Inertial effects on thermochemically driven convection and hydromagnetic dynamos in a spherical shell

Inertial effects on thermochemically driven convection and hydromagnetic dynamos in a spherical... Summary We investigate the thermochemical convection and hydromagnetic dynamos in a spherical shell using the so-called codensity formulation with different buoyancy sources: the secular cooling from the mantle, the buoyancy sources due to the solidification at the inner core boundary and the combination of the two sources. Numerical simulations of the fully non-linear problem are performed using the PARODY code. In the thermochemical regime, we find that when the Prandtl numbers are lower than Ekman numbers, inertial convection is preferred, while the large-scale columnar convection is preferred otherwise. Unlike the large-scale convection, the inertial convection is found to be almost independent of the nature of driving buoyancy source. Moreover, the codensity field evolves to a new, radially symmetric stationary state. At the Ekman numbers much smaller than the Prandtl numbers, we have obtained the westward equatorial zonal flow in the chemically driven regime, while for the other cases zonal flows are eastward near the equator. In the dynamo regime, inertial convection is preferred when the Prandtl numbers are lower than Ekman numbers and the generated dipolar magnetic fields oscillate from the polar region to the mid-latitudes and back. In this case, the generated magnetic fields are independent of the type of buoyancy source. At the Prandtl numbers greater than Ekman numbers, both dipolar and hemispherical dynamos are found. Composition and structure of the core, Dynamo: theories and simulations, Non-linear differential equations, Numerical modelling 1 INTRODUCTION Convective dynamics of the liquid cores of the Earth and some other planets is most commonly believed to be driven by the two different types of buoyancy sources (Braginsky & Roberts 1995; Christensen & Wicht 2008; Busse 2002). One is thermal buoyancy associated with a superadiabatic radial temperature gradient in the liquid core (i.e. the radial temperature gradient exceeding the adiabatic temperature gradient). The other is compositional (or chemical) buoyancy caused by a chemical differentiation of the liquid alloy constituents induced by solidification. The two sources of buoyancy, thermal and compositional, have received much attention in recent years in numerical simulations of convection-driven dynamics in rapidly rotating self-gravitating spherical fluid shells (Glatzmaier & Roberts 1997; Busse 2000; Jones 2011). The joint influence of the two driving buoyancy sources on the convective flow patterns, and the average and global physical characteristics has been investigated by Breuer et al. (2010) and Trümper et al. (2012) in a model of double-diffusive convection in a rotating spherical geometry. Both thermal and compositional fields are consistently treated as independently diffusing quantities with a mismatch in the molecular values of thermal and compositional diffusivities, in contrast to a number of previous conventional treatments in which the two buoyancy effects are treated through a single variable, called codensity, based on the assumption of equal diffusivities due to turbulent mixing (Braginsky & Roberts 1995; Lister & Buffett 1995; Shearer & Roberts 1997; Aubert et al. 2008; Christensen & Wicht 2008). Breuer et al. (2010) explored the relative influence of the two buoyancy sources for a single moderate value of the Ekman number of 10−3 and observed a considerable variation in the spatial distribution of axisymmetric zonal flow and helicity in their numerical simulations. In particular, they found that the flow changed from being nearly geostrophic for purely thermal forcing to being rather non-geostrophic and of much smaller spatial scale for purely compositional forcing. In the latter case, the zonal flows and helicity were dominated by a polar compositional plume. They concluded that these features could not be identified if the conventional codensity formulation (Braginsky & Roberts 1995) were adopted. In a related study at a lower Ekman number of 10−4, Trümper et al. (2012) similarly observed the coexistence of nearly geostrophic flow structures evoked by the thermal driving and high-Prandtl-number effects like chemical wind. Notably, the distinct features of purely thermal convection persist in the predominantly compositional regime with the thermal forcing in a subcritical regime of purely thermal convection. A model of double-diffusive convection to investigate the regime of a deep dynamo below a stably stratified layer in Mercury’s core has been employed by Manglik et al. (2010). Convection was found to penetrate to the upper, statically stable layer in the form of finger convection, enhancing the poloidal magnetic field in the initially stable layer and outside the core. A dynamo model with double-diffusive convection tuned for the conditions of Earth has been numerically studied by Takahashi (2014). The structure of the magnetic field is influenced by the thermal-to-compositional forcing ratio, with a dipolar magnetic field sustained for the forcing ratio less than 0.6 and non-dipolar field with helicity reduction otherwise. Morphological changes in the dynamo-generated magnetic field due to a presence of a thin layer of statically stable fluid at the core–mantle boundary have been recently reported by Takahashi (2016). We perform numerical simulations to elucidate which system parameters are most vital for the slow convection waves to appear in a codensity model of Boussinesq convection and hydromagnetic dynamos in a rotating spherical shell. The model and governing equations are presented in Section 2 and numerical results in Section 3. Conclusions are given in Section 4. 2 THE MODEL AND GOVERNING EQUATIONS We consider the thermochemical convection of an electrically conducting, incompressible fluid spherical shell (ri < r < ro) rotating with the angular velocity Ω. Evolution of the velocity $$\bf V$$, the magnetic field $$\bf B$$ and the codensity field C is described by the system of dimensionless equations:   \begin{eqnarray} &&{E \left( \frac{\partial {\bf V}}{\partial t} + \left( {\bf V}\cdot \nabla \right){\bf V} - \nabla ^2{\bf V} \right)+ 2 {\bf 1}_z\times {\bf V} + \nabla P}\nonumber\\ &=& R_a \frac{{\bf r}}{r_o} C + \frac{1}{P_{m}} \left( \nabla \times {\bf B} \right)\times {\bf B}, \end{eqnarray} (1)  \begin{eqnarray} \frac{\partial {\bf B}}{\partial t} = \nabla \times \left( {\bf V}\times {\bf B} \right) + \frac{1}{P_{m}} \nabla ^2{\bf B}, \end{eqnarray} (2)  \begin{eqnarray} \frac{\partial C}{\partial t}+\left({\bf V}\cdot \nabla \right) C = \frac{1}{P_{r}} \nabla ^2 C + S_{T/{\Xi }}, \end{eqnarray} (3)  \begin{eqnarray} \nabla \cdot {\bf V} = 0, \nabla \cdot {\bf B} = 0. \end{eqnarray} (4)Within the Boussinesq approximation, the deviation temperature field T and the light-element mass-fraction field Ξ are defined with respect to a reference state characterized by isentropic temperature and well-mixed mass fraction. Both buoyancy effects are used to define the codensity (density anomaly) field (Braginsky & Roberts 1995) C such that its dimensional form is   \begin{eqnarray} C = \alpha _T\rho T + \alpha _{\Xi }\rho \Xi , \end{eqnarray} (5)where αT and αΞ are the coefficients of volume expansion and ρ is the fluid density. We assume that the diffusivities of heat and light elements are the same, denoted κ, allowing for a single transport equation for the codensity field. We use the shell gap, L = ro − ri, as a length scale. The aspect ratio of the inner core, ri/ro, is set to 0.35, which is the current value for the Earth. We denote (r,  θ,  φ) the spherical coordinate system and 1z is the unit vector in the z-direction. The time is scaled by L2/ν, velocity by ν/L, magnetic induction by (ρμηΩ)1/2 and pressure, P, by ρν2/L2. The dimensionless parameters appearing in eqs (1)–(4) are the Prandtl number, Pr = ν/κ, the magnetic Prandtl number, Pm = ν/η and the Ekman number, E = ν/ΩL2, where ν is the kinematic viscosity, μ is the magnetic permeability and η is the magnetic diffusivity. The codensity is scaled with F/(4πνL), where the total buoyancy flux F is defined below. The codensity boundary conditions are treated in terms of codensity fluxes. At the inner-core boundary, the release of latent heat and light elements are the chemical sources of convection—they generate a positive mass anomaly flux Fi, which is considered uniform and imposed on