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\section{Gent/McWiliams/Redi SGS Eddy parameterization} |
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\begin{rawhtml} |
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<!-- CMIREDIR:gmredi: --> |
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\end{rawhtml} |
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|
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There are two parts to the Redi/GM parameterization of geostrophic |
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eddies. The first aims to mix tracer properties along isentropes |
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(neutral surfaces) by means of a diffusion operator oriented along the |
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local isentropic surface (Redi). The second part, adiabatically |
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re-arranges tracers through an advective flux where the advecting flow |
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is a function of slope of the isentropic surfaces (GM). |
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|
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The first GCM implementation of the Redi scheme was by Cox 1987 in the |
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GFDL ocean circulation model. The original approach failed to |
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distinguish between isopycnals and surfaces of locally referenced |
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potential density (now called neutral surfaces) which are proper |
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isentropes for the ocean. As will be discussed later, it also appears |
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that the Cox implementation is susceptible to a computational mode. |
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Due to this mode, the Cox scheme requires a background lateral |
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diffusion to be present to conserve the integrity of the model fields. |
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|
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The GM parameterization was then added to the GFDL code in the form of |
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a non-divergent bolus velocity. The method defines two |
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stream-functions expressed in terms of the isoneutral slopes subject |
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to the boundary condition of zero value on upper and lower |
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boundaries. The horizontal bolus velocities are then the vertical |
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derivative of these functions. Here in lies a problem highlighted by |
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Griffies et al., 1997: the bolus velocities involve multiple |
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derivatives on the potential density field, which can consequently |
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give rise to noise. Griffies et al. point out that the GM bolus fluxes |
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can be identically written as a skew flux which involves fewer |
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differential operators. Further, combining the skew flux formulation |
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and Redi scheme, substantial cancellations take place to the point |
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that the horizontal fluxes are unmodified from the lateral diffusion |
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parameterization. |
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|
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\subsection{Redi scheme: Isopycnal diffusion} |
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|
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The Redi scheme diffuses tracers along isopycnals and introduces a |
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term in the tendency (rhs) of such a tracer (here $\tau$) of the form: |
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\begin{equation} |
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\bf{\nabla} \cdot \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau |
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\end{equation} |
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where $\kappa_\rho$ is the along isopycnal diffusivity and |
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$\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of |
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$\tau$ onto the isopycnal surface. The unapproximated projection tensor is: |
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\begin{equation} |
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\bf{K}_{Redi} = \left( |
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\begin{array}{ccc} |
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1 + S_x& S_x S_y & S_x \\ |
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S_x S_y & 1 + S_y & S_y \\ |
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S_x & S_y & |S|^2 \\ |
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\end{array} |
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\right) |
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\end{equation} |
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Here, $S_x = -\partial_x \sigma / \partial_z \sigma$ and $S_y = |
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-\partial_y \sigma / \partial_z \sigma$ are the components of the |
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isoneutral slope. |
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|
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The first point to note is that a typical slope in the ocean interior |
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is small, say of the order $10^{-4}$. A maximum slope might be of |
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order $10^{-2}$ and only exceeds such in unstratified regions where |
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the slope is ill defined. It is therefore justifiable, and |
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customary, to make the small slope approximation, $|S| << 1$. The Redi |
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projection tensor then becomes: |
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\begin{equation} |
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\bf{K}_{Redi} = \left( |
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\begin{array}{ccc} |
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1 & 0 & S_x \\ |
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0 & 1 & S_y \\ |
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S_x & S_y & |S|^2 \\ |
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\end{array} |
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\right) |
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\end{equation} |
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|
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|
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\subsection{GM parameterization} |
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|
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The GM parameterization aims to parameterise the ``advective'' or |
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``transport'' effect of geostrophic eddies by means of a ``bolus'' |
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velocity, $\bf{u}^*$. The divergence of this advective flux is added |
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to the tracer tendency equation (on the rhs): |
