Minimal nonlinear dynamical system for the interaction between vorticity waves and shear flows

This study is a direct follow-up of the paper by Heifetz and Guha [Phys. Rev. E 100, 043105 (2019)] on a minimal nonlinear dynamical system, describing a prototype of linearized two-dimensional shear instability. In that paper, the authors describe the instability in terms of an action at a distance between two vorticity waves, each of which propagates counter to its local mean flow as well as counter to the other. Here we add to the model the effect of mutual interaction between the waves and the mean flow, where growth of the waves reduces the mean shear and vice versa. This addition yields oscillatory Hamiltonian dynamics, including states of phase slipping and libration with finite-size wave amplitude oscillations. We find that wave–mean-flow dynamics emerging from unstable normal modes in the linearized stage are doomed to librate around the antiphased neutral configuration in which the waves hinder each other's counterpropagation rate. We discuss as well how the given dynamics relates to familiar models of phase oscillators.

Figure 1

Schematic overview of the minimal model. (a) Counterpropagation mechanism of the vorticity waves $q_1\left(t\right)$ and $q_2\left(t\right)$, in regions (1,2), propagating counter to their local mean flows $U_{1,2}$. (b)–(f) Wave interaction at a distance as a function of the vorticity phase difference $\epsilon ={\epsilon }_1-{\epsilon }_2$. The far velocity field induced by one wave on the other may affect the latter's counterpropagation rate and amplitude: (b) $\epsilon =0$—the waves help each other to counterpropagate against the mean flow. (c) $\epsilon =\pi /2$—the waves amplify each other's amplitudes (the inclined lines indicate that the total vorticity perturbation is tilted against the action of the shear). (d) $\epsilon =\pi$—the waves hinder each other's counterpropagation speed. (e) $\epsilon =3\pi /2$—the waves decay each other's amplitudes (the vorticity perturbation is tilted with the action of the shear). (f) Polar phase diagram of the wave interaction mechanism. Any point on the circle represents different phase configurations (for instance, the star indicates a growing-hindering configuration as $\pi >{\epsilon }^{*}<\pi /2$). Since wave growth (decay) leads to decay (growth) of the shear $\delta U_{12}=U_1-U_2$, the latter decreases (increases) in the upper (lower) part of the circle.

Figure 2

(a)–(f) Phase plane portrait of the antisymmetric wave-action dynamics, described by Eq. (7), with $X=\mathcal{A}cos\epsilon$ and $Y=\mathcal{A}sin\epsilon$. Shown are the unit circle $\mathcal{A}=1$ (dashed) and the circle at which $\mathcal{A}=|{\mu }_0|$ (doted-dashed). Trajectories starting at angles $\epsilon \in [0,\pi /4,\pi /2,3\pi /4,\pi ,11\pi /8]$ are shown in black, blue, green, orange, red, and magenta, respectively. (a) $\mu =-2.5$, (b) $\mu =-2$, (c) $\mu =-1.5$, (d) $\mu =1.5$, (e) $\mu =2$, and (f) $\mu =2.5$. The large black dots in (b)–(e) indicate the phase difference ${\epsilon }_0$, in the unstable mode configuration (filled markers), and the decaying ones $-{\epsilon }_0$ (empty markers). Also, shown as a color map is the amplitude $\mathcal{A}$. Panels (g),(h) show phase slips of $\epsilon \left(t\right)$ and oscillations of $ln\left(\mathcal{A}\right)\left(t\right)$ for trajectories shown in panel (a). Panels (i),(j) are similar but show phase locking and amplitude growth corresponding to panel (c). Additional initial phases (gray) $\epsilon \in [5\pi /4,3\pi /2,7\pi /4]$ first decrease in amplitude before they grow.

Figure 3

Depicted are the polar vector fields corresponding to Eq. (12) (gray arrows), the fixed points, and the homoclinic orbits for different ${\gamma }_0$. $X=\tilde{\mathcal{A}}cos\left(\epsilon \right)$ and $Y=\tilde{\mathcal{A}}sin\left(\epsilon \right)$. (a) ${\gamma }_0=-2.5$, (b) ${\gamma }_0=-2.0$, (c) ${\gamma }_0=0.5$, (d) ${\gamma }_0=2.0$, (e) ${\gamma }_0=2.5$, and (f) ${\gamma }_0=5.5$. Shown as red triangles is the saddle point at $({\epsilon }^{*},{\mu }^{*})=(0,2)$. Shown as a green dot is $({\epsilon }^{*},{\mu }^{*})=(\pi ,-2)$. Orange diamonds indicate $({\epsilon }^{*},{\mu }^{*})=(\pm arccos({\gamma }_0/2),{\gamma }_0)$. Orange and red curves show the polar homoclines corresponding to Eqs. (18) and (19). The green circular line indicates a libration solution of Eq. (12). Black circular lines represent trajectories of free rotation (phase slips). The panels correspond to energies shown in Fig. 4.

Figure 4

Depicted are energy surfaces $\mathcal{E}(\tilde{\mathcal{A}},\epsilon )$ (color map) corresponding to Fig. 3. The right-hand scale represents the frequencies $\mu$. Orange and red lines show the homoclines associated with the fixed point at ${\tilde{\mathcal{A}}}^{*}=0$ and ${\epsilon }^{*}=0$, respectively. The green circular line indicates a libration solution of Eqs. (11) and (12). The top black lines [as well as the bottom black line in panel (f)] are trajectories of free rotation (phase slips).

Figure 5

Schematic illustration of the in-phase fixed point (a) and the antiphase fixed point (b). Tendencies of amplitude perturbations $\delta \tilde{\mathcal{A}}$ and phase perturbations $\delta \epsilon$ are indicated in the four quadrants surrounding the fixed points.

Figure 6

Depicted are exemplary polar trajectories of the nonlinear system Eq. (12) where $X=\tilde{\mathcal{A}}cos\left(\epsilon \right)$ and $Y=\tilde{\mathcal{A}}sin\left(\epsilon \right)$ and $\mathcal{A}\left(t\right),\epsilon \left(t\right)$. Panels (a)–(c) show results for ${\tilde{\mathcal{A}}}_0\in [2.9,3.7,4.6,5.6,6.7,7.9,9.2]$ in black, blue, green, orange, red, brown, and magenta. These trajectories depart from the initial normal mode conditions $2cos{\epsilon }_0={\mu }_0$. (a) ${\epsilon }_0=\pi /50$, ${\mu }_0=1.996$; (b) ${\epsilon }_0=\pi /4$, ${\mu }_0=1.414$; (c), ${\epsilon }_0=\pi /2$, ${\mu }_0=0$. Panel (d) depicts polar trajectories for ${\gamma }_0=5.5$ all starting at ${\epsilon }_0=3\pi /4$ (black diagonal) and onset amplitudes of $\tilde{\mathcal{A}}\in [0.1,0.2,0.3,0.4,0.5,1,2,...,15]$. Shown in red are the homocline of (19) and the in-phase saddle point. The origin and the antiphase fixed point are indicated, respectively, by orange and green. The green trajectory is associated with the normal mode initial condition as shown in Fig. 4, panel (f). Magenta trajectories indicate librated solutions for which $|{\mu }_0|<2$, and black and blue curves correspond to either libration or rotational trajectories where $|{\mu }_0|>2$. Panels (e),(g) Amplitudes $\tilde{\mathcal{A}}\left(t\right)$ and $\epsilon \left(t\right)$ for panel (a). (f),(h) Amplitudes and phases for panel (d).