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Scalars convected by a 2D incompressible flow

We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the velocity has to have uncontrolled velocity growth.


Introduction
The aim of this paper is to study the possible formation of sharp fronts in finite time for a scalar convected by a two dimensional divergence-free velocity field, with x = (x 1 , x 2 ) ∈ R 2 or R 2 /Z 2 , and t ∈ [0, T ) with T ≤ ∞. The scalar function θ(x, t) and the velocity field u(x, t) = (u 1 (x, t), u 2 (x, t)) ∈ R 2 satisfy the following set of equations where ∇ ⊥ x f = (− ∂f ∂x 2 , ∂f ∂x 1 ) for scalar functions f. The function ψ is the stream function.
There are many physical examples where the solutions satisfy the equations above, with an extra equation or operator that relates θ with the velocity field. Examples include; Passive scalars, Unsteady Prandtl equations, 2D incompressible Euler equations, Boussinesq, 2D Ideal Magnetohydrodynamics and the Quasi-geostrophic equation.
In the literature on numerical simulations for the 2D Ideal Magnetohydrodynamics (MHD) a standard candidate for a current sheet formation (see Fig. 1) is when the level sets of the magnetic stream function ( represented in (1) by θ) contain a hyperbolic saddle (an X-point configuration). The front is formed when the hyperbolic saddle closes, and becomes two Y-points configuration joined by a current sheet. (See Parker [12], Priest-Titov-Rickard [13], Friedel-Grauer-Marliani [10] and Cordoba-Marliani [8].) The same configuration was observed in numerical simulations for the Quasi-geostrophic equation (QG). In this case the geometry of the level sets of the temperature has a hyperbolic structure (See Constantin-Majda-Tabak [4], Okhitani-Yamada [11], Cordoba [6] and Constantin-Nie-Schorghofer [5]). The QG literature discusses X-points, but not Y-points. In the case of Boussinesq there is no mention, on any numerical simulation study, that a possible singularity is due to the closing of a hyperbolic saddle. In the work of Pumir-Siggia [14] there has been observed evidence for a formation of a front in finite time, across which θ varies dramatically, on a cap of a symmetric rising bubble. E-Shu [9] performed numerical simulations with the same initial data as in [14], which suggest that the thickness of the bubble decreases only exponentially.

Criterion
A singularity can be formed by collision of two particle trajectories. A trajectory X(q,t) is obtain by solving the following ordinary differential equation By this trivial argument; in order to have a collision the quantity t 0 |∇u| L ∞ ds has to diverge.
A classic criterion for formation of singularities in fluid flows is the theorem of Beale-Kato-Majda (BKM); (see [1]), which improves the estimate described above, and deals with arbitrary singularities, not just collisions. Analogues of the BKM theorem for the above 2-dimensional equations include the following results For MHD,a singularity cannot develop at a finite time T, unless we have where (See E-Shu [9].) See also Constantin-Majda-Tabak [4] and Constantin-Fefferman-Majda [3] for other conditions involving direction fields, that rule out formation of singularities in fluids.
In the case of 2D Euler, a singularity cannot develop at a finite time. From the BKM viewpoint this follows from the fact that ω is advected by the fluid, and therefore sup x |ω(x, t)| is independent of t. (See BKM [1].) Instead of looking at particle trajectories we look at level curves. Because the scalar function θ is convected by the flow, that implies that the level curves are transported by the flow. A possible singular scenario is due to level curves approaching each other very fast which will lead to a fast growth of the gradient of the scalar function. In this paper we present a variant of the BKM criterion for sharp front formation. We provide a test for numerical simulations that appear to develop sharp fronts. The BKM Theorem shows that the vorticity grows large if any singularity forms; our Theorem 1 shows that the velocity grows large if a sharp front forms.
The theorem we present in this paper was announced in [7].

Sharp Fronts
The scalar function θ is convected by the flow, therefore the level curves move with the flow. A sharp front forms when two of these level curves collapse on a single curve. We define two level curves to be two distinct time-dependent arcs Γ + (t), Γ − (t) that move with the fluid and collapse at finite time into a single arc Γ. More precisely, suppose the arcs are given by with and We call the length b-a of the interval [a,b] the length of the front. The assumption that Γ ± (t) move with the fluid means that This holds in particular for level curves of scalar functions g(x,t) that satisfy (∂ t + u · ∇ x ) g = 0. The collapse of Γ ± (t) into a single curve Γ at time T means here simply that and When (2), (3), (4), (5) and (6) hold, then we say that the fluid forms a sharp front at time T.
The standard candidates for a singularity for MHD and QG are described by the definition given for a sharp front. We investigate the possible formation of a sharp front.
The following assumption will allow us to rule out formation of sharp fronts. We say that the fluid has controlled velocity growth if we have If (7) fails, then we say that the fluid has uncontrolled velocity growth. Lemma 1. Let θ be a smooth solution of Eq.1 defined for t ∈ [0, T ). Assume there is a sharp front at time T. Then Proof: Take the derivative of the stream function with respect to x 1 along an arc Γ ± (t) by combining (9) and (5) we obtain Expression (8) follows from integrating (10) with respect to x 1 between a and b.
Theorem 1. Let u(x,t) be a divergence-free velocity field, with controlled velocity growth. Then a sharp front cannot develop at time T.
Proof: Assume there is a sharp front at time T. We define
We take the derivative of A(t) with respect to time where sup|u| = sup{|u(x 1 , x 2 , t)| : Using the definition of the stream function, the mean value theorem and (8), it is easy to check that dA(t) dt > 0 for t > t * . This contradicts (6) by the dominated convergence theorem.