Magnetohydrodynamic Waves
|
|
- Nathan Fitzgerald
- 6 years ago
- Views:
Transcription
1 Magnetohydrodynamic Waves Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 17, 2016 These slides are largely based off of 4.5 and 4.8 of The Physics of Plasmas by Boyd & Sanderson (see also Chapter 6), Wikipedia articles on the wave equation and eigenstuff, Chapter 5 of Principles of Magnetohydrodynamics by Goedbloed and Poedts, lecture notes by Steve Cranmer, and a discussion with plasma wave expert Mahboubeh Asgari-Targhi. Extensive discussion of waves beyond MHD is included in Plasma Waves by D. G. Swanson and Waves in Plasmas by T. Stix.
2 Outline The 1D wave equation Algebraic solution Eigenmode solution Sound waves Linearization of equations of hydrodynamics Derivation of dispersion relationship MHD waves Linearization of MHD equations Introduce displacement vector ξ and MHD force operator F(ξ) Derivation of dispersion relationship Shear Alfvén, fast magnetosonic, and slow magnetosonic waves Observations of MHD waves Solar corona Space plasmas Laboratory experiments
3 Why do we care about waves? Waves are ubiquitous in magnetized plasmas Just as sound waves are ubiquitous in air Waves are the simplest way that a system responds to disturbances and applied forces Waves propagate information and energy through a system Waves are closely related to shocks, instabilities, and turbulence Plasmas display a rich variety of waves within and beyond MHD
4 Applications of waves in plasma astrophysics Space physics Earth s ionosphere, magnetosphere, and solar wind environment Solar and stellar physics Coronal heating Acceleration of solar and stellar winds Molecular clouds and star formation Interstellar medium Cosmic ray acceleration and transport Accretion disks and jets Pulsar magnetospheres Whenever a plasma is disturbed, there will be waves!
5 Example: the 1D wave equation The wave equation for u in one dimension is 2 u t 2 = c2 2 u x 2 (1) where c is a real constant that represents the wave speed The solutions are waves traveling at velocities of ±c The wave equation is a hyperbolic partial differential equation Connection to conservation laws
6 The algebraic solution to the 1D wave equation Define two new variables ξ(x, t) = x ct η(x, t) = x + ct (2) Rewrite the wave equation as The solutions are then 2 u ξ η = 0 (3) u(ξ, η) = R(ξ) + L(η) (4) u(x, t) = R(x ct) + L(x + ct) (5) where R and L are arbitrary functions traveling at velocities ±c (to the right and to the left)
7 Eigenmode decomposition of the 1D wave equation Use separation of variables and look for solutions of the form Plug this solution into the wave equation u ω (x, t) = e iωt f (x) (6) 2 t 2 [ e iωt f (x) ] = c 2 2 x 2 [ e iωt f (x) ] (7) ω 2 e iωt f (x) = c 2 e iωt d2 f (x) (8) dx 2 k 2 f (x) = d2 f (x) (9) dx 2 where k = ω/c. This is an eigenvalue equation for f (x). Next: identify eigenfunctions of the differential operator d2 dx 2 with corresponding eigenvalue k 2.
8 Eigenmode decomposition of the 1D wave equation Look for solutions of the form f (x) = Ae ±ikx (10) The solution to the wave equation for this eigenmode is Recall Euler s formula u ω (x, t) = Ae ikx iωt + Be ikx iωt (11) Take the real part of Eq. 11 to get e ix = cos x + i sin x (12) u ω (x, t) = A cos (kx + ωt) + B cos (kx ωt) (13) The solutions are waves propagating in the ±x directions. Use Fourier techniques to find the full solution.
9 Definitions The lines in the u-x plane on which x ct or x + ct are constant are called characteristics The wave vector k points in the direction of wave propagation and has a magnitude of k = 2π/λ where λ is the wavelength The phase velocity is the rate at which the phase of a wave propagates through space V p = ω k (14) The group velocity is the rate at which the overall shape of the waves amplitudes propagates through space V g = ω k (15)
10 Finding the dispersion relationship for sound waves Represent variables as the sum of a background component (denoted 0 ) and a small perturbed component (denoted 1 ) ρ(r, t) = ρ 0 + ρ 1 (r, t) (16) p(r, t) = p 0 + p 1 (r, t) (17) V(r, t) = V 1 (r, t) (18) Assume the background is homogeneous, time-independent, and static (V 0 = 0) Look for solutions proportional to e i(k r ωt) Solve for a dispersion relationship that connects the wave vector k with the angular frequency ω
11 Linearizing the equations of hydrodynamics The equations of hydrodynamics are ( ρ p t ρ t + (ρv) = 0 (19) ) t + V V + p = 0 (20) + V p + γp V = 0 (21) Linearize the equations. Drop higher order terms. Use that the background is constant. ρ 1 t + ρ 0 V 1 = 0 (22) V 1 ρ 0 + p 1 t = 0 (23) p 1 t + γp 0 V 1 = 0 (24)
12 Linearizing our first equation We start out with the continuity equation ρ t + (ρv) = 0 (25) Substitute in ρ(r, t) = ρ 0 + ρ 1 (r, t) and V(r, t) = V 1 (r, t). ρ 0 t }{{} =0 + ρ 1 t + (ρ 0V 1 ) + (ρ 1 V 1 ) }{{} second order = 0 (26) ρ 1 t + (ρ 0V 1 ) = 0 (27) We dropped ρ 0 t because the background is time-independent We dropped (ρ 1 V 1 ) because ρ 1 and V 1 are both small, so the product resulting from this second order term will be negligibly small.
