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.

• Download Figure
• Download figure as PowerPoint slide

Sketch of a central seawater parcel being moved adiabatically and without change in its salinity to either the right or the left of its original position in a direction that is not neutral. When the parcel is then released it feels a vertical buoyant force and begins to move vertically (upward on the left and downward on the right) toward its original “isopycnal.”

Citation: Journal of Physical Oceanography 44, 8; 10.1175/JPO-D-13-0270.1

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

1. involve no small-scale turbulent mixing, in which case the combined process is adiabatic and isohaline and so is equivalent to epineutral dispersion, or
2. 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 ), the maximum tangent of the angle between the mixing plane and the neutral tangent plane is of order . Since the dominant cause of the observed dissipation of turbulent kinetic energy in the ocean interior is likely the breaking of internal gravity waves (MacKinnon et al. 2013), no matter whether they are forced by winds, by internal tides, or by lee waves above bottom topography, the allowable angle between the plane of mesoscale mixing and the neutral tangent plane is much less than .

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.

As a thought experiment, consider the seawater parcel at a point in the ocean and enclose it in an insulating plastic bag. Upon moving to a new location a small distance away, the parcel will experience an increment in pressure . Its in situ density will thus change by , where is the adiabatic and isohaline compressibility. At the same new location the seawater environment surrounding the enclosed parcel has an Absolute Salinity that is different than that at the original location by , a Conservative Temperature difference of , and a density difference of , where and are the appropriate saline contraction and thermal expansion coefficients, defined with respect to Absolute Salinity and Conservative Temperature (IOC et al. 2010; McDougall and Barker 2011). If at the new location the displaced parcel does not feel a buoyant (Archimedean) force, its density must be equal to that of the environment at its new location . That is,

Hence, along an infinitesimal neutral trajectory in the neutral tangent plane, the variations of and of the ocean must obey

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:

1. 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
2. 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.

a. The exact epineutral gradient

Using standard three-dimensional geometry (rather than the projected nonorthogonal approach considered in section 2b), we know that the gradient of a scalar in a surface, which in our case is the neutral tangent plane, is

In these expressions, is the unit normal to the neutral tangent plane:

The vector is parallel to the unit normal vector , but has unit length in the vertical direction (parallel to gravity), that is,

We define the projected horizontal gradient operator in section 2b, which summarizes the projected nonorthogonal approach. The slope of the neutral tangent plane is discussed in the appendix, where we find the following expression in terms of the horizontal () and vertical gradients of Absolute Salinity and Conservative Temperature:

The last parts of Eq. (4) express in terms of the spatial gradient of locally referenced potential density and of neutral density (Jackett and McDougall 1997). Both and have a positive (i.e., upward) vertical component. Equation (6) has introduced the x and y components and of the vector slope , and we will also use the shorthand notation for the inner product .

The epineutral gradient [Eq. (3)] can be written in tensor form in terms of the Cartesian gradient components as

which introduces the Redi (1982) tensor for epineutral diffusion; the full Redi diffusion tensor also includes small-scale turbulent mixing that we deliberately exclude from discussion until section 4.

The parcel-based definition of the neutral tangent plane, , of Eq. (2), can now be expressed in terms of the (exact) epineutral gradients of and as

b. The projected nonorthogonal version of the epineutral gradient

The epineutral gradient that is used in many theoretical oceanographic studies (e.g., McDougall and Jackett 1988; McDougall 1988, 1995) and in layered ocean models (Bleck 1978a,b) is based on the projected nonorthogonal coordinate system first introduced by Starr (1945), which is widely used in geophysical fluid theory and modeling. In this coordinate system the vertical coordinate is strictly vertical (parallel to the effective gravitational force); planes of constant latitude and longitude are strictly vertical cones and planes, respectively; and tracer gradients are calculated with respect to an undulating coordinate surface defined by the layer coordinate. This procedure for computing a tracer gradient is illustrated in Fig. 2. Note particularly that the lateral gradient of a tracer in the undulating isopycnal or neutral density surface is calculated using the difference in tracer values between points b and a on this surface (see Fig. 2b), but the distance increment in the denominator of the projected nonorthogonal gradient is given by , that is, the distance is measured at constant height. Hence, the projected nonorthogonal gradient in a general undulating surface is given by

Importantly, horizontal distances are measured between vertical surfaces of constant latitude x and longitude y, while values of the scalar property are evaluated on the surface (e.g., an isopycnal surface or, in the case of , a neutral tangent plane). Note that has no vertical component; it is not directed along the surface, but rather it points in the horizontal direction and thus is perpendicular to gravity.

• Download Figure
• Download figure as PowerPoint slide

(a) A three-dimensional perspective of the projected nonorthogonal coordinate system that is commonly used in layered ocean models. (b) The projected nonorthogonal gradient of a property in the neutral tangent plane, in the limit as the distances tend to zero, is equal to the tracer difference between points a and b in the above figure, divided by the exactly horizontal distance or .

Citation: Journal of Physical Oceanography 44, 8; 10.1175/JPO-D-13-0270.1

The projected nonorthogonal epineutral tracer gradient can be written in terms of the regular Cartesian gradients (e.g., is the exactly horizontal gradient of ) as (from McDougall and Jackett 1988; McDougall 1995; Griffies 2004)

using as defined in Eq. (5). Note that and are each horizontal two-dimensional vectors. In the appendix we prove that the neutral tangent definition of Eq. (8) can be expressed in terms of the corresponding projected nonorthogonal gradients as

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

The three-dimensional tracer gradient based on the projected nonorthogonal approach is

which is the flux in the neutral tangent plane (this property can be checked by showing that the scalar product of Eq. (12) with is zero) whose horizontal component is . The use of the projected nonorthogonal gradient in an ocean model or in a theoretical study is equivalent to using the three-dimensional flux (12) in Cartesian coordinates.

This three-dimensional flux, Eq. (12), can be expressed in tensor form in Eq. (13) below, which we note is the same as the commonly used small-slope approximation [first introduced by Gent and McWilliams (1990)] to the exact epineutral gradient . That is, we have found that the use of the projected nonorthogonal coordinate framework gives the same three-dimensional tracer gradient as the small-slope approximation to the exact epineutral gradient , since

This equivalence between the projected nonorthogonal coordinate version of an epineutral gradient and the small-slope approximation of epineutral diffusion was built into the small-slope approximation of Gent and McWilliams (1990), as pointed out in section 6.3 of Griffies and Greatbatch (2012). In section 3, we compare the exact epineutral tracer gradient of Eqs. (3) and (7) with the approximate small-slope form of Eqs. (12) and (13). We emphasize [from Eq. (13)] that this comparison is equivalent to comparing the projected nonorthogonal version of the epineutral gradient with its exact version .

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

One might have hoped that the two forms of the epineutral gradient and would be parallel and differ by the factor , which is the ratio of the lateral distances measured in the neutral tangent plane to those measured exactly horizontally in the direction. However, this is not the case. We expose the issues by asking whether the direction of the isoline of constant tracer within the neutral tangent plane as determined by using the projected nonorthogonal coordinate system, namely , is in fact an isoline of . That is, we ask whether is zero. Using Eqs. (10b) and (13) we find that

This result shows that unless either or are zero, the direction of the constant tracer calculated using the projected coordinate approach, or equivalently by using the small-angle approximation, is incorrect. In other words, in order for and to be parallel, either or must be zero, that is, the two-dimensional gradients and must either be parallel or perpendicular. In the general situation when and are neither parallel nor perpendicular, we note that while the two gradients and are both directed in the neutral tangent plane, they are not parallel. This misalignment means that when using to parameterize eddy fluxes, there will be a component of the tracer flux in a direction in which there is in fact no tracer gradient. In addition, even when and are parallel (say because ), these gradients differ in magnitude, but this difference is of little concern.

a. Deriving equations relating the two epineutral gradients

To help understand the relationships between the two gradients and , we begin by finding an expression for the exact epineutral gradient in terms of the projected nonorthogonal gradients and . Using Eqs. (3a) and (10b) we find

Using Eqs. (13) and (15) we can relate the following components of to those of . The magnitude of the vertical component of is

the horizontal component of is

and the magnitude of in the direction (where ) is

while the magnitude of in the direction is

Another very useful combination that we can derive from Eqs. (13) and (15) is the difference between and [recall that the factor is the ratio of the lateral distances measured along the neutral tangent plane versus horizontally for displacements in the horizontal direction ]:

The last part of this equation is obtained by first noticing that is (i) perpendicular to and (ii) is a horizontal vector, so it must be in the direction.

