DOCSLIB.ORG

Basic Features of Plasma Instabilities

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

J TRITA-EPP-72-23 BASIC FEATURES OF PLASMA INSTABILITIES B. Lehnert Stockholm, October 22, 1972 y ics and Fusion RoyaRov^Tnc^l Institute ?of Technolog^ l y Research 100 Ht* Stockholm 70, Sweden 1. BASIC FEATURES OF PLASMA INSTABILITIES B. Lehnert Royal Institute of Technology, S-10044 Stockholm 70, Sweden ABSTRACT A review is given on the present stat^i of research on plasma instabilities: (1) The basic characteristics of various instability phenomena .v own so far are outlined, including the methods of the underlying analysis, the general physical properties of unstable modes, the corresponding energy sources and coupling mechanisms, as well as applied stabilization methods. (2) A classification scheme is developed for a systematic subdivision of existing types of plasma disturbances, including both stable and unstable modes. In addition, a general description is given of each instability mode being detected so far, (3) The present state of research on plasma instabilities is summa- rized, including discussions both on linear and nonlinear modes and on associated loss mechanisms. The established classification scheme is applied in a systematic search for instability phenomena which so far have remained undiscovered. CONTENTS 1. Introduction 1 2. Basic theoretical means 2 2.1. Basic equations 2 2.1.1. Particle orbit theory 2 2.1.2. Kinetic theory 2.1.3. Macroscopic fluid theory 4 2.2. The unperturbed state 4 2.2.1. Stationary states 5 2.2.1.1. Static states 5 2.2.1.2. Non-static states 5 2.2.2. Non-stationary states 6 2.2.2.1. Laminar states 6 2.2.2.2. Non-laminar states 6 2.3. The perturbed state 6 2.3.1. Energy sources 6 2.3.1.1. Intrinsic sources of the plasma 7 2.3.1.2. Externally imposed force fields 8 2.3.1.3. External injection of energy 8 2.3.2. Coupling mechanisms 8 2.3.3. The stages of linear and nonlinear 10 analysis 2.3.4. Methods of theoretical analysis of 10 plasma disturbances 2.3.4.1. Particle orbit analysis 11 2.3.4.2. Linear normal mode analysis 11 2 .3.4.3. Nonlinear mode analysis 12 2.3.4.4. The energy principle 13 2.3.4.5. Computer analysis 13 3. General properties of plasma instabilities 14 3.1. Definitions of instability 14 3.1.1. Singe-stage processes 14 3.1.1.1. Onset of instabilities at small 14 ampl5.tudes 3.1.1.2. No onset of instabilities at small 15 amplitudes 3.1.2. Multi-stage processes 15 3.2. External effects on plasma instabilities 15 3.3. Relativistic effects 16 3.4. Linear instability properties 16 3.4.1. The restoring forces 16 3.4.2. Unstable oscillations and waves 17 3.4.2.1. Absolute and convective 17 instabilities 3.4.2.2. Negative energy waves 18 3.4.2.3. Plasma oscillations and Bernstein 19 waves 3.4.2.4. Velocity space effects 19 3.4.2.5. Parametric instabilities 19 3.4.3. Particle-wave interaction 19 3.4.3.1. Trapping instabilities 20 3.4.3.2. Drift instabilities 20 3.4.3.3. Resonance phenomena in general 21 Ill 3.4.4. Collective modes 21 3.U.U.I. Untrapped particle instabilities 21 3.4.4.2. Trapped particle instabilities 21 3.4.4.3. Resonance phenomena of collective 22 modes 3.5. Nonlinear instability properties 22 3.5.1. Nonlinear mode interaction 22 3.5.1.1. Resonant interaction 23 3.5.1.2. Non-resonant interaction 24 3.5.2. Plasma turbulence 24 Stabilization mechanisms 26 4.1. Internal plasma mechanisms 27 4.1.1. Linear mechanisms 27 4.1.1.1. "Hard" mechanisms 27 4.1.1.2. "Soft" mechanisms 28 4.1.2. Nonlinear mechanisms 29 4.2. Effects due to the boundary conditions 29 4.3. Externally applied mechanisms 30 4.3.1. Stationary mechanisms 30 4.3.2. Non-stationary mechanisms 30 4.3.2.1. Dynamic stabilization 30 31 4.3.2.2. Feedback stabilization 32 Classification of plasma disturbances 32 5.1. The rotation 32 5.2. The classification scheme External space effects 33 Relativistic modes 33 Collisional phenomena 34 The macroscopic motion in the 34 unperturbed state The unperturbed state in velocity 35 space The externally imposed magnetic field 36 Electromagnetic induction effects 37 Collective effects 37 37 5.2.10. Resonance effects 38 5.2.11. The disturbance amplitudes 39 5.2.12. The growth of the disturbances 39 6. A survey of instability subclasses 40 7. Suggestions for discoveries of new phenomena 41 7.1. Instabilities 41 7.1.1. Feedback destabilization 41 7.1.2. Relativistic instabilities 41 7.1.3. Collisional non-resonant effects 42 7.1.4. Non-static state phenomena 42 IV. 7.1.5. Magnetic induction effects 42 7.1.5.1. High-beta phenomena 43 7.1.5.2. Low-bera phenomena 43 7.1.6. Collective effects 43 7.1.7. Resonance phenomena 44 7.1.7.1. Localized phenomena 44 7.1.7.2. Collective phenomena 45 7.1.8. Nonlinear effects 45 7.1.9. Plasma interaction with other 45 immersed media 7.1.10. Ambiplasma effects 45 