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\begin{equation} |
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- \bf{\nabla} \cdot \tau \bf{u}^* |
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\end{equation} |
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|
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The bolus velocity is defined as: |
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\begin{eqnarray} |
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u^* & = & - \partial_z F_x \\ |
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v^* & = & - \partial_z F_y \\ |
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w^* & = & \partial_x F_x + \partial_y F_y |
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\end{eqnarray} |
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where $F_x$ and $F_y$ are stream-functions with boundary conditions |
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$F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the |
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bolus velocity in terms of these stream-functions is that they are |
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automatically non-divergent ($\partial_x u^* + \partial_y v^* = - |
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\partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and |
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$F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$: |
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\begin{eqnarray} |
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F_x & = & \kappa_{GM} S_x \\ |
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F_y & = & \kappa_{GM} S_y |
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\end{eqnarray} |
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This is the form of the GM parameterization as applied by Donabasaglu, |
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1997, in MOM versions 1 and 2. |
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|
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\subsection{Griffies Skew Flux} |
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|
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Griffies notes that the discretisation of bolus velocities involves |
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multiple layers of differencing and interpolation that potentially |
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lead to noisy fields and computational modes. He pointed out that the |
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bolus flux can be re-written in terms of a non-divergent flux and a |
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skew-flux: |
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\begin{eqnarray*} |
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\bf{u}^* \tau |
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& = & |
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\left( \begin{array}{c} |
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- \partial_z ( \kappa_{GM} S_x ) \tau \\ |
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- \partial_z ( \kappa_{GM} S_y ) \tau \\ |
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(\partial_x \kappa_{GM} S_x + \partial_y \kappa_{GM} S_y)\tau |
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\end{array} \right) |
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\\ |
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& = & |
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\left( \begin{array}{c} |
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- \partial_z ( \kappa_{GM} S_x \tau) \\ |
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- \partial_z ( \kappa_{GM} S_y \tau) \\ |
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\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y) \tau) |
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\end{array} \right) |
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+ \left( \begin{array}{c} |
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\kappa_{GM} S_x \partial_z \tau \\ |
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\kappa_{GM} S_y \partial_z \tau \\ |
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- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y) \partial_y \tau |
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\end{array} \right) |
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\end{eqnarray*} |
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The first vector is non-divergent and thus has no effect on the tracer |
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field and can be dropped. The remaining flux can be written: |
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\begin{equation} |
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\bf{u}^* \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau |
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\end{equation} |
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where |
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\begin{equation} |
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\bf{K}_{GM} = |
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\left( |
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\begin{array}{ccc} |
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0 & 0 & -S_x \\ |
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0 & 0 & -S_y \\ |
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S_x & S_y & 0 |
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\end{array} |
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\right) |
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\end{equation} |
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is an anti-symmetric tensor. |
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|
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This formulation of the GM parameterization involves fewer derivatives |
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than the original and also involves only terms that already appear in |
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the Redi mixing scheme. Indeed, a somewhat fortunate cancellation |
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becomes apparent when we use the GM parameterization in conjunction |
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with the Redi isoneutral mixing scheme: |
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\begin{equation} |
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\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau |
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- u^* \tau = |
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( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau |
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\end{equation} |
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In the instance that $\kappa_{GM} = \kappa_{\rho}$ then |
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\begin{equation} |
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\kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} = |
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\kappa_\rho |
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\left( \begin{array}{ccc} |
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1 & 0 & 0 \\ |
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0 & 1 & 0 \\ |
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2 S_x & 2 S_y & |S|^2 |
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\end{array} |
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\right) |
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\end{equation} |
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which differs from the variable Laplacian diffusion tensor by only |
