Introduction to MHD dynamos - François Rincon - Dynamo theories J. Plasma Phys. (Lecture Notes Series) 85, 205850401 (2019) arXiv:1903.07829
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Introduction to MHD dynamos François Rincon Bed time reading: Dynamo theories J. Plasma Phys. (Lecture Notes Series) 85, 205850401 (2019) arXiv:1903.07829 Les Houches, May 2017
Tutorial outline • Context • Short and easy (3h) • Setting the theory stage • Not too long and straightforward (4h) • Small scale MHD dynamos • Long and difficult (6h) • Large-scale MHD dynamos • Just a tad shorter, a bit less difficult (4h) • Connections between large & small-scale dynamos • Short and controversial, also difficult (2h) 2 Astroplasma, March 2021
What is dynamo theory about ? • The origin, and sustainment, of magnetic fields in the universe • on the Earth, other planets and their satellites (“planetary magnetism”) • in the Sun and other stars (“stellar magnetism”) • in galaxies, clusters and the early universe (“cosmic magnetism”) • Understanding their structural, statistical, and dynamical properties • Addressing important physics (and maths) problems • Deep connections with hydrodynamic turbulence and more generally (turbulent) transport problems • Coming up with “useful stuff” for experimentalists and observers • Warning: people strongly disagree on the definition of “useful stuff” 3 Astroplasma, March 2021
The fluid/plasma dynamo conundrum • Most astrophysical bodies, and many planetary interiors, are • in an electrically conducting fluid (MHD) or weakly-collisional plasma state • in a turbulent state • (differentially) rotating: shearing, Coriolis and precessing effects • Main questions • Can flows of electrically conducting fluid/plasma amplify magnetic fields ? • What are at the time and spatial scales on which this happens ? • At what amplitude do they saturate ? What field structure is produced ? • A complex and multifaceted problem • Requires observations, phenomenology, theory, numerics and experiments 4 Astroplasma, March 2021
A touch of history • Self-exciting fluid dynamos are now a century-old idea • First invoked by Larmor in 1919 (sunspot magnetism) • The idea took a lot of time to gain ground • Cowling’s antidynamo theorem (1933) • First examples in the 1950s (e.g. Herzenberg dynamo) • Parker’s solar dynamo phenomenology (1955) • Golden age of mathematical theory • Alpha effect / mean-field: Steenbeck, Krause, Raedler 1966, Moffatt, Roberts etc. (1970s) • Small-scale dynamo theory: Kazantsev 1967, Kraichnan, Zel’dovich et al. (70s-80s) • Numerical and experimental era • Numerical evidence of turbulent dynamos: Meneguzzi et al. 1981, flourishing since then • Experimental evidence: Riga, Karlsruhe (~2000), VKS (2007), plasma underway (2005+) • Great observational radio and spectro-polarimetric prospects too (stellar, galactic, cosmo) 5 Astroplasma, March 2021
Solar magnetism [Credits: SOHO/NASA] Global solar cycle dynamics ~ 1G-a few kG (sunspots) Small-scale surface dynamics ~ up to kG 6 [Credits: Hinode/JAXA] Astroplasma, March 2021
Planetary magnetism [Swarm/ESA] [HST/NASA] Earth’s magnetic field (2014) ~ a few G Jupiter Auroras 7 Astroplasma, March 2021
Galactic magnetism [Planck/ESA] Galactic magnetic field ~ 10 μG M51 magnetic field [Beck et al. VLA/Effelsberg] 8 Astroplasma, March 2021
1993ApJ...416..554T Galaxy clusters and cosmic magnetism [Taylor & Perley, ApJ 1993] [Fabian et al. ESA/NASA] Hydra A Lobe Perseus/NGC 1275 filaments (25 kpc) ICM fields ~10 μG [Durrer & Neronov, A&A Rev. 2013] Clusters Galaxies Primordial IGMF limits 9
Takeaway phenomenological points • Many astrophysical objects have global, ordered fields • Differential rotation, global symmetries and geometry important • Coherent structures and MHD instabilities may also be very important • Motivation for the development of “large-scale” dynamo theories • Lots of “small-scale”, random fields also discovered from the 70s • These come hand in hand with global magnetism • Simultaneous development of “small-scale dynamo” theory • Astrophysical magnetism is in a nonlinear, saturated state • Linear theory likely not the whole story (or requires non-trivial justification) • Multiple scale interactions expected to be important 10 Astroplasma, March 2021
