18 The Geostrophic Relationship 



of motion will be of the order of magnitude of 2 x 10~^ d3aies/cm.^. This 

 term sometimes approaches the Coriolis force and is not always negligible, 

 especially in meanders of the Gulf Stream. As \\'ill be showTi in Chapter 

 VIII, the inertial terms in the direction of flow of the Gulf Stream are 

 quite likely to be important, even though the cross-stream force com- 

 ponents are essentially geostrophic. 



So little is known about the nature of the turbulent shearing stresses 

 in the ocean that it is dangerous to attempt to estimate their order of 

 magnitude. The Laplacian of the horizontal velocity in the Gulf Stream 

 has been observed to reach values as high as 2 x 10~^^/cm./sec. Information 

 concerning the order of the magnitude of the horizontal-eddy viscosity is 

 scarce. We might take as a conceivable maximum value 5 x 10' cm^./sec. 

 Under these circumstances the maximum viscous force in the Gulf Stream 

 would be of the order of magnitude of 10~^ dynes/cm. 3. Slightly larger 

 eddy viscosity could make the Stream appreciably nongeostrophic. 



Under normal conditions, the stress of the wind on the ocean surface is 

 of the order of 1 dyne/cm. 2. If this stress is distributed evenly over a 

 layer of water 50 m. deep the effect of "wond stress by vertical turbulence 

 is of the order of 2 x 10~* dynes/cm.^. Thus, all these terms are small com- 

 pared to the Coriolis force and to the horizontal pressure gradients in the 

 Gulf Stream. 



One sees, therefore, that in the horizontal equations of motion there is 

 an approximate balance between the term of the Coriohs force and the 

 term of the horizontal pressure gradient. This relation, often called the 

 geostrophic equation, has been of practical use in estimating the velocities, 

 and, more especially, the transports, of the Gulf Stream. The actual 

 numerical process is somewhat involved, but is essentially based on the 

 equations obtained by cross-differentiation of the geostrophic equations 

 and the hydrostatic equation; eHmination of pressure results in the 

 following pair: 



^jufp) JP ,-. 



-dr=^dy- ^^^ 



Thus, in an ocean current flowing toward the north (positive y-direction), 

 dvldz at each level is associated with a decrease of density toward the east 

 at that level (in the Northern Hemisphere), and there is no appreciable 

 horizontal gradient of density in the direction of the stream: duj8z = u = 0. 

 The density of sea water as a function of depth and position along a 

 vertical section through the ocean thus provides us with a basis for com- 

 puting the vertical gradient of horizontal velocity normal to the section. 



