## 1. Introduction

Numerical models of ocean circulation must represent large-scale oceanic flows such as western boundary currents and the circumpolar current. In observations and in high-resolution numerical models, these currents are intimately bound to the mesoscale eddy field, which can transfer momentum vertically or horizontally. On the other hand, models of decadal to millennial climate variations are constrained by computational limitations to resolutions coarser than eddy length scales. In these models eddy effects must be parameterized.

The search for realistic parameterizations of mesoscale eddy effects has produced a number of recent studies (e.g., Gent and McWilliams 1990; Danabasoglu et al. 1994; Tandon and Garrett 1996; Visbeck et al. 1997;McDougall and McIntosh 1996; Gent and McWilliams 1996; Griffies 1998) that have focused primarily on the theoretical requirements for good mesoscale parameterizations or have described coarse-resolution fields obtained using a variety of parameterizations. The present study employs a different approach by, instead, running idealized model scenarios at a resolution high enough to resolve the mesoscale. We then examine the actual role of mesoscale eddies. Can they be represented using an adiabatic form as suggested by Gent and McWilliams (1990), and more generally, do any of the standard eddy parameterizations duplicate the eddy physics?

*ρ,*which is a dynamically active tracer governed by

**u**is the velocity,

*R*(

*ρ*) represents molecular diffusive processes that are negligible on the scales that we consider, and density sources are not included. The advection embodied in the substantial derivative

*Dρ*/

*Dt*is conservative. Water at any given location can move, and isopycnals may shift, but water parcels do not exchange density with the environment, so that the volume of water in any density class remains constant. In this sense

*Dρ*/

*Dt*will be termed adiabatic, although by strict definitions adiabatic processes should conserve heat rather than density. The operator

*R*(

*ρ*) represents diabatic processes that do alter the properties of individual water parcels. In the problems considered here

*R*(

*ρ*) is small.

*ρ*field, as in a climate model, it becomes

**u**′ and

*ρ*′. In a model without the resolution to compute eddy-scale deviations, the advective terms in

*S*(

**u**,

*ρ*) must be parameterized. The processes resolved by a coarse-resolution model represent oceanwide advection, while unresolved processes are typified by mesoscale eddies, which are often linked to baroclinic instability. Thus, a satisfactory subgrid-scale parameterization of oceanic mixing must imitate the mesoscale effects of eddies, such as those formed through instability.

Past work on parameterizations of small scales for climate modeling has relied on two hypotheses that we examine. The first hypothesis, specifically suggested by Gent and McWilliams (1990), is that evolution of the filtered large-scale density *ρ̂***u*** · **∇***ρ̂,***u*** is a parameterized, nondivergent velocity. In this study we examine how well **u*** · **∇***ρ̂**S*(**u**, *ρ*). We find that eddy-induced advection, which is adiabatic in the high-resolution model, will appear diabatic if we examine only the components resolved by coarse-resolution models. These effective diabatic processes are responsible for significant diapycnal eddy fluxes.

The second hypothesis examined is that the eddy-flux terms have a simple functional relationship to *ρ̂**S*(**u**, *ρ*) = *S*(*ρ̂*)*S*(**u**, *ρ*), though the parameterizations are more successful when results are time averaged.

These conclusions are based on using high-resolution model runs to analyze the advective terms represented by *S*(**u**, *ρ*). The model is the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model (MOM) (Pacanowski 1995), which is discussed in section 2. MOM was adapted for two physical scenarios outlined in section 3. The first is the simple example of baroclinic instability outlined by Eady (1949), and the second is a wind-forced channel model. Both cases are implemented in a reentrant channel on the *f* plane, and in both an alongchannel (zonal) average is used to define the low-resolution fields **û** and *ρ̂.*

## 2. Numerical model

The GFDL MOM is a primitive equation model that has been used extensively to investigate ocean circulation, including studies of the effects of the GM90 parameterization on coarse-resolution climate models (Danabasoglu et al. 1994; Danabasoglu and McWilliams 1995; McWilliams et al. 1996). The same model is also the heart of several near-eddy-resolving simulations of modern ocean circulation (FRAM Group 1991; Semtner and Chervin 1992).

The basic model physics and algorithms have been summarized by Semtner (1986) and Pacanowski (1995). Density evolution is described by a discretized form of (1) with *R*(*ρ*) represented by horizontal diffusion, vertical diffusion, and convective adjustment. For this study, the model domain is a reentrant channel on the *f* plane, at 50°S, extending approximately 2300 km in the zonal direction and 1150 km in the meridional direction. Zonal and meridional grid spacings are 17.7 km. A single-constituent equation of state is used in which density depends linearly on temperature. In this formulation, constraining tracer diffusion to take place along isopycnals, as formulated by Cox (1987), is equivalent to allowing no diffusion whatsoever. Table 1 summarizes key model parameters.

Horizontal and vertical viscosity and diffusion coefficients are adjusted to the minimum values needed to maintain model stability so that the mesoscale features of interest are minimally influenced by the parameterized processes. Horizontal viscosity *ν*_{H} is 10 m^{2} s^{−1} in the Eady case and 1 × 10^{3} m^{2} s^{−1} in the wind-forced case. The diffusion coefficient *κ*_{H} is 0.2 m^{2} s^{−1}. For the Eady case, these values are slightly smaller than the stability criterion values suggested by Bryan et al. (1975), who showed that to prevent grid-scale instabilities the viscosity should exceed a minimum value, *ν*_{H,crit} > |*U*|Δ/2, where *U* is the background horizontal velocity and Δ is the grid spacing. For this Eady problem, *ν*_{H,crit} = 4.1 × 10^{2} m^{2} s^{−1}. However, Bryan et al. (1975) noted that, for cases with smooth inflow (such as the Eady problem), smaller viscosities may be acceptable. They also indicated that diffusion may be several orders of magnitude smaller than viscosity. Numerical viscosity and numerical diffusion are the same: *ν*_{H,num} = *κ*_{H,num} = *kŨ*Δ^{2}/4, where *k* is the wavenumber, and *Ũ* the Fourier transform of *U.* Because the viscosity and diffusion coefficients are set to be small, at high wavenumbers numerical viscosity and diffusion may exceed the parameterized coefficients. In practical terms, this means that numerical viscosity and diffusion may attenuate energies in the upper half of the wavenumber spectrum, particularly for the Eady problem.

Since vertical diffusion and viscosity predominantly represent diapycnal effects, which are believed to be minor in much of the ocean interior, they are set to be smaller than their horizontal counterparts. For the Eady problem, where energetics are important, vertical diffusion and viscosity coefficients are set to zero in order to minimize potential energy changes during the run. For the wind-forced case, *ν*_{υ} = 1 × 10^{−4} m^{2} s^{−1} and *κ*_{υ} = 1 × 10^{−5} m^{2} s^{−1}.

Standard convective adjustment is used to mix the water column vertically where it is unstably stratified. Four convective passes through the water column are carried out at each time step. Convection is a minor process in the Eady instability, but it is important in the wind-forced model.

