Theory of zonal flow growth and propagation in toroidal geometry

Abstract

The toroidal geometry of tokamaks and stellarators is known to play a crucial role in the linear physics of zonal flows, leading to e.g. the Rosenbluth-Hinton residual and geodesic acoustic modes. However, descriptions of the nonlinear zonal flow growth from a turbulent background typically resort to simplified models of the geometry. We present a generalised theory of the secondary instability to model the zonal flow growth from turbulent fluctuations in toroidal geometry, demonstrating that the radial magnetic drift substantially affects the nonlinear zonal flow dynamics. In particular, the toroidicity gives rise to a new branch of propagating zonal flows, the toroidal secondary mode, which is nonlinearly supported by the turbulence. We present a theory of this mode and compare the theory against gyrokinetic simulations of the secondary mode. The connection with other secondary modes -- the ion-temperature-gradient and Rogers-Dorland-Kotschenreuther secondary modes -- is also examined.

Figures (12)

• Figure 1: Zonal flow velocity $v_E^\mathrm{ZF}$ (see Section \ref{['sec:review']} for definition; other quantities in the figure follow the standard conventions and are also defined in Section \ref{['sec:review']}) in real space (left) and Fourier space (right) from a nonlinear gyrokinetic simulation of tokamak ion-temperature-gradient turbulence with safety factor $q=4.2$. There are stationary ZFs and geodesic acoustic modes (GAMs) at large scales, and propagating ZFs at smaller radial scales corresponding to toroidal secondary modes. General information about the simulations shown in this study may be found in Section \ref{['sec:GK_sims']}. We note that a similar plot for a different set of parameters was previously presented in nies_saturation_2024.
• Figure 2: Illustrative case of secondary modes in the space of the primary amplitude $\mathcal{A}^P$ and secondary radial wavenumber $k_x$ (a), and qualitative picture (b). The TSM ($\omega_r \neq 0$) is found at short radial wavelengths $k_x \gtrsim \omega_\parallel / v_{Mx}$ as it is Landau damped at long radial wavelengths (Section \ref{['sec:streaming_TSM']}), and above a primary amplitude threshold $\mathcal{A}^P \gtrsim 1$ as it is otherwise subject to kinetic damping by the radial magnetic drift (Section \ref{['sec:TSM_R']}). The RDK secondary mode ($\omega_r = 0$) is found at large primary amplitudes, $k_x \tilde{v}_{g}^P \gtrsim \omega_\mathrm{GAM}$ for short wavelengths $k_x \gtrsim \omega_\parallel / v_{Mx}$ (Section \ref{['sec:TSM_NR']}), and $k_x \tilde{v}_{g}^P \gtrsim \omega_\parallel$ for long wavelengths $k_x \lesssim \omega_\parallel / v_{Mx}$ (Section \ref{['sec:streaming_TSM']}). The horizontally dashed region in (b) is where the non-resonant limit of the toroidal secondary mode exists (Section \ref{['sec:TSM_NR']}), between the TSM and RDK regions. The ISMs, discussed in Section \ref{['sec:ISM']}, are dominant at short wavelengths $k_x \gtrsim \omega_\parallel / v_{Mx}$ for weak primary drive $\mathcal{A}^P \sim 1$, and at sub-Larmor scales $k_x \rho_i \gtrsim 1$. The diagonally hashed regions in (b) have weak primary drive $k_x \tilde{v}_g^P \lesssim \omega_\parallel$; the comparatively fast parallel streaming rate then makes the choice of parallel boundary condition important and the assumptions made in our secondary mode theory may become inappropriate (Section \ref{['sec:streaming_TSM']}). Note that the size and shape of the regions can vary depending on the parameters used, e.g. the dependence of the primary drive $k_x \tilde{v}_g^P$ on velocity, $y$, and $\theta$, or the magnetic geometry.
