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W. D. Smyth

Abstract

The linear theory of double diffusive interleaving is extended to take account of baroclinic effects. This study goes beyond previous studies by including the possibility of modes with nonzero tilt in the alongfront direction, which allows for advection by the baroclinic frontal flow. This requires that the stability equations be solved numerically. The main example is based on observations of interleaving on the lower flank of Meddy Sharon, but a range of parameter values is covered, leading to conclusions that are relevant in a variety of oceanic regimes. The frontal zone is treated as infinitely wide with uniform gradients of temperature, salinity, and alongfront velocity. The stationary, vertically symmetric interleaving mode is shown to have maximum growth rate when its alongfront wavenumber is zero, providing validation for previous studies in which this property was assumed. Besides this, there exist two additional modes of instability: the ageostrophic Eady mode of baroclinic instability and a mode not previously identified. The new mode is oblique (i.e., it tilts in the alongfront direction), vertically asymmetric, and propagating. It is strongly dependent on boundary conditions, and its relevance in the ocean interior is uncertain as a result. Effects of variable diffusivity and buoyancy flux ratio are also considered.

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W. D. Smyth

Abstract

The mixing efficiency of stratified turbulence in geophysical fluids has been the subject of considerable controversy. A simple parameterization, devised decades ago when empirical knowledge was scarce, has held up remarkably well. The parameterization rests on the assumption that the flux coefficient Γ has the uniform value 0.2. This note provides a physical explanation for Γ = 0.2 in terms of the “marginal instability” property of forced stratified shear flows, and also sketches a path toward improving on that simple picture by examining cases where it fails.

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W. D. Smyth and S. Kimura

Abstract

Mixing due to sheared salt fingers is studied by means of direct numerical simulations (DNS) of a double-diffusively unstable shear layer. The focus is on the “moderate shear” case, where shear is strong enough to produce Kelvin–Helmholtz (KH) instability but not strong enough to produce the subharmonic pairing instability. This flow supports both KH and salt-sheet instabilities, and the objective is to see how the two mechanisms work together to flux heat, salt, and momentum across the layer.

For observed values of the bulk Richardson number Ri and the density ratio Rρ, the linear growth rates of KH and salt-sheet instabilities are similar. These mechanisms, as well as their associated secondary instabilities, lead the flow to a fully turbulent state. Depending on the values of Ri and Rρ, the resulting turbulence may be driven mainly by shear or mainly by salt fingering. Turbulent mixing causes the profiles of temperature, salinity, and velocity to spread; however, in salt-sheet-dominated cases, the net density (or buoyancy) layer thins over time. This could be a factor in the maintenance of the staircase and is also an argument in favor of an eventual role for Holmboe instability.

Fluxes are scaled using both laboratory scalings for a thin layer and an effective diffusivity. Fluxes are generally stronger in salt-sheet-dominated cases. Shear instability disrupts salt-sheet fluxes while adding little flux of its own. Shear therefore reduces mixing, despite providing an additional energy source. The dissipation ratio Γ is near 0.2 for shear-dominated cases but is much larger when salt sheets are dominant, supporting the use of Γ in the diagnosis of observed mixing phenomena. The profiler approximation Γz, however, appears to significantly overestimate the true dissipation ratio.

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W. D. Smyth and W. R. Peltier

Abstract

We consider the evolution of small disturbances on an inviscid, Boussinesq, stably stratified free shear layer. This flow may deliver either Kelvin-Helmholtz or Holmboe instability, depending on the details of the background stratification. Conventional one-dimensional linear analysis is employed to study the temporal and spatial structures of these instabilities and the physical mechanisms which govern their evolution. Attention is focussed upon the manner in which Kelvin-Helmboltz instability is replaced by Holmboe instability for a sequence of background flows with successively larger values of the bulk Richardson number. Unstable normal modes that exist in the transition region between the Kelvin-Heimboltz and Holmboe regimes exhibit a distinctive spatial structure, are characterized by relatively low growth rates, and are shown to occur under conditions for which overreflection of neutrally propagating internal waves apparently cannot occur because the gradient Richardson number at the steering level exceeds ¼. Detailed calculations of the propagation characteristics of internal waves in stratified shear layers the extent to which resonant overreflection theory, based upon the reflection properties of temporally neutral waves, may fail to yield physical insight into the stability characteristics of such flows.

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W. D. Smyth, H. Burchard, and L. Umlauf

Abstract

Interleaving motions on a wide, baroclinic front are modeled using a second-moment closure to represent unresolved fluxes by turbulence and salt fingering. A linear perturbation analysis reveals two broad classes of unstable modes. First are scale-selective modes comparable with interleaving as observed in oceanic fronts. These correspond well with observations in some respects but grow by a very different mechanism, which ought to be easily distinguished in hydrographic profiles. The second mode type is the so-called ultraviolet catastrophe, which is expected to lead to steppy profiles even in the absence of interleaving. Both modes are driven by positive feedbacks between interleaving and the underlying small-scale mixing processes. Contrary to expectations, use of the second-moment closure in place of earlier empirical mixing models does not lead to improved agreement with observations.

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W. D. Smyth and K. B. Winters

Abstract

Motivated by the tendency of high-Prandtl-number fluids to form sharp density interfaces, the authors investigate the evolution of Holmboe waves in a stratified shear flow through direct numerical simulation. Like their better-known cousins, Kelvin–Helmholtz waves, Holmboe waves lead the flow to a turbulent state in which rapid irreversible mixing takes place. In both cases, significant mixing also takes place prior to the transition to turbulence. Although Holmboe waves grow more slowly than Kelvin–Helmholtz waves, the net amount of mixing is comparable. It is concluded that Holmboe instability represents a potentially important mechanism for mixing in the ocean.