by global core thermodynamics on the long timescale (Aubert et al. 2009). The dimensional form of Fi is   \begin{eqnarray} F_i = - \int _{S_i} \kappa \nabla C\cdot \textrm{d}\boldsymbol{S}_{\textrm{in}}, \end{eqnarray} (6)where Si is the inner boundary surface and $$\textrm{d}\boldsymbol{S}_{\textrm{in}}$$ is the surface element oriented into the outer core. Similarly, at the outer boundary, the mass anomaly flux imposed by the mantle corresponds to the heat flux carried out from the outer core, thus representing the secular cooling of the outer core. There are no chemical contributions to Fo at the outer boundary, so the light-element mass flux is set to zero there. The dimensional form of Fo is   \begin{eqnarray} F_o = - \int _{S_o} \kappa \nabla C\cdot \textrm{d}\boldsymbol{S}_{\textrm{out}}, \end{eqnarray} (7)where So is the outer boundary surface and $$\textrm{d}\boldsymbol{S}_{\textrm{out}}$$ is the surface element oriented away from the outer core. Note that Fi corresponds to the flux of the codensity field into the outer core at the inner boundary, while Fo corresponds to the flux of the codensity field from the outer core at the outer boundary. It is worth noting that the positive flux of the codensity field generates positive buoyancy. Thus, it is the positive values of Fo that generate negative buoyancy at the outer boundary and thus promote convection. Negative values of Fo result in a stably stratified layer at the top of the outer core (Aubert et al. 2009). In general, Fo can be either positive or negative, but the total flux of convection-promoting buoyancy, F = Fi + Fo, must be positive in order for convection to occur. The Rayleigh number appearing in eq. (1) is defined as   \begin{eqnarray} R_a = \frac{F g_{0}}{4\pi \rho \nu L}, \end{eqnarray} (8)where g0 is the gravity acceleration at r = ro. We let the mass anomaly fluxes Fi and Fo be independent. As in Aubert et al. (2009), we introduce the buoyancy fraction associated with the inner boundary, fi = Fi/F. The volumetric correction term ST/Ξ in eq. (3) is defined so as the average codensity in the spherical shell does not change with time (Braginsky & Roberts 1995; Kutzner & Christensen 2002) and has the following dimensionless expression   \begin{eqnarray} S_{T/{\Xi }} = 3\frac{1 - 2f_i}{r^{3}_{o} - r^{3}_{i}}. \end{eqnarray} (9)The volumetric correction term ST/Ξ is composed of two components. The first component is given by the reference state (Davies & Gubbins 2011): its thermal part contains the sink of potential thermal buoyancy due to the adiabatic gradient and the sources of thermal buoyancy representing the secular change of the reference state with time (the surface cooling rate at the inner boundary, the latent-heat release and the radiogenic heating); its compositional part reflects the release of the lighter component at the inner boundary due to freezing of the inner core. The second component of ST/Ξ contains the remaining codensity sources, which are defined such that eq. (9) holds. Note that the effective range of fi is from 0 to ∞. In the situation with fi = 0 only the secular cooling at the outer boundary (Fo > 0) contributes to the convection; when fi < 1/2, the secular cooling dominates the chemical sources (the total codensity outflow from the outer core, Fo − Fi, is positive); when fi = 1/2, the chemical sources and the secular cooling equally contribute to convection (zero total codensity outflux); when fi > 1/2, the chemical sources dominate (negative total codensity outflux); the situation with fi = 1 corresponds to the absence of secular cooling (the heat flux at the outer boundary is adiabatic) and the convection is purely chemically driven. The cases with fi > 1 are not addressed in this study; they correspond to the cases with dominant chemical convection where the heat flow at the outer boundary is below the adiabat, that is, Fo < 0 (Aubert et al. 2009). Eqs (1)–(4) are closed by the non-penetrating and no-slip boundary conditions for the velocity field at the rigid surfaces. The outer boundary is electrically insulating, so as the magnetic field at this boundary matches with an appropriate potential field in the exterior that implies no external field sources. The inner boundary is electrically conducting, with the electric conductivity of the outer and inner cores assumed to be the same. 3 NUMERICAL RESULTS The model equations are integrated using the numerical dynamo code PARODY developed by Dormy (1997) and later used by Dormy et al. (1998), Aubert et al. (2008) and Raynaud & Dormy (2013). The code allows the integration of the non-dimensional Boussinesq equations for time-dependent thermochemical convection in a rotating spherical shell filled with an electrically conducting fluid. It is a semi-spectral code based on spherical harmonics decomposition in the lateral direction and second-order finite-difference discretization in the radial direction, which makes it suitable for parallel computation on distributed memory clusters (Dormy 1997; Dormy et al. 1998; Aubert et al. 2008; Raynaud & Dormy 2013). The usual toroidal–poloidal decomposition for velocity and magnetic fields is applied. The time integration uses the implicit second-order Crank–Nicolson scheme. Parallelization is carried out using the message-passing interface. The Matlab routines developed by Aubert et al. (2008) are used to process the numerical results. 3.1 Non-magnetic convection In the purely hydrodynamic case, we consider four distinct parametric regimes (regimes A–D, see Table 1 for full description). Regime A corresponds to the situation in which Pr ≪ E ∼ 1 and convection is expected to take the form of inertial oscillations. Regimes B and C describe the situation in which E ≪ Pr ∼ 1. These two regimes are distinguished by the value of the Rayleigh number Ra (see Table 1). Finally, regime D corresponds to E ≪ 1 ≪ Pr. To assess the effect of varying the relative contributions of chemical and secular cooling buoyancy sources, we consider the following values of fi in each regime: fi = 0 (secular cooling only), fi = 0.5 (chemical sources and secular cooling contribute equally), fi = 0.8 (chemical sources dominate) and fi = 1 (chemical sources only). Recall that fi indicates the relative contribution of chemical buoyancy forcing. In all regimes, computations were initiated with no flow field and the codensity field given by   \begin{eqnarray} C(r) = C_i + P_r\left(f_i-\frac{1}{3}S_{T/{\Xi }}\right)\left(\frac{1}{r}-\frac{1}{r_i}\right) - \frac{1}{6}Pr S_{T/{\Xi }}(r^2-r_i^2),\!\!\!\!\!\!\!\!\nonumber\\ \end{eqnarray} (10)which corresponds to the steady solution of eq. (3) with Ci prescribed at r = ri. The Rayleigh number scaled with the critical Rayleigh number for the onset of convection, $$R^c_{a}$$, is set to 2 in the case of thermally driven convection at Pr = 1. For Pr = 10−4, the supercriticality of convection is more pronounced than in the case Pr = 1 ($$R_a/R^c_{a}>2$$), while for Pr = 5, it is less pronounced ($$R_a/R^c_{a}<2$$). At Pr = 10−4, the thermal diffusion processes dominate over the viscous ones, while at Pr = 5, the viscous diffusion processes dominate over the thermal ones. At Pr = 1, both thermal and viscous diffusion processes have the same influence on the convection. In Fig. 1, we show the evolution of kinetic energy, Ek, in all regimes considered. In regime A, the kinetic energy is almost independent of the driving mode. In other regimes, the values of kinetic energy corresponding to the convection driven by secular cooling are significantly smaller than those corresponding to the cases with contribution from the chemical sources. In regimes with non-zero fi, the changes in the relative contribution of chemical sources affect the kinetic energies most significantly in case D, that is, when the viscous dissipation is large. In Table 2 (columns 3 and 4), we show the time-averaged values of the kinetic energy together with the ratio of the time-averaged toroidal component of the kinetic energy to the time-averaged total kinetic energy. Note that in all cases, the dominant part of the kinetic energy always corresponds to the toroidal component of the flow. Figure 1. View largeDownload slide Temporal evolution of the kinetic energy in regimes A, B, C and D (non-magnetic convection). Figure 1. View largeDownload slide Temporal evolution of the kinetic energy in regimes A, B, C and D (non-magnetic convection). Table 1. Specification of parametric regimes considered in non-magnetic convection.   