Surface Quasi-geostrophic Research Articles

On sharp fronts and almost-sharp fronts for singular SQG

In this paper we consider a family of active scalars with a velocity field given by u=Λ−1+α∇⊥θ, for α∈(0,1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to α=0.We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size δ). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of δ) compared to other compatible curves.

Numerical Simulations of Surface Quasi-Geostrophic Flows on Periodic Domains

We propose a novel algorithm for the approximation of surface quasi-geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomen...

Accuracy and computational efficiency of dealiasing schemes for the DNS of under resolved flows with strong gradients

In this paper, we have studied the effect of residual aliasing error of the second order Runge–Kutta (RK2) based Random Phase Shift Method (RPSM) which shows smoothing effect in the solution of under-resolved flows involving strong gradients. Firstly, we show that RPSM is almost as accurate as the fully dealiased 3/2 Padding scheme but with similar computational cost as the fast 2/3 Truncation scheme. Secondly, we show that RPSM has high accuracy in the case of under-resolved shear layer and Surface Quasi-Geostrophic (SQG) flows. Further, we show that the 2/3 Truncation scheme turns more computationally expensive than 3/2 Padding or RPSM when we try to achieve the same level of accuracy. Filtering based dealiasing schemes are found to be an inappropriate choice for a variety of flow problems because they are prone to unphysical parasitic currents. For the first time error norm based computational efficiency, i.e., high accuracy at the lower computational cost of RPSM scheme is shown. Although some artifacts of dealiasing remain due to Fourier windowing in RPSM, it is found to be numerically stable even in under-resolved conditions at later simulation time. We have validated our numerical results with the analytical ones and also with the previous literature.

Long-wave instabilities in the SQG model with two boundaries

ABSTRACT Surface quasi-geostrophic (SQG) flows with a much larger horizontal scale than the Rossby radius of deformation are considered. A new version of the SQG model with two boundaries, which is reduced to a nonlinear system of partial differential equations, is proposed to describe the dynamics of such flows. This system describes the interaction between the barotropic and baroclinic components of the stream function and generalises the two-dimensional Euler equations for flows with a vertical velocity shear. The laws of conservation of both total and surface potential energies, which follow from this system, have been formulated. The solutions of a number of problems in the theory of baroclinic instability, which are in agreement with already known solutions, have been obtained within the framework of this system. It is shown that vertical shear flows are absolutely unstable, i.e. their instability is independent of the horizontal velocity profile structure. A generalised system of the SQG model equations, which additionally takes into account the β-effect and the Ekman bottom friction, has also been proposed. The transformation of jet flows due to the bottom friction and the influence of the β-effect on the stability of shear flows have been studied based on this system.

An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains

We present a spectral element algorithm and open-source code for computing the fractional Laplacian defined by the eigenfunction expansion on finite 2D/3D complex domains with both homogeneous and nonhomogeneous boundaries. We demonstrate the scalability of the spectral element algorithm on large clusters by constructing the fractional Laplacian based on computed eigenvalues and eigenfunctions using up to thousands of CPUs. To demonstrate the accuracy of this eigen-based approach for computing the factional Laplacian, we approximate the solutions of the fractional diffusion equation using the computed eigenvalues and eigenfunctions on a 2D quadrilateral, and on a 3D cubic and cylindrical domain, and compare the results with the contrived solutions to demonstrate fast convergence. Subsequently, we present simulation results for a fractional diffusion equation on a hand-shaped domain discretized with 3D hexahedra, as well as on a domain constructed from the Hanford site geometry corresponding to nonzero Dirichlet boundary conditions. Finally, we apply the algorithm to solve the surface quasi-geostrophic (SQG) equation on a 2D square with periodic boundaries. Simulation results demonstrate the accuracy, efficiency, and geometric flexibility of our algorithm and that our algorithm can capture the subtle dynamics of anomalous diffusion modeled by the fractional Laplacian on complex geometry domains. The included open-source code is the first of its kind. Program summaryProgram title: Nektarpp_EigenMMCPC Library link to program files:https://doi.org/10.17632/whtc75rj55.1Developer’s repository link:https://github.com/paralab/Nektarpp_EigenMMLicensing provisions: MIT LicenseProgramming language: C/C++, MPINature of problem: An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains.Solution method: A distributed, sparse, iterative algorithm is developed to solve an associated integer-order Laplace eigenvalue problem for use in computing approximate solutions to the fractional diffusion equation.Additional comments including restrictions and unusual features: The code is implemented on CPUs with super-linear parallel efficiency at extreme scale.

Real-time Reconstruction of Surface Velocities from Satellite Observations in the Alboran Sea

Surface currents in the Alboran Sea are characterized by a very fast evolution that is not well captured by altimetric maps due to sampling limitations. On the contrary, satellite infrared measurements provide high resolution synoptic images of the ocean at high temporal rate, allowing to capture the evolution of the flow. The capability of Surface Quasi-Geostrophic (SQG) dynamics to retrieve surface currents from thermal images was evaluated by comparing resulting velocities with in situ observations provided by surface drifters. A difficulty encountered comes from the lack of information about ocean salinity. We propose to exploit the strong relationship between salinity and temperature to identify water masses with distinctive salinity in satellite images and use this information to correct buoyancy. Once corrected, our results show that the SQG approach can retrieve ocean currents slightly better to that of near-real-time currents derived from altimetry in general, but much better in areas badly sampled by altimeters such as the area to the east of the Strait of Gibraltar. Although this area is far from the geostrophic equilibrium, the results show that the good sampling of infrared radiometers allows at least retrieving the direction of ocean currents in this area. The proposed approach can be used in other areas of the ocean for which water masses with distinctive salinity can be identified from satellite observations.

