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\pagebreak |
%%%% \part{MIT GCM basics} |
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\part{MITgcm basics} |
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% Section: Overview |
% Section: Overview |
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\begin{itemize} |
\begin{itemize} |
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\item it can be used to study both atmospheric and oceanic phenomena; one |
\item it can be used to study both atmospheric and oceanic phenomena; one |
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hydrodynamical kernel is used to drive forward both atmospheric and oceanic |
hydrodynamical kernel is used to drive forward both atmospheric and oceanic |
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models - see fig.1% |
models - see fig% |
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\marginpar{ |
\marginpar{ |
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Fig.1 One model}\ref{fig:onemodel} |
Fig.1 One model}\ref{fig:onemodel} |
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\begin{figure} |
%% CNHbegin |
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\begin{center} |
\input{part1/one_model_figure} |
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\resizebox{!}{4in}{ |
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\rotatebox{90}{ |
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\rotatebox{180}{ |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps} |
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} |
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} |
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} |
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\end{center} |
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\label{fig:onemodel} |
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\end{figure} |
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\item it has a non-hydrostatic capability and so can be used to study both |
\item it has a non-hydrostatic capability and so can be used to study both |
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small-scale and large scale processes - see fig.2% |
small-scale and large scale processes - see fig % |
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\marginpar{ |
\marginpar{ |
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Fig.2 All scales}\ref{fig:all-scales} |
Fig.2 All scales}\ref{fig:all-scales} |
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\begin{figure} |
\input{part1/all_scales_figure} |
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\begin{center} |
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\rotatebox{90}{ |
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\rotatebox{180}{ |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps} |
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} |
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} |
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} |
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\end{center} |
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\label{fig:scales} |
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\end{figure} |
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\item finite volume techniques are employed yielding an intuitive |
\item finite volume techniques are employed yielding an intuitive |
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discretization and support for the treatment of irregular geometries using |
discretization and support for the treatment of irregular geometries using |
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orthogonal curvilinear grids and shaved cells - see fig.3% |
orthogonal curvilinear grids and shaved cells - see fig % |
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\marginpar{ |
\marginpar{ |
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Fig.3 Finite volumes}\ref{fig:Finite volumes} |
Fig.3 Finite volumes}\ref{fig:finite-volumes} |
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\item tangent linear and adjoint counterparts are automatically maintained |
\item tangent linear and adjoint counterparts are automatically maintained |
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along with the forward model, permitting sensitivity and optimization |
along with the forward model, permitting sensitivity and optimization |
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kinds of problems the model has been used to study, we briefly describe some |
kinds of problems the model has been used to study, we briefly describe some |
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of them here. A more detailed description of the underlying formulation, |
of them here. A more detailed description of the underlying formulation, |
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numerical algorithm and implementation that lie behind these calculations is |
numerical algorithm and implementation that lie behind these calculations is |
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given later. Indeed it is easy to reproduce the results shown here: simply |
given later. Indeed many of the illustrative examples shown below can be |
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download the model (the minimum you need is a PC running linux, together |
easily reproduced: simply download the model (the minimum you need is a PC |
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with a FORTRAN\ 77 compiler) and follow the examples. |
running linux, together with a FORTRAN\ 77 compiler) and follow the examples |