For this study, only examples on the *f* plane are considered. Without *β* there are no Rossby waves, so the slow spinup timescales associated with Rossby waves are not a concern. While instabilities that rely on a background vorticity gradient may exhibit some behaviors different from those found on an *f* plane, observed oceanic instabilities can closely resemble Eady instabilities, which occur on the *f* plane. For example, Samelson (1993) has noted that mixed layer fronts, such as those observed in the FASINEX experiment, are unstable with growth rates similar to the Eady model predictions.

*S*

**u**

*ρ*

**u**′ ·

**∇***ρ*′

*ρ*

*υ*′

*ρ*′

Some recent investigations have advocated investigating eddy effects by averaging thickness fluxes along isopycnals rather than density fluxes at fixed depth as used here. While isopycnally averaged fluxes might provide a clearer picture of the physical processes, any resulting parameterization would need to be converted to fixed-depth averages before being implemented in *z*-coordinate models such as MOM. Converting *z*-coordinate model output to isopycnal coordinates introduces numerical interpolation errors into the flux divergence computation. In cases where stratification is sufficiently uniform to make interpolation reliable, there is often no advantage to working on isopycnal surfaces because the *z*-coordinate density flux *υ*′*ρ*′*η,* which can be expressed to leading order as *υ*(*η*)′Δ*η*′*υ*(*z*)′*ρ*(*z*)′*dz*/*dρ**z*-average fluxes, which are natural to the model analyzed.

## 3. Model formulations

### a. The Eady problem

The Eady model (Eady 1949) exhibits one of the clearest examples of baroclinic instability, and the basic physics underlying this instability has been carefully presented by a number of authors (Gill 1982; Pedlosky 1987). Because of its simplicity, it offers a direct means to examine mesoscale eddy processes. A signature of baroclinic instability is available potential energy associated with tilting isopycnals feeding a growing disturbance. Since the Eady instability grows rapidly at a dominant wavenumber, it allows us to examine how unresolved eddies influence coarse-resolution (zonally averaged) flow on timescales too short to allow averaging of a broad spectrum of mesoscale processes. Our analysis will indicate that classic frictionless Eady instability is effectively unparameterizable because it cannot achieve a steady state.

*H.*The initial density field is

*γ*and

*α,*and Coriolis parameter

*f*are all constant, as is the Brunt–Väisälä frequency

*N*

^{2}=

*gγ.*The initial flow is in geostrophic balance, varies only with depth, and has shear ∂

*u*/∂

*z*=

*α.*

*k*and meridional wavenumber

*l*is

*H*

_{R}=

*f*/

*Nκ*and

*κ*=

*k*

^{2}+

*l*

^{2}(Gill 1982). In our numerical reentrant channel the lowest mode is approximately

*l*=

*π*/

*L*

_{y}in which case

*σ*

_{max}= 0.3093

*αf*/

*N,*which occurs when

*k*= [(0.8038

*f*/(

*NH*))

^{2}− (

*π*/

*L*

_{y})

^{2}]

^{1/2}.

In this numerical implementation of the Eady problem, the model ocean is 1500 m deep and is divided into 15 levels, each 100 m thick. Other parameters are listed in Table 1. The most unstable mode is predicted to be 16.9 wavelengths per channel length. As indicated by the solid line in Fig. 1, this mode is highly energetic and is well resolved by the 128-point zonal grid.

Numerical solutions to the Eady problem were considered by Williams (1971), who analyzed flow in an annulus, an experiment akin to what might be set up in a laboratory rotating tank. His situation differed slightly from the Eady idealization because frictional boundaries were included at *y* = 0 and *y* = *L*_{y}. The resulting instability growth was constrained by friction within the boundary layers. In contrast, in this study, free-slip boundary conditions are used on the sidewalls and on the upper and lower boundaries, and the instability grows without bound. Free-slip boundary conditions are obtained by fixing the derivative ∂*u*/∂*y* to be zero on the northern and southern boundaries. In addition, the term ∂^{2}*ρ*/∂*y*^{2} is also set to zero at the boundaries to maintain the geostrophic balance. As indicated in Fig. 2, potential energy decreases over time, as it is converted into kinetic energy. One might suppose that with the right initial parameters, the background stratification could be made to evolve until the isopycnals were flattened and all the initial available potential energy was converted to kinetic energy. This does not occur with the minimal explicit damping used here. Our three-dimensional code behaves much like the two-dimensional Eady instability analyzed by Garner et al. (1992). Because the flow is in approximately thermal wind balance, as potential energy is converted to eddy kinetic energy, the isopycnals steepen and there is a progression toward small scales. Since MOM was implemented with free-slip boundary conditions and no sidewall friction, even if the flow begins with little available potential energy, the growing instability generates progressively narrower structures that eventually vary too rapidly to be represented numerically.

Instabilities in the real ocean do not grow without bound, but the early growth stages of oceanic instability may be represented by the early growth of the Eady instability. Thus, we analyze the Eady problem results when the total kinetic energy has increased between 5% and 50% above the initial kinetic energy level, as indicated in Fig. 2. Figure 3 shows at full resolution the zonal averages of velocity *u* and density *ρ,* time-averaged over this window and over 15 model realizations. The density field in Fig. 3b has not changed perceptibly from its initial condition defined by (5). The zonal velocity in Fig. 3a shows only slight perturbations from its initial state.

### b. Wind-forced reentrant channel

In addition to the unconditionally unstable Eady problem, a wind-forced reentrant channel on the *f* plane is considered. Surface wind forcing is constant in time and eastward, with a Gaussian structure centered in the channel and an *e*-folding scale one-sixth the channel width. In contrast to the Eady case, no-slip boundary conditions are used on the northern and southern walls, a frictional drag coefficient is applied on the bottom boundary, and the boundary drag removes momentum input by the wind. The constant channel depth of 5700 m is divided into 15 levels with thicknesses varying from 30 m at the surface to 840 m at the bottom. The vertical structure matches that used in climate simulations carried out with MOM, and the use of a deep channel prevents bottom friction from controlling the ocean response to wind forcing. The initial density field is set to have a realistic vertical structure based on fits to observations by Pacanowski (1995).

The model is spun up from an initial resting state. Kinetic energy reaches equilibrium after about 1500 model days, as shown in Fig. 4. For comparison, the spinup timescale due to a frictional surface Ekman layer 2*H*/*ν*_{υ}*f*

The wind-forced flow in a channel evolves into a strong zonal-velocity jet (Fig. 5a) with correspondingly sloping isotherms (Fig. 5b). The most energetic zonal wavenumber is five wavelengths per channel, as indicated in Fig. 1.

