There are five phases in this run. First, there is a linear phase, in which the nonlinear interactions are neglected. From some initial small perturbations, unstable modes begin to grow exponentially. Nothing else interesting happens.
Next, all equilibrium gradients are turned off, the streamer is frozen in time, and the nonlinear interactions are turned on. As can be shown analytically, the streamer "equilibrium" is unstable to small perturbations. This instability is very similar to the familiar Kelvin-Helmholtz instability. It is driven by the shear in the strong poloidal gradients in the electrostatic potential of the streamer. The Kelvin-Helmholtz-like perturbations grow with a growth rate that is proportional to the amplitude of the streamer. Thus, they will always "catch up" in a real simulation, where the streamer would continue to grow, and the equilibrium gradients were not artificially turned off. The spectrum of the KH-like instability as a function of radial wavenumber has been calculated analytically and numerically -- the agreement is good. (Here, the x-axis is k_x rho, where k_x is the twisting coordinate, related to the ballooning angle theta_0 by k_x = k_theta s_hat theta_0. The y-axis is the growth rate, normalized to the velocity shear of the streamer. The curve is derived analytically; the points are from a numerical simulation. To derive this curve consistently (including at high k_x rho) one must solve an eigenvalue problem, not a three-mode problem. The assumptions are different than those used for the three-mode coupling, modulational instability problem.)
In the third phase, we begin with the large amplitude perturbations, and again turn off the nonlinear interactions. In the absence of equilibrium gradients, but in the presence of curvature, the perturbations undergo linear transit-time damping. This is described in more detail in Chapter 5 of Mike Beer's PhD thesis. The red curves -- the zonal flows -- are strictly undamped after the transit-time damping, as predicted by Rosenbluth and Hinton. Trapped particles cause the non-axisymmetric modes to be only weakly damped after the initial TTD. Rosenbluth and Hinton argued that the undamped zonal flows would be nonlinearly driven, would nonlinearly damp the linearly unstable modes, and would themselves be damped mainly by collisions. This led them to conjecture that low-collisionality tokamaks could exhibit improved confinement compared to existing tokamaks.
However, as can be seen in the fourth phase, Rosenbluth and Hinton failed to consider whether the zonal flow "equilibrium" was stable to small perturbations. In general, it is not. In this phase, we artificially reduce the modes which are not zonal flows to small amplitude, and turn the nonlinear interactions back on. Since the equilibrium gradients are still turned off, one would expect from the Rosenbluth-Hinton paper that these perturbations would remain small. Instead, they are observed to grow. The growth rate is proportional to the amplitude of the zonal flows.
In the final phase, we see that the zonal flows can actually be strongly damped by this nonlinear instability. This is the basic dynamical mechanism of turbulent viscosity in a toroidal plasma.
In this numerical experiment, one can see the important role of the collisionless nonlinear instability that regulates the accumulation of zonal flow that Rosenbluth and Hinton conjectured would be important in a low collisionality tokamak.
The collisionless nonlinear instability that breaks up the zonal flow is very interesting. In the large amplitude, flat density limit, the instability looks like this. Associated with the radial electric field from the zonal flow is a perturbed temperature gradient, shown here in red. The temperature gradient is maximum when the shear in the radial electric field is small. Interestingly, the mode requires the radial electric field shear to grow, since this localizes the mode. The real and imaginary parts of the eigenfunction of the nonlinear instability are shown. What is important to note is that the mode has a small radial extent.