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NASA NTRS1982

Pulse asymptotics of the Charney baroclinic instability problem

The asymptotic response of the Charney baroclinic instability problem to a localized perturbation is determined using the formalism of Briggs (1964) and exploiting a recently obtained highly accurate WKB approximate dispersion relation (Lindzen and Rosenthal, 1981). Comparison is made with previous results for two-level and Eady models. Small scales and rapid growth characteristic of the initial stages of cyclogenesis are found and the linear dispersion relation, which can be obtained from observation of zonal wind and stability, emerges as a forecast tool for prediction of geographically local cyclogenesis.

Farrell, B. F.↗

NASA NTRS1982

The local dispersion relation for magneto-atmospheric waves

The local dispersion relation for magneto-atmospheric waves is discussed in terms of the linearized theory of waves in a plane-stratified, inviscid, perfectly conducting atmosphere under uniform gravity. The normally used local dispersion relation is demonstrated to not be unique, depending instead on the order of derivation from the fundamental first-order perturbation equations of continuity, momentum, energy, and induction. Furthermore, it is shown that the local dispersion relation predicts that the cutoff frequency decreases with increasing magnetic field strength, while the WKB approximation method projects an increase in the cutoff frequency with increasing magnetic field strength. A new form of the local dispersion relation is developed, and consideration is given to the special case of a global dispersion relation in conditions of an isothermal atmosphere with a horizontal magnetic field.

Thomas, J. H.↗

NASA NTRS1982

Forced, stationary waves in a linear, stratified, quasi-geostrophic atmosphere

A finite difference model and an approximate WKB analysis were employed to test the validity of three of the Smagorinsky conclusions (1953) regarding the influence of diabatic heating upon the resultant stationary eddies. The conclusions comprised a necessity of including surface friction to avoid resonance, to account for stratospheric influence to accurately estimate the magnitude of the disturbances in the upper third of the troposphere, and the adequacy of the linear model for describing the gross features of sea-level normals at middle and high latitudes. Results of the WKB analysis of the effects of orography and diabatic heating demonstrate that streamfunctions are produced when orography forces a stationary solution. The surface friction is concluded to have a negligible effect on the solutions when the wind speeds are low. The solutions, however, require the presence of a lid for the middle troposphere to eliminate resonance responses.

Roads, J. O.↗

NASA NTRS1987

Observation of millimeter-wave oscillations from resonant tunneling diodes and some theoretical considerations of ultimate frequency limits

Recent observations of oscillation frequencies up to 56 GHz in resonant tunneling structures are discussed in relation to calculations by several authors of the ultimate frequency limits of these devices. It is found that calculations relying on the Wentzel-Kramers-Brillouin (WKB) approximation give limits well below the observed oscillation frequencies. Two other techniques for calculating the upper frequency limit were found to give more reasonable results. One method employs the solution of the time-dependent Schroedinger equation obtained by Kundrotas and Dargys (1986); the other uses the energy width of the transmission function for electrons through the double-barrier structure. This last technique is believed to be the most accurate since it is based on general results for the lifetime of any resonant state. It gives frequency limits on the order of 1 THz for two recently fabricated structures. It appears that the primary limitation of the oscillation frequency for double-barrier resonant-tunneling diodes is imposed by intrinsic device circuit parameters and by the transit time of the depletion layer rather than by time delays encountered in the double-barrier region.

Sollner, T. C. L. G.↗

NASA NTRS1987

Dynamics of dendritic sidebranching in the two-dimensional symmetric model of solidification

Within a WKB approximation, the evolution of time-dependent deformations of the needle crystal solution of the two-dimensional symmetric model of solidification is studied. It is found that perturbations with fixed small frequencies are initially amplified as they propagate from near the tip down the dendrite but ultimately decay. Localized wave packets behave rather differently; the packet continues to grow exponentially as it moves to arbitrarily large distances from the tip. The relevance of these results to sidebranching of dendrites is discussed.

