Editorial Type:
Article
Article Type:
Research Article

The Role of Intermittency in Internal-Wave Shear Dispersion

Eric Kunze NorthWest Research Associates, Redmond, Washington

Search for other papers by Eric Kunze in
Current site
Google Scholar
PubMed Close
and
Miles A. Sundermeyer School for Marine Science and Technology, University of Massachusetts Dartmouth, New Bedford, Massachusetts

Search for other papers by Miles A. Sundermeyer in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Dec 2015
Collections:
LatMix: Studies of Submesoscale Stirring and Mixing
DOI:
https://doi.org/10.1175/JPO-D-14-0134.1
Page(s):
2979–2990
Full access

Abstract

This paper revisits a long-standing discrepancy between (i) 1–5-km isopycnal diffusivities of O(1) m2 s−1 based on dye spreading and (ii) inferences of O(0.1) m2 s−1 from internal-wave shear dispersion Kh ~ 〈Kz〉〈〉/f2 in several studies in the stratified ocean interior, where 〈Kz〉 is the bulk average diapycnal diffusivity, 〈〉 the finescale shear variance, and f the Coriolis frequency. It is shown that, taking into account (i) the intermittency of shear-driven turbulence, (ii) its lognormality, and (iii) its correlation with unstable finescale near-inertial shear, internal-wave shear dispersion cannot necessarily be discounted based on available information. This result depends on an infrequent occurrence of turbulence bursts, as is observed, and a correlation between diapycnal diffusivity Kz and the off-diagonal vertical strain, or the vertical gradient of horizontal displacement, |χz| = |∫Vz dt|, which is not well known and may vary from region to region. Taking these factors into account, there may be no need to invoke additional submesoscale mixing mechanisms such as vortical-mode stirring or internal-wave Stokes drift to explain the previously reported discrepancies.

Corresponding author address: Eric Kunze, NorthWest Research Associates, 4126 148th Ave. NE, Redmond, WA 98052. E-mail: kunze@nwra.com

This article is included in the LatMix: Studies of Submesoscale Stirring and Mixing Special Collection.

Abstract

This paper revisits a long-standing discrepancy between (i) 1–5-km isopycnal diffusivities of O(1) m2 s−1 based on dye spreading and (ii) inferences of O(0.1) m2 s−1 from internal-wave shear dispersion Kh ~ 〈Kz〉〈〉/f2 in several studies in the stratified ocean interior, where 〈Kz〉 is the bulk average diapycnal diffusivity, 〈〉 the finescale shear variance, and f the Coriolis frequency. It is shown that, taking into account (i) the intermittency of shear-driven turbulence, (ii) its lognormality, and (iii) its correlation with unstable finescale near-inertial shear, internal-wave shear dispersion cannot necessarily be discounted based on available information. This result depends on an infrequent occurrence of turbulence bursts, as is observed, and a correlation between diapycnal diffusivity Kz and the off-diagonal vertical strain, or the vertical gradient of horizontal displacement, |χz| = |∫Vz dt|, which is not well known and may vary from region to region. Taking these factors into account, there may be no need to invoke additional submesoscale mixing mechanisms such as vortical-mode stirring or internal-wave Stokes drift to explain the previously reported discrepancies.

Corresponding author address: Eric Kunze, NorthWest Research Associates, 4126 148th Ave. NE, Redmond, WA 98052. E-mail: kunze@nwra.com

This article is included in the LatMix: Studies of Submesoscale Stirring and Mixing Special Collection.

Keywords: Circulation/ Dynamics; Mixing; Subgrid-scale processes; Physical Meteorology and Climatology; Dispersion; Mathematical and statistical techniques; Statistical techniques; Models and modeling; Parameterization; Subgrid-scale processes

1. Introduction

A number of tracer-release experiments in the ocean’s stratified interior have reported 1–5-km horizontal diffusivities Kh ~ O(1) m2 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 Kh ~ Kz|χz|2 based on Young et al. (1982), where |χz| = |∫Vz dt| and the vertical shear Vz = (uz, υz) = (∂u/∂z, ∂υ/∂z). These shear-dispersion predictions have been based on average diapycnal diffusivities 〈Kz〉, average near-inertial shear variances 〈〉, or both—that is, Kh ~ 〈〈Kz〉〉/f2 = 〈Kz〈〉〉/f2 = 〈Kz〉〈〉 ≤ 〈Kz〉4N2/(2f2) ~ O(0.1) m2 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 〈Kz|χz|2〉 = 〈Kz〉〈|χz|2〉 + 〈Kz′|χz|2′〉 will exceed 〈Kz〉〈| χz|2〉 if Kz and |χz|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(N2) and 〈 |χz|2〉 ~ O(N2/f2) but sporadically become unstable ( > 4N2) to produce nonzero turbulent diapycnal diffusivities Kz.

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

Shear dispersion in the ocean results from coupling between (i) vertically differential horizontal displacements (off-diagonal vertical strain) |χz | = |∫Vz dt| arising from vertical shear Vz = (uz, υz), and (ii) diapycnal turbulent mixing Kz (Young et al. 1982) As illustrated in Fig. 1, horizontal mixing is most effectively accomplished when diapycnal mixing coincides with maximal |χz|. This can be understood from the geometrical derivation in section 2c of Young et al. (1982). Diapycnal diffusivity Kz will not act on vertically aligned isolines—that is, θx = ∂θ/∂x (Fig. 1a)—but will act when these isolines are tilted by vertical shear Uz (Fig. 1b) to form vertical gradients—for example, θz = ∫θzt dt = −∫uzθx dt = −Xzθx (Fig. 1c), where Xz 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 Xz = ∫uz dt and Yz = ∫υz 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 Xz. Thus, the diffusive flux due to vertical diffusion can instantaneously be expressed as Kzθx, where Kz plays the role of a horizontal diffusivity, implying an average horizontal diffusivity of the form 〈Kz〉 as described in section 3a. In the Young et al. case of constant vertical diffusivity 〈Kz〉, the resulting average horizontal diffusivity is 〈Kz〉〈〉 in the oceanic limit of Kzm2 ≪ f (see below). Internal-wave shear dispersion based on average diapycnal diffusivities 〈Kz〉 and average shear variances 〈〉 = 〈〉 + 〈〉 (red lines in Fig. 2) is more completely expressed as
e1
where 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 〈Kz〉. The constraint of constant diapycnal diffusivity and shear variance will be relaxed here in section 3a.
Fig. 1.
Fig. 1.

Sequence illustrating the steps involved in internal-wave shear dispersion in a 2D vertical plane. When (a) a horizontal property gradient experiences (b) oscillating vertical shear Uz, it is (c) strained to produce vertical gradients. If (d) diapycnal mixing Kz then occurs when the off-diagonal vertical strain Xz = ∫Uz dt is maximal (c), then (e) horizontal mixing is maximized by the diapycnal dilution. At each step, the state in the preceding step is shown as dotted.

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

Fig. 2.
Fig. 2.

