1. Introduction

A key goal of large-scale ocean modeling where mesoscale eddies are permitted is to achieve simulations in which the natural tendency for oceanic flows to exhibit hydrodynamic instabilities and turbulence is not handicapped by overly strong frictional dissipation. That is, the desire is to allow dynamics at the resolved scales of motion to dominate the subgrid-scale parameterization. Such simulations impose fewer a priori assumptions on the flow associated with details of the subgrid-scale operator. Ideally, the result is a simulation that more faithfully represents the rich variety of observed oceanic flow regimes.

In practice, it is impossible to completely remove friction from a numerical simulation. In particular, ocean models require frictional dissipation in order to suppress instabilities such as those associated with the grid Reynolds number, to provide a vorticity sink at western boundaries, and to generally suppress power at unresolved scales. Indeed, in the presence of geostrophic turbulence, friction must be sufficient to absorb enstrophy upon cascading toward the smallest resolved scales. Finally, frictional dissipation can be included as a physical parameterization of the effect of unresolved scales on the resolved scales. Currently, however, there is no generally agreed upon form of a physically motivated closure for linear momentum in ocean models. Notably, Laplacian momentum friction has not been motivated from first principles at the scales resolved in large-scale ocean models.

Consequently, one can identify the two central roles of friction: numerical closure and physical parameterization. It is not uncommon for the friction to be described as though it were a physical parameterization of small scales, even though the chosen value of viscosity may be set by numerical stability considerations. However, these two very distinct considerations should be thought of separately in designing dissipation schemes for ocean models.

The focus here is on the numerical closure aspects of friction. Yet our inspiration stems from the ideas of Smagorinsky (1963, 1993), who was motivated from physically based turbulence closure arguments. The key point made in this paper is that the Smagorinsky approach, when combined with a biharmonic operator, is ideal for global ocean modeling in the presence of mesoscale variability. This combination provides an extremely effective device for use in maintaining numerical stability, without adversely affecting the resolved scales of flow.

Since the inertial range of a geostrophic turbulence cascade conserves total energy, it is desirable to have the frictional operator remove a minimal amount of kinetic energy from the scales of physical interest. That is, we wish friction to keep the flow within the bounds of numerical stability requirements without adversely affecting the resolved part of the spectrum where energy is concentrated. If set to a strength sufficient to suppress numerical instabilities, the Laplacian form of friction significantly removes both kinetic energy and enstrophy over a broad range of spatial scales. For this reason, more highly scale-selective frictional operators, such as biharmonic or higher-order operators, are often used for quasigeostrophic simulations (e.g., Holland 1978; Held et al. 1995) as well as primitive equation models (e.g., Semtner and Mintz 1977; Böning and Budich 1992). The anticipated potential vorticity approach of Sadourny and Basdevant (1985) goes one step further by formally conserving energy while dissipating enstrophy.

When using a biharmonic operator, the predominant practice in ocean modeling is to also employ a state-independent biharmonic viscosity. To maintain numerical stability, this coefficient must be set large enough to control the most vigorous anticipated motions. Such a choice may be suitable for simulations with one regime of flow, although it will lead to more dissipation than is numerically required when the flow exhibits temporal and spatial variability. When simulating realistic ocean flows that contain multiple space–time varying flow regimes and employ variable grid resolutions, such dissipation will generally be excessive in many important regions. The discussion by Gille (1997) illustrates this point in an analysis of the Southern Ocean momentum budget with an eddy-permitting global ocean simulation.

Smagorinsky (1963, 1993) proposed that the effective Laplacian viscosity due to unresolved scales should be proportional to the resolved horizontal deformation rate times the squared grid spacing. This scheme is a physically plausible parameterization of the effects of three-dimensional isotropic turbulence, and it has found much use in large-eddy simulations (e.g., see Galperin and Orszag 1993 for a compendium). For large-scale geophysical fluid simulations, however, it has little physical justification since the unresolved scales are dominated by quasi-two-dimensional geostrophic turbulence. For this reason, Leith (1968, 1996) proposed an alternative approach based on two-dimensional turbulence. Leith’s viscosity is proportional to the horizontal gradient of the relative vorticity times the cubed grid spacing. This approach has found some use in atmospheric models (e.g., Boer and Shepherd 1983), but it is not commonly used in ocean models.

