Quasilinear Theory Research Articles

Generalized quasi-linear approximation and non-normality in Taylor-Couette spiral turbulence.

Taylor-Couette flow is well known to admit a spiral turbulence state in which laminar and turbulent patches coexist around the cylinder. This flow state is quite complex, with delicate internal structure, and it can be traced into certain regimes of linear stability. This behaviour is believed to be connected to the non-normality of the linear operator, which is itself a function of the control parameters. Using spiral turbulence in both linearly stable and unstable regimes, we investigate the effectiveness of the generalized quasi-linear approximation (GQL), an extension of quasi-linear theory designed to capture the essential aspects of turbulent flows. We find that GQL performs much better in the supercritical regime than the subcritical. By including only a small number of modes in the nonlinear interactions, GQL simulations maintain a turbulent-like state when in the supercritical regime. However, a much larger number is required to avoid returning to the laminar state when in the subcritical regime. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

A new reformulation of the Muskat problem with surface tension

Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density ϑ to the L2-adjoints of these operators evaluated at the density ϑ′ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the L2-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces Wps(R) with p∈(1,∞) and s∈(1+1/p,2).

Nonlinear dynamics of kinetic fluctuations and quasi-linear relaxation in plasma

We propose an approximation of pair correlations for solving the equations of the kinetic theory of long-wave (or large-scale) fluctuations in gaseous media. The basic ones are the general nonlinear equations of the large-scale fluctuations theory at the kinetic stage of system evolution, derived from the first principles of statistical mechanics. We show that based on the equations of the long-wave fluctuations kinetics in the case of weak interaction between particles, in the approximation of pair fluctuations it is possible to reproduce the main results of the quasi-linear theory of plasma. Thus, the well-known quasi-linear theory of plasma is provided with a first-principle justification.

Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter ε=η0/h0, where η0 is the characteristic amplitude of the wave, and h0 is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered.

Account of non-standard orbits in computations of neoclassical toroidal viscous torque in the resonant plateau regime of a tokamak

This article extends theoretical details based on a short paper originally submitted to the 2022 EPS conference in plasma physics [1]. The quasilinear theory of resonant transport regimes in a tokamak is developed for the general case of orbits forming various classes separated in phase space by homoclinic orbits with infinite bounce time. Beyond standard orbits (banana and passing orbits) also all types of non-standard orbits (e.g. “potato” orbits) are taken into account. In case of a weak radial electric field, such orbits are usually present only near the magnetic axis. If the radial electric field cannot be treated as weak, there can be arbitrary many classes, located elsewhere. The present approach covers all such cases and is demonstrated on a specific example of a radial electric field profile. The resulting quasilinear kinetic equation is applicable to compute neoclassical toroidal viscous (NTV) torque in a tokamak with non-axisymmetric magnetic field perturbations. A fully non-local approach to NTV computation has been realized in the upgraded version of the code NEO-RT. Based on a generalization of magnetic flux surfaces to drift surfaces, the notion of a local thermodynamic equilibrium is extended for our purpose. We obtain an expression for the integral toroidal torque within a chosen flux surface and dicuss means to compute such integrals taking singularities in bounce and precession frequencies into account.

Using MEPED observations to infer plasma density and chorus intensity in the radiation belts

Efforts to model and predict energetic electron fluxes in the radiation belts are highly sensitive to local wave-particle interactions. In this study, we use multi-point measurements of precipitating and trapped electron fluxes to investigate the dynamic variation of chorus wave-particle interactions during the 17 March 2013 storm. Quasilinear theory characterizes the chorus wave-particle interaction as a diffusive process, with the diffusion coefficients depending on the particle energy and pitch angle, as well as the background plasma parameters such as the wave intensity and plasma density. These plasma parameters in the radiation belts are spatially localized and time-varying, so we construct event-specific diffusion coefficients using MEPED (onboard POES/MetOp) measurements of electron fluxes at low Earth orbit. This new method provides realistic diffusion coefficients for chorus waves that account for changes in the wave intensity, the plasma density, and the magnetic field strength in the outer radiation belt. We show that the inferred chorus intensity is significantly lower than previous estimates that use MEPED observations since the same amount of increased precipitation by 30–300 keV electrons can be explained by a change in the plasma density. This technique therefore allows for us to create time varying, global maps of the plasma-gyrofrequency ratio (fpe/fce), and therefore plasma density, in the outer radiation belts using the MEPED measurements. The global density estimates compare reasonably well to in situ density measurements from RBSP-B.

Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions

In the present paper, we show an improved lower bound for the lifespan of the solutions to the ideal MHD equations in the case of space dimension d=2. In particular, for small initial magnetic fields b0 of size (say) ε>0, the lifespan Tε>0 of the corresponding solution goes to +∞ in the limit ε→0+.Such a result does not follow from standard quasi-linear hyperbolic theory. For proving it, three are the crucial ingredients: first of all, to work in endpoint Besov spaces B∞,rs, under the condition s>1 and r∈[1,+∞] or s=r=1; moreover, to use the Elsässer formulation of the ideal MHD, recasted in its vorticity formulation; finally, to take advantage of the special structure of the non-linear terms.We also rigorously establish the equivalence between the original formulation of the ideal MHD and its Elsässer formulation for a large class of weak solutions. The construction of explicit counterexamples shows the sharpness of our assumptions. Related non-uniqueness issues are discussed as well.

Scattering of energetic electrons through nonlinear cyclotron resonance with coherent whistler-mode hiss emissions

Recent observations have revealed that plasmaspheric hiss consists of many discrete waves called “hiss elements.” However, the interaction of energetic electrons (10 keV to several MeV) with the plasmaspheric hiss has only been simulated by the quasilinear (QL) diffusion theory, which does not take the fine wave structure into account. The QL theory cannot address nonlinear particle motions determined by the inhomogeneity factor, which influences the scattering of electrons in pitch angle and energy. This study aims to identify differences between the nonlinear wave–particle interaction and QL theory for plasmaspheric hiss emissions. We conduct test particle simulations to demonstrate the nonlinear interactions between hiss waves and electrons. The nonlinear theory is used to model hiss elements consisting of discrete frequencies and continuous phases. Unlike the other theories, the frequency and amplitude variations in time of the hiss packet are taken into account. Frequencies of the packets are determined to satisfy the separability criterion; when the criterion is met, resonance overlapping is absent, and the electrons can generate each wave element independently. The realistic simulation model of hiss waves reproduces the scattering of electrons by both first- and second-order resonances. We also evaluate the efficiency of electron scattering by calculating nonlinear diffusion coefficients. The diffusion coefficient of equatorial pitch angle is of the same order of magnitude as those calculated by the QL diffusion theory, while we find the effective acceleration of resonant electrons by successive nonlinear trapping, which is not evaluated by the QL theory.

Inner Belt Wisp Precipitation Measured by ELFIN: Regimes of Energetic Electron Scattering by VLF Transmitter Waves

AbstractMan‐made very low frequency (VLF) transmitter waves play a critical role in energetic electron scattering and precipitation from the inner radiation belt, a type of which is called wisp precipitation. Wisps exhibit dispersive energy‐versus‐L spectra due to the evolution of electron cyclotron resonance conditions with near‐monochromatic VLF transmitter waves. Here, we report on such observations of inner belt wisp precipitation events with full pitch angle resolution in the energy range of 50 to ∼500 keV as measured by Electron Loss and Fields Investigation (ELFIN) at L < ∼2 between March 2021 and April 2022. Statistical observations (82 events) reveal occasional (18 events) wisp precipitation events with local bounce‐loss‐cone electron flux enhancements, which provide new information compared with flux enhancements measured in previous studies only in the drift loss cone. Based on magnetic field and plasmaspheric density models, quasilinear theory, and detailed pitch angle distributions of wisps from ELFIN, we have estimated the wisp electron bounce‐averaged pitch angle diffusion coefficients to be of the order of 10−4 to 10−2 s−1. These are several orders of magnitude larger than the diffusion rates calculated from models using global statistical averages of VLF transmitter wave power. When using our estimated diffusion coefficients to deduce the associated local transmitter wave amplitudes near the equator, based on quasilinear calculations from a transmitter‐induced electron diffusion model, we find these wave amplitudes to be >1 mV/m. Although probable overestimates, such inferred wave amplitudes exceed the theoretical threshold amplitude for nonlinear interactions, strongly suggesting that it is necessary to include nonlinear effects for an accurate evaluation of energetic electron scattering by transmitter waves.