A Dynamical‐Statistical Approach to Retrieve the Ocean Interior Structure From Surface Data: SQG‐mEOF‐R

AbstractCombining the dynamical surface‐trapped mode derived from the Surface Quasi‐Geostrophic (SQG) function with the statistical mode calculated from multivariate empirical orthogonal function (EOF) reconstruction (mEOF‐R) method, this paper proposes a new method, SQG‐mEOF‐R, to estimate the interior density from the sea surface density and sea surface height. This method is applied to the eddy‐resolving Ocean General Circulation Model For the Earth simulator simulation and compared with the conventional SQG, isQG (interior plus SQG), and mEOF‐R methods. Two different dynamical regimes were considered: the NorthWest Pacific, dominated by surface‐intensified eddies, and the Southeast Pacific, characterized by the occurrence of subsurface‐intensified eddies. In both regions, the proposed method proves to be robust in reconstructing mesoscale features, with reduced root‐mean square error and relatively high correlation coefficient. The superiority of SQG‐mEOF‐R is highlighted in those subsurface‐intensified eddies that are little or nothing reconstructed by SQG or isQG method.

Two-front solutions of the SQG equation and its generalizations

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.

Instability of Surface Quasigeostrophic Spatially Periodic Flows

AbstractThe surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.

A class of large solutions to the supercritical surface quasi-geostrophic equation

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.

Symmetries and Critical Phenomena in Fluids

AbstractWe study two‐dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well‐posedness result that allows for the data and solutions to be scale‐invariant. These scale‐invariant solutions are new and their study seems to have far‐reaching consequences.More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two‐dimensional Euler squation. That is, in the well‐known L1∩L∞ theory of Yudovich, the L1‐assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993.Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0‐homo‐geneous in space. Such vorticity is shown to satisfy a new one‐dimensional evolution equation on 𝕊1. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi‐periodic solutions to the two‐dimensional Euler equation exhibiting pendulum‐like behavior. Finally, using the analysis of the one‐dimensional equation, we exhibit strong solutions to the two‐dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary).A similar study can be done for the surface quasi‐geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one‐dimensional model that bears great resemblance to the so‐called De Gregorio model. We then show that finite‐time singularity formation for the one‐dimensional model implies finite‐time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support.While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite‐energy solutions for long time and have direct bearing on the global regularity problem for finite‐energy solutions. © 2019 Wiley Periodicals, Inc.

Confinement of a hot temperature patch in the modified SQG model

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.

Remarks on Well-Posedness of the Generalized Surface Quasi-Geostrophic Equation

In this paper, we are concerned with the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation in which the velocity field is expressed as $u=K\ast\omega$, where $\omega=\omega(x,t)$ is an unknown function and $K(x)=\frac{x^\perp}{|x|^{2+2\alpha}}, 0\le\alpha\le \frac12.$ When $\alpha=0$, it is the two-dimensional Euler equations. When $\alpha=\frac 12$, it corresponds to the inviscid SQG. We will prove that if the existence interval of the smooth solution to the generalized SQG for some $0<\alpha_0\le\frac12$ is $[0,T]$, then under the same initial data, the existence interval of the generalized SQG with $\alpha$ which is close to $\alpha_0$ will keep on $[0,T]$. As a byproduct, our result implies that the construction of the possible singularity of the smooth solution of the Cauchy problem to the generalized SQG with $\alpha>0$ will be subtle, in comparison with the singularity presented in [Kiselev et al 2016]. To prove our main results, the difference between the two solutions and meanwhile the approximation of the singular integrals will be dealt with. Some new uniform estimates with respect to $\alpha$ on the singular integrals and commutator estimates will be shown in this paper.

Asymptotic scale-dependent stability of surface quasi-geostrophic vortices: semi-analytic results

ABSTRACTThe scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analytically and numerically. In particular, we study the sensitivity of the stability of SQG vortices on a nondimensional number , namely the square root of the Burger number, which sets the transition scale between different dynamical regimes corresponding to local and nonlocal dynamics. We analyse the stability of two different examples. The first example is given by a Rankine vortex, characterised by constant buoyancy. For this case, asymptotic analysis suggests that the frequencies of the perturbations at scales smaller than the transition scale show a dependence. At scales larger than the transition scale, the frequencies scale instead like . The second example consists of a Rankine vortex shielded by a filament characterised by a different value of constant buoyancy. For this example we study the dispersion relation for the perturbations for the cases in which the inner vortex and the outer filament have different asymptotic properties behaviour.

Local Well-Posedness of an Approximate Equation for SQG Fronts

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes.

Collapse of generalized Euler and surface quasigeostrophic point vortices.

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.

Regularized and approximate equations for sharp fronts in the surface quasi-geostrophic equation and its generalizations**Supported by the NSF under grant number DMS-1616988.

We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (GSQG) equations. We derive a cubic approximation of the contour dynamics equation, and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.

Determining modes for the surface quasi-geostrophic equation

We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, and measures the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.

Special Issue Editorial: Small Scales and Singularity Formation in Fluid Dynamics

We review recent advances in understanding singularity and small scales formation in solutions of fluid dynamics equations. The focus is on the Euler and surface quasi-geostrophic (SQG) equations and associated models.

A Determining Form for the Subcritical Surface Quasi-Geostrophic Equation

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation.