The fastest growing modes in rotating magnetoconvection with anisotropic diffusivities and their links to the Earth's core conditions

ABSTRACT

The fastest growing modes in anisotropic rotating magnetoconvection (RMC) processes are presented. After reminding the state of the art, we present our new approach that applies to isotropic as well as anisotropic diffusivities' conditions. We describe three cases: T (only temperature perturbation is time dependent), Q (temperature and velocity field perturbations are time dependent, but magnetic field perturbation is time independent) and G (temperature, magnetic and velocity field perturbations are time dependent). Isotropies as well as anisotropies are further distinguished by values of molecular and turbulent diffusivities. We show that T case does not describe properly convection in the Earth's outer core conditions, because it implies too huge Ekman numbers for the transition between the RMC modes of weak and strong magnetic field types. In G case, the convection is usually much more facilitated than in the T case: instabilities may arise with much smaller values of Ekman number and, in general, all types of convections occur with values of the physical parameters of the Earth consistent with the most reliable estimations. We demonstrate (and indicate) that Q and G cases can be well suited for the magnetic field of the Earth (and for other planetary magnetic fields), but only G case may correspond to turbulent stay of the Earth's core. We prove that, analogously like in the marginal modes, the value of anisotropic parameter α (the ratio between horizontal and vertical diffusivities) crucially influences the convection. The cases of $\alpha <1$ ($\alpha >1$) strongly decrease (increase) the Ekman numbers at which the RMC modes of weak and strong magnetic field types change between each other. Finally, we show and stress that not all types of anisotropies in the fastest growing modes can be equally strong. More specifically, we show that a fixed Rayleigh number puts a constraint on the maximum value of α, but do not put any lower positive limit on the minimum value of α. This special constraint is given by the necessary positiveness of the growth rate of the fastest growing modes. Our RMC approach allows to easily deal with very huge wave numbers and Rayleigh numbers as well as with very small Ekman numbers, what is usually not possible in the standard geodynamo simulations.

KEYWORDS:

• Rotating magnetoconvection (RMC)
• fastest growing modes
• anisotropic diffusivities (anisotropy)
• Earth's outer core conditions

Acknowledgments

EF performed this work during several research stays at Comenius University in Bratislava of him. These stays were funded by Ministry of Education, Science, Research and Sport of the Slovak Republic and by Recovery and Resilience Plan of the Slovak Republic, i.e. co-financed by EU from the NextGeneration EU mechanism. Thus he is grateful to these Slovak and European institutions. The authors are grateful to the referees and editors because they were very helpful to improve their research. The authors like to devote this work to the memory of Prof. Paul Henry Roberts, who unfortunately passed away in the end of 2022 and who was one of the best specialists in Geomagnetism and Dynamo Theory of the last two centuries. Since 1979, JB personally knew Paul Roberts who was strongly inspiring him in his activities. Part of the recent research activities of JB and EF has been inspired by P.H. Roberts' contributions to Magnetoconvection, turbulent dynamo theory and anisotropy due to the turbulence in the Earth's core.

Notes

1 See definitions of Elsasser (${\Lambda }_z$) and Ekman ($E_z$) numbers in section 2 and definition of $\Lambda -E$ regime diagram in section 3.1.1

2 To avoid possible confusion with the pressure and dimensionless pressure, we named these variables $\tilde{P}$ and P, respectively in (1(1) $\begin{array}{rl}{R}_o{\mathrm{\partial }}_t\mathit{u}+\hat{\mathit{z}}\times \mathit{u}& =-\mathbf{\nabla }P+{\Lambda }_z(\mathbf{\nabla }\times \mathit{b})\times \hat{\mathit{y}}+\mathrm{R\vartheta }\hat{\mathit{z}}+E_z{\mathrm{\nabla }}_{\nu }^2\mathit{u},\end{array}$(1) ) and (6(6) $\begin{array}{rl}\tilde{P}& =2{\Omega }_0{\eta }_{\mathrm{zz}}{\rho }_{0}P,\tilde{\mathit{b}}=B_M\mathit{b},\tilde{\vartheta }=({\eta }_{\mathrm{zz}}\mathrm{\Delta T}/{\kappa }_{\mathrm{zz}})\vartheta \end{array}$(6) ).

3 In section 3.1.1 see definition of angle γ.

4 Several regime diagrams for the fastest growing modes will be presented in section 6.

5 It is worth to remind that the (92b(92b) $1<\frac{4\pi }{R}\mathrm{\alpha K}(R,\alpha )<2.$(92b) ) holds also for very small α, because $K\to +\mathrm{\infty }$ as $\alpha \to 0$.

6 Due to the opposite monotonocities of $E^{†}(\alpha )$ and $E^{\ast }(\alpha )$ for huge R (and α), ∃ a finite (but huge) $\tilde{\alpha }$ such that $E^{†}(\alpha )=O(E^{\ast }(\alpha ))$ for $\alpha >\tilde{\alpha }$.

7 See comment after equation (16(16) $f(x,y,z,t)=\mathrm{Re}\{F(z)\exp (\mathrm{i}\mathrm{lx}+\mathrm{i}\mathrm{my})\exp (\lambda \mathit{t}\mathit{)}\}\mathit{,}$(16) ) to the fastest growing non-stationary C, O and P modes which are related to marginal stationary SC, SO and P modes, respectively.

8 However, it is possible to prove that the difference ${\Lambda }_G^{†}-{\Lambda }_{\mathrm{Q}}^{†}$ can be very high for $R\to \mathrm{\infty }$ and $\alpha \to \mathrm{\infty }$ at $\alpha \sim R/(2{\pi }^2)$; but it is at advanced turbulence (parameterised by too strong So anisotropy) when Q approximation of G case is inconvenient as it is shown in sections 3.2 and 4.2

9 this equivalence is proven by multiplying first inequality of 90 by 3Z and putting all terms in the LHS

Additional information

Funding

This work was supported by Ministerstvo školstva, vedy, výskumu a športu Slovenskej republiky and Next Generation EU.