Modern Solar Facilities: Advanced Solar Science, Proceedings of a Workshop held at Göttingen, September 27-29, 2006

Sunspot models with bright rings

Modern Solar Facilities – Advanced Solar Science, 343–346 F. Kneer, K. G. Puschmann, A. D. Wittmann (eds.) c  Universitätsverlag Göttingen 2007 Sunspot models with bright rings L. L. Kitchatinov1,2* and G. Rüdiger2 1 Institute for Solar-Terrestrial Physics, Irkutsk, Russia 2 Astrophysikalisches Institut Potsdam, Germany * Email: lkitchatinov@aip.de Abstract. A theoretical sunspot model is provided including magnetic suppression of the diffusivities and also a strong stratification of density and temperature. Heat diffusion alone with given magnetic field and zero mean flow only produces (after a very long relaxation time) dark spots without any bright ring. Models with full dynamics of both field and flow, however, provide rings and also the observed correlation of ring temperature excess and the spot size. The rings are formed as the result of heat transport by the resulting flow system and increased thermal diffusivity due to reduced magnetic quenching around spots. 1 Introduction Measurements have shown the existence of bright rings around sunspots (Bonnet et al. 1978; Rast et al. 1999, 2001). The rings appear one spot radius beyond the spots and are reported to be about 10 K warmer than the photosphere. There is a clear trend that larger spots have brighter rings. We shall discuss in the following whether simple mean-field models of more or less flat sunspots do develop such rings or not. 2 A simple model The model concerns a horizontal layer with top and bottom at depths dtop and dbot below the photosphere. In our computations dtop = 1 Mm is fixed, slightly beneath the typical depth of Wilson depression. The fluid is assumed to be a perfect gas. Density and temperature at the top (ρtop, Ttop) and bottom (ρbot, Tbot) boundaries are prescribed with the solar structure model by Stix & Skaley (1990). The reference atmosphere is approximated by adiabatic profiles, T = Ttop + g Cp (D − z) , ρ = ρtop  T Ttop  1 γ−1 . (1) Here the gravity is g = 2.74 · 104 cm s−2 , D = dbot − dtop is the layer depth (here 15 Mm), z is the vertical coordinate, Cp and γ are specific heat and adiabaticity index defined by the condition that the bottom temperature and density are exactly reproduced, i.e. Cp = gD Tbot − Ttop , γ = 1 + ln(Tbot/Ttop) ln(ρbot/ρtop) . (2) 344 L. L. Kitchatinov and G. Rüdiger: Sunspot models with bright rings The entropy equation with S = Cv log (P/ργ ) is ρT  ∂S ∂t + u · ∇S  = ∇ · (ρTχ∇S ) . (3) The sunspot darkness is usually explained in terms of convective heat transport suppressed by magnetic field. Accordingly, the eddy diffusivities in the model depend on the magnetic field (cf. Rüdiger & Kitchatinov 2000). This dependence describes a steady decrease of the turbulent diffusivities with the magnetic field amplitude. Hence χ = χTϕ (β) , where χT is nonmagnetic diffusivity and the quenching function ϕ (β) = 3 8β2  β2 − 1 β2 + 1 + β2 + 1 β arctan β  (4) depends on the field strength normalized to the energy equipartition value, β = B/( √ μ0ρ u ), u is the rms turbulent velocity. Any anisotropy of the eddy diffusivities is neglected. The depth profiles of the equipartition field, Beq = √ μ0ρ u , and the (nonmagnetic) diffusivities are written in accordance with the mixing-length approximation, which yields χT (z) = χ0  T Ttop  3γ−4 3(γ−1) , B2 eq = B2 0  T Ttop  1 3(γ−1) , (5) where T has been defined in (1) while χ0 and B0 are the thermal diffusivity and the equipartition field on the top boundary. A marginal value for thermal convection, χ0 = 1.4·1013 cm2 /s, was taken for the diffusivity, and B0 = 500 Gauss. The density runs from 2.02 × 10−6 g/cm3 at the top to 1.8 × 10−3 g/cm3 at the bottom, and the temperature varies from 14,000 K to 120,000 K. The horizontal boundaries are stress-free and impenetrable. For the magnetic field a vacuum condition is used for the top while the field is assumed vertical on the bottom . Thermal conditions are the black-body radiation on the top and constant heat flux (F0 = 6.27 · 1010 g s−3 ) at the bottom. If wall boundaries are used we assume zero stress, zero normal velocity, and superconductor outside. Figure 1. Left: Profiles of surface brightness normalized to F0 after 1 day (solid) and after about 500 days (dotted) for vanishing heat flux across the wall boundary. Right: Time dependence of the normalized total irradiance at the top. The thermal conditions...