Setting the stage
Mathematical formulation • Compressible, viscous, resistive MHD equations @⇢ + r · (⇢u) = 0 @t External forcing (spoon, Lorentz force ✓ ◆ gravity etc.) @u j⇥B ⇢ + u · ru = rp + + r · ⌧ + F(x, t) @t c Viscous stresses ✓ ◆ @ui @uj 2 Electromotive force ⌧ ij = µ + ij r ·u @B @xj @xi 3 = r⇥(u ⇥ B) r⇥(⌘r⇥B) @t Magnetic diffusion ⌘ = c2 4⇡ c r·B=0 j= r⇥B 4⇡ ✓ ◆ Dissipation Thermal diffusion @s ⇢T + u · rs = Dµ + D⌘ + r · (rT ) @t 12 Astroplasma, March 2021
Magnetic field energetics • Magnetic energy equation Z 2 Z I Z d B (j ⇥ B) c j2 dV = u· dV (E ⇥ B) · dS dV dt 8⇡ c 4⇡ Minus the work of the Poynting flux Ohmic dissipation Lorentz force on the flow • Magnetic field is generated at the expense of kinetic energy • Simple but enlightening local equation (ideal MHD) 1 DB = b̂b̂ : ru r·u B Dt B Stretching Compression b̂ = D @ rate rate B = +u·r Dt @t 13 Astroplasma, March 2021
Conservation laws in ideal MHD • Alfvén’s theorem(s) • Magnetic field lines are “frozen into” the fluid just as material lines ✓ ◆ D B B D r = · ru = r · ru Dt ⇢ ⇢ Dt • Magnetic flux through material surfaces is conserved D (B · S) = 0 S Dt Z • Magnetic helicity Hm = A · B d3 r conservation • A measure of magnetic linkage / knottedness 1 @A = E r' c @t @ (A · B) + r · [c'B + A ⇥ (u ⇥ B)] = 0 @t 14 Astroplasma, March 2021
Simple MHD system for dynamo theory • Incompressible, resistive, viscous MHD • Captures a great deal of the dynamo problem Magnetic tension @u + u · ru = rP + B · rB + ⌫ u + f (x, t) @t Induction/stretching B2 @B P =p+ + u · rB = B · ru + ⌘ B 2 @t Advection Resistive diffusion r·u=0 r·B=0 p and B rescaled by ⇢ and (4⇡⇢)1/2 • Often paired with simple periodic boundary conditions • Can be problematic in some cases (more later) 15 Astroplasma, March 2021
Scales and dimensionless numbers • System/integral scale ℓ0, U0 • Fluid system with two dissipation channels • Dimensionless numbers: `0 U0 `0 U0 ⌫ Re = Rm = Pm = ⌫ ⌘ ⌘ • Kolmogorov viscous scale ℓν ~ Re-3/4 ℓ0 , uν ~ Re-1/4 U0 • Magnetic resistive scale ℓη (Pm-dependent) • Another important dimensionless quantity • Eddy turnover time NL ~ ℓu/u ⌧c St = Strouhal/Kubo number • Flow/eddy correlation time c ⌧NL 16 Astroplasma, March 2021
The magnetic Prandtl number landscape • Wide range of Pm in nature ⌫ Liquid metals have Pm 1 makes Plasma exp. Liquid metals exp. life easier for magnetic fields Re 1 104 108 1012 17 Astroplasma, March 2021
Large magnetic Prandtl numbers • Pm > 1: resistive cut-off scale is smaller than viscous scale • In Kolmogorov turbulence, rate of strain goes as ℓ-2/3 • Viscous eddies are the fastest at stretching B: uν / ℓν ~ Re1/2 U0 / ℓ0 • To estimate the resistive scale ℓη, balance stretching by these eddies ~ uν/ℓν with ohmic diffusion rate η/ℓη2 1/2 `⌘ ⇠ Pm `⌫ e.g. galaxies, clusters Spectral power k-5/3 forcing scales Magnetic energy spectrum tail Kinetic energy spectrum Wavenumber k k0 kν ∼ Re3/4 k0 kη ∼ Pm1/2kν >> kν 18 Astroplasma, March 2021
Low magnetic Prandtl numbers • Pm < 1: resistive cut-off falls in the turbulent inertial range • To estimate the resistive scale ℓη, balance magnetic stretching by the eddies at the same scale ~ uη/ℓη, with diffusion η/ℓη2 • i.e., Rm (ℓη) = u(ℓη) ℓη / η ~ 1 3/4 `⌘ ⇠ Pm `⌫ e.g. stellar, solar, liquid metals (Earth, experiments) k-5/3 Kinetic energy Spectral power forcing spectrum scales Magnetic energy spectrum tail Wavenumber k k0 kη ∼ Pm3/4 kν
Dynamo fundamentals • The problem of exciting a dynamo is an instability problem `0 U0 • Growth requires stretching to overcome diffusion (measured by Rm = ) ⌘ @B • Kinematic dynamo problem: + u · rB = B · ru + ⌘ B @t • Find exponentially growing solutions of the linear induction equation (velocity field is prescribed) • Dynamical problem considers effects of Lorentz force on u • Saturated state of kinematic dynamos: non-linear magnetic back reaction • Subcritical scenarios: e.g. joint excitation of u and B via MHD instabilities • Slow vs Fast • A dynamo is slow/fast if its growth rate does/doesn’t vanish as ⌘ ! 0 20 Astroplasma, March 2021
