a. Overview of basic mechanism

The first evidence that turbulence in the oceanic thermocline occurs in intermittent local patches was obtained off the west coast of Vancouver Island by Grant et al. (1968) using temperature and velocity microstructure probes mounted on the bow of a submarine. Their observations showed patches of turbulence ranging from 30 to 170 m in horizontal scale. (Precise vertical scales of patches could not be obtained due to the limited number of probes used.) Such regions of high mixing can be expected to occur as the result of fluctuating vertical shears caused by internal and inertial gravity waves. As envisioned by Phillips (1966) and by Garrett and Munk (1972), local shear instabilities are produced when the shear exceeds a critical value set by a Richardson number criterion. A random superposition of waves causes these instabilities to occur episodically in both space and time. This type of behavior was observed, for example, in the summer thermocline of the Mediterranean Sea by Woods (1968), who used divers to photograph dye streaks during internal wave breaking events.

As a result of episodic mixing, localized regions of reduced stratification are generated preferentially in regions of enhanced mixing, as described in detail by Woods and Wiley (1972) in their discussion of “billow turbulence.” Horizontal pressure gradients associated with these stratification anomalies cause the well-mixed regions to move laterally, either geostrophically or not, to form “blini,” or pancakes (Phillips 1966; Fig. 1). The types of motions that will occur will depend on the vertical and horizontal scales of the events, and the intensity and duration of the mixing, since these scales ultimately set the scales of the adjustment. If the adjustment time scale is long enough, the motion will become largely geostrophic, and the process is appropriately called geostrophic adjustment; the resulting balanced motions are sometimes referred to as semipermanent finestructure, or the vortical mode (e.g., Kunze 2001). If the time scale is short, rotation is not important and the motion will be mostly a slumping in the direction opposing the pressure gradient; a process we will refer to as “relaxation” since it tends directly toward a barotropic state. If a patch is thin, friction will play an important role in the response from the start and will limit the motion. Regardless of the dominant force balance, we shall use the general term “adjustment” to encompass all of the various motions resulting from the baroclinicity set up by a mixing event.

In all of the above cases, fluid in the weakly stratified patches will move along isopycnals relative to the surrounding fluid. For a single event, the displacement is appropriately described as advective. However, for a large number of events, the cumulative displacements of fluid parcels can be parameterized as an effective lateral diffusivity. How much diffusion will occur will depend on the nature of the adjustment and how long the resultant motions persist. Specifically, we will show that patch scales on the order of the internal deformation radius are optimal for promoting dispersion, since the resultant motions are geostrophically balanced and hence can persist for very long times, and since most of the patch is involved in the motion. In that case, radial displacements associated with the initial stages of adjustment are small compared to those associated with geostrophically balanced flows, and eddy straining dominates the dispersion (e.g., McWilliams 1988; Provenzale 1999). If the adjustment of such events occurs frequently in both space and time, the result may resemble two-dimensional geostrophic turbulence, a topic which has been discussed extensively in the literature (e.g., see reviews by Rhines 1977; McWilliams 1985; Provenzale 1999; and references therein). More generally, however, radial displacements associated with nongeostrophic adjustment may also lead to lateral dispersion, although as will be discussed in section 3b, this will be reduced compared to the geostrophic case.

b. Scope and outline

The formation of localized regions of weak stratification has been well documented in the literature, as noted above (e.g., Phillips 1966; Woods 1968; Woods and Wiley 1972; Garrett and Munk 1972). That well-mixed regions of fluid adjust laterally to form submesoscale geostrophic eddies, or vortical modes, has been firmly established both analytically (e.g., Gill 1981; McWilliams 1988) and in the laboratory (e.g., Griffiths and Hopfinger 1984; Ivey 1987; Thomas and Linden 1996). The implications of patchy diapycnal mixing in the ocean have also been explored (e.g., Vanneste and Haynes 2000; Arneborg 2002). Yet, a simple model linking these processes to lateral dispersion in the ocean is still lacking. The most recent progress in this area comes from a study by Polzin and Ferrari (2004), who examined the effects of blini or vortical motions in the open ocean in the context of the North Atlantic Tracer Release Experiment. Using an empirically derived vortical mode spectrum (Polzin et al. 2003) and theory for two-particle dispersion in two-dimensional turbulence, they showed that vortical mode stirring in the pycnocline of the open ocean can lead to diffusivities of order 1 m² s⁻¹. In the present study, the effects of vortical motions in the coastal ocean are explored. Specifically, motivated by observations from CMO, we show that the same mechanism can lead to comparable or even greater lateral diffusivities in the coastal ocean. The process we envision here is the same as that discussed by Polzin and Ferrari (2004). However, as we will show, our approach differs somewhat from theirs.