Staircase formation by resonant and non-resonant transport of potential vorticity

The E × B staircase is a quasi-periodic pattern of pressure profile corrugations. In this work, we present a new mechanism for E × B staircase formation that involves resonant transport versus non-resonant transport. We start from a potential vorticity evolution system and use quasi-linear theory, a model dispersion relation, and a bi-Lorentzian spectrum approximation, to construct the relation between the fluxes and the profiles. With these fluxes, we close the profile evolution equations and the extended turbulence intensity evolution equation, which together constitute a turbulence-profile evolution system. In this system, the Doppler effect from the E × B mean flow can cause resonance between trapped ion precession motion and the trapped ion mode, which drives a resonant transport contribution to the fluxes. The profiles will be flattened where the resonant transport is switched on. In contrast, for the regions of non-resonant transport, profiles are steeper. A quasi-periodic pattern of profile corrugation (the E × B staircase) spontaneously emerges in this system, which is the two states mentioned above, arranged as alternating layers in space. The feedback processes during the staircase pattern formation are identified. An estimate of the critical value of the boundary heat flux is obtained, above which the staircase formation will be triggered. An estimate scaling of the step size in the staircase pattern is obtained. The resonant turbulent transport is also a mechanism for collisionless saturation of zonal flow. This work is related to internal transport barrier formation and suggests some new scenarios, such as an enhanced confined L mode.

Temporal variability of quasi-linear pitch-angle diffusion

Kinetic wave-particle interactions in Earth’s outer radiation belt energize and scatter high-energy electrons, playing an important role in the dynamic variation of the extent and intensity of the outer belt. It is possible to model the effects of wave-particle interactions across long length and time scales using quasi-linear theory, leading to a Fokker-Planck equation to describe the effects of the waves on the high energy electrons. This powerful theory renders the efficacy of the wave-particle interaction in a diffusion coefficient that varies with energy or momentum and pitch angle. In this article we determine how the Fokker-Planck equation responds to the temporal variation of the quasi-linear diffusion coefficient in the case of pitch-angle diffusion due to plasmaspheric hiss. Guided by in-situ observations of how hiss wave activity and local number density change in time, we use stochastic parameterisation to describe the temporal evolution of hiss diffusion coefficients in ensemble numerical experiments. These experiments are informed by observations from three different example locations in near-Earth space, and a comparison of the results indicates that local differences in the distribution of diffusion coefficients can result in material differences to the ensemble solutions. We demonstrate that ensemble solutions of the Fokker-Planck equation depend both upon the timescale of variability (varied between minutes and hours), and the shape of the distribution of diffusion coefficients. Based upon theoretical construction of the diffusion coefficients and the results presented here, we argue that there is a useful maximum averaging timescale that should be used to construct a diffusion coefficient from observations, and that this timescale is likely less than the orbital period of most inner magnetospheric missions. We discuss time and length scales of wave-particle interactions relative to the drift velocity of high-energy electrons and confirm that arithmetic drift-averaging is can be appropriate in some cases. We show that in some locations, rare but large values of the diffusion coefficient occur during periods of relatively low number density. Ensemble solutions are sensitive to the presence of these rare values, supporting the need for accurate cold plasma density models in radiation belt descriptions.

Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence

Understanding inhomogeneous and anisotropic fluid flows requires mathematical and computational tools that are tailored to such flows and distinct from methods used to understand the canonical problem of homogeneous and isotropic turbulence. We review some recent developments in the theory of inhomogeneous and anisotropic turbulence, placing special emphasis on several kinds of quasi-linear approximations and their corresponding statistical formulations. Aspects of quasi-linear theory that have received insufficient attention in the literature are discussed, and open questions are framed.

