A numerical model of the air flow above water waves. Part 2

Further results from the nonlinear numerical model of the air flow in a deep turbulent boundary layer above water waves described in Gent & Taylor (1976) are presented. The results are calculated with the surface roughness z0 both constant and varying with position along the wave. With the form used when z0 varies, the fractional rate |ζ| of energy transfer per radian advance in phase due to the working of the pressure forces is larger than for z0 constant both when the transfer is from wind to waves and when it is from waves to wind. The latter case occurs when the waves are travelling faster than, or against, the wind. The energy transfer rates are compared with other theoretical predictions and with recent field observations.

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Field Observations of Air Flow Above the Waves

Turbulent Fluxes Through the Sea Surface, Wave Dynamics, and Prediction ◽

10.1007/978-1-4612-9806-9_32 ◽

1978 ◽

pp. 483-494 ◽

Cited By ~ 9

Author(s):

Lutz Haase ◽

Manfred Gruenewald ◽

Dieter E. Hasselmann

Keyword(s):

Air Flow ◽

Field Observations ◽

The Waves

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Peak shape analysis of diagonal and off-diagonal features in the two-dimensional electronic spectra of the Fenna–Matthews–Olson complex

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0201 ◽

2012 ◽

Vol 370 (1972) ◽

pp. 3692-3708 ◽

Cited By ~ 10

Author(s):

Dugan Hayes ◽

Gregory S. Engel

Keyword(s):

Energy Transfer ◽

Electronic Spectra ◽

Nonlinear Response ◽

Quantum Beats ◽

Two Dimensional ◽

Peak Shape ◽

Transfer Rates ◽

Prosthecochloris Aestuarii ◽

Enormous Amount ◽

Theoretical Predictions

We have recorded a series of two-dimensional electronic spectra of the Fenna–Matthews–Olson (FMO) complex from Prosthecochloris aestuarii , with several crosspeaks sufficiently resolved to permit a quantitative analysis of both the amplitude and the two-dimensional peak shape. The exponential growth and/or decay of peaks on and off the main diagonal provides information on population transfer rates between pairs of excitons. Quantum beats observed in the amplitudes and shapes of these peaks persist throughout the relaxation process, indicating that energy transfer in FMO involves both incoherent and coherent dynamics. By comparing the oscillations in the amplitude and shape of crosspeaks, we confirm theoretical predictions regarding their correlation and identify previously indistinguishable combinations of nonlinear response pathways that contribute to the signal at particular positions in the spectra. Such analysis is crucial to understanding the enormous amount of information contained in two-dimensional electronic spectra and offers a new route to uncovering a complete description of the energy transfer kinetics in photosynthetic antennae.

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Linear dynamics of wind waves in coupled turbulent air—water flow. Part 1. Theory

Journal of Fluid Mechanics ◽

10.1017/s0022112094001710 ◽

1994 ◽

Vol 271 ◽

pp. 119-151 ◽

Cited By ~ 33

Author(s):

S. E. Belcher ◽

J. A. Harris ◽

R. L. Street

Keyword(s):

Growth Rate ◽

Water Flow ◽

Water Waves ◽

Wind Waves ◽

Air Flow ◽

Local Equilibrium ◽

Outer Region ◽

Wave Growth ◽

Drift Current ◽

The Waves

When air blows over water the wind exerts a stress at the interface thereby inducing in the water a sheared turbulent drift current. We present scaling arguments showing that, if a wind suddenly starts blowing, then the sheared drift current grows in depth on a timescale that is larger than the wave period, but smaller than a timescale for wave growth. This argument suggests that the drift current can influence growth of waves of wavelength λ that travel parallel to the wind at speed c.In narrow ‘inner’ regions either side of the interface, turbulence in the air and water flows is close to local equilibrium; whereas above and below, in ‘outer’ regions, the wave alters the turbulence through rapid distortion. The depth scale, la, of the inner region in the air flow increases with c/u*a (u*a is the unperturbed friction velocity in the wind). And so we classify the flow into different regimes according to the ratio la/λ. We show that different turbulence models are appropriate for the different flow regimes.When (u*a + c)/UB(λ) [Lt ] 1 (UB(z) is the unperturbed wind speed) la is much smaller than λ. In this limit, asymptotic solutions are constructed for the fully coupled turbulent flows in the air and water, thereby extending previous analyses of flow over irrotational water waves. The solutions show that, as in calculations of flow over irrotational waves, the air flow is asymmetrically displaced around the wave by a non-separated sheltering effect, which tends to make the waves grow. But coupling the air flow perturbations to the turbulent flow in the water reduces the growth rate of the waves by a factor of about two. This reduction is caused by two distinct mechanisms. Firstly, wave growth is inhibited because the turbulent water flow is also asymmetrically displaced around the wave by non-separated sheltering. According to our model, this first effect is numerically small, but much larger erroneous values can be obtained if the rapid-distortion mechanism is not accounted for in the outer region of the water flow. (For example, we show that if the mixing-length model is used in the outer region all waves decay!) Secondly, non-separated sheltering in the air flow (and hence the wave growth rate) is reduced by the additional perturbations needed to satisfy the boundary condition that shear stress is continuous across the interface.