The rest of this paper is organized as follows. In section 2 we review the CMO dye and microstructure data, which together provide the observational basis for this study. The microstructure observations in particular, which were made concurrently with the dye-release experiments, are used to infer approximate spatial and temporal scales of localized diapycnal mixing events. In section 3, we present a simple model for lateral dispersion by small-scale geostrophic motions resulting from the adjustment of mixed patches of fluid. A random walk formulation is used to obtain order of magnitude estimates of the effective lateral diffusivity by this mechanism. A key result is a lower-bound prediction of the effective lateral diffusivity based on the scales of mixed patches and other measurable large-scale parameters. Additional considerations, including the possibility of long-lived vortices and vortex interactions, are also discussed in sections 3 and 4. Specifically, it is shown that the inclusion of these effects may increase our lower-bound estimates by an order of magnitude or more. Section 5 summarizes and concludes.

a. Microstructure patchiness

Velocity and temperature microstructure data were collected during the 1996 and 1997 CMO dye cruises on a schedule of alternating days with the dye surveys for the purpose of making direct comparisons of diapycnal diffusivities inferred from these two methods. We use these data to estimate the time and space scales of patchy mixing. A more complete description of these data can be found in a recent paper by Oakey and Greenan (2004).

The locations of the “EPSONDE” microstructure stations occupied during the 1996 CMO fluorescein dye-release experiment (Experiment 3 in Ledwell et al. 2004) are shown in Fig. 2. For each day that EPSONDE data were collected, a region relatively free of fishing gear was selected as the site of repeated station lines. Each line consisted of a series of profiles taken while the ship was underway at about 1 kt. This provided an effective horizontal resolution of 100–400 m, and an approximate station repeat interval of 2–3 h. For the 1996 fluorescein experiment, 21 EPSONDE lines were occupied, with each line consisting of between 12 and 42 profiles, giving a total of 522 profiles. A comparable number of profiles were taken during the other 1996 and 1997 dye-release experiments.

For each microstructure profile, the dissipation rates of turbulent kinetic energy

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(e.g., Oakey 1988) and temperature variance

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were computed. Here u′ and T ″ represent the high-frequency velocity and temperature fluctuations, while μ/ρ = 1.3 × 10⁻⁶ m² s⁻¹ and D = 1.4 × 10⁻⁷ m² s⁻¹ are the kinematic viscosity of water and the molecular diffusivity of heat, respectively.

Estimates of ϵ and χ for station line number 40 (occupied 1 day after survey 3 of the 1996 fluorescein dye experiment) and station line number 24 (taken between the first and second survey) are shown in Fig. 3. The data are plotted as a function of potential density, which has been mapped back to a pressure coordinate using the mean density versus pressure relationship for the survey. This was done to remove heaving and stretching of isopycnals caused by internal waves. To distinguish this modified pressure coordinate from the “raw” in situ pressure, we shall refer to the former as “stretched pressure.” Also in our analysis, station locations in the horizontal plane were mapped to an advected coordinate system in order to track the ship’s motion relative to the water. This was done by integrating the shipboard ADCP data at the depth of the target density surface relative to a common time for each experiment. Henceforth, distances in this new horizontal coordinate system are termed “advected” distance, that is, “advected ship’s log.”

Assuming that mixing events occurred where ϵ and χ were more than an order of magnitude greater than their means, the data in Fig. 3 show episodic mixing events on vertical scales of 2–10 m and horizontal scales of a few hundred meters to a few kilometers. Such patches of high dissipation rate were ubiquitous in the microstructure data during the 1996 and 1997 CMO dye experiments. (Microstructure data were not available during the 1995 pilot cruise.) Particularly noteworthy in Figs. 3a and 3b is a layer centered at approximately 35 m, which was oblique to potential density surfaces. As will be discussed in section 4b, this tilt may have affected the ensuing lateral adjustment of the well-mixed fluid. Note that numerous smaller scale events are also visible in both station lines.

b. Time scales and buoyancy flux

Considerable effort has been devoted in the literature to the problem of relating the dissipation rate, ϵ to larger-scale flow characteristics such as shear and stratification (e.g., Henyey et al. 1986; Gregg 1989; Polzin et al. 1995; Alford and Pinkel 2000; MacKinnon and Gregg 2003). The basic premise of such relationships is that the energy dissipated in mixing events is provided by shear in the internal wave field. Scalings, such as the ϵ ∝ N² relationship derived by Henyey et al. (1986) and the ϵ ∝ N obtained by MacKinnon and Gregg (2003), reflect the fact that internal wave energy, shear, and dissipation all tend to be higher where the mean stratification is greater.

Within a region of given mean stratification, local variations in N might be expected to be negatively correlated with dissipation, because of homogenization of the density profile by mixing. However, the reduction in N caused by diapycnal mixing is determined by the time integral of the buoyancy flux divergence, and not the buoyancy flux itself. For a given mixing event, we might therefore expect N within a mixing patch to vary from some nonzero value at the onset of mixing to some lower value (possibly zero, in the case of complete mixing) by the time the mixing ceases. This will be true regardless of when the maximum value of ϵ and χ occurs. As a result, regions of high ϵ and χ may be accompanied locally by either high or low N.

Nevertheless, an estimate of the amount of work done by episodic mixing events can be obtained by examining the dissipation rate and the time scales of the events themselves. Consider in one dimension the case of a single isolated mixing event of half-thickness, h (Fig. 4). For a uniformly stratified water column, the potential energy per unit area can be expressed as

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where ρ = ρ_o(1−N²z/g) is the density, g is acceleration due to gravity, N is the buoyancy frequency, and z is the vertical coordinate. From (3) it follows that the change in PE due to mixing is given by

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where ΔN² = N²_i − N²_f is the change in N² caused by mixing. Alternatively, the increase in PE due to mixing may be derived by integrating the equation for the turbulent kinetic energy over the volume of the patch and the time of the mixing event, provided buoyancy fluxes through the boundaries of the patch are ignored and the ratio of production of potential energy to dissipation of kinetic energy, Γ, is constant. In that case,

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where Δt represents the time over which mixing occurs, and we have assumed for simplicity that ϵ is also constant during the mixing event. Combining (4) and (5), it follows that the change in stratification associated with a patch of mixing is

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Equation (6) suggests that for a given buoyancy frequency N² and anomaly height h, we can estimate the change in stratification associated with a dissipation rate ϵ acting for a time Δt. For example, assuming N = 0.02 rad s⁻¹, 2h = 5 m, and Γ = 0.2, a mixing event with ϵ = 10⁻⁷ m² s⁻³ lasting for Δt = 1 h would result in ΔN²/N² = 0.1, or a 10% reduction in stratification. Alternatively, the same ϵ acting for the same period over a vertical scale of 2h = 2 m would lead to a much greater change in stratification, ΔN²/N² = 0.5, or approximately 50%. Additional estimates of ΔN²/N² for parameter values representative of the CMO dye experiments are listed in Table 1. These results suggest that dissipation rates greater than about 3 × 10⁻⁸ m² s⁻³ and lasting a few hours or longer can induce significant changes in N². This result is most pronounced for mixing events with small vertical scales, that is, h = 2 m or less.

Returning to the microstructure data, we now briefly consider the change in stratification induced by diapycnal mixing events. Figure 5 again shows a series of transects of dissipation rate, ϵ, this time in three dimensions. As in Fig. 3, a Lagrangian correction has been applied to each transect using the integrated shipboard ADCP data. However, this time we have used the ADCP data for each depth bin rather than only a single depth corresponding to the target isopycnal surface. Also, here the data are plotted as a function of raw depth rather than stretched pressure. The result is a series of curtain diagrams in a depth-dependent Lagrangian reference frame.

Inspection of Fig. 5 reveals that even with the Lagrangian correction, there was significant overlap in the locations of successive EPSONDE station lines. Furthermore, and more interesting, there is visible coherence between the different transect lines. For example, the arrows in Fig. 5 show a region of enhanced dissipation, log₁₀(ϵ) = (−7.5 to −7), approximately 5 m thick, which persists in approximately the same fluid volume throughout the three transects, a period of at least 4.3 h (the time from the middle of the first to the middle of the third transect). Based on (6), we estimate an event of this magnitude and duration would reduce the stratification by approximately ΔN²/N² = 0.1–0.4, or 10%–40%. This result demonstrates that although we may not be able to directly correlate regions of high ϵ with regions of reduced N², as discussed above, some mixing events were sufficiently energetic and long lived to cause order-1 changes in the stratification. Note that several smaller events are also visible in the transects shown in Fig. 5; however, their coherence was not nearly as pronounced. Also, estimates of the Ozmidov scale, L_O = 2π(ε/N³)^1/2 from EPSONDE profiles during experiment 3 ranged over 0.2–5 m, suggesting that this was the maximum scale of turbulent eddies in areas of enhanced dissipation. The latter implies that mixing within the large patch shown in Fig. 5 occurred at scales smaller than the observed patch scale—that is, that the patch was a collection of smaller overturns. A similar result showing patch scales many times the overturning scale was obtained by Smyth et al. (2001) based on numerical simulations of Kelvin–Helmholz billows and oceanic microstructure observations. An analogous result, although to a lesser extent, was also found by Alford and Pinkel (2000), who focused primarily on overturning regions with maximum overturning displacements greater than 2 m.

c. Stratification anomalies

We have shown that in some cases mixing events were sufficiently energetic and long lived to cause order one changes in stratification. However, we have also argued that regions of high ϵ and χ, and low N² may not necessarily be correlated. Indeed, our attempts to formally obtain such a correlation using the CMO microstructure data were unsuccessful. Nevertheless, distinct patches of reduced stratification were observed in both the tow-yo and microstructure data.

An example of such patches evident in density profiles from a transect occupied during the 1996 fluorescein dye study is plotted in Fig. 6. The profiles were all below the main pycnocline, between about 30–60-m depth. The stratification anomalies, which we define as regions where N² was reduced by more than 30% relative to smoothed versions of the same density profiles, ranged in scale from 1 to 10 m thick vertically and a few hundred meters to a few kilometers horizontally. Although larger-scale events are more readily apparent to the eye since they span many profiles, events that span only a few profiles are also evident. Also, in a number of cases anomalies were oblique to potential density surfaces, and were similar in both scale and angle of inclination to the layer seen in Figs. 3a and 3b. Similar stratification anomalies were observed in the other CMO dye-release experiments.

Density profiles obtained from EPSONDE microstructure measurements also showed stratification anomalies. Figure 7 shows an example of this from station line number 40, the same station line shown in Figs. 3a and 3b, except here the data have been smoothed to yield approximately the same resolution as the tow-yo data in Fig. 6. Again localized regions of low stratification are evident throughout the water column, particularly below the main pycnocline, which in this experiment was at about 20-m depth.

The observed layers of reduced stratification are similar to those reported by Thorpe et al. (1999) in relatively uniformly stratified waters below the seasonal thermocline of Lake Geneva. Thorpe (1999) described how such layers may be generated through localized mixing associated with internal wave groups, and how the geometry of the layers should relate to the geometry of breaking regions and the propagation characteristics of the waves. Localized mixing is not the only possible explanation for the existence of these regions, however; for example, they may be the result of internal wave strain. Unfortunately, our data do not allow us to discern between these two possibilities.