13 Deriving a wave equation for hydrodynamics Take the time derivative of Eq. 23 ρ 0 2 V 1 t + p 1 t = 0 (28) Then substitute p 1 t = γp 0 V 1 from Eq. 24 to get a wave equation 2 V 1 t 2 c2 s ( V 1 ) = 0 (29) where the sound speed is γp0 c s (30) ρ 0
14 Assume that the solution is a superposition of plane waves Assume plane wave solutions of the form V 1 (r, t) = k ˆV k e i(k r ωt) (31) Differential operators turn into multiplications with algebraic factors ik, iω (32) t The problem is linear and homogeneous, so we consider each component separately. The wave equation then becomes 2 V 1 t 2 c2 s ( V 1 ) = 0 ( iω) 2 V 1 cs 2 (ik) (ik V 1 ) = 0 ω 2 V 1 c 2 s k (k V 1 ) = 0 (33)
15 The dispersion relationship for sound waves Choose coordinates so that k = k z ẑ, which then implies that V 1 = V 1z ẑ. Eq. 33 becomes ( ω 2 kz 2 cs 2 ) Vz1 = 0 (34) The non-trivial solutions are ω = ±k z c s (35)
16 Find the phase velocity and group velocity The dispersion relationship is The phase velocity and group velocity are ω = ±k z c s (36) V p ω k z = ±c s (37) V g ω k = ±c s (38) Sound waves are compressional because V 1 0 Sound waves are longitudinal because V 1 and k are parallel
17 How do we derive the dispersion relation for MHD waves? 1 Linearize the equations of ideal MHD. Take a Lagrangian approach Partially integrate the equations with respect to time Write equations in terms of the displacement from equilibrium Assume solutions proportional to e i(k r ωt) Derive a dispersion relationship that relates k and ω Investigate the properties of the three resulting wave modes 1 Here we follow Boyd & Sanderson 4.5 and 4.8.
18 Begin with the equations of ideal MHD The continuity, momentum, induction, and adiabatic energy equations are ρ + (ρv) = 0 (39) ( t ) ρ t + V V = J B p (40) c B = (V B) (41) ( ) t t + V p = γρ V (42)
19 The linearized equations of ideal MHD The continuity, momentum, induction, and adiabatic energy equations are linearized to become ρ 1 t V 1 ρ 0 t B 1 t p 1 t = V 1 ρ 0 ρ 0 V 1 (43) = ( B 1) B 0 4π p 1 (44) = (V 1 B 0 ) (45) = V 1 p 0 γp 0 V 1 (46) Here we ignored second and higher order terms and used Ampere s law. The terms V 1 ρ 0 and V 1 p 0 vanish if we assume the background is uniform
20 The displacement vector, ξ, describes how much the plasma is displaced from the equilibrium state 2 If ξ(r, t = 0) = 0, then the displacement vector is ξ(r, t) t Its time derivative is the perturbed velocity, 0 V 1 (r, t ) dt (47) ξ t = V 1(r, t) (48) 2 A side benefit of using slides is that I do not have to try writing ξ on the chalkboard.
21 Integrate the continuity equation with respect to time Put the linearized continuity equation with a uniform background in terms of ξ ρ 1 t Integrate this with respect to time t 0 ρ 1 t dt = = V 1 ρ 0 ρ 0 V 1 (49) = ξ t ρ 0 ρ 0 ξ t t 0 (50) [ ξ t ρ 0 ρ 0 ξ ] t dt (51) which leads to a solution for ρ 1 in terms of just ξ ρ 1 (r, t) = ξ(r, t) ρ 0 ρ 0 ξ(r, t) (52)
22 We can similarly put the linearized induction and energy equations in terms of ξ Integrating the linearized equations with respect to time yields solutions for the perturbed density, magnetic field, and plasma pressure: ρ 1 (r, t) = ξ(r, t) ρ 0 ρ 0 ξ(r, t) (53) [ ] ξ(r, t) B0 (r) B 1 (r, t) = (54) c p 1 (r, t) = ξ(r, t) p 0 (r) γp 0 (r) ξ(r, t) (55) The perturbed density ρ 1 doesn t appear in the other equations, which form a closed set However, we still have the momentum equation to worry about!
23 The linearized momentum equation in terms of ξ and F[ξ] Using the solutions for ρ 1, B 1, and p 1 we arrive at 2 ξ ρ 0 = F[ξ(r, t)] (56) t2 which is reminiscent of Newton s second law The ideal MHD force operator is F(ξ) = (ξ p 0 + γp 0 ξ) + 1 4π ( B 0) [ (ξ B 0 ] + 1 4π {[ (ξ B 0)] B 0 } (57) which is a function of the displacement vector ξ and equilibrium fields, but not of V 1 = ξ t.