The Smagorinsky scheme has found notable use in large-scale ocean models (e.g., Blumberg and Mellor 1987; Rosati and Miyakoda 1988; Bleck and Boudra 1981; Bleck et al. 1992) for pragmatic reasons. First, it is more convenient to compute than the Leith viscosity due to the smaller required grid stencil, and because the deformation rate used to compute the Smagorinsky viscosity is furthermore needed to compute the stress tensor. Second, as with the Leith scheme, the Smagorinsky viscosity tailors the local dissipation to both the local flow state and local grid resolution with only a single, nondimensional adjustable parameter. If this parameter is properly chosen, the resulting viscosity ensures that the flow respects the relevant numerical stability properties, even when simulating multiple flow and grid regimes such as occur in realistic ocean simulations.

Although the use of a Smagorinsky viscosity with a Laplacian frictional operator is less dissipative than a constant viscosity, we have found that the Laplacian form of the operator remains overzealous in its dissipation of large-scale energy. As illustrated in sections 4 and 5, the result can be overdissipation of the eddies and, hence, a suboptimal use of the model’s resolved scales. For this reason, we propose the use of a Smagorinsky-like viscosity with a biharmonic operator as an appropriate numerical closure scheme for eddy-permitting ocean models. A similar idea was mentioned by Bleck and Boudra (1981) and Sadourny and Maynard (1997). In this approach, the biharmonic viscosity is proportional to the resolved deformation rate times the grid spacing to the fourth power, |D|Δ⁴, rather than |D|Δ² as used with the Laplacian operator. This biharmonic Smagorinsky scheme retains the enhanced scale selectivity of a biharmonic viscosity, as well as the pragmatic aspects of the Smagorinsky scheme mentioned above.

Even without resolved eddies, the proposed biharmonic Smagorinsky scheme will be a useful numerical closure when combined with a physically based parameterization of unresolved turbulence. Consequently, observations and/or theory can be used to specify the form and magnitude of the physical parameterization, rather than also having to be concerned with numerical stability constraints. To the extent that the physical parameterization is less scale selective than the biharmonic operator, the dominant dissipative effect on the well-resolved scales will be associated with physics, not numerics. A recent illustration of this approach is given in the paper by Kazantsev et al. (1998), in which a constant viscosity Laplacian operator provided their numerical closure, and statistical mechanics provided their physical closure.

The organization of this paper is the following. Section 2 reviews the scale selectivity and numerical stability properties of the Laplacian and biharmonic operators in support of the previous discussion. Section 3 introduces the Smagorinsky Laplacian and biharmonic viscosities and discusses their properties according to the numerical stability constraints of section 2. Section 4 presents an eddy-permitting channel experiment using the Modular Ocean Model (MOM), which is a three-dimensional, geopotential-coordinate, primitive equation, B-grid ocean model (Pacanowski and Griffies 1999). This example serves to illustrate the overdissipation resulting from Laplacian friction relative to the biharmonic friction, even when using a Smagorinsky viscosity. Section 5 presents an example of an eddy-permitting midlatitude wind-driven sector experiment using the Hallberg isopycnal model (HIM), which is a three-dimensional, isopycnal-coordinate, primitive equation, C-grid ocean model (Hallberg 1995). This example illustrates the utility of the Smagorinsky approach in a model with meridional boundaries and hence a very inhomogeneous flow field. Section 6 provides a summary and conclusions. An appendix documents the Laplacian and biharmonic friction operators valid in any set of horizontal orthogonal coordinates and nonorthogonal vertical coordinates. We also present here a discussion of the discretization choices made when implementing these ideas on the Arakawa B and C grids.

2. Scale selectivity and numerical stability

The purpose of this section is to summarize the basic ideas regarding the scale selectivity of the biharmonic friction relative to the Laplacian friction, as well as to discuss some of the numerical stability considerations that constrain the size of the viscosity. Much of the discussion here is limited to the case of constant viscosity and a uniform grid. However, the ideas correspond locally to cases with a nonconstant viscosity used on a nonuniform grid.

a. Scale selectivity of the frictional operators

Although standard, it is nonetheless useful to begin our discussion with a simple illustration of the enhanced scale selectivity of the biharmonic operator over the Laplacian. For this purpose, we proceed much as Semtner and Mintz (1977) and Holland (1978) by considering the following linear equations in one Cartesian dimension using constant viscosities

