Introduction to Magnetohydrodynamics (MHD)
|
|
- Eustacia Mitchell
- 6 years ago
- Views:
Transcription
1 Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection
2 Aim Derivation of MHD equations from conservation laws Quasi-neutrality Validity of MHD MHD equations in different forms MHD waves Alfvén s Frozen Flux Theorem Turbulence Characteristics Shocks Applications of MHD, i.e. all the interesting stuff!, will be in later lectures covering Waves, Reconnection, CMEs and Instabilities etc.
3 Derivation of MHD Possible to derive MHD from N-body problem to Klimotovich equation Louiville theorem to BBGKY hierarchy Simple fluid dynamics and control volumes First two are useful if you want to study kinetic theory along the way but all kinetics removed by the end Final method followed here so all physics is clear
4 Ideal MHD Maxwell equation Mass conservation F = ma for fluids Low frequency Maxwell Adiabatic equation for fluids Ideal Ohm s Law for fluids 8 equations with 8 unknowns
5 Mass Conservation - Continuity F (x) m = Z x+ x x dx F (x + x) x x + x Mass m in cell of width x changes due to rate of mass leaving/entering the cell F (x) x+ x dx = F (x) F (x + x = lim x!0 F (x) F (x + =0
6 Mass flux - =0 Mass flux per second through cell boundary In 3D this generalizes to F (x, t) = (x, t) v x (x, + r.( v) Z This is true for any conserved quantity so if + r.f U dx Hence applies to mass density, momentum density and energy density for example.
7 Convective Derivative In fluid dynamics the relation between total and partial derivatives is Convective derivative: Rate of change of quantity at a point moving with the fluid. Rate of change of quantity at a fixed point in space Often, and frankly for no good reason at all, write D Dt instead of d dt
8 Adiabatic energy equation If there is no heating/conduction/transport then changes in fluid element s pressure and volume (moving with the fluid) is adiabatic PV = constant Where is ration of specific heats d (PV )=0 dt Moving with a packet of fluid the mass is conserved so V / 1 d dt P =0
9 Momentum equation - Euler fluid P (x) u x x P (x + x) x x + x Total momentum in cell changes due to ( u x x) =F (x) F (x + x)+p (x) P (x + x) Now F is momentum flux per second F = u x ( = rp
10 Momentum equation - Euler fluid Use mass conservation equation to rearrange + x + x x rp Since by chain rule du x dt = rp du x (x, t) dt
11 For Euler fluid Momentum equation - MHD du dt = rp Force on charged particle in an EM field is F = q(e + v B) how does this change for MHD? Hence total EM force per unit volume on electrons is n e e(e + v B) and for ions (single ionized) is Where n e and n i densities n i e(e + v B) are the electron and ion number
12 Momentum equation - MHD Hence total EM force per unit volume e(n i n e )E +(en i u i en e u e ) B If the plasma is quasi-neutral (see later) then this is just en(u i u e ) B = j B Where j is the current density. Hence du dt = rp + j B Note jxb is the only change to fluid equations in MHD.
13 Maxwell equations Not allowed in MHD! Initial condition only Used to update B Low frequency version used to find current density j
14 Low-frequency Maxwell equations
15 Displacement current So for low velocities/frequencies we can ignore the displacement current
16 Quasi-neutrality For a pure hydrogen plasma we have Multiply each by their charge and add to get where σ is the charge density and j is the current density From Ampere's law at low frequencies Hence for low frequency processes this is quasi-nuetrality
17 MHD Maxwell equation Mass conservation Momentum conservation Low frequency Maxwell Energy conservation 8 equations with 11 unknowns! Need an equation for E
18 Ohm's Law Equations of motion for ion fluid is Assume quasi-neutrality, subtract electron equation This is called the generalized Ohm's law Note that Ohm's law for a current in a wire (V=IR) when written in terms of current density becomes When fluid is moving this becomes
19 Magnetohydrodynamics (MHD) Valid for: Low frequency Large scales If η=0 called ideal MHD Missing viscosity, heating, conduction, radiation, gravity, rotation, ionisation etc.
