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revision 1.3 by cnh, Wed Oct 24 15:21:27 2001 UTC revision 1.6 by cnh, Thu Oct 25 00:55:28 2001 UTC
# Line 37  to the problems described analytically b Line 37  to the problems described analytically b
37  In this experiment the model is configured to represent a mid-latitude  In this experiment the model is configured to represent a mid-latitude
38  enclosed sector of fluid on a sphere, $60^{\circ} \times 60^{\circ}$ in  enclosed sector of fluid on a sphere, $60^{\circ} \times 60^{\circ}$ in
39  lateral extent. The fluid is $2$~km deep and is forced  lateral extent. The fluid is $2$~km deep and is forced
40  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally  by a constant in time zonal wind stress, $\tau_{\lambda}$, that varies
41  in the north-south direction. Topologically the simulated  sinusoidally in the north-south direction. Topologically the simulated
42  domain is a sector on a sphere and the coriolis parameter, $f$, is defined  domain is a sector on a sphere and the coriolis parameter, $f$, is defined
43  according to latitude, $\varphi$  according to latitude, $\varphi$
44    
# Line 54  f(\varphi) = 2 \Omega \sin( \varphi ) Line 54  f(\varphi) = 2 \Omega \sin( \varphi )
54    
55  \begin{equation}  \begin{equation}
56  \label{EQ:taux}  \label{EQ:taux}
57  \tau_x(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})  \tau_{\lambda}(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})
58  \end{equation}  \end{equation}
59    
60  \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and  \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and
# Line 64  $\tau_0$ is set to $0.1N m^{-2}$. Line 64  $\tau_0$ is set to $0.1N m^{-2}$.
64  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:simulation_config}
65  summarises the configuration simulated.  summarises the configuration simulated.
66  In contrast to the example in section \ref{sec:eg-baro}, the  In contrast to the example in section \ref{sec:eg-baro}, the
67  current experiment simulates a spherical polar domain. However, as indicated  current experiment simulates a spherical polar domain. As indicated
68  by the axes in the lower left of the figure the model code works internally  by the axes in the lower left of the figure the model code works internally
69  in a locally orthoganal coordinate $(x,y,z)$. For this experiment description  in a locally orthoganal coordinate $(x,y,z)$. For this experiment description
70  of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous  of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous
# Line 95  linear Line 95  linear
95    
96  \noindent with $\rho_{0}=999.8\,{\rm kg\,m}^{-3}$ and  \noindent with $\rho_{0}=999.8\,{\rm kg\,m}^{-3}$ and
97  $\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1} $. Integrated forward in  $\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1} $. Integrated forward in
98  this configuration the model state variable {\bf theta} is synonomous with  this configuration the model state variable {\bf theta} is equivalent to
99  either in-situ temperature, $T$, or potential temperature, $\theta$. For  either in-situ temperature, $T$, or potential temperature, $\theta$. For
100  consistency with later examples, in which the equation of state is  consistency with later examples, in which the equation of state is
101  non-linear, we use $\theta$ to represent temperature here. This is  non-linear, we use $\theta$ to represent temperature here. This is
# Line 110  the quantity that is carried in the mode Line 110  the quantity that is carried in the mode
110  \caption{Schematic of simulation domain and wind-stress forcing function  \caption{Schematic of simulation domain and wind-stress forcing function
111  for the four-layer gyre numerical experiment. The domain is enclosed by solid  for the four-layer gyre numerical experiment. The domain is enclosed by solid
112  walls at $0^{\circ}$~E, $60^{\circ}$~E, $0^{\circ}$~N and $60^{\circ}$~N.  walls at $0^{\circ}$~E, $60^{\circ}$~E, $0^{\circ}$~N and $60^{\circ}$~N.
113  In the four-layer case an initial temperature stratification is  An initial stratification is
114  imposed by setting the potential temperature, $\theta$, in each layer.  imposed by setting the potential temperature, $\theta$, in each layer.
