## 1. Introduction

Walter Munk concluded his classic 1981 review (Munk 1981) of internal waves with a speculation on the cascade of energy through the wave spectrum and its relation to wave breaking and the diapycnal buoyancy flux in the thermocline. He divided the wavefield into intrinsic waves that propagate quasi-linearly, undisturbed by their neighbors, and compliant waves whose speed of propagation is so slow that they are strongly affected by the larger-scale wavefield. Munk felt that the boundary between these two types of waves, at vertical scale *k*_{c} and the universal spectral level. However, his nonlinearity criterion only specified the *product* of cutoff wavenumber and level of the shear or strain spectrum (which is white in vertical wavenumber), rather than identifying both of these key parameters independently. He concluded with the optimistic hope that “we were close to having the various pieces fall into place.”

Five years later, Henyey et al. (1986) published a modeling study showing that energy cascaded through the internal wave spectrum at a rate proportional to the spectral level squared.^{1} If the spectral level is not universal, small changes in level should result in large changes in dissipation. Shortly thereafter, Gregg (1989) observationally verified Henyey’s result and produced a predictive model for dissipation based on the measurement of the finescale shear. His model was subsequently confirmed and refined by Polzin et al. (1995), Hibiya et al. (2012), Polzin et al. (2014), and others and has become the accepted standard for parameterizing deep ocean mixing. Müller et al. (1992), demonstrated that the product of cutoff wavenumber and spectral level indeed appeared to be constant in vertical wavenumber spectra of shear, supporting Munk’s earlier groundwork.

In Pinkel (2020, hereafter Part I), it was demonstrated that a Poisson statistical model has great skill in replicating both probability density functions and power spectra of vertical strain in the thermocline. Indeed, the relationship between cutoff wavenumber *κ*_{0} (rad m^{−1}) and spectral level

The problem is distantly analogous to that of replacing the equations that describe the molecular dynamics of an ideal gas with a simple diffusion equation that treats the behavior of an aggregate of molecules statistically. Here, the focus is on depth variability alone, not space and time together, and the task is somewhat simpler. Can the single Poisson parameter *κ*_{0} be related to the numerous internal wave and turbulence-scale variables in a physically plausible manner?

Initially, data-derived second- and third-order structure functions are presented and compared with both Poisson (Pinkel and Anderson 1992; Part I) and internal wave models. The Poisson model structure functions begin to diverge from observations at large vertical separations, where internal wave vertical displacement, as well as strain, becomes uncorrelated. The internal wave displacement correlation scale *Z*_{corr_sL}, of order 100–300 m, defines the outer boundary of the Poisson subrange. A small modification to the Poisson vertical displacement covariance function corrects the divergent behavior, simultaneously replicating the covariance of the Garrett–Munk (GM) model of the internal wave spectrum, Munk (1981). Merging the GM and Poisson models enables the determination of non-Gaussian quantities such as finescale skewness and kurtosis from GM internal wave-scale parameters. The desired link between the many parameters of GM and the single Poisson parameter *κ*_{0} is uncovered.

Shifting attention to dissipation scales, the Poisson parameter *κ*_{0} is shown to be strongly correlated with the vertical extent of observed overturning events (the Thorpe scale; Thorpe 1977; Dillon 1982). The Gregg–Henyey parameterization can then be recast in terms of *κ*_{0} and internal-wave-scale quantities. The essential levels and dependencies of the parameterization are recovered without need for arbitrary tuning factors. The Poisson subrange thus fits consistently in the center of the energy cascade, adding a non-Gaussian perspective to our present understandings.

## 2. The structure function of the Poisson thermocline

The data considered are vertical profiles of ocean density obtained in seven Pacific Ocean experiments, as described in Part I of this work. There, it is demonstrated that the vertical straining of the thermocline appears to be well modeled as a Poisson process over vertical scales 2–200 m. In linking this non-Gaussian domain with the larger-scale world of Munk’s intrinsic wavefield, structure functions of vertical displacement prove to be a surprisingly useful metric.

*S*

_{η}(

*k*) and the autocorrelation function

^{3/2}is subtracted from the third-order structure function estimate, as well. Oddly, the corrections that seem appropriate, given the shape of the curves, are smaller than would be predicted from a reasonable estimate of our precision in measuring the depth of a density surface. It is likely that the error in isopycnal depth estimation is correlated over small vertical separations, given the 2-m low-pass filter that is applied to the raw temperature and conductivity profiles. Similarly, since isopycnals are constrained from passing through one another, the noise contribution to the mean cube must be a positive number.

^{2}The noise correction applied to the third-order structure function is negligible at vertical scales greater than 4 m.

Estimates of the structure functions (Figs. 1a and 1b) are generally consistent with the Poisson model. Note that the single constant *κ*_{0}, associated with conceptual Poisson elements of 1–2-m vertical scale, governs the behavior of the second-order structure functions to scales of 30–100 m, even though the associated probability density functions of separation visually appear to be Gaussian at much smaller separations (Part I, Fig. 3). The lack of depth variability in the second-order structure function *κ*_{0} in the open ocean. The displacement variance atop Kaena Ridge in the HOME Nearfield is similar to that in the Farfield (450 km to the southwest) in the upper ocean. However, for the Nearfield, a generation site for the baroclinic tide, both *κ*_{0} as the seafloor is approached. The structure function

The increase in

At scales less than 10 m, the observational estimates of *κ*_{0} sites. The departure of *κ*_{0} shift this “microscale” behavior to sufficiently large vertical scale that finite sensor resolution does not overly contaminate its signature.

