Tayler-Spruit dynamo in stably stratified rotating fluids: Application to proto-magnetars

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Table 1.

Theoretical and measured scaling laws of the different quantities discussed in Sects. 3.4 and 3.5, and the dimensionless normalisation factor α defined by Fuller et al. (2019) and Eq. (22).

Quantity (dimensionless)	Fuller et al. (2019)’s scaling law	Best fit exponent	α
$B_{tot}^{m \neq 0} / B_{tor}^{m \neq 0}$ $B^{m\neq0}_{\mathrm{tot}}/B^{m\neq0}_{\mathrm{tor}}$	ω_A/Ω_o	(N_eff/Ω_o)^{−0.18 ± 0.1}	—
$B_{tor}^{m = 0} / (\sqrt{4 π ρ r_{o}^{2} Ω_{o}^{2}} q^{1 / 3})$ $B^{m=0}_{\mathrm{tor}}/(\sqrt{4\pi\rho r_o^2\Omega_o^2}q^{1/3})$	α(N_eff/Ω_o)^−1/3	(N_eff/Ω_o)^{−0.11 ± 0.05}	0.017 ± 0.001
$B_{pol}^{m = 0} / (\sqrt{4 π ρ r_{o}^{2} Ω_{o}^{2}} q^{2 / 3})$ $B^{m=0}_{\mathrm{pol}}/(\sqrt{4\pi\rho r_o^2\Omega_o^2}q^{2/3})$	α²(N_eff/Ω_o)^−5/3	(N_eff/Ω_o)^{−1.1 ± 0.2}	0.009 ± 0.002
$B_{dip} / (\sqrt{4 π ρ r_{o}^{2} Ω_{o}^{2}} q^{2 / 3})$ $B_{\mathrm{dip}}/(\sqrt{4\pi\rho r_o^2\Omega_o^2}q^{2/3})$	α²(N_eff/Ω_o)^−5/3	(N_eff/Ω_o)^{−1.5 ± 0.1}	0.007 ± 0.001
$B_{pol}^{m = 0} / B_{tor}^{m = 0}$ $B^{m=0}_{\mathrm{pol}}/B^{m=0}_{\mathrm{tor}}$	ω_A/N_eff	(ω_A/N_eff)^{0.93 ± 0.2}	—
$B_{dip} / B_{tor}^{m = 0}$ $B_{\mathrm{dip}}/B^{m=0}_{\mathrm{tor}}$	ω_A/N_eff	(ω_A/N_eff)^{1.3 ± 0.1}	—
B_sB_ϕ/(4πρr_o²Ω_o²q)	α³(N_eff/Ω_o)⁻²	(N_eff/Ω_o)^{−1.8 ± 0.1}	0.016 ± 0.004
$B_{s}^{m = 0} B_{ϕ}^{m = 0} / (4 π ρ r_{o}^{2} Ω_{o}^{2} q)$ $B_s^{m=0}B_{\phi}^{m=0}/(4\pi\rho r_o^2\Omega_o^2q)$	α³(N_eff/Ω_o)⁻²	(N_eff/Ω_o)^{−1.6 ± 0.1}	0.01 ± 0.004
$v_{r}^{m \neq 0} v_{ϕ}^{m \neq 0} / (r_{o}^{2} Ω_{o}^{2} q^{5 / 3})$ $v_r^{m\neq 0}v_{\phi}^{m\neq 0}/(r_o^2\Omega_o^2q^{5/3})$	(N_eff/Ω_o)^−10/3	(N_eff/Ω_o)^{−3.5 ± 0.2}	—
$v_{s}^{m \neq 0} v_{ϕ}^{m \neq 0} / (r_{o}^{2} Ω_{o}^{2} q^{5 / 3})$ $v_s^{m\neq 0}v_{\phi}^{m\neq 0}/(r_o^2\Omega_o^2q^{5/3})$	(N_eff/Ω_o)^−10/3	(N_eff/Ω_o)^{−3.4 ± 0.2}	—
ν_mix/ν_AM	(N_eff/Ω_o)^−5/3	(N_eff/Ω_o)^{−1.2 ± 0.2}	—

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