Gravity waves, scale asymptotics, and the pseudo-incompressible equations

Multiple-scale asymptotics is used to analyze the Euler equations for the dynamical situation of a gravity wave (GW) near breaking level. A simple saturation argument in combination with linear theory is used to obtain the relevant dynamical scales. As small expansion parameter the ratio of inverse of the vertical wave number and potentialtemperature and pressure scale heights is used, which we allow to be of the same order of magnitude here. It is shown that the resulting equation hierarchy is consistent with that obtained from the pseudo-incompressible equations, both for non-hydrostatic and hydrostatic gravity waves, while this is not the case for the anelastic equations unless the additional assumption of sufficiently weak stratication is adopted. To describe vertical propagation of wave packets over several atmospheric scale heights, WKB theory is used to show that the pseudo-incompressible flow divergence generates the same amplitude equation that also obtains from the full Euler equations. This gives a mathematical justication for the use of the pseudo-incompressible equations for studies of gravity-wave breaking in the atmosphere for arbitrary background stratication. The WKB theory interestingly also holds at wave amplitudes close to static instability. In the mean-flow equations we obtain in addition to the classic wave-induced momentum-flux divergences a wave-induced correction of hydrostatic balance in the vertical-momentum equation which cannot be obtained from Boussinesq or anelastic dynamics

Stochastic Acceleration of Low Energy Electrons in Plasmas with Finite Temperature

This paper extends our earlier work on the acceleration of low-energy electrons by plasma turbulence to include the effects of finite temperature of the plasma. We consider the resonant interaction of thermal electrons with the whole transverse branch of plasma waves propagating along the magnetic field. We show that our earlier published results for acceleration of low-energy electrons can be applied to the case of finite temperature if a sufficient level of turbulence is present. From comparison of the acceleration rate of the thermal particles with the decay rate of the waves with which they interact, we determine the required energy density of the waves as a fraction of the magnetic energy density, so that a substantial fraction of the background plasma electrons can be accelerated. The dependence of this value on the plasma parameter alpha = omega_pe / Omega_e (the ratio of electron plasma frequency to electron gyrofrequency), plasma temperature, and turbulence spectral parameters is quantified. We show that the result is most sensitive to the plasma parameter alpha. We estimate the required level of total turbulence by calculating the level of turbulence required for the initial acceleration of thermal electrons and that required for further acceleration to higher energies

Stochastic Acceleration of Low Energy Electrons in Cold Plasmas

We investigate the possibility of stochastic acceleration of background low-energy electrons by turbulent plasma waves. We consider the resonant interaction of the charged particles with all branches of the transverse plasma waves propagating parallel to a uniform magnetic field. Numerical results and asymptotic analytic solutions valid at non-relativistic and ultra-relativistic energies are obtained for the acceleration and scattering times of electrons. These times have a strong dependence on plasma parameter alpha = Omega_pe / Omega_e (the ratio of electron plasma frequency to electron gyrofrequency) and on the spectral index of plasma turbulence. It is shown that particles with energies above certain critical value may interact with higher frequency electromagnetic plasma waves and this interaction is allowed only in plasmas with alpha < 1. We show that for non-relativistic and semi-relativistic electrons in low-alpha plasmas the ratio of the acceleration time to the scattering time can be less than unity for a wide range of energies. From this we conclude that the transport equation derived for cosmic rays which requires this ratio to be much larger than one is not applicable at these energies. An approximate "critical" value of particle energy above which the dynamics of charged particles may be described by this transport equation is determined as a function of plasma parameters. We propose new transport equation for the opposite limit (energies less than this critical value) when the acceleration rate is much faster than the pitch angle scattering rate. This equation is needed to describe the electron dynamics in plasmas with alpha <= 0.1.Comment: 22 pages, 13 figures, Latex, submitted to Astrophysical Journa

The interaction between synoptic-scale balanced flow and a finite-amplitude mesoscale wave field throughout all atmospheric layers: weak and moderately strong stratification

The interaction between locally monochromatic finite-amplitude mesoscale waves, their nonlinearly induced higher harmonics, and a synoptic-scale flow is reconsidered, both in the tropospheric regime of weak stratification and in the stratospheric regime of moderately strong stratification. A review of the basic assumptions of quasi-geostrophic theory on an f-plane yields all synoptic scales in terms of a minimal number of natural variables, i.e. two out of the speed of sound, gravitational acceleration and Coriolis parameter. The wave scaling is defined so that all spatial and temporal scales are shorter by one order in the Rossby number, and by assuming their buoyancy field to be close to static instability. WKB theory is applied, with the Rossby number as scale separation parameter, combined with a systematic Rossby-number expansion of all fields. Classic results for synoptic-scale-flow balances and inertia-gravity-wave (IGW) dynamics are recovered. These are supplemented by explicit expressions for the interaction between mesoscale geostrophic modes (GMs), a possibly somewhat overlooked agent of horizontal coupling in the atmosphere, and the synoptic-scale flow. It is shown that IGW higher harmonics are slaved to the basic IGW, and that their amplitude is one order of magnitude smaller than the basic-wave amplitude. GM higher harmonics are not that weak and they are in intense nonlinear interaction between themselves and the basic GM. Compressible dynamics plays a significant role in the stratospheric stratification regime, where anelastic theory would yield insufficient results. Supplementing classic derivations, it is moreover shown that, in the absence of mesoscale waves, quasi-geostrophic theory holds also in the stratospheric stratification regime

