In physical oceanography and fluid mechanics, the Miles-Phillips mechanism describes the generation of wind waves from a flat sea surface by two distinct mechanisms. Wind blowing over the surface generates tiny wavelets. These wavelets develop over time and become ocean surface waves by absorbing the energy transferred from the wind. The Miles-Phillips mechanism is a physical interpretation of these wind-generated surface waves.Both mechanisms are applied to gravity-capillary waves and have in common that waves are generated by a resonance phenomenon. The Miles mechanism is based on the hypothesis that waves arise as an instability of the sea-atmosphere system. The Phillips mechanism assumes that turbulent eddies in the atmospheric boundary layer induce pressure fluctuations at the sea surface. The Phillips mechanism is generally assumed to be important in the first stages of wave growth, whereas the Miles mechanism is important in later stages where the wave growth becomes exponential in time.
History It was Harold Jeffreys in 1925 who was the first to produce a plausible explanation for the phase shift between the water surface and the atmospheric pressure which can give rise to an energy flux between the air and the water. For the waves to grow, a higher pressure on the windward side of the wave, in comparison to the leeward side, is necessary to create a positive energy flux. Using dimensional analysis, Jeffreys showed that the atmospheric pressure can be displayed as
p = S ρ a ( U ∞ − C ) 2 ∂ η ∂ x {\displaystyle p=S\rho _{a}(U_{\infty }-C)^{2}{\frac {\partial \eta }{\partial x}}}
where S {\displaystyle S} is the constant of proportionality, also termed sheltering coefficient, ρ a {\displaystyle \rho _{a}} is the density of the atmosphere, U ∞ {\displaystyle U_{\infty }} is the wind speed, C {\displaystyle C} is the phase speed of the wave and η {\displaystyle \eta } is the free surface elevation. The subscript ∞ {\displaystyle \infty } is used to make the distinction that no boundary layer is considered in this theory. Expanding this pressure term to the energy transfer yields
∂ E ∂ t = 1 2 ρ w g S ρ a ( U ∞ − C ) 2 ( a k ) 2 C {\displaystyle {\frac {\partial E}{\partial t}}={\frac {1}{2\rho _{w}g}}S\rho _{a}(U_{\infty }-C)^{2}(ak)^{2}C}
where ρ w {\displaystyle \rho _{w}} is the density of the water, g {\displaystyle g} is the gravitational acceleration, a {\displaystyle a} is the wave amplitude and k {\displaystyle k} is the wavenumber. With this theory, Jeffreys calculated the sheltering coefficient at a value of 0.3 based on observations of wind speeds. In 1956, Fritz Ursell examined available data on pressure variation in wind tunnels from multiple sources and concluded that the value of S {\displaystyle S} found by Jeffreys was too large. This result led Ursell to reject the theory from Jeffreys. Ursell's work also resulted in new advances in the search for a plausible mechanism for wind-generated waves. These advances led a year later to two new theoretical concepts: the Miles and Phillips mechanisms.
Miles' Theory John W. Miles developed his theory in 1957 for inviscid, incompressible air and water. He assumed that air can be expressed as a mean shear flow with varying height above the surface. By solving the hydrodynamic equations for the coupled sea-atmosphere system, Miles was able to express the free surface elevation as a function of wave parameters and sea-atmosphere characteristics as
η = a exp [ 1 2 ε β k C w ( U C w ) 2 t ] exp [ i ( k x − ω t ) ] {\displaystyle \eta =a\exp \left[{\frac {1}{2}}\varepsilon \beta kC_{w}\left({\frac {U}{C_{w}}}\right)^{2}t\right]\exp[i(kx-\omega t)]}
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