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J. N. Moum, J. D. Nash, and W. D. Smyth

Abstract

Extended measurements of temperature fluctuations that include the turbulence wavenumber band have now been made using rapidly sampled fast thermistors at multiple depths above the core of the Equatorial Undercurrent on the Tropical Atmosphere Ocean (TAO) mooring at 0°, 140°W. These measurements include the signature of narrowband oscillations as well as turbulence, from which the temperature variance dissipation rate χT and the turbulence energy dissipation rate ϵχ are estimated.

The narrowband oscillations are characterized by the following:

  • groupiness—packets consist of O(10) oscillations;
  • spectral peaks of up to two orders of magnitude above background;
  • a clear day–night cycle with more intensive activity at night;
  • enhanced mixing rates;
  • frequencies of 1–2 × 10−3 Hz, close to both the local buoyancy and shear frequencies, N/2π and S/2π, which vary slowly in time;
  • high vertical coherence over at least 30-m scales; and
  • abrupt vertical phase change (π/2 over <20 m).
The abrupt vertical phase change is consistent with instabilities formed in stratified shear flows. Linear stability analysis applied to measured velocity and density profiles leads to predicted frequencies that match those of the observed oscillations. This correspondence suggests that the observed oscillation frequencies are set by the phase speed and wavelength of instabilities as opposed to the Doppler shifting of internal gravity waves with intrinsic frequency set by the local stratification N.

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W. D. Smyth, J. D. Nash, and J. N. Moum

Abstract

Direct numerical simulations are used to compare turbulent diffusivities of heat and salt during the growth and collapse of Kelvin–Helmholtz billows. The ratio of diffusivities is obtained as a function of buoyancy Reynolds number Reb and of the density ratio Rρ (the ratio of the contributions of heat and salt to the density stratification). The diffusivity ratio is generally less than unity (heat is mixed more effectively than salt), but it approaches unity with increasing Reb and also with increasing Rρ. Instantaneous diffusivity ratios near unity are achieved during the most turbulent phase of the event even when Reb is small; much of the Reb dependence results from the fact that, at higher Reb, the diffusivity ratio remains close to unity for a longer time after the turbulence decays. An explanation for this is proposed in terms of the Batchelor scaling for scalar fields. Results are interpreted in terms of the dynamics of turbulent Kelvin–Helmholtz billows, and are compared in detail with previous studies of differential diffusion in numerical, laboratory, and observational contexts. The overall picture suggests that the diffusivities become approximately equal when Reb exceeds O(102). The effect of Rρ is significant only when Reb is less than this value.

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Eric D. Skyllingstad, W. D. Smyth, and G. B. Crawford

Abstract

The role of resonant wind forcing in the ocean boundary layer was examined using an ocean large-eddy simulation (LES) model. The model simulates turbulent flow in a box, measuring ∼100–300 m on a side, whose top coincides with the ocean surface. Horizontal boundary conditions are periodic, and time-dependent wind forcing is applied at the surface. Two wind forcing scenarios were studied: one with resonant winds, that is, winds that rotated at exactly the inertial frequency (at 45°N), and a second with off-resonance winds from a constant direction. The evolution of momentum and temperature for both cases showed that resonant wind forcing produces much stronger surface currents and vertical mixing in comparison to the off-resonance case. Surface wave effects were also examined and found to be of secondary importance relative to the wind forcing.

The main goal was to quantify the main processes via which kinetic energy input by the wind is converted to potential energy in the form of changes in the upper-ocean temperature profile. In the resonant case, the initial pathway of wind energy was through the acceleration of an inertially rotating current. About half of the energy input into the inertial current was dissipated as the result of a turbulent energy cascade. Changes in the potential energy of the water column were ∼7% of the total input wind energy. The off-resonance case developed a much weaker inertial current system, and consequently less mixing because the wind acted to remove energy after ∼¼ inertial cycle. Local changes in the potential energy were much larger than the integrated values, signifying the vertical redistribution of water heated during the summer season.

Visualization of the LES results revealed coherent eddy structures with scales from 30–150 m. The largest-scale eddies dominated the vertical transport of heat and momentum and caused enhanced entrainment at the boundary layer base. Near the surface, the dominant eddies were driven by the Stokes vortex force and had the form of Langmuir cells. Near the base of the mixed layer, turbulent motions were driven primarily by the interaction of the inertial shear with turbulent Reynolds stresses. Bulk Richardson number and eddy diffusivity profiles from the model were consistent with one-dimensional model output using the K-profile parameterization.

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W. D. Smyth, J. N. Moum, and D. R. Caldwell

Abstract

The time evolution of mixing in turbulent overturns is investigated using a combination of direct numerical simulations (DNS) and microstructure profiles obtained during two field experiments. The focus is on the flux coefficient Γ, the ratio of the turbulent buoyancy flux to the turbulent kinetic energy dissipation rate ϵ. In observational oceanography, a constant value Γ = 0.2 is often used to infer the buoyancy flux and the turbulent diffusivity from measured ϵ. In the simulations, the value of Γ changes by more than an order of magnitude over the life of a turbulent overturn, suggesting that the use of a constant value for Γ is an oversimplification. To account for the time dependence of Γ in the interpretation of ocean turbulence data, a way to assess the evolutionary stage at which a given turbulent event was sampled is required. The ratio of the Ozmidov scale L O to the Thorpe scale L T is found to increase monotonically with time in the simulated flows, and therefore may provide the needed time indicator. From the DNS results, a simple parameterization of Γ in terms of L O/L T is found. Applied to observational data, this parameterization leads to a 50%–60% increase in median estimates of turbulent diffusivity, suggesting a potential reassessment of turbulent diffusivity in weakly and intermittently turbulent regimes such as the ocean interior.

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