Convection-driven dynamos in the limit of rapid rotation

Transcription

1 Convection-driven dynamos in the limit of rapid rotation Michael A. Calkins Jonathan M. Aurnou (UCLA), Keith Julien (CU), Louie Long (CU), Philippe Marti (CU), Steven M. Tobias (Leeds) *Department of Physics, University of Colorado, Boulder August 15, / 20

2 action in the highly electrically conducting metallic hydrogen mantle and in the less electrically conducting region near the base of the molecular envelope (Stanley and Glatzmaier, 2010). The exact location and extent of the dynamo region is poorly constrained. Nellis (2000) argues that the dynamo can extend up to 95% of the planet radius. Consequently, the dynamo region is expected to exhibit significant changes in density and electrical conductivity, the effects of which are relatively unknown. The upcoming Juno mission will critically improve our understanding of Jupiter by resolving the magnetic field up to spherical harmonic degree 14, planet s internal structure (Connerney, private communication). Planetary Dynamos Saturn Saturn was revealed to have a magnetic field during the Pioneer 11 flyby in 1979 (Smith et al., 1980), and subsequent observations were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 in 1981 (Ness et al., 1982). The Cassini spacecraft arrived at Saturn in 2004 and is still collecting data (Dougherty et al., 2005; Burton et al., 2009). These observations have mapped the large-scale radial magnetic field, shown in Fig. 2d, and suggest that no resolved (a) Mercury (b) Ganymede -0.6 µt µt -1.5 µt µt (c) Jupiter (d) Saturn Large-scale planetary magnetic fields are ubiquitous features of the Solar System µt µt -60 µt 0 60 µt (e) Uranus (f) Neptune -100 µt µt -100 µt µt Fig. 2. Radial magnetic field at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). Active planetary magnetic fields (Schubert and Soderlund, 2011) Please cite this article in press as: Schubert, G., Soderlund, K.M. Planetary magnetic fields: Observations and models. Phys. Earth Planet. In. (2011), doi: /j.pepi / 20

3 action in the highly electrically conducting metallic hydrogen mantle and in the less electrically conducting region near the base of the molecular envelope (Stanley and Glatzmaier, 2010). The exact location and extent of the dynamo region is poorly constrained. Nellis (2000) argues that the dynamo can extend up to 95% of the planet radius. Consequently, the dynamo region is expected to exhibit significant changes in density and electrical conductivity, the effects of which are relatively unknown. The upcoming Juno mission will critically improve our understanding of Jupiter by resolving the magnetic field up to spherical harmonic degree 14, planet s internal structure (Connerney, private communication). Planetary Dynamos Saturn Saturn was revealed to have a magnetic field during the Pioneer 11 flyby in 1979 (Smith et al., 1980), and subsequent observations were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 in 1981 (Ness et al., 1982). The Cassini spacecraft arrived at Saturn in 2004 and is still collecting data (Dougherty et al., 2005; Burton et al., 2009). These observations have mapped the large-scale radial magnetic field, shown in Fig. 2d, and suggest that no resolved (a) Mercury (b) Ganymede -0.6 µt µt -1.5 µt µt (c) Jupiter (d) Saturn µt µt -60 µt 0 60 µt (e) Uranus (f) Neptune Large-scale planetary magnetic fields are ubiquitous features of the Solar System. Only Venus remains an unknown µt µt -100 µt µt Fig. 2. Radial magnetic field at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). Active planetary magnetic fields (Schubert and Soderlund, 2011) Please cite this article in press as: Schubert, G., Soderlund, K.M. Planetary magnetic fields: Observations and models. Phys. Earth Planet. In. (2011), doi: /j.pepi / 20

