Based on an inverted one-and-one-half inviscid reduced gravity shallow water model with bottom topography representing an abyssal layer under a stagnant upper layer on the equatorial β-plane

a set of field equations governing the wave-induced Lagrangian residual currents is developed. The equations show that the wave-induced Lagrangian residual current satisfies generalized geostrophic dynamics. The relation of meridional residual current to vertical residual current resulted from the varied bottom is similar to the Sverdrup transport relation. The transport process of potential vorticity for zeroth order approximation is determined by the advection whose velocity is equal to that of the wave-induced Lagrangian residual current. A Kelvin wave solution and the related solution of Kelvin wave-induced Lagrangian residual current for the case of slowly varying topography are obtained analytically. The wave solution shows that a shoaling eastward bottom can decrease the propagation speed of the Kelvin wave and cause it to take a longer time to transmit the energy from the west to the central and eastern parts of the basin

and can also shorten the wavelength and enhance the wave amplitude. The wave-induced residual current solution reveals that the existence of a sloping bottom can result in a ε order meridional component of wave-induced residual current and that Kelvin wave-induced Lagrangian currents's responses to bottom variation are greater than those of Kelvin wave orbital currents.