Summary: Mean field theory. ASTR 7500: Solar & Stellar Magnetism. Lecture 11 Tues 26 Feb Summary: Mean field theory. Summary: Mean field theory

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1 ASTR 7500: Solar & Stellar Magnetism Summary: Mean fiel theory Hale CGEG Solar & Space Physics Average of inuction equation: ( ) = v + v η New solution properties arie from the term: E = v Assumption of scale separation in time an space: Matthias Rempel, Prof. Juri Toomre + HAO/NSO colleagues Lecture Tues 6 Feb 03 E i = a ij j + b ijk j x k zeus.colorao.eu/astr7500-toomre 53 / / 84 Summary: Mean fiel theory Summary: Mean fiel theory Some reorering of terms: E = α + γ β δ +... α, β: symmetric tensors γ, δ: vectors Symmetry constraints imply: α, δ: pseuo tensor (relate to helicity an rotation) β, γ: true tensors Assumption isotropy (non mirror-symmetric, weakly inhomogeneous): = [ α + (v + γ) (η + η t ) ] with the scalar quantities an vector 3 τ c v ( v ), η t = 3 τ c v γ = 6 τ c v = η t 55 / / 84 Turbulent iffusivity - estruction of magnetic fiel Turbulent iffusivity ominant issipation process for large scale fiel in case of large R m : η t = 3 τ c v L v rms R m η η Formally η t comes from avection term (transport term, non-issipative) Turbulent cascae transporting magnetic energy from the large scale L to the micro scale l m (avection + reconnection) ηj m η t j m l m R m L Important: The large scale etermines the energy issipation rate, l m ajusts to allow for the issipation on the microscale. Present for isotropic homogeneous turbulence Turbulent iamagnetism, turbulent pumping Expulsion of flux from regions with larger turbulence intensity iamagnetism γ = η t Turbulent pumping (stratifie convection): γ = 6 τ c v Upflows expan, ownflows converge Stronger velocity an smaller filling factor of ownflows Mean inuction effect of up- an ownflow regions oes not cancel Downwar transport foun in numerical simulations Requires inhomogeneity (stratification) 57 / / 84

2 Kinematic α-effect τc v0 ( v0 ) 3 abcock-leighton α-effect Hk = v0 ( v0 ) kinetic helicity Requires rotation + aitional preferre irection (stratification) Similar to kinetic α-effect, but riven by magnetic buoyancy Leaing polarities have larger propability to reconnect across equator with counterpart on other hemisphere Polarity of hemisphere = polarity of following sunspots 59 / 84 Fast or slow ynamo? 60 / 84 How well oes this work in practice? Turbulent inuction effects require reconnection to operate; however, the expressions vl 0 vl 0 αij = τc εikl vk 0 + εjkl vk 0 xj xi 0 0 γi = τc vv xk i k 0 βij = τc v δij vi 0 vj 0 are inepenent of η (in this approximation), inicating fast ynamo action (no formal proof since we mae strong assumptions!) From Racine et al. 0 6 / 84 How well oes this work in practice? α= 3 τc v0 ( 6 / 84 Generalize Ohm s law v0 ) What is neee to circumvent Cowling s theorem? Crucial for Cowling s theorem: Impossibility to rive a current parallel to magnetic fiel Cowling s theorem oes not apply to mean fiel if a mean current can flow parallel to the mean fiel (since total fiel non-axisymmetric this is not a contraiction!) j = σ E + v + γ + α σ contains contributions from η, β an δ. Ways to circumvent Cowling: α-effect anisotropic conuctivity (off iagonal elements + δ-effect) From Racine et al / / 84

3 α -ynamo Meanfiel energy equation V = µ0 µ0 ηj V v (j ) V + j E V Inuction of fiel parallel to current (proucing helical fiel!) Energy conversion by α-effect αj α-effect only pumps energy into meanfiel if meanfiel is helical (current helicity must have same sign as α)! = αµ0 j Dynamo action oes not necessarily require that j E is an energy source. It can be sufficient if E changes fiel topology to circumvent Cowling, if other energy sources like ifferential rotation are present (i.e. Ω j effect). Dynamo cycle: α α t p t Poloial an toroial fiel of similar strength In general stationary solutions 65 / 84 α moel for Geoynamo 66 / 84 α moel for Geoynamo between reversals: uring reversals: Strong influence of rotation τrot = ay, τc 000 years (Sun: τrot = 7 ays, τc weeks) Flow organization: Taylor columns Creit: 3D geoynamo simulation G.A. Glatzmaier (UCSC) Seconary flow along columns (bounary effect) helicity 67 / 84 αω-, α Ω-ynamo 68 / 84 αω-ynamo A = r sin p Ω + ηt (r sin θ) = α + ηt A (r sin θ) Cyclic behavior: P (α Ω ) / Dynamo cycle: α Ω, α t p t Propagation of magnetic fiel along contourlines of Ω ynamo-wave Toroial fiel much stronger that poloial fiel Direction of propagation Parker-Yoshimura-Rule : In general traveling (along lines of constant Ω) an perioic solutions s = α Ω eφ Movie α-effect Movie Ω-effect 69 / 84 Movie: αω-ynamo 70 / 84

