## 1. Introduction

A number of tracer-release experiments in the ocean’s stratified interior have reported 1–5-km horizontal diffusivities *K*_{h} ~ *O*(1) m^{2} s^{−1} (e.g., Ledwell et al. 1998; Sundermeyer and Ledwell 2001; D. A. Birch et al. 2015, unpublished manuscript). This is an order of magnitude larger than predictions for internal-wave shear dispersion *K*_{h} ~ *K*_{z}|*χ*_{z}|^{2} based on Young et al. (1982), where |*χ*_{z}| = |∫**V**_{z} *dt*| and the vertical shear **V**_{z} = (*u*_{z}, *υ*_{z}) = (∂*u*/∂*z*, ∂*υ*/∂*z*). These shear-dispersion predictions have been based on average diapycnal diffusivities 〈*K*_{z}〉, average near-inertial shear variances 〈*K*_{h} ~ 〈〈*K*_{z}〉*f*^{2} = 〈*K*_{z}〈*f*^{2} = 〈*K*_{z}〉〈*K*_{z}〉4*N*^{2}/(2*f*^{2}) ~ *O*(0.1) m^{2} s^{−1} for a canonical open-ocean internal-wave field.

To explore a possible reason for this order-of-magnitude discrepancy, this paper examines the consequences of recognizing that 〈*K*_{z}|*χ*_{z}|^{2}〉 = 〈*K*_{z}〉〈|*χ*_{z}|^{2}〉 + 〈*K*_{z}′|*χ*_{z}|^{2}′〉 will exceed 〈*K*_{z}〉〈| *χ*_{z}|^{2}〉 if *K*_{z} and |*χ*_{z}|^{2}, both of which are positive definite, are intermittent and positively correlated. This may be the case in the ocean where finescale near-inertial waves provide background variances 〈*O*(*N*^{2}) and 〈 |*χ*_{z}|^{2}〉 ~ *O*(*N*^{2}/*f*^{2}) but sporadically become unstable (*N*^{2}) to produce nonzero turbulent diapycnal diffusivities *K*_{z}.

Section 2 provides background on internal-wave shear dispersion; section 3 discusses modifications arising from intermittency, lognormality, and correlation between turbulence and unstable shear; and section 4 summarizes the results.

## 2. Background

*χ*_{z}| = |∫

**V**

_{z}

*dt*| arising from vertical shear

**V**

_{z}= (

*u*

_{z},

*υ*

_{z}), and (ii) diapycnal turbulent mixing

*K*

_{z}(Young et al. 1982) As illustrated in Fig. 1, horizontal mixing is most effectively accomplished when diapycnal mixing coincides with maximal |

*χ*_{z}|. This can be understood from the geometrical derivation in section 2c of Young et al. (1982). Diapycnal diffusivity

*K*

_{z}will not act on vertically aligned isolines—that is,

*θ*

_{x}= ∂

*θ*/∂

*x*(Fig. 1a)—but will act when these isolines are tilted by vertical shear

*U*

_{z}(Fig. 1b) to form vertical gradients—for example,

*θ*

_{z}= ∫

*θ*

_{zt}

*dt*= −∫

*u*

_{z}

*θ*

_{x}

*dt*= −

*X*

_{z}

*θ*

_{x}(Fig. 1c), where

*X*

_{z}is the

*x*component of the off-diagonal vertical strain. Since internal-wave off-diagonal vertical strain is dominated by finescale near-inertial waves, it is rotary with

*X*

_{z}= ∫

*u*

_{z}

*dt*and

*Y*

_{z}= ∫

*υ*

_{z}

*dt*out of phase by 90°. As further explained by Young et al., the vertical shear also elongates isolines (Fig. 1) so that the projection of this elongation in the vertical likewise goes as

*X*

_{z}. Thus, the diffusive flux due to vertical diffusion can instantaneously be expressed as

*K*

_{z}

*θ*

_{x}, where

*K*

_{z}

*K*

_{z}

*K*

_{z}〉, the resulting average horizontal diffusivity is 〈

*K*

_{z}〉〈

*K*

_{z}

*m*

^{2}≪

*f*(see below). Internal-wave shear dispersion based on

*average*diapycnal diffusivities 〈

*K*

_{z}〉 and

*average*shear variances 〈

*ω*is the wave frequency and

*m*is the vertical wavenumber. Derivation of (1) can be found in Young et al. (1982) under various conditions for constant 〈

*K*

_{z}〉. The constraint of constant diapycnal diffusivity and shear variance will be relaxed here in section 3a.

Schematic time series of (a) shear variance *K*_{z} assuming the usual application of internal-wave shear dispersion based on constant 〈*K*_{z}〉 as in (4) (red), and displaying the actual intermittency of turbulence *K*_{z} and its correlation with unstable instantaneous finescale shear variance *K*_{z}}_{b} levels labeled on the right axis. The horizontal time axis has been normalized with the Coriolis frequency *f* and is deliberately not quantified for lack of observational information.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Schematic time series of (a) shear variance *K*_{z} assuming the usual application of internal-wave shear dispersion based on constant 〈*K*_{z}〉 as in (4) (red), and displaying the actual intermittency of turbulence *K*_{z} and its correlation with unstable instantaneous finescale shear variance *K*_{z}}_{b} levels labeled on the right axis. The horizontal time axis has been normalized with the Coriolis frequency *f* and is deliberately not quantified for lack of observational information.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Schematic time series of (a) shear variance *K*_{z} assuming the usual application of internal-wave shear dispersion based on constant 〈*K*_{z}〉 as in (4) (red), and displaying the actual intermittency of turbulence *K*_{z} and its correlation with unstable instantaneous finescale shear variance *K*_{z}}_{b} levels labeled on the right axis. The horizontal time axis has been normalized with the Coriolis frequency *f* and is deliberately not quantified for lack of observational information.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

