On high-order perturbation expansion for the study of long–short wave interactions

In high-order analysis and simulation of long–short surface wave interaction using mode decomposition, ‘divergent’ terms of the form $k_{S}a_{L}=O(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D716})\gg 1$ appear in the high-order expansions, where $k_{L,S}$, $a_{L,S}$ are respectively the long, short modal wavenumbers and amplitudes, with $\unicode[STIX]{x1D6FE}\equiv k_{S}/k_{L}\gg 1$ and $k_{L}a_{L}\sim k_{S}a_{S}=O(\unicode[STIX]{x1D716})$ finite. We address the effect of these terms on the numerical scheme, showing numerical cancellation at all orders $m$; but increasing ill-conditioning of the numerics with $\unicode[STIX]{x1D6FE}$ and $m$, which we quantify. In the context of mode decomposition, we show theoretical exact cancellation of the divergent terms up to $m=3$, extending the existing result of Brueckner & West (J. Fluid Mech., vol. 196, 1988, pp. 585–592) and supporting the conjecture that this is obtained for all orders $m$. We show the latter by developing a theoretical proof for any $m$ using a Dirichlet–Neumann operator and mathematical induction. The implication of the theoretical proof on the numerical simulation of long–short wave interaction is discussed.