Keith Julien

The magnetorotational instability is widely believed to be responsible for outward angular momentum transport in astrophysical accretion disks. The efficiency of this transport depends on the amplitude of this instability in the saturated state. We employ an asymptotic expansion based on an explicit, astrophysically motivated timescale separation between the orbital period, Alfvén crossing time and viscous or resistive dissipation timescales, originally proposed by Knobloch and Julien, Phys. Fluids 17, 094106 (2005), to formulate a semi-analytical description of the saturated state in an incompressible disk. In our approach a Keplerian shear flow is maintained by the central mass but the instability saturates via the generation of a mean vertical magnetic field. The theory assumes that the time-averaged angular momentum flux and the radial magnetic flux are constant and determines both self-consistently. The results predict that, depending on parameters, steady saturation may be supercritical or subcritical, and in the latter case that the upper (lower) solution branch is always stable (unstable). The angular momentum flux is always outward, consistent with the presence of accretion, and for fixed wavenumber peaks in the subcritical regime. The limit of infinite Reynolds number at large but finite magnetic Reynolds number is also discussed.

A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [ Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower- and upper-branch states are characterized in detail.

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically-relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e., incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilise the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously-growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using equations obtained from a large Reynolds-number asymptotic reduction of the Navier-Stokes equations. The reduced equations employ a decomposition into streamwise-averaged (mean) and streamwise-varying (fluctuation) components and are characterized by an effective order one Reynolds number in the mean equations along with a formally higher-order diffusive regularization of the fluctuation equations. A robust numerical algorithm for computing exact coherent states is introduced. Numerical continuation of the lower branch states to lower Reynolds numbers reveals the presence of a saddle-node; the saddle-node allows access to upper branch states that, like the lower branch states, appear to be self-consistently described by the reduced equations. Both lower and upper branch states are characterized in detail.

A reduced description of shear flows consistent with the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow [J. Wang et al., Phys. Rev. Lett. 98, 204501 (2007)] is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found for Waleffe flow [F. Waleffe, Phys. Fluids 9, 883--900 (1997)]. The lower branch solution is characterized by fluctuations that vary slowly along the critical layer while the upper branch solutions display a bimodal structure and are more strongly focused on the critical layer. The reduced model provides a rational framework for investigations of subcritical spatiotemporal patterns in parallel shear flows.

Rapidly rotating Rayleigh–Bénard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet larger Rayleigh numbers by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence. The transitions are quantified by examining the properties of the horizontally and temporally averaged temperature and thermal dissipation rate. The maximum of the thermal dissipation rate is used to define the width of the thermal boundary layer. In contrast to the non-rotating Rayleigh–Bénard convection, the temperature drop across this layer decreases monotonically with increasing Rayleigh number and does not saturate. The breakdown of the convective Taylor column regime is attributed to the onset of convective instability of the thermal boundary layer and confirmed using the explicit linear stability analysis. Horizontal spectra of the vorticity, vertical velocity and temperature fluctuations are computed and their evolution with time is elucidated. A large-scale barotropic mode evolves from random initial conditions on an extremely long time scale and leads to continued evolution of the nominally saturated Nusselt number and its variance over very long times. The results are used to provide insights into the dynamics of rapidly rotating convection outside the asymptotic regime described by the reduced equations.

We present numerical calculations of the scattering of acoustic-gravity waves by a model sunspot consisting of an inhomogeneous, vertically magnetized region embedded in a polytropically stratiÐed layer. The calculations show the transformation of incoming f-mode and p-mode power into downwardpropagating, slow magnetoacoustic waves resulting in substantial absorption of incident f-modes and Ðrst-order p-modes and indicating that the slow-mode transformation model is a viable explanation for the observed absorption. In addition to absorption, we Ðnd large scattering phase shifts for the f-modes. Phase shifts are small for all p-modes. Coupling between modes of di †erent radial order is generally small. The remaining di †erences between our results and the observations may constitute evidence that, if the mode transformation picture is correct, real sunspots must deviate signiÐcantly from the simple unidirectional Ðeld geometry we use here. In particular, we suggest that a sunspot consisting of a bundle of Ðbrils, in each of which the magnetic Ñux density increases rapidly with depth, would reproduce more closely the measured strong absorption and scattering of p-modes.