2.1. Solutions of the initial value problem

As the initial temperature perturbation (2.1) gives rise to a purely poloidal flow, the instantaneous state of the flow is uniquely defined by (Davidson, Sreenivasan & Aspden Reference Davidson, Sreenivasan and Aspden2007)

(2.6)\begin{gather} \boldsymbol{u} = \boldsymbol{\nabla} \times [ (\psi/s) \hat{\boldsymbol{e}}_\phi ], \end{gather}

(2.7)\begin{gather}\nabla_*^2 \psi = \frac{\partial^2{\psi}}{\partial{z^2}}+s\frac{\partial}{\partial{s}} \left(\frac{1}{s}\frac{\partial{\psi}}{\partial{s}}\right) ={-}s \zeta_\phi, \end{gather}

where $\psi$ is the Stokes streamfunction of the velocity and $\boldsymbol {\zeta }$ is the vorticity. In a similar way, the induced magnetic field is expressed as

(2.8)\begin{gather} \boldsymbol{b} = \boldsymbol{\nabla} \times [ (\xi/s) \hat{\boldsymbol{e}}_\phi ], \end{gather}

(2.9)\begin{gather}\nabla_*^2 \xi ={-}s \mu j_\phi, \end{gather}

where $\xi$ is the Stokes streamfunction of the induced magnetic field and $\boldsymbol {j}$ is the electric current density.

Taking the $\phi$ components of the curl of (2.2) and (2.3) and using (2.7), an equation for the evolution of $\psi$ follows:

(2.10)\begin{equation} \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{z^2}}\right](\nabla_*^2\psi) =\left(\frac{\partial}{\partial{t}}-\eta\nabla^2\right) g\alpha s\frac{\partial{\theta}}{\partial{s}},\end{equation}

where $V_M=B /\sqrt {\mu \rho }$ is the magnetic (Alfvén) wave velocity. A similar approach gives the equation for the evolution of $\xi$,

(2.11)\begin{equation} \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{z^2}}\right](\nabla_*^2 \xi) = B g \alpha s \frac{\partial^2{\theta}}{\partial{z}\partial{s}}.\end{equation}

The Hankel–Fourier transforms

(2.12)\begin{gather} H_1 \{\psi(s,z)\} = \hat{\psi}(k_s,k_z) = \frac{1}{2{\rm \pi} ^2}\int_{0}^{\infty}\int_{0}^{\infty}\psi(s,z){J}_1(k_ss)\, \mbox{e}^{-\textrm{i}k_zz}\,\mbox{d}s\,\mbox{d}z, \end{gather}

(2.13)\begin{gather}H_0 \{\theta(s,z)\} = \hat{\theta}(k_s,k_z)= \frac{1}{2{\rm \pi} ^2}\int_{0}^{\infty}\int_{0}^{\infty}\theta(s,z){J}_0(k_ss) \,\mbox{e}^{-\textrm{i}k_zz}s\,\mbox{d}s\,\mbox{d}z, \end{gather}

where ${J}_0$ and ${J}_1$ are the zeroth- and first-order Bessel functions of the first kind, are applied to (2.10),

(2.14)\begin{equation} \left[\left(\frac{\partial}{\partial{t}}+\nu k^2\right) \left(\frac{\partial}{\partial{t}}+\eta k^2\right)+ V_M^2k_z^2\right]\hat{\psi}=g\alpha\frac{k_s}{k^2}\frac{\partial{\hat{\theta}}} {\partial{t}} + g\alpha\eta k_s\hat{\theta},\end{equation}

where $k^2=k_s^2+k_z^2$ and the following identities are used:

(2.15a,b)\begin{equation} H_1 \{ f^\prime(s) \} ={-}k_s H_0 \{f(s) \}, \quad H_1 \{ s^{{-}1} \nabla_*^2 [sf(s)] \}={-}k^2 H_1 \{\,f(s) \}.\end{equation}

Since the zeroth-order Hankel transform of $u_z$ is given by

(2.16)\begin{align} \hat{u}_z & = \frac{1}{2{\rm \pi} ^2} \int_{0}^{\infty}\int_{0}^{\infty} \left(\frac1{s} \frac{\partial{{\psi}}}{\partial{s}}\right) {J}_0(k_s s)\, \mbox{e}^{-\textrm{i}k_z z} \, s \, \mbox{d}s \, \mbox{d}z, \end{align}

