413.
ON THE DYNAMICS OF REVOLVING FLUIDS. [Proceedings of the Royal Society, A, Vol. xcm. pp. 148 — 154, 1916.]
So much of meteorology depends ultimately upon the dynamics of revolving fluid that it is desirable to formulate as clearly as possible such simple conclusions as are within our reach, in the hope that they may assist our judgment when an exact analysis seems impracticable. An important contribution to this subject is that recently published by Dr Aitken*. It formed the starting point of part of the investigation which follows, but I ought perhaps to add that I do not share Dr Aitken's views in all respects. His paper should be studied by all interested in these questions.
As regards the present contribution to the theory it may be well to premise that the limitation to symmetry round an axis is imposed throughout.
The motion of an inviscid fluid is governed by equations of which the first expressed by rectangular coordinates may be written
du' ,duf ,du' ,du' dP .,.
where
P=jdp/p+V,
(2)
and V ia the potential of extraneous forces. In (2) the density p is either a constant, as for an incompressible fluid, or at any rate a known function of the pressure p. Referred to cylindrical coordinates r, 6, z, with velocities u, v, w, reckoned respectively in the directions of r, 0, z increasing, these equations become f
did H- u du + v 1 du «>> !+w du dP ........ (3)
dt dv dr dv (rdB ( dv r) u\ c dz iv dr ' ....... dP (4A
dt dw dr dw (rd0 + dw r) I c dw w — iz ~ rdd' ....... dP ........ y/
dt ' dr rd6 dz dz
The Dynamics of Cyclones and Anticyclones. — Part 3," Boy. Soc. Edin. Proc. Vol. xxxvi.
p. 174 (1916).
t Compare Basset's Hydrodynamics, § 19.effect of a departure from the circular form is then to depress the pitch when the area is maintained constant (da' = 0). But for an elliptic deformation the reverse is the case.