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Table 7 Parameter dependence: power law fits and relative misfit σ r

From: Deep magnetic field stretching in numerical dynamos

Quantity

Type

Mid-shell

σ r

δ σ r

Outer boundary

Ad r /Ad h

Global

\(0.738 \cdot (E^{2} \cdot Ra)^{\frac {1}{8}}\)

0.029

0.004

 

HN

\(0.580 \cdot (E^{2} \cdot Ra)^{-\frac {1}{11}}\)

0.042

0.000

St r /St h

Global

\(2.081 \cdot (E^{3} \cdot Ra^{2} \cdot Pm)^{-\frac {1}{12}}\)

0.054

0.007

 

HN

\(2.982 \cdot (E \cdot Ra^{2} \cdot Pm)^{-\frac {1}{28}}\)

0.080

0.000

St h /Ad h

Global

\(0.628 \cdot Pm^{-\frac {1}{4}}\)

0.071

0.004

\(2.996 \cdot (E \cdot Ra \cdot Pm^{3})^{-\frac {1}{6}}\)

 

HN

\(0.765 \cdot (E^{6} \cdot Ra^{3} \cdot Pm^{5})^{-\frac {1}{11}}\)

0.140

0.005

\(10.489 \cdot (E \cdot Ra \cdot Pm^{2})^{-\frac {1}{3}}\)

St/Ad

Global

\(1.666 \cdot (E^{4} \cdot Ra^{3} \cdot Pm^{6})^{-\frac {1}{24}}\)

0.036

0.004

\(2.996 \cdot (E \cdot Ra \cdot Pm^{3})^{-\frac {1}{6}}\)

 

HN

\(0.425 \cdot (E^{9} \cdot Ra^{4} \cdot Pm^{3})^{-\frac {1}{13}}\)

0.184

0.016

\(10.489 \cdot (E \cdot Ra \cdot Pm^{2})^{-\frac {1}{3}}\)

\(\mathcal {P}/\mathcal {T}\)

Global

\(0.302 \cdot (E^{3} \cdot Ra^{2})^{\frac {1}{16}}\)

0.035

0.002

\(0.319 \cdot Pm^{\frac {1}{6}}\)

 

HN

\(0.105 \cdot (E^{-1} \cdot Pm)^{\frac {1}{8}}\)

0.089

0.002

\(0.261 \cdot Pm^{\frac {1}{6}}\)

\(\|u_{r}\|/\|\vec {u}_{h}\|\)

Global

\(0.556 \cdot (E^{3} \cdot Ra^{2})^{\frac {1}{16}}\)

0.025

0.005

 

HN

\(0.117 \cdot (Ra^{3} \cdot Pm^{5})^{\frac {1}{20}}\)

0.078

0.003

  1. The fit deterioration is defined by δσ r =σ r σ ro where σ ro is the initial best fit relative misfit. Quantities are the same as in Table 2. Also given are the power laws for the outer boundary (Peña et al. 2016)