Well-posedness for SQG sharp fronts with unbounded curvature

This paper studies the finite difference method for the surface quasi-geostrophic (SQG) equations with the fractional Laplacian and Riesz potential. A numerical discretization for the Riesz potential in the SQG equations is proposed. The second-order convergence of the method in l ∞ norm has been verified through both theoretical analysis and numerical experiments. Based on the upwind scheme, an implicit-explicit finite difference (IMEX-FD) scheme for the SQG equations is established, in which the convection term is discretized explicitly and implicitly, while the discretization of other terms employs an implicit structure. The stability of this scheme is proven when the order of fractional Laplacian is one, and the first-order convergence in time and space is also verified through numerical experiments. Finally, the evolutions of single and double vortices problems in SQG equations are simulated in this paper to verify the effectiveness of the IMEX-FD scheme.

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.

Local well-posedness for the generalized surface quasi-geostrophic equation with singular velocity in optimal space

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

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

This article is devoted to the study of the critical dissipative surface quasi-geostrophic (SQG) equation in $${\mathbb{R}^2}$$ . For any initial data $${\theta_{0}}$$ belonging to the space $${\Lambda^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$$ , we show that the critical (SQG) equation has at least one global weak solution in time for all 1/4 ≤ s ≤ 1/2 and at least one local weak solution in time for all 0 < s < 1/4. The proof for the global existence is based on a new energy inequality which improves the one obtain in Lazar (Commun Math Phys 322:73–93, 2013) whereas the local existence uses more refined energy estimates based on Besov space techniques.

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity fieldurelated to the scalarθby$u=\nabla^\perp\Lambda^{\beta-2}\theta$, where$1&lt;\beta\le 2$and$\Lambda=(-\Delta)^{1/2}$is the Zygmund operator. The borderline caseβ= 1 corresponds to the SQG equation and the situation is more singular forβ&gt; 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with$u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu&gt;0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.

The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.

In this paper, we consider weighted jump and variational inequalities of truncated singular integral operator with rough kernel [Formula: see text] where the kernel [Formula: see text] satisfies the cancellation condition and the homogeneous condition of degree [Formula: see text] and [Formula: see text]. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the [Formula: see text] estimate of the jump and variational inequalities of the families [Formula: see text] for [Formula: see text]. Moreover, one can get the weighted [Formula: see text] boundedness of the Calderón–Zygmund operator by letting [Formula: see text].

In this paper we study the time evolution of a temperature patch in $\mathbb{R}^2$ according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

We investigate the evolution of "almost sharp" fronts for the surface quasi-geostrophic equation. This equation was originally introduced in the geophysical context to investigate the formation and evolution of fronts, i.e., discontinuities between masses of hot and cold air. These almost sharp fronts are weak solutions of quasi-geostrophic with large gradient. We relate their evolution to the evolution of sharp fronts.

In this work we consider the surface quasi-geostrophic (SQG) system under location uncertainty (LU) and propose a Milstein-type scheme for these equations, which is then used in a multi-step method. The SQG system considered here consists of one stochastic partial differential equation, which models the stochastic transport of the buoyancy, and a linear operator linking the velocity and the buoyancy. In the LU setting, the Euler-Maruyama scheme converges with weak order 1 and strong order 0.5. Our aim is to develop higher order schemes in time, based on a Milstein-type scheme in a multi-step framework. First we compared different kinds of Milstein schemes. The scheme with the best performance is then included in the two-step scheme. Finally, we show how our two-step scheme decreases the error in comparison to other multi-step schemes.

The surface quasigeostrophic (SQG) model describes flows with surface buoyancy perturbations with no interior quasigeostrophic potential vorticity at small Rossby number Ro and O(1) Burger number, where quasigeostrophic dynamics are expected to hold. Numerical simulations of SQG dynamics have shown that vortices are frequently generated at small scales, which may have O(1) Rossby numbers and therefore may be beyond the limits of SQG. This paper examines the dynamics of an initially geostrophically balanced elliptical surface buoyancy perturbation in both the SQG model and the nonhydrostatic Boussinesq primitive equations (PE). In the case of very small Rossby number, it is confirmed that both models agree, as expected. For larger Ro, non-SQG effects emerge and as a result the solution of the PE deviates significantly from that of SQG. In particular, an increase in the Rossby number has the following effects: (i) the buoyancy filaments at the surface are stabilized in that they generate fewer secondary vortices; (ii) the core of the vortex experiences inertial instability, which results in a uniform buoyancy profile in its interior; (iii) the divergent part of the energy spectrum increases in magnitude; (iv) the PE model has significantly more gravity waves that are radiated from the vortex; (v) the magnitude of the vertical velocity increases; and (vi) in the mature stages of evolution, there are gravitational instabilities that develop because of the complicated dynamics inside the vortex. It is demonstrated that significant non-SQG effects are evident when the large-scale Rossby number of the initial flow is about 0.05 and the local Rossby number is O(1).

In this work we evaluate surface ocean current reconstruction at a global scale from microwave Sea Surface Temperature (SST) measurements, based on the Surface Quasi- Geostrophic (SQG) theory. One of the main advantages of microwave measurements is that they are not affected by cloud cover, which is an important aspect for global studies and operational applications. A new heuristic method that combines the benefits of microwave and altimetry measurements is also proposed and evaluated. Results showed that reconstruction of surface currents based on SQG theory can be improved if the information about the energy spectrum provided by altimeters is used.

Mixing Associated with Submesoscale Processes