Cowling’s antidynamo theorem • Axisymmetric dynamo action is impossible [Cowling, MNRAS, 1933] ez • In polar geometry, write e' Poloidal Toroidal z er • B = r ⇥ ( e' /r) + r e' • u = upol + r⌦ e' ' r ✓ ◆ @ 2 @ + upol · r = ⌘ No source term @t r @r ✓ ◆ @ 2 @ + upol · r = Bpol · r⌦ + ⌘ + @t r @r • Poloidal flow can only redistribute flux so must decay ultimately • As decays, so must the toroidal field • Note: only applies if u and B share the same symmetry axis 21 Astroplasma, March 2021
Antidynamo theorems and their implications • Many other antidynamo results can be proven • Plane two-dimensional motions cannot sustain a dynamo [Zel’dovich’s theorem, JETP 1957] • A purely toroidal flow cannot sustain a dynamo • B(x, y, t) cannot be a dynamo field • Dynamos are only possible in “complex” geometries or flows • An extra burden for both theory and numerics • A popular “minimal” configuration is 2.5D (or 2D-3C) • u(x, y, t) with all three components non-vanishing • B(x, y, z, t) = R b(x, y, t)eikz z 22 Astroplasma, March 2021
The fast dynamo paradigm [Vainshtein & Zel’dovich, SPU, 1972] • Chaotic stretching, twisting, folding and merging of field lines • For small diffusion, field doubles at each “iteration” (characteristic time) • Exponential growth with “ideal” growth rate 1 = ln 2 ~ stretching rate Stretch Twist 3D essential ! Lyapunov exponents of Galloway-Proctor flow [credits F. Cattaneo] Merge (requires a tiny bit of magnetic diffusion !) [adapted from Brandenburg & Fold Subramanian, Phys. Rep. 2005] 23 Astroplasma, March 2021
MHD dynamos II. Textbook theory & numerics François Rincon Bed time reading: Dynamo theories J. Plasma Phys. (Lecture Notes Series) 85, 205850401 (2019) arXiv:1903.07829 Les Houches, May 2017
An imperfect dichotomy • Large-scale dynamo effect • Magnetic field generated on • long system time (Ω−1, S −1), spatial scales (L) much larger than flow scales ℓ0 • also lots of magnetic fluctuations on low and sub flow scales down to the magnetic resistive scale • Small-scale dynamo effect • Magnetic field generated on short time (ℓ/u), spatial scales (ℓ) from flow scales down to the resistive scale • Each of these can be excited by laminar or turbulent flows • They have traditionally mostly been described by different theories • in all MHD astrophysical settings, large-scale dynamos are swamped by small-scale ones • this creates a lot of theoretical difficulties • MHD instabilities also play a key role in large-scale dynamos • the magneto-rotational, and other magnetoshear instabilities • Kelvin-Helmholtz instability coupled to magnetic buoyancy 25 Astroplasma, March 2021
Large vs small magnetic Prandtl numbers k-5/3 Kinetic energy e.g. stellar, solar, Spectral power forcing spectrum liquid metals (Earth, experiments) scales Magnetic energy 3/4 spectrum tail `⌘ ⇠ Pm `⌫ Wavenumber k k0 kη ∼ Pm3/4 kν > kν 26 Astroplasma, March 2021
Small-scale dynamo theory how to make magnetic fields on scales comparable to flows
Numerical evidence • Homogeneous, isotropic, non-helical, incompressible, 3D turbulent flow of conducting fluid is a small-scale dynamo 64x64x64 spectral direct numerical simulations at Pm=1 [Meneguzzi, Frisch, Pouquet, PRL, 1981] 28 Astroplasma, March 2021
Zel’dovich-Moffatt-Saffman phenomenology [Moffatt & Saffman, 7, 155 (1964); Phys. Fluids, Zel’dovich et al., JFM 144, 1 (1984)] • Consider incompressible, kinematic dynamo problem @B + u · rB = B · ru + ⌘ B r·B=0 @t • Assume that B(0, r) = B0 (r) • has finite total, energy, no singularity • lim B0 (r) = 0 r!1 • Take simplest possible model of time-evolving “smooth” velocity field • Random linear shear: u = Cr Tr C = 0 [incompressible] [think of this as being 3D] 29 Astroplasma, March 2021
Stretching and squeezing d ri • Evolution of vector connecting 2 fluid particles: = Cik rk dt • Consider constant C = diag(c1 , c2 , c3 ) c1 + c2 + c3 = 0 • Exponential stretching along first axis c 1 > 0 > c2 > c3 c 1 > c2 > 0 > c3 e1 e2 e1 e2 e3 e3 “rope” “pancake” • In ideal MHD, we thus expect B 2 ⇠ exp(2c1 t) • However, perpendicular squeezing implies that even a tiny magnetic diffusion matters…is growth still possible in that case ? 30 Astroplasma, March 2021
Magnetic field evolution Z • Decompose B(t, r) = b(t, k0 ) exp (ik(t) · r)d3 k0 db 2dk = Cb ⌘k b = CT k k·b=0 dt dt ✓ Z t ◆ • Diffusive part of evolution ~ exp ⌘ k 2 (s)ds 0 • super-exponential decay of most Fourier modes because k3 ⇠ k03 exp(|c3 |t) • surviving modes live in an exponentially narrow k-cone such that Z t ⌘ k 2 (s)ds = O(1) 0 • rope case: k02 ⇠ exp( |c2 |t) k03 ⇠ exp( |c3 |t) 31 Astroplasma, March 2021
Magnetic field evolution (ropes) • Surviving modes at time t have an initial field e1 • b1 (0, k0 ) ⇠ b2 (0, k0 )k02 /k01 ⇠ exp( |c2 |t) e2 e3 • This field is stretched along the first axis, so b(t, k0 ) ⇠ exp (c1 t) exp ( |c2 |t) • Now, estimate the magnetic field in physical space Z B(t, r) ⇠ Bk d3 k0 ⇠ exp( |c2 |t) ⇠ exp [(c1 |c2 |)t] ⇠ exp [( |c2 | |c3 |)t] Magnetic field stretches into an asymptotically-decaying rope 32 Astroplasma, March 2021
Magnetic energy evolution (ropes) • What about magnetic energy ? e1 Z e2 e3 2 3 Em = B (t, r)d r Volume ⇠ exp(c1 t) B 2 ⇠ exp ( 2|c2 |t) Important: no shrinking along axis 2 and 3 as diffusion sets a minimum scale in these directions Em ⇠ exp [(c1 2|c2 |)t] ⇠ exp [(|c3 | |c2 |)t] (3D) Total magnetic energy grows ! (in 3D) Volume occupied by the magnetic field grows faster than field decays pointwise • Similar conclusions apply in the pancake case, but Em ⇠ exp [(c1 c2 )t] 33 Astroplasma, March 2021
Generalization to random, time-dependent shear • Renovate shear flow every time-interval • Succession of random area-preserving stretches and squeezes • Introduce the matrix Tt ⌘ T(t0 , t) such that k(t, k0 ) = Tt k0 Y t ⇥ ⇤ • Volterra multiplicative integral form: Tt = CT (s)ds s=0 Product of unimodular • Formal solution random matrices Z Z t B(t, r) = exp iTt (k0 · r) ⌘ (Ts k0 )2 ds (TTt ) 1 b(0, k0 )d3 k0 0 • Hard work: calculate the properties of the multiplicative integral ! 34 Astroplasma, March 2021
Lyapunov basis of random shear flow • Zel’dovich showed that the cumulative effects of any random sequence of shears can be reduced to diagonal form • In particular there is always a net positive “stretching” Lyapunov exponent 1 lim ln k(n⌧, k0 ) ⌘ 1 >0 n!1 n⌧ • The underlying Lyapunov basis (e1 , e2 , e3 ) • is a function of the full random sequence, but is independent of time • “cristallizes” exponentially fast in time (exponents converge as 1/t) • The problem reduces to that described earlier • Magnetic energy growth is possible in a smooth, 3D chaotic velocity field in the presence of magnetic diffusion 35 Astroplasma, March 2021
Small-scale dynamo fields at Pm ≥ 1 • Pm=Rm=1250, Re=1 [from Schekochihin et al., ApJ 2004] u B • Folded field structure • Reversals at resistive scale 1/2 `⌘ ⇠ `⌫ Pm • Folds coherent over flow scale Critical Rm ~ 60 • Field strength and curvature anticorrelated 36 Astroplasma, March 2021
Small-scale dynamo at low Pm • Yes, but much harder Pm=1250, Re=1, Rm=1250 • Critical Rm~200 • More complicated than Pm=1, Re=440, Rm=440 Zel’dovich picture Pm=0.07, Re=6200, Rm=430 [Iskakov et al., PRL 2007] 37 Astroplasma, March 2021
Kazantsev-Kraichnan theory • Consider again the following kinematic dynamo problem: @B + u · rB = B · ru + ⌘ B r·B=0 @t • This problem can be solved analytically if u is • a random Gaussian process with no memory (zero-correlation time) • The so-called Kraichnan ensemble ⌦ 0 0 ↵ u (x, t)u (x , t ) = ij (r) (t i j t0 ) • Obviously, not your usual turbulent flow, but still… • Very useful to understand some properties of small-scale dynamo modes • Originally solved by Kazantsev [JETP, 1968] [and further explored by Zel’dovich, Ruzmaikin, Sokoloff, Vainshtein, Kitchatinov, Vergassola, Vincenzi, Subramanian, Boldyrev, Schekochihin etc.] 38 Astroplasma, March 2021
Equation for magnetic correlations • Goal: derive a closed equation for the two-point, single time magnetic correlator [or magnetic energy spectrum] ⌦ B i (x, t)B j (x0 , t) = H ij (x x0 , t) ✓ i j ◆ i j 0 r r r r H ij (x x , t) = HN (r, t) ij 2 + HL (r, t) 2 r r HN = HL + (rHL0 )/2 • Induction equation at (x,t) and (x’,t) gives @H ij @ ⌦ i k 0 j 0 ↵ ⌦ i j 0 k 0 ↵ = 0k B (x, t)B (x , t)u (x , t) B (x, t)B (x , t)u (x , t) @t @x Third order Second order @ ⌦ k j 0 i ↵ ⌦ i j 0 k ↵ moments + B (x, t)B (x , t)u (x, t) B (x, t)B (x , t)u (x, t) moment @x k ✓ ◆ @ @ ij + ⌘ + H @xk @x0k @ @ @ [statistical h·i = h·i = h·i @x0k @xk @rk homogeneity] 39 Astroplasma, March 2021