For internal waves, the largest contributions to (1) are from finescale near-inertial waves, which have minimum frequency *ω* = *f* and contain the bulk of the shear variance 〈*m* = *m*_{c} = 0.2*π* rad m^{−1} [corresponding to a vertical wavelength *λ*_{c} = 2*π*/*m*_{c} = 10 m and a length scale *L*_{c} = 1/*m*_{c} = *λ*_{c}/(2*π*) = 1.6 m for typical ocean conditions]. Thus, finescale near-inertial waves have the maximal internal-wave off-diagonal vertical strain variance 〈|*χ*_{z}|^{2}〉 = 〈|**V**_{z}|^{2}〉/*f*^{2}. In (1), the second term in the denominator of the ratio in parentheses—that is 〈*m*^{4}/*f*^{2}—represents diffusive smoothing of small-scale vertical structure that acts to short-circuit lateral dispersion if vertical diffusion is large. For a canonical midlatitude deep-ocean internal-wave field with 〈*K*_{z}〉 ~ 0.05 × 10^{−4} m^{2} s^{−1} (Gregg 1989) and *m*_{c} ~ 0.2*π* (Gargett et al. 1981), poleward of 0.5° latitude the second term in the denominator is three orders of magnitude smaller than the first so it can be neglected for all practical purposes.^{1}

*m*

_{c}~

*E*

^{−1}(Gargett et al. 1981; Gargett 1990; Polzin et al. 1995) and 〈

*K*

_{z}〉 ~

*E*

^{2}(Henyey et al. 1986; D’Asaro and Lien 2000), where

*E*is the nondimensional spectral energy level of the internal-wave field, 〈

*K*

_{z}〉

*m*

^{2}is invariant so that the second term remains small compared to the first for

*all*internal-wave-driven mixing. Finally, if 〈

*K*

_{z}〉

*m*

^{2}should become comparable or greater than

*f*, then the shear in wavenumber

*m*would be smoothed away in less than an inertial period to restore the 〈

*K*

_{z}〉

*m*

^{2}≪

*f*state because the shear-containing

*m*has been reduced. We conclude that the ocean is in the weak-mixing limit corresponding to (37a) in Young et al. (1982). Since the

*O*(10

^{−3}) correction associated with the parenthetic term in (1) is not significant, (1) can be approximated to within a few percent as

*χ*_{z}|

^{2}〉 = 〈

*f*

^{2}is the variance of the vertical gradient of horizontal displacement

**= (**

*χ**X*,

*Y*)—that is, the two off-diagonal components of the vertical strain, or deformation, tensor

*K*

_{z}and off-diagonal vertical strain |

*χ*_{z}| = |∫

**V**

_{z}

*dt*|, rather than vertical shear

*V*

_{z}= |

**V**

_{z}|. Thus, it might more aptly be referred to as “strain dispersion,” but we will use the term “shear dispersion” for consistency with the literature.

*K*

_{h}based on (2) in the literature use either 〈

*K*

_{z}〉, 〈

*K*

_{z}′

*N*

^{2}/2,

*δ*

_{c}is the critical gradient Froude number. For canonical midlatitude upper pycnocline values of 〈

*K*

_{z}〉 = 0.05 × 10

^{−4}m

^{2}s

^{−1},

*N*= 10

^{−2}rad s

^{−1},

*f*= 10

^{−4}rad s

^{−1}, and

*δ*

_{c}= 2,

*K*

_{h}based on (4) can be no more than ~0.1 m

^{2}s

^{−1}.

Several field studies have compared internal-wave shear dispersion [(4)] with submesoscale dye dispersion. Tracer studies in the North Atlantic subtropical gyre (Ledwell et al. 1998) used upper-bound 〈*K*_{z}〉 inferred from tracer and a shear spectrum based on moored current-meter measurements to infer internal-wave shear dispersion from (2) of *K*_{h} < 0.024 m^{2} s^{−1}, more than a factor of 2 smaller than the 1–10-km dye lateral diffusivity of 0.07 ± 0.04 m^{2} s^{−1} 14 days after release, and nearly two orders of magnitude smaller than the dye lateral diffusivity of 0.6–6 m^{2} s^{−1} estimated 5–6 months after release. Similar discrepancies were reported in 5–10-km tracer patches sampled over 3–5 days on the New England continental shelf (Sundermeyer and Ledwell 2001). Using dye-based 〈*K*_{z}〉 and relative lateral displacements with depth |*χ*_{z}| inferred from shipboard ADCP time series, shear dispersion [(2)] in two of four tracer releases was nearly an order-of-magnitude smaller than dye-based lateral diffusivities, which ranged from 0.1–0.6 to 3–7 m^{2} s^{−1}, with the other two releases showing closer agreement between dye and shear dispersion. Finally, shear dispersion using 〈*K*_{z}〉 was again an order-of-magnitude smaller than the 1–10-km lateral dye diffusivities over 6 days of 0.5–4 m^{2} s^{−1} in the summer pycnocline of the Sargasso Sea (Shcherbina et al. 2015; D. A. Birch et al. 2015, unpublished manuscript). The discrepancy between dye and shear-dispersion estimates is by no means universal as other studies ranging in duration from a few hours to 1–2 days have found closer agreement (e.g., Inall et al. 2013; Moniz et al. 2014). Common to all the aforementioned studies is that they did not account for the intermittency of diapycnal turbulent mixing events and their possible correlation with the off-diagonal vertical strain |*χ*_{z}| = |∫**V**_{z} *dt*|.