Barber, Michael N.↗

NASA NTRS1987

Dendritic sidebranching in the three-dimensional symmetric model in the presence of noise

The time-dependent behavior of sidebranching deformations in the three-dimensional symmetric model of dendritic solidification is studied within a WKB approximation. Localized wave packets generated by pulses in the neighborhood of the tip are found to grow in amplitude and to spread and stretch as they move down the sides of the dendrite. This behavior is shown to imply that noise in the solidifying medium is selectively amplified in such a way as to produce a fluctuating train of sidebranches in qualitative agreement with experimental observations. A rough estimate indicates that purely thermal noise is probably not quite strong enough to fit the data.

Langer, J. S.↗

NASA NTRS1987

Velocity selection at large undercooling in a two-dimensional nonlocal model of solidification

The formation of needle-crystal dendrites from an undercooled melt is investigated analytically, applying the method of Caroli et al. (1986) to Langer's (1980) symmetric two-dimensional nonlocal model of solidification with finite anisotropy in the limit of large undercooling. A solution based on the WKB approximation is obtained, and a saddle-point evaluation is performed. It is shown that needle-crystal solutions exist only if the capillary anisotropy is nonzero, in which case a particular value of the growth velocity can be selected. This finding and the expression for the dependence of the selected velocity on the singular perturbation parameter and the strength of the anisotropy are found to be in complete agreement with the results of a boundary-layer model (Langer and Hong, 1986).

Barbieri, Angelo↗

NASA NTRS1988

A numerical model for gravity wave dissipation in the thermosphere

Two simplified models have been developed for the internal gravity wave dissipation due to viscosity, thermal conduction, and ion-drag in a multilayered, isothermal thermosphere. Both models use the WKB approximation, ray theory, and the time-averaged equations of gravity wave energy conservation. One model uses all the equations appropriate to a dissipative atmosphere, while the other uses the dispersion equation and polarization relations applicable to a nondissipative atmosphere, neglecting the viscous and thermal conduction contributions to the energy flux. Results from these models are compared to each other and to the results obtained by Klostermeyer (1973), using a full-wave model.

Hickey, M. P.↗

NASA NTRS1987

Nonparallel stability of boundary layers

The asymptotic formulations of the nonparallel linear stability of incompressible growing boundary layers are critically reviewed. These formulations can be divided into two approaches. The first approach combines a numerical method with either the method of multiple scales, or the method of averaging, of the Wentzel-Kramers-Brillouin (WKB) approximation; all these methods yield the same result. The second approach combined a multi-structure theory with the method of multiple scales. The first approach yields results that are in excellent agreement with all available experimental data, including the growth rates as well as the neutral stability curve. The derivation of the linear stability of the incompressible growing boundary layers is explained.

Nayfeh, Ali H.↗

NASA NTRS1990

Stability analysis of the accretion line

The stability of the flow along the accretion line in the Bondi-Hoyle-Lyttleton-type accretion flow is studied. The WKB approximation is used to analyze short wavelength linear perturbations of the axisymmetric steady state flow. The viscosity terms in the gasdynamical equations are included in the analysis. The radial modes are studied both in two- and three-dimensional accretion flow. Since a displaced accretion line cannot be defined in three-dimensional flow, the tangential modes are only analyzed in the two-dimensional flow. The analysis of the perturbed gasdynamical equations are made simpler as a result of the radial and tangential modes being decoupled in the linear regime. As was shown before, the three-dimensional accretion flow is unstable against radial modes. The two-dimensional accretion flow is found to be unstable against tangential modes, as well as against radial modes. Using typical numerical parameters which are in common use, the wavelength which gives the maximum growth rate of a small perturbation is found to be about 0.5 r(a), where r(a) is the accretion radius.

