2. Discretization and Algorithm

This chapter lays out the numerical schemes that are employed in the core MITgcm algorithm. Whenever possible links are made to actual program code in the MITgcm implementation. The chapter begins with a discussion of the temporal discretization used in MITgcm. This discussion is followed by sections that describe the spatial discretization. The schemes employed for momentum terms are described first, afterwards the schemes that apply to passive and dynamically active tracers are described.

2.1. Notation

Because of the particularity of the vertical direction in stratified fluid context, in this chapter, the vector notations are mostly used for the horizontal component: the horizontal part of a vector is simply written \(\vec{\bf v}\) (instead of \({{\bf v}_h}\) or \(\vec{\mathbf{v}}_{h}\) in chapter 1) and a 3D vector is simply written \(\vec{v}\) (instead of \(\vec{\mathbf{v}}\) in chapter 1).

The notations we use to describe the discrete formulation of the model are summarized as follows.

General notation:

\(\Delta x, \Delta y, \Delta r\) grid spacing in X, Y, R directions

\(A_c,A_w,A_s,A_{\zeta}\) : horizontal area of a grid cell surrounding \(\theta,u,v,\zeta\) point

\({\cal V}_u , {\cal V}_v , {\cal V}_w , {\cal V}_\theta\) : Volume of the grid box surrounding \(u,v,w,\theta\) point

\(i,j,k\) : current index relative to X, Y, R directions

Basic operators:

\(\delta_i\) : \(\delta_i \Phi = \Phi_{i+1/2} - \Phi_{i-1/2}\)

\(~^{-i}\) : \(\overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2\)

\(\delta_x\) : \(\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi\)

\(\overline{ \nabla }\) = horizontal gradient operator : \(\overline{ \nabla } \Phi = \{ \delta_x \Phi , \delta_y \Phi \}\)

\(\overline{ \nabla } \cdot\) = horizontal divergence operator : \(\overline{ \nabla }\cdot \vec{\mathrm{{\bf f}}} = \dfrac{1}{\cal A} \{ \delta_i \Delta y \, \mathrm{f}_x + \delta_j \Delta x \, \mathrm{f}_y \}\)

\(\overline{\nabla}^2\) = horizontal Laplacian operator : \(\overline{\nabla}^2 \Phi = \overline{ \nabla } \cdot \overline{ \nabla } \Phi\)

2.2. Time-stepping

The equations of motion integrated by the model involve four prognostic equations for flow, \(u\) and \(v\), temperature, \(\theta\), and salt/moisture, \(S\), and three diagnostic equations for vertical flow, \(w\), density/buoyancy, \(\rho\)/\(b\), and pressure/geo-potential, \(\phi_{\rm hyd}\). In addition, the surface pressure or height may by described by either a prognostic or diagnostic equation and if non-hydrostatics terms are included then a diagnostic equation for non-hydrostatic pressure is also solved. The combination of prognostic and diagnostic equations requires a model algorithm that can march forward prognostic variables while satisfying constraints imposed by diagnostic equations.

Since the model comes in several flavors and formulation, it would be confusing to present the model algorithm exactly as written into code along with all the switches and optional terms. Instead, we present the algorithm for each of the basic formulations which are:

  1. the semi-implicit pressure method for hydrostatic equations with a rigid-lid, variables co-located in time and with Adams-Bashforth time-stepping;

  2. as 1 but with an implicit linear free-surface;

  3. as 1 or 2 but with variables staggered in time;

  4. as 1 or 2 but with non-hydrostatic terms included;

  5. as 2 or 3 but with non-linear free-surface.

In all the above configurations it is also possible to substitute the Adams-Bashforth with an alternative time-stepping scheme for terms evaluated explicitly in time. Since the over-arching algorithm is independent of the particular time-stepping scheme chosen we will describe first the over-arching algorithm, known as the pressure method, with a rigid-lid model in Section 2.3. This algorithm is essentially unchanged, apart for some coefficients, when the rigid lid assumption is replaced with a linearized implicit free-surface, described in Section 2.4. These two flavors of the pressure-method encompass all formulations of the model as it exists today. The integration of explicit in time terms is out-lined in Section 2.5 and put into the context of the overall algorithm in Section 2.7 and Section 2.8. Inclusion of non-hydrostatic terms requires applying the pressure method in three dimensions instead of two and this algorithm modification is described in Section 2.9. Finally, the free-surface equation may be treated more exactly, including non-linear terms, and this is described in Section 2.10.2.

2.3. Pressure method with rigid-lid

The horizontal momentum and continuity equations for the ocean ((1.98) and (1.100)), or for the atmosphere ((1.45) and (1.47)), can be summarized by:

\[\begin{split}\begin{aligned} \partial_t u + g \partial_x \eta & = G_u \\ \partial_t v + g \partial_y \eta & = G_v \\ \partial_x u + \partial_y v + \partial_z w & = 0\end{aligned}\end{split}\]

where we are adopting the oceanic notation for brevity. All terms in the momentum equations, except for surface pressure gradient, are encapsulated in the \(G\) vector. The continuity equation, when integrated over the fluid depth, \(H\), and with the rigid-lid/no normal flow boundary conditions applied, becomes:

(2.1)\[\partial_x H \widehat{u} + \partial_y H \widehat{v} = 0\]

Here, \(H\widehat{u} = \int_H u dz\) is the depth integral of \(u\), similarly for \(H\widehat{v}\). The rigid-lid approximation sets \(w=0\) at the lid so that it does not move but allows a pressure to be exerted on the fluid by the lid. The horizontal momentum equations and vertically integrated continuity equation are be discretized in time and space as follows:

(2.2)\[u^{n+1} + \Delta t g \partial_x \eta^{n+1} = u^{n} + \Delta t G_u^{(n+1/2)}\]
(2.3)\[v^{n+1} + \Delta t g \partial_y \eta^{n+1} = v^{n} + \Delta t G_v^{(n+1/2)}\]
(2.4)\[\partial_x H \widehat{u^{n+1}} + \partial_y H \widehat{v^{n+1}} = 0\]

As written here, terms on the LHS all involve time level \(n+1\) and are referred to as implicit; the implicit backward time stepping scheme is being used. All other terms in the RHS are explicit in time. The thermodynamic quantities are integrated forward in time in parallel with the flow and will be discussed later. For the purposes of describing the pressure method it suffices to say that the hydrostatic pressure gradient is explicit and so can be included in the vector \(G\).

Substituting the two momentum equations into the depth-integrated continuity equation eliminates \(u^{n+1}\) and \(v^{n+1}\) yielding an elliptic equation for \(\eta^{n+1}\). Equations (2.2), (2.3) and (2.4) can then be re-arranged as follows:

(2.5)\[u^{*} = u^{n} + \Delta t G_u^{(n+1/2)}\]
(2.6)\[v^{*} = v^{n} + \Delta t G_v^{(n+1/2)}\]
(2.7)\[\partial_x \Delta t g H \partial_x \eta^{n+1} + \partial_y \Delta t g H \partial_y \eta^{n+1} = \partial_x H \widehat{u^{*}} + \partial_y H \widehat{v^{*}}\]
(2.8)\[u^{n+1} = u^{*} - \Delta t g \partial_x \eta^{n+1}\]
(2.9)\[v^{n+1} = v^{*} - \Delta t g \partial_y \eta^{n+1}\]

Equations (2.5) to (2.9), solved sequentially, represent the pressure method algorithm used in the model. The essence of the pressure method lies in the fact that any explicit prediction for the flow would lead to a divergence flow field so a pressure field must be found that keeps the flow non-divergent over each step of the integration. The particular location in time of the pressure field is somewhat ambiguous; in Figure 2.1 we depicted as co-located with the future flow field (time level \(n+1\)) but it could equally have been drawn as staggered in time with the flow.


Figure 2.1 A schematic of the evolution in time of the pressure method algorithm. A prediction for the flow variables at time level \(n+1\) is made based only on the explicit terms, \(G^{(n+^1/_2)}\), and denoted \(u^*\), \(v^*\). Next, a pressure field is found such that \(u^{n+1}\), \(v^{n+1}\) will be non-divergent. Conceptually, the \(*\) quantities exist at time level \(n+1\) but they are intermediate and only temporary.

The correspondence to the code is as follows:

  • the prognostic phase, equations (2.5) and (2.6), stepping forward \(u^n\) and \(v^n\) to \(u^{*}\) and \(v^{*}\) is coded in timestep.F

  • the vertical integration, \(H \widehat{u^*}\) and \(H \widehat{v^*}\), divergence and inversion of the elliptic operator in equation (2.7) is coded in solve_for_pressure.F

  • finally, the new flow field at time level \(n+1\) given by equations (2.8) and (2.9) is calculated in correction_step.F

The calling tree for these routines is as follows:

Pressure method calling tree

\(\phantom{W}\) DYNAMICS
\(\phantom{WW}\) TIMESTEP \(\phantom{xxxxxxxxxxxxxxxxxxxxxx}\) \(u^*,v^*\) (2.5) , (2.6)
\(\phantom{W}\) SOLVE_FOR_PRESSURE
\(\phantom{WW}\) CALC_DIV_GHAT \(\phantom{xxxxxxxxxxxxxxxx}\) \(H\widehat{u^*},H\widehat{v^*}\) (2.7)
\(\phantom{WW}\) CG2D \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxx}\) \(\eta^{n+1}\) (2.7)
\(\phantom{WW}\) CALC_GRAD_PHI_SURF \(\phantom{xxxxxxxxxx}\) \(\nabla \eta^{n+1}\)
\(\phantom{WW}\) CORRECTION_STEP \(\phantom{xxxxxxxxxxxxw}\) \(u^{n+1},v^{n+1}\) (2.8) , (2.9)

In general, the horizontal momentum time-stepping can contain some terms that are treated implicitly in time, such as the vertical viscosity when using the backward time-stepping scheme (implicitViscosity =.TRUE.). The method used to solve those implicit terms is provided in Section 2.6, and modifies equations (2.2) and (2.3) to give:

\[\begin{split}\begin{aligned} u^{n+1} - \Delta t \partial_z A_v \partial_z u^{n+1} + \Delta t g \partial_x \eta^{n+1} & = u^{n} + \Delta t G_u^{(n+1/2)} \\ v^{n+1} - \Delta t \partial_z A_v \partial_z v^{n+1} + \Delta t g \partial_y \eta^{n+1} & = v^{n} + \Delta t G_v^{(n+1/2)}\end{aligned}\end{split}\]

2.4. Pressure method with implicit linear free-surface

The rigid-lid approximation filters out external gravity waves subsequently modifying the dispersion relation of barotropic Rossby waves. The discrete form of the elliptic equation has some zero eigenvalues which makes it a potentially tricky or inefficient problem to solve.

The rigid-lid approximation can be easily replaced by a linearization of the free-surface equation which can be written:

(2.10)\[\partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = {\mathcal{P-E+R}}\]

which differs from the depth-integrated continuity equation with rigid-lid (2.1) by the time-dependent term and fresh-water source term.

Equation (2.4) in the rigid-lid pressure method is then replaced by the time discretization of (2.10) which is:

(2.11)\[\eta^{n+1} + \Delta t \partial_x H \widehat{u^{n+1}} + \Delta t \partial_y H \widehat{v^{n+1}} = \eta^{n} + \Delta t ( {\mathcal{P-E}})\]

where the use of flow at time level \(n+1\) makes the method implicit and backward in time. This is the preferred scheme since it still filters the fast unresolved wave motions by damping them. A centered scheme, such as Crank-Nicholson (see Section 2.10.1), would alias the energy of the fast modes onto slower modes of motion.

As for the rigid-lid pressure method, equations (2.2), (2.3) and (2.11) can be re-arranged as follows:

(2.12)\[u^{*} = u^{n} + \Delta t G_u^{(n+1/2)}\]
(2.13)\[v^{*} = v^{n} + \Delta t G_v^{(n+1/2)}\]
(2.14)\[\eta^* = \epsilon_{\rm fs} ( \eta^{n} + \Delta t ({\mathcal{P-E}}) ) - \Delta t ( \partial_x H \widehat{u^{*}} + \partial_y H \widehat{v^{*}} )\]
(2.15)\[\partial_x g H \partial_x \eta^{n+1} + \partial_y g H \partial_y \eta^{n+1} - \frac{\epsilon_{\rm fs} \eta^{n+1}}{\Delta t^2} = - \frac{\eta^*}{\Delta t^2}\]
(2.16)\[u^{n+1} = u^{*} - \Delta t g \partial_x \eta^{n+1}\]
(2.17)\[v^{n+1} = v^{*} - \Delta t g \partial_y \eta^{n+1}\]

Equations (2.12) to (2.17), solved sequentially, represent the pressure method algorithm with a backward implicit, linearized free surface. The method is still formerly a pressure method because in the limit of large \(\Delta t\) the rigid-lid method is recovered. However, the implicit treatment of the free-surface allows the flow to be divergent and for the surface pressure/elevation to respond on a finite time-scale (as opposed to instantly). To recover the rigid-lid formulation, we use a switch-like variable, \(\epsilon_{\rm fs}\) (freesurfFac), which selects between the free-surface and rigid-lid; \(\epsilon_{\rm fs}=1\) allows the free-surface to evolve; \(\epsilon_{\rm fs}=0\) imposes the rigid-lid. The evolution in time and location of variables is exactly as it was for the rigid-lid model so that Figure 2.1 is still applicable. Similarly, the calling sequence, given here, is as for the pressure-method.

2.5. Explicit time-stepping: Adams-Bashforth

In describing the the pressure method above we deferred describing the time discretization of the explicit terms. We have historically used the quasi-second order Adams-Bashforth method (AB-II) for all explicit terms in both the momentum and tracer equations. This is still the default mode of operation but it is now possible to use alternate schemes for tracers (see Section 2.16), or a 3rd order Adams-Bashforth method (AB-III). In the previous sections, we summarized an explicit scheme as:

(2.18)\[\tau^{*} = \tau^{n} + \Delta t G_\tau^{(n+1/2)}\]

where \(\tau\) could be any prognostic variable (\(u\), \(v\), \(\theta\) or \(S\)) and \(\tau^*\) is an explicit estimate of \(\tau^{n+1}\) and would be exact if not for implicit-in-time terms. The parenthesis about \(n+1/2\) indicates that the term is explicit and extrapolated forward in time. Below we describe in more detail the AB-II and AB-III schemes.

