Normal form of synchronization and resonance between vorticity waves in shear flow instability

A minimal model of linearized two-dimensional shear instabilities can be formulated in terms of an action-at-a-distance, phase-locking resonance between two vorticity waves, which propagate counter to their local mean flow as well as counter to each other. Here we analyze the prototype of this interaction as an autonomous, nonlinear dynamical system. The wave interaction equations can be written in a generalized Hamiltonian action-angle form. The pseudoenergy serves as the Hamiltonian of the system, the action coordinates are the contribution of the vorticity waves to the wave action, and the angles are the phases of the vorticity waves. The term “generalized action angle” emphasizes that the action of each wave is generally time dependent, which allows instability. The synchronization mechanism between the wave phases depends on the cosine of their relative phase, rather than the sine as in the Kuramoto model. The unstable normal modes of the linearized dynamics correspond to the stable fixed points of the dynamical system and vice versa. Furthermore, the normal form of the wave interaction dynamics reveals a new type of inhomogeneous bifurcation: annihilation of a stable and an unstable star node yields the emergence of two neutral center fixed points of opposite circulation.

Figure 1

Schematic of interacting vorticity waves in a shear flow. A hyperbolic tangent shear layer and the corresponding vorticity gradient profile are shown on the left-hand side. On the right-hand side, the cross-stream displacement, the associated cross-stream velocity, and the associated sign of vorticity for each wave are shown and represented by the same color. The position of each undulating material line after a short time interval is shown by dashed line. Interaction leads to an additional cross-stream velocity (shown by a different color). Note that cross-steam velocities due to undulations of the other material line are weaker (represented by shorter arrows) than those due to the self-induced vorticity anomalies. The horizontal arrow associated with a wave indicates the intrinsic wave propagation direction. Both waves are counterpropagating, i.e., moving opposite to the background velocity at that location.

Figure 2

Schematic description of the linear interactions between counterpropagating vorticity waves. The waves depict interfacial displacement, while the horizontal and vertical arrows, respectively, denote streamwise (background) and cross-stream velocities. Note that cross-steam velocities due to undulations of the other material line are weaker (represented by shorter arrows) than those due to the self-induced vorticity anomalies. (a) Fully helping, (b) fully hindering, (c) fully growing, and (d) fully decaying configurations. (e) Depending on the phase difference $\epsilon$ between the vorticity perturbations at the upper and lower undulating material lines, different kinds of linear interactions can be expected, as shown by the “concentric circles.” The locations where the configurations (a)–(d) occur have been marked. The configuration given in Fig. 1 lies in the second quadrant (shaded in gray), which is the “growing-hindering configuration.”

Figure 3

Bifurcation diagram. Black line denotes ${\lambda }_i^{NM}$, and red line denotes ${\lambda }_r^{NM}$. The unstable normal modes (${\lambda }_r^{NM}>0$) compose the branch of stable fixed points (marked by the solid red line) of the dynamical system Eq. (7), whereas the stable modes (${\lambda }_r^{NM}<0$) compose the unstable branch (marked by the dashed red line).

Figure 4

Phase portrait with colors indicating the divergence field, $\mathcal{D}$. The green and black dots denote the fixed points. A unit circle, centered at the origin, is plotted in cyan. (a) $\mu =-3$, (b) $\mu =-2$, (c) $\mu =-1$, (d) $\mu =0$, (e) $\mu =1$, (f) $\mu =2$, and (g) $\mu =3$.

Figure 5

Phase portrait with colors indicating the curl field, $\mathcal{C}$. Everything else is the same as in Fig. 4.

Figure 6

Trace-determinant diagram of the Jacobian matrix evaluated at the fixed points. Two neutral center fixed points, respectively shown by dark blue (positive circulation) and magenta (negative circulation) circles, approach the origin (path shown by the blue solid arrow) along the determinant axis (where ${\mathcal{D}}^{*}=0$), as the bifurcation parameter, $\mu$, is decreased from a high absolute value to 2 (cf. Fig. 3). A new type of bifurcation occurs at $\left|\mu \right|=2$; as $\left|\mu \right|$ is further decreased, a source (filled orange circle with outward arrows) and sink (filled green circle with inward arrows) fixed points are born. These fixed points lie on the trace-determinant parabola (${\mathrm{trace}}^2=4\mathrm{determinant}$ where ${\mathcal{C}}^{*}=0$). The exactly opposite behavior happens when $\left|\mu \right|$ is increased beyond 2, along the path indicated by the dashed blue arrows.

Figure 7

Phase portrait showing selected trajectories in the conservative phase space ${\mathcal{A}}_1-{\mathcal{A}}_2-\epsilon$. The Rayleigh instability problem of Sec. 4 has been chosen. (a) Unstable case: $\mu =0$ and initial points located along ${\mathcal{A}}_1=-{\mathcal{A}}_2$, (b) Stable case: $\mu =3$ and initial points located along ${\mathcal{A}}_1=-\left[{\mu }^2/2-1+\mu \sqrt{{\left(\mu /2\right)}^2-1}\right]{\mathcal{A}}_2$ and (c) stable case: $\mu =3$ and initial points located along ${\mathcal{A}}_1=-\left[{\mu }^2/2-1-\mu \sqrt{{\left(\mu /2\right)}^2-1}\right]{\mathcal{A}}_2$. The red lines denote ${\mathcal{A}}_1=-\left[{\mu }^2/2-1+\pm \mu \sqrt{{\left(\mu /2\right)}^2-1}\right]{\mathcal{A}}_2$ for $\epsilon =0$. Colors show the normalized pseudoenergy, which, being a constant of motion, remains constant along each trajectory.

Figure 8

Bifurcation diagram for Rayleigh instability. Black line denotes ${\lambda }_i$, solid red line denotes ${\lambda }_r>0$, marking the unstable normal mode, and the dashed red line denotes ${\lambda }_r<0$, i.e., the stable normal mode. Bifurcation occurs when $k=0.64$ (shown by the thin vertical line). Direct comparison can be made with Fig. 3, noting that here $\mu$ is always greater than $-2$.

Figure 9

Schematic of a general shear layer, the complex instability dynamics of which can be understood using the minimal model of $N$ ($=4$ in this case) interacting vorticity waves. The color convention is same as that of Fig. 1.

Figure 10

Rayleigh's piecewise shear layer (thin solid line) and scaled shear layer, $U=tanh\left(by\right)/tanh\left(b\right)$ (thick solid line). The upper (lower) dashed line represents $y=1(-1)$. (a) Mean velocity profiles and (b) mean vorticity gradient profiles.

Figure 11

(a) Dispersion relation, i.e., growth rate versus wave number for Rayleigh (gray) and scaled hyperbolic tangent shear layer (black). (b) Normalized perturbation vorticity for Rayleigh (black and white contour lines; black denotes positive and white denotes negative) and scaled hyperbolic tangent shear layer (filled colored contours). (c) Normalized perturbation streamfunction following the same color scheme as in panel (b).