The apparent failure of internal-wave shear dispersion to explain observed isopycnal mixing in some studies has fostered a number of alternative hypotheses. Smith and Ferrari (2009) examined eddy shear dispersion in a quasigeostrophic (QG) numerical simulation in a baroclinically unstable QG model that reproduced filaments of similar *O*(1)-km horizontal scale to those observed in the ocean (Ledwell et al. 1998). Their lateral diffusivity scaled as *K*_{h} ~ *K*_{z}*N*^{2}/*f*^{2} like (4), so it still falls short of observed dye diffusivities by more than an order of magnitude, though they noted considerable scatter in the aspect ratio of filaments about *N*/*f*. The Smith and Ferrari model did not include internal waves or unbalanced subinertial flows as might arise in a primitive equation model (e.g., Molemaker et al. 2010). These non-QG flows must be present on the submesoscale based on surface drifter-pair spreading (Lumpkin and Elipot 2010; Poje et al. 2014) and horizontal wavenumber spectra of submesoscale water-mass anomalies on isopycnals (Rudnick and Ferrari 1999; Cole and Rudnick 2012; Kunze et al. 2015).

It has also been proposed that horizontal dispersion might arise from Stokes drift in a dissipative weakly nonlinear internal-wave field (Bühler et al. 2013) or from stirring by finescale potential vorticity anomalies (Lelong et al. 2002; Sundermeyer et al. 2005; Lelong and Sundermeyer 2005; Sundermeyer and Lelong 2005) termed the vortical mode (Müller 1984). Kinematic arguments and numerical simulations indicate that vortical-mode stirring should be more effective than vortical-mode shear dispersion, and that this stirring may be enhanced by upscale energy transfer of vortical-mode (PV) variance (Sundermeyer et al. 2005; Brunner-Suzuki et al. 2014). Direct measurement of finescale vortical mode in the ocean has proven challenging and inconclusive (Müller et al. 1988; Kunze et al. 1990a; Kunze and Sanford 1993; Kunze 1993; Polzin et al. 2003) because of the dominance of internal waves in dynamic (horizontal kinetic and available potential energy) signals on these scales and the resulting demands on sampling to avoid space–time aliasing. However, recently, Pinkel (2014) reported that subinertial finescale potential vorticity is dominated by finescale fluctuations in stratification *N*^{2} and that subinertial vertical shear is many orders of magnitude below near-inertial shear so unlikely to contribute significantly to shear dispersion.

## 3. Intermittency

Here, we revisit internal-wave shear dispersion, taking into account the observed intermittency of *K*_{z}, including its lognormal distribution, and its possible correlation with finescale off-diagonal components of vertical strain *χ*_{z} = ∫**V**_{z} *dt*. Ocean turbulence is sporadic (black curves in Fig. 2), typically only occupying 5%–10% of space–time in the pycnocline (Gregg and Sanford 1988). Nonzero turbulent dissipation rates are also approximately lognormally distributed (Gregg et al. 1993; Davis 1996). In the stratified interior, turbulence is largely driven by finescale *O*(1)-m internal-wave shear (Eriksen 1978; Desaubies and Smith 1982; Polzin 1996), so unstable shears are also intermittent and present 5%–10% of the time (Kunze et al. 1990b). While shears are not correlated with turbulence when they are stable, or when unstable scales are not resolved (Gregg et al. 1986; Toole and Schmitt 1987), resolved unstable instantaneous finescale shears *V*_{z} > 2*N* on *O*(1) m scales exhibit strong correlation with turbulent dissipation rates ε as described in section 3c.

### a. Theoretical shear dispersion with intermittent shear-driven turbulent mixing

*θ*(

*x*,

*z*, 0) = cos(

*kx*) from which more general realistic initial conditions can be constructed (Young et al. 1982), we model intermittent (

*r*of the time) mixing

*K*

_{z}as

*u*, defined as

*δ*

_{N}= |

**V**

_{z}|/

*N*and

*δ*

_{c}is the critical threshold value. The

*u*

_{z}term represents the

*O*(

*N*) background shear and the

*u*

_{z}′ term represents excess shear-associated turbulent bursts. Shear-driven turbulent bursts are modeled with a boxcar function Π(

*t*−

*t*

_{n}) of average duration

*δt*that is nonzero every Δ

*t*= δ

*t*/

*r*on average, that is,

*t*

_{n}~

*n*Δ

*t*. The

*z*dependence in (7) is valid in the weak-mixing limit of Young et al. (1982), which was argued in section 2 to be appropriate for midlatitude ocean internal waves, but this approximation will not hold on the equator or for finescale subinertial shears. Integrating (7), the horizontal displacement is

*x*in the tracer initial condition, yields

*K*

_{h}given by the average of the coefficient in front of

*k*

^{2}Θ on the right-hand side,

*H*(⋅) is the Heaviside function. The first term in (11) corresponds to shear dispersion associated with 〈

*K*

_{z}〉 and background shear variance

*N*

^{2}

*N*

^{2}

*N*

^{2}

*δ*

_{N}~ (2–3)

*δ*

_{c}, depending on the statistics of turbulence and the relationship between shear and turbulence as laid out in section 3d. Whether the second term dominates will depend on the duration

*δt*of the unstable shear and mixing events since it takes

*O*(

*f*

^{‒1}) for elevated shear to express itself as elevated off-diagonal vertical strain |

*χ*_{z}| = |∫

**V**

_{z}

*dt*|. Since (

*X*

_{z},

*Y*

_{z}) = ∫(

*u*

_{z},

*υ*

_{z})

*dt*, there is no guarantee that large strain |

*χ*_{z}| will coincide in time with large

*K*

_{z}. While there are no measured correlations between |

*χ*_{z}|

^{2}and

*K*

_{z}, with some observationally based caveats noted below, it may be possible to use correlations between

*K*

_{z}, that is,

*K*

_{h}~ 〈

*K*

_{z}

*f*

^{2}as described next. However, we emphasize that off-diagonal vertical strain |

*χ*_{z}| is the physically relevant variable.

