324 



Forces and their Relationship to the Structure of the Ocean 



Since the rotation of the Earth does not affect the conservation of the mass, the con- 

 tinuity equation does not contain the angular velocity of the Earth's rotation when a 

 polar co-ordinate system is used for the rotating Earth (co-ordinates: pole distance 

 '& — 90° — (f), longitude A and r along the Earth's radius R). However, there are 

 changes in the cross-section of a current for meridional motion due to the convergence 

 of the meridians and for vertical displacements of mass due to the divergence of the 

 Earth's radii. Thus in the continuity equation for polar co-ordinates, in addition to 

 the previous terms derived from flow through the volume element, there will be two 

 further terms considering these further circumstances in this special co-ordinate 

 system. These give the following equation : 



dp 1 



87 '^ R sin ^ 



dpu sin '& 8pv' 



dpw 2pw 

 + ^ + -^ = 0. (X.27) 



The effect of the convergence of the meridians is expressed in the term (puIR) cot g'& 

 which is obtained by differentiation of the first expression in the brackets and the effect 

 of the divergence of the Earth's radii is contained in the term 2pwjR. Since for vertical 

 displacements of mass in the sea, which is shallow relative to the Earth's radius, the 

 vertical velocities appearing are very small, this last term is not too important and can 

 safely be neglected. For small oceanic spaces the convergence of the meridians can also 

 be disregarded in first approximation, though not for large-scale ocean currents 

 (see Chap. XXI).t 



If the liquid has boundary surfaces either at a solid body (the sea bottom) or when 

 it is surrounded by differently stratified liquids (other water bodies) the continuity 

 equation will take special forms and must be replaced or supplemented by special 

 boundary conditions. At a solid boundary, in order to secure a reasonable state of 

 motion with no empty spaces, the component of the velocity perpendicular to the 

 surface must be zero. If /, m, n are the direction-cosines of the normal to the surface 

 then a necessary condition is 



lu -{- mv + nv — 0. 



(X.28) 



t The continuity equation which corresponds to the Lagrange equations of motion is more 

 difficult to derive and reference should be made to text-books of hydrodynamics. Taking the functional 

 determinant 



8(x, y, z) 

 8{a, b, c) ' 



the condition of constancy of mass in a volume element 8a 8b 8c will be 



8(x, y, z) 

 d{a, b, c) 



where Po 's the initial density at the point {a, b, c). For incompressible liquids p = Po the continuity 

 equation takes the form 



8{x, y, z ) 

 8(a, b, c) 



= 1. 