24 Building up intuition for the displacement vector ξ and force operator F(ξ) The displacement vector ξ gives the direction and distance a parcel of plasma is displaced from the equilibrium state The force operator F(ξ) gives the direction and magnitude of the force on a parcel of plasma when it is displaced by ξ Discussion question: What is the sign of ξ F(ξ) when the configuration is unstable? Why?
25 Deriving the dispersion relation for MHD waves Assume that the plasma is uniform and infinite Perform a Fourier analysis by assuming solutions of the form ξ (r, t) = ξ (k, ω) e i(k r ωt) (58) k,ω The linearized momentum equation, then becomes 2 ξ ρ 0 = F (ξ (r, t)), (59) t2 ρ 0 ω 2 ξ = kγp 0 (k ξ) + {k [k (ξ B 0)]} B 0 4π (60)
26 Deriving the dispersion relation for MHD waves Choose Cartesian axes such that k = k ŷ + k ẑ (61) Expanding the vector products yields ( ) ω 2 k 2 V A 2 ξ x = 0 (62) ( ω 2 k 2 c2 s k 2 VA 2 ) ξy k k cs 2 ξ z = 0 (63) k k cs 2 ξ y + ( ω 2 k cs 2 ) ξz = 0 (64) where c s is the sound speed The Alfvén speed is defined as V A B0 2 4πρ 0 (65)
27 The dispersion relation for MHD waves To get a non-trivial solution (ξ 0), we need det ω 2 k 2 V 2 A ω 2 k 2 c2 s k 2 V 2 A k k c 2 s 0 k k c 2 s ω 2 k 2 c2 s = 0 (66) Eq. 66 reduces to the dispersion relation for MHD waves ( ) [ ω 2 k 2 V A 2 ω 4 k 2 ( ] cs 2 + VA) 2 ω 2 + k 2 k 2 c2 s VA 2 = 0 (67)
28 Non-trivial solutions of the dispersion relation for MHD waves The solution corresponding to shear Alfvén waves is ω 2 = k 2 V 2 A (68) The solution corresponding to slow and fast magnetosonic waves is ω 2 = 1 ( 2 k2 cs 2 + VA) [ 2 1 ± ] 1 δ (69) where δ is δ 4k2 c2 s VA 2 k ( ) 2 cs 2 + VA 2 2, 0 δ 1 (70) All three solutions are real No growth or decay No dissipation or free energy
29 Shear Alfvén and magnetosonic waves Left: Shear Alfvén waves propagating parallel to B 0 The displacement ξ is orthogonal to B0 and k These are transverse waves Right: A magnetosonic wave propagating orthogonal to B 0 The displacement ξ is parallel to k but orthogonal to B 0 These are longitudinal waves
30 Properties of the shear Alfvén wave The dispersion relationship is ω 2 = k 2 V 2 A The wave is transverse The restoring force is magnetic tension No propagation orthogonal to B 0 The displacement vector ξ = ξ xˆx is orthogonal to both B 0 = B 0 ẑ and k = k ŷ + k ẑ Shear Alfvén waves are incompressible Since k ξ = 0, the linearized continuity and energy equations show that both ρ 1 and p 1 are 0
31 Properties of slow and fast magnetosonic waves Magnetosonic waves are analogous to sound waves modified by the presence of a magnetic field Magnetosonic waves are longitudinal and compressible The restoring force includes contributions from magnetic pressure and plasma pressure These are also known as magnetoacoustic waves and slow/fast mode waves
32 What is the difference between slow and fast magnetosonic waves? Obvious differences Fast waves are faster (or the same phase velocity) Slow waves are slower (or the same phase velocity) Plasma pressure and magnetic pressure perturbations may work together or in opposition In the slow wave, these two effects are out of phase In the fast wave, these two effects are in phase The phase velocity depends on the angle of propagation with respect to the magnetic field and plasma β Slow mode waves cannot propagate orthogonal to B0 Fast mode waves propagate quasi-isotropically
33 Phase velocity and energetics Friedrichs diagrams plot the phase speed of waves as distance from the origin as a function of angle with respect to B 0 The wave energy includes contributions from kinetic, magnetic, and thermal energy Half of wave energy is kinetic energy for all three waves Half of the shear Alfvén wave s energy is magnetic The energetics of the slow and fast waves depend on the type of wave, the angle of propagation, and plasma β
34 kz kz kz kz kz kx kx β = 0.1 β = 0.5 β = 1 β = 2 β = 10 Friedrichs diagrams for MHD waves: Phase speed plotted as radial distance, with the angle between k and B0 shown as the angle away from the y axis. Here, β = (cs/va) 2. Blue point: Alfvén speed. Black point: sound speed. Curve color-codes shown below. GREEN: SLOW-MODE RED: FAST-MODE BLUE: ALFVÉN Illustration of how MHD waves partition their total fluctuation energy into kinetic, magnetic, and thermal energy in various regimes: wavevectors parallel to B0 (top row), an isotropic distribution of wavevectors (middle row), wavevectors perpendicular to B0 (bottom row); columns denote plasma β regimes. Kinetic energy fractions are denoted vi, magnetic energy fractions are denoted Bi, and the thermal energy fraction is denoted th. Accessibility note for the top row of plots: The (green) slow mode is always the contour closest to the origin, and the (red) fast mode is always the contour furthest from the origin.
