Observations of Overturning in the Thermocline: The Context of Ocean Mixing

a. Focus

Ocean mixing in double-diffusively stable regions of the thermocline is thought to be accomplished through dynamic instability of finescale motions. Gravitationally unstable regions (“overturns”) are produced, which subsequently collapse, dissipating mechanical energy and diffusing scalar quantities. The vertical fluxes produced by this mixing are of importance in maintaining global balances. Proper parameterization of dissipation rate in general oceanic environments requires an understanding of the governing factors. With this as a goal, we present simultaneous observations of overturns and the finescale conditions that accompany them. The issue of flow context is central.

Overturns can result from Kelvin–Helmholtz (K–H) or shear instability (Miles 1961; Howard 1961), the convective instability of internal waves (Orlanski and Bryan 1969), and several other mechanisms. The objective of this study is to identify the processes that lead to instability. We focus on the phenomenon of (meter scale) overturning rather than the measurement of (centimeter scale) microstructure. Observations of overturning and of microstructure offer complementary views of the mixing process. To oversimplify, we view the overturns as an early phase of the instability and the microstructure as the aftermath. (Microconductivity data from this experiment are analyzed in a companion paper.)

Overturns are monitored over a 300-m vertical extent for nine days in March 1995 in a strongly sheared, coastal environment. Depth–time fields of 6.4-m Ri, 2-m strain γ, and a quantity termed the 2-m “effective strain rate” ∂ŵ/∂z are also measured with the overall aim of determining the relationship between these finescale quantities and overturning.

The paper is organized as follows. Section 2 provides an overview of conditions and observations made during the experiment, as well as a description of the methods and definitions used. Section 3 focuses on depth–time maps of Ri, γ, and ∂ŵ/∂z for a representative 12-h period. Locations and times of overturns are overplotted, allowing visual determination of the finescale conditions that accompany them. In section 4, probability density functions of these quantities are produced, using the complete dataset as well as subsets of the data where overturning is observed. In addition, related PDFs are computed using data sampled just before, and just after, overturns. This statistical approach corroborates the qualitative conclusions drawn from section 3. Section 5 examines time series of several indirect estimates of depth-averaged dissipation rate. Section 6 discusses the dependence of overturning, and resulting mixing, upon instantaneous and mean stratification. The relative contributions to total mixing from overturns of varying size and background stratification are presented in section 7. Section 8 is a summary. Conclusions are presented in section 9.

b. Background

A pioneering study of mixing and its finescale context was conducted by Gregg et al. (1986), who tracked a persistent mixing patch, 5–10 m thick, for several days. It appeared to be associated with a near-inertial feature. The patch was centered, during most of its lifetime, on the velocity maximum of the feature, not the shear maximum. Gregg et al. (1986) suggested that the near-inertial wave provided a preferential site for instability of smaller-scale internal waves.

Marmorino et al. (1987), using Doppler sonar and towed thermistor chains, monitored meter-scale temperature fluctuations as well as larger-scale variations (7–50 m) in shear and Richardson number. They observed a number of regions of intense temperature fluctuation similar to the one observed by Gregg et al. (1986). These also had 5–10 m thickness and existed within a near-inertial wave packet. Within the patches 1–3 m overturns were observed. The patches were associated with high shear and with values of 7-m Ri lower than one.

In addition to the near-inertial groups, Marmorino et al. (1987) observed small-scale internal wave groups (e.g., their Fig. 6a). The groups had vertical extent 5–8 m and horizontal wavelength 20–30 m. One-meter temperature inversions observed within the groups were interpreted as evidence of wave breaking. Observed 7-m Ri in this region was greater than unity.

Hebert et al. (1992) observed a 22-m overturn above the core of the equatorial undercurrent. Using towed thermistor data, they showed that it was associated with low Ri and high internal wave strain γ ≡ N²/N², where

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is the buoyancy frequency (their Figs. 2a and 3c). A packet of high frequency (ω ≥ N/2), high vertical wavenumber (m ≥ 1/100 cpm) internal waves was also present. High correlations were observed between dissipation rate and the presence of these waves (Moum et al. 1992). The authors concluded that instability associated with the waves generated the overturn. However, they were unable to determine the specific instability mechanism responsible.

In parallel with these observations, theoretical studies by Henyey et al. (1986) and Broutman and Young (1986) have explored the processes that transfer energy from the larger-scale flows to breaking scales. The vertical scale of internal waves can be reduced through refraction by the “background” flows of larger-scale waves (Henyey et al. 1986; Broutman and Young 1986). Their Eikonal approach involves injecting “test waves” into a Garrett and Munk (1975) background wavefield (Henyey et al. 1986) or into a near-inertial wavefield (Broutman and Young 1986), and tracking the resulting changes in wavenumber and frequency.

It has thus far been difficult to resolve small-scale internal waves in the ocean and to verify whether this mechanism is the dominant cascade process. However, the Eikonal approach provides an important intellectual context for this work. Observations are here presented of small-scale propagating disturbances with some of the same characteristics as these “test waves.”

a. Experiment

An eight-beam Doppler sonar system and a repeat-profiling CTD (Fig. 1) were operated from the Research Platform FLIP during the first leg of the Marine Boundary Layer Experiment (MBL I, Feb–Mar 1995). The experiment was sited approximately 30 km west of Point Argüello, California, in 1.5 km of water (Fig. 2). FLIP was secured with a one-point mooring during the first part of the experiment (Fig. 2). Five days before the end of the experiment, on 6 March (yearday 66), the mooring was released and FLIP drifted freely toward the northwest.

