Lecture 7 - Magnetohydrodynamics II o Topics in today’s lecture:
o Equation of Motion
o Lorentz Forces
o Magnetic Pressure and Tension
o MHD Equilibria
Lecture 7 - MHD II
Magnetic Forces
dv o First consider, momentum equation: " = #$p + "g dt o Moving plasma will experience the Lorentz force (j!B) => dv ! " = #$p + j% B + "g dt o How do we interpret the Lorentz Force? Using Ampere’s Law: 1 j" B = # B " ($ " B) ! µ0 $ 1 ' but using the vector identity "& B# B) = B * (" * B) + (B# ")B % 2 ( 1 & B2 ) ! => j" B = (B# $)B % $( + µ0 ' 2µ0 * ! o 1st term is magnetic tension parallel to B - important for curved fields. 2nd term is magnetic pressure - important when |B| changes along field. ! ! Lecture 7 - MHD II Magnetic Pressure and Tension
o Lorentz force can be resolved into two components:
2 o Pressure term (B /2µ0): Isotropic. Gradient of a scalar. Perpendicular to B in example.
o Tension term (B/µ0) : Directed towards centre of curvature of B. e.g., B' has larger radius of curvature than B, therefore larger restoring force due to tension.
o Magnetic pressure and tension represent two kinds of B' restoring forces => associated with distinct wave modes: Alfven waves and magnetoacoustic waves.
B
Lecture 7 - MHD II
MHD Equation of Motion
o Motion in of plasma can be described by: dv " = # $p + j % B + "g dt (1) (2) (3) (4)
o In corona, term (3) dominates.
o Along B, j || B => (3) = 0, so (2) and (4) important. (2) p / L = 0 0 >> 1 for (4) "0 g
p0 p0 L0 =><< L0H<<= = H H = Scale height "0g"0 g i.e., for length-scales smalled than the scale height. Lecture 7 - MHD II !
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! MHD Equation of Motion
o L0 < H => p ~ costant.
o L0 > H => p falls
o Pressure gradient (!p) - acts from high to low p.
o Is normal to isobars.
Pressure isobars
Lecture 7 - MHD II
MHD Equation of Motion
dv " = # $p + j% B + " g dt (1) (2) (3) (4)
(2) p (i) = ! = (3) B2 / (2µ)
o W hen ! <<1, j " B dominates. Condition applies in corona.
B (ii) (1) ! (3) " v ! vA = µ#
o Perturbations or waves propagate at a a characteristic velocity, called the Alfven speed.
Lecture 7 - MHD II
! Typical values on Sun
Photosphere Chromosphere Corona N (m-3) 1023 1020 1015
T (K) 6000 104 106
B (G) 5 - 103 100 10
" 106 - 1 10-1 10-3
3 vA (km/s) 0.05 - 10 10 10
o 1 Tesla = 104 Gauss (G)
-21 2 9 1/2 o " = 3.5 x 10 N T/B , vA = 2 x 10 B/N
Lecture 7 - MHD II
MHD Equation of Motion - Equilibria
dv " = # $p + j% B + " g dt (1) (2) (3) (4)
o If v << vA, then (1) << (3) and so
0 = " #p + j $ B + % g,
o Called equation of magnetohydrostatic equilibrium.
o where Note: j = "# B / µ, ".B = 0, $ = p / (R T)
Lecture 7 - MHD II
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! MHD Equation of Motion - Equilibria
0 = " #p + j $ B + % g (2) (3) (4)
o If Lo << H, then (4) << (2) and 0 = " #p + j $ B o Called equation of magnetostatic equilibrium.
o If Lo << 2H/", then (4) << (3) and 0 = j " B
o Called force-free configuration. ! Lecture 7 - MHD II
Potential fields
o A simple solution to the force-free equation is j = 0, which is called a potental field configuration.
o From Ampere’s Law, " ! B = 0
which has a general solution B = "#,
where # is the scalar magnetic potential.
o Substituting the general solution into the solenoid condition ("#B=0) gives,
2 = 0 ! " # =># o Potential fields therefore satisfy Laplace’s equation. General solution can be given in terms of associated Legendre polynomials.
Lecture 7 - MHD II
! ! Force-free fields
o Assume that L0<< H and "<<1, we have the force-free field equation. If the magnetic field is not potential then the general solution is that the current must be parallel to the magnetic field. Thus, µj $ B or µj = $B
where % is a scalar function of position (i.e., $ = $(r) ) o From Ampere’s Law => "!B = $B o Now since "#B=0 and "#("!B) =0, we obtain "#("!B) = "#($B) = $"#B + B#"$ o Hence, B#"$ = 0 o So that $ is constant along each field line, although it may vary from field line to field line. If $ = 0, then the magnetic field reduces to the potential case.
Lecture 7 - MHD II
MHD Equation of Motion - Equilibria o If $ = const, then "!B = $B => "!("!B) = "!($B) = $"!B = $2B o However, "!"!B = "("#B) - "2B and so, "2B + $2B = 0 o Which is the Helmholtz Equation. A field that fulfils this condition is force-free. When boundary conditions specified, can then use standard mathematical methods to solve. o E.g., consider a coronal arcade with an arc (e.g, sinusoidal) in the xz plane and uniform in y. Also require field to vanish in at high altitude (e.g., exponential).
Lecture 7 - MHD II Linear force-free model o Now from ,
o Therefore,
where l2 = k2 - $2. o The projection of the field lines onto the xy-plane are parallel straight lines:
whereas the projection onto the xz-plane are arcs (see figure above).
Lecture 7 - MHD II