V. 



ON ATMOSPHERIC MOVEMENTS.* 

 (FIRST PAPER.) 



By Prof. H. von Helmholtz. 



I. INFLUENCE OF VISCOSITY ON THE GENERAL CIRCULATION OF 



THE ATMOSPHERE. 



The influence of fluid friction in the interior of very extended regions 

 that are filled with fluid and contain no vortex motion is always rela- 

 tively very small. This can be proved from considerations that are 

 based upon the principle of mechanical similarity. If we form the 

 Eulerian hydro-dynamic equations and in them iudicate by w, v, w the 

 components of the velocity parallel to the axes of x, y, z ; by £ the den- 

 sity, by p the pressure, by P the potential of the forces that act upon a 

 unit of mass of the fluid; then if we consider P, e, p, m, v, w as func- 

 tions of x, y, z, t we have, as is well known, the following partial differ- 

 ential equations for a fluid under the influence of friction t : 



-\-u 4-v \-w .,+ .A „ .... (1) 



C )P 1 dp_du ?u , „du , _Ju ¥■ p^) 2 « ( J 2 u , J 2 u~ 



• — — — — — — — |— l\i ■ 



J30 £ J 30 ()t (j30 



de _d(£u) Mev) few) 

 dt~ dx + da dz 



(la) 



Two other equations symmetrical with regard to the other coordinates 

 are to be added to the first of these equations. If now we have found 

 any special integral whatever of these equations, which obtains for a 

 definite region, then the equations will also hold good for a second case 

 where all the linear dimensions x, y, z and also the time t aud the fric- 

 tion constant ¥ are increased by a factor n, but where P, p, e, u, v, w 

 retain for every value of the new coordinates nx, ny, nz, nt, the same values 

 as they had in the first case for the original coordinates #, y, z, t. Hence 

 it follows that when in the movement of the magnified mass the friction 

 constant can be also simultaneously and correspondingly increased, the 



* From the Siizungsberichte of the Royal Prussian Academy of Science at Berlin, 

 1888, May 31, pp. 647-663. 



[t Namely viscosity as represented by Maxwell's kinematic coefficient v or Helm- 

 , , ,k 2 0.0001878 

 holtz £ =1LOOT^3 =0 - 13417 J 



78 