### b. Properties of observed finescale shear

Finescale shear layers have low aspect ratios and cross isopycnals with gentle slopes, consistent with near-inertial waves (Itsweire et al. 1989). This is mirrored in the aspect ratios of turbulent layers (Marmorino et al. 1986; Marmorino et al. 1987), which can persist for many inertial time scales *f*^{−1} (Gregg et al. 1986; Sundermeyer et al. 2005). Because group velocities are small at the 10-m vertical wavelengths that dominate internal-wave shear, these waves cannot propagate from the surface or bottom but likely arise in situ from wave–wave or wave–mean flow interactions that transfer wave variance from lower to higher vertical wavenumber *m* on time scales such as *m*(*kU*_{z})^{−1} ~ *N*(*ω*^{2} − *f*^{2})^{−1/2}/*N* ~ (*ω*^{2} − *f*^{2})^{−1/2} based on ray-tracing theory. The most relevant interactions for production of unstable shear can occur on time scales much shorter than *f*^{−1} (Sun and Kunze 1999) so that interpreting the resulting shears from linear internal-wave theory may be inappropriate. Whether unstable shear |**V**_{z}| > *δ*_{c}*N* and mixing persist long enough to produce a correlation between large vertical gradients of horizontal displacement |*χ*_{z}| = |∫**V**_{z} *dt*| and *K*_{z} remains an open question that we discuss in the appendix and revisit in the summary.

### c. Relation between unstable shears and turbulence

*K*

_{z}to

*J. Phys. Oceanogr.*),

*z*is the vertical scale used to estimate shear

*V*

_{z}and buoyancy frequency

*N*estimates, and

*H*(·) is the Heaviside function to account for the fact that turbulent dissipations only arise for supercritical shear. Parameterization (12) is made up of two components: (i) the available horizontal kinetic energy in the unstable shear (

*N*

^{2})Δ

*z*

^{2}/24 and (ii) the Kelvin–Helmholtz growth rate in the initial shear instability (|

**V**

_{z}| −

*δ*

_{c}

*N*)/4 (Hazel 1972), which we equate to the rate at which unstable kinetic energy is lost to turbulence production. The exact model used is not critical provided the dissipation rate ε is an increasing function of

*K*

_{z}=

*γε*/〈

*N*

^{2}〉, where

*γ*is the mixing efficiency (Osborn 1980), must be based on dissipation rates

*ε*averaged over a turbulent growth and decay event on a time scale

*O*(

*N*

^{−1}) to be meaningful, and this same averaging must be applied to the unstable or reduced shear (|

**V**

_{z}| −

*δ*

_{c}

*N*) > 0. The vertical scale Δ

*z*must be small enough to resolve unstable shear but not so small to capture the outer scales of turbulence; scale separation of these (Kunze 2014) ensures this is possible provided that unstable |

**V**

_{z}|/

*N*is not too large. Polzin (1996) argued that (12) was robust to changes in Δ

*z*provided that 〈

*N*

^{2}〉 = 〈(Δ

*V*/Δ

*z*)

^{2}〉/〈

*N*

^{2}〉 ~

*O*(1); an iterative procedure might be more effective for heterogeneous turbulence environments. A critical gradient Froude number

*δ*

_{c}= 2 is consistent with laboratory (Thorpe 1971; Scotti and Corcos 1972; Rohr et al. 1988) and numerical (Itsweire et al. 1993) turbulence studies, but values of 1.7–2 give similar results (Polzin 1996).

Shear-driven inferred turbulent diapycnal diffusivity *K*_{z} from (12) as a function of gradient Froude number *δ*_{N} = |**V**_{z}|/*N* assuming a critical gradient Froude number *δ*_{c} = 2, *N* = 10^{−2} rad s^{−1}, Δ*z* = 1 m, and *γ* = 0. 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Shear-driven inferred turbulent diapycnal diffusivity *K*_{z} from (12) as a function of gradient Froude number *δ*_{N} = |**V**_{z}|/*N* assuming a critical gradient Froude number *δ*_{c} = 2, *N* = 10^{−2} rad s^{−1}, Δ*z* = 1 m, and *γ* = 0. 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Shear-driven inferred turbulent diapycnal diffusivity *K*_{z} from (12) as a function of gradient Froude number *δ*_{N} = |**V**_{z}|/*N* assuming a critical gradient Froude number *δ*_{c} = 2, *N* = 10^{−2} rad s^{−1}, Δ*z* = 1 m, and *γ* = 0. 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Peters et al. (1995) reported that 2-m estimates of ε parameterized from (12)—that is, shear |**V**_{z}|, or rather reduced shear |**V**_{z}| − *δ*_{c}*N*—both (i) reproduced average microstructure dissipation rate 〈*ε*〉 and (ii) were significantly correlated with instantaneous microstructure *ε* with *R* ~ 0.6, which is likely a lower bound given problems associated with instrument noise in the 2-m shear estimate. Polzin (1996) found that (12) was as good a predictor of average dissipation rate *ε* as a finescale parameterization based on internal wave–wave interaction theory (Henyey et al. 1986; Gregg 1989; Polzin et al. 1995). More importantly, (12) reproduced the behavior of *ε* as a function of *N* and *V*_{z} (Polzin 1996, his Figs. 2 and 3), again indicating a strong correlation between unstable shear and turbulent mixing. The observed consistency of (12) with microstructure measurements indicates that turbulence and unstable shear occupy the same 5%–10% of space/time.

*δt*~ 4/(|

**V**

_{z}| −

*δ*

_{c}

*N*) ~

*O*(

*N*

^{−1}) ≪

*O*(

*f*

^{−1}), as also inferred from the ratio of turbulent energy variance to dissipation rates (Dillon 1982; Moum 1996), this is too rapid to allow large vertical gradients of horizontal displacement |

*χ*_{z}| = |∫

**V**

_{z}

*dt*| to develop. On the other hand, microstructure time series measurements often find that more persistent turbulent patches associated with near-inertial shear dominate mixing (Gregg et al. 1986; Sundermeyer et al. 2005). This is only possible if unstable shears |

*V*

_{z}| >

*δ*

_{c}

*N*are being continuously replenished by nonlinear wave–wave interactions or wave–mean flow interactions to create a succession of

*O*(

*N*

^{−1}) turbulent bursts over longer durations

*δt*; that is,

*α*is the replenishment rate by nonlinear interactions (refer to the appendix) and

*σ*= (|

**V**

_{z}| −

*δ*

_{c}

*N*)/4 is the rate of loss to turbulence. Maintenance of unstable shears for durations of

*O*(

*f*

^{−1}) or longer would allow time for production of large |

*χ*_{z}| = |∫

**V**

_{z}

*dt*|, while diapycnal turbulent mixing

*K*

_{z}remains elevated [(11)].

