Water Bodies and Stationary Current Conditions at Boundary Surfaces 465 



Table 132. Slope of the tropospheric transition layer and the physical sea 

 level in the North and South Equatorial Current in the Atlantic Ocean. 



Theoretical value: m=26-4m; mi=— 2-58cm; m — 52Q m; m^ 

 Observed value: A?2 = 26-4m; Wi=— 2-4cm; A?7 = 52-5m; nii 



— 51 cm 

 -50 cm 



(c) Stationary Vortices in a Two-Layered Ocean 



When the water masses in a two-layered ocean are in rotation they will be subject 

 to a centrifugal force in addition to the gradient and Coriolis forces. Under stationary 

 conditions these three forces must balance. Such systems of rotating water masses 

 have been examined in detail by Exner (1917) and especially by Bjerknes (1921). 

 When the motion is symmetrical around the rotation axis, the vortices are termed 

 "circular vortices". In cylindrical co-ordinates r is the distance at right angles from the 

 axis of rotation z (positive downwards) and c is the rotational velocity (at right angles 

 to r, positive for cyclonic and negative for anticyclonic motion). For a non-accelerated 

 current (c = 0) the following quantities can be introduced in the boundary surface 

 equation (XIV. 5): 



X^=fc^+ j; 



Zl=g; A'2=/C2 + 



cz 



Zz = g. 



c'^lr is the centrifugal force, which must be taken into account for curved trajectories. 

 The slopes of the pressure surfaces, of the physical sea level for the lighter and the 

 heavier water and of the boundary surface can be determined, and it is obtained 



tan /Si 



/ 



rg 



tan ^2 



f c^ 



■L r — -^ 



'-2 



g rg 



(XVI.9) 



and 



tan y = — 



/ P2^ 

 g Pi 



PlCl 



Pi 



1 



rg 



P2C2 



pA 



The third equation can be somewhat simplified 

 Ac = C2 — Ci 



Pi Ac 



P2 — Pi 



With sufficient accuracy, when 



/ 

 tan y = 



P2 — Pi 



(• ~ ^)' 



(XVI. 10) 



On comparison with formula (XIV.8) it can be seen that the effect of the centrifugal 

 force is contained in the expression in brackets. The difference between the slope of 

 the boundary surface in a rotating flow from that in a straight current remains small; 

 assuming /= 1 X 10'* (about 45° latitude), r = 100 km and Ci + Cg = 40 cm/sec, 

 then the expression in brackets gives 1-04, that is, an increase of about 5% can scarcely 



2H 



