442 General Theory of Ocean Currents in a Homogeneous Sea 



Co is the impulse of disturbance imparted to the steady current Fo at the time r = 0. 

 If this disturbance is only applied in the direction of the steady current and if at the 

 time ? = the total velocity is denoted by V, then 



M = (J/ - Ko) sin/r and v=Vo-\-{V- V^) cos//. (XIII.62) 



If the permanent equilibrium of a steady current is disturbed, the difference between 

 the disturbance vector and the steady gradient current is transformed into an inertia 

 movement with a corresponding circle of inertia. The period of the circular movement 

 is 



T = -77 = — -. — ; = I pendulum day. 

 / oj sm <^ ^ *^ 



The amplitude of the two velocity components is the same, and the phase of the 

 >'-component precedes that in the .v-component by one-quarter of a period. These are 

 the characteristic features of a pure inertia movement. It is superimposed on the uni- 

 form gradient current and thus gives an oscillating flow, the period of which depends on 

 the Coriolis force. This period is identical with the period of one revolution around the 

 circle of inertia; numerical values for it are given in Table 1 12a (see p. 316) for differ- 

 ent latitudes. Inertia oscillations are not associated with any large transverse displace- 

 ments of the water masses, since the disturbance velocity c = V — Vq usually remains 

 small. The magnitude of these can be taken from Table 2 for different latitudes and 

 velocities. In the open ocean these transverse displacements are usually of little 

 importance but they are still characteristic phenomena which are quite noticeable in 

 current measurements. 



If pressure forces are present in a homogeneous sea due to a slope in the sea surface 

 {dijdx = 4; dijdy = iy) the equation of motion (XIII.3) will apply. A steady motion 

 (geostrophic current) is associated with a corresponding slope of the sea surface given by 

 /j. and iy so that 



- / - / 



^x=- V and 'V = - ^ ^• 



If, further at the time / = 0, there are current components Mq and i\ and slopes 

 ij, and iy o which do not correspond to the condition for a steady state, then the above 

 equations have the following general solution (Fr. Defant, 1940): 



w = t/ + ("o - U) cos ft + [vo - (glf)ix,o] sin//, 

 V = V-\-{vo- V) cos ft - [uq - (g//)/x_o] sin//, 

 ix = ix + ['x,o — 'x] cos ft — [iyo — iy] sin ft, 



iy = h + [iy — Iy] COS ft + [/^ „ " I x] siu//. 



(XIII.63) 



This set of equations shows that for a completely free initial state, both the current 

 field and the sea surface will perform inertia oscillations around their equilibrium 

 position which, however, will not correspond in all points to the conditions for pure 

 inertia waves. In the current field the amplitudes of the corresponding velocity 

 components will be equal only when the sea surface slope corresponds initially to the 

 steady state. But according to the second pair of equations the sea surface does not 



