2D Quasi-geostrophic Equation Research Articles

Global solutions to the dissipative quasi-geostrophic equation with dispersive forcing

We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces $H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) and the scaling critical space $H^{2-\alpha}(\mathbb{R}^2)$. More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data $\theta_{0} \in H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset $K \subset H^{2-\alpha}(\mathbb{R}^{2})$, which is precompact in some homogeneous Sobolev spaces.

Time periodic solutions to the 2D quasi-geostrophic equation with the supercritical dissipation

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the corresponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

Long Time Existence and Asymptotic Behavior of Solutions for the 2D Quasi-geostrophic Equation with Large Dispersive Forcing

We consider the initial value problem of the 2D dispersive quasi-geostrophic equation. We prove the long time existence of the solution for given initial data $$\theta _0 \in H^s(\mathbb {R}^2)$$ with $$s>2$$ . Moreover, we show that the solution converges to the corresponding linear dispersive solution $$e^{-AtR_1}\theta _0$$ when the size of dispersion parameter goes to infinity.

Stability of Fractionally Dissipative 2D Quasi-geostrophic Equation with Infinite Delay

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space $$H^s$$ with $$s\ge 2-2\alpha $$ and $$\alpha \in (\frac{1}{2},1)$$ . First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax–Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin–Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay.

Attractors for 2D quasi-geostrophic equations with and without colored noise in W2α−,p(ℝ2)

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in [Formula: see text] In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in [Formula: see text] Here, we do not add any modifying factor on the nonlinear term.

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

In this article, the spectral instability and the associated bifurcations of the shear flows of the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. To determine the linear instability of the basic shear flow, we write the corresponding eigenvalue problem as a system of finite difference equations whose nontrivial solutions are expressed in the form of continued fractions. By a rigorous analysis of these continued fractions, we prove the existence of a number Rc such that if the control parameter R, which is proportional to the Reynolds number and the intensity of the curl of forcing, is over Rc, then the basic shear flow loses its stability. To shed light on the bifurcation involved in the loss of stability of the basic shear flow, a natural method is used to reduce the quasi-geostrophic equation to a system of ordinary differential equations. Based on numerical experiments on the coefficients of this reduced system, we show that both supercritical and subcritical Hopf bifurcations occur depending on the frequency of the generalized Kolmogorov forcing. Moreover, we investigate the double Hopf bifurcations which occur at critical aspect ratios. Our results show that in the double Hopf bifurcation case, two periodic solutions, one stable and the other unstable, bifurcate on R>Rc.

2D Quasi-Geostrophic equation; sub-critical and critical cases

Our task here is to use a version of the ‘vanishing viscosity technique’ to study the critical 2D Quasi-Geostrophic equation. The present paper extends and specializes the results reported in Dlotko et al. (2015). We treat now in more detail the solutions of the critical problem (α=12); in particular their uniqueness, regularity and long time behavior.

Commutator estimate in terms of partial derivatives of solutions for the dissipative quasi-geostrophic equation

Through Littlewood-Paley decomposition argument, a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2D Quasi-Geostrophic equation is established. As an application of this estimate, we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical Besov spaces.

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on β-plane

We consider the 2d quasigeostrophic equation on the β-plane for the stream function ψ, with dissipation and a random force:Here . For typical values of the horizontal period L we prove that the law of the action-vector of a solution for (*) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as β → ∞, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of (*) converges to that of the effective equation. Moreover, this convergence is uniform in κ ∈ (0, 1]. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension ⩽3; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits limL=ρ→∞limβ→∞ and limκ→0limβ→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.

Long time dynamics of 2D quasi-geostrophic equations with damping in [formula omitted

The long time behavior of solutions of 2D quasi-geostrophic equations with damping on the whole space is studied. The existence of the global attractor in Lp is proved. New difficulties come from nonlocal terms and the Banach space Lp. We overcome them by using commutator estimates and positivity inequalities associated with fractional Laplacian.

On dimension of the global attractor for 2D quasi-geostrophic equations

We obtain a precise upper bound of the fractal dimension of the global attractor for 2D quasi-geostrophic equations. The upper bound is a decreasing function of the coefficient κ of dissipative term, which conforms to physical intuition. Moreover, the bound tends to infinity as κ→0 and α→12, which reflects the chaotic behavior of the QG equation without dissipative effect and in critical case, respectively. Our result gives an answer to a problem posed in Ju [N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Commun. Math. Phys., 255 (2005) 161–181].

Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing

We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.

Bifurcation of Critical Points for Solutions of the 2D Euler and 2D Quasi-geostrophic Equations

We consider the 2D Euler and 2D quasi-geostrophic equations with periodic boundary conditions. For both systems we will use the stream-function formulation and study the bifurcation problem for the critical points of the stream function. In a small neighborhood of the origin, we construct a set of initial data such that initial critical points of the stream function bifurcate from 1 to 2 and then to 3 critical points in finite time. For the quasi-geostrophic equation the whole bifurcation process takes place strictly within the lifespan of the constructed local solution.

