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. Introducing a streamfunction $\psi$ such that $u = -\partial\psi/\partial z$ and $w = \partial\psi/\partial x$ we form the evolution equation for the component of vorticity into the page.

\[\begin{equation} \left(\frac{\partial^2 ~}{\partial x^2} + \frac{\partial^2 ~}{\partial z^2}\right)\frac{\partial\psi}{\partial t} = \frac{\partial b}{\partial x} - f\frac{\partial v}{\partial z} + \frac{\partial \mathcal{F}^{(z)}}{\partial x} - \frac{\partial \mathcal{F}^{(x)}}{\partial z} \end{equation}\]

and 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 dynamics are controlled by a single second-order in time equation for the stream-function

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

where the forcing is

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

In expanding the RHS some cancellations can be made thanks to the assumption of thermal wind balance and the Sawyer-Eliassen equation is reached

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