Schematic time series of (a) shear variance and (b) diapycnal diffusivity Kz assuming the usual application of internal-wave shear dispersion based on constant 〈〉 and 〈Kz〉 as in (4) (red), and displaying the actual intermittency of turbulence Kz and its correlation with unstable instantaneous finescale shear variance as in (11) (black). Solid and dotted curves in black correspond to 10% and 5%, respectively, intermittency of turbulent events with burst average {Kz}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 〈〉 near rolloff vertical wavenumber m = mc = 0.2π rad m−1 [corresponding to a vertical wavelength λc = 2π/mc = 10 m and a length scale Lc = 1/mc = λc/(2π) = 1.6 m for typical ocean conditions]. Thus, finescale near-inertial waves have the maximal internal-wave off-diagonal vertical strain variance 〈|χz|2〉 = 〈|Vz|2〉/f2. In (1), the second term in the denominator of the ratio in parentheses—that is 〈〉m4/f2—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 〈Kz〉 ~ 0.05 × 10−4 m2 s−1 (Gregg 1989) and mc ~ 0.2π (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

Moreover, since mc ~ E−1 (Gargett et al. 1981; Gargett 1990; Polzin et al. 1995) and 〈Kz〉 ~ E2 (Henyey et al. 1986; D’Asaro and Lien 2000), where E is the nondimensional spectral energy level of the internal-wave field, 〈Kz〉m2 is invariant so that the second term remains small compared to the first for all internal-wave-driven mixing. Finally, if 〈Kz〉m2 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 〈Kz〉m2 ≪ 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
e2
or more physically as
e3
where 〈|χz|2〉 = 〈 + 〉 = 〈 + 〉/f2 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 (Kundu and Cohen 2004). Equation (3) and Fig. 1 illustrate that, physically, dispersion is better described as coupling of diapycnal diffusion Kz and off-diagonal vertical strain |χz| = |∫Vz dt|, rather than vertical shear Vz = |Vz|. Thus, it might more aptly be referred to as “strain dispersion,” but we will use the term “shear dispersion” for consistency with the literature.
Estimates of internal-wave-driven horizontal diffusivity Kh based on (2) in the literature use either 〈Kz〉, 〈〉, or both (red lines in Fig. 2) following Young et al. (1982). This amounts to assuming that fluctuations in the turbulent diapycnal diffusivity and shear variance are uncorrelated, that is, 〈Kz′′〉 = 0. Thus, for an internal-wave field that is stable on average with 〈〉 ≤ N2/2,
e4
where δc is the critical gradient Froude number. For canonical midlatitude upper pycnocline values of 〈Kz〉 = 0.05 × 10−4 m2 s−1, N = 10−2 rad s−1, f = 10−4 rad s−1, and δc = 2, Kh based on (4) can be no more than ~0.1 m2 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 〈Kz〉 inferred from tracer and a shear spectrum based on moored current-meter measurements to infer internal-wave shear dispersion from (2) of Kh < 0.024 m2 s−1, more than a factor of 2 smaller than the 1–10-km dye lateral diffusivity of 0.07 ± 0.04 m2 s−1 14 days after release, and nearly two orders of magnitude smaller than the dye lateral diffusivity of 0.6–6 m2 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 〈Kz〉 and relative lateral displacements with depth |χz| 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 m2 s−1, with the other two releases showing closer agreement between dye and shear dispersion. Finally, shear dispersion using 〈Kz〉 was again an order-of-magnitude smaller than the 1–10-km lateral dye diffusivities over 6 days of 0.5–4 m2 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 |χz| = |∫Vz 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 Kh ~ KzN2/f2 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 N2 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 Kz, including its lognormal distribution, and its possible correlation with finescale off-diagonal components of vertical strain χz = ∫Vz 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 Vz > 2N 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

Starting from the horizontal advection–diffusion equation for a conservative tracer θ,
e5
with idealized initial condition 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 Kz as
e6
due to sporadically unstable near-inertial shear u, defined as
e7
as illustrated in Fig. 1, where the gradient Froude number δN = |Vz|/N and δc is the critical threshold value. The uz term represents the O(N) background shear and the uz′ term represents excess shear-associated turbulent bursts. Shear-driven turbulent bursts are modeled with a boxcar function Π(t − tn) of average duration δt that is nonzero every Δt = δt/r on average, that is, tn ~ nΔt. 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
e8
which, when substituted in as the advective coordinate x in the tracer initial condition, yields
e9
so that the tracer evolution proceeds as
e10
This is in the form of a diffusion equation with average horizontal diffusivity Kh given by the average of the coefficient in front of k2Θ on the right-hand side,
e11
where H(⋅) is the Heaviside function. The first term in (11) corresponds to shear dispersion associated with 〈Kz〉 and background shear variance = 〈〉 = N2/2 found by Young et al. (1982). The second term arises from excess unstable shear N2 > N2 associated with turbulent bursts. The second term can be an order of magnitude larger than the first for δN ~ (2–3)δc, 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 |χz| = |∫Vz dt|. Since (Xz, Yz) = ∫(uz, υz) dt, there is no guarantee that large strain |χz| will coincide in time with large Kz. While there are no measured correlations between |χz|2 and Kz, with some observationally based caveats noted below, it may be possible to use correlations between and Kz, that is, Kh ~ 〈Kz〉/f2 as described next. However, we emphasize that off-diagonal vertical strain |χz| 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(kUz)−1 ~ N(ω2 − f2)−1/2/N ~ (ω2 − f2)−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 |Vz| > δcN and mixing persist long enough to produce a correlation between large vertical gradients of horizontal displacement |χz| = |∫Vz dt| and Kz remains an open question that we discuss in the appendix and revisit in the summary.

c. Relation between unstable shears and turbulence

To evaluate (11), we relate instantaneous Kz to using the unstable shear parameterization proposed by Kunze et al. (1990b) and observationally supported by Peters et al. (1995), Polzin (1996), and Jurisa et al. (2015, manuscript submitted to J. Phys. Oceanogr.),
eq1
e12
(Fig. 3), where Δz is the vertical scale used to estimate shear Vz 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 ( − N2)Δz2/24 and (ii) the Kelvin–Helmholtz growth rate in the initial shear instability (|Vz| − δcN)/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 —for example, Mellor and Yamada (1982) could also be used. But finescale parameterizations like Gregg et al. (2003) or MacKinnon and Gregg (2003) are not suitable because they are based on average variances. The “instantaneous” vertical diffusivity Kz = γε/〈N2〉, 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 (|Vz| − δcN) > 0. The vertical scale Δz 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 |Vz|/N is not too large. Polzin (1996) argued that (12) was robust to changes in Δz provided that 〈〉/〈N2〉 = 〈(ΔV/Δz)2〉/〈N2〉 ~ O(1); an iterative procedure might be more effective for heterogeneous turbulence environments. A critical gradient Froude number δc = 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).
Fig. 3.
Fig. 3.

Shear-driven inferred turbulent diapycnal diffusivity Kz from (12) as a function of gradient Froude number δN = |Vz|/N assuming a critical gradient Froude number δc = 2, N = 10−2 rad s−1, Δz = 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 |Vz|, or rather reduced shear |Vz| − δcN—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 Vz (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.

While (12) suggests that unstable shear should be removed on a time scale δt ~ 4/(|Vz| − δcN) ~ 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 |χz| = |∫Vz 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 |Vz| > δcN 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,
e13
where α is the replenishment rate by nonlinear interactions (refer to the appendix) and σ = (|Vz| − δcN)/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 |χz| = |∫Vz dt|, while diapycnal turbulent mixing Kz remains elevated [(11)].

d. Statistical prediction of shear dispersion with intermittency

To account for turbulence intermittency r ∈ [0, 1], instantaneous Kz is assumed to be zero for 1 − r of the time [assuming molecular diffusivity κ to be negligible compared to 〈Kz〉 (Fig. 2)] and to have burst averages {Kz}b = 〈Kz〉/r for the r of the time it is nonzero; that is, {·}b represents the average during the turbulent bursts and Kz 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 {Kz}b = 〈Kz〉/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 Kz is nonzero, Kz is taken to have a lognormal probability density function as is observed to be approximately true (e.g., Gregg et al. 1993)
e14
with two parameters: (i) the physically meaningful ln({Kz}b) = {ln(Kz)}b + /2, where {Kz}b = 〈Kz〉/r; and (ii) the standard deviation of ln(Kz), σlnK, where = ln(1 + {Kz2}b/{Kz}b2). Shear dispersion (11) can then be reexpressed as
e15
taking advantage of 〈xy′〉 = 〈(〈x〉 + x′)y′〉 = 〈〈x〉y′〉 + 〈x′y′〉 = 〈x′y′〉 in the second rhs term and letting corr represent 〈sin2(fδt)〉 as well as ocean Kz and Vz/N not lying on the curve shown in Fig. 3 from (12). It is possible that corr is a function of Vz/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(N2/2) (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 (Kz) is obtained numerically from (12) and the PDF from (14). Choosing a standard deviation σlnK 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 σlnK = 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 σlnε or σlnχ 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-Kz tail, which is not well characterized observationally, it might easily over- or underestimate Kh. 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(Kz), and (iii) correlation corr, as explored in Fig. 4 and Table 1, which show Kh 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 Kh ~ 0.2 m2 s−1 for σlnK = 1.2 and Kh ~ 0.52 m2 s−1 for σlnK = 2.1, boosted by lognormality alone from 〈Kz〉〈〉/(2f2) = 0.18 m2 s−1 corresponding to no intermittency and no lognormality (thin solid line at r = 1). As r → 0, the inferred horizontal diffusivity Kh increases sharply. For typical low ocean intermittencies, r = 0.05–0.1, σlnK = 1.2, and corr = 1 (Fig. 4); isopycnal diffusivity Kh ~ 0.4–0.6 m2 s–1, which is in the range found by dye estimates; and Kh > 1 m2 s–1 only for r < 0.035. With σlnK = 1.2 and corr = 1 but no lognormality (thick dotted curve, Fig. 4), the horizontal diffusivity is smaller, with Kh > 1 m2 s–1 only for intermittencies r < 0.01. For higher σlnK = 2.1 and corr = 1 (Fig. 4), the predicted Kh rises above dye-inferred diffusivities for r < 0.36. For observed r = 0.05–0.1, Kh has diffusivities greater than or equal to dye estimates for σlnK > 1.8. Horizontal diffusivities Kh exceed 1 for all intermittencies r if σlnK exceeds 2.4. Finally, assuming a lower-bound correlation corr = 0.6 (Peters et al. 1995) and σlnK = 1.2 (thin solid curve, Fig. 4) produces horizontal diffusivities Kh = 0.3–0.4 m2 s−1 for intermittencies r = 0.05–0.1 and Kh > 1 m2 s−1 for intermittencies r < 0.02. We caution that, for r ≤ 0.02 (shading in Fig. 4), {Kz}b = 〈Kz〉/r is sufficiently high to violate the weak-mixing assumption used to justify the chosen z dependence in (7) because {Kz}bδt > 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 Kh derived by Young et al. (1982) no longer applies, particularly as r → 0 or large σlnK.

Fig. 4.
Fig. 4.

Horizontal diffusivity Kh from (9) as a function of turbulent intermittency r assuming Δz = 1 m in (12), bulk average diapycnal diffusivity K0 = 0.05 × 10−4 m2 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 δc = 2. Finescale parameterization [(12)] is used to relate shear Vz to diapycnal diffusivity Kz numerically. Thick solid curves correspond to Kz having a lognormal distribution [(14)] during the r fraction of time turbulence is present, with σlnK labeled along the upper axis and perfect correlation (corr = 1) between Kz and unstable shear variance > N2. The thin solid curve uses a lower-bound correlation of 0.6 for σlnK = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of σlnK. Horizontal diffusivity Kh increases as intermittency r → 0, with lognormality and increasing σlnK. Gray shading for r < 0.02 indicates where {Kz}b = 〈Kz〉/r is high enough to violate the weak-mixing approximation in (7) because {Kz}bδt > 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

Table 1.

Dependence of horizontal diffusivity Kh [(15)] on intermittency r and σlnK for corr = 1. The reported range of intermittency r is boldface in the second column. The reported dye diffusivities Kh are bold in the third column.

Table 1.

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 Kh 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 |χz| = |∫Vz dt| (Fig. 4). The exact relationship [e.g., (12)] between Kz and is secondary, provided that Kz is an increasing function of . While observations have established a clear relationship between turbulence and reduced shear (Peters et al. 1995; Polzin 1996; Jurisa et al. 2015, manuscript submitted to J. Phys. Oceanogr.), the relation between Kz and |χz|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 〈Kz〉, average variance of vertical shear 〈〉, or both have been used, essentially assuming that fluctuations in these two quantities are uncorrelated (〈Kz′′〉 = 0 so that 〈Kz〉 = 〈Kz〉〈〉), resulting in a horizontal diffusivity Kh ~ 0.1 m2 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 Kz increases with finescale would produce similar results, and this particular parameterization was chosen because it has strong observational support from finescale shear and microstructure measurements (Peters et al. 1995; Polzin 1996; Jurisa et al. 2015, manuscript submitted to J. Phys. Oceanogr.) Taken together and applied to the shear version of (15), a horizontal diffusivity Kh ~ O(1) m2 s−1 was found for σlnK = 1.2 (Gregg et al. 1993), ocean intermittencies r < 0.03, and correlations corr > 0.6. These Kh 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 Kz [e.g., (14)], and the correlation between Kz and (Fig. 4; Table 1) with the exact model relating diapycnal diffusivity Kz and shear variance being secondary.

There are two major caveats. First, the PDF of Kz in the ocean is only approximately lognormal with deviations at high Kz which will impact Kh estimates from (15). While ocean sampling is sufficient to characterize the mean 〈Kz〉, standard deviations σlnK are rarely reported, let alone sufficient sampling for higher-order quantities like shear dispersion [(15)]. Second, while Kz and finescale shear variance appear to be well correlated, it is the correlation between Kz and the variance of off-diagonal vertical strain |χz|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 Kzm2/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 Kh due to shear dispersion hinge on unstable shear persisting for O(f−1) so that the off-diagonal vertical strain |χz| = |∫Vz dt| can become large while diapycnal mixing Kz is strong [Fig. 5b; (11)]. Although the unstable-shear parameterization [(12)] implies rapid loss of unstable shears to turbulence on a time scale 4/(|Vz| − δcN) ~ O(N−1), which is too short for the creation of large vertical gradients of horizontal displacement |χz| (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 |Vz| and turbulent diapycnal mixing Kz long enough to create large |χz| ~ |∫Vz dt| coincident with large Kz. Such persistent shear and mixing events should result in a strong correlation between Kz and |χz| = |∫Vz dt|, which is an underlying assumption in our results, but this remains to be verified.

Fig. 5.
Fig. 5.

