N

distance away from the boundaries (and, in general, the velocity profiles) is very sensitive to the vertical distribution of the eddy viscosity K. In deriving eqns [19] and [20], K has been presumed constant, for simplicity. However, this assumption is not strictly correct since K is known to vary rapidly in the vertical direction within 'wall layers' existing near the top and bottom boundaries and where the velocity varies logarithmically. Only outside these wall layers, but within an homogenous turbulent shear flow region, such as the top and bottom mixed shear layers, the eddy viscosity is approximately constant and proportional to the velocity and the virtual extent of the flow. In any case, the velocity profiles are very sensitive to the magnitude of K and the location (near or far from lateral boundaries) where they are monitored (Figure 9). In the limit of very shallow layers (U*/fh>> 1) or, equivalently, for very large Ekman numbers (K f_1h~2 >> 1) it can be assumed that the effects of Coriolis forces in the vertical distribution of velocities are negligible. Large Ekman numbers occur also in narrow or small lakes. In those cases, the balance between pressure and frictional forces (the last two terms in eqns. [1] and [2]) results in a velocity profile in the direction of the wind (no currents perpendicular to the wind) of the form shown in Figure 8(b). For constant K the following analytical expression can be derived that represents the vertical distribution of flow velocities

Note that at the bottom (here z = 0) the velocity is zero. Observed velocity profiles in laboratory experiments are, however, closer to a double-logarithmic curve, which corresponds to a parabolic distribution of the eddy viscosity K.

In response to a suddenly imposed wind stress on its free surface, the vertical velocity distribution in a homogenous layers of thickness h >> D, consists of a steady-state Ekman spiral and inertial oscillations (of frequency f ) that penetrate to gradually increasing depths. At time t inertial oscillations exists up to a depth of order (2Kt)1/2, which is the depth to which momentum diffusion penetrates in a nonrotating system. However, the magnitude of the inertial oscillations will decrease with time as

Thus, after a long period, the inertial oscillations are imperceptibles, even if we do not consider frictional losses at the bottom (see Figure 10).

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