## 4. Examining the role of eddies

*G*

_{z}is the vertical derivative of

*G.*

**u*** is nondivergent, and the sum of

**u**

**u*** is the residual mean circulation

**U**

*G*depends on the component of the eddy flux that is diapycnal relative to the mean, that is, the flux through mean isopycnals. In this form of the Eulerian-mean equations, the eddy terms (4) have been separated into two components:

*S*

**u**

*ρ*

**u**′ ·

**∇***ρ*′

**u**

**∇***ρ*

*G*

_{z}

*D**/

*Dt,*representing advection of

*ρ*

**U**

*ρ*

_{1}<

*ρ*

*ρ*

_{2}). The term

*G*

_{z}, on the other hand, is the eddy-flux divergence in the “mean diapycnal” direction; although locally density advection is confined within isopycnal surfaces, relative to

*ρ*

*G*

_{z}can produce or destroy filtered density. In this section, we will use the term “diabatic” to refer to processes that are diabatic in zonally averaged coordinates (though they are locally adiabatic), “adiabatic” to refer to processes that neither create nor destroy zonal-mean density, and “diapycnal” to refer to motions across zonal-mean isopycnals. Here

*G*differs from the mean diapycnal component of the eddy flux only by the factor

*ρ*

_{z}/|

**∇***ρ*|, which is nearly unity.

The merit of the transformed Eulerian-mean equations, originally developed by Andrews and McIntyre (1976) to describe the effect of atmospheric waves, is the separation of eddy effects into the advecting component **u*** in (8) and the diabatic component *G*_{z} of (9). There are, of course, myriad ways to partition *S*(**u**, *ρ*) into adiabatic and diabatic components, but the particular form selected is unique in associating all the effects of *G*_{z} with the diapycnal eddy flux, which is believed to be small in the atmosphere and ocean.

Although the processes due to *G*_{z} are strictly diapycnal, it does not follow that the eddy transport velocity **u*** advects tracers along isopycnals. In many quasi-equilibrium scenarios in which the low-resolution density field is approximately steady the effects of diapycnal advection by the Eulerian-mean **u****u*** compensate to make *D***ρ**Dt* small. If the eddy-induced and Eulerian-mean advection were directed along isopycnals, then the meridional overturning streamfunctions *ψ**ψ** corresponding to **u****u*** would be parallel to isopycnals. Figures 6a,b and 7a,b show streamfunctions for the Eady and wind-forced cases, respectively. Isopycnals are superimposed in dotted lines. In neither case is *ψ**ψ** parallel to isopycnals. In the wind-forced case, which is in quasi equilibrium, the two processes tend to compensate to minimize net diapycnal advection.

The terms in the density balance equation (7) are shown at three depths for the Eady case in Fig. 8 and for the wind-forced case in Fig. 9.

At all depths in the Eady instability case, the dominant balance is between density advection due to eddy effects (**u*** · **∇***ρ**ρ**t*): *G*_{z} is the next largest term, substantially exceeding advection by the Eulerian-mean velocity. This suggests that *G*_{z} may be important when Eady instabilities are growing. To test the role of *G*_{z}, the Eady instability scenario was rerun with an additional fictitious, zonally invariant buoyancy source that exactly cancelled the zonally averaged *G*_{z} in the density equation. In this run the kinetic energy evolved as in the original case, but potential energy remained nearly constant. In addition, the wavenumber spectral peak was broader than what is shown in Fig. 1 and was less clearly dominated by the predicted most unstable mode. This shows that the small but significant values of *G*_{z} seen in Fig. 8 play a critical role in the growth of Eady-type instability.

At the wind-forced case’s deeper levels (Figs. 9b,c) the advective effects due to mesoscale eddies (**u***) roughly balance advection by the Eulerian-mean velocity (**u**). This is analogous to the Ferrel cells in the atmosphere (Plumb and Mahlman 1987) and the Southern Ocean Deacon cell (Döös and Webb 1994) that are evident when the mean circulation is averaged in *z* coordinates but diminish when eddy effects are considered by averaging in pressure or density coordinates. The next most important contribution is from *G*_{z}. Nearer the top of the model domain (Fig. 9a) eddy advection, represented by *υ***ρ*_{y} + *w***ρ*_{z}, is balanced primarily by *G*_{z}, while mean-flow advection plays a comparatively small role.

*G*

_{z}is nonzero in both model cases considered, even a perfect parameterization of

**u*** ·

**∇***ρ*

*G*may be required. Andrews and McIntyre (1976) based their neglect of the role of

*G*

_{z}on analysis of a linearized balance equation for density variance

*ρ*′

^{2}

**∇***ρ*

*ρ*′ evolution equation by

*ρ*′ and averaging zonally, is

*R*(

*ρ*) are here expressed as they are used in the numerical model, as horizontal and vertical diffusion and convective adjustment

*C*(

*ρ*).

In the case of the linearizable wave motions considered by Andrews and McIntyre (1976), time change of variance was the dominant term on the right of (11). Alternatively, McDougall and McIntosh (1996) proposed that within the oceanic general circulation advection by the mean flow, **u****∇***ρ*′^{2}*κ*_{H} and *κ*_{υ} in (11) to yield a positive quantity representing dissipation of variance by explicit diffusion plus a divergence term that is generally small. As discussed by Davis (1994), hypotheses such as these depend on both the physical circumstances being studied (atmospheric waves versus oceanic fine scales) and the way fluctuations like *ρ*′ are separated from mean or resolved components like *ρ***u**′*ρ*′^{2}

In light of the foregoing discussion, it is perhaps not surprising that our model runs show no simple description of the variance budget that could serve as a basis for parameterizing *G.* The terms represented in (11) are shown in Fig. 10 for the Eady instability case and in ig. 11 for the wind-forced model. In the Eady case, the system is not in steady state, and the time-dependent changes in *ρ*′^{2}**u**′*ρ*′**∇***ρ***u****∇***ρ*′^{2}**u**′*ρ*′**∇***ρ**G*_{z} is unlikely to be easily parameterized.

An alternative to parameterizing *G*_{z} is to find a way to make it small by forcing *S*(**u**, *ρ*) to be nearly completely advective. We considered this alternative by seeking the optimal advective velocity **u**^{*}_{opt}*G*_{z}. This was done by finding the nondivergent **u**^{*}_{opt}**u**^{*}_{opt}**∇***ρ**S*(**u**, *ρ*) and had no flow through the boundaries. The corresponding streamfunctions *ψ*^{*}_{opt}*ψ*^{*}_{opt}*G*_{z} is smaller than the other advective terms at all depths, *ψ*^{*}_{opt}*ψ** from the Andrews and McIntyre (1976) separation (8) and (9) shown in Fig. 6b. The wind-forced case shows less apparent correlation between the *ψ*^{*}_{opt}*ψ** in Fig. 7b.

The results of these model analyses indicate that mesoscale eddies are a major factor controlling the large-scale density field. Not only are the adiabatic effects defined by **u*** · **∇***ρ**G*_{z} also matters. Thus, parameterizing both will be an important aspect of building a reliable time-evolving climate model. Since many existing subgrid-scale parameterizations (e.g., GM90; Visbeck et al. 1997) have been developed using the assumption that the cross-isopycnal advection represented by *G*_{z} is zero, the findings of this study suggest that further thought about basic parameterization requirements may be necessary in order to design models that evolve correctly in time.