Growth mechanics of the viscoelastic membranes

Living membranes can grow as a response to their environment, and they are also known as viscoelastic materials. However, the growth phenomenon and the viscoelastic behavior of these membranes have been studied separately up to now. The main goal of this paper is to develop a general formulation that can account for growth and viscoelastic behaviors in membranes, simultaneously. Therefore, we propose a formulation for growing quasi-linear viscoelastic membranes. The growth model begins with a multiplicative decomposition of the gradient deformation tensor into elastic and growth tensor, where the growth of the membrane is considered as a single scalar-valued parameter. Furthermore, the quasi-linear viscoelasticity theory is utilized in the elastic configuration as additive decomposition of the second Piola–Kirchhoff stress tensor. To solve the evolution equations for both growth and quasi-linear viscoelastic relations, two implicit second order accuracy integration methods are employed. Moreover, strain-driven and stress-driven growth phenomena are accounted for based on a Total Lagrangian (TL) finite element formulation. The developed formulations can capture growth phenomena and viscoelastic properties of the membranes, individually, and it also enables to model co-existence of growth and viscoelasticity. Eventually, to evaluate the applicability of the developed formulation, several examples are investigated and compared to those available in the literature.

Hamiltonian formulations of quasilinear theory for magnetized plasmas

Hamiltonian formulations of quasilinear theory are presented for the cases of uniform and nonuniform magnetized plasmas. First, the standard quasilinear theory of Kennel and Engelmann (Kennel, Phys. Fluids, 1966, 9, 2377) is reviewed and reinterpreted in terms of a general Hamiltonian formulation. Within this Hamiltonian representation, we present the transition from two-dimensional quasilinear diffusion in a spatially uniform magnetized background plasma to three-dimensional quasilinear diffusion in a spatially nonuniform magnetized background plasma based on our previous work (Brizard and Chan, Phys. Plasmas, 2001, 8, 4762–4771; Brizard and Chan, Phys. Plasmas, 2004, 11, 4220–4229). The resulting quasilinear theory for nonuniform magnetized plasmas yields a 3 × 3 diffusion tensor that naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space (e.g., energy and pitch angle).

Effects of alpha particles on the transport of helium ash driven by collisionless trapped electron mode turbulence

The effects of alpha (α) particles on the transport of helium ash driven by collisionless trapped electron mode (CTEM) turbulence are analytically studied using quasi-linear theory in tokamak deuterium (D) and tritium (T) plasmas. Under the parameters used in this work, the transport of helium ash is mainly determined by the diffusion due to very weak convection. It is found that the ratio between helium ash diffusivity and effective electron thermal conductivity (D He/χ eff) driven by CTEM turbulence, which is a proper normalized parameter for quantifying the efficiency of helium ash removal, is smaller than unity. This indicates the less efficient removal of helium ash through CTEM turbulence as compared with ion temperature gradient (ITG) turbulence in [Angioni et al 2009 Nucl. Fusion 49 055013]. However, the efficiency of helium ash removal is increased 55% by the presence of 3% α particles with their density gradient being equivalent to that of electrons, and this enhancement can be further strengthened by steeper profile of α particles. This is mainly because the enhancement of helium ash diffusivity by α particles is stronger than that of the effective electron thermal conductivity. Moreover, the higher fraction of T ions, higher temperature ratio between electrons and thermal ions as well as flatter electron density profile, the stronger enhancement of D He/χ eff, and α particles further strengthen the favorable effects of these parameters on the removal of helium ash.

Direct statistical simulation of the Busse annulus

We consider direct statistical simulation (DSS) of a paradigm system of convection interacting with mean flows. In the Busse annulus model, zonal jets are generated through the interaction of convectively driven turbulence and rotation; non-trivial dynamics including the emergence of multiple jets and bursting ‘predator–prey’ type dynamics can be found. We formulate the DSS by expanding around the mean flow in terms of equal-time cumulants and arrive at a closed set of equations of motion for the cumulants. Here, we present results using an expansion terminated at the second cumulant (CE2); it is fundamentally a quasilinear theory. We focus on particular cases including bursting and bistable multiple jets and demonstrate that CE2 can reproduce the results of direct numerical simulation if particular attention is given to symmetry considerations.

