K. Shafer Smith

The geography of linear baroclinic instability in Earth's oceans

Satellite observations reveal a mesoscale oceanic circulation dominated by turbulence that is correlated, in most cases, with local baroclinicity. Linear baroclinic instability theory has proved useful in understanding the time and space scales of atmospheric eddies. The question addressed here is, to what degree can the observed oceanic eddy activity be understood though a local, linear stability analysis? This question is addressed as follows. A local quasigeostrophic linear stability calculation is performed on a grid of wavenumbers, ranging inmagnitude about the local deformation wavenumber, for each vertical profile in a dataset of neutral density for the world’s oceans. The initial results show that nearly the entire ocean is unstable, but in many places, particularly in low-latitudes, the instability is dominated by surface intensified modes, resulting in very small-scale, quickly growing waves. At higher latitudes, the primary instabilities are due to thermocline depth shears, and have a broader vertical structure. For each unstable wave, at each location, the mean-to-eddy energy conversion rate is also calculated and used to select the growing waves that are both fast and have significant energetic conversion potential. This procedure removes most of the surface-instabilities, which cannot lead to significant energy conversion, and reveals the slower but more powerful thermocline-level instabilities where they exist. The time and space scales of these growing waves are compared to estimates of the Eady growth rate and deformation scale, respectively. It is found that while the timescale is wellapproximated by the Eady-estimate, the spatial scales are uniformly smaller than the deformation scale, typically by a factor of 4. The zonally averaged spatial scales are then compared to observed eddy scales. The spatial scales of maximum growth are everywhere significantly smaller than the observed eddy scales. In the Antarctic Circumpolar Current, for example, the scale of maximum growth is about 5 km, much smaller than the observed eddy scales, estimates of which range from 30-100 km. A possible, and unsurprising conclusion is that the observed eddy scales are the result of an inverse cascade, and cannot be understood by linear theory alone. ∗Corresponding author address: K. Shafer Smith, Center for Atmosphere Ocean Science, New York University, 251 Mercer Street, New York, NY 10012. E-mail: shafer@cims.nyu.edu

Scales, Growth Rates, and Spectral Fluxes of Baroclinic Instability in the Ocean

AbstractAn observational, modeling, and theoretical study of the scales, growth rates, and spectral fluxes of baroclinic instability in the ocean is presented, permitting a discussion of the relation between the local instability scale; the first baroclinic deformation scale Rdef; and the equilibrated, observed eddy scale. The geography of the large-scale, meridional quasigeostrophic potential vorticity (QGPV) gradient is mapped out using a climatological atlas, and attention is drawn to asymmetries between midlatitude eastward currents and subtropical return flows, the latter of which has westward and eastward zonal velocity shears. A linear stability analysis of the climatology, under the “local approximation,” yields the growth rates and scales of the fastest-growing modes. Fastest-growing modes on eastward-flowing currents, such as the Kuroshio and the Antarctic Circumpolar Current, have a scale somewhat larger (by a factor of about 2) than Rdef. They are rapidly growing (e folding in 1–3 weeks) and d...

Evidence for Enhanced Eddy Mixing at Middepth in the Southern Ocean

Abstract Satellite altimetric observations of the ocean reveal surface pressure patterns in the core of the Antarctic Circumpolar Current (ACC) that propagate downstream (eastward) but slower than the mean surface current by about 25%. The authors argue that these observations are suggestive of baroclinically unstable waves that have a steering level at a depth of about 1 km. Detailed linear stability calculations using a hydrographic atlas indeed reveal a steering level in the ACC near the depth implied by the altimetric observations. Calculations using a nonlinear model forced by the mean shear and stratification observed close to the core of the ACC, coinciding with a position where mooring data and direct eddy flux measurements are available, reveal a similar picture, albeit with added details. When eddy fluxes are allowed to adjust the mean state, computed eddy kinetic energy and eddy stress are close to observed magnitudes with steering levels between 1 and 1.5 km, broadly consistent with observatio...

The Production and Dissipation of Compensated Thermohaline Variance by Mesoscale Stirring

Abstract Temperature–salinity profiles from the region studied in the North Atlantic Tracer Release Experiment (NATRE) show large isopycnal excursions at depths just below the thermocline. It is proposed here that these thermohaline filaments result from the mesoscale stirring of large-scale temperature and salinity gradients by geostrophic turbulence, resulting in a direct cascade of thermohaline variance to small scales. This hypothesis is investigated as follows: Measurements from NATRE are used to generate mean temperature, salinity, and shear profiles. The mean stratification and shear are used as the background state in a high-resolution horizontally homogeneous quasigeostrophic model. The mean state is baroclinically unstable, and the model produces a vigorous eddy field. Temperature and salinity are stirred laterally in each density layer by the geostrophic velocity and vertical advection is by the ageostrophic velocity. The simulated temperature–salinity diagram exhibits fluctuations at depths ju...

The scales and equilibration of midocean eddies: Forced-dissipative flow

The statistical dynamics of midocean eddies, generated by baroclinic instability of a zonal mean flow, are studied in the context of homogeneous stratified quasigeostrophic turbulence. Existing theory for eddy scales and energies in fully developed turbulence is generalized and applied to a system with surface-intensified stratification and arbitrary zonal shear. The theory gives a scaling for the magnitude of the eddy potential vorticity flux, and its (momentum conserving) vertical structure. The theory is tested numerically by varying the magnitude and mode of the mean shear, the Coriolis gradient, and scale thickness of the stratification and found to be partially successful. It is found that the dynamics of energy in high ( m . 1) baroclinic modes typically resembles the turbulent diffusion of a passive scalar, regardless of the stratification profile, although energy in the first mode does not. It is also found that surface-intensified stratification affects the baroclinicity of flow: as thermocline thickness is decreased, the (statistically equilibrated) baroclinic energy levels remain nearly constant but the statistically equilibrated level of barotropic eddy energy falls. Eddy statistics are found to be relatively insensitive to the magnitude of linear bottom drag in the small drag limit. The theory for the magnitude and structure of the eddy potential vorticity flux is tested against a 15-layer simulation using profiles of density and shear representative of those found in the mid North Atlantic; the theory shows good skill in representing the vertical structure of the flux, and so might serve as the basis for a parameterization of eddy fluxes in the midocean. Finally, baroclinic kinetic energy is found to concentrate near the deformation scale. To the degree that surface motions represent baroclinic eddy kinetic energy, the present results are consistent with the observed correlation between surface eddy scales and the first radius of deformation.

