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2% ================================================================
3% Chapter Ñ Lateral Ocean Physics (LDF)
4% ================================================================
5\chapter{Lateral Ocean Physics (LDF)}
9$\ $\newline    % force a new ligne
11The lateral physics on momentum and tracer equations have been given in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} and \S\ref{DYN_ldf}). In this section we
12further discuss the choices that underlie each lateral physics option.
13Choosing one lateral physics means for the user defining, (1) the space and time variations of the eddy coefficients ; (2) the direction along which the lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal surfaces); and (3) the type of operator used (harmonic, or biharmonic operators, and for tracers only, eddy induced advection on tracers). These
14three aspects of the lateral diffusion are set through namelist parameters and CPP keys (see the nam\_traldf and nam\_dynldf below).
15%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
21% ================================================================
22% Lateral Mixing Coefficients
23% ================================================================
24\section{Lateral Mixing Coefficient (key\_ldftra\_c.d and key\_ldfdyn\_c.d)}
28Introducing a space variation in the lateral eddy mixing coefficients changes
29the model core memory requirement, adding up to four three-dimensional
30arrays for geopotential or isopycnal second order operator applied to
31momentum. Six cpp keys control the space variation of eddy
32coefficients: three for momentum and three for tracer. They
33allow to specify a space variation in the three space directions, in the
34horizontal plane, or in the vertical only. The default option is a constant
35value over the whole ocean on momentum and tracers.
37The number of additional arrays that have to be defined and the gridpoint
38position at which they are defined depend on both the space variation chosen
39and the type of operator used. The resulting eddy viscosity and
40diffusivity coefficients can be either single or multiple valued functions.
41Changes in the computer code when switching from one option to another have
42been minimized by introducing the eddy coefficients as statement function
43(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions are replaced by their actual meaning during the preprocessing step (cpp capability). The specification of the space variation of the coefficient is settled in \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files \textit{ldftra\_cNd.h90} and \textit{ldfdyn\_cNd.h90}, with N=1, 2 or 3. The user can change these include files following his desiderata. The way the mixing coefficient are set in the reference version can be briefly described as follows:
45\subsubsection{Constant Mixing Coefficients (default option)}
46When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are defined, a constant value over the whole ocean on momentum and tracers that is specified through
47\np{ahm0} and \np{aht0} namelist parameters.
49\subsubsection{Vertically varying Mixing Coefficients (\key{ldftra\_c1d} and \key{ldfdyn\_c1d})} 
50The 1D option is only available in $z$-coordinate with full step. Indeed in all the other type of vertical coordinate, the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{j}) and therefore, introducing depth-dependant mixing coefficients will requires 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}.  In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of the surface value, and the transition is round z=300~m with a width of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m). This profile is hard coded in \hf{ldftra\_c1d} file, but can be easily modified by users.
52\subsubsection{Horizontally Varying Mixing Coefficients (\key{ldftra\_c2d} and \key{ldfdyn\_c2d})}
54By default the horizontal variation of the eddy coefficient depend on the local mesh size and the type of operator used:
55\begin{equation} \label{Eq_title}
56  A_l = \left\{     
57   \begin{aligned}
58         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
59         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^& \text{for bilaplacian operator } 
60   \end{aligned}    \right.
61\quad \text{comments}
63where $e_{max}$ is the max of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameters. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}. %gm  not only that! stability reasons: with non uniform grid size, it is common to face a blow up of the model due to to large diffusive coefficient compare to the smallest grid size... this is especially true for bilaplacian (to be added in the text!) 
65Other formulations can be introduced by the user for a given configuration. For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian viscous operator uses \np{ahm0}~=~$4.10^4 m^2.s^{-1}$ poleward of 20$^{\circ}$ north and south and decreases to \np{aht0}~=~$2.10^3 m^2.s^{-1}$ at the equator \citep{Madec1996, Delecluse2000}. This specification can be found in \rou{ldf\_dyn\_c2d\_orca} routine defined in \mdl{ldfdyn\_c2d}. Similar specific horizontal variation can be found for Antarctic or Arctic sub-domain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
67\subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})}
69The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with a grid size dependence of the magnitude of the coefficient.
71\subsubsection{Space and time Varying Mixing Coefficients}
73There is no default specification of space and time varying mixing coefficient. The only case available is specific to ORCA2 and ORCA05 global ocean configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. This specification is actually used when a ORCA key plus \key{traldf\_eiv} plus \key{traldf\_c2d} are defined.
75A space variation in the eddy coefficient appeals several remarks:
77(1) the momentum diffusive operator acting along model level surfaces is
78written in terms of curl and divergent components of the horizontal current
79(see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values
80in these two terms, this option is not available.
