## 1. Outline

This note serves four purposes:

First, the relationship between thickness diffusion in layer and level models is developed. For level models, this provides a simple basis for the “eddy-induced transport velocity” discussed by Gent et al. (1995).

Second, many of these diffusion hypotheses rest upon profoundly uncertain flux/gradient assumptions. An alternative approach has been applied for momentum equations (Holloway 1992). The relation between this momentum parameterization and thickness diffusion is set out.

Third, an illustrative example anticipates where both thickness diffusion and momentum parameterization should fail.

Fourth, the failure points toward a more comprehensive formulation for both thickness and momentum parameterizations.

## 2. Layer and level formulations

*h*

_{i}in layer

*i*is usually given as

_{t}

*h*

_{i}

**∇**

_{i}

**u**

_{i}

*h*

_{i}

*H*

_{i}

**∇**

_{i}is two-dimensional gradient operator along layer

*i*and

**u**

_{i}is the (model-resolved, layer depth-averaged) lateral (along layer) velocity. Mixing or entrainment between layers appears as thickness source

*H*

_{i}. If

*c*

_{i}is the layer depth-average of a water property such as potential temperature or concentration of dissolved substance, then the layer burden

*c*

_{i}

*h*

_{i}is usually taken to satisfy

_{t}

*c*

_{i}

*h*

_{i}

**∇**

_{i}

**u**

_{i}

*c*

_{i}

*h*

_{i}

*C*

_{i}

**∇**

_{i}

*K*

_{i}

**∇**

_{i}

*c*

_{i}

*h*

_{i}

*C*

_{i}includes sources or sinks (e.g., radioactive decay) as well as exchange between layers, while

*K*

_{i}is a coefficient for subgrid-scale eddy diffusion along the layer. Layer models permit explicit control over exchange between layers, including the possibility that

*H*

_{i}=

*C*

_{i}= 0.

*h*but rather may prognose temperature and salinity from which density is diagnosed. Property or concentration

*c*satisfies

_{t}

*c*

**∇**

**u**

*c*

*C*

**∇**

**K**

**∇**

*c*

**∇**is a 3D gradient operator,

**u**is a 3-component velocity usually taken to satisfy

**∇**·

**u**= 0,

*C*is any body source/sink for

*c,*and

**K**

Models formulated in layers can be approximated in levels, and vice versa. Difficulty arises because of the treatment of mixing. Diffusivity *K*_{i} in layered models acts along the two-dimensional surface or layer denoted *i.* Interlayer exchanges are expressed in *C*_{i} and may represent effective diffusion among layers. Insofar as *K*_{i} and *C*_{i} together reflect diffusion along and among layers, corresponding diffusion in level models can be given by a tensor rotation of **K**

In practice, a difficulty arises when interlayer exchange *C*_{i} is made realistically small, for which the rotated **K**

## 3. Thickness diffusion in layer models

In layer models one may ask if a diffusive term like *K*_{i} should appear in the *h*_{i} equation. A problem arises when a deep layer rests on variable bottom topography. Diffusing the thickness of this layer causes unphysical interface deformations. Summed over the water column, thickness diffusion in the presence of variable bottom topography is unacceptable. Modelers have worked around this difficulty in various ways: by numerical diffusion due to nonconservative advection schemes or by occasionally smoothing elevation fields (Bleck et al. 1992, appendix C).

The problem due to bottom topography is circumvented in “*N* + ½” layer models supposing *N* “active” layers overlying a quiescent deep layer (hence without bottom topography). Thickness diffusion (not necessarily defined by isopycnals) was explicitly included by McCreary and Kundu (1988) in a 2½-layer model of the Somali Current. McCreary and Kundu used constant coefficient *K*_{i} to diffuse layer thickness while diffusing tracer concentration *c*_{i} rather than layer-integrated tracer burden, *c*_{i}*h*_{i}. Smoothing *h*_{i} helped preserve model stability. Cherniawsky and Holloway (1991, 1993; hereafter CH) included diffusion for both thickness and tracer burden, using flux-divergence form with flow-dependent *K*_{i} in a 2½-layer model of the North Pacific.

