The Pinning Control Of Two-dimensional Drift-wave Turbulence

The drift wave instability is driven by the free energy of the density or temperature gradient and occurs in all bounded, magnetized plasmas. The drift-wave turbulence is generally believed to be responsible for the anomalous cross-field transport which enhances the plasma energy losses. Therefore, it is very important to stabilize the drift wave turbulence to improve the plasma con-finement. Due to the research on the control of flow turbulence and one-dimensional drift-wave turbulence, we will discuss the pinning control of two-dimensional drift-wave turbulence.In the first section, we consider the control of two-dimensional drift-wave turbulence de-scribed by the Hasegawa-Mima equation using the uniform pinning coupling, in which we choose the targets as the unstable analytical solutions of the Hasegawa-Mima equation and the control strength as a constant. First, the viability and effectiveness of this pinning method is proved by simulation. Then, the influences of the target on the controllability of the drift-wave turbulence is discussed numerically. It is shown that there is a critical pinning strength when the target is fixed. If the pinning strength is greater than the critical pinning strength, the drift-wave turbulence can be regulated into spatiotemporal patterns, which means the turbulence is controlled. In addition, the efficiency of the pinning depends on the the pattern structure of the target. In order to explain the numerical results, we theoretically analyzed the controlled system by the perturbed method. Linearizing system evolution equation around the target, we can obtain the equation for the per-turbation. By the scheme of three-wave coupling, in which we consider a pair of perturbed modes and the target, the growth rates of the pair of coupled modes can be obtained. When the turbulence is regulated, the growth rate of the perturbed mode is negative. Then, the relationship between the turbulence controllability and the target structure is obtained:with the increase of the wave number of the target, the critical pinning strength is increased by a power-law scaling. Moreover, in both the transition and transient process of the pinning regulation, the modes of the turbulence are found to be suppressed in a hierarchical fashion, that is, by the sequence of mode wave number. The findings give insight into the dynamics of drift-wave turbulence, as well as indicative to the design of new control techniques for real-world turbulence.Noticing that the control signals can’t always be homogeneous, which might be either inho-mogeneous or localized in real applications, we consider nonuniform pinning feedback control of the drift-wave turbulence in the second section. We also choose the H-M model for dissipative drift waves. Particularly, we introduce a nonuniform pinning term in the form of a sinusoidal dis-tribution. Similarly, we prove the effectiveness of this pinning method by simulation first, and then investigate the relationship between control efficiency and the inhomogeneous signals represented by the pinning mode amplitude and the mode number numerically. It is found that the control efficiency is improved when the pinning mode number increases or the pinning mode amplitude decreases when the target is fixed. In order to analyze the mechanism of the inhomogeneous pin-ing control, we study the control system equation in the Fourier space and discuss how the pinning signal influence the modes in drift wave turbulence. There are four types of resonant relationships in Fourier space for all the modes(including drift-mode and pinning signals). If we use a node to represent the drift-mode or the mode of control, and a line to represent the coupling relationship between two modes, we can obtain a network which can describes the interaction between inter-drift-modes and the pinning signals clearly. In this network, the turbulence state being regulated means that all modes should be damped to zero except the mode whose mode vector is equal to the target, which will be oscillates at the same frequency as the target mode. Therefore, a truncation, which contain general case of the control process around target, can be made for simplification and further discussion. It is found that the results obtained form this simplified network model agree with the numerical result. Then, we explain this results form the effect of the inhomogeneous pa-rameters on the growth rate of the dominant modes. Finally, an extreme situation, the local pinning, is considered.In the last section, we suggest a control strategy which applies feedback injections by moving control signals to control two-dimensional drift-wave turbulence. The controllers are nonuniformly distributed in the computational domain and has a moving speed. The pinning signals have the form of two-dimensional travelling plane wave with the travelling speed ω0/κ0(here, ω0is the wave frequency, and kQ is the wave number). It is numerically shown that with the moving controllers, drift-wave turbulence can be controlled more efficient than the usual pinning strategy with static controllers. However, the effect of wave frequency ω0on the control efficiency is non-monotonic. With the increase of angular velocity, the control error may increase a bit to a maximum value, then decreases drastically and maintain almost a stable value. To explain this phenomenon, we study the system in the Fourier space and adopt the simplified network model used in previous section. It is found that the new introduced parameter—wave frequency can affect the growth rate of the dominant modes in the simplified model, and then affect the control efficiency. As for the non-monotonic relationship between the control error and the wave frequency, the maximum point of the error is considered. It is found that the wave frequency which corresponds to the maximum point, is equal to the frequency mismatch of the two dominant modes. Then, we can infer that when the triplet relation is fulfilled, the control error turn out to be largest.