2.1. Problem set-up and governing equations

A localized density disturbance $\rho ^\prime$ that occurs in an unstably stratified rotating fluid layer threaded by a uniform magnetic field is considered. Since $\rho ^\prime$ is related to a temperature perturbation $\varTheta$ by $\rho ^\prime = -\rho \alpha \varTheta$, where $\rho$ is the ambient density and $\alpha$ is the coefficient of thermal expansion, an initial temperature perturbation is chosen in the form

(2.1)\begin{equation} \varTheta_0=C \exp [-(x^2+y^2+z^2)/\delta^2], \end{equation}

where $C$ is a constant and $\delta$ is the length scale of the perturbation. Figure 1 shows the initial perturbation which subsequently evolves under gravity $\boldsymbol {g} = -g \hat {\boldsymbol {e}}_y$, background rotation $\boldsymbol {\varOmega } = \varOmega \hat {\boldsymbol {e}}_z$ and a uniform magnetic field $\boldsymbol {B} = B \hat {\boldsymbol {e}}_x$ in Cartesian coordinates $(x,y,z)$. In an otherwise quiescent medium, the initial temperature perturbation (2.1) gives rise to a velocity field $\boldsymbol {u}$, which in turn interacts with $\boldsymbol {B}$ to generate the induced magnetic field $\boldsymbol {b}$. The initial velocity $\boldsymbol {u}_0$ and induced field $\boldsymbol {b}_0$ are both zero. In the Boussinesq approximation, the following magnetohydrodynamic (MHD) equations give the evolution of $\boldsymbol {u}$, $\boldsymbol {b}$ and $\varTheta$:

(2.2)$$\begin{gather} \frac{\partial{\boldsymbol{u}}}{\partial{t}} ={-}\frac{1}{\rho}\boldsymbol{\nabla}{p^*} -2 \boldsymbol{\varOmega} \times \boldsymbol{u} + \frac{1}{\mu \rho}\left(\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \right) \boldsymbol{b} - \boldsymbol{g} \alpha\varTheta +\nu\nabla^2\boldsymbol{u}, \end{gather}$$

(2.3)$$\begin{gather}\frac{\partial{\boldsymbol{b}}}{\partial{t}} =\left(\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \right) {\boldsymbol{u}} +\eta \nabla^2 \boldsymbol{b}, \end{gather}$$

(2.4)$$\begin{gather}\frac{\partial{\varTheta}}{\partial{t}}={-}\gamma \boldsymbol{\hat{e}}_y \boldsymbol{\cdot} \boldsymbol{u}+ \kappa\nabla^2\varTheta, \end{gather}$$

(2.5)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b}=0, \end{gather}$$

where $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity, $\eta$ is the magnetic diffusivity, $\mu$ is the magnetic permeability, $\boldsymbol {\varOmega }= \varOmega \hat {\boldsymbol {e}}_z$, $p^*=p -(\rho /2) |{\boldsymbol {\varOmega }} \times {\boldsymbol {x}}|^2+ (\boldsymbol {B} \boldsymbol {\cdot} \boldsymbol {b})/\mu$ and $\gamma =\partial T_0/\partial y <0$ is the mean temperature gradient in the unstably stratified fluid.

Figure 1. The initial state of a density perturbation $\rho ^{\prime }$ that evolves in an unstably stratified fluid subject to a uniform magnetic field $\boldsymbol {B} = B \hat {\boldsymbol {e}}_x$, background rotation $\boldsymbol {\varOmega } = \varOmega \hat {\boldsymbol {e}}_z$ and gravity $\boldsymbol {g} = -g \hat {\boldsymbol {e}}_y$ in Cartesian coordinates $(x,y,z)$.

2.2. Solutions for the velocity field

Taking the curl of (2.2) and (2.3) and eliminating the electric current density between them, we obtain for the velocity field

(2.6)\begin{align} &\left[\left(\frac{\partial}{\partial{t}} -\nu \nabla^2\right)\left(\frac{\partial}{\partial{t}}-\eta \nabla^2\right) -V_M^2 \frac{\partial^2{}}{\partial{x^2}}\right]^2 (-\nabla^2 \boldsymbol{u})\nonumber\\ &\quad =4\varOmega^2\left(\frac{\partial}{\partial{t}} -\eta\nabla^2\right)^2\frac{\partial^2{\boldsymbol{u}}}{\partial{z^2}} -2\varOmega\alpha\frac{\partial}{\partial{z}}\left(\frac{\partial}{\partial{t}} -\eta\nabla^2\right)^2(\boldsymbol{\nabla}\times\varTheta\boldsymbol{g})\nonumber\\ &\qquad -\alpha\left(\frac{\partial}{\partial{t}} -\eta \nabla^2\right)\left[\left(\frac{\partial}{\partial{t}} -\nu \nabla^2\right)\left(\frac{\partial}{\partial{t}}-\eta \nabla^2\right) - V_M^2 \frac{\partial^2{}}{\partial{x^2}}\right] (\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\varTheta\boldsymbol{g}), \end{align}

where $V_M= B/\sqrt {\mu \rho }$ is the Alfvén velocity. The assumption of an unbounded domain facilitates the use of Fourier transforms along each coordinate direction,

(2.7) \begin{equation} \mathscr{F}(\boldsymbol{A}) = \hat{\!\!\boldsymbol{A}}= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\boldsymbol{A} {{\rm e}}^{- \mathrm{i} \, \boldsymbol{k}\ \boldsymbol{\cdot}\ \boldsymbol{x}}\,\mathrm{d}\kern0.7pt x\, \mathrm{d}y\,\mathrm{d}z, \end{equation}

and

(2.8) \begin{equation} \mathscr{F}^{{-}1}\left(\,\,\hat{\!\!\boldsymbol{A}}\right)=\boldsymbol{A}= {\frac{1}{(2{\rm \pi})^3}} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{\!\!\boldsymbol{A}} \mathrm{e}^{\mathrm{i} \,\boldsymbol{k}\ \boldsymbol{\cdot}\ \boldsymbol{x}}\, \mathrm{d}k_x\, \mathrm{d}k_y\,\mathrm{d}k_z, \end{equation}

where $\boldsymbol {k}=(k_x,k_y,k_z)$ represents the wave vector such that $|\boldsymbol {k}|=k=\sqrt {k_x^2+k_y^2+k_z^2}$. Application of the Fourier transform (2.7) to the Cartesian components of (2.6) gives

(2.9)\begin{align} &\left[\left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)+\omega_M^2\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)+\omega_C^2\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)\right]\hat{u}_x\nonumber\\ &\quad=\left[-\omega_C\omega_A^2\frac{k_zk}{k_x^2+k_z^2} \left(\frac{\partial}{\partial{t}}+\omega_\eta\right)^2\right.\nonumber\\ &\left.\qquad+\,\omega_A^2 \frac{k_xk_y}{k_x^2+k_z^2}\left(\frac{\partial}{\partial{t}} +\omega_\eta\right) \left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)+\omega_M^2\right)\right]\hat{u}_y, \end{align}

(2.10)\begin{align} &\left[\left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta \right)+\omega_M^2\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)+\omega_C^2\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)\right.\nonumber\\ &\left.\qquad +\,\omega_A^2\left(\frac{\partial}{\partial{t}} +\omega_\eta\right) \left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)+\omega_M^2\right)\right]\hat{u}_y=0, \end{align}

