Review of Zonal Flows
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iSM Q9I5-633X JP0555001 NATIONAL INSTITUTE FOR FUSION SCIENCE Review of Zonal Flows P.H. Diamond, S.-I. Itoh, K. Itoh and T.S. Hahm (Received - Sep. 9, 2004 ) NIFS-805 Oct. 2004 RESEARCH REPORT NIFS Series This report was prepared as a preprint of work performed as a collaboration research of the National Institute for Fusion Science (NIFS) of Japan. The views presented here are solely those of the authors. This document is intended for information only and may be published in a journal after some rearrangement of its contents in the future. Inquiries about copyright should be addressed to the Research Information Center, National Insitute for Fusion Science, Oroshi-cho, Toki-shi, Gifu-ken 509-5292 Japan. E-mail: [email protected] <Notice about photocopying> In order to photocopy any work from this publication, you or your organization must obtain permission from the following organizaion which has been delegated for copyright for clearance by the copyright owner of this publication. Except in the USA Japan Academic Association for Copyright Clearance (JAACC) 6-41 Akasaka 9-chome, Minato-ku, Tokyo 107-0052 Japan Phone: 81-3-3475-5618 FAX: 81-3-3475-5619 E-mail: [email protected] In the USA Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923 USA Phone: 1-978-750-8400 FAX: 1-978-646-8600 Review of Zonal Flows P. H. Diamond1, S.-I. Itoh2, K. Itoh3, T. S. Hahm4 1 University of California San Diego, La Jolla, CA 92093-0319, USA 2 Institute for Applied Mechanics, Kyushu University 87, Kasuga 816-8580, Japan 3 National Institute for Fusion Science, Toki 509-5292, Japan 4 Princeton University, Plasma Physics Laboratory, Princeton, NJ 08543, USA Abstract A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, zonal flows in nature are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave - zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research. Dedication This review article is dedicated to the memory of Professor. Marshall N. Rosenbluth. Keywords: zonal flow, geodesic acoustic mode, zonal magnetic field, drift waves, convective cells, turbulence, turbulent transport, turbulence suppression, nonlinear interaction, modulational instability, envelope formalism, nonlocal interaction in Fourier space, coherent structure, statistical property, confinement improvement, dynamo, planetary atmosphere, nonlinear theory, numerical simulation, experimental observation Contents 1. Introduction 2. Basic Physics of Zonal Flows: A Heuristic Overview 2.1 Introduction 2.2 Basic dynamics of zonal flows 2.3 Self-consistent solution and multiple states 2.4 General comments 2.5 Implications for experiments 3. Theory of zonal flows 3.1 Linear Dynamics of Zonal Flow Modes 3.2 Generation mechanism 3.3 Shearing and back reaction of flows on turbulence 3.4 Nonlinear Damping and Saturation: Low Collisionality Regimes 3.5 The Drift Wave - Zonal Flow System: Self-consistent State 3.6 Suppression of Turbulent Transport 4. Numerical Simulations of Zonal Flow Dynamics 4.1 Introduction 4.2 Ion Temperature Gradient Driven Turbulence 4.3. Electron Temperature Gradient Driven Turbulence 4.4. Fluid Simulations with Zonal Flows 4.5 Edge turbulence 4.6 Short summary of the correspondence between theoretical issues and numerical results 5. Zonal Flows in Planetary Atmospheres 5.1 Waves in rotating sphere. 5.2 Zonal Belts of Jupiter 5.3 Superrotation of the Venusian Atmosphere 6. Extensions of Theoretical Models 6.1 Streamers 6.2 Noise Effects and Probabilistic Formulations 6.3 Statistical properties 6.4 Non-Markovian Theory 6.5 Envelope Formalism 7. Laboratory Experiments on Zonal Flow Physics 7.1 Characteristics of Zonal Flows 7.2. Zonal Flow Dynamics and Interaction with Ambient Turbulence 7.3. Suggestions on future experiments and information needed from simulations and theory 8. Summary and Discussion Acknowledgements Appendix A Ray of Drift Wave Packet and Trapping Appendix B Hierarchy of Nonlinear Governing Equations Appendix C Near isomorphism between ETG and ITG 1. Introduction Zonal flows, by which we mean azimuthally symmetric band-like shear flows, are a ubiquitous phenomena in nature and the laboratory. The well-known examples of the Jovian belts and zones, and the terrestrial atmospheric jet stream are familiar to nearly everyone - the latter especially to travelers enduring long, bumpy airplane rides against strong head winds. Zonal flows are also present in the Venusian atmosphere (which rotates faster than the planet does!) and occur in the solar tachocline, where they play a role in the solar dynamo mechanism. In the laboratory, the importance of sheared Ex B flows to the development of L-mode confinement, the L-to-H transition and internal transport barriers (ITBs) is now well