Leibniz-Institut für Astrophysik PotsdamLeibniz-Institut für Astrophysik Potsdam

The existence of the Λ effectindicated by SDO data

G. Rüdiger, M. Küker

Göttingen 2014

ROTATING TURBULENCEBoussinesq (1897), Taylor (1915):

Stress-strain relation, eddy viscosity

for rotation

negative through the whole convection zone

 

Göttingen 2014

MAGNETIC OBSERVATIONSMAGNETIC OBSERVATIONS

BUT: no confirmation by observations Ward 1965, Gilman & Howard 1984, Balthasar et al. 1986, Vrsnak et al. 2004

Göttingen 2014All tracers are magnetic-defined

Nesme-Ribes et al. 1993

Q𝜃𝜙 ≈ 107 cm2/s2

SDO DATASDO DATA

Göttingen 2014

Hathaway et al. 2013

Giant cells, 200 Mm,

Persisting over months

GIANT CELLS GIANT CELLS

Göttingen 2014

Hathaway et al. 2013

ROTATING & ANISOTROPIC TURBULENCEROTATING & ANISOTROPIC TURBULENCE

Rigid rotation: two possible tensors (odd in 𝛺, even in g )

Käpylä et al. 2011

Q𝜃𝜙 >0Q𝜃𝜙 <0Q𝜃𝜙 =0

Göttingen 2014

ANGULAR MOMENTUM TRANSPORTANGULAR MOMENTUM TRANSPORT

Göttingen 2014

V, H are functions of the Coriolis number 𝛺* =2𝜏corr 𝛺

• V < 0 for small 𝜏corr (inward transport)• H > 0 always (equatorial transport)

SOCA theory:RKH 2013

THE BASIC MODELTHE BASIC MODEL

Göttingen 2014

Solution of the conservation law of angular momentum

without meridional flow div (r sin𝜃 𝜚Q )=0.equation cubic in 𝛺, H=1, V=0, 𝜚=const:

No dependence on the viscosity, inner part: W>0. outer part: W<0

IMPROVED MODELIMPROVED 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 COMPLETE MODELS WITH MERIDIONAL CIRCULATIONCIRCULATION

Göttingen 2014

With 𝜦, meridional flow(polewards ) Without 𝜦, only meridional flow(equatorwards) Kippenhahn, Köhler, Pitadella et al., Balbus et al.Küker & R. 2012

Göttingen 2014

MERIDIONAL CIRCULATIONMERIDIONAL CIRCULATION

Schad et al. ´12

One-cell

meridional circulation?

HORIZONTAL REYNOLDS STRESSHORIZONTAL REYNOLDS STRESS

Göttingen 2014

withΛ effect,W=0.14

without Λ effect,W<0

Göttingen 2014

MAIN-SEQUENCE MODELSMAIN-SEQUENCE MODELS

MOST stars: ε Eri, Prot= 11 days, k=0.11. κ1 Ceti, Prot=9 days, k=0.09

kmod =0.12kmod =0.13

Theory: Weak dependence of δ𝛺 on 𝛺

OBSERVATIONSOBSERVATIONS

Göttingen 2014

Barnes et al. 2005

Reinhold et al. 2013

Kepler data

CONCLUSIONCONCLUSION

Göttingen 2014

If a mean-field formulation is possible

and if Hathaways measurements reflect

the horizontal Reynolds stress

the Λ effect exists