The 9-day mean buoyancy frequency profile (Fig. 3, heavy line) peaks at 7 cph at 30 m, dropping approximately exponentially to 2 cph at 400 m.

Mean velocities relative to FLIP are computed over the moored portion (yearday 59–66) of the experiment (Fig. 4, heavy solid line), the drifting portion (yearday 66–68, heavy dashed line), and the 12-h period from yearday 67.5–68 (light black line). The moored averages reflect the presence of a persistent westward “jetlike” flow: at times, observed surface currents during the experiment exceed 1 m s⁻¹. While drifting, FLIP approximately follows the depth-mean current from 0 to 90 m (Smith 1998). During this period, currents relative to FLIP from 100 to 400 m are smaller than during the moored period, but still appreciable (10 cm s⁻¹).

MBL I is an energetic, coastal site. Strong shears, large baroclinic tidal displacements, and frequent overturns are observed, making it an ideal location for a mixing study. An awareness of the differences between MBL I and more typical conditions is, however, essential. To illustrate, the depth dependence of shear squared and the tidal displacements are discussed here briefly.

Garrett and Munk (1972, 1975; henceforth GM75) synthesized numerous observations to provide a characterization of “typical” open ocean conditions. To compare MBL I shears with GM75 conditions, 6.4-m S² is plotted as a function of mean buoyancy frequency N (Fig. 5). Profiles are multiplied by 1/(0.6) to account for biasing due to horizontal separation of the sonar beams (appendix A) and corrected for attenuation by the first-difference operator. At all depths, S²(N) is significantly above GM levels (dotted line). However, observed S²(N) ∼ N² for N ≥ 2.75 cph (depth ⩽ 250 m). Below this, shear increases with depth (decreasing N), departing from the GM75 standard.

Doppler sonar records often suffer from increased noise at great range as the strength of the returned signal falls. If the observed deep elevated shear were due to instrument noise with a white frequency spectrum, the shear variance should decrease by a factor of 6 and 30 as the time-averaging is increased from 4 min (right) to 24 min (middle) to 2 h (left). The decrease is, in fact, much less, indicating that the frequency spectrum of the observed shear is red, not typical of noise. Visual examination of individual velocity profiles (Fig. 9) provides additional confidence that noise is not dominant. Increasing S² with depth below 250 m is a robust feature of the MBL I site, presumably due to bottom-generated motions.

In addition, baroclinic tidal activity is much stronger in MBL I than typically seen in the open ocean. During the spring tide of early March (yearday 59–62, Fig. 6), 40-m vertical displacement of isopycnals is observed at depths of 100–200 m. Heightened potential energy due to the displacements (black contours) is associated with a discrete tidal ray emanating from nearby topography.

b. CTD observations

The CTD, a modified Sea-Bird Instruments SBE-9, profiles the upper 415 m of the water column every 4 minutes. The fall rate of the instruments is approximately 3.6 m s⁻¹. At this rate it is not necessary to pump the conductivity cell to assure adequate flushing. Data are sampled at 24 Hz, providing a measurement every 15 cm. Only the downcast of each drop is used for analysis; during the upcast the sensors are in the wake of the package. A Macintosh computer controls the CTD winch, reads and stores the data, and displays them in real time. An external clock ensures that the profiles are synchronized to the sonar, as well as to other onboard sensors.

To reduce the effects of salinity spiking, the cross-spectrum between conductivity and temperature is estimated, using data from a region of near-uniform salinity. The magnitude and phase of conductivity are then altered to match those of temperature. The data are corrected for effects of the thermal inertia of the conductivity cell, as described in Lueck and Picklo (1990) and Morison et al. (1994). Temperature and conductivity are then low-pass filtered with a 2-m cutoff, and salinity and density are computed. The matching technique is described in more detail in Anderson (1993) and Sherman (1989).

c. Overturns

The CTD data are used to form time–depth series of density, buoyancy frequency, isopycnal depth, strain, and effective strain rate as described below. They also determine the locations of overturning events.

The Thorpe scale L_T (Thorpe 1977) is a useful indicator of both the location and vertical extent of overturning regions. Within observed density inversions, data are sorted to reestablish a monotonically increasing profile. The Thorpe displacement Th(z) is the vertical distance each parcel of water must be moved to restore it to a stable position in the water column (Fig. 7). Thus Th is zero except in the vicinity of density inversions. There, it is an objective measure of vertical excursions within the overturn. Other useful overturn statistics include the maximum displacement L_max and the patch size L_P, defined as the region over which Σ (Th) = 0 (Seim and Gregg 1994).

In practice, it is often difficult to determine whether observed density inversions are in fact statically unstable overturns. There are two main challenges. First, random noise in the temperature and conductivity sensors can introduce unphysical inversions. These might appear in low-temperature-gradient regions, where small sensor-induced fluctuations are comparable in magnitude to the true background gradient. Second, salinity spiking can create spurious inversions in density. Spiking can be detected either by examining water-mass characteristics in the immediate vicinity of the inversion, and insisting on a tight T–S relation (Galbraith and Kelley 1996), or by requiring that Thorpe displacements be nonzero in both temperature and density (Peters et al. 1995).

