## 1. Introduction

The idea that ocean properties are advected and mixed predominantly along “isopycnal” surfaces dates back at least to Iselin (1939), who noted the similarity between the salinity–temperature diagram of vertical casts in the center of a subtropical gyre and that plotted from data in the winter surface mixed layer in the whole hemisphere. This concept of “isopycnal mixing” has become part of the underpinning assumptions of physical oceanography. Papers by Veronis (1975), Solomon (1971), Redi (1982), and McDougall and Church (1986) pointed out that ocean models needed to rotate their diffusion tensor to be aligned with the locally referenced potential density surface in order to avoid the ill effects of having density mixed horizontally at fixed depth.

### a. A physical argument for orienting lateral mixing along neutral tangent planes

While the strong lateral mixing achieved by the energetic mesoscale eddies is widely believed to be oriented along isopycnals (McDougall 1987a), we are aware of only one convincing argument that supports this assumption; the argument has been made in section 7.2 of Griffies (2004) and in section 2 of McDougall and Jackett (2005) and is explained here with the aid of Fig. 1. The argument involves the rather small amount of dissipation of turbulent kinetic energy that is measured in the ocean interior.

We begin by initially adopting the counterargument, so that we take the lateral mesoscale dispersion to occur along a surface that differs in slope from the neutral tangent plane. Individual fluid parcels are then transported above and below the neutral tangent plane and would need to subsequently rise or sink in order to regain a vertical position of neutral buoyancy. This situation is illustrated in Fig. 1 where a central seawater parcel is moved adiabatically and without exchange of salinity in a nonneutral direction to either the left or right and then released. The fluid parcel then feels a vertical buoyant force (upward if displaced to the left and downward if displaced to the right) and it begins to move back to its original “density” surface.

This vertical motion would either

- involve no small-scale turbulent mixing, in which case the combined process is adiabatic and isohaline and so is equivalent to epineutral dispersion, or
- the sinking and rising parcels would mix and entrain in a plumelike fashion with the ocean environment, and therefore experience irreversible mixing.

McDougall and Jackett (2005) showed that even if all the observed dissipation of turbulent kinetic energy were due to the second case above of nonneutral lateral mixing (implying no contribution from breaking internal waves to

Nycander (2011) has examined mixing in the ocean along inclined planes and has concluded that an exchange between potential and kinetic energies is required to move seawater parcels along a neutral tangent plane. He also concluded that such energetic arguments do not shed light on the question of which mixing direction is preferred by the energetic ocean mesoscale eddies. Nycander’s results confirm the long-standing practice (since the 1980s) of defining the neutral direction using parcel movement arguments in terms of the lack of vertical buoyant restoring forces, rather than in terms of the changes in gravitational potential energy.

We conclude that the only evidence to support the notion that strong lateral mixing is directed along neutral tangent planes is the measured smallness of the dissipation of turbulent kinetic energy in the ocean interior, coupled with the arguments of McDougall and Jackett (2005) and Griffies (2004). Furthermore, this evidence from measured dissipation rates is in accordance with the interpretation of tracer distributions as per isopycnal water mass analysis, with this framework originating from Iselin (1939). The relatively small dissipation of mechanical energy in the ocean interior represents a key distinction from the troposphere, where radiative damping leads to relatively large levels of dissipation and associated diabatic mixing. Consequently, oceanographers have a fundamental reason to be concerned about details of how mixing is oriented. This concern motivates our examination of the geometry of ocean interior mixing.

### b. Specification of the neutral tangent plane

To provide a mathematical foundation for later discussions, we present a physical and mathematical review of a neutral tangent plane. Physically, the neutral tangent plane is that plane in space in which a seawater parcel can be moved an infinitesimal distance without being subject to a vertical buoyant restoring force. That is, the neutral tangent plane is the plane of neutral or zero buoyancy.

