THE STRETCHING OF MAGNETIC-FLUX TUBES IN THE CONVECTIVE OVERSHOOT REGION

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, from which a portion of it may ultimately protrude to form an active region on the surface. The amount of stretching required for such " destabilization " is given approximately by l/l0 almost-equal-to [1 + gamma-ad-beta-0(H/LAMBDA-0)<del(ad) - del>/1 - 2-gamma-ad-LAMBDA-0/R(cz))]1/2, where beta-0 is the ratio of gas pressure to initial magnetic pressure in the flux tube before stretching occurs, H/LAMBDA-0 almost-equal-to 1/4 is the ratio of the thickness of the overshoot region to the pressure scale height, LAMBDA-0/R(cz) almost-equal-to 1/8 is the ratio of the pressure scale height to the radius of the base of the convection zone, Del = 1 -1/gamma, where gamma = dlnP/dln-rho describes the variation of P with rho in the overshoot region, gamma-ad almost-equal-to 5/3 and del(ad) almost-equal-to 2/5 describe the adiabatic expansion of gas in the slowly rising flux tube, and <del(ad) - del> represents a depth average of del(ad) - del through the overshoot layer. When the tube reaches the top of the overshoot region, the magnetic field strength has increased to its maximum possible value B(final), determined by B(final)2/(8-pi-P) almost-equal-to gamma-ad(H/LAMBDA-0)<del(ad) - del>/(1 - 2-gamma-ad-LAMBDA-0/R(cz)), independent of l/l0. We show how 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 surmise that these phenomena must be considered in self-consistent models of solar and stellar dynamos operating in the overshoot region. Finally, our model is valid only for initially stable flux tubes. Spruit and van Ballegooijen's stability condition shows that if LAMBDA-0/R(cz) greater-than-or-similar-to 1/4, no buoyantly stable flux tubes are possible in the convective overshoot layer. We repeat their suggestion that in those stars for which this condition is true, the nature of magnetic activity should be qualitatively different from that of solar-like stars.