A scale invariance criterion for les parametrizations

Turbulent kinetic energy cascades in fluid dynamical systems are usually characterized by scale invariance. However, representations of subgrid scales in large eddy simulations do not necessarily fulfill this constraint. So far, scale invariance has been considered in the context of isotropic, incompressible, and three-dimensional turbulence. In the present paper, the theory is extended to compressible flows that obey the hydrostatic approximation, as well as to corresponding subgrid-scale parametrizations. A criterion is presented to check if the symmetries of the governing equations are correctly translated into the equations used in numerical models. By applying scaling transformations to the model equations, relations between the scaling factors are obtained by demanding that the mathematical structure of the equations does not change. The criterion is validated by recovering the breakdown of scale invariance in the classical Smagorinsky model and confirming scale invariance for the Dynamic Smagorinsky Model. The criterion also shows that the compressible continuity equation is intrinsically scale-invariant. The criterion also proves that a scaleinvariant turbulent kinetic energy equation or a scale-invariant equation of motion for a passive tracer is obtained only with a dynamic mixing length. For large-scale atmospheric flows governed by the hydrostatic balance the energy cascade is due to horizontal advection and the vertical length scale exhibits a scaling behaviour that is different from that derived for horizontal length scales

A scale invariance criterion for les parametrizations

Turbulent kinetic energy cascades in fluid dynamical systems are usually characterized by scale invariance. However, representations of subgrid scales in large eddy simulations do not necessarily fulfill this constraint. So far, scale invariance has been considered in the context of isotropic, incompressible, and three-dimensional turbulence. In the present paper, the theory is extended to compressible flows that obey the hydrostatic approximation, as well as to corresponding subgrid-scale parametrizations. A criterion is presented to check if the symmetries of the governing equations are correctly translated into the equations used in numerical models. By applying scaling transformations to the model equations, relations between the scaling factors are obtained by demanding that the mathematical structure of the equations does not change. The criterion is validated by recovering the breakdown of scale invariance in the classical Smagorinsky model and confirming scale invariance for the Dynamic Smagorinsky Model. The criterion also shows that the compressible continuity equation is intrinsically scale-invariant. The criterion also proves that a scaleinvariant turbulent kinetic energy equation or a scale-invariant equation of motion for a passive tracer is obtained only with a dynamic mixing length. For large-scale atmospheric flows governed by the hydrostatic balance the energy cascade is due to horizontal advection and the vertical length scale exhibits a scaling behaviour that is different from that derived for horizontal length scales

Non-linear Weibel-type Soliton Modes

Discussion is given of non-linear soliton behavior including coupling between electrostatic and electromagnetic potentials for non-relativistic, weakly relativistic, and fully relativistic plasmas. For plasma distribution functions that are independent of the canonical momenta perpendicular to the soliton spatial structure direction there are, in fact, no soliton behaviors allowed because transverse currents are zero. Dependence on the transverse canonical momenta is necessary. When such is the case, it is shown that the presence or absence of a soliton is intimately connected to the functional form assumed for the particle distribution functions. Except for simple situations, the coupled non-linear equations for the electrostatic and electromagnetic potentials would seem to require numerical solution procedures. Examples are given to illustrate all of these points for non-relativistic, weakly relativistic, and fully relativistic plasmas.Comment: Accepted for publication at Journal of Physics A: Mathematical and Theoretica

Effect of temperature anisotropy on various modes and instabilities for a magnetized non-relativistic bi-Maxwellian plasma

Using kinetic theory for homogeneous collisionless magnetized plasmas, we present an extended review of the plasma waves and instabilities and discuss the anisotropic response of generalized relativistic dielectric tensor and Onsager symmetry properties for arbitrary distribution functions. In general, we observe that for such plasmas only those electromagnetic modes whose magnetic field perturbations are perpendicular to the ambient magneticeld, i.e.,B1 \perp B0, are effected by the anisotropy. However, in oblique propagation all modes do show such anisotropic effects. Considering the non-relativistic bi-Maxwellian distribution and studying the relevant components of the general dielectric tensor under appropriate conditions, we derive the dispersion relations for various modes and instabilities. We show that only the electromagnetic R- and L- waves, those derived from them and the O-mode are affected by thermal anisotropies, since they satisfy the required condition B1\perpB0. By contrast, the perpendicularly propagating X-mode and the modes derived from it (the pure transverse X-mode and Bernstein mode) show no such effect. In general, we note that the thermal anisotropy modifies the parallel propagating modes via the parallel acoustic effect, while it modifies the perpendicular propagating modes via the Larmor-radius effect. In oblique propagation for kinetic Alfven waves, the thermal anisotropy affects the kinetic regime more than it affects the inertial regime. The generalized fast mode exhibits two distinct acoustic effects, one in the direction parallel to the ambient magnetic field and the other in the direction perpendicular to it. In the fast-mode instability, the magneto-sonic wave causes suppression of the firehose instability. We discuss all these propagation characteristics and present graphic illustrations

A comparison of different solutions for the dynamic smagorinsky model applied in a GCM

A discussion of different approaches and solutions of the basic tensor equation within the Dynamic Smagorinsky Model (DSM) suitable for General Circulation Models (GCM) is presented. Particular interest is dedicated to the relationship between various approaches (i.e., the specific formulation of the tensor equation), namely a least-square approach, a time lag approach, and a simple tensor contraction approach, and the impact of the specific solution (i.e., how to solve the equation) on the Smagorinsky parameter c2S . In addition to the standard solutions, clipped solutions, absolute solutions, and tensor norm solutions are examined. The numerical results are based on calculations from a general circulation model, where the different approaches are applied to provide the turbulent horizontal momentum diffusion. Here, they are examined with focus on two issues: 1) At the beginning of the simulations, the different choices for the tensor equation result in different values for the locally distributed and zonally averaged values of the Smagorinsky parameter. These values show that for the standard solutions almost half of the values of c2S are negative, in accordance with known results from isotropic turbulence and leads to unstable simulations. In addition, the tensor norm is related to the absolute solution via the Cauchy-Schwarz inequality. 2) As the simulations proceed, the differences of the Smagorinsky parameter values diminish except for the tensor norm solutions while evolving to a stationary state in a process of self-organization such that they form a group with values comparable to isotropic three-dimensional simulations. In summary, the least-squares and time lag approaches provide reasonable results, while the simple contraction approach fluctuates more. For the solutions, it is discussed whether the clipped or the tensor norm solution is more reasonable. © 2018 The authors

A new distribution function for relativistic counterstreaming plasmas

The particle distribution function that describes two interpenetrating plasma streams is re-investigated. It is shown how, based on the Maxwell-Boltzmann-J\"uttner distribution function that has been derived almost a century ago, a counterstreaming distribution function can be derived that uses velocity space. Such is necessary for various analytical calculations and numerical simulations that are reliant on velocity coordinates rather than momentum space. The application to the electrostatic two-stream instability illustrates the differences caused by the use of the relativistic distribution function.Comment: Accepted for publication in Astrophysics & Space Scienc

Multi-scale Methods for Geophysical Flows

Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-scale and microscopic turbulence regimes. The relation of scales and flow phenomena is essential in order to validate and improve current numerical weather and climate prediction models. While regime separation is often possible on a formal level via multi-scale analysis, the systematic exploration, structure preservation, and mathematical details remain challenging. This chapter provides an entry to the literature and reviews fundamental notions as background for the later chapters in this collection and as a departure point for original research in the field