(2.11)\begin{align} &\left[\left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)+\omega_M^2\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)+\omega_C^2\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)^2\left(\frac{\partial}{\partial{t}} +\omega_\kappa\right)\right]\hat{u}_z\nonumber\\ &\quad=\left[\omega_C\omega_A^2\frac{k_x k}{k_x^2+k_z^2} \left(\frac{\partial}{\partial{t}} +\omega_\eta\right)^2\right.\nonumber\\ &\left.\qquad+\,\omega_A^2\frac{k_yk_z}{k_z^2+k_x^2}\left(\frac{\partial}{\partial{t}} +\omega_\eta\right) \left(\left(\frac{\partial}{\partial{t}} +\omega_\nu\right)\left(\frac{\partial}{\partial{t}} +\omega_\eta\right)+\omega_M^2\right)\right]\hat{u}_y, \end{align}

where we have combined the transformed temperature equation (2.4) with the transform of (2.6). In (2.9)–(2.11),

(2.12a–c)\begin{equation} \omega_C^2=4 \varOmega^2 k_z^2/k^2, \quad \omega_A^2=g\alpha \gamma (k_x^2+k_z^2)/k^2 , \quad \omega_M^2=V_M^2 k_x^2, \end{equation}

represent the squares of the frequencies of linear inertial, buoyancy and Alfvén waves, respectively (Braginsky Reference Braginsky1967; Busse et al. Reference Busse, Dormy, Simitev and Soward2007). In an unstably stratified medium, wherein $\omega _A^2 <0$, $|\omega _A|$ is a measure of the strength of buoyancy. As the present study focuses on a system where the viscous and thermal diffusion are much smaller than magnetic diffusion, the frequencies $\omega _\nu = \nu k^2$ and $\omega _\kappa = \kappa k^2$ in (2.9)–(2.11) are small compared with $\omega _\eta = \eta k^2$. In this limit, the solution of the form $\hat {u}_y \sim {{\rm e}}^{\mathrm {i} \lambda t}$ for the homogeneous equation (2.10) gives the following quintic equation in $\lambda$:

(2.13)$$\begin{gather} \lambda^5- 2 \mathrm{i}\omega_\eta \lambda^4 - (\omega_A^2+\omega_C^2+\omega_\eta^2+2\omega_M^2) \lambda^3 +2 \mathrm{i} \omega_\eta( \omega_A^2+\omega_C^2+\omega_M^2) \lambda^2\nonumber\\ +\,(\omega_A^2\omega_\eta^2+\omega_C^2\omega_\eta^2+ \omega_A^2\omega_M^2+\omega_M^4) \lambda- \mathrm{i} \omega_A^2\omega_\eta\omega_M^2=0, \end{gather}$$

the approximate roots of which were discussed in Sreenivasan & Maurya (Reference Sreenivasan and Maurya2021) for known relative orders of magnitudes of $\omega _M$, $\omega _A$, $\omega _C$ and $\omega _\eta$. In the solution for (2.10)

(2.14)\begin{equation} \hat{u}_y = \sum_{m=1}^{5}D_{m} \mbox{e}^{\mathrm{i} \lambda_{m}t}, \end{equation}

the coefficients $D_{m}$ are determined using the initial conditions for $\hat {u}_y$ and its time derivatives (§ 2.3). Of the five terms in the expansion on the right-hand side of (2.14), two terms represent oppositely travelling fast MAC waves, two other terms represent oppositely travelling slow MAC waves and the fifth term represents the overall growth of the velocity perturbation.

By substituting (2.14) in (2.9) and (2.11), the following solutions for (2.9) and (2.11) are obtained:

(2.15)\begin{gather} \hat{u}_{x} =\hat{u}^H_{x}+\hat{u}^P_{x}=\sum_{m=1}^{5}A_{m} \mathrm{e}^{\mathrm{i}\lambda_m^H t}+\sum_{m=1}^{5}M_m{\rm e}^{\mathrm{i} \lambda_m^P t}, \end{gather}

(2.16)\begin{gather} \hat{u}_{z} =\hat{u}^H_{z}+\hat{u}^P_{z}=\sum_{m=1}^{5}C_{m} \mathrm{e}^{\mathrm{i}\lambda_m^H t} +\sum_{m=1}^{5}N_m{\rm e}^{\mathrm{i}\lambda_m^P t} .\end{gather}

In (2.15) and (2.16), $\hat {u}_x^H$ and $\hat {u}_z^H$ are the homogeneous solutions of (2.9) and (2.11), respectively, $\hat {u}_x^P$ and $\hat {u}_z^P$ are the particular solutions, $\lambda _m^P$ are the roots of (2.13) and $\lambda _m^H$ are the roots of (2.13) with $\omega _A=0$

(2.17)\begin{equation} \left. \begin{aligned} \lambda^H_1 & =\frac{1}{2}\left(\omega_C+ \mathrm{i} \omega_\eta+\sqrt{\omega_C^2-2 \mathrm{i} \omega_C\omega_\eta-\omega_\eta^2+4\omega_M^2}\right),\\ \lambda^H_2 & = \frac{1}{2} \left(- \omega_C+ \mathrm{i} \omega_\eta - \sqrt{\omega_C^2 + 2 \mathrm{i} \omega_C\omega_\eta-\omega_\eta^2+4\omega_M^2}\right),\\ \lambda^H_3 & =\frac{1}{2}\left(\omega_C+ \mathrm{i} \omega_\eta-\sqrt{\omega_C^2-2 \mathrm{i}\omega_C\omega_\eta-\omega_\eta^2+4\omega_M^2}\right),\\ \lambda^H_4 & =\frac{1}{2} \left(- \omega_C+ \mathrm{i} \omega_\eta + \sqrt{\omega_C^2 +2 \mathrm{i}\omega_C\omega_\eta -\omega_\eta^2+4\omega_M^2}\right),\\ \lambda^H_5 & =0, \end{aligned} \right\} \end{equation}

which give the frequencies of the fast and slow magneto–Coriolis (MC) waves in the absence of buoyancy (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017). The coefficients $A_m$, $C_m$, $M_m$ and $N_m$ in (2.15) and (2.16) are evaluated as in § 2.3 below.

The solutions for the induced magnetic field transforms $\hat {b}_x$, $\hat {b}_y$ and $\hat {b}_z$ are obtained following a similar approach.

2.3. Evaluation of spectral coefficients

From (2.14), the initial conditions for $\hat {u}_y$ and its time derivatives are given by

(2.18)\begin{equation} \mathrm{i}^n \sum_{m=1}^{5}D_m \lambda_{m}^n =\left(\frac{\partial^n\hat{u}_y}{\partial t^n}\right)_{t=0}=a_{n+1}, \quad n=0,1,2,3,4. \end{equation}

Algebraic simplifications give the right-hand sides of (2.18) in the limit of $\nu =\kappa =0$, as follows:

(2.19)\begin{equation} \left. \begin{aligned} a_1 & =\hat{u}_y|_{t=0}=0,\\ a_2 & =\frac{\partial{\hat{u}_y}}{\partial{t}}|_{t=0}= \alpha g \left(\frac{k_z^2+k_x^2}{k^2}\right)\hat{\varTheta}_0,\\ a_3 & =\frac{\partial^2 \hat{u}_y}{\partial{t^2}}|_{t=0}=0,\\ a_4 & =\frac{\partial^3{\hat{u}_y}}{\partial{t^3}}|_{t=0}={-}(\omega_M^2+\omega_C^2+\omega_A^2) a_2,\\ a_5 & =\frac{\partial^4{\hat{u}_y}}{\partial{t^4}}|_{t=0}= \omega_M^2\omega_\eta a_2. \end{aligned} \right\} \end{equation}

The coefficients $D_m$ may now be obtained using the roots of (2.13). For example, we obtain