2.5.1. Adams-Bashforth II

The quasi-second order Adams-Bashforth scheme is formulated as follows:

(2.19)\[G_\tau^{(n+1/2)} = ( 3/2 + \epsilon_{\rm AB}) G_\tau^n - ( 1/2 + \epsilon_{\rm AB}) G_\tau^{n-1}\]

This is a linear extrapolation, forward in time, to \(t=(n+1/2+{\epsilon_{\rm AB}})\Delta t\). An extrapolation to the mid-point in time, \(t=(n+1/2)\Delta t\), corresponding to \(\epsilon_{\rm AB}=0\), would be second order accurate but is weakly unstable for oscillatory terms. A small but finite value for \(\epsilon_{\rm AB}\) stabilizes the method. Strictly speaking, damping terms such as diffusion and dissipation, and fixed terms (forcing), do not need to be inside the Adams-Bashforth extrapolation. However, in the current code, it is simpler to include these terms and this can be justified if the flow and forcing evolves smoothly. Problems can, and do, arise when forcing or motions are high frequency and this corresponds to a reduced stability compared to a simple forward time-stepping of such terms. The model offers the possibility to leave terms outside the Adams-Bashforth extrapolation, by turning off the logical flag forcing_In_AB (parameter file data, namelist PARM01, default value = .TRUE.) and then setting tracForcingOutAB (default=0), momForcingOutAB (default=0), and momDissip_In_AB (parameter file data, namelist PARM01, default value = TRUE), respectively for the tracer terms, momentum forcing terms, and the dissipation terms.

A stability analysis for an oscillation equation should be given at this point.

A stability analysis for a relaxation equation should be given at this point.


Figure 2.2 Oscillatory and damping response of quasi-second order Adams-Bashforth scheme for different values of the \(\epsilon _{\rm AB}\) parameter (0.0, 0.1, 0.25, from top to bottom) The analytical solution (in black), the physical mode (in blue) and the numerical mode (in red) are represented with a CFL step of 0.1. The left column represents the oscillatory response on the complex plane for CFL ranging from 0.1 up to 0.9. The right column represents the damping response amplitude (y-axis) function of the CFL (x-axis).

2.5.2. Adams-Bashforth III

The 3rd order Adams-Bashforth time stepping (AB-III) provides several advantages (see, e.g., Durran 1991 [Dur91]) compared to the default quasi-second order Adams-Bashforth method:

  • higher accuracy;

  • stable with a longer time-step;

  • no additional computation (just requires the storage of one additional time level).

The 3rd order Adams-Bashforth can be used to extrapolate forward in time the tendency (replacing (2.19)) as:

(2.20)\[G_\tau^{(n+1/2)} = ( 1 + \alpha_{\rm AB} + \beta_{\rm AB}) G_\tau^n - ( \alpha_{\rm AB} + 2 \beta_{\rm AB}) G_\tau^{n-1} + \beta_{\rm AB} G_\tau^{n-2}\]

3rd order accuracy is obtained with \((\alpha_{\rm AB},\,\beta_{\rm AB}) = (1/2,\,5/12)\). Note that selecting \((\alpha_{\rm AB},\,\beta_{\rm AB}) = (1/2+\epsilon_{AB},\,0)\) one recovers AB-II. The AB-III time stepping improves the stability limit for an oscillatory problem like advection or Coriolis. As seen from Figure 2.3, it remains stable up to a CFL of 0.72, compared to only 0.50 with AB-II and \(\epsilon_{\rm AB} = 0.1\). It is interesting to note that the stability limit can be further extended up to a CFL of 0.786 for an oscillatory problem (see Figure 2.3) using \((\alpha_{\rm AB},\,\beta_{\rm AB}) = (0.5,\,0.2811)\) but then the scheme is only second order accurate.

However, the behavior of the AB-III for a damping problem (like diffusion) is less favorable, since the stability limit is reduced to 0.54 only (and 0.64 with \(\beta_{\rm AB} = 0.2811\)) compared to 1.0 (and 0.9 with \(\epsilon_{\rm AB} = 0.1\)) with the AB-II (see Figure 2.4).

A way to enable the use of a longer time step is to keep the dissipation terms outside the AB extrapolation (setting momDissip_In_AB to .FALSE. in main parameter file data, namelist PARM03, thus returning to a simple forward time-stepping for dissipation, and to use AB-III only for advection and Coriolis terms.

The AB-III time stepping is activated by defining the option #define ALLOW_ADAMSBASHFORTH_3 in CPP_OPTIONS.h. The parameters \(\alpha_{\rm AB},\beta_{\rm AB}\) can be set from the main parameter file data (namelist PARM03) and their default values correspond to the 3rd order Adams-Bashforth. A simple example is provided in verification/advect_xy/input.ab3_c4.

AB-III is not yet available for the vertical momentum equation (non-hydrostatic) nor for passive tracers.


Figure 2.3 Oscillatory response of third order Adams-Bashforth scheme for different values of the \((\alpha_{\rm AB},\,\beta_{\rm AB})\) parameters. The analytical solution (in black), the physical mode (in blue) and the numerical mode (in red) are represented with a CFL step of 0.1.


Figure 2.4 Damping response of third order Adams-Bashforth scheme for different values of the \((\alpha_{\rm AB},\,\beta_{\rm AB})\) parameters. The analytical solution (in black), the physical mode (in blue) and the numerical mode (in red) are represented with a CFL step of 0.1.

2.6. Implicit time-stepping: backward method

Vertical diffusion and viscosity can be treated implicitly in time using the backward method which is an intrinsic scheme. Recently, the option to treat the vertical advection implicitly has been added, but not yet tested; therefore, the description hereafter is limited to diffusion and viscosity. For tracers, the time discretized equation is:

(2.21)\[\tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = \tau^{n} + \Delta t G_\tau^{(n+1/2)}\]

where \(G_\tau^{(n+1/2)}\) is the remaining explicit terms extrapolated using the Adams-Bashforth method as described above. Equation (2.21) can be split split into:

(2.22)\[\tau^* = \tau^{n} + \Delta t G_\tau^{(n+1/2)}\]
(2.23)\[\tau^{n+1} = {\cal L}_\tau^{-1} ( \tau^* )\]

where \({\cal L}_\tau^{-1}\) is the inverse of the operator

\[{\cal L}_\tau = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right]\]

Equation (2.22) looks exactly as (2.18) while (2.23) involves an operator or matrix inversion. By re-arranging (2.21) in this way we have cast the method as an explicit prediction step and an implicit step allowing the latter to be inserted into the over all algorithm with minimal interference.

The calling sequence for stepping forward a tracer variable such as temperature with implicit diffusion is as follows:

Adams-Bashforth calling tree

\(\phantom{W}\) THERMODYNAMICS
\(\phantom{WW}\) TEMP_INTEGRATE
\(\phantom{WWW}\) GAD_CALC_RHS \(\phantom{xxxxxxxxxw}\) \(G_\theta^n = G_\theta( u, \theta^n)\)
\(\phantom{WWW}\) either
\(\phantom{WWWW}\) EXTERNAL_FORCING \(\phantom{xxxx}\) \(G_\theta^n = G_\theta^n + {\cal Q}\)
\(\phantom{WWWW}\) ADAMS_BASHFORTH2 \(\phantom{xxi}\) \(G_\theta^{(n+1/2)}\) (2.19)
\(\phantom{WWW}\) or
\(\phantom{WWWW}\) EXTERNAL_FORCING \(\phantom{xxxx}\) \(G_\theta^{(n+1/2)} = G_\theta^{(n+1/2)} + {\cal Q}\)
\(\phantom{WW}\) TIMESTEP_TRACER \(\phantom{xxxxxxxxxx}\) \(\tau^*\) (2.18)
\(\phantom{WW}\) IMPLDIFF \(\phantom{xxxxxxxxxxxxxxxxxw}\) \(\tau^{(n+1)}\) (2.23)

In order to fit within the pressure method, the implicit viscosity must not alter the barotropic flow. In other words, it can only redistribute momentum in the vertical. The upshot of this is that although vertical viscosity may be backward implicit and unconditionally stable, no-slip boundary conditions may not be made implicit and are thus cast as a an explicit drag term.

2.7. Synchronous time-stepping: variables co-located in time


Figure 2.5 A schematic of the explicit Adams-Bashforth and implicit time-stepping phases of the algorithm. All prognostic variables are co-located in time. Explicit tendencies are evaluated at time level \(n\) as a function of the state at that time level (dotted arrow). The explicit tendency from the previous time level, \(n-1\), is used to extrapolate tendencies to \(n+1/2\) (dashed arrow). This extrapolated tendency allows variables to be stably integrated forward-in-time to render an estimate (\(*\) -variables) at the \(n+1\) time level (solid arc-arrow). The operator \({\cal L}\) formed from implicit-in-time terms is solved to yield the state variables at time level \(n+1\).

The Adams-Bashforth extrapolation of explicit tendencies fits neatly into the pressure method algorithm when all state variables are co-located in time. The algorithm can be represented by the sequential solution of the follow equations:

(2.24)\[G_{\theta,S}^{n} = G_{\theta,S} ( u^n, \theta^n, S^n )\]
(2.25)\[G_{\theta,S}^{(n+1/2)} = (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1}\]
(2.26)\[(\theta^*,S^*) = (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}\]
(2.27)\[(\theta^{n+1},S^{n+1}) = {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)\]
(2.28)\[\phi^n_{\rm hyd} = \int b(\theta^n,S^n) dr\]
(2.29)\[\vec{\bf G}_{\vec{\bf v}}^{n} = \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^n, \phi^n_{\rm hyd} )\]
(2.30)\[\vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} = (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1}\]
(2.31)\[\vec{\bf v}^{*} = \vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)}\]
(2.32)\[\vec{\bf v}^{**} = {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )\]
(2.33)\[\eta^* = \epsilon_{\rm fs} \left( \eta^{n} + \Delta t ({\mathcal{P-E}}) \right)- \Delta t \nabla \cdot H \widehat{ \vec{\bf v}^{**} }\]
(2.34)\[\nabla \cdot g H \nabla \eta^{n+1} - \frac{\epsilon_{\rm fs} \eta^{n+1}}{\Delta t^2} ~ = ~ - \frac{\eta^*}{\Delta t^2}\]
(2.35)\[\vec{\bf v}^{n+1} = \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1}\]

Figure 2.5 illustrates the location of variables in time and evolution of the algorithm with time. The Adams-Bashforth extrapolation of the tracer tendencies is illustrated by the dashed arrow, the prediction at \(n+1\) is indicated by the solid arc. Inversion of the implicit terms, \({\cal L}^{-1}_{\theta,S}\), then yields the new tracer fields at \(n+1\). All these operations are carried out in subroutine THERMODYNAMICS and subsidiaries, which correspond to equations (2.24) to (2.27). Similarly illustrated is the Adams-Bashforth extrapolation of accelerations, stepping forward and solving of implicit viscosity and surface pressure gradient terms, corresponding to equations (2.29) to (2.35). These operations are carried out in subroutines DYNAMICS, SOLVE_FOR_PRESSURE and MOMENTUM_CORRECTION_STEP. This, then, represents an entire algorithm for stepping forward the model one time-step. The corresponding calling tree for the overall synchronous algorithm using Adams-Bashforth time-stepping is given below. The place where the model geometry hFac factors) is updated is added here but is only relevant for the non-linear free-surface algorithm. For completeness, the external forcing, ocean and atmospheric physics have been added, although they are mainly optional.

Synchronous Adams-Bashforth calling tree

\(\phantom{WWW}\) DO_OCEANIC_PHYS
\(\phantom{WWW}\) CALC_GT
\(\phantom{WWWW}\) GAD_CALC_RHS \(\phantom{xxxxxxxxxxxxxlwww}\) \(G_\theta^n = G_\theta( u, \theta^n )\) (2.24)
\(\phantom{WWWW}\) EXTERNAL_FORCING \(\phantom{xxxxxxxxxxlww}\) \(G_\theta^n = G_\theta^n + {\cal Q}\)
\(\phantom{WWWW}\) ADAMS_BASHFORTH2 \(\phantom{xxxxxxxxxxxw}\) \(G_\theta^{(n+1/2)}\) (2.25)
\(\phantom{WWW}\) TIMESTEP_TRACER \(\phantom{xxxxxxxxxxxxxxxww}\) \(\theta^*\) (2.26)
\(\phantom{WWW}\) IMPLDIFF \(\phantom{xxxxxxxxxxxxxxxxxxxxxvwww}\) \(\theta^{(n+1)}\) (2.27)
\(\phantom{WW}\) DYNAMICS
\(\phantom{WWW}\) CALC_PHI_HYD \(\phantom{xxxxxxxxxxxxxxxxxxxxi}\) \(\phi_{\rm hyd}^n\) (2.28)
\(\phantom{WWW}\) MOM_FLUXFORM or MOM_VECINV \(\phantom{xxi}\) \(G_{\vec{\bf v}}^n\) (2.29)
\(\phantom{WWW}\) TIMESTEP \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxx}\) \(\vec{\bf v}^*\) (2.30), (2.31)
\(\phantom{WWW}\) IMPLDIFF \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxlw}\) \(\vec{\bf v}^{**}\) (2.32)
\(\phantom{WW}\) UPDATE_R_STAR or UPDATE_SURF_DR (NonLin-FS only)
\(\phantom{WWW}\) CALC_DIV_GHAT \(\phantom{xxxxxxxxxxxxxxxxxxxx}\) \(\eta^*\) (2.33)
\(\phantom{WWW}\) CG2D \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxi}\) \(\eta^{n+1}\) (2.34)
\(\phantom{WWW}\) CALC_GRAD_PHI_SURF \(\phantom{xxxxxxxxxxxxxx}\) \(\nabla \eta^{n+1}\)
\(\phantom{WWW}\) CORRECTION_STEP \(\phantom{xxxxxxxxxxxxxxxxw}\) \(u^{n+1},v^{n+1}\) (2.35)
\(\phantom{WWW}\) CYCLE_TRACER \(\phantom{xxxxxxxxxxxxxxxxxxxxx}\) \(\theta^{n+1}\)

2.8. Staggered baroclinic time-stepping


Figure 2.6 A schematic of the explicit Adams-Bashforth and implicit time-stepping phases of the algorithm but with staggering in time of thermodynamic variables with the flow. Explicit momentum tendencies are evaluated at time level \(n-1/2\) as a function of the flow field at that time level \(n-1/2\). The explicit tendency from the previous time level, \(n-3/2\), is used to extrapolate tendencies to \(n\) (dashed arrow). The hydrostatic pressure/geo-potential \(\phi _{\rm hyd}\) is evaluated directly at time level \(n\) (vertical arrows) and used with the extrapolated tendencies to step forward the flow variables from \(n-1/2\) to \(n+1/2\) (solid arc-arrow). The implicit-in-time operator \({\cal L}_{\bf u,v}\) (vertical arrows) is then applied to the previous estimation of the the flow field (\(*\) -variables) and yields to the two velocity components \(u,v\) at time level \(n+1/2\). These are then used to calculate the advection term (dashed arc-arrow) of the thermo-dynamics tendencies at time step \(n\). The extrapolated thermodynamics tendency, from time level \(n-1\) and \(n\) to \(n+1/2\), allows thermodynamic variables to be stably integrated forward-in-time (solid arc-arrow) up to time level \(n+1\).