4 action in the highly electrically conducting metallic hydrogen mantle and in the less electrically conducting region near the base of the molecular envelope (Stanley and Glatzmaier, 2010). The exact location and extent of the dynamo region is poorly constrained. Nellis (2000) argues that the dynamo can extend up to 95% of the planet radius. Consequently, the dynamo region is expected to exhibit significant changes in density and electrical conductivity, the effects of which are relatively unknown. The upcoming Juno mission will critically improve our understanding of Jupiter by resolving the magnetic field up to spherical harmonic degree 14, planet s internal structure (Connerney, private communication). Planetary Dynamos Saturn Saturn was revealed to have a magnetic field during the Pioneer 11 flyby in 1979 (Smith et al., 1980), and subsequent observations were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 in 1981 (Ness et al., 1982). The Cassini spacecraft arrived at Saturn in 2004 and is still collecting data (Dougherty et al., 2005; Burton et al., 2009). These observations have mapped the large-scale radial magnetic field, shown in Fig. 2d, and suggest that no resolved (a) Mercury (b) Ganymede -0.6 µt µt -1.5 µt µt (c) Jupiter (d) Saturn µt µt -60 µt 0 60 µt (e) Uranus (f) Neptune Large-scale planetary magnetic fields are ubiquitous features of the Solar System. Only Venus remains an unknown. Dynamos: fields are generated by interior fluid motions µt µt -100 µt µt Fig. 2. Radial magnetic field at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). Active planetary magnetic fields (Schubert and Soderlund, 2011) Please cite this article in press as: Schubert, G., Soderlund, K.M. Planetary magnetic fields: Observations and models. Phys. Earth Planet. In. (2011), doi: /j.pepi / 20

5 Finlay et al. Earth, Planets and Space (2016) 68:112 The Geodynamo Page 13 of 18 Radial component of the geomagnetic field in 2015 (Finlay et al., 2016). The geomagnetic field has existed for 3.5 billion years. 3 / 20

6 Finlay et al. Earth, Planets and Space (2016) 68:112 The Geodynamo Page 13 of 18 Radial component of the geomagnetic field in 2015 (Finlay et al., 2016). The geomagnetic field has existed for 3.5 billion years. The field would have decayed in 10 4 years in the absence of a regenerative process. 3 / 20

7 The Geodynamo Earth has a molten iron outer core. Earth s interior (Ed Garnero, ASU) 4 / 20

8 The Geodynamo Earth has a molten iron outer core. Buoyancy-driven motions sustain the geomagnetic field. Earth s interior (Ed Garnero, ASU) 4 / 20

9 The Geodynamo Earth has a molten iron outer core. Buoyancy-driven motions sustain the geomagnetic field. Similar processes likely occur in other planets. Earth s interior (Ed Garnero, ASU) 4 / 20

10 Modeling Planetary Dynamos Numerical simulations are the primary research tool. Dynamo simulation at E = (Aubert et al., 2013) 5 / 20

11 Modeling Planetary Dynamos Numerical simulations are the primary research tool. Many studies produce magnetic fields broadly similar to the observed fields. Dynamo simulation at E = (Aubert et al., 2013) 5 / 20

12 Modeling Planetary Dynamos Dynamo simulation at E = (Aubert et al., 2013) Numerical simulations are the primary research tool. Many studies produce magnetic fields broadly similar to the observed fields. Accessing realistic parameters remains computationally challenging. 5 / 20

13 Dimensionless Parameters ρ (D t u + 2Ω u) = p + µ 1 B B + ρg + ρν 2 u Several important dimensionless parameters characterize the dynamical state of the geodynamo. 6 / 20

14 Dimensionless Parameters ρ (D t u + 2Ω u) = p + µ 1 B B + ρg + ρν 2 u Several important dimensionless parameters characterize the dynamical state of the geodynamo. The Ekman number: E = viscous forces Coriolis force = ρν 2 u 2ρΩ u = ν 2ΩD / 20

15 Dimensionless Parameters ρ (D t u + 2Ω u) = p + µ 1 B B + ρg + ρν 2 u Several important dimensionless parameters characterize the dynamical state of the geodynamo. The Ekman number: E = viscous forces Coriolis force = The Rossby number: Ro = inertia Coriolis force = ρν 2 u 2ρΩ u = ρd tu 2ρΩ u = ν 2ΩD U 2ΩD / 20

16 Dimensionless Parameters ρ (D t u + 2Ω u) = p + µ 1 B B + ρg + ρν 2 u Several important dimensionless parameters characterize the dynamical state of the geodynamo. The Ekman number: E = viscous forces Coriolis force = The Rossby number: Ro = inertia Coriolis force = The Reynolds number: Re = ρν 2 u 2ρΩ u = ρd tu 2ρΩ u = ν 2ΩD U 2ΩD 10 6 inertia viscous forces = ρd tu ρν 2 u = Ro E / 20

