In his paper “Two Important Factors Controlling Winter-Time Precipition in the Southeastern United States”, J. Namias wrote: “It is difficult, however, to give a physical explanation of such an anticyclonic trajectory of an extensive polar air-mass. Indeed, over the Continent, polar air-currents generally possess a cyclonic curvature, with the exception of small branches which are diverted around anticyclonic eddies on the right side of dominating cold air “wake-streams”. While at present there appears no satisfactory explanation for this large-scale anticyclonic trajectory of polar air once. it has left the Continent, no experienced synoptic meteo rologist can doubt that it must be accepted as an observed fact”. It is the purpose of this report to re-examine the problem of the large-scale anticyclonic trajectory of polar air. To study this problem we shall consider a barotropic atmosphere. It will be assumed that we measure the axes of x and y in the horizontal plane, while the axis of z vertically upwards. The equations of frictionless motion are u and v being the horizontal velocity component, w the vertical component, p the pressure, ρ the density and ƒ the Coriolis parameter, 2ωsinφ, where φ is the latitude and cω the angular velocity of the earth's rotation. After cross-differentiation of the equations of motion, we have the equation of vorticity obtained by C.=G. Rossby(1) as follows: where ς is the relative vorticity around the vertical, which is given by and the total vorticity (ƒ+ς) is always positive. Through comparing the order of magnitude of the terms on both sides of Eq. (1) with the aid of Hesselberg-Friedmann's tables and after simplifications, we have which is the well-known formula bearing the name of J. Bjerknes(2). This formula, then, applied to the present problem, gives the satisfactory reason of decreasing curl (or increasing anticyclonic motion) of polar air, whose horizontal extent increases and hence whose depth decreases once it has left the polar origin and has moved southward. When a current of air meets a mountain barrier energy is required to lift it over the obstruction, and there is a tendency, often very marked, for the air to sweep round the ends of the barrier, so avoiding the ascent. When the obstruction is a large mountain range, the dimensions of the disturbance are such that the earth's rotation exercises an influence, and the winds obey Buys-Ballot's law. Moreover, theoretical results as given by J. Bjerknes(1) show that the air current over the barrier curves anticyclonically as it glides up the mountain range. The flow of the air sliding down from above the barrier would give it a cyclonic curvature, thus tending to change the wind back to the original orientation again. The result is a shallow depression to the lee of the mountains and a ridge of high pressure on the windward side. Such developments are recognized in various parts of Japan. A large proportion of the disturbances which form over southern Kwantô are formed on a cold front held up by the principal mountainrange. Then the warmer air is trapped on the lee side and in a position acts as a warm sector; the cold air sweeps round and the occluding process leads to intensive development of the depression. In every latest issue of the Monthly Weather Review, the monthly mean motion of upper air current and the isobars in the selected high levels are graphically shown. We can generally see that air enters from the west and changes to a northwest wind over the western slope of the North America and back to west again in the East.
In the second report of this problem, the author calculated the moist-labile energy in which are considered both the ascending and the descending currents. In this case the energy equation has been evaluated analytically. But, as the mass of ascending current becomes smaller in comparison with that of descending current, this analytical method requires more laborious numerical computation, Thus, in the present paper, the numerical integration has been applied to the evaluation of mass-integral (Massenintegral). The result shows that, as the mass of ascending current becomes smaller, the energy loss due to the decrease of lapse-rate becomes smaller. In the extreme case of an infinitely small mass of ascending current, the energy equation coincides with that of parcel method, if the available energy is given to the ascending mass alone.