Soker, Noam↗

NASA NTRS1990

Remarks on transport theories of interplanetary fluctuations

The structure of a transport theory presented by Zhou and Matthaeus (1989), in which coupling of 'inward' and 'outward'-type fluctuations appears in the leading order, is studied. Allowance is made for the dynamic behavior of the 'fast' scale variables, which are averaged over to obtain slow-scale transport equations. The relationship of the two-scale transport models to transport models derived from the WKB approximation (Jeffreys and Jeffreys, 1980: Weinberg 1962; Dewar, 1970) as it has been applied to the solar wind MHD wave problem (Parker, 1965; Hollweg, 1973, 1974) is discussed.

Zhou, YE↗

NASA NTRS1990

Ion cyclotron absorption at the second harmonic of the oxygen gyrofrequency

A hot ray tracing code (HOTRAY) is used to establish the ultimate demise of the wave energy generated inside the plasmasphere during storm conditions. The code uses the WKB approximation of geometric optics to calculate the paths of electromagnetic and electrostatic waves in a hot, multiion plasma. Ray tracing calculations with HOTRAY are detailed and path integrated absorption is discussed. Unducted waves which are preferentially generated in the field aligned direction are rapidly refracted to large wave normal angles. The waves are observed to reflect at frequencies between oxygen and helium gyrofrequencies once their frequency falls below the bi-ion frequency and the waves bounce back accross the equator. Either Landau resonance with plasmaspheric electrons or second harmonic cyclotron resonance with thermal oxygen ultimately absorb the wave energy. It is suggested that both processes may provide a significant heating mechanism for either population during a storm.

Horne, R. B.↗

NASA NTRS1991

Nonlinear instability of the accretion line

The instability of the flow along the accretion line in the two-dimensional Bondi-Hoyle-Lyttleton type accretion flow is studied. By means of numerical simulations, the nonlinear regime of the instability is explored, thus extending a previous work that dealt with the linear regime using the WKB approximation. In contrast to the behavior in the linear regime, in the nonlinear regime the radial and tangential modes are coupled. The radial instability, which manifests itself as large density and velocity variations on short distances, grows far beyond the linear regime. The tangential instability, on the other hand, stays in the linear regime and is dominant by long-wavelength perturbations. This results from both the influence of the radial instability and the incoming material (the material accreted onto the accretion line).

Soker, Noam↗

NASA NTRS1993

The stability of cosmic-ray-dominated shocks - A secondary instability

Previous studies in one spatial dimension showed that the precursors of cosmic-ray-dominated shocks are unstable against traveling acoustic disturbances. Here we report that a secondary, Rayleigh-Taylor type instability can exist in cosmic-ray-dominated media influenced by the acoustic instability. Using the local WKB approximation, the growth rate of the secondary instability is shown to be comparable to that of the one-dimensional acoustic instability itself in the cases we have considered. The nonlinear development has been followed numerically with a two-dimensional PPM hydrodynamics code that also incorporates the two-fluid cosmic-ray energy equation. We show that the secondary instability may cause the precursor and postshock flows to become highly turbulent.

Ryu, Dongsu↗

NASA NTRS1995

Radial energy transport by magnetospheric ULF waves: Effects of magnetic curvature and plasma pressure

