Sharp necessary and sufficient criteria for inviscid instability of inflectional (jet-type) parallel shear flows: the open core of Rayleigh's 1880 stability problem

Rayleigh's 1880 inflection-point theorem states that a two-dimensional inviscid parallel shear flow whose velocity profile $U(y)$ has no inflection point ($U''$ of one sign) is linearly stable — equivalently, an inflection point is a necessary condition for inviscid instability, and Fjørtoft (1950) strengthened this to require a vorticity extremum. Whether, and for exactly which profiles, these conditions are also sufficient for instability has never been settled in general: the converse is false in bounded domains (stable inflectional profiles can be constructed), and all known sharp necessary-and-sufficient criteria assume monotone profiles (Rosenbluth–Simon 1964; Balmforth–Morrison 1999; Hirota–Morrison–Hattori 2014) or Howard's unbounded class H, while the newest advance (Deguchi–Hirota–Dowling 2024, the 'hurdle theorem') still gives only a sufficient condition and states that no useful recipe is known for non-monotone basic flows in a finite domain. The problem posed here is the surviving open core of Rayleigh's 1880 question: obtain a sharp, profile-computable, necessary-and-sufficient criterion for the existence of an unstable eigenvalue of Rayleigh's equation, valid for general — in particular non-monotone, jet-type — profiles in bounded domains. The viscous clause of the 1880 question was answered as a framework by the Orr–Sommerfeld equation and is recorded as closed lineage rather than as the open core.

Background

Consider a two-dimensional, incompressible, inviscid ('frictionless') flow between parallel rigid walls $y=\pm 1$, whose base state is a parallel shear flow with streamwise velocity $U(y)$. Linearizing the Euler equations about $U(y)$ and seeking normal modes of streamfunction $\phi(y)e^{i\alpha(x-ct)}$, with streamwise wavenumber $\alpha>0$ and complex phase speed $c=c_r+ic_i$, yields Rayleigh's equation

$(U-c)(\phi''-\alpha^2\phi)-U''\phi=0,\qquad \phi(\pm 1)=0,$

where a prime denotes differentiation with respect to $y$. The flow is linearly unstable at wavenumber $\alpha$ exactly when this eigenvalue problem admits an eigenvalue with $c_i=\operatorname{Im}c>0$. A point where $U(y_c)=c_r$ is called a critical layer, and a point $y_I$ where $U''(y_I)=0$ is an inflection point of the velocity profile. The problem extends to unbounded domains (free shear layers and jets where $U$ tends to constants as $y\to\pm\infty$) with decay conditions replacing the wall conditions.

Rayleigh (1880) proved the inflection-point theorem: if $U''$ never changes sign, the flow is stable for every $\alpha$; equivalently, the presence of an inflection point is a necessary condition for inviscid instability. Fjørtoft (1950) strengthened this: instability additionally requires an extremum of vorticity within the flow (vorticity, the curl of the velocity field, is $-U'(y)$ for this base flow, so its interior extrema are exactly the inflection points of $U$; for a single inflection point $y_I$, $U'''(y_I)/U'(y_I)<0$). These are necessary conditions only. Tollmien (1935), in the first sufficiency theory, proved the converse for a restricted class of profiles (e.g. symmetric profiles vanishing at the walls, and boundary-layer-type profiles); Howard (1964) obtained sharp criteria for unbounded flows within the class of profiles he defined (his 'class H'). In bounded domains the converse is false in general: stable flows possessing inflection points can be constructed (accounts in Tollmien 1935, Drazin & Howard 1966, and Drazin & Reid 2004), so no pointwise sign condition on $U$ and $U''$ alone can be sharp.

