2.1. Problem set-up and governing equations

A localised density perturbation $\rho ^\prime$ is situated in an unstably stratified fluid subject to a uniform magnetic field and rapid rotation. 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{align} \varTheta _0=C \exp \big [-\big(x^2+y^2+z^2 \big)/\delta ^2 \big ], \end{align}

where $C$ is a constant and $\delta$ is the length scale of the perturbation. Figure 1 shows the initial perturbation, which evolves under the influence of gravity $\boldsymbol{g} = -g \hat {\boldsymbol{e}}_y$ , background rotation $\boldsymbol{\varOmega } = \varOmega \hat {\boldsymbol{e}}_z$ and a uniform magnetic field $\boldsymbol{B}_0 = B_0 \hat {\boldsymbol{e}}_y$ in Cartesian coordinates $(x,y,z)$ . The basic state temperature $T_0$ is assumed to have constant variations in both the vertical ( $y$ ) and horizontal ( $z$ ) directions (Teed et al. Reference Teed, Jones and Hollerbach2010), which results in a steady mean flow $\boldsymbol{u}_0$ satisfying the thermal wind equation

(2.2) \begin{align} 2\varOmega \frac {\partial \boldsymbol{u}_0}{\partial z}= -g \alpha \hat {\boldsymbol{e}}_y \times \boldsymbol{\nabla }T_0. \end{align}

Here, the mean flow corresponds to a constant shear,

(2.3) \begin{align} \boldsymbol{u}_0=u_c z\, \hat {\boldsymbol{e}}_x, \end{align}

which on substitution in (2.2) in scalar form,

(2.4) \begin{align} \begin{aligned} 2 \varOmega \,u_c = - g \alpha \dfrac {\partial T_0}{\partial z}, \end{aligned} \end{align}

gives the basic state temperature,

(2.5) \begin{align} T_0 =\beta _y y+\beta _z z, \end{align}

where $\beta _y = \partial T_0/\partial y \lt 0$ is the mean vertical temperature gradient and $\beta _z = - 2\varOmega u_c/g \alpha \gt 0$ is the mean horizontal (lateral) temperature gradient.

Figure 1.

Initial state of a density perturbation $\rho ^{\prime }$ that evolves in an unstably stratified fluid subject to a uniform magnetic field $\boldsymbol{B}_0 = B_0 \hat {\boldsymbol{e}}_y$ , background rotation $\boldsymbol{\varOmega } = \varOmega \hat {\boldsymbol{e}}_z$ and gravity $\boldsymbol{g} = -g \hat {\boldsymbol{e}}_y$ in Cartesian coordinates $(x,y,z)$ . The lateral variation in temperature produces a mean flow $\boldsymbol{u}_0=u_c z \,\hat {\boldsymbol{e}}_x$ .

The initial temperature perturbation (2.1) gives rise to a velocity field $\boldsymbol{u}$ , which interacts with the ambient magnetic field $\boldsymbol{B}_0$ to produce the induced magnetic field $\boldsymbol{b}$ . The initial velocity perturbation and induced field are both zero. Since all the variables are decomposed into their mean and perturbation parts, e.g.

(2.6a–c) \begin{align} T= T_0 + \varTheta , \quad \boldsymbol{U}= \boldsymbol{u}_0 + \boldsymbol{u}, \quad \boldsymbol{B}= \boldsymbol{B}_0 + \boldsymbol{b}, \end{align}

the following linearised equations give the evolution of $\boldsymbol{u}$ , $\boldsymbol{b}$ and $\varTheta$ :

(2.7) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} +u_c\, z \frac {\partial \boldsymbol{u}} {\partial x}+u_c u_z \hat {\boldsymbol{e}}_x +2\varOmega \hat {\boldsymbol{e}}_z\times \boldsymbol{u}= -\frac {1}{\rho }\boldsymbol{\nabla }p^\star + \frac {1}{\rho } \boldsymbol{j} \times \boldsymbol{B}_0 +g\alpha \varTheta \hat {\boldsymbol{e}}_y + \nu {\nabla} ^2 \boldsymbol{u}, \\[-28pt] \nonumber \end{align}

(2.8) \begin{align} \frac {\partial \boldsymbol{b}}{\partial t} + u_c \,z \frac {\partial \boldsymbol{b}}{\partial x} = u_c b_z\hat {\boldsymbol{e}}_x +(\boldsymbol{B}_0\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}+\eta {\nabla} ^2 \boldsymbol{b}, \\[-28pt] \nonumber \end{align}

(2.9) \begin{align} \frac {\partial \varTheta }{\partial t} +u_c z \frac {\partial \varTheta }{\partial x} +u_y\beta _y+u_z\beta _z=\kappa {\nabla} ^2 \varTheta , \\[-28pt] \nonumber \end{align}

(2.10) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} =0, \\[-28pt] \nonumber \end{align}

(2.11) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{b}=0, \\[0pt] \nonumber \end{align}

where $\boldsymbol{j}$ is the electric current density, $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity, $\eta$ is the magnetic diffusivity and $p^*=p -(\rho /2) |{\boldsymbol{\varOmega }} \times {\boldsymbol{x}}|^2$ is a modified pressure.

2.2. Solutions for the perturbation velocity field

Taking the curl of (2.7) and (2.8), and eliminating $\boldsymbol{j}$ , we obtain

(2.12) \begin{align} P_\nu P_\eta \boldsymbol{\zeta } & =2(\boldsymbol{\varOmega }\boldsymbol{\cdot }\boldsymbol{\nabla }) P_\eta \boldsymbol{u} \nonumber\\[5pt]& \quad -u_c P_\eta \bigg [\!-\frac {\partial u_y}{\partial x} \hat {\boldsymbol{e}}_x+\bigg (\frac {\partial u_x}{\partial x} +\frac {\partial u_z}{\partial z}\bigg )\hat {\boldsymbol{e}}_y -\frac {\partial u_z}{\partial y}\hat {\boldsymbol{e}}_z\bigg ] +g \alpha P_\eta \bigg ( \!-\frac {\partial \varTheta }{\partial z}\hat {\boldsymbol{e}}_x +\frac {\partial \varTheta }{\partial x}\hat {\boldsymbol{e}}_z\bigg )\nonumber\\[5pt]& \quad +u_c\frac {(\boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{\nabla })}{\rho \mu } \bigg [\frac {\partial b_y}{\partial x}\hat {\boldsymbol{e}}_x +\bigg (-\frac {\partial b_x}{\partial x} +\frac {\partial b_z}{\partial z}\bigg )\hat {\boldsymbol{e}}_y -\frac {\partial b_z}{\partial y}\hat {\boldsymbol{e}}_z\bigg ] +\frac {(\boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{\nabla })^2}{\rho \mu }\boldsymbol{\zeta }, \end{align}

where $P_\nu =({\partial }/{\partial t}) +u_c z ({\partial }/{\partial x})-\nu {\nabla} ^2$ , $P_\eta =({\partial }/{\partial t}) +u_c z ( {\partial }/{\partial x})-\eta {\nabla} ^2$ , $\boldsymbol{\zeta }=\boldsymbol{\nabla }\times \boldsymbol{u}$ is the vorticity and $\mu$ is the magnetic permeability. Successive elimination of $\boldsymbol{\zeta }$ and $\varTheta$ through the $z$ component of the curl of (2.12), the $z$ component of (2.8) and (2.9) gives

(2.13) \begin{align} P^2 P_\kappa (-{\nabla} ^2 u_z)& = P\bigg [-g \alpha P_\eta \frac {\partial ^2}{\partial y\partial z} \big(\beta _y u_y +\beta _z u_z \big)\bigg. \nonumber \\[5pt]& \quad \bigg. +u_c P_\kappa \frac {B_0}{\rho \mu } \frac {\partial }{\partial y} \bigg (-\frac {\partial ^2 b_x}{\partial x^2} +\frac {\partial ^2 b_z}{\partial x \partial z} -\frac {\partial ^2 b_y}{\partial x \partial y}\bigg )\bigg ]\nonumber \\[5pt]& \quad +2\varOmega \frac {\partial }{\partial z} P_\kappa \bigg [2 \varOmega \frac {\partial }{\partial z} P_\eta ^2u_z+u_c P_\eta ^2\frac {\partial u_z}{\partial y} -u_c\frac {B_0^2}{\rho \mu } \frac {\partial ^3 u_z}{\partial y^3}\bigg ]\nonumber\\[5pt]& \quad-2\varOmega g \alpha \frac {\partial ^2}{\partial x\partial z} P_\eta ^2 \big(\beta _y u_y+\beta _z u_z \big), \end{align}

where $P= P_\nu P_\eta -B_0^2/\rho \mu$ and $P_\kappa =({\partial }/{\partial t}) +u_c z ({\partial }/{\partial x})-\kappa {\nabla} ^2$ . For zero horizontal variation in temperature, $\beta _z, \, u_c \to 0$ and (2.13) reduces to the classical MAC wave equation (Braginsky Reference Braginsky1967; Busse et al. Reference Busse, Dormy, Simitev and Soward2007, pp. 165–168). However, since $x$ is not a preferred horizontal direction, setting $\partial (\,\,)/\partial x = 0$ in (2.13) (e.g. Hathaway, Gilman & Toomre Reference Hathaway, Gilman and Toomre1979) gives

(2.14) \begin{align} \begin{aligned} &\bigg [\bigg (\frac {\partial }{\partial t} -\nu {\nabla} _*^2\bigg )\bigg (\frac {\partial }{\partial t} -\eta {\nabla} _*^2\bigg ) -\frac {B_0^2}{\rho \mu } \, \frac {\partial ^2}{\partial y^2} \bigg ]^2 \bigg (\frac {\partial }{\partial t}-\kappa {\nabla} _*^2\bigg ) \big({-}{\nabla} _*^2 \hat {u}_z \big)\\&= -g \alpha \bigg (\frac {\partial }{\partial t} -\eta {\nabla} _*^2\bigg ) \bigg [\bigg (\frac {\partial }{\partial t} -\nu {\nabla} _*^2\bigg )\bigg (\frac {\partial }{\partial t} -\eta {\nabla} _*^2\bigg ) -\frac {B_0^2}{\rho \mu }\, \frac {\partial ^2}{\partial y^2} \bigg ] \, \frac {\partial ^2}{\partial y \partial z} \big(\beta _y \hat {u}_y+\beta _z \hat {u}_z \big)\\&+2 \varOmega \, \frac {\partial }{\partial z} \bigg (\frac {\partial }{\partial t}-\kappa {\nabla} _*^2\bigg ) \bigg [2\varOmega \frac {\partial }{\partial z}\bigg (\frac {\partial }{\partial t} -\eta {\nabla} _*^2\bigg )^2+u_c \frac {\partial }{\partial y} \bigg (\frac {\partial }{\partial t} -\eta {\nabla} _*^2\bigg )^2 -u_c \frac {B_0^2}{\rho \mu } \frac {\partial ^3}{\partial y^3}\bigg ] \hat {u}_z, \end{aligned} \end{align}

where ${\nabla} _*^2=\partial ^2/\partial y^2+\partial ^2/\partial z^2$ .