For well-stratified problems, internal gravity waves may be the limiting process for determining a stable time-step. In the circumstance, it is more efficient to stagger in time the thermodynamic variables with the flow variables. Figure 2.6 illustrates the staggering and algorithm. The key difference between this and Figure 2.5 is that the thermodynamic variables are solved after the dynamics, using the recently updated flow field. This essentially allows the gravity wave terms to leap-frog in time giving second order accuracy and more stability.

The essential change in the staggered algorithm is that the thermodynamics solver is delayed from half a time step, allowing the use of the most recent velocities to compute the advection terms. Once the thermodynamics fields are updated, the hydrostatic pressure is computed to step forward the dynamics. Note that the pressure gradient must also be taken out of the Adams-Bashforth extrapolation. Also, retaining the integer time-levels, \(n\) and \(n+1\), does not give a user the sense of where variables are located in time. Instead, we re-write the entire algorithm, (2.24) to (2.35), annotating the position in time of variables appropriately:

(2.36)\[\phi^{n}_{\rm hyd} = \int b(\theta^{n},S^{n}) dr\]
(2.37)\[\vec{\bf G}_{\vec{\bf v}}^{n-1/2} = \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )\]
(2.38)\[\vec{\bf G}_{\vec{\bf v}}^{(n)} = (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1/2} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-3/2}\]
(2.39)\[\vec{\bf v}^{*} = \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{\rm hyd}^{n} \right)\]
(2.40)\[\vec{\bf v}^{**} = {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )\]
(2.41)\[\eta^* = \epsilon_{\rm fs} \left( \eta^{n-1/2} + \Delta t ({\mathcal{P-E}})^n \right)- \Delta t \nabla \cdot H \widehat{ \vec{\bf v}^{**} }\]
(2.42)\[\nabla \cdot g H \nabla \eta^{n+1/2} - \frac{\epsilon_{\rm fs} \eta^{n+1/2}}{\Delta t^2} = - \frac{\eta^*}{\Delta t^2}\]
(2.43)\[\vec{\bf v}^{n+1/2} = \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1/2}\]
(2.44)\[G_{\theta,S}^{n} = G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )\]
(2.45)\[G_{\theta,S}^{(n+1/2)} = (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1}\]
(2.46)\[(\theta^*,S^*) = (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}\]
(2.47)\[(\theta^{n+1},S^{n+1}) = {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)\]

The corresponding calling tree is given below. The staggered algorithm is activated with the run-time flag staggerTimeStep =.TRUE. in parameter file data, namelist PARM01.

Staggered Adams-Bashforth calling tree

\(\phantom{WWW}\) DO_OCEANIC_PHYS
\(\phantom{WW}\) DYNAMICS
\(\phantom{WWW}\) CALC_PHI_HYD \(\phantom{xxxxxxxxxxxxxxxxxxxxi}\) \(\phi_{\rm hyd}^n\) (2.36)
\(\phantom{WWW}\) MOM_FLUXFORM or MOM_VECINV \(\phantom{xxi}\) \(G_{\vec{\bf v}}^{n-1/2}\) (2.37)
\(\phantom{WWW}\) TIMESTEP \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxx}\) \(\vec{\bf v}^*\) (2.38), (2.39)
\(\phantom{WWW}\) IMPLDIFF \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxlw}\) \(\vec{\bf v}^{**}\) (2.40)
\(\phantom{WW}\) UPDATE_R_STAR or UPDATE_SURF_DR (NonLin-FS only)
\(\phantom{WWW}\) CALC_DIV_GHAT \(\phantom{xxxxxxxxxxxxxxxxxxxx}\) \(\eta^*\) (2.41)
\(\phantom{WWW}\) CG2D \(\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxi}\) \(\eta^{n+1/2}\) (2.42)
\(\phantom{WWW}\) CALC_GRAD_PHI_SURF \(\phantom{xxxxxxxxxxxxxx}\) \(\nabla \eta^{n+1/2}\)
\(\phantom{WWW}\) CORRECTION_STEP \(\phantom{xxxxxxxxxxxxxxxxw}\) \(u^{n+1/2},v^{n+1/2}\) (2.43)
\(\phantom{WWW}\) CALC_GT
\(\phantom{WWWW}\) GAD_CALC_RHS \(\phantom{xxxxxxxxxxxxxlwww}\) \(G_\theta^n = G_\theta( u, \theta^n )\) (2.44)
\(\phantom{WWWW}\) EXTERNAL_FORCING \(\phantom{xxxxxxxxxxlww}\) \(G_\theta^n = G_\theta^n + {\cal Q}\)
\(\phantom{WWWW}\) ADAMS_BASHFORTH2 \(\phantom{xxxxxxxxxxxw}\) \(G_\theta^{(n+1/2)}\) (2.45)
\(\phantom{WWW}\) TIMESTEP_TRACER \(\phantom{xxxxxxxxxxxxxxxww}\) \(\theta^*\) (2.46)
\(\phantom{WWW}\) IMPLDIFF \(\phantom{xxxxxxxxxxxxxxxxxxxxxvwww}\) \(\theta^{(n+1)}\) (2.47)
\(\phantom{WWW}\) CYCLE_TRACER \(\phantom{xxxxxxxxxxxxxxxxxxxxx}\) \(\theta^{n+1}\)

The only difficulty with this approach is apparent in equation (2.44) and illustrated by the dotted arrow connecting \(u,v^{n+1/2}\) with \(G_\theta^{n}\). The flow used to advect tracers around is not naturally located in time. This could be avoided by applying the Adams-Bashforth extrapolation to the tracer field itself and advecting that around but this approach is not yet available. We’re not aware of any detrimental effect of this feature. The difficulty lies mainly in interpretation of what time-level variables and terms correspond to.

2.9. Non-hydrostatic formulation

The non-hydrostatic formulation re-introduces the full vertical momentum equation and requires the solution of a 3-D elliptic equations for non-hydrostatic pressure perturbation. We still integrate vertically for the hydrostatic pressure and solve a 2-D elliptic equation for the surface pressure/elevation for this reduces the amount of work needed to solve for the non-hydrostatic pressure.

The momentum equations are discretized in time as follows:

(2.48)\[\frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{\rm nh}^{n+1} = \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)}\]
(2.49)\[\frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{\rm nh}^{n+1} = \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)}\]
(2.50)\[\frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{\rm nh}^{n+1} = \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)}\]

which must satisfy the discrete-in-time depth integrated continuity, equation (2.11) and the local continuity equation

(2.51)\[\partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0\]

As before, the explicit predictions for momentum are consolidated as:

\[\begin{split}\begin{aligned} u^* & = u^n + \Delta t G_u^{(n+1/2)} \\ v^* & = v^n + \Delta t G_v^{(n+1/2)} \\ w^* & = w^n + \Delta t G_w^{(n+1/2)}\end{aligned}\end{split}\]

but this time we introduce an intermediate step by splitting the tendency of the flow as follows:

\[\begin{split}\begin{aligned} u^{n+1} = u^{**} - \Delta t \partial_x \phi_{\rm nh}^{n+1} & & u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1} \\ v^{n+1} = v^{**} - \Delta t \partial_y \phi_{\rm nh}^{n+1} & & v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1}\end{aligned}\end{split}\]

Substituting into the depth integrated continuity (2.11) gives

(2.52)\[\partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{\rm nh}^{n+1} \right) + \partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{\rm nh}^{n+1} \right) - \frac{\epsilon_{\rm fs}\eta^{n+1}}{\Delta t^2} = - \frac{\eta^*}{\Delta t^2}\]

which is approximated by equation (2.15) on the basis that i) \(\phi_{\rm nh}^{n+1}\) is not yet known and ii) \(\nabla \widehat{\phi}_{\rm nh} \ll g \nabla \eta\). If (2.15) is solved accurately then the implication is that \(\widehat{\phi}_{\rm nh} \approx 0\) so that the non-hydrostatic pressure field does not drive barotropic motion.

The flow must satisfy non-divergence (equation (2.51)) locally, as well as depth integrated, and this constraint is used to form a 3-D elliptic equations for \(\phi_{\rm nh}^{n+1}\):

(2.53)\[\partial_{xx} \phi_{\rm nh}^{n+1} + \partial_{yy} \phi_{\rm nh}^{n+1} + \partial_{rr} \phi_{\rm nh}^{n+1} = \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}\]

The entire algorithm can be summarized as the sequential solution of the following equations:

(2.54)\[u^{*} = u^{n} + \Delta t G_u^{(n+1/2)}\]
(2.55)\[v^{*} = v^{n} + \Delta t G_v^{(n+1/2)}\]
(2.56)\[w^{*} = w^{n} + \Delta t G_w^{(n+1/2)}\]
(2.57)\[\eta^* ~ = ~ \epsilon_{\rm fs} \left( \eta^{n} + \Delta t ({\mathcal{P-E}}) \right) - \Delta t \left( \partial_x H \widehat{u^{*}} + \partial_y H \widehat{v^{*}} \right)\]
(2.58)\[\partial_x g H \partial_x \eta^{n+1} + \partial_y g H \partial_y \eta^{n+1} - \frac{\epsilon_{\rm fs} \eta^{n+1}}{\Delta t^2} ~ = ~ - \frac{\eta^*}{\Delta t^2}\]
(2.59)\[u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1}\]
(2.60)\[v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1}\]
(2.61)\[\partial_{xx} \phi_{\rm nh}^{n+1} + \partial_{yy} \phi_{\rm nh}^{n+1} + \partial_{rr} \phi_{\rm nh}^{n+1} = \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}\]
(2.62)\[u^{n+1} = u^{**} - \Delta t \partial_x \phi_{\rm nh}^{n+1}\]
(2.63)\[v^{n+1} = v^{**} - \Delta t \partial_y \phi_{\rm nh}^{n+1}\]
(2.64)\[\partial_r w^{n+1} = - \partial_x u^{n+1} - \partial_y v^{n+1}\]

where the last equation is solved by vertically integrating for \(w^{n+1}\).

2.10. Variants on the Free Surface

We now describe the various formulations of the free-surface that include non-linear forms, implicit in time using Crank-Nicholson, explicit and [one day] split-explicit. First, we’ll reiterate the underlying algorithm but this time using the notation consistent with the more general vertical coordinate \(r\). The elliptic equation for free-surface coordinate (units of \(r\)), corresponding to (2.11), and assuming no non-hydrostatic effects (\(\epsilon_{\rm nh} = 0\)) is:

(2.65)\[\epsilon_{\rm fs} {\eta}^{n+1} - \nabla _h \cdot \Delta t^2 (R_o-R_{\rm fixed}) \nabla _h b_s {\eta}^{n+1} = {\eta}^*\]


(2.66)\[{\eta}^* = \epsilon_{\rm fs} \: {\eta}^{n} - \Delta t \nabla _h \cdot \int_{R_{\rm fixed}}^{R_o} \vec{\bf v}^* dr \: + \: \epsilon_{\rm fw} \Delta t ({\mathcal{P-E}})^{n}\]


\(u^*\) : gU ( DYNVARS.h )
\(v^*\) : gV ( DYNVARS.h )
\({\eta}^*\) : cg2d_b ( SOLVE_FOR_PRESSURE.h )
\({\eta}^{n+1}\) : etaN ( DYNVARS.h )

Once \({\eta}^{n+1}\) has been found, substituting into (2.2), (2.3) yields \(\vec{\bf v}^{n+1}\) if the model is hydrostatic (\(\epsilon_{\rm nh}=0\)):

\[\vec{\bf v}^{n+1} = \vec{\bf v}^{*} - \Delta t \nabla _h b_s {\eta}^{n+1}\]

This is known as the correction step. However, when the model is non-hydrostatic (\(\epsilon_{\rm nh}=1\)) we need an additional step and an additional equation for \(\phi'_{\rm nh}\). This is obtained by substituting (2.48), (2.49) and (2.50) into continuity:

(2.67)\[[ \nabla _h^2 + \partial_{rr} ] {\phi'_{\rm nh}}^{n+1} = \frac{1}{\Delta t} \nabla _h \cdot \vec{\bf v}^{**} + \partial_r \dot{r}^*\]


\[\vec{\bf v}^{**} = \vec{\bf v}^* - \Delta t \nabla _h b_s {\eta}^{n+1}\]

Note that \(\eta^{n+1}\) is also used to update the second RHS term \(\partial_r \dot{r}^*\) since the vertical velocity at the surface (\(\dot{r}_{\rm surf}\)) is evaluated as \((\eta^{n+1} - \eta^n) / \Delta t\).

Finally, the horizontal velocities at the new time level are found by:

(2.68)\[\vec{\bf v}^{n+1} = \vec{\bf v}^{**} - \epsilon_{\rm nh} \Delta t \nabla _h {\phi'_{\rm nh}}^{n+1}\]

and the vertical velocity is found by integrating the continuity equation vertically. Note that, for the convenience of the restart procedure, the vertical integration of the continuity equation has been moved to the beginning of the time step (instead of at the end), without any consequence on the solution.


\({\eta}^{n+1}\) : etaN ( DYNVARS.h )
\({\phi}^{n+1}_{\rm nh}\) : phi_nh ( NH_VARS.h )
\(u^*\) : gU ( DYNVARS.h )
\(v^*\) : gV ( DYNVARS.h )
\(u^{n+1}\) : uVel ( DYNVARS.h )
\(v^{n+1}\) : vVel ( DYNVARS.h )

Regarding the implementation of the surface pressure solver, all computation are done within the routine SOLVE_FOR_PRESSURE and its dependent calls. The standard method to solve the 2D elliptic problem (2.65) uses the conjugate gradient method (routine CG2D); the solver matrix and conjugate gradient operator are only function of the discretized domain and are therefore evaluated separately, before the time iteration loop, within INI_CG2D. The computation of the RHS \(\eta^*\) is partly done in CALC_DIV_GHAT and in SOLVE_FOR_PRESSURE.

The same method is applied for the non hydrostatic part, using a conjugate gradient 3D solver (CG3D) that is initialized in INI_CG3D. The RHS terms of 2D and 3D problems are computed together at the same point in the code.

2.11. Spatial discretization of the dynamical equations

Spatial discretization is carried out using the finite volume method. This amounts to a grid-point method (namely second-order centered finite difference) in the fluid interior but allows boundaries to intersect a regular grid allowing a more accurate representation of the position of the boundary. We treat the horizontal and vertical directions as separable and differently.