To assess whether the observed patches of reduced stratification could in fact have been the result of localized mixing, the stratification anomalies indicated in Fig. 7 by boldface line segments are also plotted over the ϵ and χ transects shown in Figs. 3a and 3b. The same analysis was done for station line number 20, shown in Figs. 3c and 3d. As anticipated based on the discussion of the previous section, the correspondence between regions of reduced stratification and high ϵ and χ is sporadic. However, distinct similarities are also apparent, although more so in Figs. 3a and 3b than in Figs. 3c and 3d. For example, in Figs. 3a and 3b, the layer of high ϵ and χ centered at about 35 m coincides approximately with layers of reduced stratification. Rather than being aligned with the region of highest ϵ or χ, however, the low stratification occurs near the boundaries of this region. The reason for this offset is unclear. Nevertheless, this example suggests that these layers are linked. We reiterate, however, that on the scales of interest here, localized regions of low N² would be the end product of turbulence and would not necessarily be coincident with regions of high dissipation.

a. Momentum scaling

Consider an isolated diapycnal mixing event in which mixing is much greater than the ambient background level. The fate of such an event can be understood as follows. If the timescale of the turbulent mixing event is short compared to the adjustment time of the resulting mixed patch, the process may be thought of as having two distinct phases: a turbulent mixing phase, and an adjustment phase. The mixing phase will create a local buoyancy anomaly. For perfect mixing, the final stratification within this anomaly would be zero so that ΔN²/N² = 1. However, if localized mixing causes an order-1 change in the local buoyancy frequency, but does not lead to perfect mixing, a more realistic value might be, for example, ΔN²/N² = 0.5.

For a given vertical event scale h, we can define a deformation radius associated with the induced stratification anomaly, R = ΔNh/f (e.g., Gill 1981). Taking a buoyancy frequency, N = 10 cph (0.0175 rad s⁻¹), and a vertical event scale of 2h = 5 m, as were typical for the CMO dye studies, and letting ΔN² = 0.5N², we find R = 325 m. If we assume for the moment that friction is not important during the adjustment of a mixed patch, then for a patch of horizontal scale 2L = 1.0 km (again typical for CMO) both relative acceleration and Coriolis acceleration will play a role in the adjustment, since in that case the horizontal scale is comparable to the deformation scale.

Using the above assumptions, we can employ simple scaling arguments to understand how the adjustment of mixed patches ultimately leads to an effective horizontal dispersion. Consider the x component of the horizontal momentum equation,

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where ν_B represents a background viscosity that may or may not be directly related to the mixing events. This equation can be scaled as follows:

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where U, L, and T represent velocity, length, and time scales, respectively, and the scaling for the pressure gradient term derives from the hydrostatic equation,

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Equation (7) represents the basic momentum balance associated with the adjustment of mixed patches of fluid. For a single event with L ≥ R, we envision a classic Rossby adjustment problem. The buoyancy anomaly induced by diapycnal mixing results in a horizontal pressure gradient as represented by the first term on the right-hand side of (7). If the anomaly is axisymmetric, and if the influence of friction is small, this pressure gradient will lead to an initial radial spreading of the well-mixed fluid over a distance of order R (e.g., Fig. 1). As this initial adjustment occurs, a geostrophic flow will be established in the azimuthal direction such that the mixed region rotates anticyclonically. The spreading, and thus a decrease in the PE of the well-mixed fluid, occurs as some of the available potential energy is converted to kinetic energy. There is also a radiation of energy away from the site by internal waves. However, these need not be considered here since their contribution to lateral dispersion is small and since they do not interact significantly with the balanced geostrophic motions (e.g., Reznik et al. 2001).

Whether the radial displacements directed down pressure gradient, or the azimuthal displacements associated with the geostrophic flow are larger, will depend on the scale of the mixed patch. For example, if L ≫ R, the azimuthal displacements will be much greater than the slumping displacements. Alternatively, if the effects of rotation are small (L ≪ R), azimuthal displacements will be negligible and the largest displacements will be due to the slumping motions. As will be discussed in section 3c, the greatest net displacement of fluid and hence the largest effective lateral dispersion is expected to occur for L ∼ R—that is, in a regime where the flow is ultimately geostrophic and the whole patch becomes involved in the adjustment.

b. Vertical diffusion and event time scales

The above considerations can be used to formulate a simple parameterization of the effective horizontal dispersion caused by a collection of random mixing events. However, before we proceed, it is useful to consider how the vertical scales of episodic mixing events are related to the mean vertical diffusion. (We return to the effective lateral dispersion due to episodic mixing events in the next section.)

