. Compendium of meteorology. Meteorology. 456 DYNAMICS OF THE ATMOSPHERE necess&,ry to insure the stability of the circular vortex in the present case. Equations (9) and (11) now reduce to du dt = fv v' • Vu', - (p fvby- ^ ff^w'S^ (Dv'-'Vx'gSz - (p fv'-vu'Sy. (12) (13) Let it now be assumed that the path of integration consists of the sides of a "rectangle" bounded below by the earth's surface and with the top at about tropopause height (Fig. 1). The direction of integra-. PiG. 1.—Idealized meridional circulation. tion is indicated by arrows. With subscripts from 1 to 4 denoting
. Compendium of meteorology. Meteorology. 456 DYNAMICS OF THE ATMOSPHERE necess&,ry to insure the stability of the circular vortex in the present case. Equations (9) and (11) now reduce to du dt = fv v' • Vu', - (p fvby- ^ ff^w'S^ (Dv'-'Vx'gSz - (p fv'-vu'Sy. (12) (13) Let it now be assumed that the path of integration consists of the sides of a "rectangle" bounded below by the earth's surface and with the top at about tropopause height (Fig. 1). The direction of integra-. PiG. 1.—Idealized meridional circulation. tion is indicated by arrows. With subscripts from 1 to 4 denoting average values along the sides of the rec- tangle correspondingly labelled, (13) may be written fih vi)B + g— {w2 - Wi)H dz (14) = g{v'-Vn'2 - v'-Wi)H +f{v'-vu^ - v'-vu[)B. Here, B and H are the lengths of the sides of the rec- tangle. Suppose further that the rectangular-shaped boundary is a streamline in a corresponding simple cellular meridional motion, in which v and w are derived from a stream function , . Try . ttZ i/a -^ sm -=^ sm — . 13 rt (15) This implies that for the present the atmosphere is treated as incompressible. It may be anticipated, how- ever, that the results to be obtained will also roughly apply to cellular motions whose kinematics are essen- tially the same as in this most simple cellular motion. By deriving v and w from (15) and forming the averages «^i, Wj , ©2, and w^, one obtains {Wi - Wi)B = (v, - vi)H. (16) If one does not think of the velocities in this formula as averages along the sides of the rectangular boundary, but rather as averages over the whole region of the velocities in the ascending and descending motions, and in the north and south motions, the formula simply states the well-known continuity principle that the ratio between the magnitude of the vertical and hori- zontal velocities is proportional to the ratio between the vertical and horizontal scale of the motion. By a suitable choice of the integration curve in the circul
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