Magnetic power spectrum in a dynamo model of Jupiter. Yue-Kin Tsang

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1 Magnetic power spectrum in a dynamo model of Jupiter Yue-Kin Tsang School of Mathematics, University of Leeds Chris Jones University of Leeds

2 Structure of the Earth Let s start on Earth... CRUST various types of rocks MANTLE magnesium-iron silicate OUTER CORE liquid iron + nickle INNER CORE solid iron + nickle CMB (not to scale) core-mantle boundary (CMB): sharp boundary between the non-conducting mantle and the conducting outer core location of CMB r dyn : the depth at which dynamo action starts seismic waves observation gives r dyn 3486 km another way to estimate r dyn : magnetic power spectrum

3 Gauss coefficients g lm and h lm Outside the dynamo region, r dyn < r < a: j = B = µ j = = B = Ψ Earth interior j = dynamo region r dyn a = radius of Earth B = = 2 Ψ = a Consider only internal sources, l ( a ) l+1 Ψ(r, θ, φ) = a ˆP lm (cos θ)(g lm cos mφ+h lm sin mφ) r l=1 m= ˆP lm : Schmidt s semi-normalised associated Legendre polynomials g lm and h lm are determined from magnetic field measurements on the surface (r a)

4 The Lowes spectrum Average magnetic energy over a spherical surface of radius r E B (r) = 1 1 B(r, θ, φ) 2 sin θ dθ dφ 2µ 4π Inside the source-free region r dyn < r < a, 2µ E B (r) = ( a ) 2l+4 l ( (l + 1) g 2 r lm + hlm) 2 l=1 m= Lowes spectrum (magnetic energy as a function of l): ( a ) 2l+4 l ( R l (r) = (l + 1) g 2 r lm + hlm) 2 m= ( a ) 2l+4 = Rl (a) (downward continuation) r

5 Estimate CMB using the Lowes spectrum R l (a) ( a ) 2l+4 R l (c) = Rl (a) c a = Earth s radius Figure exponential behavior, completely different from the gravity spectrum s, which is essentially a power law I, though we plot a slightly different kind of spectrum, without the multiplying quadratic in l). erpretation of this result is that the two parts of the spectrum reflect different source regions the long ds with l 13 come almost entirely from the core, those with l > 13 from the crust. This idea is given c = 3486 km.55a Figure (Robert Parker, UCSD) white noise source hypothesis: turbulence in the core leads to even distribution of magnetic energy across different scales l R l (c) independent of l = c r dyn pport from the observation that the equation of the best straight line through the core spectrum is α(r1/a) 2l thing predicted in homogeneous turbulence. A plausible argument might be made that the fluid motions in t

6 Interior structure of Jupiter ical Journal Supplement Series, 22:5 (11pp), 212 September along the Jupiter adiabat. These quantities are ot as insensitive to the stoichiometry as η, λ i,or fficients and its influence needs some assessment. lculations core? are performed for three different helium s using the PBE functional. The general trends d in Figure 7 and the correct absolute values uctivities can be obtained by scaling with the ios shown in Figure 5. In particular, we have chosen and 2 helium atoms to represent the mean helium Interior structure of Jupiter French et al R [R 1-6 J] 1-7.5R dence of the electrical (top) and the thermal (bottom) conducncentration of helium: Y (dotted line), Y et al. 212) J R J (NASA JPL) (French Y (dashed line). These calculations were performed 1 1 nctional. 1 8 σ [S/m] β [m²/s] ab initio (HSE) Liu et al. 28 Röpke & Redmer 1989 Stevenson 1977 Lee & More 1984 Gómez-Pérez et al. 21 low temperature and pressure near surface gaseous molecular H/He extremely high temperature and pressure inside liquid metallic H conductivity σ(r) varies smoothly with radius r R [R J ] Figure 8. Upper panel: electrical conductivity along Jupiter s adiabat calculated with the HSE functional assuming a constant mean He fraction of Y.275. Comparison with different models and estimations is made (Stevenson & Salpeter 1977; Lee&More1984; Röpke & Redmer 1989; Liu et al. 28; Gómez-Pérez et al. 21); see the text for further details. Lower panel:

