AAPPS-DPP2023, Nov 12-17 2023, Port Messe Nagoya, JapanVorticity wave interaction and exceptional points in shear flow instabilities
C. Meng1 and Z. B. Guo2
12Southwestern Institute of Physics,
School of Physics, Peking University
e-mail (speaker): mengcong@swip.ac.cn
Interaction of vorticity waves demonstrates the mechanismof shear flow instabilities [1], and naturally provides a nonmodal approach to study optimal growth and transient dynamics
in such instabilities [2]. In the dynamical system of counterpropagating vorticity waves [3], the onset of shear instabilitiescorresponds to a bifurcation of fixed points from two neutral
centers to a pair of stable and unstable nodes.
On the other hand, the role of symmetries and symmetrybreaking bifurcations in fluid dynamics is long recognized [4],and it is recently shown via spectral analysis that shear flow
instabilities are the result of spontaneous PT-symmetry
breaking [5]. The results, however, have not yet been related tothe dynamics of vorticity waves.
In this work, we establish a link between vorticity wave
interaction and PT-symmetry breaking in shear flowinstabilities. We show that the dynamical system of counterpropagating vorticity waves is a non-Hermitian system withsaddle-node exceptional points. Vorticity wave dynamics are
directly linked to the PT-symmetry-breaking bifurcation of
fixed points (Fig.1) and the Krein collision between eigenmodes
with opposite Krein signatures. The key parameter to control
the bifurcation is the ratio between frequency detuning andcoupling strength of the vorticity waves.

of phase dynamics from a phase-slip state to a phase-locked
state near the exceptional point corresponds to spontaneous PTsymmetry breaking and the onset of shear instabilities. (Fig.2)
The phase-locked state leads to exponential growth of the shear
instabilities, while the phase-slip dynamics lead to non-modal,transient growth of initial perturbations. Near the exceptional/shares thepoint, the phase-slip frequency
same critical exponent 1/2 with the phase rigidity of systemeigenvectors.

Fig. 1. (Left) Parameter space of two coupled vorticity waves
and bifurcation of fixed points with amplitude ratio
and relative phase. When the control, the system exhibits a PT-symmetryparameter
breaking bifurcation and the two vorticity waves become phase. The saddle-nodelocked. (Right) The eigenspectra,exceptional point locates atin the complex
space.

Fig. 4. (Left) Parameter space of the three coupled vorticity, PTwave system. When the control parameter
symmetry is spontaneously broken and the three vorticity waves
inbecome phase-locked. (Right) The eigenspectra, ,space. The saddle-node exceptional points
the complexand.
locate at

Fig. 2. Amplitude and phase dynamics of two coupled vorticity
to a phasewaves, from a phase-slip state.
locked state
The most striking feature of non-Hermitian systems is theexistence of exceptional points [6]. We show that the transition

Fig. 3. Criticalbehavior of thesystem around theexceptional point,/.
The phase rigidity
measuresbiorthogonality ofeigenvectors.
We then extend the analysis to interaction of multiple
vorticity waves [7]. Multiple exceptional points exist in thesystem, as shown in Fig. 4.

In general, it is shown that the framework of vorticity wave
interaction is naturally related to non-Hermitian physics and PTsymmetry-breaking bifurcations. Therefore, we anticipate
further applications of this framework to the analysis of various
magnetohydrodynamic instabilities in plasmas [8].
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[2] E. Heifetz et al, Phys. Fluids 17, 064107 (2005).
[3] E. Heifetz et al, Phys. Rev. E 100, 043105 (2019).
[4] J D Crawford et al, Annu. Rev. Fluid Mech. 23,341 (1991)[5] H. Qin et al, Phys. Plasmas 26, 032102 (2019).
[6] K. Ding et al, Nat. Rev. Phys. 4, 745–760 (2022).
[7] A. Guha et al, J. Fluid Mech. 816, 412–427 (2017).
[8] Y. Zhang et al, Phys. Rev. Lett. 125, 255003 (2020).