Asymptotic methods in magnetoconvection
The author examines the magnetohydrodynamic convection of an electrically and heat conducting non-viscous fluid in a uniform vertical magnetic field. The stability of a thin horizontal layer of a Boussinesq fluid is investigated in Cartesian coordinates with \(z\)-coordinate directed along the magnetic field. The effects of Ampère force and non-uniform temperature gradient on the magnetoconvection are studied analytically using Galerkin expansion. The linear stability analysis is applied, and only the marginal state is considered using a single term of the Galerkin expansion. The author proves that if the magnetic viscosity prevails over the thermal conductivity, the overstability cannot occur and the principle of exchange of stabilities is valid. The method of matched asymptotics is employed to investigate the effect of the Hartmann layers on the magnetoconvection. The critical Rayleigh number, close to that obtained by experiments, is found analytically by a regular perturbation technique. The author demonstrates that the Chandrasekhar's power law is correct only for free-free isothermal boundary conditions. Under other boundary conditions, the critical Rayleigh number obeys the power law with a different constant of proportionality. The critical Rayleigh number depends critically on the heating and does not depend on the nature of boundaries. It is shown that a non-uniform temperature gradient and magnetic field suppress or augment the magnetohydrodynamical convection.
scientific article; zbMATH DE number 1071709