## 3. The Poisson wavefield relation

The second-order structure function *open-ocean* sites each collapse (e.g., HOME Farfield, Fig. 1) to a depth-independent curve (2), in spite of large variations in buoyancy frequency with depth. For linear waves in the WKB approximation, wave amplitude grows and vertical wavenumber decreases as the buoyancy frequency decreases with depth. A perhaps underappreciated fact is that these two effects evolve in concert such that the WKB second-order structure function of vertical displacement is invariant with depth/*N*^{2}(*z*) for both the compliant wavefield and the Poisson subrange, rendering it an exceptionally useful metric of the state of the thermocline.

*Z*

_{corr_0},

*N*

_{0}are measured at a common reference depth.

*N*

^{2}is thus consistent with both the Poisson model and a linear internal wavefield under WKB scaling.

*However, in order for the Poisson model to match the correlation-based description of the structure function, the Poisson constant κ*

_{0}

*must be related to internal wave parameters*

*and Z*

_{corr_sL}

*such that*

*z*<

*Z*

_{corr_sL}the Poisson microscale and finescale predictions for

*z*increases beyond

*Z*

_{corr_sL}, the Poisson model predicts that

In Fig. 2, the accuracy of the Poisson wavefield relation is examined across the combined pan-Pacific dataset. Individual points represent a 200-m vertical depth average, with successive averages offset vertically by 100 m, such that there is 100 m of overlap between averages. The shallowest depth interval in each experiment is 100–300 m, except in Tasman Tidal Dissipation Experiment (TTIDE), where it is 1350–1550 m.

The Poisson wavefield relation. (a) The ratio of the vertical correlation scale of the wavefield to twice the displacement variance, key descriptors of the second-order wavefield, is compared with estimates of *κ*_{0} determined from the third moment of isopycnal separation. Colored dots represent a 200-m vertical average, overlapped by 100 m. (b) The variability of *Z*_{corr_sL}, *κ*_{0} for the seven cruises is given independently, with the blue mesh surface indicating the analytic relation.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The Poisson wavefield relation. (a) The ratio of the vertical correlation scale of the wavefield to twice the displacement variance, key descriptors of the second-order wavefield, is compared with estimates of *κ*_{0} determined from the third moment of isopycnal separation. Colored dots represent a 200-m vertical average, overlapped by 100 m. (b) The variability of *Z*_{corr_sL}, *κ*_{0} for the seven cruises is given independently, with the blue mesh surface indicating the analytic relation.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The Poisson wavefield relation. (a) The ratio of the vertical correlation scale of the wavefield to twice the displacement variance, key descriptors of the second-order wavefield, is compared with estimates of *κ*_{0} determined from the third moment of isopycnal separation. Colored dots represent a 200-m vertical average, overlapped by 100 m. (b) The variability of *Z*_{corr_sL}, *κ*_{0} for the seven cruises is given independently, with the blue mesh surface indicating the analytic relation.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

*Z*

_{corr_sL}are determined for Fig. 2 by finding the vertical separation

The impressive agreement in Fig. 2 is in part a consequence of basic geometry, with 2*κ*_{0} being the decay rate of an initially triangular covariance function of height *κ*_{0} has a distinctly non-Gaussian identity, and is here estimated from the third moment of isopycnal separation. This demonstrates the link between the higher-order moments of the oceanic strain field and conventional second-order metrics.

## 4. The intrinsic internal wave spectrum

*R*

_{η}

*P*for

*k*is in cycles per meter and the spectrum is presented in two-sided form, −∞ <

*k*< ∞, such that its level is half that typically stated for 0 <

*k*< ∞. The term

*E*= 6.3 × 10

^{−5}is the Garrett–Munk dimensionless energy parameter,

*b*= 1.3 km is the scale depth of the thermocline, and

*κ*

_{0}= 1–1.3, consistent with

*E*,

The displacement spectrum [Eq. (13)] can be multiplied by (2*πk*)^{2} to form an associated spectrum of vertical strain, *γ* = ∂*η*/∂*z*. Polzin (1995), Kunze (2017), and others have found the *level* of the strain spectrum to be a useful metric for modeling ocean mixing rates. Interestingly, the GM strain spectrum has magnitude *independent of all possible variations in E*, *N*(*z*), or *b*, provided the Poisson wavefield relation is maintained.

The GM_P structure function and displacement spectrum. (a) The GM_P displacement variance can change by a factor of 8, with *no change* in structure function, spectral level, or predicted dissipation rate, provided *κ*_{0} is held constant. (b) Conversely, by varying *κ*_{0} rather than displacement variance, changes are seen at all scales in the structure function and displacement spectrum. The black reference represents a *k*^{−3} wavenumber dependence, that of the Munk (1981) version of the GM spectrum at scales smaller than 10 m.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The GM_P structure function and displacement spectrum. (a) The GM_P displacement variance can change by a factor of 8, with *no change* in structure function, spectral level, or predicted dissipation rate, provided *κ*_{0} is held constant. (b) Conversely, by varying *κ*_{0} rather than displacement variance, changes are seen at all scales in the structure function and displacement spectrum. The black reference represents a *k*^{−3} wavenumber dependence, that of the Munk (1981) version of the GM spectrum at scales smaller than 10 m.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The GM_P structure function and displacement spectrum. (a) The GM_P displacement variance can change by a factor of 8, with *no change* in structure function, spectral level, or predicted dissipation rate, provided *κ*_{0} is held constant. (b) Conversely, by varying *κ*_{0} rather than displacement variance, changes are seen at all scales in the structure function and displacement spectrum. The black reference represents a *k*^{−3} wavenumber dependence, that of the Munk (1981) version of the GM spectrum at scales smaller than 10 m.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

With a white strain spectrum, the GM strain variance increases without bound as smaller scales are considered. Munk (1981), proposed that the spectrum (13) be cut off at some limiting wavenumber *k*_{u} and that a *k*^{−1} spectral form be fitted to the strain spectrum (*k*^{−3} in displacement), extending from *k*_{u} to 1 cpm. Consistency with observations led to the selection of the strain (and shear) spectral slope. Consistency also required *k*_{u} = 1/10 cpm, independent of *z*). This conflicts with the behavior of linear waves under WKB scaling. The determination of *k*_{u} plays a huge role in the GM model’s allocation of wavefield variance (see appendix in Gregg and Kunze 1991). Approximately 2/3 of the shear and strain variance in the GM spectrum lies in the *k*_{u} < *k* < 1 cpm regime.