E  Pr  Ra  A  10−2  10−4  50  B  10−4  1  103  C  10−3  1  200  D  10−4  5  103    E  Pr  Ra  A  10−2  10−4  50  B  10−4  1  103  C  10−3  1  200  D  10−4  5  103  View Large Table 2. The time-averaged kinetic/magnetic energy, $$\bar{E}_{k/m}$$, and the scaled toroidal component of the kinetic/magnetic energy, $$\bar{E}_{k,T/m,T}$$, in each regime. The superscript (H) denotes the non-magnetic convection and the superscript (M) denotes the dynamo cases. The time average is taken over 4 ≤ t ≤ 5. The last two columns show the values of the magnetic Reynolds number, Rm, and the Elsasser number, Λ, in the dynamo cases.     $$\bar{E}_k^{(H)}$$  $$\bar{E}_{k,T}^{(H)}/\bar{E}_k^{(H)}$$  $$\bar{E}_k^{(M)}$$  $$\bar{E}_{k,T}^{(M)}/\bar{E}_k^{(M)}$$  $$\bar{E}_m^{(M)}$$  $$\bar{E}_{m,T}^{(M)}/\bar{E}_m^{(M)}$$  Rm  Λ  A  fi = 0  6.84  0.95  7.3  ≈1  0.2  0.97  613  0.6    fi = 0.5  6.84  0.95  6.8  0.98  10.9  0.89  592  35    fi = 0.8  6.84  0.95  4.4  0.99  18.8  0.84  455  60    fi = 1  6.84  0.95  7  0.98  12.7  0.87  605  41  B  fi = 0  600  0.79  482  0.77  5360  0.57  77  2.7    fi = 0.5  1979  0.84  1813  0.84  409  0.74  150  0.2    fi = 0.8  2966  0.84  1443  0.80  4636  0.69  134  2.3    fi = 1  2628  0.74  1766  0.80  7775  0.63  148  3.9  C  fi = 0  18  0.82  18.4  0.82  0.1  0.60  61  0.002    fi = 0.5  117  0.85  43.2  0.76  564  0.76  93  11    fi = 0.8  122  0.82  56.9  0.76  765  0.64  107  15    fi = 1  120  0.78  75  0.74  713  0.62  122  14  D  fi = 0  212  0.72  137  0.71  3286  0.54  166  66    fi = 0.5  336  0.74  368  0.78  195  0.77  271  3.9    fi = 0.8  423  0.74  427  0.75  38  0.65  292  0.8    fi = 1  608  0.75  426  0.76  2368  0.51  291  47      $$\bar{E}_k^{(H)}$$  $$\bar{E}_{k,T}^{(H)}/\bar{E}_k^{(H)}$$  $$\bar{E}_k^{(M)}$$  $$\bar{E}_{k,T}^{(M)}/\bar{E}_k^{(M)}$$  $$\bar{E}_m^{(M)}$$  $$\bar{E}_{m,T}^{(M)}/\bar{E}_m^{(M)}$$  Rm  Λ  A  fi = 0  6.84  0.95  7.3  ≈1  0.2  0.97  613  0.6    fi = 0.5  6.84  0.95  6.8  0.98  10.9  0.89  592  35    fi = 0.8  6.84  0.95  4.4  0.99  18.8  0.84  455  60    fi = 1  6.84  0.95  7  0.98  12.7  0.87  605  41  B  fi = 0  600  0.79  482  0.77  5360  0.57  77  2.7    fi = 0.5  1979  0.84  1813  0.84  409  0.74  150  0.2    fi = 0.8  2966  0.84  1443  0.80  4636  0.69  134  2.3    fi = 1  2628  0.74  1766  0.80  7775  0.63  148  3.9  C  fi = 0  18  0.82  18.4  0.82  0.1  0.60  61  0.002    fi = 0.5  117  0.85  43.2  0.76  564  0.76  93  11    fi = 0.8  122  0.82  56.9  0.76  765  0.64  107  15    fi = 1  120  0.78  75  0.74  713  0.62  122  14  D  fi = 0  212  0.72  137  0.71  3286  0.54  166  66    fi = 0.5  336  0.74  368  0.78  195  0.77  271  3.9    fi = 0.8  423  0.74  427  0.75  38  0.65  292  0.8    fi = 1  608  0.75  426  0.76  2368  0.51  291  47  View Large The numerical results for all the investigated cases are presented in Figs 2–5. The snapshots correspond to the dimensionless time t = 5. The results for regime A are shown in Fig. 2. In this case, inertial convection is preferred. The only difference between the driving modes is in the codensity fields: when fi = 0, the region of positive codensity is thicker than in the cases with positive fi. With fi > 0, the differences between the condensity profiles are only slight. The independence of the codensity field on the driving mode when fi > 0 is consistent with eq. (3), in which the diffusive term dominates the other terms when Pr ≪ 1 and the source term ST/Ξ becomes negligible. Note that the kinetic energy in regime A is very small in comparison with regimes B, C and D. An interesting feature is the ‘breathing’ behaviour of the toroidal component of the flow field (see A-NONMAG.gif in the supplementary data, where we show the meridional section of the toroidal scalar, VT, for the case fi = 0). Another interesting feature of the numerical results is that the codensity evolves to a radially symmetric stationary state governed by   \begin{eqnarray} \nabla ^2 C = - P_r S_{T/{\Xi }} + P_r V_r\frac{\partial C}{\partial r}. \end{eqnarray} (11)with fi entering via the source term ST/Ξ as well as the boundary conditions $$\partial C/\partial r|_{r_i}=-P_r f_i/r_i^2$$ and $$\partial C/\partial r|_{r_o}=-P_r f_o/r_o^2$$ (the dimensionless versions of eqs 6 and 7). We note that small values of Pr make the stationary codensity field almost insensitive to variations in fi. Figure 2. View largeDownload slide Equatorial sections of the codensity, C, equatorial sections of the radial components of velocity, Vr, meridional sections of the poloidal scalars, VP and meridional sections of the toroidal scalars, VT, (from left to right) in regime A with fi = 0, 0.5, 0.8 and 1 (from top to bottom). Red (blue) colours indicate positive (negative) values. The snapshots are taken at t = 5. Figure 2. View largeDownload slide Equatorial sections of the codensity, C, equatorial sections of the radial components of velocity, Vr, meridional sections of the poloidal scalars, VP and meridional sections of the toroidal scalars, VT, (from left to right) in regime A with fi = 0, 0.5, 0.8 and 1 (from top to bottom). Red (blue) colours indicate positive (negative) values. The snapshots are taken at t = 5. Figure 3. View largeDownload slide Same as in Fig. 2, but for regime B. Figure 3. View largeDownload slide Same as in Fig. 2, but for regime B. Figure 4. View largeDownload slide Same as in Fig. 2, but for regime C. Figure 4. View largeDownload slide Same as in Fig. 2, but for regime C. Figure 5. View largeDownload slide Same as in Fig. 2, but for regime D. Figure 5. View largeDownload slide Same as in Fig. 2, but for regime D. The results for regime B are presented in Fig. 3. In this case, large-scale columnar convection is developed in the whole volume. The spiralling cross-section of the columns is a dominant feature at moderate Prandtl numbers, which is observable for all driving modes. The Reynolds stresses, created by the spiralling character of the convection, generate a differential rotation (Zhang 1994; Busse & Simitev 2007; Šimkanin et al. 2010). The differential rotation is probably the cause of some irregularities in the spiralling structure of columnar convection, called ‘active’ and ‘quiet’ zones in Busse & Simitev (2007) and Šimkanin et al. (2010). The poloidal flow, driven by the thermal and compositional buoyancy forces, is transformed to the zonal flow via the Coriolis force and the Reynolds stresses present in the force balance [cf. Takahashi (2014)]. Looking at the fourth column in Fig. 3, we see that the zonal flow is eastward at the equator in all the investigated cases except in the purely chemically driven convection (fi = 1), in which case it is westward. Usually, the Boussinesq convection at moderate Rayleigh numbers shows a pattern with a prograde (eastward) zonal flow near the equator and retrograde zonal flow near the poles. The results for regime C are presented in Fig. 4. At fi = 1, the large-scale columnar convection with strong spiralling nature is developed in the whole volume and partially in the case with  fi = 0.8. In the cases with fi = 0 and 0.5, the convection is not developed in the whole volume, but is concentrated to a small volume in one hemisphere. The results for regime D are presented in Fig. 5. The large-scale columnar convection is developed in the whole volume and the spiralling cross-section of the columns is still visible, although it is not so dominant compared with regime B. As Pr increases, the influence of the differential rotation, which has dominated the evolution of convection columns at Pr of unit order and smaller, gradually diminishes (Busse & Simitev 2007). 3.2 Hydromagnetic dynamos In order to study the effects of magnetic field on convection, we study the same four parametric regimes as in the previous subsection, with the magnetic Prandtl number Pm as an additional dimensionless parameter. The values of Pm used in each regime are given in Table 3. Computations were initiated with no flow field and a strong dipole-dominated field of magnitude $$B\sim \mathcal {O}(1)$$. Temporal evolution of kinetic and magnetic energies in each regime is shown in Fig. 6. Table 2 (columns 5–8) shows the time-averaged kinetic/magnetic energy. Also shown is the scaled toroidal component for both fields and the magnetic Reynolds and Elsasser numbers. We see that the kinetic/magnetic energy is concentrated in the toroidal component of the respective field, indicating that the geostrophic flows and the toroidal magnetic fields predominate in all regimes. One interpretation of the Elsasser number is that it is the ratio of the Joule damping time to the period of rotation. Small values of Λ identified in regimes A and C at fi = 0 thus indicate relatively strong Joule damping, as confirmed also by the evolution of the magnetic energy in these regimes shown in Fig. 6. Figure 6. View largeDownload slide Temporal evolution of kinetic and magnetic energies, Ek and Em, in regimes A, B, C and D (magnetic convection). Figure 6. View largeDownload slide Temporal evolution of kinetic and magnetic energies, Ek and Em, in regimes A, B, C and D (magnetic convection). Table 3. Specification of parametric regimes considered in magnetic convection.   