Now, applying the two dimensional Fourier transform defined by

(2.15) \begin{align} \mathcal{F}\big (\boldsymbol{A}\big ) = \hat {\boldsymbol{A}}= \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\boldsymbol{A} \mbox{e}^{- {i} \, (k_y \, y+k_z \, z)}\,\mathrm{d}y\, \mathrm{d}z \end{align}

to (2.14), the following homogeneous equation is obtained:

(2.16) \begin{align} \bigg [\bigg (\frac {\partial }{\partial t}& +\nu k^2\bigg )\bigg (\frac {\partial }{\partial t} +\eta k^2\bigg )+\frac {B_0^2\, k_y^2}{\rho \mu }\bigg ]^2 \bigg (\frac {\partial }{\partial t}+\kappa k^2\bigg )k^2 \, \hat {u}_z \nonumber\\[5pt]=& - g \alpha k_z^2 \bigg (\frac {\partial }{\partial t} +\eta k^2\bigg ) \bigg [\bigg (\frac {\partial }{\partial t} +\nu k^2\bigg )\bigg (\frac {\partial }{\partial t}+\eta k^2\bigg ) +\frac {B_0^2\, k_y^2}{\rho \mu }\bigg ] \left (\beta _y - \beta _z \frac {k_y}{k_z} \right )\, \hat {u}_z \nonumber\\[5pt]& \qquad +2 \varOmega \, \mathrm{i}\, k_z \bigg (\frac {\partial }{\partial t} +\kappa k^2\bigg )\bigg [2 \varOmega \, \mathrm{i}\, k_z \bigg (\frac {\partial }{\partial t}+\eta k^2\bigg )^2+u_c\mathrm{i} k_y\bigg (\frac {\partial }{\partial t}+\eta k^2\bigg )^2 \nonumber \\[5pt] & \qquad +u_c\mathrm{i} k_y\frac {B_0^2\, k_y^2}{\rho \mu } \bigg ]\, \hat {u}_z, \end{align}

where $k^2=k_y^2+k_z^2$ and the following substitution is made from the continuity (2.10):

(2.17) \begin{align} \hat {u}_y=-\frac {k_z}{k_y}\hat {u}_z. \end{align}

Seeking a plane wave solution of the form $\hat {u}_z \sim \mbox{e}^{{i} \lambda t}$ , we obtain

(2.18) \begin{align} \begin{aligned} \big [\big (\mathrm{i}\lambda +\omega _\nu \big )&\big (\mathrm{i}\lambda +\omega _{\eta }\big )+ \omega _{M}^2\big ]^2 \big (\mathrm{i}\lambda +\omega _\kappa \big )=-\omega _{A, V}^2\big (\mathrm{i}\lambda +\omega _{\eta }\big )\big [\big (\mathrm{i}\lambda +\omega _\nu \big )\big (\mathrm{i}\lambda +\omega _{\eta }\big )+ \omega _{M}^2\big ]\\ &+\omega _{A, H}^2\big (\mathrm{i}\lambda +\omega _{\eta }\big )\big [\big (\mathrm{i}\lambda +\omega _\nu \big )\big (\mathrm{i}\lambda +\omega _{\eta }\big )+\omega _{M}^2\big ]-\omega _{C}^2 \big (\mathrm{i}\lambda +\omega _\kappa \big )\big (\mathrm{i}\lambda +\omega _{\eta }\big )^2\\ &+\omega _{A, H}^2\big (\mathrm{i}\lambda +\omega _{\kappa }\big )\big [\big (\mathrm{i}\lambda +\omega _\eta \big )^2+ \omega _{M}^2\big ], \end{aligned} \end{align}

where

(2.19a–g) \begin{gather} \begin{aligned} &\omega _C^2=4 \varOmega ^2 k_z^2/k^2, \quad \omega _{A,V}^2=g\alpha \beta _y k_z^2/k^2, \quad \omega _{A,H}^2=g\alpha \beta _z k_y k_z/k^2, \\ &\omega _M^2= B_0^2\, k_y^2/ \rho \mu = V_M^2 k_y^2, \quad \omega _{\eta }^2=\eta ^2 k^4, \quad \omega _\kappa ^2 =\kappa ^2 k^4, \quad \omega _\nu ^2=\nu ^2 k^4, \end{aligned} \end{gather}

represent the squares of the frequencies of linear inertial waves, vertical and horizontal parts of the buoyancy, Alfvén waves, magnetic diffusion, thermal diffusion, and viscous diffusion, respectively. Here, $V_M= B_0/\sqrt {\rho \mu }$ is the Alfvén wave velocity. In line with a recent study (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021) where both viscous and thermal diffusion are much smaller than magnetic diffusion, the characteristic equation has the following form in the limit of $\omega _\kappa ,\omega _\nu \to 0$ :

(2.20) \begin{align} \lambda ^5- 2 \mathrm{i}\omega _\eta \lambda ^4& - \big(\omega _C^2+\omega _\eta ^2+2\omega _M^2 \big.\nonumber \\[3pt] & \quad \big. +\omega _{A,V}^2-2\omega _{A,H}^2 \big) \lambda ^3 +2 \mathrm{i} \omega _\eta \big(\omega _C^2+\omega _M^2 +\omega _{A,V}^2-2\omega _{A,H}^2 \big) \lambda ^2 \nonumber \\[3pt] & \quad +\big(\omega _C^2\omega _\eta ^2+\omega _M^4+ \big(\omega _M^2 +\omega _\eta ^2 \big) \big.\nonumber \\[3pt] & \quad \big.\big(\omega _{A,V}^2-2\omega _{A,H}^2\big)\big) \lambda -\mathrm{i}\omega _\eta \omega _M^2 \big(\omega _{A,V}^2 -\omega _{A,H}^2 \big)=0. \end{align}

The general solution for $\hat {u}_z$ is given by

(2.21) \begin{align} \hat {u}_z = \sum _{m=1}^{5}D_{m} \mbox{e}^{{i} \lambda _{m}t}, \end{align}

where the coefficients $D_m$ are evaluated from the initial conditions of $\hat {u}_z$ and its derivatives (see § 2.3). Of the five terms in the expansion on the right-hand side of (2.21), 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.

2.3. Evaluation of spectral coefficients

From (2.21), the initial conditions for $\hat {u}_z$ and its time derivatives are given by

(2.22) \begin{align} \begin{aligned} \mathrm{i}^n \sum _{m=1}^{5}D_m \lambda _{m}^n =\bigg (\frac {\partial ^n\hat {u}_z}{\partial t^n}\bigg )_{t=0} =a_{n+1}, \quad n=0,1,2,3,4. \end{aligned} \end{align}

Algebraic simplifications give the right-hand sides of (2.22) as follows:

(2.23) \begin{align} a_1&=\hat {u}_z|_{t=0}=0, \nonumber\\[3pt]a_2&=\frac {\partial {\hat {u}_z}}{\partial {t}}|_{t=0}= -g \alpha \frac {k_y k_z}{k^2}\hat {\varTheta }_0,\nonumber\\[3pt]a_3&=\frac {\partial ^2 \hat {u}_z}{\partial {t^2}}|_{t=0}=0,\nonumber\\[3pt]a_4&=\frac {\partial ^3{\hat {u}_z}}{\partial {t^3}}|_{t=0}= -\big(\omega _M^2+\omega _C^2+\omega _{A,V}^2-2\omega _{A,H}^2 \big)\,a_2,\nonumber\\[3pt]a_5&=\frac {\partial ^4{\hat {u}_z}}{\partial {t^4}}|_{t=0}= \omega _M^2\omega _\eta \,a_2. \end{align}

The coefficients $D_m$ may now be obtained using the roots of (2.20). For example, we obtain

(2.24) \begin{align} 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)}, \\[-28pt] \nonumber \end{align}

(2.25) \begin{align} 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)} \\[0pt] \nonumber \end{align}

for the forward-travelling fast and slow wave solutions, respectively. We separate the fast and slow MAC wave parts of the general solution, which is a linear superposition of the two wave solutions (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021). For example,

(2.26) \begin{align} \begin{aligned} \hat {u}_{z,f}&=D_{1} \mathrm{e}^{ {i} {\lambda }_1 t} +D_{2} \mathrm{e}^{{i}{\lambda }_2 t},\\ \hat {u}_{z,s}&=D_{3} \mathrm{e}^{{i}{\lambda }_3 t} +D_{4}\mathrm{e}^{{i} {\lambda }_4 t}, \end{aligned} \end{align}

where the subscripts $f$ and $s$ in the left-hand sides of (2.26) denote the fast and slow wave parts of the solution.