## 5. Seeking an optimal subgrid-scale parameterization

The preceding section showed the significant influence of mesoscale processes on large-scale flow in both the baroclinically unsteady Eady problem and a wind-forced channel. In this section, we examine whether any commonly used subgrid-scale parameterizations can adequately represent the eddy-flux terms from the mesoscale models introduced in section 3. A perfect parameterization would allow the time evolution of coarse-resolution model prognostic variables to track precisely the spatially filtered variables (**u***ρ*

### a. Diffusive parameterizations

^{2}

_{H}

*K*

_{H}and

*K*

_{υ}are constant horizontal and vertical eddy diffusion coefficients. Note the distinction between

*κ,*which represents the diffusion coefficients set in the numerical models, and

*K,*which represents the coefficient best estimated to represent the eddy flux as a diffusive process. In analogy with molecular diffusion,

*K*

_{H}and

*K*

_{υ}should be positive. If they are not, diffusion creates local maxima, a process that is incompatible with the original physics.

Does (12) adequately represent the influence of mesoscale eddies? The time-averaged eddy-flux divergence terms on the left side of (12) are shown in Figs. 12a and 13a. For the Eady case, the horizontal Laplacian of the density fields in Fig. 12b is of opposite sign and has a slightly different spatial structure from the eddy-flux divergence terms. Similarly, in Fig. 13b, horizontal diffusion captures some of the large-scale structure of the time-averaged eddy fluxes. The vertical second derivatives of the density field in Figs. 12c and 13c are not closely related to the eddy-flux divergences.

For a quantitative comparison, we use a least squares fitting technique (see the appendix) to estimate optimal values of *K*_{H} and *K*_{υ}. The effectiveness of the fit will be judged using the skill index *Z,* the percentage of the mean-squared eddy-flux divergence terms explained by the fit.

*η*is perpendicular to isopycnal surfaces and

*K*

_{d}is the diapycnal diffusion coefficient.

*K*

_{υ}designed to respond to varying background stratification. As a proxy for that, we will also allow

*K*

_{υ}to vary as a function of depth

*z*:

*K*

_{d}varies with depth:

*A*

_{H}, the biharmonic diffusion coefficient, which should be negative for numerically stable downgradient diffusion, and

*K*

_{υ}, the vertical diffusion coefficient.

**u*** represented in (8) is parameterized directly:

*K*

_{I}has the units of a diffusion coefficient. As in (8),

**u**

^{*}

_{GM}

Estimates of the meridional streamfunction *ψ*^{*}_{GM}*K*_{I}*ρ*_{y}/*ρ*_{z}**u**^{*}_{GM}*ψ*^{*}_{GM}*ψ**, the zonal average of the eddy-induced transport streamfunction from (8) (Figs. 6b and 7b), or *ψ*^{*}_{opt}**u**′ · **∇***ρ*′

The GM90 parameterization for the advective terms **u**^{*}_{GM}**∇***ρ**K*_{I} is shown in Figs. 12d and 13d. In the Eady instability case, there is a close resemblance between the spatial structures of the GM90 parameterization terms (Fig. 12d) and of the horizontal diffusion terms (Fig. 12b). This is not surprising since the GM90 parameterization and horizontal diffusion are indistinguishable to leading order in Rossby number when standard quasigeostrophic scalings are applied. In the wind-forced case (Figs. 13b, d), the fields do not resemble each other as closely but show the same large-scale features.

*K*

_{I}. For a zonally averaged scenario, they suggest

*α*is constant and

*l*is a characteristic length scale over which particles are transferred. In regions of small growth rate,

*l*is determined by the Rossby radius; where the growth rate is larger,

*l*increases to approach the width of the high-growth rate region.

We evaluate the success of these parameterizations at representing the zonal and time-averaged eddy-flux divergences in section 5b. In section 5c we assess the effectiveness of the parameterizations at representing the model output when less averaging is allowed.

### b. Parameterization results

For each of the parameterizations, we sought coefficients *K, A,* or *α* that minimized the least square error in Eqs. (12)–(17) for time-averaged and zonally averaged fields. For the Eady case, presented in Table 2, time snapshots spaced at 5-day intervals from day 110 to 135 were averaged, and fits were calculated for each of 15 model realizations. Table 3 shows least squares fits of the various parameterizations of **u**′ · **∇***ρ*′

For the Eady instability, the fraction *Z* of mean-squared **u**′ · **∇***ρ*′*K* and *α* imply that eddies tend to concentrate tracer distributions into narrower regions. In a practical model implementation, using diffusion coefficients to achieve upgradient density transfers is unstable and will lead to unphysical model behavior. Thus, if we constrain ourselves to stable positive values of *K,* for Eady’s prototypical example of baroclinic instability, the instantaneous effects of eddies are diametrically different from both horizontal diffusion and from the GM90 parameterization, which was designed to represent instability. For the other Eady examples, *Z* is less than 1%, indicating that the least squares fits capture a negligibly small fraction of the eddy-flux divergence. The failure of the parameterizations to represent the average Eady instability is not surprising. Because the Eady instability has only a single growing mode, all eddies transfer density in the same direction, and an ensemble of eddies will not balance each other out to provide a net diffusive flux.

In the wind-forced time-averaged case, the most successful of the parameterizations shown in Table 3 are the quasi-adiabatic parameterizations based on GM90 and Visbeck et al. (1997). Both of these capture 43% ± 5% of the spatial variance in **u**′ · **∇***ρ*′*K*_{I} and *K*_{H} are nearly the same within error bars, and the increased skill in the GM90 case apparently comes from the improved relationship it allows between horizontal and vertical diffusion. These findings are consistent with those of Visbeck et al. (1997), who also found that horizontal diffusion was less successful than the GM90 parameterization or any of its variations.

Although one might hope to find universally valid diffusion coefficients, the best fits for the Eady instability and wind-forced cases differ by several orders of magnitude. As shown in both Tables 2 and 3, those parameters that are effective in describing the flux divergence are reasonably well determined. Uncertainty in the Eady case is particularly small because the growing instability is essentially deterministic in form, if not phase. Nonetheless, for the Eady problem, the fitted coefficients are not universal parameters: changing the background stratification by 30% results in substantially different coefficients.

Even for long time averages, the parameterizations are unable to capture the entire eddy-flux divergence. Although *G*_{z} is large in the upper ocean, below 300 m over 90% of the time-mean **u**′ · **∇***ρ*′**u*** · **∇***ρ***u*** computed based on (8). In contrast, in the same depth range **u**^{*}_{GM}**∇***ρ***u**′ · **∇***ρ*′**u**^{*}_{GM}**u*** incompletely, and this may explain why *Z* is as small as 40%.