E  Pr  Ra  Pm  A  10−2  10−4  50  160  B  10−4  1  103  2.5  C  10−3  1  200  10  D  10−4  5  103  2.5    E  Pr  Ra  Pm  A  10−2  10−4  50  160  B  10−4  1  103  2.5  C  10−3  1  200  10  D  10−4  5  103  2.5  View Large A typical spatial distribution of the radial magnetic field component, Br, is displayed in Fig. 7. The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. The generated magnetic field is dipolar in all regimes except regime C where the thermochemically driven dynamos are hemispherical. In regime A, the magnetic flux spots are concentrated in the polar regions, in which case the magnetic fields oscillate from polar regions to the mid-latitudes, but remain mostly in the polar regions. The temporal evolution of the radial magnetic field in regime A can be found in the files A-DYNAMO-fi0.gif, A-DYNAMO-fi05.gif, A-DYNAMO-fi08.gif and A_DYNAMO-fi1.gif in the supplementary data. Figure 7. View largeDownload slide Spatial distribution (Hammer projection) of radial magnetic field component, Br, at r = ro in regimes A, B, C and D (from left to right) for thermochemically driven dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 7. View largeDownload slide Spatial distribution (Hammer projection) of radial magnetic field component, Br, at r = ro in regimes A, B, C and D (from left to right) for thermochemically driven dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Regimes B and D are qualitatively similar to the results presented by Šimkanin and Hejda (2011, 2013) and Šoltis and Šimkanin (2014). The results for regime A are shown in Fig. 8 and those for regime C in Fig. 9. In regime A, convection takes the form of inertial oscillations similar to the non-magnetic case A. In regime C, the large-scale columnar convection with spiralling pattern is developed in the whole volume in the case of purely chemically driven dynamos, while convection is confined to a single hemisphere in the dynamos forced by the combined chemical and secular-cooling sources. Even though the dynamos at fi = 0.5 and 0.8 are hemispherical, a further increase in Ra can be expected to result in the dynamos of dipolar nature. Figure 8. View largeDownload slide Equatorial sections of the radial magnetic field and velocity components, Br and Vr, respectively; meridional sections of the poloidal and toroidal scalars for the magnetic field, BP and BT, respectively, and the meridional sections of the poloidal and toroidal scalars for the velocity, VP and VT, respectively (from left to right), in regime A for dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 8. View largeDownload slide Equatorial sections of the radial magnetic field and velocity components, Br and Vr, respectively; meridional sections of the poloidal and toroidal scalars for the magnetic field, BP and BT, respectively, and the meridional sections of the poloidal and toroidal scalars for the velocity, VP and VT, respectively (from left to right), in regime A for dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 9. View largeDownload slide Same as in Fig. 8, but for regime C. Figure 9. View largeDownload slide Same as in Fig. 8, but for regime C. 4 CONCLUSIONS We have investigated numerically the non-linear thermochemical convection and dynamos using the so-called codensity formulation. In the non-magnetic regime A (Fig. 2), the results suggest the existence of a steady branch of solutions that bifurcates supercritically from the basic state given by eq. (11). In our numerical simulations, the dominant balance in the limit of small Ekman numbers is between the Coriolis force, pressure and buoyancy. The time dependence of the velocity field is weak, as demonstrated by the weak oscillations of the toroidal field in the vicinity of the inner core’s equator. The results have shown that the codensity field is almost independent of whether the convection is driven purely by secular cooling or whether the secular cooling is combined with the chemical sources due to the phase change at the inner core boundary. In regime A (see Table 1), we have found that the large-scale columnar convection developed in the whole fluid volume, with the spiralling cross-section of the convection columns as a dominant feature. Results for regimes B, C and D are consistent with existing numerical investigations (e.g. Christensen et al. 1999). Nevertheless, we point out that convection in regime C is not developed in the whole volume, but instead localized to one hemisphere (see Fig. 4). As a result of geostrophy, the toroidal component of the velocity field dominates the poloidal one in all regimes considered. The chemically driven regime B has displayed the westward equatorial flow. The Boussinesq convection at moderate Rayleigh numbers typically shows a pattern of prograde (eastward) zonal flows near the equator and retrograde zonal flows near the poles. Gastine et al. (2013) note that many anelastic and fully compressible models of solar and stellar convection have observed a transition between the ‘solar-like’ (i.e. eastward equatorial zonal flow) and the ‘anti-solar’ (i.e. westward equatorial zonal flow) differential rotation profiles when buoyancy dominates the force balance. Yet, no systematic parameter study has been made to investigate the influence of density stratification on the transition between the rotation-dominated and the buoyancy-dominated zonal flow regimes. Though our flows are Boussinesq, they could potentially be used to explain such a transition. Whether the westward equatorial zonal flows can exist in the Boussinesq case or are exclusively a feature of the fully compressible regimes is still unclear. In the magnetic regime A, where the Prandtl number was smaller than the Ekman number, the inertial convection was preferred. Magnetic flux spots oscillated from the polar region to the mid-latitudes and back and the generated magnetic fields had dipolar oscillating structure. In regimes B and D, we studied the situations with very small Ekman numbers and Prandtl numbers of order unity. The generated magnetic fields were dipolar and had a similar structure when the sources of convection were either thermochemical or purely chemical. On the other hand, the structure of the thermally driven dynamos differed from those with thermochemical and chemical sources. In regime C, the generated magnetic fields were dipolar for thermally and chemically driven dynamos, while for the thermochemical driving, the dynamos were hemispherical. As a result of geostrophy, the toroidal component of the velocity field dominated the flow in all regimes considered. A similar result holds in the magnetic case, where the toroidal field dominates the magnetic field, which is in agreement with the results of Takahashi (2014). Relaxation of the codensity approximation and incorporation of the double diffusive effects into the model, similar to Manglik et al. (2010) in non-magnetic case and Takahashi (2014) in the magnetic case, is a task left for the future investigation. ACKNOWLEDGEMENTS This study was supported by the Ministry of Education, Youth and Sports through the project No. LG13042, by the Slovak Research and Development Agency under the contract No. APVV-14-0378, and by the VEGA grants 1/0319/15 and 2/0115/16. We would like to thank the Institute of Geophysics, Academy of Sciences of the CR, Prague for CPU time on the NEMO cluster (SGI). REFERENCES Aubert J., Amit H., Hulot G., Olson P., 2008. Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity, Nature , 454, 758– 761. 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Dynamos driven by double diffusive convection with a stably stratified layer and inhomogeneous core-mantle boundary heat flow, In 15th Symposium of Study of the Earth Deep Interior – SEDI 2016 , Meeting Booklet, Nantes, France. Trümper T., Breuer M., Hansen U., 2012. Numerical study on double–diffusive convetion in the Earth’s core, Phys. Earth planet. Inter. , 194–195, 55– 63. Google Scholar CrossRef Search ADS   Zhang K., 1994. On coupling between the Poincaré equation and the heat equation, J. Fluid Mech. , 268, 211– 229. Google Scholar CrossRef Search ADS   SUPPORTING INFORMATION Supplementary data are available at GJI online. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper. © The Author(s) 2017. 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Inertial effects on thermochemically driven convection and hydromagnetic dynamos in a spherical shell