35 Limitations of this analysis We linearized the equations of ideal MHD and combined them to derive the dispersion relationship for shear Alfvén waves, fast magnetosonic waves, and slow magnetosonic waves for a uniform, static, and infinite background Discussion questions: In what ways do our assumptions limit the applicability of these results? What are some situations where these assumptions are invalid?
36 In situ measurements of waves in space plasmas I Spacecraft observations provide highly detailed localized information I Anticorrelations between δb and δv in Wind data are due to Alfve n waves in the solar wind near 1 AU (Shi et al. 2015)
37 Observations of plasma waves in the solar corona Alfvén waves are a leading mechanism for heating solar & stellar coronae and accelerating solar & stellar winds Power spectra of Doppler velocity observations show counter-propagating waves, which are necessary for the development of turbulence (Morton et al. 2015)
38 Laboratory experiments on plasma waves Laboratory experiments offer an opportunity to study plasma waves in detail Left: The Large Plasma Device at UCLA which is used to study Alfvén waves, interacting magnetic flux ropes, and other phenomena Right: Polarized shear Alfvén waves detected in the experiment (shown are isosurfaces of field-aligned current and perturbed magnetic field vectors)
39 X-ray stripes in Tycho s supernova remnant are interpreted as cosmic ray acceleration sites I I Accelerated particles around supernova remnant shock waves generate Alfve n waves Laming (2015) proposed that the interaction between these Alfve n waves and the shock may result in these stripes
40 Summary Waves are ubiquitous in astrophysical, laboratory, space, and heliospheric plasmas The three principal wave modes for ideal MHD are the shear Alfvén wave, the slow magnetosonic wave, and the fast magnetosonic wave The shear Alfvén wave is a transverse wave that propagates along the magnetic field Slow and fast magnetosonic waves are longitudinal waves that may propagate obliquely Plasma waves are well-studied in solar, space, and laboratory plasmas and play important roles in a variety of astrophysical plasmas
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit Rough breakdown of MHD shocks Jump conditions: flux in = flux out mass flux: ρv n magnetic flux: B n Normal momentum flux: ρv n
More informationThe Virial Theorem, MHD Equilibria, and Force-Free Fields
The Virial Theorem, MHD Equilibria, and Force-Free Fields Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 10 12, 2014 These lecture notes are largely
More informationIdeal Magnetohydrodynamics (MHD)
Ideal Magnetohydrodynamics (MHD) Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 1, 2016 These lecture notes are largely based on Lectures in Magnetohydrodynamics
More informationReduced MHD. Nick Murphy. Harvard-Smithsonian Center for Astrophysics. Astronomy 253: Plasma Astrophysics. February 19, 2014
Reduced MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 19, 2014 These lecture notes are largely based on Lectures in Magnetohydrodynamics by Dalton
More informationSolar Physics & Space Plasma Research Center (SP 2 RC) MHD Waves
MHD Waves Robertus vfs Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK) What are MHD waves? How do we communicate in MHD? MHD is kind! MHD waves are propagating perturbations
More informationMacroscopic plasma description
Macroscopic plasma description Macroscopic plasma theories are fluid theories at different levels single fluid (magnetohydrodynamics MHD) two-fluid (multifluid, separate equations for electron and ion
More informationTheoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions
Theoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions Tom Elsden 1 Andrew Wright 1 1 Dept Maths & Stats, University of St Andrews DAMTP Seminar - 8th May 2017 Outline Introduction Coordinates
More informationSOLAR MHD Lecture 2 Plan
SOLAR MHD Lecture Plan Magnetostatic Equilibrium ü Structure of Magnetic Flux Tubes ü Force-free fields Waves in a homogenous magnetized medium ü Linearized wave equation ü Alfvén wave ü Magnetoacoustic
More informationMagnetohydrodynamic waves in a plasma
Department of Physics Seminar 1b Magnetohydrodynamic waves in a plasma Author: Janez Kokalj Advisor: prof. dr. Tomaž Gyergyek Petelinje, April 2016 Abstract Plasma can sustain different wave phenomena.
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationLinear stability of MHD configurations
Linear stability of MHD configurations Rony Keppens Centre for mathematical Plasma Astrophysics KU Leuven Rony Keppens (KU Leuven) Linear MHD stability CHARM@ROB 2017 1 / 18 Ideal MHD configurations Interested
More informationPLASMA ASTROPHYSICS. ElisaBete M. de Gouveia Dal Pino IAG-USP. NOTES: (references therein)
PLASMA ASTROPHYSICS ElisaBete M. de Gouveia Dal Pino IAG-USP NOTES:http://www.astro.iag.usp.br/~dalpino (references therein) ICTP-SAIFR, October 7-18, 2013 Contents What is plasma? Why plasmas in astrophysics?
