FOURTH ORDER EVOLUTION EQUATIONS WITH SLOWLY VARYING DEPTH AND CURRENT AND STABILITY OF STOKES WAVES

Fourth order evolution equations with slowly varying depth and current have been derived for two-dimension-al modulation Stokes waves in arbitrary water depth.In the absence of current and variation of topography the e-quations reduce to those of Brinch-Nielsen & Jonsson,and the third order terms agree with those of Turpin & Ben-moussa & Mei.Stokes wave in the slowly varying depth and current is specified,and its instability to small side-bands is studied in detail.A general criterion of instability with growth rate 0(e~2)is given,and the maximumgrowth rate has also been obtained.An interesting result is that in the fourth-order theory the existence of currentand varying depth leads the Stokes wave to be always unstable to small sidebands disturbance.