The 'radial' transport of energy by internal ULF waves, stimulated by dayside magnetospheric boundary oscillations, is analyzed in the framework of one-fluid magnetohydrodynamics. (the term radial is used here to denote the direction orthogonal to geomagnetic flux surfaces.) The model for the inhomogeneous magnetospheric plasma and background magnetic field is axisymmetric and includes radial and parallel variations in the magnetic field, magnetic curvature, plasma density, and low but finite plasma pressure. The radial mode structure of the coupled fast and intermediate MHD waves is determined by numerical solution of the inhomogeneous wave equation; the parallel mode structure is characterized by a Wentzel-Kramer-Brillouin (WKB) approximation. Ionospheric dissipation is modeled by allowing the parallel wave number to be complex. For boudnary oscillations with frequencies in the range from 10 to 48 mHz, and using a dipole model for the background magnetic field, the combined effects of magnetic curvature and finite plasma pressure are shown to (1) enhance the amplitude of field line resonances by as much as a factor of 2 relative to values obtained in a cold plasma or box-model approximation for the dayside magnetosphere; (2) increase the energy flux delivered to a given resonance by a factor of 2-4; and (3) broaden the spectral width of the resonance by a factor of 2-3. The effects are attributed to the existence of an 'Alfven buoyancy oscillation,' which approaches the usual shear mode Alfven wave at resonance, but unlike the shear Alfven mode, it is dispersive at short perpendicular wavelengths. The form of dispersion is analogous to that of an internal atmospheric gravity wave, with the magnetic tension of the curved background field providing the restoring force and allowing radial propagation of the mode. For nominal dayside parameters, the propagation band of the Alfven buoyancy wave occurs between the location of its (field line) resonance and that of the fast mode cutoff that exists at larger radial distances.

Kouznetsov, Igor↗

NASA NTRS1998

Calculation of the Bulk Electromagnetic Properties of Thunderclouds Using a Two-Space Scattering Formalism

A two-space scatterer formalism and equivalent medium approach of Twersky are used to obtain formulae for the bulk electromagnetic parameters of a thundercloud medium that is illuminated by lightning optical emissions. A modified WKB approximation is applied to derive the two-space scattering amplitude. Optical constants of dry air, moist atmosphere, and water,.are inserted into the formulae to generate numerical values via iteration. The two-space scatterer formalism results are close to those obtained from free- or sin-le-space formalisms for the dilute case. The numerical values of the bulk parameters are required to successfully transform the scattering problem to that of an equivalent obstacle excited by an incident wave traveling in K - space but radiating in k(sub 1) -space.

Phanord, Dieudonne D.↗

NASA NTRS2003

Three-Dimensional MHD Modeling of The Solar Corona and Solar Wind: Comparison with The Wang-Sheeley Model

We present simulation results from a tilted-dipole steady-state MHD model of the solar corona and solar wind and compare the output from our model with the Wang-Sheeley model which relates the divergence rate of magnetic flux tubes near the Sun (inferred from solar magnetograms) to the solar wind speed observed near Earth and at Ulysses. The boundary conditions in our model specified at the coronal base and our simulation region extends out to 10 AU. We assumed that a flux of Alfven waves with amplitude of 35 km per second emanates from the Sun and provides additional heating and acceleration for the coronal outflow in the open field regions. The waves are treated in the WKB approximation. The incorporation of wave acceleration allows us to reproduce the fast wind measurements obtained by Ulysses, while preserving reasonable agreement with plasma densities typically found at the coronal base. We find that our simulation results agree well with Wang and Sheeley's empirical model.

Usmanov, A. V.↗

NASA NTRS2011

Three-Dimensional Modeling of the Solar Wind: From the Coronal Base to the Outer Heliosphere

We have developed a global fully three-dimensional magnetohydrodynamic solar wind model for the region that extends from the coronal base to 100 AU. The simulation domain consists of tree spherical shell subdomains with computational boundaries between them placed at 20 solar radii and 0.3 AU. The location of the first boundary ensures that the flow at the boundary is both supersonic and super-Alfvenic. A steady-state solution in the innermost (coronal) region is obtained by the time-relaxation method. The solution uses a tilted dipole model or solar magnetograms as the boundary condition at the coronal base and includes a flux of Alfven waves in the WKB approximation which provide additional acceleration for the coronal outflow in the open field regions. The intermediate region solution is constructed by the integration of steady-state equations along radius using a marching scheme. The outer region solution (0.3-100 AU) is obtained again by the time relaxation and takes into account turbulence transport and heating as well as heating, flow deceleration, and other effects due to the interstellar pickup protons treated as a separate fluid. We use the model to simulate the global steady-state structure of the solar wind from the coronal base to the heliospheric boundary and compare the results with Ulysses and Voyager observations.

Usmanov, A. V↗