(2.20)$$\begin{gather} D_{1}=\dfrac{a_{5}- \mathrm{i} \, a_{4}(\lambda_2+\lambda_3+\lambda_4+\lambda_5) +\mathrm{i} \, a_{2}(\lambda_2\lambda_4\lambda_5+\lambda_3\lambda_4\lambda_5 +\lambda_2\lambda_3\lambda_4+\lambda_2\lambda_3\lambda_5)}{(\lambda_1-\lambda_2) (\lambda_1-\lambda_3)(\lambda_1-\lambda_4)(\lambda_1-\lambda_5)}, \end{gather}$$

(2.21)$$\begin{gather}D_{3}=\dfrac{a_{5}- \mathrm{i} \, a_{4}(\lambda_1+\lambda_2+\lambda_4+\lambda_5) + \mathrm{i} \,a_{2}(\lambda_1\lambda_4\lambda_5+\lambda_2\lambda_4\lambda_5 +\lambda_1\lambda_2\lambda_4+\lambda_1\lambda_2\lambda_5)}{(\lambda_3-\lambda_1) (\lambda_3-\lambda_2)(\lambda_3-\lambda_4)(\lambda_3-\lambda_5)}, \end{gather}$$

for the forward-travelling fast and slow wave solutions, respectively. The coefficients $A_m$ and $C_m$ are determined in a similar way. The coefficients $M_m$ and $N_m$ in equations (2.15) and (2.16) are obtained using the method of undetermined coefficients, as follows:

(2.22)\begin{align} M_m &= \frac{D_{m}}{T_m}\left[-\omega_C\omega_A^2 \left(\frac{kk_z}{k_x^2+k_z^2}\right) \big(-(\lambda^P_m)^2+\omega_\eta^2 +2 \mathrm{i} \omega_\eta\lambda^P_m\big)\right.\nonumber\\ &\quad \left.+\,\omega_A^2\frac{k_xk_y}{k_x^2+k_z^2} \big(- \mathrm{i}(\lambda^P_m)^3-2\omega_\eta(\lambda^P_m)^2 + \mathrm{i}(\omega_\eta^2+\omega_M^2)\lambda^P_m +\omega_M^2\omega_\eta\big)\right], \end{align}

and

(2.23)\begin{align} N_m& = \frac{D_{m}}{T_m}\left[\omega_C\omega_A^2 \left(\frac{kk_x}{k_x^2+k_z^2}\right) \big(-(\lambda^P_m)^2+\omega_\eta^2 +2 \mathrm{i}\omega_\eta\lambda^P_m\big)\right.\nonumber\\ &\left.\quad+\,\omega_A^2\frac{k_zk_y}{k_x^2+k_z^2} \big(- \mathrm{i}(\lambda^P)^3-2\omega_\eta(\lambda^P_m)^2 + \mathrm{i}(\omega_\eta^2+\omega_M^2)\lambda^P_m +\omega_M^2\omega_\eta\big)\right], \end{align}

with

(2.24)$$\begin{gather} T_m = \mathrm{i} (\lambda^P_m)^5+2(\lambda^P_m)^4\omega_\eta - \mathrm{i}(\lambda^P_m)^3(\omega_C^2+\omega_\eta^2 +2\omega_M^2)-2\omega_\eta(\lambda^P_m)^2(2\omega_C^2+\omega_M^2)\nonumber\\ +\, \mathrm{i}\lambda^P_m(\omega_C^2\omega_\eta^2+\omega_M^4), \quad m=1,2,\ldots,5. \end{gather}$$

2.4. Fast and slow MAC waves in unstable stratification

From the solution of the initial value problem, the velocity and induced magnetic field can be obtained at discrete points in time. The analysis of the solutions is limited to times much shorter than the time scale for the exponential increase of the perturbations. When the buoyancy force is small compared with the Lorentz force ($|\omega _A/\omega _M| \ll 1$), the parameter regime is determined by the Lehnert number $Le$ and the magnetic Ekman number $E_\eta$

(2.25a,b)\begin{equation} Le= \dfrac{V_M}{2 \varOmega \delta}, \quad E_\eta= \dfrac{\eta}{2 \varOmega \delta^2}, \end{equation}

both based on the length scale $\delta$ of the initial buoyancy perturbation (2.1).

Figure 2 shows the evolution of the kinetic helicity $\boldsymbol {u} \boldsymbol {\cdot} \boldsymbol {\zeta }$ at $y=0$. The real-space fields are obtained from the transforms $\hat {\boldsymbol {u}}$ and $\hat {\boldsymbol {\zeta }}$ via the inverse Fourier transform (2.8). Here, a truncation value of $\pm 10/\delta$ is used for the three wavenumbers in the integrals since the initial wavenumber $k_0= \sqrt {3}/\delta$ (Appendix A). Apart from the segregation of oppositely signed helicity between the two halves about the mid-plane $z=0$, the evolution of blobs into columnar structures through the propagation of damped waves is evident.

Figure 2. Evolution of the kinetic helicity on the $x$–$z$ plane at $y=0$ with time (measured in units of the magnetic diffusion time $t_\eta$) for $Le=0.03$ and $E_\eta =2\times 10^{-5}$. The snapshots are at (a) $t/t_\eta =1\times 10^{-4}$, (b) $t/t_\eta =2.5\times 10^{-3}$ and (c) $t/t_\eta = 1\times 10^{-2}$. The ratio $|\omega _A/\omega _M| =0.05$ at times after the formation of the waves.

As we seek to understand the role of MAC waves in the dipolar and multipolar dynamo regimes, we may separate the fast and slow MAC wave parts of the general solution, which is a linear superposition of the two wave solutions (see Sreenivasan & Maurya Reference Sreenivasan and Maurya2021). For example

(2.26)\begin{equation} \left. \begin{aligned} \hat{u}_{x,f} & =M_1 \mathrm{e}^{\mathrm{i}\lambda^P_1 t}+ M_2 \mathrm{e}^{\mathrm{i}\lambda^P_2 t} + A_{1} \mathrm{e}^{\mathrm{i}{\lambda}_1^H t}+ A_{2} \mathrm{e}^{\mathrm{i} {\lambda}_2^H t},\\ \hat{u}_{y,f} & = D_1 \mathrm{e}^{\mathrm{i}\lambda_1 t}+ D_2 \mathrm{e}^{\mathrm{i}\lambda_2 t},\\ \hat{u}_{z,f} & =N_1 \mathrm{e}^{\mathrm{i} \lambda^P_1 t}+ N_2 \mathrm{e}^{\mathrm{i} \lambda^P_2 t} +C_{1} \mathrm{e}^{\mathrm{i} {\lambda}_1^H t}+C_{2} \mathrm{e}^{\mathrm{i}{\lambda}_2^H t}, \end{aligned} \right\} \end{equation}

and

(2.27)\begin{equation} \left. \begin{aligned} \hat{u}_{x,s} & = M_3 \mathrm{e}^{\mathrm{i} \lambda^P_3 t} + M_4 \mathrm{e}^{\mathrm{i}\lambda^P_4 t} +A_{3} \mathrm{e}^{\mathrm{i}{\lambda}_3^H t} + A_{4} \mathrm{e}^{\mathrm{i}{\lambda}_4^H t},\\ \hat{u}_{y,s} & = D_3 \mathrm{e}^{\mathrm{i}\lambda_3 t}+D_4 \mathrm{e}^{\mathrm{i}\lambda_4 t},\\ \hat{u}_{z,s} & =N_3 \mathrm{e}^{\mathrm{i}\lambda^P_3 t}+ N_4 \mathrm{e}^{\mathrm{i}\lambda^P_4 t}+C_{3} \mathrm{e}^{\mathrm{i}{\lambda}_3^H t}+C_{4} \mathrm{e}^{\mathrm{i} {\lambda}_4^H t}, \end{aligned} \right\} \end{equation}

where the subscripts $f$ and $s$ on the left-hand sides of (2.26) and (2.27) denote the fast and slow wave parts of the solution. Figure 3(a) shows the variation of fundamental frequencies with $Le$. To compute the frequencies, the mean wavenumbers are first calculated through ratios of $L^2$ norms; e.g.

