# Ocean Mixing

## Abstract and Keywords

Oceanic mixing is one of the major determinants of the ocean circulation and its climatological influences. Existing distributions of mixing properties determine the rates of storage and redistribution within the climate system of fundamental scalar tracers including heat, fresh water, oxygen, carbon, and others. Observations have overturned earlier concepts that mixing rates might be approximately uniform throughout the ocean volume, with profound implications for determining the circulation and its properties. Inferences about past and potential future oceanic circulations and the resulting climate influence require determination of changed energy inputs and the expected consequent adjustment of mixing processes and their influence.

Keywords: mixing, stirring, balanced eddies, internal waves, oceanic general circulation, submesoscale

Mixing and Stirring

The ocean stores, redistributes, and exchanges with the atmosphere a wide variety of substances essential to the climate system including heat (enthalpy), fresh water, kinetic and potential energies, carbon, oxygen, and nutrients of all sorts. An understanding of climate, and its past and potential future changes, thus requires a complete depiction of the distribution and redistribution of these scalar properties once they have entered the ocean. In other words, the oceanic general circulation in all of the elements that control it and change it must be described if varying climate states are to be understood.

In a laminar flow, physics dictate that the movement of scalar properties is cleanly divisible into advective components—proportional to the velocity field, ν, and diffusive ones dependent upon the molecular diffusion coefficients, κ, arising from the internal energy of the fluid. In a turbulent fluid like the ocean, the distinction is no longer clear-cut: the inability to analyze and cleanly represent the flow field from the largest scales down to the molecular ones has led historically to representations of the unresolved scales as, at least in part, as “diffusion-like.” A complete discussion of mixing encompasses the entirety of oceanic fluid physics including fully turbulent flows in the presence of stratification, rotation, external forcing, complex boundaries, and global-scale fluid flows. A systematic account of many of these elements can be found in the book by Thorpe (2005). Mixing is simultaneously a consequence of, and a cause of, the large-scale general circulation of the ocean.

Several important elements of ocean mixing have been clarified recently: (1) processes and effective rates vary by three and four orders of magnitude over the ocean volume; (2) the physics of mixing is deeply intertwined with the oceanic energy budget; (3) all of the enormous range of scales of oceanic motions from 10,000 km to 1 mm are both affected by, and in turn influence, variations in mixing. The consequences have proven profound for the basic understanding of the oceanic general circulation and its climate impacts. To limit this article, the focus is primarily on the interior ocean, touching only tangentially on the special regions of the mixed layers at the sea surface and sea floor and the variety of boundary layers occurring over sidewalls and general topographic features. The focus here is the vertical or “diapycnal” problem involving transfers across the stable stratification, which differs fundamentally both in magnitude and physics from so-called isopycnal mixing—easier motions along layers of near-constant density—nearly perpendicular to local gravity.

Discussion of the causes and numerical values of mixing, apart from being an extremely challenging observational problem, is at the very edge of modeling and theoretical understanding of fluid physics. Much of the context of this discussion can be understood from the conventional advection-diffusion equation for a tracer, $C$, written for an ordinary fluid, e.g.,

${\text{\kappa}}_{C}\text{}$ is a Newtonian diffusion coefficient of molecular origin, assumed constant in both space and time, and $\nabla $ is the three-dimensional gradient operator in any suitable coordinate system. The most common tracers in oceanic studies are temperature, *T*, and salinity, *S*. These two fields determine the density, $\rho $, structure of the ocean as,

where **r** is a three-dimensional position ($(x,y,z)$ in Cartesian coordinates) and $p$ is the pressure. (Variations in $\rho $ are here considered to be “small” in the mass balance but important in the momentum equations.) The function in Eq. (2) involves a complicated empirical rule rendering the density field a nonlinear function of the variables $T,S,p$. For many purposes, however, it can be linearized in the form,

where ${\rho}_{0},\alpha ,\beta ,\gamma $ are treated as locally constant. Often $\gamma $ is set to 0, in the Boussinesq approximation, treating the fluid as incompressible. If Eq. (3) is adequate,$\rho $ itself can be treated as a tracer, $C=\rho $.

In the simplest cases, $v$, ${\kappa}_{C}$ are independent of tracer concentration $C$ (called a “passive” tracer), but if *T*, *S*, or *p* is being considered, they influence $v$ (“active” tracers). Typical numerical values are ${\kappa}_{C}\approx {10}^{-7}{\text{m}}^{2}/\text{s}$ for temperature and ${10}^{-9}{\text{m}}^{2}/\text{s}$ for salt. Discussion of tracer mixing in the ocean is intimately tied to momentum mixing (frictional processes), and they are best treated together. Thorpe (2007) has a clear introduction. For present purposes, $v$ is taken as a “given.”

The derivatives in the tracer “diffusion” term on the right-hand side only become important when the length scales over which they are taken become very small, of the order of centimeters or less. An attractive step for an oceanographer attempting to understand the behavior of a tracer in the ocean is to argue that this diffusion term can be neglected, setting the left-hand-side equal to zero so that,

As with much of fluid dynamics, most interest and difficulty arises from the overall behavior of the advection-term, $v\cdot \nabla C$. Estimates of the spatial variation of velocity, $v$, show it carrying laminar-like structures on the largest scale (e.g., the overall gyre structure of the circulation of the Pacific Ocean) through an entire stochastic continuum of spatial scales down to 1 mm. As far as can be determined, no spatial scales are absent in the range between these rough limits (or in the jargon, there is no spatial “spectral gap”). Much, if not all, of the flow contains powerful random (stochastic) elements—that is, it is turbulent—and is best perceived from one of the many animations of the ocean circulation now available on the web. Figure 1 shows a snapshot of estimated surface speed in the ocean.

