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revision 1.5 by adcroft, Mon Oct 15 19:34:28 2001 UTC revision 1.6 by cnh, Wed Oct 24 15:21:27 2001 UTC
# Line 601  discussion: Line 601  discussion:
601  \left.  \left.
602  \begin{tabular}{l}  \begin{tabular}{l}
603  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
604  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}\right\} $  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
605  \\  \\
606  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $  $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
607  \\  \\
608  $+\mathcal{F}_{u}$  $+\mathcal{F}_{u}$
609  \end{tabular}  \end{tabular}
# Line 621  $+\mathcal{F}_{u}$ Line 621  $+\mathcal{F}_{u}$
621  \left.  \left.
622  \begin{tabular}{l}  \begin{tabular}{l}
623  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
624  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
625  $ \\  $ \\
626  $-\left\{ -2\Omega u\sin lat\right\} $ \\  $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
627  $+\mathcal{F}_{v}$  $+\mathcal{F}_{v}$
628  \end{tabular}  \end{tabular}
629  \ \right\} \left\{  \ \right\} \left\{
# Line 642  $+\mathcal{F}_{v}$ Line 642  $+\mathcal{F}_{v}$
642  \begin{tabular}{l}  \begin{tabular}{l}
643  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
644  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
645  ${+}\underline{{2\Omega u\cos lat}}$ \\  ${+}\underline{{2\Omega u\cos \varphi}}$ \\
646  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
647  \end{tabular}  \end{tabular}
648  \ \right\} \left\{  \ \right\} \left\{
# Line 656  $\underline{\underline{\mathcal{F}_{\dot Line 656  $\underline{\underline{\mathcal{F}_{\dot
656  \end{equation}  \end{equation}
657  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
658    
659  In the above `${r}$' is the distance from the center of the earth and `$lat$  In the above `${r}$' is the distance from the center of the earth and `$\varphi$
660  ' is latitude.  ' is latitude.
661    
662  Grad and div operators in spherical coordinates are defined in appendix  Grad and div operators in spherical coordinates are defined in appendix
# Line 694  computed at all other levels by integrat Line 694  computed at all other levels by integrat
694    
695  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
696  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
697  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic  \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
698  contribution to the pressure field: only the terms underlined twice in Eqs. (  contribution to the pressure field: only the terms underlined twice in Eqs. (
699  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
700  and, simultaneously, the shallow atmosphere approximation is relaxed. In  and, simultaneously, the shallow atmosphere approximation is relaxed. In
# Line 703  variation of the radial position of a pa Line 703  variation of the radial position of a pa
703  vertical momentum equation (\ref{eq:mom-w}) becomes:  vertical momentum equation (\ref{eq:mom-w}) becomes:
704    
705  \begin{equation*}  \begin{equation*}
706  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
707  \end{equation*}  \end{equation*}
708  making a small correction to the hydrostatic pressure.  making a small correction to the hydrostatic pressure.
709    
# Line 789  stepping forward the vertical momentum e Line 789  stepping forward the vertical momentum e
789  %%CNHend  %%CNHend
790    
791  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
792  course, some complication that goes with the inclusion of $\cos \phi \ $  course, some complication that goes with the inclusion of $\cos \varphi \ $
793  Coriolis terms and the relaxation of the shallow atmosphere approximation.  Coriolis terms and the relaxation of the shallow atmosphere approximation.
794  But this leads to negligible increase in computation. In \textbf{NH}, in  But this leads to negligible increase in computation. In \textbf{NH}, in
795  contrast, one additional elliptic equation - a three-dimensional one - must  contrast, one additional elliptic equation - a three-dimensional one - must
# Line 1045  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} Line 1045  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt}
1045  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
1046  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1047  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1048  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is  derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
1049  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1050  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1051  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
# Line 1377  In spherical coordinates, the velocity c Line 1377  In spherical coordinates, the velocity c
1377  and vertical direction respectively, are given by (see Fig.2) :  and vertical direction respectively, are given by (see Fig.2) :
1378    
1379  \begin{equation*}  \begin{equation*}
1380  u=r\cos \phi \frac{D\lambda }{Dt}  u=r\cos \varphi \frac{D\lambda }{Dt}
1381  \end{equation*}  \end{equation*}
1382    
1383  \begin{equation*}  \begin{equation*}
1384  v=r\frac{D\phi }{Dt}\qquad  v=r\frac{D\varphi }{Dt}\qquad
1385  \end{equation*}  \end{equation*}
1386  $\qquad \qquad \qquad \qquad $  $\qquad \qquad \qquad \qquad $
1387    
# Line 1389  $\qquad \qquad \qquad \qquad $ Line 1389  $\qquad \qquad \qquad \qquad $
1389  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
1390  \end{equation*}  \end{equation*}
1391    
1392  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial  Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
1393  distance of the particle from the center of the earth, $\Omega $ is the  distance of the particle from the center of the earth, $\Omega $ is the
1394  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
1395    
# Line 1397  The `grad' ($\nabla $) and `div' ($\nabl Line 1397  The `grad' ($\nabla $) and `div' ($\nabl
1397  spherical coordinates:  spherical coordinates:
1398    
1399  \begin{equation*}  \begin{equation*}
1400  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }  \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
1401  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}  ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
1402  \right)  \right)
1403  \end{equation*}  \end{equation*}
1404    
1405  \begin{equation*}  \begin{equation*}
1406  \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial  \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
1407  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1408  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
1409  \end{equation*}  \end{equation*}
1410    
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