View Expanded

with A > 0 the Laplacian viscosity, B > 0 the biharmonic viscosity, and ψ a model field, such as velocity. The evolution of a monochromatic grid wave ψ = c(t)e^ikx_n (where k is a wavenumber, x_n = nΔ are the grid points, and Δ is the uniform grid spacing) takes the form of a damped exponential. Using centered differences in space, the spindown times

View Expanded

have a ratio τ_L/τ_B that falls off in a roughly quadratic fashion as the wavenumber decreases and reaches a maximum at the highest resolved wavenumber k = π/Δ. Consequently, the longer waves, which are also better resolved, have longer spindown times with the biharmonic operator, given the same spindown time for the shorter waves. The wavenumber where the two operators exhibit the same spindown times depends on the values chosen for the viscosities. As discussed below, for reasons of diffusive stability we are motivated to consider B = A(Δ²/8) when relating the Smagorinsky Laplacian and biharmonic viscosities. With this choice, for example, grid-scale waves, with k = π/Δ, are damped twice as fast with the Laplacian operator than with the biharmonic operator, whereas 10Δ waves, with k = π/(10Δ), are damped 80 times faster, and 20Δ waves are damped 320 times faster. This behavior illustrates what is meant by scale selectivity—the biharmonic operator more strongly selects the small scales to dissipate and leaves the large scales relatively untouched.

c. Stability of the diffusion equation

Sufficient conditions for numerical stability of the linear equations (1) and (2) can be found using a von Neumann analysis (e.g., Haltiner and Williams 1980). For centered differences in space and forward differences in time, one finds for one spatial dimension, A < Δ²/(2Δt), whereas for two dimensions,

View Expanded

where Δ² = 2[(Δx)⁻² + (Δy)⁻²]⁻¹. Note that use of Δ = min(Δx, Δy) is conservative and is used in the following. For the biharmonic operator in two dimensions, a sufficient condition for stability is

View Expanded

where A_max = Δ²/(4Δt).

For global ocean models, the convergence of the meridians can make this constraint more limiting on the available time steps than the Courant–Freidrichs–Lewy (CFL) constraint. This situation occurs especially when one attempts to use a globally constant viscosity relevant for resolving the western boundary layer in the midlatitudes. A common practice to overcome this constraint is to taper the viscosity according to the grid spacing (e.g., Smith et al. 2000). The local grid dependence of the Smagorinsky viscosity naturally accounts for the convergence of the meridians, as well as other nonuniform grid configurations such as enhanced equatorial resolution.

d. Grid Reynolds number constraint

Instead of considering the balance between time tendency and friction, consider now a steady-state balance between advection and Laplacian friction

Uψ_xAψ_xx(8)

where U is a constant advection velocity. Using centered differences in space, Bryan et al. (1975) showed that the finite-difference counterpart of this equation will suppress a computational mode if

View Expanded

That is, the Reynolds number¹ based on the grid scale must be less than two. Similar considerations for the balance Uψ_x = −Bψ_xxxx leads to UΔ³/B < α, where α is generally tedious to determine, yet with α = 16 motivated by combining the constraints (7) and (9).

In coarse-resolution models, not satisfying the grid Reynolds number constraint typically ensures a noisy solution. In the presence of an eddying flow, independent of the discretization choice, a viscosity satisfying a similar constraint is necessary to prevent velocity self-advection from steepening velocity gradients into grid-scale velocity shocks, as described in one dimension by Burgers’s equation u_t + uu_x = Au_xx. Failure to do so can result in an unbounded solution.

3. The Smagorinsky viscosity

A length scale and timescale are sufficient to determine a kinematic viscosity. For the timescale, Smagorinsky (1963, 1993) chose the deformation rate

View Expanded

In Cartesian coordinates, the horizontal tension D_T = u_x − υ_y, and the horizontal shearing strain D_S = u_y + υ_x. Generalizations to arbitrary coordinates, as well as further discussion, are given in the appendix. For the length scale, Smagorinsky chose the maximum wavenumber admitted by the horizontal grid

L⁻¹π(11)

For generally anisotropic grids, we choose to set Δ ≡ min(Δx, Δy). The resulting viscosity is given by

A_smagCπ²D(12)

where C is a dimensionless scaling parameter. Smagorinsky, and various other researchers, have provided physically based arguments for specifying the value of C (see in particular Smagorinsky 1993 and references therein). For our purposes, it is empirically determined based on details of the numerical model.