The visually conspicuous “swirling” flows carry tracer properties with them, stretching their boundaries out to enormous lengths. These motions are commonly referred to, somewhat vaguely, as the oceanic “mesoscale eddies,” or “balanced eddies.” The latter terminology emphasizes their tendency to satisfy near-geostrophic balance (dominated by the Earth’s rotation), although at the shortest scales, below about 10 km, they become significantly non-geostrophic and are then commonly labeled “submesoscale” eddies. Below visual detection, molecular processes are also at work, preventing the structures from becoming arbitrarily elongated.

Suppose a patch of dye with a circumferential distance of order ${D}_{0}$ is introduced into such a flow in two dimensions (Fig. 2).

At the initial time, at the patch edge, ${\kappa}_{C}{\nabla}^{2}C$ may not be negligible, but the initial $D={D}_{0}$ is sufficiently small that an integral of the last term in Eq. (1),

taken around its boundary, is negligible compared to other integrated term from Eq. (1) Through time, the patch is stretched out and extended in a tortuous pathway. Even in comparatively simple, non-turbulent flows, pathways commonly have a chaotic behavior (see, e.g., Ottino, 1989). The boundary demarcating the edge of the path becomes extended in time, growing to values $D\gg {D}_{0}$. At this boundary, the gradients, $\nabla C$, are extremely large, and the integrals in Eq. (5) taken around the long boundary $D$ are very important and cannot be ignored and indeed prevent the fluid from developing arbitrarily elongated and tangled structures. Indefinite growth is prevented both by these molecular processes and those acting to control particle separations within a dye patch (Salmon, 1998). Thus one must distinguish the statistics of turbulent motion insofar as they affect the center of mass of the dye patch from those controlling the separation of fluid particles within the patch (see, especially, Garrett, 2006).

Eckart (1948) divided oceanic processes into three parts, supposing there exists a large-scale flow carrying tracer patterns of interest. Superimposed upon those large-scale flows are smaller-scale motions, ${\text{L}}_{\text{s}}$, which stretch the tracer out into the complex patterns with intricate and extended boundaries. He called those motions “stirring.” Finally, the very smallest scales at both patch edges, and within the interior of the patch, the molecular scale ${\text{l}}_{\text{m}}$ rarely observed directly, are where molecular processes dominate and act to homogenize the fluid. (Note that Figure 2 depicts an essentially two-dimensional fluid movement; in three dimensions the elongated, serpentine, structure is less pronounced.)

Although the molecular scale was recognized early on as essential to understanding the distribution of properties in the oceans, the observational tools available before about 1965 did not permit discussion of how these mixing processes actually operated. A great deal had been learned about the physics of conventional turbulence in unstratified, non-rotating fluids, but the oceanic applicability remained obscure. An oceanographic exception was the growth of interest in so-called double-diffusive processes, although most of that discussion was focused on laboratory-scale experiments and theory (e.g., Radko, 2013).

For roughly the first 100 years of physical oceanography (to about 1965), observational capabilities on the largest scales were limited to ship-borne measurements of temperature and salinity, resulting in the now-familiar and classical charts of oceanic gyres and major boundary currents, and which are known to be quasi-permanent features. These are identified with scale $\text{L}$ (see Fig. 3), represented as a large-scale, fundamentally laminar flow.

On the other hand, oceanographers were well aware of the existence of smaller scales of flow (see Helland-Hansen & Nansen, 1909) and realized that they had both a dynamical and kinematical influences on the larger scales, ones that ultimately had to be treated statistically.

Assuming that these motions were turbulent, in the sense of being stochastic, and in a fluid-dynamics approach dating back to the 19th century, it was hypothesized that both $v,C$ could be broken up into two pieces: $v=\overline{v}+{v}^{\prime}$, $C=\overline{C}+C\prime $, where the overbar represents an averaging process, in space, or time, or both. If space was involved (a useful choice in a spatially homogeneous flow) the averaging interval was taken to be larger than the stirring scale ${L}_{s}$, but less than $L$, such that $\overline{{v}^{\prime}}=0$, $\overline{C\prime}=0$ but $\overline{{v}^{\prime}C\prime}\ne 0$. (See especially Tennekes & Lumley, 1972, or any book on turbulence, e.g., Batchelor, 1953.) Alternatively, the assumption might be made that a time average would remove all of the small scales—although that involves a strong and difficult-to-justify assumption about space-time statistics of the ocean. In particular, the longest available records do not support the existence of any kind of spectral gap in frequency.^{1} Eq. (4) then becomes, when averaged,

where $\overline{v}$, $\overline{C}$ are the quasi-laminar fields with everything else to be handled statistically. The problem—the “turbulence closure”—is what to do with the divergence of $\overline{v\prime C\prime}$, the covariance of the spreading scales, (In conventional homogeneous, isotropic turbulence, this divergence vanishes.) Following Osborne Reynolds and many others, it can be supposed that these spreading scales act as an analogue of the suppressed diffusive ones, but operating entirely locally on the $L$-scale, so that,

In a turbulent flow, it is often argued that all tracers are stirred by eddies at the same rate so that $K$ would be appropriate for all $C$. Because of known anisotropies in oceanic flows and features, anisotropic rules were used,

(in a Cartesian system) or in the most general form,

where

has become a tensor (written above both in the special Cartesian and a more general index forms) with spatial or even temporal structure whose physical form must obey certain rules (e.g., Griffies, 2004).

Use of these “eddy” coefficients (and their analogues for viscosity in the momentum equations) has had a long and useful history. Extensive discussions can be found in Defant (1961). In both analytic and numerical modeling, the symmetric elements of $K$ destroy any gradients long before they reach the ${l}_{m}$ scale and thus function *implicitly* to dissipate tracer variance. That mixing depends only on the *local* properties of $C$ has been assumed (for a contrasting approach, see Canuto, Cheng, & Howard, 2007). Anti-symmetric elements can act as stirring motions, increasing the gradients (see e.g., Flierl & McGillicuddy, 2002; Griffies, 2004).

