Anisotropic truncation for turbulent transport in the Hasegawa-Wakatani system

Abstract

Reduced models based on an anisotropic truncation of the Fourier space, retaining only a few poloidal wave-numbers while keeping the full radial resolution, are developed and applied to the Hasegawa-Wakatani system. The impact of the truncation is studied first by considering the fixed-gradient formulation, and by comparing to direct numerical simulations (DNS). The turbulent particle flux, and the transition from the quasi-two dimensional turbulence to the zonal flow (ZF) dominated state, are used as the main criteria for validation. Then, similar reduced models are developed in a flux-driven formulation and compared to the DNS, focusing on two cases far from the non-linear threshold of the transition from turbulence to zonal dominated states of the fixed gradient formulation. In both fixed gradient and flux driven cases, it is found that at least 4 poloidal modes, distributed around the most unstable mode, are needed to reproduce the DNS results reasonably. In the flux-driven case, about 10 modes are needed to recover the probability distribution function of the particle flux of the DNS. Considering the role played by different poloidal scales in the turbulent cascade, it is observed that in the turbulent state, an inverse energy cascade in radial wave-numbers takes place at large poloidal scales, while a forward enstrophy cascade in radial wave-numbers is observed to occur at smaller poloidal scales. Moreover, when they form, ZFs feed on poloidal scales that are around and slightly smaller than the injection scale, while giving their energy to the larger poloidal scales. In that case, there is an anisotropic inverse energy transfer, akin to inverse cascade, from the energy injection to the large poloidal scales through ZFs, while the forward enstrophy cascade seems to stay isotropic.

Figures (17)

• Figure 1: Linear growth rate $\gamma_{k}^{+}$ of the HW system as a function of the poloidal wavenumber $k_{y}$ with $k_{x}=0$, $C=1$ and $\kappa=1$ (in the inviscid case $\nu=D=0$). Dashed lines show the maximum growth rate $\gamma_{max}^{+}$ and the associated most unstable wavenumber $k_{y0}$.
• Figure 2: Schematic of the modes kept in PTMs, compared to a matching box DNS Fourier grid (grey dots). The modes with poloidal wavenumber equal to the most unstable mode $k_{y0}$ are shown in blue squares. Zonal modes (red triangles, $k_{y}=0$) are always kept. Blue, green and orange: PTMs whith 1, 3 and 6 poloidal modes kept. The latter is a LES along the poloidal direction (pLES).
• Figure 3: Selected wavenumbers $k_{y}^{j}$ (coloured crosses), normalised by the most unstable wavenumber $k_{y0}$, and their coverage of the growth rate $\gamma_{k}^{+}(k_{y})$ (black line), for the 5 PTMs with poloidal resolution $n_{y}$ given in Table \ref{['tab:PTRMselect']}. The most unstable mode $k_{y0}$ is shown by the dashed line, and the modes in the DNS grid are shown by the black dots. Coloured solid lines show how the injection is approximated in PTMs.
• Figure 4: Spatiotemporal evolution of the zonal velocity profile $\overline{v}_{y}(x,t)$. Top row: $C=0.01$ (turbulent regime), bottom row: $C=2$ (ZF regime). DNS (leftmost column) results compared to PTMs. The radial axis ($x$ axis) is normalised by $k_{y0}(C)$, and time is normalised by $\gamma_{max}^{+}(C)$.
• Figure 5: ZF level $\Xi_{\mathcal{K}}=\mathcal{K}_{ZF}/\mathcal{K}$ (a) and mean radial particle flux $\Gamma$ (b) versus adiabaticity parameter $C$ for DNS (black), and PTMs with $n_{y}=1$ (blue), $n_{y}=2$ (green), $n_{y}=4$ (orange), $n_{y}=10$ (red) and $n_{y}=20$ (violet). Both quantities are averaged over $100<\gamma_{max}^{+}t<200$, with errorbars corresponding to the standard deviation.