Given these expressions for the Smagorinsky viscosity, it is useful to summarize how it obeys the three numerical stability considerations just described. It is sufficient to consider the case of a Laplacian operator in one spatial dimension. The three constraints can be summarized as

View Expanded

If the deformation rate scales as U/Δ, the Smagorinsky viscosity then has the scale

A_smagCπ²U(17)

The grid Reynolds number constraint (14) then implies that the Smagorinsky dimensionless scaling coefficient must satisfy C > π/2 ≈ 2.2. Taking the minimum value yields the Smagorinsky viscosity A_smag = UΔ/2, which satisfies the linear diffusion equation constraint (16) provided that the advection velocity satisfies the advective CFL constraint UΔt/Δ < 1. Note that ocean models are often more stringently constrained by Doppler-shifted gravity wave speeds U_g > U. In our applications, we have found values of C ≈ 3–4 to be suitable. Note that these values are larger than those suggested by Smagorinsky (1993) based on arguments from three-dimensional turbulence.

The constraints (14) and (15) suggest three distinct regimes for large-scale ocean models. When resolution is coarse and flows are weak, the need to resolve western boundary currents is the most stringent constraint on the level of viscosity. In this regime, we have found the Laplacian Smagorinsky scheme, using a minimum allowable viscosity set by the Munk boundary layer constraint, to be of use. In general, such a minimum set by frictional boundary layer constraints is necessary in the more quiet deeper ocean, where both the Laplacian and biharmonic Smagorinsky viscosities can be insufficient to resolve boundary currents. With more vigorous flows allowed with higher resolutions, but resolutions that are still too coarse to permit a well-represented geostrophic energy/enstrophy cascade, the Reynolds number constraint becomes more important than the western boundary layer constraint. In this case the biharmonic Smagorinsky scheme is an ideal numerical closure when combined with a physically based parameterization of the effects of the unresolved scales. With a Smagorinsky biharmonic scheme satisfying both the Reynolds number and western boundary constraints, the physically based viscosity can be set by theory or observations, rather than numerics. The third regime is one in which the resolution is sufficiently high, compared to the deformation radius, that the enstrophy cascade to small scales is well represented. In this case, the Smagorinsky biharmonic scheme alone is suitable.

4. A thermally forced channel model

This section illustrates how the use of a Laplacian friction operator, even with a Smagorinsky viscosity, can overdamp otherwise physically unstable and numerically well-resolved waves. The configuration we employ is a zonally reentrant, flat-bottom channel configuration using the primitive equation, geopotential-coordinate ocean model MOM (Pacanowski and Griffies 1999). The experiment is initialized from rest with a stratification achieved by running a coarse-resolution version for 4000 yr, at which time the vertical stratification is stationary and there is a large store of available potential energy. Thereafter, the solution is interpolated to a higher horizontal grid resolution, and a small amount of grid noise is added in order to initiate the growth of baroclinically unstable modes. A similar model configuration and experimental procedure has been used by Griffies et al. (2000). Further details are given in the caption to Fig. 1.

Shown in Fig. 1 are snapshots of the temperature field at 300 m and day 400. The left panel in Fig. 1 is the result when using the biharmonic Smagorinsky scheme, and the right panel in Fig. 1 uses a Laplacian Smagorinsky scheme. Each simulation used a Smagorinsky scaling coefficient C = 2. Day 400 occurs in the middle of the growing phase of the most unstable wave. By day 800, each solution has reached a peak in kinetic energy.

As suggested by the analysis in section 2a, we see that the Laplacian operator heavily dissipates the growing waves. Indeed, there is relatively little nonlinear wave activity for the Laplacian solution, whereas the biharmonic solution has reached a stage of strong nonlinear turbulent mixing. By day 800, the biharmonic solution has 18 times more kinetic energy than the Laplacian solution, and it has almost completed the familiar quasigeostrophic turbulence process of nonlinear transfer from the baroclinic modes to the barotropic mode (e.g., Rhines 1977; Salmon 1998).

Other experiments with the Laplacian operator were conducted to further investigate what was necessary to allow for a stronger growth of the unstable waves. To do so, we ran cases with much smaller values of the Smagorinsky scaling coefficient (as small as C = 0.5), as well as with various levels of constant horizontal viscosity. Consistently, for those cases that did generate nontrivial turbulence, the Laplacian solution was underdissipated and, so, was characterized by a substantial amount of power at the grid scale.