7.2. Stabilization effects 46 8. Conclusions 47 9. References 48 9.1. General references on plasma disturbance 48 phenomena 9.1.1. Plasma instabilities 48 9.1.2. Plasma waves 49 9.2. References to special classes and 49 properties of plasma instabilities 9.2.1. Absolute and convective instabilities 49 9.2.2. Relativistic instabilities 50 9.2.3. Resistive instabilities 50 9.2.4. Drift instabilities 50 9.2.5. Two-stream and beam-plasma 51 instabilities 9.2.6. Collective instabilities 51 9.2.7. Trapped particle instabilities 51 9.2.8. Cyclotron instabilities 51 9.2.9. Parametric instabilities 51 9.2.10. Nonlinear phenomena and turbulence 51 9.2.11. Moving striations 52 9.2.12. Numerical simulation by means of 52 computers 9.3. Stabilization methods 53 9.3.1. Dynamic stabilization 53 9.3.2. Feedback stabilization 53 9.4. Special references 53 10. Index of instabilities 55 10.1, Classification index 55 10.2. Name index 62 Tables on instability subclasses T1-T20 Figure 1. Simple mechanical analogies of Fl stability problem. Figure 2. Classification scheme of plasma F2 instabilities. Figure 3. General division in subclasses. F3 Appendix. Basic properties of individual A0-A127 instability modes and subclasses 1. 1. INTRODUCTION A magnetized plasma constitutes a complex system with many degrees of freedom, within which various energy forms interact with each other. Such an interaction is provided by the numerous types of coupling mechanisms being involved in the dynamics of plasma distur- bances. As a result, the latter may develop either into instabilities or into wave phenomena and other stable perturbations. Since the end of the 1950:s, the number of detected basic types of plasma instabilities has been steadily increasing, up to the present date where most elementary modes appear to have been explored. This development has earlier been summarized in a number of general reviews [jL - SFJ of which the contribution by F. Cap and collaborators dl»?Zl represents the most extensive and updated one. The latter in- cludes abstracts of 5112 papers as well as extensive lists of references. Reviews on specific classes of instabilities have further been given by a number of investigators Q-J> - B9^\ . The large efforts made in the study of plasma instabilities are mainly due to their importance to plasma confinement in fusion research. Nevertheless there is also a widespread general interest in such phenomena within basic plasma physics, as well as in other fields of application such as cosmical and space physics, magnetohydrodynamic power generation, and in the fields represented by some special tech- nical problems. Thus, in addition to the confinement problem, the instability analysis plays a central role in the investigations of collisionless heating mechanisms, collisionless shock transitions, anomalous transport phenomena, nonlinear plasma phenomena, and turbu- lence. Within the various fields of application just mentioned, there is further a considerable interest in plasma wave phenomena, as des- cribed in several contexts J£lO - 1*C] • This review is mainly con- centrated on the basic features of plasma instabilities. Plasma wave phenomena will only be treated as far as they are connected with growing unstable plasma disturbances. We shall further restrict ourselves to pure plasma *phenomena; i,e. the semiconductor effects earlier reviewed by Hartnagel C5~j are not included in the present discussions. The review serves four purposes, namely to outline the basic physical features of plasma instabilities, to make an attempt to systematize the rich flora of various types of existing modes, to summarize the present state of research on instabilities, and to give hints to the possible detection of new modes and plasma disturbance phenomena. 2. BASIC THEORETICAL MEANS I i Starting with the basic means of plasma theory, we shall now j attempt a systematic study of plasma disturbances and their associated I physical effects. i i 2.1. Basic Equations The electromagnetic field is governed by Maxwell's equations, where curlE = - 3B/3t ; divB = 0 (1) represents the law of electromagnetic induction of the electric and magnetic fields £ and JB, and curlB/v = j + £ 3E/3t (2) — o **- o — expresses the magnetic field in terms of its sources. Here j[ denotes the electric current density and p and e the magnetic permeability and dielectric constant in vacuo, given in MKS units. Conservation of the electric charge density a further yields divi = - 3a/3t ; divE = a/eQ (3) The Equations (1) - (3) of the electromagnetic field have to be coupled with those determining the motion of the charged particles in the plasma.
Recommended publications
  • Principles of Instability and Stability in Digital PID Control Strategies and Analysis for a Continuous Alcoholic Fermentation Tank Process Start-Up

    Principles of Instability and Stability in Digital PID Control Strategies and Analysis for a Continuous Alcoholic Fermentation Tank Process Start-Up

    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 July 2019 doi:10.20944/preprints201809.0332.v2 Principles of Instability and Stability in Digital PID Control Strategies and Analysis for a Continuous Alcoholic Fermentation Tank Process Start-up Alexandre C. B. B. Filhoa a Faculty of Chemical Engineering, Federal University of Uberlândia (UFU), Av. João Naves de Ávila 2121 Bloco 1K, Uberlândia, MG, Brazil. [email protected] Abstract The art work of this present paper is to show properly conditions and aspects to reach stability in the process control, as also to avoid instability, by using digital PID controllers, with the intention of help the industrial community. The discussed control strategy used to reach stability is to manipulate variables that are directly proportional to their controlled variables. To validate and to put the exposed principles into practice, it was done two simulations of a continuous fermentation tank process start-up with the yeast strain S. Cerevisiae NRRL-Y-132, in which the substrate feed stream contains different types of sugar derived from cane bagasse hydrolysate. These tests were carried out through the resolution of conservation laws, more specifically mass and energy balances, in which the fluid temperature and level inside the tank were the controlled variables. The results showed the facility in stabilize the system by following the exposed proceedings. Keywords: Digital PID; PID controller; Instability; Stability; Alcoholic fermentation; Process start-up. © 2019 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 July 2019 doi:10.20944/preprints201809.0332.v2 1.
  • Control Theory

    Control Theory

    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
  • Plasma Stability in a Dipole Magnetic Field Andrei N. Simakov MS Physics, Moscow Institute of Physics and Technology, Russia

    Plasma Stability in a Dipole Magnetic Field Andrei N. Simakov MS Physics, Moscow Institute of Physics and Technology, Russia

    Plasma Stability in a Dipole Magnetic Field by Andrei N. Simakov M.S. Physics, Moscow Institute of Physics and Technology, Russia, 1997 Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2001 @ Massachusetts Institute of Technology 2001. All ri0h FTECNOOG 8cience JUN 2 2 2001 LIBRARIES Author................................................. "-partment of Physics April 26, 2001 C ertified by ..... ................... Peter J. Catto Senior Research Scientist, Plasma Science and Fusion Center Thesis Supervisor Certified by Miklos Porkolab Professor, Department of Physics Thesis Supervisor Accepted by ........ .......... Thomas J. Greytak Professor, Associate Depart nt Head for Education Ii 2 Plasma Stability in a Dipole Magnetic Field by Andrei N. Simakov M.S. Physics, Moscow Institute of Physics and Technology, Russia, 1997 Submitted to the Department of Physics on April 26, 2001, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract The MHD and kinetic stability of an axially symmetric plasma, confined by a poloidal magnetic field with closed lines, is considered. In such a system the stabilizing effects of plasma compression and magnetic field compression counteract the unfavorable field line curvature and can stabilize pressure gradient driven magnetohydrodynamic modes provided the pressure gradient is not too steep. Isotropic pressure, ideal MHD stability is studied first and a general interchange stability condition and an integro-differential eigenmode equation for ballooning modes are derived, using the MHD energy principle. The existence of plasma equilibria which are both interchange and ballooning stable for arbitrarily large beta = plasma pressure / magnetic pressure, is demonstrated.
  • Local and Global Instability of Buoyant Jets and Plumes

    Local and Global Instability of Buoyant Jets and Plumes

    XXIV ICTAM, 21-26 August 2016, Montreal, Canada LOCAL AND GLOBAL INSTABILITY OF BUOYANT JETS AND PLUMES Patrick Huerre1a), R.V.K. Chakravarthy1 & Lutz Lesshafft 1 1Laboratoire d’Hydrodynamique (LadHyX), Ecole Polytechnique, Paris, France Summary The local and global linear stability of buoyant jets and plumes has been studied as a function of the Richardson number Ri and density ratio S in the low Mach number approximation. Only the m = 0 axisymmetric mode is shown to become globally unstable, provided that the local absolute instability is strong enough. The helical mode of azimuthal wavenumber m = 1 is always globally stable. A sensitivity analysis indicates that in buoyant jets (low Ri), shear is the dominant contributor to the growth rate, while, for plumes (large Ri), it is the buoyancy. A theoretical prediction of the Strouhal number of the self-sustained oscillations in helium jets is obtained that is in good agreement with experimental observations over seven decades of Richardson numbers. INTRODUCTION Buoyant jets and plumes occur in a wide variety of environmental and industrial contexts, for instance fires, accidental gas releases, ventilation flows, geothermal vents, and volcanic eruptions. Understanding the onset of instabilities leading to turbulence is a research challenge of great practical and fundamental interest. Somewhat surprisingly, there have been relatively few studies of the linear stability properties of buoyant jets and plumes, in contrast to the related purely momentum driven classical jet. In the present study, local and global stability analyses are conducted to account for the self-sustained oscillations experimentally observed in buoyant jets of helium and helium-air mixtures [1].
  • Astrophysical Fluid Dynamics: II. Magnetohydrodynamics

    Astrophysical Fluid Dynamics: II. Magnetohydrodynamics

    Winter School on Computational Astrophysics, Shanghai, 2018/01/30 Astrophysical Fluid Dynamics: II. Magnetohydrodynamics Xuening Bai (白雪宁) Institute for Advanced Study (IASTU) & Tsinghua Center for Astrophysics (THCA) source: J. Stone Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 2 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 3 What is a plasma? Plasma is a state of matter comprising of fully/partially ionized gas. Lightening The restless Sun Crab nebula A plasma is generally quasi-neutral and exhibits collective behavior. Net charge density averages particles interact with each other to zero on relevant scales at long-range through electro- (i.e., Debye length). magnetic fields (plasma waves). 4 Why plasma astrophysics? n More than 99.9% of observable matter in the universe is plasma. n Magnetic fields play vital roles in many astrophysical processes. n Plasma astrophysics allows the study of plasma phenomena at extreme regions of parameter space that are in general inaccessible in the laboratory. 5 Heliophysics and space weather l Solar physics (including flares, coronal mass ejection) l Interaction between the solar wind and Earth’s magnetosphere l Heliospheric
  • Symmetric Stability/00176