Munk (1981) explored various parameters of nonlinearity, referring to *k*_{u} as the *compliant* wave cutoff and suggesting that *k*_{u} varies inversely with wave energy, such that a universal value of *Ek*_{u} exists. In subsequent observations of shear (e.g., Müller et al. 1992) and strain (Fig. 6 in Part I), this appears to hold. The proper prediction of the relationship between *E* and *k*_{u} is considered a test of the veracity of dynamical models (e.g., Allen and Joseph 1989; Chunchuzov 1996, 2002; Lvov et al. 2004).

At high wavenumber, *k* > *k*_{u}, can the Poisson model wavenumber spectrum *Ek*_{u} is universal? This is potentially challenging given that the Poisson spectrum is based on the statistical behavior of a stack of conceptual Poisson elements. But the low-wavenumber form of the Poisson strain spectrum is white and of level *π*/*κ*_{0}= 6–10 m in an isopycnal-following frame, approximating Munk’s *k*^{−1} cutoff. In an Eulerian frame the Poisson strain spectrum cuts off at a slightly larger scale (Fig. 5 in Part I). Furthermore, the Poisson wavefield relation requires that *Z*_{corr_sL}/*κ*_{0} vary with wavefield energy, *k*_{u} will indeed vary inversely with energy, *but only if the vertical correlation scale of the wavefield remains fixed.*

This modification to Munk’s hypothesis is significant, but also in line with observational experience. As a thought experiment, one can take a typical open-ocean wavefield (e.g., PATCHEX) and add a very energetic mode-1 internal tide (e.g., HOME Farfield) that interacts minimally with the broadband wavefield. The vertical correlation scale *Z*_{corr_sL} and wavefield potential energy will both increase relative to the nontidal site, but *k*_{u}, *κ*_{0}, and the non-Gaussian nature of the thermocline will be minimally altered. Alternatively, one can uniformly increase the overall level of the internal wave spectrum, keeping *Z*_{corr_sL} fixed. Here wavefield energy and *k*_{u}, *κ*_{0} vary inversely, in accord with Munk’s hypothesis.

*κ*

_{0}

*S*

_{γSL}(

*k*) [Eq. (3) in Part I]. The resulting semi-Lagrangian (sL) GM_Poisson (GM_P) spectrum is

Regarding the observational requirement that *k*_{u} be independent of depth, in violation of the WKB approximation, Eq. (10) gives *κ*_{0} as the ratio of *Z*_{corr_sL} to *N*_{0}/*N*(*z*) in accord with WKB scaling. Thus *κ*_{0} and the associated spectral cutoff *k*_{u} = *κ*_{0}/2*π* is depth independent.

The associated GM_P vertical wavenumber spectrum for strain, *S*_{γGM_P}(*k*) is again related to the displacement spectrum *S*_{ηGM_P}(*k*) by the factor (2*πk*)^{2}. Note that no matter how the parameters of the GM model are varied, the GM_P strain variance is very close to unity, provided the Poisson wavefield relation (10) is maintained.

*κ*

_{u}=

*κ*

_{0}/(2

*π*) cpm.

*λ*is far from micro. Given that strain variance is always unity in the GM_P model, it is not surprising that the Taylor microscale is a metric of displacement variance.

*R*

_{ηP}≈

*R*

_{ηGM}, is identical to Poisson microscale result (appendix C, section a in Part I), derived from the behavior of a vertical stack of hypothetical Poisson elements, with thickness governed by the exponential PDF.

The behavior of the GM_P structure function and displacement spectrum is illustrated in Fig. 3a as *κ*_{0} held constant, and in Fig. 3b at fixed *κ*_{0}. At fixed *κ*_{0}, the GM_P structure function is insensitive to variations in both *N*^{2}(*z*) at separations Δ*z* < *Z*_{corr_sL}. The associated wavenumber spectral level also remains fixed for *N*^{2}(*z*) appears through changes in

The mathematical forms of the GM_P strain and displacement spectra are essentially unchanged from the Munk (1981) version of the GM model. However, there is now the implication that skewness and other moments of the strain field can be determined at all vertical scales from outer scale quantities like spectral level. Also, the low-wavenumber spectral bandwidth *k*_{u} = 1/(2*πκ*_{0}) through the Poisson wavefield relation. These conceptual advances are associated with the underlying Poisson structure of the thermocline.

## 5. Ocean turbulence and the Poisson subrange

Through the Poisson wavefield relation, the Poisson constant *κ*_{0} is seen to play many roles, including setting the level of the internal wave spectrum. The appearance of this meter-scale, non-Gaussian parameter in an internal-wave-scale spectral model makes sense in the context of a cross-scale energy cascade where the Poisson deformation of the thermocline plays an integral role. Small-scale turbulent dissipation must lie at the end of the cascade, somehow coexisting with a Poisson microscale that is here based on the concept of reversible fine structure, an adiabatic concept. The challenge is to link these seemingly disparate processes.

