Quasi-geostrophic Equation Research Articles

On analyticity up to the boundary for critical quasi-geostrophic equation in the half space

<p style='text-indent:20px;'>We study the Cauchy problem for the surface quasi-geostrophic equation with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show a natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.</p>

Vortex collapses for the Euler and Quasi-Geostrophic models

<p style='text-indent:20px;'>This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.</p>

Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces.

Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces.

Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations

We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.

An Initiation Process of Tropical Depression–Type Disturbances under the Influence of Upper-Level Troughs

Abstract In this study, the statistical relationship between tropical upper-tropospheric troughs (TUTTs) and the initiation of summertime tropical depression–type disturbances (TDDs) over the western and central North Pacific is investigated. By applying a spatiotemporal filter to the 34-yr record of brightness temperature and using JRA-55 products, TDD-event initiations are detected and classified as trough-related (TR) or non-trough-related (non-TR). The conventional understanding is that TDDs originate primarily in the lower troposphere; our results refine this view by revealing that approximately 30% of TDDs in the 10°–20°N latitude ranges are generated under the influence of TUTTs. Lead–lag composite analysis of both TR- and non-TR-TDDs clarifies that TR-TDDs occur under relatively dry and less convergent large-scale conditions in the lower troposphere. This result suggests that TR-TDDs can form in a relatively unfavorable low-level environment. The three-dimensional structure of the wave activity flux reveals southward and downward propagation of wave energy in the upper troposphere that converges at the midtroposphere around the region where TR-TDDs occur, suggesting the existence of extratropical forcing. Further, the role of dynamic forcing associated with the TUTT on the TR-TDD initiation is analyzed using the quasigeostrophic omega equation. The result reveals that moistening in the mid- to upper troposphere takes place in association with the sustained dynamical ascent at the southeast side of the TUTT, which precedes the occurrence of deep convective heating. Along with a higher convective available potential energy due to the destabilizing effect of TUTTs, the moistening in the mid- to upper troposphere also helps to prepare the environment favorable to TDDs initiation.

(2+1)-dimensional coupled Boussinesq equations for Rossby waves in two-layer cylindrical fluid* *Supported by the Nature Science Foundation of Shandong Province of China (ZR2018MA017).

In this paper, the existence and propagation characteristics of Rossby waves in a two-layer cylindrical fluid are studied. Firstly, based on the dimensionless baroclinic quasi-geostrophic vortex equations including exogenous and dissipative, we derive new (2+1)-dimensional coupled Boussinesq equations describing wave propagation in polar coordinates by employing a multiscale analysis and perturbation method. Then, the Lie symmetries and conservation laws of the coupled Boussinesq equations are analyzed. Subsequently, by using the -expansion method, the exact solutions of the (2+1)-dimensional coupled Boussinesq equations are obtained. Finally, the effects of coupling term coefficients on the propagation characteristics of Rossby waves are analyzed.

Global solutions of a surface quasigeostrophic front equation

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.

Gevrey regularity and finite time singularities for the Kakutani–Matsuuchi model

In this paper, we deal with the Kakutani–Matsuuchi model which describes the surface elevation η of the water-waves under the effect of viscosity. We first derive the decay rate of weak solutions. This can be used to obtain the decay rate of ‖η(t)‖Ḣ1 when initial data is sufficiently small in Ḣ1. We next show the existence, uniqueness, Gevrey regularity and decay rates of η with sufficiently small initial data in B2,11. To do so, we derive a commutator estimate involving Gevrey operator. We then apply our method to the supercritical quasi-geostrophic equations. We finally show the formation of singularities of smooth solutions in finite time for a certain class of initial data.

