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NASA NTRS1981

Mixing of a passive scalar in isotropic and sheared homogeneous turbulence

In order to calculate the velocity and scalar fields, the three dimensional, time-dependent equations of motion and the diffusion equation were solved numerically. The following cases were treated: isotropic, homogeneous turbulence with decay of a passive scalar; and homogeneous turbulent shear flow with a passive scalar whose mean varies linearly in the spanwise direction. The solutions were obtained at relatively low Reynolds numbers so that all of the turbulent scales could be resolved without modeling. Turbulent statistics such as integral length scales, Taylor microscales, Kolmogorov length scale, one- and two-point correlations of velocity-velocity and velocity-scalar, turbulent Prandtl/Schmidt number, r.m.s. values of velocities, the scalar quantity and pressure, skewness, decay rates, and decay exponents were calculated. The results are compared with the available expermental results, and good agreement is obtained.

Shirani, E.↗
NASA NTRS1981

Centrifugally driven diffusion of Iogenic plasma

The plasma distribution around Io as measured by Voyager 1 displays an asymmetric discontinuity at Io's orbit that has been suggested to be the signature of centrifugally driven interchange diffusion fed by plasma derived from Io. This hypothesis is explored further and found to be valid. The particular form for the diffusion coefficient appropriate to centrifugally driven turbulence is derived. The nonlinear character of this kind of diffusion is thereby made explicit. Solutions to the nonlinear, time-independent and linearized, time-dependent diffusion equations are given. These display a markedly conservative behavior. The nonlinear, steady state solutions are identical in form to the solutions of the previously studied equation of linear, atmospherically driven diffusion. The linearized, time-dependent solutions exhibit a negative feed-back quality that buffers the response of the density to changes in the source strength. Estimates of the source strength, the diffusion coefficient, and the signal propagation speed are also given.

Siscoe, G. L.↗
NASA NTRS1986

The propagation of solar flare particles in a coronal loop

A time-dependent diffusion equation with velocity-dependent diffusion and energy-loss coefficients was solved for the case where energetic solar particles are injected into a coronal loop and then diffuse out the ends of the loop into the lower corona/chromosphere. The solution yields for the case of relativistic electrons, precipitation rates and populations which are necessary for calculating thick and thin target X-ray emission. It follows that the thick target emission is necessarily delayed with respect to the particle acceleration on injection by more than the mere travel time of the particle over the loop length. In addition the time-dependent electron population at the top of the loop is calculated. This is useful in estimating the resulting micron-wave emission. The results show relative timing differences in the different emission processes which are functions of particle species, energy and the point of injection of the particles into the loop. Equivalent quantities are calculated for non-relativistic protons.

Ryan, J. M.↗
NASA NTRS1987

The thermal stability of coronal loops by nonlinear diffusion asymptotics

A nonlinear reaction-diffusion equation and some additional constraints are derived which describe the time-dependent behavior of the temperature structure of the plasma in coronal loops. The equation is analyzed using nonlinear diffusion asymptotics, in particular singular perturbation techniques, and the results are interpreted in the context of the physical problem of the thermal stability and temporal behavior of the plasma. The results are consistent with the possibility of cyclic thermal behavior of the plasma, as suggested by Kuin and Martens (1982).

Pakkert, J. W.↗
NASA NTRS1994

Technical report series on global modeling and data assimilation. Volume 2: Direct solution of the implicit formulation of fourth order horizontal diffusion for gridpoint models on the sphere

High order horizontal diffusion of the form K Delta(exp 2m) is widely used in spectral models as a means of preventing energy accumulation at the shortest resolved scales. In the spectral context, an implicit formation of such diffusion is trivial to implement. The present note describes an efficient method of implementing implicit high order diffusion in global finite difference models. The method expresses the high order diffusion equation as a sequence of equations involving Delta(exp 2). The solution is obtained by combining fast Fourier transforms in longitude with a finite difference solver for the second order ordinary differential equation in latitude. The implicit diffusion routine is suitable for use in any finite difference global model that uses a regular latitude/longitude grid. The absence of a restriction on the timestep makes it particularly suitable for use in semi-Lagrangian models. The scale selectivity of the high order diffusion gives it an advantage over the uncentering method that has been used to control computational noise in two-time-level semi-Lagrangian models.