2.12. Continuity and horizontal pressure gradient term

The core algorithm is based on the “C grid” discretization of the continuity equation which can be summarized as:

(2.80)\[\partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{\rm nh}}{\Delta x_c} \delta_i \Phi_{\rm nh}' = G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h'\]
(2.81)\[\partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{\rm nh}}{\Delta y_c} \delta_j \Phi_{\rm nh}' = G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h'\]
(2.82)\[\epsilon_{\rm nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{\rm nh}' \right) = \epsilon_{\rm nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}'\]
(2.83)\[\delta_i \Delta y_g \Delta r_f h_w u + \delta_j \Delta x_g \Delta r_f h_s v + \delta_k {\cal A}_c w = {\cal A}_c \delta_k (\mathcal{P-E})_{r=0}\]

where the continuity equation has been most naturally discretized by staggering the three components of velocity as shown in Figure 2.7. The grid lengths \(\Delta x_c\) and \(\Delta y_c\) are the lengths between tracer points (cell centers). The grid lengths \(\Delta x_g\), \(\Delta y_g\) are the grid lengths between cell corners. \(\Delta r_f\) and \(\Delta r_c\) are the distance (in units of \(r\)) between level interfaces (w-level) and level centers (tracer level). The surface area presented in the vertical is denoted \({\cal A}_c\). The factors \(h_w\) and \(h_s\) are non-dimensional fractions (between 0 and 1) that represent the fraction cell depth that is “open” for fluid flow.

The last equation, the discrete continuity equation, can be summed in the vertical to yield the free-surface equation:

(2.84)\[{\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal A}_c(\mathcal{P-E})_{r=0}\]

The source term \(\mathcal{P-E}\) on the rhs of continuity accounts for the local addition of volume due to excess precipitation and run-off over evaporation and only enters the top-level of the ocean model.

2.13. Hydrostatic balance

The vertical momentum equation has the hydrostatic or quasi-hydrostatic balance on the right hand side. This discretization guarantees that the conversion of potential to kinetic energy as derived from the buoyancy equation exactly matches the form derived from the pressure gradient terms when forming the kinetic energy equation.

In the ocean, using z-coordinates, the hydrostatic balance terms are discretized:

(2.85)\[\epsilon_{\rm nh} \partial_t w + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots\]

In the atmosphere, using p-coordinates, hydrostatic balance is discretized:

(2.86)\[\overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0\]

where \(\Delta \Pi\) is the difference in Exner function between the pressure points. The non-hydrostatic equations are not available in the atmosphere.

The difference in approach between ocean and atmosphere occurs because of the direct use of the ideal gas equation in forming the potential energy conversion term \(\alpha \omega\). Because of the different representation of hydrostatic balance between ocean and atmosphere there is no elegant way to represent both systems using an arbitrary coordinate.

The integration for hydrostatic pressure is made in the positive \(r\) direction (increasing k-index). For the ocean, this is from the free-surface down and for the atmosphere this is from the ground up.

The calculations are made in the subroutine CALC_PHI_HYD. Inside this routine, one of other of the atmospheric/oceanic form is selected based on the string variable buoyancyRelation.

2.14. Flux-form momentum equations

The original finite volume model was based on the Eulerian flux form momentum equations. This is the default though the vector invariant form is optionally available (and recommended in some cases).

The “G’s” (our colloquial name for all terms on rhs!) are broken into the various advective, Coriolis, horizontal dissipation, vertical dissipation and metric forces:

(2.87)\[G_u = G_u^{\rm adv} + G_u^{\rm Cor} + G_u^{h- \rm diss} + G_u^{v- \rm diss} + G_u^{\rm metric} + G_u^{\rm nh-metric}\]
(2.88)\[G_v = G_v^{\rm adv} + G_v^{\rm Cor} + G_v^{h- \rm diss} + G_v^{v- \rm diss} + G_v^{\rm metric} + G_v^{\rm nh-metric}\]
(2.89)\[G_w = G_w^{\rm adv} + G_w^{\rm Cor} + G_w^{h- \rm diss} + G_w^{v- \rm diss} + G_w^{\rm metric} + G_w^{\rm nh-metric}\]

In the hydrostatic limit, \(G_w=0\) and \(\epsilon_{\rm nh}=0\), reducing the vertical momentum to hydrostatic balance.

These terms are calculated in routines called from subroutine MOM_FLUXFORM and collected into the global arrays gU, gV, and gW.


\(G_u\) : gU ( DYNVARS.h )
\(G_v\) : gV ( DYNVARS.h )
\(G_w\) : gW ( NH_VARS.h )

2.14.1. Advection of momentum

The advective operator is second order accurate in space:

(2.90)\[{\cal A}_w \Delta r_f h_w G_u^{\rm adv} = \delta_i \overline{ U }^i \overline{ u }^i + \delta_j \overline{ V }^i \overline{ u }^j + \delta_k \overline{ W }^i \overline{ u }^k\]
(2.91)\[{\cal A}_s \Delta r_f h_s G_v^{\rm adv} = \delta_i \overline{ U }^j \overline{ v }^i + \delta_j \overline{ V }^j \overline{ v }^j + \delta_k \overline{ W }^j \overline{ v }^k\]
(2.92)\[{\cal A}_c \Delta r_c G_w^{\rm adv} = \delta_i \overline{ U }^k \overline{ w }^i + \delta_j \overline{ V }^k \overline{ w }^j + \delta_k \overline{ W }^k \overline{ w }^k\]

and because of the flux form does not contribute to the global budget of linear momentum. The quantities \(U\), \(V\) and \(W\) are volume fluxes defined:

(2.93)\[U = \Delta y_g \Delta r_f h_w u\]
(2.94)\[V = \Delta x_g \Delta r_f h_s v\]
(2.95)\[W = {\cal A}_c w\]

The advection of momentum takes the same form as the advection of tracers but by a translated advective flow. Consequently, the conservation of second moments, derived for tracers later, applies to \(u^2\) and \(v^2\) and \(w^2\) so that advection of momentum correctly conserves kinetic energy.


\(uu, vu, wu\) : fZon, fMer, fVerUkp ( local to MOM_FLUXFORM.F )


\(uv, vv, wv\) : fZon, fMer, fVerVkp ( local to MOM_FLUXFORM.F )

2.14.2. Coriolis terms

The “pure C grid” Coriolis terms (i.e. in absence of C-D scheme) are discretized:

(2.96)\[{\cal A}_w \Delta r_f h_w G_u^{\rm Cor} = \overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i - \epsilon_{\rm nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i\]
(2.97)\[{\cal A}_s \Delta r_f h_s G_v^{\rm Cor} = - \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j\]
(2.98)\[{\cal A}_c \Delta r_c G_w^{\rm Cor} = \epsilon_{\rm nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k\]

where the Coriolis parameters \(f\) and \(f'\) are defined:

\[\begin{split}\begin{aligned} f & = 2 \Omega \sin{\varphi} \\ f' & = 2 \Omega \cos{\varphi}\end{aligned}\end{split}\]

where \(\varphi\) is geographic latitude when using spherical geometry, otherwise the \(\beta\)-plane definition is used:

\[\begin{split}\begin{aligned} f & = f_o + \beta y \\ f' & = 0\end{aligned}\end{split}\]

This discretization globally conserves kinetic energy. It should be noted that despite the use of this discretization in former publications, all calculations to date have used the following different discretization:

(2.99)\[G_u^{\rm Cor} = f_u \overline{ v }^{ji} - \epsilon_{\rm nh} f_u' \overline{ w }^{ik}\]
(2.100)\[G_v^{\rm Cor} = - f_v \overline{ u }^{ij}\]
(2.101)\[G_w^{\rm Cor} = \epsilon_{\rm nh} f_w' \overline{ u }^{ik}\]

where the subscripts on \(f\) and \(f'\) indicate evaluation of the Coriolis parameters at the appropriate points in space. The above discretization does not conserve anything, especially energy, but for historical reasons is the default for the code. A flag controls this discretization: set run-time integer selectCoriScheme to two (=2) (which otherwise defaults to zero) to select the energy-conserving conserving form (2.96), (2.97), and (2.98) above.


\(G_u^{\rm Cor}, G_v^{\rm Cor}\) : cF ( local to MOM_FLUXFORM.F )

2.14.3. Curvature metric terms

The most commonly used coordinate system on the sphere is the geographic system \((\lambda,\varphi)\). The curvilinear nature of these coordinates on the sphere lead to some “metric” terms in the component momentum equations. Under the thin-atmosphere and hydrostatic approximations these terms are discretized:

(2.102)\[{\cal A}_w \Delta r_f h_w G_u^{\rm metric} = \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i\]
(2.103)\[\begin{split}{\cal A}_s \Delta r_f h_s G_v^{\rm metric} = - \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\\end{split}\]
(2.104)\[G_w^{\rm metric} = 0\]

where \(a\) is the radius of the planet (sphericity is assumed) or the radial distance of the particle (i.e. a function of height). It is easy to see that this discretization satisfies all the properties of the discrete Coriolis terms since the metric factor \(\frac{u}{a} \tan{\varphi}\) can be viewed as a modification of the vertical Coriolis parameter: \(f \rightarrow f+\frac{u}{a} \tan{\varphi}\).

However, as for the Coriolis terms, a non-energy conserving form has exclusively been used to date:

\[\begin{split}\begin{aligned} G_u^{\rm metric} & = \frac{u \overline{v}^{ij} }{a} \tan{\varphi} \\ G_v^{\rm metric} & = \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\varphi}\end{aligned}\end{split}\]

where \(\tan{\varphi}\) is evaluated at the \(u\) and \(v\) points respectively.


\(G_u^{\rm metric}, G_v^{\rm metric}\) : mT ( local to MOM_FLUXFORM.F )

2.14.4. Non-hydrostatic metric terms

For the non-hydrostatic equations, dropping the thin-atmosphere approximation re-introduces metric terms involving \(w\) which are required to conserve angular momentum:

(2.105)\[{\cal A}_w \Delta r_f h_w G_u^{\rm metric} = - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i\]
(2.106)\[{\cal A}_s \Delta r_f h_s G_v^{\rm metric} = - \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j\]
(2.107)\[{\cal A}_c \Delta r_c G_w^{\rm metric} = \overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k\]

Because we are always consistent, even if consistently wrong, we have, in the past, used a different discretization in the model which is:

\[\begin{split}\begin{aligned} G_u^{\rm metric} & = - \frac{u}{a} \overline{w}^{ik} \\ G_v^{\rm metric} & = - \frac{v}{a} \overline{w}^{jk} \\ G_w^{\rm metric} & = \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )\end{aligned}\end{split}\]


\(G_u^{\rm metric}, G_v^{\rm metric}\) : mT ( local to MOM_FLUXFORM.F )

2.14.5. Lateral dissipation

Historically, we have represented the SGS Reynolds stresses as simply down gradient momentum fluxes, ignoring constraints on the stress tensor such as symmetry.

(2.108)\[{\cal A}_w \Delta r_f h_w G_u^{h- \rm diss} = \delta_i \Delta y_f \Delta r_f h_c \tau_{11} + \delta_j \Delta x_v \Delta r_f h_\zeta \tau_{12}\]
(2.109)\[{\cal A}_s \Delta r_f h_s G_v^{h- \rm diss} = \delta_i \Delta y_u \Delta r_f h_\zeta \tau_{21} + \delta_j \Delta x_f \Delta r_f h_c \tau_{22}\]

The lateral viscous stresses are discretized:

(2.110)\[\tau_{11} = A_h c_{11\Delta}(\varphi) \frac{1}{\Delta x_f} \delta_i u -A_4 c_{11\Delta^2}(\varphi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u\]
(2.111)\[\tau_{12} = A_h c_{12\Delta}(\varphi) \frac{1}{\Delta y_u} \delta_j u -A_4 c_{12\Delta^2}(\varphi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u\]
(2.112)\[\tau_{21} = A_h c_{21\Delta}(\varphi) \frac{1}{\Delta x_v} \delta_i v -A_4 c_{21\Delta^2}(\varphi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v\]
(2.113)\[\tau_{22} = A_h c_{22\Delta}(\varphi) \frac{1}{\Delta y_f} \delta_j v -A_4 c_{22\Delta^2}(\varphi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v\]

where the non-dimensional factors \(c_{lm\Delta^n}(\varphi), \{l,m,n\} \in \{1,2\}\) define the “cosine” scaling with latitude which can be applied in various ad-hoc ways. For instance, \(c_{11\Delta} = c_{21\Delta} = (\cos{\varphi})^{3/2}\), \(c_{12\Delta}=c_{22\Delta}=1\) would represent the anisotropic cosine scaling typically used on the “lat-lon” grid for Laplacian viscosity.

It should be noted that despite the ad-hoc nature of the scaling, some scaling must be done since on a lat-lon grid the converging meridians make it very unlikely that a stable viscosity parameter exists across the entire model domain.

The Laplacian viscosity coefficient, \(A_h\) (viscAh), has units of \(m^2 s^{-1}\). The bi-harmonic viscosity coefficient, \(A_4\) (viscA4), has units of \(m^4 s^{-1}\).


\(\tau_{11}, \tau_{12}\) : vF, v4F ( local to MOM_FLUXFORM.F )


\(\tau_{21}, \tau_{22}\) : vF, v4F ( local to MOM_FLUXFORM.F )

Two types of lateral boundary condition exist for the lateral viscous terms, no-slip and free-slip.

The free-slip condition is most convenient to code since it is equivalent to zero-stress on boundaries. Simple masking of the stress components sets them to zero. The fractional open stress is properly handled using the lopped cells.

The no-slip condition defines the normal gradient of a tangential flow such that the flow is zero on the boundary. Rather than modify the stresses by using complicated functions of the masks and “ghost” points (see Adcroft and Marshall (1998) [AM98]) we add the boundary stresses as an additional source term in cells next to solid boundaries. This has the advantage of being able to cope with “thin walls” and also makes the interior stress calculation (code) independent of the boundary conditions. The “body” force takes the form:

(2.114)\[G_u^{\rm side-drag} = \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j \left( A_h c_{12\Delta}(\varphi) u - A_4 c_{12\Delta^2}(\varphi) \nabla^2 u \right)\]
(2.115)\[G_v^{\rm side-drag} = \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i \left( A_h c_{21\Delta}(\varphi) v - A_4 c_{21\Delta^2}(\varphi) \nabla^2 v \right)\]

In fact, the above discretization is not quite complete because it assumes that the bathymetry at velocity points is deeper than at neighboring vorticity points, e.g. \(1-h_w < 1-h_\zeta\)


\(G_u^{\rm side-drag}, G_v^{\rm side-drag}\) : vF ( local to MOM_FLUXFORM.F )

2.14.6. Vertical dissipation

Vertical viscosity terms are discretized with only partial adherence to the variable grid lengths introduced by the finite volume formulation. This reduces the formal accuracy of these terms to just first order but only next to boundaries; exactly where other terms appear such as linear and quadratic bottom drag.

(2.116)\[G_u^{v- \rm diss} = \frac{1}{\Delta r_f h_w} \delta_k \tau_{13}\]
(2.117)\[G_v^{v- \rm diss} = \frac{1}{\Delta r_f h_s} \delta_k \tau_{23}\]
(2.118)\[G_w^{v- \rm diss} = \epsilon_{\rm nh} \frac{1}{\Delta r_f h_d} \delta_k \tau_{33}\]

represents the general discrete form of the vertical dissipation terms.