More informationMHD Modes of Solar Plasma Structures
PX420 Solar MHD 2013-2014 MHD Modes of Solar Plasma Structures Centre for Fusion, Space & Astrophysics Wave and oscillatory processes in the solar corona: Possible relevance to coronal heating and solar
More informationThe Magnetorotational Instability
The Magnetorotational Instability Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics March 10, 2014 These slides are based off of Balbus & Hawley (1991), Hawley
More informationSpace Physics. ELEC-E4520 (5 cr) Teacher: Esa Kallio Assistant: Markku Alho and Riku Järvinen. Aalto University School of Electrical Engineering
Space Physics ELEC-E4520 (5 cr) Teacher: Esa Kallio Assistant: Markku Alho and Riku Järvinen Aalto University School of Electrical Engineering The 6 th week: topics Last week: Examples of waves MHD: Examples
More informationThe Euler Equation of Gas-Dynamics
The Euler Equation of Gas-Dynamics A. Mignone October 24, 217 In this lecture we study some properties of the Euler equations of gasdynamics, + (u) = ( ) u + u u + p = a p + u p + γp u = where, p and u
More informationPlasma waves in the fluid picture I
Plasma waves in the fluid picture I Langmuir oscillations and waves Ion-acoustic waves Debye length Ordinary electromagnetic waves General wave equation General dispersion equation Dielectric response
More informationNONLINEAR MHD WAVES THE INTERESTING INFLUENCE OF FIREHOSE AND MIRROR IN ASTROPHYSICAL PLASMAS. Jono Squire (Caltech) UCLA April 2017
NONLINEAR MHD WAVES THE INTERESTING INFLUENCE OF FIREHOSE AND MIRROR IN ASTROPHYSICAL PLASMAS Jono Squire (Caltech) UCLA April 2017 Along with: E. Quataert, A. Schekochihin, M. Kunz, S. Bale, C. Chen,
More informationSW103: Lecture 2. Magnetohydrodynamics and MHD models
SW103: Lecture 2 Magnetohydrodynamics and MHD models Scale sizes in the Solar Terrestrial System: or why we use MagnetoHydroDynamics Sun-Earth distance = 1 Astronomical Unit (AU) 200 R Sun 20,000 R E 1
More informationRecapitulation: Questions on Chaps. 1 and 2 #A
Recapitulation: Questions on Chaps. 1 and 2 #A Chapter 1. Introduction What is the importance of plasma physics? How are plasmas confined in the laboratory and in nature? Why are plasmas important in astrophysics?
More informationIntroduction to Magnetohydrodynamics (MHD)
Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection Aim Derivation of MHD equations from conservation laws Quasi-neutrality
More informationDispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson
2017-02-14 Dispersive Media, Lecture 7 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasmas as a coupled system Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas
More informationWaves in plasma. Denis Gialis
Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous.
More informationJet Stability: A computational survey
Jet Stability Galway 2008-1 Jet Stability: A computational survey Rony Keppens Centre for Plasma-Astrophysics, K.U.Leuven (Belgium) & FOM-Institute for Plasma Physics Rijnhuizen & Astronomical Institute,
More informationWaves and characteristics: Overview 5-1
Waves and characteristics: Overview 5-1 Chapter 5: Waves and characteristics Overview Physics and accounting: use example of sound waves to illustrate method of linearization and counting of variables
More informationMHD turbulence in the solar corona and solar wind
MHD turbulence in the solar corona and solar wind Pablo Dmitruk Departamento de Física, FCEN, Universidad de Buenos Aires Motivations The role of MHD turbulence in several phenomena in space and solar
More informationMagnetohydrodynamic Waves
Magnetohydrodynamic Waves Magnetohydrodynamic waves are found in a wide variety of astrophysical plasmas. They have been measured in plasma fusion devices and detected in the MAGNETOSPHERE OF EARTH, the
More informationPlasma Physics for Astrophysics
- ' ' * ' Plasma Physics for Astrophysics RUSSELL M. KULSRUD PRINCETON UNIVERSITY E;RESS '. ' PRINCETON AND OXFORD,, ', V. List of Figures Foreword by John N. Bahcall Preface Chapter 1. Introduction 1
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More information2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson
2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasma physics Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas Transverse waves
More informationLecture # 3. Introduction to Kink Modes the Kruskal- Shafranov Limit.
Lecture # 3. Introduction to Kink Modes the Kruskal- Shafranov Limit. Steve Cowley UCLA. This lecture is meant to introduce the simplest ideas about kink modes. It would take many lectures to develop the
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationSolar Wind Turbulence
Solar Wind Turbulence Presentation to the Solar and Heliospheric Survey Panel W H Matthaeus Bartol Research Institute, University of Delaware 2 June 2001 Overview Context and SH Themes Scientific status
More information20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:
Phys780: Plasma Physics Lecture 20. Alfven Waves. 1 20. Alfven waves ([3], p.233-239; [1], p.202-237; Chen, Sec.4.18, p.136-144) We have considered two types of waves in plasma: 1. electrostatic Langmuir
More informationMHD WAVES AND GLOBAL ALFVÉN EIGENMODES
MHD WVES ND GLOBL LFVÉN EIGENMODES S.E. Sharapov Euratom/CCFE Fusion ssociation, Culham Science Centre, bingdon, Oxfordshire OX14 3DB, UK S.E.Sharapov, Lecture 3, ustralian National University, Canberra,
More informationPhysics Dec Time Independent Solutions of the Diffusion Equation
Physics 301 10-Dec-2004 33-1 Time Independent Solutions of the Diffusion Equation In some cases we ll be interested in the time independent solution of the diffusion equation Why would be interested in
More informationAnalysis of Jeans Instability of Partially-Ionized. Molecular Cloud under Influence of Radiative. Effect and Electron Inertia
Adv. Studies Theor. Phys., Vol. 5, 2011, no. 16, 755-764 Analysis of Jeans Instability of Partially-Ionized Molecular Cloud under Influence of Radiative Effect and Electron Inertia B. K. Dangarh Department
More informationWhy study plasma astrophysics?