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Abstract

Summary We investigate the thermochemical convection and hydromagnetic dynamos in a spherical shell using the so-called codensity formulation with different buoyancy sources: the secular cooling from the mantle, the buoyancy sources due to the solidification at the inner core boundary and the combination of the two sources. Numerical simulations of the fully non-linear problem are performed using the PARODY code. In the thermochemical regime, we find that when the Prandtl numbers are lower than Ekman numbers, inertial convection is preferred, while the large-scale columnar convection is preferred otherwise. Unlike the large-scale convection, the inertial convection is found to be almost independent of the nature of driving buoyancy source. Moreover, the codensity field evolves to a new, radially symmetric stationary state. At the Ekman numbers much smaller than the Prandtl numbers, we have obtained the westward equatorial zonal flow in the chemically driven regime, while for the other cases zonal flows are eastward near the equator. In the dynamo regime, inertial convection is preferred when the Prandtl numbers are lower than Ekman numbers and the generated dipolar magnetic fields oscillate from the polar region to the mid-latitudes and back. In this case, the generated magnetic fields are independent of the type of buoyancy source. At the Prandtl numbers greater than Ekman numbers, both dipolar and hemispherical dynamos are found. Composition and structure of the core, Dynamo: theories and simulations, Non-linear differential equations, Numerical modelling 1 INTRODUCTION Convective dynamics of the liquid cores of the Earth and some other planets is most commonly believed to be driven by the two different types of buoyancy sources (Braginsky & Roberts 1995; Christensen & Wicht 2008; Busse 2002). One is thermal buoyancy associated with a superadiabatic radial temperature gradient in the liquid core (i.e. the radial temperature gradient exceeding the adiabatic temperature gradient). The other is compositional (or chemical) buoyancy caused by a chemical differentiation of the liquid alloy constituents induced by solidification. The two sources of buoyancy, thermal and compositional, have received much attention in recent years in numerical simulations of convection-driven dynamics in rapidly rotating self-gravitating spherical fluid shells (Glatzmaier & Roberts 1997; Busse 2000; Jones 2011). The joint influence of the two driving buoyancy sources on the convective flow patterns, and the average and global physical characteristics has been investigated by Breuer et al. (2010) and Trümper et al. (2012) in a model of double-diffusive convection in a rotating spherical geometry. Both thermal and compositional fields are consistently treated as independently diffusing quantities with a mismatch in the molecular values of thermal and compositional diffusivities, in contrast to a number of previous conventional treatments in which the two buoyancy effects are treated through a single variable, called codensity, based on the assumption of equal diffusivities due to turbulent mixing (Braginsky & Roberts 1995; Lister & Buffett 1995; Shearer & Roberts 1997; Aubert et al. 2008; Christensen & Wicht 2008). Breuer et al. (2010) explored the relative influence of the two buoyancy sources for a single moderate value of the Ekman number of 10−3 and observed a considerable variation in the spatial distribution of axisymmetric zonal flow and helicity in their numerical simulations. In particular, they found that the flow changed from being nearly geostrophic for purely thermal forcing to being rather non-geostrophic and of much smaller spatial scale for purely compositional forcing. In the latter case, the zonal flows and helicity were dominated by a polar compositional plume. They concluded that these features could not be identified if the conventional codensity formulation (Braginsky & Roberts 1995) were adopted. In a related study at a lower Ekman number of 10−4, Trümper et al. (2012) similarly observed the coexistence of nearly geostrophic flow structures evoked by the thermal driving and high-Prandtl-number effects like chemical wind. Notably, the distinct features of purely thermal convection persist in the predominantly compositional regime with the thermal forcing in a subcritical regime of purely thermal convection. A model of double-diffusive convection to investigate the regime of a deep dynamo below a stably stratified layer in Mercury’s core has been employed by Manglik et al. (2010). Convection was found to penetrate to the upper, statically stable layer in the form of finger convection, enhancing the poloidal magnetic field in the initially stable layer and outside the core. A dynamo model with double-diffusive convection tuned for the conditions of Earth has been numerically studied by Takahashi (2014). The structure of the magnetic field is influenced by the thermal-to-compositional forcing ratio, with a dipolar magnetic field sustained for the forcing ratio less than 0.6 and non-dipolar field with helicity reduction otherwise. Morphological changes in the dynamo-generated magnetic field due to a presence of a thin layer of statically stable fluid at the core–mantle boundary have been recently reported by Takahashi (2016). We perform numerical simulations to elucidate which system parameters are most vital for the slow convection waves to appear in a codensity model of Boussinesq convection and hydromagnetic dynamos in a rotating spherical shell. The model and governing equations are presented in Section 2 and numerical results in Section 3. Conclusions are given in Section 4. 2 THE MODEL AND GOVERNING EQUATIONS We consider the thermochemical convection of an electrically conducting, incompressible fluid spherical shell (ri < r < ro) rotating with the angular velocity Ω. Evolution of the velocity $$\bf V$$, the magnetic field $$\bf B$$ and the codensity field C is described by the system of dimensionless equations:   \begin{eqnarray} &&{E \left( \frac{\partial {\bf V}}{\partial t} + \left( {\bf V}\cdot \nabla \right){\bf V} - \nabla ^2{\bf V} \right)+ 2 {\bf 1}_z\times {\bf V} + \nabla P}\nonumber\\ &=& R_a \frac{{\bf r}}{r_o} C + \frac{1}{P_{m}} \left( \nabla \times {\bf B} \right)\times {\bf B}, \end{eqnarray} (1)  \begin{eqnarray} \frac{\partial {\bf B}}{\partial t} = \nabla \times \left( {\bf V}\times {\bf B} \right) + \frac{1}{P_{m}} \nabla ^2{\bf B}, \end{eqnarray} (2)  \begin{eqnarray} \frac{\partial C}{\partial t}+\left({\bf V}\cdot \nabla \right) C = \frac{1}{P_{r}} \nabla ^2 C + S_{T/{\Xi }}, \end{eqnarray} (3)  \begin{eqnarray} \nabla \cdot {\bf V} = 0, \nabla \cdot {\bf B} = 0. \end{eqnarray} (4)Within the Boussinesq approximation, the deviation temperature field T and the light-element mass-fraction field Ξ are defined with respect to a reference state characterized by isentropic temperature and well-mixed mass fraction. Both buoyancy effects are used to define the codensity (density anomaly) field (Braginsky & Roberts 1995) C such that its dimensional form is   \begin{eqnarray} C = \alpha _T\rho T + \alpha _{\Xi }\rho \Xi , \end{eqnarray} (5)where αT and αΞ are the coefficients of volume expansion and ρ is the fluid density. We assume that the diffusivities of heat and light elements are the same, denoted κ, allowing for a single transport equation for the codensity field. We use the shell gap, L = ro − ri, as a length scale. The aspect ratio of the inner core, ri/ro, is set to 0.35, which is the current value for the Earth. We denote (r,  θ,  φ) the spherical coordinate system and 1z is the unit vector in the z-direction. The time is scaled by L2/ν, velocity by ν/L, magnetic induction by (ρμηΩ)1/2 and pressure, P, by ρν2/L2. The dimensionless parameters appearing in eqs (1)–(4) are the Prandtl number, Pr = ν/κ, the magnetic Prandtl number, Pm = ν/η and the Ekman number, E = ν/ΩL2, where ν is the kinematic viscosity, μ is the magnetic permeability and η is the magnetic diffusivity. The codensity is scaled with F/(4πνL), where the total buoyancy flux F is defined below. The