(2.28a,b)\begin{equation} \bar{k}_x = \frac{\|k_x \hat{u}\| }{\| \hat{u}\|}, \quad \bar{k} = \frac{\| k \hat{u}\|}{\| \hat{u}\|}, \end{equation}

which are based on the velocity field. The values of $|\omega _M/\omega _C|$, given in brackets below the horizontal axis of figure 3(b), are systematically higher than that of $Le$, which is essentially the initial value of this ratio. The enhanced instantaneous value of $|\omega _M/\omega _C|$ is due to the anisotropy of the columnar flow, and would not be evident if $\boldsymbol {\varOmega }$ were aligned with $\boldsymbol {B}$ (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021). For $E_\eta = 2 \times 10^{-5}$, all calculations for $Le > 2 \times 10^{-3}$ satisfy the inequality $|\omega _C| > |\omega _M| > |\omega _A|>|\omega _\eta |$, thought to be essential for axial dipole formation in convective dynamos. Figure 3(b) shows the variation of the dimensionless helicity $h^*$ of the fast and slow MAC waves, obtained by summing the helicity at all points in $(x,z)$ for $z<0$ and $y=0$ and then normalizing this value by the non-magnetic helicity. For $Le > 2 \times 10^{-3}$ ($|\omega _M/\omega _C| > 0.045$), the slow wave helicity increases dramatically, and for $Le \sim 10^{-2}$ ($|\omega _M/\omega _C| \sim 0.1$), the slow wave helicity is greater than the fast wave helicity. This result is consistent with the higher intensity of the slow wave motions relative to that of the fast wave motions for $|\omega _M/\omega _C| \sim 0.1$, inferred from the fast Fourier transform (FFT) spectra in dynamo simulations (Varma & Sreenivasan Reference Varma and Sreenivasan2022).

Figure 3. (a) Variation of absolute values of frequencies with $Le$. (b) Variation with $Le$ of the total kinetic helicity in the region $z<0$ normalized by the non-magnetic helicity. The values of $|\omega _M/\omega _C|$ that correspond to the values of $Le$ are given in brackets below the horizontal axis. All calculations are performed for $E_\eta = 2 \times 10^{-5}$. The slow wave frequency, $\omega _s$, takes non-zero values for $Le > 2 \times 10^{-3}$, when $|\omega _M| > |\omega _A|$.

In figure 4, the contours of the fast and slow MAC wave helicities are shown at two times for $|\omega _M/\omega _C|=0.13$, which lies in the region of slow wave dominance in figure 3(b). The fast waves split in two and propagate rapidly along $z$. The slow waves do not propagate as far as the fast waves at the same time due to their lower group velocity. Yet, as indicated by the colour bars, the slow waves are markedly more intense than the fast waves. Both fast and slow wave columns propagate along $x$ at the Alfvén velocity (§ 2.5 below).

Figure 4. Helicity on the section $y=0$ of fast (a,b) and slow (c,d) MAC waves at two times, $t/t_\eta = 2.5\times 10^{-3}$ (a,c) and $t/t_\eta =1 \times 10^{-2}$ (b,d). Here, $E_\eta = 2 \times 10^{-5}$ and $Le=0.03$ ($|\omega _M/\omega _C| =0.13$).

The effect of progressively increasing the buoyant forcing on the fast and slow waves is examined in the following section.

2.5. Effect of progressively increasing buoyancy

For the regime given by $|\omega _C| \gg |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$, the roots of the homogeneous equation (2.13) are approximated by (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021)

(2.29)$$\begin{gather} \lambda_{1,2} \approx{\pm} \left(\omega_C+ \frac{\omega_M^2}{\omega_C} \right) + \mathrm{i} \, \frac{\omega_M^2\omega_\eta}{\omega_C^2}, \end{gather}$$

(2.30)$$\begin{gather}\lambda_{3,4} \approx{\pm} \left(\frac{\omega_M^2}{\omega_C}+ \frac{\omega_A^2}{2\omega_C}\right) + \mathrm{i} \, \omega_\eta \, \left(1-\frac{\omega_A^2}{2\omega_M^2}\right), \end{gather}$$

(2.31)$$\begin{gather}\lambda_{5} \approx \mathrm{i} \, \frac{\omega_A^2\omega_\eta}{\omega_M^2}. \end{gather}$$

As $|\omega _A|$ increases relative to $|\omega _M|$, the fast MAC waves, given by frequencies $\lambda _{1,2}$, are nearly unaffected. However, the slow waves, whose real frequencies are approximated by

(2.32)\begin{equation} \mbox{Re}(\lambda_{3,4}) \approx{\pm} \left(\frac{\omega_M^2}{\omega_C}+ \frac{\omega_A^2}{2\omega_C}\right) \approx{\pm} \frac{\omega_M^2}{\omega_C} \, \left(1+ \frac{\omega_A^2}{\omega_M^2} \right)^{1/2}, \end{equation}

for $|\omega _C| \gg |\omega _M|, |\omega _A|$ (Braginsky Reference Braginsky1967), would be significantly attenuated in an unstably stratified fluid with $\omega _{A}^2<0$ as $|\omega _A|$ nears $|\omega _M|$. We see below that the decrease of the slow wave frequency translates into the marked decrease of the slow wave helicity relative to the fast wave helicity.

Figure 5(a) indicates that both fast and slow MAC waves propagate along the mean-field direction $x$ such that $x/\delta = t/t_a$, where $t_a$ is the Alfvén wave travel time. Contrary to that in the inertial–Alfvén wave system discussed in Bardsley & Davidson (Reference Bardsley and Davidson2017), the Alfvén waves propagating in the direction orthogonal to the rotation axis are simply the degenerate form of the MAC waves (see also Varma & Sreenivasan Reference Varma and Sreenivasan2022). For small $|\omega _A/\omega _M|$, the helicity of slow wave motions is greater than that of fast waves, but as $|\omega _A/\omega _M|$ approaches unity, the slow wave helicity weakens considerably and falls below that of the fast wave. The effect of increasing buoyancy forcing on the fast and slow wave helicity is shown graphically in figure 6. The fast waves are practically unaffected by the strength of forcing as their intensity and $z$ propagation rate are nearly invariant for $|\omega _A/\omega _M|$ in the range 0.1–1 (figure 6a–c). The slow wave helicity, on the other hand, is substantially weakened as $|\omega _A/\omega _M|$ increases in the same range (figure 6d–f). For $|\omega _A/\omega _M| \approx 1$, a state is reached where the slow wave helicity is nearly zero. The induced magnetic field $b_z$ also weakens considerably with increasing $|\omega _A/\omega _M|$ (figure 6g–i), which indicates that only the slow waves have a direct bearing on field generation.

Figure 5. (a) Variation of the fast wave (dashed line) and slow wave (solid line) helicity for $z<0$, normalized by the non-magnetic helicity, with $x/\delta$ and $|\omega _A/\omega _M|$ at different times (measured in units of the Alfvén wave travel time $t_a$). Here, $E_\eta =2\times 10^{-5}$ and $Le=0.03$. (b) Variation of the normalized slow wave helicity with the local Rayleigh number $Ra_\ell$ (defined in (2.33)) and $|\omega _A/\omega _M|$ for different $Le$.