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two non-zero elements in the $z$-row. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F}) |
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|
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$\sigma_x$: {\bf SlopeX} (argument on entry) |
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|
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$\sigma_y$: {\bf SlopeY} (argument on entry) |
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|
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$\sigma_z$: {\bf SlopeY} (argument) |
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|
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$S_x$: {\bf SlopeX} (argument on exit) |
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|
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$S_y$: {\bf SlopeY} (argument on exit) |
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|
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\end{minipage} } |
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|
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|
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|
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\subsection{Variable $\kappa_{GM}$} |
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|
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Visbeck et al., 1996, suggest making the eddy coefficient, |
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$\kappa_{GM}$, a function of the Eady growth rate, |
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$|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant, |
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$\alpha$, and a length-scale $L$: |
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\begin{displaymath} |
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\kappa_{GM} = \alpha L^2 \overline{ \frac{|f|}{\sqrt{Ri}} }^z |
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\end{displaymath} |
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where the Eady growth rate has been depth averaged (indicated by the |
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over-line). A local Richardson number is defined $Ri = N^2 / (\partial |
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u/\partial z)^2$ which, when combined with thermal wind gives: |
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\begin{displaymath} |
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\frac{1}{Ri} = \frac{(\frac{\partial u}{\partial z})^2}{N^2} = |
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\frac{ ( \frac{g}{f \rho_o} | {\bf \nabla} \sigma | )^2 }{N^2} = |
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\frac{ M^4 }{ |f|^2 N^2 } |
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\end{displaymath} |
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where $M^2$ is defined $M^2 = \frac{g}{\rho_o} |{\bf \nabla} \sigma|$. |
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Substituting into the formula for $\kappa_{GM}$ gives: |
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\begin{displaymath} |
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\kappa_{GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z = |
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\alpha L^2 \overline{ \frac{M^2}{N^2} N }^z = |
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\alpha L^2 \overline{ |S| N }^z |
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\end{displaymath} |
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|
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|
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\subsection{Tapering and stability} |
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|
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Experience with the GFDL model showed that the GM scheme has to be |
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matched to the convective parameterization. This was originally |
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expressed in connection with the introduction of the KPP boundary |
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layer scheme (Large et al., 97) but in fact, as subsequent experience |
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with the MIT model has found, is necessary for any convective |
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parameterization. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GMREDI\_SLOPE\_LIMIT} ({\em |
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pkg/gmredi/gmredi\_slope\_limit.F}) |
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|
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$\sigma_x, s_x$: {\bf SlopeX} (argument) |
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|
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$\sigma_y, s_y$: {\bf SlopeY} (argument) |
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|
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$\sigma_z$: {\bf dSigmadRReal} (argument) |
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|
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$z_\sigma^{*}$: {\bf dRdSigmaLtd} (argument) |
| 239 |
|
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\end{minipage} } |
| 241 |
|
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\begin{figure} |
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\begin{center} |
| 244 |
\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
| 245 |
\end{center} |
| 246 |
\caption{Taper functions used in GKW99 and DM95.} |
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\label{fig:tapers} |
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\end{figure} |
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|
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\begin{figure} |
| 251 |
\begin{center} |
| 252 |
\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}} |
| 253 |
\end{center} |
| 254 |
\caption{Effective slope as a function of ``true'' slope using Cox |
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slope clipping, GKW91 limiting and DM95 limiting.} |
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\label{fig:effective_slopes} |
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\end{figure} |
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|
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|
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\subsubsection{Slope clipping} |
| 261 |
|
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Deep convection sites and the mixed layer are indicated by |
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homogenized, unstable or nearly unstable stratification. The slopes in |
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such regions can be either infinite, very large with a sign reversal |
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or simply very large. From a numerical point of view, large slopes |
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lead to large variations in the tensor elements (implying large bolus |
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flow) and can be numerically unstable. This was first recognized by |
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Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing |
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tensor. Here, the slope magnitude is simply restricted by an upper |
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limit: |
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\begin{eqnarray} |
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|\nabla \sigma| & = & \sqrt{ \sigma_x^2 + \sigma_y^2 } \\ |
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S_{lim} & = & - \frac{|\nabla \sigma|}{ S_{max} } |
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\;\;\;\;\;\;\;\; \mbox{where $S_{max}$ is a parameter} \\ |
