Wave Kinetic Equation Research Articles

On the available free energy in drift wave turbulence

We report a variational formalism to calculate how much free energy can be released in a drift wave turbulence system, through which estimating the upper limit of the zonal flow intensity becomes possible. This formalism is rooted in the same structure between the wave kinetic equation and the particle kinetic equation, i.e., both following a Liouville equation. The minimal energy state is approached by rearranging the “quasi-particles” (i.e., wave packets) in their phase space. Taking the Charney–Hasegawa–Mima system as a prototype, we analytically derived an upper limit of the available free energy for a Gaussian initial distribution of the wave action. For more general scenarios, by developing an algorithm of rearrangement, we numerically calculated the available free energy. Through direct numerical simulations, it is further verified that the total energy of the zonal flow does not exceed the upper bound set by the variational principle.

Loop diagrams in the kinetic theory of waves

Recent work has given a systematic way for studying the kinetics of classical weakly interacting waves beyond leading order, having analogies with renormalization in quantum field theory. An important context is weak wave turbulence, occurring for waves which are small in magnitude and weakly interacting, such as those on the surface of the ocean. Here we continue the work of perturbatively computing correlation functions and the kinetic equation in this far-from-equilibrium state. In particular, we obtain the next-to-leading-order kinetic equation for waves with a cubic interaction. Our main result is a simple graphical prescription for the terms in the kinetic equation, at any order in the nonlinearity.

Weak magnetohydrodynamic turbulence theory revisited

Two recent papers, P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021) and Yoon et al., Phys. Plasmas 29, 112303 (2022), utilized in the derivation of the kinetic equation for the intensity of turbulent fluctuations the assumption that the wave spectra are isotropic, that is, the ensemble-averaged magnetic field tensorial fluctuation intensity is given by the isotropic diagonal form, ⟨δBiδBj⟩k=⟨δB2⟩kδij. However, it is more appropriate to describe the incompressible magnetohydrodynamic turbulence involving shear Alfvénic waves by modeling the turbulence spectrum as being anisotropic. That is, the tensorial fluctuation intensity should be different in diagonal elements across and along the direction of the wave vector, ⟨δBiδBj⟩k=12 ⟨δB⊥2⟩k(δij−kikj/k2)+⟨δB∥2⟩k(kikj/k2). In the present paper, we thus reformulate the weak magnetohydrodynamic turbulence theory under the assumption of anisotropy and work out the form of nonlinear wave kinetic equation.

Verification of wave turbulence theory in the kinetic limit

Using the 1D Majda-McLaughlin-Tabak model as an example, we develop numerical experiments to study the validity of the wave kinetic equation (WKE) at the kinetic limit (i.e., small nonlinearity and large domain). We show that the dynamics converge to the WKE prediction, in terms of the closure model and energy flux, when the kinetic limit is approached. When the kinetic limit is combined with a process of widening the inertial range, the theoretical Kolmogorov constant can be recovered numerically to a very high precision. Published by the American Physical Society 2024

Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation

AbstractWe introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.

On discrete models of Boltzmann-type kinetic equations

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables \(F(x,y;
 v,w)\). The function \(F\) is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function \(F\). Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann \(H\)-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

Nonlinear susceptibilities for weakly turbulent magnetized plasma: Electrostatic approximation

The plasma weak turbulence theory is a perturbative nonlinear theory, which has been proven to be quite valid in a number of applications. However, the standard weak turbulence theory found in the literature is fully developed for highly idealized unmagnetized plasmas. As many plasmas found in nature and laboratory are immersed in a background static magnetic field, it is necessary to extend the existing discussions to include the effects of ambient magnetic field. Such a task is quite formidable, however, which has prevented fundamental and significant progresses in the subject matter. The central difficulty lies in the formulation of the complete nonlinear response functions for magnetized plasmas. The present paper derives the nonlinear susceptibilities for weakly turbulent magnetized plasmas up to the third order nonlinearity, but in doing so, a substantial reduction in mathematical complexity is achieved by the use of Bessel function addition theorem (or sum rule). The present paper also constructs the weak turbulence wave kinetic equation in a formal sense. For the sake of simplicity, however, the present paper assumes the electrostatic interaction among plasma particles. Fully electromagnetic generalization is a subject of a subsequent paper.

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this end we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

On the convergence rates of discrete solutions to the Wave Kinetic Equation

On the convergence rates of discrete solutions to the Wave Kinetic Equation

Self-similar evolution of wave turbulence in Gross-Pitaevskii system.

We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation(GPE) and the associated wave kinetic equation(WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.

Energy cascade in the Garrett–Munk spectrum of internal gravity waves

We study the spectral energy transfer due to wave–triad interactions in the Garrett–Munk spectrum of internal gravity waves based on a numerical evaluation of the collision integral in the wave kinetic equation. Our numerical evaluation builds on the reduction of the collision integral on the resonant manifold for a horizontally isotropic spectrum. We evaluate directly the downscale energy flux available for ocean mixing, whose value is in close agreement with the finescale parameterization. We further decompose the energy transfer into contributions from different mechanisms, including local interactions and three types of non-local interactions, namely parametric subharmonic instability, elastic scattering (ES) and induced diffusion (ID). Through analysis on the role of each mechanism, we resolve two long-standing paradoxes regarding the mechanism for forward cascade in frequency and zero ID flux for the GM76 spectrum. In addition, our analysis estimates the contribution of each mechanism to the energy transfer in each spectral direction, and reveals new understanding of the importance of local interactions and ES in the energy transfer.