Figure 6. Fast MAC wave helicity (a–c), slow MAC wave helicity (d–f) and $z$-component of the induced magnetic field, $b_z$ (g–i) for $|\omega _A/\omega _M| = 0.1$ (a,d,g), 0.6 (b,e,h), 0.95 (c,f,i). The plots are generated at time $t/t_\eta =0.01$ for the parameters $Le=0.03$ and $E_\eta =2 \times 10^{-5}$.

Figure 5(b) shows the normalized slow wave helicity against a ‘local’ Rayleigh number based on the length scale of the initial perturbation

(2.33)\begin{equation} Ra_\ell = \dfrac{g \alpha |\gamma| \delta^2}{2 \varOmega \eta}, \end{equation}

as well as $|\omega _A/\omega _M|$. Evidently, the forcing needed to suppress the slow waves increases with $Le$, although the total suppression of these waves occurs universally at $|\omega _A/\omega _M| \approx 1$. This result prompts us to look at the condition for vanishing slow wave helicity through a relation between $Ra_\ell$ and a parameter $\varLambda$ defined by

(2.34)\begin{equation} \varLambda=\left(\dfrac{\omega_M^2}{\omega_C \omega_\eta}\right)_{\!0} \sim \dfrac{V_M^2}{2 \varOmega \eta}, \end{equation}

which measures the initial ratio of the slow MC wave frequency to the magnetic diffusion frequency. Figure 7 shows that the same linear relation between $Ra_\ell$ and $\varLambda$, whose values are tabulated in table 1, holds for any $E_\eta$. Both $Ra_\ell$ and $\varLambda$ are measurable in dynamo models, the latter being of the same order of magnitude as the Elsasser number – the square of the scaled magnetic field – in many models. If the state of vanishing slow wave helicity is taken as a proxy for polarity transitions, then we may expect a self-similar relationship between $Ra_\ell$ and $\varLambda$ in low-inertia dynamos, where the nonlinear inertial force is small compared with the Coriolis force. This idea is explored further in § 3.

Figure 7. Variation of the local Rayleigh number $Ra_\ell$, defined in (2.33), shown against $\varLambda$, defined in (2.34), for the state of approximately zero slow MAC wave helicity. Three values of $E_\eta$ are considered – the diamonds represent $E_\eta =6\times 10^{-7}$, circles represent $E_\eta =6\times 10^{-6}$ and triangles represent $E_\eta =2\times 10^{-5}$.

Table 1. Values of $Ra_\ell$, defined in (2.33), at different $\varLambda$, defined in (2.34), for suppression of slow MAC waves in the linear magnetoconvection calculations. The dimensionless parameters $Le$ and $E_\eta$ are defined in (2.25a,b).

3.1. Dipolar, multipolar and polarity-reversing dynamos

Figure 8 shows the magnetic colatitude of the dipole field, $\theta$ at the upper boundary obtained from spherical harmonic Gauss coefficients, as follows:

(3.7a,b)\begin{equation} \cos\theta = g_1^0/|\boldsymbol{m}|, \quad \boldsymbol{m}= (g_1^0, g_1^1, h_1^1), \end{equation}

where $g_1^0$, $g_1^1$ and $h_1^1$ are obtained from the Schmidt-normalized expansion for the scalar potential of the field (Glatzmaier Reference Glatzmaier2013, pp. 142–143). For $E=6 \times 10^{-5}$ and $Pm=Pr=5$, the evolution of $\theta$ is shown for runs at three closely spaced values of $Ra$ in strongly supercritical convection. For $Ra=20\,000$, the run that begins from a dipole seed field enters a chaotic multipolar state and subsequently regains dipolarity (figure 8a), a behaviour first noted by Sreenivasan & Kar (Reference Sreenivasan and Kar2018). For the same $Ra$, the run that begins from the saturated (strong field of mean square intensity $B^2 = O(1)$) state of $Ra=18\,000$ gives a stable dipole field throughout (figure 8d). Both these runs saturate at nearly identical dipolar states. For $Ra=21\,000$, the run that begins from a seed field is multipolar throughout (figure 8b). At the same $Ra$, the run that begins from a strong field produces occasional polarity reversals separated by well-defined periods of normal and reversed polarity (figure 8e), as in earlier reversing simulations (Glatzmaier et al. Reference Glatzmaier, Coe, Hongre and Roberts1995; Kutzner & Christensen Reference Kutzner and Christensen2002) and experiments (Monchaux et al. Reference Monchaux2009). The radial magnetic fields at the outer boundary before and after a reversal are shown in figure 9. For still higher forcing ($Ra=24\,000$), both seed field and strong field runs give multipolar solutions (figure 8c,f). Therefore, polarity reversals happen in a narrow range of $Ra$ that lies between dipolar and multipolar regimes, possibly due to variations in outer boundary heat flux.

Figure 8. Evolution of dipole colatitude with magnetic diffusion time for three closely spaced values of $Ra$ in strongly driven dynamo simulations. The top panels (a)–(c) are for simulations starting from a seed magnetic field while the bottom panels (d)–(f) are for simulations starting from a strong magnetic field. The dynamo parameters are $E = 6 \times 10^{-5},Pm=Pr=5$; (a) $Ra=20\,000$, (b) $Ra=21\,000^{{{\dagger}} }$, (c) $Ra=24\,000$, (d) $Ra=20\,000$, (e) $Ra=21\,000^{\ddagger}$, (f) $Ra=24\,000$. For $Ra=21\,000$, the superscript ${{\dagger}}$ denotes a seed field start whereas the superscript $\ddagger$ denotes a strong field start.

Figure 9. The contours of the radial magnetic field at the outer boundary for $Ra=21\,000$ at magnetic diffusion times (a) $t_d=1.1$ and (b) $t_d=2.3$ shown in figure 8(e). The other dynamo parameters are $E = 6 \times 10^{-5}$, $Pm=Pr=5$.

The range of spherical harmonic degrees $l \leq l_E$ is of particular interest since kinetic helicity is known to be generated in the nonlinear dynamo in this range of energy-containing scales (see § 1). A relative helicity is defined that measures the augmentation of lower-hemisphere helicity in the nonlinear dynamo (magnetic) run relative to that in the equivalent non-magnetic simulation

(3.8)\begin{equation} H_r = \dfrac{h_{m} - h_{nm}}{h_{nm}}, \end{equation}

for $l \leq l_E$. Here, the subscripts $m$ and $nm$ denote the magnetic (dynamo) and non-magnetic values, respectively. The variation of $H_r$ with time in dynamos at $E=6 \times 10^{-5}$ and progressively increasing $Ra$ is given in figure 10. The runs that begin from a seed field and produce an axial dipole show an approximately twofold increase in helicity (figure 10a,b). For $Ra=21\,000$, the run that begins from a seed field and produces a multipolar solution shows no noticeable increase in helicity over the non-magnetic state (figure 10c). At the same $Ra$, the run that begins from a strong field and produces polarity reversals undergoes a decrease in helicity from its initial value to the non-magnetic value (figure 10d). We emphasize that the effect of the magnetic field on helicity generation can be experienced only in the energy-containing range of harmonic degrees $l \leq l_E$. If the entire spectrum is considered (Ranjan et al. Reference Ranjan, Davidson, Christensen and Wicht2020), the dipolar dynamo helicity would be lower than the non-magnetic helicity.

Figure 10. Relative helicity, $H_r$, as defined in (3.8), in dynamo simulations for (a) $Ra=3000$ (seed field start), (b) $Ra=18\,000$ (seed field start), (c) $Ra=21\,000^{{\dagger}}$ (seed field start), (d) $Ra=21\,000^{\ddagger}$ (strong field start). The superposed red lines are polynomial fits showing the trend of the evolution. The dotted vertical lines in (a,b) mark the approximate times of formation of the axial dipole from a multipolar field at the outer boundary in the run. The other dynamo parameters are $E = 6 \times 10^{-5}$, $Pm=Pr=5$.

The results in figure 8 and figure 10 prompt us to examine the nature of wave motions in dipolar and reversing dynamos. An important aim of this study is to show through the analysis of MAC waves that the formation of the dipole from a multipolar state and the onset of polarity reversals lie in similar regimes. Such an argument is essential to place a constraint on the Rayleigh number that admits polarity reversals (see § 3.3).

3.2. The suppression of slow MAC waves in polarity-reversing dynamos

Isolated density disturbances in a rotating stratified fluid layer excite MAC waves whose frequencies depend on the fundamental frequencies $\omega _M$, $\omega _A$ and $\omega _C$, representing Alfvén waves, internal gravity waves and linear inertial waves, respectively. In unstable stratification that drives planetary core convection, $\omega _A^2 <0$, where $|\omega _A|$ is simply a measure of the strength of buoyancy. Since the dimensional frequencies $\omega _M^2$, $-\omega _A^2$ and $\omega _C^2$ in the dynamo are given by (Varma & Sreenivasan Reference Varma and Sreenivasan2022)

(3.9a–c)\begin{equation} \omega_{M}^2 =\frac{(\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{k})^2}{\mu \rho},\quad -\omega_{A}^2= g\alpha\beta\left(\frac{ k_{z}^2+ k_{\phi}^2}{k^2}\right),\quad \omega_{C}^2 = \frac{4(\boldsymbol{\varOmega} \boldsymbol{\cdot} \boldsymbol{k})^2}{ k^2}, \end{equation}

and scaling the frequencies by $\eta /L^2$, we obtain, in dimensionless units,

(3.10a–c)\begin{equation} \omega_M^2 = \frac{Pm}{E} (\boldsymbol{B} \boldsymbol{\cdot} {\boldsymbol k})^2, \quad -\omega_A^2 = \frac{Pm^2 Ra}{Pr \,E} \, \left(\frac{{k_z}^2 + k_{\phi}^2}{k^2}\right), \quad \omega_C^2 = \frac{Pm^2}{E^2} \frac{k_z^2}{k^2}, \end{equation}

where $k_s$, $k_{\phi }$ and $k_z$ are the radial, azimuthal and axial wavenumbers in cylindrical coordinates $(s,\phi,z)$, $k_\phi =m/s$, where $m$ is the spherical harmonic order, and $k^2=k_s^2+k_\phi ^2+k_z^2$. Here, $\omega _A$ is evaluated on the equatorial plane where the buoyancy force is maximum. The magnetic (Alfvén) wave frequency $\omega _M$ is based on the three components of the measured magnetic field at the peak-field location. The wavenumber $k_\phi$ is evaluated at $s=1$, approximately mid-radius of the spherical shell.

In the limit of zero magnetic diffusion ($\omega _\eta =0$), the characteristic equation (2.13) reduces to

(3.11)\begin{equation} \lambda^4-\lambda^2(\omega_A^2+\omega_C^2+2\omega_M^2)+\omega_A^2\omega_M^2+\omega_M^4=0, \end{equation}

the roots of which are (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021)

(3.12)$$\begin{gather} \lambda_{1,2} ={\pm}\frac{1}{\sqrt{2}} \sqrt{\omega_A^2+\omega_C^2+2\omega_M^2+\sqrt{\omega_A^4+2\omega_A^2\omega_C^2+4\omega_M^2\omega_C^2+\omega_C^4}}, \end{gather}$$

(3.13)$$\begin{gather}\lambda_{3,4} ={\pm}\frac{1}{\sqrt{2}} \sqrt{\omega_A^2+\omega_C^2+2\omega_M^2-\sqrt{\omega_A^4+2\omega_A^2\omega_C^2+4\omega_M^2\omega_C^2+\omega_C^4}}. \end{gather}$$

For the inequality $|\omega _C| > |\omega _M| > |\omega _A|$, $\lambda _{1,2}$ and $\lambda _{3,4}$ give the frequencies of the fast ($\omega _f$) and slow ($\omega _s$) MAC waves, respectively. While the fast waves are linear inertial waves weakly modified by the magnetic field and buoyancy, the slow waves are magnetostrophic.

In figure 11, the magnitudes of the fundamental frequencies are shown as a function of the spherical harmonic order $m$ in the saturated state of the dynamo run at $E=6 \times 10^{-5}$, and $Pr=Pm=5$. The frequencies are computed from (3.10a–c) using the mean values of the $s$ and $z$ wavenumbers. For example, real-space integration over $(s,\phi )$ gives the kinetic energy as a function of $z$, the Fourier transform of which gives the one-dimensional spectrum $\hat {u}^2 (k_z)$. Subsequently, we obtain

(3.14)\begin{equation} \bar{k}_z = \frac{\varSigma k_z \hat{u}^2 (k_z)} {\varSigma \hat{u}^2 (k_z)}. \end{equation}

A similar approach gives $\bar {k}_s$. The computed frequencies in figure 11(a–c), shown for dynamos with $Ra= 6000\unicode{x2013} 18\,000$, satisfy the inequality $|\omega _C| > |\omega _M| > |\omega _A|$ in a range of the spherical harmonic order $m$. The dashed vertical lines show the value of $m$ below which the helicity in the nonlinear dynamo is greater than that in the non-magnetic run at the same parameters. The frequency root $\lambda _3$ in (3.13), shown by the black lines, represents the slow MAC waves of frequency $\omega _s$ when the inequality $|\omega _C| > |\omega _M| > |\omega _A|$ holds true. Evidently, the scales of helicity generation in the nonlinear dynamo overlaps with the scales where the slow MAC waves are generated. The range of $m$ over which the above frequency inequality holds narrows down with $Ra$, and for the polarity-reversing dynamo with $Ra=21\,000$, this inequality does not exist at any $m$ (figure 11d).

Figure 11. Panels (a i,b i,c i,d i) show the absolute values of wave frequencies plotted for the saturated state of the dynamo. The dashed vertical lines show the upper boundary of the range of wavenumbers $m$ for which the helicity of the dynamo run is greater than that of the equivalent non-magnetic run. Panels of (a ii,b ii,c ii,d ii) show the spectral distribution of the power supplied to the axial dipole, defined in (3.15). The dynamo parameters are $E=6 \times 10^{-5}$ and $Pm=Pr=5$; (a) $Ra=6000$, (b) $Ra=18\,000$, (c) $Ra=20\,000$, (d) $Ra=21\,000$.

Figure 11(a–d) also shows the spectral distribution of the power supplied to the poloidal part of the axial dipole field $B^P_{\mathrm {10}}$, given by (e.g. Buffett & Bloxham Reference Buffett and Bloxham2002)

(3.15)\begin{equation} P_{\mathrm{10}}= \int_{V} \boldsymbol {B}^{P}_{\mathrm{10}} \boldsymbol{\cdot} [\boldsymbol{\nabla} \times(\boldsymbol {u} \times \boldsymbol{ B})_{m}] \, {\rm d}V, \end{equation}

where $\boldsymbol {u}$ and $\boldsymbol {B}$ share the same value of $m$ (Bullard & Gellman Reference Bullard and Gellman1954). The axial dipole is predominantly generated in the wavenumbers where the MAC waves are generated, except in the dynamo close to reversals (figure 11c) where the peak of $P_{10}$ lies outside the MAC wave window. In the reversing dynamo without the MAC wave window, the power supplied to the dipole is small.