At the time of this writing, a major field experiment, the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES; Sheen et al., 2013), is underway to better understand the structure of $K$, although the physics of the Southern Ocean flow are often untypical of the global ocean.

Isopycnals and Neutral Surfaces

Most of the ocean is stably stratified (Fig. 3), and the fluid has difficulty moving tracers across that stratification, which requires it to work against gravity. In contrast, movement perpendicular to gravity involves much less work and energy (except for internal viscous stresses), and substantial evidence supports the inference that usually ${K}_{33}\ll {K}_{11},{K}_{22}$. In practice, the surfaces of constant density, ${x}_{3}=\sigma \left({x}_{1},{x}_{2}\right)$, in many places deviate from the horizontal (Fig. 3 at the southern end), and the expression representing vertical diffusion can become inaccurate. ${K}_{33}$ is better expressed as ${K}_{\sigma \sigma}$ (a “diapycnal” diffusion), and ${K}_{11},{K}_{22}$ and the various cross-terms become expressions for diffusion along the isopycnal surfaces (“isopycnal” diffusion). Because the equation of state of seawater, representing the density as a function of temperature, salinity, pressure, and location, $\rho \left(T\left({x}_{1},{x}_{2},{x}_{3}\right),S\left(s,{x}_{2},{x}_{3}\right),p\left({x}_{1},{x}_{2},{x}_{3}\right)\right)$, is significantly non-linear, many authors replace isopycnals with “neutral surfaces” along which the work done against gravity by movement along the surfaces is minimal. Here the “vertical” coordinate will differ from the geocentric vertical in the coordinate system $\left(x,y,z\right)\to \left({x}_{1},{x}_{2},{x}_{3}\right)$, where it is understood that ${x}_{3}$ is normal to an isopycnal or neutral surface and is nonetheless sometimes most simply written as $z$. The term “isopycnal” will be used here without distinguishing the various representations of surfaces of minimal work against gravity. ${K}_{33}$ becomes a “diapycnal” coefficient, and ${K}_{11,22}$ are “isopycnal” coefficients. Over much of the ocean, the distinction between the geodetic vertical and the isopycnal normal direction can be ignored, but the difference is crucial in regions of isopycnal slope as seen in Figure 3 in the Southern Ocean. The non-linearity of the equation of state gives rise to a number of sometimes puzzling behaviors of mixed fluids under the rubrics of “cabbeling” and “thermobaricity.” For example, two fluid parcels with different temperature and salinity but the same density will, when mixed, have a different density (see e.g., McDougall & Garrett, 1992; Schanze & Schmitt, 2013.)

By definition, in an isopycnal coordinate system, mixing cannot move fluid parcels of different density across the reference surfaces, but that is not true of passive tracers. This distinguishing behavior leads to detailed complexity in numerical models of the ocean.

Scales

Oceanic flows do not occupy discretely separable spatial scales, $L,{L}_{S},{l}_{m}$, but consist instead of a continuum of near-geostrophic global horizontal scale structures (basin and larger), changing significantly in time only slowly (many decades and longer): a geostrophically balanced eddy field, with spatial scales of a few hundred kilometers and changing over many months and longer; an internal-inertial wave field on spatial scales of tens of kilometers and changing over hours to days; a “submesoscale” where rotation no longer dominates the eddies, and a small-scale turbulence with features of tens of meters to millimeters tending to be three-dimensionally isotropic, and changing over seconds to days. These motions overlap in a continuous occupation of frequency/wavenumber spaces. Derivatives of tracers and velocity, taken over the smallest scales, usually greatly exceed those taken on the largest scales.

A small extension of Eckart’s (1948) schematic description of oceanic mixing is that tracers varying on scale $L$ are stirred by the balanced (geostrophic) and submesoscale eddies (scale ${L}_{BE}$). Then localized breaking by the internal wave field (${L}_{IW}$) generates the turbulence that is dissipated in a traditional turbulent cascade to the molecular scale ${l}_{m}$, dominating the final conversion into molecular scales. Although a useful conceptual structure, it can be very inaccurate and misleading. For example, the gyre scales contain many features such as the western boundary currents, the equatorial current system, etc., which lie on the same scale (of the order of 100 km) as do the balanced eddies. Balanced eddies appear to exist even at very long periods—much longer than a year. They include such diverse physics as linear and non-linear Rossby and Kelvin waves, isolated vortices, coupled waves, instability structures, etc. Internal waves overlap the high wavenumber part of the submesoscale eddy field and also contain balanced “vortical” structures. This listing remains only a sketch, ignoring many different boundary layer types at the surface, sidewalls, and bottom. All of these physical processes are coupled to a greater or lesser degree. Sometimes, as in resonant interactions of wave-like motions, energy can move from one space- and timescale to radically different ones. In conventional turbulence, interactions take place only between nearest-neighbor wavenumber scales, leading to the discussion of power laws, of which the Kolmogoroff ${k}^{-5/3}$ law is the best known, and where energy “cascades” from large to the smallest scales. But in the ocean, some processes produce cascades in the opposite direction with differing power laws. Wave-like motions are often invoked statistically in a “wave-turbulence” approach (e.g., Polzin & Lvov, 2011).

Generalizing, ${K}_{1j},{K}_{2j},{K}_{j1},{K}_{j2}$ are dominated by the balanced eddies in the open ocean, with the submesoscale becoming most important where isopycnal slopes are steep. ${K}_{33},{K}_{3j},{K}_{j3}$ are usually controlled by breaking internal waves and provide the ultimate link between the stirring-dominated isopycnal terms and the ultimate dissipation. Generalizations about regions near lateral boundaries are not justified at the present time.