### c. Parameterizing time-varying phenomena

Subgrid-scale parameterizations are intended primarily to represent the spatially and temporally averaged effects of mesoscale eddies. An ideal parameterization would accurately duplicate the effects of eddies over small spatial and temporal scales. We now examine how well the parameterizations discussed above apply to shorter timescales. The Eady problem represents an evolution far from equilibrium, and averaging flux potentials over shorter intervals does not change the results. However, without changing the zonal averaging of the wind-forced case, we can vary the degree of temporal averaging to see how the success of the parameterizations changes.

Figure 14 shows *Z* as a function of averaging time for horizontal diffusion, biharmonic diffusion, the GM90 parameterization, and the Visbeck et al. parameterization for the wind-forced case. Regardless of the parameterization selected, *Z* is small for short averaging times and asymptotes to its maximum value at long averaging times, with an *e*-folding scale of about 100 days. Over the shorter time periods indicated in Fig. 14, the density field undergoes greater variability, and eddy-flux divergences are less well represented by the parameterizations.

Ultimately the success of subgrid-scale parameterizations depends on the timescales of the processes studied. For this model, the frequency spectrum of density is red for time periods shorter than 200 days and then levels off. Apparently the parameterizations are most successful when applied to fields that have been averaged over time periods that are long compared with the decorrelation time of the detailed field. This suggests that parameterization of cases like the Eady problem may be effectively impossible because the average state evolves significantly over the the minimum averaging time required for parameterizations to be effective. For a system that has an inherently red frequency spectrum, without a low-frequency cutoff the minimum averaging time required for parameterizations to be effective may be too long to be useful.

For these model runs, parameterizations appear consistently better able to represent low-frequency effects than high-frequency fluctuations. Similarly in a different idealized model, Lee et al. (1997) found that only after many years of model integration did tracer distributions appear to be controlled by the eddy-induced advective velocity **u***, which the GM90 parameterization is designed to represent. In a more realistic high-resolution North Atlantic model, Rix and Willebrand (1996) binned upper-ocean data into 4° boxes and temporally averaged over 19 years in order to obtain accurate statistical moments, but they did not consider the temporal evolution of their model. Their equivalent to *Z,* the squared correlation coefficient between **u**^{*}_{GM}**u*** computed based on the eddy thickness flux, was between 38% and 52%, about the same as *Z* for the GM90 parameterization in Table 3.

A number of variations to the GM90 parameterization have been suggested but were not considered here. In these least squares fits, we avoided the domain boundaries where the second derivatives required for diffusion are ill-defined. However, Treguier et al. (1997) treated the eddy-flux divergence as diffusion of potential vorticity in the ocean interior and as diffusion of buoyancy at the boundaries; their analysis of the energetics and boundary requirements suggested constraints on the horizontal and vertical structure of the diffusivity. Killworth (1997) has also considered top and bottom boundary conditions along with the *β* effect, but in the simplified boundaryless *f*-plane examples used here, his parameterization is the same as the GM90 parameterization. Gent and McWilliams (1996) have argued that climate models would produce better results if the eddy-induced transport velocity **u*** were also included in the momentum equations, but exploration of this effect is beyond the scope of the current study.

The results shown here do not offer definitive evaluations of eddy parameterizations for all possible situations. We have not included topography and the related physical process of form stress, which would alter the impact of mesoscale eddies and the corresponding eddy parameterizations. In addition, one might expect the dynamics of closed basins with clear continental boundaries to be somewhat different from the conceptually simple channel models examined in this paper. Finally, passive tracers, such as oxygen or nutrients, might show different types of eddy–mean flow interactions than the dynamically active density examined in this study and might have different parameterizations. What these results do show is that common parameterizations do not capture the instantaneous character of unstable eddies even when extensive zonal averaging is applied, but parameterizations that conserve the volume of water between isopycnal pairs, as suggested by GM90 or Visbeck et al. (1997), can sometimes credibly represent a portion of the time-averaged eddy-flux divergence.

## 6. Summary

Numerical models of slowly evolving climatic processes are typically run at coarse resolution and rely on parameterizations to represent the effects of mesoscale eddies. In this study two time-evolving model cases have been run in a zonal channel with sufficient resolution to examine the influence of mesoscale effects on the larger scale. The first case, an Eady model, is unstable; the second, a wind-forced channel, evolves to a statistically steady state. The zonal average of the model variables was used to represent the coarse-resolution mean field, and the deviations from the zonal average were used to examine the influence of eddies on the mean density field. We specifically examined whether these eddy advective terms are adiabatic in the zonal mean (so that they conserve mean density) and whether they can be represented by any common eddy parameterizations.

The results show that eddy density fluxes dominate the zonal-mean density flux. Most eddy density advection neither creates nor destroys zonal-mean density. However a portion of the advection, *G*_{z}, is determined by the local deviations of the density field from the zonal-mean isopycnals, and at coarse resolution it appears diabatic. Thus, a reliable parameterization of subgrid-scale processes must account for the high-resolution eddy processes represented by *G*_{z}. As suggested by Tandon and Garrett (1996) these processes might include eddy diffusion over short length scales as well as interactions of small-scale features with the large-scale field. Although *G*_{z} can be determined from the variance budget, in the examples that we have considered, there is no simple dominant balance in the variance budget that might allow us to predict *G*_{z}.

Efforts to represent the eddy-flux processes using a number of common eddy parameterizations indicated that for averages over time periods long enough to filter out the most energetic fluctuations, the GM90 and Visbeck et al. (1997) parameterizations have skill indices of about 40%, in comparison with the 10% skill index of simple horizontal and vertical diffusion. Thus, for the modeling scenarios examined here, the quasi-adiabatic formulation suggested by GM90 is not perfect, but it represents an improvement over more traditional subgrid-scale parameterizations, for steady-state long-term model runs that do not depend on time-varying processes to represent any critical portion of their physics. In contrast, even with optimally selected diffusion coefficients, no parameterization reliably captures instantaneous or short time period eddy-induced processes.

## Acknowledgments

Support for this work was provided by the National Oceanic and Atmospheric Administration (NOAA Award NA47GP0188 to the Lamont/Scripps Consortium for Climate Research.) Comments from Dan Rudnick and from the anonymous reviewers helped to improve the presentation.

