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Mode splitting

To be able to represent gravity waves and the effects of these correctly, the time step must be chosen such that the Courant number become less than unity. To avoid this restriction when propagating the 3-D fields, mode splitting very similar to the splitting described by Berntsen et al. [3] and applied by Slagstad [38] and later described in [1] is applied.

The 3-D velocity field is split into its baroclinic part and its depth integrated part according to:

$\displaystyle \hspace*{-7mm}(U(x,y,\sigma),V(x,y,\sigma))$ $\textstyle =$ $\displaystyle (U_A(x,y) +
U_B(x,y,\sigma) ,
V_A(x,y) + V_B(x,y,\sigma))$ (50)

where

\begin{eqnarray*}
(U_A(x,y),V_A(x,y)) & = (\int_{-1}^{0} U(x,y,\sigma) d\sigma ,
\int_{-1}^{0} U(x,y),\sigma d\sigma ) . \nonumber
\end{eqnarray*}



Depth integrating the moment equations and the continuity equation gives, neglecting for a moment the atmospheric pressure terms and the vertical viscosity terms, gives:


\begin{displaymath}
\frac{\partial U_A D}{\partial x} + \frac{\partial V_A D}{\partial y} +
\frac{\partial \eta}{\partial t} = 0 ,
\end{displaymath} (51)


$\displaystyle {\frac{\partial U_A D}{\partial t} -
f V_A D +
gD \frac{\partial \eta}{\partial x} = }$
    $\displaystyle \frac{\partial}{\partial x}
( A_{M2D} \frac{\partial U_A D}{\part...
...c{\partial U_A D}{\partial y} ) + \frac{1}{\rho_0}(\tau_{0x} - \tau_{bx}) + A_x$ (52)

and
$\displaystyle {\frac{\partial V_A D}{\partial t} +
f U_A D +
gD \frac{\partial \eta}{\partial y} = }$
    $\displaystyle \frac{\partial}{\partial y}
( A_{M2D} \frac{\partial V_A D}{\part...
...\partial V_A D}{\partial x} ) + \frac{1}{\rho_0}(\tau_{0y} - \tau_{by}) + A_y ,$ (53)

where


$\displaystyle (A_x , A_y)$ $\textstyle =$ $\displaystyle ( \int_{-1}^{0} \Delta U d \sigma , \int_{-1}^{0}
\Delta V
d \sigma) ,$ (54)
$\displaystyle \Delta U$ $\textstyle =$ $\displaystyle - \frac{g D^2}{\rho_0} \int_{\sigma}^0
\left( \frac{\partial \rho...
...+
\frac{\partial UVD}{\partial y} + \frac{\partial U \omega}{\partial
\sigma} ,$ (55)
$\displaystyle \Delta V$ $\textstyle =$ $\displaystyle - \frac{g D^2}{\rho_0} \int_{\sigma}^0
\left( \frac{\partial \rho...
...\frac{\partial V^2 D}{\partial y} + \frac{\partial V \omega}{\partial
\sigma} .$ (56)

The non-linear and the internal pressure terms are represented through $A_x$ and $A_y$. The vertical integration is exact except for the horizontal viscosity terms. These terms take mainly care of small scale oscillations, and the 2-D $A_{M2D}$ field is computed according to (11) where $(U,V)$ is replaced by $(U_A,V_A)$ or chosen to be constant in space and time.

The equations for the baroclinic fields $U_B$ and $V_B$ become after subtracting (53) from (38) and (54) from (39) :


$\displaystyle {\frac{\partial U_B D}{\partial t} -
f V_B D + \frac{1}{\rho_0}(\tau_{0x} - \tau_{bx}) + A_x = }$
    $\displaystyle \frac{\partial}{\partial x}
( A_{M} \frac{\partial U_B D}{\partia...
...partial \sigma}(\frac{K_M}{D} \frac{\partial
U_B}{\partial \sigma} ) + \Delta U$ (57)

and
$\displaystyle {\frac{\partial V_B D}{\partial t} +
f U_B D + \frac{1}{\rho_0}(\tau_{0y} - \tau_{by}) + A_y = }$
    $\displaystyle \frac{\partial}{\partial y}
( A_{M} \frac{\partial V_B D}{\partia...
...rtial \sigma}(\frac{K_M}{D} \frac{\partial
V_B}{\partial \sigma} ) + \Delta V .$ (58)


next up previous contents
Next: The numerical -coordinate model Up: report Previous: Vertical boundary conditions   Contents
Helge Avlesen 2004-05-10