Remarks on the weak–strong uniqueness for the 2D quasi-geostrophic equation in BMO space

This paper is concerned with the weak–strong uniqueness of the Cauchy problem for the 2D dissipative quasi-geostrophic equation. We proved the weak–strong uniqueness when one of the two weak solutions lies in the regular class ∇ θ ∈ L 1 ( ( 0 , T ) ; B M O ) .

A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation

In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces \({\mathring{B}^{1-\alpha}_{\infty,q}}\), and global well-posedness of the critical quasi-geostrophic equation in \({\mathring{B}^{0}_{\infty,q}}\) for all 1 ≤ q < ∞. Here \({\mathring{B}^{s}_{\infty,q} }\) is the closure of the Schwartz functions in the norm of \({B^{s}_{\infty,q}}\).

Changes in Jupiter’s Great Red Spot (1979–2006) and Oval BA (2000–2006)

We analyze velocity fields of the Great Red Spot (GRS) and Oval BA that were previously extracted from Cassini, Galileo, and Hubble Space Telescope images (Asay-Davis, X.S., Marcus, P.S., Wong, M., de Pater, I. [2009]. Icarus 203, 164–188). Our analyses use reduced-parameter models in which the GRS, Oval BA, and surrounding zonal (east–west) flows are assumed to have piece-wise-constant potential vorticity (PV), but with finite-sized transition regions between the pieces of constant PV rather than sharp steps. The shapes of the regions of constant PV are computed such that the flow is a steady, equilibrium solution of the 2D quasigeostrophic equations when viewed in a frame translating uniformly in the east–west direction. All parameter values of the models, including the magnitudes of the PV, areas of the regions with constant PV, locations of the transition regions, widths of the transition regions, and the value of the Rossby deformation radius, are found with a genetic algorithm such that the velocity produced by the equilibrium solution is a “best-fit” to the observed velocity fields. A Monte Carlo method is used to estimate the uncertainties in the best-fit parameter values. The best-fit results show that there were significant changes (greater than the uncertainties) in the PV of the GRS between Galileo in 1996 and Hubble in 2006. In particular, the shape of the PV anomaly of the GRS became rounder, and the area of the PV anomaly of the GRS decreased by 18%, although the magnitudes of PV in the anomaly remained constant. In contrast, neither the area nor the magnitude of the PV anomaly of the Oval BA changed from 2000, when its cloud cover was white, to 2006, when its cloud cover was red. The best-fit results also show that the areas of the PV anomalies of the GRS and of the Oval BA are smaller than the areas of their corresponding cloud covers at all times. Using the best-fit values of the Rossby deformation radius, we show that the Brunt–Väisälä frequency is 15% larger at 33°S than at 23°S. As expected ( Marcus, 1993), the best-fit results show that the PV of the zonal flow has “jumps” at the latitudes of the maxima of the eastward-going jet streams. However, a surprising result is that a large “jump” in the PV of the zonal flow occurs at the location of a maximum of the westward going jet stream neighboring the GRS. Another surprise is that the jumps in the PV of the zonal flow do not all have the same sign, which implies that there is not a monotonic “staircase” of zonal PV from north to south as was anticipated ( Marcus, 1993; McIntyre, 2008).

Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions

We consider the 2D surface quasi-geostrophic equation with fractional dissipation on . We obtain local and global results for plane wave type initial conditions. In some cases an explicit series expansion is constructed which represents the solution as a sum of delta functions in the Fourier space.

On the a priori estimates for the Euler, the Navier–Stokes and the quasi-geostrophic equations

We prove new a priori estimates for the 3D Euler, the 3D Navier–Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.

Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation (−Δ) γ/2, 0 < γ ≤ 1). We show in this paper that if $${\theta \in C((0, T); C^{1-\gamma})}$$ , or $${\theta \in L^{r}((0, T); C^\alpha)}$$ with $${\alpha = 1 - \gamma + \frac{\gamma}{r}}$$ is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in $${(0, T] \times {\mathbb R}^2}$$ . This result improves our previous result in [18].

A regularity criterion for the dissipative quasi-geostrophic equations

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations (with dissipation (−Δ)γ/2, 0<γ⩽1). More precisely, we show that if θ∊Ltr0((0,T);Bp,∞α(R2)) with α=2p+1−γ+γr0 is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in (0,T]×R2. This result extends the regularity result of Constantin and Wu [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. I. H. Poincaré – AN (2007), doi:10.1016/j.anihpc.2007.10.001] to scaling invariant spaces.