Ocean Mixing by Kelvin-Helmholtz Instability

aBStr ac t. Kelvin-Helmholtz (KH) instability, characterized by the distinctive finite-amplitude billows it generates, is an important mechanism in the development of turbulence in the stratified interior of the ocean. In particular, it is often assumed that the onset of turbulence in internal waves begins in this way. Clear recognition of the importance of KH instability to ocean mixing arises from recent observations of the phenomenon in a broad range of oceanic environments. KH instability is a critical link in the chain of events that leads from internal waves to mixing. After 150 years of research, identifying the prevalence of KH instability in the ocean and defining useful parameterizations that quantify its contribution to ocean mixing in numerical models remain first-order problems.

finite-amplitude billows it generates, is an important mechanism in the development of turbulence in the stratified interior of the ocean.In particular, it is often assumed that the onset of turbulence in internal waves begins in this way.Clear recognition of the importance of KH instability to ocean mixing arises from recent observations of the phenomenon in a broad range of oceanic environments.KH instability is a critical link in the chain of events that leads from internal waves to mixing.After 150 years of research, identifying the prevalence of KH instability in the ocean and defining useful parameterizations that quantify its contribution to ocean mixing in numerical models remain first-order problems.

INtrODuctION
A regime of strongly nonlinear fluid motions exists at scales smaller than can be resolved by global ocean models.They include a broad range of phenomena that exhaust at least some of their energy to turbulence.While these organized motions are generally well understood, there is as yet no deterministic theory for the resulting turbulence.In practice, we understand turbulence through the statistics of its density, velocity, and vorticity fluctuations.Turbulence stirs the ocean, stretching material surfaces and locally increasing gradients to the point that they rapidly and irreversibly diffuse at molecular scales.This process is ultimately responsible for mixing the ocean.
The transition from organized flow to turbulence occurs through a sequence of instability processes, beginning with a primary instability.Different primary instabilities dominate under different cir- Seen from the side, such clouds often take shapes similar to those of surface waves breaking on a beach.Figure 1 shows vivid examples from the atmosphere and from a low-level shear flow in the Canadian Arctic.Increasingly highfidelity ocean measurements have led to clear observations of the presence and structure of KH billows in the ocean.
In this article, we review the history and current status of research into KH instability with a focus on its role in the energy cascade from oceanic internal waves to small-scale turbulence.

SOMe HIStOry
Named in honor of the pioneering investigators William Thomson (Lord Kelvin, 1824-1907)  Theoretical studies of the homogenous shear layer by Lord Rayleigh (1880) were extended to include the effects of stable density stratification by Taylor (1927Taylor ( , 1931)), who also demonstrated stratification effects in the laboratory, and Goldstein (1931).The equation that describes the instability in the absence of viscosity and diffusion is called the Taylor-Goldstein equation in their honor (Thorpe, 1969).Miles (1961) and Howard (1961) showed the critical value of Ri to be ¼.
The occurrence of KH billows in the ocean was first revealed when scuba divers conducted dye-release experiments in the stratified thermocline of the Mediterranean Sea (Woods, 1968, shown here in Figure 3).Photographs of the resulting dye patterns revealed billow trains associated with internal  (Thorpe, 2002) and thin striations consistent with convection rolls (Klaassen and peltier, 1991).
gravity waves (IGWs) of much greater wavelength, as well as small-scale streaks suggestive of secondary instability (Figure 3b).The spatial and temporal scales of the billows compared favorably with predictions based on the theory of KH instability.The photos also showed the role of KH billows in IGW breaking as, following the instability, the dye quickly mixed away.Hazel (1972)  The importance of KH instability has been documented observationally in a variety of oceanic regimes, for example, within strongly sheared estuarine flows (Geyer and Smith, 1987;Geyer et al., 2010) and at the edges of gravity currents (Wesson and Gregg, 1994).Van Haren and Gostiaux (2010; Figure 4) have observed a train of about 10 billows in a highly sheared zone associated with tidal flow at 560 m depth over Great Meteor Seamount.