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described in detail in the documentation. |
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\subsection{Global atmosphere: `Held-Suarez' benchmark} |
\subsection{Global atmosphere: `Held-Suarez' benchmark} |
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Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height |
A novel feature of MITgcm is its ability to simulate both atmospheric and |
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field obtained using a 5-level version of the atmospheric pressure isomorph |
oceanographic flows at both small and large scales. |
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run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies |
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along the northern hemisphere storm track. There are no mountains or |
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land-sea contrast in this calculation, but you can easily put them in. The |
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model is driven by relaxation to a radiative-convective equilibrium profile, |
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following the description set out in Held and Suarez; 1994 designed to test |
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atmospheric hydrodynamical cores - there are no mountains or land-sea |
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contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to |
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descretize the globe permitting a uniform gridding and obviated the need to |
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fourier filter. |
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Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal |
Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$ |
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wind and meridional overturning streamfunction from the 5-level model. |
temperature field obtained using the atmospheric isomorph of MITgcm run at |
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2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole |
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(blue) and warm air along an equatorial band (red). Fully developed |
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\begin{figure} |
baroclinic eddies spawned in the northern hemisphere storm track are |
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\begin{center} |
evident. There are no mountains or land-sea contrast in this calculation, |
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\resizebox{!}{4in}{ |
but you can easily put them in. The model is driven by relaxation to a |
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\rotatebox{90}{ |
radiative-convective equilibrium profile, following the description set out |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps} |
in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores - |
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} |
there are no mountains or land-sea contrast. |
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} |
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\end{center} |
%% CNHbegin |
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\label{fig:hscs} |
\input{part1/cubic_eddies_figure} |
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\end{figure} |
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As described in Adcroft (2001), a `cubed sphere' is used to discretize the |
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globe permitting a uniform gridding and obviated the need to fourier filter. |
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The `vector-invariant' form of MITgcm supports any orthogonal curvilinear |
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grid, of which the cubed sphere is just one of many choices. |
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Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal |
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wind and meridional overturning streamfunction from a 20-level version of |
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the model. It compares favorable with more conventional spatial |
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discretization approaches. |
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A regular spherical lat-lon grid can also be used. |
A regular spherical lat-lon grid can also be used. |
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\begin{figure} |
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\begin{center} |
\input{part1/hs_zave_u_figure} |
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\resizebox{!}{4in}{ |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps} |
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} |
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} |
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\end{center} |
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\label{fig:hslatlon} |
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\end{figure} |
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\subsection{Ocean gyres} |
\subsection{Ocean gyres} |
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Baroclinic instability is a ubiquitous process in the ocean, as well as the |
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atmosphere. Ocean eddies play an important role in modifying the |
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hydrographic structure and current systems of the oceans. Coarse resolution |