As suggested by Garrett and Munk (1972), and following section 2b, the mean vertical diffusivity κ_z in a stratified fluid can be related to the scales of episodic mixing events and their frequency of occurrence using PE arguments. Consider a single isolated mixing event of half thickness h, as sketched in Fig. 4. If Q(h)dh represents the expected number of events of half-height between h ± ½dh per unit time, per unit depth, then from (3) it follows that the average rate of change in PE density—that is, the buoyancy flux—can be written as

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Alternatively, writing the buoyancy flux as a diffusivity, κ_z, times a buoyancy gradient,

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Combining (10) and (11), the vertical diffusivity can be expressed as the sum of isolated mixing events,

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Recent work by Vanneste and Haynes (2000) suggests this result is valid to within an order-1 coefficient that depends on the probability distribution of the mixing events. (Note, however, that our notation differs slightly from theirs in that we define the frequency of events per unit depth, not over the full water column. Meanwhile our event frequency, ϕ, defined below, is for a given location in the water column, not per unit depth.)

Equation (12) provides a simple relation between the effective vertical diffusivity κ_z, the average frequency of mixing events of a given scale Q(h)dh, and the buoyancy anomaly associated with these events ΔN². For simplicity, let us assume that all events have the same half-thickness, h, and buoyancy anomaly, ΔN². Then (12) reduces to

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where ϕ represents the average frequency of events. In the present context, (13) is most useful if the vertical diffusivity is known, as from the dye experiments or microstructure. In this case the frequency of events, ϕ, can be estimated from observations of the vertical scale h and the buoyancy anomaly ΔN²; that is,

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Equations (13) and (14) show that for a given vertical diffusivity, κ_z, the larger the thickness of the mixing events, the less frequently they must occur. For example, for the second of the 1996 CMO dye experiments, the vertical diffusivity in the lower part of the patch was κ_z = 0.9 × 10⁻⁵ m² s⁻¹. Assuming a typical buoyancy anomaly, ΔN² = N² and a typical vertical scale, 2h = 2 m, (14) implies an average event recurrence time of 1/ϕ = 0.2 days. Alternatively, mixing events of vertical scale 2h = 5 m would result in a larger event recurrence time, 1/ϕ = 0.3 days. Similar calculations for a variety of combinations of patch thicknesses and vertical diffusivities are listed in Table 2. For a reasonable range of vertical scales h and vertical diffusivities κ_z, the inferred event recurrence times range from 1/ϕ = 0.2 days to 1/ϕ = 48 days.

Relevant to this range of recurrence times, is that according to (13) and (14), ϕ also depends on ΔN²/N², which we assumed here has a nominal value of 0.5. All else being equal, however, (6) suggests that ΔN²/N² may itself vary, so that the actual range of recurrence times may not be as wide spread as Table 2 suggests. Alternatively, however, if ϵ, Δt, and h are not independent—for example, if the vertical scale of the mixed patches is related to the dissipation rate through the Ozmidov scale, L_O = 2π(ε/N³)^1/2 (e.g., Smyth et al. 2001)—then (6) implies that any such reduction in the range of recurrence time scales may be negated. Regardless of the actual value of ΔN²/N², however, the important points in Table 2 are 1) to show approximately how the event recurrence time scale varies for different vertical scales, and 2) to show that for the same vertical patch scale h, the recurrence time scale is always less than the frictional time scale, as we shall now explain.

The frictional time scale, h²/ν_B associated with vertical viscosity is also important to the adjustment of diapycnal mixing events. Here we imagine that diapycnal mixing events and their resultant stratification anomalies occur and relax amidst some background vertical viscosity, ν_B, which may be molecular viscosity or may represent processes acting on small scales at times outside of the mixing events. For purely molecular viscosity acting on a very thin layer, the horizontal pressure gradients associated with a density anomaly may be balanced by friction, and a frictional adjustment may occur. In the case of an eddy viscosity, we might imagine that as small-scale eddies redistribute momentum, they would also mix away the density anomaly associated with the mixing event, halting the adjustment. Although it is conceivable that internal waves can transfer momentum vertically independent of turbulent mixing events, thus enhancing ν_B somewhat, we shall assume that ν_B ≤ κ_z—that is, the background viscosity is less than the diffusivity due to mixing events. Oakey and Greenan (2004) estimated eddy viscosities during CMO from 4 × 10⁻⁵ to 1 × 10⁻⁴, averaged over regions of both high and low ε. However, if event-like mixing dominated over background mixing, which we believe to have been the case, the background viscosity would have been much less than this and hence may still have been less than or of the same order as the total vertical diffusivity. In either case, a short frictional time scale would limit the amount of time that diapycnally mixed fluid would have to be displaced laterally during adjustment.

Whether friction is likely to affect the adjustment of mixed layers can be determined. Table 3 lists the frictional time scales, h²/ν_B for molecular and modest eddy viscosities over a range of vertical event scales typical of CMO. As the resulting times are generally of order a day or more, viscous accelerations are unlikely to have been important relative to the Coriolis acceleration for these events. Furthermore, for ν_B = κ_z, (and especially for ν_B < κ_z) and comparable vertical scales h, the inferred frictional time scale, Tν_B = h²/ν_B is always larger than the recurrence time scale, 1/ϕ. Note that this is true independent of the value of ΔN²/N². This follows directly from (14) and the definition of the frictional time scale; that is,

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since by definition, ΔN²/N² ≥ 1 and by assumption ν_B ≤ κ_z. This suggests that successive anomalies will likely overlap with their predecessors. In an extreme case, if a density anomaly is completely mixed away by a new event, the recurrence of events will thus halt (or at least significantly modify) the adjustment of previous events long before friction can play a significant role.

c. Effective lateral dispersion

The size of the deformation radius estimated in section 3a and the time scales estimated in section 3b are all consistent with the idea that episodic mixing events observed during the CMO dye-release experiments may have resulted in the geostrophic adjustment of mixed regions. We now consider the effective lateral diffusivity associated with a collection of such events. First, we consider dispersion by a field of independent adjustment events, that is, assuming that vortices do not interact. We then consider how the interaction of vortices might modify our results.