In the interior the vertical stresses are discretized:

\[\begin{split}\begin{aligned} \tau_{13} & = A_v \frac{1}{\Delta r_c} \delta_k u \\ \tau_{23} & = A_v \frac{1}{\Delta r_c} \delta_k v \\ \tau_{33} & = A_v \frac{1}{\Delta r_f} \delta_k w\end{aligned}\end{split}\]

It should be noted that in the non-hydrostatic form, the stress tensor is even less consistent than for the hydrostatic (see Wajsowicz (1993) [Waj93]). It is well known how to do this properly (see Griffies and Hallberg (2000) [GH00]) and is on the list of to-do’s.


\(\tau_{13}\) : fVrUp, fVrDw ( local to MOM_FLUXFORM.F )
\(\tau_{23}\) : fVrUp, fVrDw ( local to MOM_FLUXFORM.F )

As for the lateral viscous terms, the free-slip condition is equivalent to simply setting the stress to zero on boundaries. The no-slip condition is implemented as an additional term acting in conjunction with the interior and free-slip stresses. Bottom drag represents additional friction, in addition to that imposed by the no-slip condition at the bottom. The drag is cast as a stress expressed as a linear or quadratic function of the mean flow in the layer above the topography:

(2.119)\[\tau_{13}^{\rm bottom-drag} = \left( 2 A_v \frac{1}{\Delta r_c} + r_b + C_d \sqrt{ \overline{2 \mathrm{KE}}^i } \right) u\]
(2.120)\[\tau_{23}^{\rm bottom-drag} = \left( 2 A_v \frac{1}{\Delta r_c} + r_b + C_d \sqrt{ \overline{2 \mathrm{KE}}^j } \right) v\]

where these terms are only evaluated immediately above topography. \(r_b\) (bottomDragLinear) has units of m s-1 and a typical value of the order 0.0002 m s-1. \(C_d\) (bottomDragQuadratic) is dimensionless with typical values in the range 0.001–0.003.

After defining ALLOW_BOTTOMDRAG_ROUGHNESS in MOM_COMMON_OPTIONS.h, the quadratic drag coefficient can be computed by the logarithmic law of the wall:

(2.121)\[u(z) = \left(\frac{\tau}{\rho}\right)^{\frac{1}{2}}\frac{1}{0.4} \ln{\frac{z+z_r}{z_r}}\]

where \(z_r\) is the roughness length (runtime parameter zRoughBot). Here, \(z\) is the height from the seafloor and \(\tau\) is the bottom stress (and stress in the log-layer). The velocity is computed at the center of the bottom cell \(z_b=\frac{1}{2}\Delta r_f h_w\), so stress on the bottom cell is \(\tau / \rho = C_d u_b^2\), where \(u_b = u(z_b)\) is the bottom cell velocity and:

(2.122)\[C_d = \left(\frac{0.4}{ \ln{\frac{\frac{1}{2} \Delta r_f h_w + z_{r} }{z_{r}}}}\right)^2\]

This formulation assumes that the bottommost cell is sufficiently thin that it is in a constant-stress log layer. A value of zRoughBot of 0.01 m yields a quadratic drag coefficient of 0.0022 for \(\Delta r_f =\) 100 m, or a quadratic drag coefficient of 0.0041 for \(\Delta r_f =\) 10 m.

For \(z_r = 0\), \(C_d\) defaults to the value of bottomDragQuadratic.


\(\tau_{13}^{\rm bottom-drag} / \Delta r_f , \tau_{23}^{\rm bottom-drag} / \Delta r_f\) : vF ( local to MOM_FLUXFORM.F )

2.14.7. Derivation of discrete energy conservation

These discrete equations conserve kinetic plus potential energy using the following definitions:

(2.123)\[\mathrm{KE} = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j + \epsilon_{\rm nh} \overline{ w^2 }^k \right)\]

2.14.8. Mom Diagnostics

<-Name->|Levs|<-parsing code->|<--  Units   -->|<- Tile (max=80c)
VISCAHZ | 15 |SZ      MR      |m^2/s           |Harmonic Visc Coefficient (m2/s) (Zeta Pt)
VISCA4Z | 15 |SZ      MR      |m^4/s           |Biharmonic Visc Coefficient (m4/s) (Zeta Pt)
VISCAHD | 15 |SM      MR      |m^2/s           |Harmonic Viscosity Coefficient (m2/s) (Div Pt)
VISCA4D | 15 |SM      MR      |m^4/s           |Biharmonic Viscosity Coefficient (m4/s) (Div Pt)
VAHZMAX | 15 |SZ      MR      |m^2/s           |CFL-MAX Harm Visc Coefficient (m2/s) (Zeta Pt)
VA4ZMAX | 15 |SZ      MR      |m^4/s           |CFL-MAX Biharm Visc Coefficient (m4/s) (Zeta Pt)
VAHDMAX | 15 |SM      MR      |m^2/s           |CFL-MAX Harm Visc Coefficient (m2/s) (Div Pt)
VA4DMAX | 15 |SM      MR      |m^4/s           |CFL-MAX Biharm Visc Coefficient (m4/s) (Div Pt)
VAHZMIN | 15 |SZ      MR      |m^2/s           |RE-MIN Harm Visc Coefficient (m2/s) (Zeta Pt)
VA4ZMIN | 15 |SZ      MR      |m^4/s           |RE-MIN Biharm Visc Coefficient (m4/s) (Zeta Pt)
VAHDMIN | 15 |SM      MR      |m^2/s           |RE-MIN Harm Visc Coefficient (m2/s) (Div Pt)
VA4DMIN | 15 |SM      MR      |m^4/s           |RE-MIN Biharm Visc Coefficient (m4/s) (Div Pt)
VAHZLTH | 15 |SZ      MR      |m^2/s           |Leith Harm Visc Coefficient (m2/s) (Zeta Pt)
VA4ZLTH | 15 |SZ      MR      |m^4/s           |Leith Biharm Visc Coefficient (m4/s) (Zeta Pt)
VAHDLTH | 15 |SM      MR      |m^2/s           |Leith Harm Visc Coefficient (m2/s) (Div Pt)
VA4DLTH | 15 |SM      MR      |m^4/s           |Leith Biharm Visc Coefficient (m4/s) (Div Pt)
VAHZLTHD| 15 |SZ      MR      |m^2/s           |LeithD Harm Visc Coefficient (m2/s) (Zeta Pt)
VA4ZLTHD| 15 |SZ      MR      |m^4/s           |LeithD Biharm Visc Coefficient (m4/s) (Zeta Pt)
VAHDLTHD| 15 |SM      MR      |m^2/s           |LeithD Harm Visc Coefficient (m2/s) (Div Pt)
VA4DLTHD| 15 |SM      MR      |m^4/s           |LeithD Biharm Visc Coefficient (m4/s) (Div Pt)
VAHZSMAG| 15 |SZ      MR      |m^2/s           |Smagorinsky Harm Visc Coefficient (m2/s) (Zeta Pt)
VA4ZSMAG| 15 |SZ      MR      |m^4/s           |Smagorinsky Biharm Visc Coeff. (m4/s) (Zeta Pt)
VAHDSMAG| 15 |SM      MR      |m^2/s           |Smagorinsky Harm Visc Coefficient (m2/s) (Div Pt)
VA4DSMAG| 15 |SM      MR      |m^4/s           |Smagorinsky Biharm Visc Coeff. (m4/s) (Div Pt)
momKE   | 15 |SM      MR      |m^2/s^2         |Kinetic Energy (in momentum Eq.)
momHDiv | 15 |SM      MR      |s^-1            |Horizontal Divergence (in momentum Eq.)
momVort3| 15 |SZ      MR      |s^-1            |3rd component (vertical) of Vorticity
Strain  | 15 |SZ      MR      |s^-1            |Horizontal Strain of Horizontal Velocities
Tension | 15 |SM      MR      |s^-1            |Horizontal Tension of Horizontal Velocities
UBotDrag| 15 |UU   129MR      |m/s^2           |U momentum tendency from Bottom Drag
VBotDrag| 15 |VV   128MR      |m/s^2           |V momentum tendency from Bottom Drag
USidDrag| 15 |UU   131MR      |m/s^2           |U momentum tendency from Side Drag
VSidDrag| 15 |VV   130MR      |m/s^2           |V momentum tendency from Side Drag
Um_Diss | 15 |UU   133MR      |m/s^2           |U momentum tendency from Dissipation
Vm_Diss | 15 |VV   132MR      |m/s^2           |V momentum tendency from Dissipation
Um_Advec| 15 |UU   135MR      |m/s^2           |U momentum tendency from Advection terms
Vm_Advec| 15 |VV   134MR      |m/s^2           |V momentum tendency from Advection terms
Um_Cori | 15 |UU   137MR      |m/s^2           |U momentum tendency from Coriolis term
Vm_Cori | 15 |VV   136MR      |m/s^2           |V momentum tendency from Coriolis term
Um_Ext  | 15 |UU   137MR      |m/s^2           |U momentum tendency from external forcing
Vm_Ext  | 15 |VV   138MR      |m/s^2           |V momentum tendency from external forcing
Um_AdvZ3| 15 |UU   141MR      |m/s^2           |U momentum tendency from Vorticity Advection
Vm_AdvZ3| 15 |VV   140MR      |m/s^2           |V momentum tendency from Vorticity Advection
Um_AdvRe| 15 |UU   143MR      |m/s^2           |U momentum tendency from vertical Advection (Explicit part)
Vm_AdvRe| 15 |VV   142MR      |m/s^2           |V momentum tendency from vertical Advection (Explicit part)
ADVx_Um | 15 |UM   145MR      |m^4/s^2         |Zonal      Advective Flux of U momentum
ADVy_Um | 15 |VZ   144MR      |m^4/s^2         |Meridional Advective Flux of U momentum
ADVrE_Um| 15 |WU      LR      |m^4/s^2         |Vertical   Advective Flux of U momentum (Explicit part)
ADVx_Vm | 15 |UZ   148MR      |m^4/s^2         |Zonal      Advective Flux of V momentum
ADVy_Vm | 15 |VM   147MR      |m^4/s^2         |Meridional Advective Flux of V momentum
ADVrE_Vm| 15 |WV      LR      |m^4/s^2         |Vertical   Advective Flux of V momentum (Explicit part)
VISCx_Um| 15 |UM   151MR      |m^4/s^2         |Zonal      Viscous Flux of U momentum
VISCy_Um| 15 |VZ   150MR      |m^4/s^2         |Meridional Viscous Flux of U momentum
VISrE_Um| 15 |WU      LR      |m^4/s^2         |Vertical   Viscous Flux of U momentum (Explicit part)
VISrI_Um| 15 |WU      LR      |m^4/s^2         |Vertical   Viscous Flux of U momentum (Implicit part)
VISCx_Vm| 15 |UZ   155MR      |m^4/s^2         |Zonal      Viscous Flux of V momentum
VISCy_Vm| 15 |VM   154MR      |m^4/s^2         |Meridional Viscous Flux of V momentum
VISrE_Vm| 15 |WV      LR      |m^4/s^2         |Vertical   Viscous Flux of V momentum (Explicit part)
VISrI_Vm| 15 |WV      LR      |m^4/s^2         |Vertical   Viscous Flux of V momentum (Implicit part)

2.15. Vector invariant momentum equations

The finite volume method lends itself to describing the continuity and tracer equations in curvilinear coordinate systems. However, in curvilinear coordinates many new metric terms appear in the momentum equations (written in Lagrangian or flux-form) making generalization far from elegant. Fortunately, an alternative form of the equations, the vector invariant equations are exactly that; invariant under coordinate transformations so that they can be applied uniformly in any orthogonal curvilinear coordinate system such as spherical coordinates, boundary following or the conformal spherical cube system.

The non-hydrostatic vector invariant equations read:

(2.124)\[\partial_t \vec{\bf v} + ( 2\vec{\boldsymbol{\Omega}} + \vec{\boldsymbol{\zeta}}) \times \vec{\bf v} - b \hat{\bf r} + \nabla B = \nabla \cdot \vec{\boldsymbol{\tau}}\]

which describe motions in any orthogonal curvilinear coordinate system. Here, \(B\) is the Bernoulli function and \(\vec{\boldsymbol{\zeta}}= \nabla \times \vec{\bf v}\) is the vorticity vector. We can take advantage of the elegance of these equations when discretizing them and use the discrete definitions of the grad, curl and divergence operators to satisfy constraints. We can also consider the analogy to forming derived equations, such as the vorticity equation, and examine how the discretization can be adjusted to give suitable vorticity advection among other things.

The underlying algorithm is the same as for the flux form equations. All that has changed is the contents of the “G’s”. For the time-being, only the hydrostatic terms have been coded but we will indicate the points where non-hydrostatic contributions will enter:

(2.125)\[G_u = G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B} + G_u^{\partial_z \tau^x} + G_u^{h- \rm diss} + G_u^{v- \rm diss}\]
(2.126)\[G_v = G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B} + G_v^{\partial_z \tau^y} + G_v^{h- \rm diss} + G_v^{v- \rm diss}\]
(2.127)\[G_w = G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B} + G_w^{h- \rm diss} + G_w^{v- \rm diss}\]


\(G_u\) : gU ( DYNVARS.h )
\(G_v\) : gV ( DYNVARS.h )
\(G_w\) : gW ( NH_VARS.h )

2.15.1. Relative vorticity

The vertical component of relative vorticity is explicitly calculated and use in the discretization. The particular form is crucial for numerical stability; alternative definitions break the conservation properties of the discrete equations.

Relative vorticity is defined:

(2.128)\[\zeta_3 = \frac{\Gamma}{A_\zeta} = \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u )\]

where \({\cal A}_\zeta\) is the area of the vorticity cell presented in the vertical and \(\Gamma\) is the circulation about that cell.


\(\zeta_3\) : vort3 ( local to MOM_VECINV.F )

2.15.2. Kinetic energy

The kinetic energy, denoted \(\mathrm{KE}\), is defined:

(2.129)\[\mathrm{KE} = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j + \epsilon_{\rm nh} \overline{ w^2 }^k )\]


\(\mathrm{KE}\) : KE ( local to MOM_VECINV.F )

2.15.3. Coriolis terms

The potential enstrophy conserving form of the linear Coriolis terms are written:

(2.130)\[G_u^{fv} = \frac{1}{\Delta x_c} \overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i\]
(2.131)\[G_v^{fu} = - \frac{1}{\Delta y_c} \overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j\]

Here, the Coriolis parameter \(f\) is defined at vorticity (corner) points.

