Orr-Sommerfeld equation and complex deformation

TL;DR

This work develops a complex deformation framework to define and analyze resonances for Rayleigh and Orr–Sommerfeld equations in a 2D channel, linking inviscid Rayleigh resonances to viscous Orr–Sommerfeld spectra as $R\to\infty$ (equivalently $\epsilon=R^{-1/2}\to0$). The authors establish a rigorous description of how resonances arise from the high-Reynolds-number limit, prove uniform Fredholm bounds for the deformed operators, and formulate a perturbation theory that yields first-order shifts for simple resonances. They also connect these resonances to generalized embedded eigenvalues and provide a detailed resonance-state description via singular ODE analysis and wave-front set considerations. The paper includes a thorough treatment of the circle and segment geometries, along with numerical illustrations for several shear profiles, highlighting boundary-layer effects in the segment case and the absence of boundary layers on the circle. Together, these results advance the understanding of stability and dissipation mechanisms in shear flows and provide a robust framework for analyzing inviscid limits in hydrodynamic spectral problems.

Abstract

For shear flows in a 2D channel, we define resonances near regular values of the shear profile for the Rayleigh equation under an analyticity assumption. This is done via complex deformation of the interval on which Rayleigh equation is considered. We show such resonances are inviscid limits of the eigenvalues of the corresponding Orr--Sommerfeld equation.

Figures (10)

• Figure 1: Shear profile $U(x)=\cos(0.7\pi x)$, $x\in [-1,1]$ with $\alpha=\frac{\sqrt{6}\pi}{5}$. This shear flow has a simple resonance $c_0=0$ for this choice of $\alpha$ (see Theorem \ref{['theorem:limit']} for the notion of resonance and § \ref{['subsection:example_segment']} for a discussion of this example). (A) Numerical computation of the viscous perturbations $c(\epsilon)$ of the resonance (snowflake) for $\epsilon:=R^{-\frac{1}{2}}\in [0, 0.1]$ predicted by Theorem \ref{['theorem:limit']}. For small $\epsilon$, numerics suggest that the resonance becomes an unstable eigenvalue (which is confirmed in § \ref{['subsection:example_segment']}). (B) Imaginary parts of $c(\epsilon)$, for $\epsilon\in [0, 0.01]$. Both figures are colored according to $\epsilon$. The method used for computation is described in Appendix \ref{['section:matlab']}.
• Figure 2: Shear profile $U(x)=\sin(3x)$, $x\in \mathbb R/2\pi\mathbb Z$, $\alpha=3$. For this flow and this choice of $\alpha$, $c_0=0$ is a simple resonance (see § \ref{['subsection:example_circle']}). There are also two numerically computed resonances near $0$. (A) Numerical computation of the viscous perturbations of the resonances (snowflakes) with $\epsilon\in [0,0.2]$. (B) Imaginary parts of $c(\epsilon)$, the viscous perturbation of $0$, for $\epsilon\in [0, 0.01]$. Both figures are colored according to $\epsilon$.
• Figure 3: Illustration of the choice of perturbations. Blue shadows represent complex values. Functions $\chi_{1}$ and $\chi_2$ are used in the proof of Lemma \ref{['lemma:invertibility_full_circle']}.
• Figure 4: Numerical computation of $\mathcal{R}$ with different choices of $\tau$ in the complex deformation (see § \ref{['section:escape_function']} for the meaning of this parameter and Appendix \ref{['section:matlab']} for the numerical methods). Near $0$, the set $\mathcal{R}$ remains unchanged for different $\tau$. Lower curves in all colors are outside the scope of Theorem \ref{['theorem:limit']}. They correspond to values of the parameter $c$ for which Rayleigh equation is not elliptic on the spaces defined by complex deformation (this is the range of $U$ on the complex deformation). (A) $U(x)=\cos(3\pi x)$, $x\in [-1,1]$, $\alpha=\frac{\sqrt{35} \pi}{2}$. (B) $U(x)=\sin(3x)$, $x\in \mathbb R/2\pi\mathbb Z$, $\alpha=3$.
• Figure 5: Complex deformations used in the proof of Theorem \ref{['theorem:resonances']}. Arrows indicate the wave front sets of resonant states. Starting with an element of $\Omega(c)$, we use Lemmas \ref{['lemma:singularity_positive_derivative']} and \ref{['lemma:singularity_negative_derivative']} to extend it as a holomorphic function on the interior of the orange rectangles. We use then Lemma \ref{['lemma:elliptic_extension']} to extend this holomorphic function to the green region. The resulting function may be restricted to the blue lines, and thus define a smooth function on $M_{a,b}$

Theorems & Definitions (84)

• Remark 1.1
• Theorem 1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Definition 1.5
• Theorem 2
• Remark 1.6
• Remark 1.7
• Lemma 2.1