Forces and their Relationship to the Structure of the Ocean 329 



where m, d, w are the mean values of these components and u', v', w' are the compo- 

 nents of the superimposed turbulent motion for which by definition 



u' = 0, V = 0, w' = 0. (X.40) 



The bar over these symbols indicates mean values considered over a sufficiently long 

 time. It should further be noted that the mean values of the squares and products of 

 «', v', w' of course must not necessarily vanish. 



If the impulse equations (X. 36) are apphed to such a turbulent flow it is not suffi- 

 cient to consider the equations for the mean steady flow alone, since also the turbulent 

 parts of the velocity changes are involved in the relationship between the mean steady 

 current and the forces acting on the masses. This can be derived directly from the 

 impulse theorem. Considering, for example, a part of the "control surface" that is at 

 one time vertical to the x-axis and at another time vertical to the jv-axis, then in the 

 first case a mass pu will pass through a unit area in unit time; the impulse transport 

 due to the x-component u of the velocity is then pun and its mean value over a longer 

 period puu. Now 



uu — {it -\- u'Y + «^ + 2wm' + u'^. 



In deriving the mean value uu it should be noted that u is already a mean value of u 

 and w' = 0, so that 



puu — pu^ = pu'^. 



To the impulse of the steady mean current a turbulence contribution is added in form 

 of the square of the turbulent variation in velocity, which when inserted in equation 

 (X. 38) has the effect on the mean motion of an additional pressure. 



Similarly, a mass pv will pass through unit area of the control surface perpendicular 

 to the >^-axis in unit time. The x-component of the impulse transferred through the 

 surface is thus, in this case, puv and taking an average gives puv per unit time. With 



uv — uv ■}- u'v -\- uv' + u'v', 



puv = puv + pu'v'. 



In addition to the impulse of the steady mean current puv must be added a turbulence 

 contribution which in general does not vanish; because positive values of m' are mostly 

 correlated with positive values of v' and vice versa, so that the products are preferably 

 positive. In the opposite case the products are mostly negative. 



If this turbulent contribution of the impulse transport is transferred to the right- 

 hand side of equations (X. 36) it can be taken as a force acting along the .v-axis, which 

 in all cases will be perpendicular to the >'-axis. It can therefore also be considered an 

 apparent shearing stress 



r = -pTv' (X.41) 



arising from the turbulence of the current and was previously regarded (see pp. 3 1 7-3 1 9) 

 as an apparent internal frictioH. Equation (X. 41) mediates between this viewpoint 

 and the equation (X. 10) which defines the turbulent viscosity coefficient r]. 



5. Circulation and Vorticity 



The Bjerknes theorem concerning the formation of vortices and circulation accelera- 

 tion (1898, 1900, 1901) has been found very useful in the theoretical treatment of 