For monotone profiles the question is now essentially settled. Rosenbluth & Simon (1964) used a Nyquist-type argument (a contour-encirclement test, imported from control theory, for the zeros of an analytic function) to give a necessary and sufficient condition for monotone profiles with a single inflection point. Balmforth & Morrison (1999) removed the single-inflection restriction: for monotone real-analytic profiles, instability at wavenumber $\alpha$ occurs if and only if a Nyquist function $\epsilon(c_r)$ encircles the origin as $c_r$ traverses the range of $U$, reducible to a sign test at the inflection-point speeds $c_I=U(y_I)$. Hirota, Morrison & Hattori (2014) recast this variationally: for monotone analytic profiles, stability is equivalent to the definiteness of an explicitly computable quadratic form $Q$. The modern viewpoint behind these results is that the linearized Euler operator is Hamiltonian, and instability arises not by passage through zero frequency but by a Kreĭn collision — the resonance of two counter-propagating neutral modes (eigenmodes with real phase speed, neither growing nor decaying) of opposite energy signature (Kreĭn signature, the sign of the wave energy in the Hamiltonian formulation) drawn from the continuous spectrum (the band of real phase speeds $c$ lying in the range of $U$, arising from the critical-layer singularities).

For non-monotone profiles — alternating or free jets, where $U(y)$ has interior maxima and minima and several inflection points — no sharp criterion is known: neutral modes may possess several critical layers, singular neutral modes (neutral modes whose streamfunction has a logarithmic singularity at a critical layer) can occur, and the Rayleigh-quotient/test-function strategy (bounding the operator's spectrum by substituting trial functions into its quadratic form) has, in the words of Deguchi, Hirota & Dowling (2024), 'no useful recipe … for non-monotonic basic flows in a finite domain'. Their hurdle theorem supplies a new sufficient condition for instability (a threshold on a profile-derived quantity over an interval of the domain) covering wide classes that include many non-monotone jet profiles, and shows that within their classes the true stability boundary is bracketed between their unstable regions and the Kelvin–Arnol'd stable regions (the energetic sufficient conditions for stability, anticipated by Thomson (Kelvin) in 1880 and systematized by Arnol'd in 1966); the bracketing is not sharp. Lin (2003) proved rigorous instability results around neutral modes for non-monotone ideal plane flows, again only partially.

The viscous clause of the 1880 question — Rayleigh's analysis was frictionless, and including viscosity was left open — was answered at the level of formulation by the Orr–Sommerfeld equation (the viscous generalization of Rayleigh's equation, written down independently in 1907–1908), whose eigenvalue problem, with the Reynolds number (the ratio of inertial to viscous forces) as parameter, became the central object of twentieth-century linear stability theory: the asymptotic theory of the viscous Tollmien–Schlichting instability waves, begun by Heisenberg (1924) and developed through the 1930s, with accurate computation beginning with Orszag (1971), who computed the critical Reynolds number $5772.22$ for plane Poiseuille flow (pressure-driven flow between parallel plates); standard reference Drazin & Reid (2004). Viscous spectral theory still generates new mathematics — e.g. Li & Zhao (2025) constructed inviscidly spectrally stable shear flows that viscous dissipation renders spectrally unstable without boundary effects — but the inclusion of viscosity as asked in 1880 is a settled framework, and the surviving open core of the 1880 question is the inviscid characterization problem stated below.

Problem Statement

For the inviscid linear stability problem of two-dimensional incompressible parallel shear flow, obtain a sharp characterization of instability: a criterion, computable directly from the velocity profile $U(y)$ (its analytic properties, derivatives, and associated global or variational quantities) without solving the full non-selfadjoint eigenvalue problem (the Rayleigh operator is not equal to its adjoint, so its eigenvalues need not be real and classical Sturm–Liouville theory does not apply directly), that is both necessary and sufficient for the existence of an eigenvalue $c$ with $\operatorname{Im}c>0$ of Rayleigh's equation

$(U-c)(\phi''-\alpha^2\phi)-U''\phi=0,\qquad \phi(\pm 1)=0,$

for each streamwise wavenumber $\alpha>0$, valid for general — in particular non-monotone, jet-type — profiles having multiple extrema and multiple inflection points in the bounded channel. A complete answer includes proofs of sufficiency (criterion implies an unstable eigenvalue exists) and necessity (instability implies the criterion holds), and an explicit statement of the regularity assumptions under which the proof is carried out. The criterion must be necessary and sufficient for every profile in the general non-monotone class above, and may not be narrowed to a structural subclass of it (e.g. symmetric profiles, or a prescribed number of extrema or inflection points).

References

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