Three-dimensional non-linear rotating surface waves in channels of variable depth in the presence of formation of a small perturbation of atmospheric pressure across the channel

We consider three-dimensional free-boundary problem on the propagation of incompressible, homogeneous and inviscid fluid with zero surface tension confined in a channel of variable depth. Since for large-scale flows the fluid motion is affected by the rotation of the earth, the model is considered in rotating reference frame. Additionally, small atmospheric pressure variations across the channel are taken into account. It is shown that the non-trivial solution to the problem represents three-dimensional solitary wave which is given by the rotation modified Korteweg-de Vries equation (fKdV): b 1 ξ xxx + b 2 ξ ξ x + b 3 ( f ) ξ x = 0 , where x is the down-channel coordinate and the coefficients b i ( i = 1 , 2 , 3 ) of the resulting fKdV equation depend on the transverse topography of the channel and, additionally, b 3 depends on the Coriolis parameter f. It is also shown that if the vertical profile of the channel is symmetric about the vertical axis, the small atmospheric variations will not appear in the resulting fKdV equation. The effects of channel’s cross-sectional geometry on the shape of the resulting three-dimensional wave profile in a longitudinal direction are studied numerically. Additionally, to better understand the effects of the Earth rotation, the above analysis is performed at different latitudes.