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## APPENDIX

### Fitting Diffusion Coefficients

*K*

_{H}and

*K*

_{υ}are determined by fitting the parameterizations to the zonally averaged eddy-flux terms

**u**′ ·

**∇***ρ*′

**A**

*ρ*

*y–z*plane, and

**b**contains the zonally averaged eddy-flux terms at the same

*y–z*points. Only interior points are used because boundary values can dominate the least squares fit without being representative of domain-wide conditions. This is particularly true for the GM90 parameterization (Gent et al. 1995). Here

**A**

*ρ*

**u**′ ·

**∇***ρ*′

*Z,*which is closely related to the correlation of the eddy-flux terms to the model terms in

**A**

*Y*represents the observed eddy-flux terms

**u**′ ·

**∇***ρ*′

*X*

_{i}are columns of

**A**

*Y*with coefficients

*K*

_{i}, and

*Ỹ*is the total functional fit. (If

*Y*and

*Ỹ*had zero mean,

*Z*would be the correlation coefficient between

*Y*and

*Ỹ.*) For comparison, if

*Y*and

*X*

_{i}were statistically independent variables with zero mean and normal distribution, then the mean of

*Z*would be

*M*/

*N,*where

*M*is the number of parameters used in the fit (here 2) and

*N*is the number of independent samples over which the fit is tested (here 780). Our data are correlated in time and space so estimating the effective value of

*N*is difficult. In practice, the samples are not independent, and we expect

*Z*to be greater than

*M*/

*N.*

To estimate the uncertainty in coefficients, we assume that the independently initialized Eady runs are statistically independent and that averages over 500-day segments of the wind-forced run are effectively independent. Thus, we can estimate the uncertainty of the average values from the variability. Standard error decreases as 1/*P**P* is the number of assumed independent samples included in the average.

Mean kinetic and potential energy (in cm^{2} s^{−2}) for the Eady problem run at quarter-degree resolution as a function of time (in days) from the beginning of the model run. The initial value of potential energy (872 cm^{2} s^{−2}) is removed to better illustrate the transfer of potential energy to kinetic energy. The time period analyzed in this paper is delineated by vertical gray lines indicating days 110 and 135, when kinetic energy has increased between 5% and 50% above its initial value.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Mean kinetic and potential energy (in cm^{2} s^{−2}) for the Eady problem run at quarter-degree resolution as a function of time (in days) from the beginning of the model run. The initial value of potential energy (872 cm^{2} s^{−2}) is removed to better illustrate the transfer of potential energy to kinetic energy. The time period analyzed in this paper is delineated by vertical gray lines indicating days 110 and 135, when kinetic energy has increased between 5% and 50% above its initial value.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Mean kinetic and potential energy (in cm^{2} s^{−2}) for the Eady problem run at quarter-degree resolution as a function of time (in days) from the beginning of the model run. The initial value of potential energy (872 cm^{2} s^{−2}) is removed to better illustrate the transfer of potential energy to kinetic energy. The time period analyzed in this paper is delineated by vertical gray lines indicating days 110 and 135, when kinetic energy has increased between 5% and 50% above its initial value.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 110 to 135 at 5-day intervals for the evolving Eady instability. Panels indicate (a) the zonal velocity *u*^{−3}m s^{−1} and (b) the density anomaly *δ**ρ*^{−4} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ**f* negative. Readers who prefer to think about systems with positive *f* should interpret meridional distance as extending from north to south.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 110 to 135 at 5-day intervals for the evolving Eady instability. Panels indicate (a) the zonal velocity *u*^{−3}m s^{−1} and (b) the density anomaly *δ**ρ*^{−4} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ**f* negative. Readers who prefer to think about systems with positive *f* should interpret meridional distance as extending from north to south.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 110 to 135 at 5-day intervals for the evolving Eady instability. Panels indicate (a) the zonal velocity *u*^{−3}m s^{−1} and (b) the density anomaly *δ**ρ*^{−4} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ**f* negative. Readers who prefer to think about systems with positive *f* should interpret meridional distance as extending from north to south.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Mean kinetic energy (in cm^{2} s^{−2}) of wind-forced channel model, as a function of time (in days) from the beginning of the model run. Output is analyzed from day 3000 to 7000 at quarter-day intervals.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Mean kinetic energy (in cm^{2} s^{−2}) of wind-forced channel model, as a function of time (in days) from the beginning of the model run. Output is analyzed from day 3000 to 7000 at quarter-day intervals.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Mean kinetic energy (in cm^{2} s^{−2}) of wind-forced channel model, as a function of time (in days) from the beginning of the model run. Output is analyzed from day 3000 to 7000 at quarter-day intervals.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 3000 to day 7000 at quarter-day increments for the evolving wind-forced channel at quarter-degree resolution. Panels indicate (a) the zonal velocity *u*^{−2} m s^{−1} and (b) the density anomaly *δ**ρ*^{−2} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ*

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 3000 to day 7000 at quarter-day increments for the evolving wind-forced channel at quarter-degree resolution. Panels indicate (a) the zonal velocity *u*^{−2} m s^{−1} and (b) the density anomaly *δ**ρ*^{−2} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ*

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Cross sections showing zonal and time averages of variables from day 3000 to day 7000 at quarter-day increments for the evolving wind-forced channel at quarter-degree resolution. Panels indicate (a) the zonal velocity *u*^{−2} m s^{−1} and (b) the density anomaly *δ**ρ*^{−2} kg m^{−3} and density defined as *ρ*_{0} − *δ**ρ*

