Determining the depth of Jupiter’s dynamo region Yue-Kin Tsang School of Mathematics, University of Leeds Chris Jones ( Leeds)
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 one way to deduce r dyn from observation on the surface: spectrum of magnetic energy
Gauss coefficients g lm and h lm Outside the dynamo region, r > r dyn : j = 0 j = 0 dynamo region ∇ × B = µ 0 j = 0 = ⇒ B = −∇ Ψ r dyn ⇒ ∇ 2 Ψ = 0 ∇ · B = 0 = a a = radius of Earth Consider only internal sources, ∞ l � l +1 ˆ � a � � Ψ( r, θ, φ ) = a P lm (cos θ )( g lm cos mφ + h lm sin mφ ) r m =0 l =1 ˆ P lm : Schmidt’s semi-normalised associated Legendre polynomials g lm and h lm can be determined from magnetic field measured on the planetary surface ( r ≈ a )
The Lowes spectrum Average magnetic energy over a spherical surface of radius r 1 1 � | B ( r, θ, φ ) | 2 sin θ d θ d φ E B ( r ) = 2 µ 0 4 π Inside the source-free region r dyn < r < a , ∞ l � � a � � � 2 l +4 � � g 2 lm + h 2 � 2 µ 0 E B ( r ) = ( l + 1) lm r l =1 m =0 Lowes spectrum (magnetic energy as a function of l ): l � a � 2 l +4 � g 2 lm + h 2 � � R l ( r ) = ( l + 1) lm r m =0 � a � 2 l +4 = R l ( a ) (downward continuation) r
Estimate location of CMB using the Lowes spectrum � a 2 l +4 � R l ( a ) R l ( r dyn ) = R l ( a ) r dyn a = Earth’s radius (Robert Parker, UCSD) downward continuation through the j = 0 region from a to r dyn : � r dyn � r dyn � � ln R l ( a ) = 2 ln l + 4 ln + ln R l ( r dyn ) a a white source hypothesis : turbulence in the core leads to an even distribution of magnetic energy across different scales l , R l ( r dyn ) is independent of l r dyn ≈ 0 . 55 a ≈ 3486 km agrees very well with results from seismic waves observations
Interior structure of Jupiter 8 10 7 10 6 10 5 10 4 10 3 10 a b i n itio (HSE) 2 10 σ [S/m] Li u et a l. 2008 1 Röpke & Redmer 1989 10 Stevenson 1977 0 10 Lee & More 1984 -1 10 Gómez- P érez et a l. 2010 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 0 0.5 R J R J (NASA JPL) theoretical σ ( r ) (French et al. 2012) low temperature and pressure near surface ⇒ gaseous molecular H/He extremely high temperature and pressure inside ⇒ liquid metallic H core? conductivity σ ( r ) varies smoothly with radius r At what depth does dynamo action start?
Lowes spectrum from the Juno mission Juno’s spacecraft reached Jupiter on 4th July, 2016 currently in a 53-day orbit, measuring Jupiter’s magnetic field (and other data) latest results give R l ( r J ) up to l = 10 suggesting r dyn ≈ 0 . 85 r J ( r J = Jupiter’s radius) (Connerney et al. 2018)
Lowes spectrum from the Juno mission Juno’s spacecraft reached Jupiter on 4th July, 2016 currently in a 53-day orbit, measuring Jupiter’s magnetic field (and other data) latest results give R l ( r J ) up to l = 10 suggesting r dyn ≈ 0 . 85 r J ( r J = Jupiter’s radius) Questions : given the conductivity profile σ ( r ) is smoothly varying, estimation of r dyn using Lowes spectrum the right approach? white source hypothesis valid? concept of “dynamo radius” r dyn (Connerney et al. 2018) well-defined?