7 .122 yr 1 is 3 times greater for Jupiter than Earth, which has an average rate of change ical 41 yr Journal Lowes 1. Supplement spectrum Series, 22:5 (11pp), for212 Jupiter: September observations Interior structure of Jupiter ab initio (HSE) 1 1 Liu et al. 28 Röpke & Redmer Stevenson Lee & More 1984 Gómez-Pérez et al R [R 1-6 J] 1-7.5R dence of the electrical (top) and the thermal (bottom) conducncentration tic field spectra of helium: for Earth Y(IGRF21), plotted (dotted at line), core-mantle Y boundary.275 (CMB), and for Jupiter 1 12 et al. 212) J R J (Ridley & Holme 216) (French ed Y 2 at depths.311 of (dashed.8,.85, line). and These.9 R J. All calculations spectra normalized were performed to unit power for the dipole (l 1 = 1) 1 term. nctional. g lm and h lm for l max 4 7 computed from magnetic measurements of 1 8 JOVIMAGNETIC various SECULAR missions VARIATION (e.g. Pioneer 1 & 11, Voyager & 2, JUNO: l max = 12) 1 4 downward continuation to a flat Lowes spectrum.73r J r dyn.9r J along the Jupiter adiabat. These quantities are ot as insensitive to the stoichiometry as η, λ i,or fficients and its influence needs some assessment. lculations are performed for three different helium s using the PBE functional. The general trends d in Figure 7 and the correct absolute values uctivities can be obtained by scaling with the ios shown in Figure 5. In particular, we have chosen and 2 helium atoms to represent the mean helium σ [S/m] β [m²/s] For a continuous conductivity profile σ(r): dynamo action in transition region? dynamo radius r dyn well-defined? estimation of r dyn from Lowes spectrum reliable? R [R J ] French et al. Figure 8. Upper panel: electrical conductivity along Jupiter s adiabat calculated with the HSE functional assuming a constant mean He fraction of Y.275. Comparison with different models and estimations is made (Stevenson & Salpeter 1977; Lee&More1984; Röpke & Redmer 1989; Liu et al. 28; Gómez-Pérez et al. 21); see the text for further details. Lower panel:

8 A numerical model of Jupiter spherical shell of radius ratio r in /r out =.963 (small core) anelastic: linearise about a hydrostatic adiabatic basic state ( ρ, T, p,... ) rotating fluid with electrical conductivity σ(r) forced by buoyancy convection driven by secular cooling of the planet dimensionless numbers: Ra, Pm, Ek, Pr ( ρu) = [ ] Ek u + (u )u + 2ẑ u = Π + 1 ρ ( ) EkRaPm Pm t ( B) B S d T ˆr + Ek Pr dr Fν ρ ( S ρ T t + u S B = (u B) (η B) t ) + Pm Pr F Q = Pr ( Q ν + 1 ) RaPm Ek Q J + Pm Pr H S Boundary conditions: no-slip at r in and stress-free at r out, S(r in ) = 1 and S(r out) =, electrically insulating outside r in < r < r out (Jones 214)

9 A numerical model of Jupiter Density (kg/m 3 ) spherical shell of radius ratio r in /r out =.963 (small core) anelastic: linearise about a hydrostatic adiabatic basic state ( ρ, T, p,... ) rotating fluid with electrical conductivity σ(r) forced by buoyancy convection driven by secular cooling of the planet dimensionless numbers: Ra, Pm, Ek, Pr a Jupiter basic state: Data points from French et al Rational polynomial fit radius (metres) (a) C.A. Jones / Icarus 241 (214) ρ(r) η(r) = 1 14 µ σ(r) cut off x 17 log 1 η (m 2 /s) Data points from French et al Hyperbolic fit cut off x 1 radius (metres) 7 (b)

10 Ra = 2 1 7, Ek = , Pm = 3, Pr =.1 radial magnetic field r = r out r =.75r out zonal velocity r = r out

11 slope Dynamo radius from Lowes spectrum 1 12 R l (r) in nt r J.93r J.9r J r J.83r J l r dyn =.87r J -.1 R l(r) B r at the surface r = r out = g lm, h lm = R l (r out ) downward continuation into source-free j = region: ( rout ) 2l+4 R l (r) = Rl (r out ) r r dyn =.87r J, how reliable is this estimate?

12 slope Magnetic power spectrum, F l (r) 1 12 R l (r) in nt r J.93r J.9r J r J.83r J l r dyn =.87r J -.1 R l(r) µ E B (r) = 1 B(r, θ, φ) 2 sin θ dθ dφ 4π = F l (r) l=1 j = exactly = R l (r) = F l (r)

13 slope R l (r) versus F l (r) 1 12 F l (r) (!) and R l (r) (- -) in nt r J.93r J.9r J r J.83r J l j (R l deviates from F l ) starting at about.9r J r dyn =.87r J R l(r) F l(r) Lowes spectrum R l prediction deeper than actual r dyn numerical model produces r dyn consistently with observations transition layer (moderate σ) not contributes to dynamo action F l (r) flat in a large range.5r J < r <.9r J : white-noise source