On Landau damping

In a collisionless, magnetized plasma, particles may stream freely along magnetic field lines, leading to "phase mixing" of their distribution function and consequently, to smoothing out of any "compressive" fluctuations (of density, pressure, etc.). This rapid mixing underlies Landau damping of these fluctuations in a quiescent plasma-one of the most fundamental physical phenomena that makes plasma different from a conventional fluid. Nevertheless, broad power law spectra of compressive fluctuations are observed in turbulent astrophysical plasmas (most vividly, in the solar wind) under conditions conducive to strong Landau damping. Elsewhere in nature, such spectra are normally associated with fluid turbulence, where energy cannot be dissipated in the inertial-scale range and is, therefore, cascaded from large scales to small. By direct numerical simulations and theoretical arguments, it is shown here that turbulence of compressive fluctuations in collisionless plasmas strongly resembles one in a collisional fluid and does have broad power law spectra. This "fluidization" of collisionless plasmas occurs, because phase mixing is strongly suppressed on average by "stochastic echoes," arising due to nonlinear advection of the particle distribution by turbulent motions. Other than resolving the long-standing puzzle of observed compressive fluctuations in the solar wind, our results suggest a conceptual shift for understanding kinetic plasma turbulence generally: rather than being a system where Landau damping plays the role of dissipation, a collisionless plasma is effectively dissipationless, except at very small scales. The universality of "fluid" turbulence physics is thus reaffirmed even for a kinetic, collisionless system.

Soil moisture (SM) in the topsoil horizon is a key variable in terrestrial ecosystems, regulating water and energy exchange at the interface between the soil and the atmosphere. In forest ecosystems, SM exhibits pronounced spatio-temporal variability within the active root zone as a result of complex interactions among multiple factors including soil properties, topography, climate and vegetation. Formulating broadly reliable statements about spatio-temporal soil moisture characteristics and its effects remains challenging. Existing research is limited and sometimes contradictory regarding when and to what extent controls such as tree species influence topsoil SM variability across space and time.This study investigates spatio-temporal dynamics of topsoil SM and their controls in different forest ecosystems. The objectives are to quantify the spatial variability at the plot scale and its temporal evolution from event to seasonal scale, thereby improving our understanding of SM dynamics between wet and dry states and across different mono- and mixed-species forest stands. We investigate the variables governing spatial soil moisture variability and how they modulate SM patterns over time, with a focus on how ecohydrological processes amplify or mitigate SM variability.SM was recorded in four stands in the ECOSENSE forest in southwestern Germany, using 400 time domain transmissometry sensors (SMT100, Truebner GmbH, Germany) installed at 12 cm depth in a tree-centered design across stands of mixed and pure Douglas fir, Beech and Silver fir. A continuous 2.5-year dataset (2023 – 2025) was analysed using statistical and geostatistical approaches to identify dominating spatial SM patterns during wet and dry periods. Temporal stability was evaluated to determine the pattern persistence. Spatial SM differed significantly among plots during most of the observation period. Mean SM followed a similar annual cycle throughout all plots, with typical maxima in late winter and minima in early fall. Despite comparable soil properties and topography, the pure Beech and Douglas fir plots revealed significant seasonal differences in mean SM. Beech has more prominent autumn wetting, while Douglas fir has stronger spring drying, likely reflecting changes in evapotranspiration dynamics. The individual probability density functions of spatial soil moisture distribution transitioned between wet- or dry-preferential unimodal states and intermediate bimodal states. Across plots, spatial mean SM and the coefficient of variation exhibited an upward-convex relationship: variability was low under dry and wet ( 30% mean SM) and high under intermediate moisture conditions. The geostatistical variogram analyses showed short autocorrelation lengths, and pronounced spatial variability at few meters’ distance. Temporal stability of SM varied across plots with a range of persistently wet and dry spots. Individual locations deviated by up to 50% from temporally averaged SM at the Beech, Douglas fir and mixed plot, and up to 75% for the Silver fir plot.Combined with LiDAR derived canopy structure metrics, micro topographic maps, soil properties, and continuous ecohydrological- and meteorological observation, the presented soil moisture dataset provides a unique framework to investigate jointly modulating factors of soil moisture. It enables detailed analysis of wetting-drying cycles, including seasonal and species-specific differences.