An effective semilocal model for wave turbulence in two-dimensional nonlinear optics

The statistical evolution of ensembles of random, weakly interacting waves is governed by wave kinetic equations (WKEs). To simplify the analysis, one frequently works with reduced differential models of the wave kinetics. However, the conditions for deriving such reduced models are seldom justified self-consistently. Here, we derive a reduced model for the wave kinetics of the Schrödinger–Helmholtz equations in two spatial dimensions, which constitute a model for the dynamics of light in a spatially nonlocal, nonlinear optical medium. This model has the property of sharply localizing the frequencies of the interacting waves into two pairs, allowing for a rigorous and self-consistent derivation of what we term the semilocal approximation model (SLAM) of the WKE. Using the SLAM, we study the stationary spectra of Schrödinger–Helmholtz wave turbulence, and characterize the spectra that carry energy downscale, and waveaction upscale, in a forced-dissipated setup. The latter involves a nonlocal transfer of waveaction, in which waves at the forcing scale mediate the interactions of waves at every larger scale. This is in contrast to the energy cascade, which involves local scale-by-scale interactions, familiar from other wave turbulent systems and from classical hydrodynamical turbulence.

Full derivation of the wave kinetic equation

Full derivation of the wave kinetic equation

Direct and Inverse Cascades in Turbulent Bose-Einstein Condensates.

When a Bose-Einstein condensate (BEC) is driven out of equilibrium, density waves interact nonlinearly and trigger turbulent cascades. In a turbulent BEC, energy is transferred toward small scales by a direct cascade, whereas the number of particles displays an inverse cascade toward large scales. In this work, we study analytically and numerically the direct and inverse cascades in wave-turbulent BECs. We analytically derive the Kolmogorov-Zakharov spectra, including the log correction to the direct cascade scaling and the universal prefactor constants for both cascades. We test and corroborate our predictions using high-resolution numerical simulations of the forced-dissipated Gross-Pitaevskii model in a periodic box and the corresponding wave-kinetic equation. Theoretical predictions and data are in excellent agreement, without adjustable parameters. Moreover, in order to connect with experiments, we test and validate our theoretical predictions using the Gross-Pitaevskii model with a confining cubic trap. Our results explain previous experimental observations and suggest new settings for future studies.

Turbulence-driven vortex-flow around a magnetic island

We predict a new state of plasma self-organization, in the presence of magnetic islands, namely turbulence-driven island vortex-flows, which are non-axisymmetric flows. The interaction between an E × B sheared vortex-flow and the drift-wave (DW) turbulence driving it, is derived in the presence of a coherent static magnetic island. The turbulence is driven by a 3D density profile due to quasi-linear island-induced profile flattening, and the DWs thus follow the local electron diamagnetic drift along the island. The metric tensor is introduced, making the analysis easier in island geometry. An extended Charney–Hasegawa–Mima equation describes the DW turbulence—flow interaction, from which a wave-kinetic equation is obtained in island geometry. This yields a turbulence-vortex-flow predator–prey model which predicts a nonlinear threshold for island vortex-flow formation. The vortex-flow threshold decreases with increasing island-width as , which shows that wider islands may more easily drive vortex-flows.

Уравнение Больцмана и волновые кинетические уравнения

The well-known nonlinear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim – Uehling – Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the generalized kinetic equation that depends on a function of four real variables F(x1; x2; x3; x4). The function F is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the above mentioned kinetic equations correspond to different forms of the function (polynomial) F. Then the problem of discretization of the generalized kinetic equation is considered on the basis of ideas which are similar to those used for construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models possses a monotone functional similar to Boltzmann H-function. The behaviour of solutions of the simplest Broadwell model for the wave kinetic equation is discussed in detail.

Paraxial beams in fluctuating fusion plasmas: Diffusive limit and beyond

A paraxial expansion of the (ensemble-averaged) Wigner function in the relevant wave kinetic equation for electron cyclotron waves in fluctuating plasmas allows the derivation of phase-space equations similar to the equations for the Gaussian beam parameters in the paraxial WKB method [G.V. Pereverzev, Phys. Plasmas 5, 3529 (1998)]. This is relatively straightforward when the scattering of the wave field by density fluctuations can be described by a diffusion operator in refractive-index space. The general case is rather more complicated, yet we could find a heuristic construction of a paraxial Wigner function. Here we use a simple model, which has an analytical solution, to test both the theoretical validity of the diffusion approximation and the heuristic paraxial approach beyond the diffusion approximation.

One property of discrete models of wave kinetic equation

One property of discrete models of wave kinetic equation

A deep learning approximation of non-stationary solutions to wave kinetic equations

We present a deep learning approximation, stochastic optimization based, method for wave kinetic equations. To build confidence in our approach, we apply the method to a Smoluchowski coagulation equation with multiplicative kernel for which an analytic solution exists. Our deep learning approach is then used to approximate the non-stationary solution to a 3-wave kinetic equation corresponding to acoustic wave systems. To validate the neural network approximation, we compare the decay rate of the total energy with previously obtained theoretical results. A finite volume solution is presented and compared with the present method.