### d. Statistical prediction of shear dispersion with intermittency

*r*∈ [0, 1], instantaneous

*K*

_{z}is assumed to be zero for 1 −

*r*of the time [assuming molecular diffusivity

*κ*to be negligible compared to 〈

*K*

_{z}〉 (Fig. 2)] and to have burst averages {

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*for the

*r*of the time it is nonzero; that is, {·}

_{b}represents the average during the turbulent bursts and

*K*

_{z}is composed of two populations, one is uniformly zero for 1 −

*r*fraction of the time and the other has a lognormal distribution and mean {

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*for

*r*of the time (Fig. 2b). Bursts are not necessarily limited to be of

*O*(

*N*

^{−1}) duration but may persist for many inertial time scales (Gregg et al. 1986; Sundermeyer et al. 2005). For the

*r*of the time

*K*

_{z}is nonzero,

*K*

_{z}is taken to have a lognormal probability density function as is observed to be approximately true (e.g., Gregg et al. 1993)

*K*

_{z}}

_{b}) = {ln(

*K*

_{z})}

_{b}+

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*; and (ii) the standard deviation of ln(

*K*

_{z}),

*σ*

_{lnK}, where

*K*

_{z}

^{2}}

_{b}/{

*K*

_{z}}

_{b}

^{2}). Shear dispersion (11) can then be reexpressed as

*xy*′〉 = 〈(〈

*x*〉 +

*x*′)

*y*′〉 = 〈〈

*x*〉

*y*′〉 + 〈

*x*′

*y*′〉 = 〈

*x*′

*y*′〉 in the second rhs term and letting corr represent 〈sin

^{2}(

*fδt*)〉 as well as ocean

*K*

_{z}and

*V*

_{z}/

*N*not lying on the curve shown in Fig. 3 from (12). It is possible that corr is a function of

*V*

_{z}/

*N*so it should be inside the integral, but there is insufficient observational information to assess this at this time. As before, the first rhs term in (15) represents horizontal mixing associated with the background ubiquitous shear variance 〈

*O*(

*N*

^{2}/2) (Young et al. 1982), while the second is due to the turbulent bursts that are present

*r*of the time and which we are assuming to have correlation corr with unstable shears (Fig. 2a). For the second term, shear variance

*K*

_{z}) is obtained numerically from (12) and the PDF from (14). Choosing a standard deviation

*σ*

_{lnK}is more challenging as this is rarely reported. Theoretically, it should have an upper bound of 2.57 for random internal-wave breaking (Gregg et al. 1993). But ocean turbulence is neither stationary nor homogeneous. After correcting for instrument noise, Gregg et al. (1993) found

*σ*

_{lnK}= 1.2 in a dataset with a canonical [Garrett–Munk (GM)] level internal-wave field and 1.5 in a dataset 4 times more energetic. Baker and Gibson (1987) reported

*σ*

_{lnε}or

*σ*

_{lnχ}ranging from 1.30 to 2.53 in seven off-equatorial microstructure measurement datasets. We caution that ocean turbulence is unlikely to be perfectly lognormal because of the nonstationarity of the background environment and forcing, both failing to satisfy underlying homogeneity assumptions in statistical theories of turbulence (Davis 1996); however, insufficient information is available to determine a robust observationally based PDF. Since (15) is sensitive to the high-

*K*

_{z}tail, which is not well characterized observationally, it might easily over- or underestimate

*K*

_{h}. Thus, the results reported below are highly speculative and require observational testing.

The principal variables affecting (16) are (i) the intermittency *r*, (ii) assumed lognormality PDF(*K*_{z}), and (iii) correlation corr, as explored in Fig. 4 and Table 1, which show *K*_{h} depending roughly as *r*^{–1/2} on intermittency *r*. For perfect shear-turbulence correlation (corr = 1; thick solid curves, Fig. 4) in the continuous turbulence limit (*r* = 1), the horizontal diffusivity *K*_{h} ~ 0.2 m^{2} s^{−1} for *σ*_{lnK} = 1.2 and *K*_{h} ~ 0.52 m^{2} s^{−1} for *σ*_{lnK} = 2.1, boosted by lognormality alone from 〈*K*_{z}〉〈*f*^{2}) = 0.18 m^{2} s^{−1} corresponding to no intermittency and no lognormality (thin solid line at *r* = 1). As *r* → 0, the inferred horizontal diffusivity *K*_{h} increases sharply. For typical low ocean intermittencies, *r* = 0.05–0.1, *σ*_{lnK} = 1.2, and corr = 1 (Fig. 4); isopycnal diffusivity *K*_{h} ~ 0.4–0.6 m^{2} s^{–1}, which is in the range found by dye estimates; and *K*_{h} > 1 m^{2} s^{–1} only for *r* < 0.035. With *σ*_{lnK} = 1.2 and corr = 1 but no lognormality (thick dotted curve, Fig. 4), the horizontal diffusivity is smaller, with *K*_{h} > 1 m^{2} s^{–1} only for intermittencies *r* < 0.01. For higher *σ*_{lnK} = 2.1 and corr = 1 (Fig. 4), the predicted *K*_{h} rises above dye-inferred diffusivities for *r* < 0.36. For observed *r* = 0.05–0.1, *K*_{h} has diffusivities greater than or equal to dye estimates for *σ*_{lnK} > 1.8. Horizontal diffusivities *K*_{h} exceed 1 for all intermittencies *r* if *σ*_{lnK} exceeds 2.4. Finally, assuming a lower-bound correlation corr = 0.6 (Peters et al. 1995) and *σ*_{lnK} = 1.2 (thin solid curve, Fig. 4) produces horizontal diffusivities *K*_{h} = 0.3–0.4 m^{2} s^{−1} for intermittencies *r* = 0.05–0.1 and *K*_{h} > 1 m^{2} s^{−1} for intermittencies *r* < 0.02. We caution that, for *r* ≤ 0.02 (shading in Fig. 4), {*K*_{z}}_{b} = 〈*K*_{z}〉/*r* is sufficiently high to violate the weak-mixing assumption used to justify the chosen *z* dependence in (7) because {*K*_{z}}_{b}*δt* > *m*^{–2} for *δt* ~ *f*^{−1}, so that the turbulent events can wipe out the vertical structure on scales *m*^{−1}, short-circuiting horizontal mixing. The intermittency and lognormality introduced in (15) imply that the upper bound for shear-dispersion-induced *K*_{h} derived by Young et al. (1982) no longer applies, particularly as *r →* 0 or large *σ*_{lnK}.

