| Abstract: | We consider the Oberbeck-Boussinesq system without dissipation (ideal convection) in a horizontal layer and in a “barrel” with flat bottom and flat cover. It is shown that the velocity circulation along a fluid contour consisting of two fluid curves on the bottom and on the cover connected by two isothermic fluid curves can be calculated explicitly and is a linear function of time. The serre result stating that the azimuthal component of vorticity in rotationally symmetric ideal fluid flows between coaxial cylinders increases linearly is generalized to the case of stratified fluids. It is proved that all plane and axially symmetric isothermic flows in a layer or in a barrel are unstable with respect to nonisothermic perturbations and in the case of a homogenous fluid all axially symmetric flows between coaxial cylinders are unstable in the sense of Lyapunov with respect to perturbations of the azimuthal component of the velocity in any metric including the maximum of magnitude of vorticity. Translated fromMaternaticheskie Zametki, Vol. 68, No. 4, pp. 627–636, October, 2000. |
