As discussed in Section 5.7, for buoyant flux tubes with significantly super-equipartition field strength , where is in equipartition with the kinetic energy density of the convective downflows, the magnetic buoyancy of the tubes dominates the hydrodynamic force from the convective downflows and the flux tubes can rise unaffected by convection. On the other hand if the field strength of the flux tubes is comparable to or smaller than the equipartition value , the magnetic buoyancy is weaker than the hydrodynamic force from the convective downflows and the evolution of the tubes becomes largely controlled by the convective flows. In this regime of convection dominated evolution, due to the strong asymmetry between up- and downflows characteristic of stratified convection, it is found that magnetic flux is preferentially transported downward against its magnetic buoyancy out of the turbulent convection zone into the stably stratified overshoot region below. This process of “turbulent pumping” of a magnetic field has been demonstrated by several high resolution 3D compressible MHD simulations (see Tobias et al., 1998, 2001; Dorch and Nordlund, 2001).

Tobias et al. (2001) carried out a series of 3D MHD simulations to investigate the turbulent pumping of
a magnetic field by stratified convection penetrating into a stably stratified layer. A thin slab of a
unidirectional horizontal magnetic field is introduced into the middle of an unstably stratified convecting
layer, which has a stable overshoot layer attached below. It is found that the fast, isolated downflow plumes
efficiently pump flux from the convecting layer into the stable layer on a convective time scale. Tobias
et al. (2001) quantify this flux transport by tracking the amount of flux in the unstable layer and
that in the stable layer, normalized by the total flux. Within a convective turnover time, the
flux in the stable layer is found to increase from the initial value of 0 to a steady value that is
greater than 50% of the total flux (reaching 80% in many cases), i.e. more than half of the
total flux is settled into the stable overshoot region in the final steady state. Moreover the
stable overshoot layer is shown to be an effective site for the storage of a toroidal magnetic
field. If the initial horizontal magnetic slab is put in the stable overshoot layer, the penetrative
convection is found to be effective in pinning down the majority of the flux against its magnetic
buoyancy, preventing it from escaping into the convecting layer. It should be noted however that the
fully compressible simulations by Tobias et al. (2001) assumed a polytropic index of
for the unstable layer which corresponds to a value for the non-dimensional superadiabaticity
of 0.1. Thus the convective flow speed for the strong downdrafts is on the order of
times the sound speed, i.e. not very subsonic. On the other hand, the range of initial
magnetic field strength considered in the simulations corresponds to a plasma ranging from
2 × 10^{3} to 2 × 10^{7}. Thus comparing to the kinetic energy density of the convective flows, the
strongest initial field considered is of the order , i.e. below equipartition, although a
field with strength up to may be generated as a result of amplification by the strong
downflows during the evolution. Therefore the above results obtained with regard to the efficient
turbulent pumping of a magnetic field out of the convection zone into the stable overshoot
region against its magnetic buoyancy apply only to fields weaker than or at most comparable to
.

Abbett et al. (2004) found that turbulent pumping is very weak and ineffective in an MHD convection model without a stable overshoot layer at the bottom. Considering both the results of Tobias et al. (2001) and Abbett et al. (2004) it appears that the presence of the stable overshoot layer below the convection zone is an essential ingredient for effective turbulent pumping.Update

The turbulent pumping of magnetic flux with field strength out of the convection zone into the stable overshoot region demonstrated by the high resolution 3D MHD simulations has profound implications for the working of the interface mean field dynamo models (Parker, 1993; Charbonneau and MacGregor, 1997) as discussed by Tobias et al. (2001). The interface dynamo models require efficient transport of the large scale poloidal field generated by the -effect of the cyclonic convection out of the convection zone into the stably stratified tachocline region where strong rotational shear generates and amplifies the large scale toroidal magnetic field. Turbulent pumping is shown to enhance both the transport of magnetic flux into the stable shear layer and the storage of the toroidal magnetic field there. It further implies that the transport of magnetic flux by turbulent pumping should not be simply treated as an enhanced isotropic turbulent diffusivity in the convection zone as is typically assumed in the mean field dynamo models. Tobias et al. (2001) noted that a better treatment would be to add an extra advective term to the mean field equation characterizing the effect of turbulent pumping, which would correspond to including the antisymmetric part of the -tensor known as the -effect (see Moffatt, 1978).

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