In the ocean, a defining property of turbulent mixing is its intermittency. Active overturning occurs between two (mid-gyre thermocline sites) and 30 (near-seafloor tidal conversion sites) percent of the time. When a turbulent event develops, is it aware of the preexisting steppiness of the thermocline? A growing convective instability developing on a small-scale wave might expand vertically through the low-gradient layer (wave crest) with further expansion inhibited by bounding high gradient sheets. The pioneering simulations of Orlanski and Bryan (1969) illustrate this process. Alternatively, a growing Kelvin–Helmholtz instability, perhaps originating on a sheet, might expand until limited by adjacent sheets.^{3}

To establish a link between Poisson-scale and turbulent processes, one can compare the correlation scale

For the more recent cruises, a routine developed by J. Klymak (2016, personal communication), incorporating the Galbraith and Kelly (1996) overturn quality metric as well as other refinements, has been applied to the profiling datasets. Strain statistics obtained during periods of stable stratification can be compared with Thorpe scales derived from the imbedded turbulent events.^{4} A linear relationship is found between the correlation scale *κ*_{0} is nearly independent of depth/

A comparison of the observed Thorpe scale of overturning events with the inverse Poisson constant estimated from 4-m strain variance determined during periods when the stratification is stable. The individual points represent averages over 20-m depth intervals from 100 to 800 m and over 1500 (AESOP) to 8100 (Nearfield) profiles (Part I, Table 1). Averages are formed in isopycnal coordinates. A similar pattern is seen in Eulerian frame averages. The reference line is

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

A comparison of the observed Thorpe scale of overturning events with the inverse Poisson constant estimated from 4-m strain variance determined during periods when the stratification is stable. The individual points represent averages over 20-m depth intervals from 100 to 800 m and over 1500 (AESOP) to 8100 (Nearfield) profiles (Part I, Table 1). Averages are formed in isopycnal coordinates. A similar pattern is seen in Eulerian frame averages. The reference line is

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

A comparison of the observed Thorpe scale of overturning events with the inverse Poisson constant estimated from 4-m strain variance determined during periods when the stratification is stable. The individual points represent averages over 20-m depth intervals from 100 to 800 m and over 1500 (AESOP) to 8100 (Nearfield) profiles (Part I, Table 1). Averages are formed in isopycnal coordinates. A similar pattern is seen in Eulerian frame averages. The reference line is

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The 2006 Assessing the Effects of Submesoscale Ocean Parameterizations (AESOP) cruise was sited in 1100-m water off the coast of Monterey, California. The lowest modes of an open-ocean wavefield are not present at this coastal site. Both *Z*_{corr_sL} are smaller than in the Farfield. There is weak tidal generation as well as reflection from the irregular seafloor. Strain variance increases, as does

The HOME Nearfield Experiment took place atop Kaena Ridge, Hawaii, at 1100-m depth in a surrounding 4800-m ocean. The Ridge is a site of strong barotropic to baroclinic conversion. The upper 800 m of the water column were sampled, with strain variance and

In preparing Fig. 4, it was necessary to specify a threshold on the Thorpe scale, to discriminate between events and nonevents. A threshold of 0.1 m is used here and in Fig. 5. The smaller values of average Th are sensitive to this choice of threshold. Similarly, an assumed strain noise variance of 0.125 in AESOP and Farfield, 0.375 in the Nearfield is subtracted from the observed strain variance before estimates of *κ*_{0} are formed. This corresponds to an uncertainty in estimating the depths of individual isopycnals of 1 m (and 1.7 m in the HOME Nearfield). This correction weakly affects the constant of proportionality between Th and

In the 2015 TTIDE experiment, eight sites were occupied for 1–3 days. (a) A strong correlation is seen between *κ*_{0} and the Thorpe scale of observed mixing events. An energetic baroclinic tide shoals and reflects at the TTIDE site. (b),(c) Both the fractional duration of overturning *ϕ* and the climatological mixing rate ⟨*ε*⟩ increase as the seafloor is approached, in contrast to classical open-ocean behavior, where

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

In the 2015 TTIDE experiment, eight sites were occupied for 1–3 days. (a) A strong correlation is seen between *κ*_{0} and the Thorpe scale of observed mixing events. An energetic baroclinic tide shoals and reflects at the TTIDE site. (b),(c) Both the fractional duration of overturning *ϕ* and the climatological mixing rate ⟨*ε*⟩ increase as the seafloor is approached, in contrast to classical open-ocean behavior, where

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

In the 2015 TTIDE experiment, eight sites were occupied for 1–3 days. (a) A strong correlation is seen between *κ*_{0} and the Thorpe scale of observed mixing events. An energetic baroclinic tide shoals and reflects at the TTIDE site. (b),(c) Both the fractional duration of overturning *ϕ* and the climatological mixing rate ⟨*ε*⟩ increase as the seafloor is approached, in contrast to classical open-ocean behavior, where

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

The robustness of this result is not sensitive to the use of “4-m strain” to estimate *κ*_{0}, as opposed to estimates made at some other vertical scale. In Figs. 1 and A1, it is seen that a measurement of strain variance or mean cube strain anywhere within the Poisson subrange *κ*_{0}.

It is of value to inquire whether this Poisson–Ozmidov relationship emerges only in long-term averages of wave and turbulent patch statistics in the thermocline or whether it is maintained on a day-to-day basis. If the latter, is it because the short-term variability in the wavefield is very small, reflecting a residence time for energy of weeks to months? Or do Th and

In the 2015 TTIDE experiment, eight sites on the east coast of Tasmania were visited, each for a period of 1–2 days. A remotely generated baroclinic tide shoals and reflects on this coast, leading to extreme variability in the near-seafloor wave and turbulent fields. With an acoustic altimeter on the profiling CTD, the density field was monitored to within 15 m of the sea floor. Profiles could be repeated at 8–15-min intervals, slightly faster than the buoyancy period at the observed depths. Collectively, the observations spanned depths 10–2000 m. While these single-site datasets are a factor of 10–30 smaller than the others reported here, the range of variability in *κ*_{0} and Thorpe scale in TTIDE is very large. Again, a strong correlation is seen between Th and

*ε*

_{event}⟩ and the climatological mean dissipation rate ⟨

*ε*⟩, as related by

*ϕ*is an intermittency factor, the percentage of time that a nonzero Thorpe scale is detected on any given density surface.

