--- manual/s_examples/baroclinic_gyre/fourlayer.tex 2001/10/24 19:43:07 1.4 +++ manual/s_examples/baroclinic_gyre/fourlayer.tex 2001/10/24 23:14:44 1.5 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/baroclinic_gyre/fourlayer.tex,v 1.4 2001/10/24 19:43:07 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/baroclinic_gyre/fourlayer.tex,v 1.5 2001/10/24 23:14:44 cnh Exp $ % $Name: $ \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates} @@ -37,8 +37,8 @@ In this experiment the model is configured to represent a mid-latitude enclosed sector of fluid on a sphere, $60^{\circ} \times 60^{\circ}$ in lateral extent. The fluid is $2$~km deep and is forced -by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally -in the north-south direction. Topologically the simulated +by a constant in time zonal wind stress, $\tau_{\lambda}$, that varies +sinusoidally in the north-south direction. Topologically the simulated domain is a sector on a sphere and the coriolis parameter, $f$, is defined according to latitude, $\varphi$ @@ -54,7 +54,7 @@ \begin{equation} \label{EQ:taux} -\tau_x(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}}) +\tau_{\lambda}(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}}) \end{equation} \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and @@ -64,7 +64,7 @@ Figure \ref{FIG:simulation_config} summarises the configuration simulated. In contrast to the example in section \ref{sec:eg-baro}, the -current experiment simulates a spherical polar domain. However, as indicated +current experiment simulates a spherical polar domain. As indicated by the axes in the lower left of the figure the model code works internally in a locally orthoganal coordinate $(x,y,z)$. For this experiment description of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous @@ -95,7 +95,7 @@ \noindent with $\rho_{0}=999.8\,{\rm kg\,m}^{-3}$ and $\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1} $. Integrated forward in -this configuration the model state variable {\bf theta} is synonomous with +this configuration the model state variable {\bf theta} is equivalent to either in-situ temperature, $T$, or potential temperature, $\theta$. For consistency with later examples, in which the equation of state is non-linear, we use $\theta$ to represent temperature here. This is @@ -110,7 +110,7 @@ \caption{Schematic of simulation domain and wind-stress forcing function for the four-layer gyre numerical experiment. The domain is enclosed by solid walls at $0^{\circ}$~E, $60^{\circ}$~E, $0^{\circ}$~N and $60^{\circ}$~N. -In the four-layer case an initial temperature stratification is +An initial stratification is imposed by setting the potential temperature, $\theta$, in each layer. The vertical spacing, $\Delta z$, is constant and equal to $500$m. } @@ -119,14 +119,16 @@ \subsection{Equations solved} -The implicit free surface form of the -pressure equation described in Marshall et. al \cite{Marshall97a} is -employed. +The implicit free surface {\bf HPE} form of the +equations described in Marshall et. al \cite{Marshall97a} is +employed. The flow is three-dimensional with just temperature, $\theta$, as +an active tracer. The equation of state is linear. A horizontal laplacian operator $\nabla_{h}^2$ provides viscous -dissipation. The wind-stress momentum input is added to the momentum equation -for the ``zonal flow'', $u$. Other terms in the model +dissipation and provides a diffusive sub-grid scale closure for the +temperature equation. A wind-stress momentum forcing is added to the momentum +equation for the zonal flow, $u$. Other terms in the model are explicitly switched off for this experiement configuration (see section -\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 solved in this configuration, written in spherical polar coordinates as follows @@ -136,7 +138,7 @@ \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} - A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}} & = & -\cal{F} +\cal{F}_{\lambda} \\ \frac{Dv}{Dt} + fu + \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} - @@ -144,40 +146,64 @@ & = & 0 \\ -\frac{\partial \eta}{\partial t} + \frac{\partial H \hat{u}}{\partial \lambda} + -\frac{\partial H \hat{v}}{\partial \varphi} +\frac{\partial \eta}{\partial t} + \frac{\partial H \widehat{u}}{\partial \lambda} + +\frac{\partial H \widehat{v}}{\partial \varphi} &=& 0 +\label{eq:fourl_example_continuity} \\ \frac{D\theta}{Dt} - K_{h}\nabla_{h}^2\theta - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}} & = & 0 +\label{eq:eg_fourl_theta} \\ p^{\prime} & = & g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz \\ \rho^{\prime} & = &- \alpha_{\theta}\rho_{0}\theta^{\prime} \\ -{\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}} +{\cal F}_{\lambda} |_{s} & = & \frac{\tau_{\lambda}}{\rho_{0}\Delta z_{s}} \\ -{\cal F} |_{i} & = & 0 +{\cal F}_{\lambda} |_{i} & = & 0 \end{eqnarray} \noindent where $u$ and $v$ are the components of the horizontal flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$). -The terms $H\hat{u}$ and $H\hat{v}$ are the components of the term -integrated in eqaution \ref{eq:free-surface}, as descirbed in section +The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical +integral term given in equation \ref{eq:free-surface} and +explained in more detail in section \ref{sect:pressure-method-linear-backward}. +However, for the problem presented here, the continuity relation (equation +\ref{eq:fourl_example_continuity}) differs from the general form given +in section \ref{sect:pressure-method-linear-backward}, +equation \ref{eq:linear-free-surface=P-E+R}, because the source terms +${\cal P}-{\cal E}+{\cal R}$ +are all $0$. -The suffices ${s},{i}$ indicate surface and interior of the domain. -The forcing $\cal F$ is only applied at the surface. The pressure field, $p^{\prime}$, is separated into a barotropic part due to variations in sea-surface height, $\eta$, and a hydrostatic -part due to variations in density, $\rho^{\prime}$, over the water column. +part due to variations in density, $\rho^{\prime}$, integrated +through the water column. + +The suffices ${s},{i}$ indicate surface and interior of the domain. +The windstress forcing, ${\cal F}_{\lambda}$, is applied in the surface layer +by a source term in the zonal momentum equation. In the ocean interior +this term is zero. + +In the momentum equations +lateral and vertical boundary conditions for the $\nabla_{h}^{2}$ +and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified +when the numerical simulation is run - see section +\ref{SEC:eg_fourl_code_config}. For temperature +the boundary condition is ``zero-flux'' +e.g. $\frac{\partial \theta}{\partial \varphi}= +\frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$. + + \subsection{Discrete Numerical Configuration} - The model is configured in hydrostatic form. The domain is discretised with + The domain is discretised with a uniform grid spacing in latitude and longitude $\Delta \lambda=\Delta \varphi=1^{\circ}$, so that there are sixty grid cells in the zonal and meridional directions. @@ -218,10 +244,16 @@ As described in \ref{sec:tracer_equations}, the time evolution of potential temperature, -$\theta$, equation is solved prognostically. -The pressure forces that drive the fluid motions, ( +$\theta$, (equation \ref{eq:eg_fourl_theta}) +is evaluated prognostically. The centered second-order scheme with +Adams-Bashforth time stepping described in section +\ref{sec:tracer_equations_abII} is used to step forward the temperature +equation. The pressure forces that drive the fluid motions, ( $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface -elevation $\eta$ and the hydrostatic pressure. +elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the +pressure is evaluated explicitly by integrating density. The sea-surface +height, $\eta$, is solved for implicitly as described in section +\ref{sect:pressure-method-linear-backward}. \subsubsection{Numerical Stability Criteria} @@ -302,7 +334,7 @@ stability limit of 0.25. \subsection{Code Configuration} -\label{SEC:code_config} +\label{SEC:eg_fourl_code_config} The model configuration for this experiment resides under the directory {\it verification/exp1/}. The experiment files