tokens: time is dominated by mid latitudes where R ∼ r , so magnetic energy can only be dissipated at the pole as fast as it migrates from mid latitudes. Assuming loops of azimuthal field are totally dissipated near the pole, their effective damping rate is equal to their migration rate γ mig . Because the loop migration is essentially a random walk process, the migration time-scale is where N step is the migration length divided by a step length, and t step is the time it takes to complete each step. The number of steps is N step ∼ r /ξ ⊥ . The time of each step is a magnetic diffusion time across a radial wavelength, . In order for the the instability to grow, this diffusion rate must be smaller than the growth rate . So we have . Then using equations ( B3 ) and ( B4 ), we have This maximum migration rate will be realized when ω A ∼ ω c , with the critical field strength ω c defined by equation ( 1 ). When ω A ∼ ω c , field loops can reconnect with loops of opposite oscillation phase (i.e. those separated by radial distance ∼1/ k r ) after ∼1 oscillation cycle, such that they can migrate by a distance ∼ξ ⊥ each oscillation cycle. When ω A > ω c , reconnection requires many oscillation cycles, the migration rate will depend on the magnetic diffusivity, and it will be smaller than equation ( B5 ). Why do we still think the instability damping rate is δ v A / r ? Let’s consider whether migration of magnetic loops toward the pole can destroy them at faster rates. Assuming a loop reconnects with its neighbour after being displaced horizontally by ξ ⊥ at a rate , the two loops have moved apart from one another at a speed . Then the maximum rate at which the instability can be damped due to loops migrating to the pole is But since ω A Ω, γ diss is smaller than the damping rate δ v A / r . The actual destruction rate is likely much slower due to the random walk process discussed above, and is given by equation ( B5 ). This means that unstable perturbations will damp faster by weak turbulence than they will by migrating toward the pole. So the instability is still limited by weak turbulence, and setting the growth rate equal to the damping rate still implies The migration rate of equation ( B5 ) implies that the background field, whose energy density is , is destroyed at this rate. The assosiated energy damping rate per unit volume is The turbulent energy damping discussed in the paper will operate regardless of the loop migration, and will be more important when the reconnection time-scale is longer than an oscillation times scale. Hence, while damping from loop migration may be relevant when ω A ∼ ω c , we do not expect any of our scaling arguments or results to be altered. APPENDIX C: EFFECTIVE STRATIFICATION As discussed in Spruit ( 2002 ), the effective Brunt–Väisälä frequency N eff depends on the thermal diffusion time-scale across the Tayler instability length-scale. This thermal diffusion time-scale in turn depends on N eff . Spruit ( 2002 )
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