2.4. Complementarity of vertical and horizontal buoyancies

For the rapidly rotating regime given by $|\omega _C| \gg |\omega _M| \gg |\omega _{A,V}|,|\omega _{A,H}| \gg |\omega _\eta |$ , the roots of the characteristic (2.20) are approximated by

(2.27) \begin{align} \lambda _{1,2} \approx \pm \bigg (\omega _C + \frac {\omega _M^2}{\omega _C} \bigg ) +\mathrm{i} \, \frac {\omega _M^2\omega _\eta }{\omega _C^2}, \\[-28pt] \nonumber \end{align}

(2.28) \begin{align} \lambda _{3,4} \approx \pm \bigg (\frac {\omega _M^2}{\omega _C}+ \frac {\omega _{A,V}^2-2\omega _{A,H}^2}{2\omega _C}\bigg ) + \mathrm{i} \, \omega _\eta \, \bigg (1-\frac {\omega _{A,V}^2-\omega _{A,H}^2}{2\omega _M^2}\bigg ), \\[-28pt] \nonumber \end{align}

(2.29) \begin{align} \lambda _{5} \approx \mathrm{i}\, \frac {\omega _\eta \big(\omega _{A,V}^2-\omega _{A,H}^2 \big)}{\omega _M^2}, \\[0pt] \nonumber \end{align}

following the procedure of Sreenivasan & Maurya (Reference Sreenivasan and Maurya2021). Here, the slow MAC waves of frequency $\lambda _{3,4}$ are damped on the time scale $(\omega _\eta )_0^{-1} = (\eta k_0^2)^{-1} \sim \delta ^2/\eta$ , where the subscript ‘0’ represents the initial state of the buoyancy disturbance (see also Sreenivasan & Maurya Reference Sreenivasan and Maurya2021).

From (2.28), the resultant buoyancy frequency is given by

(2.30) \begin{align} \begin{aligned} {\omega _A}^2=&\omega _{A,V}^2-2\omega _{A,H}^2 =\omega _{A,V}^2\bigg (1 - 2\,\beta ^\star \frac {k_y}{k_z}\bigg ), \end{aligned} \end{align}

where $\beta ^\star =\beta _z/\beta _y$ . If $\beta ^\star =0$ , the approximate classical MAC wave roots are recovered. In the limit of $|\omega _C| \gg |\omega _M|$ , $|\omega _A|$ , the real parts of $\lambda _{3,4}$ may be expressed as (Braginsky Reference Braginsky1967)

(2.31) \begin{align} \begin{aligned} \mbox{Re}(\lambda _{3,4}) \approx \pm \frac {\omega _M^2}{\omega _C}\, \bigg (1+ \frac {{\omega _A}^2}{\omega _M^2} \bigg )^{1/2}, \end{aligned} \end{align}

which implies that slow MAC waves would vanish in an unstably stratified fluid with ${\omega _A}^2\lt 0$ as $|\omega _A|$ nears $|\omega _M|$ . Since $\beta _y \lt 0$ and $\beta _z \gt 0$ in (2.5), $\omega _{A,V}^2 \lt 0$ and $\omega _{A,H}^2 \gt 0$ in (2.30). These two buoyancy frequencies contribute to the resultant buoyancy frequency. An increase in either $|\omega _{A,H}|$ or $|\omega _{A,V}|$ results in an increase in $|\omega _A|$ , which may match $|\omega _M|$ and thereby suppress the slow MAC waves. While a large vertical buoyancy requires a small horizontal buoyancy to suppress these waves, a small vertical buoyancy together with a large horizontal buoyancy would produce the same result. As we shall see later in § 4, the complementarity of the buoyancy frequencies places a bound on the magnitude of the horizontal buoyancy in two-component convection.

2.5. Fast and slow MAC waves under unstable stratification and a thermal wind

The evolution of velocity and the induced magnetic field is obtained from the solution of the initial value problem. The solution to the problem is obtained for times much shorter than the time scale for the exponential increase of the perturbations. The Lehnert number $Le$ and the magnetic Ekman number $E_\eta$ based on the length scale of the perturbation are used to describe the parameter regime,

(2.32a,b) \begin{align} Le= \dfrac {V_M}{2 \varOmega \delta } \approx \bigg (\frac {\omega _{M}}{\omega _{C}}\bigg )_{\!0}, \quad E_\eta = \dfrac {\eta }{2 \varOmega \delta ^2} \approx \bigg (\frac {\omega _{\eta }}{\omega _{C}}\bigg )_{\!0}. \end{align}

The relative intensity of horizontal buoyancy is measured by the ratio $ \textit{Ra}_{\ell ,H}/Ra_\ell$ , where $ \textit{Ra}_\ell$ is the resultant local Rayleigh number given by the sum of the local vertical and horizontal Rayleigh numbers,

(2.33a,b) \begin{align} \textit{Ra}_{\ell ,V} = \dfrac {g \alpha |\beta _y| \delta ^2}{2 \varOmega \eta }, \quad \textit{Ra}_{\ell ,H} = 2 \dfrac {g \alpha |\beta _z| \delta ^2}{2 \varOmega \eta } \frac {k_y}{k_z}, \end{align}

so that

(2.34) \begin{align} \textit{Ra}_{\ell } = \dfrac {g \alpha |\beta | \delta ^2}{2 \varOmega \eta }, \end{align}

based on the resultant temperature gradient,

(2.35) \begin{align} \beta = \beta _y - 2 \beta _z k_y/k_z. \end{align}

Figure 2.

Evolution of $u_z^2$ on the $y$ – $z$ plane at $x=0$ 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) $t/t_\eta =1\times 10^{-3}$ and (c) $t/t_\eta = 1\times 10^{-2}$ for $ \textit{Ra}_{\ell , {H}}/Ra_\ell =0.4$ and $|\omega _{A}^2/\omega _M^2|=0.1$ .

Figure 2 shows the evolution of the axial kinetic energy density of the perturbation. Here, the real-space $z$ velocity is obtained from the inverse Fourier transform of (2.21),

(2.36) \begin{align} u_z =\frac {1}{(2\pi )^2} \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\hat {u}_z \mbox{e}^{ {i} \, (k_y \, y+k_z \, z)}\,\mathrm{d}k_y\, \mathrm{d}k_z, \end{align}

where a truncation value of $\pm 5/\delta$ is used for the two wavenumbers in the computed integrals since the initial wavenumber $k_0 =\sqrt {3}/\delta$ (Majumder et al. Reference Majumder, Sreenivasan and Maurya2024). The evolution of blobs into columnar structures through the propagation of damped waves is evident in the $y{-}z$ plane at $x=0$ .

Figure 3.

Variation of the squares of the fundamental frequencies with $ \textit{Ra}_{\ell ,H}/Ra_\ell$ under constant vertical buoyancy conditions. Panels (a, c) and (b, d) correspond to cases with $ -\omega _{A,V}^2/\omega _{M}^2 = 0.8$ and $ -\omega _{A,V}^2/\omega _{M}^2 = 0.3$ , respectively. In panels (a) and (b), $\omega _{C}^2$ , $\omega _{M}^2$ and $\omega _{A}^2$ are plotted, while panels (c) and (d) decompose $\omega _{A}^2$ into its vertical ( $\omega _{A,V}^2$ ) and horizontal ( $\omega _{A,H}^2$ ) parts. The dotted vertical lines indicate the values of $ \textit{Ra}_{\ell ,H}/Ra_\ell$ where the slow MAC wave is suppressed as $ |\omega _M| \approx |\omega _A|$ . Panel (e) shows the suppression points of the slow wave for various strengths of $ -\omega _{A,V}^2/\omega _{M}^2$ . The blue and red points in panel (e) correspond to the slow wave suppression points in panels (a) and (b), respectively. (f) Axial kinetic energy $E_{k,z}$ of the fast and slow MAC waves versus $ \textit{Ra}_{\ell ,H}/Ra_\ell$ where $ -\omega _{A,V}^2/\omega _{M}^2 = 0.3$ . The parameters used are $ E_\eta = 2 \times 10^{-5}$ , $ Le = 0.03$ ( $\omega _{M}/\omega _{C}=0.18$ ) and $ t/t_\eta = 1 \times 10^{-2}$ .

In figure 3(a–d), the fundamental frequencies are plotted against the relative intensity of horizontal buoyancy. Both $ \textit{Ra}_{\ell ,V}$ and $Le$ are kept constant. The frequencies and the local Rayleigh numbers are based on the mean wavenumbers obtained from ratios of $L^2$ norms; e.g.

(2.37a–c) \begin{gather} \bar {k}_y = \frac {|\!|k_y \, \hat {u}_z|\!| }{|\!| \hat {u}_z|\!|}, \quad \bar {k}_z = \frac {|\!|k_z \, \hat {u}_z|\!| }{|\!| \hat {u}_z|\!|}, \quad \bar {k} = \frac {|\!| k \, \hat {u}_z|\!|}{|\!| \hat {u}_z|\!|}. \end{gather}

By progressively increasing $ \textit{Ra}_{\ell ,H}$ for a fixed $ \textit{Ra}_{\ell ,V}$ and $Le$ , $\omega _{A,H}$ increases, and the resultant buoyancy frequency $\omega _{A}$ crosses the Alfvén frequency $\omega _{M}$ at certain $ \textit{Ra}_{\ell ,H}/Ra_{\ell }$ , where the slow MAC waves of frequency $\omega _s$ disappear ( $\omega _s = \omega _{3,4}$ ; see (2.28)). The same exercise performed with a different value of $ \textit{Ra}_{\ell ,V}$ reveals the complementarity between $ \textit{Ra}_{\ell ,V}/Ra_{\ell }$ and $ \textit{Ra}_{\ell ,H}/Ra_{\ell }$ (figure 3 e). The red and blue points in figure 3(e) indicate the value of $ \textit{Ra}_{\ell ,H}/Ra_{\ell }$ from figures 3(a) and 3(b), where $|\omega _{A}| \approx |\omega _{M}|$ so that $\omega _{s}$ approaches zero. In § 3, it is shown that the inequality $|\omega _M| \gt |\omega _A|$ represents the dipole-dominated regime, while $|\omega _M| \approx |\omega _A|$ represents the transition to a multipolar state, with polarity reversals occurring in a narrow range of $ \textit{Ra}_{\ell ,H}/Ra_{\ell }$ that lies between the dipolar and multipolar states.