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\sigma_z^\star & = & \min( \sigma_z , S_{lim} ) \\ |
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{[s_x,s_y]} & = & - \frac{ [\sigma_x,\sigma_y] }{\sigma_z^\star} |
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\end{eqnarray} |
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Notice that this algorithm assumes stable stratification through the |
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``min'' function. In the case where the fluid is well stratified ($\sigma_z < S_{lim}$) then the slopes evaluate to: |
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\begin{equation} |
| 281 |
{[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z} |
| 282 |
\end{equation} |
| 283 |
while in the limited regions ($\sigma_z > S_{lim}$) the slopes become: |
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\begin{equation} |
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{[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{|\nabla \sigma|/S_{max}} |
| 286 |
\end{equation} |
| 287 |
so that the slope magnitude is limited $\sqrt{s_x^2 + s_y^2} = |
| 288 |
S_{max}$. |
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|
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The slope clipping scheme is activated in the model by setting {\bf |
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GM\_tap\-er\_scheme = 'clipping'} in {\em data.gmredi}. |
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|
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Even using slope clipping, it is normally the case that the vertical |
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diffusion term (with coefficient $\kappa_\rho{\bf K}_{33} = |
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\kappa_\rho S_{max}^2$) is large and must be time-stepped using an |
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implicit procedure (see section on discretisation and code later). |
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Fig. \ref{fig-mixedlayer} shows the mixed layer depth resulting from |
| 298 |
a) using the GM scheme with clipping and b) no GM scheme (horizontal |
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diffusion). The classic result of dramatically reduced mixed layers is |
| 300 |
evident. Indeed, the deep convection sites to just one or two points |
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each and are much shallower than we might prefer. This, it turns out, |
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is due to the over zealous re-stratification due to the bolus transport |
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parameterization. Limiting the slopes also breaks the adiabatic nature |
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of the GM/Redi parameterization, re-introducing diabatic fluxes in |
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regions where the limiting is in effect. |
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|
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\subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991} |
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|
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The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99}) |
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addressed two issues with the clipping method: the introduction of |
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large vertical fluxes in addition to convective adjustment fluxes is |
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avoided by tapering the GM/Redi slopes back to zero in |
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low-stratification regions; the adjustment of slopes is replaced by a |
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tapering of the entire GM/Redi tensor. This means the direction of |
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fluxes is unaffected as the amplitude is scaled. |
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|
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The scheme inserts a tapering function, $f_1(S)$, in front of the |
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GM/Redi tensor: |
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\begin{equation} |
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f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{|S|}\right)^2 \right] |
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\end{equation} |
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where $S_{max}$ is the maximum slope you want allowed. Where the |
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slopes, $|S|<S_{max}$ then $f_1(S) = 1$ and the tensor is un-tapered |
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but where $|S| \ge S_{max}$ then $f_1(S)$ scales down the tensor so |
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that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 = |
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\kappa S_{max}^2$. |
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|
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The GKW tapering scheme is activated in the model by setting {\bf |
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GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
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|
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\subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
| 332 |
|
| 333 |
The tapering scheme used by Danabasoglu and McWilliams, 1995, |
| 334 |
\cite{dm:95}, followed a similar procedure but used a different |
| 335 |
tapering function, $f_1(S)$: |
| 336 |
\begin{equation} |
| 337 |
f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right) |
| 338 |
\end{equation} |
| 339 |
where $S_c = 0.004$ is a cut-off slope and $S_d=0.001$ is a scale over |
| 340 |
which the slopes are smoothly tapered. Functionally, the operates in |
| 341 |
the same way as the GKW91 scheme but has a substantially lower |
| 342 |
cut-off, turning off the GM/Redi SGS parameterization for weaker |
| 343 |
slopes. |
| 344 |
|
| 345 |
The DM tapering scheme is activated in the model by setting {\bf |
| 346 |
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
| 347 |
|
| 348 |
\subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
| 349 |
|
| 350 |
The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the |
| 351 |
DM95 tapering scheme, but also tapers the scheme with an additional |
| 352 |
function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are |
| 353 |
reduced near the surface: |
| 354 |
\begin{equation} |
| 355 |
f_2(S) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \pi/2)\right) |
| 356 |
\end{equation} |
| 357 |
where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with |
| 358 |
$c=2$~m~s$^{-1}$. This tapering with height was introduced to fix |
| 359 |
some spurious interaction with the mixed-layer KPP parameterization. |
| 360 |
|
| 361 |
The LDD tapering scheme is activated in the model by setting {\bf |
| 362 |
GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}. |
| 363 |
|
| 364 |
|
| 365 |
|
| 366 |
|
| 367 |
\begin{figure} |
| 368 |
\begin{center} |
| 369 |
%\includegraphics{mixedlayer-cox.eps} |
| 370 |
%\includegraphics{mixedlayer-diff.eps} |
| 371 |
Figure missing. |
| 372 |
\end{center} |
| 373 |
\caption{Mixed layer depth using GM parameterization with a) Cox slope |
| 374 |
clipping and for comparison b) using horizontal constant diffusion.} |
| 375 |
\label{fig-mixedlayer} |
| 376 |
\end{figure} |
| 377 |
|
| 378 |
\subsection{Package Reference} |
| 379 |
% DO NOT EDIT HERE |
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|
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|
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