Max J. Suarez↗
NASA NTRS1999

A Textbook for a First Course in Computational Fluid Dynamics

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

Zingg, D. W.↗
NASA NTRS1975

Cratering and cosmogenic nuclides

A simple probabilistic model was constructed for the average value of a cosmogenic nuclide as a function of depth in a regolith. An arbitrary function was chosen for the size distribution of craters. The resulting integro-differential equation was found to reduce in limiting cases to the marching equation with a characteristic residence time and to the diffusion equation. The regolith diffusion constant is shown to be a simple integral of the cratering rate weighted by geometrical terms. This formal treatment provides a direct and general connection between cosmogenic nuclides and cratering rates and crater population in a simple analytical form. The validity of this model remains to be tested.

Blake, M. L.↗
NASA NTRS1991

Diffusion of a multi-species component and its role in oxygen and water transport in silicates

The diffusion of a multispecies component is complicated by the different diffusion coefficient of each species and the interconversion reactions among the species. A diffusion equation is derived that incorporates the diffusive fluxes of all species contributing to the component's concentration. The effect of speciation on diffusion is investigated experimentally by measuring concentration profiles of all species developed during diffusion experiments. Data on water diffusion in rhyolitic glasses indicate that H2O molecules predominate over OH groups as the diffusing species at very low to high water concentrations. A simple theoretical relationship is drawn between the effective total oxygen diffusion coefficient and the total water concentration of silicates at low water content.

Zhang, Youxue↗
NASA NTRS1994

A unifying comparison of nearly scatter free transport models

Gombosi et al. (1993) recently derived a modified telegrapher's equation for charged particle transport under the influence of isotropic scattering. This equation obeys causality and disallows upstream diffusion for particles with random velocities smaller than the bulk flow velocity. The acausal diffusion equation was obtained to lowest order in the expansion of smallness prameters. The paper by Gombosi et al. (1993) prompted responses from Pauls et al. (1993) and Earl (1993). This paper is written to explain the differences between the methods, assumptions, and results of Gombosi et al. (1993), Pauls et al. (1993), and Earl (1993) and presents a new method of obtaining approximate solutions. It is shown that the assumptions used by Gombosi et al. (1993) and Pauls et al. (1993) are physically equivalent. In our solution method, the solution of the modified telegrapher's equation is obtained as the casual limit of solutions accurate to second order in the smallness parameter expansion. In order to investigate the coherent velocity, we have also developed `wavenumber eigenfunctions' which account for all the pitch angle dependence in our Boltzmann equation. Using truncation, Earl (1993) obtains approximations for the wavenumber dependence of the lowest two frequency modes, which correspond to two of the wavenumber eigenmodes. We find that a consequence of including only two wavenumber eigenmodes is that one obtains solutions which disobey causality at sufficiently short times. Furthermore, the coherent velocity of the two eigenmodes is strongly dependent on wavenumber and approaches the particle velocity in the limit of large wavenumber for both isotropic and anisotropic scattering processes. We conclude that Earl's (1993) solutions and solutions obtained using the new solution method implicitly assume weak acausality and reasonable behavior in the temporal regime, t less than 4 tau. The solutions are not strictly consistent with the behavior of the lowest two frequency modes but have similar behavior in the regime of low wavenumber.

Schwadron, N. A.↗
NASA NTRS2001

Modelling of Trapped Radiation Near Jupiter

Energetic (62 to approx. 130 MeV) proton fluxes measured with the Galileo Probe inside Jupiter's main ring (radii 122,500 to 128,940 km) show a modest increase as Jupiter is approached. Solutions of a reduced (1-d) equatorial diffusion equation with-constant losses described by a lifetime tau match those data for tau greater than or equal to 10(exp 9) s for diffusion coefficient D(sub LL) = 10(exp -9) L(exp 4)/s, 10(exp -9) L(exp -3)/s, or 10(exp -10) L(exp 4)/s (this particle population may be undergoing nearly loss-free inward radial diffusion). Tau would increase as Jupiter is approached if its value were determined by pitch angle scattering due to EM wave-particle interactions. if the absolute amplitudes of the waves' magnetic fluctuations did not vary with radial distance. Exploration of numerical solutions of the diffusive transport equation for Jupiter's magnetospheric ions at die planetary ring and closer to Jupiter has been done. Explicit range-energy relationships for energy loss in ring matter (SiO2), modeled as a continuous disk, are incorporated. A spatial resolution of approx. 1000 km in the radial direction is used in the vicinity of the main ring; off-equatorial ion fluxes have been included (assuming the atmosphere is a perfect absorber). For protons the maximum effect of energy loss in the microscopic size range of ring particles is expected at energies approx. 0.1 Mev. Presumably there could be sufficiently large, unobserved bodies within the ring that would have to be modeled as discrete objects, as planetary satellites would be. Within limitations of the model and computational resources it's planned to characterize solutions for transport of magnetospheric ions past the planetary ring; the procedures that are used are fairly well known and may be applied generally to the simpler magnetospheric models.