The closed equation • Using properties of the Kraichnan ensemble, the problem reduces to a closed equation for magnetic correlator HL (r, t) ✓ ◆ ✓ ◆ @HL 4 4 0 = HL00 + 0 0 00 + H L + + HL @t r r (r) = 2⌘ + L (0) L (r) “Turbulent diffusivity” (twice) • Schrödinger equation with imaginary time 2 1/2 • Change variables: HL (r, t) = (r, t)r (r) @ 00 = (r) V (r) @t 1 • Wave function of quantum particle of variable m(r) = in potential 2(r) 2 1 00 2 0 0 (r)2 V (r) = 2 (r) (r) (r) r 2 r 4(r) 40 Astroplasma, March 2021
Solutions 00 1 @ m(r) = [2(r)] = V (r) @t 2m(r) (r) = 2⌘ + L (0) L (r) 2 1 00 2 0 0 (r)2 V (r) = 2 (r) (r) (r) r 2 r 4(r) Et • Look for solutions of the form = E (r)e • Growing dynamo modes correspond to discrete bound states: E> 1 and Pm
Different regimes ⌦ 00 00 ↵ • Recall u (x, t)u (x , t ) = i` (x i ` x00 ) (t t00 ) • So (r) ⇠ u(r)2 ⌧ (r) ⇠ r u(r) is a turbulent diffusivity • Consider the scaling law L (0) L (r) ⇠ r⇠ Roughness exponent • Smooth flow: u ⇠ r ) ⇠=2 [“large Pm”] • “Kolmogorov” turbulence: u ⇠ r1/3 ) ⇠ = 1 + 1/3 = 4/3 [“low Pm”] • Potential as a function of ⇠ Ve↵ (r) • Ve↵ (r) = 2/r2 , r ⌧ `⌘ p ⇠ < ⇠c = 11/3 1 3 3 2 2 `⌘ • Ve↵ (r) = (2 ⇠ ⇠ )/r , r `⌘ r 2 4 ⇠ > ⇠c • Growing bound modes for ⇠ > 1 • includes both Pm >> 1 and Pm
A few important results at large Pm • Consider the so called Batchelor regime `⌘ ⌧ `⌫ • The magnetic field is stretched and transported by a viscous flow 2 ✓ i j ◆ ij ij r ij 1r r • The velocity field is smooth: (r) = 0 2 2 + ··· 2 2 r • Spectral view at scales much smaller than the viscous scale • Work under Kazantsev-Kraichnan assumptions • Fokker-Planck type equation for the magnetic spectrum M(k) ✓ 2 ◆ @M 2@M @M = k 2 + 6M 2⌘k 2 M @t 5 @k 2 @k • Typical growth rate of the order of the shearing rate at viscous scales 5 5 [Kazantsev JETP 1968; = 2 = |00L (0)| ⇠ u⌫ /`⌫ 4 2 Kulsrud and Anderson, ApJ 1992; Schekochihin et al., ApJ 2002] 43 Astroplasma, March 2021
Large Pm, diffusion-free regime • Magnetic diffusion negligible if magnetic field only has k ⌧ k⌘ • If we excite a given k0 initially, the spectrum spreads towards small-scales ✓ ◆3/2 r k 5 5 ln2 (k/k0 ) M (k, t) / e3/4 t exp k0 4⇡ t 4 t • The energy of each mode grows at rate 3 /4 • Total energy grows at rate 2 as the number of excited mode also grows • The magnetic field develops the so-called k 3/2 Kazantsev spectrum M (k) E(k) k 3/2 Stirring k k⌫ k⌘ 44 Astroplasma, March 2021
Large Pm, resistive regime • After the spectrums hits k ⇠ k⌘ , the long-time asymptotics is ✓ ◆ 3/2 k 3/4 t p M (k, t) / k K0 e k⌘ = /(10⌘) ⇠ Pm1/2 k⌫ k⌘ • The spectrum peaks at the resistive scale [falls off exponentially beyond] • The asymptotic total energy growth rate is now also approximately 3 /4 [weak dependence on boundary condition at small k] k 3/2 M (k) E(k) Stirring k k⌫ k⌘ 45 Astroplasma, March 2021
Magnetic pdf in the diffusion-free regime • One can derive a Fokker-Planck equation for the pdf of B @ 2 ij k @ ` @ P [B] = Tk` B i B j P [B] @t 2 @B @B • Simplifies in the isotropic case as 1D diffusion equation with drift @ 2 1 @ 4 @ Z 1 pdf defined as P [B] = 2 B P [B] @t 4 B @B @B hB n (t)i = 4⇡ dBB 2+n P [B] 0 • Lognormal solution Z ! 1 2 1 dB 0 0 [ln(B/B 0 ) + (3/4)2 t] P [B](t) = p 0 P 0 [B ] exp ⇡2 t 0 B 2 t • The magnetic field is strongly intermittent / non-Gaussian 1 n • Magnetic moments grow as hB (t)i / exp n(n + 3)2 t 4 46 Astroplasma, March 2021
Saturation of small-scale dynamo • As B gets large-enough, Lorentz force saturates dynamo [Meneguzzi et al., PRL 1981] • What is “large-enough “? • How does it work ? • Historical ideas • Batchelor argument [PRSL,1950]: • magnetic field is similar to hydrodynamic vorticity • should peak at viscous scale, hence saturation for B 2 ⇠ u2⌫ ⌦ 2↵ 1/2 ⌦ 2↵ B ⇠ Re u Sub-equipartition unless Re=1 • Schlüter-Biermann argument [Z. Naturforsch.,1950]: ⌦ 2 ↵ ⌦ 2 ↵ • equipartition at all scales B ⇠ u 47 Astroplasma, March 2021
Saturation phenomenology • Geometric structure and orientation of the field matters • Magnetic tension B · rB encodes magnetic curvature • Reduction of stretching Lyapunov exponents • A field realization can only saturate itself Kinematic Saturated Saturated magnetic field i c field et magn um my D [Cattaneo et al., PRL 1996] [Cattaneo & Tobias, JFM 2009] • Saturation at low Pm • Pretty much Terra incognita (almost no published simulation) 48 Astroplasma, March 2021