Horizontal diffusivity *K*_{h} from (9) as a function of turbulent intermittency *r* assuming Δ*z* = 1 m in (12), bulk average diapycnal diffusivity *K*_{0} = 0.05 × 10^{−4} m^{2} s^{−1}, buoyancy frequency *N* = 0.01 rad s^{−1}, Coriolis frequency *f* = 8 × 10^{−5} rad s^{−1}, mixing efficiency *γ* = 0.2, and a critical gradient Froude number *δ*_{c} = 2. Finescale parameterization [(12)] is used to relate shear *V*_{z} to diapycnal diffusivity *K*_{z} numerically. Thick solid curves correspond to *K*_{z} having a lognormal distribution [(14)] during the *r* fraction of time turbulence is present, with *σ*_{lnK} labeled along the upper axis and perfect correlation (corr = 1) between *K*_{z} and unstable shear variance *N*^{2}. The thin solid curve uses a lower-bound correlation of 0.6 for *σ*_{lnK} = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of *σ*_{lnK}. Horizontal diffusivity *K _{h}* increases as intermittency

*r*→ 0, with lognormality and increasing

*σ*

_{lnK}. Gray shading for

*r*< 0.02 indicates where {

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*is high enough to violate the weak-mixing approximation in (7) because {

*K*

_{z}}

_{b}

*δt*>

*m*

^{−2}for durations

*δt*~

*f*

^{−1}, but the impact of this remains to be assessed.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Horizontal diffusivity *K*_{h} from (9) as a function of turbulent intermittency *r* assuming Δ*z* = 1 m in (12), bulk average diapycnal diffusivity *K*_{0} = 0.05 × 10^{−4} m^{2} s^{−1}, buoyancy frequency *N* = 0.01 rad s^{−1}, Coriolis frequency *f* = 8 × 10^{−5} rad s^{−1}, mixing efficiency *γ* = 0.2, and a critical gradient Froude number *δ*_{c} = 2. Finescale parameterization [(12)] is used to relate shear *V*_{z} to diapycnal diffusivity *K*_{z} numerically. Thick solid curves correspond to *K*_{z} having a lognormal distribution [(14)] during the *r* fraction of time turbulence is present, with *σ*_{lnK} labeled along the upper axis and perfect correlation (corr = 1) between *K*_{z} and unstable shear variance *N*^{2}. The thin solid curve uses a lower-bound correlation of 0.6 for *σ*_{lnK} = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of *σ*_{lnK}. Horizontal diffusivity *K _{h}* increases as intermittency

*r*→ 0, with lognormality and increasing

*σ*

_{lnK}. Gray shading for

*r*< 0.02 indicates where {

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*is high enough to violate the weak-mixing approximation in (7) because {

*K*

_{z}}

_{b}

*δt*>

*m*

^{−2}for durations

*δt*~

*f*

^{−1}, but the impact of this remains to be assessed.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Horizontal diffusivity *K*_{h} from (9) as a function of turbulent intermittency *r* assuming Δ*z* = 1 m in (12), bulk average diapycnal diffusivity *K*_{0} = 0.05 × 10^{−4} m^{2} s^{−1}, buoyancy frequency *N* = 0.01 rad s^{−1}, Coriolis frequency *f* = 8 × 10^{−5} rad s^{−1}, mixing efficiency *γ* = 0.2, and a critical gradient Froude number *δ*_{c} = 2. Finescale parameterization [(12)] is used to relate shear *V*_{z} to diapycnal diffusivity *K*_{z} numerically. Thick solid curves correspond to *K*_{z} having a lognormal distribution [(14)] during the *r* fraction of time turbulence is present, with *σ*_{lnK} labeled along the upper axis and perfect correlation (corr = 1) between *K*_{z} and unstable shear variance *N*^{2}. The thin solid curve uses a lower-bound correlation of 0.6 for *σ*_{lnK} = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of *σ*_{lnK}. Horizontal diffusivity *K _{h}* increases as intermittency

*r*→ 0, with lognormality and increasing

*σ*

_{lnK}. Gray shading for

*r*< 0.02 indicates where {

*K*

_{z}}

_{b}= 〈

*K*

_{z}〉/

*r*is high enough to violate the weak-mixing approximation in (7) because {

*K*

_{z}}

_{b}

*δt*>

*m*

^{−2}for durations

*δt*~

*f*

^{−1}, but the impact of this remains to be assessed.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Dependence of horizontal diffusivity *K*_{h} [(15)] on intermittency *r* and *σ*_{lnK} for corr = 1. The reported range of intermittency *r* is boldface in the second column. The reported dye diffusivities *K*_{h} are bold in the third column.