The axial kinetic energy of the MAC waves is given by

(2.38) \begin{align} E_{k,z}=\frac {1}{(2\pi )^2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }|\hat {u}_z(k_y,k_z)|^2 \,\mbox{d}k_y\, \mbox{d}k_z, \end{align}

where the limits of the integration are set to $\pm 5/\delta$ . While the energy of the fast MAC waves is practically unaffected by increasing horizontal buoyancy, the slow MAC wave energy goes to zero when $|\omega _A|$ approximately matches $|\omega _M|$ (figure 3 f).

Figure 4.

Variation of $ \textit{Ra}_{\ell }$ with Elsasser number $\varLambda = (\omega _M^2/\omega _C\omega _\eta )_0$ for the state of vanishing slow wave axial kinetic energy. Three values of $E_\eta$ are considered: diamonds represent $E_\eta =1\times 10^{-6}$ , circles represent $E_\eta =2\times 10^{-5}$ and triangles represent $E_\eta =5\times 10^{-6}$ .

The relation between the resultant local Rayleigh number $ \textit{Ra}_\ell$ and the ratio of frequencies given by

(2.39) \begin{align} \bigg (\dfrac {\omega _M^2}{\omega _C \omega _\eta }\bigg )_{\!0} \sim \dfrac {V_M^2}{2 \varOmega \eta } \end{align}

is shown in figure 4 for the condition of vanishing slow wave axial kinetic energy for three different values of $E_\eta$ . Each point is obtained by increasing the forcing through $ \textit{Ra}_\ell$ and noting the transition from the regime $|\omega _{M}|\gt |\omega _{A}|$ to the regime $|\omega _{A}|\gtrsim |\omega _{M}|$ . The resultant forcing needed to suppress the slow MAC waves increases with $(\omega _M^2/\omega _C \omega _\eta )_0$ , which is a measure of the square of the peak dimensionless magnetic field in the dynamo (see § 3.4). The linear relation obtained in figure 4 reflects the parity between $|\omega _{M}|$ and $|\omega _{A}|$ at the state of vanishing slow MAC waves. Since the relative orders of magnitude of the basic frequencies do not depend on the orientation of the gravity, rotation and magnetic field axes, the Cartesian model serves as the basis for the study of the role of equatorially anti-symmetric boundary heat flux in the nonlinear dynamo in § 3, where the state of vanishing slow waves is shown to be a proxy for polarity transitions. We anticipate a self-similar behaviour for the dipole–multipole transition in the dynamo, where both $ \textit{Ra}_\ell$ and the square of the peak magnetic field are measurable quantities.

3.1. Effect of equatorially symmetric and anti-symmetric heat flux patterns on polarity transitions

We begin our simulations with a strong field dynamo ( $B^2_{\textit{rms}} \gt 1$ ) and increase the heterogeneity of the heat flux at the outer boundary in steps while keeping all other parameters fixed. Figure 5(a–d) gives examples of the equatorially anti-symmetric, symmetric and composite boundary conditions whose effects on polarity transitions are analysed in this study. The simulations are performed for at least one magnetic diffusion time in the saturated state.

In figure 6, the colatitude of the axial dipole field, $\theta$ , at the outer boundary is obtained from the Gauss coefficients of the spherical harmonics, as follows:

(3.8) \begin{align} \cos \theta = g_1^0/|\boldsymbol{m}|, \quad \boldsymbol{m} = \big(g_1^0, g_1^1, h_1^1 \big), \end{align}

where $g_1^0$ , $g_1^1$ and $h_1^1$ are derived from the Schmidt-normalised expansion for the scalar potential of the field (Glatzmaier Reference Glatzmaier2013, pp. 142–143).

Figure 5.

Distribution of heterogeneous radial temperature gradient $\partial T/\partial r$ at the outer boundary for different conditions: (a) $Y_2^1$ ; (b) $Y^2_2$ ; (c) $Y_2^2:Y_2^1=2:1$ ; and (d) tomographic condition derived from the seismic shear wave velocity variation in the Earth’s lower mantle (Masters et al. Reference Masters, Johnson, Laske and Bolton1996).

For $E = 1.2 \times 10^{-5}$ , $ \textit{Pm} = \textit{Pr} = 1$ and $ \textit{Ra}_V = 2500$ , a strong dipole-dominated dynamo undergoes well-defined polarity reversals, as shown in figure 6(b), by progressively increasing $q^*$ for the equatorially anti-symmetric $Y_2^1$ heat flux pattern. With a further increase in $q^*$ , the dynamo exhibits a multipolar solution (figure 6 c). However, the equatorially symmetric $Y_2^2$ heat flux pattern does not produce polarity transitions even with a strong heterogeneity (see figure 6 d). For the same parameters, the Earth-like composite heat flux pattern in figure 5(d), which consists of both symmetric and anti-symmetric parts, produces polarity transitions. From the evolution of the dipole axis tilt (figure 17, Appendix A), well-defined polarity reversals are noted for $q^*=13$ . For the anti-symmetric $Y_2^1$ boundary heat flux variation, the radial magnetic fields in the dipolar, reversing and multipolar states are shown in figure 7 for two series with different vertical thermal forcing. In table 1, the runs for two values of $ \textit{Ra}_V$ , $2500$ and $20000$ , are presented with different heat flux patterns. In the reversing runs, the ratio of magnetic to kinetic energy $E_m/E_k$ falls below unity, as noted in earlier studies (Kutzner & Christensen Reference Kutzner and Christensen2002; Tassin, Gastine & Fournier Reference Tassin, Gastine and Fournier2021) although a recent study (Frasson et al. Reference Frasson, Schaeffer, Nataf and Labrosse2025) suggests that multipolar solutions can exist for $E_m/E_k\gt 1$ . Reversing dynamo models in general have not realised $E_m/E_k \gt 1$ probably because their magnetic Ekman number $E_\eta$ is much higher than that in the Earth’s core in the relatively small length scales. In contrast, the criterion of vanishing slow MAC wave helicity considered in this study applies to the energy-containing scales where density perturbations are continually produced and is independent of the choice of $E_\eta$ in the low-inertia limit that rapidly rotating planets operate in.

Figure 6.

Evolution of dipole colatitude with magnetic diffusion time for (a) $q^*=17$ , $Y_2^1$ (stable dipolar), (b) $q^*=18$ , $Y_2^1$ (reversing), (c) $q^*=20$ , $Y_2^1$ (multipolar), and (d) $q^*=18$ and $30$ , $Y_2^2$ (stable dipolar). The other dynamo parameters are $ \textit{Ra}_V = 2500$ , $E = 1.2 \times 10^{-5}$ , $ \textit{Pm} = \textit{Pr} = 1$ .

Figure 7.

Contours of the radial magnetic field at the outer boundary for $ \textit{Ra}_V = 2500$ at $q^* =\,(a)\, 17$ , (b) 18, (c) 20; and for $ \textit{Ra}_V =25\,000$ at $q^* =\,(d)\, 5$ , (e) 6, (f) 7. The other dynamo parameters are $E = 1.2 \times 10^{-5}$ , $ \textit{Pm} = \textit{Pr} = 1$ . A $Y_2^1$ heat flux heterogeneity is applied at the outer boundary.

The main output parameters of the polarity-reversing dynamo runs with the anti-symmetric $Y_2^1$ heat flux boundary condition are reported in table 2. The ratio of anti-symmetric to total kinetic energy remains nearly constant even at high $q^*$ (figure 8), so equatorially anti-symmetric boundary conditions do not induce polarity transitions by breaking the equatorial symmetry of the convection columns. Before analysing the role of wave motions in polarity transitions, we examine the mean temperature and velocity fields produced by the heterogeneous boundary heat flux, which give useful comparisons with the linear convection model.

Figure 8.

Ratio of anti-symmetric to total kinetic energy for $q^*=0$ (black), $q^*=15$ (red), $q^*=18$ (blue) and $q^*=20$ (green) with the $Y_2^1$ heat flux heterogeneity at the outer boundary. The other dynamo parameters are $ \textit{Ra}=2500,\ E = 1.2 \times 10^{-5},\ \textit{Pm}=Pr=1$ .

3.2. Mean resultant temperature gradient and velocity field

In line with that obtained in (2.35) in the Cartesian model, the mean (superadiabatic) resultant temperature gradient is given by

(3.9) \begin{align} \beta = \beta _s - 2 \beta _z k_s/k_z, \end{align}

where $\beta _s = \partial T_0/\partial s \lt 0$ in an unstably stratified layer with homogeneous boundary heat flux (see figure 9 a). In figure 9(b), $\beta _z = \partial T_0/\partial z$ is plotted for an equatorially antisymmetric $Y_2^1$ heat-flux heterogeneity at the outer boundary. The temperature gradients are calculated from a steady state just below non-magnetic convective onset ( $ \textit{Ra}_V = 30$ , whereas onset occurs at $ \textit{Ra}_V = 31$ for $q^* = 18$ ) so that the perturbations do not affect the calculation of the mean gradients. An averaged $\beta _z$ , obtained by taking the average of the positively signed $\partial T_0/\partial z$ within the spherical shell, is proportional to the intensity of the anti-symmetric heterogeneity. Likewise, the averaged $\beta _s$ is obtained by taking the average of $\partial T_0/\partial s$ within the shell. The averaged resultant gradient $\beta$ is then evaluated from (3.9) using the average $s$ and $z$ wavenumbers defined in (3.13) (see figure 9 d). The resultant $\beta$ calculated at $ \textit{Ra}_V = 30$ and the resultant $\beta$ averaged over ${\sim}0.5$ diffusion time from the strongly supercritical state at $ \textit{Ra}_V=310$ exhibit similar structures as well as nearly equal maximum and minimum values (see supplementary figure S4). Convection is suppressed in stably stratified regions where $\partial T/\partial r \gt 0$ , whereas regions with $\partial T/\partial r \lt 0$ support convection (see figure 9 f). Within the convective region, $\beta _z \gt 0$ below the equatorial plane, while $\beta _z \lt 0$ above the equator. Consistent with the Cartesian model in § 2, where the positively signed $\beta _z$ is symmetric about the mid-plane $z = 0$ , the equatorially anti-symmetric boundary condition yields a $\beta _z$ that is symmetric about the equator, which is then used to compute the resultant $\beta$ . The unstably stratified region is in focus in the analysis in § 3.3 – at moderately positive values of $\beta _z$ , both fast and slow MAC waves are present, giving the dipolar dynamo regime; at large positive values of $\beta _z$ , the slow MAC waves are selectively suppressed whereas the fast MAC waves persist, giving the reversing and multipolar dynamo regimes.