## 4. Thickness diffusion in level models

**u**

_{i},

*H*

_{i}, and

*C*

_{i}(which can be restored later), CH have

_{t}

*h*

_{i}

**∇**

_{i}

*K*

_{i}

**∇**

_{i}

*h*

_{i}

_{t}

*c*

_{i}

*h*

_{i}

**∇**

_{i}

*K*

_{i}

**∇**

_{i}

*c*

_{i}

*h*

_{i}

The terms **∇**_{i}·(*K*_{i}*h*_{i}**∇**_{i}*c*_{i}) or *h*_{i}**∇**_{i}·(*K*_{i}**∇**_{i}*c*_{i}) operate as diffusion on *c*_{i}. Depending upon which diffusion form is preferred, remaining terms on right sides appear as advection with velocity −*K*_{i}*h*^{−1}_{i}**∇**_{i}*h*_{i} or −2*K*_{i}*h*^{−1}_{i}**∇**_{i}*h*_{i}. For the present purpose we appreciate that a factor of 2 will be readily absorbed against the uncertainty of isopycnal diffusivity *K*_{i}. Thus, we anticipate apparent advection along the isopycnal surface “roughly” as −*K*_{i} *h*^{−1}_{i}**∇**_{i}*h*_{i}.

*σ*have derivatives ∂

_{z}

*σ*and

**∇**

_{x}

*σ*, where

**∇**

_{x}denotes horizontal gradient in the level coordinates. Then

*h*=

*σ*

_{o}/∂

_{z}

*σ*and

^{−1}

**∇**

_{i}

*h*

**∇**

_{x}

_{z}

*σ*

_{z}

*σ*

_{zz}

*σ*

**∇**

_{x}

*σ*

_{z}

*σ*

^{2}

_{z}

**S**

**S**is the slope −

**∇**

_{x}

*σ*/∂

_{z}

*σ*and

*σ*

_{o}is a reference density. Apparent advection is given by

**U*** = −

**K**

_{z}

**S**. This is the form of “eddy-induced transport velocity,” which Gent et al. (1995) describe as “exact Fickian diffusion of thickness.” Horizontal divergence of

**U*** renders this form awkward for use in level model tracer advection, requiring nonlocal determination of a vertical velocity

*W** from 3D nondivergence:

**∇**

_{x}

**U**

_{z}

*W*

**U*** = −∂

_{z}(

**K**

**S**). It is important to recognize that this choice is not compelled from physics. Rather, it is convenient from the view of admitting locally determined

*W*

**∇**

_{x}

**K**

**S**

**U*** = −∂

_{z}Ψ*,

*W*=

**∇**

_{x}·Ψ* then define a “quasi-Stokes streamfunction” Ψ* in analogy with atmospheric studies by Andrews and McIntyre (1978), Andrews et al. (1987), Dunkerton (1978), Danielsen and Hipskind (1980), Holton (1981), or Tung (1982). On the other hand, Treguier et al. (1997) argue that a consistent Fickian assumption based on potential vorticity mixing implies the

**U*** = −

**K**

_{z}

**S**form.

We see that a simple derivation of **U*** results from amending Redi (1982) to account for thickness diffusion. Alternatively, one may follow discussions given by GM, Gent et al. (1995), MM, or the atmospheric literature cited above.

## 5. Statistical dynamical alternative to Fickian assumption

Any of the above-referenced schemes depend upon making a Fickian flux/gradient assumption. This is deeply suspect, as deciding which fluxes are forced by which gradients (of what properties) is uncertain. Holloway (1992, 1996) pursued a different approach based upon statistical dynamics. Although this approach was applied initially with respect to horizontal momentum, there is a complementarity (shown below) between the momentum result and thickness diffusion. The alternative approach, including momentum, allows us to anticipate where present schemes fail and to sketch a possible resolution of these failures. Briefly, Holloway (1992, 1996) recognizes that fields we seek to characterize (tracers, momenta, thicknesses, etc.) represent moments of an underlying probability distribution *P* for possible ocean states. Evolution equations for moments of probability should include forcing terms due to gradients of entropy, *S* = − ∫ ln*P dP* (where integration is over all possible ocean states), with respect to the realized moment fields. This concept replaces Fickian hypotheses. The problem is that careful estimation of entropy gradients seems enormously difficult and is, to date, unsolved with respect to oceans. Holloway tried simply taking the gradient to be proportional to displacement of realized moments from a reference state of weak entropy gradient, characterized by a depth-independent velocity field **u*** given by *H***u*** = −**z** × **∇**(*fL*^{2}*H*), where *H* is depth of ocean, **z** is unit vertical, *f* is Coriolis, and *L* is a length associated with eddy vorticity. A parameterization scheme consisted of replacing ordinary lateral eddy viscosity by a “neptune” term *N*(**u*** − **u**), where *N* is an operator such as Laplacian diffusion. It is not so important whether *N* is Laplacian or something else. The key is that *N*(**u*** − **u**) is centered about **u** = **u*** rather than the state of rest **u** = 0.