The energy flux of three-dimensional waves in the atmosphere: Exact expression for a basic model diagnosis with no equatorial gap

AbstractA model diagnosis for the energy flux of off-equatorial Rossby waves in the atmosphere has previously been done using quasi-geostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space-time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux which can indicate the direction of the group velocity for linear shallow water waves at all latitudes. This analytical result is extended in the present study to a height-dependent framework for three-dimensional waves in the atmosphere. This is achieved by investigating the classical analytical solution of both equatorial and off-equatorial waves in a Boussinesq fluid. For the horizontal component of the energy flux, the same expression has been obtained between equatorial waves and off-equatorial waves in the height-dependent framework, which is linked to a scalar quantity inverted from the isentropic perturbation of Ertel’s potential vorticity. The expression of the vertical component of the energy flux requires computation of another scalar quantity that may be obtained from the meridional integral of geopotential anomaly in a wavenumber-frequency space. The exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized Madden-Julian Oscillation forcing in a basic model experiment. The zonal and vertical fluxes manifest the energy transfer of both equatorial Kelvin waves and off-equatorial Rossby waves with a smooth transition at around 10°S and around 10°N. The meridional flux of wave energy represents connection between off-equatorial divergence regions and equatorial convergence regions.

Stability for the 2D anisotropic surface quasi-geostrophic equation with horizontal dissipation

The stability of the surface quasi-geostrophic (SQG) equation with either horizontal or vertical dissipation is still an open problem. In this paper, we give a partial answer to this problem. More precisely, in the case when the domain is Ω=T×R, we obtain the stability by decomposing the solution into the horizontal averages and the oscillating parts.

An extension of Calderón-Zygmund type singular integral with non-smooth kernel

In the present paper, we consider a kind of singular integralTf(x)=p.v.∫RnΩ(y)|y|n−βf(x−y)dy which can be viewed as an extension of the classical Calderón-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the Lq space for 1<q<∞ and a weak (1,1) type of the singular integral when 0<β<(q−1)nq without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong (q,q) type estimate and weak (1,1) type estimate of the Calderón-Zygmund type singular integral can be recovered when β→0 from our obtained estimates.

Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T2, perturbed by multiplicative transport noise. The equation admits the white noise measure on T2 as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space–time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski’s approach in Gubinelli and Perkowski (2020).

On the Existence, Uniqueness, and Smoothing of Solutions to the Generalized SQG Equations in Critical Sobolev Spaces

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux so as to preserve the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes.

Non-decaying solutions to the critical surface quasi-geostrophic equations with symmetries

We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying as $|x| \to +\infty$, which leads to ambiguity in the velocity $\vec{R}^\perp \theta$. This ambiguity is corrected by imposing $m$-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential non-uniqueness, due to symmetry-breaking bifurcations, in analogy with work \cite{jiasverakillposed,guillodsverak} on the Navier-Stokes equations.

On the generalized eigenvalue problem of Rossby waves vertical velocity under the condition of zonal mean flow and topography

Tailleux (2003, 2012) proposed the influence of background mean flow and topography on the propagation of Rossby waves through a generalized eigenvalue problem, but the mean flow in his paper did not take latitude into account and was not ideal enough. In the middle and high latitudes, the latitude secondary shear flow (ūyy) is close to the β scale, which cannot be ignored in barotropic instability. It is an important factor affecting the propagation of Rossby waves. In this paper, the effects of mean flow and topography on Rossby wave propagation in the barotropic atmosphere are studied based on the quasi-geostrophic vortex equation. To make the physical process more involved in line with the actual atmospheric conditions, we consider both vertical and zonal mean flows. Under the semi geostrophic concept, we obtain the eigenvalue problem of vertical velocity. In the atmosphere, the dispersion relation generally has symmetry for a certain latitude. We study the northern hemisphere, so we only consider the case of negative wavenumber. By comparing the stability of Rossby waves without zonal mean current and with zonal mean current, it is found that zonal mean current has a non-ignored effect on the stability of Rossby waves, it speeds up the phase velocity, with the increase of wavenumber, the frequency decreases.

Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , \begin{equation*} \left \{ \begin {aligned} &amp; \partial _t \theta +u\cdot \nabla \theta =0, \quad \text {in } \mathbb {R}^2 \times (0,T), \\ &amp; u=\nabla ^\perp \psi , \quad \psi = (-\Delta )^{-s}\theta \quad \text {in } \mathbb {R}^2 \times (0,T) , \end{aligned} \right . \end{equation*} 0 &gt; s &gt; 1 0&gt;s&gt;1 , we consider for k ≥ 1 k\ge 1 the problem of finding a family of k k -vortex solutions θ ε ( x , t ) \theta _\varepsilon (x,t) such that as ε → 0 \varepsilon \to 0 θ ε ( x , t ) ⇀ ∑ j = 1 k m j δ ( x − ξ j ( t ) ) \begin{equation*} \theta _\varepsilon (x,t) \rightharpoonup \sum _{j=1}^k m_j\delta (x-\xi _j(t)) \end{equation*} for suitable trajectories for the vortices x = ξ j ( t ) x=\xi _j(t) . We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem ( − Δ ) s W = ( W − 1 ) + γ , in R 2 , 1 &gt; γ &gt; 1 + s 1 − s \begin{equation*} (-\Delta )^sW = (W-1)^\gamma _+ , \quad \text {in } \mathbb {R}^2, \quad 1&gt;\gamma &gt; \frac {1+s}{1-s} \end{equation*} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].

A C1 virtual element method for the stationary quasi-geostrophic equations of the ocean

In this present paper, we propose and analyze a C1-conforming virtual element method to solve the so-called one-layer stationary quasi-geostrophic equations (QGE) with applications in the large scale wind-driven ocean circulation, formulated in terms of the stream-function. This problem corresponds to a nonlinear fourth order partial differential equation. The C1 virtual space and the discrete scheme are built in a straightforward way due to the flexibility of the virtual approach. Under the assumption of small data, we prove well-posedness of the discrete problem by using a fixed-point strategy and under standard assumptions on the computational domain, we establish error estimates in H2-norm for the stream-function. Finally, we report four numerical experiments that illustrate the behavior of the proposed scheme and confirm our theoretical results on different families of polygonal meshes.

A new three dimensional dissipative Boussinesq equation for Rossby waves and its multiple soliton solutions

A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a three dimensional dissipative Boussinesq equation by the reduced perturbation method, i.e.utt+e1uxx+e2(u2)xx+e3utxy+e4uxxxx+e5uxxyy=0. It is emphasized that the new equation is different from the existing Boussinesq equations, which describe the three dimensional nonlinear Rossby waves in the atmosphere. Moreover, we explore the dispersion relation of the linear wave through the new equation. Using the travelling wave method and simplest equation method, the general solution and soliton solutions of the equation are obtained successfully respectively. Finally, the formation mechanism of Rossby waves is discussed by multiple soliton solutions.

Seesawing of Winter Temperature Extremes between East Asia and North America

AbstractIn recent winters, there have been repeated observations of extreme warm and cold spells in the midlatitude countries. This has evoked questions regarding how winter temperature extremes are induced. In this study, we demonstrate that abnormally warm winter weather in East Asia can drive the onset of extremely cold weather in North America approximately one week forward. These seesawing extremes across the basin are mediated by the North Pacific Oscillation (NPO), one of the recurrent atmospheric patterns over the North Pacific. Budget analysis of the quasigeostrophic geopotential tendency equation shows that intense thermal advection over East Asia is able to trigger the growth of the NPO. Vorticity fluxes associated with the upper-level stationary trough then strengthen and maintain the NPO against thermal damping following the onset of the NPO. Differential diabatic heating accompanied by changes in circulation also positively contribute to the growth and maintenance of the NPO. These results imply that recurrent cold extremes, seemingly contrary to global warming, may be an inherent feature resulting from strengthening warm extremes.

Nonlinear Instability for the Surface Quasi-Geostrophic Equation in the Supercritical Regime

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.