470 Water Bodies and Stationary Current Conditions at Boundary Surfaces 



occur at arbitrarily inclined discontinuity surfaces. Taking horizontal accelerations 

 into account but neglecting the very small vertical accelerations (w^ = W2 = 0) and if 

 the boundary surface is parallel to the >'-axis having an inclination tan y then equation 

 (XIV.5) gives the relations (^-positive upwards) : 



(Pi^i — P2W2) =f{pii\ — P2V2) — Sipi — P2) tan € and Pii\ — p.iV2 =f(piUi — p^Uo) 



(XIV. 11) 



Near the boundary surface the velocity in each of the water bodies will be tangential 

 to it : Wi = Ml tan e and vt-g = Mo tan e, so that 



PiH'i — P2H'2 = (pith — P2W2) tan e. 



(XIV. 12) 



These equations form the basis of the dynamics of up- and down-ghding surfaces. 

 If in the first of these equations e = y (stationary boundary surface condition), then 

 P2W2 — P2W2 = and from (XIV. 12) it follows that p^Wi = P2^2- On the other hand, 

 according to the second part of the equation (XIV. 1 1) 



PlVl — P2«2 ^0. 



This implies that : tip- and down-gliding can also occur at stationary boundary surfaces 

 if the currents are accelerated also in the direction parallel to the gliding plane. If the 

 mutual adjustment between current velocities and stable position of the boundary 

 layer gets disturbed by changes in the velocities, then up- and down-gliding motions 

 must occur along the boundary surface in order to preserve a stationary state of its 

 inclination. Thus when 



(0 Pi'"i — />2i'2 < 0: piMi — P2W2 > and p^w\ — P2**'2 > 



and when 



(2) pjt'i — /Da^a < 0: piu^ — p^Uo < and p^w^ — p<^<2, < 0. 



In the first case where there is a stronger acceleration in the lower water mass along 

 the positive j'-axis than in the upper, an up-gliding surface is to be expected. In the 

 second case, however, where there is a stronger relative acceleration along the positive 

 j-axis in the upper water mass, there will be a down-gliding surface. These two cases 

 are illustrated in Fig. 213; they apply for the Northern Hemisphere. In the Southern 



Fig. 213. Stationary up-gliding (to the left) and down-gliding surfaces (to the right) 



(Northern Hemisphere). 