To maximize confidence in our determination of overturning, temperature, conductivity, and density profiles are independently sorted and Thorpe displacements computed for each. Only regions for which the conductivity- and density-based Thorpe displacements are comparable to the temperature-based Thorpe displacement are classified “overturns.” These regions are characterized using the temperature-based Thorpe displacement, rather than the density-based one, to exclude noise introduced in the density calculation. By requiring that inversions be present in both temperature and conductivity records, we minimize the influence of random noise in the sensors. In fact, nearly every observed inversion in temperature is accompanied by one in conductivity. Insisting that both temperature and density invert ensures neither lateral intrusions nor salinity spiking contaminates the overturn record. Given this cautious approach, we may not have caught all of the overturns but are confident that we have not introduced spurious ones.

A sample (large) overturn, which passed these tests, is shown in Fig. 7. Unfiltered (15 cm) conductivity and temperature are converted to equivalent densities and plotted on the same axis as density. The sorted density profile is overplotted. Calculated Thorpe displacements are shown to the right. Despite the higher resolution of the unsmoothed temperature and conductivity relative to density, all three profiles show similar Thorpe signatures.

Stable (salinity compensated) temperature inversions are common at the MBL site, particularly above 200 m. We restrict quantitative studies to the range 200–350 m to minimize concerns of intrusive contamination.

To produce depth–time maps of overturning, profiles of Th are initially formed. A list of time–depth pairs is then created, documenting each region in which the maximum Thorpe displacement L_max exceeds 2 m. In all, 2227 overturns with L_max ≥ 2 m were detected over the 9-day period (yearday 59–68), between 200 and 350 m. These occupy 3.06% of the sampled water column (Table 1). During the 12-h period (yearday 67.5–68.0) highlighted in this work, 330 overturns with L_max ≥ 2 m were found (Table 2). They occupy 2.96% of the water column.

The probability density of L_T for the 2227 overturns is presented in Fig. 8. The PDF falls off for L_T ⩽ 1.5 m, reflecting our minimum size cutoff. For L_T ≥ 1.5 m, the distribution is consistent with the tail of a lognormal distribution (dashed line), with σ_log10LT = 0.37, = −0.1. Lognormal distributions of L_T are typically observed (e.g., Wijesekera et al. 1993; Moum et al. 1992).

d. Sonar observations

The Doppler sonar, designed and constructed at the Marine Physical Laboratory of Scripps, was mounted on the stern of FLIP at approximately 87-m depth (Fig. 1). It consists of four upward-looking (170 kHz) and four downward-looking (140 kHz) beams directed 30° from vertical. Only the downward-looking portion of the sonar data is used here. A repeat sequence code (Pinkel and Smith 1992) consisting of four repeats of a 7-bit Barker code is transmitted. The bandwidth of the transmitted code is 16.6 kHz. Vertical resolution is 3.2 m (however, data are sampled every 1.6 m). Transmissions are made every 1 s, and estimates recorded every minute. To improve precision and to enable integration with CTD data, 4-min averages are subsequently formed. These are aligned in time with the downward section of each CTD profile. The rms precision of these 4-min averages, estimated from frequency spectra, is 1.1 cm s⁻¹. The usable range of depths sampled by the downward-looking sonar is 100–500 m. Horizontal velocity estimates are computed by averaging back-to-back beam velocities. Shear magnitude, S, is computed by differencing velocities over 6.4 m and then smoothing with a 6.4-m convolution filter.

A sample of full (3.2 m)-resolution velocity data is displayed in Fig. 9. Twenty profiles of zonal velocity spanning 80 min near yearday 67.9 are plotted, offset by 5 cm s⁻¹. Note the reproducibility of small-scale velocity features from profile to profile.

The high quality of the MBL sonar enables the present examination of finescale Ri. Still, two aspects of the sonar affect S² (and therefore Ri) estimates. The horizontal separation of the beams biases S² low since shear on horizontal scales comparable to the beam separation is underrepresented: Instrument noise biases S² high. These competing effects are discussed in appendix A.

e. Finescale quantities

Estimates of the 6.4-m Richardson number are formed from the combined CTD and sonar data. Profiles of buoyancy frequency N are formed by differencing the sorted density profiles, then smoothing with a 6.4-m Butterworth filter. This procedure ensures that the two constituents of Ri have similar bandwidth. Variations in N are comparable to those in S in producing 6.4-m Ri fluctuations.

We wish to monitor fluctuations in finescale quantities in both Eulerian and isopycnal following frames. The Eulerian time series of a thermocline variable reflects temporal and spatial variability; that is,

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When ∇θ has finescale irregularities, the time series is termed “fine-structure contaminated.” With a continuous depth–time view, we can alter the concept of an Eulerian time series to avoid “contamination.” Specifically, we define the Eulerian time fluctuation of a variable at depth z_o relative to its time mean, where the mean is taken along the isopycnal, which is passing through z_o at that instant.

As an example, consider an Eulerian temperature record under this definition:

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where T̂_Eul indicates whether the various isopycnal surfaces passing through z_o are warmer or colder than their respective long-term average values. In the absense of lateral T–S variability, mixing, and radiative transport an Eulerian temperature record is constant. The signature of vertical advection is absent.

