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We use a non-local maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field u is obtained from the active scalar . by a Fourier multiplier with symbol iζ⊥|ζ|-1m(|ζ|), wheremis a smooth increasing function that grows slower than log log |ζ| as |ζ| → ∞. [ABSTRACT FROM AUTHOR]

The weak-strong uniqueness of the dissipative surface quasi-geostrophic equation is studied. It is proved that if &thetas; and &thetas;˜ are two weak solutions of the quasi-geostrophic equation initially from the same function &thetas;(0) = &thetas;˜(0) ∈ L2(…2) and the weak solution . is in the regular class ∇&thetas; ∈ Lr (0, T; B0p,∞(…2)) for 2/p + α/r =α, 2/α < p < ∞, 0 < α ≤2, then &thetas; = &thetas;˜ on …2 × [0,T]. [ABSTRACT FROM AUTHOR]

We study solutions of the surface quasi-geostrophic (SQG) equation which are locally constant outside a thin neighbourhood of a curve that evolves with time. To such an SQG solution we associate a distinguished curve (the 'spine'). If the above thin neighbourhood has thickness δ, then we prove that the spine satisfies its own evolution equation (equal to the sharp-front equation) modulo errors O(δ2| log δ|). [ABSTRACT FROM AUTHOR]

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincare Anal. Non Lineaire 28 283-301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field. [ABSTRACT FROM AUTHOR]

We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalized viscosity of the form ([?]^)a, where global regularity has been proved mathematically for the subcritical parameter range a [?] 1/2. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a very slow and oscillatory damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary a = 1/2 is of technical nature. [ABSTRACT FROM AUTHOR]

We consider the surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove the global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity. [ABSTRACT FROM AUTHOR]

In this work, we give a complete description of the asymptotic behaviour of quasi-geostrophic equations in the subcritical range \gamma\in(\frac{1}{2},1] . We first show that its solutions simplify asymptotically as t - [?]. More precisely, solutions behave as a particular self-similar solution normalized by the mass as t - [?] and when the initial data belong to L^{1}(\mathbb{R}^{2})\cap L^{\frac{2}{2\gamma-1}}(\mathbb{R}^{2}) . On the other hand, we show that solutions with initial data in L^{\frac{2}{2\gamma-1}}(\mathbb{R}^{2}) decay towards zero as t - [?] in this space. All results are obtained regardless of the size of the initial condition. [ABSTRACT FROM AUTHOR]

It is shown that the non-homogeneous dissipative quasi-geostrophic equation ∂θ∂t+u⋅∇θ+κ(−Δ)αθ=sin⁡x2,   u=(−∂x2, ∂x1)(−Δ)−β/2θ with and losses stability at a critical value and this instability gives rise to a circle of steady-state solutions. [ABSTRACT FROM AUTHOR]

We revisit the discussion of the energetics of quasi-geostrophic flows from a geometric perspective based on the introduction of an effective metric, built in terms of the flow stratification and the Coriolis parameter. In particular, an appropriate notion of normal modes is defined through a spectral geometry problem in the ocean basin (a compact manifold with boundary) for the associated Laplace–Beltrami scalar operator. This spectral problem can be used to systematically encode non-local aspects of stratification and topography. As examples of applications we revisit the isotropy assumption in geostrophic turbulence, identify (a patch of) the hyperbolic space as the leading-order term in the effective geometry for the deep mesoscale ocean and, finally, discuss some diagnostic tools based on a simple statistical mechanics toy-model to be used in numerical simulations and/or observations of quasi-geostrophic flows. [ABSTRACT FROM AUTHOR]

We study the three-dimensional active scalar equation called the magneto-geostropic equation, which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth’s fluid core. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity u(x, t) and the scalar temperature produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive () active scalar equation, initial data implies the existence of unique, global weak solutions. If with s  >  0, then the solution for all time. In the case of positive diffusivity (), even for singular initial data , the global solution is instantaneously -smoothed and satisfies the drift-diffusion equation classically for all t  >  0. We demonstrate, via a particular example, that the viscous magneto-geostrophic equation permits exponentially growing ‘dynamo type’ instabilities. [ABSTRACT FROM AUTHOR]