Power Integrals

The most direct connection that can be made across the scales is that based upon “power integrals” as introduced by Stern (1975) and discussed by Joyce (1980). In a turbulent flow field, any structures in the $C$ field would be dissipated at a rate,

(the derivatives in Eq. (11) are negligible for scales larger than ${l}_{m}$) where the overbar denotes a space and/or time-average. Then in a statistically steady-state ocean, the input of this variance, integrated over the ocean is,

and the integrals are taken over the ocean boundary area, particularly the surface, but including all generation boundaries and the volume, respectively, so that the generation-rate balances the dissipation rate. ${Q}_{C}$ is a measure of the transfer of $C$ to the ocean volume from the atmosphere and other boundaries under the assumption, in the steady state, that

If ${Q}_{C}$ and $C$ are covarying, the resulting variance generated must be dissipated at the molecular scale, all physics of scale transformations being integrated out in Eq. (12). Generated variances on the left must be consumed in the term on the right at the molecular level, otherwise the spatial variance of $C$ would grow without bound. (See Schanze & Schmitt, 2013, for the complications introduced by various nonlinearities.) The special case of $C=T$ corresponds to entropy conservation. Much of the dissipation, like the generation, occurs within surface the mixed layer, but significant structures remain in the oceanic interior that must be removed at the molecular scale.

These arguments can be generalized to certain types of oceanic regional subvolumes. Using Eqs. (11) and (12) to solve for ${\kappa}_{C}$ gives values (for temperature and salinity) roughly consistent with those obtained from completely different methods (and with very different error bars). All observations do, however, remain highly uncertain (see Schneider & Bhatt, 2000).

Consider now the analogue of Eq. (11) for the velocity field, defined as $v=\left({u}_{1},\text{}{u}_{2},\text{}{u}_{3}\right)=\left(u,\text{}v,\text{}w\right)$, with the latter being the Cartesian special case, but the first form permitting the use of indices. The dissipation of kinetic energy in the ocean, ultimately at similar molecular scales, is usually written as

where $\nu $ is the ordinary molecular kinematic viscosity, which would again be dominated by the ${l}_{m}$ spatial scale. If the turbulence lies mainly in isotropic motions at the smallest scales, the above simplifies to

for any $i,\text{}j$. The connection between $\epsilon ,\text{\chi}$, is exploited in the measurements described below. The “Kolmogoroff length-scale,” ${l}_{K}={\left({\nu}^{3}/\epsilon \right)}^{1/4}$ is a measure of the smallest turbulence scale present in the fluid, and complete measurement of dissipation in the ocean would have to reach to distances of the order of 1 cm and less.

Large-scale forces acting on the ocean (e.g., winds and tides with timescales ranging from the mean to short-lived squalls) are communicated to the ocean through processes that generate motions on all of the observed flow-scales. The wind and other generation processes can also destroy previously produced structures, and thus $Q$ must be interpreted as the net production in the boundary regions.

Vertical Mixing

## Quasi-Analytic Model Estimates

The most influential attempt at estimating an element of $K$ was for the vertical (diapycnal) component ${K}_{33}$, with the advection-diffusion Eq. (9) written as the special case of a one-dimensional steady-state balance,

For constant $w$ and ${\text{K}}_{33}$ the solution is a simple constant plus exponential,

In a widely influential paper, Munk (1966), noting the generally horizontal nature of isopycnals in the open Pacific Ocean, inferred $w/{K}_{33}$ from fitting the Eq. (17) profiles of temperature and salinity below 1000 m. Earlier, Wyrtki (1962) used passive oxygen and nutrient distributions, which, however, involve interior sources and sinks having significant uncertainties. By also employing radiocarbon as a tracer with a decay term on the right-hand side, Munk was able to separately distinguish $w\approx {10}^{-7}\text{m}/\text{s},{K}_{33}\approx {10}^{-4}{\text{m}}^{2}/\text{s}$, assuming the parameters applied identically to temperature, salinity (or linearized density), and radiocarbon, as would be true in conventional turbulence. In the then-convention of centimeter-gram-second (CGS) units, the result was the memorable (“canonical”) value ${K}_{33}=1{\text{cm}}^{2}/\text{s}$.

Earlier “thermocline models” (e.g., Stommel & Webster, 1962; Welander, 1971) implicitly included Eq. (16) as a special case. These remarkable nonlinear solutions showed, however, that “perfect fluid” (adiabatic, $K=0$) models gave a gross structure in the thermocline (roughly 300–1000 m depth) similar to the observed temperatures and salinities and visually indistinguishable from those with significant vertical mixing. Munk’s “abyssal recipe” was widely misapplied in that upper ocean region, where he had made no claims about ${K}_{33}$, and there arose a sense that a contradiction existed between the one-dimensional balance and a possible near-adiabatic upper ocean. The seeming paradox was only resolved with the development of direct field measurements, taken up below.

Lateral Mixing

Lateral analogues in the horizontal dimension for determining ${K}_{11,22}$ using large-scale observed horizontal gradients have a long history. An example is discussed by Needler and Heath (1975) and Hogg (1987), who, using the conspicuous feature of the Mediterranean Salt Tongue, did a simultaneous least-squares fit for both vertical and horizontal values. The latter found ${K}_{11}={K}_{22}\approx 500{\text{m}}^{2}/\text{s},{K}_{33}\approx 5\times {10}^{-5}{\text{m}}^{2}/\text{s}$.

In more recent years, with the availability of near-global altimetric measurements of surface geostrophic flows, a number of attempts have been made to estimate ${K}_{11,22}$ directly from the eddy variability. Starting with Holloway (1986) and Stammer (1997), estimates range from about 250 m^{2}/s in the quiescent eastern parts of the ocean to nearly 3500 m^{2}/s in the eddy-intensified areas of the low-latitude western-boundary regions (Fig. 4). Numerous regional estimates have been published subsequently, with a particular emphasis in recent years on the eddy-rich Southern Ocean.

The lateral terms ${K}_{1\text{*}},\text{}{K}_{2\text{*}}$, where the * denotes any of the 3 indices 1, 2, or 3, are commonly identified primarily with the powerful kinetic energy of the balanced eddies and within a modeling context, have been the subject of a wide variety of proposals and tests. Recall, however, that the quasi-time average flow field has numerous spatial scales far smaller than the largest scale defining the general circulation. Boundary currents, such as the Kuroshio and Gulf Stream on the west, the Equatorial Undercurrents and their associated shear fields, as well as flows on eastern boundaries, are all capable of stretching and greatly extending the mixing volumes of tracers. But because the modeling community has focused on parameterizing the mesoscale elements of $K$, comparatively little is known about the mixing processes in strongly sheared boundary currents. More generally, the presence of strong horizontal flows (boundary currents, the circumpolar current) implies that the interaction of the balanced eddy field with those motions cannot be ignored. Note in particular (Ferrari & Nikurashin, 2010; Klocker, Ferrari, & LaCasce, 2012) that the horizontal elements of $K$ are apparently suppressed near the surface at the core of the Antarctic Circumpolar Current and presumably in other strongly sheared currents. In the same currents, excess mixing can occur at depth in the “critical levels” where the phase velocity of the eddy disturbance coincides with the velocity of the mean shear (Abernathey, Marshall, Mazloff, & Shuckburgh, 2010).

Lateral mixing has also been studied via the separation (dispersion) of clusters of pseudo-Lagrangian floats (Lacasce, 2008) as well as from data arising from the purposeful dye experiments. As with ${K}_{33}$, realistic numerical models of the ocean must accommodate a complex and highly detailed spatial structure in the lateral components of the mixing tensor. Whether the temporal variations resulting from variations in time of energy inputs to the ocean are significant and, if so, on what timescales, remains largely unknown.

Vertical Mixing: Direct Field Measurements

The discovery of what is now called “microstructure” on scales of centimeters in vertical profiles of temperature and salinity in the middle 1960s, through newly available profiling instruments (see Baker, 1981), led to the realization that the observed structures were an indicator of the direct actions of molecular processes in the oceanic circulation. (“Fine structure” is commonly defined as appearing on the larger, meter, scales.)

Profiling instruments that measure turbulent velocities have led to numerous direct estimates of diapycnal or vertical mixing rates. Osborn (1980) proposed that the relationship between the turbulent mixing coefficient and the measured dissipation could be calculated as

where $\epsilon $ is defined in Eq. (12), brackets indicate an average, and $N=\sqrt{-\left(g/\rho \right)\partial \rho /\partial {x}_{3}}$ is the buoyancy frequency. $\text{\Gamma}$ is the so-called mixing efficiency, which is widely taken to be approximately 0.2, but a growing body of evidence suggests that it can vary considerably (Inoue & Smyth, 2009; Lozovatsky & Fernando, 2013). Its physical interpretation is as the fraction of the turbulent energy that goes into mixing, noting that it must vanish in an unstratified fluid. Variations in $\text{\Gamma}$ by factors of two or three are not usually the limiting uncertainty in calculations involving Eq. (18), but eventually it must not be treated as a universal constant.

An alternative measure (Osborn & Cox, 1972) is based directly on measurements of ${\chi}_{{\rm T}}$ in the form,

The two methods should produce identical results where only temperature controls the stratification. Merrifield, Laurent, Owens, Thurnherr, and Toole (2016) discuss the differences when salinity becomes a factor. A major issue with Eqs. (18) and (19) is that few instruments are capable of measuring the full spectrum of motions down to the Kolmogoroff scale, ${l}_{K}$ in $\epsilon $, or the even smaller equivalent Batchelor-scale, ${l}_{B}=\sqrt{{\kappa}^{2}\nu /\epsilon}$ for temperature. Thus the unresolved contributions to $\epsilon $ or ${\chi}_{{\rm T}}$ must come from extrapolation. Some of these extrapolation formulas are quite complex. Waterhouse et al. (2014) have compiled a global calculation of ${K}_{33}$ from a very large variety of velocity profile measurements and extrapolations, taken over the years, and a fixed value of $\text{\Gamma}$ in Eq. (18) (shown in Fig. 5).

Upper ocean values are generally of the order of ${10}^{-5}{\text{m}}^{2}/\text{s}$, with higher values in the very high latitudes. Values at depth derive from very sparse sampling and tend to be higher over topographic features, e.g., the Mid-Atlantic Ridge in the South Atlantic Ocean.

Figure 6 shows an approximate breakdown of their results by topographic roughness type.

Abyssal plain values are nearly uniform with depth, while those over rough topography, and particularly over the mid-ocean ridges, show a pronounced increase toward the bottom. Any calculation today of global mean abyssal values from these measurements would be highly uncertain.

Vertical Mixing: Volumetric Inverse Methods

Profiling instruments produce measurements at single points, and in Figure 5 they remain sparse in both space and time, conspicuously so at depth. A different approach to determination of ${K}_{33}$ comes from integrating geostrophic flows over large volumes lying between isopycnals—including boundary effects—in geostrophic box models. A closely related approach is through the $\beta $-spiral (e.g., Fukumori, 1991; Olbers, Wenzel, & Willebrand, 1985).