82(2) with a horizontal varying viscosity, the quadratic integral constraints
83on enstrophy and on the square of the horizontal divergence for operators
84acting along model-surfaces are no more satisfied (\colorbox{yellow}{Appendix C}).
86(3) for isopycnal diffusion on momentum or tracers, an additional purely
87horizontal background diffusion with uniform coefficient can be added by
88setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal eddy
89viscosity or diffusivity coefficient (\textbf{namelist parameters} which default value are $0$). Nevertheless, the technique used to compute the isopycnal slopes allows to get rid of such a background diffusion which introduces spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
91(4) when an eddy induced advection is used (\key{trahdf\_eiv}), $A^{eiv}$ , the eddy induced coefficient has to be defined. Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient (i.e. \textbf{key\_traldf\_cNd}).
93(5) the eddy coefficient associated to a biharmonic operator must be set to a \emph{negative} value.
96% ================================================================
97% Direction of lateral Mixing
98% ================================================================
99\section{Direction of Lateral Mixing (\mdl{ldfslp})}
102%gm%  we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme.
105A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. This occurs when $(a)$ horizontal mixing is required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$ or mixed $s$-$z$-coordinate, and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the quantity to be diffused. For tracer, this leads to the following four slopes : $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while for momentum the slopes are  $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and  $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$.
106%gm% add here afigure of the slope in i-direction
108\subsection{slopes for tracer geopotential mixing in $s$-coordinate}
110In $s$-coordinates, geopotential mixing ($i.e.$ horizontal one) $r_1$ and $r_2$ are the slopes between the geopotential and computational surfaces. Their discrete formulation is found by locally
111vanishing the diffusive fluxes when $T$ is horizontally uniform, i.e. by replacing in \eqref{Eq_tra_ldf_iso} $T$ by $z_T$, the depth of $T$-point, and setting to zero the diffusive fluxes in the three directions. This leads to the following expression for the slopes:
113\begin{equation} \label{Eq_ldfslp_geo}
115 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
116           \;\delta_{i+1/2}[z_T]
117      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_T]
119 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
120           \;\delta_{j+1/2} [z_T]
121      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_T]
123 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_T]}}^{\,i,\,k+1/2}
124      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
125 \\
126 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_T]}}^{\,j,\,k+1/2}
127      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
128 \\
132%gm%  caution I'm not sure the simplification was a good idea!
133These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=T and \np{ln\_traldf\_hor}=T or \np{ln\_dynldf\_hor}=T.
135\subsection{slopes for tracer iso-neutral mixing}
136In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational
137surfaces. Their formulation does not depend on the vertical coordinate used. Their discrete formulation is found using the fact that the diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting to zero diffusive fluxes in the three directions leads to the following definition for the neutral slopes:
139\begin{equation} \label{Eq_ldfslp_iso}
141 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
142                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
144 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
145                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
147 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
148         \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
149             {\delta_{k+1/2}[\rho]}
151 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
152         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
153             {\delta_{k+1/2}[\rho]}
158%gm% rewrite this as the explanation in not very clear !!!
159%In practice, \eqref{Eq_ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \eqref{Eq_ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.
161%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \eqref{Eq_ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).
163%In $z$-coordinate, the derivative of the  \eqref{Eq_ldfslp_iso} numerator is evaluated at a same depth ($T$-level which is also $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.
165As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \eqref{Eq_ldfslp_iso} have to be evaluated at the same local pressure (which, in decibars, is approximated by the depth in meters in the model). Therefore \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is needed depending on the vertical coordinate used:
169\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus the $in situ$ density can be used. it is not the case for the vertical derivatives. $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987} (see \S\ref{TRA_bn2}).
171\item[$z$-coordinate with partial step : ] the technique is identical to the full step case except that at partial step level, the \emph{horizontal} density gradient is evaluated as described in \S\ref{TRA_zpshde}.
173\item[$s$- or hybrid $s$-$z$ coordinate : ] in the current release of \NEMO, there is no specific treatment for iso-neutral mixing in $s$-coordinate. In other word, iso-neutral mixing will only be accurately represented with a linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} will include a pressure dependent part, leading to a wrong evaluation of the neutral slopes.
176Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for the constraint on iso-neutral fluxes. Following \citet{Griffies2004}, instead of specifying directly that there is a zero neutral diffusive flux of locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between the neutral direction diffusive fluxes of potential temperature and salinity:
178\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
181This constraint leads to the following definition for the slopes:
183\begin{equation} \label{Eq_ldfslp_iso2}
185 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
186      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
187      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
188       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
190 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
191      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
192      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
193       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
195 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
196      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
197       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
198      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
200 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
201      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
202       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
203      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
207where $\alpha$ and $\beta$, the thermal expansion and saline contracion coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three velocity point. Inorder to save computation time, they should be approximated by the mean of their values at $T$-points (for example in the case of $\alpha$$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
209Note that such a formulation could be also used in $z$ and $zps$ cases.
214This implementation is a rather old one. It is similar to the one proposed by Cox [1987], except for
215the background horizontal diffusion. Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires a minimum background horizontal diffusion for numerical stability reasons. To overcome this problem, several techniques have been proposed in which the numerical
216schemes of the OGCM are modified \citep{Weaver1997, Griffies1998}.
217Here, another strategy has been chosen \citep{Lazar1997}: a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of grid point noise generated by the iso-neutral diffusive operator (Fig.~\ref{Fig_LDF_ZDF1}).
218This allows an iso-neutral diffusion scheme without additional background horizontal mixing. This technique can be viewed as a diffusive operator that acts along large-scale (2~$\Delta$x) iso-neutral surfaces. The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of a horizontal background mixing.
220Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, contrary to the \citet{Griffies1998} operator which has that property.
223\begin{figure}[!ht] \label{Fig_LDF_ZDF1}  \begin{center}
225\caption {averaging procedure for isopycnal slope computation.}
226\end{center}   \end{figure}
229%There is three additional questions about the slope calculation. First the expression of the rotation tensor used have been obtain assuming the "small slope" approximation, so a bound has to be specified on slopes. Second, numerical stability issues also require a bound on slopes. Third, the question of boundary condition spefified on slopes...
231%from griffies: chapter 13.1....
235In addition and also for numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly to zero fom $70$ meters depth
236and the surface (the fact that the eddies "feel" the surface motivates this
237flattening of isopycnals near the surface).
240\begin{figure}[!ht] \label{Fig_eiv_slp}  \begin{center}
242\caption {Vertical profile of the slope used for lateral mixing in the mixed layer : \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior and their have to adjust to the surface boundary (i.e. tend to zero at the surface as there is no mixing across the air-sea interface: wall boundary condition). Nevertheless, the profile between surface zero value and interior iso-neutral one is unknown, and especially the value at the based of the mixed layer ; \textit{(b)} profile of slope using a linear tapering of the slope near the surface and imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actuelly used in \NEMO: linear decrease of the slope from zero at the surface to its ocean interior value computed just below the mixed layer. Note the huge change in the slope at the based of the mixed layer between  \textit{(b)}  and \textit{(c)}.
244\end{center}   \end{figure}
247\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
249\subsection{slopes for momentum iso-neutral mixing}
251The diffusive iso-neutral operator on momentum is the same as the on used on tracer but applied to each component of the velocity (see \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the surface along which the diffusive operator acts and the surface of computation ($z$- or $s$-surfaces) are defined at $T$-, $f-$, and \textit{uw-}points for the $u$-component, and $f-T$-, \textit{vw}-points for the $v$-component. They are computed as follows from the slopes used for tracer diffusion, i.e. \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
253\begin{equation} \label{Eq_ldfslp_dyn}
255&r_{1T}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
256&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2T}\ &= \overline{r_{2v}}^{\,j} \\
257&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
258&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
262The major issue remains in the specification of the boundary conditions. The
263choice made consists in keeping the same boundary conditions as for lateral
264diffusion along model level surfaces, i.e. using the shear computed along
265the model levels and with no additional friction at the ocean bottom (see
269% ================================================================
270% Eddy Induced Mixing
271% ================================================================
272\section{Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
276When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), an eddy induced tracer advection term is added, the formulation of which depends on the slopes of iso-neutral surfaces. Contrary to iso-neutral mixing, the slopes use here are referenced to the geopotential surfaces, i.e. \eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}  + \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
277\begin{equation} \label{Eq_ldfeiv}
279 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
280v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
281w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
284where $A^{eiv}$ is the eddy induced velocity coefficient set through \np{aeiv}, a \textit{nam\_traldf} namelist parameter.
285The three components of the eddy induced velocity are computed and add to the eulerian velocity in the mdl{traadv\_eiv}. This has been preferred to a separate computation of the advective trends associated to the eiv velocity as it allows to take advantage of all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not only the $2^{nd}$ order advection scheme as in previous release of OPA \citep{Madec1998}. This is particularly useful for passive tracers where \emph{positivity}of the advection scheme is of paramount importance.
287At surface, lateral and bottom boundaries, the eddy induced velocity and thus the
288advective eddy fluxes of heat and salt are set to zero.
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