Studies On Spatiotemporal Chaos And Two-dimensional Turbulence Control

Spatiotemporal chaos and turbulence occur in a variety of nonlinear dynamic systems such as hydrodynamics, plasma systems, laser systems, chemical reactions, Josephson junction arrays and biological networks. In many practical situations such behaviors are considered to be harmful. For instance, drift-wave turbulence which is caused by pressure-driven instability in magnetized plasma is generally believed to be responsible for anomalous cross-field particle transport that causes undesirable energy loss. Flow turbulence also has undesirable consequences: it enhances energy consumption of pipelines, aircrafts, ships, and automobiles;it is an element to be reckoned with in air-travel safety;and so forth. Therefore, spatiotemporal chaos and turbulence control in these systems is of crucial importance.Turbulence in fluid and magnetized plasma can be considered to be a type of high-dimensional spatiotemporal chaos. It is expected that flow turbulence control may be benefited from those strategies developed in controlling high-dimensional chaotic systems. However, flow turbulence control based on the applications of nonlinear dynamics analysis and chaos control methods is just at its beginning. Some results have been already achieved. Guan Shuguang et al have applied global and local feedback control method developed in spatiotemporal chaos control to control flow turbulence described by two-dimensional incompressible Navier-Stokes equation. By applying control signals to all components of flow velocity field, they found that turbulence can be controlled to desirable ordered target states. However, turbulence cannot be completely controlled to the target by global feedback when only one component of the velocity vector is uni-directionally coupled to a target state, while the other component is uncoupled. Wu Shunguang et al have studied controlling spatiotemporal chaos in a system described by a one-dimensional nonlinear drift-wave equation by applying small external periodic force. Gravier et al have studied experimentally nonlinear drift-wave in a cylindrical magnetized laboratory plasma and applied the time-delay auto-synchronization method to control drift-waves chaos. However, many difficult and challenging problems are stillremaining in turbulence control so far.In this paper, we study turbulence and spatiotemporal chaos control, focusing on the problems of control efficiency, optimization of control results and the properties of the asymptotic states of the systems under control. We suggest some new methods for controlling turbulence. The physical mechanisms underlying those control schemes are heuristically analyzed.In chapter 1, the basic chaotic phenomena, chaos control methods, flow turbulence, drift-wave turbulence and direct numerical simulation methods of turbulence are briefly introduced.In Chapter 2, we investigate flow turbulence control. The model studied is the two-dimensional incompressible Navier-Stokes equation. Two works have been done. We firstly consider how to improve the efficiency of flow turbulence control with the feedback signals being applied to both x and y components of the velocity field. We suggest a control strategy which applies local feedback injections with moving controllers (called moving control). It is shown that with the moving controllers, flow turbulence can be controlled more efficiently than the usual pinning strategy with static controllers (called static control). The moving control method can entrain the system from turbulence to ordered targets faster than the static control. In particular, the moving control can successfully suppress turbulence with a small number of controllers in a certain control time, with which the static control fails to suppress turbulence. The physical mechanism underlying this high control efficiency is heuristically analyzed. The advantages and difficulties of the proposed control strategy in practical applications are discussed.It is generally accepted that in experiments controlling a single component of velocity field could be easier than controlling the whole velocity vector. In this regard, we investigate how to enhance the efficiency of flow turbulence control with applying global feedback signals only to a single component of flow velocity field. We suggest a control strategy which applies global feedback sporadically. It is found that this control strategy can significantly enhance the control precision as the optimal fractionof control period is chosen, both larger and smaller control time fractions may reduce the control precision. We further investigate optimization and controllability of flow turbulence control by applying local pinning feedback signals to control a single velocity component. It is found that with a certain number of controllers there exists an optimal control strength at which the control error reduces to the minimum value, and larger and smaller control strengths give worse control effects. Moreover, given a fixed control strength there may exist an optimal number of controllers achieving the best result, and larger and smaller numbers of controllers again provide worse control effects. We analyse this strange and interesting feature based on the mode-mode interactions of the turbulent systems.In Chapter 3, we study spatiotemporal chaos control in drift-wave systems. We consider a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave. We firstly apply time-delay and space-shift feedback signals to suppress spatiotemporal chaos. By using global and local feedback strategies, we show numerically that the spatiotemporally chaotic state can be effectively controlled to periodic states if suitable time delay length and space shift distance are chosen. It is the first time, to our knowledge, to show that spatiotemporal chaos can be suppressed by space-shift feedback only. The obvious advantage for this space-shift feedback over the time-delay feedback is that the former needs much smaller storage of data than the latter. Furthermore, we obtain an analytical expression of the controllability condition by combining partly analytical and partly numerical computations. It is found that an optimal combination of time-delay length r and space-shift distance Ia can distribute the phases of the modes with large amplitude such that the control signal can effectively drives the system to the asymptotic state of minimum energy, and thus successfully suppress turbulence. We further investigate the influence of system size on the efficiency of spatiotemporal chaos control. It is found that the efficiency of the control approach is practically not influenced by changing the system size. This result indirectly shows that our control method can work for different boundary condition.The variable measurement in magnetized plasma is usually difficult. The non-feedback control is more convenient for drift-wave turbulence control. We apply an additional sinusoidal wave to suppress the drift-wave spatiotemporal chaos, and the system becomes a double-sinusoidal-wave (the original driving wave and the additional controlling wave) driven system. It is shown that with some proper choices of the frequency Q, and control strength g of the controlling wave spatiotemporal chaos can be successfully suppressed by the controlling wave. By transforming the drift-wave equation to a set of amplitude and phase equations of space modes, we find an interesting phenomenon of frequency entrainment. The quasi-periodic motion of the system can be decomposed such that the phases of modes rotate at high frequency with small modulation of frequency equal to the difference of the two frequencies of the diving and controlling waves. The amplitudes of modes and the energy of the system however oscillate periodically at this difference frequency. This frequency entrainment directly associated with the controllability transition of the system from spatiotemporal chaos to quasi-periodicity.