Vorticity wave interaction, Krein collision, and exceptional points in shear flow instabilities

We relate the model of vorticity wave interaction to Krein collision, $\mathcal{PT}$-symmetry breaking, and the formation of exceptional points in shear flow instabilities. We show that the dynamical system of coupled vorticity waves is a pseudo-Hermitian system with nonreciprocal coupling terms. Krein signatures of the eigenvalues are illustrated as the signs of the action of the vorticity waves. Interaction between positive-action and negative-action vorticity waves then corresponds to the Krein collision between eigenvalues with opposite Krein signatures, the spontaneous breaking of $\mathcal{PT}$ symmetry, and the formation of exceptional points. The control parameter of the $\mathcal{PT}$-symmetry-breaking bifurcation is the ratio between frequency detuning and coupling strength of the vorticity waves. The critical behavior near the exceptional points is described as a transition between phase-locking and phase-slip dynamics of the vorticity waves. The phase-slip dynamics correspond to nonmodal, transient growth of perturbations in the regime of unbroken $\mathcal{PT}$ symmetry, and the phase-slip frequency $\mathrm{\Omega }\propto |k-k_c{|}^{1/2}$ shares the same critical exponent with the phase rigidity of system eigenvectors.

Figure 1

Examples of two-interface shear layer profiles $U\left(x\right)$, with the dispersion relations of isolated vorticity waves ${\omega }_r\left(k\right)$ (dashed lines) and the real part of the corresponding eigenvalue branches $\mathrm{Re}\left(E\right)-k$ (solid lines). (a) A piecewise profile that supports two copropagating vorticity waves, with $\mathrm{\Delta }{\overline{q}}_1=1$ and $\mathrm{\Delta }{\overline{q}}_2=0.1$. (b) A piecewise profile that supports two counterpropagating vorticity waves, with $\mathrm{\Delta }{\overline{q}}_1=1$ and $\mathrm{\Delta }{\overline{q}}_2=-1$. The eigenvalue branches are labeled with their Krein signatures.

Figure 2

Graphical interpretation of the Krein signatures in Fig. 1, with two eigenvalue branches of the pencil $\mathcal{L}$ at (a) $k=0.4$, (b) $k=k_c\simeq 0.64$, and (c) $k=0.8$. The zeros of the eigenvalue branches correspond to purely real eigenvalues of $\mathbf{H}$.

Figure 3

Examples of three-interface piecewise profiles $U\left(x\right)$, the dispersion relations of isolated vorticity waves, ${\omega }_r\left(k\right)$ (dashed lines), and the real part of the corresponding eigenvalue branches, $\mathrm{Re}\left(E\right)-k$ (solid lines). (a) A piecewise profile that supports three copropagating vorticity waves, with $\mathrm{\Delta }{\overline{q}}_1=1/2$, $\mathrm{\Delta }{\overline{q}}_2=3/2$, and $\mathrm{\Delta }{\overline{q}}_3=1/2$. (b) A symmetric jet profile that supports three counterpropagating vorticity waves, with $\mathrm{\Delta }{\overline{q}}_1=2$, $\mathrm{\Delta }{\overline{q}}_2=-2$, and $\mathrm{\Delta }{\overline{q}}_3=2$. (c) An asymmetric jet profile with $\mathrm{\Delta }{\overline{q}}_1=2$, $\mathrm{\Delta }{\overline{q}}_2=-3/2$, and $\mathrm{\Delta }{\overline{q}}_3=3/2$. The eigenvalue branches are labeled with their Krein signatures.

Figure 4

Graphical interpretation of the Krein signatures in Fig. 3, with three eigenvalue branches of the pencil $\mathcal{L}$ at (a) $k=0.4$, (b) $k=0.8$, (c) $k=1.2$, (d) $k=1.5$, and (e) $k=1.8$. The zeros of the eigenvalue branches correspond to purely real eigenvalues of $\mathbf{E}$.

Figure 5

The eigenvalues $E_1\left(k\right)$ (red) and $E_2\left(k\right)$ (blue) in the complex $\left[\mathrm{Re}\right(E$), $\mathrm{Im}\left(E\right),k]$ space. The saddle-node exceptional point locates at $k=k_c$.

Figure 6

Parameter space of the two vorticity wave system and bifurcation of fixed points with the control parameter $k$. The black vertical dashed line is the critical $k=k_c\simeq 0.64$. When $k\ge k_c$, the system has two neutrally stable fixed points, located at ${\theta }_{12}=0$, with unbroken $\mathcal{PT}$ symmetry. When $0<k<k_c$, the system has a pair of stable (solid red line, $-\pi <{\theta }_{12}<0$) and unstable (dashed red line, $0<{\theta }_{12}<\pi$) fixed points, and $\mathcal{PT}$ symmetry is spontaneously broken.

Figure 7

Plot of the phase rigidity $\left|r\right|$ as a function of $k$. At the EP, $r=0$ and the eigenvectors are identical. In the vicinity of the EP, $\left|r\right|\propto |k-k_c{|}^{1/2}$ and the eigenvectors are extremely nonorthogonal. When $k\gg k_c$, the coupling is weak and $\left|r\right|\to 1$. Note that at $k=0.5$, the frequency mismatch of two vorticity waves vanishes and $\left|r\right|\simeq 1$.

Figure 8

The amplitude and phase dynamics of two coupled counterpropagating vorticity waves, calculated from Eqs. (44a) and (44b). The initial perturbations are random with amplitude $\sim {10}^{-5}$. The control parameter of the system is (a) $k=0.4$, (b) $k=0.639\sim k_c^{-}$, (c) $k=0.641\sim k_c^{+}$, and (d) $k=1$.

Figure 9

The critical exponent of phase-slip frequency $\mathrm{\Omega }$ near the EP is $s=1/2$. The red dots are numerically calculated from Eqs. (44a) and (44b), and the solid blue line is $\propto |k-k_c{|}^{1/2}$.