The Tropospheric Circulation 635 



wide current in a stratified ocean in which he showed that the stability of the waves 

 depends on whether 



C/2 > g:^D. (XIX. 50) 



P 



Here f/is the velocity of the basic current, D is the thickness of the upper moving layer 

 and J p is the density difference between the lower, homogeneous and motionless layer 

 and the homogeneous upper layer. The upper inequality sign results only in stable 

 waves and the lower one only in unstable waves. For W = g(Aplp)D there is a single 

 "just unstable" wave, the wave-number of which is given by k =f/{U\/2). This 

 wave always remains stationary. 



Choosing a surface layer 200 m thick moving at 200 cm sec^^ and having a density 

 ratio zJp//3 = 2 x 10~^ the wavelength of the "-ust unstable" perturbation is 180 km. 

 All other wavelengths are stable and do not grow. It is remarkable that this wave- 

 length corresponds closely to that observed. Some objections can be raised against 

 the application of the Stommel perturbation theory to the meanders actually observed 

 in the Gulf Stream and it would be desirable to test the Stommel model somewhat 

 more closely and to specialize some of his assumptions. 



In order to handle the problem of the meandering behaviour of the Gulf Stream in 

 a more comprehensive way, the problem may be looked upon as intimately connected 

 with the way in which the stability of a narrow geostropliic current is changed when this 

 flow is subjected to external perturbations. In a deeply penetrating way the latter 

 question has been dealt with by van Mieghem (1951) for atmospheric currents. He 

 assumed a straight geostrophic flow in hydrodynamic equilibrium in any direction on 

 the rotating earth allowing for horizontal (transversal) and vertical wind shear. On this 

 current he imposed a disturbance acting in lateral (transverse) as well as vertical 

 direction and attempted to find the conditions under which the disturbance decreased 

 in time (stable state) or increased in time (unstable state). In the stable case the chance 

 disturbances vanish with time; in the unstable case they grow into meanders and may 

 even degenerate into independent vortices. If the positive x-axis is chosen in eastward 

 direction, the >'-axis normal to it (to the north) and the r-axis positive towards the 

 zenith and if the geostrophic current flows along the j'-axis (w^ = 0, iiy ^ u(x,z), 

 u^ = 0), then the equilibrium values of the pressure P = P(x,z) and the specific 

 volume a = a(x,z) are only functions of jc and z and the equation of motion as well as 

 the quasihydrostatic equation leads to the Margules equilibrium condition of the 

 geostrophic current : 



cPca_cPca^^^ (XIX.51) 



ox cz dz ex 



where oj^ and a»y are the horizontal and vertical components of earth rotation vector 

 (coj. = ojy = oj cos </) and cu^ = cm sin (/>) and N is the number of solenoids in the cross- 

 section {x,z) (baroclinicity). For a small fluid particle in the interior of the water mass 

 which is at the co-ordinate origin at time t^ and at that instant is subject to a transverse 

 impulse, its velocity components relative to the earth at the same instant will then be 



V, = u-\- r„ Vy = Vy, V, = V,. '. (XIX.52) 