20 Solar scales and times For a coronal plasma with: Density 10 9 cm -3 Temperature 1MK Magnetic field 10 G D = v th! pe ' 1mm r L = m iv th eb mfp = v th i = ' 10 cm ' 10 km c! pi ' 10 km Observational limits Cadence ~ seconds Pixel ~ 100 km
21 Validity of MHD Assumed quasi-neutrality therefore must be low frequency and speeds << speed of light Assumed scalar pressure therefore collisions must be sufficient to ensure the pressure is isotropic. In practice this means: mean-free-path << scale-lengths of interest collision time << time-scales of interest Larmor radii << scale-lengths of interest However as MHD is just conservation laws plus lowfrequency MHD it tends to be a good first approximation to much of the physics even when some of these conditions are not met.
22 Eulerian form of @t = r.( v) = v.r.(v) 1 r.p + 1 = r (v B) j = 1 µ 0 r B Final equation can be used to eliminate current density so 8 equations in 8 unknowns
23 Lagrangian form of MHD equations D Dt = DP Dt = Dv Dt = r.v P r.v 1 r.p + 1 j B DB Dt =(B.r)v B(r.v) D j = 1 r B Dt µ 0 Alternatives D Dt = Specific internal energy density P = ( 1) B P r.v = B.rv
24 = r. r.( v) vv + I(P + B2 2 ) = r. + P + B2 v 2µ = r(vb Bv) E = P 1 + v2 2 + B2 2µ 0 The total energy density
25 Plasma beta A key dimensionless parameter for ideal MHD is the plasma-beta It is the ratio of thermal to magnetic pressure = 2µ 0P B 2 Low beta means dynamics dominated by magnetic field, high beta means standard Euler dynamics more important / c2 s v 2 A
26 Univorm B field MHD Waves Constant density, pressure Zero initial velocity Apply small perturbation to system Assume initially in stationary equilibrium r.p 0 = j 0 B 0 Simplify to easiest case with 0,P 0, B 0 = B 0 ẑ constant and no equilibrium current or velocity Apply perturbation, e.g. P = P 0 + P 1
27 MHD Waves Ignore quadratic terms, e.g. P 1 r.v 1 Linear equations so Fourier decompose, e.g. P 1 (r,t)=p 1 exp i(k.r!t) Gives linear set of equations of the form Ā.ū = ū Where ū =(P 1, 1,v 1,B 1 ) Solution requires det Ā Ī =0
28 Dispersion relation (Fast and slow magnetoacoustic waves) B k α (Alfvén waves) Alfvén speed Sound speed
29 Alfvén Waves Incompressible no change to density or pressure Group speed is along B does not transfer energy (information) across B fields
30 Fast magneto-acoustic waves For zero plasma-beta no pressure Compresses the plasma c.f. a sound wave Propagates energy in all directions
31 Magnetic pressure and tension Magnetic pressure Magnetic tension
32 Pressure perturbations
33 Phase and group speeds Phase speeds: : Group speeds
34 MHD Waves Movie Throwing a pebble into a plasma lake... For low plasma beta v A >> c s transv. velocity density Three types of MHD waves Alfvén waves magnetic tension (ω=v A k װ ) Fast magnetoacoustic waves magnetic with plasma pressure (ω V A k) Slow magnetoacoustic waves magnetic against plasma pressure (ω C S k װ )
35 MHD Waves Movie Throwing a pebble into a plasma lake... For low plasma betav A >> c s transv. velocity density Three types of MHD waves Alfvén waves magnetic tension (ω=v A k װ ) Fast magnetoacoustic waves magnetic with plasma pressure (ω V A k) Slow magnetoacoustic waves magnetic against plasma pressure (ω C S k װ )
36 Linear MHD for uniform media 1. The perturbations are waves 2. Waves are dispersionless 3. ω and k are always real 4. Waves are highly anisotropic 5. There are incompressible - Alfvén waves - and compressible - magnetoacoustic modes However, natural plasma systems are usually highly structured and often unstable See later lectures on waves & instabilities
37 Resistivity Electron-ion collisions dissipate current If we assume the resistivity is = r (v B)+ µ 0 r 2 B Ratio of advective to diffusive - magnetic Reynolds number R m = µ 0L 0 v 0 R m >> 1 Usually in space physics ( ).
38 Alfvén s theorem Rate of change of flux through a surface moving with fluid d dt Z S n.b ds = = Z ds v B.dl l Z r (E + v B).n ds S S Magnetic flux through a surface moving with the fluid is conserved with ideal MHD Ohm s law, i.e. no resistivity Often stated as- the flux is frozen in to the fluid
39 Line Conservation x(x + X,t) x(x + X, 0) X x x(x,t) x(x, 0) = X Consider two points which move with the fluid x i j D Dt x j X j X j j X j =( x.r)u
40 Line Conservation -2 Equation for evolution of the vector between two points D Dt Also for ideal MHD D Dt x =( x.r)u B Hence if we choose x to be along the magnetic field at t = 0 then it will remain aligned with the magnetic field. Two points moving with the fluid which are initially on the same field-line remain on the same field line in ideal MHD Reconnection not possible in ideal MHD = B.r v
41 Shown that Implies B and Cauchy Solution x x i j B i i Bj j 0 satisfy the same equation hence X j Where superscript zero refers to initial values B i j B 0 j 0 Cauchy solution 0 = 1,x 2,x 3 1,X 2,X 3 ) B i j B 0 j
42 MHD based on Cauchy B i j B 0 j 0 = 1,x 2,x 3 1,X 2,X 3 ) P = const Dv Dt = 1 r.p + 1 µ 0 (r B) B dx dt = v Only need to know position of fluid elements and initial conditions for full MHD solution
43 Non-ideal terms in MHD Ideal MHD is a set of conservation laws Non-ideal terms are dissipative and entropy producing Resistivity Viscosity Radiation transport Thermal conduction
44 Non-ideal MHD Resistivity Coriolis Gravity Other Thermal Conduction Radiation Ohmic heating Other
45 MHD + r.f (U) =RHS ln(e k ) Energy input RHS = driver Inertial Range RHS = 0 ln(k) Dissipation RHS = viscous
46 Kolmogorov 1941 (K41) Energy transfer rate through inertial range is constant " local dissipation rate Fluctuations in flow moves energy between scales Eddie turn over time l l v l Energy transfer from scale l to smaller scales (l) v2 l l = v3 l l
47 K41-2 (l) v2 l l = v3 l l but (l) =" v l l 1/3 " 1/3 Energy associated with flow lv 2 l E l " 2/3 l 5/3 More commonly E k " 2/3 k 5/3 Ignores anisotropy of MHD S 3 (l) =< (u(x + l) u(x)) 3 >= 4 5 "l
48 Iroshmikov-Kraichnan (IK) Alfvén waves interact weakly and pass by in time Eddie turn-over time is l v l Time-scale for energy transfer is now l v A l l v l v A v l Repeat K41 but now get E k ("v A ) 1/2 k 3/2
49 Goldreich-Sridah (G-S) IK model is isotropic G-S extended with critical balance k k v A k? v k Turbulence then only really in k? E k k 5/3 Parallel spectrum, i.e. Alfvén wave, passively fixed by critical balance due to perpendicular cascades This is for strong turbulence when B v B
50 MHD Characteristics Sets of ideal MHD equations can be written as All equations sets of this types share the same properties they express conservation laws can be decomposed into waves non-linear solutions can form shocks satisfy L1 contraction, TVD constraints
51 Characteristics is called the Jabobian matrix For linear systems can show that Jacobian matrix is a function of equilibria only, e.g. function of p 0 but not p 1
52 Properties of the Jacobian Left and right eigenvectors/eigenvalues are real Diagonalisable:
53 Characteristic waves This example is for linear equations with constant A But A = R R 1 so R 1 A = w t R 1 R 1 U =0 w + = = x 1 Λ. 0 with w R U w is called the characteristic field
54 Riemann problems is diagonal so all equations decouple i.e. characteristics w i propagate with speed In MHD the characteristic speeds are v x,v x ± c f,v x ± v A,v x ± c s Solution in terms of original variables U This analysis forms the basis of Riemann decomposition used for treating shocks, e.g. Riemann codes in numerical analysis
55 Temperature T = T (x c s t) Basic Shocks c 2 s = P/ x Without dissipation any 1D traveling pulse will eventually, i.e. in finite time, form a singular gradient. These are shocks and the differentially form of MHD is not valid. Also formed by sudden release of energy, e.g. flare, or supersonic flows.
56 Rankine-Hugoniot relations U U L U R S(t) x x l x r Integrate equations from x l to x r across moving discontinuity S(t)
57 Jump Conditions Use Let x l and x r tend to S(t) and use conservative form to get Rankine-Hugoniot conditions for a discontinuity moving at speed v s All equations must satisfy these relations with the same v s
58 The End
Macroscopic 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 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 informationFluid equations, magnetohydrodynamics
Fluid equations, magnetohydrodynamics Multi-fluid theory Equation of state Single-fluid theory Generalised Ohm s law Magnetic tension and plasma beta Stationarity and equilibria Validity of magnetohydrodynamics
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 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 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 informationComputational Astrophysics
16 th Chris Engelbrecht Summer School, January 2005 3: 1 Computational Astrophysics Lecture 3: Magnetic fields Paul Ricker University of Illinois at Urbana-Champaign National Center for Supercomputing
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 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 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 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 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 informationProf. 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 informationRadiative & Magnetohydrodynamic Shocks
Chapter 4 Radiative & Magnetohydrodynamic Shocks I have been dealing, so far, with non-radiative shocks. Since, as we have seen, a shock raises the density and temperature of the gas, it is quite likely,
More informationMagnetohydrodynamic Waves
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
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 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 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 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 informationp = nkt p ~! " !" /!t + # "u = 0 Assumptions for MHD Fluid picture !du/dt = nq(e + uxb) " #p + other Newton s 2nd law Maxwell s equations
Intro to MHD Newton s 2nd law Maxwell s equations Plasmas as fluids Role of magnetic field and MHD Ideal MHD What do we need to know to understand the sun, solar wind&shocks, magnetospheres? Some material
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 informationAccretion Disks I. High Energy Astrophysics: Accretion Disks I 1/60
Accretion Disks I References: Accretion Power in Astrophysics, J. Frank, A. King and D. Raine. High Energy Astrophysics, Vol. 2, M.S. Longair, Cambridge University Press Active Galactic Nuclei, J.H. Krolik,
More informationAdvection, Conservation, Conserved Physical Quantities, Wave Equations
EP711 Supplementary Material Thursday, September 4, 2014 Advection, Conservation, Conserved Physical Quantities, Wave Equations Jonathan B. Snively!Embry-Riddle Aeronautical University Contents EP711 Supplementary
More informationTransition From Single Fluid To Pure Electron MHD Regime Of Tearing Instability
Transition From Single Fluid To Pure Electron MHD Regime Of Tearing Instability V.V.Mirnov, C.C.Hegna, S.C.Prager APS DPP Meeting, October 27-31, 2003, Albuquerque NM Abstract In the most general case,
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 informationIntroduction to Plasma Physics
Introduction to Plasma Physics Hartmut Zohm Max-Planck-Institut für Plasmaphysik 85748 Garching DPG Advanced Physics School The Physics of ITER Bad Honnef, 22.09.2014 A simplistic view on a Fusion Power
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 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 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 information36. TURBULENCE. Patriotism is the last refuge of a scoundrel. - Samuel Johnson
36. TURBULENCE Patriotism is the last refuge of a scoundrel. - Samuel Johnson Suppose you set up an experiment in which you can control all the mean parameters. An example might be steady flow through
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 informationThe Hopf equation. The Hopf equation A toy model of fluid mechanics
The Hopf equation A toy model of fluid mechanics 1. Main physical features Mathematical description of a continuous medium At the microscopic level, a fluid is a collection of interacting particles (Van
More informationAstrophysical Fluid Dynamics (2)
Astrophysical Fluid Dynamics (2) (Draine Chap 35;;Clarke & Carswell Chaps 1-4) Last class: fluid dynamics equations apply in astrophysical situations fluid approximation holds Maxwellian velocity distribution
More informationFundamentals of Turbulence
Fundamentals of Turbulence Stanislav Boldyrev (University of Wisconsin - Madison) Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas What is turbulence? No exact definition.
More informationCreation and destruction of magnetic fields
HAO/NCAR July 30 2007 Magnetic fields in the Universe Earth Magnetic field present for 3.5 10 9 years, much longer than Ohmic decay time ( 10 4 years) Strong variability on shorter time scales (10 3 years)
More informationSimple examples of MHD equilibria
Department of Physics Seminar. grade: Nuclear engineering Simple examples of MHD equilibria Author: Ingrid Vavtar Mentor: prof. ddr. Tomaž Gyergyek Ljubljana, 017 Summary: In this seminar paper I will
More information26. Non-linear effects in plasma
Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 1 26. Non-linear effects in plasma Collisionless shocks ([1], p.405-421, [6], p.237-245, 249-254; [4], p.429-440) Collisionless
More informationPROBLEM SET. Heliophysics Summer School. July, 2013
PROBLEM SET Heliophysics Summer School July, 2013 Problem Set for Shocks and Particle Acceleration There is probably only time to attempt one or two of these questions. In the tutorial session discussion
More informationMHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION
MHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION Marty Goldman University of Colorado Spring 2017 Physics 5150 Issues 2 How is MHD related to 2-fluid theory Level of MHD depends
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 informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
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 informationResistive MHD, reconnection and resistive tearing modes
DRAFT 1 Resistive MHD, reconnection and resistive tearing modes Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK (This version is of 6 May 18 1. Introduction
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 informationNatalia Tronko S.V.Nazarenko S. Galtier
IPP Garching, ESF Exploratory Workshop Natalia Tronko University of York, York Plasma Institute In collaboration with S.V.Nazarenko University of Warwick S. Galtier University of Paris XI Outline Motivations:
More information2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
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 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 informationTwo Fluid Dynamo and Edge-Resonant m=0 Tearing Instability in Reversed Field Pinch
1 Two Fluid Dynamo and Edge-Resonant m= Tearing Instability in Reversed Field Pinch V.V. Mirnov 1), C.C.Hegna 1), S.C. Prager 1), C.R.Sovinec 1), and H.Tian 1) 1) The University of Wisconsin-Madison, Madison,
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 informationIV. Compressible flow of inviscid fluids
IV. Compressible flow of inviscid fluids Governing equations for n = 0, r const: + (u )=0 t u + ( u ) u= p t De e = + ( u ) e= p u+ ( k T ) Dt t p= p(, T ), e=e (,T ) Alternate forms of energy equation
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 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 informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More information2 GOVERNING EQUATIONS
2 GOVERNING EQUATIONS 9 2 GOVERNING EQUATIONS For completeness we will take a brief moment to review the governing equations for a turbulent uid. We will present them both in physical space coordinates
More informationMAGNETOHYDRODYNAMICS
Chapter 6 MAGNETOHYDRODYNAMICS 6.1 Introduction Magnetohydrodynamics is a branch of plasma physics dealing with dc or low frequency effects in fully ionized magnetized plasma. In this chapter we will study
More informationAstrophysical Hydrodynamics
Astrophysical Hydrodynamics i Teacher: Saleem Zaroubi a Room 282, phone 4055, email: saleem@astro.rug.nl b Office hours: You are always welcome to come to my office for short questions. It is better to
More informationMagnetohydrodynamics (MHD) Philippa Browning Jodrell Bank Centre for Astrophysics University of Manchester
Magnetohydrodynamics (MHD) Philippa Browning Jodrell Bank Centre for Astrophysics University of Manchester MagnetoHydroDynamics (MHD) 1. The MHD equations. Magnetic Reynolds number and ideal MHD 3. Some
More informationMagnetohydrodynamics (MHD)
Magnetohydrodynamics (MHD) Robertus v F-S Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK) The Outline Introduction Magnetic Sun MHD equations Potential and force-free fields
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 informationDissipation Scales & Small Scale Structure
Dissipation Scales & Small Scale Structure Ellen Zweibel zweibel@astro.wisc.edu Departments of Astronomy & Physics University of Wisconsin, Madison and Center for Magnetic Self-Organization in Laboratory
More informationHybrid Simulations: Numerical Details and Current Applications
Hybrid Simulations: Numerical Details and Current Applications Dietmar Krauss-Varban and numerous collaborators Space Sciences Laboratory, UC Berkeley, USA Boulder, 07/25/2008 Content 1. Heliospheric/Space
More informationThe Effect of Magnetic Turbulence Energy Spectra and Pickup Ions on the Heating of the Solar Wind
The Effect of Magnetic Turbulence Energy Spectra and Pickup Ions on the Heating of the Solar Wind C. S. Ng Geophysical Institute, University of Alaska Fairbanks A. Bhattacharjee, P. A. Isenberg, D. Munsi,
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 informationPlasmas as fluids. S.M.Lea. January 2007
Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider
More informationScope of this lecture ASTR 7500: Solar & Stellar Magnetism. Lecture 9 Tues 19 Feb Magnetic fields in the Universe. Geomagnetism.
Scope of this lecture ASTR 7500: Solar & Stellar Magnetism Hale CGEG Solar & Space Physics Processes of magnetic field generation and destruction in turbulent plasma flows Introduction to general concepts
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationICMs and the IPM: Birds of a Feather?
ICMs and the IPM: Birds of a Feather? Tom Jones University of Minnesota 11 November, 2014 KAW8: Astrophysics of High-Beta Plasma in the Universe 1 Outline: ICM plasma is the dominant baryon component in
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 informationCoronal Heating Problem
PHY 690C Project Report Coronal Heating Problem by Mani Chandra, Arnab Dhabal and Raziman T V (Y6233) (Y7081) (Y7355) Mentor: Dr. M.K. Verma 1 Contents 1 Introduction 3 2 The Coronal Heating Problem 4
More informationMAGNETIC NOZZLE PLASMA EXHAUST SIMULATION FOR THE VASIMR ADVANCED PROPULSION CONCEPT
MAGNETIC NOZZLE PLASMA EXHAUST SIMULATION FOR THE VASIMR ADVANCED PROPULSION CONCEPT ABSTRACT A. G. Tarditi and J. V. Shebalin Advanced Space Propulsion Laboratory NASA Johnson Space Center Houston, TX
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationCreation and destruction of magnetic fields
HAO/NCAR July 20 2011 Magnetic fields in the Universe Earth Magnetic field present for 3.5 10 9 years, much longer than Ohmic decay time ( 10 4 years) Strong variability on shorter time scales (10 3 years)
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 informationTHE INTERACTION OF TURBULENCE WITH THE HELIOSPHERIC SHOCK
THE INTERACTION OF TURBULENCE WITH THE HELIOSPHERIC SHOCK G.P. Zank, I. Kryukov, N. Pogorelov, S. Borovikov, Dastgeer Shaikh, and X. Ao CSPAR, University of Alabama in Huntsville Heliospheric observations
More informationPhysics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18
Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018 Properties of Sound Sound Waves Requires medium for propagation Mainly
More information0.2. CONSERVATION LAW FOR FLUID 9
0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is
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 informationEntropy generation and transport
Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
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 informationSpace Plasma Physics Thomas Wiegelmann, 2012
Space Plasma Physics Thomas Wiegelmann, 2012 1. Basic Plasma Physics concepts 2. Overview about solar system plasmas Plasma Models 3. Single particle motion, Test particle model 4. Statistic description
More informationMagnetohydrodynamics (MHD) Philippa Browning Jodrell Bank Centre for Astrophysics University of Manchester
Magnetohydrodynamics (MHD) Philippa Browning Jodrell Bank Centre for Astrophysics University of Manchester MagnetoHydroDynamics (MHD) 1. The MHD equations. Magnetic Reynolds number and ideal MHD 3. Some
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 informationSelected Topics in Plasma Astrophysics
Selected Topics in Plasma Astrophysics Range of Astrophysical Plasmas and Relevant Techniques Stellar Winds (Lecture I) Thermal, Radiation, and Magneto-Rotational Driven Winds Connections to Other Areas
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 informationTokamak Fusion Basics and the MHD Equations
MHD Simulations for Fusion Applications Lecture 1 Tokamak Fusion Basics and the MHD Equations Stephen C. Jardin Princeton Plasma Physics Laboratory CEMRACS 1 Marseille, France July 19, 21 1 Fusion Powers
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
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 informationGas Dynamics: Basic Equations, Waves and Shocks
Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks Susanne Höfner Susanne.Hoefner@fysast.uu.se Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks
More informationρ c (2.1) = 0 (2.3) B = 0. (2.4) E + B
Chapter 2 Basic Plasma Properties 2.1 First Principles 2.1.1 Maxwell s Equations In general magnetic and electric fields are determined by Maxwell s equations, corresponding boundary conditions and the
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 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 informationTwo-scale numerical solution of the electromagnetic two-fluid plasma-maxwell equations: Shock and soliton simulation
Mathematics and Computers in Simulation 76 (2007) 3 7 Two-scale numerical solution of the electromagnetic two-fluid plasma-maxwell equations: Shock and soliton simulation S. Baboolal a,, R. Bharuthram
More informationA Three-Fluid Approach to Model Coupling of Solar Wind-Magnetosphere-Ionosphere- Thermosphere
A Three-Fluid Approach to Model Coupling of Solar Wind-Magnetosphere-Ionosphere- Thermosphere P. Song Center for Atmospheric Research University of Massachusetts Lowell V. M. Vasyliūnas Max-Planck-Institut
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 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 informationSound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8
References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that
More informationRandom Walk on the Surface of the Sun
Random Walk on the Surface of the Sun Chung-Sang Ng Geophysical Institute, University of Alaska Fairbanks UAF Physics Journal Club September 10, 2010 Collaborators/Acknowledgements Amitava Bhattacharjee,
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 information