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The effects of radiation on the structure of shocks in a fully-ionized plasma are investigated by solving the steady-state fluid equations for ions, electrons, and radiation. The electrons and ions are assumed to have the same bulk velocity but separate temperatures, and the radiation is modeled with the gray-diffusion approximation. Both electron and ion conduction are included, as well as ion viscosity. When the material is optically thin, three-temperature behavior occurs. When the diffusive flux of radiation is important but radiation pressure is not, two-temperature behavior occurs, with the electrons strongly coupled to the radiation. Since the radiation heats the electrons on length scales that are much longer than the electron-ion Coulomb coupling length scale, these solutions resemble radiative shock solutions rather than plasma shock solutions that neglect radiation. When radiation pressure is important, all three components are strongly coupled. Results with constant values for the transport and coupling coefficients are compared to a full numerical simulation with a good match between the two, demonstrating that steady shock solutions constitute a straightforward and comprehensive verification test methodology for multi-physics numerical algorithms., Comment: Accepted for publication in Shock Waves. The final publication is available at Springer via http://dx.doi.org/10.1007/s00193-016-0649-9

I consider the hydrodynamic stability of imploding gases as a model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes under a homologous flow, a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor (2C)^(|N0| ti)$, where C is the convergence ratio of the implosion, N0 is the initial buoyancy frequency and ti is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor exp(pi |N0| ts), where N0 is the buoyancy frequency at stagnation and ts is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. By evolving the complete set of linear equations, it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode l > 100 for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of ~30 zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of ~(12/l) degrees, or < 0.1 degree to resolve the fastest-growing modes., Comment: 10 pages, 3 figures, accepted for publication in the Journal of Fluid Mechanics Rapids

It is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier-Stokes equations for an ideal gas. For a constant dynamic viscosity and thermal conductivity, and at particular values of the shock Mach number, the velocity can be expressed in terms of a polynomial root. For a constant kinematic viscosity, independent of Mach number, the velocity can be expressed in terms of a hyperbolic tangent function. The remaining fluid variables are related to the velocity through simple algebraic expressions. The solutions derived here make excellent verification tests for numerical algorithms, since no source terms in the evolution equations are approximated, and the closed-form expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the non-linear character of the Navier-Stokes equations and may stimulate further analytical developments., Comment: 11 pages, 9 figures, accepted for publication in the Journal of Fluid Mechanics Rapids

The modeling of turbulence, whether it be numerical or analytical, is a difficult challenge. Turbulence is amenable to analysis with linear theory if it is subject to rapid distortions, i.e., motions occurring on a time scale that is short compared to the time scale for non-linear interactions. Such an approach could prove useful for understanding aspects of astrophysical turbulence, which is often subject to rapid distortions, such as supernova explosions or the free-fall associated with gravitational instability. As a proof of principle, a particularly simple problem is considered here: the evolution of vorticity due to a planar rarefaction in an ideal gas. Vorticity can either grow or decay in the wake of a rarefaction front, and there are two competing effects that determine which outcome occurs: entropy fluctuations couple to the mean pressure gradient to produce vorticity via baroclinic effects, whereas vorticity is damped due to the conservation of angular momentum as the fluid expands. In the limit of purely entropic fluctuations in the ambient fluid, a strong rarefaction generates vorticity with a turbulent Mach number on the order of the root-mean square of the ambient entropy fluctuations. The analytical results are shown to compare well with results from two- and three-dimensional numerical simulations. Analytical solutions are also derived in the linear regime of Reynolds-averaged turbulence models. This highlights an inconsistency in standard turbulence models that prevents them from accurately capturing the physics of rarefaction-turbulence interaction. Finally, dimensional analysis of the equations indicates that rapid distortion of turbulence can give rise to two distinct regimes in the turbulent spectrum: a distortion range at large scales where linear distortion effects dominate, and an inertial range at small scales where non-linear effects dominate., Comment: 19 pages, 25 figures, accepted for publication in The Astrophysical Journal

