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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:
which must satisfy the discrete-in-time depth integrated continuity,
equation 2.16 and the local continuity equation
![$\displaystyle \partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0$](img517.png) |
(2.56) |
As before, the explicit predictions for momentum are consolidated as:
but this time we introduce an intermediate step by splitting the
tendancy of the flow as follows:
Substituting into the depth integrated continuity
(equation 2.16) gives
![$\displaystyle \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}...
...ght) - \frac{\epsilon_{fs}\eta^{n+1}}{\Delta t^2} = - \frac{\eta^*}{\Delta t^2}$](img528.png) |
(2.59) |
which is approximated by equation
2.20 on the basis that i)
is not yet known and ii)
. If 2.20 is
solved accurately then the implication is that
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
:
![$\displaystyle \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + \...
...ft( \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \right) / \Delta t$](img532.png) |
(2.60) |
The entire algorithm can be summarized as the sequential solution of
the following equations:
![$\displaystyle u^{*}$](img422.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle u^{n} + \Delta t G_u^{(n+1/2)}$](img417.png) |
(2.61) |
![$\displaystyle v^{*}$](img423.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle v^{n} + \Delta t G_v^{(n+1/2)}$](img419.png) |
(2.62) |
![$\displaystyle w^{*}$](img533.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle w^{n} + \Delta t G_w^{(n+1/2)}$](img534.png) |
(2.63) |
![$\displaystyle \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)$](img535.png) |
![$\displaystyle -$](img489.png) |
![$\displaystyle \Delta t \left( \partial_x H \widehat{u^{*}}
+ \partial_y H \widehat{v^{*}} \right)$](img536.png) |
(2.64) |
![$\displaystyle \partial_x g H \partial_x \eta^{n+1}
+ \partial_y g H \partial_y \eta^{n+1}$](img537.png) |
![$\displaystyle -$](img489.png) |
![$\displaystyle \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
~ = ~ - \frac{\eta^*}{\Delta t^2}$](img490.png) |
(2.65) |
![$\displaystyle u^{**}$](img538.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle u^{*} - \Delta t g \partial_x \eta^{n+1}$](img427.png) |
(2.66) |
![$\displaystyle v^{**}$](img539.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle v^{*} - \Delta t g \partial_y \eta^{n+1}$](img429.png) |
(2.67) |
![$\displaystyle \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +
\partial_{rr} \phi_{nh}^{n+1}$](img540.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle \left(
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}
\right) / \Delta t$](img541.png) |
(2.68) |
![$\displaystyle u^{n+1}$](img426.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle u^{**} - \Delta t \partial_x \phi_{nh}^{n+1}$](img542.png) |
(2.69) |
![$\displaystyle v^{n+1}$](img428.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle v^{**} - \Delta t \partial_y \phi_{nh}^{n+1}$](img543.png) |
(2.70) |
![$\displaystyle \partial_r w^{n+1}$](img544.png) |
![$\displaystyle =$](img176.png) |
![$\displaystyle - \partial_x u^{n+1} - \partial_y v^{n+1}$](img545.png) |
(2.71) |
where the last equation is solved by vertically integrating for
.
Next: 2.10 Variants on the
Up: 2. Discretization and Algorithm
Previous: 2.8 Staggered baroclinic time-stepping
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