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\subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization} |
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\label{sec:pkg:gmredi} |
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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, the Redi scheme \citep{re82}, 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. The second part, GM \citep{gen-mcw:90,gen-eta:95}, 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. |
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|
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The first GCM implementation of the Redi scheme was by \cite{Cox87} 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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\cite{gretal:98}: 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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\subsubsection{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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\subsubsection{GM parameterization} |
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|
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The GM parameterization aims to represent the ``advective'' or |
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``transport'' effect of geostrophic eddies by means of a ``bolus'' |
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velocity, $\bf{u}^\star$. 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}^\star |
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\end{equation} |
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|
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The bolus velocity $\bf{u}^\star$ is defined as the rotational of a |
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streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$: |
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\begin{equation} |
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\bf{u}^\star = \nabla \times \bf{F}^\star = |
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\left( \begin{array}{c} |
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- \partial_z F_y^\star \\ |
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+ \partial_z F_x^\star \\ |
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\partial_x F_y^\star - \partial_y F_x^\star |
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\end{array} \right), |
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\end{equation} |
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and thus is automatically non-divergent. In the GM parameterization, the streamfunction is |
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specified in terms of the isoneutral slopes $S_x$ and $S_y$: |
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\begin{eqnarray} |
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F_x^\star & = & -\kappa_{GM} S_y \\ |
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F_y^\star & = & \kappa_{GM} S_x |
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\end{eqnarray} |
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with boundary conditions $F_x^\star=F_y^\star=0$ on upper and lower boundaries. |
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In the end, the bolus transport in the GM parameterization is given by: |
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\begin{equation} |
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\bf{u}^\star = \left( |
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\begin{array}{c} |
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u^\star \\ |
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v^\star \\ |
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w^\star |
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\end{array} |
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\right) = \left( |
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\begin{array}{c} |
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- \partial_z (\kappa_{GM} S_x) \\ |
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- \partial_z (\kappa_{GM} S_y) \\ |
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\partial_x (\kappa_{GM} S_x) + \partial_y (\kappa_{GM} S_y) |
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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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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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Note that in the MITgcm, the variables containing the GM bolus streamfunction are: |
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\begin{equation} |
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\left( |
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\begin{array}{c} |
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GM\_PsiX \\ |
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GM\_PsiY |
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\end{array} |
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\right) = \left( |
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\begin{array}{c} |
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\kappa_{GM} S_x \\ |
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\kappa_{GM} S_y |
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\end{array} |
