Surface Quasi-geostrophic Research Articles

Variation on a theme of caffarelli and vasseur

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L2 gain H¨older continuity, provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on the BMO norm of a smooth velocity implies a uniform bound on the Cβ norm of the solution for some β > 0. We apply elementary tools involving the control of H¨older norms by using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasigeostrophic (SQG) equation in addition to the two proofs obtained earlier. Bibliography: 6 titles.

Instability of Surface Quasigeostrophic Vortices

Abstract The instability of circular vortices is studied numerically in the surface quasigeostrophic (SQG) model, and their evolutions are compared with those of barotropically unstable 2D vortices. The growth rates in the SQG model evidence similarity with their barotropic counterparts for moderate radial gradients of temperature (or of vorticity in the 2D model). For stronger gradients, SQG vortices are more unstable than 2D vortices. The nonlinear, finite-amplitude evolutions of perturbed vortices provide evidence that moderately unstable, elliptically perturbed vortices form tripoles. When they are more unstable, they break into two dipoles. Weakly unstable vortices with triangular perturbations form transient quadrupoles that break; they stabilize only for large gradients of mean temperature. Finally, with square perturbations, pentapoles degenerate into dipoles, at least for the range of mean temperature gradients explored here. The analysis of nonlinear stabilizations reveals that the deformation of the vortex core and the leak of its temperature anomaly to the periphery are essential ingredients to stabilize the perturbation at finite amplitude. In conclusion, SQG vortex instability exhibits considerable similarity to the barotropic instability of 2D vortices.

Two Comments on the Surface Quasigeostrophic Model for the Atmospheric Energy Spectrum

Abstract The horizontal wavenumber spectra of wind and temperature in the upper troposphere and lower stratosphere display a narrow k−3 range at scales on the order of 1000 km and a broad k−5/3 range at mesoscales on the order of 1 to 500 km. Recently, Tulloch and Smith suggested that a surface quasigeostrophic (SQG) turbulence model can explain the observed spectra. Here, it is first argued that the mesoscale spectra are not likely to be explained by any quasigeostrophic model because the Rossby number corresponding to the mesoscale dynamics is on the order of unity or larger. Then it is argued that the SQG model in particular cannot explain the observations because its mesoscale spectrum displays a k−5/3 dependence only in a very thin layer just below the tropopause. The thickness of this layer can be estimated to be of the order of 10 m, whereas aircraft measurements are typically performed several hundred meters away from the tropopause.

Upper Ocean Turbulence from High-Resolution 3D Simulations

Abstract The authors examine the turbulent properties of a baroclinically unstable oceanic flow using primitive equation (PE) simulations with high resolution (in both horizontal and vertical directions). Resulting dynamics in the surface layers involve large Rossby numbers and significant vortical asymmetries. Furthermore, the ageostrophic divergent motions associated with small-scale surface frontogenesis are shown to significantly alter the nonlinear transfers of kinetic energy and consequently the time evolution of the surface dynamics. Such impact of the ageostrophic motions explains the emergence of the significant cyclone–anticyclone asymmetry and of a strong restratification in the upper layers, which are not allowed by the quasigeostrophic (QG) or surface quasigeostrophic (SQG) theory. However, despite this strong ageostrophic character, some of the main surface properties are surprisingly still close to the surface quasigeostrophic equilibrium. They include a noticeable shallow (≈k−2) velocity spectrum as well as a conspicuous local spectral relationship between surface kinetic energy, sea surface height, and density variance over a large range of scales (from 400 to 4 km). Furthermore, surface velocities can be remarkably diagnosed from only the surface density using SQG relations. This suggests that the validity of some specific SQG relations extends to dynamical regimes with large Rossby numbers. The interior dynamics, on the other hand, strongly differ from the surface dynamics, involving a small Rossby number, a steep (≈k−4) velocity spectrum, and a somewhat steeper density spectrum. The compensation of the surface restratification by a destratification at depth confirms a connection between the surface and the interior induced by the small-scale divergent motions.

An inviscid regularization for the surface quasi‐geostrophic equation

AbstractInspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite‐time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. © 2007 Wiley Periodicals, Inc.

Nodal lines in turbulence

The statistics of the nodal lines of scalar fields in two-dimensional (2d) turbulence is found to be conformal invariant and equivalent to that of cluster boundaries in critical phenomena. That allows for a rich variety of exact analytic results, first time in turbulence studies. In particular, the statistics of zero-vorticity lines in Navier-Stokes turbulence is found to be equivalent to that of critical percolation. The statistics of the zero-temperature lines in surface quasi-geostrophic (SQG) turbulence is found to be equivalent to that of the isolines of a Gaussian (free) field.

Inverse Turbulent Cascades and Conformally Invariant Curves

We offer a new example of conformal invariance (local scale invariance) far from equilibrium-the inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a one-dimensional Brownian walk (called Schramm-Loewner evolution or SLEkappa). The diffusivity is close to kappa=4, that is, isotemperature curves belong to the same universality class as domain walls in the O(2) spin model. Several statistics of temperature clusters and isolines are shown to agree with the theoretical expectations for such a spin system at criticality. We also show that the direct cascade in two-dimensional Navier-Stokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.

Potential use of microwave sea surface temperatures for the estimation of ocean currents

In this paper, we examine the emerging potential offered by satellite microwave radiometer SST measurements to complement altimeter data to quantitatively derive surface ocean currents. The proposed methodology does not follow standard sequential temporal analysis but follows the application of the Surface Quasi‐Geostrophic (SQG) theory. Accordingly, under favourable environmental conditions, the implementation for this methodology is simple and robust, and most importantly, solely requires a single SST image. For the present demonstration, altimetric measurements are used to infer a necessary adjustment to match the kinetic energy level for length scales smaller than 300 km. This helps to derive a regional effective Brunt‐Väisälä frequency to produce SQG surface current estimates. As demonstrated, the results are very encouraging and strongly invite to consider the systematic use of satellite microwave radiometer measurements.

Large-scale energy spectra in surface quasi-geostrophic turbulence

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation \[\partial_t(-\Delta)^{1/2}\psi+J\big(\psi,(-\Delta)^{1/2}\psi\big)=\mu\Delta\psi+f\] is studied. The nonlinear transfer of this system conserves the two quadratic quantities but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.

Surface quasigeostrophic turbulence: The study of an active scalar.

We study the statistical and geometrical properties of the potential temperature (PT) field in the surface quasigeostrophic (SQG) system of equations. In addition to extracting information in a global sense via tools such as the power spectrum, the g-beta spectrum, and the structure functions we explore the local nature of the PT field by means of the wavelet transform method. The primary indication is that an initially smooth PT field becomes rough (within specified scales), though in a qualitatively sparse fashion. Similarly, initially one-dimensional iso-PT contours (i.e., PT level sets) are seen to acquire a fractal nature. Moreover, the dimensions of the iso-PT contours satisfy existing analytical bounds. The expectation that the roughness will manifest itself in the singular nature of the gradient fields is confirmed via the multifractal nature of the dissipation field. Following earlier work on the subject, the singular and oscillatory nature of the gradient field is investigated by examining the scaling of a probability measure and a sign singular measure, respectively. A physically motivated derivation of the relations between the variety of scaling exponents is presented, the aim being to bring out some of the underlying assumptions which seem to have gone unnoticed in previous presentations. Apart from concentrating on specific properties of the SQG system, a broader theme of the paper is a comparison of the diagnostic inertial range properties of the SQG system with both the two- and three-dimensional Euler equations. (c) 2002 American Institute of Physics.