and widely appreciated. While many mechanism can act to trigger and stimulate the growth of sheared electric fields (i.e. profile evolution and transport bifurcation, neoclassical effects, external momentum injection, etc.) certainly one possibility is via the self-generation and amplification of Ex B flows by turbulent stresses (i.e. the turbulent transport of momentum). Of course, this is the same mechanism as that responsible for zonal flow generation. However, it should be emphasized that it is now widely recognized and accepted that zonal flows are a key constituent in virtually all cases and regimes of drift wave turbulence - indeed, so much so that this classic problem is now frequently referred to as "drift wave-zonal flow turbulence". This paradigm shift occurred on account of the realization that zonal flows are ubiquitous in dynamical models used for describing fusion plasmas (i.e. ITG, TEM, ETG, resistive ballooning, and interchange, etc.) in all geometries and regimes (i.e. core, edge, etc.), and because of the realization that zonal flows are a critical agent of self-regulation for drift wave transport and turbulence. Both theoretical work and numerical simulation made important contributions to this paradigm shift. Indeed, for the case of low collisionality plasmas, a significant portion of the available free energy is ultimately deposited in the zonal flows. Figure 1.1 presents energy flow charts which illustrate the classic paradigm of drift wave turbulence and the new paradigm of drift wave - zonal flow turbulence. The study of zonal flow has had a profound impact on fusion research. For instance, the proper treatment of the zonal flow physics has partly resolved the confusion concerning the prospect of burning plasma as has been discussed by Rosenbluth and collaborators in conjunction with the design of the International Thermonuclear Experimental Reactor (ITER). At the same time, the understanding of the turbulence-zonal flow system has advanced the understanding of self-organization process in nature. We note here that, while zonal flows have a strong influence on the formation of transport barriers, the dynamics of barriers and transitions involve evolutions of both the mean ExB flow as well as the zonal Ex B flow. The topics of mean Er dynamics, transport barriers, and confinement regime transitions are beyond the scope of this review. In the context of tokamak plasmas, zonal flows are n = 0 electrostatic potential fluctuations with finite radial wave number. Zonal flows are elongated, asymmetric vortex modes, and thus have zero frequency. They are predominantly poloidally symmetric as well, though some coupling to \ow-nt sideband modes may occur. On account of their symmetry, zonal flows cannot access expansion free energy stored in temperature, density gradients, etc., and are not subject to Landau damping. These zonal flows are driven exclusively by nonlinear interactions, which transfer energy from the finite-n drift waves to the n = 0 flow. Usually, such nonlinear interactions are three- wave triad couplings between two high k drift waves and one low q = q/ zonal flow excitation. In position space, this energy transfer process is simply one whereby Reynolds work is performed on the flow by the wave stresses. Two important consequences of this process of generation follow directly. First, since zonal flow production is exclusively via nonlinear transfer from drift waves, zonal flows must eventually decay and vanish if the underlying drift wave drive is extinguished. Thus, zonal flows differ in an important way from mean Ex B flows, which can be sustained (and are, in strong H-mode and ITB regimes) in the absence of turbulence. Second, since zonal flows are generated by nonlinear energy transfer from drift waves, their generation naturally acts to reduce the intensity and level of transport caused by the primary drift wave turbulence. Thus, zonal flows necessarily act to regulate and partially suppress drift wave turbulence and transport. This is clear from numerical simulations, which universally show that turbulence and transport levels are reduced when the zonal flow generation is (properly) allowed. Since zonal flows cannot tap expansion free energy, are generated by nonlinear coupling from drift waves, and damp primarily (but not exclusively) by collisional processes, they constitute a significant and benign (from a confinement viewpoint) reservoir or repository for the available free energy of the system. Another route to understanding the effects of zonal flow on drift waves is via the shearing paradigm. From this standpoint, zonal flows produce a spatio-temporally complex shearing pattern, which naturally tends to distort drift wave eddies by stretching them, and in the process generates large kr .