In like manner, we consider the anomaly in strain

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and squared shear

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The Eulerian Richardson number is simply related to the strain and shear anomalies:

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It is also instructive to examine time variability in a semi-Lagrangian (isopycnal following, s-L) reference frame. The s-L frame is established by a set of reference isopycnals, {ρ_i}, whose mean depths are separated by 1 m. The depth z(t, ρ_i), and displacement η(t, ρ_i) ≡ z(t, ρ_i) − z(ρ_i), of each isopycnal are computed from each density profile (after sorting to remove inversions) by linear interpolation. Observations are expressed in an s-L frame by linear interpolation of the Eulerian data to the depths of the specific reference isopycnals.

It is convenient to define s-L strain slightly differently than in an Eulerian frame (Pinkel et al. 1991). Between any two time series of isopycnal depth z(t, ρ_i) and z(t, ρ_j), the normalized depth difference is given by

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where z is the mean depth of the pair and Δz is the mean separation. The finite difference strain is defined in terms of the normalized isopycnal separation:

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Positive values of strain (normalized isopycnal separation greater than one) refer to isopycnals farther apart than average.

An “effective strain rate” can be defined as

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To estimate this quantity, strain observations are differenced in time and divided by the normalized separation. Precision of γ and ∂ŵ/∂z estimates is discussed in appendix B.

The lateral advection of horizontal structure complicates interpretation of this strain rate. In fact,

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Specific acknowledgment of the advective presence in these data distinguishes our “effective strain rate” from the true “vertical divergence” ∂w/∂z as considered in Sun and Kunze (1999), for example.

a. Eulerian maps

Representative 12-h depth–time series of Eulerian 2-m strain γ_Eul, 6.4-m shear squared anomaly Ŝ²_Eul, and 6.4-m Ri Ri_Eul are presented in Fig. 10. Twenty-three such 12-h periods have been examined. One hundred and eighty 4-min profiles compose each 12-h map.

Two-meter Eulerian γ_Eul (Fig. 10a) represents the distortion of the water column by passing waves. (It is also affected to a smaller degree by lateral intrusions—see appendix C.) By and large, strain features advect vertically with the displacement field. However, some cross-isopycnal migration is seen (this aspect of the strain field will be seen more clearly in the s-L view). Notice the tendency of overturns (yellow–green dots) to lie in regions of high strain, low N.

The 6.4-m Ŝ²_Eul is characterized by “layers” of several hours duration (Fig. 10b). These are advected vertically by the isopycnal displacement field, as observed by Anderson (1993). High frequency features are reproduced in the shear and displacement fields (e.g., the “wiggles” at yearday 67.63–67.65, 150 m).

Contributions to Ri may be examined separately in Figs. 10a and 10b; jointly in Fig. 10c. Layers of low Ri advect vertically with time with an apparent (advection affected) timescale of a few hours. By and large, these layers follow isopycnal surfaces. However, there are notable exceptions, which will be more apparent in the close-ups and s-L maps, which follow.

Many overturns (black dots) lie in regions of low 6.4-m Ri (e.g., yearday 67.93, 350 m, yearday 67.93, 350 m and yearday 67.94, 300 m). Larger-scale internal waves vertically advect the latter region and associated overturns. The large overturning region beginning at yearday 67.75, 375 m is centered on a low Ri, high shear layer but moves relative to it.

Some layers of 6.4-m Ri remain low for hours without any observed 2-m overturning (e.g., yearday 67.8–67.9, 350 m). The Richardson number Ri must remain subcritical long enough for instability to develop, a time of order 4/(|S| − 2N) (Hazel 1972). Therefore a one-to-one relationship between low Ri and overturning is not expected. For example, Kunze et al. (1990) noted a lack of 1-m overturns in unstable 1-m Ri regions. But the lack of 2-m overturns in low 6.4-m Ri regions is surprising. Assuming that there is active overturning, which we do not resolve, a significant separation between the scale of the unstable Ri and that of the resulting instability is implied. Alternately, Ri ⩽ Ri_c (where Ri_c is a critical Richardson number) may be necessary, but not sufficient, for instability, as suggested by linear theory (Miles 1961; Howard 1961).

Seventy-four percent of overturns lie in regions of 6.4-m Ri ≥ 1, 50% where 6.4-m Ri ≥ 2. Regions at (yearday 67.75, 150 m), (yearday 67.95, 160 m), and (yearday 67.73, 340 m) are examples. Many such overturns lie in regions of either high strain or high effective strain rate magnitude (described below).

Recall that shear variance estimates (on which 6.4-m Ri is based) are increased by instrument noise and decreased by the finite beam separation of the sonar (appendix A). Our 6.4-m Ri estimates might be large by as much as a factor of 1/(0.6) relative to a “true” 6.4-m Ri estimate.

In addition, vertical resolution is a concern: Ri decreases with decreasing vertical scale (e.g., Kunze et al. 1990). Assuming a “canonical” shear spectrum white out to vertical wavenumber m = 0.1 cpm and falling off as m⁻¹ thereafter (Gargett et al. 1981), 6.4-m S² is only 55% of 2-m S². Nevertheless, the observed large-scale patterns are intriguing. Is (unresolved) shear between 6.4 m and 2 m sufficient to trigger 2-m overturns? The observations of extended regions of low 6.4-m Ri with no 2-m overturning, by contrast, are robust with respect to instrument concerns.