Equation (20) is a convenient starting point to derive an expression for the cross product of and (which would be zero if these three-dimensional gradients were parallel):

This result confirms that found in Eq. (14). Namely, in order for the exact and the approximate versions of the two epineutral tracer gradients, and , to be parallel, either or must be zero. That is, the two-dimensional gradients and must either be parallel or perpendicular.

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, and , is sketched in Fig. 3, using what we have learnt from the above Eqs. (18)–(20). The slope of the neutral tangent plane is taken to be directed due south (having the Southern Ocean in mind). Because both and lie in the neutral tangent plane (i.e., both and are zero), it is sufficient to examine the horizontal components of and in order to understand their differences.

• Download Figure
• Download figure as PowerPoint slide

(a) Sketch of the horizontal components of the projected nonorthogonal epineutral tracer gradient and the exact epineutral tracer gradient . The horizontal components of these gradients are and , respectively. The components of these horizontal fluxes in the direction are equal [see Eq. (19)], while the components in the direction are in the ratio [see Eq. (18)]. The vector points in the direction and is equal to [see Eq. (20)]. Panel (a) has been drawn with . (b) Illustrates the special case when both and point due north (or due south) and these gradients are in the ratio .

Citation: Journal of Physical Oceanography 44, 8; 10.1175/JPO-D-13-0270.1

From Eq. (19), we see that the components of and in the direction are equal, while from Eq. (18) the components of and in the direction are in the ratio , that is, . From Eq. (20), the difference between the three-dimensional gradients and is exactly the horizontal vector , and this vector is shown in Fig. 3a. This vector points in the direction , and its magnitude is . In the special case where (illustrated in Fig. 3b), points due north and , showing that the projected nonorthogonal version of the epineutral tracer gradient is now parallel to the exact epineutral gradient , but that these gradients differ in magnitude by the expected factor . This special case of being zero provides excellent motivation for emphasizing the comparison between and in Eq. (20) and Fig. 3a in the general case when is not zero.

This geometric discussion serves to illustrate that the relevant comparison between the exact and the small-slope approximated versions of the epineutral tracer gradient is between and [or equivalently between and ]. In the diffusion tensor form, the relevant difference between these fluxes is

The fact that the use of the small-slope approximation leads to gradients in the and directions that differ in relative magnitude (compared to the corresponding components of ) by the factor has been pointed out in section 14.1.4.3 of Griffies [2004; see his Eq. (14.26) in which this result is couched in terms of an amplified isopycnal diffusivity by the factor in the direction]. However, the fact that the small-slope approximation involves the flux of tracer in a direction in which there is no actual tracer gradient has, to our knowledge, not been noticed heretofore.

c. How different are the two epineutral gradients?

Since the horizontal component of is , it would seem that the relative error in neglecting the difference between and would scale [using Eq. (20)] as

However, the relative error is actually significantly less than this scaling suggests because the magnitude of in the ocean is usually much less than . Here we develop expressions for and make the connection to neutral helicity.

Since , is also equal to . Using the expression Eq. (A5) for the neutral tangent plane slope, we find

Consider now the case where the tracer is Conservative Temperature (we could equally well have chosen Absolute Salinity for this purpose since their epineutral gradients are parallel), so that Eq. (24) becomes

To within the Boussinesq approximation we may take and to be equal to the corresponding gradients in an isobaric surface, and , so that

where the last three parts of this equation have used the results of McDougall and Jackett (1988, 2007) and section 3.13 of IOC et al. (2010) that relate these various triple scalar products to neutral helicity , defined as , where is the thermobaric parameter (McDougall 1987b) that expresses the nonlinear dependence of specific volume on both Conservative Temperature and pressure; a nonlinear property that does not concern us in this section of the paper.