It is often assumed that quantum effects will be significant only in the low temperature and high density plasmas. In this paper this assumption is challenged by considering the quantum contribution to the Landau damping of electron plasma waves in normal temperature and high density plasmas. Starting from the linearized Vlasov equation which contains the Bohm quantum potential, the dispersion relation of electron plasma waves propagating in a quantum plasma is derived. A linear Landau damping rate and equations for this process are also deduced. Result indicates that quantum effects enlarge effective frequency of plasmas, which is attributed to an increase in charge or number density of plasma electrons. As a result, Debye length is reduced, and the Debye screening effect becomes obvious. So the quantum behavior appears screening effect here. Landau damping rate is reduced by quantum effects and the exchange of energy between particles and waves is retarded.

We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

The analytic function of relativistic nonextensive distribution is given, and employed to solve the Landau damping of longitudinal oscillation in ultra-relativistic plasmas. The unified expression of Landau damping which reduces to the result in relativistic Maxwellian distributed plasmas in the extensive limit is obtained, and find that Landau damping is relevant to both the number and energy of resonant particles, which described by temperature and nonextensive parameters in relativistic nonextensive distribution.

Space plasmas generally posses distribution functions that exhibit high or super thermal energy tails in velocity space which can have different temperatures, a dense cold population and a hot population. Moreover, in laboratory plasmas when a laser or electron beam is passed through a dense plasma, hot low density electron populations can be generated. Presence of such low density electron distributions can act to increase the magnitude of the wave damping rate. In this paper we employ non-Maxwellian distribution function such as the generalized (r, q) distribution function with two electron temperatures to study the Landau damping of electrostatic waves. The results show that the Landau damping increases significantly when the percentage of high energy particles increases and with the increase of the high energy tail and less pronounced shoulders in the profile of the distribution function.

In this paper, we consider the higher dimensional second order differential equations of the formẍ + ∇F(x,t) = 0,x ∈R n with a class of weakly coupled potentials F( x, t ), periodically depending on t. We prove the existence of infinitely many quasi-periodic solutions for such equations via the KAM theorem.

In this paper, we consider the concentration-compactness principle in [Formula: see text], which is applied to treat the problems associated with the determination of extremal functions in functional inequalities. A more precise equality of [Formula: see text] is obtained under the conditions, that, [Formula: see text] (as stated in Theorem ??). It is an improvement of the limit inequality “[Formula: see text]” originally proposed by [P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoam. 1(1) (1985) 145–201] and [D. Smets, A concentration-compactness lemma with applications to singular eigenvalue problems, J. Funct. Anal. 167(2) (1999) 463–480]. Moreover, we establish that the additional term [Formula: see text] may only possess the “oscillation”, excluding the “concentration” and “translation to infinity”. Consequently, due to Brézis–Lieb Lemma, we reach [Formula: see text] if [Formula: see text] belongs to some regular space (e.g. [Formula: see text]). This follows the solvability of the minimization problem (2.2).

The spatial distribution of frozen soil and snow cover at the start of the spring melt season plays an important role in the generation of spring runoff and in the exchange of energy between the land surface and the atmosphere. Field observations were made at the University of Minnesota's Rosemount Agricultural Experiment Station to identify statistical distributions that can be used to describe the spatial variability of frozen soil and snow in macroscale hydrology models. These probability distributions are used to develop algorithms that simulate the subgrid spatial variability of snow and soil ice content for application within the framework of the variable infiltration capacity macroscale hydrologic model. Point simulations show that the new snow algorithm increases the melt rate for thin snowpacks, and the new soil frost algorithm allows more drainage through the soil during the winter. Simulations of the Minnesota River show that the new snow algorithm makes little difference to regional streamflow but does play an important role in the regional energy balance, especially during the spring snowmelt season. The new soil frost algorithm has a larger impact on spring streamflow and plays a minor role in the surface energy balance during the spring soil thaw season.