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models of the oceans cannot resolve the eddy field and yield rather broad, |
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diffusive patterns of ocean currents. But if the resolution of our models is |
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increased until the baroclinic instability process is resolved, numerical |
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solutions of a different and much more realistic kind, can be obtained. |
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Fig. ?.? shows the surface temperature and velocity field obtained from |
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MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$ |
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grid in which the pole has been rotated by 90$^{\circ }$ on to the equator |
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(to avoid the converging of meridian in northern latitudes). 21 vertical |
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levels are used in the vertical with a `lopped cell' representation of |
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topography. The development and propagation of anomalously warm and cold |
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eddies can be clearly been seen in the Gulf Stream region. The transport of |
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warm water northward by the mean flow of the Gulf Stream is also clearly |
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visible. |
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\input{part1/ocean_gyres_figure} |
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\subsection{Global ocean circulation} |
\subsection{Global ocean circulation} |
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Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$ |
Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$ |
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global ocean model run with 15 vertical levels. The model is driven using |
global ocean model run with 15 vertical levels. Lopped cells are used to |
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monthly-mean winds with mixed boundary conditions on temperature and |
represent topography on a regular $lat-lon$ grid extending from 70$^{\circ |
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salinity at the surface. Fig.E2b shows the overturning (thermohaline) |
}N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with |
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circulation. Lopped cells are used to represent topography on a regular $% |
mixed boundary conditions on temperature and salinity at the surface. The |
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lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$. |
transfer properties of ocean eddies, convection and mixing is parameterized |
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in this model. |
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\begin{figure} |
Fig.E2b shows the meridional overturning circulation of the global ocean in |
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\begin{center} |
Sverdrups. |
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\resizebox{!}{4in}{ |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps} |
\input{part1/global_circ_figure} |
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% } |
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} |
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\end{center} |
\subsection{Convection and mixing over topography} |
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\label{fig:horizcirc} |
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\end{figure} |
Dense plumes generated by localized cooling on the continental shelf of the |
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ocean may be influenced by rotation when the deformation radius is smaller |
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\begin{figure} |
than the width of the cooling region. Rather than gravity plumes, the |
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\begin{center} |
mechanism for moving dense fluid down the shelf is then through geostrophic |
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\resizebox{!}{4in}{ |
eddies. The simulation shown in the figure (blue is cold dense fluid, red is |
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\rotatebox{90}{ |
warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to |
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\rotatebox{180}{ |
trigger convection by surface cooling. The cold, dense water falls down the |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps} |
slope but is deflected along the slope by rotation. It is found that |
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entrainment in the vertical plane is reduced when rotational control is |
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strong, and replaced by lateral entrainment due to the baroclinic |
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instability of the along-slope current. |
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\end{center} |
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\label{fig:moc} |
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\end{figure} |
\input{part1/convect_and_topo} |
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\subsection{Flow over topography} |
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\subsection{Ocean convection} |