Mihalov, John D.↗
NASA NTRS1990

Asymptotic analysis of dissipative waves with applications to their numerical simulation

Various problems involving the interplay of asymptotics and numerics in the analysis of wave propagation in dissipative systems are studied. A general approach to the asymptotic analysis of linear, dissipative waves is developed. It was applied to the derivation of asymptotic boundary conditions for numerical solutions on unbounded domains. Applications include the Navier-Stokes equations. Multidimensional traveling wave solutions to reaction-diffusion equations are also considered. A preliminary numerical investigation of a thermo-diffusive model of flame propagation in a channel with heat loss at the walls is presented.

Hagstrom, Thomas↗
NASA NTRS1973

Differential equation of exospheric lateral transport and its application to terrestrial hydrogen

The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.

Hodges, R. R., Jr.↗
NASA NTRS1979

Interfacial geometry and D-variation effects in two-phase systems

Numerical solutions of the governing diffusion equation for two-phase concentration dependent diffusion coefficients are examined. Solutions were also calculated for planar, cylindrical, and spherical geometries to compare the effect of interface geometries with those caused by concentration-dependent diffusion coefficients, and two methods of averaging D were considered to determine the best averaging method for different types of D-variations. The effects of interface-location criteria on mass conservation and convergence of interface location, diffusion coefficient variation in the alpha and beta-phases of a two-phase binary alloy system, effect of D(alpha) variation in a cylindrical couple on beta-phase thickness, and geometry and D-variation effects on the degree of homogenization were determined. It is concluded that typical D(alpha)-variations can have a greater influence on the kinetics of interdiffusion than the geometry.

Tenney, D. R.↗
NASA NTRS1975

A cubic spline approximation for problems in fluid mechanics

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1975

Viscous flow solutions with a cubic spline approximation

A cubic spline approximation is used for the solution of several problems in fluid mechanics. This procedure provides a high degree of accuracy even with a nonuniform mesh, and leads to a more accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several typical integration schemes are presented. For two-dimensional flows a spline-alternating-direction-implicit (SADI) method is evaluated. The spline procedure is assessed and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1976

The absorption of trapped particles by the inner satellites of Jupiter and the radial diffusion coefficient of particle transport

The process of trapped particle absorption by the inner Jovian satellites is considered in detail taking into account both the particle and satellite motions in a magnetic dipole field which is displaced from the center of the planet and tilted with respect to the planetary rotation axis. An expression is derived for computing the sweeping time at a given satellite, defined as the time required for the satellite to sweep up a given fraction of the trapped particles within its sweeping region. By making use of the sweeping time and the radial diffusion equation of particle transport approximate expressions for the diffusion coefficient are derived. Measurements obtained by Pioneer 10 are then used to obtain estimates of the diffusion coefficient at the orbits of Io and Europa. We find that the diffusion coefficient is a function of energy and magnetic latitude for electrons in the energy range 0.7-14 MeV.

Mogro-Campero, A.↗
NASA NTRS1986

Monte Carlo study of a model of diffusion-controlled reactions

The Monte Carlo method is used to perform averages over sink configurations in the present study of diffusion-controlled reactions occurring between solute particles and immobile spherical sinks. In order to determine the average steady state solute concentration profile in a locally perturbed solution for sink volume fractions phi of less than 0.3, the diffusion equation in the monopolar plus dipolar approximation of diffusive couplings between the sinks is solved numerically. The Monte Carlo method is shown to be the most accurate and efficient in the phi = 0.001-0.1 region, where a system of only 25 sinks suffices and the monopolar approximation alone is sufficiently accurate.

Beenakker, C. W. J.↗