Large Pm phenomenology • Plausible (but not definitive) scenario from simulations [Schekochihin et al., ApJ 2002, 2004] • Lorentz force first suppresses stretching at viscous scales B · rB ⇠ u · ru ⇠ u2⌫ /`⌫ ⌦ ↵ 1/2 ⌦ ↵ 2 2 2 B ⇠ Re u ⇠ B /`⌫ (folded structure) • From there, slower, larger-scale eddies take over stretching • B keeps growing and acts on increasingly more energetic eddies… ⌦ 2↵ • Secular growth regime: B ⇠ "t ⌦ 2 ↵ ⌦ 2 ↵ • Final state: B ⇠ u after “suppression” of full inertial range • “Isotropic MHD turbulence”, folded structure is preserved • P[B] not log-normal anymore (likely exponential) 49 Astroplasma, March 2021
But…what about reconnection ? • New challenges… (a) (a) (b)(b) Dynamotheories Dynamo theories 4545 Courtesy from Iskakov & Schekochihin (unpublished) (c) (c) (d)(d) Figure Figure23.23.2D |u| snapshotsofof|u| 2Dsnapshots |u|(left) and|B| (left)and |B|(right) (right)inina anonlinear |B| nonlinearsimulation simulationof ofsmall-scale small-scale dynamo dynamodriven drivenbybyturbulence turbulenceforced forcedatatthe theboxboxscale scaleatatReRe==290 290Rm Rm=Re=290, 2900,P Rm=2900, =2900, P m= =1010 Pm=10 m 3 3 (the (themagnetic magneticenergy energyspectra spectrafor forthis this512 512 spectral spectralsimulation simulationsuggest suggestthat thatit it is is reasonably reasonably well well • More resolved). resolved). in AtAtsuch suchmy largefinal large Rm,seminar Rm, the thedynamo next dynamo fiedtime… fiedbecomes becomesweaklyweaklysupercritical supercriticaltotoa asecondary secondary fast-reconnection fast-reconnectioninstability instabilityininregions regionsofofreversing reversingfield fieldpolarities polaritiesassociated associatedwith withstrong strong 50 electrical electricalcurrents. currents.TheTheinstability instabilitygenerates generatesmagnetic magneticplasmoids plasmoidsandandoutflows, Astroplasma, leavinga a March 2021 outflows,leaving
Large-scale dynamo theory how to build-up magnetic fields on system scales much larger than flow scales
Differential rotation: the Omega effect • Shearing of magnetic field by differential rotation (shear) • In polar geometry, consider the initial axisymmetric configuration • a purely poloidal magnetic field: Bpol = Br (r, z)er + Bz (r, z)ez • a toroidal, shearing velocity field (differential rotation): u = r⌦(r, z)e' ✓ ◆ @B' 1 = r(Bpol · r)⌦ + ⌘ B' @t r2 • On short times, B' can grow linearly in time • Ultimately, diffusion always dominates • This effect alone cannot produce a dynamo (Cowling) • But it can transiently make strong toroidal field out of weak poloidal field 52 Astroplasma, March 2021
Turbulence: Parker’s mechanism • Effect of a localized cyclonic swirl on a straight magnetic field [Parker, ApJ 1955] [Moffatt, Les Houches lectures 1973] • In polar geometry, this mechanism can produce axisymmetric poloidal field out of axisymmetric toroidal field — and the converse • Kinetic helicity in the swirl is essential • This “alpha effect” can mediate statistical dynamo action • Ensemble of turbulent helical swirls should have a net effect of this kind • Cowling’s theorem does not apply as each swirl is localized (“non-axisymmetric”) 53 Astroplasma, March 2021
Numerical evidence • Small-scale helical turbulence can generate large-scale field • Critical Rm is O(1), lower than that of the small-scale dynamo [Brandenburg, ApJ 2001] [Meneguzzi et al., PRL 1981 — again !] • Helicity seemingly key for large-scale dynamos (but see later) 54 Astroplasma, March 2021
Twisting and magnetic helicity • Assume conservation of magnetic helicity (up to resistive effects) • Systematic twisting produces • negative large-scale magnetic helicity (large-scale writhe) • positive small-scale magnetic helicity (small-scale twist) [adapted from Mininni, ARFM 2011] • Consequences • Interpretation of large-scale helical dynamo as “inverse transfer” of helicity [Frisch et al., JFM 1975] • Transfer of helicity at small scales 55 Astroplasma, March 2021
Mean-field approach • Incompressible, kinematic problem with uniform diffusivity @B = r⇥(u ⇥ B) + ⌘ B @t r·u=0 r·B=0 • Split fields into large-scale (` > `0 ) and fluctuating part (` < `0 ) B = B + B̃ u = u + ũ @B ⇣ ⌘ + u · rB = B · ru + r ⇥ ũ ⇥ B̃ + ⌘ B @t • To determine the evolution of B we need to know E = ũ ⇥ B̃ • We cannot just sweep fluctuations under the rug: closure problem 56 Astroplasma, March 2021