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the Eady instability case represented with solid contours and shading, calculated from zonally and time-averaged velocity and density fields, with *υ* = −*ψ*_{z} and *w* = *ψ*_{y}. Dotted contours indicate isopycnals from Fig. 3b. Panels show (a) *ψ***u**^{−4} m^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} corresponding to **u*** as defined in (8) (with contour interval 1.67 × 10^{−2} m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{−2} m^{2} s^{−1}) found by least squares fitting **u**^{*}_{opt}*G*_{z} (see text), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, the parameterized streamfunction suggested by GM90, with *K*_{I} = 60 m^{2} s^{−1} and contour interval 1.67 × 10^{−2} m^{2} s^{−1}. Negative streamfunction values are shaded gray, and by definition the streamfunctions are zero on the boundaries.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the Eady instability case represented with solid contours and shading, calculated from zonally and time-averaged velocity and density fields, with *υ* = −*ψ*_{z} and *w* = *ψ*_{y}. Dotted contours indicate isopycnals from Fig. 3b. Panels show (a) *ψ***u**^{−4} m^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} corresponding to **u*** as defined in (8) (with contour interval 1.67 × 10^{−2} m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{−2} m^{2} s^{−1}) found by least squares fitting **u**^{*}_{opt}*G*_{z} (see text), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, the parameterized streamfunction suggested by GM90, with *K*_{I} = 60 m^{2} s^{−1} and contour interval 1.67 × 10^{−2} m^{2} s^{−1}. Negative streamfunction values are shaded gray, and by definition the streamfunctions are zero on the boundaries.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the Eady instability case represented with solid contours and shading, calculated from zonally and time-averaged velocity and density fields, with *υ* = −*ψ*_{z} and *w* = *ψ*_{y}. Dotted contours indicate isopycnals from Fig. 3b. Panels show (a) *ψ***u**^{−4} m^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} corresponding to **u*** as defined in (8) (with contour interval 1.67 × 10^{−2} m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{−2} m^{2} s^{−1}) found by least squares fitting **u**^{*}_{opt}*G*_{z} (see text), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, the parameterized streamfunction suggested by GM90, with *K*_{I} = 60 m^{2} s^{−1} and contour interval 1.67 × 10^{−2} m^{2} s^{−1}. Negative streamfunction values are shaded gray, and by definition the streamfunctions are zero on the boundaries.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the wind-forced case, calculated from zonally and time-averaged velocity and density fields (solid contours and shading), as in Fig. 6. Dotted contours indicate isopycnals from Fig. 5b. Panels show (a) *ψ*^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} (with contour interval 1.25 m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{2} s^{−1}; values within 0.01 m^{2} s^{−1} of zero are plotted as positive values), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, (with *K*_{I} = 2.5 × 10^{2} m^{2} s^{−1} and contour interval 1.25 m^{2} s^{−1}).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the wind-forced case, calculated from zonally and time-averaged velocity and density fields (solid contours and shading), as in Fig. 6. Dotted contours indicate isopycnals from Fig. 5b. Panels show (a) *ψ*^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} (with contour interval 1.25 m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{2} s^{−1}; values within 0.01 m^{2} s^{−1} of zero are plotted as positive values), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, (with *K*_{I} = 2.5 × 10^{2} m^{2} s^{−1} and contour interval 1.25 m^{2} s^{−1}).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Meridional streamfunctions *ψ* for the wind-forced case, calculated from zonally and time-averaged velocity and density fields (solid contours and shading), as in Fig. 6. Dotted contours indicate isopycnals from Fig. 5b. Panels show (a) *ψ*^{2} s^{−1}), (b) *ψ** = *υ*′*ρ*′*ρ*_{z} (with contour interval 1.25 m^{2} s^{−1}), (c) *ψ*^{*}_{opt}^{2} s^{−1}; values within 0.01 m^{2} s^{−1} of zero are plotted as positive values), and (d) *ψ*_{GM} = −*K*_{I}*ρ*_{y}/*ρ*_{z}, (with *K*_{I} = 2.5 × 10^{2} m^{2} s^{−1} and contour interval 1.25 m^{2} s^{−1}).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density balance (in kg m^{−3} s^{−1} × 10^{6}) averaged over 15 model realizations, zonally and temporally at 5-day intervals from day 110 to 135 for (a) level 2, (b) level 5, and (c) level 8. Because of the vertical symmetry of the Eady problem, balances below 750 m are similar to those above 750 m. The terms from the density balance (7) are eddy-induced advection *υ***ρ*_{y} + *w***ρ*_{z} (black solid line); mean advection *υ**ρ*_{y} + *w**ρ*_{z} (black dotted line); *G*_{z} (black dashed line); negative diffusion −*κ*_{H}*ρ*_{yy} − *κ*_{υ}*ρ*_{zz} (gray solid line); and density tendency ∂*ρ**t* (gray dotted line).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density balance (in kg m^{−3} s^{−1} × 10^{6}) averaged over 15 model realizations, zonally and temporally at 5-day intervals from day 110 to 135 for (a) level 2, (b) level 5, and (c) level 8. Because of the vertical symmetry of the Eady problem, balances below 750 m are similar to those above 750 m. The terms from the density balance (7) are eddy-induced advection *υ***ρ*_{y} + *w***ρ*_{z} (black solid line); mean advection *υ**ρ*_{y} + *w**ρ*_{z} (black dotted line); *G*_{z} (black dashed line); negative diffusion −*κ*_{H}*ρ*_{yy} − *κ*_{υ}*ρ*_{zz} (gray solid line); and density tendency ∂*ρ**t* (gray dotted line).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density balance (in kg m^{−3} s^{−1} × 10^{6}) averaged over 15 model realizations, zonally and temporally at 5-day intervals from day 110 to 135 for (a) level 2, (b) level 5, and (c) level 8. Because of the vertical symmetry of the Eady problem, balances below 750 m are similar to those above 750 m. The terms from the density balance (7) are eddy-induced advection *υ***ρ*_{y} + *w***ρ*_{z} (black solid line); mean advection *υ**ρ*_{y} + *w**ρ*_{z} (black dotted line); *G*_{z} (black dashed line); negative diffusion −*κ*_{H}*ρ*_{yy} − *κ*_{υ}*ρ*_{zz} (gray solid line); and density tendency ∂*ρ**t* (gray dotted line).