codensity boundary conditions are treated in terms of codensity fluxes. At the inner-core boundary, the release of latent heat and light elements are the chemical sources of convection—they generate a positive mass anomaly flux Fi, which is considered uniform and imposed on by global core thermodynamics on the long timescale (Aubert et al. 2009). The dimensional form of Fi is   \begin{eqnarray} F_i = - \int _{S_i} \kappa \nabla C\cdot \textrm{d}\boldsymbol{S}_{\textrm{in}}, \end{eqnarray} (6)where Si is the inner boundary surface and $$\textrm{d}\boldsymbol{S}_{\textrm{in}}$$ is the surface element oriented into the outer core. Similarly, at the outer boundary, the mass anomaly flux imposed by the mantle corresponds to the heat flux carried out from the outer core, thus representing the secular cooling of the outer core. There are no chemical contributions to Fo at the outer boundary, so the light-element mass flux is set to zero there. The dimensional form of Fo is   \begin{eqnarray} F_o = - \int _{S_o} \kappa \nabla C\cdot \textrm{d}\boldsymbol{S}_{\textrm{out}}, \end{eqnarray} (7)where So is the outer boundary surface and $$\textrm{d}\boldsymbol{S}_{\textrm{out}}$$ is the surface element oriented away from the outer core. Note that Fi corresponds to the flux of the codensity field into the outer core at the inner boundary, while Fo corresponds to the flux of the codensity field from the outer core at the outer boundary. It is worth noting that the positive flux of the codensity field generates positive buoyancy. Thus, it is the positive values of Fo that generate negative buoyancy at the outer boundary and thus promote convection. Negative values of Fo result in a stably stratified layer at the top of the outer core (Aubert et al. 2009). In general, Fo can be either positive or negative, but the total flux of convection-promoting buoyancy, F = Fi + Fo, must be positive in order for convection to occur. The Rayleigh number appearing in eq. (1) is defined as   \begin{eqnarray} R_a = \frac{F g_{0}}{4\pi \rho \nu L}, \end{eqnarray} (8)where g0 is the gravity acceleration at r = ro. We let the mass anomaly fluxes Fi and Fo be independent. As in Aubert et al. (2009), we introduce the buoyancy fraction associated with the inner boundary, fi = Fi/F. The volumetric correction term ST/Ξ in eq. (3) is defined so as the average codensity in the spherical shell does not change with time (Braginsky & Roberts 1995; Kutzner & Christensen 2002) and has the following dimensionless expression   \begin{eqnarray} S_{T/{\Xi }} = 3\frac{1 - 2f_i}{r^{3}_{o} - r^{3}_{i}}. \end{eqnarray} (9)The volumetric correction term ST/Ξ is composed of two components. The first component is given by the reference state (Davies & Gubbins 2011): its thermal part contains the sink of potential thermal buoyancy due to the adiabatic gradient and the sources of thermal buoyancy representing the secular change of the reference state with time (the surface cooling rate at the inner boundary, the latent-heat release and the radiogenic heating); its compositional part reflects the release of the lighter component at the inner boundary due to freezing of the inner core. The second component of ST/Ξ contains the remaining codensity sources, which are defined such that eq. (9) holds. Note that the effective range of fi is from 0 to ∞. In the situation with fi = 0 only the secular cooling at the outer boundary (Fo > 0) contributes to the convection; when fi < 1/2, the secular cooling dominates the chemical sources (the total codensity outflow from the outer core, Fo − Fi, is positive); when fi = 1/2, the chemical sources and the secular cooling equally contribute to convection (zero total codensity outflux); when fi > 1/2, the chemical sources dominate (negative total codensity outflux); the situation with fi = 1 corresponds to the absence of secular cooling (the heat flux at the outer boundary is adiabatic) and the convection is purely chemically driven. The cases with fi > 1 are not addressed in this study; they correspond to the cases with dominant chemical convection where the heat flow at the outer boundary is below the adiabat, that is, Fo < 0 (Aubert et al. 2009). Eqs (1)–(4) are closed by the non-penetrating and no-slip boundary conditions for the velocity field at the rigid surfaces. The outer boundary is electrically insulating, so as the magnetic field at this boundary matches with an appropriate potential field in the exterior that implies no external field sources. The inner boundary is electrically conducting, with the electric conductivity of the outer and inner cores assumed to be the same. 3 NUMERICAL RESULTS The model equations are integrated using the numerical dynamo code PARODY developed by Dormy (1997) and later used by Dormy et al. (1998), Aubert et al. (2008) and Raynaud & Dormy (2013). The code allows the integration of the non-dimensional Boussinesq equations for time-dependent thermochemical convection in a rotating spherical shell filled with an electrically conducting fluid. It is a semi-spectral code based on spherical harmonics decomposition in the lateral direction and second-order finite-difference discretization in the radial direction, which makes it suitable for parallel computation on distributed memory clusters (Dormy 1997; Dormy et al. 1998; Aubert et al. 2008; Raynaud & Dormy 2013). The usual toroidal–poloidal decomposition for velocity and magnetic fields is applied. The time integration uses the implicit second-order Crank–Nicolson scheme. Parallelization is carried out using the message-passing interface. The Matlab routines developed by Aubert et al. (2008) are used to process the numerical results. 3.1 Non-magnetic convection In the purely hydrodynamic case, we consider four distinct parametric regimes (regimes A–D, see Table 1 for full description). Regime A corresponds to the situation in which Pr ≪ E ∼ 1 and convection is expected to take the form of inertial oscillations. Regimes B and C describe the situation in which E ≪ Pr ∼ 1. These two regimes are distinguished by the value of the Rayleigh number Ra (see Table 1). Finally, regime D corresponds to E ≪ 1 ≪ Pr. To assess the effect of varying the relative contributions of chemical and secular cooling buoyancy sources, we consider the following values of fi in each regime: fi = 0 (secular cooling only), fi = 0.5 (chemical sources and secular cooling contribute equally), fi = 0.8 (chemical sources dominate) and fi = 1 (chemical sources only). Recall that fi indicates the relative contribution of chemical buoyancy forcing. In all regimes, computations were initiated with no flow field and the codensity field given by   \begin{eqnarray} C(r) = C_i + P_r\left(f_i-\frac{1}{3}S_{T/{\Xi }}\right)\left(\frac{1}{r}-\frac{1}{r_i}\right) - \frac{1}{6}Pr S_{T/{\Xi }}(r^2-r_i^2),\!\!\!\!\!\!\!\!\nonumber\\ \end{eqnarray} (10)which corresponds to the steady solution of eq. (3) with Ci prescribed at r = ri. The Rayleigh number scaled with the critical Rayleigh number for the onset of convection, $$R^c_{a}$$, is set to 2 in the case of thermally driven convection at Pr = 1. For Pr = 10−4, the supercriticality of convection is more pronounced than in the case Pr = 1 ($$R_a/R^c_{a}>2$$), while for Pr = 5, it is less pronounced ($$R_a/R^c_{a}<2$$). At Pr = 10−4, the thermal diffusion processes dominate over the viscous ones, while at Pr = 5, the viscous diffusion processes dominate over the thermal ones. At Pr = 1, both thermal and viscous diffusion processes have the same influence on the convection. In Fig. 1, we show the evolution of kinetic energy, Ek, in all regimes considered. In regime A, the kinetic energy is almost independent of the driving mode. In other regimes, the values of kinetic energy corresponding to the convection driven by secular cooling are significantly smaller than those corresponding to the cases with contribution from the chemical sources. In regimes with non-zero fi, the changes in the relative contribution of chemical sources affect the kinetic energies most significantly in case D, that is, when the viscous dissipation is large. In Table 2 (columns 3 and 4), we show the time-averaged values of the kinetic energy together with the ratio of the time-averaged toroidal component of the kinetic energy to the time-averaged total kinetic energy. Note that in all cases, the dominant part of the kinetic energy always corresponds to the toroidal component of the flow. Figure 1. View largeDownload slide Temporal evolution of the kinetic energy in regimes A, B, C and D (non-magnetic convection). Figure 1. View largeDownload slide Temporal evolution of the kinetic energy in regimes A, B, C and D (non-magnetic convection). Table 1. Specification of parametric regimes considered in non-magnetic convection.   