Suppose that patches of diapycnal mixing occur on horizontal scales comparable to or greater than the deformation radius, and that the recurrence time scale of events, 1/ϕ is greater than or equal to the rotation time scale, 1/f so that events adjust under the influence of rotation before being disturbed by subsequent events. In this case, after sufficient time has elapsed, the dominant balance in the horizontal momentum Eq. (7) will ultimately be between the pressure gradient and Coriolis acceleration. The horizontal velocity associated with the adjustment of a mixed patch thus scales as

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Now consider the effect of many such events on the dispersal of a passive tracer. The effective horizontal diffusivity associated with the events can be thought of as a random walk of fluid parcels with horizontal step size S and frequency Φ—that is, κ_H ∼ ½S²Φ. In the case of geostrophic adjustment, the step size S is at least of order the geostrophic velocity (16) times the adjustment time scale T ∼ 1/f. An alternative choice of time scale for this purpose is the eddy turnover time. However, for Rossby number R² ⩽ 1, which we assume to be the case here, the eddy turnover time is greater than or equal to an inertial period, so that 1/f gives a lower bound on the displacement; that is,

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Meanwhile, in the simplest case, the frequency of taking a step is given by the frequency of the diapycnal mixing events themselves. Thus, the effective horizontal diffusivity is

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or, substituting for ΔN²h²ϕ in terms of N²κ_z from (14), and for the deformation radius, R = ΔNh/f,

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Equations (18) and (19) provide an estimate of dispersion by vortices generated through the adjustment of mixed patches. However, this estimate is conservative for at least two reasons. First, we used 1/f rather than the (larger) eddy turnover time to estimate the step size S. Second, if motions are long lived, they may interact, enhancing the dispersion more than (18) and (19) imply.

Consider a single geostrophic vortex generated by the adjustment of a mixed patch of fluid. If the vortex is long lived, the displacement or step size experienced by a fluid parcel within the vortex will increase until it completes half of a revolution around the vortex, or about one-half of an eddy turnover time. Assuming an axisymmetric Gaussian-shaped vortex (e.g., McWilliams 1988), a parcel 1 deformation radius from the vortex center will take 1–2 inertial periods to do this. Meanwhile, at distances greater than a deformation radius, but still within the influence of the vortex, parcels will take longer to achieve the same angular displacement. As a result, even after many inertial periods, some parcels will not have completed even one-half of a revolution. Once they do, however, their corresponding linear displacements will be greater than parcels nearer to the vortex center. The rms displacement of all fluid parcels combined will thus continue to grow for multiple eddy turnover times, that is, several inertial periods. Eventually the rms displacement will asymptotically approach a finite value (e.g., Rhines and Young 1983). Using a simple analytical/numerical model for an idealized inviscid Gaussian vortex, we have found the rms displacement of fluid parcels to grow linearly for small times, asymptotically approaching a fixed value for very large times (i.e., many tens of inertial periods), with a transition period in between. For the size and strength of the vortices envisioned here, we thus find that the mean squared displacement of parcels within a few deformation radii from the vortex center may be as much as 5–10 times as large as (18) suggests.

A second means by which long-lived vortices can lead to an increased dispersion is by interacting nonlinearly. In that case, the implicit assumption that closed streamlines will limit vortices’ ability to disperse fluid, breaks down. Instead, vortices will mutually advect, as well as merge and/or destroy one another, and the mixing becomes similar to two-dimensional turbulence. Pasquero et al. (2001) and Pasquero et al. (2002) found that in this realm, Lagrangian decorrelation time scales range from 4–18 times the eddy turnover time.

In the case of interacting vortices, and possibly also in the case of isolated vortices but before the mixing is complete within closed streamlines, the viscous time scale Tν_B or, nondimensionally, the Ekman number (fh²/ν_B) provides a measure of the time that a given vortex will contribute to lateral dispersion. For interacting vortices, in analogy with two-dimensional turbulence, this can be seen by combining simple energetics arguments with Taylor’s (1921) eddy diffusion formulation, κ_H ≈ T_IKE, where KE is the kinetic energy of the turbulent flow field, and T_I is the integral time scale of the motion. Assuming the KE of the eddy field is given by a balance between the production of eddies by the adjustment of mixed patches and their decay by viscous forces, the diffusivity can be expressed as

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Let the production rate of KE be given by the frequency of mixing events ϕ times some fraction of the PE generated by individual events per (3); that is,

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where the rhs has now been normalized by h to obtain KE per unit volume. Assuming that the integral time scale T_I is approximately equal to the eddy turnover time—that is, a few inertial periods—and again using (14), it then follows that the horizontal diffusivity for nonlinearly interacting vortices scales as

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where α is an unknown coefficient, presumed to be of order 1.