KH INStaBIlIt y aND INterNal WaVe BreaKING
In uniform stratification (a useful approximation for the main thermocline), shear instability is dominant in waves of near-inertial frequency, where motions are nearly parallel and primarily horizontal.This is true even when wave amplitude is large enough that isopycnals are overturned (Dunkerton 1997;Lelong and Dunkerton, 1998a,b).(However, instability differs from the standard KH model in that the mean flow structure is a sinusoid rather than a shear layer.) Higher-frequency waves are more likely to break via convective instability.
Internal waves in the thermocline may also break via parametric subharmonic instability (e.g., Hibiya et al., 1998), or by a resonance with small-scale instability such as salt fingering (Stern, 1969).
Nonlinear interfacial waves, as may occur at the base of a surface mixed layer, break primarily via KH instability (e.g., Moum et al., 2003;Lamb and Farmer, 2011).Internal waves encoun- Figure 2. tilting tank laboratory experiment in which a dense (here, dark) layer of fluid underlies a lighter fluid initially at rest.When the tank is tilted, buoyant forces accelerate the denser fluid down and the lighter fluid up the slope, thereby creating a velocity gradient across the interface and subsequent billow formation (Thorpe, 1971).
Many classes of ocean models implicitly assume mixing via shear instability by applying mixing in regimes of low Ri, for example, the one-dimensional mixed layer models of Mellor and Yamada (1982) and Price et al. (1986).
The Gregg-Henyey-Polzin scaling of dissipation due to fine-scale internal wave interactions (Henyey et al., 1986;Gregg, 1989;Polzin et al., 1995) rests on a similar assumption.A direct connection between turbulence and low Ri in the North Atlantic thermocline has provided quantitative evidence for the prevalence of shear instability (Polzin, 1996).

MecHaNIcS OF tHe eNerGy caScaDe
The tendency of vorticity in a parallel shear layer to accumulate into evenly spaced maxima is the primary driver of (e.g., Klaassen andPeltier 1985, 1991).
This perturbation is identified as a secondary instability.In the example given in Figure 6, the secondary instability is represented by its vorticity field, which describes a series of counter-rotating convection cells in regions of the core where the density field is overturned.
Similar motions have been seen in laboratory experiments (e.g., Thorpe 1985), in the ocean (Figure 3b), and in clouds (Figure 1c).Other examples include the secondary KH instability (Corcos and Sherman, 1976;Staquet, 1995;Smyth, 2003; Figure 7d), the "knot" instability that causes localized pairing (Thorpe 1985(Thorpe , 2002; Figure 1c), and the "stagna-  The first numerical simulations of KH instability were restricted to two dimensions due to memory limitations (e.g., Patnaik et al., 1976;Klaassen and Peltier, 1985).These simulations confirmed the primary instability but could not resolve the subsequent transition to three-dimensional motion.Threedimensional simulations became possible in the 1990s (Caulfield and Peltier, 1994;Scinocca, 1995) and have been used in numerous studies since then as  The structure located between the two largest billows may be an example of the stagnation point instability (Mashayek and peltier, 2011).
computational capacity has increased.
Early numerical studies assumed that momentum and mass diffuse at the same rate, as is approximately true in air.Increasing computer size has also allowed simulations of instability in seawater, which is more difficult due to the slow diffusion rates of heat and salt (Smyth, 1999;Kimura and Smyth, 2007;Smyth and Kimura, 2011).
In the example shown in Figure 7, the diffusivities are characteristic of thermally stratified seawater (see Smyth and Thorpe, 2012, for further details).Linear stability analysis provides compelling evidence that the oscillations found in strongly sheared equatorial currents are, in fact, a KH instability.
Predicted wavelengths of computed instabilities (Sun et al., 1998; Figure 10a) agree with directly measured wavelengths (Moum et al., 1992), and predicted frequencies (Smyth et al., 2011; Figure 10b) also agree well with directly measured frequencies (Moum et al., 2011).These analyses show that the con- the atmosphere (Busack and Brummer, 1988) and in the ocean (Mack and Hebert, 1997;Sun et al., 1998), and is     Oscillations in the upper equatorial pacific, thought to arise from KH instability, show a striking tendency to have frequency close to N, the buoyancy frequency.This is not a property of individual KH billows, but may follow from the statistics of a random ensemble of instabilities due to sporadic shear amplification by interacting gravity waves.consider a random ensemble of instabilities growing on small shear layers within a larger sheared zone.phase velocities lie approximately within the range of the mean current (Howard, 1961; Figure B1), U min < c < U max .The observer's velocity is assumed to lie in the range of the mean flow, so that U min < 0 and U max > 0. Wavelengths lie between zero and about seven times the thickness of the largest shear layer (e.g., Hazel, 1972; Figure B2).
treating phase velocity and wavelength as independent random variables yields a probability distribution function for the frequency (Figure B3), shown in blue for representative values U min = -1 ms -1 , U max = 1 ms -1 , and λ 0 = 700 m.For comparison, gray bars show the frequency distribution of instabilities computed via linear stability analysis of the observed velocity and density profiles.The theoretical peak frequency, f peak = max (|U min |, U max )/λ 0 , is consistent with that of computed instabilities for this regime, and also with observations (Figure 10).
Given that the maximum possible wavelength λ 0 is about 2π times the thickness D of the sheared zone, the peak angular frequency ω peak = 2π f peak is within a factor two of the mean shear.(Since U min < 0, the mean shear would be (U max + |U min |)/D, and the sum of two positive numbers is, at most, twice the greater.)The mean flow is characterized by richardson number of order unity, hence N and the shear are nearly equal, implying that ω peak ~ N.