In figure 12, the upper panels show the absolute values of the frequencies for a range of spherical harmonic order $m$ at three different times during the evolution of the dynamo from a seed field. The lower panels show the $m$-distribution of the power supplied to the axial dipole. As the magnetic field increases from a seed, the slow MAC waves of frequency $\omega _s$ are absent at early times but are subsequently excited in the scales where the inequality $|\omega _{C}|>|\omega _{M}|>|\omega _{A}|$ is satisfied. Kinetic helicity is generated in these scales, where a peak of the axial dipole power $P_{10}$ is also noted. In the growth phase of the dynamo where the field is multipolar, the dominant contribution to $P_{10}$ comes from higher wavenumbers of $\boldsymbol {u}$ and $\boldsymbol {B}$ within the MAC wave window (figure 12b); here, helicity is generated via slow MAC waves at the peak-field locations (figures S1 and S2 in the supplementary material available at https://doi.org/10.1017/jfm.2024.1). Once the axial dipole is established, it can hold itself up (Sreenivasan & Jones Reference Sreenivasan and Jones2011) by inducing the generation of slow waves (figure 12c).

Figure 12. Panels of (a i–c i) show the absolute values of wave frequencies plotted for three different times in the growth phase of a dynamo starting from a seed field. The shaded grey area shows the range of scales where the helicity of the dynamo run is greater than that of the equivalent non-magnetic run. Panels (a ii–c ii) show the spectral distribution of the power supplied to the axial dipole, defined in (3.15). The dynamo parameters are $Ra= 18\,000, E=6 \times 10^{-5}$ and $Pm=Pr=5$. The plots are shown at times (a) $t_d=0.073$, (b) $t_d=0.168$, (c) $t_d=0.3$.

In figure 13, the dynamo frequencies are computed from (3.10a–c) using the mean values of the $s$, $\phi$ and $z$ wavenumbers in the saturated state of the dynamo in the range $l \le l_E$, which represents the energy-containing scales. The mean spherical harmonic order $\bar {m}$ is evaluated through a weighted average as in (3.14), but over the range of $m$ within $l \le l_E$. As the field increases from a small seed value in the dipolar dynamo run at $Ra=3000$ (figure 13a), slow MAC waves of frequency $\omega _s$ are first excited when $|\omega _M| > |\omega _A|$. The formation of the axial dipole from a chaotic field, marked by the dotted vertical line, follows slow wave excitation. In the run at $Ra=21\,000$ that begins from a seed field and produces a multipolar solution, $|\omega _M|$ remains lower than $|\omega _A|$ throughout (figure 13b), so the slow waves are never excited. Since polarity reversals are found in strongly driven dynamos starting from a strong field (see figure 8e), the variation of the frequencies with strength of forcing in dynamos starting from a strong field is studied in figure 13(c). Here, we note that $|\omega _M|$ falls below $|\omega _A|$ at $Ra \approx 21\,000$, which indicates that polarity reversals would indeed onset in the regime $|\omega _M| \approx |\omega _A|$ when slow MAC waves disappear. Further increase in forcing places the dynamo in a multipolar regime, the transition to which is marked by the dotted vertical line in figure 13(c). Thus, polarity reversals take place in a narrow range of $Ra$ situated between the dipolar and multipolar regimes. The volume-averaged mean square value of the axial dipole field is much smaller in the multipolar regime of $Ra=21\,000$ than in the stable dipole regime of $Ra=3000$ (figure 13d), which suggests that the slow MAC waves have an important role in the formation of the axial dipole. In the run at $Ra=21\,000$ that begins from a strong field and produces polarity reversals, the dipole intensity decreases and eventually reaches the same order of magnitude as that in the multipolar run. This is consistent with the suppression of the slow waves at this $Ra$, and reflected in the decrease in kinetic helicity to the non-magnetic value (figure 10d).

Figure 13. (a,b) Absolute values of the measured frequencies $\omega _M$, $\omega _A$, $\omega _C$ and $\omega _s$ plotted against time (measured in units of the magnetic diffusion time, $t_d$) for $Ra=3000$ (a) and $Ra=21\ 000$ (b). These simulations study the evolution of the dynamo starting from a small seed magnetic field. The dotted vertical line in (a) marks the time of formation of the axial dipole from a multipolar field. (c) Frequencies shown against $Ra$ for saturated dynamos starting from a strong field. The dashed vertical line indicates the value of $Ra$ at which the slow MAC wave frequency $\omega _s$ goes to zero as $\omega _M \approx \omega _A$. This marks the transition from the dipolar regime to the polarity-reversing regime. The dotted vertical line marks the beginning of the multipolar regime. (d) The Elsasser number of the axial dipole field component, based on its root mean square value, for $Ra=3000$, $Ra=21\,000^{{\dagger}}$ (seed field start) and $Ra=21\,000^{\ddagger}$ (strong field start). The dynamo parameters are $E = 6 \times 10^{-5}$, $Pm=Pr=5$. The colour codes in (a) are also used in (b,c).

Figure 14 shows the measurement of wave motion in the saturated state of dynamos at $E=6 \times 10^{-5}$ and $Pr=Pm=5$ and three values of $Ra$ spanning the dipole-dominated regime and reversals. Contours of $\dot {u}_z$ at cylindrical radius $s=1$ are plotted over small time windows in which the ambient magnetic field and wavenumbers are approximately constant. These contours show the propagation paths of the fluctuating $z$ velocity. In line with the discussion so far, the measurement of axial motions is limited to the energy-containing scales $l \leq l_E$, with no restriction on the wavenumber. The measured axial group velocity of the waves, $U_{g,z}$ – obtained from the slope of the black lines in figure 14 – is compared with the estimated fast ($U_f$) and slow ($U_s$) group velocities obtained by taking the derivatives of the respective frequencies in (3.12) and (3.13) with respect to $k_z$ (table 3). The theoretical frequencies $\omega _f$ and $\omega _s$ are estimated using the three components of the magnetic field at the peak-field location and the mean values of $k_s$, $k_z$ and $m$ over the range of energy-containing scales, $l \leq l_E$. In the dipole-dominated run at $Ra=6000$, the slow and fast waves coexist, but the intensity of slow wave motions measured at the peak-field locations is at least as high as that of the fast wave motions. At $Ra=20\,000$, the increasing occurrence of the fast waves alongside the slow waves is noted from the nearly vertical propagation paths (figure 14b). The measured slow wave group velocity at $Ra=20\,000$ is greater than that at $Ra=6000$, which reflects the larger self-generated field at the higher Rayleigh number. In the reversing dynamo at $Ra=21\,000$, slow waves are totally absent while the fast waves are abundant (figure 14c).

Figure 14. Contour plots of $\partial {u}_z/\partial t$ at cylindrical radius $s=1$ for $l\leq l_E$ and small intervals of time in the saturated state of three dynamo simulations. The parallel black lines indicate the predominant direction of travel of the waves and their slope gives the group velocity $U_{g,z}$. The dynamo parameters are $E=6 \times 10^{-5}$, $Pr=Pm=5$; (a) $Ra=6000$, (b) $Ra=20\ 000$, (c) $Ra= 21\ 000$. The estimated group velocity of the fast and slow MAC waves ($U_f$ and $U_s$ respectively) and $U_{g,z}$ are given in table 3.

