Magnetohydrodynamics (MHD)

Transcription

1 Magnetohydrodynamics (MHD) Robertus v F-S Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK)

2 The Outline Introduction Magnetic Sun MHD equations Potential and force-free fields Selected applications (sunspots, prominences) Conclusions

3 Why Bother? Modern Physics è Plasma Physics Solar and stellar interiors are composed of ionised plasma and hence are excellent conductors of electricity In fact, 99.9% matter of Universe is in plasma state! ST system is a natural plasma laboratory à geo-, astro- and tokamak physics, To explore space plasmas waves, in general, are excellent diagnostic tools!!!

4 Why bother: Big questions What is the basis of stability and dynamics of MHD systems from liquid metals through fusion plasma or solar-terrestrial structures? What mechanisms are responsible for heating in MHD astrophysical plamsa up to several million K? What accelerates the solar wind up to measured speeds exceeding 700 km/s? What are the physical processes behind the enormous energy releases (e.g. solar flares, megnetospheric substorms, energisation of ULF waves)?

5 What is the MHD model? Single fluid (continuum) approximation, macroscopic description Locally charged, globally neutral close to LTE MHD: perturbations of magnetic field, plasma velocity and plasma mass density, described by the MHD ( single fluid approximation) set of equations, which connects the magnetic field B, plasma velocity v, kinetic pressure p and density ρ. Simplified Maxwell s eqs + classical fluid dynamics

6 Why to study MHD? MHD plays a crucial approximation in the description of dynamics and structure of the solar interior, the entire solar atmosphere (sunspots, chromosphere, TR, corona, solar wind) and in Earth magnetosphere. MHD approximation is adequately describes the evolution and development of plasma perturbations, the transfer of plasma energy and momentum, plasma heating / acceleration, helioseismology, solar atmospheric (magneto) seismology, magnetosphere seismology. Also, we use it because it is relatively simple when compared to other approaches (e.g., kinetic theory)!

7 Is the Sun magnetic?

8 Is the Sun magnetic? Ca II emission Extreme ultra-violet

9 Is the Sun magnetic?

10 Governing equations t ρ + ( ρv) 0 (1.1) v t + ( v ) v B t 1 ρ p E + 1 ρ j B + (1.3) F v ( ) + s L (1.4) ρt s v t (1.)

11 Governing equations Notation: ρ is density, v velocity, p pressure, B magnetic induction, E electric field, j electric current, T temperature, s entropy per unit mass, F ν viscosity force, and L energy loss function 1 µ Ampere s law: j B (1.5) µ is magnetic permeability of empty space Viscous force in isotropic plasmas F v ( ω i τ i <<1) : 1 ν v + v 3 ν is kinematic viscosity, ρν const is dynamic viscosity (1.6)

12 Governing equations Ohm s law: m ( E + v B) j+ i σ σ j B ρe (1.7) σ is conductivity, m i ion mass, e proton electric charge. Last term is Hall current. Clapeiron law ( R gas constant, ~ µ mean atomic weight): Entropy: p ( R / ~ µ ) ρt γ s c ln( p / ρ ) + v (1.8) const (1.9) c ν is specific heat at constant density, γ is adiabatic index (usually 5/3) Energy loss function: L q ρν 1 3 v j v k σ j 1 j, k 1 x k + x j ( v) 3 (1.10)

13 Force-balance in magnetised plasmas Solar Physics & Space Plasma A magnetic field in a conducting fluid exerts a force per unit volume F mag!!!!! ( B) B F mag j B where j is the current and B the magnetic induction (often referred to as magnetic field strength). This is the sum of Lorentz forces on the particles. The equation of motion of an element of material inside a flux tube in a conducting fluid is µ o " " " ( B) B p + ρ g +.S + ρv! µ where g is the local gravitational acceleration, p the gas pressure, ρ the density and S a tensor describing viscous stresses. Setting v! 0 Governing equations we have the equation of magnetohydrostatic equilibrium. o

14 Force-balance in magnetised plasmas Solar Physics & Space Plasma A magnetic field in a conducting fluid exerts a force per unit volume F mag!!!!! ( B) B F mag j B where j is the current and B the magnetic induction (often referred to as magnetic field strength). This is the sum of Lorentz forces on the particles. The equation of motion of an element of material inside a flux tube in a conducting fluid is µ o " " " ( B) B p + ρ g +.S + ρv! µ where g is the local gravitational acceleration, p the gas pressure, ρ the density and S a tensor describing viscous stresses. Setting v! 0 Governing equations we have the equation of magnetohydrostatic equilibrium. o

