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

It has been recently 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 links to the MRI by the use of the local shearing box model, typically used in the study of accretion discs. In such limit the growth rates of the two instability types coincide for any power-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 disk 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.Comment: 15 pages, 3 figures. Accepted for publication in the Journal of Fluid Mechanic

Anisotropic turbulent viscosity and large-scale motive force in thermally driven turbulence at low Prandtl number

The fully developed turbulent Boussinesq convection is known to form large-scale rolls, often termed the 'large-scale circulation' (LSC). It is an interesting question how such a large-scale flow is created, in particular in systems when the energy input occurs at small scales, when inverse cascade is required in order to transfer energy into the large-scale modes. Here, the small-scale driving is introduced through stochastic, randomly distributed heat source (say radiational). The mean flow equations are derived by means of simplified renormalization group technique, which can be termed 'weakly nonlinear renormalization procedure' based on consideration of only the leading order terms at each step of the recursion procedure, as full renormalization in the studied anisotropic case turns out unattainable. The effective, anisotropic viscosity is obtained and it is shown, that the inverse energy cascade occurs via an effective 'motive force' which takes the form of transient negative, vertical diffusion

Foundations of Convection with Density Stratification

The phenomenon of thermal and compositional (chemical) convection is very common in nature and therefore of great importance from the point of view of understanding of many fundamental aspects of the environment and universe. A number of books have been written on the topic, such as e.g. the seminal work of Chandrasekhar (1961) on Hydrodynamic and Hydromagnetic Stability, a large portion of which is devoted to the convective instability near its onset or the outstanding book of Getling (1998) where systematization of the knowledge on convection has been continued with a thorough description of the weakly nonlinear stages. Most of the works, however, considered weakly stratified, that is the so-called Boussinesq convection. It is the aim of this book to continue the process of systematization. It seems important to put the current knowledge on weakly and strongly stratified convection in order and provide a comprehensive description of the marginal, weakly nonlinear and fully developed stages of convective flow in both cases. To that end the book provides a short compendium of knowledge on the linear and weakly nonlinear limits of the Boussinesq convection, as a useful reference for a reader and than proceeds with a review of the theory on fully developed, weakly stratified convection. The entire third chapter is devoted to a detailed derivation and a study of the three aforementioned stages of stratified (anelastic) convection. The description of stratified convection requires extreme care, since many aspects have to be considered simultaneously for full consistency. Detailed and systematic explanations are therefore provided. This book is meant as a textbook for courses on hydrodynamics and convective flows, for the use of lecturers and students, however, it may also be of use for the entire scientific community as a practical reference.Comment: Textboo

Generalization of the

Rotne-Prager-Yamakawa mobility and shear disturbance tensor