*event*Ozmidov scale

*c*

_{0}is a dimensionless constant of order unity. If

*κ*

_{0}is independent of ⟨

*N*

^{2}⟩ in the open-ocean thermocline (Part I, Fig. 9), then the event dissipation rate varies as

*ϕ*is given by some ratio of the triggering time scale for turbulent events and the individual event duration. Examples might be

*f*is the local inertial frequency, M

_{2}is the frequency of the semidiurnal tide, and

*c*

_{1}is a dimensionless constant. Noting the

*ε*(25), we can invoke the Poisson wavefield relation to see

*ϕ*=

*c*

_{1}

*f*/

*N*and presented the result in terms of Poisson scale (28a), internal wave scale (28b), and GM parameters (28c), as expressed in (14) and (15).

*c*

_{0}and

*c*

_{1}are set to unity and ⟨

*ε*⟩

_{IW_P}is evaluated at 30° latitude and at the Garrett–Munk energy level, we find

*ϵ*⟩

_{IW_Gregg}= 7 × 10

^{−10}W kg

^{−1}and ⟨

*ϵ*⟩

_{IW_Polzin}= 8 × 10

^{−10}W kg

^{−1}from the Gregg 1989 synthesis of data and theory and the more recent study by Polzin et al. (2014).

Equation (28a) is the Poisson version of the Gregg–Henyey relation. The original Gregg–Henyey relation has been modified over the years (e.g., Polzin et al. 1995; Hibiya et al. 2012; Polzin et al. 2014) to include adjustments for both latitude and shear-to-strain ratio. These data from HOME Nearfield, AESOP, and TTIDE demonstrate the validity of the Poisson–Ozmidov relation (24) (Figs. 4 and 5a), even in the absence of a continuous energy cascade through the internal wave spectrum. In these regions of intensified near-seafloor mixing, the variability of *κ*_{0} and ⟨*ε*⟩ with *ϕ*(*z*) (Figs. 5b and 5c), not in the relationship between Th and *κ*_{0}. One can conjecture that the Poisson–Ozmidov relation (24) remains valid universally and that site-dependent environmental and wavefield forcing factors primarily influence intermittancy.

## 6. Discussion

A Poisson subrange governed by the single parameter *κ*_{0} is consistent with existing understanding of the larger-scale intrinsic internal wavefield and with the established parameterization of ocean turbulence. Given the linkage established between *κ*_{0} and the vertical wavenumber spectrum of strain, global maps of non-Gaussian parameters such as skewness can now be produced for any vertical scale

*κ*

_{0}, the metric for deformation, generally varies inversely with wavefield energy. But it is also linked with the vertical correlation scale of the wavefield

*Z*

_{corr_sL}as specified by the Poisson wavefield relation [inverse of Eq. (10)],

*and*the vertical wavenumber bandwidth across which this energy is distributed. The Relation is demonstrated in Fig. 2 here and in Fig. 6 in Part I, where vertical wavenumber spectra of strain and displacement are rescaled using values of

*κ*

_{0}estimated from 8 m strain skewness. The nondimensionalized spectra are seen to coincide to within a factor of 2 over vertical scales 10–200 m. In excess of 30 000 km of downcast CTD profile data contribute to these results.

A *k*^{−2} spectral form is known to describe the internal wave displacement spectrum over a range of scales *level* of this hybrid GM_P spectrum is given by *k*^{−2} spectral region. Also, if both displacement variance *k*_{u} = *κ*_{0}/(2*π*) is independent of *N*(*z*). These linkages make sense in the context of an energy cascade.

*κ*

_{0}estimates are based on the 70%–95% of the observations occurring when overturning is

*not*present. The Thorpe scale estimates are averaged only over the overturning events. Given this Poisson–Ozmidov relation, it is straightforward to recast the Gregg–Henyey mixing model in terms of the Poisson formalism (28), leading to an expression for the GM dimensionless energy parameter

*ϕ*.

The GM_P restatement of the Gregg–Henyey model has ⟨*ε*_{event}⟩ varying with wavefield energy level and correlation scale, while *ϕ* remains strictly a function of the background ocean. This conflicts with common experience, in that overturning events occur more frequently at intense mixing sites. More reasonably, *ϕ* might be defined as the fractional time of existence of the *X*% most energetic overturns, the rare events that dominate a long-term average. Given that observations of ⟨*ε*_{event}⟩ tend to vary over orders of magnitude at a single site, it makes sense that the model should focus on the dominant overturns that establish the climatological average.

The present custom is to report the climatological average ⟨*ε*⟩, rather than ⟨*ε*_{event}⟩ and *ϕ* separately. Going forward, it is clear that both event-averaged dissipation rates and the intermittency of the mixing events have distinct stories to tell. It is critical that a useful definition of *ϕ* be agreed upon and that both *ϕ* and ⟨*ε*_{event}⟩ be reported in observational campaigns.

It is useful to present a phenomenological hypothesis for the energy cascade. At the largest scales, *k*_{c} ~ *N*(*z*)/*U*_{rms}.

