THE EQUATIONS GOVERNING THE MOTION OF A PERFECT-GAS ATMOSPHERE

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Abstract
The equations of motion, of continuity, and of heat-transfer by bodily motion are derived for a frictionless perfect-gas atmosphere on a smooth rotating spherical Earth of radius a. The coordinates of a point P are the vertical distance (Z) from P to the Earth's surface, the great circle are (S) from a fixed point O on the Earth's surface to the vertical through P, and the angle (θ) between the great circle through O and P and the meridian of O. The equations are obtained correct to terms of order 1/a and so can be applied to wind-systems extending to about 1, 000 km. from O. In the case when the motion is steady (i.e. in the operator d/dt the term in δ/δt contributes nothing) and always parallel to the Earth's surface, dimensionless variables are introduced which are the ratios of the velocity, pressure, temperature, etc., at any point to their values at a point on the Earth's surface situated at the ‘outer boundary’ of the wind-system. By considering the magnitudes of the constants in the resulting equations, rotary motions of the atmosphere are divided into ‘small-scale’, with radii up to 200 km., and ‘large-scale’ with radii up to 1, 200 km. The equations are solved by approximation for some small-scale motions (gradient-wind, revolving fluid, etc.) in which the heat-content of unit mass of air is constant, and the vertical distribution of temperature is adiabatic. For small-scale motions of convergence or divergence, the solution breaks down near the axis of rotation. For large-scale motions no vortical solutions are found, but there is a solution in which the wind blows perpendicularly to the meridian through O. It is concluded that a possible way of escape from these difficulties is to abandon the assumption that the motion is always parallel to the Earth's surface.

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