474 Water Bodies and Stationary Current Conditions at Boundary Surfaces 



z = /?2 + S cos (77// )x, (XIV. 1 3) 



5 f (p^ih— Pi«i) 



where o — 



g Pi — Pi 



(Margules boundary surface slope). 



If small periodic variations (disturbance values) are imposed on this equilibrium 

 system in the currents Ui and u.^, then the boundary surface will oscillate about 

 its steady-state position. As a consequence in the most simple case these oscillations 

 will give rise to upward and downward movements in the central part of the vortex 

 with a phase exactly opposite to that of the outer vortex portions. To the equation 

 (XIV. 13) will thus be added an additional periodic term of the form 



Z — A COS -J- COS a J, (XIV.14) 



whereby C7„ is the frequency of the free ("Eigen") oscillation (period T = Irrjan', 

 n = 1, 2, 3, ... , gives the number of node-points in the oscillating system). 



When corresponding boundary conditions are taken into account the equations of 

 motion give an equation for the determination of the frequency a„ of the "Eigen" 

 oscillations of the oscillating vortex as a function of the dimensions of the system. 

 The following equation is obtained 



where //^ and Ju are the thicknesses of the two layers and 2/ is the total horizontal 

 extent of the vortex. These "Eigen" frequencies depend in a characteristic way on the 

 angular velocity of the Earth. If the Earth were not rotating (/= 0) then the period 

 of the free oscillation would be given by 



277 _ 2/ /// 



cTr ~ n \]\ 



In 11 ll pilh + Pilh 



g(p2 - Pi) 



(XIV. 16) 



This is a period for an internal standing wave in a two-layered water mass of an 

 extent / (see Vol. II). 



If for large dimensions of the oscillating system the period Tr for a non-rotating 

 Earth is large, then the second term in the equation(XIV.l 5) will be so small as compared 

 with/2 tjj^t jt can be neglected and the longest "Eigen" period of the system will be 

 equal to the inertia period. 



Ti = ha pendulum day = — ^ . (XIV. 17) 



If the second expression accompanying/^ in the equation (XIV. 1 5) cannot be neglected, 



when (r^ > Ti), 



when (Tr < T,). 



