Instability of two-dimensional Taylor-Green Vortices

TL;DR

The paper develops a general, nonperturbative instability criterion for linear Hamiltonian operators and applies it to the 2D Taylor–Green vortex in the Euler equations. By decomposing the linearized operator into finite‑rank negative directions and invariant subspaces, the authors reduce the problem to a finite‑dimensional holomorphic determinant Φ(λ) whose zeros determine unstable modes; the Taylor–Green vortex exhibits four simple unstable eigenvalues embedded in its essential spectrum on the imaginary axis, with a complex pair λ_* in a precise region, located via a rigorous computer‑assisted winding argument. The analysis also yields stability in several invariant subspaces (notably odd perturbations) and provides an orbital stability criterion, along with a Navier–Stokes corollary: for small viscosity, unstable eigenvalues persist as ν → 0^+. The framework extends to generalized (m,n) Taylor–Green vortices, predicting real instabilities for certain parameter ranges and giving a detailed spectral picture, including the fully characterized (2,2) case that reveals a richer spectrum beyond simple scaling from (1,1).

Abstract

For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.

Figures (7)

• Figure 1: On the left, real part of the (nonsmooth) eigenfunction $\omega_\star$, normalized so that $\| \omega_\star \|_{L^2 ([0, 2\pi]^2)} = \| \omega_E \|_{L^2 ([0, 2\pi]^2)} = 2\pi$. On the right, the real part of the perturbation $\omega_E + \delta \omega_\star$ with $\delta = 0.1$. Such eigenfunction and perturbation break the original symmetries of the cellular flow (odd with respect to both variables).
• Figure 2: Visual representation of the spaces $\mathrm{V}_s$, $\mathrm{V}_u$, $\mathrm{Ran}(P)$ and $\mathrm{Ker}(P)$ in Proposition \ref{['prop:orbstabgeneral']}, whose hypotheses ensure that the space $\overline{\mathrm{Ran}(\mathcal{J})}$, represented by the blue line and containing all the possible growing modes, lies entirely in the stable cone $\langle \mathcal{H} \cdot, \cdot\rangle > b_1$.
• Figure 3: Visualization of the $\mathcal{J}$ acting on the basis $\{ E_{j, k} \}_{(j, k) \in \mathcal{I}}$ of our invariant subspace $(\mathrm{Ev}_x \mathrm{Ev}_y)^{[\mathrm{odd}]}_+$. The gridpoint $(j, k)$ represents $E_{j, k}$ and it has arrows to each component of $\mathcal{J} E_{j, k}$. Gray arrows correspond to the general case, while coloured arrows correspond to the special cases arising from reflections.
• Figure 4: Visual representation of the instability semicircle on the rotated $\lambda$-plane. The chosen value of $\lambda_\star$ is compatible with Proposition \ref{['prop:EvEvstability']}.
• Figure 5: Visual representation of the winding number argument. The left panel shows the covering of the contour $\Gamma$, with each pair of consecutive $\Gamma_i, \Gamma_{i+1}$ intersecting over the contour $\Gamma$. The right panel illustrates a rigorous enclosure of $\Phi(\Gamma_i)$ . The small circular regions represent the image sets $\Phi(\Gamma_i)$, which satisfy the $\theta_i$-oriented condition (lying in a half-plane away from the origin). In our computer-assisted application, the enclosures will be balls, and the $\theta_i$ will point to the center of those balls, although this is not necessary for the validity of the Lemma.

Theorems & Definitions (49)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 1.4: Hamiltonian Operator
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Remark 2.2: Negative and kernel directions of $\mathcal{H}$
• proof : Proof of Theorem \ref{['thm:instability_criterion']}