Leibniz-Institut für Astrophysik Potsdam The existence of the Λ effect indicated by SDO data G. Rüdiger, M. Küker Göttingen 2014

ROTATING TURBULENCE Boussinesq (1897), Taylor (1915): Stress-strain relation, eddy viscosity for rotation negative through the whole convection zone Göttingen 2014

MAGNETIC OBSERVATIONS BUT: no confirmation by observations Ward 1965, Gilman & Howard 1984, Balthasar et al. 1986, Vrsnak et al Göttingen 2014 All tracers are magnetic-defined Nesme-Ribes et al Q ≈ 10 7 cm 2 /s 2

SDO DATA Göttingen 2014 Hathaway et al Giant cells, 200 Mm, Persisting over months

GIANT CELLS Göttingen 2014 Hathaway et al. 2013

ROTATING & ANISOTROPIC TURBULENCE Rigid rotation: two possible tensors (odd in, even in g ) Käpylä et al Q >0 Q <0 Q =0 Göttingen 2014

ANGULAR MOMENTUM TRANSPORT Göttingen 2014 V, H are functions of the Coriolis number * =2 corr V < 0 for small corr (inward transport) H > 0 always (equatorial transport) SOCA theory: RKH 2013

THE BASIC MODEL Göttingen 2014 Solution of the conservation law of angular momentum without meridional flow div (r sin Q )=0. equation cubic in, H=1, V=0, =const: No dependence on the viscosity, inner part: W>0. outer part: W<0

IMPROVED MODEL Göttingen 2014 V=-0.1, H= 1 for x<0.95, strong density gradient W>0 in the bulk of the convection zone

COMPLETE MODELS WITH MERIDIONAL CIRCULATION Göttingen 2014 With, meridional flow (polewards ) Without, only meridional flow (equatorwards) Kippenhahn, Köhler, Pitadella et al., Balbus et al. Küker & R. 2012

Göttingen 2014 MERIDIONAL CIRCULATION Schad et al. ´12 One-cell meridional circulation?

HORIZONTAL REYNOLDS STRESS Göttingen 2014 with Λ effect, W=0.14 without Λ effect, W<0

Göttingen 2014 MAIN-SEQUENCE MODELS MOST stars: ε Eri, P rot = 11 days, k=0.11. κ 1 Ceti, P rot =9 days, k=0.09 k mod =0.12 k mod =0.13 Theory: Weak dependence of δ on

OBSERVATIONS Göttingen 2014 Barnes et al Reinhold et al Kepler data

CONCLUSION Göttingen 2014 If a mean-field formulation is possible and if Hathaways measurements reflect the horizontal Reynolds stress  the Λ effect exists