Currents in a Strait 535 



the depth, which is usually the case, then this critical velocity of propagation reduces 

 to the value Vgh. The stationary current waves in moving water will have exactly 

 the same form throughout the entire water layer as the bottom wave if c > ■\/sf^'-> 

 if, however, c < \/gh then above a certain height it will be inverted, that is, above a 

 rise in the bottom there will be a depression of the water level and above a depression 

 in the bottom there will be a lift of the water level. If the velocity c is exactly the velocity 

 of free waves then resonance will occur and in this case the frictional forces will be 

 decisive. In all cases occurring in nature \/gh is always several times larger than c 

 and the stream lines show the wave-form of the bottom with decreasing amplitude 

 and only up to a certain height ; above this level of no horizontal motion the wave is 

 inverted, but the amphtude is so small that these waves will scarcely be noticeable. It 

 cannot be excluded that many of the vertical displacements in isotherms and iso- 

 halines, which are always found at the same place, may be due to effects of this type 

 produced by bottom disturbances. 



In stratified water conditions are different, especially when there are well- 

 developed transition layers. Under certain conditions the disturbance by the bottom 

 relief may be shown in amplified form at a boundary layer; it may even be larger than 

 the disturbance causing it, while the surface of the water remains almost entirely 

 unaffected. Theoretical treatment is also possible in this case (Defant, 1923). If the 

 thickness of the upper layer is hi and that of the lower layer /zg, resonance (enlarged 

 amplitude of the stream-line waves) will occur at two values of the current velocity. 

 If the total depth of water h^ + Ag is small as compared with the wavelength of the 

 bottom disturbance these values are given by the equations 



c,-Vk(h + h.)} and ., = y[(l-^j)j^J. (XVI.I6) 



The first value Cj already for small depth is many times larger than any values found 

 in nature. Cg is the velocity of propagation of internal waves at the internal boundary 

 surface (see Vol. II) and may be so small that it can be quite close to the observed 

 current velocities. At these values the boundary surface will show the greatest varia- 

 tions while the sea surface remains almost undisturbed. For example, choosing pg — Pi 

 = 10~^, P2 = 1-028 then for larger h^ and hi = 50 m Cg will be 0-7 m/sec. Values of this 

 order are frequently found in sea straits and it can be expected that at corresponding 

 current velocities there will be large stationary vertical displacements in the density 

 transition layer. 



The currents in the two water masses in sea straits usually have different velocity 

 values and are of opposite directions. This case can also be treated theoretically. If 

 the thickness of the upper and lower layer is small compared with the wavelength of 

 the bottom wave and their velocities are c„ and Ci, then the conditions for large 

 stationary boundary waves is given with sufficient accuracy by 



c] hi -{-clhl=(^l- ^^ g hi h,. (XVL17) 



A good example of this case is shown in the longitudinal density section through the 

 Bosphorus in Fig. 241. The isopycnals clearly follow the outline of the bottom. 



