Spectra of magnetic energy and secular variation in a dynamo model of Jupiter 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 ⇒ dynamo action entirely confined within the outer core dynamo radius r dyn : top of the dynamo region ≈ r cmb one way to deduce r cmb from observation at the surface : magnetic energy spectrum
Gauss coefficients g lm and h lm Outside the dynamo region, r > r dyn : j = 0 j = 0 dynamo region r dyn ∇ × B = µ 0 j = 0 = ⇒ B = −∇ Ψ ⇒ ∇ 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 at 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 2 l +4 � � a R l ( a ) R l ( r cmb ) = R l ( a ) r cmb a = Earth’s radius (Robert Parker, UCSD) downward continuation from a to r cmb through the mantle ( j = 0 ): � r cmb � r cmb � � ln R l ( a ) = 2 ln l + 4 ln + ln R l ( r cmb ) a a white source hypothesis : turbulence in the core leads to an even distribution of magnetic energy across different scales l , R l ( r cmb ) is independent of l r cmb ≈ 0 . 55 a ≈ 3486 km agrees very well with results from seismic waves observations
Interior structure of Jupiter (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? transition from molecular to metallic hydrogen is continuous 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) R l ( r J ) up to l = 10 from latest measurement (8 flybys) Lowes’ radius: r lowes ≈ 0 . 85 r J ( r J = 6.9894 × 10 7 m) (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) R l ( r J ) up to l = 10 from latest measurement (8 flybys) Lowes’ radius: r lowes ≈ 0 . 85 r J ( r J = 6.9894 × 10 7 m) Questions : with the conductivity profile σ ( r ) varying smoothly, meaning of r lowes ? r lowes = r dyn ? 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 ) driven by buoyancy convection forced 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 ¯ � z × u = −∇ Π ′ + 1 � Ek 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 ) driven by buoyancy convection forced 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) radius (metres) (b) 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 energy 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 energy spectrum at different depth r F l ( r ): solid lines R l ( r ): circles r > 0 . 9 r J : slope of F l ( r ) decreases rapidly with r r < 0 . 9 r J : F l ( r ) maintains the same shape and slope ⇒ a shift in the dynamics of the system r > 0 . 9 r J : F l ( r ) ≈ R l ( r ) r < 0 . 9 r J : F l ( r ) deviates from R l ( r ) ⇒ electric current becomes important below 0 . 9 r J suggests a dynamo radius r dyn ≈ 0 . 9 r J
Spectral slope of F l ( r ) and R l ( r ) log 10 F l ( r ) ∼ − α ( r ) l log 10 R l ( r ) ∼ − β ( r ) l � r out � 2 l +4 R l ( r out ) R l ( r ) = r r out β ( r ) = β ( r out ) − 2 log 10 r sharp transition in α ( r ) indicates r dyn = 0 . 907 r J F l ( r ) inside dynamo region is not exactly flat ( α dyn = 0 . 024): white source assumption is only approximate r lowes provides a lower bound to r dyn : β = 0 at r lowes = 0 . 883 General picture: α ( r out ) and α dyn control r dyn and r lowes
Comparison with Juno data noise in Juno data ⇒ results depend on fitting range larger Pm gives smaller α dyn , however α ( r out ) also becomes smaller ⇒ r dyn remains roughly the same R l ( r J ) is shallower in the numerical model than from Juno observation the metallic hydrogen layer could be deeper than predicted by theoretical calculation the existence of a stably stratified layer below the molecular layer our numerical model has more small-scale forcing than Jupiter does
Time variation of B r at the surface Pm=10 dipole is slowly varying compared to other modes B = ∂ B ˙ secular variation: ∂t for j = 0 , Lowes spectrum for secular variation: l � a � 2 l +4 � � � g 2 lm + ˙ h 2 R ˙ B ( l, r ) = ( l + 1) ˙ lm r m =0
Secular variation spectrum F ˙ B ( l, r ) ∞ B ( l, r ) = 1 � B ( r, θ, φ ) | 2 sin θ d θ d φ � | ˙ F ˙ 4 π l =1 geomagnetic SV surface (Holme & Olsen 2006) 10000 ufm (1980) 2 ) –1 ) undamped Mean square surface SV ( (nT yr 1000 CO2003 100 10 1 0.1 0.01 1 3 5 7 9 11 13 15 Spherical harmonic degree l Figure 1. Spectra of SV models at Earth’s surface. CMB (Holme et al. 2011) Figure 2. Spectra of the CHAOS-4 SV at the CMB, r = c . Green line gives theoretical model, dashed lines approximate 1 σ error bounds.
A spectral correlation time τ l a correlation time for different mode l : �� � F ( l, r ) τ l ( r ) = F ˙ B ( l, r ) t for j = 0 , τ l becomes independent of r : �� � l � g 2 lm + h 2 � � � m =0 lm τ l = � � � � lm + ˙ � l g 2 h 2 ˙ m =0 lm t two-parameter power law (e.g. Holme & Olsen 2006): τ l = τ SV · l − γ , 1 . 32 < γ < 1 . 45 τ SV ∼ a secular variation time scale one-parameter power law (e.g. Christensen & Tilgner 2004): τ l = τ SV · l − 1
Example of τ l in geodynamo τ l = τ SV · l − 1 (Lhuillier et al. 2011)
Spectral correlation time τ l in Jupiter dynamo model
γ and τ SV in Jupiter dynamo model τ l = τ SV · l − γ Pm = 10 Pm = 3
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