Philip S. Marcus

Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment

We present a numerical method that allows us to solve the Navicr-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties ; the bestmeasured speeds have fractional uncertainties of less than 0.2 o/o.

Simulation of Taylor-Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave

We use a numerical method that was described in Part 1 (Marcus 1984 a ) to solve the time-dependent Navier-Stokes equation and boundary conditions that govern Taylor-Couette flow. We compute several stable axisymmetric Taylor-vortex equilibria and several stable non-axisymmetric wavy-vortex flows that correspond to one travelling wave. For each flow we compute the energy, angular momentum, torque, wave speed, energy dissipation rate, enstrophy, and energy and enstrophy spectra. We also plot several 2-dimensional projections of the velocity field. Using the results of the numerical calculations, we conjecture that the travelling waves are a secondary instability caused by the strong radial motion in the outflow boundaries of the Taylor vortices and are not shear instabilities associated with inflection points of the azimuthal flow. We demonstrate numerically that, at the critical Reynolds number where Taylor-vortex flow becomes unstable to wavy-vortex flow, the speed of the travelling wave is equal to the azimuthal angular velocity of the fluid at the centre of the Taylor vortices. For Reynolds numbers larger than the critical value, the travelling waves have their maximum amplitude at the comoving surface, where the comoving surface is defined to be the surface of fluid that has the same azimuthal velocity as the velocity of the travelling wave. We propose a model that explains the numerically discovered fact that both Taylor-vortex flow and the one-travelling-wave flow have exponential energy spectra such that In [E(k)] ∝ k 1 , where k is the Fourier harmonic number in the axial direction.

THREE-DIMENSIONAL VORTICES IN STRATIFIED PROTOPLANETARY DISKS

We present the results of high-resolution, three-dimensional hydrodynamic simulations of the dynamics and formation of coherent, long-lived vortices in stably stratified protoplanetary disks. Tall, columnar vortices that extend vertically through many scale heights in the disk are unstable to small perturbations; such vortices cannot maintain vertical alignment over more than a few scale heights and are ripped apart by the Keplerian shear. Short, finite-height vortices that extend only 1 scale height above and below the midplane are also unstable, but for a different reason: we have isolated an antisymmetric (with respect to the midplane) eigenmode that grows with an e-folding time of only a few orbital periods; the nonlinear evolution of this instability leads to the destruction of the vortex. Serendipitously, we observe the formation of three-dimensional vortices that are centered not in the midplane, but at 1‐3 scale heights above and below. Breaking internal gravity waves create vorticity; anticyclonic regions of vorticity roll up and coalesce into new vortices, whereas cyclonic regions shear into thin azimuthal bands. Unlike the midplane-centered vortices that were placed ad hoc in the disk and turned out to be linearly unstable,theoff-midplanevorticesformnaturallyoutofperturbationsinthediskandarestableandrobustformany hundreds of orbits. Subject headings: accretion, accretion disks — hydrodynamics — planetary systems: protoplanetary disks Online material: mpeg animations

Simulation of flow between concentric rotating spheres. Part 1. Steady states

Axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed numerically as an initial-value problem. The time-independent spherical Couette flows with zero, one and two Taylor vortices computed in our simulations are found to be reflection-symmetric about the equator despite the fact that our pseudospectral numerical method did not impose these properties. Our solutions are examined for self-consistency, compared with other numerical calculations, and tested against laboratory experiments. At present, the most precise laboratory measurements are those that measure Taylor-vortex size as a function of Reynolds number, and our agreement with these results is within a few per cent. We analyse our flows by plotting their meridional circulations, azimuthal angular velocities, and energy spectra. At Reynolds numbers just less than the critical value for the onset of Taylor vortices, we find that pinches develop in the flow in which the meridional velocity redistributes the angular momentum. Taylor vortices are easily differentiated from pinches because the fluid in a Taylor vortex is isolated from the rest of the fluid by a streamline that extends from the inner to the outer sphere, whereas the fluid in a pinch mixes with the rest of the flow.

Wave speeds in wavy Taylor-vortex flow

The speed of travelling azimuthal waves on Taylor vortices in a circular Couette system (with the inner cylinder rotating and the outer cylinder at rest) has been determined in laboratory experiments conducted as a function of Reynolds number R, radius ratio of the cylinders η, average axial wavelength

Depth of a strong Jovian jet from a planetary-scale disturbance driven by storms

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Vortex dynamics in a shearing zonal flow

, number of waves m 1 and the aspect ratio Γ (the ratio of the fluid height to the gap between the cylinders). Wave speeds have also been determined numerically for axially periodic flows in infinite-length cylinders by solving the Navier-Stokes equation with a pseudospectral technique where each Taylor-vortex pair is represented with 32 axial modes, 32 azimuthal modes (in an azimuthal angle of 2π/m 1 ) and 33 radial modes. Above the onset of wavy-vortex flow the wave speed for a given η decreases with increasing R until it reaches a plateau that persists for some range in R. In the radius-ratio range examined in our experiments we find that the wave speed in the plateau region increases monotonically from 0.14Ω at η = 0.630 to 0.45Ω at η = 0.950 (where the wave speed is expressed in terms of the rotation frequency Ω of the inner cylinder). There is a much weaker dependence of the wave speed on

Simulation of flow between concentric rotating spheres. Part 2. Transitions

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On Green's functions for small disturbances of plane Couette flow

, m 1 and Γ. For three sets of parameter values ( R ,

Effects of truncation in modal representations of thermal convection

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