3.9.2 Equations solved

$\displaystyle \frac{Du}{Dt} - fv + \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} - A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}$	$\displaystyle =$	$\displaystyle \cal{F}_{\lambda}$	(3.14)
$\displaystyle \frac{Dv}{Dt} + fu + \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} - A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}$	$\displaystyle =$	0	(3.15)
$\displaystyle \frac{\partial \eta}{\partial t} + \frac{\partial H \widehat{u}}{\partial \lambda} + \frac{\partial H \widehat{v}}{\partial \varphi}$	$\displaystyle =$	0	(3.16)
$\displaystyle \frac{D\theta}{Dt} - K_{h}\nabla_{h}^2\theta - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}}$	$\displaystyle =$	0	(3.17)
$\displaystyle p^{\prime}$	$\displaystyle =$	$\displaystyle g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz$	(3.18)
$\displaystyle \rho ^{\prime }$	$\displaystyle =$	$\displaystyle - \alpha_{\theta}\rho_{0}\theta^{\prime}$	(3.19)
$\displaystyle {\cal F}_{\lambda} \vert _{s}$	$\displaystyle =$	$\displaystyle \frac{\tau_{\lambda}}{\rho_{0}\Delta z_{s}}$	(3.20)
$\displaystyle {\cal F}_{\lambda} \vert _{i}$	$\displaystyle =$	0	(3.21)

where

and

are the components of the horizontal flow vector $\vec{u}$ on the sphere ( $u=\dot{\lambda},v=\dot{\varphi}$ ). The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical integral term given in equation 1.35 and explained in more detail in section 2.3. However, for the problem presented here, the continuity relation (equation 3.16) differs from the general form given in section 2.3, equation

, because the source terms ${\cal P}-{\cal E}+{\cal R}$ are all 0.

The pressure field, $p^{\prime }$ , is separated into a barotropic part due to variations in sea-surface height, $\eta$ , and a hydrostatic part due to variations in density, $\rho ^{\prime }$ , integrated through the water column.

The suffices

indicate surface layer and the interior of the domain. The windstress forcing, ${\cal F}_{\lambda}$ , is applied in the surface layer by a source term in the zonal momentum equation. In the ocean interior this term is zero.

In the momentum equations lateral and vertical boundary conditions for the $\nabla_{h}^{2}$ and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified when the numerical simulation is run - see section 3.9.4. For temperature the boundary condition is ``zero-flux'' e.g. $\frac{\partial \theta}{\partial \varphi}= \frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$ .