QuaNtIF yING MIxING By KH INStaBIlIt y
I l l I a M D .S M y t H a N D J a M e S N .M O u M S p e c I a l I S S u e O N I N t e r N a l WaV e S aBStr act.Kelvin-Helmholtz (KH) instability, characterized by the distinctive Figures 1a,b, 2, 3, and 4).The maximum shear primarily determines the instability's growth rate, while wavelength is typically an order of magnitude greater than the thickness of the sheared layer.Visual evidence for KH instability frequently can be seen at the top of the atmospheric boundary layer in late afternoon and evening, where it appears as patches of banded clouds (occasionally seen in Web videos of flows near Marys Peak taken from the top of our building at Oregon State University and shared at http://marycam.coas.oregonstate.edu).
of Glasgow and Hermann von Helmholtz (1821-1894) of Berlin, "Kelvin-Helmholtz instability" referred originally to the instability of two adjacent fluid layers in relative motion.Thomson (1871) used this instability as a model for the generation of ocean surface waves by wind, while von Helmholtz (1890) sought to explain the banded clouds discussed above.In real fluids, a transition layer of nonzero thickness-a shear layer-always separates the layers.Disturbances between moving fluid layers were first documented via experiments in a tilting tube by Reynolds (1883) 1 .In these experiments, a long horizontal tube containing two fluid layers of different densities was tipped slightly, so that the buoyancy difference set the layers in motion relative to one another, resulting in the formation of waves at the interface.Thorpe (1971; Figure 2) documented the largeamplitude, two-dimensional structure in which vorticity creates billows and intermingles the fluids from adjacent layers.
Figure 3. underwater snapshots made by divers from laboriously executed dye release experiments in the Mediterranean thermocline (Woods, 1968).(a) Side view of the rollup of a Kelvin-Helmholtz billow.(b) Shear instabilities viewed in the context of larger-scale waves.
KH instability.Given an initial wavelike perturbation, mass conservation requires accelerated horizontal flow above the crests and below the troughs (Figure 5a).The resulting current anomalies advect vorticity toward the center of the figure.The vorticity concentration induces vertical velocity perturbations that amplify the original wave (Figure 5b), resulting in positive feedback and exponential growth of the perturbation. 2The most-amplified wavelength depends on the details of the initial profiles, but typically ranges from 6 to 11 times the initial transition-layer thickness.Stable stratification tends to slow the growth of the billows, but also to accelerate breaking once the billows reach sufficient amplitude to overturn.Internal waves in the ocean contain regions of strong shear and stratification that are well described by this scenario, particularly in cases where two or more wave trains interfere constructively.The downscale energy cascade from IGW to turbulence often begins with KH instability, but it does not end there.We can identify at least one further step, that being one of several secondary instabilities that grow on mature KH billows.The classic review article of Thorpe (1987) describes several such instabilities; today, several more are known.To compute these secondary instabilities, a twodimensional KH billow is first simulated numerically.Further analysis then determines which three-dimensional perturbation, if applied to the finite-amplitude billow, would grow most rapidly tion point" instability ofMashayek and Peltier (2011).The latter may explain the form of KH billows observed in the Connecticut River estuary(Geyer et al.,    2 a more modern view of shear-driven instabilities is phrased in terms of resonances between vorticity and gravity waves (e.g.,Baines and Mitsudera, 1994).

