Scaling laws for dynamos in rotating spherical shells and applications to planetary magnetic fields

Transcription

1 Scaling laws fo dynamos in otating spheical shells and applications to planetay magnetic fields Ulich Chistensen Max-Planck-Institute fo Sola System Reseach, Katlenbug-Lindau, Gemany In collaboation with Julien Aubet, Casten Kutzne, Pete Olson, Andeas Tilgne, Johannes Wicht

2 Outline of geodynamo models Solve equations of themal / compositional convection and magnetic induction in a otating and electically conducting spheical shell Fundamental MHD equations Some paametes not Eath-like Diffeences between models: - paamete values - bounday conditions - mode of diving convection Fluid oute coe Solid inne coe Tangent cylinde

3 Successes of geodynamo modeling Eath s adial magnetic field at CMB Dipole tilt 1 million yeas Tilt of dipole axis vs. time in dynamo model Dynamo model field at full esolution and filteed to l < 13

4 Contol paametes Name Foce balance Eath value Model values Ra* E P Pm Rayleigh numbe Ekman numbe Pandtl numbe Magnetic Pandtl # Buoyancy Retading foces Viscosity Coiolis foce Viscosity Themal diffusion 5000 x citical < 50 x citical x Viscosity Magnetic diffusion

5 Models all wong? Geodynamo models successfully epoduced the popeties of the geomagnetic field This is supising, because seveal contol paametes ae fa fom Eath values (viscosity and themal diffusivity too lage) Pessimistic view: Models give ight answe fo wong easons Optimistic assumption: Models ae aleady in egime whee diffusion plays no fist-ode ole

6 Scaling laws Use many dynamo models, coveing decent ange in contol paametes. Ty to find laws that elate the chaacteistic velocity, magnetic field stength and heat flow to the contol paametes If diffusion plays no ole, these popeties must be chaacteised by non-dimensional numbes unelated to any diffusivity

7 Contol paametes Thee diffusivities ente into the dynamo poblem: ν: viscous κ: themal λ: magnetic Paametes descibing diffusive effects: E = ν/ωd 2 Ekman numbe P = ν/κ Pandtl numbe Pm = ν/λ magnetic Pandtl numbe Contol paamete independent of any diffusivity: Ra* = αgδt/(ω 2 D) modified Rayleigh numbe Choose basic scales fo time (Ω -1 ) and magnetic field ([ρµ] 1/2 ΩD) that do not depend on diffusivities

8 Boussinesq equations B B T Ra* u E P u e u u t u o z v v + + = ) ( 2 ) ( 2. Inetia Coiolis Viscosity Buoyancy Loentz ) ( 2 P E E T E T u t T = = + κ κ Advection Diffusion ) ( 2 Pm E E B E u B B u t B = + = + λ λ Advection Induction Diffusion

9 Output paametes Ro = U/(DΩ) Rossby numbe non-dim velocity Lo =B/[ρµ] ½ ΩD Loentz numbe non-dim field stength Nu* = γq adv /(ρc p ΔTΩD 3 ) Modified Nusselt numbe Relations to conventional paametes Rayleigh numbe Ra Ra* = Ra E 2 /P Nusselt numbe Nu Nu* = (Nu-1) E/P Elsasse numbe Λ Lo = (Λ E/Pm) 1/2 magn. Reynolds # Rm Ro = Rm E/Pm

10 Data basis and case selection Total of 162 model cases (127 dynamos, 35 failed dynamos) Diven by fixed ΔT, no-slip boundaies All paametes vaied by at least two odes of magnitude Each un fo at least 50 advection times (some much longe) Symmety in longitude assumed fo E 10-5 Selection citeia fo scaling analysis: Self-sustained dynamo Dipole-dominated magnetic field (f dip > 0.35) E Fully developed convection (Nusselt-# > 2.0) 70 model cases pass these citeia

11 Dynamo egimes (at P=1) Powe Hamonic degee Typical magnetic specta on oute bounday As the Ekman numbe is loweed, dipola dynamos occupy a boade egion and ae found at lowe magnetic Pandtl #

12 Minimum magnetic Pandtl numbe Pm min 450 E 3/4 Eath values: E Pm m 2x10-8

13 Citical magnetic Reynolds numbe Field decay Self-sustained dynamos The mimimum magnetic Reynolds numbe fo self-sustained dynamos is 40, iespective of the value of the magnetic Pandtl numbe.

14 Dynamo egimes (0.1 P 10) Inetial vs. Coiolis foce: Relative dipole stength dipola Pm colo-coded Pm>2.5 Pm<0.5 nondipola Local Rossby numbe Ro L calculated with mean length scale l in the kinetic enegy spectum Ro L = U/(lΩ) When inetia dominates the dipola egime beaks down

15 Flux-based Rayleigh numbe Modified Rayleigh # based on buoyancy flux: Ra Q * = γg o Q buoy / (ρc p Ω 3 D 2 ) Ra Q *= Ra*Nu* = Ra (Nu-1)E 3 P -2 To vey good appoximation Ra Q * is equivalent to the powe geneated by buoyancy foces γ = (1-η 2 )/4πη : geomety facto

16 Scaling of Nusselt numbe Use of modified diffusionless paametes allows to collapse the data and expess the dependence by a single powe-law. Compaed to non-otating convection, the exponent is vey lage ( 0.5).

17 Dimensional heat flow Fo an exponent of 0.5: Q adv ~ γαgρc p D2 ΔT 2 / Ω Advected heat flow is independent of themal conductivity. Eath s coe: Q adv = 2 TW ΔT supeadiab 1 mk Requiement fo validity is pobably that the themal bounday laye thickness is lage than the Ekman laye thickness, which in tems of non-dimensional paametes equies Ra* Q < 400 E P -2. Satisfied in the numeical models and pehaps fo themal convection in the Eath s coe.