    Symmetric Stability/00176

    Encyclopedia of Atmospheric Sciences (2nd Edition) - CONTRIBUTORS’ INSTRUCTIONS PROOFREADING The text content for your contribution is in its final form when you receive your proofs. Read the proofs for accuracy and clarity, as well as for typographical errors, but please DO NOT REWRITE. Titles and headings should be checked carefully for spelling and capitalization. Please be sure that the correct typeface and size have been used to indicate the proper level of heading. Review numbered items for proper order – e.g., tables, figures, footnotes, and lists. Proofread the captions and credit lines of illustrations and tables. Ensure that any material requiring permissions has the required credit line and that we have the relevant permission letters. Your name and affiliation will appear at the beginning of the article and also in a List of Contributors. Your full postal address appears on the non-print items page and will be used to keep our records up-to-date (it will not appear in the published work). Please check that they are both correct. Keywords are shown for indexing purposes ONLY and will not appear in the published work. Any copy editor questions are presented in an accompanying Author Query list at the beginning of the proof document. Please address these questions as necessary. While it is appreciated that some articles will require updating/revising, please try to keep any alterations to a minimum. Excessive alterations may be charged to the contributors. Note that these proofs may not resemble the image quality of the final printed version of the work, and are for content checking only.
  • The Counter-Propagating Rossby-Wave Perspective on Baroclinic Instability

    The Counter-Propagating Rossby-Wave Perspective on Baroclinic Instability

    Q. J. R. Meteorol. Soc. (2005), 131, pp. 1393–1424 doi: 10.1256/qj.04.22 The counter-propagating Rossby-wave perspective on baroclinic instability. Part III: Primitive-equation disturbances on the sphere 1∗ 2 1 3 CORE By J. METHVEN , E. HEIFETZ , B. J. HOSKINS and C. H. BISHOPMetadata, citation and similar papers at core.ac.uk 1 Provided by Central Archive at the University of Reading University of Reading, UK 2Tel-Aviv University, Israel 3Naval Research Laboratories/UCAR, Monterey, USA (Received 16 February 2004; revised 28 September 2004) SUMMARY Baroclinic instability of perturbations described by the linearized primitive equations, growing on steady zonal jets on the sphere, can be understood in terms of the interaction of pairs of counter-propagating Rossby waves (CRWs). The CRWs can be viewed as the basic components of the dynamical system where the Hamil- tonian is the pseudoenergy and each CRW has a zonal coordinate and pseudomomentum. The theory holds for adiabatic frictionless flow to the extent that truncated forms of pseudomomentum and pseudoenergy are globally conserved. These forms focus attention on Rossby wave activity. Normal mode (NM) dispersion relations for realistic jets are explained in terms of the two CRWs associated with each unstable NM pair. Although derived from the NMs, CRWs have the conceptual advantage that their structure is zonally untilted, and can be anticipated given only the basic state. Moreover, their zonal propagation, phase-locking and mutual interaction can all be understood by ‘PV-thinking’ applied at only two ‘home-bases’— potential vorticity (PV) anomalies at one home-base induce circulation anomalies, both locally and at the other home-base, which in turn can advect the PV gradient and modify PV anomalies there.
  • Effects of Asymmetric Gas Distribution on the Instability of a Plane Power-Law Liquid Jet