Why study plasma astrophysics? Nick Murphy and Xuening Bai Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics January 25, 2016 Today s plan Definition of a plasma Plasma astrophysics:
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More information6.1. Linearized Wave Equations in a Uniform Isotropic MHD Plasma. = 0 into Ohm s law yields E 0
Chapter 6. Linear Waves in the MHD Plasma 85 Chapter 6. Linear Waves in the MHD Plasma Topics or concepts to learn in Chapter 6:. Linearize the MHD equations. The eigen-mode solutions of the MHD waves
More informationProf. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit Central concepts: Phase velocity: velocity with which surfaces of constant phase move Group velocity: velocity with which slow
More informationTwo ion species studies in LAPD * Ion-ion Hybrid Alfvén Wave Resonator
Two ion species studies in LAPD * Ion-ion Hybrid Alfvén Wave Resonator G. J. Morales, S. T. Vincena, J. E. Maggs and W. A. Farmer UCLA Experiments performed at the Basic Plasma Science Facility (BaPSF)
More informationIncompressible MHD simulations
Incompressible MHD simulations Felix Spanier 1 Lehrstuhl für Astronomie Universität Würzburg Simulation methods in astrophysics Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 1 / 20 Outline
More informationSound Waves Sound Waves:
3//18 Sound Waves Sound Waves: 1 3//18 Sound Waves Linear Waves compression rarefaction Inference of Sound Wave equation: Sound Waves We look at small disturbances in a compressible medium (note: compressible
More informationThe Physics of Fluids and Plasmas
The Physics of Fluids and Plasmas An Introduction for Astrophysicists ARNAB RAI CHOUDHURI CAMBRIDGE UNIVERSITY PRESS Preface Acknowledgements xiii xvii Introduction 1 1. 3 1.1 Fluids and plasmas in the
More informationCHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY
1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic
More informationMulti-D MHD and B = 0
CapSel DivB - 01 Multi-D MHD and B = 0 keppens@rijnh.nl multi-d MHD and MHD wave anisotropies dimensionality > 1 non-trivial B = 0 constraint even if satisfied exactly t = 0: can numerically generate B
More information2 The incompressible Kelvin-Helmholtz instability
Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect
More informationSummary PHY101 ( 2 ) T / Hanadi Al Harbi
الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force
More informationIdeal MHD Equilibria
CapSel Equil - 01 Ideal MHD Equilibria keppens@rijnh.nl steady state ( t = 0) smoothly varying solutions to MHD equations solutions without discontinuities conservative or non-conservative formulation
More information9 Fluid Instabilities
9. 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 important
More informationBeyond Ideal MHD. Nick Murphy. Harvard-Smithsonian Center for Astrophysics. Astronomy 253: Plasma Astrophysics. February 8, 2016
Beyond Ideal MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 8, 2016 These lecture notes are largely based on Plasma Physics for Astrophysics by
More informationMagnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium
Mechanics and Mechanical Engineering Vol. 12, No. 3 (2008) 267 274 c Technical University of Lodz Magnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium Emad E. Elmahdy Mathematics
More informationMagnetic Reconnection in Laboratory, Astrophysical, and Space Plasmas
Magnetic Reconnection in Laboratory, Astrophysical, and Space Plasmas Nick Murphy Harvard-Smithsonian Center for Astrophysics namurphy@cfa.harvard.edu http://www.cfa.harvard.edu/ namurphy/ November 18,
More informationHeating and current drive: Radio Frequency
Heating and current drive: Radio Frequency Dr Ben Dudson Department of Physics, University of York Heslington, York YO10 5DD, UK 13 th February 2012 Dr Ben Dudson Magnetic Confinement Fusion (1 of 26)
More informationConservation Laws in Ideal MHD
Conservation Laws in Ideal MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 3, 2016 These lecture notes are largely based on Plasma Physics for Astrophysics
More information(TRAVELLING) 1D WAVES. 1. Transversal & Longitudinal Waves
(TRAVELLING) 1D WAVES 1. Transversal & Longitudinal Waves Objectives After studying this chapter you should be able to: Derive 1D wave equation for transversal and longitudinal Relate propagation speed
More informationMagnetospheric Physics - Final Exam - Solutions 05/07/2008
Magnetospheric Physics - Final Exam - Solutions 5/7/8. Dipole magnetic field a Assume the magnetic field of the Earth to be dipolar. Consider a flux tube with a small quadratic cross section in the equatorial
More informationAST242 LECTURE NOTES PART 5
AST242 LECTURE NOTES PART 5 Contents 1. Waves and instabilities 1 1.1. Sound waves compressive waves in 1D 1 2. Jeans Instability 5 3. Stratified Fluid Flows Waves or Instabilities on a Fluid Boundary
More informationHydrodynamic modes of conducting liquid in random magnetic field
arxiv:1602.08543v1 [physics.plasm-ph] 27 Feb 2016 Hydrodynamic modes of conducting liquid in random magnetic field A. A. Stupka November 7, 2017 Oles Honchar Dnipropetrovs k National University, Gagarin
More informationAmplification of magnetic fields in core collapse
Amplification of magnetic fields in core collapse Miguel Àngel Aloy Torás, Pablo Cerdá-Durán, Thomas Janka, Ewald Müller, Martin Obergaulinger, Tomasz Rembiasz Universitat de València; Max-Planck-Institut
More informationBasic plasma physics
Basic plasma physics SPAT PG Lectures Jonathan Eastwood 10-14 October 2016 Aims Provide new PhD students in SPAT and the SPC section with an overview of the most important principles in space plasma physics,
More informationSOLAR WIND ION AND ELECTRON DISTRIBUTION FUNCTIONS AND THE TRANSITION FROM FLUID TO KINETIC BEHAVIOR
SOLAR WIND ION AND ELECTRON DISTRIBUTION FUNCTIONS AND THE TRANSITION FROM FLUID TO KINETIC BEHAVIOR JUSTIN C. KASPER HARVARD-SMITHSONIAN CENTER FOR ASTROPHYSICS GYPW01, Isaac Newton Institute, July 2010
More informationLinear and non-linear evolution of the gyroresonance instability in Cosmic Rays
Linear and non-linear evolution of the gyroresonance instability in Cosmic Rays DESY Summer Student Programme, 2016 Olga Lebiga Taras Shevchenko National University of Kyiv, Ukraine Supervisors Reinaldo
More information7.2.1 Seismic waves. Waves in a mass- spring system
7..1 Seismic waves Waves in a mass- spring system Acoustic waves in a liquid or gas Seismic waves in a solid Surface waves Wavefronts, rays and geometrical attenuation Amplitude and energy Waves in a mass-
More informationFluid Dynamics. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/14
Fluid Dynamics p.1/14 Fluid Dynamics Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/14 The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic). Use
More informationSpace Physics. An Introduction to Plasmas and Particles in the Heliosphere and Magnetospheres. May-Britt Kallenrode. Springer
May-Britt Kallenrode Space Physics An Introduction to Plasmas and Particles in the Heliosphere and Magnetospheres With 170 Figures, 9 Tables, Numerous Exercises and Problems Springer Contents 1. Introduction
More information1 Energy dissipation in astrophysical plasmas
1 1 Energy dissipation in astrophysical plasmas The following presentation should give a summary of possible mechanisms, that can give rise to temperatures in astrophysical plasmas. It will be classified
More informationGravitational Collapse and Star Formation
Astrophysical Dynamics, VT 010 Gravitational Collapse and Star Formation Susanne Höfner Susanne.Hoefner@fysast.uu.se The Cosmic Matter Cycle Dense Clouds in the ISM Black Cloud Dense Clouds in the ISM
More informationFlow: Waves and instabilities in stationary plasmas
Waves and instabilities in stationary plasmas: Overview F-1 Flow: Waves and instabilities in stationary plasmas Overview Introduction: theoretical themes for a complete MHD description of laboratory and
More informationMomentum transport from magnetic reconnection in laboratory an. plasmas. Fatima Ebrahimi
Momentum transport from magnetic reconnection in laboratory and astrophysical plasmas Space Science Center - University of New Hampshire collaborators : V. Mirnov, S. Prager, D. Schnack, C. Sovinec Center
More informationMAGNETOHYDRODYNAMICS - 2 (Sheffield, Sept 2003) Eric Priest. St Andrews
MAGNETOHYDRODYNAMICS - 2 (Sheffield, Sept 2003) Eric Priest St Andrews CONTENTS - Lecture 2 1. Introduction 2. Flux Tubes *Examples 3. Fundamental Equations 4. Induction Equation *Examples 5. Equation
More information2 u 1-D: 3-D: x + 2 u
c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function
More informationApplying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models
0-0 Applying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models B. Srinivasan, U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle,
More informationwhere G is Newton s gravitational constant, M is the mass internal to radius r, and Ω 0 is the
Homework Exercise Solar Convection and the Solar Dynamo Mark Miesch (HAO/NCAR) NASA Heliophysics Summer School Boulder, Colorado, July 27 - August 3, 2011 PROBLEM 1: THERMAL WIND BALANCE We begin with
More informationTurbulent Origins of the Sun s Hot Corona and the Solar Wind
Turbulent Origins of the Sun s Hot Corona and the Solar Wind Steven R. Cranmer Harvard-Smithsonian Center for Astrophysics Turbulent Origins of the Sun s Hot Corona and the Solar Wind Outline: 1. Solar
More informationLecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Dispersion Introduction - An electromagnetic wave with an arbitrary wave-shape
More informationAstrofysikaliska Dynamiska Processer
Astrofysikaliska Dynamiska Processer VT 2008 Susanne Höfner hoefner@astro.uu.se Aims of this Course - understanding the role and nature of dynamical processes in astrophysical contexts and how to study
More informationFigure 1: Surface waves
4 Surface Waves on Liquids 1 4 Surface Waves on Liquids 4.1 Introduction We consider waves on the surface of liquids, e.g. waves on the sea or a lake or a river. These can be generated by the wind, by
More informationSchool and Conference on Analytical and Computational Astrophysics November, Angular momentum transport in accretion disks
2292-13 School and Conference on Analytical and Computational Astrophysics 14-25 November, 2011 Angular momentum transport in accretion disks Gianluigi Bodo Osservatorio Astronomico, Torino Italy Angular
More informationIntroduction. Chapter Plasma: definitions
Chapter 1 Introduction 1.1 Plasma: definitions A plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behaviour. An equivalent, alternative definition: A plasma is a
More informationFundamentals of Magnetohydrodynamics (MHD)
Fundamentals of Magnetohydrodynamics (MHD) Thomas Neukirch School of Mathematics and Statistics University of St. Andrews STFC Advanced School U Dundee 2014 p.1/46 Motivation Solar Corona in EUV Want to
More informationGreg Hammett Imperial College, London & Princeton Plasma Physics Lab With major contributions from:
Greg Hammett Imperial College, London & Princeton Plasma Physics Lab With major contributions from: Steve Cowley (Imperial College) Bill Dorland (Imperial College) Eliot Quataert (Berkeley) LMS Durham
More informationPhysical Processes in Astrophysics
Physical Processes in Astrophysics Huirong Yan Uni Potsdam & Desy Email: hyan@mail.desy.de 1 Reference Books: Plasma Physics for Astrophysics, Russell M. Kulsrud (2005) The Physics of Astrophysics, Frank
More informationAlfvénic Turbulence in the Fast Solar Wind: from cradle to grave
Alfvénic Turbulence in the Fast Solar Wind: from cradle to grave, A. A. van Ballegooijen, and the UVCS/SOHO Team Harvard-Smithsonian Center for Astrophysics Alfvénic Turbulence in the Fast Solar Wind:
More information13.1 Ion Acoustic Soliton and Shock Wave
13 Nonlinear Waves In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (ie, nonlinear terms) in wave amplitude In such a
More informationKinetic, Fluid & MHD Theories
Lecture 2 Kinetic, Fluid & MHD Theories The Vlasov equations are introduced as a starting point for both kinetic theory and fluid theory in a plasma. The equations of fluid theory are derived by taking
More information13. ASTROPHYSICAL GAS DYNAMICS AND MHD Hydrodynamics
1 13. ASTROPHYSICAL GAS DYNAMICS AND MHD 13.1. Hydrodynamics Astrophysical fluids are complex, with a number of different components: neutral atoms and molecules, ions, dust grains (often charged), and
More informationIntroductory Review on Magnetohydrodynamic (MHD) Simulations in Astrophysics
Asian winter school on numerical astrophysics 13 March 2006 @Chiba Univ Introductory Review on Magnetohydrodynamic (MHD) Simulations in Astrophysics Kazunari Shibata ( Kwasan and Hida Observatories, Kyoto
More informationOverview spherical accretion
Spherical accretion - AGN generates energy by accretion, i.e., capture of ambient matter in gravitational potential of black hole -Potential energy can be released as radiation, and (some of) this can
More informationFUNDAMENTALS OF MAGNETOHYDRODYNAMICS (MHD)
FUNDAMENTALS OF MAGNETOHYDRODYNAMICS (MHD) Dana-Camelia Talpeanu KU Leuven, Royal Observatory of Belgium Basic SIDC seminar ROB, 7 March 2018 CONTENTS 1. Ideal MHD 2. Ideal MHD equations (nooooooo.) 2.1
More informationLet s consider nonrelativistic electrons. A given electron follows Newton s law. m v = ee. (2)
Plasma Processes Initial questions: We see all objects through a medium, which could be interplanetary, interstellar, or intergalactic. How does this medium affect photons? What information can we obtain?
More information8.2.2 Rudiments of the acceleration of particles
430 The solar wind in the Universe intergalactic magnetic fields that these fields should not perturb them. Their arrival directions should thus point back to their sources in the sky, which does not appear
More informationPlasma Processes. m v = ee. (2)
Plasma Processes In the preceding few lectures, we ve focused on specific microphysical processes. In doing so, we have ignored the effect of other matter. In fact, we ve implicitly or explicitly assumed
More informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More informationNovember 2, Monday. 17. Magnetic Energy Release
November, Monday 17. Magnetic Energy Release Magnetic Energy Release 1. Solar Energetic Phenomena. Energy Equation 3. Two Types of Magnetic Energy Release 4. Rapid Dissipation: Sweet s Mechanism 5. Petschek
More informationCoronal Heating versus Solar Wind Acceleration
SOHO 15: Coronal Heating, 6 9 September 2004, University of St. Andrews, Scotland Coronal Heating versus Solar Wind Acceleration Steven R. Cranmer Harvard-Smithsonian Center for Astrophysics, Cambridge,
More informationWave Equation in One Dimension: Vibrating Strings and Pressure Waves
BENG 1: Mathematical Methods in Bioengineering Lecture 19 Wave Equation in One Dimension: Vibrating Strings and Pressure Waves References Haberman APDE, Ch. 4 and Ch. 1. http://en.wikipedia.org/wiki/wave_equation
More informationThe Basic Properties of Surface Waves
The Basic Properties of Surface Waves Lapo Boschi lapo@erdw.ethz.ch April 24, 202 Love and Rayleigh Waves Whenever an elastic medium is bounded by a free surface, coherent waves arise that travel along
More information