Apparent time variability in Fig. 10 and subsequent figures is due to true temporal modulation, mixing, and the advection of features past FLIP. Relative currents are 20 cm s⁻¹ toward the south-southwest during this period (Fig. 4). In the absense of intrinsic temporal changes, an observed hour-long feature would correspond to a 720-m-long structure drifting past at 20 cm s⁻¹. Attempts in later sections (3d and 4b) to assess the relative contributions of intrinsic and advective variability suggest that neither can be ignored.

b. Case studies of instabilities

It is of value to examine individual overturning events in detail. In Fig. 11, we present close-ups in depth (⩽50 m) and time (⩽3 h) centered around three different overturning events seen in Fig. 10.

Figure 11a shows a series of overturning events associated with low 6.4-m Ri. Shear is high in this region, as evident from Fig. 10b. In addition, the buoyancy frequency is low, as indicated by a larger than average spacing between isopycnals. The region is sandwiched between higher gradient regions above and below. The feature crosses five isopycnals (moving toward less dense water) over the 3-h period. It is also advected vertically with the background flow. Intermittent overturning is approximately centered on the shear maximum. We interpret Fig. 11a as a “classic” shear instability, caught in the act.

Figure 11b depicts a second event, where 6.4-m Ri is not uniformly low (the color scale has been extended so that white indicates Ri ≥ 2, instead of Ri ≥ 1 as in Figs. 11a and 11c). Scale differences (6.4 m for Ri, 2 m for overturns) and/or beam separation effects (section a of appendix A) may be responsible for the poor correspondence. However, the event is also associated with the passage of a strain feature across these isopycnals, as seen by the large isopycnal separation. High-wavenumber, high-frequency motions make the isopycnals appear “jittery” in this region. These motions will be explored in detail in section 3d. Overturning ceases when the feature propagates (or is advected) away.

A third example of an instability event is presented in Fig. 11c. It is associated with low 6.4-m Ri, as in Fig. 11a, and with the passage of a small-scale disturbance, as in Fig. 11b. Very large overturns are colocated with a high strain feature, which “peels off” isopycnals from above as it migrates vertically. Reminiscent of a spilling surface wave, overturning follows the wave feature as it migrates upward.

c. Semi-Lagrangian maps

In Fig. 12, semi-Lagrangian maps of strain, shear squared anomaly, and Ri are presented from the same time and depth range as the Eulerian fields of Fig. 10. Each map has been averaged over 24 min (six profiles) in time prior to plotting (longer time averaging is possible in the s-L frame since smearing of features by vertical advection is avoided). The ordinate is not depth, but rather the isopycnal whose mean depth is indicated. Vertical excursions on a s-L plot imply cross-isopycnal migration, while horizontal variations indicate change at fixed density.

Contours surrounding regions of 6.4-m Ri ⩽ 1 have been plotted in Figs. 12a and 12b. Low-Ri regions are formed by both high shear and low strain; ignoring variations in either constituent is not justifiable.

The 2-m γ_s-L features (Fig. 12a) slowly migrate across isopycnal surfaces; e.g., at (yearday 67.77, 330 m) and (yearday 67.52, 250 m). They persist for several hours on average. Lateral advection affects the perceived cross-iospycnal migration: observed vertical excursions can represent true propagation or the sampling of spatially tilted features as they drift past.

Overturns, marked with yellow–green spots, demonstrate a clear tendency to lie in regions of high strain. Persistent, high strain features that exhibit overturning are reminiscent of the low-gradient, high-dissipation “patches” observed by Gregg et al. (1986) and by Marmorino et al. (1987). Relative to 6.4-m shear (Fig. 10b) and 6.4-m Ri (Fig. 10c), changing (propagating) strain is often the best indicator of overturn locations.

d. Effective strain rate

The effective strain rate, ∂ŵ/∂z (Fig. 13a), involves both the depth and time derivatives of isopycnal displacement, emphasizing higher-frequency, higher-wavenumber constituents in the resolved field. In the absence of horizontal advection, it is equivalent to the “vertical divergence” ∂w/∂z [Eq. (8)], which appears in the ray-tracing equation governing the time rate of change of the vertical wavenumber of a Henyey/Broutman “test wave” (Sun and Kunze 1999).

Imprecision in density estimates introduces rms |∂ŵ/∂z|_noise = 3.2 × 10⁻⁴ s⁻¹ in our estimates (appendix B). The color scale of Fig. 13 is such that strong features are well above the noise level.

Wavelike motions are seen in the ∂ŵ/∂z field. Anticipated horizontal phase speeds (0.5–5 cm s⁻¹, appendix D) are lower than typical advective velocities (Fig. 4, 20 cm s⁻¹). Sloping features may therefore represent the advection of spatially slanted wave crests, in addition to true vertical propagation.

To focus on the wavelike aspects of the ∂ŵ/∂z field, components of downward-propagating (Fig. 13b) and upward-propagating phase (Fig. 13c) are separated. These are formed by two-dimensional Fourier transform filtering of the depth–time series in Fig. 13a.

Clear “groups,” localized regions of large amplitude, may be seen in both upward and downward fields. Notice the presence of upward-traveling groups, with downward-traveling phases (consistent with linear internal wave propagation): for example, Fig. 13b (yearday 67.73, 300 m), (yearday 67.53, 240 m), and (yearday 67.57, 170 m). Also common, however, are regions where group and phase are both up (e.g., yearday 67.72, 200 m) or both down (e.g., yearday 67.73, 250 m). These apparent nonpropagating features are consistent with the lateral advection of internal wave groups.

