The rotation rates of six weakly magnetic neutron stars accreting in low-mass X-ray binaries have most likely been measured by Type I X-ray burst observations with the Rossi X-Ray Timing Explorer Proportional Counter Array. The phenomenology of the nearly coherent oscillations detected during the few seconds of thermonuclear burning is most simply understood as rotational modulation of brightness asymmetries on the neutron star surface. We show that, as suggested by Strohmayer and colleagues, the frequency changes of 1-2 Hz observed during bursts are consistent with angular momentum conservation as the burning shell hydrostatically expands and contracts during the burst. We calculate how vertical heat propagation through the radiative outer layers of the atmosphere and convection affect the coherence of the oscillation. We show that the evolution and coherence of the rotational profile depends strongly on whether the burning layers are composed of pure helium or mixed hydrogen/helium. Our results help explain the absence (presence) of oscillations from hydrogen-burning (helium-rich) bursts that was found by Muno and collaborators. We also investigate angular momentum transport within the burning layers and address the recoupling of the burning layers with the star. We show that the Kelvin-Helmholtz instability is quenched by the strong stratification, and that mixing between the burning fuel and underlying ashes by the baroclinic instability does not occur. However, the baroclinic instability may have time to operate within the differentially rotating burning layer.

http://iopscience.iop.org/article/10.1086/317191/pdf

Rotational Evolution during Type I X-Ray Bursts

We analyse the effects of small, circularly symmetric topography on the slow flow of an inviscid, incompressible, diffusionless, horizontally uniform, baroclinic current and show that the vertical influence depends primarily on three parameters: a stratification measure S (the square of the ratio of buoyancy frequency times height scale to Coriolis parameter times length scale), a topographic parameter β (ratio of scaled topographic height multiplied by scaled bottom current to Rossby number e) and the scaled upstream shear u′0(z) (the dimensional upstream shear divided by the ratio of the r.m.s. upstream flow speed to height scale).Investigating a linear stratification model we find that the topographic effect is depth independent if S [lsim ] e and a Taylor column, as indicated by the appearance of closed streamlines above the bump, exists when β > 2. Moderate stratification (S ∼ 1) causes the flow to be fully three-dimensional and the Taylor column to be a conical vortex whose height depends on β S and u′0). The results are compared with Davies's (1971, 1972) experiments.Our results tend to support the Taylor column theory of Jupiter's Great Red Spot but effects due to variations in the Coriolis parameter with latitude have been (unjustifiably) ne glected. Using typical values for the earths oceans we find that Taylor columns of significant height could be found there. Some pertinent observations from the ocean are discussed.

On the stratified Taylor column

The unsteady boundary layer of a rotating, stratified, viscous, and diffusive flow along an insulating slope is investigated using theory, numerical simulation, and laboratory experiment. Previous work in this field has focused either on steady flow, or flow over a conducting boundary, both of which yield Ekman-type solutions. After the onset of a circulation directed along constant-depth contours, Ekman-type flux up or down the slope is opposed by buoyancy forces. In the unsteady, insulating case, it is found that the cross-slope transport decreases in time as (t/τ)−½ where
 \[
 \tau = \frac{1}{S^2f\cos\alpha}\left(\frac{1/\sigma + S}{1+S}\right),
 \]
 may be called the ‘shut-down’ time. Here S = (N sin α/f cos α)2, f is the Coriolis frequency, α is the slope angle, N is the buoyancy frequency, and σ is the Prandtl number. Subsequently the along-slope flow, , approximately obeys a simple diffusion equation
 \[
 \frac{\partial\hat{v}}{\partial t} = \nu\left(\frac{1/\sigma + S}{1+S}\right)\frac{\partial^2\hat{v}}{\partial\hat{z}^2},
 \]
 where t is time, ν is the kinematic viscosity, and , is nevertheless close to thermal wind balance, and the much-reduced cross-slope transport is balanced by stress on the boundary. For a slope of infinite extent the steady asymptotic state is the diffusively driven ‘boundary-mixing’ circulation of Thorpe (1987). By inhibiting the cross-slope transport, buoyancy virtually eliminated the boundary stress and hence the ‘ fast’ spin-up of classical theory in laboratory experiments with a bowl-shaped container of stratified, rotating fluid.

Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up

We consider the manner in which a container filled with viscous fluid adjusts to changes in its rotation rate. We begin with homogeneous flows involving small departures in rotation rate from an initial state of solid-body rotation in an axisymmetric container. This is followed by a summary of other more recent developments, including weakly and fully nonlinear calculations and comparison with experiment and the question of spin-down. The question of "spin-over" is addressed, followed by a brief synopsis of free-surface effects, and a discussion of nonaxisymmetric spin-up. The second part of the review focuses on the effects of stratification on the spin-up process. Linearized (low Rossby number) spin-up within a cylindrical container is described. Thereafter, both experimental and nonlinear computational results are described and compared. The final section focuses on stratified spin-up and spin-down in conical geometries, and a number of comparisons between theory and experiment are given.

/pdf/spin-up-of-homogeneous-and-stratified-fluids-3cxxl0di71.pdf

Spin-Up of Homogeneous and Stratified Fluids

Perturbation series are developed and mathematically justified, using a straightforward perturbation formalism (that is more widely applicable than those given in standard textbooks), for the case of the two-dimensional inviscid Orr-Sommerfeld-like eigenvalue problem describing quasi-geostrophic wave instabilities of parallel flows in rotating stratified fluids. The results are first used to examine the instability properties of the perturbed Eady problem, in which the zonal velocity profile has the form u = z + μ u 1 ( y , z ) where, formally, μ [Lt ] 1. The connexion between baroclinic instability theories with and without short wave cutoffs is clarified. In particular, it is established rigorously that there is instability at short wavelengths in all cases for which such instability would be expected from the ‘critical layer’ argument of Bretherton. (Therefore the apparently conflicting results obtained earlier by Pedlosky are in error.) For the class of profiles of form u = z + μ u 1 ( y ) it is then shown from an examination of the O (μ) eigenfunction correction why, under certain conditions, growing baroclinic waves will always produce a counter-gradient horizontal eddy flux of zonal momentum tending to reinforce the horizontal shear of such profiles. Finally, by computing a sufficient number of the higher corrections, this first-order result is shown to remain true, and its relationship to the actual rate of change of the mean flow is also displayed, for a particular jet-like form of profile with finite horizontal shear. The latter detailed results may help to explain at least one interesting feature of the mean flow found in a recent numerical solution for the wave regime in a heated rotating annulus.

On the non-separable baroclinic parallel flow instability problem

Time-dependent motion of a rotating stratified fluid is analyzed within the quasigeostrophic approximation. A few examples of mechanically driven flow are analyzed. It is found that the motion is characterized by the ratio B of the stability frequency and the Coriolis parameter. Thus the ratio of the horizontal and vertical characteristic scale is in general O(B). In particular the decay process caused by a horizontal boundary will penetrate a distance B−1L into the fluid, L denoting the horizontal scale of the motion.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112069001662

GÖUsta Walin

Papers

Some aspects of time-dependent motion of a stratified rotating fluid