3.2.1. Explicit Expressions in the Limit *τS* 1

Equation (1c) for Θ can most easily be solved in the limit of vanishing *ω*0*τS*. In this limit, we obtain for Θ,

$$\Theta = P\_m^m(\cos \theta) \exp(i m \varphi + i \omega \tau t) q(r), \tag{17}$$

with

$$q(r) = \hat{\mathcal{R}} \left( \frac{m(m+1)r^{m+4}}{(m+5)(m+4) - (m+1)m} - \frac{m(m+1)r^{m+2}}{(m+3)(m+2) - (m+1)m} - cr^m \right), \tag{18}$$

where the coefficient *c* is given by

$$c = \begin{cases} \frac{m(m+1)}{(m+5)(m+4)-(m+1)m} - \frac{m(m+1)}{(m+3)(m+2)-(m+1)m'}, & \text{case A,} \\ \frac{(m+4)(m+1)}{(m+5)(m+4)-(m+1)m} - \frac{(m+2)(m+1)}{(m+3)(m+2)-(m+1)m'}, & \text{case B.} \end{cases} \tag{19}$$

As Θ and the left hand side of Equation (16) is imaginary, the real parts of the two terms on the right hand side must balance. We thus obtain for *R*ˆ the result

$$\hat{R} = s \sqrt{\frac{\pi}{2|\omega\_0|}} \frac{\gamma^2 (m - \omega\_0)}{(\omega\_0 - m\gamma^2)} \left| \frac{m(m+2)}{\frac{\omega\_0^2 - m^2 \gamma^2}{\omega\_0 - m\gamma^2} (m+1)(m+2) - m} \right|^2 \tag{20}$$

$$\times (2m+9)(2m+7)(2m+5)^2 (2m+3) \frac{m+2}{m+1} \frac{1}{b'}$$

where the coefficient *b* assumes the values

$$b = \begin{cases} m(10m+27) & \text{case A.} \\ 14m^2 + 59m + 63 & \text{case B.} \end{cases} \tag{21}$$

Obviously the lowest value of Rˆ is usually reached for *m* = 1, but the fact that there are four different possible values of the frequency *ω*<sup>0</sup> complicates the determination of the critical value Rˆ <sup>c</sup>. Expression (20) is also of interest, however, in the case of spherical fluid shells when the (*m* = 1)-mode is affected most strongly by the presence of the inner boundary. Convection modes corresponding to higher values of *m* may then become preferred at onset as their *r*-dependence decays more rapidly with distance from the outer boundary according to relationships (7b).

3.2.2. Solution of the Heat Equation in the General Case

For the solution of Equation (1c), in the general case it is convenient to use the Green's function method. The Green's function *G*(*r*, *a*) is obtained as solution of the equation

$$
\left[\partial\_r r^2 \partial\_r + \left(-i\omega\_0 \tau S \, r^2 - m(m+1)\right)\right] G(r, a) = \delta(r-a),\tag{22}
$$

which can be solved in terms of the spherical Bessel functions *jm*(*µr*) and *ym*(*µr*),

$$G(r,a) = \begin{cases} G\_1(r,a) = A\_1 j\_m(\mu r) & \text{for } 0 \le r < a, \\ G\_2(r,a) = A j\_m(\mu r) + B y\_m(\mu r) & \text{for } a < r \le 1, \end{cases} \tag{23}$$

where

$$\mu \equiv \sqrt{-i\omega\_0 \tau S}, \quad A\_1 = \mu \left( y\_m(\mu a) - j\_m(\mu a) \frac{y\_m(\mu)}{j\_m(\mu)} \right), \tag{24a}$$

$$A = -\mu j\_m(\mu a) \frac{y\_m(\mu)}{j\_m(\mu)}, \quad B = \mu j\_m(\mu a). \tag{24b}$$

A solution of Equation (1c) can be obtained in the form

$$q(r) = -m(m+1)\hat{\mathbf{R}} \left( \int\_0^r \mathbf{G}\_2(r, a) \left( a^m - a^{m+2} \right) a^2 \mathbf{d}a + \int\_r^1 \mathbf{G}\_1(r, a) \left( a^m - a^{m+2} \right) a^2 \mathbf{d}a \right). \tag{25}$$

Evaluations of these integrals for *m* = 1 yield the expressions

$$q(r) = \begin{cases} \frac{2\hat{\mathbb{R}}}{\left(\omega\_0 \tau S\right)^2} \left(r(\mu^2 + 10) - \mu^2 r^3 - \frac{10\left(\mu r \cos(\mu r) - \sin(\mu r)\right)}{r^2 \left(\mu \cos \mu - \sin \mu\right)}\right) & \text{case A.}\\\frac{2\hat{\mathbb{R}}}{\left(\omega\_0 \tau S\right)^2} \left(r(2\mu^2 + 10) - \mu^2 r^3 - \frac{\left(\mu^2 - 10\right)\left(\mu r \cos(\mu r) - \sin(\mu r)\right)}{r^2 \left(2\mu \cos \mu - \left(2 - \mu^2\right) \sin \mu\right)}\right) & \text{case B.} \end{cases} \tag{26}$$

