A Formula of Ekman Pumping for a Meso-scale Straining Flow

Abstract

   Vertical velocity induced by a meso-scale straining flow in the free atmosphere in the presence of a viscous boundary layer (i.e., Ekman pumping) is analytically investigated. Meso-scale in this note means that the Rossby number Ro is less (but not extremely less) than unity.
   Assuming that the horizontal velocity in the free atmosphere is nondivergent at the lower boundary, a meso-scale formula of Ekman pumping is derived. In this formula, the induced vertical velocity is expressed in terms only of the horizontal velocity and its horizontal derivatives at the lower boundary of the free atmosphere, and other terms (e.g., geostrophic velocity) do not enter the formula.
   In order to derive the formula, an inviscidly balanced approximation is made. This approximation means a replacement of the horizontal velocity in momentum equation, which is materially differentiated, by the velocity of the free atmosphere. In this approximation, the momentum equation in the free atmosphere remains primitive.
   The derived formula shows, e.g., the following. For a horizontally uniform straining flow, the induced vertical velocity is less than that in Ekman's formula. For a purely deformational flow, the induced vertical velocity becomes negative.