Quasigeostrophic Turbulence with Explicit Surface Dynamics: Application to the Atmospheric Energy Spectrum

Abstract The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior. The system exhibits three f...

A local model for planetary atmospheres forced by small-scale convection

An equivalent-barotropic fluid on the b plane, forced at small scales by random stirring and dissipated by linear heat and vorticity drag, is considered as a local model for flow in the weather layer of internally forced planetary atmospheres. The combined presence of b, a finite deformation scale, and large-scale dissipation produce novel dynamics with possible relevance to the giant gas planets, which are apparently driven by smallscale convective stirring. It is shown that in order for anisotropy to form, one must have b(el 5) 21/3 * 3.9, where e is the (convectively driven) energy generation rate, l is the deformation wavenumber, and b is the Coriolis gradient. The critical value above is not equivalent to the barotropic stability criterion, and numerical simulations demonstrate that anisotropic flow with average zonal velocities that are supercritical with respect to the latter can form. The formation of jets (a different matter) is not implied by the excess of zonal kinetic energy, and is instead sensitive to the relevant stability criterion for the flow geometry at hand. When b is sufficiently large that anisotropy does form, the flow scale and rms zonal velocity are set by a combination of Rossby wave cascade inhibition, the total energy constraint imposed by the large-scale dissipation, and the partitioning between available potential and kinetic energies. The resulting theory demonstrates that a relatively narrow range of parameters will allow for the formation of anisotropic flow with scale larger than the deformation scale. This is consistent with observations that indicate little separation between the jet scales and deformation scales on Jupiter and Saturn.

The latmix summer campaign: Submesoscale stirring in the upper ocean

AbstractLateral stirring is a basic oceanographic phenomenon affecting the distribution of physical, chemical, and biological fields. Eddy stirring at scales on the order of 100 km (the mesoscale) is fairly well understood and explicitly represented in modern eddy-resolving numerical models of global ocean circulation. The same cannot be said for smaller-scale stirring processes. Here, the authors describe a major oceanographic field experiment aimed at observing and understanding the processes responsible for stirring at scales of 0.1–10 km. Stirring processes of varying intensity were studied in the Sargasso Sea eddy field approximately 250 km southeast of Cape Hatteras. Lateral variability of water-mass properties, the distribution of microscale turbulence, and the evolution of several patches of inert dye were studied with an array of shipboard, autonomous, and airborne instruments. Observations were made at two sites, characterized by weak and moderate background mesoscale straining, to contrast diff...

Diagnosing Lateral Mixing in the Upper Ocean with Virtual Tracers: Spatial and Temporal Resolution Dependence

Several recent studies diagnose lateral stirring and mixing in the upper ocean using altimetry-derived velocity fields to advect ‘‘virtual’’ particles and fields offline. However, the limited spatiotemporal resolution of altimetric maps leads to errors in the inferred diagnostics, because unresolved scales are necessarily imperfectly modeled. The authors examine a range of tracer diagnostics in two models of baroclinic turbulence: the standard Phillips model, in which dispersion is controlled by large-scale eddies, and the Eady model, where dispersion is determined by local scales of motion. These models serve as a useful best- and worst-case comparison and a valuable test of the resolution sensitivity of tracer diagnostics. The effect of unresolved scales is studied by advecting tracers using model velocity fields subsampled in space and time and comparing the derived tracer diagnostics with their ‘‘true’’ value obtained from the fully resolved flow. The authors find that eddy diffusivity and absolute dispersion, which are governed by largescaledynamics,areinsensitivetospatialsamplingerrorin eitherflow.Measuresthatdependstronglyonsmall scales, such as relative dispersion andfinite-timeLyapunov exponents, are highly sensitive to spatial sampling in the Eady model. Temporal sampling error is found to have a more complicated behavior because of the onset of particle overshoot leading to scrambling of Lagrangian diagnostics. This leads to a potential restrictionontheutilityofrawaltimetrymapsforstudyingmixingintheupperocean.Theauthorsconcludethat offline diagnostics of mixing in ocean flows with an energized submesoscale should be viewed with some caution.

New Methods for Estimating Ocean Eddy Heat Transport Using Satellite Altimetry

AbstractAttempts to monitor ocean eddy heat transport are strongly limited by the sparseness of available observations and the fact that heat transport is a quadratic, sign-indefinite quantity that is particularly sensitive to unresolved scales. In this article, a suite of stochastic filtering strategies for estimating eddy heat transport are tested in idealized two-layer simulations of mesoscale oceanic turbulence at high and low latitudes under a range of observation scenarios. A novel feature of these filtering strategies is the use of computationally inexpensive stochastic models to forecast the underlying nonlinear dynamics. The stochastic model parameters can be estimated by regression fitting to climatological energy spectra and correlation times or by adaptively learning these parameters “on-the-fly” from the observations themselves.The authors show that, by extracting high-wavenumber information that has been aliased into the low wavenumber band, “stochastically super-resolved” velocity fields wi...