Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

Abstract

Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the θ-dependent radial and angular velocity fields with the optimal rates u r (t) t −1 and u θ (t) t −2 in the appropriate radially weighted L 2 spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as t → ∞, a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the θ-dependent angular Fourier modes in the vorticity are ejected from the origin as t → ∞, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. Our work appears to be the first to find vorticity depletion relevant for the dynamics of vortices.Remark 1.2. By density we can extend the results to cover any ω in k ∈ L 2 (satisfying (1.7)) for which the norms appearing on the right-hand sides above are finite.Remark 1.3. The L 2 norms we are using in Theorem 1.1, namely (1.10), are natural in light of (1.8) and are wellsuited for studying vorticity depletion. However, these norms are quite strong at the origin (and infinity). Note that≈ k −1 (as opposed to k −2 as one might expect), which explains why some of the powers of k in Theorem 1.1 are slightly higher than might be at first expected. Similarly, note that F k contains information about the second term in the expansion (1.8).Remark 1.4. The correct analogue of propagation of regularity for mixing problems is the regularity of: e iktu(r) ω k (t, r), the object which measures the difference between the passive scalar and full linearized (or nonlinear) dynamics. Regularity of this object is often studied in dispersive equations and it is sometimes called the 'profile'; see e.g.[14] for more discussions (note that regularity of this type was called 'gliding regularity' in [52]). Understanding regularity of the profile plays a major role in all of the works involving nonlinear inviscid/Landau damping [5-8, 10, 14, 15, 19, 52] including those which obtain results in Sobolev spaces [9,31]. Theorem 1.1 deduces higher regularity of the vorticity profile than is necessary to prove the (1.11), at least for k ≥ 3. However, as regularity plays a crucial role in the nonlinear theory, it seems appropriate to study it as carefully as possible in the linear problem. This goal has motivated many of the primary aspects of our approach.Remark 1.5. Because in this work we were only concerned with obtaining finite Sobolev regularity of the vorticity profile, we have not carefully quantified how the constants in (1.13) depend on n. This is...

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