A particularly simple example is that of Hogg et al. (1982), who found that the deep Brazil Basin had to have a value of ${K}_{33}$ of about ${10}^{-4}{\text{m}}^{2}/\text{s}$, not inconsistent with the abyssal recipes canonical value. Later profiling measurements (Toole et al., 1997) show the volumetric average there as being dominated by much higher values over and near the Mid-Atlantic Ridge and lower values over the abyssal plain. Various box inversions (e.g., Ganachaud, 2003; Lumpkin & Speer, 2007; Macdonald et al., 2009) have produced basin-scale estimates ranging from about 2×10^{−5} to 3×10^{−4}, depending upon the location and depth as well as assumptions about detailed rates of flow. These results combine the effects of interior and lateral boundary/topography mixing rates. All have significant stated uncertainties. Many places (Ganachaud, 2003) show an increasing ${K}_{zz}$ towards the bottom. A number of studies (e.g., Sloyan & Rintoul, 2000) have estimated the net diapycnal flux.

Of particular interest have been experiments with purposeful dye releases in the open ocean (e.g., Watson et al., 2013), which tend to give higher mixing rates than have been inferred from local profiling measurements—again probably best interpreted as an indication of higher intensity mixing near boundary topographies of various sorts (Mashayek et al., 2016).

Boundary Mixing

Munk (1966) noted in passing that oceanic mixing could be greatly enhanced at the sidewall boundaries of the ocean. Stimulated by the intense motions generated on sloping boundaries by the internal wave field, Phillips (1970) and Wunsch (1970) examined the flows at the boundaries where no flux into the solid is permitted. Later examinations include that of Dell and Pratt (2015). No simple separation into vertical and horizontal effects is possible in those regions.

The peculiar properties of internal waves on sloping boundaries, leading to intense breaking, and boundary layer instabilities (Ivey, Winters, & DeSilva, 2000) seen in the laboratory led to fine and microstructure measurements, first near islands and seamounts (Eriksen, 1998; Hogg, Katz, & Sanford, 1978; Wunsch, 1972) and to work on continental margins (e.g., Lamb, 2014). Direct experiments with injected artificial dyes (e.g., Ledwell et al., 2016) have confirmed that excess mixing does occur through various processes at continental boundaries. The collection of papers found in Imberger (1998) provides a useful overview. Oceanic canyons, both on continental margins and in the mid-ocean ridges, also show strongly increased mixing, likely associated with internal wave instabilities of various kinds (see Kunze, Rosenfeld, Carter, & Gregg, 2002, Thurnherr, St. Laurent, Speer, Toole, & Ledwell, 2005). Armi (1978) calculated the effect of ambient flow and associated turbulence on sloping topography, which gave rise to debate (see Garrett, Maccready, & Rhines, 1993) concerning the influence of that mixing on the interior stratification. At the present time it does appear that sloping boundary turbulence and consequent mixing are dominated by the complex motions and breaking from the internal wave field, including the internal tide. Whether the maximum effective mixing rates occur in the boundary layers or in the interior region above the boundary layer is not clear. Garrett et al. (1993) have argued that boundary layers tend to mix already-mixed fluid and so are relatively ineffective compared to the enhanced breaking of internal waves just above the bottom.

A sweeping generalization would be that the Munk canonical value for a vertical mixing coefficient of about ${10}^{-4}{\text{m}}^{2}/\text{s}$ applies primarily to the deep ocean (somewhere deeper than the 1 km depth he focused on), as an average of much smaller interior values (below ${10}^{-5}{\text{m}}^{2}/\text{s}$) and much higher values over topographic features, with generally increasing values as the topography (on many scales) is approached from above (see Munk & Wunsch, 1998; Toole, Polzin, & Schmitt, 1994). Regions with very small values of inferred ${K}_{33}$ imply correspondingly very small values of the vertical velocity, $w$, in those places (see the next section), with major consequences for the behavior of the large-scale circulation. Furthermore, the strong increase of ${K}_{33}$ toward the oceanic floor, away from the abyssal plains, gives rise to somewhat unexpected overall behavior through terms such as $\partial {K}_{33}/\partial {x}_{3}$ (Ferrari, Mashayek, McDougall, Nikurashin, & Campin, 2016), which act in the advection-diffusion equation as an equivalent abyssal vertical velocity.

Boundary mixing is included in the inverse box model inversions where hydrographic lines terminate in topographic features but not in most of the profiling measurements. Comparisons of volume averages from box models with point measurements from profilers are thus difficult.

In the region above 1–2 km in the water column, the dominant values of ${K}_{33}\le {10}^{-5}{\text{m}}^{2}/\text{s}$ are consistent with the thermocline theories, treating the motion as nearly non-diffusive. Much recent attention has thus been focused on the outcrop regions of the isopycnals/neutral surfaces in the Southern Ocean, where direct wind mixing, within the surface mixed-layer/Ekman layer, likely provides most of the transformation of water properties required by the observed upper-ocean circulation.

Exchanges between the oceanic interior and fluid in the boundary layers of the sidewalls (recalling that their slope is usually less than 3°) requires special treatment in its own right (e.g., Dell & Pratt, 2015; Garrett et al., 1993; Phillips, Shyu, & Salmun, 1986) and may (Ferrari et al., 2016) control the interior circulation. Attention in the next several years is likely to further focus on the interaction of ocean flows on all scales with boundary features of all types.

Consequences of Spatial Variations

The existence of regions of strongly enhanced values of ${K}_{ij}$, particularly the orders of magnitude found in ${K}_{33}$, greatly complicates quantitative calculations of effective rates of mixing. Not only does the spatial variation have important consequences for the ocean circulation on the largest, $L$, scale, but the residence time of fluid within those “hot spots” is a function of the large-scale circulation itself (Mashayek et al., 2016).

The simplest dependence on the spatial structure can be seen from the important dynamical relationship in a mainly geostrophically balanced ocean in the linear vorticity equation,

where $\nu $ is the meridional velocity and $\beta $ is the local meridional derivative of the Coriolis parameter, $f$. To have a meridional flow requires a finite $w$ (the conventional Eulerian vertical component) with finite derivative. On the other hand, in regions of nearly horizontal isopycnals, Eq. (16) shows that the existence of $w$ depends directly upon the existence of a finite ${K}_{33}$. Weak values of vertical mixing, in the absence of steeply sloping isopycnals, imply weak or non-existent $w$, and hence a strong limit on any meridional flow.

In the “abyssal recipe” of Eq. (16), a steady state (e.g., of temperature, $C=T$) is maintained with a constant ${K}_{33}$ as a balance between the downward diffusion of heat from the high temperatures near the surface and the upward advection cold water from below. But if the assumption of a vertically constant ${K}_{33}$ is abandoned, the situation can be very different. Suppose by way of example that the abyssal ocean has a constant temperature gradient, $\partial T/\partial {x}_{3}={T}_{0}^{\prime}$, in which case Eq. (16) becomes

and if, as now seems clear (Fig. 5), ${K}_{33}$ is often largest near the ocean bottom, $w<0$ and the balance is now that of the *downward advection* of temperature against the *upward diffusion* of cold water.^{2} An implication is the reversal of the standard picture of an upward movement of water balancing high-latitude convection of dense, deep water to the abyss. The product of the density profile with that of ${K}_{33}$ determines the sign of $w$. A more general derivation of this result, with a discussion of the potential consequences for the general circulation, is given by Ferrari et al. (2016).

The implied values of $w$ from Eq. (21) will have a lateral variability that is as large or larger than that seen in ${K}_{33}$ itself. Figure 7 displays a 20-year average and its standard deviation obtained from a state estimate (an ocean general circulation model fit to the data by constrained least-squares).

Among other implications is that the Eulerian $w$ field in the ocean is expected to be extremely noisy; Figure 7 displays an estimate of $w$ at three depths. The spatial structures, particularly at depth, are complex even with 20 years of averaging. Liang, Wunsch, Heimbach, and Forget (2016) showed equivalent maps of the net time-averaged ${w}^{*}=w+{w}_{eddy}$ arising from both the Eulerian and eddy-induced values. No global generalization, even about just the open ocean values of $w$ or ${w}^{*}$, is likely to be accurate. No reconciliation yet exists of the inferences from the Munk (1966) model and those deriving from the intense spatial variability.

Upper Ocean/Mixed Layer

The surface layers of the ocean in direct contact with the atmosphere have a specialized mixing literature of their own (D’Asaro, 2014). Surface outcrop areas of isopycnals visible in Figure 3 and in other oceans are of particular importance. Exchanges between the mixed layer and the ocean below have focused on the influence of the submesoscale, particularly as it influences the nutrient transfers and hence the biology (e.g., Mahadevan, 2016).

Vertical Mixing: The Energy Problem

In the underlying Navier-Stokes equations such as Eq. (1), ${\kappa}_{C}$ derives from the vibrational and other motions of molecules powered by the internal energy of the fluid. Consider that an isolated thermally stratified fluid will, after a sufficiently long time, become uniform in temperature, with a consequent increase in its potential energy—the center of mass having been moved upward and with the internal energy correspondingly reduced. In numerical models, where the coefficients such as $K$ are commonly introduced as ad hoc parameters representing the action of eddies on the larger scale $L$, no explicit source of energy has been specified. Beginning in the late 1990s, attention began to turn toward understanding of the complete oceanic energy budget including, specifically, the source of energy powering the stirring that carries the mixing the final step toward the molecular scale.

Energy suppliers for the turbulent stirring and mixing are identical to those powering the overall general circulation, and thus a clean separation of sources is not possible. Such overall power sources include: (1) the wind. (2) tides. (3) precipitation and evaporation. (4) heat exchange with the atmosphere. and (5) atmospheric pressure changes. Attaching numbers with useful accuracy to these various processes has proved challenging, and the reader is referred for more detail to the article on ocean energetics. A summary would be that (3) and (5) are minor contributors, (1) and (2) are very important, and that evaluating (4) has proven problematic, as it appears to depend strongly on the flows established by (1) as well. Input of energy from the wind is thought to take place primarily on the largest scales and on the scale of oceanic inertial motions (near the Coriolis frequency, $f$). Much of the energy input appears first as an increase in oceanic potential energy, prior to its release into kinetic energy by a variety of pathways. Winds are believed to be the major power source for the balanced eddies via baroclinic instabilities but then can act to destroy them. Tidal input of energy is believed to occur primarily through the conversion at topography of the ordinary surface tides into internal tides on scales far smaller than $L$.

Understanding the energetics of mixing then depends upon determination of the pathways by which the energy inputs carry energy down to the point where molecular processes can begin to work efficiently. These pathways and their importance and locales remain poorly quantified. Attention has focused on internal waves, both interior breaking (see, e.g., Munk, 1981; Thorpe, 2010) and the very intense motions they induce on and over slopes of all kinds.

Most of the kinetic energy of the ocean circulation lies in balanced eddies, and they are an obvious source for conversion into smaller scales. As compared to the internal wave field, their vertical and horizontal scales tend to be larger, but in topographic interactions they are capable of generating a large variety of much smaller scales including internal waves and boundary layers. The lifetime of the eddy field has been estimated as roughly 6 months (Ferrari & Wunsch, 2009), but where and how the corresponding dissipation occurs has not been fully quantified (Clément, Frajka-Williams, Sheen, Brearley, & Garabato, 2016; Zhai, Johnson, & Marshall, 2010). The direct coupling between geostrophically balanced motions and internal waves is weak (Vanneste, 2013). Also, a strong tendency exists in rotationally dominated turbulence to drive energy “upscale,” that is to larger rather than smaller spatial scales (e.g., Vallis, 2006). But turbulent bottom friction, topographic interactions, and flow separation at bottom features all remain viable dissipation candidates.