This thought experiment is typical of our thinking about turbulent fluxes. We imagine the adiabatic and isohaline movement of fluid parcels and then let these parcels mix molecularly with their surroundings. Central to this way of thinking about turbulent fluxes are the following two properties of the tracer that is being mixed:

- It must be a “potential” fluid property, for otherwise its value will change during the displacement even though the displacement is done without exchange of heat or mass, and
- it should be close to being a “conservative” fluid property so that when it does mix intimately (i.e., molecularly) with its surroundings, we can be sure that there is negligible production or destruction of the property.

The present paper is concerned with the mixing of tracers by both epineutral mixing and by small-scale mixing processes. The processes we examine are different from the separation of diffusion from advection (or the separation between symmetric and antisymmetric diffusion) that are crucial in the temporal residual mean theory and its parameterization (McDougall and McIntosh 2001; Gent et al. 1995; Griffies 1998). Rather, our focus concerns two basic geometric aspects of ocean interior mixing.

In sections 2 and 3, we show that the use of the projected nonorthogonal coordinate system gives the same diffusive tracer fluxes as the small-slope approximation to the Redi (1982) diffusion tensor. Thereafter, we show that this tracer flux has a component that is nonzero in a direction in which there is no tracer gradient. We show that for temperature and salinity, this unphysical aspect of the small-slope diffusive flux is proportional to neutral helicity. In section 4, we turn from epineutral diffusion to consider the mixing achieved by small-scale mixing processes such as breaking internal gravity waves. As a three-dimensional isotropic diffusion process, small-scale mixing should not be included in ocean mixing parameterizations as a one-dimensional “dianeutral” or “vertical” diffusion. This realization leads to a simplification of the Redi (1982) diffusion tensor.

## 2. The small-slope approximation to epineutral diffusion

### a. The exact epineutral gradient

*x*and

*y*components

### b. The projected nonorthogonal version of the epineutral gradient

*x*and longitude

*y*, while values of the scalar property

### c. Equivalence of the small-slope approximation and the projected coordinate approach

This equivalence between the projected nonorthogonal coordinate version of an epineutral gradient

### d. The two versions of the epineutral gradient are not parallel

## 3. Comparing the two versions of the epineutral gradient

### a. Deriving equations relating the two epineutral gradients

### b. Visualizing the difference between the two epineutral gradients

The geometry of the horizontal components of the projected and exact forms of the epineutral gradients,

From Eq. (19), we see that the components of

The fact that the use of the small-slope approximation leads to gradients in the

### c. How different are the two epineutral gradients?

The triple scalar product

- the ill-defined nature of neutral surfaces and the empty nature of ocean hydrographic data in
space (McDougall and Jackett 1988, 2007); - the mean vertical downwelling advection achieved by the helical nature of neutral trajectories (Klocker and McDougall 2010);
- the close connection between
and the spiraling of epineutral contours when the ocean is not motionless (Zika et al. 2010); and now - the difference between the projected nonorthogonal epineutral gradient of
, , and its exact epineutral gradient counterpart .

The quantification of the magnitude of neutral helicity in the ocean is far from complete, yet the above studies have shown that the angle between

The error made by neglecting this fraction of the epineutral tracer gradient can be compared with the very small errors that are made by taking Conservative Temperature

### d. The component of the small-slope gradient in a direction in which there is no actual gradient

We conclude that the relative magnitude of the component of

### e. Incorporating the full epineutral gradient into ocean models

This same suggestion of replacing the horizontal component of the projected epineutral gradient