4 αω-ynamo with meriional flow αω-ynamo with meriional flow A + ( r r (rv r ) + ) θ (v θ) = r sin p Ω ) + η t ( + (r sin θ) r sin θ v p (r sin θa) = α + η t If η t is sufficiently small, such that: ( τ = D C /η t > D C /v m η t < v m D C ) (r sin θ) A the meriional flow v m can control the cycle perio an propagation of the magnetic activity Aitional avection like effects can arise from the γ-effect, they can be accounte for by formally substituting: v m v m + γ 7 / 84 Meriional flow: Polewar at top of convection zone Equatorwar at bottom of convection zone Effect of avection: Equatorwar propagation of activity Correct phase relation between poloial an toroial fiel Circulation time scale of flow sets ynamo perio Requirement: Sufficiently low turbulent iffusivity Movie: Flux-transport-ynamo (M. Dikpati, HAO) 7 / 84 Ω J ynamo Dynamos an magnetic helicity = [δ ( )] (Ω j) j z similar to α-effect, but aitional z-erivative of current couples poloial an toroial fiel δ ynamo is not possible: Magnetic helicity (integral measure of fiel topology): H m = A V has following conservation law (no helicity fluxes across bounaries): A V = µ 0 η j V j E = j (δ j) = 0 δ-effect is controversial (not all approximations give a non-zero effect) in most situations α ominates Decomposition into small an large scale part: A V = + E V µ 0 η A V = E V µ 0 η j V j V 73 / / 84 Dynamos an magnetic helicity Non-kinematic effects Dynamos have helical fiels: α-effect inuces magnetic helicity of same sign on large scale α-effect inuces magnetic helicity of opposite sign on small scale Asymptotic staturation (time scale R m τ c ): j = j L l c Time scales: j = α µ 0 η + η t η j Galaxy: 0 5 years (R m 0 8, τ c 0 7 years) Sun: 0 8 years Earth: 0 6 years Proper way to treat them: 3D simulations Still very challenging Has been successful for geoynamo, but not for solar ynamo Semi-analytical treatment of Lorentz-force feeback in mean fiel moels: Macroscopic feeback: Change of the mean flow (ifferential rotation, meriional flow) through the mean Lorentz-force f = j + j Mean fiel moel incluing mean fiel representation of full MHD equations: Movie: Non-kinematic flux-transport ynamo Microscopic feeback: Change of turbulent inuction effects (e.g. α-quenching) 75 / / 84

5 Microscopic feeback Feeback of Lorentz force on small scale motions: Intensity of turbulent motions significantly reuce if µ 0 > ϱv rms. Typical expression use α k + eq with the equipartition fiel strength eq = µ 0 ϱv rms Similar quenching also expecte for turbulent iffusivity Aitional quenching of α ue to topological constraints possible (helicity conservation) Controversial! Microscopic feeback Symmetry of momentum an inuction equation v / µ 0 ϱ: v = µ 0 ϱ ( ) +... = ( )v +... E = v Strongly motivates magnetic term for α-effect (Pouquet et al. 976): ( ) 3 τ c ϱ j ω v Kinetic α: + v E Magnetic α: + v E 77 / / 84 Microscopic feeback From helicity conservation one expects leaing to algebraic quenching j α α k + g eq With the asymptotic expression (steay state) Microscopic feeback Catastrophic α-quenching (R m!) in case of steay state an homogeneous : α k + R m eq If j 0 (ynamo generate fiel) an η t unquenche: α η t µ 0 j η t L η t l c l c L α l c k L we get j = α µ 0 η + η t η j α k + η t µ 0j η eq + ηt η eq In general α-quenching ynamic process: linke to time evolution of helicity ounary conitions matter: Loss of small scale current helicity can alleviate catastrophic quenching Catastrophic α-quenching turns large scale ynamo into slow ynamo 79 / / 84 3D simulations Why not just solving the full system to account for all non-linear effects? Most systems have R e R m, requiring high resolution Large scale ynamos evolve on time scales τ c t τ η, requiring long runs compare to convective turn over 3D simulations successful for geoynamo R m 300: all relevant magnetic scales resolvable Incompressible system Solar ynamo: Ingreients can be simulate Compressible system: ensity changes by 0 6 through convection zone ounary layer effects: Tachocline, ifficult to simulate (strongly subaiabatic stratification, large time scales) How much resolution require? (C about 0 9 Mm 3, Mm resolution numerical problem) Small scale ynamos can be simulate (for P m ) Where i the first magnetic fiel come from? Meanfiel inuction equation linear in : possible solution. = [ α + (v + γ) (η + η t ) ] = 0 is always a vali solution! Generalize Ohm s law with electron pressure term: E = v + σ j ϱ e p e. leas to inuction equation with inhomogeneous source term: = (v η ) + ϱ ϱ e p e. e 8 / 84 8 / 84

6 Where i the first magnetic fiel come from? Early universe: Ionization fronts from point sources (quasars) riven through an inhomogeneous meium: /ϱ e ϱ e p e can lea to about 0 3 G Collapse of intergalactic meium to form galaxies leas to 0 0 G Galactic ynamo (growth rate 3Gy ) leas to 0 6 G after 0 Gy (toay) Summarizing remarks Destruction of magnetic fiel: Turbulent iffusivity: cascae of magnetic energy from large scale to issipation scale (avection+reconnection) Enhances issipation of large fiel by a factor R m Creation of magnetic fiel: Small scale ynamo (non-helical) Amplification of fiel at an below energy carrying scale of turbulence Stretch-twist-fol-(reconnect) Prouces non-helical fiel an oes not require helical motions Controversy: behavior for P m Large scale ynamo (helical) Amplification of fiel on scales larger than scale of turbulence Prouces helical fiel an oes require helical motions Requires rotation + aitional symmetry irection (controversial Ω J effect oes not require helical motions) Controversy: catastrophic vs. non-catastrophic quenching 83 / / 84