The conclusions above suggest that, within constraints of available observations, internal-wave shear dispersion cannot yet be ruled out as a possible explanation for the horizontal diffusivities *K*_{h} inferred from 1–5-km open-ocean dye studies. The key features for this argument are (i) the intermittency of ocean turbulence *r*, (ii) its probability density function [(14)], and (iii) its degree of correlation with finescale off-diagonal vertical strain |*χ*_{z}| = |∫**V**_{z} *dt*| (Fig. 4). The exact relationship [e.g., (12)] between *K*_{z} and *K*_{z} is an increasing function of *J. Phys. Oceanogr.*), the relation between *K*_{z} and |*χ*_{z}|^{2} is unknown and likely varies from one dynamic regime to another.

## 4. Summary

In the traditional application of internal-wave shear dispersion, either average turbulent diapycnal diffusivity 〈*K*_{z}〉, average variance of vertical shear 〈*K*_{z}′*K*_{z}*K*_{z}〉〈*K*_{h} ~ 0.1 m^{2} s^{−1}. Here, we have explored the consequences of accounting for (i) the intermittency of ocean turbulence *r*, (ii) its lognormality [(14)], and (iii) its correlation with unstable finescale shear. An established parameterization [(12)] was used to relate vertical shear and turbulent mixing; any scaling where *K*_{z} increases with finescale *J. Phys. Oceanogr.*) Taken together and applied to the shear version of (15), a horizontal diffusivity *K*_{h} ~ *O*(1) m^{2} s^{−1} was found for *σ*_{lnK} = 1.2 (Gregg et al. 1993), ocean intermittencies *r* < 0.03, and correlations corr > 0.6. These *K*_{h} are consistent with dye estimates in the stratified ocean interior (Ledwell et al. 1998; Sundermeyer and Ledwell 2001; D. A. Birch et al. 2015, unpublished manuscript). The conclusions are most sensitive to the intermittency *r*, probability density function for *K*_{z} [e.g., (14)], and the correlation between *K*_{z} and *K*_{z} and shear variance

There are two major caveats. First, the PDF of *K*_{z} in the ocean is only approximately lognormal with deviations at high *K*_{z} which will impact *K*_{h} estimates from (15). While ocean sampling is sufficient to characterize the mean 〈*K*_{z}〉, standard deviations *σ*_{lnK} are rarely reported, let alone sufficient sampling for higher-order quantities like shear dispersion [(15)]. Second, while *K*_{z} and finescale shear variance *K*_{z} and the variance of off-diagonal vertical strain |*χ*_{z}|^{2} that is relevant for shear dispersion, and this has not been characterized in the ocean. A third caveat is that the argument presented following (1) for the smallness of *K*_{z}*m*^{2}/*f* in the ocean need not hold for intermittent mixing; horizontal dispersion can be short-circuited by elevated vertical mixing persisting for *δt* ~ *O*(*f*^{−1}) or longer as *r* → 0 (shading in Fig. 4) as illustrated by (1).

Elevated horizontal diffusivities *K*_{h} due to shear dispersion hinge on unstable shear persisting for *O*(*f*^{−1}) so that the off-diagonal vertical strain |*χ*_{z}| = |∫**V**_{z} *dt*| can become large while diapycnal mixing *K*_{z} is strong [Fig. 5b; (11)]. Although the unstable-shear parameterization [(12)] implies rapid loss of unstable shears to turbulence on a time scale 4/(|**V**_{z}| − *δ*_{c}*N*) ~ *O*(*N*^{−1}), which is too short for the creation of large vertical gradients of horizontal displacement |*χ*_{z}| (Fig. 5a), microstructure observations often find that the strongest turbulent mixing can persist for days associated with near-inertial shear (Gregg et al. 1986; Sundermeyer et al. 2005). This implies that unstable shears must be continuously replenished by nonlinear wave–wave or wave–mean interactions (appendix) in order to maintain unstable shears |**V**_{z}| and turbulent diapycnal mixing *K*_{z} long enough to create large |*χ*_{z}| ~ |∫**V**_{z} *dt*| coincident with large *K*_{z}. Such persistent shear and mixing events should result in a strong correlation between *K*_{z} and |*χ*_{z}| = |∫**V**_{z} *dt*|, which is an underlying assumption in our results, but this remains to be verified.

Cartoon of two scenarios for shear dispersion, where the thick dotted curves represent unstable reduced shear |**V**_{z}| − 2*N*, thick solid curves diapycnal diffusivity *K*_{z}, and thin solid curves the vertical gradient of horizontal displacement |*χ*_{z}| = |∫**V**_{z} *dt*|. (a) The unstable shear and mixing decay over a buoyancy time scale *δt* ~ *N*^{−1} so that |*χ*_{z}| remains small. (b) Unstable shear and mixing persist over an inertial time scale *δt* ~ *f*^{−1}, allowing |*χ*_{z}| to become large in conjunction with strong mixing.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Cartoon of two scenarios for shear dispersion, where the thick dotted curves represent unstable reduced shear |**V**_{z}| − 2*N*, thick solid curves diapycnal diffusivity *K*_{z}, and thin solid curves the vertical gradient of horizontal displacement |*χ*_{z}| = |∫**V**_{z} *dt*|. (a) The unstable shear and mixing decay over a buoyancy time scale *δt* ~ *N*^{−1} so that |*χ*_{z}| remains small. (b) Unstable shear and mixing persist over an inertial time scale *δt* ~ *f*^{−1}, allowing |*χ*_{z}| to become large in conjunction with strong mixing.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Cartoon of two scenarios for shear dispersion, where the thick dotted curves represent unstable reduced shear |**V**_{z}| − 2*N*, thick solid curves diapycnal diffusivity *K*_{z}, and thin solid curves the vertical gradient of horizontal displacement |*χ*_{z}| = |∫**V**_{z} *dt*|. (a) The unstable shear and mixing decay over a buoyancy time scale *δt* ~ *N*^{−1} so that |*χ*_{z}| remains small. (b) Unstable shear and mixing persist over an inertial time scale *δt* ~ *f*^{−1}, allowing |*χ*_{z}| to become large in conjunction with strong mixing.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

