Time-dependent response to cooling in a beta-plane basin

Author Posting. © American Meteorological Society, 2006. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Journal of Physical Oceanography 36 (2006): 2185-2198, doi:10.1175/JPO2967.1.The time-dependent response of an ocean basin to the imposition of cooling (or heating) is examined in the context of a quasigeostrophic, two-layer model on the beta plane. The focus is on the structure and magnitude of the vertical motion and its response to both a switch-on forcing and a periodic forcing. The model employed is a time-dependent version of an earlier model used to discuss the intensification of sinking in the region of the western boundary current. The height of the interface of the two-layer model serves as an analog of temperature, and the vertical velocity at the interface consists of a cross-isopycnal velocity modeled in terms of a relaxation to a prescribed interface height, an adiabatic representation of eddy thickness fluxes parameterized as lateral diffusion of thickness, and the local vertical motion of the interface itself. The presence of time dependence adds additional dynamical features to the problem, in particular the emergence of low-frequency, weakly damped Rossby basin modes. If the buoyancy forcing is zonally uniform the basin responds to a switch-on of the forcing by coming into steady-state equilibrium after the passage of a single baroclinic Rossby wave. If the forcing is nonuniform in the zonal direction, a sequence of Rossby basin modes is excited and their decay is required before the basin achieves a steady state. For reasonable parameter values the boundary layers, in which both horizontal and vertical circulations are closed, are quasi-steady and respond to the instantaneous state of the interior. As in the steady problem the flow is sensitive to small nonquasigeostrophic mass fluxes across the perimeter of the basin. These fluxes generally excite basin modes as well. The basin modes will also be weakly excited if the beta-plane approximation is relaxed. The response to periodic forcing is also examined, and the sensitivity of the response to the structure of the forcing is similar to the switch-on problem.This research was supported in part by NSF Grant OCE-9901654

Floats and ƒ/H

The sensitivity of oceanic float dispersion to f/H, where H is a spatially filtered representation of the water depth, is examined with floats from the Atlantic and Pacific oceans. The first and second moments of the displacements relative to f/H were found and compared to those for zonal and meridional displacements. In all cases, the moments relative to f/H display equal or greater anisotropy than those relative to geographical coordinates, suggesting a preferred tendency for spreading along f/H. In regions where the topography is flat (at the equator, in the interior South Atlantic and in the North Pacific), transport is much greater along than across latitude lines, and the moments relative to f/H are essentially the same. But the results differ where the topography is steep (the North Atlantic and near the western boundary in the Equatorial and South Atlantic), where anisotropic spreading relative to f/H occurs even though the geographical moments are isotropic or meridionally-enhanced. Only in the North Pacific, where the topography is smaller scale and less steep, is the spreading more anisotropic in geographical coordinates. The present method is tested using trajectories from a stochastic model, and correctly shows that no such tendency for spreading along f/H exists. Mean and eddy effects are discussed, but are not believed to be well resolved

Corrigendum to Diffusivity and viscosity dependence in the linear thermocline (J. Mar. Res., 62, 743–769)

The statement in LaCasce (2004) that the amplitude, A, of the western boundary layer solution (Eq. 25 in the article) is determined by matching at the northern wall to the streamfunction from the northern boundary inner layer solution, ϕin, is incorrect. The correct procedure is to match to the full boundary layer solution at the northern wall, with contributions from both the inner and outer layers. Doing this ensures that the full transport in the northern layer is fed into the western layer..

On turbulence and normal modes in a basin

The problem of forced, geostrophic turbulence in a basin is revisited. The primary focus is the time dependent field, which is shown to be approximately isotropic (in contrast to the strongly zonally anisotropic fields seen in periodic domains). It is also approximately homogeneous, away from the boundaries. Phenomenological arguments suggest the isotropy occurs because the inverse cascade of energy is arrested by basin normal modes rather than by free Rossby waves. Peaks in the velocity spectra at modal frequencies are consistent with basin modes, as has been noted previously. We discuss which modes would be excited and whether dissipation or the mean flow would be expected to alter the modes and their frequencies. A relatively novel feature is the use of Eulerian velocity statistics to quantify the wave and turbulence characteristics. These measures are more suitable to this environment than measures like wavenumber spectra, given the inhomogeneities associated with the boundaries. With regards to the mean, we observe a linear 〈q〉 - 〈ψ〉 relation in the region of the mean gyres (at the northern and southern boundaries), consistent with previous theories. This is of interest because our numerical advection scheme has implicit rather than explicit small scale dissipation, and requires no boundary conditions on the vorticity. The gyre structure is however somewhat different than in an (inviscid) Fofonoff-type solution, suggesting dissipation cannot be neglected

Diffusivity and viscosity dependence in the linear thermocline

We re-visit the linear model of a thermally-driven ocean in a single hemisphere, to understand dependencies on viscosity and mixing. We focus in particular on the vertical and surface velocities. In all cases, the meridional and vertical flows are intensified near the boundaries; as a result, the overturning depends on viscosity. The two types of viscosity we examine (Rayleigh damping and diffusion with the no-slip condition) yield significantly different boundary transports. This in turn can cause large changes in the surface velocities. We observe a single-gyre surface circulation with a diffusive viscosity but two gyres with the Rayleigh. We also examine how the solutions change when the vertical mixing itself is intensified near the boundaries. With (spatially) constant mixing, a significant fraction of the vertical transport occurs in the thermocline interior where viscosity is unimportant. But the localized mixing increases the boundary intensification of the upwelling, making viscosity even more important. There is evidence of similar boundary intensification and viscosity dependence in numerical models

