Elasticity in drift-wave–zonal-flow turbulence

We present a theory of turbulent elasticity, a property of drift-wave–zonal-flow (DW-ZF) turbulence, which follows from the time delay in the response of DWs to ZF shears. An emergent dimensionless parameter ${|\langle v\rangle }^{\prime }|/\Delta {\omega }_k$ is found to be a measure of the degree of Fickian flux-gradient relation breaking, where ${|\langle v\rangle }^{\prime }|$ is the ZF shearing rate and $\Delta {\omega }_k$ is the turbulence decorrelation rate. For ${|\langle v\rangle }^{\prime }|/\Delta {\omega }_k>1$, we show that the ZF evolution equation is converted from a diffusion equation, usually assumed, to a telegraph equation, i.e., the turbulent momentum transport changes from a diffusive process to wavelike propagation. This scenario corresponds to a state very close to the marginal instability of the DW-ZF system, e.g., the Dimits shift regime. The frequency of the ZF wave is ${\Omega }_{\mathrm{ZF}}=\pm {\gamma }_d^{1/2}{\gamma }_{\mathrm{modu}}^{1/2}$, where ${\gamma }_d$ is the ZF friction coefficient and ${\gamma }_{\mathrm{modu}}$ is the net ZF growth rate for the case of the Fickian flux-gradient relation. This insight provides a natural framework for understanding temporally periodic ZF structures in the Dimits shift regime and in the transition from low confined mode to high confined mode in confined plasmas.

Figure 1

Sketch of the propagation mechanism for the ZF wave. Here $F_d$ is the restoring force (friction force), $F_{\mathrm{RS}}$ is the repulsive force (Reynolds force), and the black arrow denotes the direction of the DW momentum flux.

Figure 2

(a) The direct DW-DW collision is dominant: ${\tau }_N\simeq \Delta {\omega }_k^{-1}$. (b) The indirect DW-DW collision is dominant: ${\tau }_N\simeq {\left|{\langle v\rangle }^{\prime }\right|}^{-1}$.