MHD 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 physics (plasmas) stars, Sun, solar corona, solar wind, interstellar space, magnetospheres, Earth interior

Motivations The role of MHD turbulence in several phenomena in space and solar physics (plasmas) stars, Sun, solar corona, solar wind, interstellar space, magnetospheres, Earth interior Coronal heating magnetic loops Coronal heating coronal holes, solar wind acceleration (origin) Particle acceleration by turbulence

Basic structure of the Sun and the corona T core 14 10 6 K T radiative 10 5 K T photosphere 6 10 3 K T corona 2 10 6 K

Corona: external layer of Sun, atmosphere (heliosphere), from solar surface into interplanetary space. Plasma, electrons and ions gas, in motion, electric currents and magnetic fields. Corona emits radiation, however its emision in the visible range is small as compared to the solar surface. Corona in white light (in an eclipse)

Corona in UV (TRACE)

Magnetic loops loops Very structured magnetic field, in a broad range of length scales.

Magnetic loop: simple model Z=L Z B 0 Z=0 2 πl 2πl

Photosphere granulation (movements in the loop footpoints)

Velocity at the loop footpoints l p 1000 km t p 1000 s

MagnetoHydroDynamics Macroscopic description of a plasma: flows + electric currents Fluid equations (Navier-Stokes) + Electrodynamics (Maxwell eqs, Ohm s law)

MagnetoHydroDynamics Macroscopic description of a plasma: flows + electric currents Fluid equations (Navier-Stokes) + Electrodynamics (Maxwell eqs, Ohm s law) u = u(x, y, z, t) = plasma velocity B = B(x, y, z, t) = magnetic field ρ( u t + u u) = p + J B + ρν 2 u B t = (u B) + η 2 B J = B = current density, B = 0, u = 0

B t = (u B) + η 2 B Magnetic field tied to velocity field (frozen-in condition)

B t = (u B) + η 2 B Magnetic field tied to velocity field (frozen-in condition) Torsion (curl) of magnetic field lines means electric currents J = B

B t = (u B) + η 2 B Magnetic field tied to velocity field (frozen-in condition) Torsion (curl) of magnetic field lines means electric currents J = B Electric currents dissipate by Joule effect B t = (u B) + η 2 B

PROBLEM: t diss l 2 /η but η << 1 in the corona and l 1000 km in the loop footpoints producing the twist of the magnetic field t diss 10 4 years!!

PROBLEM: t diss l 2 /η but η << 1 in the corona and l 1000 km in the loop footpoints producing the twist of the magnetic field t diss 10 4 years!! We need a mechanism to accelerate the energy dissipation High frequency waves, directly injected at the footpoints (not observed). Turbulence: transfer of energy from large to small scales, where it can be easily dissipated. transfer non-linear terms.

Turbulence

Turbulence Energy transfer from large to small scales, which enhances dissipative processes and mixing: turbulent cascade. u = k u k e i k r, B = k B k e i k r non linear terms : u u, J B, (u B) mode mode coupling

E( k ) TURBULENT CASCADE energy cascade E(k) ~ k -5/3 energy input inertial range dissipation ηj 2, ν w 2 large scales k Turbulence transfers energy from large to small scales small scales k 1 l, large k small l If l small enough t diss hs

Reynolds number: turbulence R l non linear dissipative (u )u ν 2 u In the corona, R l 10 10 13 >> 1 turbulence u l l ν

Reynolds number: turbulence R l non linear dissipative (u )u ν 2 u u l l ν In the corona, R l 10 10 13 >> 1 turbulence Kolmogorov: dissipation rate injection rate transfer rate ɛ (independent of ν, η) u l ɛ 1/3 l 1/3 l d = l for which non-linear term dissipative term, that is R ld 1 l d ν ν3/4 u ld ɛ 1/4 l l d R 3/4 If R 10 10 l >> l d (several orders of magnitude)

3D MHD turbulence Direct Numerical Simulation B

But something happens if there is already a background magnetic field B 0, large compared to magnetic field fluctuations b, that is B = B 0 + b, B 0 >> b, anisotropy.

