456 Water Bodies and Stationary Current Conditions at Boundary Surfaces 

 When To = it follows 



tany 



Pi 



tan/3i 



P2 — Pi 



and since pa ~ Pi is of the order of 10~^, the slope of the boundary surface will be 

 about 1000 times greater than that of the isobaric surfaces; it has, moreover, the 

 reverse inclination as compared with that of the isobaric surfaces in the upper water 

 body; the physical sea level has thus the opposite inclination in comparison to that of 

 the boundary surface underneath. 



Table 131. Slope of the boundary surface and the isobaric surfaces for moving water 

 masses. <^ = 45° N. pi = 1-027, p^ = 1-028; ^3 = 0, ^Sg = 



The slope of the Margules boundary surface can also be derived quite readily from the equations 

 of motion. This will be given here since it will be required later. We consider two water bodies 

 1 and 2, one above the other, the upper limit of the lower being the boundary surface and the upper 

 limit of the lighter above it being the physical sea level ; furthermore, we allow only slopes along the 

 X-axis. The position of the two boundary surfaces can be defined by the deviations ii and i^ from 

 their equilibrium position at rest (level surfaces). The pressures at an arbitrary point A in the water 

 body 1 and at a similar point B in the water body 2 will then be: 



P\ = (fh + h - =)Pig - Pig^i 

 and 



P2= - Plg^l + (f'l + QPlg + Vh - ^2 - 2)p2g- 



Then for stationary state the equations of motion will take the form 



1 



fvr+g'^ 



= 0; 2 fv^ + 



■?■ + 



8x 



P2 



P2 ^X 



0. 



The first equation gives immediately the slope of the physical sea level 



tan i3i = 



^^1 



g 



Elimination of c^j/?x from the second gives the slope of the boundary surface 



tan y = £^2 = - / ^^^'^ ~ Pi^i 



^X g P2- Pi 



which are the same equations as before. 



It might be mentioned here that equation (Xrv.7) can also be written 



tan y = Pztan^o - Pitan^i 



P2 - Pi 



which gives the slope of the boundary surface directly from the slopes of the isobaric surfaces. Further 

 the equations of motion give a relationship between the horizontal pressure gradients on either side 

 of the surface 



8p2 _ 8pi 



dx dx 



g(p2 - Pi) tan y. 



