Nonlinear interaction and turbulence transition in the limiting regimes of plasma edge turbulence

Abstract

We study the nonlinear coupling mechanism and turbulent transition in magnetically confined plasma flows based on two representative limiting regime dynamics. The two-field flux-balanced Hasegawa–Wakatani (BHW) model is taken as a simplified approximation to the key physical processes in the energy-conserving nonlinear plasma flows. The limiting regimes separate the effects of finite non-adiabatic resistivity and extreme non-normal dynamics to enable a more detailed investigation on each individual aspect with the help of various mathematical tools. We adopt the strategy from the selective decay theory used for the simpler one-field system to develop new crucial a priori estimations in the two-field model framework. The competing effects from model dissipation, finite particle resistivity, as well as the nonlinear interaction with a zonal mean state to induce dual direction energy transports are characterized from the systematic analysis. Non-normal dynamics with aligned eigendirections is also shown to go through a sharp transition from turbulence to regularized zonal flows. The diverse phenomena implied from the limiting regime analysis are further confirmed from direct numerical simulations of the BHW model.

A The most likely state of the two-field BHW model from the variational principle

The physicists’ selective decay principle [15] states that the long time behavior of the system reaches the states with minimized energy in a given constant energy. However, from the enstrophy and energy Eqs. (23) and (24) for the limiting model $\kappa =0$ implies that the energy will decay in a much faster rate compared with a relatively constant enstrophy in the case with a dominant adiabaticity factor $\alpha$. For both cases, the selective decay state can be discovered by computing the critical solution under the variational principle. Here, we propose to compute the selective decay states directly through the Lagrangian multiplier method. This may offer us some intuition about the structures in the group of admissible solutions. The strategy is generalized from the basic variational method approach detailed in Section 4 of [21] for the one-field HM model.

Directly from the definition (12), the total energy and enstrophy for the BHW model can be found as

$$\begin{aligned} \begin{aligned}E\left( \varphi ,n\right) =&\frac{1}{2}\int \left( \left| \nabla \varphi \right| ^{2}+n^{2}\right) ,\\ W\left( \varphi ,\tilde{n}\right) =&\frac{1}{2}\int \left( \varDelta \varphi -\tilde{n}\right) ^{2}. \end{aligned} \end{aligned}$$

In the two-field BHW model, energy and enstrophy should be determined by both electrostatic potential $\varphi$ and density fluctuation n. The variational principle here is to find the extrema of the energy E given a constant enstrophy W. According to the Lagrangian multiplier method, we need to solve the following variational relations with $\varGamma$ as the Lagrangian multiplier

$$\begin{aligned} \begin{aligned} \frac{\delta E}{\delta \varphi }&=\varGamma \frac{\delta W}{\delta \varphi },\\ \frac{\delta E}{\delta \tilde{n}}&=\varGamma \frac{\delta W}{\delta \tilde{n}}. \end{aligned} \end{aligned}$$

(A.1)

The variational derivatives for the quadratic functionals W and E can be first computed directly as

$$\begin{aligned} \begin{aligned}\frac{\delta W}{\delta \varphi }&=\left( \varDelta ^{2}\varphi -\varDelta \tilde{n}\right) , \quad \frac{\delta W}{\delta \tilde{n}}=-\left( \varDelta \varphi -\tilde{n}\right) ,\\ \frac{\delta E}{\delta \varphi }&=-\varDelta \varphi , \quad \frac{\delta E}{\delta \tilde{n}}=\tilde{n}. \end{aligned} \end{aligned}$$

Substituting the above derivatives back to the Euler–Lagrangian equations (A.1), we find the critical state solution $\left( \bar{\varphi }^{*},\tilde{\varphi }^{*},\tilde{n}^{*}\right)$ in the following relations:

$$\begin{aligned} \begin{aligned} \partial _{x}^{2}\bar{\varphi }^{*}&=-\varGamma \partial _{x}^{4}\bar{\varphi }^{*}=0,\\ \varDelta \tilde{\varphi }^{*}&=-\varGamma \varDelta \left( \varDelta \tilde{\varphi }^{*}-\tilde{n}^{*}\right) ,\\ \tilde{n}^{*}&=-\varGamma \left( \varDelta \tilde{\varphi }^{*}-\tilde{n}^{*}\right) . \end{aligned} \end{aligned}$$

For convenience, we separate the zonal and fluctuation components in the state variables. The first equation above for the zonal state implies a constant zonal velocity $\bar{v}=V_{0}=\mathrm {const.}$ Notice that the additional constant zonal profile will not alter the solution structure since the BHW model is Galilean invariant [24]. Then, the critical solution goes to a purely fluctuation state satisfying

$$\begin{aligned} \begin{aligned} \varDelta \tilde{\varphi }^{*}&=\varDelta \tilde{n}^{*},\\ \varDelta ^{2}\tilde{\varphi }^{*}&=\left( 1-\varGamma ^{-1}\right) \varDelta \tilde{\varphi }^{*}. \end{aligned} \end{aligned}$$

(A.2)

This gives the HM state (18) with the density fluctuation converging to the electrostatic potential in a single constant wavenumber $k=\left| \mathbf {k}\right|$

$$\begin{aligned} \tilde{\varphi }^{*}=\tilde{n}^{*}=\sum _{k^{2}=\varGamma ^{-1}-1}\hat{\varphi }_{\mathbf {k}}^{+}e^{i\mathbf {k\cdot x}}, \end{aligned}$$

(A.3)

and the permitted Lagrangian multiplier $\varGamma <1$. The critical exact solution also implies the corresponding Dirichlet quotient (25)

$$\begin{aligned} \varLambda =\frac{W}{E}=\varGamma ^{-1}. \end{aligned}$$

Combined with the monotonically decreasing $\varLambda$ due to the resistivity effect (29), we see the solution is driven to the critical point (A.3) with the minimum permitted value of $\varGamma$. The final permitted selective decay state is the single-mode solution with the largest resolved wavenumber. The variational principle predicts the same smallest scale state from the reversed selective decay due to the particle resistivity.