Haugen Brandenburg Dobler 2003 Ap J Nonhelical MHD

Haugen, Brandenburg, & Dobler (2003, Ap. J) Non-helical MHD at 10243 6

Inverse cascade of magnetic helicity argument due to Frisch et al. (1975) and Initial components fully helical: and k is forced to the left

Magnetic helicity Maxwell eqns Vector potential Uncurled induction eqn 13

Magnetic helicity 14

Brandenburg (2001, Ap. J 550, 824) Slow saturation 15

Periodic box, no shear: resistively limited saturation Brandenburg & Subramanian Phys. Rep. (2005, 417, 1 -209) Significant field already after kinematic growth phase followed by slow resistive adjustment Blackman & Brandenburg (2002, Ap. J 579, 397) 16

Magnetic helicity conservation Steady state, closed box Early times 17

Slow-down explained by magnetic helicity conservation molecular value!! 18

Slow-down explained by magnetic helicity conservation 19

With hyperdiffusivity Brandenburg & Sarson (2002, PRL) for ordinary hyperdiffusion 20

Evidence from different simulations: strong fields only with helicity flux 3 -D simulations, no mean-field modeling Forced turbulence in domain with solar -like shear Brandenburg (2005, Ap. J 625, 539) Convective dynamo in a box with shear and rotation Käpylä, Korpi, Brandenburg (2008, A&A 491, 353) Only weak field if box is closed

Nonlinear stage: consistent with … Brandenburg (2005, Ap. J) 28

Best if W contours ^ to surface Example: convection with shear need small-scale helical exhaust out of the domain, not back in on the other side Käpylä et al. (2008, A&A) Magnetic Buoyancy? Tobias et al. (2008, Ap. J)

Käpylä, Korpi, Brandenburg (2008, A&A) To prove the point: convection with vertical shear and open b. c. s Magnetic helicity flux Käpylä, Korpi, & myself (2008, A&A 491, 353) Effects of b. c. s only in nonlinear regime

Implications of tau approximation 1. MTA does not a priori break down at large Rm. (Strong fluctuations of b are possible!) 2. Extra time derivative of emf with 3. hyperbolic eqn, oscillatory behavior possible! 4. t is not correlation time, but relaxation time

Kinetic and magnetic contributions

Connection with a effect: writhe with internal twist as by-product a effect produces helical field W clockwise tilt (right handed) left handed internal twist both for thermal/magnetic buoyancy 33

… the same thing mathematically Two-scale assumption Production of large scale helicity comes at the price of producing also small scale magnetic helicity 34

Revised nonlinear dynamo theory (originally due to Kleeorin & Ruzmaikin 1982) Two-scale assumption Dynamical quenching Kleeorin & Ruzmaikin (1982) Steady limit algebraic quenching: ( selective decay) 35

General formula with magnetic helicity flux Rm also in the numerator 36

Mean field theory is predictive • Open domain with shear – Helicity is driven out of domain (Vishniac & Cho) – Mean flow contours perpendicular to surface! • Excitation conditions – Dependence on angular velocity – Dependence on b. c. : symmetric vs antisymmetric 37

Calculate full aij and hij tensors Original equation (uncurled) Mean-field equation fluctuations Response to arbitrary mean fields

Test fields Example:

Validation: Roberts flow SOCA result normalize Brandenburg, Rädler, Schrinner (2009, A&A) SOCA

Kinematic a and ht independent of Rm (2… 200) Sur et al. (2008, MNRAS)

Scale-dependence: nonlocality cf talk by Alexander Nepomnyashchy 42

Time-dependent case Hubbard & Brandenburg (2009, Ap. J) 43

Importance of time-dependence 44

From linear to nonlinear Brandenburg et al. (2008, Ap. J) Mean and fluctuating U enter separately Use vector potential 45

Nonlinear aij and hij tensors Consistency check: consider steady state to avoid da/dt terms Expect: l=0 (within error bars) consistency check! 46

ht(Rm) dependence for B~Beq (i) (iii) (iv) l is small consistency a 1 and a 2 tend to cancel to decrease a h 2 is small

Application to passive vector eqn cf. Cattaneo & Tobias (2009) Verified by test-field method Tilgner & Brandenburg (2008) 48

Is the field in the Sun fibril? Käpylä et al (2008) with rotation without rotation

Takes many turnover times Rm=121, By, 512^3 LS dynamo not always excited 50

Deeply rooted sunspots? Hindman et al. (2009, Ap. J) • Solar activity may not be so deeply rooted • The dynamo may be a distributed one • Near-surface shear important

Near-surface shear layer Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999)

Origin of sunspot Theories for shallow spots: (i) Collapse by suppression of turbulent heat flux (ii) Negative pressure effects from <bibj>-<uiuj> vs Bi. Bj 53

Formation of flux concentrations Recent work with Kleeorin & Rogachevskii (ar. Xiv: 0910. 1835) 54