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density balance (in kg m^{−3} s^{−1} × 10^{5}) zonally and time-averaged from model output at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12. Terms of (7) are shown as in Fig. 8. Also included is the net influence of convective adjustment.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density balance (in kg m^{−3} s^{−1} × 10^{5}) zonally and time-averaged from model output at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12. Terms of (7) are shown as in Fig. 8. Also included is the net influence of convective adjustment.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density balance (in kg m^{−3} s^{−1} × 10^{5}) zonally and time-averaged from model output at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12. Terms of (7) are shown as in Fig. 8. Also included is the net influence of convective adjustment.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density variance evolution as in (11) (in kg^{2} m^{−6} s^{−1} × 10) averaged zonally and temporally for 15 model realizations from model snapshots at 5-day intervals from day 110 to 135 for (a) level 2 and (b) level 5. Terms represented are from (11) and show *ρ*_{y}*υ*′*ρ*′*w*′*ρ*′*ρ*_{z} (black solid line); the eddy triple correlation term *υ*′/2)∂*ρ*′^{2}/∂*y**w*′/2)∂*ρ*′^{2}/∂*z**υ**ρ*′^{2}*y* + (*w**ρ*′^{2}*z* (gray solid line); and the time evolution of density variance ∂*ρ*′^{2}*t* (gray dotted line). Friction and convective adjustment are negligible.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density variance evolution as in (11) (in kg^{2} m^{−6} s^{−1} × 10) averaged zonally and temporally for 15 model realizations from model snapshots at 5-day intervals from day 110 to 135 for (a) level 2 and (b) level 5. Terms represented are from (11) and show *ρ*_{y}*υ*′*ρ*′*w*′*ρ*′*ρ*_{z} (black solid line); the eddy triple correlation term *υ*′/2)∂*ρ*′^{2}/∂*y**w*′/2)∂*ρ*′^{2}/∂*z**υ**ρ*′^{2}*y* + (*w**ρ*′^{2}*z* (gray solid line); and the time evolution of density variance ∂*ρ*′^{2}*t* (gray dotted line). Friction and convective adjustment are negligible.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eady model density variance evolution as in (11) (in kg^{2} m^{−6} s^{−1} × 10) averaged zonally and temporally for 15 model realizations from model snapshots at 5-day intervals from day 110 to 135 for (a) level 2 and (b) level 5. Terms represented are from (11) and show *ρ*_{y}*υ*′*ρ*′*w*′*ρ*′*ρ*_{z} (black solid line); the eddy triple correlation term *υ*′/2)∂*ρ*′^{2}/∂*y**w*′/2)∂*ρ*′^{2}/∂*z**υ**ρ*′^{2}*y* + (*w**ρ*′^{2}*z* (gray solid line); and the time evolution of density variance ∂*ρ*′^{2}*t* (gray dotted line). Friction and convective adjustment are negligible.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density variance (in kg^{2} m^{−6} s^{−1}) zonally and time-averaged at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12, which has been scaled up by 10^{3}. Terms represented are based on (11) as in Fig. 10, but convective adjustment (gray dashed line) is also included.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density variance (in kg^{2} m^{−6} s^{−1}) zonally and time-averaged at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12, which has been scaled up by 10^{3}. Terms represented are based on (11) as in Fig. 10, but convective adjustment (gray dashed line) is also included.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Wind-forced model density variance (in kg^{2} m^{−6} s^{−1}) zonally and time-averaged at quarter-day intervals from day 3000 to 7000 for (a) level 4, (b) level 8, and (c) level 12, which has been scaled up by 10^{3}. Terms represented are based on (11) as in Fig. 10, but convective adjustment (gray dashed line) is also included.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eddy flux divergences, diffusion terms (12), and GM90 parameterization (18) for the Eady instability case, averaged over 15 realizations and time-averaged from samples taken every 5 days from day 110 to day 135. Panels show (a) the eddy-flux divergence term −**u**′ · **∇***ρ*′^{−8} kg m^{−3} s^{−1}, (b) the horizontal eddy diffusion ^{2}_{H}*ρ*^{−12} kg m^{−5}, (c) the vertical eddy diffusion ∂^{2}*ρ**z*^{2} with contour interval 1.9 × 10^{−8} in kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u***, with contour interval 3.7 × 10^{−12} kg m^{−5}. Gray regions indicate negative values, and boundary values for the diffusion terms are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eddy flux divergences, diffusion terms (12), and GM90 parameterization (18) for the Eady instability case, averaged over 15 realizations and time-averaged from samples taken every 5 days from day 110 to day 135. Panels show (a) the eddy-flux divergence term −**u**′ · **∇***ρ*′^{−8} kg m^{−3} s^{−1}, (b) the horizontal eddy diffusion ^{2}_{H}*ρ*^{−12} kg m^{−5}, (c) the vertical eddy diffusion ∂^{2}*ρ**z*^{2} with contour interval 1.9 × 10^{−8} in kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u***, with contour interval 3.7 × 10^{−12} kg m^{−5}. Gray regions indicate negative values, and boundary values for the diffusion terms are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Eddy flux divergences, diffusion terms (12), and GM90 parameterization (18) for the Eady instability case, averaged over 15 realizations and time-averaged from samples taken every 5 days from day 110 to day 135. Panels show (a) the eddy-flux divergence term −**u**′ · **∇***ρ*′^{−8} kg m^{−3} s^{−1}, (b) the horizontal eddy diffusion ^{2}_{H}*ρ*^{−12} kg m^{−5}, (c) the vertical eddy diffusion ∂^{2}*ρ**z*^{2} with contour interval 1.9 × 10^{−8} in kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u***, with contour interval 3.7 × 10^{−12} kg m^{−5}. Gray regions indicate negative values, and boundary values for the diffusion terms are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Terms in (12) and (18) for the wind-forced model, time-averaged from samples taken every quarter-day from day 3000 to day 7000. As in Fig. 12, results show (a) −**u**′ · **∇***ρ*′^{−7} kg m^{−3} s^{−1}, (b) ^{2}_{H}*ρ*^{−9} kg m^{−5}, (c) ∂^{2}*ρ**z*^{2} with contour interval 7.1 × 10^{−5} kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u*** with contour interval 1.4 × 10^{−9} kg m^{−5}.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Terms in (12) and (18) for the wind-forced model, time-averaged from samples taken every quarter-day from day 3000 to day 7000. As in Fig. 12, results show (a) −**u**′ · **∇***ρ*′^{−7} kg m^{−3} s^{−1}, (b) ^{2}_{H}*ρ*^{−9} kg m^{−5}, (c) ∂^{2}*ρ**z*^{2} with contour interval 7.1 × 10^{−5} kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u*** with contour interval 1.4 × 10^{−9} kg m^{−5}.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Terms in (12) and (18) for the wind-forced model, time-averaged from samples taken every quarter-day from day 3000 to day 7000. As in Fig. 12, results show (a) −**u**′ · **∇***ρ*′^{−7} kg m^{−3} s^{−1}, (b) ^{2}_{H}*ρ*^{−9} kg m^{−5}, (c) ∂^{2}*ρ**z*^{2} with contour interval 7.1 × 10^{−5} kg m^{−5}, and (d) −**u*** · **∇***ρ**K*_{I} using the GM90 parameterization of **u*** with contour interval 1.4 × 10^{−9} kg m^{−5}.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

The skill index *Z* as a function of averaging time, for days 3000 to 5000 of the wind-forced run for (gray line) horizontal and vertical diffusion, (dotted line) biharmonic diffusion, (solid line) GM90 parameterization, and (dashed line) Visbeck et al. (1997). Error bars represent standard deviations of *Z* divided by the square-root of the number of samples. Error bars for the Visbeck et al. case are approximately the same as GM90 error bars and are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

The skill index *Z* as a function of averaging time, for days 3000 to 5000 of the wind-forced run for (gray line) horizontal and vertical diffusion, (dotted line) biharmonic diffusion, (solid line) GM90 parameterization, and (dashed line) Visbeck et al. (1997). Error bars represent standard deviations of *Z* divided by the square-root of the number of samples. Error bars for the Visbeck et al. case are approximately the same as GM90 error bars and are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

The skill index *Z* as a function of averaging time, for days 3000 to 5000 of the wind-forced run for (gray line) horizontal and vertical diffusion, (dotted line) biharmonic diffusion, (solid line) GM90 parameterization, and (dashed line) Visbeck et al. (1997). Error bars represent standard deviations of *Z* divided by the square-root of the number of samples. Error bars for the Visbeck et al. case are approximately the same as GM90 error bars and are not shown.

Citation: Journal of Physical Oceanography 29, 6; 10.1175/1520-0485(1999)029<1109:TIOMEO>2.0.CO;2

Parameters applied to the Eady instability in the middle column and the wind-forced channel in the right column. Some of the parameters define the initial conditions for the Eady background state and do not apply to the wind-forced case.

Eady model fits for *K, A,* and *α* calculated for time-averaged model output sampled at 5-day intervals from day 110 to 135. Mean coefficients and uncertainties are based on 15 model runs. Values of *K* and *A* are in MKS units. The quantity *Z* and its standard deviation, here expressed as percentages, represent the fraction of the mean-squared flux divergence, (**u**′ · **∇***ρ*′^{2} explained by the parameterization (see appendix). Vertically varying cases are not included for the Eady model, since the small derivatives of the vertical density shear make the calculations unreliable.

Wind-forced channel model fits for *K, A,* and *α* calculated by averaging over days 3000 to 7000 in the model run. Means and uncertainties are based on averages over eight blocks of 500 days that were treated as uncorrelated parameter estimates.