The potential enstrophy conserving form of the non-linear Coriolis terms are written:

(2.132)\[G_u^{\zeta_3 v} = \frac{1}{\Delta x_c} \overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i\]
(2.133)\[G_v^{\zeta_3 u} = - \frac{1}{\Delta y_c} \overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j\]

The Coriolis terms can also be evaluated together and expressed in terms of absolute vorticity \(f+\zeta_3\). The potential enstrophy conserving form using the absolute vorticity is written:

(2.134)\[G_u^{fv} + G_u^{\zeta_3 v} = \frac{1}{\Delta x_c} \overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i\]
(2.135)\[G_v^{fu} + G_v^{\zeta_3 u} = - \frac{1}{\Delta y_c} \overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j\]

The distinction between using absolute vorticity or relative vorticity is useful when constructing higher order advection schemes; monotone advection of relative vorticity behaves differently to monotone advection of absolute vorticity. Currently the choice of relative/absolute vorticity, centered/upwind/high order advection is available only through commented subroutine calls.


\(G_u^{fv} , G_u^{\zeta_3 v}\) : uCf ( local to MOM_VECINV.F )
\(G_v^{fu} , G_v^{\zeta_3 u}\) : vCf ( local to MOM_VECINV.F )

2.15.4. Shear terms

The shear terms (\(\zeta_2w\) and \(\zeta_1w\)) are are discretized to guarantee that no spurious generation of kinetic energy is possible; the horizontal gradient of Bernoulli function has to be consistent with the vertical advection of shear:

(2.136)\[G_u^{\zeta_2 w} = \frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{ \overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{\rm nh} \delta_i w ) }^k\]
(2.137)\[G_v^{\zeta_1 w} = \frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{ \overline{ {\cal A}_c w }^j ( \delta_k v - \epsilon_{\rm nh} \delta_j w ) }^k\]


\(G_u^{\zeta_2 w}\) : uCf ( local to MOM_VECINV.F )
\(G_v^{\zeta_1 w}\) : vCf ( local to MOM_VECINV.F )

2.15.5. Gradient of Bernoulli function

(2.138)\[G_u^{\partial_x B} = \frac{1}{\Delta x_c} \delta_i ( \phi' + \mathrm{KE} )\]
(2.139)\[G_v^{\partial_y B} = \frac{1}{\Delta x_y} \delta_j ( \phi' + \mathrm{KE} )\]


\(G_u^{\partial_x \mathrm{KE}}\) : uCf ( local to MOM_VECINV.F )
\(G_v^{\partial_y \mathrm{KE}}\) : vCf ( local to MOM_VECINV.F )

2.15.6. Horizontal divergence

The horizontal divergence, a complimentary quantity to relative vorticity, is used in parameterizing the Reynolds stresses and is discretized:

(2.140)\[D = \frac{1}{{\cal A}_c h_c} ( \delta_i \Delta y_g h_w u + \delta_j \Delta x_g h_s v )\]


\(D\) : hDiv ( local to MOM_VECINV.F )

2.15.7. Horizontal dissipation

The following discretization of horizontal dissipation conserves potential vorticity (thickness weighted relative vorticity) and divergence and dissipates energy, enstrophy and divergence squared:

(2.141)\[G_u^{h- \rm diss} = \frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*) - \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* )\]
(2.142)\[G_v^{h- \rm diss} = \frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* ) + \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* )\]


\[\begin{split}\begin{aligned} D^* & = \frac{1}{{\cal A}_c h_c} ( \delta_i \Delta y_g h_w \nabla^2 u + \delta_j \Delta x_g h_s \nabla^2 v ) \\ \zeta^* & = \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c \nabla^2 v - \delta_j \Delta x_c \nabla^2 u )\end{aligned}\end{split}\]


\(G_u^{h-dissip}\) : uDissip ( local to MOM_VI_HDISSIP.F )
\(G_v^{h-dissip}\) : vDissip ( local to MOM_VI_HDISSIP.F )

2.15.8. Vertical dissipation

Currently, this is exactly the same code as the flux form equations.

(2.143)\[G_u^{v- \rm diss} = \frac{1}{\Delta r_f h_w} \delta_k \tau_{13}\]
(2.144)\[G_v^{v- \rm diss} = \frac{1}{\Delta r_f h_s} \delta_k \tau_{23}\]

represents the general discrete form of the vertical dissipation terms.

In the interior the vertical stresses are discretized:

\[\begin{split}\begin{aligned} \tau_{13} & = A_v \frac{1}{\Delta r_c} \delta_k u \\ \tau_{23} & = A_v \frac{1}{\Delta r_c} \delta_k v\end{aligned}\end{split}\]


\(\tau_{13}, \tau_{23}\) : vrf ( local to MOM_VECINV.F )

2.16. Tracer equations

The basic discretization used for the tracer equations is the second order piece-wise constant finite volume form of the forced advection-diffusion equations. There are many alternatives to second order method for advection and alternative parameterizations for the sub-grid scale processes. The Gent-McWilliams eddy parameterization, KPP mixing scheme and PV flux parameterization are all dealt with in separate sections. The basic discretization of the advection-diffusion part of the tracer equations and the various advection schemes will be described here.

2.16.1. Time-stepping of tracers: ABII

The default advection scheme is the centered second order method which requires a second order or quasi-second order time-stepping scheme to be stable. Historically this has been the quasi-second order Adams-Bashforth method (ABII) and applied to all terms. For an arbitrary tracer, \(\tau\), the forced advection-diffusion equation reads:

(2.145)\[\partial_t \tau + G_{\rm adv}^\tau = G_{\rm diff}^\tau + G_{\rm forc}^\tau\]

where \(G_{\rm adv}^\tau\), \(G_{\rm diff}^\tau\) and \(G_{\rm forc}^\tau\) are the tendencies due to advection, diffusion and forcing, respectively, namely:

(2.146)\[G_{\rm adv}^\tau = \partial_x (u \tau) + \partial_y (v \tau) + \partial_r (w \tau) - \tau \nabla \cdot {\bf v}\]
(2.147)\[G_{\rm diff}^\tau = \nabla \cdot \left ( {\bf K} \nabla \tau \right )\]

and the forcing can be some arbitrary function of state, time and space.

The term, \(\tau \nabla \cdot {\bf v}\), is required to retain local conservation in conjunction with the linear implicit free-surface. It only affects the surface layer since the flow is non-divergent everywhere else. This term is therefore referred to as the surface correction term. Global conservation is not possible using the flux-form (as here) and a linearized free-surface (Griffies and Hallberg (2000) [GH00] , Campin et al. (2004) [CAHM04]).

The continuity equation can be recovered by setting \(G_{\rm diff}=G_{\rm forc}=0\) and \(\tau=1\).

The driver routine that calls the routines to calculate tendencies are CALC_GT and CALC_GS for temperature and salt (moisture), respectively. These in turn call a generic advection diffusion routine GAD_CALC_RHS that is called with the flow field and relevant tracer as arguments and returns the collective tendency due to advection and diffusion. Forcing is add subsequently in CALC_GT or CALC_GS to the same tendency array.


\(\tau\) : tau ( argument )
\(G^{(n)}\) : gTracer ( argument )
\(F_r\) : fVerT ( argument )

The space and time discretization are treated separately (method of lines). Tendencies are calculated at time levels \(n\) and \(n-1\) and extrapolated to \(n+1/2\) using the Adams-Bashforth method:

(2.148)\[G^{(n+1/2)} = (\tfrac{3}{2} + \epsilon) G^{(n)} - (\tfrac{1}{2} + \epsilon) G^{(n-1)}\]

where \(G^{(n)} = G_{\rm adv}^\tau + G_{\rm diff}^\tau + G_{\rm src}^\tau\) at time step \(n\). The tendency at \(n-1\) is not re-calculated but rather the tendency at \(n\) is stored in a global array for later re-use.


\(G^{(n+1/2)}\) : gTracer ( argument on exit )
\(G^{(n)}\) : gTracer ( argument on entry )
\(G^{(n-1)}\) : gTrNm1 ( argument )
\(\epsilon\) : ABeps ( PARAMS.h )

The tracers are stepped forward in time using the extrapolated tendency:

(2.149)\[\tau^{(n+1)} = \tau^{(n)} + \Delta t G^{(n+1/2)}\]


\(\tau^{(n+1)}\) : gTracer ( argument on exit )
\(\tau^{(n)}\) : tracer ( argument on entry )
\(G^{(n+1/2)}\) : gTracer ( argument )
\(\Delta t\) : deltaTtracer ( PARAMS.h )

Strictly speaking the ABII scheme should be applied only to the advection terms. However, this scheme is only used in conjunction with the standard second, third and fourth order advection schemes. Selection of any other advection scheme disables Adams-Bashforth for tracers so that explicit diffusion and forcing use the forward method.

2.17. Advection schemes

2.18. Shapiro Filter

The Shapiro filter (Shapiro 1970) [Sha70] is a high order horizontal filter that efficiently remove small scale grid noise without affecting the physical structures of a field. It is applied at the end of the time step on both velocity and tracer fields.

Three different space operators are considered here (S1,S2 and S4). They differ essentially by the sequence of derivative in both X and Y directions. Consequently they show different damping response function specially in the diagonal directions X+Y and X-Y.

Space derivatives can be computed in the real space, taking into account the grid spacing. Alternatively, a pure computational filter can be defined, using pure numerical differences and ignoring grid spacing. This later form is stable whatever the grid is, and therefore specially useful for highly anisotropic grid such as spherical coordinate grid. A damping time-scale parameter \(\tau_{shap}\) defines the strength of the filter damping.

The three computational filter operators are :

\[\begin{split}\begin{aligned} & \mathrm{S1c:}\hspace{2cm} [1 - 1/2 \frac{\Delta t}{\tau_{\rm Shap}} \{ (\frac{1}{4}\delta_{ii})^n + (\frac{1}{4}\delta_{jj})^n \} ]\\ & \mathrm{S2c:}\hspace{2cm} [1 - \frac{\Delta t}{\tau_{\rm Shap}} \{ \frac{1}{8} (\delta_{ii} + \delta_{jj}) \}^n]\\ & \mathrm{S4c:}\hspace{2cm} [1 - \frac{\Delta t}{\tau_{\rm Shap}} (\frac{1}{4}\delta_{ii})^n] [1 - \frac{\Delta t}{\tau_{\rm Shap}} (\frac{1}{4}\delta_{jj})^n]\end{aligned}\end{split}\]

In addition, the S2 operator can easily be extended to a physical space filter:

\[\mathrm{S2g:}\hspace{2cm} [1 - \frac{\Delta t}{\tau_{\rm Shap}} \{ \frac{L_{\rm Shap}^2}{8} \overline{\nabla}^2 \}^n]\]

with the Laplacian operator \(\overline{\nabla}^2\) and a length scale parameter \(L_{\rm Shap}\). The stability of this S2g filter requires \(L_{\rm Shap} < \mathrm{Min}^{(\rm Global)}(\Delta x,\Delta y)\).

2.18.1. SHAP Diagnostics

<-Name->|Levs|parsing code|<-Units->|<- Tile (max=80c)
SHAP_dT |  5 |SM      MR  |K/s      |Temperature Tendency due to Shapiro Filter
SHAP_dS |  5 |SM      MR  |g/kg/s   |Specific Humidity Tendency due to Shapiro Filter
SHAP_dU |  5 |UU   148MR  |m/s^2    |Zonal Wind Tendency due to Shapiro Filter
SHAP_dV |  5 |VV   147MR  |m/s^2    |Meridional Wind Tendency due to Shapiro Filter

2.19. Nonlinear Viscosities for Large Eddy Simulation

In Large Eddy Simulations (LES), a turbulent closure needs to be provided that accounts for the effects of subgridscale motions on the large scale. With sufficiently powerful computers, we could resolve the entire flow down to the molecular viscosity scales (\(L_{\nu}\approx 1 \rm cm\)). Current computation allows perhaps four decades to be resolved, so the largest problem computationally feasible would be about 10m. Most oceanographic problems are much larger in scale, so some form of LES is required, where only the largest scales of motion are resolved, and the subgridscale effects on the large-scale are parameterized.

To formalize this process, we can introduce a filter over the subgridscale L: \(u_\alpha\rightarrow \overline{u_\alpha}\) and L: \(b\rightarrow \overline{b}\). This filter has some intrinsic length and time scales, and we assume that the flow at that scale can be characterized with a single velocity scale (\(V\)) and vertical buoyancy gradient (\(N^2\)). The filtered equations of motion in a local Mercator projection about the gridpoint in question (see Appendix for notation and details of approximation) are:

(2.155)\[{\frac{{ \overline{D} {{\tilde {\overline{u}}}}}} {{\overline{Dt}}}} - \frac{{{\tilde {\overline{v}}}} \sin\theta}{{\rm Ro}\sin\theta_0} + \frac{{M_{Ro}}}{{\rm Ro}} \frac{\partial{\overline{\pi}}}{\partial{x}} = -\left({\overline{\frac{D{\tilde u}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{u}}}}}}{{\overline{Dt}}} }\right) +\frac{\nabla^2{{\tilde {\overline{u}}}}}{{\rm Re}}\]
(2.156)\[{\frac{{ \overline{D} {{\tilde {\overline{v}}}}}} {{\overline{Dt}}}} - \frac{{{\tilde {\overline{u}}}} \sin\theta}{{\rm Ro}\sin\theta_0} + \frac{{M_{Ro}}}{{\rm Ro}} \frac{\partial{\overline{\pi}}}{\partial{y}} = -\left({\overline{\frac{D{\tilde v}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{v}}}}}}{{\overline{Dt}}} }\right) +\frac{\nabla^2{{\tilde {\overline{v}}}}}{{\rm Re}}\]
(2.157)\[\frac{{\overline{D} \overline w}}{{\overline{Dt}}} + \frac{ \frac{\partial{\overline{\pi}}}{\partial{z}} - \overline b}{{\rm Fr}^2\lambda^2} = -\left(\overline{\frac{D{w}}{Dt}} - \frac{{\overline{D} \overline w}}{{\overline{Dt}}}\right) +\frac{\nabla^2 \overline w}{{\rm Re}}\]
(2.158)\[\frac{{\overline{D} \bar b}}{{\overline{Dt}}} + \overline w = -\left(\overline{\frac{D{b}}{Dt}} - \frac{{\overline{D} \bar b}}{{\overline{Dt}}}\right) +\frac{\nabla^2 \overline b}{\Pr{\rm Re}}\]
(2.159)\[\mu^2\left({\frac{\partial{\tilde {\overline{u}}}}{\partial{x}}} + {\frac{\partial{\tilde {\overline{v}}}}{\partial{y}}} \right) + {\frac{\partial{\overline w}}{\partial{z}}} = 0\]

Tildes denote multiplication by \(\cos\theta/\cos\theta_0\) to account for converging meridians.

The ocean is usually turbulent, and an operational definition of turbulence is that the terms in parentheses (the ’eddy’ terms) on the right of (2.155) - (2.158)) are of comparable magnitude to the terms on the left-hand side. The terms proportional to the inverse of , instead, are many orders of magnitude smaller than all of the other terms in virtually every oceanic application.

