Subdominant Modes in Zonal-Flow-Regulated Turbulence

From numerical solutions of a gyrokinetic model for ion temperature gradient turbulence it is shown that nonlinear coupling is dominated by three-wave interactions that include spectral components of the zonal flow and damped subdominant modes. Zonal flows dissipate very little energy injected by the instability, but facilitate its transfer from the unstable mode to dissipative subdominant modes, in part due to the small frequency sum of such triplets. Although energy is transferred to higher wave numbers, consistent with shearing, a large fraction is transferred to damped subdominant modes within the instability range. This is a new aspect of regulation of turbulence by zonal flows.

Figure 1

Time-averaged and $k_x^{\prime }$-summed nonlinear transfer spectrum, $\sum _{k_x^{\prime }}⟨T_{k,k^{\prime }}^{(n)}{⟩}_t$, as a function of $k_y^{\prime }$. First 1000 POD modes are considered. (a) is for $n=1$, $k=(0,0.25)$; (b) is for sum over all $n=1,2,3,\dots ,1000$, $k=(0.1,0)$. Positive values of $T_{k,k^{\prime }}$ indicate energy flowing into mode $k$ whereas negative values show energy flowing out of $k$.

Figure 2

Energy transfer in three wave interactions, e.g., in block 1, $\sum _t{T}_{k,k^{\prime }}^{(n)}=-119$ for $n=1$, $k=(0,0.25)$, and $k^{\prime }=(0.1,0)$. Similarly, in block 2, $\sum _{t,n}{T}_{k,k^{\prime }}^{(n)}=24$ for $n=2,3,\dots$, $k=(-0.2,0.25)$, and $k^{\prime }=(-0.1,0.25)$. See the text for discussion.

Figure 3

(Color online) Frequency sum, $|{\widehat{\omega }}_k^{(l)*}+{\widehat{\omega }}_{k^{\prime }}^{(m)}+{\widehat{\omega }}_{k-k^{\prime }}^{(n)}|$, scanned over $k_y^{\prime }$ for $k=(0,0.2)$, $k_x^{\prime }=0.1$.