Exact one-dimensional solutions to the equations of fluid dynamics are derived in the large-Pr and small-Pr limits (where Pr is the Prandtl number). The solutions are analogous to the Pr = 3/4 solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-Pr solution is very similar to Becker's solution, differing only by a scale factor. The small-Pr solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-Pr equations is O(1/Pr) for large Pr and O(Pr) for small Pr., Comment: 11 pages, 6 figures. Accepted for publication in Journal of Fluid Mechanics Rapids

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article., Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of the conjectures have been proved

We numerically evolve turbulence driven by the magnetorotational instability (MRI) in a 3D, unstratified shearing box and study its structure using two-point correlation functions. We confirm Fromang and Papaloizou's result that shearing box models with zero net magnetic flux are not converged; the dimensionless shear stress $\alpha$ is proportional to the grid scale. We find that the two-point correlation of the magnetic field shows that it is composed of narrow filaments that are swept back by differential rotation into a trailing spiral. The correlation lengths along each of the correlation function principal axes decrease monotonically with the grid scale. For mean azimuthal field models, which we argue are more relevant to astrophysical disks than the zero net field models, we find that: $\alpha$ increases weakly with increasing resolution at fixed box size; $\alpha$ increases slightly as the box size is increased; $\alpha$ increases linearly with net field strength, confirming earlier results; the two-point correlation function of the magnetic field is resolved and converged, and is composed of narrow filaments swept back by the shear; the major axis of the two-point increases slightly as the box size is increased; these results are code independent, based on a comparison of ATHENA and ZEUS runs. The velocity, density, and magnetic fields decorrelate over scales larger than $\sim H$, as do the dynamical terms in the magnetic energy evolution equations. We conclude that MHD turbulence in disks is localized, subject to the limitations imposed by the absence of vertical stratification, the use of an isothermal equation of state, finite box size, finite run time, and finite resolution, Comment: 25 pages, 8 figures, accepted by ApJ

In the dynamic diffusion limit of radiation hydrodynamics, advection dominates diffusion; the latter primarily affects small scales and has negligible impact on the large scale flow. The radiation can thus be accurately regarded as an ideal fluid, i.e., radiative diffusion can be neglected along with other forms of dissipation. This viewpoint is applied here to an analysis of simple waves in an ideal radiating fluid. It is shown that much of the hydrodynamic analysis carries over by simply replacing the material sound speed, pressure and index with the values appropriate for a radiating fluid. A complete analysis is performed for a centered rarefaction wave, and expressions are provided for the Riemann invariants and characteristic curves of the one-dimensional system of equations. The analytical solution is checked for consistency against a finite difference numerical integration, and the validity of neglecting the diffusion operator is demonstrated. An interesting physical result is that for a material component with a large number of internal degrees of freedom and an internal energy greater than that of the radiation, the sound speed increases as the fluid is rarefied. These solutions are an excellent test for radiation hydrodynamic codes operating in the dynamic diffusion regime. The general approach may be useful in the development of Godunov numerical schemes for radiation hydrodynamics., Comment: 16 pages, 10 figures, accepted for publication in The Astrophysical Journal

In numerical models of thin astrophysical disks that use an Eulerian scheme, gas orbits supersonically through a fixed grid. As a result the time step is sharply limited by the Courant condition. Also, because the mean flow speed with respect to the grid varies with position, the truncation error varies systematically with position. For hydrodynamic (unmagnetized) disks an algorithm called FARGO has been developed that advects the gas along its mean orbit using a separate interpolation substep. This relaxes the constraint imposed by the Courant condition, which now depends only on the peculiar velocity of the gas, and results in a truncation error that is more nearly independent of position. This paper describes a FARGO-like algorithm suitable for evolving magnetized disks. Our method is second order accurate on a smooth flow and preserves the divergence-free constraint to machine precision. The main restriction is that the magnetic field must be discretized on a staggered mesh. We give a detailed description of an implementation of the code and demonstrate that it produces the expected results on linear and nonlinear problems. We also point out how the scheme might be generalized to make the integration of other supersonic/super-fast flows more efficient. Although our scheme reduces the variation of truncation error with position, it does not eliminate it. We show that the residual position dependence leads to characteristic radial variations in the density over long integrations., Comment: 32 pages, 18 figures, accepted for publication in The Astrophysical Journal. Contains an additional appendix providing more details for some of the test problems (to be published as an addendum in the ApJS December 2008, v179n2 issue)