Munk was hoping to identify *k*_{c} with *k*_{u}, the 10-m cutoff, but for reasonable values of *U*_{rms} and *N*(*z*), the compliant wave boundary appeared to be at much larger scale, more like 60 m [Munk 1981, Eq. (9.30)]. Holloway (1980) also found that the wavefield becomes strongly nonlinear at scales below ~40 m. While there is no *spectral* signature of this boundary, the non-Gaussian aspect of the motion field begins to emerge at this scale. In the present data, estimates of the third-order structure function

Regarding the sequence of events associated with this process (Figs. 7d and 8e in Part I), suggest that the short-wave packets that locally dominate thermocline strain are ones whose intrinsic frequencies have become small, such that their signatures are *effectively embedded* in the local mean flow. At vertical scales smaller than *k*_{c} and *κ*_{0} is associated with the nonsinusoidal waveforms of these waves, as is the non-Gaussian (Poisson) structure of scalar profiles in the thermocline. The vertical extent of the isopycnal “layers” and the resulting 10-m cutoff at *k*_{u} are associated with wave displacement *amplitude*, not vertical wavelength (Figs. 7d–f and 8e in Part I). This is consistent with the scale separation between

The development of nonsinusoidal vertical waveforms can be seen in numerical simulations of a wave encountering a critical layer. Excellent examples are presented by Winters and D’Asaro (1994, their Figs. 3–5). As wave amplitude builds, the vertical waveform becomes highly steppy. The “sheets and layers” are short lived in the Winters and D’Asaro (1994) Eulerian-frame simulation because their small-scale wave is given a high frequency and this remains constant throughout the encounter.

In terms of spectral descriptions, there is a significant difference between the cascade of energy through the internal wavefield and the cascade associated with isotropic turbulence. Increasing the energy flux through a turbulent field drives velocity variability to progressively smaller scales and eventual destruction by molecular viscosity. Increasing the energy flux through the internal wave spectrum causes compliant waves to become nonsinusoidal at *larger* vertical scale. Thermocline steps become larger and the correlation scale of the strain field, *grows*. This behavior is demonstrated in the Poisson simulation in Fig. 4 of Part I. It is seen observationally in Fig. 10 of Part I where 2-m strain variance, directly proportional to

One of the puzzles associated with the Gregg–Henyey parameterization of ocean turbulence is that it appears to give accurate predictions at geographically localized energetic sites such as TTIDE, as well as in regions of spatial homogeneity. In localized sites, a weakly nonlinear cascade through the full internal wave spectrum does not have the spatial extent to occur. Perhaps the thermocline steppiness that sets strain spectral level is most sensitive to the end-state of the cascade, the extreme nonlinear distortion of compliant waves that are on the edge of breaking. The state of these waves and the corresponding mixing rate vary together. The relationship between strain spectral level and observed wave breaking remains fixed while both fluctuate on a more rapid/more local scale.

Figures 7 and 8 in Part I show the role of larger-scale near-inertial shear, as well as near-inertial and tidal strain (Sun and Kunze 1999) in modulating the propagation of short waves. When inertial shear inhibits the vertical propagation of a wave packet, local wave amplitude grows and an irregular thermocline potentially develops. If the packet’s intrinsic frequency remains above *f*, it can resume propagation a quarter inertial period later, returning to a more sinusoidal waveform. The formation of these adiabatic irregularities in conjunction with a causal background shear (and strain) must play a key role in orchestrating the presence of overturning and resultant ocean mixing.

Recently, Kunze (2019) presented a differing view of the end stages of the energy cascade. Focusing on a stationary homogeneous spectral description, Kunze argues that a field of anisotropic stratified turbulence lies between the wavenumbers of the shortest internal waves and the climatological Ozmidov scale. Such a model is not inconsistent with the Poisson structure of the thermocline. It does contrast with the view presented here, which is based on observations of internal waves directly breaking (e.g., Alford and Pinkel 2000). A brief comparison of the two views is presented in appendix B.

As the vertical resolution of numerical models improves and small-scale internal waves and submesoscale motions are admitted to the motion fields, the Poisson nature of the thermocline should be reproduced. Realistic models of diapycnal diffusion will require knowledge of the co-occurrence of the various types of wave breaking and the strain and shear state of the evolving thermocline. Both observational and theoretical guidance is necessary here if we are to capitalize on ongoing computational progress.

## Acknowledgments

The author thanks Eric Slater, Lloyd Green, Mike Goldin, Tony Aja, Chris Neely, Mai Bui, Tyler Hughen, San Nguyen, Jonathan Ladner, and Sara Goheen for their participation in the development of the instrument systems described here and for the operation of these systems at sea. The seagoing leadership of Captains DeWitt Efird and Tom Golfinos (R/P *FLIP*) and Tom Desjardins and Dave Murline (R/V *Roger Revelle*) is greatly appreciated. Mike Gregg, Matthew Alford, Jennifer MacKinnon, Andrew Lucas, and Gregory Wagner provided valuable comments on drafts of this work. Jody Klymak processed the HOME and AESOP CTD information used to estimate the Thorpe scales in Fig. 4. Support from the National Science Foundation and the Office of Naval Research is gratefully acknowledged.

## APPENDIX A

### Structure Functions of Isopycnal Separation from all Cruises

While oceanographers are most familiar with spectral descriptions of the motion field at internal wave scales, higher-order descriptors are necessary to describe the increasingly non-Gaussian nature of the motions at smaller vertical scales. The second order structure function (Fig. A1) is a useful bridge between spectral and non-Gaussian descriptors, and the third order structure function (Fig. A2) is an excellent metric of the Poisson nature of the compliant wavefield and its transition to the Gaussian intrinsic regime. Here, data from Fig. 1 are repeated, along with results from the corresponding results from the other five sites used in this analysis.

Second-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals. For the open-ocean observations, MILDEX, PATCHEX, Farfield, and even the coastal AESOP data, the independence of the structure function with measurement depth/

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

Second-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals. For the open-ocean observations, MILDEX, PATCHEX, Farfield, and even the coastal AESOP data, the independence of the structure function with measurement depth/

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

Second-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals. For the open-ocean observations, MILDEX, PATCHEX, Farfield, and even the coastal AESOP data, the independence of the structure function with measurement depth/

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

Third-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

Third-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

Third-order structure functions of vertical displacement for the seven cruises. Reference lines give predictions based on the Poisson model (black) and the Poisson microscale model (red) (Part I, appendix B), using a single value of *κ*_{0} derived from the PDF of 8-m strain (Part I, Fig. 3), averaged over all 200-m measurement intervals.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0287.1

## APPENDIX B

### Compliant Waves, Vortical Motions, and Anisotropic Turbulence

*L*

_{o}, the motion field is asserted to be steady, isotropic turbulence. The distinction between “event” and “climatological” averages is absent in his steady-state model.