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\right)= \left( |
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\begin{array}{c} |
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F_y^\star \\ |
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-F_x^\star |
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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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\subsubsection{Griffies Skew Flux} |
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|
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\cite{gr:98} 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}^\star \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}^\star \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^\star \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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\subsubsection{Variable $\kappa_{GM}$} |
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|
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\cite{visbeck:97} 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 |
| 242 |
over-line). A local Richardson number is defined $Ri = N^2 / (\partial |
| 243 |
u/\partial z)^2$ 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} |
| 249 |
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 |
| 255 |
\end{displaymath} |
| 256 |
|
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|
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\subsubsection{Tapering and stability} |
| 259 |
|
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Experience with the GFDL model showed that the GM scheme has to be |
| 261 |
matched to the convective parameterization. This was originally |
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expressed in connection with the introduction of the KPP boundary |
| 263 |
layer scheme \citep{lar-eta:94} but in fact, as subsequent experience |
| 264 |
with the MIT model has found, is necessary for any convective |
| 265 |
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) |
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|
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\end{minipage} } |
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|
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\begin{figure} |
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\begin{center} |
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\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}} |
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\end{center} |
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\caption{Taper functions used in GKW91 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} |
| 290 |
\begin{center} |
| 291 |
\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/effective_slopes.eps}} |
| 292 |
\end{center} |
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\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} |
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|
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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 |
| 306 |
flow) and can be numerically unstable. This was first recognized by |
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\cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing |
| 308 |
tensor. Here, the slope magnitude is simply restricted by an upper |
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limit: |
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\begin{eqnarray} |
| 311 |
|\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} |
| 316 |
\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: |
| 319 |
\begin{equation} |
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{[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z} |
| 321 |
\end{equation} |
| 322 |
while in the limited regions ($\sigma_z > S_{lim}$) the slopes become: |
| 323 |
\begin{equation} |
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{[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{|\nabla \sigma|/S_{max}} |
| 325 |
\end{equation} |
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so that the slope magnitude is limited $\sqrt{s_x^2 + s_y^2} = |
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S_{max}$. |
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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 |
| 337 |
a) using the GM scheme with clipping and b) no GM scheme (horizontal |
| 338 |
diffusion). The classic result of dramatically reduced mixed layers is |
| 339 |
evident. Indeed, the deep convection sites to just one or two points |
| 340 |
each and are much shallower than we might prefer. This, it turns out, |
| 341 |
is due to the over zealous re-stratification due to the bolus transport |
| 342 |
parameterization. Limiting the slopes also breaks the adiabatic nature |
| 343 |
of the GM/Redi parameterization, re-introducing diabatic fluxes in |
| 344 |
regions where the limiting is in effect. |
| 345 |
|
| 346 |
\subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991} |
| 347 |
|
| 348 |
The tapering scheme used in \cite{gkw:91} |
| 349 |
addressed two issues with the clipping method: the introduction of |
| 350 |
large vertical fluxes in addition to convective adjustment fluxes is |
| 351 |