In practice, most applications of neptune and of thickness diffusion have been made in models with coarse grid spacing. On grid spacing coarser than the first internal radius of deformation, the thickness and momentum parameterizations act nearly independently. Thickness diffusion **U*** adjusts the density field without directly rearranging barotropic velocity, while neptune forces the ocean toward a barotropic reference **u*** without directly modifying density. Altered density and velocity fields then interact according to their entwined equations of motion.

Acting alone, thickness diffusion would achieve level density surfaces with no implication for barotropic flow. In particular, thickness diffusion **U*** does not imply neptune **u***. On the other hand, neptune alone would achieve barotropic **u** = **u***, implying level density surfaces. To accomplish that leveling by adiabatic adjustments (consistently with the basis for neptune), thickness flux terms such as used by CH or GM are implied.

## 6. An illustrative failure

While Fickian hypotheses for thickness diffusion are dubious, the idea of entropy gradient forcing of moments of probable oceans is novel. We may remain skeptical. Various authors cite post hoc justifications. McCreary and Kundu (1988), CH, New et al. (1995), and Hu (1997) find encouraging results with thickness diffusion in layer models, while Danabasoglu et al. (1994), Gent et al. (1995), and Danabasoglu and McWilliams (1995) find good results in level models. Although Robitaille and Weaver (1995) show improved simulation of freon uptake, England (1995) and England and Holloway (1996) show a mixed picture with improvements in the Southern Ocean but deteriorated results in the North Atlantic. Duffy et al. (1995) also find mixed results, concluding that thickness diffusion does not “significantly compromise” radiocarbon uptake while improving modeled temperature.

Likewise, Alvarez et al. (1994), Eby and Holloway (1994), Fyfe and Marinone (1995), Holloway et al. (1995), Pal and Holloway (1996), and Sou et al. (1996) find encouraging results from neptune. Quantitative appraisal against a global inventory of long-term current meters is examined by Holloway and Sou (1996), while England and Holloway (1996) find neptune recovering some freon uptake skill in the North Atlantic. Examined in detail however, results are not uniformly positive.

The danger is that we may obtain right-appearing answers for wrong reasons. How strong is the a priori justification? Gent et al. (1995) remark that “the most obvious and simplest parameterization, downgradient Fickian, has been demonstrated to be inadequate” (for tracers) but then adopt the same downgradient Fickian assumption for thickness. On the other hand, entropy gradient forcing is novel for ocean dynamics. Moreover, approximations made by Holloway for evaluating entropy gradients are dubious, waiting to be improved.

A thought experiment illustrates where the parameterizations may fail. Consider an ocean full of eddies but at rest in the mean, hence with level mean isopycnals. Examine the vicinity of a seamount (for this example). Thickness diffusion, whether in layers or levels, would recognize level isopycnals and predict no change. Neptune predicts evolution toward mean **u***, but only as barotropic flow, anticipating a mean Taylor column with positive pressure anomaly above the seamount. Either prediction (no mean flow or a barotropic Taylor column) is an allowed solution for ideal flow. Is either prediction likely to be right?

Tides, boundary mixing, etc., complicate circumstances at any actual seamount. However, whether from observations (e.g., Brink 1995) or from numerical experiments (e.g., Chapman and Haidvogel 1992; Beckmann and Haidvogel 1993, 1996; Haidvogel et al. 1993; Beckmann 1995), we expect elevated isopycnals (“cold dome”) in the mean above the seamount. Isopycnal layers lying entirely above the seamount crest develop negative thickness anomalies. To achieve this state, thickness fluxes are systematically countergradient, providing a source rather than sink for available potential energy. (Pumping heat away from the cold dome, eddies power a natural regrigerator.) Although flow emerges with the sense of a neptune-forced Taylor column, barotropic **u*** is wrong. The cold dome implies baroclinic shear. While this thought experiment only addresses a seamount, the aim is to discover that something is wrong—something that can turn up unexpectedly elsewhere.

What has gone wrong? Simply, thickness flux and neptune ideas were applied outside their intended range of application in order to anticipate how they fail. We then see which underlying assumptions are wrong. For neptune, the error is clear. Holloway approximated the condition of weak entropy gradient for cases of model grids coarser than internal deformation radii, whereas the cold dome on a seamount is at finer scale.

It is known since Salmon et al. (1976) that maximum entropy mean flow is more bottom-trapped at these smaller scales and the simplified treatment by Holloway should be extended. For thickness flux however, the error is less clear. Nothing warns us that a Fickian assumption should be wrong (fluxes run backward!) simply because internal deformation radii are resolved. This haphazard character of Fickian assumptions is alarming, seeming to succeed in one circumstance then unexpectedly failing in another.