Mean-field approach @ B̃ h ⇣ ⌘ ⇣ ⌘ ⇣ ⌘i = r ⇥ ũ ⇥ B + u ⇥ B̃ + ũ ⇥ B̃ ũ ⇥ B̃ + ⌘ B̃ @t Tangling/shearing Tricky bit — closure problem ! of mean field [also known as the “pain in the neck” term] • Assume linear relation between B̃ and B [Warning: hard to justify if there is small-scale dynamo !] • Expand (ũ ⇥ B̃)i = ↵ij Bj + ijk rk Bj + · · · • Simplest pseudo-isotropic case: ↵ij = ↵ ij, ijk = ✏ijk • For u = 0 , we obtain a closed “↵2“ dynamo equation (⌘ ⌧ ) @B = r ⇥ ↵B + B @t alpha effect beta effect (“turbulent” diffusion) • Exponentially growing solutions with real eigenvalues = |↵|k k2 • Max growth rate max = ↵2 /(4 ) at scale `max = 2 /↵ `0 57 Astroplasma, March 2021
Mean-field dynamo with Omega effect • Add large-scale differential rotation to MF equation: u = r⌦(r, z)e' @B = e' r (Bpol · r) ⌦ + r ⇥ ↵B + B @t Omega effect Alpha effect • Growing, oscillatory solutions leading to field reversals: Parker waves 2 • This is called the ↵⌦ dynamo ( ⌦ if ↵ acts both ways) ↵ ⌦ effect (+↵ effect) Poloidal field Toroidal field ↵ effect • Remarks • Many other couplings possible: pumping effects, non-diagonal terms etc. • 3Dness of the dynamo is hidden in mean-field coefficients 58 Astroplasma, March 2021
Calculation of mean-field coefficients • We only know how to calculate ↵ and perturbatively for • small correlation times (low Strouhal number ⌧c /⌧NL, random waves) • low magnetic Reynolds number Rm ⇠ ⌧⌘ /⌧NL ⌧ 1 @ B̃ h ⇣ ⌘ ⇣ ⌘i (u=0) ˙ = r ⇥ ũ ⇥ B + ũ ⇥ B̃ ũ ⇥ B̃ + ⌘ B̃ @t O(B̃rms /⌧c ) O(B/⌧NL ) O(B̃rms /⌧NL ) O(B̃rms /⌧⌘ ) ⌧NL = `u /urms tricky “pain in the neck” term G ⌧⌘ = `2u /⌘ • In both cases we can justify neglecting the tricky term • First Order Smoothing Approximation (FOSA, SOCA, Born, quasilinear…) [Steenbeck et al., Astr. Nach. 1966; see H. K. Moffatt’s textbook, CUP 1978; Brandenburg & Subramanian, Phys. Rep. 2005] 59 Astroplasma, March 2021
Calculation of mean-field coefficients • Let’s see how the calculation proceeds for ⌧c /⌧NL ⌧ 1 • Neglecting the tricky term and assuming small resistivity, Z t ⇥ ⇤ ũ(t) ⇥ B̃(t) = ũ(t) ⇥ r⇥ ũ(t0 ) ⇥ B(t0 ) dt0 0 Z th i = ˆ (t t0 )B(t0 ) ↵ ˆ(t t0 )r ⇥ B dt0 (isotropic case) 0 1 0 ˆ 1 ˆ = ũ(t) · !(t ↵ ˜ ) = ũ(t) · ũ(t0 ) ˜ = r ⇥ ũ ! 3 3 • For slowly varying B and short-correlated velocities, this simplifies as ũ(t) ⇥ B̃(t) = ↵B r⇥B 1 1 ↵' ⌧c (ũ · !) ˜ ' ⌧c ũ2 3 3 • The role of kinetic helicity is explicit 1 • At low Rm, we have the similar result ↵ ' ⌧⌘ (ũ · !) ˜ 3 60 Astroplasma, March 2021
Dynamical regime of large-scale dynamos • When B gets “large enough”, the Lorentz force back-reacts • Big questions: what happens then, and what is “large-enough” ? [Brandenburg & Subramanian, Phys. Rep. 2005, and refs. therein: Proctor, 2003; Diamond et al. 2005] 2 2 • Equipartition argument: saturation when B ⇠ 4⇡⇢ ũ2 ⌘ Beq , but • B and ũ have very different scales • Large-scale dynamos alone produce plenty of small-scale field 2 • Equipartition of small-scale fields: b̃2 ⇠ 2 Beq , with b̃2 ⇠ Rm Bp 2 • 2 Not very astro-friendly: B ⇠ Beq /Rmp ⌧ Beq 2 for p=O(1) • Possibility of “catastrophic” alpha quenching ↵0 ↵(B) = q 2 ) 1 + Rm (B /Beq 2 q = O(1) 61 Astroplasma, March 2021
The quenching issue • Physical origin of quenching debated: • Magnetized fluid has “memory”: possible drastic reduction of statistical effects compared to random walk estimates [see review by Diamond et al., 2005] • Magnetic helicity conservation argument: • in “closed” systems, large-scale field can only reach equipartition on slow, large-scale resistive timescales [e.g. Brandenburg, ApJ 2001] • Possible way out of problem is to ”evacuate” magnetic helicity [Blackman & Field, ApJ 2000; see discussion by Brandenburg, Space Sci. Rev. (2009)] 78 F. Rincon d [Bhat et al., MNRAS 2016] hA · BiV = 2⌘ h (r(a)⇥ B) · BiV hr · FHm i (b) dt • Open boundary conditions (periodic b.c. not ok) • Internal fluxes of helicity [Kleeorin et al., Vishniac-Cho etc.] • More on this in my final seminar next time 62 F IGURE 31. (a) Magnetic (full black lines) and kinetic energy (dotted blueMarch Astroplasma, lines) 2021 spectr
Large vs small-scale growth: who wins ?