    Effects of Asymmetric Gas Distribution on the Instability of a Plane Power-Law Liquid Jet

    energies Article Effects of Asymmetric Gas Distribution on the Instability of a Plane Power-Law Liquid Jet Jin-Peng Guo 1,† ID , Yi-Bo Wang 2,†, Fu-Qiang Bai 3, Fan Zhang 1 and Qing Du 1,* 1 State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China; [email protected] (J.-P.G.); [email protected] (F.Z.) 2 Wuxi Weifu High-Technology Co., Ltd., No. 5 Huashan Road, Wuxi 214031, China; [email protected] 3 Internal Combustion Engine Research Institute, Tianjin University, Tianjin 300072, China; [email protected] * Correspondence: [email protected]; Tel.: + 86-022-2740-4409 † These authors contributed equally to this work. Received: 17 May 2018; Accepted: 27 June 2018; Published: 16 July 2018 Abstract: As a kind of non-Newtonian fluid with special rheological features, the study of the breakup of power-law liquid jets has drawn more interest due to its extensive engineering applications. This paper investigated the effect of gas media confinement and asymmetry on the instability of power-law plane jets by linear instability analysis. The gas asymmetric conditions mainly result from unequal gas media thickness and aerodynamic forces on both sides of a liquid jet. The results show a limited gas space will strengthen the interaction between gas and liquid and destabilize the power-law liquid jet. Power-law fluid is easier to disintegrate into droplets in asymmetric gas medium than that in the symmetric case. The aerodynamic asymmetry destabilizes para-sinuous mode, whereas stabilizes para-varicose mode. For a large Weber number, the aerodynamic asymmetry plays a more significant role on jet instability compared with boundary asymmetry.
  • 1 Jp1.14 a Preferred Scale for Warm Core Instability in A

    1 Jp1.14 a Preferred Scale for Warm Core Instability in A

    JP1.14 APREFERREDSCALEFORWARMCOREINSTABILITYIN ANON-CONVECTIVEMOISTBASICSTATE BrianH.Kahn JetPropulsionLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA DouglasM.Sinton * DepartmentofMeteorology,SanJoséStateUniversity,SanJosé,CA ABSTRACT 1.INTRODUCTION The existence, scale, and growth rates of sub- Tropical cyclones, mesoscale convective systems synoptic scale warm core circulations are investigated (MCS), and polar lows require moisture supplied by withasimpleparameterizationforlatentheatreleasein warmcore,sub-synoptic-scalecirculationsthatdevelop a non-convective basic state using a linear two-layer in generally saturated, near neutral conditions over a shallow water model. For a range of baroclinic flows range of weak to moderate vertical shears (Xu and frommoderatetohighRichardsonnumber,conditionally Emanuel 1989; Emanuel 1989; Houze 2004). stable lapse rates approaching saturated adiabats Accountingforthescaleofsuchwarmcorecirculations consistentlyyieldthemostunstablemodeswithawarm- has proven to be elusive. In a recent paper on the corestructureandaRossbynumber ~O (1)withhigher relationship between extra-tropical precursors and Rossby numbers stabilized. This compares to the ensuing tropical depressions, Davis and Bosart (2006) corresponding most unstable modes for the dry cases state: whichhavecold-corestructuresandRossbynumbers~ O(10 -1) or in the quasi-geostrophic range. The “Notheoryexiststhataccuratelydescribesthe maximumgrowthratesof0.45oftheCoriolisparameter scale contraction from the precursor are an order
  • The Rayleigh-Taylor and Flute Instabilities

