Wind Setup

The response of a homogenous layer, initially at rest (i.e., with its free surface being horizontal), to a suddenly imposed wind of constant speed and direction consists of two parts: (1) a steady response, manifested by the piling up of water against a leeward coast (referred to as wind setup) and, as consequence, by tilted isopycnals and free surfaces, and (2) a transient or oscillatory response (seiching) of those same surfaces, as a consequence of the interplay between inertial and gravity forces. The period of the oscillatory responses will vary from lake to lake according to their geometry, and for stratified lakes, depending upon the vertical density distribution. The seiching motion of isopycnals (internal waves) in stratified waters and will not be discussed here. As for the surface seiches, their frequency a can be, in most cases estimated using Merian's formula npc , ,

where c is the speed of propagation of long surface waves in a basin of depth equal to the averaged depth of the basin H, L is the length of the basin along the direction of the wind and n is the mode of the oscillation. In lakes of complex geometry, such as those with multiple basins, the frequency of the surface seiches needs to be determined through more complicated procedures, but its magnitude is, in any case, on the same order as given by eq. 10. For example, in a lake with H k 10 m and L k 10 km, the frequency a1 of the gravest first mode is 0(10~3). Being their frequency much larger than f, surface seiches are seldom affected by rotational effects. They are subject, though, to large dissipative losses. Velocity observations collected in straights (for example connecting large lakes and embayments or several basins in multibasin lakes) are often contaminated and even dominated by large-amplitude oscillations affecting the whole water column, which are associated to the seiching motion of the free surface. The wind setup is characterized by a constant-slope free surface whose magnitude follows from a simple stress pressure-gradient balance (the two terms in the right hand side in eqns [1] and [2]). For a lake of constant depth H acted upon a uniform wind stress in the y-direction, the free surface displacement can be estimated as

where y = 0 at the center of the lake and u* is the shear velocity at the surface of the lake. Equation [11] is obtained by presuming that the lake is sufficiently narrow (i.e., b/R0 << 1, where b is the width of the basin and the Rossby radius is evaluated for long surface waves) so that both components of the transport (U, V) vanish everywhere in the basin and Coriolis effects can be effectively neglected. Coriolis forces will only affect the wind setup solution in the limit of b/ R0 >> 1.

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