Before quitting, the seamount example supports another comment. Entropy gradient forcing can be confused with potential vorticity mixing, insofar as either can drive mean flow. However, countergradient thickness fluxes in layers entirely overlying the seamount will be associated also with countergradient potential vorticity fluxes (including a relative vorticity part). Thus, potential vorticity systematically “unmixes” in this circumstance while entropy gradient forcing proceeds.

## 7. Outlook

A thought experiment anticipating how thickness diffusion and neptune will fail is only a thought experiment. Briefly, let us guess what may follow.

Entropy gradient forcing of probable oceans offers an alternative to Fickian assumptions. At the larger scales to which thickness diffusion has been applied, entropy gradients force downgradient thickness flux either in layer models (CH) or in level models (GM). Complementing this, neptune forces model **u** toward **u***.

At scales smaller than internal deformation radii, simple neptune (barotropic **u***) should be corrected for baroclinic effects. Within the idealization of quasigeostrophy, this correction can be made. Difficulties arise applying results from stratified quasigeostrophy when realistic topography is required.

The problem of backward thickness fluxes on small scales is more grievous. However, entropy gradient forcing resolves this insofar as “cold doming” is already characteristic of quasigeostrophic statistical mechanics. A quasi-Stokes streamfunction **Ψ***, such as MM propose, can be useful. If we identify **Ψ*** with **K****S** of GM, we return to the problem of backward fluxes on small scales. However, if **S*** denotes mean slopes of isopycnals at the maximum entropy under ideal quasigeostrophy, then a simple characterization **Ψ*** = **K****S** − **S***) yields the countergradient thickness fluxes and eddy source of available potential energy as foreseen from the thought experiment. Tracer transports by **Ψ*** accompany momentum forcing toward baroclinic **u***, as components of Eliassen–Palm flux [Gill 1982, Eq. (12.9.14)]. At larger scales for which **S*** = 0 is a reasonable approximation, the tracer transport goes to CH or GM, while neptune forces barotropic **u***.

Is this concern at small scales misplaced? From a practical view, modeling near an individual seamount might admit sufficiently fine resolution to deal with eddies explictly. However, first, the condition of “sufficiently” fine resolution is not well understood. Second, greater computing power invites modeling on larger domains to resolve smaller scales. As internal deformation radii are resolved but eddies are not yet “sufficiently” resolved, the problem of inappropriate parameterizations at small scales will appear. Third, we seek parameterizations based on fundamentals to avoid haphazard failures. Fourth, the conceptual principle of entropy forcing provides an overarching framework for both thickness and momenta fluxes, etc.

Practical difficulties and opportunities present themselves. Finite amplitude of real topography in the stratified ocean means that ideal quasigeostrophy will provide only a starting point toward estimating baroclinic **u*** and **S***. Although estimation will be imprecise, it should suffice for modest progress. An opportunity occurs for layer modeling to overcome the difficulty of thickness diffusion above topography. Insofar as **S*** already must take account of finite amplitude topography, thickness flux should not be problematic.

Further, we recognize that *K,* appearing in **Ψ*** = **K****S** − **S***), is a coefficient of entropy gradient forcing rather than a diffusion coefficient. The importance of this is that research into alternative numerical advection schemes (e.g., Gerdes et al. 1991; Farrow and Stevens 1995; Hecht et al. 1995) may motivate models in which explicit diffusion is minimized. A nearly advective ocean model would be consistent both with neptune and with a parameterization such as **Ψ*** = **K****S** − **S***), perhaps with associated (Olbers et al. 1985; Greatbatch and Lamb 1990) vertical viscosity *ν*_{v} = (*f*^{2}/*N*^{2})**K***N* is stability frequency. By focusing representation of eddy effects into viscosity operators and **Ψ***, models could better preserve water mass properties (cf. England 1993). Physically motivated mixing could then be introduced over the minimal numerical diffusion necessary for realizable advection.

Finally, admitting ideas such as entropy forcing of moments of probable oceans into traditional areas of ocean theory exposes the possibility that ongoing research, for example, as Kazantsev et al. (1997, manuscript submitted to *J. Phys. Oceanogr.*), may bring even stronger statistical mechanical tools to bear upon the practical enterprise of ocean modeling.

## Acknowledgments

This note is based on research supported in parts by the Office of Naval Research (N00014-96-I-0518) and the Panel on Energy Research and Development (6B2027). I have appreciated helpful discussions with Peter Gent and Dingming Hu, as well as valued suggestions from reviewers.

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