In warm climates in particular, sea breeze can induce substantial diurnal motion in
water bodies. Because wind forcing is a quadratic function of its speed, response
harmonics generated by sea breeze are present in the water level and current, in addition to
the fundamental forcing frequency (Zetler 1971). Nonlinear interactions within the water
body also transfer energy into harmonic frequencies, as shown in numerous studies of tidal
motion. In two-dimensional, depth-averaged horizontal flow, the quadratic bottom stress,
advection, and nonlinear continuity terms generate response harmonics because they are
nonlinear with respect to the current velocity, water-surface elevation, or both.
A central consideration in understanding wind-induced water motion and its harmonics
is that a water body is locally forced over its entire surface. In contrast, the tide must
propagate from a connection to the ocean and is damped by friction as it traverses the bay
or estuary. Thus, a distinction between wind and tide is that wind is a local forcing whereas
the tide is a boundary forcing. The relative strength of terms in the equations of motion is,
The sea breeze fluctuates with a frequency of 1 cpd (cycle per day) that is close to
frequencies of the diurnal tidal constituents (K1 O1, S1, and others). Similarly, higher
harmonics of the water motion induced by sea breeze (wind harmonics) lie at frequencies
near the higher harmonics of the diurnal tidal frequencies. Thus, wind harmonics can be
obscured by tidal motion and not easily detected. Conversely, tidal constituents must be
calculated carefully if wind harmonics are present because they introduce similar motion
not of gravitational origin. In embayments where the tidal amplitude is small, the sea
breeze can contribute significantly to the diurnal variance of the water surface and current.
This situation is common along the coast of Texas, where the strong predominant southeast
wind and sea breeze can dominate the tide in producing setup and setdown in its numerous
shallow estuaries and bays (Collier and Hedgpeth 1950). Militello (2000) and Militello and
Kraus (2001) examined sea-breeze-induced motion at Baffin Bay, Texas, a large, non-tidal
water body. Kraus and Militello (1999) document along-axis oscillations in water level
exceeding 0.6 m in response to periodic fronts passing East Matagorda Bay, Texas.
This paper introduces a new closed-form analytical solution of the one-dimensional
(1D), depth-averaged linearized momentum and continuity equations that incorporates
linear bottom friction and the non-linear wind stress. The analytic solution describes
linearized wind-forced motion in a 1D basin with horizontal bottom as governed by water
depth, basin length, bottom friction coefficient, wind speed, and fundamental frequency of
the oscillatory wind.
ORIGIN OF WIND HARMONICS
For focus of discussion and development of the analytic solution, a spatially uniform
oscillatory wind blowing parallel to the x-axis is specified. The wind speed is then given as
W = w0 + w sin( σt )
where w0 = speed of the steady wind, w = amplitude of the oscillatory wind, and σ =2π/T ,
in which T = period of the oscillatory wind. A sinusoidal representation for the wind with
T = 24 hr is a reasonable description of sea breeze and is implemented below.
Kraus & Militello