The statistics of both shear and turbulence may vary in different regions of the ocean. Thus, the effects discussed here will be of varying importance depending on the degree of correlation between *K*_{z} and finescale off-diagonal vertical strain |*χ*_{z}| = |∫**V**_{z} *dt*|, so these must be resolved to correctly evaluate shear dispersion. In locales where turbulence and finescale shear are more uniform, such as seamount summits (Kunze and Toole 1997) and banks (Palmer et al. 2013), intermittency *r* ~ 1, so traditional shear dispersion 〈*K*_{z}〉〈*f*^{2} will better reproduce true isopycnal mixing on the submesoscale. Likewise, traditional shear dispersion is expected to hold where *K*_{z} and |*χ*_{z}|^{2} are uncorrelated or unstable shears do not persist long enough to produce significant |*χ*_{z}| = |∫**V**_{z} *dt*| coincident with elevated diapycnal mixing. However, where turbulence is intermittent (*r* ≪ 1) and turbulence production is dominated by near-inertial wave packets, we anticipate that intermittency effects will be important to shear dispersion.

This paper has found that internal-wave shear dispersion is more subtle than previously recognized, depending not just on average diapynal diffusivities 〈*K*_{z}〉 and average shear variances 〈*K*_{h} are sensitive to the assumed lognormal probability distribution for *K*_{z}, which may not be a good description of ocean turbulence at the high *K*_{z} weighted by (15) (Gregg et al. 1993; Davis 1996). For these reasons, only observational assessment of the distribution of *K*_{z} and its correlation with the off-diagonal vertical strain |*χ*_{z}| = |∫**V**_{z} *dt*| will be able to address whether the proposed mechanism is important for shear dispersion in the ocean.

## Acknowledgments

A. Sykulski, M. Inall, J. Ledwell, and an anonymous reviewer provided helpful comments. E. Kunze was supported by ONR Grant N00014-12-1-0942 and M. Sundermeyer by ONR Grant N00014-09-1-0194. In memoriam Murray Levine.

## APPENDIX

### Replenishment of Unstable Shear

*N*

^{2}by increasing the vertical wavenumber

*m*or increasing the wave horizontal kinetic energy HKE,

*O*(

*f*). We caution that the linear hydrostatic dispersion relation was used in conversion from horizontal wavenumber

*k*and vertical group velocity

*Cg*

_{z}to vertical wavenumber

*m*and frequency

*ω*. This may not apply on the finescale; however, it should provide reasonable ballpark estimates. Comparing (A8) to the shear instability growth rate in (12) implies balance when

*dm/dt*in the first term on the rhs of (A1) and neglecting the second term. This suggests the details of finescale wave dynamics are negligible compared to the kinematics of vertical shearing. For unstable near-inertial shears (

*ω*< 2

*f*), a balance requires unstable shears

*δ*

_{N}= |

**V**

_{z}|

*/N*= 2–3, consistent with finescale observations of unstable shear (Kunze et al. 1990b; Peters et al. 1995; Polzin 1996; Jurisa et al. 2015, manuscript submitted to

*J. Phys. Oceanogr.*).

Buoyancy-frequency-normalized shear |*V*_{z}|*/N* vs Coriolis-frequency-normalized frequency *ω*/*f* to achieve a balance (solid curve) between shear variance amplification (4*ω*^{2} + *f*^{2})(*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|/(*Nω*^{2}) [(A8)] and loss to turbulence (|**V**_{z}| − 2*N*)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|*/N* vs (|**V**_{z}| − 2*N*)/4 without wave dynamics [(A3)–(A7)], that is, omitting the second term on the right-hand side of (A1). The thin dotted horizontal line corresponds to the assumed critical *δ*_{c} = |**V**_{z}|*/N* = 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Buoyancy-frequency-normalized shear |*V*_{z}|*/N* vs Coriolis-frequency-normalized frequency *ω*/*f* to achieve a balance (solid curve) between shear variance amplification (4*ω*^{2} + *f*^{2})(*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|/(*Nω*^{2}) [(A8)] and loss to turbulence (|**V**_{z}| − 2*N*)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|*/N* vs (|**V**_{z}| − 2*N*)/4 without wave dynamics [(A3)–(A7)], that is, omitting the second term on the right-hand side of (A1). The thin dotted horizontal line corresponds to the assumed critical *δ*_{c} = |**V**_{z}|*/N* = 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

Buoyancy-frequency-normalized shear |*V*_{z}|*/N* vs Coriolis-frequency-normalized frequency *ω*/*f* to achieve a balance (solid curve) between shear variance amplification (4*ω*^{2} + *f*^{2})(*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|/(*Nω*^{2}) [(A8)] and loss to turbulence (|**V**_{z}| − 2*N*)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (*ω*^{2} − *f*^{2})^{1/2}|**V**_{z}|*/N* vs (|**V**_{z}| − 2*N*)/4 without wave dynamics [(A3)–(A7)], that is, omitting the second term on the right-hand side of (A1). The thin dotted horizontal line corresponds to the assumed critical *δ*_{c} = |**V**_{z}|*/N* = 2.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-14-0134.1

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^{1}

This term plays a role in limiting horizontal mixing for subinertial (*ω* ≪ *f*, e.g., vortical mode) shears to 〈*K*_{z}*m*^{4}) ~ *N*^{2}*δ*_{g}^{2}/(*K*_{z}*m*^{4}), where *δ*_{g} is the geostrophic gradient Froude number. For the abovementioned values of *K*_{z} and *m*, and vortical-mode gradient Froude number *δ*_{g} ≪ 1 as reported by Pinkel (2014), subinertial shear dispersion is much less than 1 m^{2} s^{−1}.