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% $Name$ |
% $Name$ |
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\section{Flux-form momentum equations} |
\section{Flux-form momentum equations} |
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\label{sec:flux-form_momentum_eqautions} |
\label{sect:flux-form_momentum_equations} |
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\begin{rawhtml} |
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<!-- CMIREDIR:flux-form_momentum_eqautions: --> |
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\end{rawhtml} |
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The original finite volume model was based on the Eulerian flux form |
The original finite volume model was based on the Eulerian flux form |
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momentum equations. This is the default though the vector invariant |
momentum equations. This is the default though the vector invariant |
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For the non-hydrostatic equations, dropping the thin-atmosphere |
For the non-hydrostatic equations, dropping the thin-atmosphere |
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approximation re-introduces metric terms involving $w$ and are |
approximation re-introduces metric terms involving $w$ and are |
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required to conserve anglular momentum: |
required to conserve angular momentum: |
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\begin{eqnarray} |
\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
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- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\ |
- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\ |
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``lat-lon'' grid for Laplacian viscosity. |
``lat-lon'' grid for Laplacian viscosity. |
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\marginpar{Need to tidy up method for controlling this in code} |
\marginpar{Need to tidy up method for controlling this in code} |
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It should be noted that dispite the ad-hoc nature of the scaling, some |
It should be noted that despite the ad-hoc nature of the scaling, some |
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scaling must be done since on a lat-lon grid the converging meridians |
scaling must be done since on a lat-lon grid the converging meridians |
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make it very unlikely that a stable viscosity parameter exists across |
make it very unlikely that a stable viscosity parameter exists across |
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the entire model domain. |
the entire model domain. |
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The no-slip condition defines the normal gradient of a tangential flow |
The no-slip condition defines the normal gradient of a tangential flow |
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such that the flow is zero on the boundary. Rather than modify the |
such that the flow is zero on the boundary. Rather than modify the |
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stresses by using complicated functions of the masks and ``ghost'' |
stresses by using complicated functions of the masks and ``ghost'' |
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points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as |
points (see \cite{adcroft:98}) we add the boundary stresses as |
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an additional source term in cells next to solid boundaries. This has |
an additional source term in cells next to solid boundaries. This has |
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the advantage of being able to cope with ``thin walls'' and also makes |
the advantage of being able to cope with ``thin walls'' and also makes |
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the interior stress calculation (code) independent of the boundary |
the interior stress calculation (code) independent of the boundary |
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|
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In fact, the above discretization is not quite complete because it |
In fact, the above discretization is not quite complete because it |
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assumes that the bathymetry at velocity points is deeper than at |
assumes that the bathymetry at velocity points is deeper than at |
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neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$ |
neighboring vorticity points, e.g. $1-h_w < 1-h_\zeta$ |
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F}) |
{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F}) |
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to the variable grid lengths introduced by the finite volume |
to the variable grid lengths introduced by the finite volume |
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formulation. This reduces the formal accuracy of these terms to just |
