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CONFINEMENT OF A HOT TEMPERATURE PATCH IN THE MODIFIED SQG MODEL

. In this paper we study the time evolution of a temperature patch in 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.


1.
Introduction. In the present paper we study the time evolution of a patch of temperature in R 2 according to the modified surface quasi-geostrophic (SQG) equation, considered in recent years by different authors (see e.g. [14] and the references therein). We recall the basic equations of the α-patch model: let θ(x, t), x ∈ R 2 be the solution of the equation where u = (u 1 , u 2 ) = ∂ψ ∂x 2 , − ∂ψ ∂x 1 , θ(x, t) = Λ 2−α ψ(x, t), and Λ = √ −∆ . We denote this system as active scalar f low, where the scalar field θ(x, t) plays the role of a temperature field. In this paper we consider the evolution of a positive compactly supported initial temperature θ 0 ∈ L ∞ (R 2 ) according to (1) for α ∈ (0, 1) . Here we assume that the support of θ 0 is contained in the ball of bounded diameter 2R 0 and centred at the origin. The main aim of this paper is to give an estimate on the growth rate of the diameter of the support of a hot patch of temperature evolving according to (1)-(2)- (3). From the physical point of view, the problem here considered is the confinement in time of a hot patch of temperature in absence of an external forcing, under the quasi-geostrophic approximation. Mathematically, we are considering the long time behavior of the solution of the modified SQG patch model, corresponding to the evolution of an initial datum concentrated in a compact set with smooth boundaries.
We briefly recall the main results obtained about the growth rate of the diameter of the support of a vortex patch in the case of the Euler equation in 2D (that is (3) for α = 0). The first result, obtained by Marchioro in [18], gives a bound of O(t 1/3 ) on the diameter of the support. An improvement of this result was given by Iftimie et al. in [11], where a bound of O (t ln t) 1/4 was proved. For the sake of completness we also recall that a similar estimate was independently obtained at the same time by Serfati in [22].
In the paper by Iftimie et al. [11], the main result was obtained in two different ways. Indeed a first proof of Proposition 2.1 in [11] was based on the application of sharp estimates on the mass of vorticity near the boundary by using a mollifier (similarly to the proof given in [18]). The second proof in the Appendix of the same paper, was given by Gamblin and is based on estimates of higher order momenta In this paper we will generalize the last result, for the estimate of the growth of the support of a temperature patch according to the α-patch model. We will show that the proof is based on non-trivial adaptation of the tecniques used in [11], due to the different Green function of the problem.
2. The physical motivation: confinement of an hot patch of temperature in the SQG model. Here we give an outline of the physical problem considered in this paper, firstly recalling the derivation of the model equations and their meaning. The behavior of the atmosphere in the mid-latitudes is described by the equations of rotating fluids and in many cases geophysicists adopt the so-called quasigeostrophic approximation, that is a perturbative expansion around a uniformly rotating state (see [19]). Under this approximation, the long-scale dynamics of the atmosphere in mid-latitudes is governed by the balance between the Coriolis force and the pressure gradient, i.e.
where we denoted with u ≡ (u 1 , u 2 ) the velocity field, p(x, y, t) the pressure field and f is the Coriolis parameter (assumed constant). We observe that, by means of (5), we can consider the pressure field as the stream function in this kind of problems in fluid mechanics. In order to obtain the evolution equations for the pressure field, a perturbative expansion with respect to the zeroth-order geostrophic field should be done. At the first order of this expansion, the equation of conservation of the potential vorticity arise, that is where In equation (7), q(·) is the potential vorticity and N the buoyancy frequency. The further assumption in the mathematical derivation of the surface quasi-geostrophic equation (SQG) is to take (f /N ) 2 = 1 and q = const. (for simplicity null), therefore obtaining from (7) Is moreover usual in geophysics to apply the Boussinesq approximation, for which the vertical derivative of the pressure field is proportional to the temperature field θ, that is Moreover, from the energy balance law, we have that also the temperature field is advected by the fluid (∂ t + (u · ∇)) θ = 0. (10) By using equations (8) and (9) we finally have where We recall that the operator appearing in (11) is a pseudo-differential operator, defined by the Fourier symbol as where F(·) is the Fourier transform of the operator. This is the operator mapping Neumann to Dirichlet boundary value problems (for a discussion about this point see e.g. [2]). Assuming that the fluid is incompressible, i.e. that the velocity field is divergence free, we finally obtain the basic equations of the SQG model where θ = (−∆) 1/2 ψ. Starting from the seminal paper of Constantin, Majda and Tabak [6] about the singular front formation in a model for a quasi-geostrophic flow, a great number of works about the SQG equation and its generalizations with dissipation appeared in literature (see e.g. [4], [5], [15], [16], [21] and the references therein). The modified SQG equation was recently studied in different papers as a family of active scalar flows interpolating the 2D Euler and SQG equations. Indeed, we observe that, equations (1)-(2)-(3), for α = 0 describe the flow of an ideal incompressible fluid according to the Euler equation in two dimensions in which θ(x, t) has the meaning of vorticity, while for α = 1 the Surface Quasi-Geostrophic equation (SQG) is recovered in which θ(x, t) has the meaning of temperature. This last system has relevance in geophysics as just shown (see for instance [19]). The modified SQG equation was firstly studied by Pierrhaumbert et.al [20], in relation to the spectrum of turbulence in 2D, beginning the research on α−turbulence (see also [10]). On the other hand, from a mathematical point of view, this dynamical system was deeply studied (see [6], [16]) in relation to the problem of formation of singular fronts. The authors observed a formal analogy with the 3D Euler equation and studied the singular behavior of the solutions in the more suitable framework of the 2D quasigeostrophic equation.
The patch problem for SQG-type equations has been considered by different authors in recent papers and leads to nontrivial problems. Local existence results were proved by Gancedo in [9] for the SQG and modified SQG patches with boundaries which are H 3 closed curves. Computational studies have been also performed in [8] and [17], suggesting a finite time singularity with two patches touching each other. In the recent paper [14], Kiselev et al. have studied the modified SQG patch dynamics in domains with boundaries. Some results in this direction have been also proved by Cavallaro et al. in [3] where the dynamics of N concentrated temperature fields has been considered. Finally, uniqueness results for the SQG patch equation have been just proved by Cordoba et al. in [7].
As far as we know, a detailed analysis of the growth in time of the radius of a single patch of temperature evolving according to the modified SQG equation in the whole plane is still missing in the literature, this will be the main object of our paper.
We also briefly add some considerations on the modified SQG that will be useful in the next sections.
It is possible to introduce a weak form of eq.s (1.1)-(1.4): where f (x, t) is a bounded smooth function and In (14) the velocity field u is given by where K(x−y) = ∇ ⊥ G(x−y), being ∇ ⊥ = (∂ x2 , −∂ x1 ) and G(x) is the fundamental solution of the fractional Laplacian (−∆) (1−α/2) with vanishing boundary condition at infinity. Moreover the divergence-free of the velocity field and the fact that θ is transported by the flow show that the Lebesgue measure and the maximum of θ are conserved during the motion. We remember that the Green function of (−∆) (1−α/2) in R 2 with vanishing boundary conditions at infinity is (see for instance [1] and [13]): where Γ(·) denotes the Euler Gamma function. Finally, we recall, for the utility of the reader, that several quantities are conserved in the time evolution for the α-patch model (see e.g. [3]), that is • the maximum norm • the center of mass 1 m 0 x θ(x, t)dx = 1 m 0 x θ 0 (x)dx = c 0 .
• the moment of inertia These conservation laws play a central role for the proof of the main theorem in the next section. The aim of this paper is to study the confinement of an hot patch of temperature in absence of an external forcing . We are therefore interested in proving some rigorous results on the concentration of a temperature field under the quasi-geostrophic approximation, in the more flexible formulation of the modified SQG equation in which the parameter α (that appears in the fractional Laplacian of equation (3)) describes the difference in the behavior of the air mass between the description given by the ideal incompressible fluid (according to the Euler equation) and the one provided by the SQG equation.

