Inviscid Damping Research Articles

On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on $$\mathbb {T}\times I$$ T×I. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $$H^{s}, s>-1$$ Hs,s>-1. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier–Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of Deng and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).

Inviscid Damping Near the Couette Flow in a Channel

We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. The vorticity, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel, will be driven to higher frequencies by the linear flow, and will converge weakly to $0$ as $t\to\infty$, modulo the shear flows (zero mode in $x$).

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

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 $$\theta $$ -dependent radial and angular velocity fields with the optimal rates $$\left\| u^r(t)\right\| \lesssim \langle t \rangle ^{-1}$$ and $$\left\| u^\theta (t)\right\| \lesssim \langle t \rangle ^{-2}$$ in the appropriate radially weighted $$L^2$$ spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as $$t \rightarrow \infty $$ , a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the $$\theta $$ -dependent angular Fourier modes in the vorticity are ejected from the origin as $$t \rightarrow \infty $$ , 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.

Linear Inviscid Damping and Vorticity Depletion for Shear Flows

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.

Metastability of Kolmogorov Flows and Inviscid Damping of Shear Flows

First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study of 2D turbulence. For both rectangular and square tori, we prove that the non-shear part of perturbations near Kolmogorov flow decays in a time scale much shorter than the viscous time scale. The results are obtained for both the linearized NS equations with any initial vorticity in L^2, and the nonlinear NS equation with initial L^2 norm of vorticity of the size of viscosity. In the proof, we use the Hamiltonian structure of the linearized Euler equation and RAGE theorem to control the low frequency part of the perturbation. Second, we consider two classes of shear flows for which a sharp stability criterion is known. We show the inviscid damping in a time average sense for non-shear perturbations with initial vorticity in L^2. For the unstable case, the inviscid damping is proved on the center space. Our proof again uses the Hamiltonian structure of the linearized Euler equation and an instability index theory recently developed by Lin and Zeng for Hamiltonian PDEs.

On the De Gregorio Modification of the Constantin–Lax–Majda Model

We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM) model $\omega_t = \omega H\omega$ on the unit circle. The De Gregorio equation is $\omega_t+u \omega_x-u_x\omega =0, u_x = H\omega.$ In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when $t\to\pm\infty$, in a way resembling inviscid damping. We prove that such a behavior takes place near a manifold of equilibria.

Experimental study of the stability and dynamics of a two-dimensional ideal vortex under external strain

The dynamics of two-dimensional (2-D) ideal fluid vortices is studied experimentally in the presence of an irrotational strain flow. Laboratory experiments are conducted using strongly magnetized pure electron plasmas, a technique which is made possible by the isomorphism between the drift–Poisson equations describing plasma dynamics transverse to the field and the 2-D Euler equations describing an ideal fluid. The electron plasma system provides an excellent opportunity to study the dynamics of a 2-D Euler fluid due to weak dissipation and weak 3-D effects, simple diagnosis and precise control. The plasma confinement apparatus used here was designed specifically to study vortex dynamics under the influence of external flow by applying boundary conditions in two dimensions. Additionally, vortex-in-cell simulations are carried out to complement the experimental results and to extend the parameter range of the studies. It is shown that the global dynamics of a quasi-flat vorticity profile is in good quantitative agreement with the theory of a piecewise-constant elliptical patch of vorticity, including the equilibria, dynamical orbits and stability properties. Deviations from the elliptical patch theory are observed for non-flat vorticity profiles; they include inviscid damping of the orbits and modified stability limits. The dependence of these phenomena on the flatness of the initial profile is discussed. The relationship of these results to other theoretical, numerical and experimental studies is also discussed.

Linear Inviscid Damping for Couette Flow in Stratified Fluid

We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear.

Landau Damping: Paraproducts and Gevrey Regularity

We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on $${\mathbb {T}}^d$$ in Gevrey $$-\frac{1}{s}$$ regularity ( $$s > 1/3$$ ) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.

Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity

In a previous article, \cite{Zill3}, we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution. As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times t \lesssim Re^(1/3), and in particular exhibits inviscid damping (e.g. the vorticity weakly approaches a shear flow). For times t \gtrsim Re^(1/3), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterward, the remaining shear flow decays on very long time scales t \gtrsim Re back to the Couette flow. When properly defined, the dissipative length-scale in this setting is L_D \sim Re^(-1/3), larger than the scale L_D \sim Re^(-1/2) predicted in classical Batchelor-Kraichnan 2D turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re^(-1)) $L^2$ function.

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L 2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

Dynamical pattern formation in two-dimensional fluids and Landau pole bifurcation.

A phenomenological theory is proposed to analyze the asymptotic dynamics of perturbed inviscid Kolmogorov shear flows in two dimensions. The phase diagram provided by the theory is in qualitative agreement with numerical observations, which include three phases depending on the aspect ratio of the domain and the size of the perturbation: a steady shear flow, a stationary dipole, and four traveling vortices. The theory is based on a precise study of the inviscid damping of the linearized equation and on an analysis of nonlinear effects. In particular, we show that the dominant Landau pole controlling the inviscid damping undergoes a bifurcation, which has important consequences on the asymptotic fate of the perturbation.

Impact of Asymmetric Dynamical Processes on the Structure and Intensity Change of Two-Dimensional Hurricane-Like Annular Vortices

AbstractIn this study, a simple two-dimensional (2D) unforced barotropic model is used to study the asymmetric dynamics of the hurricane inner-core region and to assess their impact on the structure and intensity change. Two sets of experiments are conducted, starting with stable and unstable annular vortices, to mimic intense mature hurricane-like vortices. The theory of empirical normal modes (ENM) and the Eliassen–Palm flux theorem are then applied to extract the dominant wave modes from the dataset and diagnose their kinematics, structure, and impact on the primary vortex.From the first experiment, it is found that the evolution and the lifetime of an elliptical eyewall, described by a stable annular vortex perturbed by an external wavenumber-2 impulse, may be controlled by the inviscid damping of sheared vortex Rossby waves (VRWs) or the decay of an excited quasimode. The critical radius and structure of the quasimode obtained by the ENM analysis are shown to be consistent with the predictions of a linear eigenmode analysis of small perturbations. From the second experiment, it is found that the outward-propagating VRWs that arise due to barotropic instability and the inward mixing of high vorticity in the unstable annular vortex affect the primary circulation and create a secondary ring of enhanced vorticity that contains a secondary wind maximum. Sensitivity tests performed on the spatial extent of the initial external impulse verifies the robustness of the results. That the secondary eyewall occurs close to the critical radius of some of the dominant modes emphasizes the important role played by the VRWs.

Inviscid Dynamical Structures Near Couette Flow

Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.

On the Dynamics of Two-Dimensional Hurricane-Like Vortex Symmetrization

Abstract Despite the fact that asymmetries in hurricanes, such as spiral rainbands, polygonal eyewalls, and mesovortices, have long been observed in radar and satellite imagery, many aspects of their origin, space–time structure, and dynamics still remain unsolved, particularly their role on the vortex intensification. The underlying inner-core dynamics need to be better understood to improve the science of hurricane intensity forecasting. To fill this gap, a simple 2D barotropic “dry” model is used to perform two experiments starting respectively with a monopole and a ring of enhanced vorticity to mimic hurricane-like vortices during incipient and mature stages of development. The empirical normal mode (ENM) technique, together with the Eliassen–Palm (EP) flux calculations, are used to isolate wave modes from the model datasets to investigate their space–time structure, kinematics, and the impact on the changes in the structure and intensity of the simulated hurricane-like vortices. From the ENM diagnostics, it is shown in the first experiment how an incipient storm described by a vortex monopole intensifies by “inviscid damping” of a “discrete-like” vortex Rossby wave (VRW) or quasi mode. The critical radius, the structure, and the propagating properties of the quasi mode are found to be consistent with predictions of the linear eigenmode analysis of small perturbations. In the second experiment, the fastest growing wavenumber-4 unstable VRW modes of a vortex ring reminiscent of a mature hurricane are extracted, and their relation with the polygonal eyewalls, mesovortices, and the asymmetric eyewall contraction are established in consistency with results previously obtained from other authors.

