The dynamics of revolving fluid on a rotating earth

Abstract

In a paper on the dynamics of revolving fluids, the late Lord Rayleigh considered the special case of fluid revolving about a fixed vertical axis, neglecting the rotation of the earth. The object of the present paper is to investigate the modifications of Rayleigh’s results which are brought about by the rotation of the earth, and by translation in a vertical plane of the axis of symmetry. Air is treated as an incompressible non-viscous fluid. Let the motion be referred to rectangular axes, x, y, z, rotating with the earth, the axis of z being vertical and the axes of x and y in the horizontal plane. At a point x, y, z , the components of velocity are u, v, w, the pressure is p , and density p . If gravity is the only impressed force, the equations of motion are D u /D t = —1/ p dp / dx + lv , (1) D v /D t = —1/ p dp / dy — lu , (2) D w /D t = —1/ p dp / dz + g , (3) where D/D t = d / dt + ud / dx + vd / dy + wd / dz and the equation of continuity is du / dx + dv / dy + dw / dz = 0, (4) where l — 2ω sin Φ, ω being the angular velocity of rotation of the earth, and Φ the latitude. The terms lv , — lu , in the first two equations represent components of the deviating force due to the earth’s rotation, and the inclusion of these terms takes complete account of the rotation of the earth.