Krzysztof A. Mizerski

Generalization of the Rotne-Prager-Yamakawa mobility and shear disturbance tensors

The Rotne–Prager–Yamakawa approximation is one of the most commonly used methods of including hydrodynamic interactions in modelling of colloidal suspensions and polymer solutions. The two main merits of this approximation are that it includes all long-range terms (i.e. decaying as

The magnetoelliptic instability of rotating systems

{R}^{- 3}

On the connection between the magneto-elliptic and magneto-rotational instabilities

or slower in interparticle distances) and that the diffusion matrix is positive definite, which is essential for Brownian dynamics modelling. Here, we extend the Rotne–Prager–Yamakawa approach to include both translational and rotational degrees of freedom, and derive the regularizing corrections to account for overlapping particles. Additionally, we show how the Rotne–Prager–Yamakawa approximation can be generalized for other geometries and boundary conditions.

Large-scale convective dynamos in a stratified rotating plane layer

We address the question of stability of the Euler flow with elliptical streamlines in a rotating frame, interacting with uniform external magnetic field perpendicular to the plane of the flow. Our motivation for this study is of astrophysical nature, since many astrophysical objects, such as stars, planets and accretion discs, are tidally deformed through gravitational interaction with other bodies. Therefore, the ellipticity of the flow models the tidal deformations in the simplest way. The joint effect of the magnetic field and the Coriolis force is studied here numerically and analytically in the limit of small elliptical (tidal) deformations (ζ � 1), using the analytical technique developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). We find that the effect of background rotation and external magnetic field is quite complex. Both factors are responsible for new destabilizing resonances as the vortex departs from axial symmetry (ζ � 1); however, just like in the non-rotating case, there are three principal resonances causing instability in the leading order. The presence of the magnetic field is very likely to destabilize the system with respect to perturbations propagating in the direction of the magnetic field if the basic vorticity and the background rotation have opposite signs (i.e. for anticyclonic background rotation). We present the dependence of the growth rates of the modes on various parameters describing the system, such as the strength of the magnetic field (h), the inverse of the Rossby number (Rv), the ellipticity of the basic flow (� ) and the direction of propagation of modes (ϑ). Our analytical predictions agree well with the numerical calculations.

The effect of stratification and compressibility on anelastic convection in a rotating plane layer

It has recently been suggested that the magneto-rotational instability (MRI) is a limiting case of the magneto-elliptic instability (MEI). This limit is obtained for horizontal modes in the presence of rotation and an external vertical magnetic field, when the aspect ratio of the elliptic streamlines tends to infinite. In this paper we unveil the link between these previously unconnected mechanisms, explaining both the MEI and the MRI as different manifestations of the same magneto-elliptic-rotational instability (MERI). The growth rates are found and the influence of the magnetic and rotational effects is explained, in particular the effect of the magnetic field on the range of negative Rossby numbers at which the horizontal instability is excited. Furthermore, we show how the horizontal rotational MEI in the rotating shear flow limit is linked to the MRI by the use of the local shearing box model, typically used in the study of accretion discs. In such a limit the growth rates of the two instability types coincide for any power-law-type background angular velocity radial profile with negative exponent corresponding to the value of the Rossby number of the rotating shear flow. The MRI requirement for instability is that the background angular velocity profile is a decreasing function of the distance from the centre of the disc, which corresponds to the horizontal rotational MEI requirement of negative Rossby numbers. Finally a physical interpretation of the horizontal instability, based on a balance between the strain, the Lorentz force and the Coriolis force, is given.

The Rotne-Prager-Yamakawa approximation for periodic systems in a shear flow.

We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ ≫ 1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.

Dynamo generation of a magnetic field by decaying Lehnert waves in a highly conducting plasma

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ−1/12 ≪ |θ| ≪ 1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.

The short-wavelength instability of magnetically buoyant layer

Rotne-Prager-Yamakawa approximation is a commonly used approach to model hydrodynamic interactions between particles suspended in fluid. It takes into account all the long-range contributions to the hydrodynamic tensors, with the corrections decaying at least as fast as the inverse fourth power of the interparticle distances, and results in a positive definite mobility matrix, which is fundamental in Brownian dynamics simulations. In this communication, we show how to construct the Rotne-Prager-Yamakawa approximation for the bulk system under shear flow, which is modeled using the Lees-Edwards boundary conditions.

Compressible Ekman–Hartmann boundary layers

Abstract Random waves in a uniformly rotating plasma in the presence of a locally uniform seed magnetic field and subject to weak kinematic viscosity and resistivity are considered. These “Lehnert” waves may have either positive or negative helicity, and it is supposed that waves of a single sign of helicity are preferentially excited by a symmetry-breaking mechanism. A mean electromotive force proportional to is derived, demonstrating the conflicting effects of the two diffusive processes. Attention is then focussed on the situation , relevant to conditions in the universe before and during galaxy formation. An -effect, axisymmetric about the rotation vector, is derived, decaying on a time-scale proportional to ; this amplifies a large-scale seed magnetic field to a level independent of , this field being subsequently steady and having the character of a “fossil field”. Subsequent evolution of this fossil field is briefly discussed.

Turbulence induced by magnetic fields

We revisit the problem introduced by Gilman (1970) and Acheson (1979) of linear stability of a plane layer of compressible fluid permeated by a horizontal magnetic field of magnitude decreasing with height with respect to short-wavelength two-dimensional perturbations varying in the directions perpendicular to the applied field. We show, that in the limit of large horizontal wave numbers the perturbations become strongly localised in the vertical direction. The motiavtion for this study is of astrophysical nature and comes from the common belief, that the magnetic buoyancy effects produce short-wavelength instabilities in the solar tachocline. We analyse the solar tachocline parameter regime to speculate about the strength of the magnetic field at the base of the solar convective zone and the time scales of the field variations induced by the magnetic buoyancy instability on the Sun.