Merging of zonal flows in gyrofluid resistive drift-wave turbulence

Abstract

Non-linear dynamics of zonal flows is investigated in the context of the gyrofluid modified Hasegawa-Wakatani model. Merging of zonal flows and the chaotic developement of the initial zonal flow pattern is explored. Conservation equations for zonal flow momentum and energy with consistent finite Larmor radius (FLR) effects are derived and used for a quantitative analysis of zonal flow mergers in numerical simulations. The nonlinear local Reynolds stress transfer as opposed to (hyper)viscous dissipation is found to be the main cause of merging. The applicability of the concept of a phase transition in the strict thermodynamical sense is discussed in context of zonal flow transition hysteresis.

Figures (7)

• Figure 1: Zonal velocity profile for $\tau=0$ and $C=1.7$. After the development of a seemingly stable zonal flow profile up to $\sim 500 L_\perp/c_0$ two streams vanish at $x_1 \approx 6\rho_0$ and $x_2 \approx 42 \rho_0$ (left and middle vertical lines) at $t_1\approx 620$ (lower horizontal line). Another flow vanishes at $x_3 \approx 104 \rho_0$ and $t_2 \approx 1500$ (upper horizontal and right hand vertical line).
• Figure 2: . Merging of a zonal flow for the same simulation as presented in Figure \ref{['fig:what_is_merging']} with parameters $\tau_i=0, C=1.7$. The top panel depicts the zonal flow velocity profile $\langle v_y(x,t) \rangle_y$ from $t_0=800$ up to $t_1=2000L_\perp/c_0$. The dotted vertical lines at $x=104\rho_0$ show the location of the merger in the top and the bottom panel. The inset shows the vorticity profile at $t=850 L_\perp/c_0$. Non-zonal vorticity structures are present in the vicinity of the merger (red, vertical, dotted line) and also in other radial locations. The bottom panel shows the total cold-ion zonal flow momentum drive (the right-hand side of equation \ref{['zf_momentum_cold']}) integrated over the whole depicted time-frame (red, solid line) and the hyperviscous contribution to the total drive (black, dash-dotted line). Also the zonal flow profile at $t=800 L_\perp/c_0$ (blue, dashed line) is shown in the bottom panel. At the radial location of the merger $x = 104\rho_0$ the momentum drive is destructively interfering with the zonal flow momentum. The momentum drive is also interfering with a downwards flow at $x \approx 20\rho_0$. However, in this case it only leads to a weakening of the flow and not to a merger. Note that the hyperviscous contribution tends to be small but not negligible compared to the radial gradient of the Reynolds stress in equation \ref{['zf_momentum_cold']}.
• Figure 3: The time-averaged, normalized spectrum of the zonal flow profile before and after the merger for the simulation shown in Figure \ref{['fig:why_zf_merge']} ($\tau_i=0, C=1.7$). The blue dash-dotted line is the spectrum averaged over the time before the merger $t=800$ to $t=1550L_\perp/c_0$. The red, solid line is the spectrum after the merger $t=1750$ to $t=2000L_\perp/c_0$. One peak at $q_r = 12$ exists in the spectrum only before the merger. The peak at $q_r = 9$ appears amplified after the merger. The dominant radial mode-numbers are $q_r^d = 6$ initially and $q_r^d = 9$ after the merger. For higher wave-numbers the spectra are virtually indistinguishable.
• Figure 4: The distribution of the dominant radial mode numbers $q_r^d$ after $t=300 L_\perp/c_0$ (top panel) and $t=3000 L_\perp/c_0$ for 100 simulations respectively with $\tau=0$ and $C=0.4$. Each of the simulations was initialized with a random perturbation added to a Gaussian blob. Top panel: About 40 % of the simulations show a dominant radial mode-number of 7, which corresponds to a wavelength of $\lambda_r \approx 18 \rho_0$. Bottom panel: Approximately 50% of the simulations have the dominant radial mode-number of 7. In both cases other dominant mode-numbers are also present for $5 \leq q_r \leq 8$.
• Figure 5: Mean values of the dominant radial mode numbers $q_r^d$ for different values of domain-size (top) and resolution (bottom) respectively of at least twenty simulations for each value. The error-bars are the standard deviations.