Other interpretations are of course possible. However, if the observed ∂ŵ/∂z features result, for example, from the advection of quasi-permanent density structure, it is hard to account for the regular crest/trough patterns, groupiness and presence of up/down features in the same region. If the signals do indeed represent waves, what can be conjectured about their intrinsic frequency and magnitude?

The intrinsic frequency ω_int is difficult to determine, given the Doppler shifting of the waves by background currents and larger waves (Doppler shifting due to vertical motions is absent in a s-L frame). Still, simple estimates of ω_int and horizontal wavenumber k may be made assuming a linear internal wave dispersion relation and advection by observed horizontal currents (appendix D). In a current of 20 cm s⁻¹, a typical wave of vertical wavenumber 0.08 cpm (12 m) and observed frequency 4 cph (like the feature at yearday 67.73, 300 m) would have ω_int near 0.14 cph (7 h timescale) and k near 5.7 × 10⁻³ cpm (180-m wavelength), for a variety of propagation directions relative to the current. Advection can shift such waves from intrinsic timescales of hours to observed frequencies near N and higher. Uncertainties in velocity and encounter angle strongly affect these estimates.

It is also instructive to consider the magnitude of fluctuations that results from a single internal wave with displacement η = a sin(kx + mz − ω_intt); k and m are horizontal and vertical wavenumbers, and ω_int is the intrinsic frequency. Then, |∂ŵ/∂z| = amω_int. To prevent isopycnals from crossing, am = |γ̂| ⩽ 1. Here ω ⩽ N implies |∂ŵ/∂z| ⩽ N = 5 × 10⁻³ s⁻¹ (3 cph), representing an upper bound on single-wave “intrinsic” contributions to ∂ŵ/∂z. Observed values are in this range (Fig. 13, Figs. 19–20). On the other hand, the same wave swept past FLIP at velocity U would produce |∂ŵ/∂z|_adv [the second term on the rhs of Eq. (8)] = Ukma. Taking U = 20 cm s⁻¹ and k = 1/(300 m), |∂ŵ/∂z|_adv ⩽ 4 × 10⁻³ s⁻¹—the same order as the intrinsic term.

Based on these simplistic considerations, the observed ∂ŵ/∂z features appear consistent with Doppler-shifted internal waves of intermediate frequencies. Still, their short duration and limited vertical propagation range suggest that they are significantly nonlinear, violating the assumptions of the intrinsic frequency calculation. A more detailed study of the dynamics of these waves is clearly called for and will be the subject of a future paper.

Whatever their nature, overturns are much more likely to occur in regions of high |∂ŵ/∂z| (Fig. 14). Here the envelopes of the small-scale wave packets are highlighted. For example, the instability event given in close-up (Fig. 11c) appears on this plot as a feature that begins at (yearday 67.5, isopycnal number 250) and migrates upward. Most overturns lie inside the packet envelopes (yellow regions): 80% of observed overturns occur where |∂ŵ/∂z| ≥ 1.73 × 10⁻⁶ s⁻², the 9-day mean value from 200 to 350 m. Regions with 6.4-m Ri ⩽ 1 are also contoured in blue in each panel. Almost every overturn is encircled by either a blue or a yellow contour. Based on these observations, two types of phenomena appear to result in 2-m overturns: high 6.4-m shear layers, which are well resolved by the sonar, and internal wave groups, which are better indexed by CTD-inferred |∂ŵ/∂z|.

a. Finescale statistics: Overturns

To quantify the relationship between the various finescale fields and the overturns, probability density functions (PDFs) of Richardson number, strain, and effective strain rate are estimated. Separate PDFs of the total field (259 856 Eulerian data; 339 599 s-L data) and the subset of observed overturns (10 422 Eulerian data;5522 s-L data) are created. The overturning totals are greater than the number of observed overturning events (2227) since the events can span multiple isopycnals and depth bins.

PDFs of inverse Richardson number in the s-L frame (Fig. 15a) and the Eulerian frame (Fig. 15b) are in qualitative agreement with results of Pinkel and Anderson (1997a). In both reference frames, high values of Ri⁻¹ are more likely when overturns are present (light black lines): 37% of overturns, compared with 30% of all observations, have Ri⁻¹_s-L ≥ 1. In the Eulerian frame, 23% of overturns have Ri⁻¹_Eul ≥ 1, compared with 16% of all observations.

PDFs of inverse 2-m strain for all data (Fig. 15c, heavy line) are consistent with the gamma distribution of Pinkel and Anderson (1992). When only overturning data are sampled (light line), the distribution is shifted toward lower values of inverse strain: overturns are much more likely to lie in regions that are less stratified than average. This is consistent with the conclusion drawn visually from the depth–time maps (Figs. 10a and 12a).

The PDF of log₁₀[(∂ŵ/∂z)²] (Fig. 15d) is indistinguishable from lognormal (dashed line, obscured by heavy line). The PDF of (∂ŵ/∂z)², considering overturning data only, is also roughly lognormal, but peaks at a higher value, demonstrating the tendency of overturns to occur where the effective strain rate magnitude is large.