Numerical MHD simulations play an increasingly important role for understanding the mechanisms of stellar magnetism. We present simulations of convection and dynamos in density-stratified rotating spherical fluid shells. We employ a new 3D simulation code for obtaining the solution of a physically consistent anelastic model of the process with a minimum number of parameters. The reported dynamo simulations extend into a “buoyancy-dominated” regime where the buoyancy forcing is dominant while the Coriolis force is no longer balanced by pressure gradients, and strong anti-solar differential rotation develops as a result. We find that the self-generated magnetic fields, despite being relatively weak, are able to reverse the direction of differential rotation from anti-solar to solar-like. We also find that convection flows in this regime are significantly stronger in the polar regions than in the equatorial region, leading to non-oscillatory dipole-dominated dynamo solutions, and to a concentration of magnetic field in the polar regions. We observe that convection has a different morphology in the inner and the outer part of the convection zone simultaneously such that organized geostrophic convection columns are hidden below a near-surface layer of well-mixed highly chaotic convection. While we focus our attention on the buoyancy-dominated regime, we also demonstrate that conical differential rotation profiles and persistent regular dynamo oscillations can be obtained in the parameter space of the rotation-dominated regime even within this minimal model. [ABSTRACT FROM AUTHOR]

The general aspects of zonal flow physics, their formation, damping and interplay with quasi two dimensional turbulence are explained in the context of magnetized plasmas and quasi-geostrophic fluids with an emphasis on formation and selection of spatial patterns. General features of zonal flows as they appear in planetary atmospheres, rotating convection experiments and fusion plasmas are reviewed. Detailed mechanisms for excitation and damping of zonal flows, and their effect on turbulence via shear decorrelation is discussed. Recent results on nonlocality and staircase formation are outlined. [ABSTRACT FROM AUTHOR]

The stability of circular vortices to normal mode perturbations is studied in a multi-layer quasi-geostrophic model. The stratification is fitted on the Gulf of Cadiz where many Mediterranean Water (MW) eddies are generated. Observations of MW eddies are used to determine the parameters of the reference experiment; sensitivity tests are conducted around this basic case. The objective of the study is two-fold: (a) determine the growth rates and nonlinear evolutions of unstable perturbations for different three-dimensional (3D) velocity structures of the vortices, (b) check if the different structure of our idealized vortices, mimicking MW cyclones and anticyclones, can induce different stability properties in a model that conserves parity symmetry, and apply these results to observed MW eddies. The linear stability analysis reveals that, among many 3D distributions of velocity, the observed eddies are close to maximal stability, with instability time scales longer than 100 days (these time scales would be less than 10 days for vertically more sheared eddies). The elliptical deformation is most unstable for realistic eddies (the antisymmetric one dominates for small eddies and the triangular one for large eddies); the antisymmetric mode is stronger for cyclones than for anticyclones. Nonlinear evolutions of eddies with radii of about 30 km, and elliptically perturbed, lead to their re-organization into 3D tripoles; smaller eddies are stable and larger eddies break into 3D dipoles. Horizontally more sheared eddies are more unstable and sustain more asymmetric instabilities. In summary, few differences were found between cyclone and anticyclone stability, except for strong horizontal velocity shears. [ABSTRACT FROM AUTHOR]

We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on a rapidly rotating planet or star. The presence of a background toroidal magnetic field is assumed, and the “shallow water” beta-plane approximation is used. We derive a single equation for the slow large length scale dynamics. The range of validity of this equation fits the MHD of the lighter fluid at the top of Earth's outer core. The form of this equation is similar to the quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the parameters are essentially different. Our equation also implies the inverse cascade; but contrary to the usual Q-G situation, the energy cascades to smaller length scales, while the enstrophy cascades to the larger scales. We find the Kolmogorov-type spectrum for the inverse cascade. The spectrum indicates the energy accumulation in larger scales. In addition to the energy and enstrophy, the obtained equation possesses an extra (adiabatic-type) invariant. Its presence implies energy accumulation in the 30° sector around zonal direction. With some special energy input, the extra invariant can lead to the accumulation of energy in zonal magnetic field; this happens if the input of the extra invariant is small, while the energy input is considerable. [ABSTRACT FROM AUTHOR]