Cartoon of two scenarios for shear dispersion, where the thick dotted curves represent unstable reduced shear |Vz| − 2N, thick solid curves diapycnal diffusivity Kz, and thin solid curves the vertical gradient of horizontal displacement |χz| = |∫Vz dt|. (a) The unstable shear and mixing decay over a buoyancy time scale δt ~ N−1 so that |χz| remains small. (b) Unstable shear and mixing persist over an inertial time scale δt ~ f−1, allowing |χz| 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 Kz and finescale off-diagonal vertical strain |χz| = |∫Vz 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 〈Kz〉〈〉/f2 will better reproduce true isopycnal mixing on the submesoscale. Likewise, traditional shear dispersion is expected to hold where Kz and |χz|2 are uncorrelated or unstable shears do not persist long enough to produce significant |χz| = |∫Vz 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 〈Kz〉 and average shear variances 〈〉, but also on the time dependences and interdependences of these quantities. Our model horizontal diffusivities Kh are sensitive to the assumed lognormal probability distribution for Kz, which may not be a good description of ocean turbulence at the high Kz weighted by (15) (Gregg et al. 1993; Davis 1996). For these reasons, only observational assessment of the distribution of Kz and its correlation with the off-diagonal vertical strain |χz| = |∫Vz 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

Nonlinear wave–wave or wave–mean flow interactions can maintain unstable shear > N2 by increasing the vertical wavenumber m or increasing the wave horizontal kinetic energy HKE,
ea1
where ray-tracing theory (Olbers 1981) suggests that
ea2
and
ea3
for finescale near-inertial waves. This can be rewritten as
ea4
where the vertical group velocity is
ea5
and its derivative
ea6
so that
ea7
Combining (A1), (A2) and (A7)
ea8
with a growth rate of O(f). We caution that the linear hydrostatic dispersion relation was used in conversion from horizontal wavenumber k and vertical group velocity Cgz 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
ea9
(Fig. A1). The curve is nearly identical if internal-wave dynamics is neglected, that is, only including the kinematics of 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 < 2f), a balance requires unstable shears δN = |Vz|/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.).
Fig. A1.
Fig. A1.

Buoyancy-frequency-normalized shear |Vz|/N vs Coriolis-frequency-normalized frequency ω/f to achieve a balance (solid curve) between shear variance amplification (4ω2 + f2)(ω2 − f2)1/2|Vz|/(Nω2) [(A8)] and loss to turbulence (|Vz| − 2N)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (ω2 − f2)1/2|Vz|/N vs (|Vz| − 2N)/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 δc = |Vz|/N = 2.

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

REFERENCES

  • Baker, M. A., and C. H. Gibson, 1987: Sampling turbulence in the stratified ocean: Statistical consequences of strong intermittency. J. Phys. Oceanogr., 17, 1817–1836, doi:10.1175/1520-0485(1987)017<1817:STITSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Brunner-Suzuki, A.-M. E. G., M. A. Sundermeyer, and M.-P. Lelong, 2014: Upscale energy transfer induced by the vortical mode and internal waves. J. Phys. Oceanogr., 44, 2446–2469, doi:10.1175/JPO-D-12-0149.1.

    • Search Google Scholar
    • Export Citation

  • Bühler, O., N. Grisouard, and M. Holmes-Certon, 2013: Strong particle dispersion by weakly dissipative random internal waves. J. Fluid Mech., 719, R4, doi:10.1017/jfm.2013.71.

    • Search Google Scholar
    • Export Citation

  • Cole, S. T., and D. L. Rudnick, 2012: The spatial distribution and annual cycle of upper ocean thermohaline structure. J. Geophys. Res., 117, C02027, doi:10.1029/2011JC007033.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., and R.-C. Lien, 2000: The wave–turbulence transition for stratified flows. J. Phys. Oceanogr., 30, 1669–1678, doi:10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Davis, R. E., 1996: Sampling turbulent dissipation. J. Phys. Oceanogr., 26, 341–358, doi:10.1175/1520-0485(1996)026<0341:STD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Desaubies, Y., and W. K. Smith, 1982: Statistics of Richardson number and instability in oceanic internal waves. J. Phys. Oceanogr., 12, 1245–1259, doi:10.1175/1520-0485(1982)012<1245:SORNAI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov scales. J. Geophys. Res., 87, 9601–9613, doi:10.1029/JC087iC12p09601.

    • Search Google Scholar
    • Export Citation

  • Eriksen, C. C., 1978: Measurements and models of finestructure, internal gravity waves and wave breaking in the deep ocean. J. Geophys. Res., 83, 2989–3009, doi:10.1029/JC083iC06p02989.

    • Search Google Scholar
    • Export Citation

  • Gargett, A. E., 1990: Do we really know how to scale the turbulent kinetic energy dissipation rate ε due to breaking of oceanic internal waves? J. Geophys. Res., 95, 15 971–15 974, doi:10.1029/JC095iC09p15971.

    • Search Google Scholar
    • Export Citation

  • Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. J. Williams III, 1981: A composite spectrum of vertical shear in the upper ocean. J. Phys. Oceanogr., 11, 1258–1271, doi:10.1175/1520-0485(1981)011<1258:ACSOVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94, 9686–9698, doi:10.1029/JC094iC07p09686.

  • Gregg, M. C., and T. B. Sanford, 1988: The dependence of turbulent dissipation on stratification in a diffusively stable thermocline. J. Geophys. Res., 93, 12 381–12 392, doi:10.1029/JC093iC10p12381.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., E. A. D’Asaro, T. J. Shay, and N. Larson, 1986: Observations of persistent mixing and near-inertial internal waves. J. Phys. Oceanogr., 16, 856–885, doi:10.1175/1520-0485(1986)016<0856:OOPMAN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., H. E. Seim, and D. B. Percival, 1993: Statistics of shear and turbulent dissipation profiles in random internal wave fields. J. Phys. Oceanogr., 23, 1777–1799, doi:10.1175/1520-0485(1993)023<1777:SOSATD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial waters. Nature, 422, 513–515, doi:10.1038/nature01507.

    • Search Google Scholar
    • Export Citation

  • Hazel, P., 1972: Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech., 51, 39–61, doi:10.1017/S0022112072001065.

    • Search Google Scholar
    • Export Citation

  • Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91, 8487–8495, doi:10.1029/JC091iC07p08487.

    • Search Google Scholar
    • Export Citation

  • Inall, M. E., D. Aleynik, and C. Neil, 2013: Horizontal advection and dispersion in a stratified sea: The role of inertial oscillations. Prog. Oceanogr., 117, 25–36, doi:10.1016/j.pocean.2013.06.008.

    • Search Google Scholar
    • Export Citation

  • Itsweire, E. C., T. R. Osborn, and T. P. Stanton, 1989: Horizontal distribution and characteristics of shear layers in the seasonal thermocline. J. Phys. Oceanogr., 19, 301–320, doi:10.1175/1520-0485(1989)019<0301:HDACOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Itsweire, E. C., J. R. Koseff, D. A. Briggs, and J. H. Ferziger, 1993: Turbulence in stratified shear flows: Implications for interpreting shear-induced mixing in the ocean. J. Phys. Oceanogr., 23, 1508–1522, doi:10.1175/1520-0485(1993)023<1508:TISSFI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kundu, P. K., and I. M. Cohen, 2004: Fluid Mechanics. Elsevier, 759 pp.