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Fig.E3 shows convection over a slope using the non-hydrostatic ocean |
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isomorph and lopped cells to respresent topography. .....The grid resolution |
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is |
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\subsection{Boundary forced internal waves} |
\subsection{Boundary forced internal waves} |
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\subsection{Carbon outgassing sensitivity} |
The unique ability of MITgcm to treat non-hydrostatic dynamics in the |
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presence of complex geometry makes it an ideal tool to study internal wave |
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Fig.E4 shows.... |
dynamics and mixing in oceanic canyons and ridges driven by large amplitude |
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barotropic tidal currents imposed through open boundary conditions. |
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\begin{figure} |
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\begin{center} |
Fig. ?.? shows the influence of cross-slope topographic variations on |
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\resizebox{!}{4in}{ |
internal wave breaking - the cross-slope velocity is in color, the density |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps} |
contoured. The internal waves are excited by application of open boundary |
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} |
conditions on the left.\ They propagate to the sloping boundary (represented |
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\end{center} |
using MITgcm's finite volume spatial discretization) where they break under |
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\label{fig:co2mrt} |
nonhydrostatic dynamics. |
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\end{figure} |
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\subsection{Parameter sensitivity using the adjoint of MITgcm} |
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Forward and tangent linear counterparts of MITgcm are supported using an |
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`automatic adjoint compiler'. These can be used in parameter sensitivity and |
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data assimilation studies. |
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As one example of application of the MITgcm adjoint, Fig.E4 maps the |
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gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude |
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of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $% |
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\mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is |
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sensitive to heat fluxes over the Labrador Sea, one of the important sources |
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of deep water for the thermohaline circulations. This calculation also |
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yields sensitivities to all other model parameters. |
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\subsection{Global state estimation of the ocean} |
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An important application of MITgcm is in state estimation of the global |
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ocean circulation. An appropriately defined `cost function', which measures |
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the departure of the model from observations (both remotely sensed and |
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insitu) over an interval of time, is minimized by adjusting `control |
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parameters' such as air-sea fluxes, the wind field, the initial conditions |
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etc. Figure ?.? shows an estimate of the time-mean surface elevation of the |
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ocean obtained by bringing the model in to consistency with altimetric and |
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in-situ observations over the period 1992-1997. |
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\input{part1/globes_figure} |
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\subsection{Ocean biogeochemical cycles} |
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MITgcm is being used to study global biogeochemical cycles in the ocean. For |
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example one can study the effects of interannual changes in meteorological |
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forcing and upper ocean circulation on the fluxes of carbon dioxide and |
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oxygen between the ocean and atmosphere. The figure shows the annual air-sea |
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flux of oxygen and its relation to density outcrops in the southern oceans |
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from a single year of a global, interannually varying simulation. |
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\subsection{Simulations of laboratory experiments} |
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Figure ?.? shows MITgcm being used to simulate a laboratory experiment |
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enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An |