Figure 4 .
Figure 4. temperature variations in a downslope tidal flow (van Haren and Gostiaux, 2010).This time series was constructed using data from multiple temperature sensors spaced vertically on a chain above the seafloor at 550 m depth.typical wavelengths are inferred to be 75 m.Graphic courtesy H. van Haren

Figure 6 .
Figure 6.Secondary instability of a Kelvin-Helmholtz billow calculated using perturbation analysis.colors show the perturbation vorticity field.yellow indicates perturbations of the spanwise (y) vorticity that defines the primary instability.red and blue show opposite signs of the streamwise (x) vorticity.

Figure 7 .Figure 8 .
Figure 7. Direct numerical simulations of the density field at successive time in the life cycle of a Kelvin-Helmholtz billow train.colors show density in the transition layer; upper and lower homogeneous layers are rendered transparent.(a) The initial state is a two-layer flow, with a lower (dense) layer flowing to the left and an upper layer to the right.a small perturbation is applied.(b) two wavelengths of the primary Kelvin-Helmholtz instability.(c) Kelvin-Helmholtz billows are beginning to pair.Secondary instability is visible in a cutaway at upper right, taking the form of shear-aligned convection rolls.(d) Secondary shear instability forms on the braids.(e) The fully turbulent state.(f) turbulence decays to form sharp layers and random small-scale waves.

Figure
Figure 7a shows the initial state-two layers separated by a thin transition layer.A small perturbation grows to form a train of KH billows, two of which are shown in Figure 7b.The billows subsequently merge and develop two secondary instabilities in sequence: convective secondary instability in overturned regions of the billow cores (Figure 7c, cf. Figure 6) and secondary shear instability in the braids (Figure 7d).These instabilities, and a complex combination of tertiary instabilities, lead the flow to a fully turbulent state (Figure 7e).Ultimately, turbulence decays, leaving a field of random IGWs propagating on a layer that has been dramatically thickened by irreversible mixing (Figure 7f, compare with 7a).The statistical properties of simulated mixing events like those shown here closely resemble those extracted from observations of turbulent patches in the ocean thermocline(Smyth et al., 2001).
for the growth of KH instability occur sporadically, which in turn suggests that random internal wave interactions add to the ambient shear to drive Ri to subcritical values, thereby generating instability at random points in space and time.The similarity between the dominant frequency of the oscillations and N is consistent with this scenario, even though the frequency of individual KH events is independent of stratification (see Box 1).The relationship between KH instability and small-scale turbulence is complex: the instability causes turbulence, and turbulence already present in the environment affects its growth.Liu et al. (2011) have recently extended the classical method of stability analysis to include the effects of ambient turbulence.In strongly forced flows, such as exist in the upper equatorial ocean, this relationship can lead to a cycle in which shear is continually reinforced, but new instabilities cannot grow until turbulence generated by previous instabilities decays.As a result, current shear is large and, despite strong stratification, Ri is typically near-critical in the mean. 3It has long been suspected that turbulence generated beneath the equatorial mixed layer (which does not generally extend as far as it does beneath mid-latitude mixed layers) is due to shear instability.Recent measurements from a vertical array of fast temperature sensors moored for extended periods in the upper equatorial ocean have confirmed the basic structure of KH instability in the small-scale fluctuations that appear on a daily cycle at frequencies near the local buoyancy frequency, N (Figure 9; Moum et al., 2011).The frequency and intermittency of the fluctuations can be seen in the Figure 9a spectrogram.The phase of the oscillations varies vertically in a manner consistent with KH instability.Oscillations typically comprise O(10) wavelengths.Potential energy stored in these motions (Figure 9b) also varies on a daily basis and is correlated with the turbulent kinetic energy dissipation rate (Figure 9c), indicating that turbulent mixing at small scales coincides with KH billows (Figure 9d).The interpretation of these fluctuations as KH billows has been tested via application to oceanic data of Rayleigh's (1880) method of linear stability analysis.That method has been used in the interpretation of billows observed in

