Roll waves in mud

The stability of a viscoplastic fluid film falling down an inclined plane is explored, with the aim of determining the critical Reynolds number for the onset of roll waves. The Herschel–Bulkley constitutive law is adopted and the fluid is assumed two-dimensional and incompressible. The linear stability problem is described for an equilibrium in the form of a uniform sheet flow, when perturbed by introducing an infinitesimal stress perturbation. This flow is stable for very high Reynolds numbers because the rigid plug riding atop the fluid layer cannot be deformed and the free surface remains flat. If the flow is perturbed by allowing arbitrarily small strain rates, on the other hand, the plug is immediately replaced by a weakly yielded ‘pseudo-plug’ that can deform and reshape the free surface. This situation is modelled by lubrication theory at zero Reynolds number, and it is shown how the fluid exhibits free-surface instabilities at order-one Reynolds numbers. Simpler models based on vertical averages of the fluid equations are evaluated, and one particular model is identified that correctly predicts the onset of instability. That model is used to describe nonlinear roll waves

Inclusion of turbulence in solar modeling

The general consensus is that in order to reproduce the observed solar p-mode oscillation frequencies, turbulence should be included in solar models. However, until now there has not been any well-tested efficient method to incorporate turbulence into solar modeling. We present here two methods to include turbulence in solar modeling within the framework of the mixing length theory, using the turbulent velocity obtained from numerical simulations of the highly superadiabatic layer of the sun at three stages of its evolution. The first approach is to include the turbulent pressure alone, and the second is to include both the turbulent pressure and the turbulent kinetic energy. The latter is achieved by introducing two variables: the turbulent kinetic energy per unit mass, and the effective ratio of specific heats due to the turbulent perturbation. These are treated as additions to the standard thermodynamic coordinates (e.g. pressure and temperature). We investigate the effects of both treatments of turbulence on the structure variables, the adiabatic sound speed, the structure of the highly superadiabatic layer, and the p-mode frequencies. We find that the second method reproduces the SAL structure obtained in 3D simulations, and produces a p-mode frequency correction an order of magnitude better than the first method.Comment: 10 pages, 12 figure

Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model

Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the Ginzburg--Landau equation is a normal form that describes pattern formation due to the appearance of a single mode of instability in a wide variety of dissipative problems. In a similar vein, a certain "single-wave model" arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the single-wave model is a kind of Hamiltonian mean-field theory describing the patterns that form in phase space. The single-wave model was originally derived in the context of nonlinear plasma theory, where it describes the behavior near threshold and subsequent nonlinear evolution of unstable plasma waves. However, the single-wave model also arises in fluid mechanics, specifically shear-flow and vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter physics, and other Hamiltonian theories characterized by mean field interaction. We demonstrate, by a suitable asymptotic analysis, how the single-wave model emerges from a large class of nonlinear advection-transport theories. An essential ingredient for the reduction is that the Hamiltonian system has a continuous spectrum in the linear stability problem, arising not from an infinite spatial domain but from singular resonances along curves in phase space whereat wavespeeds match material speeds (wave-particle resonances in the plasma problem, or critical levels in fluid problems). The dynamics of the continuous spectrum is manifest as the phenomenon of Landau damping when the system is ... Such dynamical phenomena have been rediscovered in different contexts, which is unsurprising in view of the normal-form character of the single-wave model

Instability of sheared density interfaces

Of the canonical flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield) of stratified shear flow, the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the current study. First, a diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Second, the nonlinear dynamics of TCI is studied across its range of unstable horizontal wavenumbers and bulk Richardson numbers using numerical simulation. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin–Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Third, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations

Instabilities in a staircase stratified shear flow

We study stratified shear flow instability where the density profile takes the form of a staircase of interfaces separating uniform layers. Internal gravity waves riding on density interfaces can resonantly interact due to a background shear flow, resulting in the Taylor-Caulfield instability. The many steps of the density profile permit a multitude of interactions between different interfaces, and a rich variety of Taylor-Caulfield instabilities. We analyse the linear instability of a staircase with piecewise-constant density profile embedded in a background linear shear flow, locating all the unstable modes and identifying the strongest. The interaction between nearest-neighbour interfaces leads to the most unstable modes. The nonlinear dynamics of the instabilities are explored in the long-wavelength, weakly stratified limit (the defect approximation). Unstable modes on adjacent interfaces saturate by rolling up the intervening layer into a distinctive billow. These nonlinear structures coexist when stacked vertically and are bordered by the sharp density gradients that are the remnants of the steps of the original staircase. Horizontal averages remain layer-like

Solar-like oscillations of semiregular variables

Oscillations of the Sun and solar-like stars are believed to be excited stochastically by convection near the stellar surface. Theoretical modeling predicts that the resulting amplitude increases rapidly with the luminosity of the star. Thus one might expect oscillations of substantial amplitudes in red giants with high luminosities and vigorous convection. Here we present evidence that such oscillations may in fact have been detected in the so-called semiregular variables, extensive observations of which have been made by amateur astronomers in the American Association for Variable Star Observers (AAVSO). This may offer a new opportunity for studying the physical processes that give rise to the oscillations, possibly leading to further information about the properties of convection in these stars.Comment: Astrophys. J. Lett., in the press. Processed with aastex and emulateap

Does the Sun shrink with increasing magnetic activity?

It has been demonstrated that frequencies of f-modes can be used to estimate the solar radius to a good accuracy. These frequencies have been used to study temporal variations in the solar radius with conflicting results. The variation in f-mode frequencies is more complicated than what is assumed in these studies. If a careful analysis is performed then it turns out that there is no evidence for any variation in the solar radius.Comment: To appear in Astrophys.