    The Rayleigh-Taylor and Flute Instabilities

    Chapter 19 The Rayleigh-Taylor and flute instabilities In Chapter 9, we learned that magnetohydrodynamic plasma equilibria must be determined self-consistently, i.e. the presence of currents flowing in the plasma modifies the magnetic configuration in which the plasma rests. A static magnetohydrodynamic equilibrium (plasma fluid velocity U = 0, hence electric field E = 0) occurs when the plasma pressure gradients are balanced by magnetic (j x B) forces. However, even if a magnetohydrodynamic equilibrium exists in some particular case, the lack of plasma stability can lead to the spontaneous generation of E fields and associated plasma velocities U. For if the plasma is disturbed slightly, its motion can deform the magnetic field in such a way as to produce magnetic forces that tend to amplify the original disturbance. This type of phenomenon is called a ‘magnetohydrodynamic(MHD) instability’. Because of the complexity of the magnetohydrodynamic equations, we are generally only able to treat analytically the case of linear stability, i.e. stability against injinitesimally small disturbances, in relatively simple geometries. For spatially uniform plasmas, infinitesimal perturbations will generally have a wave- like spatial structure. In such cases, as was discussed in Chapter 15, a plane wave with a single wave-vector k will generally have a single frequency w. Thus, for a uniform plasma, this plane wave will be a ‘normal mode’. For non-uniform plasmas, such as those considered in the present Chapter, it will be necessary to find the ‘eigenfunctions’, describing the spatial structure in the direction of non-uniformity, of the normal modes of perturbations, i.e.
  • 9 Fluid Instabilities

    9 Fluid Instabilities

    9 Fluid Instabilities 9.1 Stability of a shear flow In many situations, gaseous flows can be subject to fluid instabilities in which small perturbations can rapidly flow, thereby tapping a source of free energy. An impor- tant example for this are Kelvin-Helmholtz and Rayleigh-Taylor instabilities, which we discuss in this chapter. We consider a flow in the x-direction, which in the lower half-space z < 0 has velocity U1 and density ρ1, whereas in the upper half-space the gas streams with U2 and has density ρ2. In addition there can be a homogeneous gravitational field g pointing into the negative z-direction. Let us assume the flow can, at least approximately, be treated as an incompressible potential flow. Let the velocity field in the upper and lower halves be given by upper half: v2 = rΦ2 for z > 0 (9.1) lower half: v1 = rΦ1 for z < 0 (9.2) The equation of motion for an incompressible gas with constant density can be written as @v P + (v · r)v = g − r : (9.3) @t ρ If we use the identity (v · r)v = rv2=2 − r × v, together with the assumption of a potential flow v = rΦ (and hence r × v = 0), we can write the equation of motion as @Φ 1 P r + r v2 − g + r = 0 (9.4) @t 2 ρ 1 9 Fluid Instabilities Writing the gravitational acceleration as g = −g^ez, this implies @Φ 1 P + v2 + gz + = const: (9.5) @t 2 ρ which is Bernoulli's theorem, a useful result that we will exploit later on.
  • Causes of Reduced North Atlantic Storm Activity in a CCSM3

    Causes of Reduced North Atlantic Storm Activity in a CCSM3

    15 SEPTEMBER 2009 D O N O H O E A N D B A T T I S T I 4793 Causes of Reduced North Atlantic Storm Activity in a CAM3 Simulation of the Last Glacial Maximum AARON DONOHOE AND DAVID S. BATTISTI Department of Atmospheric Sciences, University of Washington, Seattle, Washington (Manuscript received 7 August 2008, in final form 6 April 2009) ABSTRACT The aim of this paper is to determine how an atmosphere with enhanced mean-state baroclinity can support weaker baroclinic wave activity than an atmosphere with weak mean-state baroclinity. As a case study, a Last Glacial Maximum (LGM) model simulation previously documented to have reduced baroclinic storm activity, relative to the modern-day climate (simulated by the same model), despite having an enhanced midlatitude temperature gradient, is considered. Several candidate mechanisms are evaluated to explain this apparent paradox. A linear stability analysis is first performed on the jet in the modern-day and the LGM simulation; the latter has relatively strong barotropic velocity shear. It was found that the LGM mean state is more unstable to baroclinic disturbances than the modern-day mean state, although the three-dimensional jet structure does stabilize the LGM jet relative to the Eady growth rate. Next, feature tracking was used to assess the storm track seeding and temporal growth of disturbances. It was found that the reduction in LGM eddy activity, relative to the modern-day eddy activity, is due to the smaller magnitude of the upper-level storms entering the North Atlantic domain in the LGM. Although the LGM storms do grow more rapidly in the North Atlantic than their modern-day counterparts, the storminess in the LGM is reduced because storms seeding the region of enhanced baroclinity are weaker.