J/ApJ/774/6 Empirical scaling laws for shell dynamos (Yadav+, 2013)
Consistent scaling laws in anelastic spherical shell dynamos.
Yadav R.K., Gastine T., Christensen U.R., Duarte L.D.V.
<Astrophys. J., 774, 6 (2013)>
=2013ApJ...774....6Y 2013ApJ...774....6Y
ADC_Keywords: Models ; Magnetic fields
Keywords: brown dwarfs; convection; methods: numerical; stars: interiors;
stars: low-mass; stars: magnetic field
Abstract:
Numerical dynamo models always employ parameter values that differ by
orders of magnitude from the values expected in natural objects.
However, such models have been successful in qualitatively reproducing
properties of planetary and stellar dynamos. This qualitative
agreement fuels the idea that both numerical models and astrophysical
objects may operate in the same asymptotic regime of dynamics. This
can be tested by exploring the scaling behavior of the models. For
convection-driven incompressible spherical shell dynamos with constant
material properties, scaling laws had been established previously that
relate flow velocity and magnetic field strength to the available
power. Here we analyze 273 direct numerical simulations using the
anelastic approximation, involving also cases with radius-dependent
magnetic, thermal, and viscous diffusivities. These better represent
conditions in gas giant planets and low-mass stars compared to
Boussinesq models. Our study provides strong support for the
hypothesis that both mean velocity and mean magnetic field strength
scale as a function of the power generated by buoyancy forces in the
same way for a wide range of conditions.
File Summary:
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FileName Lrecl Records Explanations
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ReadMe 80 . This file
table1.dat 89 11 Simulation setups
table2.dat 130 273 Simulation dataset used in the article to infer
various scaling laws
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Byte-by-byte Description of file: table1.dat
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Bytes Format Units Label Explanations
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1- 3 A3 --- SID Subset name (A=Anelastic; B=Boussinesq)
5- 6 I2 --- N [1/58] Number of cases
8- 17 A10 --- Core Core description ("Insulating" or "Conducting")
19- 23 A5 --- BC Mechanical boundary condition (1)
25- 28 F4.2 --- eta [0.1/0.8] Aspect ratio η (inner radius to
outer radius ratio of shell with dynamo action)
30- 35 A6 --- g(r) Gravity profile g(r) (2)
37- 39 F3.1 --- m [0/2] Polytropic index m (2)
41- 46 A6 --- Pr Prandtl number Pr range
48- 53 A6 --- Pm Magnetic Prandtl number Pm range (ratio of
viscosity ν and thermal diffusivity κ)
55- 58 E4.1 --- E Ekman number E (or lower range)
59 A1 --- --- [-]
60- 63 E4.1 --- E2 ? Ekman number (upper range)
65- 70 A6 --- Nrho Number of density scale heights inside the
shell Nρ
72- 89 A18 --- Var Varying inside the shell (3)
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Note (1): Mechanical boundary conditions are:
SF = shear stress free on both boundaries
Mixed = rigid inner and stress-free outer boundary
Note (2): Gravity profile as in equation (1):
g(r)=g1(r/r0)+g2(r20/r2)
and by using g1 or g2 appropriately we can either choose a linear
gravity profile (g1=1, g2=0), approximately representing a
self-gravitating body with weak density variation, or an r-2 gravity
profile (g1=0, g2=1), exemplifying objects with a massive core.
Assuming an ideal gas equation of state leads to a polytropic
reference state defined by ρ=Tm, where m is the polytropic
index. See section 2.1.
Note (3): Transport properties which vary along the shell radius, where
ν is the kinematic viscosity, κ the thermal diffusivity and
λ the magnetic diffusivity. In groups A2b and B1b the variation
of λ is obtained from Equation (11) and in group A4 the
variation follows Equation (12).
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Byte-by-byte Description of file: table2.dat
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Bytes Format Units Label Explanations
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1- 3 A3 --- SID Simulation identifier (as in table 1)
5- 12 E8.2 --- E Ekman number E=ν/(ΩD2)
14- 17 F4.1 --- Pr [0/10] Prandtl number Pr=ν/κ
19- 22 F4.1 --- Pm [0/20] Magnetic Prandtl number Pm=ν/λ
24- 31 E8.2 --- Ra Rayleigh number Ra
33- 36 F4.2 --- Nrho [0/5.5] Number of density scale heights Nρ
38- 46 F9.5 --- Mass [8/630] Non-dimensional mass of spherical shell
48- 51 F4.2 --- chim [0.7/1]? Fraction of total radius (1)
53- 63 E11.5 --- P [2.e-8/3.e-2] Non-dimensional buoyancy power (2)
65- 74 F10.8 --- Ro [0.001/0.44] Rossby number defined by Eq.13
76- 85 F10.8 --- Roconv [0.001/0.3] Rossby number (3)
87- 97 F11.9 --- Lo [0/0.1] Lorentz number defined by Eq.14
99-107 F9.7 --- fohm [0/0.7] Ohmic fraction defined by Eq.17
109-118 F10.6 --- taumag [0.7/135] Magnetic dissipation time τmag
defined by Eq.18
120-130 E11.5 --- fdip Dipolarity (4)
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Note (1): After which exponential decay of electrical conductivity starts.
Blanks indicated Not Applicable.
Note (2): Per unit mass defined by Eq.15.
Note (3): Defined by excluding zonal and meridional flow components.
Note (4): The dipolarity fdip is the magnetic energy in the axisymmetric
dipole normalized by the cumulated magnetic energy in harmonic degrees
up to 12, both evaluated at the outer boundary. Dynamos with fdip>0.3
are considered dipolar. See section 3.1.
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History:
From electronic version of the journal
(End) Greg Schwarz [AAS], Emmanuelle Perret [CDS] 25-Feb-2015