To demonstrate how harmonics are generated through wind forcing, for the special case
W ≥ 0, the quadratic wind velocity is
1
1
W W = W 2 = w0 + w2 + 2w0 w sin ( σt ) - w2 cos ( 2σt )
2
(2)
2
2
Eq. 2 contains three forcing components as a steady part, a fundamental diurnal frequency
oscillatory wind, w0 = 0, and the Fourier expansion of the quadratic wind velocity produced
by Eq. (2) is
∞
∑ A sin[(2 j - 1)σt ]
W W =w
2
(3)
j
j =1
in which
-8
Aj =
(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.
ANALYTICAL SOLUTION
Equations of Motion
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
h
h
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 CfL = (8/(3π)) um Cf , where um is a representative value of the
magnitude of the current. The quantity CfL has dimensions of velocity.
3
Kraus & Militello
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