  • Kunze, E., 1993: Submesocale dynamics near a seamount. Part II: The partition of energy between internal waves and geostrophy. J. Phys. Oceanogr., 23, 2589–2601, doi:10.1175/1520-0485(1993)023<2589:SDNASP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., 2014: The relation between unstable shear layer thicknesses and turbulence lengthscales. J. Mar. Res., 72, 95–104, doi:10.1357/002224014813758977.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., and T. B. Sanford, 1993: Submesoscale dynamics near a seamount. Part I: Measurements of Ertel vorticity. J. Phys. Oceanogr., 23, 2567–2588, doi:10.1175/1520-0485(1993)023<2567:SDNASP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., and J. M. Toole, 1997: Tidally driven vorticity, diurnal shear, and turbulence atop Fieberling Seamount. J. Phys. Oceanogr., 27, 2663–2693, doi:10.1175/1520-0485(1997)027<2663:TDVDSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., M. G. Briscoe, and A. J. Williams III, 1990a: Interpreting shear and strain finestructure from a neutrally buoyant float. J. Geophys. Res., 95, 18 111–18 125, doi:10.1029/JC095iC10p18111.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., A. J. Williams III, and M. G. Briscoe, 1990b: Observations of shear and vertical stability from a neutrally buoyant float. J. Geophys. Res., 95, 18 127–18 142, doi:10.1029/JC095iC10p18127.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., J. M. Klymak, R.-C. Lien, R. Ferrari, C. M. Lee, M. A. Sundermeyer, and L. Goodman, 2015: Submesoscale water-mass spectra in the Sargasso Sea. J. Phys. Oceanogr., 45, 1325–1338, doi:10.1175/JPO-D-14-0108.1.

    • Search Google Scholar
    • Export Citation

  • Ledwell, J. R., A. J. Watson, and C. S. Law, 1998: Mixing of a tracer in the pycnocline. J. Geophys. Res., 103, 21 499–21 529, doi:10.1029/98JC01738.

    • Search Google Scholar
    • Export Citation

  • Lelong, M.-P., and M. A. Sundermeyer, 2005: Geostrophic adjustment of an isolated diapycnal mixing event and its implications for small-scale lateral dispersion. J. Phys. Oceanogr., 35, 2352–2367, doi:10.1175/JPO2835.1.

    • Search Google Scholar
    • Export Citation

  • Lelong, M.-P., M. A. Sundermeyer, and E. Kunze, 2002: Numerical simulations of lateral dispersion by the relaxation of diapycnal mixing events. Eos, Trans. Amer. Geophys. Union, 83 (Meeting Suppl.), Abstract OS31O-08.

  • Lumpkin, R., and S. Elipot, 2010: Surface drifter pair spreading in the North Atlantic. J. Geophys. Res., 115, C12017, doi:10.1029/2010JC006338.

    • Search Google Scholar
    • Export Citation

  • MacKinnon, J. A., and M. C. Gregg, 2003: Mixing on the late-summer New England shelf—Solibores, shear, and stratification. J. Phys. Oceanogr., 33, 1476–1492, doi:10.1175/1520-0485(2003)033<1476:MOTLNE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marmorino, G. O., J. P. Dugan, and T. E. Evans, 1986: Horizontal variability of microstructure in the vicinity of a Sargasso Sea front. J. Phys. Oceanogr., 16, 967–986, doi:10.1175/1520-0485(1986)016<0967:HVOMIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marmorino, G. O., L. J. Rosenblum, and C. L. Trump, 1987: Finescale temperature variability: The influence of near-inertial waves. J. Geophys. Res., 92, 13 049–13 069, doi:10.1029/JC092iC12p13049.

    • Search Google Scholar
    • Export Citation

  • Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys., 20, 851–875, doi:10.1029/RG020i004p00851.

    • Search Google Scholar
    • Export Citation

  • Molemaker, M. J., J. C. McWilliams, and X. Capet, 2010: Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech., 654, 35–63, doi:10.1017/S0022112009993272.

    • Search Google Scholar
    • Export Citation

  • Moniz, R. J., D. A. Fong, C. B. Woodson, S. K. Willis, M. T. Stacey, and S. G. Monismith, 2014: Scale-dependent dispersion within the stratified interior on the shelf of northern Monterey Bay. J. Phys. Oceanogr., 44, 1049–1064, doi:10.1175/JPO-D-12-0229.1.

    • Search Google Scholar
    • Export Citation

  • Moum, J. N., 1996: Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res., 101, 14 095–14 109, doi:10.1029/96JC00507.

    • Search Google Scholar
    • Export Citation

  • Müller, P., 1984: Small-scale vortical motions. Internal Gravity Waves and Small Scale Turbulence: Proc. ‘Aha Huliko‘a Hawaiian Winter Workshop, Honolulu, HI, University of Hawai‘i at Mānoa, 249–262.

  • Müller, P., R.-C. Lien, and R. Williams, 1988: Estimates of potential vorticity at small scales in the ocean. J. Phys. Oceanogr., 18, 401–416, doi:10.1175/1520-0485(1988)018<0401:EOPVAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Olbers, D. J., 1981: The propagation of internal waves in a geostrophic current. J. Phys. Oceanogr., 11, 1224–1233, doi:10.1175/1520-0485(1981)011<1224:TPOIWI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., 1980: Estimates of the local rate of diffusion form dissipation measurements. J. Phys. Oceanogr., 10, 83–89, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Palmer, M. R., M. E. Inall, and J. Sharples, 2013: The physical oceanography of Jones Bank: A mixing hotspot in the Celtic Sea. Prog. Oceanogr., 117, 9–24, doi:10.1016/j.pocean.2013.06.009.

    • Search Google Scholar
    • Export Citation

  • Peters, H., M. C. Gregg, and T. B. Sanford, 1995: On the parameterization of equatorial turbulence: Effect of finescale variations below the range of the diurnal cycle. J. Geophys. Res., 100, 18 333–18 348, doi:10.1029/95JC01513.

    • Search Google Scholar
    • Export Citation

  • Pinkel, R., 2014: Vortical and internal wave shear and strain. J. Phys. Oceanogr., 44, 2070–2092, doi:10.1175/JPO-D-13-090.1.

  • Poje, A. C., and Coauthors, 2014: Submesoscale dispersion in the vicinity of the Deepwater Horizon spill. Proc. Natl. Acad. Sci. USA, 111, 12 693–12 698, doi:10.1073/pnas.1402452111.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., 1996: Statistics of the Richardson number: Mixing models and finestructure. J. Phys. Oceanogr., 26, 1409–1425, doi:10.1175/1520-0485(1996)026<1409:SOTRNM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterization of turbulent dissipation. J. Phys. Oceanogr., 25, 306–328, doi:10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., E. Kunze, J. M. Toole, and R. W. Schmitt, 2003: The partition of finescale energy into internal waves and subinertial motions. J. Phys. Oceanogr., 33, 234–248, doi:10.1175/1520-0485(2003)033<0234:TPOFEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Rohr, J. J., E. C. Itsweire, K. N. Helland, and C. W. Van Atta, 1988: Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech., 195, 77–111, doi:10.1017/S0022112088002332.

    • Search Google Scholar
    • Export Citation

  • Rudnick, D., and R. Ferrari, 1999: Compensation of horizontal temperature and salinity gradients in the ocean mixed layer. Science, 283, 526–529, doi:10.1126/science.283.5401.526.

    • Search Google Scholar
    • Export Citation

  • Scotti, R. S., and G. M. Corcos, 1972: An experiment on the stability of small disturbances in a stratified free-shear layer. J. Fluid Mech., 52, 499–528, doi:10.1017/S0022112072001569.

    • Search Google Scholar
    • Export Citation

  • Shcherbina, A. Y., and Coauthors, 2015: The LatMix summer campaign: Submesoscale stirring in the upper ocean. Bull. Amer. Meteor. Soc., 96, 1257–1279, doi:10.1175/BAMS-D-14-00015.1.

    • Search Google Scholar
    • Export Citation

  • Smith, S., and R. Ferrari, 2009: The production and dissipation of compensated thermohaline variance by mesoscale stiriring. J. Phys. Oceanogr., 39, 2477–2501, doi:10.1175/2009JPO4103.1.