General Circulation Models (GCMs)

Oceanic kinetic energy (what moves tracers around) is dominated by the balanced eddies. As modeling power has grown, the greatest efforts have been directed at the detailed numerical representation of the balanced eddies in models. In particular, many global models fail to resolve the eddy field, and the numerical parameterization of the effects of balanced eddies has generated a voluminous literature (see the collection of papers in Hecht & Hasumi, 2008). In numerical models the mixing tensor (Eq. 10) is generally formulated in isopycnal coordinates, reflecting the anisotropy of diapycnal and isopycnal mixing. In so-called $z$−coordinate models the resulting tensor is then transformed approximately back to geodetic coordinates. The retransformation is not required for models written in isopycnal coordinates.

Because lateral motions are much more intense than vertical ones, numerical difficulties arise in preventing artificial vertical diffusivities arising from the horizontal motions, distinguishing between those fields (temperature and salinity) that determine the isopycnals and the passive tracers that make no such contribution. This complex subject is covered in Griffies (2004), Haidvogel and Beckmann (1999), Olbers, Willebrand, and Eden (2012), and Vallis (2006).

The largest literature focuses on the best numerical representation of $K$—in all three dimensions—and its equivalents (the problem of “sub-grid scale parameterization”). In the process, the distinction between stirring and mixing has largely been lost, with the molecular scales assumed to take care of themselves—between the actions of $K$ and with the inescapable numerical dissipation becoming a surrogate for the true molecular mixing. Much (not all) of the eddy effect is represented in terms of an equivalent lateral velocity (sometimes called the bolus velocity) and is calculated from the anti-symmetric components of tensor $K$ (the “skew-flux”; see especially Flierl & McGillicuddy, 2002), and is related closely to the transformed Eulerian mean (TEM). A common numerical method relies upon extensions and modifications of the proposal of Gent and McWilliams (1990). Because the method, and most of its variants, is focused on baroclinic instability processes, it removes potential energy from the fluid by flattening isopycnals rather than increasing it as in mixing. It is thus a scheme for spreading and not mixing. An explicit dissipation picture for the energy released is lacking.

Many of the isopycnals in the upper 1000 m of the ocean—above the domain of Munk’s “abyssal recipes”—outcrop at the sea surface in the Southern Ocean (Fig. 3). There the intense turbulence existing in the surface boundary layer, derived directly from the locally powerful wind field, can provide an easy route for mixing the fluid across these density surfaces (see, e.g., the modeling study of von Storch, Sasaki, & Marotzke, 2007). In the abyss, where the isopycnals do not outcrop to the surface, direct wind-forcing does not act, and diapycnal mixing is dependent upon the wave breaking mechanisms.

In a hypothetical ocean model that was capable of directly resolving the centimeter and smaller scales, $l$, important in ${\kappa}_{C}{\nabla}^{2}C$, no parameterizations would be required. Taking the requisite length arbitrarily as 1 mm, vertically and horizontally, a global model would require of the order of 10^{27} grid points or Fourier components, and thus the need for statistical mechanics and parameterizations will exist indefinitely. Global models in which the balanced eddy field is adequately resolved are on the horizon, but the mixing scales, ${l}_{m}$, remain far beyond direct reach.

Summary Discussion

The remarkable spatial variations now known to exist in ocean mixing processes have many implications for understanding and potential prediction of the ocean circulation and all of its climate consequences. With the coupling among motions at all time and space scales, discussion of mixing can hardly be done without a long excursion into all ocean flows and energetics. Spatial variations in effective mixing have powerful implications for the general circulation itself (defined as a time average over several decades), implying a very different picture than was accepted until quite recently.

In seeking gross generalizations, it appears that the tide(directly generating internal waves)- and wind-generated instabilities and inertial motions produce the turbulence that mixes the ocean. Roughly speaking, the equivalent vertical or diapycnal or dianeutral mixing coefficients range from comparatively low average values in the upper 1000 m of about ${10}^{-5}{\text{m}}^{2}/\text{s}$ to an order of magnitude larger below that depth. Very much higher values appear in the surface mixed layer with distinct physics. Values increase in the abyss, over certain types of topographic features, by two or more additional orders of magnitude. The implications are that much of the upper ocean can be treated as a near-perfect fluid, with the necessary mixing occurring at the sea surface at high latitudes. In contrast, the very intense mixing over topographic features implies that vertical exchanges and hence the meridional oceanic motions are confined to very special regions, including lateral boundaries of all types. A general increase in diapycnal mixing toward the sea floor in the open ocean implies a net downward flux of temperature generally, with part of the upward return flow being a lateral boundary phenomenon, which represents a radical change in understanding of the deep circulation.

Determination of the influence of the ocean on future climate change requires quantitative prediction of the ways in which energy exchanges with the atmosphere will shift and in turn how those translate into modifications of the general circulation, its storage and transfers of heat, freshwater, carbon, oxygen, etc. The gross ocean stratification will also change, along with the mixing energetics and rates, as will meteorological interactions. A priori calculation of the overall stratification and resulting movement of climate properties in the ocean past, present, and future remains a major challenge.

Acknowledgment

I am grateful to R. Ferrari and C. Garrett for numerous comments and corrections and to the ECCO Consortium for support.

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## Notes:

(1.)
Davis (1994) emphasized, absent a spatial or temporal gap in scales, the assumption that such mixed averages such as $\overline{v\text{\u2032}\overline{C}\text{\u2032}}\text{}=\text{}0$ can fail. Few oceanic measurements exist such
that averages taken over any long timescale are truly stable: e.g., doubling the record length usually measurably changes the calculated average and may well be covarying with the variability of the fluctuations. The trough in spectra at frequencies just below the inertial frequency (usually denoted *f*) is sometimes regarded as representing a spectral gap, but it is probably best regarded as the effect of a peak rather than a minimum. Recent theoretical arguments have also shown how energy can be transferred across this frequency interval.

(2.) The argument that true mixing can only increase the potential energy applies only to the net effects over the entire volume.