avoided by tapering the GM/Redi slopes back to zero in |
| 352 |
low-stratification regions; the adjustment of slopes is replaced by a |
| 353 |
tapering of the entire GM/Redi tensor. This means the direction of |
| 354 |
fluxes is unaffected as the amplitude is scaled. |
| 355 |
|
| 356 |
The scheme inserts a tapering function, $f_1(S)$, in front of the |
| 357 |
GM/Redi tensor: |
| 358 |
\begin{equation} |
| 359 |
f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{|S|}\right)^2 \right] |
| 360 |
\end{equation} |
| 361 |
where $S_{max}$ is the maximum slope you want allowed. Where the |
| 362 |
slopes, $|S|<S_{max}$ then $f_1(S) = 1$ and the tensor is un-tapered |
| 363 |
but where $|S| \ge S_{max}$ then $f_1(S)$ scales down the tensor so |
| 364 |
that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 = |
| 365 |
\kappa S_{max}^2$. |
| 366 |
|
| 367 |
The GKW91 tapering scheme is activated in the model by setting {\bf |
| 368 |
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
| 369 |
|
| 370 |
\subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
| 371 |
|
| 372 |
The tapering scheme used by \cite{dm:95} followed a similar procedure but used a different |
| 373 |
tapering function, $f_1(S)$: |
| 374 |
\begin{equation} |
| 375 |
f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right) |
| 376 |
\end{equation} |
| 377 |
where $S_c = 0.004$ is a cut-off slope and $S_d=0.001$ is a scale over |
| 378 |
which the slopes are smoothly tapered. Functionally, the operates in |
| 379 |
the same way as the GKW91 scheme but has a substantially lower |
| 380 |
cut-off, turning off the GM/Redi SGS parameterization for weaker |
| 381 |
slopes. |
| 382 |
|
| 383 |
The DM95 tapering scheme is activated in the model by setting {\bf |
| 384 |
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
| 385 |
|
| 386 |
\subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
| 387 |
|
| 388 |
The tapering used in \cite{ldd:97} is based on the |
| 389 |
DM95 tapering scheme, but also tapers the scheme with an additional |
| 390 |
function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are |
| 391 |
reduced near the surface: |
| 392 |
\begin{equation} |
| 393 |
f_2(z) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \frac{\pi}{2})\right) |
| 394 |
\end{equation} |
| 395 |
where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with |
| 396 |
$c=2$~m~s$^{-1}$. This tapering with height was introduced to fix |
| 397 |
some spurious interaction with the mixed-layer KPP parameterization. |
| 398 |
|
| 399 |
The LDD97 tapering scheme is activated in the model by setting {\bf |
| 400 |
GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}. |
| 401 |
|
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|
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|
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|
| 405 |
\begin{figure} |
| 406 |
\begin{center} |
| 407 |
%\includegraphics{mixedlayer-cox.eps} |
| 408 |
%\includegraphics{mixedlayer-diff.eps} |
| 409 |
Figure missing. |
| 410 |
\end{center} |
| 411 |
\caption{Mixed layer depth using GM parameterization with a) Cox slope |
| 412 |
clipping and for comparison b) using horizontal constant diffusion.} |
| 413 |
\label{fig-mixedlayer} |
| 414 |
\end{figure} |
| 415 |
|
| 416 |
\subsubsection{Package Reference} |
| 417 |
\label{sec:pkg:gmredi:diagnostics} |
| 418 |
|
| 419 |
{\footnotesize |
| 420 |
\begin{verbatim} |
| 421 |
------------------------------------------------------------------------ |
| 422 |
<-Name->|Levs|<-parsing code->|<-- Units -->|<- Tile (max=80c) |
| 423 |
------------------------------------------------------------------------ |
| 424 |
GM_VisbK| 1 |SM P M1 |m^2/s |Mixing coefficient from Visbeck etal parameterization |
| 425 |
GM_Kux | 15 |UU P 177MR |m^2/s |K_11 element (U.point, X.dir) of GM-Redi tensor |
| 426 |
GM_Kvy | 15 |VV P 176MR |m^2/s |K_22 element (V.point, Y.dir) of GM-Redi tensor |
| 427 |
GM_Kuz | 15 |UU 179MR |m^2/s |K_13 element (U.point, Z.dir) of GM-Redi tensor |
| 428 |
GM_Kvz | 15 |VV 178MR |m^2/s |K_23 element (V.point, Z.dir) of GM-Redi tensor |
| 429 |
GM_Kwx | 15 |UM 181LR |m^2/s |K_31 element (W.point, X.dir) of GM-Redi tensor |
| 430 |
GM_Kwy | 15 |VM 180LR |m^2/s |K_32 element (W.point, Y.dir) of GM-Redi tensor |
| 431 |
GM_Kwz | 15 |WM P LR |m^2/s |K_33 element (W.point, Z.dir) of GM-Redi tensor |
| 432 |
GM_PsiX | 15 |UU 184LR |m^2/s |GM Bolus transport stream-function : X component |
| 433 |
GM_PsiY | 15 |VV 183LR |m^2/s |GM Bolus transport stream-function : Y component |
| 434 |
GM_KuzTz| 15 |UU 186MR |degC.m^3/s |Redi Off-diagonal Tempetature flux: X component |
| 435 |
GM_KvzTz| 15 |VV 185MR |degC.m^3/s |Redi Off-diagonal Tempetature flux: Y component |
| 436 |
\end{verbatim} |
| 437 |
} |
| 438 |
|
| 439 |
\subsubsection{Experiments and tutorials that use gmredi} |
| 440 |
\label{sec:pkg:gmredi:experiments} |
| 441 |
|
| 442 |
\begin{itemize} |
| 443 |
\item{Global Ocean tutorial, in tutorial\_global\_oce\_latlon verification directory, |
| 444 |
described in section \ref{sec:eg-global} } |
| 445 |
\item{ Front Relax experiment, in front\_relax verification directory.} |
| 446 |
\item{ Ideal 2D Ocean experiment, in ideal\_2D\_oce verification directory.} |
| 447 |
\end{itemize} |
| 448 |
|
| 449 |
% DO NOT EDIT HERE |
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