115  The vertical spacing, $\Delta z$, is constant and equal to $500$m.  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
116  }  }
# Line 119  The vertical spacing, $\Delta z$, is con Line 119  The vertical spacing, $\Delta z$, is con
119    
120  \subsection{Equations solved}  \subsection{Equations solved}
121    
122  The implicit free surface form of the  The implicit free surface {\bf HPE} form of the
123  pressure equation described in Marshall et. al \cite{Marshall97a} is  equations described in Marshall et. al \cite{Marshall97a} is
124  employed.  employed. The flow is three-dimensional with just temperature, $\theta$, as
125    an active tracer.  The equation of state is linear.
126  A horizontal laplacian operator $\nabla_{h}^2$ provides viscous  A horizontal laplacian operator $\nabla_{h}^2$ provides viscous
127  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation and provides a diffusive sub-grid scale closure for the
128  for the ``zonal flow'', $u$. Other terms in the model  temperature equation. A wind-stress momentum forcing is added to the momentum
129    equation for the zonal flow, $u$. Other terms in the model
130  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiement configuration (see section
131  \ref{SEC:code_config} ). This yields an active set of equations in  \ref{SEC:eg_fourl_code_config} ). This yields an active set of equations
132  solved in this configuration, written in spherical polar coordinates as  solved in this configuration, written in spherical polar coordinates as
133  follows  follows
134    
135  \begin{eqnarray}  \begin{eqnarray}
136  \label{EQ:model_equations}  \label{EQ:model_equations}
137  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
138    \frac{1}{\rho}\frac{\partial p^{'}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
139    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
140  & = &  & = &
141  \cal{F}  \cal{F}_{\lambda}
142  \\  \\
143  \frac{Dv}{Dt} + fu +  \frac{Dv}{Dt} + fu +
144    \frac{1}{\rho}\frac{\partial p^{'}}{\partial \varphi} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} -
145    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
146  & = &  & = &
147  0  0
148  \\  \\
149  \frac{\partial \eta}{\partial t} + \frac{\partial H \hat{u}}{\partial \lambda} +  \frac{\partial \eta}{\partial t} + \frac{\partial H \widehat{u}}{\partial \lambda} +
150  \frac{\partial H \hat{v}}{\partial \varphi}  \frac{\partial H \widehat{v}}{\partial \varphi}
151  &=&  &=&
152  0  0
153    \label{eq:fourl_example_continuity}
154  \\  \\
155  \frac{D\theta}{Dt} -  \frac{D\theta}{Dt} -
156   K_{h}\nabla_{h}^2\theta  - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}}   K_{h}\nabla_{h}^2\theta  - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}}
157  & = &  & = &
158  0  0
159    \label{eq:eg_fourl_theta}
160  \\  \\
161  p^{'} & = &  p^{\prime} & = &
162  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz  g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz
163  \\  \\
164  \rho^{'} & = &- \alpha_{\theta}\rho_{0}\theta^{'}  \rho^{\prime} & = &- \alpha_{\theta}\rho_{0}\theta^{\prime}
165  \\  \\
166  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}  {\cal F}_{\lambda} |_{s} & = & \frac{\tau_{\lambda}}{\rho_{0}\Delta z_{s}}
167  \\  \\
168  {\cal F} |_{i} & = & 0  {\cal F}_{\lambda} |_{i} & = & 0
169  \end{eqnarray}  \end{eqnarray}
170    
171  \noindent where $u$ and $v$ are the components of the horizontal  \noindent where $u$ and $v$ are the components of the horizontal
172  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
173  The suffices ${s},{i}$ indicate surface and interior of the domain.  The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical
174  The forcing $\cal F$ is only applied at the surface.  integral term given in equation \ref{eq:free-surface} and
175  The pressure field $p^{'}$ is separated into a barotropic part  explained in more detail in section \ref{sect:pressure-method-linear-backward}.