Figure 9.

Horizontal section plots at $z = 0.4$ below the equator showing (a) $\beta _s$ for homogeneous boundary heating, (b) $\beta _z$ for $q^* = 18$ , (c) $\beta _\phi$ for $q^* = 18$ and (d) resultant gradient $\beta$ for $q^* = 18$ . The temperature gradients are calculated at vertical Rayleigh number $ \textit{Ra}_V = 30$ , which is a state near the onset of non-magnetic convection. A $Y_2^1$ heterogeneity in outer boundary heat flux is applied. The orange-green coloured strip at the periphery of panel(b) represents $\partial T_0 / \partial r$ at the outer boundary. Panels (e) and (f) show snapshots of the axial velocity $u_z$ at $q^* = 0$ and $q^* = 18$ , respectively, for dynamo simulations at $ \textit{Ra}_V = 2500$ . The other parameters are $ E = 1.2 \times 10^{-5}$ and $ \textit{Pm} = \textit{Pr} = 1$ .

Figure 10.

(a) Values of $\beta _s$ (blue), $\beta _z$ (red) and the resultant $\beta$ (black) at cylindrical radius $s = 1$ at the section $z = 0.4$ below the equator for $q^* = 18$ . (b) Meridional sections plot of the $\phi$ -component of the velocity, $u_0$ for $q^* = 18$ . The plot shows $\phi$ -averaged values of $u_0$ in the unstably stratified region where $\beta \lt 0$ ( $\phi = 1.31 \, \pi$ to $0.31 \, \pi$ through $\phi =0$ ) in the grey-shaded region of panel (a). (c) Values of $u_0$ along a vertical line passing through cylindrical radius $s = 1$ (circles) are compared with the theoretical dimensionless thermal wind in (3.10). The parameters are $ \textit{Ra}_V = 30$ (below onset), $E = 1.2 \times 10^{-5}$ , and $ \textit{Pm} = \textit{Pr} = 1$ . A $Y_2^1$ heat flux heterogeneity is applied at the outer boundary.

Equation (3.9) is derived from the theoretical model described in § 2, which predicts the presence or absence of slow MAC waves subject to the frequency inequality $|\omega _{C}| \gt |\omega _{M}|,|\omega _{A}| \gt |\omega _{\eta }|$ , with slow MAC waves present when $|\omega _{M}| \gt |\omega _{A}|$ and absent when $|\omega _{M}| \leqslant |\omega _{A}|$ . The previous frequency inequality does not predict the localised suppression of convection observed in figure 9(f) and can only be applied to regions where convection is present.

Interestingly, $\beta _\phi = \partial T_0/\partial \phi$ (figure 9 c) has practically no influence on $\beta$ , which justifies our original assumption of ignoring variations with respect to the non-preferred horizontal direction (see (2.14), § 2.2). The convective region produced by $\beta _s \lt 0$ and $\beta _z\gt 0$ with the equatorially anti-symmetric heat flux at the boundary corresponds to the unstably stratified system modelled by the equatorial radial configuration of the Cartesian linear model in § 2.1.

Figure 10(a) shows the azimuthal variation of the mean temperature gradients at cylindrical radius $s = 1$ and at the section $z = 0.4$ below the equator for the $Y_2^1$ heterogeneity with $q^* = 18$ . The grey-shaded region indicates $\beta _z \gt 0$ , where the resultant $\beta$ is calculated from (3.9). Figure 10(b) gives the meridional plot of the steady mean flow $u_0$ ( $\phi$ -component of the velocity) averaged over the unstably stratified longitudes in figure 10(a) ( $ \phi = 1.31 \, \pi$ to $0.31 \, \pi$ through $\phi =0$ ). In figure 10(c), the magnitude of this equatorially anti-symmetric flow measured at $s=1$ shows a fair agreement with the theoretical value of the thermal wind calculated from (2.3) in dimensionless units,

(3.10) \begin{align} u_0 = -\frac {g \alpha \beta _z L}{2\varOmega \eta } \,z =-Ra_V \, \beta ^* z, \end{align}

where $\beta ^* =\beta _z/\beta _s$ is obtained from the respective averaged temperature gradients.

3.3. Role of slow MAC waves in the dipole–multipole transition

In § 2, we examined the evolution of an isolated density perturbation in an unstably stratified fluid subject to background rotation, a uniform magnetic field and a horizontal temperature gradient. Perturbations of this kind excite MAC waves in the dynamo, the frequencies of which depend on the fundamental frequencies given as follows:

(3.11) \begin{align} \omega _{C}^2 = \frac {4(\boldsymbol{\varOmega } \boldsymbol{\cdot }\boldsymbol{k})^2}{ k^2},\quad \omega _{M}^2 =\frac {(\boldsymbol{B} \boldsymbol{\cdot }\boldsymbol{k})^2}{\mu \rho }, \quad -\omega _{A,V}^2= g\alpha \beta _s \, \frac { k_{h}^2}{k^2}, \quad \omega _{A,H}^2= g\alpha \beta _z \frac { k_{h}^2}{k^2}\frac {k_s}{k_z} \end{align}

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

(3.12) \begin{align} \omega _C^2 = \frac {\textit{Pm}^2}{E^2} \frac {k_z^2}{k^2}, \!\!\quad \omega _M^2 = \frac {\textit{Pm}}{E} (\boldsymbol{B} \boldsymbol{\cdot }{\boldsymbol k})^2, \!\!\quad -\omega _{A,V}^2 = \frac {\textit{Pm} \textit{Ra}_V}{E} \,\frac {{k_h}^2}{k^2}, \!\!\quad \omega _{A,H}^2 = \frac {\textit{Pm} \textit{Ra}_H}{2 \,E} \,\frac {{k_h}^2}{k^2}, \!\!\quad \end{align}

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, $k^2=k_s^2+k_\phi ^2+k_z^2$ and $k_h$ is the horizontal wavenumber in the equatorial region defined by $k_h^2=k_\phi ^2+k_z^2$ . The $s$ , $\phi$ and $z$ wavenumbers are calculated in the saturated state of the dynamo runs in the energy-containing scales ( $l\leqslant l_E$ ). 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

(3.13) \begin{align} \bar {k}_z = \frac {\varSigma k_z \, \hat {u}^2 (k_z)} {\varSigma \hat {u}^2 (k_z)}. \end{align}

A similar approach yields $\bar {k}_s$ and $\overline {m}$ . 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. The horizontal Rayleigh number is given by

(3.14) \begin{align} \textit{Ra}_H=2 \, \frac {g \alpha \beta _z L^2}{2 \varOmega \eta }\frac {k_s}{k_z}. \end{align}

The local vertical and horizontal Rayleigh numbers in the dynamo based on the flow length scale are given by

(3.15) \begin{align} \textit{Ra}_{\ell ,V}=\frac {g \alpha |\beta _s|}{2 \varOmega \eta } \left (\frac {\pi }{k_s}\right )^2,\quad \textit{Ra}_{\ell ,H}=2 \, \frac {g \alpha |\beta _z|}{2 \varOmega \eta } \frac {k_s}{k_z} \left (\frac {\pi }{k_s}\right )^2, \end{align}

and the resultant local Rayleigh number is then

(3.16) \begin{align} \textit{Ra}_{\ell }=\textit{Ra}_{\ell ,V} +\textit{Ra}_{\ell ,H}. \end{align}

Figure 11.

Variation of the squares of the fundamental frequencies with $q^*$ and $ \textit{Ra}_{\ell ,H}/Ra_\ell$ (within brackets) for (a) $ \textit{Ra}_V=2500$ , $Y_2^1$ heat flux heterogeneity, and (b) $ \textit{Ra}_V=2500$ , $Y_2^2$ heat flux heterogeneity. The dotted vertical line marks the polarity-reversing state that lies between the dipolar and multipolar regimes. The other dynamo parameters are $E = 1.2 \times 10^{-5}$ , $ \textit{Pm}=Pr=1$ .

In figure 11, the magnitudes of fundamental frequencies are shown as a function of $q^*$ and $ \textit{Ra}_{\ell , H}/Ra_\ell$ for heterogeneous heat flux boundary conditions. The frequencies and local Rayleigh numbers are computed from (3.12), (3.15) and (3.16) using the mean values of the wavenumbers as shown in (3.13). In figure 11(a), we begin with a strong field dynamo and increase the $Y_2^1$ heat flux heterogeneity, which is reflected in the horizontal buoyancy frequency $\omega _{A,H}$ , while keeping the vertical buoyancy constant at $ \textit{Ra}_V=2500$ . The slow MAC waves are present at relatively low $q^*$ when $|\omega _{M}| \gt |\omega _A|$ . In the polarity-reversing state at $q^* = 18$ ( $ \textit{Ra}_{\ell , {H}}/Ra_\ell \approx 0.62$ ), the magnitude of the resultant buoyancy frequency $|\omega _A|$ approximately matches that of the Alfvén frequency $|\omega _{M}|$ , causing the slow wave frequency $\omega _{s}$ to go to zero. While the generation of slow MAC waves ceases, the existing slow MAC waves are damped on the time scale $\delta ^2 / \eta$ (see § 2.4), which is much shorter than the magnetic diffusion time $L^2 / \eta$ because the ratio of the core depth to the length scale of buoyancy disturbances would be $L / \delta \sim 10^2$ (Sreenivasan & Maurya Reference Sreenivasan and Maurya2021). Whether the rapid damping of the slow MAC waves causes the rapid decay of the axial dipole field during a polarity reversal (see figure 6 b) is at present an open question that requires further investigation. The equatorially symmetric $Y_2^2$ heat flux boundary condition has zero horizontal buoyancy at the equator, and even at large $q^*$ , the condition $|\omega _A| \gt |\omega _{M}|$ is never met (figure 11 b).

