389.
NOTE ON THE FORMULA FOB THE GRADIENT WIND.
\_Advisory Committee for Aeronautics. Reports and Memoranda. No. 147. January, 1915.]
AN instantaneous derivation of the formula for the " gradient wind " has been given by Gold*. " For the steady horizontal motion of air along a path whose radius of curvature is r, we may write directly the equation
(<or sin \ ± vf __ I dp (tor sin X)2 r p dr r
expressing the fact that the part of the centrifugal force arising from the motion of the wind is balanced by the effective gradient of pressure.
"In the equation p is atmospheric pressure, p density, v velocity of moving air, \ is latitude, and o> is the angular velocity of the earth about its axis." Gold deduces interesting consequences relating to the motion and pressure of air in anti-cyclonic regionsf.
But the equation itself is hardly obvious without further explanations, unless we limit it to the case where sinX=l (at the pole) and where the relative motion of the air takes place about the same centre as the earth's rotation. I have thought that it may be worth while to take the problem avowedly in two dimensions, but without further restriction upon the character of the relative steady motion.
The axis of rotation is chosen as axis of z. The axes of x and y being supposed to rotate in their own plane with angular velocity co, we denote by n, v, the velocities at time t, relative to these axes, of the particle which then occupies the position x, y. The actual velocities of the same particle, parallel to the instantaneous positions of the axes, will be u ~ wy, v + cax, and the accelerations in the same directions will be
du du du _.
~j- + u -7- + v -,----2a>v — to-x
' dt dx dy
* Proc. Roy. Soc. Vol. LXSXA. p. 436 (1908).
t See also Shaw's Forecasting Weather, Chapter n. 1|. At the walls rj takes opposite signs.