The probability density of ∂ŵ/∂z, computed using all data, is symmetric about ∂ŵ/∂z = 0 (Fig. 16). Extreme values of ∂ŵ/∂z are more common than in Gaussian distributions (dashed and light lines).

b. Finescale statistics: Before and after overturns

The previous section examined differences in statistics of finescale variables under all conditions and under “unstable” conditions. That analysis is here extended to include not only overturns but periods just before and just after overturns as well. The goal is to use statistics of many overturns to construct a schematic picture of finescale conditions before and after a representative “event.” Can differences be detected between “leadup” and “aftermath” of overturns? Relevant issues are the degree to which observed before/after data represent the time history of events (rather than just upstream/downstream samples of drifting features), and the amount by which the existing flow is altered by the overturning process.

Before/after PDFs of Ri⁻¹ and γ⁻¹ and (∂ŵ/∂z)² are presented in Figs. 17 and 18. “Leadup” PDFs of Ri⁻¹ (Fig. 17a), γ⁻¹ (Fig. 17c), and ∂ŵ/∂z (Fig. 18a) are computed from data sampled 4 min (magenta line), 8 min (blue line), and 24 min (approximately one buoyancy period 1/N, green line) before observed overturning events. PDFs of all data and overturning data (last section) are replotted in black and red, respectively. Figures 17b,d and Fig. 18b are the corresponding “aftermath” PDFs, computed using data sampled after overturning events. The number of observations used in forming the various PDFs (noted in the legends of fig. 17a,b) differs slightly since overturns at the very beginning/end of the record cannot be used to form before/after PDFs, respectively.

Ri⁻¹ and γ⁻¹ PDFs (Fig. 17) each constitute a family of curves leading smoothly from “normal” to “unstable” conditions (before: Fig. 17a,c), and back again (after: Fig. 17b,d). Conditions one buoyancy period before overturns (magenta line) are already distinguishable from “normal” conditions (black line): the probability of higher Ri⁻¹ and lower γ⁻¹ values is elevated. Here γ⁻¹ PDFs continue a steady progress toward “unstable” conditions. By contrast, the 4-min and 8-min before Ri⁻¹ PDFs are similar to the overturning PDF. Apparently, Ri⁻¹ reaches the overturning value 4–8 min before a 2-m overturn is detectable.

No asymmetry is distinguishable between “before” and “after” in the Ri⁻¹ and γ⁻¹ curves. Two interpretations are possible. If intrinsic time dependence is not being observed, “before” and “after” represent spatially offset “upstream” and “downstream” samples. In an isotropic ocean, no asymmetry would be seen. Alternately, the similarity between the curves can mean that overturns affect the larger-scale flow only slightly, as in spilling surface waves. Overturns occur without drastically affecting the surrounding Ri⁻¹ and γ⁻¹ fields.

Effective strain rate, in contrast, exhibits differences between before (Fig. 18a) and after (Fig. 18b) periods. Relative to the curve representing normal conditions, the 24-min before curve is shifted to the right. The 4, 8-min before curves overlie the overturning curve, indicating that “overturning” conditions are present 8 min before overturns actually occur, as in the Ri⁻¹ curves. After overturns, however, the behavior is different. A more rapid transition back to normal conditions is seen:the PDFs 4 min and 8 min after are shifted slightly relative to the overturning curve. That is, |∂ŵ/∂z| drops immediately after overturning occurs. The difference between before and after effective strain rate PDFs suggests that perceived variability in ∂ŵ/∂z is due to true temporal evolution, in addition to horizontal advection. Evidently, the ∂ŵ/∂z field is affected by overturning. Does instability in the small-scale waves result in their destruction, leaving the larger-scale flow relatively unaffected?

c. Joint statistics

We observe correspondence between overturns and low Ri, high γ and high |∂ŵ/∂z|. The codependence of these quantities is thus of interest. To investigate, joint PDFs of (log_10γ, ∂ŵ/∂z) and (Ri⁻¹, ∂ŵ/∂z) (Figs. 19 and 20) are formed. As in the one-dimensional study (section 4a), probability density is computed for all data over 9 days and 200–350 m, as well as for overturning observations only.

The γ and ∂ŵ/∂z are not independent quantities (Fig. 19a). Isopycnals that are far apart (large γ) are more likely to be converging (negative ∂ŵ/∂z). Conversely, isopycnals closer together than average display a slight tendency to diverge. However, at extreme (large or small) values of γ, large values (positive or negative) of ∂ŵ/∂z are unlikely.

Considering only overturns (Fig. 19b), the PDF is shifted toward higher values of γ and exhibits more spread in ∂ŵ/∂z than the PDF computed from all data. These features are consistent with the one-dimensional PDFs (Fig. 15) and the depth–time maps (Figs. 10a, 12a, and 14). The probability density is again asymmetric in the sense that low gradient regions are associated with converging isopycnals.

The log of the ratio of the two PDFs is presented in Fig. 19c. Values greater than zero indicate higher probability density for overturning conditions than for the entire population. Regions where both the numerator and the denominator are small are left white. From 19c, overturning is much more likely for large values of γ. However, if it is to occur at smaller values of γ, then |∂ŵ/∂z| must be high. It is much less likely for overturns to be associated with low γ and low |∂ŵ/∂z| than for the general population.

Joint probability density functions of 6.4-m Ri⁻¹ and ∂ŵ/∂z are presented in Fig. 20. Simultaneous large values of general population Ri⁻¹ and ∂ŵ/∂z are unlikely (20a). This may be due to the small implied horizontal wavelength of the ∂ŵ/∂z waves (appendix D), which causes their shear to be underestimated due to the separation of the sonar beams (section a in appendix A). PDF_overturns (Fig. 20b) extends to higher values of Ri⁻¹ and |∂ŵ/∂z| than PDF_all.

