The effects of magnetic stretching in the convective overshoot region

Abstract

We consider the fate of a magnetic flux tube lying initially near the bottom of the solar convective overshoot region, which is the weakly stable boundary layer at the base of the convection zone perturbed by convective motions. Stretching of the flux tube, for instance by differential rotation, decreases its density, causing it to rise quasi-statically (we refer to this as “vertical flux drift”) until it reaches the top of the overshoot region and enters the buoyantly unstable convection zone. We refer to this as “destabilization” of the flux tube. The amount of stretching required for destabilization is given approximately by l/l ₀ ≈ [1 -I- γ _ad β ₀ (H/Λ ₀) < ∇_ad − ∇ > /(1 − 2γ _ad Λ ₀/R _cz)]^1/2, where β ₀ is the ratio of gas pressure to initial magnetic pressure in the flux tube before stretching occurs, H/Λ ₀ is the ratio of the depth of the overshoot region to the pressure scale height, Λ ₀/R _cz is the ratio of the pressure scale height to the radius of the base of the convection zone, Λ ≡ 1 − 1/γ, where γ ≡ dlnP/dlnp describes the variation of P with ϱ in the overshoot region, γ _ad ;:≈ 5/3 and ∇ _ad ≈ 2/5 describe the adiabatic expansion of gas in the slowly rising flux tube, and < ∇_ad − ∇ > represents a depth-average of ∇_ad − ∇ through the overshoot layer. When the tube reaches the top of the overshoot region, the magnetic field strength has increased to B _final, determined by B ^fina₂ /(8πP _e ) ≈ γ _ad(H/Λ ₀) < ∇_ad − ∇ > /(1 − 2γ _ad Λ ₀/R _cz), independent of l/l ₀. This constraint can be used to place limits on conditions in the convective overshoot region. We conclude that vertical flux drift and flux destabilization are inevitable consequences of field amplification, and must be considered in self-consistent models of solar and stellar dynamos. Finally, we find that the direction of vertical flux drift switches from outward to inward for those later-type stars for which 2γ _ad Λ ₀/R _cz ≳ 1. We speculate that magnetic activity on these stars is qualitatively different than for the more solar-like stars.