Changeset 6303

Timestamp:

2016-02-10T17:18:53+01:00 (8 years ago)

Author:

acc

Message:

Branch nemo_v3_6_STABLE (Documentation). Update Chap_LDF.tex to include Smagorinsky options. Also add a couple of references to Biblio.bib

Location:

branches/2015/nemo_v3_6_STABLE/DOC/TexFiles

Files:

• Biblio/Biblio.bib (modified) (2 diffs)
• Chapters/Chap_LDF.tex (modified) (6 diffs)

branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Biblio/Biblio.bib

@@ -1263 +1263 @@
 }
 
+@ARTICLE{Griffies_Hallberg_MWR00,
+  author = {S.M. Griffies and R. Hallberg},
+  title = {Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale eddy-permitting ocean models},
+  journal = MWR,
+  year = {2000},
+  volume = {128(8)},
+  pages = {2935--2946}
+}
+
 @ARTICLE{Guilyardi_al_JC04,
   author = {E. Guilyardi and S. Gualdi and J. M. Slingo and A. Navarra and P. Delecluse 
@@ -2766 +2775 @@
   year = {2004},
   pages = {245--263},
+}
+
+@INBOOK{Smagorinsky_93,
+  author = {Smagorinsky, J.},
+  chapter = {Some historical remarks on the use of non-linear viscosities},
+  title = {Large Eddy Simulation of Complex Engineering and Geophysical Flows},
+  pages = {3--36},
+  year = {1993},
+  publisher = {Cambridge University Press, B. Galperin and S. A. Orszag (eds.)},
 }
 

branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Chapters/Chap_LDF.tex

@@ -68 +68 @@
 When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are 
 defined, a constant value is used over the whole ocean for momentum and 
-tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist 
+tracers, which is specified through the \np{rn\_ahm\_0\_lap} and \np{rn\_aht\_0} namelist 
 parameters.
 
@@ -77 +77 @@
 mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation 
 of the lateral mixing coefficient is introduced in which the surface value is 
-\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 
+\np{rn\_aht\_0} (\np{rn\_ahm\_0\_lap}), the bottom value is 1/4 of the surface value, 
 and the transition takes place around 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). 
@@ -93 +93 @@
 \end{equation}
 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked 
-ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) 
+ocean domain, and $A_o^l$ is the \np{rn\_ahm\_0\_lap} (momentum) or \np{rn\_aht\_0} (tracer) 
 namelist parameter. 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 
@@ -105 +105 @@
 Other formulations can be introduced by the user for a given configuration. 
 For example, in the ORCA2 global ocean model (see Configurations), the laplacian 
-viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 
-north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s 
+viscosity operator uses \np{rn\_ahm\_0\_lap}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 
+north and south and decreases linearly to \np{rn\_aht\_0}~= 2.10$^3$ m$^2$/s 
 at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification 
 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 
@@ -120 +120 @@
 \subsubsection{Space and Time Varying Mixing Coefficients}
 