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initially homogeneous tank of water ($1m$ in diameter) is driven from its |
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free surface by a rotating heated disk. The combined action of mechanical |
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and thermal forcing creates a lens of fluid which becomes baroclinically |
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unstable. The stratification and depth of penetration of the lens is |
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arrested by its instability in a process analogous to that whic sets the |
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stratification of the ACC. |
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% $Header$ |
% $Header$ |
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% $Name$ |
% $Name$ |
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vertical coordinate `$r$' is interpreted as pressure, $p$, if we are |
vertical coordinate `$r$' is interpreted as pressure, $p$, if we are |
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modeling the atmosphere and height, $z$, if we are modeling the ocean. |
modeling the atmosphere and height, $z$, if we are modeling the ocean. |
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\input{part1/zandpcoord_figure.tex} |
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The state of the fluid at any time is characterized by the distribution of |
The state of the fluid at any time is characterized by the distribution of |
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velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a |
velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a |
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`geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may |
`geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may |
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\marginpar{ |
\marginpar{ |
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Fig.5 The vertical coordinate of model}: |
Fig.5 The vertical coordinate of model}: |
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\begin{figure} |
%%CNHbegin |
| 343 |
\begin{center} |
\input{part1/vertcoord_figure.tex} |
| 344 |
\resizebox{!}{4in}{ |
%%CNHend |
|
\rotatebox{90}{ |
|
|
\rotatebox{180}{ |
|
|
\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps} |
|
|
} |
|
|
} |
|
|
} |
|
|
\end{center} |
|
|
\label{fig:vertcoord} |
|
|
\end{figure} |
|
| 345 |
|
|
| 346 |
\begin{equation*} |
\begin{equation*} |
| 347 |
\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}% |
\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}% |
| 361 |
\end{equation} |
\end{equation} |
| 362 |
|
|
| 363 |
\begin{equation*} |
\begin{equation*} |
| 364 |
b=b(\theta ,S,r)\text{ equation of state} |
b=b(\theta ,S,r)\text{ equation of state} |
| 365 |
\end{equation*} |
\end{equation*} |
| 366 |
|
|
| 367 |
\begin{equation*} |
\begin{equation*} |
| 368 |
\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} |
\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} |
| 369 |
\end{equation*} |
\end{equation*} |
| 370 |
|
|
| 371 |
\begin{equation*} |
\begin{equation*} |
| 372 |
\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} |
\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} |
| 373 |
\end{equation*} |
\end{equation*} |
| 374 |
|
|
| 375 |
Here: |
Here: |
| 376 |
|
|
| 377 |
\begin{equation*} |
\begin{equation*} |
| 378 |
r\text{ is the vertical coordinate} |
r\text{ is the vertical coordinate} |
| 379 |
\end{equation*} |
\end{equation*} |
| 380 |
|
|
| 381 |
\begin{equation*} |
\begin{equation*} |
| 382 |
\frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{ |
\frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{ |
| 383 |
is the total derivative} |
is the total derivative} |
| 384 |
\end{equation*} |
\end{equation*} |
| 385 |
|
|
| 386 |
\begin{equation*} |
\begin{equation*} |
| 387 |
\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}% |
\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}% |
| 388 |
\text{ is the `grad' operator} |
\text{ is the `grad' operator} |
| 389 |
\end{equation*} |
\end{equation*} |
| 390 |
with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}% |
with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}% |
| 391 |
\frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$ |
\frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$ |
| 392 |
is a unit vector in the vertical |
is a unit vector in the vertical |
| 393 |
|
|
| 394 |
\begin{equation*} |
\begin{equation*} |
| 395 |
t\text{ is time} |
t\text{ is time} |
| 396 |
\end{equation*} |
\end{equation*} |
| 397 |
|
|
| 398 |
\begin{equation*} |
\begin{equation*} |
| 399 |
\vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the |
\vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the |
| 400 |
velocity} |
velocity} |
| 401 |
\end{equation*} |
\end{equation*} |
| 402 |
|
|
| 403 |
\begin{equation*} |
\begin{equation*} |
| 404 |
\phi \text{ is the `pressure'/`geopotential'} |
\phi \text{ is the `pressure'/`geopotential'} |
| 405 |
\end{equation*} |
\end{equation*} |
| 406 |
|
|
| 407 |
\begin{equation*} |
\begin{equation*} |
| 408 |
\vec{\Omega}\text{ is the Earth's rotation} |
\vec{\Omega}\text{ is the Earth's rotation} |
| 409 |
\end{equation*} |
\end{equation*} |
| 410 |
|
|
| 411 |
\begin{equation*} |
\begin{equation*} |
| 412 |
b\text{ is the `buoyancy'} |
b\text{ is the `buoyancy'} |
| 413 |
\end{equation*} |
\end{equation*} |
| 414 |
|
|
| 415 |
\begin{equation*} |
\begin{equation*} |
