v}{\partial t} & = -\mathcal{J}(\psi,V + fx) + \mathcal{F}^{(y)} \\ \frac{\partial b}{\partial t} & = -\mathcal{J}(\psi,B) + \mathcal{B} \end{align*}\]

where $\mathcal{J}(\psi, \cdot) \equiv (\partial{\psi}/\partial x) \partial/ \partial z - (\partial{\psi}/\partial z) \partial/ \partial x$ is advection by the perturbations. The vorticity is related to the streamfunction via Poisson's equation $\nabla^2\psi = -\zeta$.

Taking a time derivative of \eqref{eq:vorticity-evolution}, the evolution of the vorticity can be written in of the streamfunction,

\[\begin{equation*} \frac{\partial^2\zeta}{\partial t^2} = \frac{\partial ~}{\partial x}\mathcal{J}(\psi,B) - f\frac{\partial ~}{\partial z}\mathcal{J}(\psi,V + fx) + \mathfrak{F}(x,z,t), \end{equation*}\]

where the forcing is

\[\begin{equation} \mathfrak{F}(x,z,t) = \frac{\partial^2 \mathcal{F}^{(x)}}{\partial z\partial t} - \frac{\partial^2 \mathcal{F}^{(z)}}{\partial x\partial t} + f\frac{\partial\mathcal{F}^{(y)}}{\partial z} - \frac{\partial\mathcal{B}}{\partial x}. \end{equation}\]

Expanding the RHS, during which some cancellations can be made thanks to the assumption of thermal wind balance, the Sawyer-Eliassen equation can be written as

\[\begin{equation} \frac{\partial^2\zeta}{\partial t^2} = \frac{\partial B}{\partial z}\frac{\partial^2\psi}{\partial x^2} - 2\frac{\partial B}{\partial x}\frac{\partial^2\psi}{\partial x\partial z} + f\left(f + \frac{\partial V}{\partial x}\right)\frac{\partial^2\psi}{\partial z^2} + \mathfrak{F}(x,z,t), \\ \end{equation}\]

where

\[\begin{equation} \nabla^2\psi = -\zeta. \end{equation}\]