## 4. The isotropic diffusivity of small-scale mixing processes

*x*,

*y*,

*z*Cartesian coordinates at the rate

The above treatment of dianeutral diffusion in Eqs. (30)–(32) is different from what has been done previously in ocean modeling (e.g., Redi 1982; Hirst et al. 1996; Gent and McWilliams 1990). In the traditional treatment of dianeutral diffusion, mixing is taken to occur exactly in the dianeutral direction with a zero component along the neutral tangent plane. However, this orientation is not in accord with how the ocean works. Instead, a blob of dye is diffused by small-scale turbulent mixing in a spherical manner. In contrast, as illustrated in Fig. 4, the dianeutral diffusion tensor of Redi (1982) diffuses this blob only in the dianeutral direction. Hence, if dianeutral diffusion is allowed to operate in the absence of epineutral diffusion, an initial blob of tracer will be diffused dianeutrally into a thin line in space, as in Fig. 4. This one-dimensional dianeutral diffusion is not how small-scale turbulent mixing works; rather the diffusion should be in three dimensions. Likewise, in the now common small-slope approximation of the full Redi diffusion tensor, the dianeutral diffusion becomes purely vertical diffusion, which is again a purely one-dimensional diffusion. In this case, a small spherical blob of dye, when acted upon by only this small-slope approximated dianeutral diffusion will spread vertically in a thin vertical line, and it will never be any wider in the *x* and *y* directions than its initial size in these directions.

We contend that an ocean model should be able to increase the dianeutral diffusivity and decrease the epineutral diffusivity (even to zero) and achieve three-dimensional spherical diffusion. For this purpose, Eq. (31) is the proper form of the full diffusion tensor and Eq. (32) is its small-slope approximation. The existing Redi (1982) and Gent and McWilliams (1990) versions of these tensors are not appropriate for this purpose.

## 5. Conclusions

The points raised in this note are of a conceptual nature and heretofore incompletely explored in the literature. We anticipate that their importance for ocean modeling practice, though yet to be tested, may in fact be negligible. That is, the niceties exposed in this paper may have little impact on large-scale simulation integrity, particularly compared to uncertainties associated with other aspects of diffusive closures (e.g., values for the diffusivities). We are nonetheless compelled to raise the ideas here to clarify basic notions regarding geometric aspects of ocean interior mixing parameterized by diffusion. In this way this note complements the warnings of Young (2012) regarding the unconventional (but not incorrect) use of projected nonorthogonal coordinates in atmospheric and oceanic science.

Specifically, the key points made in this note are the following:

- The epineutral diffusion achieved by the small-slope approximation is not down the correct epineutral tracer gradient
. That is, under the small-slope approximation, there is a small gradient of tracer in a direction in which there is no actual epineutral gradient of tracer. This is also an undesirable property of the projected nonorthogonal coordinate system that is used in layered ocean models and in theoretical oceanographic studies. - The difference between the correct epineutral tracer gradient
and the small-slope approximation to it, , is explained geometrically (see Fig. 3). The relevant difference between the tracer gradients, , is equal to [from Eq. (20)] and points in the direction of the thermal wind (here is the square of the slope of the neutral tangent plane). - For (the tracer) Conservative Temperature, the difference between the correct (Redi) epineutral gradient
and the small-slope approximation to it is proportional to neutral helicity and to the square of the slope of the neutral tangent plane. - While it is uncomfortable to realize that the small-angle approximation to epineutral diffusion (and equivalently, the use of the projected nonorthogonal coordinate system) gives rise to an epineutral property flux in a direction in which there is actually no epineutral tracer gradient, it must be said that the fraction of the epineutral flux in this direction is very small and is negligible for all foreseeable applications. For example, we have shown that for an epineutral slope
as large as , the error in using Conservative Temperature (which is not in fact 100% conservative) is greater than the error discussed in this paper of having a lateral tracer flux in a direction in which there is no gradient. - Small-scale mixing processes act to diffuse tracers isotropically (i.e., directionally uniformly in space); hence, it is a misnomer to call this process dianeutral diffusion. This realization simplifies the diffusion tensor that is used in ocean models. This is illustrated by a thought experiment in which a tiny blob of tracer is diffused by small-scale mixing processes (see Fig. 4). The blob should diffuse spherically, whereas the full Redi (1982) tensor has it diffusing as a line (a pencil) normal to the neutral tangent plane, and the small-slope approximation has this line being vertical.

The authors thank the anonymous reviewers for help in clarifying elements of this note. SMG wishes to thank Peter Gent for discussions on matters related to this work during July 2012.

# APPENDIX

## The Slope of the Neutral Tangent Plane

Here we first prove that even though the two-dimensional gradients

*x*and

*y*components

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