In particular, the experiments discussed above showed that the average ratio between the cross-wind and up-wind mean square slope components varies from 0.75 to 1.03 on slick-covered ocean surfaces. Such a large value of the cross-wind slope component cannot be explained satisfactorily with the unimodal directional distribution, and better agreement is obtained when,

for wave components shorter than the dominant wavelength, the bimodality of directional LY2109761 spreading is taken into account ( Hwang & Wang 2001). In particular, the ATM data (airborne topographic mapper) of the 3D surface topography and bimodal directional function showed that the average ratio between cross-wind and up-wind slope components can reach a value of 0.88 ± 1.0. In this section we will follow mainly the ideas developed by Massel (2007). Thus, let us define ε as a module of the local surface slope in the direction θ1 against the x axis. For two slope components along the x and y axes we have equation(30) εx=∂ζ∂x=εcosθ1,εy=∂ζ∂y=εsinθ1,in which angle θ1 increases

anticlockwise from the x axis. We assume that the x axis is the wind direction. Therefore, the up- and cross-wind of waves surface slopes are εu = εx and εc = εy respectively. To determine the statistical characteristics of wave slopes, we express the two-dimensional probability density function Anti-cancer Compound Library f(ε, θ1) for the module slope ε and direction θ1 in the form suggested by Longuet-Higgins (1957): equation(31) f(ε,θ1)=ε2πΔ××exp−ε2(σy2cos2θ1−2σxy2sinθ1cosθ1+σx2sin2θ1)2Δ,in Racecadotril which the corresponding mean square slopes are equation(32) σx2=(∂ζ∂x)2¯,σy2=(∂ζ∂y)2,¯σxy2=∂ζ∂x∂ζ∂y¯and equation(33) Δ=|σx2σxy2σxy2σy2|.The bar symbolizes the statistical averaging for a given time series of slopes. For the x   axis, parallel to the main wave direction,

the mean square slope σζxy2 is equal to zero, and eq. (31) becomes equation(34) f(ε,θ1)=ε2πΔexp−−ε2(σc2cos2θ1+σu2sin2θ1)2Δ,with Δ=σu2σc2. The subscripts c and u refer to the cross-wind and up-wind components respectively. In order to compare the theoretical distribution of slopes with the Cox & Munk (1954) experiment, we rewrite eq. (24) as a function of two slope components εu and εc, i.e. equation(35) f(εu,εc)=f(ε,θ1)⋅J=f(ε,θ1)|∂ε∂εu,∂ε∂εc∂θ1∂εu,∂θ1∂εc|.Using the fact that equation(36) ε=εu2+εc2andθ1=arctan(εcεu),we obtain equation(37) J=1εu2+εc2.Substituting the above expression in eq. (35) yields equation(38) f(ξ,η)=12πσuσcexp[−12(ξ2+η2)],where equation(39) ξ=εuσu,η=εcσc. Equation (38) takes the form of a two-dimensional Gaussian distribution. It should be noted that Cox & Munk (1954) used the two-dimensional, slightly modified Gaussian distribution (the so-called Gram-Charlier distribution) to fit their experimental data. The Gram-Charlier distribution takes the general form (Massel 1996) equation(40) f=[Gaussain distribution]×[1+∑i,jcijHiHj],in which Hi(x) are Hermite polynomials.