We investigate the role of velocity shears in producing strong phase mixing and damping of Alfven waves. We show that phase mixing damps waves in regions of strong velocity shear, thus providing a possible source of heat to the solar corona and solar wind. We compare the linear solutions with direct numerical solution of the three-dimensional equations of compressible (resistive) magnetohydrodynamics (MHD). In regions far removed from the velocity shear, the simulations show that the radial evolution of the wave amplitudes follows closely WKB theory. In the shear layer, strong damping occurs in the numerical simulations, quantitatively close to that computed from the linearized analysis.

A simple method is proposed to extend a low-order turbulence scheme including a subgrid-scale cloudiness scheme to represent not only nonconvective (stratiform) cloudiness and turbulence but also shallow, nonprecipitating cumulus convection by utilizing an appropriate subgrid-scale distribution function. A simple approach is chosen that avoids the knowledge of the skewness parameter. All cloud water-related variables (cloud water content, partial cloudiness, liquid water flux) are computed by interpolating linearly as a function of the saturation deficit between two limit cases: the stratocumulus case, which can be well represented by a Gaussian distribution function, and the trade wind cumulus case, characterized by a positively skewed distribution function with a skewness of 2. Comparisons of the scheme with a third-order turbulence scheme and large-eddy simulations (LES) for the Puerto Rico Field Experiment show satisfying results. It is shown that the liquid water flux term, which is strongly dependent on the distribution chosen, is the crucial parameter of the scheme.

We use high cadence observations of velocity distribution functions (VDFs) from the Magnetospheric Multiscale Mission (MMS) to empirically estimate diffusion coefficients and heating rates in a crossing of the Earth’s bow shock. We approximate the observed VDFs using non-parametric representations and evaluate the gradients of the modeled VDFs (GPR-VDF) to empirically estimate diffusion coefficients. This allows us to have a better representation of the non-thermal features of the distribution functions. We invert the proton guiding center equation to get estimates of diffusion coefficients and proton heating rates. We compare these results with theoretical models and simulations of stochastic heating, heating via cyclotron or Landau damping, and other heating methods to constrain the heating mechanism(s) at work across the shock. Our approach allows for an estimate from observations of collisionless heating rates within a kinetic framework and discussion of the mechanism(s) at work. This methodology could be applied to future multipoint measurements in the magnetosphere (e.g. Plasma Observatory) to study heating across shocks and other regions of interest. 

A numerical splitting scheme technique is applied to the integration of the Vlasov equation in two-space and three velocity dimensions. Calculations are performed for a cold, stationary ion, mobile electron plasma in the presence of a constant external magnetic field. Results for Landau damping and the two-stream instability are presented in both one and two velocity dimensions. The Landau damping results show a plateau structure in the electron velocity distribution function in both the parallel and perpendicular directions formed as a result of the damping of a high initial electric field propagating at 45° to the parallel velocity axis. The two-stream velocity distribution function shows pitch angle scattering and velocity diffusion for an electron beam initially propagating at 45° to the magnetic field direction through a low-temperature equal density background. The two-stream distribution function evolves to a stable quasi-isotropic plateau structure similar to that found in auroral sounding rocket data.

In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces on $\mathbb T^n_x \times \mathbb R^n_v $ is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. However, for the sake of brevity, this review is certainly not comprehensive.

The possibility of energy transfer from fusion alpha particles to externally driven wave fields for the purpose of driving current in a tokamak reactor was originally suggested by Fisch and Rax (Phys. Rev. Lett. 69 (1992) 612). The idea is reconsidered here for a class of alpha particle distribution functions that is nearly isotropic, except for the presence of diamagnetic flow due to radial inhomogeneity. The transfer of energy from alpha particles to high frequency waves (far above the ion gyrofrequency) requires that the driving term, which is proportional to the radial gradient of the alpha particle distribution function, be large enough to dominate the usual Landau damping. For a classic slowing down distribution, Landau damping dominates and the alpha particles absorb energy from the waves. Alternatively, if the alpha particle distribution is peaked near the birth speed, then the radial gradient driving term tends to dominate and energy can be extracted by the waves. Radial diffusion is a natural mechanism which displaces alpha particles outward before they can slow down (locally). However, a solution of a model Fokker-Planck equation, including radial diffusion and collisional slowing down, shows that the alpha particle distribution function still tends to absorb energy from the waves. Within the context of linear theory, high frequency waves may not be optimal as a means of extracting energy from alpha particles