17 Dimensionless Parameters ~10 9 Earth s core Reynolds number, Re ~10 3 Ro ~10-6 Ro ~ 0.1 Lab DNS ~10-15 ~10-8 ~10-5 Ekman number, E 7 / 20

18 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. 8 / 20

19 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. An alternative, or complimentary, approach is the development of reduced models. 8 / 20

20 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. An alternative, or complimentary, approach is the development of reduced models. Some classic examples: Oberbeck-Boussinesq approximation: asymptotic limit that removes stiffness associated with compressibility. 8 / 20

21 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. An alternative, or complimentary, approach is the development of reduced models. Some classic examples: Oberbeck-Boussinesq approximation: asymptotic limit that removes stiffness associated with compressibility. Large Prandtl number approximation: asymptotic limit that removes stiffness associated with weak inertial effects. 8 / 20

22 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. An alternative, or complimentary, approach is the development of reduced models. Some classic examples: Oberbeck-Boussinesq approximation: asymptotic limit that removes stiffness associated with compressibility. Large Prandtl number approximation: asymptotic limit that removes stiffness associated with weak inertial effects. For both of these models: dynamically unimportant phenomena have been filtered from the governing equations, leading to significant computational savings. 8 / 20

23 Reduced Models An obvious fact: modeling rapidly rotating dynamos is difficult. An alternative, or complimentary, approach is the development of reduced models. Some classic examples: Oberbeck-Boussinesq approximation: asymptotic limit that removes stiffness associated with compressibility. Large Prandtl number approximation: asymptotic limit that removes stiffness associated with weak inertial effects. For both of these models: dynamically unimportant phenomena have been filtered from the governing equations, leading to significant computational savings. What about dynamos? 8 / 20

24 Insight from Numerical Simulations Simulations with (E, Ro) 1 are geostrophically balanced to leading order: 2ρΩ u p Forces vs. buoyancy forcing for E = 10 4 (Soderlund et al., 2012) 9 / 20

25 Insight from Numerical Simulations Simulations with (E, Ro) 1 are geostrophically balanced to leading order: 2ρΩ u p Small deviations from this balance are termed quasi-geostrophic (QG). Forces vs. buoyancy forcing for E = 10 4 (Soderlund et al., 2012) 9 / 20

26 Insight from Numerical Simulations Simulations with (E, Ro) 1 are geostrophically balanced to leading order: 2ρΩ u p Small deviations from this balance are termed quasi-geostrophic (QG). Convective motions are spatially anisotropic: Meridional plane view for E = (Marti et al., 2016) height width = H L = 1 Ro 1 9 / 20

27 Local-area Reduced Models To simplify the problem we focus on the rotating plane layer geometry. 10 / 20

28 Local-area Reduced Models To simplify the problem we focus on the rotating plane layer geometry. Childress & Soward (1972) first demonstrated dynamo action in this geometry. 10 / 20

29 Local-area Reduced Models To simplify the problem we focus on the rotating plane layer geometry. Childress & Soward (1972) first demonstrated dynamo action in this geometry. They assumed flows were just supercritical (weakly nonlinear) and P m = ν/η = O(1). 10 / 20

30 Local-area Reduced Models To simplify the problem we focus on the rotating plane layer geometry. Childress & Soward (1972) first demonstrated dynamo action in this geometry. They assumed flows were just supercritical (weakly nonlinear) and P m = ν/η = O(1). They showed rapidly rotating convection easily drives large-scale dynamos. 10 / 20

31 The Quasi-geostrophic Dynamo Model The Childress-Soward approach can be generalized to fully nonlinear flows with P m 1 (Calkins et al., 2015). 11 / 20

32 The Quasi-geostrophic Dynamo Model The Childress-Soward approach can be generalized to fully nonlinear flows with P m 1 (Calkins et al., 2015). A brief summary: Multiscale asymptotics are employed. 11 / 20