The variation of the parameter $f_{\textit{dip}}$ against $|\omega _A/\omega _M|$ in figure 12 shows that reversing (R) and multipolar (M) states exist at values of $f_{\textit{dip}} \leqslant 0.35$ , the proposed lower bound for the existence of dipole-dominated numerical dynamos in the literature.

Figure 12.

Variation of $f_{\textit{dip}}$ with $|\omega _{A}/\omega _{M}|$ for different heat flux boundary conditions at two values of $ \textit{Ra}_V$ . The vertical dotted line indicates $|\omega _{A}/\omega _{M}| = 1$ , while the horizontal dotted line marks $f_{\textit{dip}} = 0.35$ , which has been proposed as the lower bound for the existence of dipole-dominated numerical dynamos (Christensen & Aubert Reference Christensen and Aubert2006). Types ‘D’, ‘R’ and ‘M’ denote dipolar, reversing and multipolar dynamo states, respectively. The other dynamo parameters are $E = 1.2 \times 10^{-5}$ and $ \textit{Pm} = \textit{Pr} = 1$ .

Figure 13.

Panels (a–c) (i) show the absolute values of wave frequencies plotted for the saturated state of the dynamo run: (a) $q^*= 0$ , (b) $q^*= 10$ and (c) $q^*= 18$ for the $Y_2^1$ heat flux heterogeneity at the outer boundary. 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–c) (ii) show the spectral distribution of the power supplied to the axial dipole, defined in (3.17). The other dynamo parameters are $ \textit{Ra}_V=2500$ , $E = 1.2 \times 10^{-5}$ , $ \textit{Pm}=Pr=1$ .

The top row of figure 13 displays the square of the fundamental frequencies for a range of spherical harmonic order $m$ for (a) $q^*=0$ , (b) $q^*=10$ , and (c) $q^*=18$ in the saturated state for the equatorially anti-symmetric $Y_2^1$ heat flux pattern. Dynamo simulations with $q^* = 0$ and $q^* = 10$ exhibit stable axial dipoles while the simulation with $q^* = 18$ gives a reversing dynamo. The shaded area indicates the range of $m$ where the kinetic helicity in the dynamo exceeds that of the equivalent non-magnetic run. (For the governing equations of equivalent non-magnetic run, see Majumder et al. Reference Majumder, Sreenivasan and Maurya2024). The black line represents the square of the slow MAC wave frequency under the condition $|\omega _C| \gt |\omega _M| \gt |\omega _A|$ . As $q^*$ increases, the range of $m$ satisfying this condition narrows, and for $q^*=18$ , this condition is not satisfied for any $m$ . The bottom row of figure 13 gives the spectral distribution of the power supplied to the poloidal component of the axial dipole field $B_{10}^P$ (e.g. Buffett & Bloxham Reference Buffett and Bloxham2002) for the three $q^*$ values,

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

where $\boldsymbol{u}$ and $\boldsymbol{B}$ have the same $m$ (Bullard & Gellman Reference Bullard and Gellman1954). The dipole power peaks at $q^* = 0$ and $q^* = 10$ within the range of $m$ where slow MAC waves are generated and the helicity of the dynamo is greater than that of the equivalent non-magnetic simulation. In contrast, the run with $q^* = 18$ shows no such peak since there is no helicity generation at any $m$ . The $f_{\textit{dip}}$ values for $ \textit{Ra} = 2500$ decrease with increasing heterogeneity, with a sharp drop observed at the polarity transition for $q^* = 18$ (table 1). These results indicate the crucial role of the slow MAC waves in helicity generation in the energy-containing scales, and in turn, dipole formation.

Table 3.

Summary of the data for MAC wave measurement in the dynamo models. The sampling frequency $\omega _n$ is selected to ensure that the fast MAC waves are captured when measuring group velocity. The values of $\omega _M^2$ , $-\omega _{A}^2$ and $\omega _C^2$ are computed using (3.12), based on the averaged wavenumbers $m$ , $k_s$ and $k_z$ in the energy-containing scales where $l \leqslant l_E$ . The group velocity in the $z$ direction ( $U_{g,z}$ ) is then compared with the estimated velocities of the fast ( $U_{\kern-1pt f}$ ) and slow ( $U_s$ ) MAC waves.

Figure 14.

(a–b) Contour plots of $\partial {u}_z/\partial t$ at a cylindrical radius of $s=1$ are shown for the scales $l \leqslant l_E$ over short time intervals in the saturated state of two dynamo simulations with (a) $q^*=17$ and (b) $q^*=18$ for the equatorially anti-symmetric $Y_2^1$ heat flux boundary condition. The parallel black lines represent the primary wave travel direction, with their slope giving the group velocity $U_{g,z}$ . Table 3 lists the estimated group velocities for the fast and slow MAC waves ( $U_{\kern-1pt f}$ and $U_s$ , respectively) and $U_{g,z}$ . The other dynamo parameters are $ \textit{Ra}_V=2500, E = 1.2 \times 10^{-5}$ and $ \textit{Pm}=Pr=1$ .

Figure 14 shows the measurement of wave velocities from their propagation paths in the saturated state of dynamos at $q^* = 17$ and $q^* = 18$ for the equatorially anti-symmetric $Y_2^1$ heat flux pattern at the outer boundary. Contours of the fluctuating $z$ -velocity, represented by $\partial u_z / \partial t$ at cylindrical radius $s = 1$ , are plotted over short time intervals where the ambient magnetic field and wavenumbers in the energy-containing scales $l \leqslant l_E$ remain approximately constant. By examining the slope of the black lines, we determine the axial group velocity $U_{g,z}$ , and compare it with the estimated axial group velocities of the fast ( $U_{\kern-1pt f}$ ) and slow ( $U_s$ ) waves. These velocities are derived from the respective frequencies in the diffusionless limit by taking their derivative with respect to $k_z$ (Varma & Sreenivasan Reference Varma and Sreenivasan2022). In the dipole-dominated run at $q^* = 17$ , the slow MAC waves are predominant although the fast MAC waves also exist (figure 14 a). In contrast, at $q^* = 18$ , the slow MAC waves are nearly absent while the fast waves are abundant (figure 14 b).

3.4. Complementarity of vertical and horizontal buoyancies in the dipole–multipole transition

In the presence of rapid rotation and a magnetic field, the buoyant forcing generates fast and slow MAC waves when the inequality $ |\omega _{C}| \gt |\omega _{M}| \gt |\omega _{A}| \gt |\omega _{\eta }|$ is satisfied. The magnetic field intensity increases with the vigour of convection in this dipole-dominated regime; however, a sufficiently strong forcing suppresses the slow MAC waves when $ |\omega _{A}| \approx |\omega _{M}|$ , leading to dipole collapse. The nature of buoyancy could be either vertical, measured by $ \textit{Ra}_{V}$ , or horizontal, measured by $ \textit{Ra}_{H}$ . In a dynamo with homogeneous boundary heat flux, the dipolar regime transitions in succession to polarity-reversing and multipolar states with increasing $ \textit{Ra}_{V}$ (Majumder et al. Reference Majumder, Sreenivasan and Maurya2024). In this study, $ \textit{Ra}_H$ is progressively increased for a fixed $ \textit{Ra}_{V}$ until the dynamo undergoes the polarity transition (table 1). The square of the peak magnetic field measured at the polarity transition, $B^2_{\textit{peak}}$ , is plotted against $ \textit{Ra}_V$ for two different Ekman numbers in figure 15(a). The approximately linear variation of $ \textit{Ra}_V$ with $B^2_{\textit{peak}}$ is different in the two cases. However, the resultant local Rayleigh number $ \textit{Ra}_\ell$ , defined by (3.16), exhibits a self-similar variation with $B^2_{\textit{peak}}$ (figure 15 b). This self-similar line separates the dipolar and multipolar regimes. The filled symbols represent the evolution path of the dynamo with increasing order of heat flux heterogeneity for two vertical Rayleigh numbers ( $ \textit{Ra}_V=2500$ and $ \textit{Ra}_V=20\,000$ ). The filled black symbols here represent the states where $ |\omega _{M}| \gt |\omega _{A}|$ , while the filled red symbols represent $ |\omega _{M}| \lesssim |\omega _{A}|$ . In figure 15(c), all the polarity-reversing states are plotted in the $ \textit{Ra}_{\ell ,H}$ versus $ \textit{Ra}_{\ell , V}$ space, normalised by the value of $ \textit{Ra}_\ell$ . This plot also demarcates the dipolar and multipolar regimes, and furthermore demonstrates the complementarity between the horizontal and vertical buoyancies in producing the polarity transition through the condition of vanishing slow MAC waves, as suggested by figure 3(e). For $E=1.2 \times 10^{-5}$ , reversals occur in a strongly driven dynamo at $ \textit{Ra}_V=27000$ and $ \textit{Ra}_{\ell ,H}/Ra_\ell =0.2$ (blue diamond) as well as a relatively weakly driven dynamo at $ \textit{Ra}_V=1500$ and $ \textit{Ra}_{\ell ,H}/Ra_\ell =0.64$ (red diamond); see also table 2. Since the self-generated magnetic field depends on the vertical Rayleigh number $ \textit{Ra}_{\ell ,V}$ , only weak-field dynamos of $B^2_{\textit{rms}} \lt O(1)$ exist for $ \textit{Ra}_{\ell ,V}/Ra_\ell \lt 0.25$ . Therefore, solutions for $ \textit{Ra}_{\ell ,V}/Ra_\ell \lt 0.25$ (i.e. $ \textit{Ra}_{\ell ,H}/Ra_\ell \gt 0.75$ ) are not considered. Figure 15(d) relates the two measures of heterogeneity, $q^*$ and $ \textit{Ra}_{\ell , H}/Ra_\ell$ , and indicates that values of $ \textit{Ra}_{\ell , H}/Ra_\ell \gt 0.5$ correspond to values of $q^*$ of $O(10)$ .

Figure 15.