We conclude from this experiment that the analysis presented in section 2a has a great deal of relevance to the ability of a model to admit geostrophic turbulence through the growth of baroclinically unstable waves. The use of Laplacian friction, even when employing a Smagorinsky viscosity, suppresses the growth of the unstable waves. It either yields trivial levels of turbulence, or fails to dissipate the enstrophy cascade as it reaches the grid scale. Only by moving toward much higher grid resolution will the dissipation levels at the relevant wavelengths be small enough for the Laplacian friction to admit turbulence in this model, while still absorbing the associated enstrophy cascade.

For the current experiment, the biharmonic friction required roughly 10% more total computational time on a Cray T90 processor than the case with the Laplacian. This cost should be compared to that engendered by adding the higher grid resolution needed to permit an analogous level of turbulence with the Laplacian operator.

5. A wind-driven sector model

We now follow up the thermally forced channel simulations with wind-driven sector experiments using the primitive equation, isopycnal ocean model HIM (Hallberg 1995). Notably, the presence of meridional boundaries introduces strong inhomogeneities that are always present in realistic ocean simulations. These inhomogeneities provide the key difference from the channel experiments.

There are no diapycnal mass fluxes in these simulations, so the mean stratification in each simulation is identical. The western boundary current spawns a vigorous eddy field, even with a Laplacian viscosity. The wind forcing has a strength typical for the North Atlantic, and the Ekman pumping pattern drives a familiar two-gyre circulation. All of these simulations are run to 10 000 days, with a statistically steady state attained at about 4000 days. The resolution of ¹/₆° × ¹/₅° (a square grid of 18.5 km at 33.5°N) is adequate for representing the first baroclinic deformation radius of about 40 km. The vertical resolution of five isopycnal layers permits the development of baroclinic eddies with a reasonable vertical gyre structure.

The Smagorinsky biharmonic viscosity exhibits strong spatial variations (Fig. 2). The flow deformation rate (and hence the viscosity) is intense near the surface in the western boundary current region and around the separated western boundary current. Strong eddies are shed from the separated western boundary current, and these give rise to a broad region of enhanced time mean viscosity. In the eastern portion of the basin, and especially in the abyssal shadow region (Luyten et al. 1983), the viscosity is substantially smaller. In this simulation, there is a variation of over two orders of magnitude in the viscosity between the most and least energetic regions. In each region, the viscosity is adequate to maintain numerical integrity (i.e., satisfy the grid Reynolds number and frictional boundary layer constraints) without causing more dissipation than is necessary.

The same simulation was repeated with a constant biharmonic viscosity replacing the Smagorinsky viscosity. To determine the constant viscosity, we tried taking both the volumetric average of the Smagorinsky viscosity as well as the largest time mean value that was observed anywhere in the model domain. Use of either value resulted in a numerically unstable simulation. This experience illustrates that numerical stability requires a constant biharmonic viscosity that is able to control the largest deformation rates that occur at any place and time. Consequently, the time mean Smagorinsky biharmonic viscosity may be everywhere less than the required constant coefficient biharmonic viscosity! Further experimentation found that a constant viscosity of 2 × 10¹⁰ m⁴ s⁻¹ gave a numerically stable simulation.

The flow has slightly (3%) less kinetic energy and available potential energy averaged over 1000 days with the constant biharmonic viscosity than with the Smagorinsky viscosity. However, the two cases are qualitatively similar, and the difference is not statistically significant. This is reassuring, since there is a long tradition of using a constant coefficient biharmonic viscosity in eddy-permitting ocean model simulations. In these simulations, the turbulence and the eddy kinetic energy are dominated by the western boundary region. The differences between simulations using constant-coefficient and Smagorinsky biharmonic viscosities will be more pronounced when those simulations include multiple, independent turbulent regimes such as for the World Ocean. Also, it should be reemphasized that the appropriate value of the constant coefficient viscosity was only known after actually running several simulations. The Smagorinsky viscosity avoids the need for an a priori guess at this value.

