Modeling of the solar interface dynamo Artyushkova M. E. Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences Based on “Popova H., Artyushkova M. and Sokoloff D. The WKB approximation for the interface dynamo // Geophysical & Astrophysical Fluid Dynamics V.104, № 5. P “ Vienna, EGU-2012

Parker, E.N., Hydromagnetic dynamo models// Astrophysical Journal. 1955, 122, 293–314. Single-layer dynamo model proposed by Parker in toroidal component of vectorial potential, poloidal magnetic field - dynamo number combined with α-effect amplitude, differential rotation Ω and coefficient of diffusion. So the equation system is dimensionless. - toroidal magnetic field - latitude (so corresponds to the poles).

WKB-approach for solving migratory dynamo model The WKB solution is looked for in the form of waves traveling in the -direction ~ where corresponds to the wave vector, or impulse action - complex growth rate, gives the length of the activity cycle derived from the Hamilton-Jacobi equation According this method the solution must decay towards the boundaries of the domain - the poles and equator Kuzanyan, K.M. and Sokoloff, D.D., A dynamo wave in an inhomogeneous medium. Geophys. Astrophys.Fluid Dyn. 1995, 81, 113–129. Bassom, A.P., Kuzanyan, K.M., Sokoloff, D. and Soward, A.M., Non-axisymmetric 2-dynamo waves in thin stellar shells// Geophys. Astrophys. Fluid Dyn. 2005, 2, 309–336.

Two-layers dynamo model - magnetic field, potential in the first layer - magnetic field, potential in the second layer - dynamo number combined with α-effect amplitude, Ω Parker’s system of equations : ratio of the turbulent diffusivity coefficients in the first and second layers α-effect localized first layer and the differential rotation Ω localized in the second layer Boundary conditions for

Applying the WKB-approach for two-layers model slowly varying functions The Hamilton–Jacobi equation - Hamilton–Jacobi equation for one-layer dynamo for comparison

Squared the Hamilton–Jacobi equation with Solving the Hamilton–Jacobi equation condition that k has to be continuous from pole to equator, particulary k has to be continuous at a turning point, where two roots of the Hamilton–Jacobi equation coincide where

Roots of the Hamilton–Jacobi equation for various values of as points in the complex k plane Re {k} Numbers 1,2,3,4 enumerate the branches of the various roots. Im {k} Single-layer model (Kuzanyan, K.M. and Sokoloff, D.D, 1995) Two-layers model,