Physical regimes of electrostatic wave-wave nonlinear interactions generated by an electron beam propagating in a background plasma.

Electron-beam plasma interaction has long been a topic of great interest. Despite the success of the quasilinear and weak turbulence theories, their validities are limited by the requirements of a sufficiently dense mode spectrum and a small wave amplitude. In this paper, we extensively study the collective processes of a mono-energetic electron beam emitted from a thermionic cathode propagating through a cold plasma by performing high-resolution two-dimensional particle-in-cell simulations and using analytical theories. We confirm that, during the initial stage of two-stream instability between the beam and background cold electrons, it is saturated due to the well-known wave-trapping mechanism. Further evolution occurs due to strong wave-wave nonlinear processes. We show that the beam-plasma interaction can be classified into four different physical regimes in the parameter space for the plasma and beam parameters. The differences between the regimes are analyzed in detail. We identify a new regime in the strong Langmuir turbulence featured by what we call electron modulational instability (EMI) that could create a local Langmuir wave packet growing faster than the ion plasma frequency. Ions do not have time to respond to EMI in the initial growing stage. On a longer timescale, the action of the ponderomotive force produces very strong ion density perturbations, and eventually, the beam-plasma wave interaction stops being resonant due to the strong ion density perturbations. Consequently, in this EMI regime, electron beam-plasma interaction occurs in a repetitive (intermittent) process. The beam electrons are strongly scattered by waves, and the Langmuir wave spectrum is significantly broadened, which in turn gives rise to strong heating of bulk electrons. Associated energy transfer from the beam to the background plasma electrons has been studied. A resulting kappa (κ) distribution and a wave-energy spectrum E^{2}(k)∼k^{-5} are observed in the strong turbulent regime.

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. The inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

Kinetic theory of collisionless relaxation for systems with long-range interactions

We develop the kinetic theory of collisionless relaxation for systems with long-range interactions in relation to the statistical theory of Lynden-Bell. We treat the multi-level case. We make the connection between the kinetic equation obtained from the quasilinear theory of the Vlasov equation and the relaxation equation obtained from a maximum entropy production principle. We propose a method to close the infinite hierarchy of kinetic equations for the phase level moments and obtain a kinetic equation for the coarse-grained distribution function in the form of a generalized Landau, Lenard–Balescu or Kramers equation associated with a generalized form of entropy (Chavanis, 2004). This allows us to go beyond the two-level case associated with a Fermi–Dirac-type entropy. We discuss the numerous analogies with two-dimensional turbulence. We also mention possible applications of the present formalism to fermionic and bosonic dark matter halos.

Effect of viscoelasticity on inversion in axial deformation and on the volume of pressurized thin-walled tubes

Axial prestretching can significantly affect the mechanical response of an inflated tube. This phenomenon is known, for example in the biomechanics of the cardiovascular system, where arteries, if excised from a body, manifest non-negligible shortening. Previous studies have shown that axially prestretched tubes are more easily distensible by internal pressure than their non-prestretched counterparts. It has even been found that, in the case of the material described by the Gent strain energy density function, the volume attained by a tube at a given internal pressure can be maximized by the initial elongation of the tube to the prestretch point referred to as inversion axial prestretching. However, to the best of our knowledge, all studies dealing so far with the effect of axial prestretching on the mechanical response of a pressurized tube have assumed elastic behavior. Nevertheless, soft tissues as well as elastomer materials are always more or less viscoelastic. The present study shifts our attention from purely theoretical elastic models to a more realistic assumption of viscoelastic behavior, which is modeled by means of the Quasilinear Theory of Viscoelasticity (QLV). The adopted model assumes the elastic part of the material response to be given by the Gent model. Results suggest that the viscoelastic tube loses the property of inversion in axial deformation but can still preserve the potential to optimize the inflation volume by means of axial prestretching. This result can be helpful in designing elastomer pipelines and pumps, as well as contributing to the understanding of the principals of blood circulation physiology.