Phase mixing and echoes in a pure electron plasma

The two-dimensional (2D) fluid echo is a spontaneous appearance of a diocotron wave after two externally excited waves have damped away, explicitly demonstrating the reversible nature of spatial Landau damping. The inviscid damping, or phase mixing, is directly imaged by a low-noise charge-coupled device camera, which shows the spiral wind-up of the density perturbation. Surprisingly, the basic echo characteristics agree with a simple nonlinear ballistic theory that neglects all collective (i.e., mode) effects. Also, the simple 2D picture is violated by end confinement fields that cause vz-dependent θ drifts, so the observed echo must be interpreted as a superposition of separately damping and separately echoing velocity classes. The maximal echo lifetimes agree with a theory describing weak collisional velocity scattering between velocity classes. In addition, large second wave excitations degrade the echo up to 5× faster than collisions.

A New Look at the Problem of Tropical Cyclones in Vertical Shear Flow: Vortex Resiliency

A new paradigm for the resiliency of tropical cyclone (TC) vortices in vertical shear flow is presented. To elucidate the basic dynamics, the authors follow previous work and consider initially barotropic vortices on an f plane. It is argued that the diabatically driven secondary circulation of the TC is not directly responsible for maintaining the vertical alignment of the vortex. Rather, an inviscid damping mechanism intrinsic to the dry adiabatic dynamics of the TC vortex suppresses departures from the upright state. Recent work has demonstrated that tilted quasigeostrophic vortices consisting of a core of positive vorticity surrounded by a skirt of lesser positive vorticity align through projection of the tilt asymmetry onto vortex Rossby waves (VRWs) and their subsequent damping (VRW damping). This work is extended here to the finite Rossby number (Ro) regime characteristic of real TCs. It is shown that the VRW damping mechanism provides a direct means of reducing the tilt of intense cyclonic vortices (Ro > 1) in unidirectional vertical shear. Moreover, intense TC-like, but initially barotropic, vortices are shown to be much more resilient to vertical shearing than previously believed. For initially upright, observationally based TC-like vortices in vertical shear, the existence of a “downshear-left” tilt equilibrium is demonstrated when the VRW damping is nonnegligible. On the basis of these findings, the axisymmetric component of the diabatically driven secondary circulation is argued to contribute indirectly to vortex resiliency against shear by increasing Ro and enhancing the radial gradient of azimuthal-mean potential vorticity. This, in addition to the reduction of static stability in moist ascent regions, increases the efficiency of the VRW damping mechanism.

Vortex dynamics of 2D electron plasmas

Electron columns confined magnetically in vacuum evolve in ( r, θ) as N∼10 9 field-aligned rods of charge, or point vortices. Neglecting discreteness, the column evolves as would vorticity in an inviscid, incompressible fluid, governed by the Euler equations. The macroscopic flow dynamics is readily imaged, including effects such as surface waves and inviscid damping, two vortex merger, and gradient-driven vortex motion. Turbulent initial states are observed to relax to “vortex crystal” meta-equlibria, due to vortex “cooling” from entropic mixing of background vorticity; and characteristics of this process are predicted by theory. The microscopic discreteness gives rise to point-vortex diffusion, which is strongly affected by the overall flow shear. Macroscopically and microscopically, the vortex dynamics depends critically on whether the vortex is prograde or retrograde with respect to the flow shear.

Inviscid damping of asymmetries on a two-dimensional vortex

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity (quadrupole moment) of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparable to the vortex rotation frequency, the vorticity perturbation does not resemble a quasimode; rather, it is rapidly dominated by spiral filaments. Over longer times, linear theory predicts algebraic decay of ellipticity; however, nonlinear oscillations of ellipticity emerge in the experiment before a transition to algebraic decay would occur.