320 Forces and their Relationship to the Structure of the Ocean 



velocity of this middle layer is greater, the adjacent layers will exert, due to the 

 transfer of their flow momenta, a retardation on the current maximum in the middle 

 and will eventually eliminate it. The middle layer in case III will be accelerated by the 

 equalization of velocity in the turbulent flow. Equation (X. 12) also shows that for a 

 constant internal friction the vertical profile must take the form of a parabola. 



2. The Basic Hydrodynamic Equations 



For a complete description of the water movement in ocean currents, it is necessary 

 to know on the one hand the path of each small element of water in it, and on the other 

 hand the position of such a small element along this path at any time; i.e. it is neces- 

 sary to know the co-ordinates of a small element of water as a function of time. The 

 basic hydrodynamic equations of motion in their most general form are the mathe- 

 matical-physical tool for dealing with and for a theoretical understanding of the 

 different successive states of a water mass. 



The motion can be looked at from two different view-points. The different mass 

 elements may be followed as they pass a. fixed point in space and particular attention 

 may be paid to the changes in the state of motion of the water mass which occur at 

 this point. Alternatively, the changes of state of individual small elements moving 

 along their track may be followed, and thereby a description of the conditions in the 

 current in the course of their displacement can be obtained. The first approach gives 

 the Eulerian basic hydrodynamic equations of motion (Euler, 1755) and the second 

 leads to the equations of motion of Lagrange (Lagrange, 1781); both of these con- 

 cepts are applied in oceanography according to the type of problem to be solved. 



For a small element of water in the point .v, y, z the components of the velocity are 

 denoted u, v, w in the directions of the co-ordinate-axes of a left-hand system (xy- 

 plane horizontal, x-axis positive to the east, >'-axis positive to the north, z-axis posi- 

 tive towards the centre of the Earth), They will be functions of x, y, z and of the time 

 /. First of all the basic Newtonian relationship of mechanics is applied: 



Mass X acceleration = sum of all forces. 



The individual accelerations dujdt, dvjdt and dwjdt are made up of two parts. The first 

 part arises from changes in the state of motion at the point; it is given by dujdt, 

 dvjdt and dwjdt (local change). The second arises since after a small time dt the water 

 elements under consideration are no longer found at the initial point (.v, y, z) but are 

 displaced by udt, vdt and wdt, respectively (advective change). Thus to the local part 

 must be added an advective part, so that the total individual acceleration in the x- 

 direction of the small elements of the liquid under consideration will be 



du du du du du 



Similar equations apply for dvjdt and dwjdt. It may be emphasized here that the partial 

 derivative djdt always represents the change in the quantity under consideration at a 

 fixed point, while the total derivative djdt represents the individual change in a quantity 

 for one and the same element (which changes its position with time). 

 Taking the mass of unit volume as p, and considering that since the only external 