• Figure 3: Secondary mode frequency (left) and growth rate (right) of the fastest growing modes as a function of $k_x \rho_i$ (a) and phase shift $\delta$ (b) (see definition of $\delta$ below in this caption). We calculate the frequency and growth rate from the generalised secondary mode dispersion relation \ref{['eq:D_TSM']} (solid lines), the ISM dispersion relation \ref{['eq:D_ISM']} (dashed lines), and from gyrokinetic simulations of the secondary mode (markers). Different primary amplitude values $\mathcal{A}^P$ (colors) are examined. The primary drive has the form \ref{['eq:model_gP_M']} with $v_{Ex}^P \propto \sin(k_y^P y)$ and $\eta_\perp^P = \eta_\parallel^P = 3 \sin(k_y^P y + \delta) / \sin(k_y^P y)$. The primary drive varies along the magnetic field as $g_i^P \propto e^{-(\theta/\pi)^2}$. The gyrokinetic simulations of the secondary mode have a safety factor value $q=20$ and employ the phase-shift-periodic boundary condition \ref{['eq:BC_phase-shift-periodic']} with $\Theta = 0$ (circles) and $\Theta = 0.288572618$ (crosses); the squares correspond to gyrokinetic simulations where the parallel streaming and binormal magnetic drift were artificially switched off in the simulations. The three markers overlap in those cases where the secondary mode is unaffected by the choice of boundary condition and by the inclusion of in-surface derivatives.
• Figure 4: Real frequency and growth rate of ISMs for varying primary amplitudes $\mathcal{A}^P$ (left to right), with each red circle corresponding to the solution of \ref{['eq:D_ISM']} at a point $(y,\theta)$ on the flux surface. The simulations correspond to the $k_x \rho_i = 0.2$ case in Figure \ref{['fig:validation_D_TSM_scan_kx']}, i.e. the primary drive varies as $g_i^P \propto e^{-(\theta/\pi)^2}$ and has $\eta_\parallel^P=\eta_\perp^P =3$. The TSM solution obtained from \ref{['eq:D_TSM']} is represented by the black triangle, while the mode frequencies from gyrokinetic simulations for a safety factor $q=20$ with phase shifts $\Theta=0$ and $\Theta=0.288572618$ are indicated by blue circles and crosses, respectively. The blue squares correspond to GK simulations with the parallel streaming and binormal magnetic drift set to zero.
• Figure 5: Spatial variation of the nonlinear contribution $\partial_x Q_\mathrm{NL}$\ref{['eq:QSW_NL']} to the SW force evolution \ref{['eq:SW-force_time_der']} in the toroidal secondary mode simulations presented in Figure \ref{['fig:validation_D_TSM_scan_kx']} for $\mathcal{A}^P=4$ (left column) and $\mathcal{A}^P=1$ (right column) at $k_x \rho_i = 0.1$. The $y$-integrated value of $\partial_x Q_\mathrm{NL}$ is shown in the top row as a function of $x$ for three values of $\theta$, $\theta \in \{\pi/2, 0, -\pi/2\}$ (above, on, and below the tokamak midplane, respectively). The second, third, and fourth rows show the variation of $\partial_x Q_\mathrm{NL}$ in $(x,y)$ for these three $\theta$ values, as well as contours of the nonzonal pressure fluctuations $P^\mathrm{NZ}$ (orange contours) and electrostatic potential fluctuations $\varphi^\mathrm{NZ}$ (green contours), with negative values indicated by dashed lines. For $\mathcal{A}^P=4$, the $P^\mathrm{NZ}$ and $\varphi^\mathrm{NZ}$ contours overlap at $\theta=0$, and have phase shifts of opposite signs at $\theta=\pm \pi/2$ due to the radial magnetic drift advection. Moreover, the outward-propagating ($\omega/k_x > 0$) TSM shown here is strongly localised to $\theta=-\pi/2$. As a result, $\langle \partial_x Q_\mathrm{NL}\rangle_y$ has a large up-down asymmetry, see first row. In the $\mathcal{A}^P=1$ case, $\partial_x Q_\mathrm{NL}$ exhibits contributions from higher $y$-harmonics, but its $y$-averaged value still has a similar up-down asymmetry.