E  Pr  Ra  A  10−2  10−4  50  B  10−4  1  103  C  10−3  1  200  D  10−4  5  103    E  Pr  Ra  A  10−2  10−4  50  B  10−4  1  103  C  10−3  1  200  D  10−4  5  103  View Large Table 2. The time-averaged kinetic/magnetic energy, $$\bar{E}_{k/m}$$, and the scaled toroidal component of the kinetic/magnetic energy, $$\bar{E}_{k,T/m,T}$$, in each regime. The superscript (H) denotes the non-magnetic convection and the superscript (M) denotes the dynamo cases. The time average is taken over 4 ≤ t ≤ 5. The last two columns show the values of the magnetic Reynolds number, Rm, and the Elsasser number, Λ, in the dynamo cases.     $$\bar{E}_k^{(H)}$$  $$\bar{E}_{k,T}^{(H)}/\bar{E}_k^{(H)}$$  $$\bar{E}_k^{(M)}$$  $$\bar{E}_{k,T}^{(M)}/\bar{E}_k^{(M)}$$  $$\bar{E}_m^{(M)}$$  $$\bar{E}_{m,T}^{(M)}/\bar{E}_m^{(M)}$$  Rm  Λ  A  fi = 0  6.84  0.95  7.3  ≈1  0.2  0.97  613  0.6    fi = 0.5  6.84  0.95  6.8  0.98  10.9  0.89  592  35    fi = 0.8  6.84  0.95  4.4  0.99  18.8  0.84  455  60    fi = 1  6.84  0.95  7  0.98  12.7  0.87  605  41  B  fi = 0  600  0.79  482  0.77  5360  0.57  77  2.7    fi = 0.5  1979  0.84  1813  0.84  409  0.74  150  0.2    fi = 0.8  2966  0.84  1443  0.80  4636  0.69  134  2.3    fi = 1  2628  0.74  1766  0.80  7775  0.63  148  3.9  C  fi = 0  18  0.82  18.4  0.82  0.1  0.60  61  0.002    fi = 0.5  117  0.85  43.2  0.76  564  0.76  93  11    fi = 0.8  122  0.82  56.9  0.76  765  0.64  107  15    fi = 1  120  0.78  75  0.74  713  0.62  122  14  D  fi = 0  212  0.72  137  0.71  3286  0.54  166  66    fi = 0.5  336  0.74  368  0.78  195  0.77  271  3.9    fi = 0.8  423  0.74  427  0.75  38  0.65  292  0.8    fi = 1  608  0.75  426  0.76  2368  0.51  291  47      $$\bar{E}_k^{(H)}$$  $$\bar{E}_{k,T}^{(H)}/\bar{E}_k^{(H)}$$  $$\bar{E}_k^{(M)}$$  $$\bar{E}_{k,T}^{(M)}/\bar{E}_k^{(M)}$$  $$\bar{E}_m^{(M)}$$  $$\bar{E}_{m,T}^{(M)}/\bar{E}_m^{(M)}$$  Rm  Λ  A  fi = 0  6.84  0.95  7.3  ≈1  0.2  0.97  613  0.6    fi = 0.5  6.84  0.95  6.8  0.98  10.9  0.89  592  35    fi = 0.8  6.84  0.95  4.4  0.99  18.8  0.84  455  60    fi = 1  6.84  0.95  7  0.98  12.7  0.87  605  41  B  fi = 0  600  0.79  482  0.77  5360  0.57  77  2.7    fi = 0.5  1979  0.84  1813  0.84  409  0.74  150  0.2    fi = 0.8  2966  0.84  1443  0.80  4636  0.69  134  2.3    fi = 1  2628  0.74  1766  0.80  7775  0.63  148  3.9  C  fi = 0  18  0.82  18.4  0.82  0.1  0.60  61  0.002    fi = 0.5  117  0.85  43.2  0.76  564  0.76  93  11    fi = 0.8  122  0.82  56.9  0.76  765  0.64  107  15    fi = 1  120  0.78  75  0.74  713  0.62  122  14  D  fi = 0  212  0.72  137  0.71  3286  0.54  166  66    fi = 0.5  336  0.74  368  0.78  195  0.77  271  3.9    fi = 0.8  423  0.74  427  0.75  38  0.65  292  0.8    fi = 1  608  0.75  426  0.76  2368  0.51  291  47  View Large The numerical results for all the investigated cases are presented in Figs 2–5. The snapshots correspond to the dimensionless time t = 5. The results for regime A are shown in Fig. 2. In this case, inertial convection is preferred. The only difference between the driving modes is in the codensity fields: when fi = 0, the region of positive codensity is thicker than in the cases with positive fi. With fi > 0, the differences between the condensity profiles are only slight. The independence of the codensity field on the driving mode when fi > 0 is consistent with eq. (3), in which the diffusive term dominates the other terms when Pr ≪ 1 and the source term ST/Ξ becomes negligible. Note that the kinetic energy in regime A is very small in comparison with regimes B, C and D. An interesting feature is the ‘breathing’ behaviour of the toroidal component of the flow field (see A-NONMAG.gif in the supplementary data, where we show the meridional section of the toroidal scalar, VT, for the case fi = 0). Another interesting feature of the numerical results is that the codensity evolves to a radially symmetric stationary state governed by   \begin{eqnarray} \nabla ^2 C = - P_r S_{T/{\Xi }} + P_r V_r\frac{\partial C}{\partial r}. \end{eqnarray} (11)with fi entering via the source term ST/Ξ as well as the boundary conditions $$\partial C/\partial r|_{r_i}=-P_r f_i/r_i^2$$ and $$\partial C/\partial r|_{r_o}=-P_r f_o/r_o^2$$ (the dimensionless versions of eqs 6 and 7). We note that small values of Pr make the stationary codensity field almost insensitive to variations in fi. Figure 2. View largeDownload slide Equatorial sections of the codensity, C, equatorial sections of the radial components of velocity, Vr, meridional sections of the poloidal scalars, VP and meridional sections of the toroidal scalars, VT, (from left to right) in regime A with fi = 0, 0.5, 0.8 and 1 (from top to bottom). Red (blue) colours indicate positive (negative) values. The snapshots are taken at t = 5. Figure 2. View largeDownload slide Equatorial sections of the codensity, C, equatorial sections of the radial components of velocity, Vr, meridional sections of the poloidal scalars, VP and meridional sections of the toroidal scalars, VT, (from left to right) in regime A with fi = 0, 0.5, 0.8 and 1 (from top to bottom). Red (blue) colours indicate positive (negative) values. The snapshots are taken at t = 5. Figure 3. View largeDownload slide Same as in Fig. 2, but for regime B. Figure 3. View largeDownload slide Same as in Fig. 2, but for regime B. Figure 4. View largeDownload slide Same as in Fig. 2, but for regime C. Figure 4. View largeDownload slide Same as in Fig. 2, but for regime C. Figure 5. View largeDownload slide Same as in Fig. 2, but for regime D. Figure 5. View largeDownload slide Same as in Fig. 2, but for regime D. The results for regime B are presented in Fig. 3. In this case, large-scale columnar convection is developed in the whole volume. The spiralling cross-section of the columns is a dominant feature at moderate Prandtl numbers, which is observable for all driving modes. The Reynolds stresses, created by the spiralling character of the convection, generate a differential rotation (Zhang 1994; Busse & Simitev 2007; Šimkanin et al. 2010). The differential rotation is probably the cause of some irregularities in the spiralling structure of columnar convection, called ‘active’ and ‘quiet’ zones in Busse & Simitev (2007) and Šimkanin et al. (2010). The poloidal flow, driven by the thermal and compositional buoyancy forces, is transformed to the zonal flow via the Coriolis force and the Reynolds stresses present in the force balance [cf. Takahashi (2014)]. Looking at the fourth column in Fig. 3, we see that the zonal flow is eastward at the equator in all the investigated cases except in the purely chemically driven convection (fi = 1), in which case it is westward. Usually, the Boussinesq convection at moderate Rayleigh numbers shows a pattern with a prograde (eastward) zonal flow near the equator and retrograde zonal flow near the poles. The results for regime C are presented in Fig. 4. At fi = 1, the large-scale columnar convection with strong spiralling nature is developed in the whole volume and partially in the case with  fi = 0.8. In the cases with fi = 0 and 0.5, the convection is not developed in the whole volume, but is concentrated to a small volume in one hemisphere. The results for regime D are presented in Fig. 5. The large-scale columnar convection is developed in the whole volume and the spiralling cross-section of the columns is still visible, although it is not so dominant compared with regime B. As Pr increases, the influence of the differential rotation, which has dominated the evolution of convection columns at Pr of unit order and smaller, gradually diminishes (Busse & Simitev 2007). 