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A numerical model of the air flow above water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112076001158 ◽

1976 ◽

Vol 77 (1) ◽

pp. 105-128 ◽

Cited By ~ 107

Author(s):

P. R. Gent ◽

P. A. Taylor

Keyword(s):

Numerical Model ◽

Water Waves ◽

Energy Input ◽

Phase Speed ◽

Nonlinear Effects ◽

Turbulent Energy ◽

Mean Value ◽

Fractional Rate ◽

Order Of Magnitude ◽

Wave Slope

A numerical model is proposed for the flow in a deep turbulent boundary layer over water waves. The momentum equations are closed by the use of an isotropic eddy viscosity and the turbulent energy equation. For small amplitudes the results are similar to those of Townsend's (1972) linear model, but nonlinear effects become important as the ratio of wave height to wavelength increases. With uniform surface roughness zo, the predicted fractional rate of energy input per radian advance in phase, ζ, decreases slightly with increasing amplitude and is of the same order of magnitude as in Miles’ (1957, 1959) and Townsend's linear theories. If zo is allowed to vary with position along the wave, however, the fractional rate of energy input can be significantly increased for small amplitude waves. If the variation in zo is half the mean value and the maximum wave slope zak is 0.01, we find ζ ≈ 60 (ρair/ρwater) (uo/c)2, where uo is the friction velocity and c the wave phase speed. Comparison is also made with recent laboratory and field data.

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Experiments on the generation of small water waves by wind

Journal of Fluid Mechanics ◽

10.1017/s0022112069001352 ◽

1969 ◽

Vol 35 (4) ◽

pp. 625-656 ◽

Cited By ~ 31

Author(s):

E. J. Plate ◽

P. C. Chang ◽

G. M. Hidy

Keyword(s):

Water Waves ◽

Wind Waves ◽

Air Flow ◽

Wave Spectrum ◽

Direct Interaction ◽

Turbulence Structure ◽

Constant Frequency ◽

Equilibrium Limit ◽

Small Water ◽

The Waves

The generation and growth of small water waves by a turbulent wind has been investigated in a laboratory channel. The evolution of these oscillations with fetch was traced from their inception with amplitudes in the micron range under conditions of steady air flow. The experiments revealed that the waves are generated at all air velocities in small bursts consisting of groups of waves of nearly constant frequency. After travelling for some distance downstream, these wavelets attain sufficient amplitude to become visible. For this condition, a wind speed critical to raise waves is well defined. After the first wavelets appear, two new stages of growth are identified at longer fetches if the air speed remains unchanged. In the first of these, the wave component associated with the spectral peak grows faster with fetch than any other part of the wave spectrum of the initial waves until its amplitude attains an upper limit consistent with Phillips's equilibrium range, which appears to be universal for wind waves on any body of water. If the air flow is not changed, then the frequency of this dominant wave remains constant with fetch up to equilibrium. This frequency tends to decrease, however, with increasing wind shear on the water. In the second stage of growth, only the energy of wave components with spectral densities lower than the equilibrium limit tend to increase with fetch so that the wave spectrum is maintained near equilibrium in the high-frequency range of the spectrum.The origin of the first waves and the rate of their subsequent growth was examined in the light of possible generating mechanisms. There was no indication that they were produced by direct interaction of the water surface with the air turbulence. Neither could any significant feedback of the waves into the turbulence structure be detected. The growth of the waves was found to be in better agreement with theoretical predictions. Under the shearing action of the wind, the first waves were found to grow exponentially. The growth rates agreed with the estimates from the viscous shearing mechanism of Miles (1962a) to a fractional error of 61% or less. A slight improvement was obtained with the viscous theory of Drake (1967) in which Miles’ model is extended to include the effect of the drift current induced by the wind in the water. Since the magnitude of the water currents observed in the tunnel is very small, this improvement is not significant.

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Quasilinear approximation for the spectrum of wind-generated water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112082001736 ◽

1982 ◽

Vol 117 ◽

pp. 493-506 ◽

Cited By ~ 32

Author(s):

Peter A. E. M. Janssen

Keyword(s):

Energy Transfer ◽

Water Waves ◽

Wind Profile ◽

Wave Spectrum ◽

Water System ◽

Multiple Time ◽

Critical Height ◽

Profile Changes ◽

Quasilinear Approximation ◽

The Waves

According to Miles’ theory of wind-wave generation, water waves grow if the curvature of the wind profile at the critical height is negative. As a result, the wind profile changes in time owing to the transfer of energy to the waves. In the quasilinear approximation (where the interaction of the waves with one another is neglected) equations for the coupled air–water system are obtained by means of a multiple-time-scale analysis. In this way the validity of Miles’ calculations is extended, thereby allowing a study of the large-time behaviour.While the water waves grow owing to the energy transfer from the air flow, the waves in turn modify the flow in such a way that for large times the curvature of the velocity profile vanishes. The amplitude of the waves is then limited because the energy transfer is quenched.In the high-frequency range the asymptotic wave spectrum is given by a ‘–4’ law in the frequency domain rather than the ‘classical’ ‘–5’ law.

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