15 Hydrostatic pressure balance Solar Physics & Space Plasma MHS equilibrium dp p Suppose: uniform vertical magnetic field B B z, 0 z MHS equation becomes dp dz ρ( z) g Separate variables g g j 0 g~ µ RT ( z) p( z) p( z), Λ( z) 1, log p n( z) + logp(0), Λ( z) i.e., no Lorenz force where where RT ( z) Λ( z) ~ µ g $!!#!!" pressure scale height 1 n( z) du ( ) 0 Λ u $!!#!!" integrated z number of scale heights

16 Hydrostatic pressure balance Solar Physics & Space Plasma MHS equilibrium dp p Suppose: uniform vertical magnetic field B B z, 0 z MHS equation becomes dp dz ρ( z) g Separate variables g g j 0 g~ µ RT ( z) p( z) p( z), Λ( z) 1, log p n( z) + logp(0), Λ( z) i.e., no Lorenz force where where RT ( z) Λ( z) ~ µ g $!!#!!" pressure scale height 1 n( z) du ( ) 0 Λ u $!!#!!" integrated z number of scale heights

17 Hydrostatic pressure balance Solar Physics & Space Plasma MHS equilibrium Solution p ( z) p(0)exp[-n( z)] Isothermal atmosphere (i.e. T, and Λ are const) p( z) p(0)exp(- z / Λ) ρ( z) ρ(0)exp(- z / Λ) Task: Mark the curves of Λ1, and 3 Image: V. Nakariakov

18 Hydrostatic pressure balance Typical examples (R8.3x10 3 J K -1 mol -1 ) Photosphere (g74 m/s; µ1.3, T6000) Corona (g74 m/s; µ0.6, T>10 6 ) Solar Physics & Space Plasma MHS equilibrium Earth s atmosphere (g9.81 m/s; µ09, T300) Λ RT ~ 140 km µ g Λ $! 50!!!!.5T m # 50.5!!!!!! T[MK] Mm " Comapre to loop size! Λ 8.7 km

19 Photosphere: structure of sunspots Sunspots are cooler than their surroundings because their strong magnetic field inhibits convection below the level of the photosphere. Hence, internal heat flux F i, is reduced compared to external heat flux F e Sunspot field structure determined by lateral pressure balance P i + i o e B B Pe + µ µ o

20 Solar Physics & Space Plasma Research Centre (SPRC) Prominences/filaments Filaments - called prominences when they appear in emission at the limb - are cool (0,000K) dense (101m-3) gas which is thermally isolated from the surrounding corona. They appear in active regions and in the quiet sun, and overlay magnetic neutral lines. AR filaments tend to erupt within a few days, QS filaments can last and grow for weeks.

21 Prominences/filaments Filament support comes from the magnetic tension force in dipped magnetic fields or flux ropes. This opposes the downwards force of gravity. In static equilibrium (neglecting pressure gradients and viscous terms), the force-balance equation is: Image from Gilbert et al. 001!!! ( B) B ρg + µ o! ρg - B + µ o Magnetic pressure!! ( B. ) B µ o Magnetic tension These upward-curving field lines can be envisaged in a number of geometries. 0

22 Force-free and non-force-free fields Solar Physics & Space Plasma Governing equations In the case of where all forces are negligible, except for the j x B force, the MHS equation reduces to! j B! This is known as the force-free condition. The gas pressure has virtually no influence (low-β plasma). Note: There are no cross-field currents in a force-free plasma. All currents are field-aligned. The photosphere is not force free. Moving outwards in the atmosphere the gas pressure and viscosity decrease, and the force-free condition becomes a good approximation (from ~500km above τ 500nm 1) Above a few tenths of a solar radius, the field is again not force-free. 0

23 Force-free and non-force-free fields Solar Physics & Space Plasma Governing equations In the case of where all forces are negligible, except for the j x B force, the MHS equation reduces to! j B! This is known as the force-free condition. The gas pressure has virtually no influence (low-β plasma). Note: There are no cross-field currents in a force-free plasma. All currents are field-aligned. The photosphere is not force free. Moving outwards in the atmosphere the gas pressure and viscosity decrease, and the force-free condition becomes a good approximation (from ~500km above τ 500nm 1) Above a few tenths of a solar radius, the field is again not force-free. 0