Regime of Validity of Sound-Proof Atmospheric Flow Models

Ogura and Phillips (1962) derived their original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model which would simultaneously represent internal gravity waves and the effects of advection, they had to adopt a distinguished limit stating that the dimensionless stability of the background state be of the order of the Mach number squared. For typical flow Mach numbers of M = 1/30 this amounts to total variations of potential temperature across the troposphere of less than one Kelvin, i.e., to unrealistically weak stratication. Various generalizations of Ogura and Phillips' anelastic model have been proposed to remedy this issue, e.g., by Dutton & Fichtl (1969), and Lipps & Hemler (1982). Following the same goals, but a somewhat different route of argumentation, Durran proposed the pseudoincompressible model in 1989. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths of order of the Mach number to the power 2/3, which corresponds to realistic variations of potential temperature across the pressure scale height of about 30 K

The transient IDEMIX model as a nonorographic gravity wave parameterization in an atmospheric circulation model

The Internal wave Dissipation, Energy and Mixing (IDEMIX) model presents a novel way of parameterizing internal gravity waves in the atmosphere. Using a continuous full wave spectrum in the energy balance equation and integrating over all vertical wavenumbers and frequencies results in prognostic equations for the energy density of gravity waves in multiple azimuthal compartments. It includes their non-dissipative interaction with the mean flow, allowing for an evolving and local description of momentum flux and gravity wave drag. A saturation mechanism maintains the wave field within convective stability limits, and an energetically consistent closure for critical-layer effects controls how much wave flux propagates from the troposphere into the middle atmosphere. IDEMIX can simulate zonal gravity wave drag around the mesopause, similar to a traditional gravity wave parameterization and to a state-of-the-art wave ray tracing model in an atmospheric circulation model. In addition, IDEMIX shows a reversal of the gravity wave drag around the mesopause region due to changes in the momentum flux there. When compared to empirical model data, IDEMIX captures well the summer hemisphere flow reversal, the cold summer mesospheric pole and the alternate positive and negative structures in the meridional mean flow.Comment: 21 pages, 19 figure

Generalised Ornstein-Uhlenbeck processes

We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.Comment: 24 pages, 2 figure

Multi-scale waves in sound-proof global simulations with EULAG

EULAG is a computational model for simulating flows across a wide range of scales and physical scenarios. A standard option employs an anelastic approximation to capture nonhydrostatic effects and simultaneously filter sound waves from the solution. In this study, we examine a localized gravity wave packet generated by instabilities in Held-Suarez climates. Although still simplified versus the Earth’s atmosphere, a rich set of planetary wave instabilities and ensuing radiated gravity waves can arise. Wave packets are observed that have lifetimes ≤ 2 days, are negligibly impacted by Coriolis force, and do not show the rotational effects of differential jet advection typical of inertia-gravity waves. Linear modal analysis shows that wavelength, period, and phase speed fit the dispersion equation to within a mean difference of ∼ 4%, suggesting an excellent fit. However, the group velocities match poorly even though a propagation of uncertainty analysis indicates that they should be predicted as well as the phase velocities. Theoretical arguments suggest the discrepancy is due to nonlinearity — a strong southerly flow leads to a critical surface forming to the southwest of the wave packet that prevents the expected propagation

Acoelomorpha: earliest branching bilaterians or deuterostomes?

The Acoelomorpha is an animal group comprised by nearly 400 species of misleadingly inconspicuous flatworms. Despite this, acoelomorphs have been at the centre of a heated debate about the origin of bilaterian animals for 150 years. The animal tree of life has undergone major changes during the last decades, thanks largely to the advent of molecular data together with the development of more rigorous phylogenetic methods. There is now a relatively robust backbone of the animal tree of life. However, some crucial nodes remain contentious, especially the node defining the root of Bilateria. Some studies situate Acoelomorpha (and Xenoturbellida) as the sister group of all other bilaterians, while other analyses group them within the deuterostomes which instead suggests that the last common bilaterian ancestor directly gave rise to deuterostomes and protostomes. The resolution of this node will have a profound impact on our understanding of animal/bilaterian evolution. In particular, if acoelomorphs are the sister group to Bilateria, it will point to a simple nature for the first bilaterian. Alternatively, if acoelomorphs are deuterostomes, this will imply that they are the result of secondary simplification. Here, we review the state of this question and provide potential ways to solve this long-standing issue. Specifically, we argue for the benefits of (1) obtaining additional genomic data from acoelomorphs, in particular from taxa with slower evolutionary rates; (2) the development of new tools to analyse the data; and (3) the use of metagenomics or metatranscriptomics data. We believe the combination of these three approaches will provide a definitive answer as to the position of the acoelomorphs in the animal tree of life