(a) Variation of the modified vertical Rayleigh number $ \textit{Ra}_V$ with the square of the peak magnetic field, $B^2_{\textit{peak}}$ , at the suppression of slow MAC waves. (b) Variation of the local Rayleigh number $ \textit{Ra}_\ell$ , defined in (3.15), with $B^2_{\textit{peak}}$ . The values of $ \textit{Ra}_V$ , $ \textit{Ra}_\ell$ and $B^2_{\textit{peak}}$ in the plots are given in table 2. The parameters of the two dynamo series and their symbolic representations are as follows: $E=6\times 10^{-5},\textit{Pm}=Pr=5$ (circles); $E=1.2\times 10^{-5},\textit{Pm}=Pr=1$ (diamonds). The filled symbols show the evolution path of the dynamo with increasing heat flux heterogeneity. Black symbols represent the dipolar state and red symbols represent the reversing or multipolar state. (c) The states of polarity reversals are shown in a plot of local relative vertical versus relative horizontal Rayleigh numbers, normalised by the resultant local Rayleigh number $ \textit{Ra}_\ell$ . For $E=1.2 \times 10^{-5}$ , reversals occur at $ \textit{Ra}_V=27\,000$ and $ \textit{Ra}_{\ell ,H}/Ra_\ell =0.2$ (blue diamond) as well as at $ \textit{Ra}_V=1500$ and $ \textit{Ra}_{\ell ,H}/Ra_\ell =0.64$ (red diamond). (d) Variation of $q^*$ with respect to $ \textit{Ra}_{\ell ,H}/Ra_{\ell }$ . The horizontal line at $q^* = 10$ and the vertical line at $ \textit{Ra}_{\ell ,H}/Ra_{\ell } = 0.5$ indicate that $ \textit{Ra}_{\ell ,H}/Ra_{\ell } \gt 0.5$ corresponds to $q^* = O(10)$ .

3.5. Combination of symmetric and anti-symmetric boundary heat flux

Since the heat flux heterogeneity in the lower mantle is a combination of both equatorially symmetric and anti-symmetric variations, it is instructive to consider the effect of composite heat flux boundary conditions on polarity transitions. The analysis of the heat flux distribution based on the seismic shear wave velocity in the Earth’s lower mantle (Masters et al. Reference Masters, Johnson, Laske and Bolton1996) suggests a complex heat flux pattern at the CMB which features a well-defined $Y_2^2$ component. That said, the analysis of peak-to-peak heat flux variations reveals that the ratio of the symmetric to anti-symmetric heat flux variation is approximately 1.65. This ratio is obtained by calculating the peak-to-peak variation in heat flux for the symmetric and anti-symmetric contributions separately, within the seismic tomography pattern. With a composite heat flux heterogeneity at the outer boundary, a dipole-dominated solution is observed when the inequality $|\omega _{C}| \gt |\omega _{M}| \gt |\omega _{A}|$ is satisfied in the presence of the slow MAC waves. For a given vertical buoyant forcing ( $ \textit{Ra}_V \sim 10^2$ times its value for non-magnetic onset of convection), the progressive increase of horizontal buoyancy produces polarity transitions for composite patterns where the equatorially symmetric and anti-symmetric variations are comparable. For example, a heat flux pattern with $Y_2^2:Y_2^1 = 2:1$ and the Earth-like heat flux pattern derived from seismic tomography both produce reversals for $q^*$ of $O(10)$ (table 1). For the same $ \textit{Ra}_V$ , the values of $ \textit{Ra}_{\ell ,H}/Ra_\ell$ at which the transition occurs in the two cases are nearly equal, which points to the comparable ratios of the symmetric to anti-symmetric heat flux variation. Notably, a heterogeneity consisting of comparable magnitudes of symmetric and anti-symmetric variations induces the polarity transition at a value of $ \textit{Ra}_{\ell ,H}/Ra_\ell$ (and $q^*$ ) which is of the same order as that for the transition induced by a purely anti-symmetric variation.

The polarity transition with the boundary heat flux derived from seismic tomography is analysed further in figure 18 of Appendix A. In the dipole-dominated state at $q^*=10$ , $|\omega _M|^2 \gt |\omega _A|^2$ in several regions, where $\omega _A^2$ is based on the resultant basic state temperature gradient $\beta$ , defined in (3.9). In these regions, the Alfvén wave velocity is greater than the velocity of the buoyancy perturbations and the slow MAC waves are generated. However, the polarity-reversing state at $q^*=13$ shows approximate parity between $|\omega _M|^2$ and $|\omega _A|^2$ so that the slow waves are suppressed. This analysis also explains why a composite boundary heterogeneity made up of a dominant equatorially symmetric variation does not induce polarity transitions. For example, the heterogeneity with the ratio $Y_2^2:Y_2^1 = 5:1$ does not admit polarity transitions even at high $q^*$ (table 1) since the condition $|\omega _{A}|^2 \gtrsim |\omega _{M}|^2$ is not met.

4.1. Evolution of a density disturbance in two-component magnetoconvection

In addition to the variables $\boldsymbol{u}$ , $\boldsymbol{b}$ and $\varTheta$ considered in (2.6a–c ), the composition $C$ is also decomposed into its mean and perturbation parts, $C= C_0+\gamma$ . Here, the lengths are scaled by the perturbation size $\delta$ and time is scaled by magnetic diffusion time $\delta ^2/\eta$ . The velocity $\boldsymbol{u}$ and magnetic field $\boldsymbol{B}$ are scaled by $\eta /\delta$ and $(2\varOmega \rho \mu \eta )^{1/2}$ , respectively. The temperature is scaled by $\beta _y^T \delta$ and composition is scaled by $\beta _y^C \delta$ , where $\beta _y^T$ and $\beta _y^C$ are the vertical temperature and composition gradients, respectively. The dimensionless equations for $\boldsymbol{u}$ , $\boldsymbol{b}$ , $\varTheta$ and $\gamma$ are given by

(4.1) \begin{align} E_\eta \bigg (\frac {\partial \boldsymbol{u}}{\partial t} +u_c\, z \frac {\partial \boldsymbol{u}} {\partial x} +u_c u_z \hat {\boldsymbol{e}}_x\bigg ) +\hat {\boldsymbol{e}}_z\times \boldsymbol{u}= -\boldsymbol{\nabla }p^\star + (\boldsymbol{\nabla }\times \boldsymbol{b}) \times \boldsymbol{B}_0 \nonumber \\ +Ra_{\ell ,V}^T \varTheta \hat {\boldsymbol{e}}_y + \textit{Ra}_{\ell ,V}^C \gamma \hat {\boldsymbol{e}}_y+E {\nabla} ^2 \boldsymbol{u}, \\[-28pt] \nonumber \end{align}

(4.2) \begin{align} \frac {\partial \boldsymbol{b}}{\partial t} + u_c \,z \frac {\partial \boldsymbol{b}}{\partial x} = u_c b_z\hat {\boldsymbol{e}}_x +(\boldsymbol{B}_0\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}+\eta {\nabla} ^2 \boldsymbol{b}, \\[-28pt] \nonumber \end{align}

(4.3) \begin{align} \frac {\partial \varTheta }{\partial t} +u_c z \frac {\partial \varTheta }{\partial x} +u_y+u_z\beta ^\star =q^T{\nabla} ^2 \varTheta , \\[-28pt] \nonumber \end{align}

(4.4) \begin{align} \frac {\partial \gamma }{\partial t}+u_c z \frac {\partial \gamma }{\partial x} +u_y=q^C {\nabla} ^2 \gamma , \\[-28pt] \nonumber \end{align}

(4.5) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{b}=0. \\[0pt] \nonumber \end{align}

The dimensionless parameters in (4.1), based on the length scale of the density disturbance, are the Ekman number $E=\nu /2\varOmega \delta ^2$ , magnetic Ekman number $E_\eta =\eta /2\varOmega \delta ^2$ , and local vertical compositional and thermal Rayleigh numbers, $ \textit{Ra}_{\ell ,V}^C = g \alpha ^C |\beta _{y}^C| \delta ^2/2 \varOmega \eta$ and $ \textit{Ra}_{\ell ,V}^T = g \alpha ^T |\beta _{y}^T| \delta ^2/2 \varOmega \eta$ , respectively. Here, $\alpha ^T$ and $\alpha ^C$ are the thermal and compositional expansion coefficients, respectively. Additionally, $q^T=\kappa ^T/\eta$ , $q^C=\kappa ^C/\eta$ and $\beta ^\star =\beta _{z}/\beta _{y}^C$ , where $\kappa _T$ and $\kappa _C$ are the diffusivities of temperature and composition, and $\beta _z$ is the horizontal temperature gradient.