As with the thermally driven channel, a Laplacian Smagorinsky scheme substantially suppresses the eddies relative to the biharmonic Smagorinsky scheme, as is readily seen by comparing instantaneous relative vorticities of an interior layer (Fig. 3). The eddy kinetic energy averaged over 1000 days is smaller by a factor of 4 with the Laplacian Smagorinsky scheme compared with the biharmonic Smagorinsky scheme, relative to the total kinetic energy and available potential energy of each simulation. The kinetic energy of the mean flow is reduced by less than 20% with the Laplacian, while the available potential energy is slightly (3%) larger with the Laplacian compared with the biharmonic. The difference in the levels of eddy activity between these simulations is due to suppression of the eddies by the Laplacian, not to a change in the baroclinic instability of the mean flow.

A simulation with half the resolution, but using the same nondimensional coefficient with the Smagorinsky biharmonic viscosity, gave perfectly sensible results; that is, the admitted scales remained resolved. The same would not occur with a constant coefficient biharmonic viscosity, which generally requires that the viscosity be increased as the resolution decreases. This illustrates the very convenient property of the Smagorinsky biharmonic scheme: only a single nondimensional coefficient needs to be set, and this coefficient is independent of resolution, so long as the same physical regime is present. It is only a function of the model’s numerics.

The CPU time required for the simulation with the Smagorinsky biharmonic scheme was only 3% more than with the constant coefficient biharmonic scheme and 4% more than with the Smagorinsky Laplacian scheme. Note that this difference is for a model without thermodynamics. For more realistic models, the relative cost of the scheme would be even less.

6. Summary and conclusions

This paper argues for the utility of a Smagorinsky-like viscosity for use with a biharmonic friction operator as a numerical closure scheme for eddy-permitting large-scale ocean models. This proposal is natural for one aiming to simulate geostrophic eddies in a realistic inhomogeneous environment such as the World Ocean. That is, the biharmonic operator, being more scale selective than the Laplacian operator, acts less on the large scales of motion, which are of direct interest and are better resolved, than the small scales, where enstrophy cascades. As an example of this scale selectivity, we ran a thermally forced channel that showed that the Laplacian operator, even when using a Smagorinsky viscosity, can unnecessarily suppress physically unstable modes that are well resolved by the model grid.

The utility of the Smagorinsky approach to determining viscosity follows from its providing a local deformation rate and grid-size-dependent viscosity that naturally adapts to the inhomogeneities present in realistic ocean simulations. An eddy-permitting sector model provided a vivid illustration of the ability of the Smagorinsky scheme to maintain the high levels of viscosity in the western boundary current, needed for numerical stability reasons, while allowing much lower levels in the more quiescent interior. In contrast, experiments run with constant biharmonic viscosities either underdissipated the flow, and tended to go numerically unstable, or dissipated the flow more than with the Smagorinsky scheme.

Ideally, the beneficial properties of the Smagorinsky scheme are realized by setting a single nondimensional constant. Sector experiments with different horizontal resolutions illustrated that the same value of this constant is appropriate regardless of the resolution, so long as similar physical features are present. Consequently, the Smagorinsky scheme effectively reduces the number of parameters that must be tuned when designing a new experiment. This aspect of the scheme can be quite important especially for large-scale eddy-permitting models. In practice, one should be aware of the need to resolve boundary currents, especially where the flow is less vigorous, since the Smagorinsky viscosity might be too small in these regions. This issue is more relevant for coarse models, but in general it may be necessary to set a minimum allowable viscosity determined by the relevant flow-independent frictional boundary layer scaling.

The biharmonic Smagorinsky scheme has little physical justification for large-scale geophysical simulations. Indeed, for that matter, neither does the more traditional Laplacian scheme. Yet, as argued here, the biharmonic scheme is an ideal numerical closure to which a physically based parameterization of the effects of unresolved scales may be added, if necessary. Assuming that the physical parameterization involves some form of a Laplacian, the enhanced scale selectivity of the biharmonic operator guarantees that at sufficiently large scales, the effects of resolved fluctuations and the physical subgrid parameterization will dominate the simulation, and the numerically based Smagorinsky biharmonic closure will be negligible.

On a Cray T90, the Smagorinsky scheme in our idealized examples required roughly 2%–4% more for the total model runtime relative to a constant horizontal viscosity. When using a biharmonic operator, the models ran some 5%–10% slower than with the Laplacian. In either case, the added cost should be gauged against the cost necessary to increase the grid resolution enough to admit comparably rich geostrophic eddy activity with the Laplacian operator and/or a constant viscosity. From this perspective, the biharmonic Smagorinsky scheme is very economical.