Basic Plasma Concepts and Models

Transcription

1 Basic Plasma Concepts and Models Amitava Bha5acharjee University of New Hampshire 2011 Heliophysics Summer School

2 Goal of this lecture Review a few basic plasma concepts and models that underlie the lectures later in the week. There are several excellent text books in plasma physics: Chen, Nicholson (out of print), Goldston and Rutherford, Boyd and Sanderson, Bellan. The book I am most familiar with is by Gurne5 and Bha5acharjee, from which most of the material is taken.

3 What is a Plasma? Plasma is an ensemble of charged parrcles, capable of exhibirng collecrve interacrons. Levels of DescripRon: Single- parrcle dynamics in prescribed electric and magnerc fields Plasmas as fluids in 3D configuraron space moving under the influence of self- consistent electric and magnerc fields Plasmas as kinerc fluids in 6D µ- space (that is, configuraron and velocity space), coupled to self- consistent Maxwell s equarons.

4 Single-Particle Orbit Theory Newton s law of moron for charged parrcles m dv dt = q E + v " B ( ) Guiding- Center: A very useful concept

5 ExB Dri] Single-Particle Orbit Theory m dv dt = q ( E + v " B ) Consider E = const., B = const. The charged parrcles experience a dri] velocity, perpendicular to both E and B, and independent of their charge and mass. V E = E " B B 2

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7 Gradient B dri] V G = w " qb % ' & B # $B B 2 w " = c 2, c 2 ( *, )

8 Curvature dri] V C = 2w qb 2 # % $ w = 1 2 mv 2 R C " B R C 2 & (, '

9 The Ring Current in Earth s Magnetosphere: An Example

10 AdiabaRc Invariants Albert Einstein and the AdiabaRc Pendulum (1911) Einstein suggested that while both the energy E and the frequency ν change, the raro E/ν remains approximately invariant.

11 AdiabaRc Invariants Harmonic oscillator d 2 x dt 2 +"2 (#t)x = 0, # <<1 The adiabarc invariant is J = " pdq #J / J ~ exp($c /%)

12 AdiabaRc Invariants Three types of quasi- periodic moron

13 AdiabaRc Invariants Three types of bounce moron First adiabarc invariant µ = w " / B Second adiabarc invariant J = m " v ds Third adiabarc invariant " = #R 2 B

14 KineRc DescripRon of Plasmas DistribuRon funcron f (r,v,t) NormalizaRon N = "" dxdvf (r,v,t) phase space Example: Maxwell distriburon funcron # f = n 0 exp " mv2 % $ 2kT & (, n 0 = N /V '

15 Boltzmnn- Vlasov EquaRon MoRon of an incompressible phase fluid in µ space! f s!t + v.! f s!r + q ( m E + v! B ).! f s!v = 0, s = e,i In the presence of collisions! f s!t + v.! f s!r + q m E + v! B ( ).! f s!v = (" f s "t ) c

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17 Vlasov- Poisson equarons: requirements of self- consistency in an electrostarc plasma "f s "t + v # "f s "r $ q s m s %& # "f s "v = 0 E = "#$ " # E = $" 2 % = 4&' = 4& ) q s ( dvf s s

18 Quasilinear theory: applicaron to sca5ering due to wave- parrcle interacrons Consider electrostarc Vlasov equaron "f s "t + v."f s "t # q m $%."f s "v = 0. Split every dependent variable into a mean and a fluctuaron f s = f s + f s1, f s1 = 0

19 Quasilinear Diffusion It follows a]er some algebra that the mean or average distriburon funcron obeys a diffusion equaron: " "t f s = " "v # $ & D # " "v f s % Here D is a diffusion tensor, dependent on wave fluctuarons. ' ) (

20 Lectures for which this material is directly perrnent Jokipii : Cosmic rays in the heliosphere Sojka : The Earth s Ionosphere MathemaRcal procedure of separarng physical quanrres into mean fields and fluctuarons known as mean- field theory- - - is relevant to the lectures of Rempel, Bagenal, and Miesch.

21 Fluid Models The primary fluid model of focus in this summer school is Magnetohydrodynamics (MHD) It treats the plasma as a single fluid, without disrnguishing between electrons or protons, moving under the influence of self- consistent electric and magnerc fields. It can be derived from kinerc theory by taking moments (integrarng over velocity space), and making some drasrc approximarons.

22 MHD EquaRons The la5er form is valid in the absence of rotaron.

23 MHD EquaRons (conrnued) O]en the weakest link in MHD theory, such as the isothermal or adiabarc assumpron, p /! " = const.

24 MHD EquaRons (conrnued) O]en the weakest link in MHD theory, such as the isothermal or adiabarc assumpron, p /! " = const. More sophisrcated version (cf. Rempel lecture) Remark: DistribuRon funcrons in heliophysical plasmas are most o]en non- Maxwellian, and the systems are far from thermodynamic equilibrium. Much care needs to be given to the definiron of temperature.

25 From Magne6c Reconnec6on by E. Priest and T. Forbes Frozen Flux/Field Theorem (Alfven s Theorem)

26 Magne&c Reconnec&on: Working Defini&on If a plasma is perfectly conducrng, that is, it obeys the ideal Ohm s law, E + v " B = 0 B- lines are frozen in the plasma. Departures from ideal behavior, represented by E + v " B = R, # " R $ 0 break ideal topological invariants, allowing field lines to break and reconnect.in generalized Ohm s law for collisionless plasmas, R contains resisrvity, Hall current, electron inerra, and pressure.

27 Why is magne,c reconnec,on important? MagneRc reconnecron enables a system to access states of lower energy by topological relaxaron of the magnerc field. energy liberated can be converted to flows, hearng, and parrcle acceleraron.