For the rapidly rotating system considered in § 2, the limit $\nu , \kappa _C, \kappa _T \ll \eta$ results in $E, q^C, q^T \to 0$ , while $E_\eta \ll 1$ retains a small but finite value. In this limit, a plane wave solution of the form $\hat {u}_z \sim \mbox{e}^{\mathrm{i} \lambda t}$ gives the characteristic equation,

(4.6) \begin{align} \begin{aligned} \lambda ^5&- 2 \mathrm{i}\omega _\eta \lambda ^4 - \big(\omega _{C}^2+2 \omega _{M}^2+{\omega ^C_{A,V}}^2 +{\omega ^T_{A,V}}^2-2{\omega _{A,H}^2}+\omega _{\eta }^2 \big) \lambda ^3\\[5pt]& +2 \mathrm{i} \omega _\eta \big(\omega _C^2+\omega _M^2 +{\omega ^C_{A,V}}^2+{\omega ^T_{A,V}}^2-2{\omega _{A,H}^2}\big) \lambda ^2\\[5pt]& +\big (\omega _C^2\omega _\eta ^2+\omega _M^4+\big({\omega ^C_{A,V}}^2 +{\omega ^T_{A,V}}^2-2{\omega _{A,H}^2}\big)\big(\omega _{M}^2 +\omega _{\eta }^2\big)\big ) \lambda \\[5pt]& -\mathrm{i} \omega _\eta \omega _M^2 \big({\omega ^C_{A,V}}^2 +{\omega ^T_{A,V}}^2-{\omega _{A,H}^2}\big)=0, \end{aligned} \end{align}

where the dimensionless fundamental frequencies are

(4.7a–f) \begin{align} &\omega _{C}^2=\frac {1}{E_\eta ^2}\frac {k_z^2}{k^2},\quad \omega _{M}^2=\frac {(\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{k})^2}{E_\eta }, \quad {\omega _{A,V}^T}^2=\frac {Ra_{\ell ,V}^T}{E_\eta } \frac {k_z^2}{k^2},\quad {\omega _{A,V}^C}^2=\frac {Ra_{\ell ,V}^C}{E_\eta } \frac {k_z^2}{k^2}, \quad \nonumber\\[4pt]&\omega _{A,H}^2=\frac {Ra_{\ell ,H}}{2 E_\eta } \frac {k_z^2}{k^2}, \quad \omega _{\eta }^2=k^4. \end{align}

The local horizontal Rayleigh number $ \textit{Ra}_{\ell ,H}$ has the same definition as in (2.33). For the inequality $|\omega _C| \gg |\omega _M| \gg |\omega _{A,V}^C|, \, |\omega _{A,V}^T|, \, |\omega _{A,H}| \gg |\omega _\eta |$ , the roots of the characteristic (4.6) are approximated by

(4.8) \begin{align} \lambda _{1,2} \approx \pm \bigg (\omega _C + \frac {\omega _M^2}{\omega _C} \bigg ) +\mathrm{i} \, \frac {\omega _M^2\omega _\eta }{\omega _C^2}, \\[-28pt] \nonumber\end{align}

(4.9) \begin{align} \lambda _{3,4} \approx \pm \bigg (\frac {\omega _M^2}{\omega _C}+ \frac {{\omega _{A,V}^C}^2+{\omega _{A,V}^T}^2 -2\omega _{A,H}^2}{2\omega _C}\bigg ) + \mathrm{i} \, \omega _\eta \, \bigg (1-\frac {{\omega _{A,V}^C}^2+{\omega _{A,V}^T}^2 -\omega _{A,H}^2}{2\omega _M^2}\bigg ), \\[-28pt] \nonumber\end{align}

(4.10) \begin{align} \lambda _{5} \approx \mathrm{i}\,\omega _{\eta } \frac {{\omega _{A,V}^C}^2+{\omega _{A,V}^T}^2-\omega _{A,H}^2}{\omega _M^2}. \\[0pt] \nonumber\end{align}

Here, $\lambda _{1,2}$ and $\lambda _{3,4}$ give the approximate frequencies of the oppositely travelling fast ( $\omega _f$ ) and slow ( $\omega _s$ ) MAC waves, respectively. From (4.9), the resultant buoyancy frequency is given by

(4.11) \begin{align} \omega _{A}^2={\omega _{A,V}^C}^2+{\omega _{A,V}^T}^2-2\omega _{A,H}^2, \end{align}

which indicates the complementarity between the vertical buoyancies of composition and temperature, and the horizontal buoyancy of temperature. The resultant local Rayleigh number is then given by

(4.12) \begin{align} \textit{Ra}_\ell =\frac {g \big|\alpha ^C \beta _y^C+\alpha ^T \beta _y^T-2 \alpha ^T \beta _z k_y/k_z\big |\delta ^2}{2 \varOmega \eta }. \end{align}

The solutions for $\hat {u}_z$ , $\hat {\varTheta }$ and $\hat {\gamma }$ are

(4.13) \begin{align} \big [\hat {u}_z,\hat {\varTheta },\hat {\gamma }\big ] = \sum _{m=1}^{5}\big [D_{m},G_{m},Q_{m}\big ] \mbox{e}^{{i} \lambda _{m}t}, \end{align}

where the coefficients $D_{m}$ , $G_{m}$ and $Q_{m}$ are evaluated from the initial conditions for $\hat {u}_z$ , $\hat {\varTheta }$ , $\hat {\gamma }$ and their time derivatives. For instance, the initial conditions for $\hat {u}_z$ and its time derivatives are given by

(4.14) \begin{align} \begin{aligned} \mathrm{i}^n \sum _{m=1}^{5}D_m \lambda _{m}^n =\bigg (\frac {\partial ^n\hat {u}_z}{\partial t^n}\bigg )_{t=0} =d_{n+1}, \quad n=0,1,2,3,4, \end{aligned} \end{align}

where

(4.15) \begin{align} d_1&=\hat {u}_z\big |_{t=0}=0, \nonumber\\[4pt]d_2&=\frac {\partial {\hat {u}_z}}{\partial {t}}\big |_{t=0}= \big ({\omega _{A,V}^C}^2 \hat {\varTheta }_0+{\omega _{A,V}^T}^2 \hat {\gamma }_0\big )\frac {k_y}{k_z},\nonumber\\[4pt]d_3&=\frac {\partial ^2 \hat {u}_z}{\partial {t^2}}\big |_{t=0}=0,\nonumber\\[4pt]d_4&=\frac {\partial ^3{\hat {u}_z}}{\partial {t^3}}\big |_{t=0} =-d_2 \big(\omega _M^2+\omega _C^2+\omega _{A}^2 \big),\nonumber\\[4pt]d_5&=\frac {\partial ^4{\hat {u}_z}}{\partial {t^4}}\big |_{t=0} =d_2 \omega _{\eta }\omega _{M}^2, \end{align}

where $\hat {\varTheta }_0$ and $\hat {\gamma }_0$ are the initial perturbations of temperature and composition, respectively, both of which have the Gaussian distribution as in (2.1) in Cartesian coordinates. The coefficients of the fast and slow wave parts of $\hat {u}_z$ are then obtained from the roots of (4.6), as follows:

(4.16) \begin{align} D_{1}=\dfrac {d_{5}- \mathrm{i} \, d_{4}(\lambda _2 +\lambda _3+\lambda _4+\lambda _5) +\mathrm{i} \, d_{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)}, \\[-28pt] \nonumber\end{align}

(4.17) \begin{align} D_{3}=\dfrac {d_{5}- \mathrm{i} \, d_{4}(\lambda _1 +\lambda _2+\lambda _4+\lambda _5) + \mathrm{i} \,d_{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)}. \\[0pt] \nonumber\end{align}

Further, $\hat {u}_y$ is calculated from (2.17). The spectral coefficients of $\hat {\varTheta }$ and $\hat {\gamma }$ are obtained in a similar way (Appendix B).

In two-component convection, both thermal and compositional buoyancies drive the convection, and are responsible for magnetic field generation. The fraction of thermal power relative to the total convective power is given by

(4.18) \begin{align} f^T =\frac {P^T}{P^T+P^C}\times 100 \,\%, \end{align}

where $P^T$ and $P^C$ are the dimensionless thermal and compositional powers, respectively, defined by

(4.19) \begin{align} P^T= \textit{Ra}_{\ell ,V}^T \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \boldsymbol{u} \boldsymbol{\cdot }\varTheta \hat {\boldsymbol{e}}_y \, \mbox{d}y\,\mbox{d}z,\quad P^C= \textit{Ra}_{\ell ,V}^C \int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\boldsymbol{u}\boldsymbol{\cdot }\gamma \hat {\boldsymbol{e}}_y \, \mbox{d}y\,\mbox{d}z. \end{align}

The integrals in (4.19) are computed at $x=0$ for the limits $\pm 20$ in $(y,z)$ (the integrals of $u_y \Delta T_0$ and $u_y \Delta C_0$ vanish).

4.2. Complementarity of vertical and horizontal buoyancies in two-component convection

In § 2.4, the complementarity between vertical and horizontal buoyancies at the state of vanishing slow MAC waves was noted. In two-component convection, a complementarity exists between the vertical buoyancies of composition and temperature, and the horizontal buoyancy of temperature in suppressing the slow MAC waves (figure 16). Figure 16(a) shows the variation of the squares of the fundamental frequencies and the slow MAC wave frequency with relative horizontal buoyancy, whereas figure 16(b) presents the decomposition of $\omega _{A}^2$ into its three constituent parts. While the compositional Rayleigh number $ \textit{Ra}^C_{\ell ,V}$ is held constant (see later), the thermal Rayleigh number is determined by the power ratio $f^T$ , given by (4.18). In figure 16(a), $f^T$ is set to 10 % and the horizontal buoyancy is progressively increased. When the resultant $\omega _{A}$ obtained from (4.11) matches $\omega _{M}$ , the slow MAC waves disappear.

Table 4.

Calculation of the relative intensity of horizontal buoyancy (last column) in two-component linear magnetoconvection for states where the slow MAC waves disappear. The thermal power ratio $f^T$ is defined in (4.18); $ \textit{Ra}^C_{\ell ,V}$ , $ \textit{Ra}^T_{\ell ,V}$ are the local vertical compositional and thermal Rayleigh numbers, respectively. In addition, $ \textit{Ra}_{\ell ,H}$ is the local horizontal Rayleigh number. The resultant local thermal Rayleigh number is $ \textit{Ra}_\ell ^T=Ra^T_{\ell ,V}+Ra_{\ell ,H}$ . The parameters are $E_\eta =2\times 10^{-5}$ and $t=0.01$ .

Figure 16.

Variation of the squares of frequencies with relative horizontal buoyancy in two-component magnetoconvection. The dotted vertical line in panel (a) corresponds to $|\omega _A| \approx |\omega _M|$ , when the slow wave frequency $\omega _s$ goes to zero. Panel (b) gives the decomposition of $\omega _A^2$ into its three parts consisting of the vertical buoyancies of composition and temperature ( ${\omega ^C_{A,V}}^2$ , ${\omega ^T_{A,V}}^2$ ), and the horizontal buoyancy of temperature ( $\omega _{A,H}^2$ ). The parameters used are $E_\eta = 2 \times 10^{-5}$ , $B^2_{\textit{peak}}=200$ and $f^T= 10$ % at time $t/t_\eta = 10^{-2}$ .

The analysis in figure 16 is then performed for a range of $f^T$ and the results are summarised in table 4.