But something happens if there is already a background magnetic field B 0, large compared to magnetic field fluctuations b, that is B = B 0 + b, B 0 >> b, anisotropy. For instance, in the induction equation the non-linear term is of the form In k-space, (B )u = B 0 u + (b )u k B 0 b k + k b k b k The first term is a wave-like term, the second term is a non-linear term. Non-linearities dominate for wave vectors for which k b k k B 0 (the equal corresponds to the so-called critical balance ).

Another way of looking at this is recalling mode-mode couplings imply wavenumber triad interactions k = k + k But if we think on term of interacting Alfven waves (which we will see later have to be of opposite travelling direction), with e iwt time dependence, we also must have ±w(k) = w(k ) w(k ) where the dispersion relation for Alven waves is w(k) = k B 0 = k B 0 This imply that one of either k or k have zero parallel component along the background magnetic field direction and so the k-parallel component of k can not increase. In other words, the (energy) transfer in the k-parallel direction is inhibited. k Bo Energy transfer (k space) k k

Reduced MHD (RMHD) B = B 0 ẑ + b(x, y, z, t), B 0 >> b b = b (x, y, z, t), u = u (x, y, z, t) and z << x,y and we also eliminate fast t variations (acoustic waves)

Reduced MHD (RMHD) B = B 0 ẑ + b(x, y, z, t), B 0 >> b b = b (x, y, z, t), u = u (x, y, z, t) and z << x,y and we also eliminate fast t variations (acoustic waves) We can put b = a(x, y, z, t) ẑ = y a ˆx x a ŷ u = ψ(x, y, z, t) ẑ = y ψ ˆx x ψ ŷ with a(x, y, z, t) and ψ(x, y, z, t) scalar potentials. The 3D MHD equations become: t a = B 0 z ψ + [ψ, a] + η 2 a t ω = B 0 j + [ψ, w] [a, j] + ν 2 ω with j = J z the current density, ω = ω z the vorticity, and [ψ, a] = x ψ y a y ψ x a.

There is a clear numerical advantage on using RMHD equations vs the full 3D equations (for instance, we can increase the resolution in the perpendicular directions vs the parallel direction). Z=L Z B 0 Z=0 2 πl 2πl

Current density inside a magnetic loop (numerical simulation)

Current density and perpendicular magnetic field in a cross section of the magnetic loop moviehd

Energy spectrum 10 0 10 2 k 5/3 E k 10 4 10 6 1 10 100 k Consistent with Kolmogorov spectrum.

E b k, Eu k (dashed) k α ( E bk + Eu k ) E b k, Eu k (dashed) k α ( E bk + Eu k ) E b k, Eu k (dashed) k α ( E bk + Eu k ) Energy spectrum depends on t A /t p (not universal) 10 0 10 2 k 1 k 3 10.0 α = 2.4 10 4 10 6 10 8 t A / t p = 0.1 1 10 100 k 1.0 0.1 1 10 100 k 10 0 10 2 k 2 10.0 α = 1.9 10 4 10 6 10 8 t A / t p = 0.5 1 10 100 k 1.0 0.1 1 10 100 k 10 0 10 2 k 2 10.0 α = 1.8 10 4 10 6 10 8 t A / t p = 1.0 1 10 100 k 1.0 0.1 1 10 100 k

k α ( E bk + Eu k ) E b k, Eu k (dashed) 10 0 10 2 k 5/3 10 4 10 6 10 8 S = 800 t A / t p = 2.0 1 10 100 k 10.0 α = 1.66 1.0 0.1 1 10 100 k

10 0 E b k + Eu k 10 2 10 4 128 2 x 64 256 2 x 64 512 2 x 64 10 6 1024 2 x 64 10 8 1 10 100 k

dissipation rate t A = L /v A = Alfven time, 20 s (v A = B 0 / 4πρ=Alfven velocity) dissipation rate consistent with observational value 10 7 erg/cm 2 /s Initial transient 50 t A 1000 s Fluctuations timescale 10 t A 200 s

Scaling law ɛ l2 p t 3 A ( ta t p ) q, q 3 2

Statistics of dissipation events (nanoflares) E 10 24 erg

N(E) E 1.5 Self-organized criticality (SOC)