*L*

_{O}are 0.12, 0.07, and 0.04 m, much smaller than the height of the typical (intermittent) overturns seen in the thermocline. Increasing

*K*

_{ρ}to 10

^{−4}increases the corresponding

*L*

_{0}by a factor of 4.5, still a very small value.

Kunze’s anisotropic turbulence at ~0.1–10-m vertical scales, is forced by short internal waves and it is this turbulence that ultimately breaks. Scaling arguments are used to infer the spectral forms of velocity and density variability in this subrange, with numerous plausible alternatives considered. This scaling-based approach goes beyond the present statistical exploration in that, with both *f* and *N* available as time scales, the velocity field can be considered as well as displacements and strains.

Vortical motions are indeed found at 0.1–10-m vertical scales in the sea, and they can be distinguished from internal waves in wavenumber frequency spectra of strain, Pinkel (2014). Are these motions primarily forced by small scale waves, perhaps when a wave packet locally deposits momentum to the mean flow due to a critical layer encounter? A hallmark of the vortical signal seen in wavenumber frequency spectra is that vortical variance is found in a spectral ridge centered at zero frequency. The ridge is clearly broadened by Doppler shifting associated with lateral advection, but its subinertial origin is clear. In Figs. 7b and 7d of Part I, the strain field is presented both with and without the vortical ridge present. One might think that Kunze’s wave-forced anisotropic turbulence is not strongly concentrated at subinertial frequencies. If so, it should be clearly apparent in Fig. 7d. Numerical simulations of a laterally as well as vertically localized wave packet encountering a critical layer will be instructive in addressing this concern.

The Poisson model presented here has only a single parameter available for adjustment. It leaves virtually no room for subjective choice in considering both spectral descriptions and higher-order statistics. The fact that the model seems equally applicable at energetic sites such as TTIDE and the HOME Nearfield, where the local baroclinic tide is breaking directly on topography, and at the HOME Farfield, PATCHEX, and MILDEX sites, where a classical open-ocean energy cascade is established, suggests that it is a robust descriptor of the non-Gaussian behavior of the thermocline. Thermocline irregularities formed by the nonsinusoidal vertical waveforms of nonlinear internal waves develop the same high aspect ratio, *k*_{z}/*k*_{x}, as does Kunze’s stratified turbulence. In terms of wavenumber spectra, the spectral signatures might not be very different.

The dynamics of the high-wavenumber end of the energy cascade is best explored by observing systems that document the progression of events leading to intermittent mixing (e.g., Alford and Pinkel 2000).

## REFERENCES

Alford, M. H., and R. Pinkel, 2000: Observations of overturning in the thermocline: The context of ocean mixing, Part I.

,*J. Phys. Oceanogr.***30**, 805–832, https://doi.org/10.1175/1520-0485(2000)030<0805:OOOITT>2.0.CO;2.Allen, K. R., and R. I. Joseph, 1989: A canonical statistical theory of oceanic internal waves.

,*J. Fluid Mech.***204**, 185–228, https://doi.org/10.1017/S0022112089001722.Chunchuzov, I. P., 1996: The spectrum of high-frequency internal waves in the atmospheric wave-guide.

,*J. Atmos. Sci.***53**, 1798–1814, https://doi.org/10.1175/1520-0469(1996)053<1798:TSOHFI>2.0.CO;2.Chunchuzov, I. P., 2002: On the high-wavenumber form of the Eulerian internal wave spectrum in the atmosphere.

,*J. Atmos. Sci.***59**, 1753–1774, https://doi.org/10.1175/1520-0469(2002)059<1753:OTHWFO>2.0.CO;2.Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov length scales.

,*J. Geophys. Res.***87**, 9601–9613, https://doi.org/10.1029/JC087iC12p09601.Galbraith, P. S., and D. E. Kelly, 1996: Identifying overturns in CTD profiles.

,*J. Atmos. Oceanic Technol.***13**, 688–702, https://doi.org/10.1175/1520-0426(1996)013<0688:IOICP>2.0.CO;2.Garrett, C. J. R., and W. H. Munk, 1972: Space-time scales of internal waves.

,*Geophys. Fluid Dyn.***3**, 225–264, https://doi.org/10.1080/03091927208236082.Garrett, C. J. R., and W. H. Munk, 1975: Space-time scales of internal waves: A progress report.

,*J. Geophys. Res.***80**, 291–297, https://doi.org/10.1029/JC080i003p00291.Gregg, M., 1989: Scaling turbulent dissipation in the thermocline.

,*J. Geophys. Res.***94**, 9686–9698, https://doi.org/10.1029/JC094iC07p09686.Gregg, M. C., and E. Kunze, 1991: Shear and strain in Santa Monica basin.

,*J. Geophys. Res.***96**, 16 709–16 719, https://doi.org/10.1029/91JC01385.Henyey, F. S., and J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An Eikonal approach.

,*J. Geophys. Res.***91**, 8487, https://doi.org/10.1029/JC091iC07p08487.Hibiya, T., N. Furuichi, and R. Robertson, 2012: Assessment of fine-scale parameterizations of turbulent dissipation rates near mixing hotspots in the deep ocean.

,*Geophys. Res. Lett.***39**, L24601, https://doi.org/10.1029/2012GL054068.Holloway, G., 1980: Oceanic internal waves are not weak waves.