The log of the ratio of the two PDFs (Fig. 20c) suggests two distinct conditions that accompany overturning: high Ri⁻¹ (associated with shear instability) and high |∂ŵ/∂z| (associated with the presence of wave groups). As noted, however, the ∂ŵ/∂z waves likely contain shear unresolved by the sonar. The probability of overturning is increased for either low Ri⁻¹ and high |∂ŵ/∂z| (K–H stable, waves present) or high Ri⁻¹ and low |∂ŵ/∂z| (K–H unstable, no waves present). A ridge connects the two regimes.

a. Variability as a function of instantaneous stratification

We first investigate statistical relationships between L_T, ε_T and the instantaneous bouyancy frequency N(t, z). Considering only the overturns, L_T and ε_T from discrete events are averaged with corresponding values from other events at like N. Forty averaging bins, evenly spaced in log₁₀N, are established between 0.1 and 10 cph.

Over instantaneous bouyancy frequency ranges, 0.8 ⩽ N ⩽ 3 cph, L_T ∼ N^−1/2 (Fig. 23a) and ε_T ∼ N² (Fig. 23b). There are very few realizations at the more extreme values of N. (The mean buoyancy frequency does not fall below 2 cph in this domain, Fig. 3.) Sampling error reduces the precision of the relationship at these extreme N. We emphasize that decreasing local N does not imply increasing depth since strains at the MBL site are sufficient that all values of N are deformed into a wide range in N. We also note that a fixed rms error in density, Δρ = 3 × 10⁻⁴ kg m⁻³ (appendix B), establishes apparent overturns of magnitude

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This function (Fig. 23, dash–dot line) is much steeper than, and well below, the observed relation, indicating that that instrument noise does not dominate these overturn estimates.

b. Variability as a function of mean stratification

Typically, the N dependence of ε is investigated in terms of long-term Eulerian averages, where average dissipation rate estimates are compared with mean stratification, N. The MBL observations are here viewed in this format. To convert from averages over “event” to averages over time, the overturning fraction p_o(z) is introduced. To compute p_o, an overturn index matrix is formed, with bins every meter in depth, every 4 min in time, over the 9-day data record. The matrix is initially zero filled.

Iterating through each of the 2227 overturns, a value of unity is stored in all bins spanned by that overturn. Averaging in time then produces the fraction p_o(z) of the water column sampled (at depth z), which is occupied by overturns. (The depth mean of p_o from 200 to 350 m yields the overall overturning fraction, 3.06%.)

The overturning fraction p_o is then plotted versus the 9-day mean N (Fig. 24). Unfortunately, N exhibits only a small range of variation (2–3 cph) over the 200–350 m observation window. To widen the range of N to 4 cph the study is extended upward, including overturns to depths of 120 m (thin line). For N ≥ 2.5 cph p_o(N) ∼ N⁻³, N⁻² (dashed lines). At lower N, p_o appears constant.

A similar procedure can be used to compute the Eulerian average dissipation, ε_mean(z). Instead of storing unity in the overturn index matrix, ε_T = 0.64L²_TN³ is stored for each overturn. Averaging across time produces ε_mean(z), which is plotted versus N(z) in Fig. 25a.

For N ≥ 2.5 cph, ε_mean ∼ N², consistent with previous observations ε_mean ∼ N¹ − N² (Gargett and Osborn 1981; Gargett and Holloway 1984; Lueck et al. 1983; Gregg et al. 1986) and with theoretical predictions ε_mean ∼ N² in terms of wave–wave interactions (Henyey et al. 1986; McComas and Müller 1981; Gregg 1989; Polzin et al. 1995). At greater depths, N ⩽ 2.5 cph, a different regime is encountered. Dissipation increases with increasing depth, decreasing N. In this depth range, rms shear also begins to increase with depth (Fig. 5), suggesting the influence of seafloor/coastal wave generation. The Gregg (1989) dissipation parameterization [Eq. (E1)] explicitly accounts for both shear and stratification (Fig. 25, light line). For the regime N ≥ 2.5 cph, the two dissipation rate estimates show ε ∼ N². For N ⩽ 2.5 cph, both show the increase in dissipation with depth. Though agreement in this range is not perfect (the Gregg parameterization exhibits steeper slope), the similarity in general form of the two estimates (one involving only CTD observations and the other involving primarily sonar data) is significant.

Under both parameterizations, strong diapycnal diffusivities are implied by these results (Fig. 25b). Estimates of K_ρ increase with depth for N ⩽ 3 cph, exceeding 10⁻⁴ m² s⁻¹ at the bottom of the record, still 1000 m above the seafloor. Polzin et al. (1997) observed heightened K_ρ values (up to 10⁻³ m² s⁻¹) above the mid-Atlantic Ridge, decreasing upward into the water column. Upward-propagating breaking waves generated by flow over the ridge topography are assumed to be responsible. Given the proximity of the MBL site to topography and the observed ∂ŵ/∂z waves that appear to be breaking, the similarity in magnitude and depth dependence of MBL-inferred K_ρ values and those of Polzin et al. (1997) is interesting. Are the same processes at work in both locations?