    • Search Google Scholar
    • Export Citation

  • Sun, H., and E. Kunze, 1999: Internal wave–wave interactions. Part I: The role of internal wave vertical divergence. J. Phys. Oceanogr., 29, 2886–2904, doi:10.1175/1520-0485(1999)029<2886:IWWIPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., and J. R. Ledwell, 2001: Lateral dispersion over the continental shelf: Analysis of dye-release experiments. J. Geophys. Res., 106, 9603–9621, doi:10.1029/2000JC900138.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., and M.-P. Lelong, 2005: Numerical simulations of lateral dispersion by the relaxation of diapycnal mixing events. J. Phys. Oceanogr., 35, 2368–2386, doi:10.1175/JPO2834.1.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., J. R. Ledwell, N. S. Oakey, and B. J. W. Greenan, 2005: Stirring by small-scale vortices caused by patchy mixing. J. Phys. Oceanogr., 35, 1245–1262, doi:10.1175/JPO2713.1.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., 1971: Experiments on the instability of stratified shear flows: Miscible flows. J. Fluid Mech., 46, 299–319, doi:10.1017/S0022112071000557.

    • Search Google Scholar
    • Export Citation

  • Toole, J. M., and R. W. Schmitt, 1987: Small-scale structures in the north-west Atlantic sub-tropical front. Nature, 327, 47–49, doi:10.1038/327047a0.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., P. B. Rhines, and C. J. R. Garrett, 1982: Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr., 12, 515–527, doi:10.1175/1520-0485(1982)012<0515:SFDIWA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
1

This term plays a role in limiting horizontal mixing for subinertial (ω ≪ f, e.g., vortical mode) shears to 〈〉/(Kzm4) ~ N2δg2/(Kzm4), where δg is the geostrophic gradient Froude number. For the abovementioned values of Kz and m, and vortical-mode gradient Froude number δg ≪ 1 as reported by Pinkel (2014), subinertial shear dispersion is much less than 1 m2 s−1.

Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Baker, M. A., and C. H. Gibson, 1987: Sampling turbulence in the stratified ocean: Statistical consequences of strong intermittency. J. Phys. Oceanogr., 17, 1817–1836, doi:10.1175/1520-0485(1987)017<1817:STITSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Brunner-Suzuki, A.-M. E. G., M. A. Sundermeyer, and M.-P. Lelong, 2014: Upscale energy transfer induced by the vortical mode and internal waves. J. Phys. Oceanogr., 44, 2446–2469, doi:10.1175/JPO-D-12-0149.1.

    • Search Google Scholar
    • Export Citation

  • Bühler, O., N. Grisouard, and M. Holmes-Certon, 2013: Strong particle dispersion by weakly dissipative random internal waves. J. Fluid Mech., 719, R4, doi:10.1017/jfm.2013.71.

    • Search Google Scholar
    • Export Citation

  • Cole, S. T., and D. L. Rudnick, 2012: The spatial distribution and annual cycle of upper ocean thermohaline structure. J. Geophys. Res., 117, C02027, doi:10.1029/2011JC007033.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., and R.-C. Lien, 2000: The wave–turbulence transition for stratified flows. J. Phys. Oceanogr., 30, 1669–1678, doi:10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Davis, R. E., 1996: Sampling turbulent dissipation. J. Phys. Oceanogr., 26, 341–358, doi:10.1175/1520-0485(1996)026<0341:STD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Desaubies, Y., and W. K. Smith, 1982: Statistics of Richardson number and instability in oceanic internal waves. J. Phys. Oceanogr., 12, 1245–1259, doi:10.1175/1520-0485(1982)012<1245:SORNAI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov scales. J. Geophys. Res., 87, 9601–9613, doi:10.1029/JC087iC12p09601.

    • Search Google Scholar
    • Export Citation

  • Eriksen, C. C., 1978: Measurements and models of finestructure, internal gravity waves and wave breaking in the deep ocean. J. Geophys. Res., 83, 2989–3009, doi:10.1029/JC083iC06p02989.

    • Search Google Scholar
    • Export Citation

  • Gargett, A. E., 1990: Do we really know how to scale the turbulent kinetic energy dissipation rate ε due to breaking of oceanic internal waves? J. Geophys. Res., 95, 15 971–15 974, doi:10.1029/JC095iC09p15971.

    • Search Google Scholar
    • Export Citation

  • Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. J. Williams III, 1981: A composite spectrum of vertical shear in the upper ocean. J. Phys. Oceanogr., 11, 1258–1271, doi:10.1175/1520-0485(1981)011<1258:ACSOVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94, 9686–9698, doi:10.1029/JC094iC07p09686.

  • Gregg, M. C., and T. B. Sanford, 1988: The dependence of turbulent dissipation on stratification in a diffusively stable thermocline. J. Geophys. Res., 93, 12 381–12 392, doi:10.1029/JC093iC10p12381.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., E. A. D’Asaro, T. J. Shay, and N. Larson, 1986: Observations of persistent mixing and near-inertial internal waves. J. Phys. Oceanogr., 16, 856–885, doi:10.1175/1520-0485(1986)016<0856:OOPMAN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., H. E. Seim, and D. B. Percival, 1993: Statistics of shear and turbulent dissipation profiles in random internal wave fields. J. Phys. Oceanogr., 23, 1777–1799, doi:10.1175/1520-0485(1993)023<1777:SOSATD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial waters. Nature, 422, 513–515, doi:10.1038/nature01507.

    • Search Google Scholar
    • Export Citation

  • Hazel, P., 1972: Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech., 51, 39–61, doi:10.1017/S0022112072001065.

    • Search Google Scholar
    • Export Citation

  • Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91, 8487–8495, doi:10.1029/JC091iC07p08487.

    • Search Google Scholar
    • Export Citation

  • Inall, M. E., D. Aleynik, and C. Neil, 2013: Horizontal advection and dispersion in a stratified sea: The role of inertial oscillations. Prog. Oceanogr., 117, 25–36, doi:10.1016/j.pocean.2013.06.008.

    • Search Google Scholar
    • Export Citation

  • Itsweire, E. C., T. R. Osborn, and T. P. Stanton, 1989: Horizontal distribution and characteristics of shear layers in the seasonal thermocline. J. Phys. Oceanogr., 19, 301–320, doi:10.1175/1520-0485(1989)019<0301:HDACOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Itsweire, E. C., J. R. Koseff, D. A. Briggs, and J. H. Ferziger, 1993: Turbulence in stratified shear flows: Implications for interpreting shear-induced mixing in the ocean. J. Phys. Oceanogr., 23, 1508–1522, doi:10.1175/1520-0485(1993)023<1508:TISSFI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kundu, P. K., and I. M. Cohen, 2004: Fluid Mechanics. Elsevier, 759 pp.