Relative displacement probability distribution functions from balloons and drifters

The focus of relative (pair) dispersion studies in the atmosphere and ocean is often on the mean square particle separation or the Finite Scale Lyapunov Exponent. Much less attention has been paid to the probability density function (PDF) of pair separations, despite that this determines the dispersion. In two-dimensional (2-D), nondivergent, homogeneous flows, the PDF is governed by a Fokker-Planck equation. Analytical solutions exist for the turbulent inertial ranges, but these have rarely been compared to observations.We consider the analytical PDFs for the turbulent inertial ranges and derive a new solution for the 2-D energy range. We then compare the analytical PDFs with those generated with data from three in situ sets: one from a balloon experiment in the stratosphere and two from surface drifter experiments in the ocean. For comparison, we also consider PDFs from a numerical simulation of 2-D turbulence forced at intermediate scales. The results suggest that dispersion at sub-deformation scales is nonlocal, with pair separations growing exponentially in time. This implies the kinetic energy spectra at these scales are at least as steep as κ−3. The dispersion at larger scales is harder to characterize because of the uncertainty in the PDF at larger separations, but the results are consistent with previous inferences. In general the PDF provides useful information on the spreading which can be difficult to discern from the dispersion alone

Estimating subsurface horizontal and vertical velocities from sea-surface temperature

We examine a dynamical method for estimating subsurface fields (density, pressure, horizontal and vertical velocities) in the upper ocean using sea-surface temperature (SST) and a climatological estimate of the stratification. The method derives from the surface quasi-geostrophic (SQG) approximation. The SST is used to generate a potential vorticity (PV) field which is then inverted for the pressure. We examine first the standard SQG model, in which the PV is assumed trapped in a delta-function layer at the surface. We then modify the model by introducing a subsurface PV which is proportional to the surface density and decays exponentially with depth. We derive the subsurface density from the hydrostatic relation, the horizontal velocities from geostrophy and the subsurface vertical velocities from the quasi-geostrophic omega equation.We compare the predicted densities and velocities with those from a three-dimensional (3D) ocean model, and from in situ measurements in the Mediterranean, Eastern Pacific and the Azores Current. In most cases the standard SQG model predicts the qualitative structure of the subsurface flow. But it also underestimates its strength. The modified model yields better estimates of both the strength and vertical structure of the subsurface flow

Relative dispersion at the surface of the Gulf of Mexico

We examine the relative motion of pairs and triplets of surface drifters in the Gulf of Mexico. The mean square pair separations grow exponentially in time from the smallest resolved scale (1 km) to 40-50 km, with an e-folding time of 2-3 days. Thereafter, the dispersion exhibits a power law dependence on time with an exponent of between 2 and 3 (depending on the measure used) up to scales of several hundred kilometers. The straining is for the most part isotropic, with only weak regional variations. But there are suggestions of anisotropy in the western basin, probably due to boundary current advection. The pair velocities are correlated during the early phase and a portion of the late phase. The relative displacement distributions during the early phase are, after an initial adjustment, non-Gaussian and approximately constant, suggestive of local straining. The triplet results likewise suggest two growth phases. During the early phase, the mean area and the longest triangle leg grow exponentially in time, the latter with a rate consistent with the two-particle results. Most triangles are drawn out during this time. During the late period, the triangles grow and their aspect ratios systematically decrease, suggesting an evolution to an equilateral shape. Although surface divergences should affect these statistics, they nevertheless strongly resemble those found with two-dimensional turbulent flows. If so, we would infer an enstrophy cascade at scales below the deformation radius (40-50 km) which is probably spectrally local. The latter implies that growth in particle separations comes from flow features the same size as the separations. It is also possible there is an inverse energy cascade to scales larger than the deformation radius, driven possibly by baroclinic instability. However, the late period statistics may also reflect dispersion by a large scale shear. We do not resolve an upper bound on the late time power law growth (i.e. we do not observe an ultimate diffusive stage). This may reflect shear dispersion. But it may also stem from surface convergences which can cause long time particle correlations, as seen in recent numerical simulations of particles on a surface bounding an interior turbulent flow

Relative dispersion in the subsurface North Atlantic

Pair statistics are calculated for subsurface floats in the North Atlantic. The relative diffusivity (the derivative of the mean square particle separation) is approximately constant at large scales in both eastern and western basins, though the implied scale of the energy-containing eddies is greater in the west. But the behavior at times soon after pair deployment is quite different in the two basins; in the west the diffusivity grows approximately as distance to the 4/3 power, consistent with an inverse turbulent cascade of energy (or possibly of mixing superimposed on a mean shear), but in the east the diffusivity grows more slowly, as for instance in simple stochastic systems. Exponential stretching, expected in an enstrophy cascade, is not resolved in any region; however, this may reflect only that the present pair separations are too large initially

Lagrangian statistics in unforced barotropic flows

We consider the dispersion of particles in potential vorticity (PV)-conserving flows. Because particle drift is preferentially along the mean PV contours, Lagrangian dispersion is strongly anisotropic. If the mean PV field moreover is spatially variable, as when there is topography, the anisotropy is more clearly visible in the dispersion of displacements along and across the mean PV field itself. We examine several numerical examples of unforced barotropic flows; in all cases, this projected dispersion is more anisotropic than that in cartesian (x, y) coordinates. What differs is the rate at which spreading occurs, both along and across contours. The method is applicable to real data, as is illustrated with float data from the deep North Atlantic. The results suggest a preferential spreading along contours of (barotropic) f/H