A numerical model of Jupiter spherical shell of radius ratio r in /r out = 0 . 0963 (small core) rotating fluid with electrical conductivity σ ( r ) forced by buoyancy convection driven by secular cooling of the planet ρ, ¯ anelastic:linearise about a hydrostatic adiabatic basic state(¯ T, ¯ p, . . . ) dimensionless numbers: Ra, Pm, Ek, Pr ∇ · (¯ ρ u ) = 0 � ∂ u � EkRaPm S d ¯ Ek � z × u = −∇ Π ′ + 1 � T r + Ek F ν ∂t + ( u · ∇ ) u + 2ˆ ρ ( ∇ × B ) × B − d r ˆ ¯ ¯ Pm Pr ρ ∂ B ∂t = ∇ × ( u × B ) − ∇ × ( η ∇ × B ) � ∂S � + Pm Pr � 1 � + Pm ρ ¯ ¯ T ∂t + u · ∇ S Pr ∇ · F Q = Q ν + Ek Q J Pr H S RaPm Boundary conditions: no-slip at r in and stress-free at r out , S ( r in ) = 1 and S ( r out ) = 0, electrically insulating outside r in < r < r out . (Jones 2014)
A numerical model of Jupiter spherical shell of radius ratio r in /r out = 0 . 0963 (small core) rotating fluid with electrical conductivity σ ( r ) forced by buoyancy convection driven by secular cooling of the planet ρ, ¯ anelastic:linearise about a hydrostatic adiabatic basic state(¯ T, ¯ p, . . . ) dimensionless numbers: Ra, Pm, Ek, Pr a Jupiter basic state: C.A. Jones / Icarus 241 (2014) 148–159 1 ρ ( r ) ¯ η ( r ) = 5000 14 µ 0 σ ( r ) 4500 12 4000 10 3500 Density (kg/m 3 ) log 10 η (m 2 /s) 3000 8 2500 6 2000 cut−off 4 + Data points from French et al. 2012 1500 2 Hyperbolic fit 1000 + Data points from French et al. 2012 0 500 Rational polynomial fit cut−off 0 −2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 x 10 7 x 10 7 (a) (b) radius (metres) radius (metres)
Ra = 2 × 10 7 , Ek = 1 . 5 × 10 − 5 , Pm = 10 , Pr = 0 . 1 radial magnetic field, B r ( r, θ, φ ) r = r out dipolar r = 0 . 75 r out small scales
Where does the current start flowing? 10 6 10 5 10 4 10 3 10 2 10 1 10 0 < ( r ) 10 ! 1 j rms ( r ) 10 ! 2 0.5 0.6 0.7 0.8 0.9 r=r J average current over a spherical surface of radius r µ 0 j = ∇ × B � 2 π � π rms ( r, t ) ≡ 1 | j | 2 sin θ d θ d φ j 2 4 π 0 0 j rms drops quickly but smoothly in the transition region, not clear how to define a characteristic “dynamo radius”
Magnetic power spectrum, F l ( r ) average magnetic energy over a spherical surface: 1 1 � | B ( r, θ, φ ) | 2 sin θ d θ d φ E B ( r ) = 2 µ 0 4 π Lowes spectrum: recall that if j = 0 , we solve ∇ 2 Ψ = 0, then ∞ l ∞ � � a � � � 2 l +4 � � � g 2 lm + h 2 � 2 µ 0 E B ( r ) = ( l + 1) = R l ( r ) lm r m =0 l =1 l =1 generally, for the numerical model, B ∼ � lm b lm ( r ) Y lm ( θ, φ ), ∞ 2 µ 0 E B ( r ) = 1 � | B ( r, θ, φ ) | 2 sin θ d θ d φ = � F l ( r ) 4 π l =1 j ( r, θ, φ ) = 0 exactly = ⇒ R l ( r ) = F l ( r )
Magnetic power spectrum at different depth r F l ( r ) : solid lines 10 12 R l ( r ) : dashed lines (in nT 2 ) 10 11 10 10 0.959 r J 0.925 r J 10 9 0.907 r J 0.885 r J 10 8 0.565 r J 0 20 40 60 80 100 l near the surface ( r out > r > 0 . 9 r J ) F l ( r ) ≈ R l ( r ) slope of F l ( r ) decreases with r interior and away from the core (0 . 9 r J > r > 0 . 5 r J ) F l ( r ) different from R l ( r ) F l ( r ) is shallow and maintains roughly the same shape
Spectral slope of F l ( r ) 0 F l spectral slope -0.02 R l -0.04 -0.06 -0.08 -0.1 -0.12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r=r J F l ( r ) indicates a clear transition in dynamics: | slope | minimum at r dyn = 0 . 889 r J F l ( r ) in the interior is not flat but dependence on l is weak: | slope | ∼ 0 . 02 downward continuation from spectrum at the surface F l ( r out ) predicts: r dyn = 0 . 885 r J
Summary of results from numerical model 10 6 10 4 10 2 10 0 10 ! 2 < ( r ) j rms ( r ) 10 ! 4 " ( r ) 10 ! 6 0.5 0.6 0.7 0.8 0.9 r=r J In our numerical model of Jupiter, we find that: the magnetic power spectrum provides a characteristic radius of the dynamo action in the interior and away from the core, white source hypothesis is approximately valid the dynamo radius can be predicted using the magnetic spectrum at the surface However, . . .
Comparison with Juno data 10 0 Juno, r dyn = 0.827 r J numerical model, r dyn = 0 : 885 r J magnetic spectrum at r = r J 10 ! 1 10 ! 2 10 ! 3 2 4 6 8 10 12 14 l The dynamo radius in the numerical model is too shallow compared to the prediction using the Juno data. The discrepancy suggests: the metallic hydrogen layer could be deeper than predicted by theoretical calculation the existence of a stably stratified layer above the dynamo region our numerical model has more small-scale forcing than Jupiter does
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