2.19.1. Eddy Viscosity

A turbulent closure provides an approximation to the ’eddy’ terms on the right of the preceding equations. The simplest form of LES is just to increase the viscosity and diffusivity until the viscous and diffusive scales are resolved. That is, we approximate (2.155) - (2.158):

(2.160)\[\left({\overline{\frac{D{\tilde u}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{u}}}}}}{{\overline{Dt}}} }\right) \approx\frac{\nabla^2_h{{\tilde {\overline{u}}}}}{{\rm Re}_h} +\frac{{\frac{\partial^2{{\tilde {\overline{u}}}}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.161)\[\left({\overline{\frac{D{\tilde v}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{v}}}}}}{{\overline{Dt}}} }\right) \approx\frac{\nabla^2_h{{\tilde {\overline{v}}}}}{{\rm Re}_h} +\frac{{\frac{\partial^2{{\tilde {\overline{v}}}}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.162)\[\left(\overline{\frac{D{w}}{Dt}} - \frac{{\overline{D} \overline w}}{{\overline{Dt}}}\right) \approx\frac{\nabla^2_h \overline w}{{\rm Re}_h} +\frac{{\frac{\partial^2{\overline w}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.163)\[\left(\overline{\frac{D{b}}{Dt}} - \frac{{\overline{D} \bar b}}{{\overline{Dt}}}\right) \approx\frac{\nabla^2_h \overline b}{\Pr{\rm Re}_h} +\frac{{\frac{\partial^2{\overline b}}{{\partial{z}}^2}}}{\Pr{\rm Re}_v}\] Reynolds-Number Limited Eddy Viscosity

One way of ensuring that the gridscale is sufficiently viscous (i.e., resolved) is to choose the eddy viscosity \(A_h\) so that the gridscale horizontal Reynolds number based on this eddy viscosity, \({\rm Re}_h\), is O(1). That is, if the gridscale is to be viscous, then the viscosity should be chosen to make the viscous terms as large as the advective ones. Bryan et al. (1975) [BMP75] notes that a computational mode is squelched by using \({\rm Re}_h<\)2.

MITgcm users can select horizontal eddy viscosities based on \({\rm Re}_h\) using two methods. 1) The user may estimate the velocity scale expected from the calculation and grid spacing and set viscAh to satisfy \({\rm Re}_h<2\). 2) The user may use viscAhReMax, which ensures that the viscosity is always chosen so that \({\rm Re}_h<\) viscAhReMax. This last option should be used with caution, however, since it effectively implies that viscous terms are fixed in magnitude relative to advective terms. While it may be a useful method for specifying a minimum viscosity with little effort, tests Bryan et al. (1975) [BMP75] have shown that setting viscAhReMax =2 often tends to increase the viscosity substantially over other more ’physical’ parameterizations below, especially in regions where gradients of velocity are small (and thus turbulence may be weak), so perhaps a more liberal value should be used, e.g. viscAhReMax =10.

While it is certainly necessary that viscosity be active at the gridscale, the wavelength where dissipation of energy or enstrophy occurs is not necessarily \(L=A_h/U\). In fact, it is by ensuring that either the dissipation of energy in a 3-d turbulent cascade (Smagorinsky) or dissipation of enstrophy in a 2-d turbulent cascade (Leith) is resolved that these parameterizations derive their physical meaning. Vertical Eddy Viscosities

Vertical eddy viscosities are often chosen in a more subjective way, as model stability is not usually as sensitive to vertical viscosity. Usually the ’observed’ value from finescale measurements is used (e.g. viscAr\(\approx1\times10^{-4} m^2/s\)). However, Smagorinsky (1993) [Sma93] notes that the Smagorinsky parameterization of isotropic turbulence implies a value of the vertical viscosity as well as the horizontal viscosity (see below). Smagorinsky Viscosity

Some suggest (see Smagorinsky 1963 [Sma63]; Smagorinsky 1993 [Sma93]) choosing a viscosity that depends on the resolved motions. Thus, the overall viscous operator has a nonlinear dependence on velocity. Smagorinsky chose his form of viscosity by considering Kolmogorov’s ideas about the energy spectrum of 3-d isotropic turbulence.

Kolmogorov supposed that energy is injected into the flow at large scales (small \(k\)) and is ’cascaded’ or transferred conservatively by nonlinear processes to smaller and smaller scales until it is dissipated near the viscous scale. By setting the energy flux through a particular wavenumber \(k\), \(\epsilon\), to be a constant in \(k\), there is only one combination of viscosity and energy flux that has the units of length, the Kolmogorov wavelength. It is \(L_\epsilon(\nu)\propto\pi\epsilon^{-1/4}\nu^{3/4}\) (the \(\pi\) stems from conversion from wavenumber to wavelength). To ensure that this viscous scale is resolved in a numerical model, the gridscale should be decreased until \(L_\epsilon(\nu)>L\) (so-called Direct Numerical Simulation, or DNS). Alternatively, an eddy viscosity can be used and the corresponding Kolmogorov length can be made larger than the gridscale, \(L_\epsilon(A_h)\propto\pi\epsilon^{-1/4}A_h^{3/4}\) (for Large Eddy Simulation or LES).

There are two methods of ensuring that the Kolmogorov length is resolved in MITgcm. 1) The user can estimate the flux of energy through spectral space for a given simulation and adjust grid spacing or viscAh to ensure that \(L_\epsilon(A_h)>L\); 2) The user may use the approach of Smagorinsky with viscC2Smag, which estimates the energy flux at every grid point, and adjusts the viscosity accordingly.

Smagorinsky formed the energy equation from the momentum equations by dotting them with velocity. There are some complications when using the hydrostatic approximation as described by Smagorinsky (1993) [Sma93]. The positive definite energy dissipation by horizontal viscosity in a hydrostatic flow is \(\nu D^2\), where D is the deformation rate at the viscous scale. According to Kolmogorov’s theory, this should be a good approximation to the energy flux at any wavenumber \(\epsilon\approx\nu D^2\). Kolmogorov and Smagorinsky noted that using an eddy viscosity that exceeds the molecular value \(\nu\) should ensure that the energy flux through viscous scale set by the eddy viscosity is the same as it would have been had we resolved all the way to the true viscous scale. That is, \(\epsilon\approx A_{h \rm Smag} \overline D^2\). If we use this approximation to estimate the Kolmogorov viscous length, then

(2.164)\[L_\epsilon(A_{h \rm Smag})\propto\pi\epsilon^{-1/4}A_{h \rm Smag}^{3/4}\approx\pi(A_{h \rm Smag} \overline D^2)^{-1/4}A_{h\rm Smag}^{3/4} = \pi A_{h\rm Smag}^{1/2}\overline D^{-1/2}\]

To make \(L_\epsilon(A_{h\rm Smag})\) scale with the gridscale, then

(2.165)\[A_{h\rm Smag} = \left(\frac{{\sf viscC2Smag}}{\pi}\right)^2L^2|\overline D|\]

Where the deformation rate appropriate for hydrostatic flows with shallow-water scaling is

(2.166)\[|\overline D|=\sqrt{\left({\frac{\partial{\overline {\tilde u}}}{\partial{x}}} - {\frac{\partial{\overline {\tilde v}}}{\partial{y}}}\right)^2 + \left({\frac{\partial{\overline {\tilde u}}}{\partial{y}}} + {\frac{\partial{\overline {\tilde v}}}{\partial{x}}}\right)^2}\]

The coefficient viscC2Smag is what an MITgcm user sets, and it replaces the proportionality in the Kolmogorov length with an equality. Others (Griffies and Hallberg, 2000 [GH00]) suggest values of viscC2Smag from 2.2 to 4 for oceanic problems. Smagorinsky (1993) [Sma93] shows that values from 0.2 to 0.9 have been used in atmospheric modeling.

Smagorinsky (1993) [Sma93] shows that a corresponding vertical viscosity should be used:

(2.167)\[A_{v \rm Smag} = \left(\frac{{\sf viscC2Smag}}{\pi}\right)^2H^2 \sqrt{\left({\frac{\partial{\overline {\tilde u}}}{\partial{z}}}\right)^2 + \left({\frac{\partial{\overline {\tilde v}}}{\partial{z}}}\right)^2}\]

This vertical viscosity is currently not implemented in MITgcm. Leith Viscosity

Leith (1968, 1996) [Lei68] [Lei96] notes that 2-D turbulence is quite different from 3-D. In 2-D turbulence, energy cascades to larger scales, so there is no concern about resolving the scales of energy dissipation. Instead, another quantity, enstrophy, (which is the vertical component of vorticity squared) is conserved in 2-D turbulence, and it cascades to smaller scales where it is dissipated.

Following a similar argument to that above about energy flux, the enstrophy flux is estimated to be equal to the positive-definite gridscale dissipation rate of enstrophy \(\eta\approx A_{h \rm Leith} |\nabla\overline \omega_3|^2\). By dimensional analysis, the enstrophy-dissipation scale is \(L_\eta(A_{h \rm Leith})\propto\pi A_{h \rm Leith}^{1/2}\eta^{-1/6}\). Thus, the Leith-estimated length scale of enstrophy-dissipation and the resulting eddy viscosity are

(2.168)\[L_\eta(A_{h \rm Leith})\propto\pi A_{h \rm Leith}^{1/2}\eta^{-1/6} = \pi A_{h \rm Leith}^{1/3}| \nabla \overline \omega_3|^{-1/3}\]
(2.169)\[A_{h \rm Leith} = \left(\frac{{\sf viscC2Leith}}{\pi}\right)^3L^3| \nabla \overline\omega_3|\]
(2.170)\[| \nabla \omega_3| \equiv \sqrt{\left[{\frac{\partial{\ }}{\partial{x}}} \left({\frac{\partial{\overline {\tilde v}}}{\partial{x}}} - {\frac{\partial{\overline {\tilde u}}}{\partial{y}}}\right)\right]^2 + \left[{\frac{\partial{\ }}{\partial{y}}}\left({\frac{\partial{\overline {\tilde v}}}{\partial{x}}} - {\frac{\partial{\overline {\tilde u}}}{\partial{y}}}\right)\right]^2}\]

The runtime flag useFullLeith controls whether or not to calculate the full gradients for the Leith viscosity (.TRUE.) or to use an approximation (.FALSE.). The only reason to set useFullLeith = .FALSE. is if your simulation fails when computing the gradients. This can occur when using the cubed sphere and other complex grids. Modified Leith Viscosity

The argument above for the Leith viscosity parameterization uses concepts from purely 2-dimensional turbulence, where the horizontal flow field is assumed to be non-divergent. However, oceanic flows are only quasi-two dimensional. While the barotropic flow, or the flow within isopycnal layers may behave nearly as two-dimensional turbulence, there is a possibility that these flows will be divergent. In a high-resolution numerical model, these flows may be substantially divergent near the grid scale, and in fact, numerical instabilities exist which are only horizontally divergent and have little vertical vorticity. This causes a difficulty with the Leith viscosity, which can only respond to buildup of vorticity at the grid scale.

MITgcm offers two options for dealing with this problem. 1) The Smagorinsky viscosity can be used instead of Leith, or in conjunction with Leith – a purely divergent flow does cause an increase in Smagorinsky viscosity; 2) The viscC2LeithD parameter can be set. This is a damping specifically targeting purely divergent instabilities near the gridscale. The combined viscosity has the form:

(2.171)\[A_{h \rm Leith} = L^3\sqrt{\left(\frac{{\sf viscC2Leith}}{\pi}\right)^6 | \nabla \overline \omega_3|^2 + \left(\frac{{\sf viscC2LeithD}}{\pi}\right)^6 | \nabla ( \nabla \cdot \overline {\tilde u}_h)|^2}\]
(2.172)\[| \nabla ( \nabla \cdot \overline {\tilde u}_h)| \equiv \sqrt{\left[{\frac{\partial{\ }}{\partial{x}}}\left({\frac{\partial{\overline {\tilde u}}}{\partial{x}}} + {\frac{\partial{\overline {\tilde v}}}{\partial{y}}}\right)\right]^2 + \left[{\frac{\partial{\ }}{\partial{y}}}\left({\frac{\partial{\overline {\tilde u}}}{\partial{x}}} + {\frac{\partial{\overline {\tilde v}}}{\partial{y}}}\right)\right]^2}\]

Whether there is any physical rationale for this correction is unclear, but the numerical consequences are good. The divergence in flows with the grid scale larger or comparable to the Rossby radius is typically much smaller than the vorticity, so this adjustment only rarely adjusts the viscosity if viscC2LeithD = viscC2Leith. However, the rare regions where this viscosity acts are often the locations for the largest vales of vertical velocity in the domain. Since the CFL condition on vertical velocity is often what sets the maximum timestep, this viscosity may substantially increase the allowable timestep without severely compromising the verity of the simulation. Tests have shown that in some calculations, a timestep three times larger was allowed when viscC2LeithD = viscC2Leith. Quasi-Geostrophic Leith Viscosity

A variant of Leith viscosity can be derived for quasi-geostrophic dynamics. This leads to a slightly different equation for the viscosity that includes a contribution from quasigeostrophic vortex stretching (Bachman et al. 2017 [BFKP17]). The viscosity is given by

(2.173)\[\nu_{*} = \left(\frac{\Lambda \Delta s}{\pi}\right)^{3} \left| \nabla _{h}(f\mathbf{\hat{z}}) + \nabla _{h}( \nabla \times \mathbf{v}_{h*}) + \partial_{z}\frac{f}{N^{2}} \nabla _{h} b \right|\]

where \(\Lambda\) is a tunable parameter of \(\mathcal{O}(1)\), \(\Delta s = \sqrt{\Delta x \Delta y}\) is the grid scale, \(f\mathbf{\hat{z}}\) is the vertical component of the Coriolis parameter, \(\mathbf{v}_{h*}\) is the horizontal velocity, \(N^{2}\) is the Brunt-Väisälä frequency, and \(b\) is the buoyancy.

However, the viscosity given by (2.173) does not constrain purely divergent motions. As such, a small \(\mathcal{O}(\epsilon)\) correction is added

(2.174)\[\nu_{*} = \left(\frac{\Lambda \Delta s}{\pi}\right)^{3} \sqrt{\left| \nabla _{h}(f\mathbf{\hat{z}}) + \nabla _{h}( \nabla \times \mathbf{v}_{h*}) + \partial_{z} \frac{f}{N^{2}} \nabla _{h} b\right|^{2} + | \nabla ( \nabla \cdot \mathbf{v}_{h})|^{2}}\]

This form is, however, numerically awkward; as the Brunt-Väisälä Frequency becomes very small in regions of weak or vanishing stratification, the vortex stretching term becomes very large. The resulting large viscosities can lead to numerical instabilities. Bachman et al. (2017) [BFKP17] present two limiting forms for the viscosity based on flow parameters such as \(Fr_{*}\), the Froude number, and \(Ro_{*}\), the Rossby number. The second of which,

(2.175)\[\begin{split}\begin{aligned} \nu_{*} = & \left(\frac{\Lambda \Delta s}{\pi}\right)^{3} \\ & \sqrt{\min \left( \left| \nabla _{h}q_{2*} + \partial_{z} \frac{f^{2}}{N^{2}} \nabla _{h} b \right|, \left( 1 + \frac{Fr_{*}^{2}}{Ro_{*}^{2}} + Fr_{*}^{4}\right) | \nabla _{h}q_{2*}|\right)^{2} + | \nabla ( \nabla \cdot \mathbf{v}_{h}) |^{2}} \end{aligned}\end{split}\]

has been implemented and is active when #define ALLOW_LEITH_QG is included in a copy of MOM_COMMON_OPTIONS.h in a code mods directory (specified through -mods command line option in genmake2).

