To demonstrate how harmonics are generated through wind forcing, for the special case

1

1

2

(2)

2

2

Eq. 2 contains three forcing components as a steady part, a fundamental diurnal frequency

oscillatory wind, *w*0 = 0, and the Fourier expansion of the quadratic wind velocity produced

by Eq. (2) is

∞

∑ A sin[(2 *j *- 1)σ*t *]

2

(3)

in which

-8

(4)

π(2 *j *- 3)(2 *j *- 1)(2 *j *+ 1)

Eq. 4 shows that harmonic frequencies generated by a pure oscillatory wind are odd

1/35.

For a basin of uniform width and water depth *h *>> η (deviation of water surface from

still water), the continuity and momentum equations of depth-averaged motion are

∂η

∂*u*

+*h*

=0

(5)

∂*t*

∂*x*

∂η *C * f u u ρa CDW W

∂*u*

∂*u*

+*u*

+*g*

+

+

=0

(6)

∂*t*

∂*x*

∂*x*

ρ

wind-drag coefficient, and *W *= wind velocity. Militello and Kraus (2001) showed by

scaling analysis of Eqs. 5 and 6 that the pressure gradient term is of the same order as the

wind forcing and bottom friction terms, whereas the inertia and advective terms are 2-3

orders of magnitude smaller than the wind forcing term for the stated conditions.

A 1D basin of length *L *with vertical walls and uniform still-water depth is considered

(Fig. 1), over which an along-axis sinusoidal wind blows with spatial uniformity. The

governing equations are linearized, including omission of the advective term (which was

shown to be small in the scaling analysis), to allow closed-form solution and to eliminate

generation of response harmonics by nonlinear terms. Although it is not the intent to

compare the linear and non-linear models, the Lorentz approximation for estimating the

and Harleman 1966) gives *C*fL = (8/(3π)) *u*m Cf , where *u*m is a representative value of the

magnitude of the current. The quantity *C*fL has dimensions of velocity.

3

Kraus & Militello

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