3.2 Hydromagnetic dynamos In order to study the effects of magnetic field on convection, we study the same four parametric regimes as in the previous subsection, with the magnetic Prandtl number Pm as an additional dimensionless parameter. The values of Pm used in each regime are given in Table 3. Computations were initiated with no flow field and a strong dipole-dominated field of magnitude $$B\sim \mathcal {O}(1)$$. Temporal evolution of kinetic and magnetic energies in each regime is shown in Fig. 6. Table 2 (columns 5–8) shows the time-averaged kinetic/magnetic energy. Also shown is the scaled toroidal component for both fields and the magnetic Reynolds and Elsasser numbers. We see that the kinetic/magnetic energy is concentrated in the toroidal component of the respective field, indicating that the geostrophic flows and the toroidal magnetic fields predominate in all regimes. One interpretation of the Elsasser number is that it is the ratio of the Joule damping time to the period of rotation. Small values of Λ identified in regimes A and C at fi = 0 thus indicate relatively strong Joule damping, as confirmed also by the evolution of the magnetic energy in these regimes shown in Fig. 6. Figure 6. View largeDownload slide Temporal evolution of kinetic and magnetic energies, Ek and Em, in regimes A, B, C and D (magnetic convection). Figure 6. View largeDownload slide Temporal evolution of kinetic and magnetic energies, Ek and Em, in regimes A, B, C and D (magnetic convection). Table 3. Specification of parametric regimes considered in magnetic convection.   E  Pr  Ra  Pm  A  10−2  10−4  50  160  B  10−4  1  103  2.5  C  10−3  1  200  10  D  10−4  5  103  2.5    E  Pr  Ra  Pm  A  10−2  10−4  50  160  B  10−4  1  103  2.5  C  10−3  1  200  10  D  10−4  5  103  2.5  View Large A typical spatial distribution of the radial magnetic field component, Br, is displayed in Fig. 7. The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. The generated magnetic field is dipolar in all regimes except regime C where the thermochemically driven dynamos are hemispherical. In regime A, the magnetic flux spots are concentrated in the polar regions, in which case the magnetic fields oscillate from polar regions to the mid-latitudes, but remain mostly in the polar regions. The temporal evolution of the radial magnetic field in regime A can be found in the files A-DYNAMO-fi0.gif, A-DYNAMO-fi05.gif, A-DYNAMO-fi08.gif and A_DYNAMO-fi1.gif in the supplementary data. Figure 7. View largeDownload slide Spatial distribution (Hammer projection) of radial magnetic field component, Br, at r = ro in regimes A, B, C and D (from left to right) for thermochemically driven dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 7. View largeDownload slide Spatial distribution (Hammer projection) of radial magnetic field component, Br, at r = ro in regimes A, B, C and D (from left to right) for thermochemically driven dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Regimes B and D are qualitatively similar to the results presented by Šimkanin and Hejda (2011, 2013) and Šoltis and Šimkanin (2014). The results for regime A are shown in Fig. 8 and those for regime C in Fig. 9. In regime A, convection takes the form of inertial oscillations similar to the non-magnetic case A. In regime C, the large-scale columnar convection with spiralling pattern is developed in the whole volume in the case of purely chemically driven dynamos, while convection is confined to a single hemisphere in the dynamos forced by the combined chemical and secular-cooling sources. Even though the dynamos at fi = 0.5 and 0.8 are hemispherical, a further increase in Ra can be expected to result in the dynamos of dipolar nature. Figure 8. View largeDownload slide Equatorial sections of the radial magnetic field and velocity components, Br and Vr, respectively; meridional sections of the poloidal and toroidal scalars for the magnetic field, BP and BT, respectively, and the meridional sections of the poloidal and toroidal scalars for the velocity, VP and VT, respectively (from left to right), in regime A for dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 8. View largeDownload slide Equatorial sections of the radial magnetic field and velocity components, Br and Vr, respectively; meridional sections of the poloidal and toroidal scalars for the magnetic field, BP and BT, respectively, and the meridional sections of the poloidal and toroidal scalars for the velocity, VP and VT, respectively (from left to right), in regime A for dynamos with fi = 0, 0.5, 0.8 and 1 (from top to bottom). The red (blue) colour indicates positive (negative) values. The snapshots are taken at t = 5. Figure 9. View largeDownload slide Same as in Fig. 8, but for regime C. Figure 9. View largeDownload slide Same as in Fig. 8, but for regime C. 4 CONCLUSIONS We have investigated numerically the non-linear thermochemical convection and dynamos using the so-called codensity formulation. In the non-magnetic regime A (Fig. 2), the results suggest the existence of a steady branch of solutions that bifurcates supercritically from the basic state given by eq. (11). In our numerical simulations, the dominant balance in the limit of small Ekman numbers is between the Coriolis force, pressure and buoyancy. The time dependence of the velocity field is weak, as demonstrated by the weak oscillations of the toroidal field in the vicinity of the inner core’s equator. The results have shown that the codensity field is almost independent of whether the convection is driven purely by secular cooling or whether the secular cooling is combined with the chemical sources due to the phase change at the inner core boundary. In regime A (see Table 1), we have found that the large-scale columnar convection developed in the whole fluid volume, with the spiralling cross-section of the convection columns as a dominant feature. Results for regimes B, C and D are consistent with existing numerical investigations (e.g. Christensen et al. 1999). Nevertheless, we point out that convection in regime C is not developed in the whole volume, but instead localized to one hemisphere (see Fig. 4). As a result of geostrophy, the toroidal component of the velocity field dominates the poloidal one in all regimes considered. The chemically driven regime B has displayed the westward equatorial flow. The Boussinesq convection at moderate Rayleigh numbers typically shows a pattern of prograde (eastward) zonal flows near the equator and retrograde zonal flows near the poles. Gastine et al. (2013) note that many anelastic and fully compressible models of solar and stellar convection have observed a transition between the ‘solar-like’ (i.e. eastward equatorial zonal flow) and the ‘anti-solar’ (i.e. westward equatorial zonal flow) differential rotation profiles when buoyancy dominates the force balance. Yet, no systematic parameter study has been made to investigate the influence of density stratification on the transition between the rotation-dominated and the buoyancy-dominated zonal flow regimes. Though our flows are Boussinesq, they could potentially be used to explain such a transition. Whether the westward equatorial zonal flows can exist in the Boussinesq case or are exclusively a feature of the fully compressible regimes is still unclear. In the magnetic regime A, where the Prandtl number was smaller than the Ekman number, the inertial convection was preferred. Magnetic flux spots oscillated from the polar region to the mid-latitudes and back and the generated magnetic fields had dipolar oscillating structure. In regimes B and D, we studied the situations with very small Ekman numbers and Prandtl numbers of order unity. The generated magnetic fields were dipolar and had a similar structure when the sources of convection were either thermochemical or purely chemical. On the other hand, the structure of the thermally driven dynamos differed from those with thermochemical and chemical sources. In regime C, the generated magnetic fields were dipolar for thermally and chemically driven dynamos, while for the thermochemical driving, the dynamos were hemispherical. As a result of geostrophy, the toroidal component of the velocity field dominated the flow in all regimes considered. A similar result holds in the magnetic case, where the toroidal field dominates the magnetic field, which is in agreement with the results of Takahashi (2014). Relaxation of the codensity approximation and incorporation of the double diffusive effects into the model, similar to Manglik et al. (2010) in non-magnetic case and Takahashi (2014) in the magnetic case, is a task left for the future investigation. ACKNOWLEDGEMENTS This study was supported by the Ministry of Education, Youth and Sports through the project No. LG13042, by the Slovak Research and Development Agency under the contract No. APVV-14-0378, and by the VEGA grants 1/0319/15 and 2/0115/16. We would like to thank the Institute of Geophysics, Academy of Sciences of the CR, Prague for CPU time on the NEMO cluster (SGI). REFERENCES Aubert J., Amit H., Hulot G., Olson P., 2008. Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity, Nature , 454, 758– 761. 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Published: Mar 1, 2018

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