Figure
Figure 9. temperature fluctuations observed in the sheared zone above the pacific equatorial undercurrent.(a) temperature spectrogram (variance preserving), (b) wave potential energy, N T 2 ζ 2 , and (c) turbulence ε χ , for the period December 20, 2006-February 10, 2007, derived from a single fast thermistor deployed at 29 m on NOaa's taO (tropical Ocean-atmosphere) mooring at 0°, 140°W.The relationship between the potential energy of the instabilities and the turbulence is established in the scatter plot (d).Blue points plotted represent the two-dimensional histogram; darker points occur more frequently.

Figure 10 .Figure
Figure10.linear stability analyses of flow above the pacific equatorial undercurrent.(a) Growth rate versus zonal and meridional wavenumbers.colors represent mode families focused in different shear layers between the undercurrent core and the surface(Sun et al., 1998).Graphic courtesy C. Sun.(b) Histogram of absolute cyclic frequency for 155 unstable modes computed numerically from hourly averaged profiles of velocity and density(Smyth et al., 2011).curves show normalized, variance-preserving spectra of four observed temperature time series.Sensor depths are as shown in the legend.

Figure
Figure B1.probability distribution of phase velocity.

Figure
Figure B2.probability distribution of wavelength.
To determine the net contribution of KH instability to global mixing, two critical challenges must be addressed.The first is to quantify the prevalence of KH instability in thermocline internal wave fields.Other forms of instability contribute to thermocline mixing.How much?One way this quantity is being measured is with long time series that include sufficient detail to diagnose KH instability in the signal such as those made in the upper equatorial ocean by Moum et al. (2011; Figure 9).Can thermocline observations with the same fidelity be obtained that permit an evaluation of the prevalence of KH instability in thermocline internal wave fields?Glider measurements are promising in this respect (e.g., Thorpe 2012; Smyth and Thorpe 2012.)The second challenge is to parameterize KH-induced mixing (i.e., to approximate the rate of mixing in terms of other quantities that are more easily measured or modeled).For example, Kunze et al. (1990) used a simple analytical model of a shear layer to estimate the mean flow kinetic energy released in an instability event and combined the result with an estimate of the turbulence production time to arrive at the kinetic energy dissipation rate.Future parameterizations will incorporate more complete models of instability physics, and they will be calibrated using advanced laboratory techniques, comprehensive microstructure observations, and direct numerical simulations of increasing scale and realism.

acKNOWleDGeMeNtS
Our work on KH instability has been funded by the National Science Foundation (1030772 and 1129419 to WDS, and 0728357 and 1059055 to JNM) and by the Office of Naval Research (N00014-09-1-0280, N000014-10-1-2098 to JNM).Helpful critiques of this article were provided by Steve Thorpe, Aurelie Moulin, Elizabeth McHugh and two anonymous referees.