| 416 |
\theta \text{ is potential temperature} |
\theta \text{ is potential temperature} |
| 417 |
\end{equation*} |
\end{equation*} |
| 418 |
|
|
| 419 |
\begin{equation*} |
\begin{equation*} |
| 420 |
S\text{ is specific humidity in the atmosphere; salinity in the ocean} |
S\text{ is specific humidity in the atmosphere; salinity in the ocean} |
| 421 |
\end{equation*} |
\end{equation*} |
| 422 |
|
|
| 423 |
\begin{equation*} |
\begin{equation*} |
| 426 |
\end{equation*} |
\end{equation*} |
| 427 |
|
|
| 428 |
\begin{equation*} |
\begin{equation*} |
| 429 |
\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }% |
\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta |
|
\theta |
|
| 430 |
\end{equation*} |
\end{equation*} |
| 431 |
|
|
| 432 |
\begin{equation*} |
\begin{equation*} |
| 455 |
Here |
Here |
| 456 |
|
|
| 457 |
\begin{equation*} |
\begin{equation*} |
| 458 |
R_{moving}=R_{o}+\eta |
R_{moving}=R_{o}+\eta |
| 459 |
\end{equation*} |
\end{equation*} |
| 460 |
where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on |
where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on |
| 461 |
whether we are in the atmosphere or ocean) of the `moving surface' in the |
whether we are in the atmosphere or ocean) of the `moving surface' in the |
| 523 |
At the top of the atmosphere (which is `fixed' in our $r$ coordinate): |
At the top of the atmosphere (which is `fixed' in our $r$ coordinate): |
| 524 |
|
|
| 525 |
\begin{equation*} |
\begin{equation*} |
| 526 |
R_{fixed}=p_{top}=0 |
R_{fixed}=p_{top}=0 |
| 527 |
\end{equation*} |
\end{equation*} |
| 528 |
In a resting atmosphere the elevation of the mountains at the bottom is |
In a resting atmosphere the elevation of the mountains at the bottom is |
| 529 |
given by |
given by |
| 530 |
\begin{equation*} |
\begin{equation*} |
| 531 |
R_{moving}=R_{o}(x,y)=p_{o}(x,y) |
R_{moving}=R_{o}(x,y)=p_{o}(x,y) |
| 532 |
\end{equation*} |
\end{equation*} |
| 533 |
i.e. the (hydrostatic) pressure at the top of the mountains in a resting |
i.e. the (hydrostatic) pressure at the top of the mountains in a resting |
| 534 |
atmosphere. |
atmosphere. |
| 638 |
\textit{Coriolis} \\ |
\textit{Coriolis} \\ |
| 639 |
\textit{\ Forcing/Dissipation}% |
\textit{\ Forcing/Dissipation}% |
| 640 |
\end{tabular}% |
\end{tabular}% |
| 641 |
\ \right. \qquad \label{eq:gu-speherical} |
\ \right. \qquad \label{eq:gu-speherical} |
| 642 |
\end{equation} |
\end{equation} |
| 643 |
|
|
| 644 |
\begin{equation} |
\begin{equation} |
| 657 |
\textit{Coriolis} \\ |
\textit{Coriolis} \\ |
| 658 |
\textit{\ Forcing/Dissipation}% |
\textit{\ Forcing/Dissipation}% |
| 659 |
\end{tabular}% |
\end{tabular}% |
| 660 |
\ \right. \qquad \label{eq:gv-spherical} |
\ \right. \qquad \label{eq:gv-spherical} |
| 661 |
\end{equation}% |
\end{equation}% |
| 662 |
\qquad \qquad \qquad \qquad \qquad |
\qquad \qquad \qquad \qquad \qquad |
| 663 |
|
|
| 664 |
\begin{equation} |
\begin{equation} |
| 665 |
\left. |
\left. |
| 676 |
\textit{Coriolis} \\ |
\textit{Coriolis} \\ |
| 677 |
\textit{\ Forcing/Dissipation}% |
\textit{\ Forcing/Dissipation}% |
| 678 |
\end{tabular}% |
\end{tabular}% |
| 679 |
\ \right. \label{eq:gw-spherical} |
\ \right. \label{eq:gw-spherical} |
| 680 |
\end{equation}% |
\end{equation}% |
| 681 |
\qquad \qquad \qquad \qquad \qquad |
\qquad \qquad \qquad \qquad \qquad |
| 682 |
|
|
| 683 |
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 `$lat$% |
| 684 |
' is latitude. |
' is latitude. |
| 688 |
\marginpar{ |
\marginpar{ |
| 689 |
Fig.6 Spherical polar coordinate system.} |
Fig.6 Spherical polar coordinate system.} |
| 690 |
|
|
| 691 |
\begin{figure} |
%%CNHbegin |
| 692 |
\begin{center} |
\input{part1/sphere_coord_figure.tex} |
| 693 |
\resizebox{!}{4in}{ |
%%CNHend |
|
\rotatebox{90}{ |
|
|
\rotatebox{180}{ |
|
|
\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps} |
|
|
} |
|
|
} |
|
|
} |
|
|
\end{center} |
|
|
\label{fig:spcoord} |
|
|
\end{figure} |
|
|
|
|
| 694 |
|
|
| 695 |
\subsubsection{Shallow atmosphere approximation} |
\subsubsection{Shallow atmosphere approximation} |
| 696 |
|
|
| 700 |
Coriolis force is treated approximately and the shallow atmosphere |
Coriolis force is treated approximately and the shallow atmosphere |
| 701 |
approximation is made.\ The MITgcm need not make the `traditional |
approximation is made.\ The MITgcm need not make the `traditional |
| 702 |
approximation'. To be able to support consistent non-hydrostatic forms the |
approximation'. To be able to support consistent non-hydrostatic forms the |
| 703 |
shallow atmosphere approximation can be relaxed - when dividing through by $r |
shallow atmosphere approximation can be relaxed - when dividing through by $% |
| 704 |
$ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, |
r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, |
| 705 |
the radius of the earth. |
the radius of the earth. |
| 706 |
|
|
| 707 |
\subsubsection{Hydrostatic and quasi-hydrostatic forms} |
\subsubsection{Hydrostatic and quasi-hydrostatic forms} |
| 727 |
vertical momentum equation (\ref{eq:mom-w}) becomes: |
vertical momentum equation (\ref{eq:mom-w}) becomes: |
| 728 |
|
|
| 729 |
\begin{equation*} |
\begin{equation*} |
| 730 |
\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat |
\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat |
| 731 |
\end{equation*} |
\end{equation*} |
| 732 |
making a small correction to the hydrostatic pressure. |
making a small correction to the hydrostatic pressure. |
| 733 |
|
|
| 808 |
stepping forward the horizontal momentum equations; $\dot{r}$ is found by |