176    However, for the problem presented here, the continuity relation (equation
177    \ref{eq:fourl_example_continuity}) differs from the general form given
178    in section \ref{sect:pressure-method-linear-backward},
179    equation \ref{eq:linear-free-surface=P-E+R}, because the source terms
180    ${\cal P}-{\cal E}+{\cal R}$
181    are all $0$.
182    
183    The pressure field, $p^{\prime}$, is separated into a barotropic part
184  due to variations in sea-surface height, $\eta$, and a hydrostatic  due to variations in sea-surface height, $\eta$, and a hydrostatic
185  part due to variations in density, $\rho^{'}$, over the water column.  part due to variations in density, $\rho^{\prime}$, integrated
186    through the water column.
187    
188    The suffices ${s},{i}$ indicate surface and interior of the domain.
189    The windstress forcing, ${\cal F}_{\lambda}$, is applied in the surface layer
190    by a source term in the zonal momentum equation. In the ocean interior
191    this term is zero.
192    
193    In the momentum equations
194    lateral and vertical boundary conditions for the $\nabla_{h}^{2}$
195    and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified
196    when the numerical simulation is run - see section
197    \ref{SEC:eg_fourl_code_config}. For temperature
198    the boundary condition is ``zero-flux''
199    e.g. $\frac{\partial \theta}{\partial \varphi}=
200    \frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$.
201    
202    
203    
204  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
205    
206   The model is configured in hydrostatic form.  The domain is discretised with   The domain is discretised with
207  a uniform grid spacing in latitude and longitude  a uniform grid spacing in latitude and longitude
208   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so
209  that there are sixty grid cells in the zonal and meridional directions.  that there are sixty grid cells in the zonal and meridional directions.
210  Vertically the  Vertically the
211  model is configured with a four layers with constant depth,  model is configured with four layers with constant depth,
212  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
213  variables $x$ and $y$ are initialised from the values of  variables $x$ and $y$ are initialised from the values of
214  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
# Line 215  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, { Line 244  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
244    
245  As described in \ref{sec:tracer_equations}, the time evolution of potential  As described in \ref{sec:tracer_equations}, the time evolution of potential
246  temperature,  temperature,
247  $\theta$, equation is solved prognostically.  $\theta$, (equation \ref{eq:eg_fourl_theta})
248  The pressure forces that drive the fluid motions, (  is evaluated prognostically. The centered second-order scheme with
249    Adams-Bashforth time stepping described in section
250    \ref{sec:tracer_equations_abII} is used to step forward the temperature
251    equation. The pressure forces that drive the fluid motions, (
252  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface
253  elevation $\eta$ and the hydrostatic pressure.  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
254    pressure is evaluated explicitly by integrating density. The sea-surface
255    height, $\eta$, is solved for implicitly as described in section
256    \ref{sect:pressure-method-linear-backward}.
257    
258  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
259    
# Line 233  M_{w} = \pi ( \frac { A_{h} }{ \beta } ) Line 268  M_{w} = \pi ( \frac { A_{h} }{ \beta } )
268  \noindent  of $\approx 100$km. This is greater than the model  \noindent  of $\approx 100$km. This is greater than the model
269  resolution in mid-latitudes $\Delta x$, ensuring that the frictional  resolution in mid-latitudes $\Delta x$, ensuring that the frictional
270  boundary layer is well resolved.  boundary layer is well resolved.
271    
272  \\  \\
273    
274  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
# Line 299  S_{c} = \frac{c_{g} \delta t}{ \Delta x} Line 335  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
335  stability limit of 0.25.  stability limit of 0.25.
336        
337  \subsection{Code Configuration}  \subsection{Code Configuration}
338  \label{SEC:code_config}  \label{SEC:eg_fourl_code_config}
339    
340  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
341  directory {\it verification/exp1/}.  The experiment files  directory {\it verification/exp1/}.  The experiment files
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