Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation

Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conventional quantum particles. This causes unusual phase-space dynamics that is typically not captured by the WKE. We demonstrate how to correctly model this dynamics with the WME instead. Specifically, we report full-wave phase-space simulations of the zonal-flow formation (zonostrophic instability), deterioration (tertiary instability), and the so-called predator-prey oscillations. We also show how the WME facilitates analysis of these phenomena, namely, (i) we show that full-wave effects critically affect the zonostrophic instability, particularly its nonlinear stage and saturation; (ii) we derive the tertiary-instability growth rate; and (iii) we demonstrate that, with full-wave effects retained, the predator-prey oscillations do not require zonal-flow collisional damping, contrary to previous studies. We also show how the famous Rayleigh-Kuo criterion, which has been missing in wave-kinetic theories of drift-wave turbulence, emerges from the WME.

Figure 1

Plots of ${\gamma }_{\mathrm{ZI}}\left(q\right)$ at $\beta =1$ for two equilibria: (a) ${\mathcal{W}}_1$ with $\mathcal{N}=50$ and $p_f=1$ and (b) ${\mathcal{W}}_2$ with $k_x=2$, $k_y=1$, and $\mathcal{N}=100/{\left(2\pi \right)}^2$. Shown are the analytical results obtained from the WM (blue line), IWKE (red line), and TWKE (dashed line) models and the corresponding numerical results obtained from the WM (triangles) and IWKE (circles) simulations. The two blue lines in (b) correspond to two branches of $\text{Re}{\gamma }_{\mathrm{TI}}$. Only the fastest-growing mode is observed numerically.

Figure 2

Contour plots of $\mathcal{H}$ from the the IWKE for $U=u_{0}cosqy$ at $\beta =1$, $q=0.5$, and $p_x=0.5$: (a) regime 1, $u_0=0.1$; (b) regime 2, $u_0=2$; and (c) regime 3, $u_0=10$. The arrows show the phase-space velocity given by Eqs. (7) and (8). The labels $P$, $T$, and $R$ denote passing, trapped, and runaway trajectories. The vertical dashed lines in (c) denote the locations where $U^{″}=\beta$.

Figure 3

Nonlinear simulations of the ZI with the same initialization as in Fig. 1. (a) The ZF energy $E_{\mathrm{ZF}}\doteq \int U^{2}dy/2$ versus $t$ for various $q$: the IWKE model (dashed line) and the WM model (solid line). At $q\lesssim 1$, the IWKE and WM models produce similar results. At $q\gtrsim 1$, WM simulations predict oscillations of $E_{\mathrm{ZF}}$. Also shown are snapshots of $W$ from WM simulations ($q=0.4$) for (b) $t=8.0$, (c) $t=10.0$, and (d) $t=12.0$. The shape of the $\cap$ and $\cup$ structures is determined by the $R$ trajectories [Fig. 2]. Also see the movies in the Supplemental Material [44].

Figure 4

Plots of (a) ${\gamma }_{\mathrm{TI}}\left(k_x\right)$ at $\beta =0.5$ and (b) ${\gamma }_{\mathrm{TI}}\left(\beta \right)$ at $k_x=0.4$. In both cases, $U(t=0,y)=u_{0}cosqy$, $u_0=1$, $q=1.6$, and $\overline{q}=0$. Shown are the analytical approximations (10) (red line) and (11) (green line). Also shown are numerical solutions of the eigenvalue equation for $C$ (blue line) and the results of WM simulations (circles) with $W(t=0,y,\mathbf{p})=W_1\delta (p_x-k_x)e^{-p_y^2}$ (with small $W_1$).

Figure 5

Nonlinear simulations of the TI with the same initialization as in Fig. 4 ($\beta =1$ and $k_x=0.4$). (a) Energy of the ZF (blue line), DWs (green line) [19], and the total energy (red line) versus $t$. Also shown are snapshots of the normalized Wigner function $\overline{W}$ for (b) $t=8.0$, (c) $t=33.0$, and (d) $t=58.0$. Panels (c) and (d) show the presence of multiple harmonics in the $p_y$ spectrum, which is because DWs are Floquet modes rather than point particles. Also, substantial regions of negative $\overline{W}$ are present. Hence, unlike in GO, $W$ cannot be understood as the probability distribution. This shows the importance of full-wave effects. The energy oscillations seen in (a) are correlated with the horizontal shifts of the phase-space structures; compare (c) and (d). Also see the movie in the Supplemental Material [44].