I consider the nonaxisymmetric linear theory of a rotating, isothermal magnetohydrodynamic (MHD) shear flow. The analysis is performed in the shearing box, a local model of a thin disk, using a decomposition in terms of shearing waves, i.e., plane waves in a frame comoving with the shear. These waves do not have a definite frequency as in a normal mode decomposition, and numerical integration of a coupled set of amplitude equations is required to characterize their time dependence. Their generic time dependence, however, is oscillatory with slowly-varying frequency and amplitude, and one can construct accurate analytical solutions by applying the Wentzel-Kramers-Brillouin method to the full set of amplitude equations. The solutions have the following properties: 1) Their accuracy increases with wavenumber, so that most perturbations that fit within the disk are well-approximated as modes with time-dependent frequencies and amplitudes. 2) They can be broadly classed as incompressive and compressive perturbations, the former including the nonaxisymmetric extension of magnetorotationally unstable modes, and the latter being the extension of fast and slow modes to a differentially-rotating medium. 3) Wave action is conserved, implying that their energy varies with frequency. 4) Their shear stress is proportional to the slope of their frequency, so that they transport angular momentum outward (inward) when their frequency increases (decreases). The complete set of solutions constitutes a comprehensive linear test suite for numerical MHD algorithms that incorporate a background shear flow. I conclude with a brief discussion of possible astrophysical applications., Comment: 14 pages, 4 figures, accepted for Publication in the Astrophysical Journal. Primary changes: added discussion of wave-action conservation as well as a section describing the energy and angular momentum transport properties of the solutions

We quantify the effects of electron thermal conduction on the properties of hot accretion flows, under the assumption of spherical symmetry. Electron heat conduction is important for low accretion rate systems where the electron cooling time is longer than the conduction time of the plasma, such as Sgr A* in the Galactic Center. For accretion flows with density profiles similar to the Bondi solution (n[r] ~ r^[-3/2]), we show that heat conduction leads to super-virial temperatures, implying that conduction significantly modifies the dynamics of the accretion flow. We then self-consistently solve for the dynamics of spherical accretion in the presence of saturated conduction and electron heating. We find that the accretion rate onto the central object can be reduced by ~1-3 orders of magnitude relative to the canonical Bondi rate. Electron conduction may thus be an important ingredient in explaining the low radiative efficiencies and low accretion rates inferred from observations of low-luminosity galactic nuclei. The solutions presented in this paper may also describe the nonlinear saturation of the magnetothermal instability in hot accretion flows., Comment: 17 pages, 9 figures, accepted for publication in The Astrophysical Journal. Changes: additional order-of-magnitude estimates of the importance of conductive heating, discussion of pre-heating instabilities, estimate of magnetothermal instability growth rate in a hot accretion flow

We consider the formation and evolution of vortices in a hydrodynamic shearing-sheet model. The evolution is done numerically using a version of the ZEUS code. Consistent with earlier results, an injected vorticity field evolves into a set of long-lived vortices, each of which has a radial extent comparable to the local scale height. But we also find that the resulting velocity field has a positive shear stress, . This effect appears only at high resolution. The transport, which decays with time as t^-1/2, arises primarily because the vortices drive compressive motions. This result suggests a possible mechanism for angular momentum transport in low-ionization disks, with two important caveats: a mechanism must be found to inject vorticity into the disk, and the vortices must not decay rapidly due to three-dimensional instabilities., Comment: 8 pages, 10 figures (high resolution figures available in ApJ electronic edition)