33 The Quasi-geostrophic Dynamo Model The Childress-Soward approach can be generalized to fully nonlinear flows with P m 1 (Calkins et al., 2015). A brief summary: Multiscale asymptotics are employed. Variables can be both slow (mean) and fast (fluctuating): u = u + u. 11 / 20

34 The Quasi-geostrophic Dynamo Model The Childress-Soward approach can be generalized to fully nonlinear flows with P m 1 (Calkins et al., 2015). A brief summary: Multiscale asymptotics are employed. Variables can be both slow (mean) and fast (fluctuating): u = u + u. Because Ro 1, asymptotic expansions are used, e.g. u = u 0 + Rou 1 + Ro 2 u / 20

35 The Quasi-geostrophic Dynamo Model The Childress-Soward approach can be generalized to fully nonlinear flows with P m 1 (Calkins et al., 2015). A brief summary: Multiscale asymptotics are employed. Variables can be both slow (mean) and fast (fluctuating): u = u + u. Because Ro 1, asymptotic expansions are used, e.g. u = u 0 + Rou 1 + Ro 2 u 2 + The equations are separated into mean and fluctuating components, we plug the above expansions into the governing equations and collect terms of equal magnitude. 11 / 20

36 The Quasi-geostrophic Dynamo Model At leading order we have geostrophy ẑ u 0 = p 0 12 / 20

37 The Quasi-geostrophic Dynamo Model At leading order we have geostrophy ẑ u 0 = p 0 The QG dynamics are given by D t u 0 + ẑ u 1 = p 1 + RaT 0 ẑ + MB B + 2 u / 20

38 The Quasi-geostrophic Dynamo Model At leading order we have geostrophy ẑ u 0 = p 0 The QG dynamics are given by D t u 0 + ẑ u 1 = p 1 + RaT 0 ẑ + MB B + 2 u 0. The term ẑ u 1 represents an ageostrophic effect (vortex stretching) this is required for convection to occur, and thus magnetic field to be generated. 12 / 20

39 The Quasi-geostrophic Dynamo Model At leading order we have geostrophy ẑ u 0 = p 0 The QG dynamics are given by D t u 0 + ẑ u 1 = p 1 + RaT 0 ẑ + MB B + 2 u 0. The term ẑ u 1 represents an ageostrophic effect (vortex stretching) this is required for convection to occur, and thus magnetic field to be generated. The (squared) inverse Alfvén number determines the relative sizes of the magnetic to kinetic energy densities: M = B2 ρµu 2 = E mag E kin 12 / 20

40 The Quasi-geostrophic Dynamo Model Two distinct QG models can be developed based on the magnitude of the small-scale magnetic Reynolds number: Rm = magnetic induction magnetic diffusion = UL η 13 / 20

41 The Quasi-geostrophic Dynamo Model Two distinct QG models can be developed based on the magnitude of the small-scale magnetic Reynolds number: Rm = magnetic induction magnetic diffusion = UL η Case 1: Rm 1, E mag E kin (planetary regime). 13 / 20

42 The Quasi-geostrophic Dynamo Model Two distinct QG models can be developed based on the magnitude of the small-scale magnetic Reynolds number: Rm = magnetic induction magnetic diffusion = UL η Case 1: Rm 1, E mag E kin (planetary regime). Case 2: Rm 1, E mag E kin (simulation regime). 13 / 20

43 The Quasi-geostrophic Dynamo Model Two distinct QG models can be developed based on the magnitude of the small-scale magnetic Reynolds number: Rm = magnetic induction magnetic diffusion = UL η Case 1: Rm 1, E mag E kin (planetary regime). Case 2: Rm 1, E mag E kin (simulation regime). For both cases, Rm H 1, hence large-scale dynamo action is possible. 13 / 20

44 The Quasi-geostrophic Dynamo Model Two distinct QG models can be developed based on the magnitude of the small-scale magnetic Reynolds number: Rm = magnetic induction magnetic diffusion = UL η Case 1: Rm 1, E mag E kin (planetary regime). Case 2: Rm 1, E mag E kin (simulation regime). For both cases, Rm H 1, hence large-scale dynamo action is possible. These two limits are rather distinct and lead to quite different magnetic saturation and induction mechanisms. 13 / 20

