Forces and their Relationship to the Structure of the Ocean 317 



components of the tide-generating forces, it is of particular importance in the dynamics 

 of periodic phenomena.! 



{P) In addition, /r/c?/o« is also of considerable importance in all oceanic movements. 

 Like all liquids, sea-water has a viscosity which for deformation manifests itself as an 

 internal friction. The friction of water over a solid and rough sea bottom is primarily 

 an external boundary surface friction. This type of friction represents a retardation of 

 the flow of the current but only in relatively shallow water can be taken as a measure 

 of it; the most simple assumption describing the frictional mechanism is that the 

 gliding flow of the water over the solid bottom meets a tangential resistance which is 

 assumed proportional to the velocity of the current V. The frictional force in this case 

 would correspond to a vector with a direction opposite to that of the velocity vector 

 and has the absolute magnitude kpV, The quantity k is termed the coefficient of 

 gliding friction. Hydraulic investigations on the dissipation of the kinetic energy of a 

 river due to friction on the river bed have shown that the frictional force per cm^ of 

 the bottom surface is proportional not to the first power but rather to the square of the 

 flow velocity. It can be expected that the dependence of the boundary surface friction 

 on the velocity will also be of the same kind for shallow ocean currents. Taylor 

 (1920) attempted to apply the conditions found in natural channels to coastal oceanic 

 currents in shelf areas. In the friction formula the coefficient k for a normal sea bottom 

 has the value 0-0026 for depths of about 50-100 m so that 



R = -2-6 X \{)-^pV\ (X.9) 



At more shallow depths with an especially irregular sea-bottom topography k may 

 increase considerably (100 times the above value or even more). These frictional as- 

 sumptions refer always to the mean tangential resistance exerted over the whole of a 

 column of unit cross-section from the bottom to the sea surface due to the eff'ect of 

 boundary friction at the bottom. However, these assumptions do not specify the nature 

 of the friction in the interior of the total water column above the sea bottom. The 

 internal friction appears as a tangential shearing stress r between individual layers of 

 water gliding one above the other with different velocities. This stress per unit area is 

 proportional to the velocity gradient perpendicular to the direction of the flow 

 dVjdn, so that 



dV 



T=)Lt^-. X.IO 



dn 



The quantity ju is the coefficient o^ dynamic viscosity and has the dimensions [g cm^^ 

 sec"^].J 



t The inertia movement has the form of a circle only if the Coriolis force is constant (mostly 

 assumed as the mean value for the meridional width of the inertia circle). A general derivation for 

 varying latitude has been given by Wipple (1917) but this was confined, however, to movements near 

 the equator, since sin (f) was replaced by the arc 4> of latitude and cos (^ by 1. Inertia movements super- 

 imposed on horizontal and zonal currents play a large part in the dynamics of ocean currents especially 

 the occurrence of long waves and in vortical disturbances. (See in this connection Defant (1956) and 

 Vol. I, Pt. II, Chap. XIII, 6.) 



X The origin of viscosity can be sought in the continuous equalization of velocity between super- 

 imposed layers of water gliding over each other in a moving water mass. This equalization is due to 

 the interchange of individual molecules and the consequent transfer of velocity from one layer to the 

 next. This viewpoint is, however not entirely correct since the molecules in a hquid are so closely packed 

 that usually they can only oscillate within the small intermolecular spaces present and therefore only 