Understanding the mechanism that drives accretion has been the primary challenge in accretion disk theory. Turbulence provides a natural means of dissipation and the removal of angular momentum, but firmly establishing its presence in disks proved for many years to be difficult. The realization in the 1990s that a weak magnetic field will destabilize a disk and result in a vigorous turbulent transport of angular momentum has revolutionized the field. Much of accretion disk research now focuses on understanding the implications of this mechanism for astrophysical observations. At the same time, the success of this mechanism depends upon a sufficient ionization level in the disk for the flow to be well-coupled to the magnetic field. Many disks, such as disks around young stars and disks in binary systems that are in quiescence, are too cold to be sufficiently ionized, and so efforts to establish the presence of turbulence in these disks continues. This dissertation focuses on several possible mechanisms for the turbulent transport of angular momentum in weakly-ionized accretion disks: gravitational instability, radial convection and vortices driving compressive motions. It appears that none of these mechanisms are very robust in driving accretion. A discussion is given, based on these results, as to the most promising directions to take in the search for a turbulent transport mechanism that does not require magnetic fields. Also discussed are the implications of assuming that no turbulent transport mechanism exists for weakly-ionized disks., Comment: 146 pages, 47 figures, PhD dissertation

We perform local numerical experiments to investigate the nonlinear stability of thin, radially-stratified disks. We demonstrate the presence of radial convective instability when the disk is nearly in uniform rotation, and show that the net angular momentum transport is slightly inwards, consistent with previous investigations of vertical convection. We then show that a convectively-unstable equilibrium is stabilized by differential rotation. Convective instability is determined by the Richardson number Ri = N_r^2/(q\Omega)^2, where N_r is the radial Brunt-Vaisala frequency and q\Omega is the shear rate. Classical convective instability in a nonshearing medium (Ri -> -infinity) is suppressed when Ri > -1, i.e. when the shear rate becomes greater than the growth rate. Disks with a nearly-Keplerian rotation profile and radial gradients on the order of the disk radius have Ri > -0.01 and are therefore stable to local nonaxisymmetric disturbances. One implication of our results is that the ``baroclinic'' instability recently claimed by Klahr & Bodenheimer is either global or nonexistent. We estimate that our simulations would detect any genuine growth rate > 0.0025\Omega., Comment: 31 pages, 15 figures, accepted for publication in the Astrophysical Journal

We consider the nonaxisymmetric linear theory of radially-stratified disks. We work in a shearing-sheet-like approximation, where the vertical structure of the disk is neglected, and develop equations for the evolution of a plane-wave perturbation comoving with the shear flow (a shearing wave, or ``shwave''). We calculate a complete solution set for compressive and incompressive short-wavelength perturbations in both the stratified and unstratified shearing-sheet models. We develop expressions for the late-time asymptotic evolution of an individual shwave as well as for the expectation value of the energy for an ensemble of shwaves that are initially distributed isotropically in k-space. We find that: (i) incompressive, short-wavelength perturbations in the unstratified shearing sheet exhibit transient growth and asymptotic decay, but the energy of an ensemble of such shwaves is constant with time (consistent with Afshordi, Mukhopadhyay & Narayan 2004); (ii) short-wavelength compressive shwaves grow asymptotically in the unstratified shearing sheet, as does the energy of an ensemble of such shwaves; (iii) incompressive shwaves in the stratified shearing sheet have density and azimuthal velocity perturbations \delta \Sigma,\delta v_y ~ t^{-Ri} (for |Ri| << 1), where Ri = N_x^2/(q \Omega)^2 is the Richardson number, N_x^2 is the square of the radial Brunt-Vaisala frequency and q\Omega is the effective shear rate; (iv) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Ri t^{1-4Ri} for |Ri| << 1. For Keplerian disks with modest radial gradients, |Ri| is expected to be << 1, and there will therefore be weak growth in a single shwave for Ri < 0 and near-linear growth in the energy of an ensemble of shwaves, independent of the sign of Ri., Comment: 28 pages, 1 figure, submitted to the Astrophysical Journal