30 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



In the case of the atmosphere we have seen already that the changes of volume 

 of the moving masses of air are too great to allow us to consider the field of velocity 

 solenoidal. But the field of mass is not very far from being stationary, the changes 

 in this field being caused exclusively by the gradual changes in the fields of tempera- 

 ture and of pressure ; we may therefore try to derive the vertical motions, supposing 

 the field of specific momentum to be solenoidal in the first approximation. 



In order to see the errors which can then arise, we can consider a cylinder going 

 from the ground up to a certain height in the atmosphere. Calculating the vertical 

 motion through a horizontal section at the top of the cylinder, we set the transport 

 of mass up through this section equal to the transport of mass in through the walls 

 of the cylinder. The vertical motion thus found will be erroneous, inasmuch as the 

 temperature or pressure within the cylinder is changing. To find the error we shall 

 estimate the additional vertical motion produced by the local changes of temperature 

 and pressure. 



First let there be an increase of temperature within the cylinder of i C. per 

 hour, i. e., of tAh C. per second. This will give a cubic meter of air the velocity of 

 expansion of tts-'tjVtt, or less than one-millionth of a cubic meter per second. The 

 corresponding linear velocity of expansion of the air in the cylinder will be less than 

 one micron per meter in the second. There will thus arise a vertical velocity not 

 exceeding i mm. per second at the height of iooo meters, and not exceeding i 

 cm. per second at the height of 10,000 meters. We can only as an exception expect 

 to get changes of temperature greater in average than a few degrees centigrade 

 per hour for columns of air of this height. Therefore, neglecting the local change 

 of temperature, we shall get errors in the vertical velocities not exceeding a few milli- 

 meters per second at the height of 1000 meters, and a few centimeters per second at 

 the height of 10,000 meters. 



For the corresponding influence of local change of pressure, we can suppose 



temperature to be constant. For the column of air contained in the cylinder we 



have then pK = const, p being the average pressure in the cylinder and K its volume. 



As only the height z of the cylinder is variable, we can write pz = const. Differ- 



dz 



entiating with respect to time and solving with respect to , we get 



at 



dz _ z dp 

 di~ p~~dt 



Supposing the change of pressure -^ to be one m-bar per hour, i. e., tbVt m-bar per 



second, setting the height of the cylinder equal to 1000 meters, and the average 



pressure between sea-level and this height equal to 900 m-bars, we get the vertical 



dz 

 velocity j- smaller than a third of a millimeter per second. Setting the height of 



the cylinder equal to 10,000 meters and the average pressure between this level 

 and sea-level equal to 600 m-bars, we get the vertical velocity due to the variation of 

 the pressure smaller than half a centimeter per second. Thus in both cases the 