stepping forward the horizontal momentum equations; $\dot{r}$ is found by |
| 809 |
stepping forward the vertical momentum equation. |
stepping forward the vertical momentum equation. |
| 810 |
|
|
| 811 |
\begin{figure} |
%%CNHbegin |
| 812 |
\begin{center} |
\input{part1/solution_strategy_figure.tex} |
| 813 |
\resizebox{!}{4in}{ |
%%CNHend |
|
\rotatebox{90}{ |
|
|
\rotatebox{180}{ |
|
|
\includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps} |
|
|
} |
|
|
} |
|
|
} |
|
|
\end{center} |
|
|
\label{fig:solnstart} |
|
|
\end{figure} |
|
|
|
|
| 814 |
|
|
| 815 |
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 |
| 816 |
course, some complication that goes with the inclusion of $\cos \phi \ $% |
course, some complication that goes with the inclusion of $\cos \phi \ $% |
| 838 |
|
|
| 839 |
\begin{equation*} |
\begin{equation*} |
| 840 |
\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}% |
\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}% |
| 841 |
\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr |
\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr |
| 842 |
\end{equation*} |
\end{equation*} |
| 843 |
and so |
and so |
| 844 |
|
|
| 860 |
|
|
| 861 |
\begin{equation*} |
\begin{equation*} |
| 862 |
\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}% |
\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}% |
| 863 |
}_{h}+\partial _{r}\dot{r}\right) dr=0 |
}_{h}+\partial _{r}\dot{r}\right) dr=0 |
| 864 |
\end{equation*} |
\end{equation*} |
| 865 |
|
|
| 866 |
Thus: |
Thus: |
| 868 |
\begin{equation*} |
\begin{equation*} |
| 869 |
\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta |
\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta |
| 870 |
+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}% |
+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}% |
| 871 |
_{h}dr=0 |
_{h}dr=0 |
| 872 |
\end{equation*} |
\end{equation*} |
| 873 |
where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $% |
where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $% |
| 874 |
r $. The above can be rearranged to yield, using Leibnitz's theorem: |
r $. The above can be rearranged to yield, using Leibnitz's theorem: |
| 884 |
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can |
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can |
| 885 |
be written |
be written |
| 886 |
\begin{equation} |
\begin{equation} |
| 887 |
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right) |
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right) |
| 888 |
\label{eq:phi-surf} |
\label{eq:phi-surf} |
| 889 |
\end{equation}% |
\end{equation}% |
| 890 |
where $b_{s}$ is the buoyancy at the surface. |
where $b_{s}$ is the buoyancy at the surface. |
| 943 |
presenting inhomogeneous Neumann boundary conditions to the Elliptic problem |
presenting inhomogeneous Neumann boundary conditions to the Elliptic problem |
| 944 |
(\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can |
(\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can |
| 945 |
exploit classical 3D potential theory and, by introducing an appropriately |
exploit classical 3D potential theory and, by introducing an appropriately |
| 946 |
chosen $\delta $-function sheet of `source-charge', replace the inhomogenous |
chosen $\delta $-function sheet of `source-charge', replace the |
| 947 |
boundary condition on pressure by a homogeneous one. The source term $rhs$ |
inhomogeneous boundary condition on pressure by a homogeneous one. The |
| 948 |
in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$ |
source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $% |
| 949 |
By simultaneously setting $% |
\vec{\mathbf{F}}.$ By simultaneously setting $% |
| 950 |
\begin{array}{l} |
\begin{array}{l} |
| 951 |
\widehat{n}.\vec{\mathbf{F}}% |
\widehat{n}.\vec{\mathbf{F}}% |
| 952 |
\end{array}% |
\end{array}% |
| 953 |
=0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following |
=0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following |
| 954 |
self-consistent but simpler homogenised Elliptic problem is obtained: |
self-consistent but simpler homogenized Elliptic problem is obtained: |
| 955 |
|
|
| 956 |
\begin{equation*} |
\begin{equation*} |
| 957 |
\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad |
\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad |
| 958 |
\end{equation*}% |
\end{equation*}% |
| 959 |
where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such |
where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such |
| 960 |
that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref% |
that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref% |
| 1029 |
\begin{equation} |
\begin{equation} |
| 1030 |
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}% |
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}% |
| 1031 |
+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla % |
+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla % |
| 1032 |
\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] |
\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] |
| 1033 |
\label{eq:vi-identity} |
\label{eq:vi-identity} |
| 1034 |
\end{equation}% |
\end{equation}% |
| 1035 |
This permits alternative numerical treatments of the non-linear terms based |
This permits alternative numerical treatments of the non-linear terms based |
| 1042 |
|
|
| 1043 |
\subsection{Adjoint} |
\subsection{Adjoint} |
| 1044 |
|
|
| 1045 |
Tangent linear and adoint counterparts of the forward model and described in |
Tangent linear and adjoint counterparts of the forward model and described |
| 1046 |
Chapter 5. |
in Chapter 5. |
| 1047 |
|
|
| 1048 |