24 Limits of applicability Speeds are much less than the speed of light. (In the solar corona: v < a few thousand km/s). Characteristic times are much longer than the Larmour rotation period and the plasma period. In the solar corona: f MHD < 1 Hz. E.g., for B10 G, n e 5x10 14 m -3 f gyro 1.5x10 3 xb(g) 1.5x10 4 Hz f plasma 9x n e 1/ (m -3 ) x10 8 Hz

25 Limits of applicability Speeds are much less than the speed of light. (In the solar corona: v < a few thousand km/s). Characteristic times are much longer than the Larmour rotation period and the plasma period. In the solar corona: f MHD < 1 Hz. E.g., for B10 G, n e 5x10 14 m -3 f gyro 1.5x10 3 xb(g) 1.5x10 4 Hz f plasma 9x n e 1/ (m -3 ) x10 8 Hz

26 Limits of applicability Speeds are much less than the speed of light. (In the solar corona: v < a few thousand km/s). Characteristic times are much longer than the Larmour rotation period and the plasma period. In the solar corona: f MHD < 1 Hz. E.g., for B10 G, n e 5x10 14 m -3 f gyro 1.5x10 3 xb(g) 1.5x10 4 Hz f plasma 9x n e 1/ (m -3 ) x10 8 Hz

27 Important limits Cold plasma ß0 Incompressible plasma γ

28 Important limits Cold plasma ß0 Incompressible plasma γ

29 Important limits Cold plasma ß0 Incompressible plasma γ

30 Frozen-in fields The magnetic field is for the most part frozen-in to the coronal plasma. This is the same as saying that the plasma is highly conducting. We can demonstrate this by looking at field advection and diffusion. Start! with Ohm s law:!!! j E + v B σ conductivity 1/ηµ o σ!! Take the curl of this equation, and use E ( B/ t) to eliminate E. where we have also used Expanding the last term, and using B!!! (v B) η ( B)! t!! µ j B 0. B! 0 B!! (v B)! + η B! t we arrive at the induction equation

31 Frozen-in fields The two terms on the left hand side represent the advection of field by the flow, and the dissipation of field due to resistivity B!! (v B)! + η B! t 1 Normally in the solar atmosphere (e.g. corona), conductivity σ is very high, so η 1/µ o σ is very small. In this case, term is negligible in comparison with term 1. So the equation becomes! B! (v B)! t v x B is the component of flow perpendicular to the magnetic field. So perpendicular flows distort B, and vice versa. The field is locked to the plasma. (In fact one must prove that the total magnetic flux through a surface remains constant as the field is deformed).

32 Potential and force-free fields If magnetic field B(B x, B y, B z ) is known as a fnc of position, then the field lines are defined by dx B x dy B y dz B z ds B In parametric form in terms of the parameter s, the field lines satisfy dx ds Bx dy By dz Bz,,, B ds B ds B where the parameter s is the distance along the field line.

33 Potential and force-free fields Example: BB 0 (y/a, x/a, 0), where B 0 and a are cnst dx y / a dy è xdx ydy è x y ± c conts x / a Field lines: hyperbolae This is a neutral point, or X-point Image: V. Nakariakov

34 Potential and force-free fields Plasma-β If characteristic length (L) << scale height (Λ) è neglect gravity 1 0 p + j B j µ B p Ratio of pressure gradient and Lorenz force: β p L 1 µ gas pressure magnetic pressure B : B B p / µ B µ L

35 Potential and force-free fields Plasma-β Can be evaluated by the formula β 3.5x10 1 ntb, where n [m 3 ], T [K], and B [G]. Solar corona: T10 6 K, n m -3, B10 G è β3.5x10-3 Solar photospheric magnetic flux tubes: T6x10 3 K, n 10 3 m -3, B1000 G è β Solar wind near Earth: Tx10 5 K, n 10 7 m -3, B6x10-5 G è β

36 Potential and force-free fields Plasma-β Can be evaluated by the formula β 3.5x10 1 ntb, where n [m 3 ], T [K], and B [G]. Solar corona: T10 6 K, n m -3, B10 G è β3.5x10-3 Solar photospheric magnetic flux tubes: T6x10 3 K, n 10 3 m -3, B1000 G è β Solar wind near Earth: Tx10 5 K, n 10 7 m -3, B6x10-5 G è β