Equation (22) is very similar to (19), the differences being the factor (R²/L²) in (19), which we assumed to be of order 1, and the appearance of the Ekman number in (22), which sets the overall strength of the turbulent eddy field. Although these two expressions were derived in much different ways, the process they represent is similar. Still, they represent two extremes for the expected lateral dispersion. Specifically, (19) represents the case in which vortices do not interact and the dispersion is limited to one or two inertial periods, while, (22) assumes that vortices are long-lived and do interact. For the case of the interacting vortices, we presume that the Ekman number is greater than 1. In fact, for CMO, based on Table 3, we believe it was closer to 10. Also, for the case of noninteracting vortices, we have argued above that (19) may still underestimate the dispersion by up to an order of magnitude.

Both of the above scalings provide estimates of the effective horizontal diffusivity due to the adjustment of episodic mixing events when events are of large enough horizontal scale to adjust geostrophically. Furthermore, they both show that the effective horizontal diffusivity by the adjustment of mixed patches of fluid can be written in terms of the vertical diffusivity times a scaling factor, which depends on known and/or measurable parameters.

The above parameterizations are similar to one obtained by Young et al. (1982) for shear dispersion due to internal waves. However, the motions modeled here are entirely isopycnal; we have not yet considered the interaction between vertical shear and vertical diffusion. That process is compared with the present one in the appendix. The diapycnal diffusivity enters both (19) and (22) through the rate of production of vortices, not through a shear dispersion model. In fact, SL showed that shear dispersion by internal waves was insufficient to explain the lateral dispersion observed in CMO.

The diffusivity predicted by (19) and (22) can be compared directly with results of SL from the CMO dye studies. Their “irreversible” lateral diffusivities represent the effective diffusion along isopycnals, excluding “reversible” processes such as oscillatory shears, but including irreversible effects such as small-scale stirring along isopycnals and shear dispersion. Using typical values from CMO, N²/f² ∼ (0.1–1) × 10⁵, and assuming mixing events of horizontal scale of order the deformation radius (R²/L²), (19) yields

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which we have argued is a conservative estimate of the dispersion. Alternatively, assuming a lower-bound estimate of the Ekman number based on a 2-m event and a background viscosity of ν_β = 10⁻⁵ m² s⁻¹, that is, (fh²/ν_B) = 10, (22) yields

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For the range of vertical diffusivities estimated from the CMO dye studies, κ_z = (0.1–1.0) × 10⁻⁵ m² s⁻¹, the above expressions thus give horizontal diffusivities of κ_H ∼ (0.1–10) m² s⁻¹. Comparing these estimates with the values of κ_H estimated during CMO (Table 4), we find that the predicted dispersion spans the range of the observed dye dispersion. Analogous predictions using diapycnal diffusivities estimated from microstructure measurements by Oakey and Greenan (2004) and by MacKinnon and Gregg (2003) give similar results.

a. Random walks and nonlinear interactions

A fundamental issue in the above scaling is the question of how well our assumption of a random walk diffusivity applies to a situation such as CMO in which the duration of the dye experiments is only a little longer than the time between diapycnal mixing events, that is, 3–5-day experiments, in comparison with approximately an inertial period, that is, 18 h. Indeed, if both the event frequency and the frictional time scale (i.e., the lifetime of an individual event) are small, a random walk scaling would not be appropriate. However, if the frictional time scale is long relative to the time between events (i.e., large Ekman number), the number of vortices stirring the tracer at any given time, and hence the effective frequency with which an individual fluid parcel takes a step, may be many times larger than the actual generation frequency of events. In addition, as the dye patch grows, the area of the patch and hence the number of vortices acting on it will grow geometrically, thus increasing the effective number of independent Lagrangian trajectories within the patch. These two factors combined can lead to the diffusive spreading of a dye patch after a surprisingly short time.

Related to the above, an interesting aspect of the random walk scaling given by (19) is that it does not require that vortical motions generated by mixing events interact nonlinearly as a turbulent flow field. In fact, by neglecting the effect of such interactions, (19) implicitly assumes that lateral stirring by vortices can occur even if mixing events are sparse in space and time. In that case, the stirring may look less like turbulence and more like, for example, the “blinking vortex” model of Aref (1983). (In our case, however, a given vortex would “blink” on and remain so for many inertial periods rather than switching on and off in the same location indefinitely.) For CMO at least, we estimated that the recurrence time scale is shorter than the frictional time scale, so that vortices likely do interact. This was taken into account to some degree in our revised scaling given by (22). However, given our uncertainty about the frequency of mixing events and the background viscosity, we cannot say for certain which of these scalings is most appropriate for which experiment, only that they appear to be of the right order of magnitude compared with the observations.

b. The geometry of mixed patches

The scalings given by (19) and (22) are for the idealized case of an axisymmetric vortex, assuming that mixed patches of fluid adjust geostrophically. The size of the deformation radius estimated in section 3a and the time scales estimated in section 3b suggest that, regarding the latter at least, this was indeed the case during CMO. Mixed patches that adjust geostrophically are optimal for generating lateral dispersion because they provide the greatest stirring for a given amount of vertical mixing. Indeed, as discussed by Sundermeyer (1998) it can be shown that if L = R, the patches will relax with relatively small displacements. Similarly, if patches are thin in the vertical in the sense that the frequency of mixing events is large when compared with f [e.g., see (14)], but not so thin that friction dominates their initial adjustment, and further, if new events completely interrupt the flow of previous events, then the displacements will again be relatively small. Last, if the mixed patches are thin enough that the frictional time scale is short when compared with 1/f, then again the resulting displacements will be relatively small.