  • Kunze, E., 1993: Submesocale dynamics near a seamount. Part II: The partition of energy between internal waves and geostrophy. J. Phys. Oceanogr., 23, 2589–2601, doi:10.1175/1520-0485(1993)023<2589:SDNASP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., 2014: The relation between unstable shear layer thicknesses and turbulence lengthscales. J. Mar. Res., 72, 95–104, doi:10.1357/002224014813758977.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., and T. B. Sanford, 1993: Submesoscale dynamics near a seamount. Part I: Measurements of Ertel vorticity. J. Phys. Oceanogr., 23, 2567–2588, doi:10.1175/1520-0485(1993)023<2567:SDNASP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., and J. M. Toole, 1997: Tidally driven vorticity, diurnal shear, and turbulence atop Fieberling Seamount. J. Phys. Oceanogr., 27, 2663–2693, doi:10.1175/1520-0485(1997)027<2663:TDVDSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., M. G. Briscoe, and A. J. Williams III, 1990a: Interpreting shear and strain finestructure from a neutrally buoyant float. J. Geophys. Res., 95, 18 111–18 125, doi:10.1029/JC095iC10p18111.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., A. J. Williams III, and M. G. Briscoe, 1990b: Observations of shear and vertical stability from a neutrally buoyant float. J. Geophys. Res., 95, 18 127–18 142, doi:10.1029/JC095iC10p18127.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., J. M. Klymak, R.-C. Lien, R. Ferrari, C. M. Lee, M. A. Sundermeyer, and L. Goodman, 2015: Submesoscale water-mass spectra in the Sargasso Sea. J. Phys. Oceanogr., 45, 1325–1338, doi:10.1175/JPO-D-14-0108.1.

    • Search Google Scholar
    • Export Citation

  • Ledwell, J. R., A. J. Watson, and C. S. Law, 1998: Mixing of a tracer in the pycnocline. J. Geophys. Res., 103, 21 499–21 529, doi:10.1029/98JC01738.

    • Search Google Scholar
    • Export Citation

  • Lelong, M.-P., and M. A. Sundermeyer, 2005: Geostrophic adjustment of an isolated diapycnal mixing event and its implications for small-scale lateral dispersion. J. Phys. Oceanogr., 35, 2352–2367, doi:10.1175/JPO2835.1.

    • Search Google Scholar
    • Export Citation

  • Lelong, M.-P., M. A. Sundermeyer, and E. Kunze, 2002: Numerical simulations of lateral dispersion by the relaxation of diapycnal mixing events. Eos, Trans. Amer. Geophys. Union, 83 (Meeting Suppl.), Abstract OS31O-08.

  • Lumpkin, R., and S. Elipot, 2010: Surface drifter pair spreading in the North Atlantic. J. Geophys. Res., 115, C12017, doi:10.1029/2010JC006338.

    • Search Google Scholar
    • Export Citation

  • MacKinnon, J. A., and M. C. Gregg, 2003: Mixing on the late-summer New England shelf—Solibores, shear, and stratification. J. Phys. Oceanogr., 33, 1476–1492, doi:10.1175/1520-0485(2003)033<1476:MOTLNE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marmorino, G. O., J. P. Dugan, and T. E. Evans, 1986: Horizontal variability of microstructure in the vicinity of a Sargasso Sea front. J. Phys. Oceanogr., 16, 967–986, doi:10.1175/1520-0485(1986)016<0967:HVOMIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marmorino, G. O., L. J. Rosenblum, and C. L. Trump, 1987: Finescale temperature variability: The influence of near-inertial waves. J. Geophys. Res., 92, 13 049–13 069, doi:10.1029/JC092iC12p13049.

    • Search Google Scholar
    • Export Citation

  • Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys., 20, 851–875, doi:10.1029/RG020i004p00851.

    • Search Google Scholar
    • Export Citation

  • Molemaker, M. J., J. C. McWilliams, and X. Capet, 2010: Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech., 654, 35–63, doi:10.1017/S0022112009993272.

    • Search Google Scholar
    • Export Citation

  • Moniz, R. J., D. A. Fong, C. B. Woodson, S. K. Willis, M. T. Stacey, and S. G. Monismith, 2014: Scale-dependent dispersion within the stratified interior on the shelf of northern Monterey Bay. J. Phys. Oceanogr., 44, 1049–1064, doi:10.1175/JPO-D-12-0229.1.

    • Search Google Scholar
    • Export Citation

  • Moum, J. N., 1996: Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res., 101, 14 095–14 109, doi:10.1029/96JC00507.

    • Search Google Scholar
    • Export Citation

  • Müller, P., 1984: Small-scale vortical motions. Internal Gravity Waves and Small Scale Turbulence: Proc. ‘Aha Huliko‘a Hawaiian Winter Workshop, Honolulu, HI, University of Hawai‘i at Mānoa, 249–262.

  • Müller, P., R.-C. Lien, and R. Williams, 1988: Estimates of potential vorticity at small scales in the ocean. J. Phys. Oceanogr., 18, 401–416, doi:10.1175/1520-0485(1988)018<0401:EOPVAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Olbers, D. J., 1981: The propagation of internal waves in a geostrophic current. J. Phys. Oceanogr., 11, 1224–1233, doi:10.1175/1520-0485(1981)011<1224:TPOIWI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., 1980: Estimates of the local rate of diffusion form dissipation measurements. J. Phys. Oceanogr., 10, 83–89, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Palmer, M. R., M. E. Inall, and J. Sharples, 2013: The physical oceanography of Jones Bank: A mixing hotspot in the Celtic Sea. Prog. Oceanogr., 117, 9–24, doi:10.1016/j.pocean.2013.06.009.

    • Search Google Scholar
    • Export Citation

  • Peters, H., M. C. Gregg, and T. B. Sanford, 1995: On the parameterization of equatorial turbulence: Effect of finescale variations below the range of the diurnal cycle. J. Geophys. Res., 100, 18 333–18 348, doi:10.1029/95JC01513.

    • Search Google Scholar
    • Export Citation

  • Pinkel, R., 2014: Vortical and internal wave shear and strain. J. Phys. Oceanogr., 44, 2070–2092, doi:10.1175/JPO-D-13-090.1.

  • Poje, A. C., and Coauthors, 2014: Submesoscale dispersion in the vicinity of the Deepwater Horizon spill. Proc. Natl. Acad. Sci. USA, 111, 12 693–12 698, doi:10.1073/pnas.1402452111.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., 1996: Statistics of the Richardson number: Mixing models and finestructure. J. Phys. Oceanogr., 26, 1409–1425, doi:10.1175/1520-0485(1996)026<1409:SOTRNM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterization of turbulent dissipation. J. Phys. Oceanogr., 25, 306–328, doi:10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., E. Kunze, J. M. Toole, and R. W. Schmitt, 2003: The partition of finescale energy into internal waves and subinertial motions. J. Phys. Oceanogr., 33, 234–248, doi:10.1175/1520-0485(2003)033<0234:TPOFEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Rohr, J. J., E. C. Itsweire, K. N. Helland, and C. W. Van Atta, 1988: Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech., 195, 77–111, doi:10.1017/S0022112088002332.

    • Search Google Scholar
    • Export Citation

  • Rudnick, D., and R. Ferrari, 1999: Compensation of horizontal temperature and salinity gradients in the ocean mixed layer. Science, 283, 526–529, doi:10.1126/science.283.5401.526.

    • Search Google Scholar
    • Export Citation

  • Scotti, R. S., and G. M. Corcos, 1972: An experiment on the stability of small disturbances in a stratified free-shear layer. J. Fluid Mech., 52, 499–528, doi:10.1017/S0022112072001569.

    • Search Google Scholar
    • Export Citation

  • Shcherbina, A. Y., and Coauthors, 2015: The LatMix summer campaign: Submesoscale stirring in the upper ocean. Bull. Amer. Meteor. Soc., 96, 1257–1279, doi:10.1175/BAMS-D-14-00015.1.

    • Search Google Scholar
    • Export Citation

  • Smith, S., and R. Ferrari, 2009: The production and dissipation of compensated thermohaline variance by mesoscale stiriring. J. Phys. Oceanogr., 39, 2477–2501, doi:10.1175/2009JPO4103.1.

