390 General Theory of Ocean Currents in a Homogeneous Sea 



This represents a rather simple connection between the friction velocity and the 

 actual velocity distribution above the bottom. The integration constant Cq can be 

 related to a roughness length or parameter k. It has been found that for small bottom 

 irregularities such as occur on a flat bottom, sand or snow surfaces or surfaces with 

 not too large plants Cq can be given the value Cq = (A:/7'35), where k is the average 

 roughness parameter corresponding to the irregularities. If the bottom irregularities 

 are very large, it is difficult to determine the position of the point where z = for 

 which the mixing length should vanish. It is then best to shift the zero point upwards 

 by a distance Zq and to use z -\- z^m place of r in equation (XIII. 19). This will then 

 mean that in the space within the major irregularities the mean height of which is Zq 

 the turbulent mixing length falls very rapidly to zero. 



The turbulent eddy viscosity coefficient -q can be obtained from equations (XIII. 16 

 and 17) 



rj = phl^ = pU^KZ. (XIII.20) 



In the lowest bottom layers it will at first increase linearly with distance from the 

 bottom; but above a certain height it is generally assumed to remain a constant. 



There are very few oceanic observations with which it would be possible to test this logarithmic 

 law for ocean currents above the sea bottom. This would require measurements at close intervals from 

 just above the bottom to a considerable height above it. The measurements made by Merz (Moller, 

 1928) in the southern entrance to the Dardanelles, which is sufficiently wide for the current to be un- 

 affected by the lateral boundaries, are probably suitable for this. Only the layers just above the bottom 

 need to be considered. Here the rather strongly scattered individual values of the three series of measure- 

 ments gave the following distribution: 



Height above the bottom (m) . . 2 7 12 17 22 27 

 II (cm/sec) 0-3 2-8 4-6 5-5 6-5 7-2 



These values follow a logarithmic law rather well and lead to the equation 



It z 



-= 5-75 log j--;z. 



u^ 1-32 



The representation of the observations by this equation is entirely satisfactory. It is of interest that in 

 spite of the certainly rather pronounced unevenness of the bottom (hence a large value for Cq) 

 the quantity Zq introduced above is apparently zero. This may be because the heights z above the bot- 

 tom are already heights above a "mean" sea bottom and in actual fact already represent z -1- Zq. 

 This dependence of velocity on height appears to apply only up to 25 m above the bottom. As 

 shown by observation the behaviour of u is then higher up completely different. 



Current measurements near the sea bottom have been made by Mosby (1947) in order to study tur- 

 bulence and friction in the bottom layers. Using a special apparatus he has measured the direction 

 and intensity of the current in the Avaerstrommen (near Bergen, Norway) up to 2 m from the bottom 

 over a period of 3^ h; this gave the following mean vertical distribution of the horizontal velocity: 



z (cm above the bottom) . . 25 50 75 100 125 150 200 



M(cmsec-i) 16 23 27 29 31 31-7 32-5 



These values can be represented rather well by the equation 



^^ = 5-75 log ^^^. 



It does not seem to be necessary to consider Zq in the formula. Later measurements (1949) did not show 

 such simple conditions; in the bottom layer (just above the sea bed) the velocity fell off very rapidly 

 to small values. The changes in the «-values with time at different heights above the bottom show clearly 

 the turbulence of the current; it appears to decrease only very slowly towards the bottom. 