Large-scale dynamos with Kasantsev [Vainshtein & Kitchatinov, JFM 1986, Berger & Rosner, GAFD 1995, • Consider turbulence with net helicity Subramanian, PRL 1999, Boldyrev et al., PRL 2005] • Add a mirror symmetry-breaking term to the correlators ✓ i j ◆ ij ij rr ri rj (r) = N (r) + L (r) 2 + g(r)"ijk rk r2 r ✓ i j ◆ rr ri rj H ij (x x0 , t) = HN (r, t) ij 2 + H L (r, t) 2 + K(r)" ijk k r r r 2 • r ! 1 asymptotics of model gives mean-field ↵ equation @B L (0) = r ⇥ ↵B + (⌘ + ) B g(0) = ↵, = @t 2 • Full calculation leads to coupled equations for HL and K ✓ ◆ @HL 1 @ @HL 4 = 4 r + GHL 4hK h(r) = g(0) g(r) @t r @r @r @K 1 @ 4 @ G(r) = 00 + 40 /r = 4 r (K + hHL ) @t r @r @r 64 Astroplasma, March 2021
Self-adjoint spinorial form @W p = R̃T J˜R̃W A(r) = p 2 [2⌘ + N (0) N (r)] @t B(r) = 2 [2⌘ + L (0) L (r)] p p ! ✓ ◆ C(r) = 2 [g(0) g(r)] r 2/r 0 Ê C 1 @ @ 1 R̃ = 1 @ 2 J̃ = Ê = r B r + p (A rA0 ) 0 r C B 2 @r @r 2 2 r @r 2 p p p 30 p 0 1 2 2 2 @ 2 1 2 WH Ê C(r) r WH HL = 2 W H @ @ A=6 r rp r3 @r 7@ A rp 4 5 @t 2 @ 2 2 @ B(r) @ 2 2 @ 2 WK r C(r) 3 r r WK K= 4 (r WK ) @r r 4 @r r @r r @r • Therefore, the generalized helical case can be diagonalized g 2 (0) 2↵2 • Bound “small-scale” modes: n > 0 0 = = L (0) + 2⌘ 4( + ⌘) • Free “mean-field” modes: 0 < < 0 Twice the maximum ↵ 2 mean-field dynamo growth rate ! 65 Astroplasma, March 2021
Growing helical modes • Helicity allows growing large-scale Ve↵ (r) modes 2 2 2 `⌘ • Ve↵ (r) = 2/r ↵ /( + ⌘) , `0 ⌧ r r `0 • Bound modes ( > 0) dominate the [Malyshkin & Boldyrev, ApJ 2009] kinematic stage • As ! 0 , their spectrum peak shifts towards that of “mean-field” modes • Further hints that quantitative large-scale dynamo theory should factor in the small-scale dynamo 66 Astroplasma, March 2021
A few words on “test field”-like methods • Pragmatic strategies have been devised for “astrophysical applications” • postulate generalised mean-field form for E(B) (convolution integrals) • Measure effective transport coefficients in local simulations [Sur et al., MNRAS 2008, • Use the results in simpler 2D mean-field models Brandenburg, Space Sci. Rev. 2009] • Such procedures • produce converged values of transport coefficients • reproduce exact results in perturbative kinematic limits • TFM-based modelling may be useful, but: • no rigorous justification as to why it should be accurate/appropriate (Rm>>1 !) • dynamical, tensorial convolution relations E(B) can fit complex dynamics, but could well be degenerate with more physically-grounded nonlinear models • it can obfuscate the underlying physics, e.g. when MHD instabilities are involved 67 Astroplasma, March 2021
Situation so far • Historically, mean-field models have been at the core of modelling of • solar and stellar dynamos — “alpha” provided by cyclonic convection • galactic dynamos — “alpha” provided by supernova explosions • But classical mean-field theory faces strong limitations • Astro turbulence typically has ⌧c /⌧NL ⇠ 1 and Rm 1 • “Co-existence” with fast, small-scale dynamo for Rm 1 • pain in the neck term exponentially growing…then what ? • linear relation between b̃ and B doubtful • Non-linear quenching • Large-scale dynamos are “real” — independently of our limited theories • We have to think harder ! (and ask good questions to computers) 68 Astroplasma, March 2021
Tomorrow’s fundamental theory challenges • Turbulent large and small-scale MHD dynamos • Unified, self-consistent nonlinear multiscale statistical dynamo theory • Requires physically justified closures • Description of asymptotic regimes (very high Re and Rm, low Pm, strong rotation) • Interactions of different physical processes and geometrical effects • MHD instabilities combined to shear (magnetic buoyancy, MRI etc.) • Coherent structures (vortices, zonal flows, convection columns, tangent cylinders) • Reconnection in dynamos • Plasma effects (batteries, pressure anisotropies, partial ionization etc.) • History of cosmic magnetism • from the pre-CMB era to stellar and planetary magnetic fields 69 Astroplasma, March 2021
Today & tomorrow’s frontiers…next time ! • Dynamos meet reconnection • Asymptotic nonlinear dynamos • Plasma (kinetic) dynamo Next presentation will be essentially equation-free
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