A Supplementary Note to Ertel's Equations of Motion in Turbulent Flow

Abstract

We define with x_i (i=1, 2, 3) a cartesian coordinate system: i.e. horizontal axes x₁=x, x₂=y, and vertical x₃=z.
Then the fundamental equations of non-viscous fluid motion are where the index κ (and every index that is found twice in a term) shows the summation from 1 to 3 after the Einstein's expression, and C_ik=-C_ki is the coriolis-axiator. When the elements p, ρ and ui in the above equation are compounded of the temporal mean values and the fluctuations, i.e. ρ=_??_+ρ'_??_p=_??_+p', and ui=_??_+ui', we obtain by introducing these values into the equation (1) by using the equation of continuity Sometimes we can find the use of the terms which contain ρ', in the above equation. Next we must consider in detail about the 4th term in (2). For the horizontal turbulence Ertel as sumed where the mean values of ξ_i are considered as the components of the Prandtl's mixing-length. But according to the Prandtl's conception we must add to the equation (3) terms which depend on ∂ _??_/∂_xi, that are the components of the direction of gradient of another velocity components.
Here we consider that the above idea is insufficient, because the turbulence in the atmosphare arises in the direction, which even if the velocity gradient component does not exist in, parpendicular to the both directions of the mean flow and its velocity gradient. Moreover it is considered that the turbulence occurs not only from the velocity gradient, but also from the thermal convection, the eddy diffusion etc. So that for the most natural expression of u_i' we assume where the additional terms ε_i represent the turbulence arising from agency other than velocity gradient. By using the above expression we can obtain the new equation which have a very expanded physical meaning for the eddy stresses that is The more extension in detail about the equation (4) will be found dlse where.