Lengthier expressions are obtained for *m* > 1. This first-order approximation of the temperature perturbation is illustrated in Figure 2 for the preferred modes of inertial magnetoconvection. The preferred modes of convection at onset are determined by minimizing the values of the critical Rayleigh number *R*ˆ at given values of the other parameters. The critical Rayleigh number *R*ˆ and frequency *ω*<sup>1</sup> are calculated on the basis of Equation (16) using expressions (26). In the case *m* = 1 we obtain

$$\hat{\mathbf{R}} = \frac{189}{20} \frac{s\sqrt{2\pi}\gamma^2(\omega\_0 - 1)}{\sqrt{|\omega\_0|}(\omega\_0 - \gamma^2)(6\lambda\_0 - 1)^2} \tag{27}$$

$$\{\{\omega\_1, \dots, \dots, \dots\}, \dots, \dots\} \qquad \sin\omega \qquad \qquad \{\} \tag{28}$$

$$\times \left\{ \begin{pmatrix} \mu^{-4} - 525\mu^{-8} - 175 \operatorname{Re} \left\{ \frac{\sin \mu}{\mu^{6} (\mu \cos \mu - \sin \mu)} \right\} \\\\ \mu^{-4} + 231\mu^{-8} + 7 \operatorname{Re} \left\{ \frac{(\mu^{5} - 8\mu^{3} + 9\mu)\cos \mu - 9\sin \mu}{\mu^{8} ((\mu^{2} - 2)\sin \mu + 2\mu \cos \mu)} \right\} \end{pmatrix} \right\}^{-1} \text{ case B.}$$

where Re{} indicates the real part of the term enclosed by {}.

**Figure 2.** Contour plots of the (normalized) temperature perturbation Θ(*r*) of the preferred mode given by Equations (17) and (26) in case A (top row) and case B (bottom row) with values of *S* and *γ* as specified in the panels and *τ* = 10<sup>4</sup> , *m* = 1 and frequency *ω*01. Expressions (17) and (18) for the limit *τS* 1 appear identical to the plots in the first column.

#### **4. Discussion**

Expressions (27) have been plotted as functions of *S* in Figures 3c and 4c for cases A and B, respectively. Four distinct curves appear as there are four possible values of *ω*<sup>0</sup> for each *m*. For values *S* of the order 10−<sup>2</sup> or less, expressions (20) are well approached. The retrograde mode corresponding to the positive sign in (7c) always yields the lower value of *R*ˆ, but it loses its preference to the progradely traveling modified Alfven mode corresponding to the upper sign in (10b) as *S* becomes of the order 10−<sup>1</sup> or larger. This transition can be understood on the basis of the increasing difference in phase between Θ and *u<sup>r</sup>* with increasing *S*. While the mode with the largest absolute value of *ω* is preferred as long as Θ and *u<sup>r</sup>* are in phase, the mode with the minimum absolute value of *ω* becomes preferred as the phase difference increases as the latter is detrimental to the work done by the buoyancy force. The frequency perturbation *ω*<sup>1</sup> usually makes only a small contribution to *ω* which tends to decrease the absolute value of *ω*. This transition shifts towards smaller values of *S* and *γ* as *τ* is increased as illustrated in Figure 5. The magneto-inertial convective modes corresponding to higher values of *m* = 1 . . . 8 exhibit similar behavior as Figures 3d and 4d demonstrate for the cases A and B, respectively. The value *m* = 1 is always the preferred value of the wavenumber, except possibly in a very narrow range near *γ* = 0.03, as indicated by Figure 3a,b in the case A, and possibly near *γ* = 0.02 in the case B and Figure 4a,b. The axisymmetric mode *m* = 0, given for comparison in Figures 3c,d and 4c,d, is never preferred in contrast to the purely non-magnetic case where it becomes the critical one near the transition from retrograde to prograde inertial convection modes as seen in Figure 6.

**Figure 3.** Case A. (**a**) The critical Rayleigh number Rˆ <sup>c</sup> as a function of the wave number *m* for *γ* = 0.1 and *τ* = 10<sup>2</sup> . . . 10<sup>6</sup> increasing from bottom with log-scale decades given by the five thick lines. (**b**) The critical Rayleigh number Rˆ c as a function of *γ* for *S* = 1 and *m* = 1 . . . 8 increasing from bottom. (**c**) Competition of modes with increasing *S* for *γ* = 0.1 and *m* = 1. Explicit expressions (20) in the limit *<sup>τ</sup><sup>S</sup>* <sup>1</sup> are shown by broken lines. (**d**) The critical Rayleigh number Rˆ c as a function of *S* for *γ* = 0.1 and *m* = 1 . . . 8 increasing from bottom. The axisymmetric mode *m* = 0 is given for comparison in panels (**c**,**d**) by a dot-dashed line. In panels (**b**–**d**) *τ* = 10<sup>4</sup> .