,*J. Phys. Oceanogr.***10**, 906–914, https://doi.org/10.1175/1520-0485(1980)010<0906:OIWANW>2.0.CO;2.Kunze, E., 2017: Internal-wave-driven mixing: Global geography and budgets.

,*J. Phys. Oceanogr.***47**, 1325–1345, https://doi.org/10.1175/JPO-D-16-0141.1.Kunze, E., 2019: A unified model for anisotropic stratified and isotropic turbulence in the ocean and atmosphere.

,*J. Phys. Oceanogr.***49**, 385–407, https://doi.org/10.1175/JPO-D-18-0092.1.Lvov, Y. V., K. L. Polzin, and E. G. Tabak, 2004: Energy spectra of the ocean’s internal wave field: Theory and observations.

,*Phys. Rev. Lett.***92**, 128501, http://doi.org/10.1103/physrevlett.92.128501.McComas, C., and P. Muller, 1981a: The dynamic balance of internal waves.

,*J. Phys. Oceanogr.***11**, 970–986, https://doi.org/10.1175/1520-0485(1981)011<0970:TDBOIW>2.0.CO;2.McComas, C., and P. Muller, 1981b: Time scales of resonant interactions among oceanic internal waves.

,*J. Phys. Oceanogr.***11**, 139–147, https://doi.org/10.1175/1520-0485(1981)011<0139:TSORIA>2.0.CO;2.Müller, P., and D. Olbers, 1975: On the dynamics of internal waves in the deep ocean.

,*J. Geophys. Res.***80**, 3848–3860, https://doi.org/10.1029/JC080i027p03848.Müller, P., E. A. D’Asaro, and G. Holloway, 1992: Internal gravity waves and mixing.

,*Eos, Trans. Amer. Geophys. Union***73**, 25 and 31–32, http://doi.org/10.1029/91eo00018.Munk, W. H., 1981: Internal waves and small-scale processes.

*Evolution of Physical Oceanography*, B. A. Warren and C. Wunsch, Eds., MIT Press, 264–291.Orlanski, I., and K. Bryan, 1969: The formation of thermocline step structure by large-amplitude internal gravity waves.

,*J. Geophys. Res.***74**, 6975–6983, https://doi.org/10.1029/JC074i028p06975.Pinkel, R., 2014: Vortical and internal wave shear and strain.

,*J. Phys. Oceanogr.***44**, 2070–2092, https://doi.org/10.1175/JPO-D-13-090.1.Pinkel, R., 2020: The Poisson link between internal wave and dissipation scales in the thermocline. Part I: Probability density functions and the Poisson modeling of vertical strain.

,*J. Phys. Oceanogr.***50**, 3403–3424, https://doi.org/10.1175/JPO-D-19-0286.1.Pinkel, R., and S. Anderson, 1992: Toward a statistical description of finescale strain in the thermocline.

,*J. Phys. Oceanogr.***22**, 773–795, https://doi.org/10.1175/1520-0485(1992)022<0773:TASDOF>2.0.CO;2.Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation.

,*J. Phys. Oceanogr.***25**, 306–328, https://doi.org/10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.Polzin, K. L., A. C. N. Garabato, T. N. Huussen, B. M. Sloyan, and S. Waterman, 2014: Finescale parameterizations of turbulent dissipation.

,*J. Geophys. Res. Oceans***119**, 1383–1419, https://doi.org/10.1002/2013JC008979.Stommel, H., and K. N. Fedorov, 1967: Small scale structure in temperature and salinity near Timor and Mindanao.

,*Tellus***19**, 306–325, https://doi.org/10.3402/tellusa.v19i2.9792.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, https://doi.org/10.1175/1520-0485(1999)029<2886:IWWIPI>2.0.CO;2.Thorpe, S. A., 1977: Turbulence and mixing in a Scottish loch.

,*Philos. Trans. Roy. Soc. London***A286**, 125–181, https://doi.org/10.1098/rsta.1977.0112.Whalen, C. B., L. D. Talley, and J. A. MacKinnon, 2012: Spatial and temporal variability of global ocean mixing inferred from Argo profiles.

,*Geophys. Res. Lett.***39**, L18612, https://doi.org/10.1029/2012GL053196.Winters, K. B., and E. A. D’Asaro, 1994: Three-dimensional wave instability near a critical level.

,*J. Fluid Mech.***272**, 255–284, https://doi.org/10.1017/S0022112094004465.

^{1}

Müller and Olbers (1975) were the first to derive a nonlinear transfer rate through the spectrum that was proportional to the square of the spectral energy level [their Eq. (37) and preceding]. The cascade was one of a number of interaction processes they considered. McComas and Muller (1981a,b) both considered scenarios where the downscale transfer of energy was proportional to spectral level squared. I am indebted to a reviewer for pointing this out.

^{2}

If isopycnals are constrained to not cross, negative strain errors are bounded by −1. Positive strain errors can be arbitrarily large, in principle.

^{3}

The causal relationship between low-gradient regions in the thermocline and the occurrence of turbulence is a subject of longstanding debate. Stommel and Fedorov (1967), for example, conjectured that such layers represented the “scars” of preexisting patches of intense turbulence. In the present observations, where vertical density profiles are repeated rapidly in time, overturning is not seen to establish thermocline structure in abrupt events. Even in sites of massive (~50+ m) convective overturns, the low-gradient crests develop and recede on internal wave time scales, with breaking abruptly occurring long after the local density gradient has begun to decrease.

^{4}

The profile segments from periods of active overturning can be Thorpe sorted, and “strain” can be calculated. The results are a slightly noisier version of the strain profiles that immediately precede and follow the event, given the short profile repeat intervals (4–15 min) employed here. Including these profiles along with the wave-strained profiles changes the overall averages very little.