Forces and their Relationship to the Structure of the Ocean 3 1 5 



as well as in the direction of the equatorial plane. A complete derivation for the 

 Coriolis force, unobjectionable in every respect, has been given by Bjerknes and 

 co-workers (1933). If at any point on the surface of the Earth a co-ordinate system is 

 chosen with the (A:_y)-plane coinciding with the tangential plane to the Earth (x- 

 positive to the east, j^-positive to the north, z-positive towards the Earth's interior), 

 the components of the Coriolis force acting on a material particle on the Earth moving 

 in any direction with a total velocity V (ii, v, w) can readily be calculated using equa- 

 tion (X. 4). This gives the components in the three co-ordinate-directions 



Cx — 2ctjr sin </> — 2ww cos (f), Cy = —2wu sin 0, C^ — —Icou cos ^. (X. 5) 



From these it can be shown that every movement in the tangential plane to the surface 

 of the Earth will be deflected by the Coriolis force to the right in the Northern Hemis- 

 phere and to the left in the Southern Hemisphere. The terms cum sole and contra 

 solem suggested by Ekman can be used respectively for rotation in the direction of the 

 azimuthal movement of the sun, i.e., to the right in the Northern Hemisphere and 

 to the left in the Southern Hemisphere {cum sole), and for rotations in the opposite 

 direction to the azimuthal movement of the sun, i.e. to the left in the Northern 

 Hemisphere and to the right in the Southern Hemisphere {contra solem).'\ Thus every 

 movement in a horizontal direction is deflected cum sole by the Coriolis force. It also 

 follows from equation (X. 5) that there is a vertical component of the deflecting force 

 only for zonal movements (.v-component u) and not for meridional movements. The 

 importance of the vertical component for the dynamics of moving masses is quite 

 small since it acts in the same direction as gravity relative to which it is vanishingly 

 small. ^ 



The horizontal component is very important, however; at the poles (<^ = 90°) it 

 amounts to 1-46 x 10~^ cm sec"^ for m = 1 cm sec"^ and is thus of the same magnitude 

 as other forces acting in the same direction (gradient forces, tidal forces); it is zero 

 at the equator and reaches the above maximum at the poles. Since it acts at right angles 

 tothedirectionofmovementitisunabletoproducechangesin velocity and is incapable of 

 doing work; it can only produce changes in the ^//-ecZ/o/iofmovement, but these changes 

 are of decisive importance for the finally established patterns of motion. Due to the 

 effect of the Coriolis force a mass particle moving freely in a horizontal plane with a 

 velocity V will follow a curved track. Since the deflecting force acts at right angles to 

 the velocity (apart from the effect of change in latitude) and its absolute value is 

 constant, this path will describe a circle which is known as the circle of inertia. The 



t Another terminology uniform for both hemispheres is that customary in meteorology: cyclonic 

 = contra solem and anticyclonic = cum sole. 



X The usual statement, that the vertical component of the Coriolis force need not be taken into 

 consideration, since it is small by comparison with the gravitational acceleration is not entirely correct. 

 In the static equilibrium state of the sea, gravity and the vertical pressure gradient neutralize each other. 

 However, under quasi-static conditions the difference between gravity and vertical pressure gradient 

 is so small that it may be of the same order of magnitude as the vertical component of the Coriolis 

 force. Nonetheless, it is customary to neglect the latter in calculations ; it can be regarded as an increase 

 or decrease of gravity so that the acceleration towards the centre of the Earth is now g + Imi cos (p. 

 It can also be regarded as causing a small change in density in the ratio {g + Iwu cos (p) : g. 



At the equator, when m = 30 cm sec-^ it may amount to 5 units in the sixth decimal place in p 

 or 5 units in the third decimal place in at, which can be disregarded. 



