Scaling laws for the cutoff wavenumber of the short-wavelength ion-temperature-gradient mode in a Z-pinch

TL;DR

This work addresses the scaling of the SWITG cutoff in a curvature-driven ITG setting by introducing a minimal electrostatic GK model in a Z-pinch and performing both analytic limits and a linear gyrokinetic solver validation. It derives a fluid-like dispersion with a short-wavelength cutoff $k^o_\perp\rho_i$ that scales as $k^o_\perp\rho_i \sim L_B/(12L_T^{\text{eff}})$ for large gradients and shows a weaker $(L_B/L_T^{\text{eff}})^{1/3}$ scaling at intermediate drives, while in the $\eta\to\infty$ limit two ITG branches emerge and separate with increasing drive. Direct GK solutions confirm the predicted scalings and reveal isotropy in the perpendicular plane, providing a consistent picture across regimes and a smooth transition between drift-kinetic and finite-\(FLR\) effects. By coupling these linear SWITG scalings to a simple diffusive transport estimate and invoking critical balance, the paper predicts ion heat flux trends and eddy aspect ratios that illuminate SWITG-driven turbulence in Z-pinch-like curvature systems. These results offer analytic benchmarks for nonlinear GK studies in more realistic geometries and help identify the parameter regimes where SWITG turbulence could dominate transport when long-wavelength ITG is stabilised.

Abstract

We use a heuristic fluid model to predict the dependence of the cutoff wave number for the short-wavelength ion temperature gradient (SWITG) mode on ion density gradient, ion temperature gradient (ITG) and ion-electron temperature ratio. In particular, we predict that the cutoff wave number increases linearly with increasing ITG for sufficiently large values of the ITG. Direct numerical solutions of the gyrokinetic dispersion relation using a purpose-built solver confirm the predicted scalings at large ITG values and find a weaker power-law scaling for intermediate ITG values. Combining these wave number scalings with a simple diffusive estimate for turbulent fluxes produces a scaling prediction for the ITG heat flux in SWITG-driven turbulence. Applying the critical balance conjecture additionally provides scalings for the aspect ratio of the SWITG turbulent eddies.

Figures (11)

• Figure 1: An illustration of the Z-pinch geometry adopted in our analysis. We use orthonormal coordinates $(x,y,z)$ such that a constant curvature magnetic field exists along the $\boldsymbol{\hat{z}}$ direction with all equilibrium gradients taken to be along the $\boldsymbol{\hat{x}}$ direction.
• Figure 2: 2-D scan in $L_B/L_T$ and $k_y \rho_i$ of the (a) growth rate and (b) real frequency obtained with the solver for $\tau = 1$ and no density gradient ($\eta \to \infty$). Note the presence of two distinct peaks in the growth rate spectrum and a non-monotonic frequency spectrum, both key features of the SWITG. The real frequency is not shown in a narrow region at small drive and very short-wavelength where no growing modes exist and it is numerically challenging to calculate.
• Figure 3: Plots of the (a) growth rate and (b) real frequency spectra for $L_B/L_T=50$ and $200$ with $\tau=1$ and $\eta \to \infty$. The growth-rate spectra show both the long-wavelength ITG mode and the short-wavelength SWITG branch driven by FLR effects. Despite the two instability regions, the real frequency remains continuous and always in the ion diamagnetic direction.
• Figure 4: 2-D scan in $L_B/L_T$ and $k_y \rho_i$ of the (a) growth rate and (b) real frequency obtained with the solver for $\tau = \eta =1$. We note that the cutoff wave numbers where the growth rates pass through zero increase with $L_B/L_T$. Note that we plot the negative of the real frequencies here, so that the modes are propagating in the electron diamagnetic direction in contrast to the $\eta\to\infty$ case shown in Fig. \ref{['fig:eta_infty_2D_growthrate']}.
• Figure 5: Plot of $\eta_{\mathrm{crit}}$ against $L_B/L_T$ obtained from the semi-analytical solver (blue circles), the gyrokinetic code stella, (blue crosses), and the analytical prediction of Eq. \ref{['eq:etacrit']} at $k_y \rho_i = 0.1$.