Surface Quasi-geostrophic Equation Research Articles

Existence of solutions to fluid equations in Hölder and uniformly local Sobolev spaces

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Hölder spaces Cr(R2) for r>1; to avoid an integrability assumption (such as membership of the data in an Lq space) we introduce a generalization of the SQG constitutive law. As an application of the Hölder theory, we use these solutions when forming an approximation sequence in the proof of existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Huls(R2) for s≥3. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.

Global well-posedness of slightly supercritical SQG equations and gradient estimate

We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic (SQG) equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation’s criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. Combining the nonlinear maximum principle with a modulus of continuity, we prove an uniform-in-time gradient estimate for the critical and slightly supercritical SQG equation. It improves the previous double exponential bound by Kiselev–Nazarov–Volberg to the single exponential. In addition, we prove the eventual exponential decay of the solutions.

On the fractional Laplacian of some positive definite kernels with applications in numerically solving the surface quasi-geostrophic equation as a prominent fractional calculus model

This paper provides the Riesz potential and fractional Laplacian (−Δ)s, s∈R of the famous radial kernels, including the Gaussian, multiquadric, Sobolev spline, and mainly focuses on Wendland kernels. We show that (−Δ)s maps these kernels into the kernels constructed by the generalized hypergeometric functions or the Meijer G–functions. An essential application of the results applies to developing recent meshless methods for solving equations involving fractional Laplacian. Compared to the recently proposed Gaussian meshless method, our numerical results show that working with other kernels to solve fractional equations can bring better results. As a real word application, we consider the numerical study of the surface quasi-geostrophic equations.

Oceanic dipoles in a surface quasi-geostrophic model

Analysis of satellite altimetry and Argo float data leads Ni et al. (J. Geophys. Res., 125, 2020, e2020JC016479) to argue that mesoscale dipoles are widespread features of the global ocean having a relatively uniform three-structure that can lead to strong vertical exchanges. Almost all the features of the composite dipole they construct can be derived from a model for multipoles in the surface quasi-geostrophic equations for which we present a straightforward novel solution in terms of an explicit linear algebraic eigenvalue problem, allowing simple evaluation of the higher radial modes that appear to be present in the observations and suggesting that mass conservation may explain the observed frontogenetic velocities.

Metastability for the Dissipative Quasi-Geostrophic Equation and the Non-local Enhancement

In this paper, we study the linear metastability for the linearized 2D dissipative surface quasi-geostrophic equation with small viscosity $$\nu $$ around the quasi-steady state $$\Theta _{\sin }=-e^{-\nu t}\sin y$$ . We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term $$\cos y\partial _x(-\Delta )^{-\frac{1}{2}}\theta $$ re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.

Long time behavior for the critical modified surface quasi-geostrophic equation

This paper considers the dynamics of the forced critical modified surface quasi-geostrophic equation on T2 as the index α of the dissipative operator Λα belongs to [1,2). At first, when the forcing term f∈Lp(T2) with p>2α, we prove the existence of the attractor in L2(T2). Then, when f∈L∞(T2), we show the existence of determining modes and obtain a uniform upper bound about the number of the determining modes.

Existence of traveling asymmetric vortex pairs in an ideal fluid

In this paper we study the existence of counter-rotating vortex pairs with different scales and different distributions for the Euler equations. We construct a family of traveling asymmetric vortex pairs using the variational method. The quasi-geostrophic shallow water equations and the generalized surface quasi-geostrophic equations are also discussed in a similar framework.

Global solutions for the generalized SQG equation and rearrangements

In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with N N -fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating N N point vortices and counter-rotating pairs. Some other quantitative properties are also established.

Scalar auxiliary variable approache for the surface quasi-geostrophic equation

We present a numerical method for approximating the two-dimensional surface quasi-geostrophic equation. We first reformulate the equation into an equivalent system by using the scalar auxiliary variable approach. Then we propose first-order and second-order discretization schemes for solving the new surface quasi-geostrophic system in time. Furthermore, we prove that both of these schemes based on the scalar auxiliary variable approach satisfy an unconditionally energy stability property. Since the method gives two linear equations with constant coefficients in each time step. We have an efficient approach and accurate for solving these spacial fractional diffusion equations. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for inviscid problems and the problems with small viscosity.

Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation

Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation

Well-posedness for SQG sharp fronts with unbounded curvature

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. Boundedness of curvature plays an important role in the theoretical [F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014) 635–639] and numerical [D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005) 5949–5952; R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett. 112 (2014) 144505] study of singularity formation. In this paper, we establish local well-posedness for SQG sharp fronts of low Sobolev regularity, [Formula: see text] for arbitrarily small [Formula: see text]. This is the first construction for SQG sharp front solutions with unbounded curvature.

Active vector models generalising 3D Euler and electron–MHD equations

We introduce an active vector system, which generalises both the 3D Euler equations and the electron–magnetohydrodynamic equations (E–MHD). We may as well view the system as singularised systems for the 3D Euler equations, in which case the equations of (E–MHD) correspond to the order two more singular one than the 3D Euler equations. The generalised surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in Chae et al (2012 Commun. Pure Appl. Math. 65 1037–66). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform Lp−Lq estimates with respect to a parameter β. From this one can cover the Lp-boundedness of the Calderón-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space Bp,qs and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.

On white noise solutions of mSQG equations on [formula omitted

In this paper, we show existence of white noise solutions for weak formulations of modified Surface Quasi-Geostrophic (mSQG) equations. Based on previous results ([10]) on white noise solutions for mSQG equations on the torus T2, we show a similar result for the whole space R2 by letting the volume of the torus go to infinity and applying compactness methods (Skorokhod's theorem).

Hölder regularity in time of solutions to the generalized surface quasi-geostrophic equation

In this paper, we are concerned with the Hölder regularity in time of both solutions and their energy for the generalized surface quasi-geostrophic equation with the velocity v given by v=R⊥Λγ−1θ. In the first part, it is shown that the regularity in time of solutions θ is Ctα for 0<α<1 with 0<γ≤1 and Ctα/γ for γ−1<α<1 with 1≤γ<2 , as long as θ belongs to Lt∞Cxα. In the second part, we address the Hölder regularity in time of their energy and establish the corresponding criteria for the temperature in Onsager-subcritical spaces.

Large-time behaviour of solutions to the surface quasi-geostrophic equation

Large-time behaviour of solutions to the surface quasi-geostrophic equation

Stability result for the surface quasi-geostropic equations with horizontal dissipation in anisotropic Sobolev space

The stability of smooth solutions of an anisotropic surface quasi-geostrophic equation with horizontal dissipation remains an open problem. In this work, we present a partial answer to this problem in a rougher function space. More precisely, if the initial data θ0 belong to anisotropic Sobolev space H0,s with 12&amp;lt;s&amp;lt;1, then there exists a global small solution, provided that the initial data are small in H0,s. The stability problem in the space H0,1 or more regular spaces remains open. The main tools used are the Littlewood–Paley theory and standard techniques.

Asymptotic study of supercritical surface Quasi-Geostrophic equation in critical space

In this paper we prove, if θ∈C([0,∞),H2−2α(R2)) is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then ‖θ(t)‖H2−2α decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

Non existence and strong ill-posedness in Ck and Sobolev spaces for SQG

We construct solutions in R2 with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in Ck (k≥2) but that are not in Ck for t>0. We prove a similar result also for Hs in the range s∈(32,2). Moreover, we prove strong ill-posedness in the critical space H2.

Formation of small-scale vortices in the core of a large merged vortex

The merger of two surface quasi-geostrophic vortices is examined in detail. As the two vortices collapse towards each other in the merging process, they trap their external fronts between them; these fronts are inserted into the final merged vortex, where they form a central, nearly parallel, sheared velocity strip, sensitive to barotropic instability. As a result, this strip breaks up into an alley of small vortices. Subsequently, these small vortices may undergo merger and grow in size in the core of the large merged vortex. The number of small trapped vortices decreases correspondingly. Finally, a single or two small vortices remain. These processes are analysed using a numerical model of the surface quasi-geostrophic equations. The sensitivity of this process to the initial vortex characteristics is explored. A parallel is drawn between this problem and the instability of a rectilinear strip of temperature with a central gap. The application of this problem to the Ocean is discussed.