LeithQG viscosity is designed to work best in simulations that resolve some mesoscale features. In simulations that are too coarse to permit eddies or fine enough to resolve submesoscale features, it should fail gracefully. The non-dimensional parameter viscC2LeithQG corresponds to \(\Lambda\) in the above equations and scales the viscosity; the recommended value is 1.

There is no reason to use the quasi-geostrophic form of Leith at the same time as either standard Leith or modified Leith. Therefore, the model will not run if non-zero values have been set for these coefficients; the model will stop during the configuration check. LeithQG can be used regardless of the setting for useFullLeith. Just as for the other forms of Leith viscosity, this flag determines whether or not the full gradients are used. The simplified gradients were originally intended for use on complex grids, but have been shown to produce better kinetic energy spectra even on very straightforward grids.

To add the LeithQG viscosity to the GMRedi coefficient, as was done in some of the simulations in Bachman et al. (2017) [BFKP17], #define ALLOW_LEITH_QG must be specified, as described above. In addition to this, the compile-time flag ALLOW_GM_LEITH_QG must also be defined in a (-mods) copy of GMREDI_OPTIONS.h when the model is compiled, and the runtime parameter GM_useLeithQG set to .TRUE. in data.gmredi. This will use the value of viscC2LeithQG specified in the data input file to compute the coefficient. Courant–Freidrichs–Lewy Constraint on Viscosity

Whatever viscosities are used in the model, the choice is constrained by gridscale and timestep by the Courant–Freidrichs–Lewy (CFL) constraint on stability:

\[\begin{split}\begin{aligned} A_h & < \frac{L^2}{4\Delta t} \\ A_4 & \le \frac{L^4}{32\Delta t}\end{aligned}\end{split}\]

The viscosities may be automatically limited to be no greater than these values in MITgcm by specifying viscAhGridMax \(<1\) and viscA4GridMax \(<1\). Similarly-scaled minimum values of viscosities are provided by viscAhGridMin and viscA4GridMin, which if used, should be set to values \(\ll 1\). \(L\) is roughly the gridscale (see below).

Following Griffies and Hallberg (2000) [GH00], we note that there is a factor of \(\Delta x^2/8\) difference between the harmonic and biharmonic viscosities. Thus, whenever a non-dimensional harmonic coefficient is used in the MITgcm (e.g. viscAhGridMax \(<1\)), the biharmonic equivalent is scaled so that the same non-dimensional value can be used (e.g. viscA4GridMax \(<1\)). Biharmonic Viscosity

Holland (1978) [Hol78] suggested that eddy viscosities ought to be focused on the dynamics at the grid scale, as larger motions would be ’resolved’. To enhance the scale selectivity of the viscous operator, he suggested a biharmonic eddy viscosity instead of a harmonic (or Laplacian) viscosity:

(2.176)\[\left({\overline{\frac{D{\tilde u}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{u}}}}}}{{\overline{Dt}}} }\right) \approx \frac{-\nabla^4_h{{\tilde {\overline{u}}}}}{{\rm Re}_4} + \frac{{\frac{\partial^2{{\tilde {\overline{u}}}}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.177)\[\left({\overline{\frac{D{\tilde v}}{Dt} }} - {\frac{{\overline{D} {{\tilde {\overline{v}}}}}}{{\overline{Dt}}} }\right) \approx \frac{-\nabla^4_h{{\tilde {\overline{v}}}}}{{\rm Re}_4} + \frac{{\frac{\partial^2{{\tilde {\overline{v}}}}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.178)\[\left(\overline{\frac{D{w}}{Dt}} - \frac{{\overline{D} \overline w}}{{\overline{Dt}}}\right) \approx\frac{-\nabla^4_h\overline w}{{\rm Re}_4} + \frac{{\frac{\partial^2{\overline w}}{{\partial{z}}^2}}}{{\rm Re}_v}\]
(2.179)\[\left(\overline{\frac{D{b}}{Dt}} - \frac{{\overline{D} \bar b}}{{\overline{Dt}}}\right) \approx \frac{-\nabla^4_h \overline b}{\Pr{\rm Re}_4} +\frac{{\frac{\partial^2{\overline b}}{{\partial{z}}^2}}}{\Pr{\rm Re}_v}\]

Griffies and Hallberg (2000) [GH00] propose that if one scales the biharmonic viscosity by stability considerations, then the biharmonic viscous terms will be similarly active to harmonic viscous terms at the gridscale of the model, but much less active on larger scale motions. Similarly, a biharmonic diffusivity can be used for less diffusive flows.

In practice, biharmonic viscosity and diffusivity allow a less viscous, yet numerically stable, simulation than harmonic viscosity and diffusivity. However, there is no physical rationale for such operators being of leading order, and more boundary conditions must be specified than for the harmonic operators. If one considers the approximations of (2.160) - (2.163) and (2.176) - (2.179) to be terms in the Taylor series expansions of the eddy terms as functions of the large-scale gradient, then one can argue that both harmonic and biharmonic terms would occur in the series, and the only question is the choice of coefficients. Using biharmonic viscosity alone implies that one zeros the first non-vanishing term in the Taylor series, which is unsupported by any fluid theory or observation.

Nonetheless, MITgcm supports a plethora of biharmonic viscosities and diffusivities, which are controlled with parameters named similarly to the harmonic viscosities and diffusivities with the substitution h \(\rightarrow 4\) in the MITgcm parameter name. MITgcm also supports biharmonic Leith and Smagorinsky viscosities:

(2.180)\[A_{4 \rm Smag} = \left(\frac{{\sf viscC4Smag}}{\pi}\right)^2\frac{L^4}{8}|D|\]
(2.181)\[A_{4 \rm Leith} = \frac{L^5}{8}\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^6 | \nabla \overline \omega_3|^2 + \left(\frac{{\sf viscC4LeithD}}{\pi}\right)^6 | \nabla ( \nabla \cdot \overline {\bf {\tilde u}}_h)|^2}\]

However, it should be noted that unlike the harmonic forms, the biharmonic scaling does not easily relate to whether energy-dissipation or enstrophy-dissipation scales are resolved. If similar arguments are used to estimate these scales and scale them to the gridscale, the resulting biharmonic viscosities should be:

(2.182)\[A_{4 \rm Smag} = \left(\frac{{\sf viscC4Smag}}{\pi}\right)^5L^5 |\nabla^2\overline {\bf {\tilde u}}_h|\]
(2.183)\[A_{4 \rm Leith} = L^6\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^{12} |\nabla^2 \overline \omega_3|^2 + \left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12} |\nabla^2 ( \nabla \cdot \overline {\bf {\tilde u}}_h)|^2}\]

Thus, the biharmonic scaling suggested by Griffies and Hallberg (2000) [GH00] implies:

\[\begin{split}\begin{aligned} |D| & \propto L|\nabla^2\overline {\bf {\tilde u}}_h|\\ | \nabla \overline \omega_3| & \propto L|\nabla^2 \overline \omega_3|\end{aligned}\end{split}\]

It is not at all clear that these assumptions ought to hold. Only the Griffies and Hallberg (2000) [GH00] forms are currently implemented in MITgcm. Selection of Length Scale

Above, the length scale of the grid has been denoted \(L\). However, in strongly anisotropic grids, \(L_x\) and \(L_y\) will be quite different in some locations. In that case, the CFL condition suggests that the minimum of \(L_x\) and \(L_y\) be used. On the other hand, other viscosities which involve whether a particular wavelength is ’resolved’ might be better suited to use the maximum of \(L_x\) and \(L_y\). Currently, MITgcm uses useAreaViscLength to select between two options. If false, the square root of the harmonic mean of \(L^2_x\) and \(L^2_y\) is used for all viscosities, which is closer to the minimum and occurs naturally in the CFL constraint. If useAreaViscLength is true, then the square root of the area of the grid cell is used.

2.19.2. Mercator, Nondimensional Equations

The rotating, incompressible, Boussinesq equations of motion (Gill, 1982) [Gil82] on a sphere can be written in Mercator projection about a latitude \(\theta_0\) and geopotential height \(z=r-r_0\). The nondimensional form of these equations is:

(2.184)\[{\rm Ro} \frac{D{\tilde u}}{Dt} - \frac{{\tilde v} \sin\theta}{\sin\theta_0}+M_{Ro}{\frac{\partial{\pi}}{\partial{x}}} + \frac{\lambda{\rm Fr}^2 M_{Ro}\cos \theta}{\mu\sin\theta_0} w = -\frac{{\rm Fr}^2 M_{Ro} {\tilde u} w}{r/H} + \frac{{\rm Ro} {\bf \hat x}\cdot\nabla^2{\bf u}}{{\rm Re}}\]
(2.185)\[{\rm Ro} \frac{D{\tilde v}}{Dt} + \frac{{\tilde u}\sin\theta}{\sin\theta_0} + M_{Ro}{\frac{\partial{\pi}}{\partial{y}}} = -\frac{\mu{\rm Ro} \tan\theta({\tilde u}^2 + {\tilde v}^2)}{r/L} - \frac{{\rm Fr}^2M_{Ro} {\tilde v} w}{r/H} + \frac{{\rm Ro} {\bf \hat y}\cdot\nabla^2{\bf u}}{{\rm Re}}\]
(2.186)\[{\rm Fr}^2\lambda^2\frac{D{w}}{Dt} - b + {\frac{\partial{\pi}}{\partial{z}}} -\frac{\lambda\cot \theta_0 {\tilde u}}{M_{Ro}} = \frac{\lambda\mu^2({\tilde u}^2+{\tilde v}^2)}{M_{Ro}(r/L)} + \frac{{\rm Fr}^2\lambda^2{\bf \hat z}\cdot\nabla^2{\bf u}}{{\rm Re}}\]
(2.187)\[\frac{D{b}}{Dt} + w = \frac{\nabla^2 b}{\Pr{\rm Re}}\nonumber\]
(2.188)\[\mu^2\left({\frac{\partial{\tilde u}}{\partial{x}}} + {\frac{\partial{\tilde v}}{\partial{y}}} \right)+{\frac{\partial{w}}{\partial{z}}} = 0\]


\[\mu\equiv\frac{\cos\theta_0}{\cos\theta},\ \ \ {\tilde u}=\frac{u^*}{V\mu},\ \ \ {\tilde v}=\frac{v^*}{V\mu}\]
\[f_0\equiv2\Omega\sin\theta_0,\ \ \ %,\ \ \ \BFKDt\ \equiv \mu^2\left({\tilde u}\BFKpd x\ %+{\tilde v} \BFKpd y\ \right)+\frac{\rm Fr^2M_{Ro}}{\rm Ro} w\BFKpd z\ \frac{D}{Dt} \equiv \mu^2\left({\tilde u}\frac{\partial}{\partial x} +{\tilde v} \frac{\partial}{\partial y} \right) +\frac{\rm Fr^2M_{Ro}}{\rm Ro} w\frac{\partial}{\partial z}\]
\[x\equiv \frac{r}{L} \phi \cos \theta_0, \ \ \ y\equiv \frac{r}{L} \int_{\theta_0}^\theta \frac{\cos \theta_0 {\,\rm d\theta}'}{\cos\theta'}, \ \ \ z\equiv \lambda\frac{r-r_0}{L}\]
\[t^*=t \frac{L}{V},\ \ \ b^*= b\frac{V f_0M_{Ro}}{\lambda}\]
\[\pi^* = \pi V f_0 LM_{Ro},\ \ \ w^* = w V \frac{{\rm Fr}^2 \lambda M_{Ro}}{\rm Ro}\]
\[{\rm Ro} \equiv \frac{V}{f_0 L},\ \ \ M_{Ro}\equiv \max[1,\rm Ro]\]
\[{\rm Fr} \equiv \frac{V}{N \lambda L}, \ \ \ {\rm Re} \equiv \frac{VL}{\nu}, \ \ \ {\rm Pr} \equiv \frac{\nu}{\kappa}\]

Dimensional variables are denoted by an asterisk where necessary. If we filter over a grid scale typical for ocean models:

1m \(< L <\) 100km
0.0001 \(< \lambda <\) 1
0.001m/s \(< V <\) 1 m/s
\(f_0 <\) 0.0001 s -1
0.01 s -1 \(< N <\) 0.0001 s -1

these equations are very well approximated by

(2.189)\[\begin{split}{\rm Ro}\frac{D{\tilde u}}{Dt} - \frac{{\tilde v} \sin\theta}{\sin\theta_0}+M_{Ro}{\frac{\partial{\pi}}{\partial{x}}} = -\frac{\lambda{\rm Fr}^2M_{Ro}\cos \theta}{\mu\sin\theta_0} w + \frac{{\rm Ro}\nabla^2{{\tilde u}}}{{\rm Re}} \\\end{split}\]
(2.190)\[\begin{split}{\rm Ro}\frac{D{\tilde v}}{Dt} + \frac{{\tilde u}\sin\theta}{\sin\theta_0}+M_{Ro}{\frac{\partial{\pi}}{\partial{y}}} = \frac{{\rm Ro}\nabla^2{{\tilde v}}}{{\rm Re}} \\\end{split}\]
(2.191)\[{\rm Fr}^2\lambda^2\frac{D{w}}{Dt} - b + {\frac{\partial{\pi}}{\partial{z}}} = \frac{\lambda\cot \theta_0 {\tilde u}}{M_{Ro}} +\frac{{\rm Fr}^2\lambda^2\nabla^2w}{{\rm Re}}\]
(2.192)\[\frac{D{b}}{Dt} + w = \frac{\nabla^2 b}{\Pr{\rm Re}}\]
(2.193)\[\mu^2\left({\frac{\partial{\tilde u}}{\partial{x}}} + {\frac{\partial{\tilde v}}{\partial{y}}} \right)+{\frac{\partial{w}}{\partial{z}}} = 0\]
\[\nabla^2 \approx \left(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\lambda^2\partial z^2}\right)\]

Neglecting the non-frictional terms on the right-hand side is usually called the ’traditional’ approximation. It is appropriate, with either large aspect ratio or far from the tropics. This approximation is used here, as it does not affect the form of the eddy stresses which is the main topic. The frictional terms are preserved in this approximate form for later comparison with eddy stresses.