On the exchange of stability for the subcritical laminar flow

TL;DR

This paper investigates the stability exchange in steady rotational water waves with vorticity, analyzing eigenvalues to determine bifurcation behavior and stability properties near critical flow conditions.

Contribution

It introduces a detailed eigenvalue analysis of bifurcating Stokes waves in rotational flows, revealing how stability and period change depend on flow parameters.

Findings

Second eigenvalue sign determines stability exchange and period variation.

Critical depth $d_0(a)$ separates regimes of positive and negative eigenvalues.

Numerical results illustrate the dependence of eigenvalues on flow parameters.

Abstract

We consider steady water waves in a two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. Surface tension is neglected, and the flow is rotational with constant vorticity $a$. We analyze an analytic branch of Stokes waves bifurcating from a subcritical laminar flow, with the wave period serving as the bifurcation parameter. Along this branch, the first eigenvalue of the Fr\'{e}chet derivative remains negative. Our main focus is the second eigenvalue; its sign plays a crucial role in the analysis of subharmonic bifurcations. This small eigenvalue determines the validity of the principle of exchange of stabilities: a positive sign confirms it, while a negative sign indicates its violation. Furthermore, a positive second eigenvalue corresponds to an increasing period along the bifurcation curve near the critical point, whereas a negative sign implies period decrease. We investigate how the sign of the second eigenvalue depends on the Bernoulli constant $R$ (equivalently, the laminar flow depth $d$) and the vorticity $a$. We show that for each $a$ there exists a critical depth $d_0(a)$ such that the second eigenvalue is positive for $d<d_0(a)$ and negative for $d>d_0(a)$. In the laminar flow, a stagnation point forms when the depth exceeds a threshold $d_s(a)$. We demonstrate that $d_0(a) < d_s(a)$ for $a > a_0 \approx -1.01803$, whereas $d_0(a) > d_s(a)$ for $a < a_0$. We also verify the property of formal stability by a description of the domain in $(a,d)$ variables, where this property holds. Numerical illustrations of these properties are presented in the paper.