Emergence of advection-diffusion transport structure and nonlinear amplitude evolution of strongly driven instabilities

Abstract

Instabilities driven by strong gradients appear in a wide variety of physical systems, including plasmas, neutral fluids, and self-gravitating systems. This work develops an analytic formulation to describe the transport structure and nonlinear amplitude evolution of a discrete, strongly driven instability in the presence of energy sources and sinks. Initially, the mode is found to evolve linearly until the gradient in the distribution has been exhausted. It then transitions to a nonlinear phase governed by a Bernoulli differential equation, for which a closed-form analytic solution is found, and continues to evolve until the energy sources and sinks reach equilibrium. During the nonlinear phase, the leading order distribution function is found to persistently satisfy an advection-diffusion equation in time and energy coordinates. These analytical results are shown to agree closely with nonlinear kinetic simulations and to be readily applicable in the study of resonant transport in plasmas, galaxies and viscous shear flows.

Figures (3)

• Figure 1: Comparison of theoretical predictions of amplitude (left) and nonlinear growth rate (right), Eqs. \ref{['eq: amplitude_solution']} and \ref{['eq: gammaNL']}, with nonlinear kinetic simulations (using the BOT code) for three example cases: (a) $\hat{\nu}_{\text{eff}} = 0.7$ and $\hat{\gamma}_d = 0.08$, (b) $\hat{\nu}_{\text{eff}} = 0.7$ and $\hat{\gamma}_d = 0.3$, and (c) $\hat{\nu}_{\text{eff}} = 8.0$ and $\hat{\gamma}_d = 0.02$. The amplitude saturation level $|A_{\text{sat}}| = (1.756\hat{\nu}_{\text{eff}}^3/\hat{\gamma}_{d})^{2/3}$petviashvili1999, the collisionless saturation level $|A_c|= (3.2)^2$fried1971, and the final growth rate of $\hat{\gamma}_{NL}=\hat{\gamma}_d$ are shown in dotted black. Phase I is shown in pink and Phase II is shown in green.
• Figure 2: The angle-averaged $\langle \delta f(z,\Omega) \rangle=\langle f_0(z,\Omega)+f_1(z,\Omega) - F_0(\Omega) \rangle$ computed from Eqs. \ref{['eq: f0']} and \ref{['eq: f_1_passing_ellip']} at time steps ranging from $\tau = \tau_0 =18$ (in red) through $\tau = 125$ (in magenta) plotted as solid lines. The maximum width of the resonance is given by $\Delta \Omega = 4\sqrt{|A|}$. The saturated $\langle \delta f(z,\Omega)\rangle$ of the BOT simulation is shown in dashed black, and the initial deviation (at $\tau = 0)$ from $F_0$ is shown in dotted red.
• Figure 3: Comparison of theoretical predictions of Eqs. \ref{['eq: amplitude_solution_krook']} and \ref{['eq: gammaNL_krook']} for the amplitude and nonlinear growth rate with nonlinear kinetic simulations (using the BOT code) for three example cases: (a) $\hat{\nu}_{\text{K}} = 0.1$ and $\hat{\gamma}_d = 0.1$, (b) $\hat{\nu}_{\text{K}} = 1.0$ and $\hat{\gamma}_d = 0.2$, and (c) $\hat{\nu}_{\text{K}} = 0.5$ and $\hat{\gamma}_d = 0.5$. The left column shows the amplitude and the right column shows the nonlinear growth rate for each case. The amplitude saturation level $|A_{\text{sat}}|=(1.9\hat{\nu}_{K}/\hat{\gamma}_{d})^2$berk1990, the collisionless saturation level $|A_c|=(3.2)^2$fried1971, and the final growth rate of $\hat{\gamma}_{NL}=\hat{\gamma}_d$ are shown in dotted black.