**Figure 4.** Case B. (**a**) The critical Rayleigh number Rˆ <sup>c</sup> as a function of the wave number *m* for *γ* = 0.1 and *τ* = 10<sup>2</sup> . . . 10<sup>6</sup> increasing from bottom with log-scale decades given by the five thick lines. (**b**) The critical Rayleigh number Rˆ c as a function of *γ* for *S* = 1 and *m* = 1 . . . 8 increasing from bottom. (**c**) Competition of modes with increasing *S* for *γ* = 0.1 and *m* = 1. Explicit expressions (20) in the limit *<sup>τ</sup><sup>S</sup>* <sup>1</sup> are shown by broken lines. (**d**) The critical Rayleigh number Rˆ c as a function of *S* for *γ* = 0.1 and *m* = 1 . . . 8 increasing from bottom. The axisymmetric mode *m* = 0 is given for comparison in panels (**c**,**d**) by a dot-dashed line. In panels (**b**–**d**) *τ* = 10<sup>4</sup> .

**Figure 5.** The boundary where the transition from modes characterised by *ω*<sup>01</sup> to modes characterised by *ω*<sup>03</sup> occurs in various cross-sections of the parameter space. The value of the parameters are *m* = 1, *S* = 1, *γ* = 0.1, and *τ* = 5000 where they are not varied on the axes. Case A is denoted by a solid lines and Case B by broken lines.

**Figure 6.** Competition of modes with increasing *τP* in the non-magnetic case discussed in [26]. The Rayleigh number R as a function of *τP* for *m* = 0 (thick dash-dotted lines) and *m* = 1 (thin lines). Results based on the explicit expressions (4.6) and (3.4) from [26] are shown in solid lines and broken lines, respectively, in the case *m* = 1. (**a**) Case A, fixed temperature boundary conditions. (**b**) Case B, insulating thermal boundary conditions.

For very large values of *τ* and *S*, the Rayleigh number *R*ˆ increases in proportion to √ *τ*(*τS*) 2 for fixed *m*. In spite of this strong increase, Θ remains of the order *τ* 3/2*S* on the right hand side of Equation (1a). The perturbation approach thus continues to be valid for *τ* −→ ∞ as long as *S* 1 can be assumed. For any fixed low value of *S*, however, the onset of convection in the form of prograde inertial modes will be replaced with increasing *τ* at some point by the onset in the form of columnar magneto-convection because the latter obeys an approximate asymptotic relationship for *R* of the form *τ* 4/3 (see, for example, Eltayeb et al. [28]). This second transition depends on the value of *S* and will occur at higher values of *τ* and *R* for lower values of *S*. There is little chance that magneto-inertial convection occurs in the Earth's core, for instance, as *S* is of the order 30,000 while the usual estimate for *τ* is 1015, but it might be relevant for understanding of rapidly rotating stars with strong magnetic fields.

#### **5. Conclusions**

A main result of the analysis of this paper is that for small values of the magnetic Prandtl number *P<sup>m</sup>* and *γ* an azimuthal magnetic field exerts a stabilizing influence on the onset of convection in the form of sectorial magneto-inertial modes. As a consequence, magneto-convection with azimuthal wave number *m* = 1 is generally preferred at onset for both thermally-infinitely conducting and thermally-insulating boundaries. In contrast, in the absence of a magnetic field, inertial modes with azimuthal wave number *m* = 1 are preferred, but only in the case of thermally-insulating boundaries, while in the case with infinitely conducting thermal boundaries large azimuthal wave numbers are preferred soon after moderately large rotation is reached [26] and magnetic field is absent. Axisymmetric magneto-convection is never the preferred mode at onset while in the non-magnetic case it appears to be realized in a minute region of the parameter space only. These results are also in contrast to previous magnetoconvection results obtained for larger values of *P<sup>m</sup>* where a destabilizing role of the azimuthal magnetic field has been found.

The region of the parameter space investigated in the present paper differs considerably from those analyzed in previous work. Most authors have emphasized regimes of high magnetic flux density where the magnetic field exerts a destabilizing influence and strongly decreases the critical Rayleigh number for onset of convection (see, for example, in [28,29]). Unfortunately, no explicitly analytical results are possible in that region of the parameter space. Moreover, the choice of parameter values has often been motivated by applications to the problem of the geodynamo in which case the parameter *S* is large, perhaps as large

as 10<sup>5</sup> , when molecular diffusivities are used. On the other hand, small values of *S* may be relevant for magneto-convection in stars where a high thermal diffusivity is generated by radiation.

**Author Contributions:** Conceptualization, F.H.B. and R.D.S.; formal analysis, F.H.B. and R.D.S.; data curation, R.D.S.; writing—original draft preparation, F.H.B.; writing—review and editing, F.H.B. and R.D.S.; visualization, R.D.S. funding acquisition, R.D.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research of R.S. was funded by the Leverhulme Trust grant number RPG-2012-600.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are plotted from the analytical expressions printed here.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