formulation. This reduces the formal accuracy of these terms to just |
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first order but only next to boundaries; exactly where other terms |
first order but only next to boundaries; exactly where other terms |
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appear such as linar and quadratic bottom drag. |
appear such as linear and quadratic bottom drag. |
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\begin{eqnarray} |
\begin{eqnarray} |
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G_u^{v-diss} & = & |
G_u^{v-diss} & = & |
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\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
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\tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w |
\tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w |
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\end{eqnarray} |
\end{eqnarray} |
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It should be noted that in the non-hydrostatic form, the stress tensor |
It should be noted that in the non-hydrostatic form, the stress tensor |
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is even less consistent than for the hydrostatic (see Wazjowicz |
is even less consistent than for the hydrostatic (see |
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\cite{Waojz}). It is well known how to do this properly (see Griffies |
\cite{wajsowicz:93}). It is well known how to do this properly (see |
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\cite{Griffies}) and is on the list of to-do's. |
\cite{griffies:00}) and is on the list of to-do's. |
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|
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F}) |
{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F}) |
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\epsilon_{nh} \overline{ w^2 }^k \right) |
\epsilon_{nh} \overline{ w^2 }^k \right) |
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\end{equation} |
\end{equation} |
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\subsection{Mom Diagnostics} |
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\label{sec:pkg:mom_common:diagnostics} |
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\begin{verbatim} |
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|
------------------------------------------------------------------------ |
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|
<-Name->|Levs|<-parsing code->|<-- Units -->|<- Tile (max=80c) |
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|
------------------------------------------------------------------------ |
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|
VISCAHZ | 15 |SZ MR |m^2/s |Harmonic Visc Coefficient (m2/s) (Zeta Pt) |
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|
VISCA4Z | 15 |SZ MR |m^4/s |Biharmonic Visc Coefficient (m4/s) (Zeta Pt) |
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|
VISCAHD | 15 |SM MR |m^2/s |Harmonic Viscosity Coefficient (m2/s) (Div Pt) |
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|
VISCA4D | 15 |SM MR |m^4/s |Biharmonic Viscosity Coefficient (m4/s) (Div Pt) |
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|
VAHZMAX | 15 |SZ MR |m^2/s |CFL-MAX Harm Visc Coefficient (m2/s) (Zeta Pt) |
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|
VA4ZMAX | 15 |SZ MR |m^4/s |CFL-MAX Biharm Visc Coefficient (m4/s) (Zeta Pt) |
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|
VAHDMAX | 15 |SM MR |m^2/s |CFL-MAX Harm Visc Coefficient (m2/s) (Div Pt) |
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|
VA4DMAX | 15 |SM MR |m^4/s |CFL-MAX Biharm Visc Coefficient (m4/s) (Div Pt) |
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|
VAHZMIN | 15 |SZ MR |m^2/s |RE-MIN Harm Visc Coefficient (m2/s) (Zeta Pt) |
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|
VA4ZMIN | 15 |SZ MR |m^4/s |RE-MIN Biharm Visc Coefficient (m4/s) (Zeta Pt) |
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|
VAHDMIN | 15 |SM MR |m^2/s |RE-MIN Harm Visc Coefficient (m2/s) (Div Pt) |
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|
VA4DMIN | 15 |SM MR |m^4/s |RE-MIN Biharm Visc Coefficient (m4/s) (Div Pt) |
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|
VAHZLTH | 15 |SZ MR |m^2/s |Leith Harm Visc Coefficient (m2/s) (Zeta Pt) |
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|
VA4ZLTH | 15 |SZ MR |m^4/s |Leith Biharm Visc Coefficient (m4/s) (Zeta Pt) |
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|
VAHDLTH | 15 |SM MR |m^2/s |Leith Harm Visc Coefficient (m2/s) (Div Pt) |
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|
VA4DLTH | 15 |SM MR |m^4/s |Leith Biharm Visc Coefficient (m4/s) (Div Pt) |
| 432 |
|
VAHZLTHD| 15 |SZ MR |m^2/s |LeithD Harm Visc Coefficient (m2/s) (Zeta Pt) |
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|
VA4ZLTHD| 15 |SZ MR |m^4/s |LeithD Biharm Visc Coefficient (m4/s) (Zeta Pt) |
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|
VAHDLTHD| 15 |SM MR |m^2/s |LeithD Harm Visc Coefficient (m2/s) (Div Pt) |
