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1.1 Introduction

1.3 Continuous equations in `r' coordinates

2. Discretization and Algorithm

4. Software Architecture

5. Automatic Differentiation

6. Physical Parameterization and Packages

7. Diagnostics and tools

8. Interface with ECCO

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2.9 Non-hydrostatic formulation

The non-hydrostatic formulation re-introduces the full vertical momentum equation and requires the solution of a 3-D elliptic equations for non-hydrostatic pressure perturbation. We still integrate vertically for the hydrostatic pressure and solve a 2-D elliptic equation for the surface pressure/elevation for this reduces the amount of work needed to solve for the non-hydrostatic pressure.

The momentum equations are discretized in time as follows:

$\displaystyle \frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{nh}^{n+1}$	$\displaystyle =$	$\displaystyle \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)}$	(2.53)
$\displaystyle \frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1}$	$\displaystyle =$	$\displaystyle \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)}$	(2.54)
$\displaystyle \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}$	$\displaystyle =$	$\displaystyle \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)}$	(2.55)

which must satisfy the discrete-in-time depth integrated continuity, equation 2.16 and the local continuity equation

$\displaystyle \partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0$

(2.56)

As before, the explicit predictions for momentum are consolidated as:

$\displaystyle u^*$	$\displaystyle =$	$\displaystyle u^n + \Delta t G_u^{(n+1/2)}$
$\displaystyle v^*$	$\displaystyle =$	$\displaystyle v^n + \Delta t G_v^{(n+1/2)}$
$\displaystyle w^*$	$\displaystyle =$	$\displaystyle w^n + \Delta t G_w^{(n+1/2)}$

but this time we introduce an intermediate step by splitting the tendancy of the flow as follows:

$\displaystyle u^{n+1} = u^{**} - \Delta t \partial_x \phi_{nh}^{n+1}$		$\displaystyle u^{*} = u^{} - \Delta t g \partial_x \eta^{n+1}$	(2.57)
$\displaystyle v^{n+1} = v^{**} - \Delta t \partial_y \phi_{nh}^{n+1}$		$\displaystyle v^{*} = v^{} - \Delta t g \partial_y \eta^{n+1}$	(2.58)

Substituting into the depth integrated continuity (equation 2.16) gives

$\displaystyle \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}... ...ght) - \frac{\epsilon_{fs}\eta^{n+1}}{\Delta t^2} = - \frac{\eta^*}{\Delta t^2}$

(2.59)

which is approximated by equation 2.20 on the basis that i) $\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh} << g \nabla \eta$ . If 2.20 is solved accurately then the implication is that $\widehat{\phi}_{nh} \approx 0$ so that the non-hydrostatic pressure field does not drive barotropic motion.

The flow must satisfy non-divergence (equation 2.56) locally, as well as depth integrated, and this constraint is used to form a 3-D elliptic equations for $\phi_{nh}^{n+1}$ :

$\displaystyle \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + \... ...ft( \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \right) / \Delta t$

(2.60)

The entire algorithm can be summarized as the sequential solution of the following equations:

$\displaystyle u^{*}$	$\displaystyle =$	$\displaystyle u^{n} + \Delta t G_u^{(n+1/2)}$	(2.61)
$\displaystyle v^{*}$	$\displaystyle =$	$\displaystyle v^{n} + \Delta t G_v^{(n+1/2)}$	(2.62)
$\displaystyle w^{*}$	$\displaystyle =$	$\displaystyle w^{n} + \Delta t G_w^{(n+1/2)}$	(2.63)
$\displaystyle \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)$	$\displaystyle -$	$\displaystyle \Delta t \left( \partial_x H \widehat{u^{}} + \partial_y H \widehat{v^{}} \right)$	(2.64)
$\displaystyle \partial_x g H \partial_x \eta^{n+1} + \partial_y g H \partial_y \eta^{n+1}$	$\displaystyle -$	$\displaystyle \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} ~ = ~ - \frac{\eta^*}{\Delta t^2}$	(2.65)
$\displaystyle u^{**}$	$\displaystyle =$	$\displaystyle u^{*} - \Delta t g \partial_x \eta^{n+1}$	(2.66)
$\displaystyle v^{**}$	$\displaystyle =$	$\displaystyle v^{*} - \Delta t g \partial_y \eta^{n+1}$	(2.67)
$\displaystyle \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + \partial_{rr} \phi_{nh}^{n+1}$	$\displaystyle =$	$\displaystyle \left( \partial_x u^{} + \partial_y v^{} + \partial_r w^{*} \right) / \Delta t$	(2.68)
$\displaystyle u^{n+1}$	$\displaystyle =$	$\displaystyle u^{**} - \Delta t \partial_x \phi_{nh}^{n+1}$	(2.69)
$\displaystyle v^{n+1}$	$\displaystyle =$	$\displaystyle v^{**} - \Delta t \partial_y \phi_{nh}^{n+1}$	(2.70)
$\displaystyle \partial_r w^{n+1}$	$\displaystyle =$	$\displaystyle - \partial_x u^{n+1} - \partial_y v^{n+1}$	(2.71)