On the Motion of Air in the Viscous Atmosphere. (1st Paper)

Abstract

The famous studies of the motion of air and its vortex motion in the atmosphere⁽¹⁾ have been treated by the late Prof. Dr. D. Kitao about forty-five years ago. His treating was based in non viscous atmosphere which included the surf_??_ce frictional resistance terms by C. M. Guldberg and H. Mohn. Therofore, in this paper we retre_??_ted the same problems in the viscous atmosphere, starting from the following equation of motion with customary notations, In which, the axis of x to be taken southwards, y be eastwards in a horizontal surface and z to be drawn upwards in the direction opposite to that of the acceleration of gravity g only, and Ω be the potential of the extraneous forces, viz. Ω=gz, W is the vector of the angular velocity of earth's rotation, v is the Kinetic eddy viscosity and the symbol θ is the divergence of the vecter V, viz.,
We use to symbols £, η, ξ to denote the components of vorticity, ω is the angular velocity of the eartr's rotation, and φ be the latitude, then
In these equations, the first and second terms on the right-hand sides express the component angular velocities of the mean rotation of an element of air and the third terms express the component angular velocity effected by the earth's rotation.
In equation (1), we put where φ may be c_??_lled the ‘dynamen’ by Prof. D. Kitao, the surface of constant φ be the equal energy surface of air motion or this is so called ‘Isodynamical pressure surface’. From above equations we reduced to the convenient forms to calculate the relation of air motion or its vortex motion by using the auxiliary function.
Firstly, we showed the balancing problem of the air motion in the viscous atmospherre. If we write the component of the frictional resistance R_x=-v∇²u, etc. in the stendy state. the equation (1) may be shown as follows; next, we assume to be the axis of n normal to the isodynamical pressure surface, α and β are the angle between wing direction and n axis and the frictional acting axis, respectively. Then we have, from (5) the angle (Vσ) be denoted between wind direction and the axis of the vorticity, and (δ) be the angle between the axis of n and the frictional resistance. Hence, from above equations (6), the wind velocity be shown as follows;
At a great height we have this is the ‘dynamical pressure gradient wind’ parallel to the isodynamical pressure, which would prevail if there were no frictional resistance, viz. R=0.
When the acceleration of velocities are existed in three dimensions, we put etc., therefore the total acceleration of velocity is Heuce, the equation of motion are The angles _??_, γ and ε are made the direction of acceleration of velocity with R, n, and V, respectively. Therefore, from equation (8), we have thence, after reduction, we have This formula (10) is just the same shape as equation (7) in the steady state, where to use β and δ in place of β_ε and δ_γ, respectively.
In the case of the motion of air in two dimensions, we have in steady and in generaly Therefore, substituting from (11) in (7), we have If we put that the acceleration of veloeity may be linear change with time, viz. ∂²V/∂t²=0, hence, we have
In the last section, we showed the atmospheric current due to the effect of the earth's rotation only, viz, In these, cases σ=2ω, sin (Vω)=sin (Vω), and the direction of axis of the deflecting air mass be determinded with the axis of earth's rotation.