A second consideration regarding the geometry of mixed patches arises in the more general case when mixing events adjust geostrophically but are not horizontally symmetric; in this case, the dispersion will be further modified. Consider an event which is of order the deformation scale in one horizontal dimension, but much larger in the other. In this case, the mixed region will again adjust geostrophically, spreading laterally while setting up a geostrophic flow. However, instead of a symmetric vortex, the adjusted state will be a long band with geostrophic flow in the direction of the longitudinal axis of the mixed region (Fig. 8c). The displacements associated with this geostrophic flow will be larger than the azimuthal displacements associated with a symmetric patch that is of order of R (Fig. 8a). This is because in the case of an elongated patch of mixed fluid, the geostrophic flows can transport fluid much further than a deformation radius in the direction of the major axis of the mixed region (this assumes again that the geostrophic flows persist longer than the initial adjustment time before being interrupted by a new mixing event or arrested by friction). The case of symmetric mixed regions with L ≫ R is similar (Fig. 8b), except that only edges (with width of order R) are set into motion.

A third geometrical consideration for mixed patches of fluid is that, as noted in section 2c, density anomalies observed during the CMO dye studies were frequently oblique rather than parallel to potential density surfaces. The effective horizontal scales of the mixed layers, L′ ≤ L, was therefore reduced relative to the case where layers are not tilted. This is shown schematically in Fig. 9. According to the scaling given by (19), the predicted horizontal diffusivity would thus increase for large tilted layers, since the scale factor R²/L′² ≥ R²/L². Note, this assumes their effective horizontal scale, L′, was still of order R or greater.

Last, relating to our discussion in section 2b, is the question whether the adjustment of patches consisting of multiple overturns can be modeled collectively as a single adjustment event, as we have done; or alternatively, whether individual overturns should be considered individually, each going through its own adjustment, and possibly interacting and merging as a result of the adjustment. Considering (6) and (14), we believe that whether mixed patches are one-half as large with 2 times the change in stratification, or vice versa, has little impact on our final result, since both of these expressions suggest that the more important quantity is the change in PE induced by diapycnal mixing—that is, the net diapycnal diffusivity. Specifically, given the integral constraint that κ_z imposes on the buoyancy flux, it follows from the final scaling for κ_H given by (19) and (22) that any change in the vertical scale of the events h would necessarily be offset by a change in ΔN², leaving κ_H relatively unchanged. The one exception is the Ekman number scaling, (fh²/ν_B) in (22). However, our estimate of this parameter in section 3c is already quite low because of our assumption of a 2-m event scale and a conservative estimate of the eddy viscosity. Furthermore, if smaller events do interact nonlinearly and possibly merge, as discussed in the previous section, then the effective Ekman number may also remain relatively unchanged.

c. Additional considerations

One place where we may have overestimated the dispersion by mixed patches of fluid is in our use of a simplified random walk scaling, κ_H ∼ ½S²ϕ, which implicitly assumes that S (and hence S²) and ϕ are uncorrelated. However, if S² and ϕ are correlated, then this κ_H may be an underestimate because it neglects the cross-correlation term. Conversely, if S² and ϕ are negatively correlated, then by the same reasoning, κ_H may be an overestimate. Indeed, comparing the step size estimated in (17) with the step frequency estimated in (14), and assuming f, L, and κ_z are independent variables, we anticipate that, if anything, S (and hence S²) and ϕ would be inversely (i.e., negatively) correlated through their dependence on ΔN²h².

Another factor affecting the dispersion by mixed patches of fluid is that if layers do adjust geostrophically and then subsequently decay by viscous damping, additional spreading of the mixed fluid may occur during the spindown of the geostrophic flow (e.g., Gill 1981). This will again increase the effective lateral diffusivity associated with the relaxation of mixing events compared to (19).

The adjustment of diapycnal mixing events may also contribute to lateral dispersion by vertical shear dispersion. Given the relatively small lateral scale of the events compared to the scale of the dye surveys, it is likely that the relatively low-frequency vertical shears associated with them would have been aliased as high-frequency shears by shipboard ADCP observations and would therefore not be accurately portrayed in the analysis of SL. However, in the appendix, we show that shear dispersion associated with the adjustment of mixed patches is less than the effective dispersion due to purely isopycnal motions. Thus we believe that shear dispersion is only of secondary importance to the process described here.

We have argued that the adjustment of patches created by diapycnal mixing events would manifest itself in the form of small-scale stirring on the same scales at which the diapycnal mixing occurs—that is, 0.5–10 m vertically and a few hundred meters to a few kilometers horizontally. Alternatively, however, within the context of this same mechanism, suppose that vertical mixing during a particular dye experiment was due primarily to one (or very few) large event(s). In the extreme case of a single large mixing event, the dye would mix rapidly in the vertical for a brief time and then not at all. Meanwhile, in the horizontal direction the dye would appear to spread continuously, possibly as a single or very few streaks, as the well-mixed fluid adjusted laterally and ultimately relaxed due to friction. In that case, the horizontal dispersion might more appropriately be described as an advective rather than a diffusive process. However in practical terms, without additional information about the larger-scale flow, this advection may be indistinguishable from other baroclinic and/or barotropic shearing or straining processes.