    • Search Google Scholar
    • Export Citation

  • Sun, H., and E. Kunze, 1999: Internal wave–wave interactions. Part I: The role of internal wave vertical divergence. J. Phys. Oceanogr., 29, 2886–2904, doi:10.1175/1520-0485(1999)029<2886:IWWIPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., and J. R. Ledwell, 2001: Lateral dispersion over the continental shelf: Analysis of dye-release experiments. J. Geophys. Res., 106, 9603–9621, doi:10.1029/2000JC900138.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., and M.-P. Lelong, 2005: Numerical simulations of lateral dispersion by the relaxation of diapycnal mixing events. J. Phys. Oceanogr., 35, 2368–2386, doi:10.1175/JPO2834.1.

    • Search Google Scholar
    • Export Citation

  • Sundermeyer, M. A., J. R. Ledwell, N. S. Oakey, and B. J. W. Greenan, 2005: Stirring by small-scale vortices caused by patchy mixing. J. Phys. Oceanogr., 35, 1245–1262, doi:10.1175/JPO2713.1.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., 1971: Experiments on the instability of stratified shear flows: Miscible flows. J. Fluid Mech., 46, 299–319, doi:10.1017/S0022112071000557.

    • Search Google Scholar
    • Export Citation

  • Toole, J. M., and R. W. Schmitt, 1987: Small-scale structures in the north-west Atlantic sub-tropical front. Nature, 327, 47–49, doi:10.1038/327047a0.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., P. B. Rhines, and C. J. R. Garrett, 1982: Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr., 12, 515–527, doi:10.1175/1520-0485(1982)012<0515:SFDIWA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Sequence illustrating the steps involved in internal-wave shear dispersion in a 2D vertical plane. When (a) a horizontal property gradient experiences (b) oscillating vertical shear Uz, it is (c) strained to produce vertical gradients. If (d) diapycnal mixing Kz then occurs when the off-diagonal vertical strain Xz = ∫Uz dt is maximal (c), then (e) horizontal mixing is maximized by the diapycnal dilution. At each step, the state in the preceding step is shown as dotted.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Schematic time series of (a) shear variance and (b) diapycnal diffusivity Kz assuming the usual application of internal-wave shear dispersion based on constant 〈〉 and 〈Kz〉 as in (4) (red), and displaying the actual intermittency of turbulence Kz and its correlation with unstable instantaneous finescale shear variance as in (11) (black). Solid and dotted curves in black correspond to 10% and 5%, respectively, intermittency of turbulent events with burst average {Kz}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.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Shear-driven inferred turbulent diapycnal diffusivity Kz from (12) as a function of gradient Froude number δN = |Vz|/N assuming a critical gradient Froude number δc = 2, N = 10−2 rad s−1, Δz = 1 m, and γ = 0. 2.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Horizontal diffusivity Kh from (9) as a function of turbulent intermittency r assuming Δz = 1 m in (12), bulk average diapycnal diffusivity K0 = 0.05 × 10−4 m2 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 δc = 2. Finescale parameterization [(12)] is used to relate shear Vz to diapycnal diffusivity Kz numerically. Thick solid curves correspond to Kz having a lognormal distribution [(14)] during the r fraction of time turbulence is present, with σlnK labeled along the upper axis and perfect correlation (corr = 1) between Kz and unstable shear variance > N2. The thin solid curve uses a lower-bound correlation of 0.6 for σlnK = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of σlnK. Horizontal diffusivity Kh increases as intermittency r → 0, with lognormality and increasing σlnK. Gray shading for r < 0.02 indicates where {Kz}b = 〈Kz〉/r is high enough to violate the weak-mixing approximation in (7) because {Kz}bδt > m−2 for durations δt ~ f−1, but the impact of this remains to be assessed.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

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

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Buoyancy-frequency-normalized shear |Vz|/N vs Coriolis-frequency-normalized frequency ω/f to achieve a balance (solid curve) between shear variance amplification (4ω2 + f2)(ω2 − f2)1/2|Vz|/(Nω2) [(A8)] and loss to turbulence (|Vz| − 2N)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (ω2 − f2)1/2|Vz|/N vs (|Vz| − 2N)/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 δc = |Vz|/N = 2.

Fig. 1.

Sequence illustrating the steps involved in internal-wave shear dispersion in a 2D vertical plane. When (a) a horizontal property gradient experiences (b) oscillating vertical shear Uz, it is (c) strained to produce vertical gradients. If (d) diapycnal mixing Kz then occurs when the off-diagonal vertical strain Xz = ∫Uz dt is maximal (c), then (e) horizontal mixing is maximized by the diapycnal dilution. At each step, the state in the preceding step is shown as dotted.
Fig. 2.

Schematic time series of (a) shear variance and (b) diapycnal diffusivity Kz assuming the usual application of internal-wave shear dispersion based on constant 〈〉 and 〈Kz〉 as in (4) (red), and displaying the actual intermittency of turbulence Kz and its correlation with unstable instantaneous finescale shear variance as in (11) (black). Solid and dotted curves in black correspond to 10% and 5%, respectively, intermittency of turbulent events with burst average {Kz}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.
Fig. 3.

Shear-driven inferred turbulent diapycnal diffusivity Kz from (12) as a function of gradient Froude number δN = |Vz|/N assuming a critical gradient Froude number δc = 2, N = 10−2 rad s−1, Δz = 1 m, and γ = 0. 2.
Fig. 4.

Horizontal diffusivity Kh from (9) as a function of turbulent intermittency r assuming Δz = 1 m in (12), bulk average diapycnal diffusivity K0 = 0.05 × 10−4 m2 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 δc = 2. Finescale parameterization [(12)] is used to relate shear Vz to diapycnal diffusivity Kz numerically. Thick solid curves correspond to Kz having a lognormal distribution [(14)] during the r fraction of time turbulence is present, with σlnK labeled along the upper axis and perfect correlation (corr = 1) between Kz and unstable shear variance > N2. The thin solid curve uses a lower-bound correlation of 0.6 for σlnK = 1.2 (Peters et al. 1995). The thick dotted curve assumes intermittency and perfect correlation but no lognormality, so it is independent of σlnK. Horizontal diffusivity Kh increases as intermittency r → 0, with lognormality and increasing σlnK. Gray shading for r < 0.02 indicates where {Kz}b = 〈Kz〉/r is high enough to violate the weak-mixing approximation in (7) because {Kz}bδt > m−2 for durations δt ~ f−1, but the impact of this remains to be assessed.
Fig. 5.

Cartoon of two scenarios for shear dispersion, where the thick dotted curves represent unstable reduced shear |Vz| − 2N, thick solid curves diapycnal diffusivity Kz, and thin solid curves the vertical gradient of horizontal displacement |χz| = |∫Vz dt|. (a) The unstable shear and mixing decay over a buoyancy time scale δt ~ N−1 so that |χz| remains small. (b) Unstable shear and mixing persist over an inertial time scale δt ~ f−1, allowing |χz| to become large in conjunction with strong mixing.
Fig. A1.

Buoyancy-frequency-normalized shear |Vz|/N vs Coriolis-frequency-normalized frequency ω/f to achieve a balance (solid curve) between shear variance amplification (4ω2 + f2)(ω2 − f2)1/2|Vz|/(Nω2) [(A8)] and loss to turbulence (|Vz| − 2N)/4 [(12)] as shown by (A9). The thick dotted curve shows the balance for only vertical shearing effects (ω2 − f2)1/2|Vz|/N vs (|Vz| − 2N)/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 δc = |Vz|/N = 2.
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 4440 3043 1609
PDF Downloads 280 98 9