# The modulational instability in deep water under the action of wind and dissipation

##### Citations

96 citations

61 citations

### Cites background or methods from "The modulational instability in dee..."

...0004 (the same value has been used in [15])....

[...]

...Note that we use a different definition of the surface elevation from the one in [15] where the 1/2 factor is not included (the consequence is that the coefficient in the nonlinear term in equation (2) differs by a factor of 4 from the one in equation (3....

[...]

...In [15] the equation is written in a nondimensional form and the coefficient K = Γ/ω0 is introduced)....

[...]

...We considered the NLS equation discussed in [15]...

[...]

...More recently, the role of dissipation and wind in the modulational instability has been considered together within the NLS equation, [15] (then confirmed by fully nonlinear simulations, [16])....

[...]

58 citations

### Cites background from "The modulational instability in dee..."

...The case ΓM/ f = O( 2) gives rise to the following damped/forced nonlinear Schrödinger equation [11, 8, 12]...

[...]

...For a logarithmic velocity profile in the boundary layer, the Miles growth rate ΓM results in [4, 7, 8]...

[...]

...The pressure P induced at the water surface then depends on the surface elevation η as follows [4, 8] 1 ρw P(x, t) = ΓM f c(2)p 2π ηx(x, t) (2)...

[...]

58 citations

36 citations

### Cites background from "The modulational instability in dee..."

... (2014). New terms which appear in that modification of the NLS equation may be eliminated after appropriate change of variables, and then the equation turns to the form considered in (Leblanc, 2007; Kharif et al, 2010; Onorato & Proment, 2012). Numerical simulations of individual wave groups affected by wind were performed within different frameworks in (Yan and Ma, 2010; Adcock and Taylor, 2011). Laboratory a...

[...]

##### References

8,801 citations

4,773 citations

2,177 citations

1,903 citations

### "The modulational instability in dee..." refers background in this paper

...Later on, Benjamin & Feir (1967) showed analytically that Stokes waves of moderate amplitude are unstable to small long-wave perturbations travelling in the same direction....

[...]

1,303 citations

### "The modulational instability in dee..." refers background or methods in this paper

...Following Miles (1957), but within the framework of modulated wave trains, we assume the atmospheric pressure at the interface due to wind and the water wave slope to be in phase....

[...]

...In § 2, the governing equations and the wave amplification theory of Miles (1957) are briefly presented....

[...]

...Miles (1957) developed an alternative derivation of the rate of growth of the wave energy based on a linear stability analysis of the parallel shear flows....

[...]

...We remind the reader that Miles (1957) studied a linear and uniform monochromatic wave train, whereas we are considering a weakly nonlinear modulated wave train....

[...]

...Within the framework of linearized equations, Miles (1957) assumed that the surface elevation and aerodynamic pressure are η(x, t) = aeik(x−ct) and Pa =(α + iβ)ρaU 2 1 kη, respectively, where a denotes the amplitude, k is the wavenumber, c is the phase velocity, α and β are two coefficients…...

[...]