322 Forces and their Relationship to the Structure of the Ocean 



Since the depth of the sea is always very small as compared with the dimensions of the 

 Earth, the term {R + z) in the first equation of (X. 1 8) can be replaced in good approxi- 

 mation by R. 



In the Lagrange equations of motion the co-ordinates x, ^, z of a small mass element of the liquid 

 are viewed as functions of the independent variables a, b, c and of the time t; a, b, c are the initial 

 co-ordinates of the particle under consideration, i.e., values of x, y, z at the time / = 0. These functions 



;c = /i (a, b, c, t), y = fz (a, b, c, t), z = f^ (a, b, c, t), 



thus describe the history of each small element of the liquid vi'ithin the current. If only the time t 

 is altered, they give the path of the element under consideration; if on the other hand t is constant 

 and only a, b, c are allowed to change, this gives the positions of the different elements at one and the 

 same instant of time. Since the accelerations of the element a, b, c at the time / are given by 



du _ d^x dv d^y dw _ d^z 

 'dt ~ dt^' ~dt ^ dl^' IJi ~ dF^ 



the equation (X.15) can also be written in another form 



^ - X= --^ ^^' _ y = _ ' ^ ^!' _ Z = - 1 ^^ 

 dt^ P dx dt^ p dy dt^ p dz' 



To eliminate at the right-hand sides the derivatives with respyect to .v, v, z these equations can be 

 multiplied at first with 



dx dy dz 

 8a 8a 8a 



then with 



dx dy dz , dx dy dz 



-7> T' —,■ 3nd --, ^, ~, 

 db db db dc dc dc 



respectively, and finally can be added. If the forces have a potential Q, the Lagrange form of the equa- 

 tions of motion is obtained 



Idhc \ 8x id^ \ a V IdH ^Y^ ,^ ^P_^ 



\dl''~ ^) aa + U/2~ ^ ) da + \dl^~^)da^~p 8'a~ ^' 



\dt^ J db^\dt^ I db ^ W/2 ^ I 8b^ p db 



Id^x \ dx id^y \ dv (d^z \ dz I 8p 



b/2 - ^} e-c + \d^ - ^ } Tc^ \dt^~ ^ } 8c^ 



0, 



p cc 



The hydrodynamic equations of motion form a very complex set of equations. They 

 have to be solved in order to obtain a complete description of the state of motion but 

 only in very rare and in the most simple cases it is possible to arrive at a final and 

 definite solution. In most cases it is considered sufficient to determine, if possible, the 

 state of motion at each place and at each time without paying any attention to the 

 further history of the individual water elements. There is a considerable simplification 

 possible when dealing with so-called stationary currents. These are currents in which 

 the state of motion at each point does not change with time and is thus completely 

 fixed by specifying its direction and velocity. The condition for a steady state in the 

 current is thus 



8u dv dw 



8", = a, = a? = 0- <'^-2«> 