18 Powe Velocity: Rossby numbe Velocity Ro ~ Ra Q * 2/5 Dependence on magnetic Pandtl numbe?

19 Velocity scaling II Ro ~ Ra* Q 0.43 Pm Two-paamete egession educesmisfitbyfacto2.5. But is dependence on the magnetic Pandtl numbe definitely equied?

20 Scaling ohmic dissipation time τ diss = Magnetic enegy / Ohmic dissipation τ diss ~ Rm -1 τ diss ~ Rm -7/6 Pm 1/6 Chistensen & Tilgne, 2004 Which law is ight?

21 Kalsuhe laboatoy dynamo Dynamo onset Magnetic Pandtl numbe 1 in numeical models 10-5 fo liquid sodium

22 Scaling ohmic dissipation time Kalsuhe laboatoy dynamo τ diss ~ Rm -1 τ diss ~ Rm -7/6 Pm 1/6 Bette ageement with the simple scaling law

23 Velocity Scaling: stay simple Let us assume that in geneal the magnetic Pandtl numbe has no influence (at least in the limit Pm << 1) Ro = 0.85 Ra* Q 2/5

24 Rayleigh numbe of the coe Coe velocity estimated fom secula vaiation: Lage-scale flow ~ 0.5 mm/sec Total ms-flow ~ 1 mm/sec Ro ~ 6 x 10-6 Coe Rayleigh numbe Ra* Q ~ 3 x The conventional Rayleigh numbe is (5000xRa c )

25 Light element flux and inne coe gowth ate Ra* Q ~ Q buoy ~ kg s -1 If convection is diven entiely by compositional flux associated with inne coe gowth, the gowth ate is obtained as dr ic /dt = Q buoy / (4π i2 Δρ ic ). dr ic /dt ~ 0.1 mm/y Implication: Inne coe is old: 3.5 ± 1.5 Gy

26 What contols the stength of the magnetic field? Magnetostophic balance often associated with an Elsasse numbe Λ = B 2 /μηρω ~ O(1) In the numeical models, the Elsasse numbe vaies widely. Λ Foce balance not magnetostophic, o Λ not good measue. Altenative scaling: Magnetic field stength based on available powe?

27 Powe-contolled field stength Powe diving convection ~ Ra Q * Magnetic enegy density ~ 1/2 Lo 2 Ratio magn. enegy / dissipation Pediction: = τ diss If Ro ~ Ra Q * 2/5 and τ diss ~ Rm -1 ~ Pm (Ro E) -1 then Lo / f ohm ~ Ra* Q 0.3 whee f ohm is the elative faction of enegy dissipated by ohmic losses (30 85% in the models)

28 Magnetic Field Scaling Magnetic field stength Powe Lo ~ Ra* Q 0.34 Empiical fit is close to pedicted dependence

29 Magnetic Field Scaling II Assume f ohm 1 in the coe In dimensional fom the magnetic field stength is B ~ µ 1/2 ρ 1/6 (gq buoy /D) 1/3 B independent of conductivity and otation ate

30 Stength of coe magnetic field With the estimated buoyancy flux of kg/s the pedicted magnetic field stength in the coe is B 1.2 mt Compae to: Obseved field at CMB: 0.39 mt (l < 13) B s 0.4 mt inside coe fom tosional oscillations (Zatman & Bloxham, 1997)

31 How obust ae the scaling laws? Tests: Exclude cases with E 10-4 fom fit Results vitually unchanged Exclude cases with Ro l > 0.05 (high inetia) fom fit Results vitually unchanged Attempt geneal least-squaes fit of the fom: Y = A Ra Q * α Pm β E γ P δ whee Y stands fo Nu*, Ro, o Lo/ f ohm Exponents fo E and P vey small (<0.03)

32 Othe planets: Jupite Fo the obseved excess heat flow (5.4 W/m 2 ) and easonable estimates fo othe elevant paametes, the magnetic field in the dynamo egion is B 8 mt. This is in ageement with Jupite s suface field stength oughly 10 times the Eath value. Suface field Metallic hydogen

33 Mecuy Suface field stength 1/100 Eath value. Cannot be explained by low buoyancy flux, because it would imply magnetic Reynold numbe below citical.

34 Conclusions Numeical dynamo models epoduce obseved popeties of the geomagnetic field, even though viscous and themal diffusivity ae fa too lage. Scaling laws that ae independent of diffusivities fit the model esults well and lead to easonable pedictions fo the Eath. This suppots the view that the models ae aleady in a egime simila to that of the Eath s coe. The ange of validity of laws must be exploed. Is thee a dependence on the magnetic Pandtl numbe? Laboatoy dynamo expeiments will help to decide.

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36 Scaling of local Rossby numbe Decent fit possible, but involves all fou contol paametes Ro L ~ Ra Q *1/2 E -1/3 P 1/5 Pm -1/5 Coe Pedicted Eath value is Ro L , vey close to the tansition point between dipola and nondipola dynamos.

37 Enstophy balance ω ω ω ω ω ω ω ω v v = ] ) [( 2 ] [ * 2 2 B B T Ra E z u u o t Inetia Coiolis Viscosity Buoyancy Loentz Voticity ω Enstophy = ω ω Enegy of voticity

38 Souces & sinks of enstophy Buoyancy is main souce and Loentz foce and viscosity is main sink

39 Enstophy I C B LV I C B LV I C B L V budget I C B LV I C B LV I C B L V Coiolis ~ buoyancy Loentz vaiable Inetia lage in nondipola case