Table 3. Summary of the data for MAC wave measurement in the dynamo models. The sampling frequency $\omega _n$ is chosen to ensure that the fast MAC waves are not missed in the measurement of group velocity. The values of $\omega _M^2$, $-\omega _A^2$ and $\omega _C^2$ are calculated from (3.10a–c) using the mean values of $m$, $k_s$ and $k_z$ over the range of energy-containing scales, $l \leq l_E$. The measured group velocity in the $z$ direction ($U_{g,z}$) is compared with the estimated fast ($U_f$) or slow ($U_s$) MAC wave velocity.

The dominance of the slow MAC waves in the dipole-dominated dynamo and the fast MAC waves in the reversing dynamo is further evident in figure 15, where the FFT of $\dot {u}_z$ is shown. The flow largely consists of waves of frequency $\omega \sim \omega _s$ in the dipolar dynamo (figure 15a), whereas in the reversing dynamo, waves of much higher frequency $\omega \sim \omega _f$ are dominant (figure 15b). The coloured vertical lines in figure 15(a,b) give the estimated magnitudes of $\omega _C$ and $\omega _M$ normalized by $\omega _f$, where $\omega _M$ is based on the peak magnetic field.

Figure 15. (a) The FFT spectrum of $\partial u_z/\partial t$ at cylindrical radius $s = 1$ for the scales $l \le l_E$. The spectra are computed at discrete $\phi$ points and then averaged azimuthally. The solid black vertical lines in (a) and (b) correspond to $\omega /\omega _f=1$ and the dotted vertical line in (a) corresponds to $\omega _s^\star =\omega _s/\omega _f$, where $\omega _f$ and $\omega _s$ are the estimated fast and slow MAC wave frequencies. The coloured vertical lines give $\omega _C$ and $\omega _M$, both normalized by $\omega _f$. The dynamo parameters are $E = 6 \times 10^{-5}$ and $Pm=Pr=\,$5; (a) $Ra=6000$, (b) $Ra=21\,000$.

3.3. Self-similarity of the dipole–multipole transition

In the presence of small but finite magnetic diffusion, the slow MAC waves are known to disappear in an unstably stratified medium for $|\omega _A/\omega _M| \approx 1$. The same condition must hold for the appearance of slow waves in a medium where the magnetic field progressively increases from a small value. In simulations starting from a small seed field, the earliest time of excitation of the slow MAC waves is noted from group velocity measurements at closely spaced times during the growth phase of the dynamo. The peak Elsasser number $\varLambda =B_{peak}^2$ at this time is obtained from the three components of the field at the peak-field location, and presented in the last column of table 2. In each of the three dynamo series considered in this study, the last run is a polarity-reversing dynamo, for which $\varLambda$ is the measured peak Elsasser number when slow MAC waves cease to exist in the run starting from the saturated (strong field) state of the penultimate run in that series. Figure 16(a) shows that the variation of $Ra$ with $\varLambda$ in the three dynamo series is nearly linear. The Rayleigh number corresponding to reversals (at which slow MAC waves disappear) also lies on this line, indicating that the appearance and disappearance of MAC waves are in similar regimes.

Figure 16. (a) Variation of the modified Rayleigh number $Ra$ with the peak Elsasser number $\varLambda$ (square of the peak magnetic field) at both excitation and suppression of slow MAC waves. The hollow symbols represent the states where slow waves are first excited as the dynamos evolve from a small seed field; the filled symbols represent the states where slow waves are suppressed in the polarity-reversing dynamos. The parameters of the three dynamo series and their symbolic representations are as follows: $E=3\times 10^{-4},Pm=Pr=20$ (circles), $E=6\times 10^{-5},Pm=Pr=5$ (triangles), $E=1.2\times 10^{-5},Pm=Pr=1$ (diamonds). (b) Variation of the local Rayleigh number $Ra_\ell$, defined in (3.16), with $\varLambda$. The values of $Ra$, $Ra_\ell$ and $\varLambda$ in the plots are given in table 2. The sections 1 and 2 marked on the self-similar line are analysed further in figures 17 and 18 below.

Following the analysis in figure 7, where the Rayleigh number based on the length scale of the buoyant perturbation was studied, we define a local Rayleigh number in the dynamo

(3.16)\begin{equation} Ra_\ell = \dfrac{g \alpha \beta}{2 \varOmega \eta}\left(\dfrac{2 {\rm \pi}}{\bar{m}}\right)^2, \end{equation}

where $\bar {m}$ is the mean spherical harmonic order evaluated over $m$ within the energy-containing scales $l \leq l_E$. The behaviour of $Ra_\ell$, which is defined for the scales where the MAC waves are excited by buoyancy, is approximately self-similar (figure 16b). The values of $Ra_\ell$ at the onset of polarity reversals, where the slow MAC waves disappear, also lie on the same self-similar branch. While the conventional Rayleigh numbers $Ra$ at the onset of reversals in the three dynamo series lie far apart (see the filled symbols in figure 16a), the respective local Rayleigh numbers $Ra_\ell$ are remarkably close, and $\sim 10^4$ (see the filled symbols in figure 16(b) and table 2). The magnetic Ekman number based on $\bar {m}$ in the three dynamo series takes values of $E_\eta \sim 10^{-5}$ for a wide range of $Ra$, which indicates an energy-containing length scale $\sim$ 10 km for Earth (see § 1).

The fact that the linear variation of $Ra_\ell$ with $\varLambda$ demarcates the boundary between dipolar and multipolar states is evident by traversing the sections 1 and 2 marked on the $Ra_\ell$ line from left to right (figure 16b). In practice, this is done by following the evolution of the dynamo from a small seed field. Figure 17(a) shows the section 1 within dashed vertical lines, where $|\omega _M|$ crosses $|\omega _A|$. The variation of the dipole colatitude $\theta$, shown in figure 17(b), indicates a multipolar field until this crossing, and a stable dipole thereafter. While the flow is predominantly made up of fast MAC waves of frequency $\omega \sim \omega _f$ before the transition, the slow waves of frequency $\omega \sim \omega _s$ are dominant after the transition (figure 17c,d). The multipole–dipole transitions are further evident in the contour plots of the radial magnetic field at the outer boundary, given in figure 18 for sections 1 and 2 marked in figure 16(b).

Figure 17. (a) Evolution the dynamo frequencies in a simulation beginning from a small seed magnetic field at $E = 1.2 \times 10^{-5}$, $Pm=Pr=1$, $Ra=4000$. The two dashed vertical lines at $t_d=0.55$ and $t_d=0.66$ represent the end points of section 1 marked in figure 16(b) with peak Elsasser numbers $\varLambda = 32$ and 56, respectively. (b) Evolution of the dipole colatitude in the above simulation. (c,d) FFT spectra of $\partial u_z/\partial t$ at cylindrical radius $s = 1$ for the scales $l \le l_E$ at $\varLambda = 32$ and 56, respectively. The spectra are computed at discrete $\phi$ points and then averaged azimuthally. In (d), $\omega _s^* = \omega _s/\omega _f$, where $\omega _f$ and $\omega _s$ are the estimated fast and slow MAC wave frequencies.

Figure 18. Shaded contours of the radial magnetic field at the outer boundary in two dynamo simulations: (a,b) $Ra_\ell =2505$, corresponding to section 1 in figure 16(b), at $\varLambda = 32$ and 56, respectively. The dynamo parameters are $E=1.2\times 10^{-5}$, $Ra=4000$, $Pm=Pr=1$; (c,d) $Ra_\ell =11\,740$, corresponding to section 2 in figure 16(b), at $\varLambda =$ 187 and 216, respectively. The dynamo parameters are $E=6\times 10^{-5}$, $Ra=18\,000$, $Pm=Pr=5$.