-There is no default specification of space and time varying mixing coefficient. 
-The only case available is specific to the ORCA2 and ORCA05 global ocean 
-configurations. 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 an ORCA key 
+There are no default specifications of space and time varying mixing coefficient.  One
+available case is specific to the ORCA2 and ORCA05 global ocean configurations. 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 an ORCA key
 and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
+
+\subsubsection{Smagorinsky viscosity (\key{dynldf\_c3d} and \key{dynldf\_smag})}
+
+The \key{dynldf\_smag} key activates a 3D, time-varying viscosity that depends on the
+resolved motions. Following \citep{Smagorinsky_93} the viscosity coefficient is set
+proportional to a local deformation rate based on the horizontal shear and tension,
+namely:
+
+\begin{equation}
+A_{m_{Smag}} = \left(\frac{{\sf CM_{Smag}}}{\pi}\right)^2L^2\vert{D}\vert
+\end{equation}
+
+\noindent where the deformation rate $\vert{D}\vert$ is given by 
+
+\begin{equation}
+\vert{D}\vert=\sqrt{\left({\frac{\partial{u}} {\partial{x}}}
+                         -{\frac{\partial{v}} {\partial{y}}}\right)^2
+                 +  \left({\frac{\partial{u}} {\partial{y}}}
+                         +{\frac{\partial{v}} {\partial{x}}}\right)^2} 
+\end{equation}
+
+\noindent and $L$ is the local gridscale given by:
+
+\begin{equation}
+L^2 = \frac{2{e_1}^2 {e_2}^2}{\left ( {e_1}^2 + {e_2}^2 \right )}
+\end{equation}
+
+\citep{Griffies_Hallberg_MWR00} suggest values in the range 2.2 to 4.0 of the coefficient
+$\sf CM_{Smag}$ for oceanic flows. This value is set via the \np{rn\_cmsmag\_1} namelist
+parameter. An additional parameter: \np{rn\_cmsh} is included in NEMO for experimenting
+with the contribution of the shear term. A value of 1.0 (the default) calculates the
+deformation rate as above; a value of 0.0 will discard the shear term entirely.
+
+For numerical stability, the calculated viscosity is bounded according to the following:
+
+\begin{equation}
+{\rm MIN}\left ({ L^2\over {8\Delta{t}}}, rn\_ahm\_m\_lap\right ) \geq A_{m_{Smag}} 
+                                                                  \geq rn\_ahm\_0\_lap
+\end{equation}
+
+\noindent with both parameters for the upper and lower bounds being provided via the
+indicated namelist parameters.
+
+\bigskip When $ln\_dynldf\_bilap = .true.$, a biharmonic version of the Smagorinsky
+viscosity is also available which sets a coefficient for the biharmonic viscosity as:
+
+\begin{equation}
+B_{m_{Smag}} = - \left(\frac{{\sf CM_{bSmag}}}{\pi}\right)^2 {L^4\over 8}\vert{D}\vert
+\end{equation}
+
+\noindent which is bounded according to:
+
+\begin{equation}
+{\rm MAX}\left (-{ L^4\over {64\Delta{t}}}, rn\_ahm\_m\_blp\right ) \leq B_{m_{Smag}} 
+                                                                    \leq rn\_ahm\_0\_blp
+\end{equation}
+
+\noindent Note the reversal of the inequalities here because NEMO requires the biharmonic
+coefficients as negative numbers. $\sf CM_{bSmag}$ is set via the \np{rn\_cmsmag\_2}
+namelist parameter and the bounding values have corresponding entries in the namelist too.
+
+\bigskip The current implementation in NEMO also allows for 3D, time-varying diffusivities
+to be set using the Smagorinsky approach. Users should note that this option is not
+recommended for many applications since diffusivities will tend to be largest near
+boundaries (where shears are greatest) leading to spurious upwellings
+(\citep{Griffies_Bk04}, chapter 18.3.4). Nevertheless the option is there for those
+wishing to experiment. This choice requires both \key{traldf\_c3d} and \key{traldf\_smag}
+and uses the \np{rn\_chsmag} (${\sf CH_{Smag}}$), \np{rn\_smsh} and \np{rn\_aht\_m}
+namelist parameters in an analogous way to \np{rn\_cmsmag\_1}, \np{rn\_cmsh} and
+\np{rn\_ahm\_m\_lap} (see above) to set the diffusion coefficient:
+
+\begin{equation}
+A_{h_{Smag}} = \left(\frac{{\sf CH_{Smag}}}{\pi}\right)^2L^2\vert{D}\vert
+\end{equation}
+
+ 
+For numerical stability, the calculated diffusivity is bounded according to the following:
+
+\begin{equation}
+{\rm MIN}\left ({ L^2\over {8\Delta{t}}}, rn\_aht\_m\right ) \geq A_{h_{Smag}} 
+                                                             \geq rn\_aht\_0
+\end{equation}
+
+
 
 $\ $\newline    % force a new ligne
@@ -144 +227 @@
 (3) for isopycnal diffusion on momentum or tracers, an additional purely 
 horizontal background diffusion with uniform coefficient can be added by 
-setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal 
+setting a non zero value of \np{rn\_ahmb\_0} or \np{rn\_ahtb\_0}, a background horizontal 
 eddy viscosity or diffusivity coefficient (namelist parameters whose default 
 values are $0$). However, the technique used to compute the isopycnal