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|
VA4DLTHD| 15 |SM MR |m^4/s |LeithD Biharm Visc Coefficient (m4/s) (Div Pt) |
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|
VAHZSMAG| 15 |SZ MR |m^2/s |Smagorinsky Harm Visc Coefficient (m2/s) (Zeta Pt) |
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VA4ZSMAG| 15 |SZ MR |m^4/s |Smagorinsky Biharm Visc Coeff. (m4/s) (Zeta Pt) |
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|
VAHDSMAG| 15 |SM MR |m^2/s |Smagorinsky Harm Visc Coefficient (m2/s) (Div Pt) |
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|
VA4DSMAG| 15 |SM MR |m^4/s |Smagorinsky Biharm Visc Coeff. (m4/s) (Div Pt) |
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|
momKE | 15 |SM MR |m^2/s^2 |Kinetic Energy (in momentum Eq.) |
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|
momHDiv | 15 |SM MR |s^-1 |Horizontal Divergence (in momentum Eq.) |
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|
momVort3| 15 |SZ MR |s^-1 |3rd component (vertical) of Vorticity |
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|
Strain | 15 |SZ MR |s^-1 |Horizontal Strain of Horizontal Velocities |
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|
Tension | 15 |SM MR |s^-1 |Horizontal Tension of Horizontal Velocities |
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|
UBotDrag| 15 |UU 129MR |m/s^2 |U momentum tendency from Bottom Drag |
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|
VBotDrag| 15 |VV 128MR |m/s^2 |V momentum tendency from Bottom Drag |
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|
USidDrag| 15 |UU 131MR |m/s^2 |U momentum tendency from Side Drag |
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|
VSidDrag| 15 |VV 130MR |m/s^2 |V momentum tendency from Side Drag |
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|
Um_Diss | 15 |UU 133MR |m/s^2 |U momentum tendency from Dissipation |
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|
Vm_Diss | 15 |VV 132MR |m/s^2 |V momentum tendency from Dissipation |
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|
Um_Advec| 15 |UU 135MR |m/s^2 |U momentum tendency from Advection terms |
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|
Vm_Advec| 15 |VV 134MR |m/s^2 |V momentum tendency from Advection terms |
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|
Um_Cori | 15 |UU 137MR |m/s^2 |U momentum tendency from Coriolis term |
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|
Vm_Cori | 15 |VV 136MR |m/s^2 |V momentum tendency from Coriolis term |
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|
Um_Ext | 15 |UU 137MR |m/s^2 |U momentum tendency from external forcing |
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|
Vm_Ext | 15 |VV 138MR |m/s^2 |V momentum tendency from external forcing |
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|
Um_AdvZ3| 15 |UU 141MR |m/s^2 |U momentum tendency from Vorticity Advection |
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|
Vm_AdvZ3| 15 |VV 140MR |m/s^2 |V momentum tendency from Vorticity Advection |
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|
Um_AdvRe| 15 |UU 143MR |m/s^2 |U momentum tendency from vertical Advection (Explicit part) |
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|
Vm_AdvRe| 15 |VV 142MR |m/s^2 |V momentum tendency from vertical Advection (Explicit part) |
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|
ADVx_Um | 15 |UM 145MR |m^4/s^2 |Zonal Advective Flux of U momentum |
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|
ADVy_Um | 15 |VZ 144MR |m^4/s^2 |Meridional Advective Flux of U momentum |
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|
ADVrE_Um| 15 |WU LR |m^4/s^2 |Vertical Advective Flux of U momentum (Explicit part) |
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|
ADVx_Vm | 15 |UZ 148MR |m^4/s^2 |Zonal Advective Flux of V momentum |
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|
ADVy_Vm | 15 |VM 147MR |m^4/s^2 |Meridional Advective Flux of V momentum |
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|
ADVrE_Vm| 15 |WV LR |m^4/s^2 |Vertical Advective Flux of V momentum (Explicit part) |
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|
VISCx_Um| 15 |UM 151MR |m^4/s^2 |Zonal Viscous Flux of U momentum |
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|
VISCy_Um| 15 |VZ 150MR |m^4/s^2 |Meridional Viscous Flux of U momentum |
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|
VISrE_Um| 15 |WU LR |m^4/s^2 |Vertical Viscous Flux of U momentum (Explicit part) |
| 470 |
|
VISrI_Um| 15 |WU LR |m^4/s^2 |Vertical Viscous Flux of U momentum (Implicit part) |
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|
VISCx_Vm| 15 |UZ 155MR |m^4/s^2 |Zonal Viscous Flux of V momentum |
| 472 |
|
VISCy_Vm| 15 |VM 154MR |m^4/s^2 |Meridional Viscous Flux of V momentum |
| 473 |
|
VISrE_Vm| 15 |WV LR |m^4/s^2 |Vertical Viscous Flux of V momentum (Explicit part) |
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|
VISrI_Vm| 15 |WV LR |m^4/s^2 |Vertical Viscous Flux of V momentum (Implicit part) |
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|
\end{verbatim} |
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