45 The Low-Rm Model In this limit the magnetic field is given by B = B + Ro 1/2 b. 14 / 20

46 The Low-Rm Model In this limit the magnetic field is given by B = B + Ro 1/2 b. The small-scale induction equation is quasi-static and shows the dynamo must be multiscale: 0 = B u + 1 P m 2 b 14 / 20

47 The Low-Rm Model In this limit the magnetic field is given by B = B + Ro 1/2 b. The small-scale induction equation is quasi-static and shows the dynamo must be multiscale: 0 = B u + 1 P m 2 b The mean magnetic field evolves on a slow timescale relative to convection (τ = Ro 3/2 t): τ B = ẑ Z E + 1 P m 2 ZB 14 / 20

48 The Low-Rm Model Some details / 20

49 The Low-Rm Model Some details... No ad-hoc assumptions about the form of E = u b are assumed: E i = αb j. 15 / 20

50 The Low-Rm Model Some details... No ad-hoc assumptions about the form of E = u b are assumed: E i = αb j. These models are asymptotic α 2 mean field models. 15 / 20

51 The Low-Rm Model Some details... No ad-hoc assumptions about the form of E = u b are assumed: E i = αb j. These models are asymptotic α 2 mean field models. α can be written in terms of the velocity field as α = P m ( w 2 xv + x v 2 w ). 15 / 20

52 Low-Rm Results: Kinematic Regime (a) (c) I In the absence of the Lorentz force, four convection regimes are observed (Julien et al., GAFD 2012): (a) cellular; (b) columnar; (c) plume; and (d) turbulence. I For the dynamo we are mainly interested in the behavior of α. (b) (d) Axial vorticity (Calkins et al., 2016) 16 / 20

53 Low-Rm Results: Kinematic Regime Z 0.4 Increasing Ra One of the main kinematic results: the structure of α depends only weakly on the convective flow regime. 0.2 Pr = α/ Pm (Calkins et al., 2016) 17 / 20

54 Low-Rm Results: Kinematic Regime One of the main kinematic results: the structure of α depends only weakly on the convective flow regime. As a result, the mean magnetic field depends only weakly on the convective flow regime. (Calkins et al., 2016) 17 / 20

55 Low-Rm Results: Kinematic Regime (Calkins et al., 2016) One of the main kinematic results: the structure of α depends only weakly on the convective flow regime. As a result, the mean magnetic field depends only weakly on the convective flow regime. These results may suggest why DNS studies can produce magnetic fields that look similar to observed planetary magnetic fields. 17 / 20

56 Low-Rm Results: Kinematic Regime (a) Axial vorticity (b) Axial current density Ra = RaE 4/3 = 100, P r = 1 (Calkins et al., 2016) Turbulent regime: inverse energy cascade organizes small-scale electromagnetic fields. 18 / 20

57 Summary The main point: asymptotic models can be developed for local-area geometries in a planetary-like parameter regime: (Ro, E) 0, Rm 1, E mag E kin 19 / 20

58 Summary The main point: asymptotic models can be developed for local-area geometries in a planetary-like parameter regime: (Ro, E) 0, Rm 1, E mag E kin Reduced model development is a complimentary approach to numerical simulations: they can access regimes beyond those of current DNS, and provide additional physical insight. 19 / 20

59 Summary The main point: asymptotic models can be developed for local-area geometries in a planetary-like parameter regime: (Ro, E) 0, Rm 1, E mag E kin Reduced model development is a complimentary approach to numerical simulations: they can access regimes beyond those of current DNS, and provide additional physical insight. Future goal: understand the link between local small-scale models and the global-scale (spherical) dynamics. 19 / 20

60 Relevant work Calkins, M.A., K. Julien, S. M. Tobias and J. M. Aurnou. A multiscale dynamo model driven by quasi-geostrophic convection. J. Fluid Mech., 780, p (2015). Calkins, M.A., K. Julien, S. M. Tobias, J.M. Aurnou and P. Marti. Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions. Phys. Rev. E, 93, (2016). Calkins, M.A., L. Long, D. Nieves, K. Julien and S. M. Tobias. Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers. Preprint: 20 / 20