Burgers Turbulence with Large-scale Forcing

Burgers turbulence supported by white-in-time random forcing at low wavenumbers is studied analytically and by computer simulation. It is concluded that the probability density Q of velocity gradient displays four asymptotic regimes at very large Reynolds number: (A) a region of large positive gradient where Q decays rapidly (reduction of gradient by stretching); (B) an intermediate region of negative gradient where Q falls off as the inverse third power of gradient (transient inviscid steepening of negative gradient); (C) an outer power-law region of negative gradient where Q falls off as the reciprocal of gradient (shoulders of mature shocks); (D) a final region of large gradient where Q decays very rapidly (interior of mature shocks). The probability density of velocity difference across an interval r, divided by r, lies on Q throughout regions A and B and into the middle of C, for small enough inertial-range r.Comment: Revtex (8 pages) with 11 postscript figures (separate file

Phase transitions of barotropic flow coupled to a massive rotating sphere - derivation of a fixed point equation by the Bragg method

The kinetic energy of barotropic flow coupled to an infnitely massive rotating sphere by an unresolved complex torque mechanism is approximated by a discrete spin-lattice model of fluid vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained energy functional is a function of spin-spin coupling and spin coupling with the rotation of the sphere. A mean field approximation similar to the Curie-Weiss theory, modeled after that used by Bragg and Williams to treat a two dimensional Ising model of ferromagnetism, is used to find the barotropic vorticity states at thermal equilibrium for given temperature and rotational frequency of the sphere. A fixed point equation for the most probable barotropic flow state is one of the main results.Comment: 31 pages, 6 figure

Reduction of mean-square advection in turbulent passive scalar mixing

Direct numerical simulation data show that the variance of the coupling term in passive scalar advection by a random velocity field is smaller than it would be if the velocity and scalar fields were statistically independent. This effect is analogous to the "depression of nonlinearity" in hydrodynamic turbulence. We show that the trends observed in the numerical data are qualitatively consistent with the predictions of closure theories related to Kraichnan's direct interaction approximation. The phenomenon is demonstrated over a range of Prandtl numbers. In the inertial-convective range the depletion is approximately constant with respect to wavenumber. The effect is weaker in the Batchelor range

Passive Scalar: Scaling Exponents and Realizability

An isotropic passive scalar field $T$ advected by a rapidly-varying velocity field is studied. The tail of the probability distribution $P(\theta,r)$ for the difference $\theta$ in $T$ across an inertial-range distance $r$ is found to be Gaussian. Scaling exponents of moments of $\theta$ increase as $\sqrt{n}$ or faster at large order $n$, if a mean dissipation conditioned on $\theta$ is a nondecreasing function of $|\theta|$. The $P(\theta,r)$ computed numerically under the so-called linear ansatz is found to be realizable. Some classes of gentle modifications of the linear ansatz are not realizable.Comment: Substantially revised to conform with published version. Revtex (4 pages) with 2 postscript figures. Send email to [email protected]

Cascades, backscatter and conservation in numerical models of two‐dimensional turbulence

The equations governing atmospheric flow imply transfers of energy and potential enstrophy between scales. Accurate simulation of turbulent flow requires that numerical models, which have finite resolution and truncation errors, adequately capture these interscale transfers, particularly between resolved and unresolved scales. It is therefore important to understand how accurately these transfers are modelled in the presence of scale‐selective dissipation or other forms of subgrid model. Here, the energy and enstrophy cascades in numerical models of two‐dimensional turbulence are investigated using the barotropic vorticity equation. Energy and enstrophy transfers in spectral space due to truncated scales are calculated for a high‐resolution reference solution and for several explicit and implicit subgrid models at coarser resolution. The reference solution shows that enstrophy and energy are removed from scales very close to the truncation scale and energy is transferred (backscattered) into the large scales. Some subgrid models are able to capture the removal of enstrophy from small scales, though none are scale‐selective enough; however, none are able to capture accurately the energy backscatter. We propose a scheme that perturbs the vorticity field at each time step by the addition of a particular vorticity pattern derived by filtering the predicted vorticity field. Although originally conceived as a parametrization of energy backscatter, this scheme is best interpreted as an energy ‘fixer’ that attempts to repair the damage to the energy spectrum caused by numerical truncation error and an imperfect subgrid model. The proposed scheme improves the energy and enstrophy behaviour of the solution and, in most cases, slightly reduces the root mean square vorticity errors.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106663/1/2166_ftp.pd

A systematic correlation between two-dimensional flow topology and the abstract statistics of turbulence

Velocity differences in the direct enstrophy cascade of two-dimensional turbulence are correlated with the underlying flow topology. The statistics of the transverse and longitudinal velocity differences are found to be governed by different structures. The wings of the transverse distribution are dominated by strong vortex centers, whereas, the tails of the longitudinal differences are dominated by saddles. Viewed in the framework of earlier theoretical work this result suggests that the transfer of enstrophy to smaller scales is accomplished in regions of the flow dominated by saddles.Comment: 4 pages, 4 figure

Oceanic rings and jets as statistical equilibrium states

Equilibrium statistical mechanics of two-dimensional flows provides an explanation and a prediction for the self-organization of large scale coherent structures. This theory is applied in this paper to the description of oceanic rings and jets, in the framework of a 1.5 layer quasi-geostrophic model. The theory predicts the spontaneous formation of regions where the potential vorticity is homogenized, with strong and localized jets at their interface. Mesoscale rings are shown to be close to a statistical equilibrium: the theory accounts for their shape, their drift, and their ubiquity in the ocean, independently of the underlying generation mechanism. At basin scale, inertial states presenting mid basin eastward jets (and then different from the classical Fofonoff solution) are described as marginally unstable states. These states are shown to be marginally unstable for the equilibrium statistical theory. In that case, considering a purely inertial limit is a first step toward more comprehensive out of equilibrium studies that would take into account other essential aspects, such as wind forcing.Comment: 15 pages, submitted to Journal of Physical Oceanograph

Statistical mechanics of Fofonoff flows in an oceanic basin

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h=by which represents either a real bottom topography or the beta-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function psi. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or beta-effect).Comment: 26 pages, 28 figure

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure