Geometrical properties of Sine-Gordon Abelian Higgs sunspots
Abstract We discuss a slightly modified Abelian Higgs model of sunspots in which the potential term V(Φ) has a ‘Sine-Gordon’-like form, V( Φ ) = −( α β 2 )[1 − sin β( ΦΦ ∗ ) κ ] . The model accounts not only for the observed diversity of the absolute dimensions of sunspots, but is also compatible with a more-or-less constant value of their penumbra-to-umbra radius ratios provided the parameter κ is close to unity.
The main objective of this article is to study dynamic of the ::: three-dimensional Boussinesq equations with the periodic boundary ::: condition.We prove that when the Rayleigh number $R$ crosses the ::: first critical Rayleigh number $R_c$, the Rayleigh-Benard problem ::: bifurcates from the basic state to an global attractor $\Sigma$, which is homeomorphic to $S^3$.