The triple scalar product of neutral helicity has arisen in the context of

1. the ill-defined nature of neutral surfaces and the empty nature of ocean hydrographic data in space (McDougall and Jackett 1988, 2007);
2. the mean vertical downwelling advection achieved by the helical nature of neutral trajectories (Klocker and McDougall 2010);
3. the close connection between and the spiraling of epineutral contours when the ocean is not motionless (Zika et al. 2010); and now
4. 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 and is quite small in most of the ocean, even in the Southern Ocean where the influence of processes related to neutral helicity seem to be the largest (Klocker and McDougall 2010). These results for neutral helicity were obtained in both a smooth ocean atlas and in an ocean model without eddies, but neutral helicity has not yet been examined in the context of a high-resolution eddy-permitting ocean model. Nevertheless, if we take these results of McDougall and Jackett (2007) and Klocker and McDougall (2010) at face value, then we would conclude that the relative magnitude of the error in the tracer flux due to using the small-slope approximation may be no more than a few percent of the magnitude estimated by the scale analysis of Eq. (23), that is, no more than a few percent of . It must be said that the reason why neutral helicity is as small as it is in the ocean is far from clear; see McDougall and Jackett (2007) and Klocker and McDougall (2010) regarding neutral helicity and the consequent requirement that the ocean is “thin” in space.

Specifically, if we let be the (small) angle between the two-dimensional gradients and , then from Eq. (23) we see that the relative error in neglecting the difference between and is actually

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 to be 100% conservative. From McDougall (2003) and Graham and McDougall (2013), we know that Conservative Temperature is approximately two orders of magnitude more conservative than is potential temperature [see Figs. 2a and 4 of McDougall (2003) and Fig. 8 of Graham and McDougall (2013)], while from Fig. 11a of Graham and McDougall (2013) and appendix A.14 of IOC et al. (2010) we see that the epineutral gradient of potential temperature is often about 1% different than that of Conservative Temperature . This 1% difference is due to the nonconservative nature of potential temperature. We conclude that the relative error in using the epineutral gradient of Conservative Temperature is the product of these two factors of , namely, . This can be compared with the relative error, Eq. (27), involved in fluxing a tracer in a direction in which there is no gradient. For an epineutral slope of as large as this relative error is , which is smaller than the relative error in using Conservative Temperature by the factor , this factor representing the influence of neutral helicity. Since we expect to be no larger than 0.05 (McDougall and Jackett 2007), we conclude that this effect is absolutely tiny, even when compared with the use of Conservative Temperature, which itself is an improvement by two orders of magnitude on oceanographic practice under the International Equation of State of Seawater—1980 (EOS-80).

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

The component of the small-slope approximated gradient in the direction in three-dimensional space in which there is no tracer gradient is . The magnitude of this component can be calculated with the use of Eq. (21) as follows:

where the approximations are (i) so that , and (ii) that is close to being parallel to so that the sine of the angle between these two-dimensional vectors is small.

We conclude that the relative magnitude of the component of in the direction in which there is no tracer gradient is the same as Eq. (27), namely, . In other words, the salient approximation in using the projected nonorthogonal approach to calculating tracer gradients manifests itself in the direction in space in which there is no tracer gradient. This result can be understood from Eq. (20) and Fig. 3a; for the usual situation where is close to being parallel to , the error vector of Eq. (20), , is close to being aligned in the direction , in which there is no epineutral tracer gradient.

e. Incorporating the full epineutral gradient into ocean models

It is possible for a layered ocean model to incorporate the full epineutral tracer flux while retaining the projected nonorthogonal nature of its coordinate system. This correction can be achieved by replacing the existing horizontal component of the projected epineutral flux with the horizontal component of the full epineutral flux [from Eq. (15)]:

The projected nonorthogonal coordinate system of the layered model will treat this gradient as the horizontal component of its projected epineutral gradient whose three-dimensional expression has the additional exactly vertical component whose magnitude is the scalar product of with Eq. (29), namely, , exactly as it should be, from Eq. (15). So this approach provides a straightforward fix for layered ocean models. For Cartesian coordinate models, the solution is to simply use the full diffusion tensor [Eq. (7)] rather than its small-slope approximation [Eq. (13)].

This same suggestion of replacing the horizontal component of the projected epineutral gradient with Eq. (29) applies to theoretical studies that use the projected nonorthogonal coordinate system in order to accurately represent epineutral diffusion.