Small Scale Creation in Active Scalars

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.

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.

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.

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 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.

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].

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.

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.

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 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 noise present in infrared satellite measurements of sea surface temperature (SST) hampers the use of surface quasi‐geostrophic (SQG) equations to diagnose ocean dynamics at high resolutions. Here we propose a methodology to reduce the contribution of noise when diagnosing surface vorticity, divergence, and vertical velocity from SST able to retain the dynamics at scales of a few kilometers. It is based on the use of denoising techniques with curvelets as basis functions and the application of a selective low‐pass filters to improve the reconstruction of surface upwelling/downwelling patterns. First, it is tested using direct numerical simulations of SQG turbulence and then applied to diagnose low‐frequency vertical velocity patterns from real MODIS (Moderate Resolution Imaging Spectroradiometer) images. The methodology here presented, which is not tied to the validity of SQG equations nor to the use of SST, is quite general and can be applied to a wide range of measurements and dynamical frameworks.

Dynamical modeling of sub-grid scales in 2D turbulence

Sub-Symmetries and Conservation Laws