3.
Estimates on the growth of the support of a temperature patch. We now give an estimate on the growth of the support of a temperature patch, assuming that the support of the initial temperature field θ 0 is contained in the ball of radius R 0 and centered at the origin .
Proof of Theorem 3.1. The proof of the theorem is strongly based on the use of the conserved quantities and in particular of the moment of inertia i 0 and the total mass m 0 . The main difference and technical difficulty with respect to the case α = 0 is due to the different Green function given by (17).
Without loss of generality, we will take the center of mass located at the origin. We should prove that the radial component of the velocity field satisfies the following estimate for all |x| ≥ 4R 0 + C 0 [t ln(2 + t)] 1 4+α . The bound (19) implies that the vector field (1, u(x, t)) points inward along the boundary of the region D since it tells us that the radial component of the velocity field is negligible outside the domain D. This means that the region D is invariant for the flow implying that the support of the scalar field is contained in the compact support domain D. Therefore, the proof of the theorem is essentially based on the proof of the estimate (19). The radial component of the velocity field in the case of the active scalar flow is given by where (x 1 , x 2 ) ⊥ = (−x 2 , x 1 ). Let us divide the last integral in two regions: The integral over the region A 1 is simply bounded by By using the fact that x · (x − y) ⊥ = −x · y ⊥ and the conservation of the center of the mass (that is located at the origin), we have The first integral can be bounded as follows A2 x · y ⊥ |x| where we have used the conservation of the moment of inertia.
At this stage, we have obtained the following estimate for the radial component of the velocity field We now introduce the following Lemma Lemma 3.2. Let S ⊂ R 2 and h : S → R + a function belonging to L 1 (S) L ∞ (S). Then with β ∈ (0, 2).
The proof is essentially the same of Lemma 2.1 in [11], details of the proof are given in the Appendix. By using the fact that {y : |x − y| < |x|/2} ⊂ {y : |y| > |x|/2} and the Lemma 3.2 we have the following estimate on the radial component of the velocity In the next proposition we prove that also the last integral is of order O(|x| −3−α ) for |x| large, so that inequality (19) holds and the proof of the theorem is achieved.
Proof of Proposition 1. The proof of this proposition is based on the following technical Lemma: then, there exists a constant C 0 such that, for any n ≥ 1 We postpone the proof of this Lemma and use it to prove Proposition 3.1.
Fix k ≥ 1 suppose that taking n ≥ k 4+α such that then we have from Lemma 3.3 that Since by (33) we have that we obtain that the right hand side of (34) can be bounded above by C(i 0 , k, R 0 , m 0 ) /r k when (32) holds, as claimed.
We consider the following partition Then, we have that We first consider the region A 1 , where |x − y| ≥ |x|/2n and we have the following bound such that (4 + α)n 2π where we used also the fact that |y| 2 /|x| 2 ≤ ( 1 2n − 1) 2 in A 1 and the conservation of the moment of inertia.
Remark 1. We have proved the theorem for classical solutions but the general result, for weak solutions, follows immediately since the conserved quantities are stable under the passage to the weak limit.
4. Appendix: Proof of Lemma 3.2. For the sake of completness, we here provide the detailed proof of Lemma 3.2. In particular, we here recall a more general Lemma (see [12]) that includes Lemma 3.2 as a special case Proof. Let k > 0, by using the Hölder inequality, we have that Finally observe that the Lemma 3.2 is a particular case, for h ∈ L 1 (S) L ∞ (S).