37 Potential and force-free fields Plasma-β Can be evaluated by the formula β 3.5x10 1 ntb, where n [m 3 ], T [K], and B [G]. Solar corona: T10 6 K, n m -3, B10 G è β3.5x10-3 Solar photospheric magnetic flux tubes: T6x10 3 K, n 10 3 m -3, B1000 G è β Solar wind near Earth: Tx10 5 K, n 10 7 m -3, B6x10-5 G è β

38 Potential and force-free fields Plasma-β Can be evaluated by the formula β 3.5x10 1 ntb, where n [m 3 ], T [K], and B [G]. Solar corona: T10 6 K, n m -3, B10 G è β3.5x10-3 Solar photospheric magnetic flux tubes: T6x10 3 K, n 10 3 m -3, B1000 G è β Solar wind near Earth: Tx10 5 K, n 10 7 m -3, B6x10-5 G è β

39 Potential and force-free fields Force-free fields If β<<1, gas pressure neglected w.r.t. magnetic pressure 0 j B Magnetic field called force-free Potential force-free fields Suppose j:0 j B 0 Magnetic field called potential Most general solution: B ϕ, where ϕ is the scalar magnetic Solenoidal condition (s B0) has to be satisfied, i.e. ϕ ϕ ϕ ϕ x y z potential Laplace equation

40 Potential and force-free fields Potential fields Solution: -dimensional plane [xz], method of separataion of variables φ:x(x)y(y) X ʹ ʹ k X X ( x) X X ʹ ʹ a X ʹ ʹ Y sin Y Y + ʹ ʹ kx Y ʹ ʹ k Y Y ( y) c exp( ky ) + + XY b ʹ ʹ k d cos 0 kx exp( ky ) const Boundary conditions: φ(x,0)f(x), φ(0,y) φ(l,y)0, φ 0 as y, giving bd0 and sin kl 0 k nπ l

41 Potential and force-free fields Potential fields Solution: -dimensional plane [xz], method of separataion of variables φ:x(x)y(y) X ʹ ʹ k X X ( x) X X ʹ ʹ a X ʹ ʹ Y sin Y Y + ʹ ʹ kx Y ʹ ʹ k Y Y ( y) c exp( ky ) + + XY b ʹ ʹ k d cos 0 kx exp( ky ) const Boundary conditions: φ(x,0)f(x), φ(0,y) φ(l,y)0, φ 0 as y, giving bd0 and sin kl 0 k nπ l

42 Potential and force-free fields Potential fields Solution: -dimensional plane [xz], method of separataion of variables φ:x(x)y(y) X ʹ ʹ k X X ( x) X X ʹ ʹ a X ʹ ʹ Y sin Y Y + ʹ ʹ kx Y ʹ ʹ k Y Y ( y) c exp( ky ) + + XY b ʹ ʹ k d cos 0 kx exp( ky ) const Boundary conditions: φ(x,0)f(x), φ(0,y) φ(l,y)0, φ 0 as y, giving bd0 and sin kl 0 k nπ l

43 Potential and force-free fields Potential fields Full solution obtained by summing over all possible solutions. Let A k ac, ϕ( x, Example B y) Ak sinkx exp( ky) where F ( x) k πx F( x) : sin A 1, A 0, n l 1 n πx ϕ( x, y) sin exp( πy / l) l ϕ πx ϕ B B cos exp( πy / l), x x 0 l B y ϕ y B 0 πx sin l k exp( πy A k / l) sinkx

44 Potential and force-free fields Potential fields Model for magnetic field lines in coronal arcades Image: V. Nakariakov

45 Potential and force-free fields Non-potential force-free fields Suppose β<<1 and L << Λ but j 0 from Constrains on α 0 j B µ j αb (current parallel to magnetic field) B α B, α α ( r, t) ( B) ( αb) $!#!" identically 0 0 ( αb) α! B + B α B α 0 0 (αconst along magnetic field lines!)

46 Potential and force-free fields Non-potential force-free fields α:0, force-free potential fields α:const ( B) ( α B) α B α ( B) ( B) B $!#!" α:α(r) 0 B B B α B Helmholtz equation ( B ) ( αb) α B + α B α B + α B ( B) B α B + B B α [ B α 0]

47 Photospheric magnetic fields Force-free and non-force-free fields From the same photospheric field distributions, one can extrapolate the coronal magnetic field (by solving.b0 and BαB, with appropriate upper boundary conditions) The extra energy stored in non-potential fields is exhibited as twist. It is this excess of energy which can be released in the form of a solar flare or coronal mass ejection. MDI/SXT potential nonlinear FF

48 The end