Elements

The Equations of Motion in Three Dimensions.

The mo tion of an element of mass situated in latitude 4, longitude X, at a distance r from the centre of the earth, may be represented by the following equations: Thus the rate of decrease of temperature of air ascending adia batically is proportional to the ratio of the temperatures of the moving air and of the environment at the same level. In all practical problems, this ratio may be taken as unity. Then drag = — C/ioo metres, or almost exactly I° C per dz ioo metres.

Stability of Dry Air.

If a small mass of air originally at level z be displaced vertically to height z+dz, its temperature will change to T-EdT' , where dT' = dz. In its displaced posi c tion it will be lighter or heavier than itsenvironment according d dT . dT ag — is less or greater than according as is > or < dz dz Hence the air is stable or unstable according as the lapse rate s less or greater than the adiabatic lapse rate of I° C per Ioo metres. It can be readily verified that the condition for stability is that the potential temperature shall increase with height.

where ca is the angular velocity of the earth, g is the acceleration due to gravity, and X, Y, Z, are the components of the external forces resolved along axes drawn to east, to north, and vertically, respectively, but not rotating with the earth. In practice it is frequently more convenient to use cartesian co-ordinates referred to axes drawn as above, but rotating with the earth. The appro priate equations are: This equation simply states that when there are no forces the angular momentum about the axis of the earth is constant. If therefore we know the east to west component of velocity of a moving mass when in one latitude, we can compute the east to west component of velocity of that mass when it has moved to any other latitude. Care is needed in the use of this equation, since it represents only one aspect of the motion of the mass in question. By its use some highly questionable results have been derived in meteorology.

When there is no vertical velocity, there are only two equations The effect of the rotation of the earth is therefore allowed for by including in the statement of accelerations a term 20) sin4) X ve locity, and it is readily seen that this acceleration is at right angles to the direction of motion. This is known as the "deviating force due to the earth's rotation." The complete set of equations (4), (5), and (6) includes in addition an acceleration along the east-west line, which is positive to east when the motion is down ward, and a vertical component of acceleration proportional to the east-west component of velocity. Equations (8) and (9) hold for

any rectangular axes in the horizontal plane. The forces X, Y, Z include all forces other than gravitation, such as viscous or fric tional forces, or the components of the gradient of pressure. The part of these forces due to the pressure distribution may be written where the radius of curvature is r, the accelerations relative to 172 V dV the earth are — normal to the path, and along the tangent to the path. Observational evidence points to the second of these terms being small by comparison with the other (Durward, Pro fessional Note No. 24, M.O. London, and Shaw and Lempfert, Life History of Surface Air Currents) , and it is usually left out of further consideration. Equations ( to) and (I I) then state that the centrifugal acceleration balances the sum of (a) the pres sure gradient, which acts at right angles to the isobar, and is di rected towards low pressure; (b) the deviating acceleration 2co Vsin0 at right angles to the path; and (c) the viscous or turbulent resistance.

In the first consideration of the problem, we neglect the effect of friction and turbulence which are considered later. The cen trifugal acceleration which acts at right angles to the path must then balance (a) and (b) above. Since however the centrifugal acceleration is along the same line as (b), it follows that (a), which balances the two, must act along the same line, and the motion under balanced forces must be along the isobars. The equation of motion then becomes The upper sign is used if the concave side of the isobar has low pressure, and the lower sign in the opposite case. These two cases correspond to cyclonic and anticyclonic curvatures of isobars.

It is convenient to represent the pressure gradient symbolically by 2copGsin4, where p is the density of the air. Equation (12) then becomes The solution of equation (t3) is called the gradient wind, and equation (13) is generally referred to as the gradient wind equa tion. The gradient wind is the wind which, blowing around the isobars, calls into play a centrifugal force and a deviating force exactly sufficient to balance the pressure gradient. When the isobars are not much curved, the term is small by comparison with 2c...)Vsin4, and may therefore be neglected, and V =G. This particular solution is known as the geostrophic wind. Equation (t3) may also be written in which the upper sign is taken when the curvature of the isobars is anticyclonic. the lower sign when the curvature is cyclonic. It is readily seen from this that the geostrophic wind is an underestimate of the gradient wind in an anticyclone, and an overestimate in a cyclone.