% $Header$ |
% $Header$ |
| 1049 |
% $Name$ |
% $Name$ |
| 1058 |
The hydrostatic primitive equations (HPEs) in p-coordinates are: |
The hydrostatic primitive equations (HPEs) in p-coordinates are: |
| 1059 |
\begin{eqnarray} |
\begin{eqnarray} |
| 1060 |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
| 1061 |
_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} |
_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} |
| 1062 |
\label{eq:atmos-mom} \\ |
\label{eq:atmos-mom} \\ |
| 1063 |
\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ |
\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ |
| 1064 |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% |
| 1065 |
\partial p} &=&0 \label{eq:atmos-cont} \\ |
\partial p} &=&0 \label{eq:atmos-cont} \\ |
| 1066 |
p\alpha &=&RT \label{eq:atmos-eos} \\ |
p\alpha &=&RT \label{eq:atmos-eos} \\ |
| 1067 |
c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat} |
c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat} |
| 1068 |
\end{eqnarray}% |
\end{eqnarray}% |
| 1069 |
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 |
| 1110 |
The heat equation is obtained by noting that |
The heat equation is obtained by noting that |
| 1111 |
\begin{equation*} |
\begin{equation*} |
| 1112 |
c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta |
c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta |
| 1113 |
\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} |
\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} |
| 1114 |
\end{equation*} |
\end{equation*} |
| 1115 |
and on substituting into (\ref{eq-p-heat-interim}) gives: |
and on substituting into (\ref{eq-p-heat-interim}) gives: |
| 1116 |
\begin{equation} |
\begin{equation} |
| 1189 |
&=&\epsilon _{nh}\mathcal{F}_{w} \\ |
&=&\epsilon _{nh}\mathcal{F}_{w} \\ |
| 1190 |
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}% |
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}% |
| 1191 |
_{h}+\frac{\partial w}{\partial z} &=&0 \\ |
_{h}+\frac{\partial w}{\partial z} &=&0 \\ |
| 1192 |
\rho &=&\rho (\theta ,S,p) \\ |
\rho &=&\rho (\theta ,S,p) \\ |
| 1193 |
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\ |
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\ |
| 1194 |
\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq} |
\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq} |
| 1195 |
\end{eqnarray}% |
\end{eqnarray}% |
| 1271 |
pressure in the EOS by splitting the pressure into a reference function of |
pressure in the EOS by splitting the pressure into a reference function of |
| 1272 |
height and a perturbation: |
height and a perturbation: |
| 1273 |
\begin{equation*} |
\begin{equation*} |
| 1274 |
\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) |
\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) |
| 1275 |
\end{equation*} |
\end{equation*} |
| 1276 |
Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from |
Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from |
| 1277 |
differentiating the EOS, the continuity equation then becomes: |
differentiating the EOS, the continuity equation then becomes: |
| 1278 |
\begin{equation*} |
\begin{equation*} |
| 1279 |
\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{% |
\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{% |
| 1280 |
Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+% |
Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+% |
| 1281 |
\frac{\partial w}{\partial z}=0 |
\frac{\partial w}{\partial z}=0 |
| 1282 |
\end{equation*} |
\end{equation*} |
| 1283 |
If the time- and space-scales of the motions of interest are longer than |
If the time- and space-scales of the motions of interest are longer than |
| 1284 |
those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},% |
those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},% |
| 1401 |
and vertical direction respectively, are given by (see Fig.2) : |
and vertical direction respectively, are given by (see Fig.2) : |
| 1402 |
|
|
| 1403 |
\begin{equation*} |
\begin{equation*} |
| 1404 |
u=r\cos \phi \frac{D\lambda }{Dt} |
u=r\cos \phi \frac{D\lambda }{Dt} |
| 1405 |
\end{equation*} |
\end{equation*} |
| 1406 |
|
|
| 1407 |
\begin{equation*} |
\begin{equation*} |
| 1408 |
v=r\frac{D\phi }{Dt}\qquad |
v=r\frac{D\phi }{Dt}\qquad |
| 1409 |
\end{equation*} |
\end{equation*} |
| 1410 |
$\qquad \qquad \qquad \qquad $ |
$\qquad \qquad \qquad \qquad $ |
| 1411 |
|
|
| 1412 |
\begin{equation*} |
\begin{equation*} |
| 1413 |
\dot{r}=\frac{Dr}{Dt} |
\dot{r}=\frac{Dr}{Dt} |
| 1414 |
\end{equation*} |
\end{equation*} |
| 1415 |
|
|
| 1416 |
Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial |
Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial |
| 1423 |
\begin{equation*} |
\begin{equation*} |
| 1424 |
\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }% |
\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }% |
| 1425 |
,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}% |
,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}% |
| 1426 |
\right) |
\right) |
| 1427 |
\end{equation*} |
\end{equation*} |
| 1428 |
|
|
| 1429 |
\begin{equation*} |
\begin{equation*} |
| 1430 |
\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial |
\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial |
| 1431 |
\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\} |
\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\} |
| 1432 |
+\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} |
| 1433 |
\end{equation*} |
\end{equation*} |
| 1434 |
|
|
| 1435 |
%%%% \end{document} |
%%%% \end{document} |
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