28 Magnetic Island Formation Due To Reconnection

29 Fluxes of energetic electrons peak within magnetic islands [Chen et al., Nature Physics, 2008]

30 e bursts & bipolar Bz & Ne peaks ~10 islands within 10 minutes

31 Classical (2D) Steady-State Models of Reconnection Sweet-Parker [Sweet 1958, Parker 1957] Geometry of reconnection layer : Y-points Length of the reconnection layer is of the order of the system size >> width! Reconnection time scale " SP = (" A " R ) 1/2 = S 1/2 " A Solar flares: S ~ 10 12, "! SP 6 ~ 10 " A ~ 1s Too long to account for solar flares! s

32 Q. Why is Sweet-Parker reconnection so slow? A. Geometry Conservation relations of mass, energy, and flux V in L = V out ", V = V out A V in = " L V A, " L = S#1/2 Petschek [1964] Geometry of reconnection layer: X-point Length " (<< L) is of the order of the width "! =! PK A ln S Solar flares:! PK ~ 10 2 s

33 Computa6onal Tests of the Petschek Model [Sato and Hayashi 1979, Ugai 1984, Biskamp 1986, Forbes and Priest 1987, Scholer 1989, Yan, Lee and Priest 1993, Ma et al. 1995, Uzdensky and Kulsrud 2000, Breslau and Jardin 2003, Malyshkin, Linde and Kulsrud 2005] Conclusions Petschek model is not realizable in high- S plasmas, unless the resisrvity is locally strongly enhanced at the X- point. In the absence of such anomalous enhancement, the reconnecron layer evolves dynamically to form Y- points and realize a Sweet- Parker regime.

34 2D coronal loop : high- Lundquist number resisrve MHD simularon T = 0 T = 30

35 Hall MHD Model and the Generalized Ohm s Law In high-s plasmas, when the width satisfies! " of the thin current sheet " # < c /$ pi collisionless terms in the generalized Ohm s law cannot be ignored. Generalized Ohm s law (dimensionless form) Electron skin depth Ion skin depth Electron beta ( ) E + v " B = 1 S J + d e 2 dj dt + d i n J " B # $! ( ) ( ) d e! L "1 c /# pe d i! L "1 c /# pi! e p e

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38 MagneRc ReconnecRon: Sisyphus of the Plasma Universe The struggle itself is enough to fill a man s heart. One must imagine Sisyphus happy Albert Camus in The Myth of Sisyphus (Le Myth de Sisyphe, 1942) Ti6an, 1549

39 Lectures for which this material is directly perrnent Rempel : MHD Dynamos Bagenal : Dynamos and Magnetospheres Gosling: Solar Wind and Heliosphere TofoleIo : Solar Wind- Magnetosphere Coupling Miesch: Solar ConvecRon and Dynamo

40 What is Turbulence? Webster s 1913 Dictionary: The quality or state of being turbulent; a disturbed state; tumult; disorder, agitation. Alexandre J. Chorin, Lectures on Turbulence Theory (Publish or Perish, Inc., Boston 1975): The distinguishing feature of turbulent flow is that its velocity field appears to be random and varies unpredictably. The flow does, however, satisfy a set of equations, which are not random. This contrast is the source of much of what is interesting in turbulence theory.

41 Leonardo da Vinci (1500)

42 Hydrodynamic turbulence Kolmogorov (1941): can get energy spectrum by dimensional analysis Assumptions: Energy cascade rate:!(k) " k # E(k) $! " = 5/ 2, # = 3/ 2 Kolmogorov spectrum: (energy moves from one k-shell to the next) { Energy spectrum: Kolmogorov constant: C K ~ 1.4! 2 = constant E(k) ~ C K! 2 / 3 k "5/ 3! E(k) dk = total energy

43 Solar wind turbulence Observation: power law in magnetic energy spectrum Kolmogorov law From Goldstein & Roberts (1999)

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45 MHD turbulence AddiRonal dependence on the Alfvén speed V A Energy cascade rate:!(k) " k # E(k) $ V A %! " = (5 #$ ) / 2, % = (3 #$ ) / 2 Spectral index: = constant! = 0 : Kolmogorov spectrum E(k) ~ C K! 2/3 k!5/3! B /!! =!1 : IK spectrum E(k) ~ C IK! 1/2 V A 1/2 k!3/2 Iroshnikov (1963), Kraichnan (1965) C IK ~ 1.6! 2.2 Dimensionless parameter:! " k 1/ 2 E k 1/ 2 V A #1 = v k / V A << 1

46 overlapping spectra over a period of large scale Alfvén Rme k - 3/2 energy spectra

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49 Anisotropy due to magne&c field Energy spectrum E(k,k! ) relarve to a background B 0 Dimensional analysis cannot fix the form of E(k, k! ) Energy cascade faster in k than k Alfvén Rme! A ~ 1/ k V A, " # k $ v k / k V A Critical balance cascade! ~ 1 Goldreich & Sridhar 1995) recover Kolmogorov spectrum: E(k! ) ~ " 2/ 3 #5/ 3 k! 2 scale- dependent anisotropy: k ~ k /3! L "1/3

50 Kinetic vs magnetic spectrum -- observations (a) (b) From [Podesta, Roberts, & Goldstein 2006], power spectra for the total kinerc energy (a) and for the total magnerc energy (b) The best fit straight line over the interval is indicated by the short red line segment.

51 In physics the truth is rarely perfectly clear, and that is certainly universally the case in human affairs. Hence, what is not surrounded by uncertainty cannot be the truth. From Perfectly Reasonable Deviations From the Beaten Track---The Letters of Richard P. Feynman, edited by Michelle Feynman (Perseus, 2005)