Long-wave transport along a depth step ($c_T=fd{R}_D/H_0$)

The long step-trapped topographic Rossby wave has equal phase velocity and group velocity, $c_T=fd{R}_D/H_0$. For an outer surface profile $\eta ={\eta }_0\mathrm{sgn}(x)-A(x,t)e^{-|y|/R_D}$, cross-step matching gives $A_t+c_T{A}_x=2c_T{\eta }_0\delta (x)$. With $A(x,0)=0$, its weak solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-c_{T}t)]$. This describes the slowly varying outer response; the fast adjustment region at the origin is unresolved.