(2.17)\begin{align} &= \frac{1}{2{\rm \pi} ^2} \int_{0}^{\infty}\left[\int_{0}^{\infty} \left(\frac{\partial{{\psi}}}{\partial{s}}\right) {J}_0(k_s s) \, \mbox{d}s\right] \, \mbox{e}^{-\textrm{i}k_z z} \, \mbox{d}z, \end{align}

integration by parts using the identity ${J}_0^\prime (s)= -{J}_1(s)$ gives

(2.18)\begin{equation} \hat{u}_z = \frac{k_s}{2{\rm \pi} ^2} \int_{0}^{\infty}\int_{0}^{\infty} \psi(s,z) {J}_1(k_s s) \, \mbox{e}^{-\textrm{i}k_z z} \, \mbox{d}s \, \mbox{d}z= k_s \hat{\psi}, \end{equation}

from (2.12). Consequently, the transform of (2.4),

(2.19)\begin{equation} \frac{\partial{\hat{\theta}}}{\partial{t}}={-}\beta k_s \hat{\psi} - \kappa k^2\hat{\theta},\end{equation}

is used in (2.14) to obtain the equation for the evolution of $\hat {\psi }$,

(2.20)\begin{equation} \left[\left(\frac{\partial}{\partial{t}}+\nu k^2\right) \left(\frac{\partial}{\partial{t}}+\eta k^2\right) +V_M^2k_z^2+\frac{g\alpha \beta k_s^2}{k^2}\right] \left(\frac{\partial}{\partial{t}} +\kappa k^2\right)\hat{\psi}={-}g\alpha k_s^2(\eta-\kappa)\beta\hat{\psi}.\end{equation}

Seeking plane wave solution of the form $\hat {\psi } \sim \mbox {e}^{\textrm {i}\lambda t}$ for (2.20), we obtain the relation

(2.21)\begin{equation} (\textrm{i}\lambda+\omega_\nu)(\textrm{i}\lambda+\omega_\eta)(\textrm{i}\lambda+\omega_\kappa)+ \omega_M^2(\textrm{i}\lambda+\omega_\kappa)+\omega_A^2(\textrm{i}\lambda+\omega_\eta)=0,\end{equation}

which consists of the fundamental frequencies $\omega _M=V_M k_z$ (Alfvén wave), $\omega _A=\sqrt {g \alpha \beta } k_s/k$ (internal gravity wave), $\omega _\eta =\eta k^2$ (magnetic diffusion), $\omega _\nu =\nu k^2$ (viscous diffusion) and $\omega _\kappa =\kappa k^2$ (thermal diffusion). The subscripts $M$ and $A$ above represent waves of magnetic and Archimedean origin.

Equation (2.21) is of the form

(2.22)\begin{equation} \lambda^3+a_2\lambda^2+a_1\lambda+a_0=0,\end{equation}

where

(2.23)\begin{equation} \left.\begin{array}{c@{}} a_2 ={-}\textrm{i}(\omega_\eta+\omega_\kappa+\omega_\nu), \\ a_1 ={-}(\omega_A^2+\omega_\eta\omega_\kappa+ \omega_\eta\omega_\nu+\omega_\kappa\omega_\nu+\omega_M^2), \\ a_0 = \textrm{i}(\omega_A^2\omega_\eta+\omega_\kappa\omega_M^2+\omega_\eta\omega_\kappa\omega_\nu). \end{array}\right\} \end{equation}

To solve (2.22), we use the cubic formula (Dickson Reference Dickson1898; Dunham Reference Dunham1990). The solutions of (2.21) are then given by

(2.24a)\begin{gather} \lambda_{1,2} ={\pm} \frac{\sqrt{3}}{12}\left(2^{2/3} Q+ \frac{2\sqrt[3]{2}P}{Q}\right)+\frac{\textrm{i}}{12} \left(4(\omega_\eta+\omega_\kappa+\omega_\nu)- 2^{2/3} Q+\frac{2\sqrt[3]{2}P}{Q}\right), \end{gather}

(2.24b)\begin{gather}\lambda_3=\frac{\textrm{i}}{6}\left(2^{2/3}Q-\frac{2\sqrt[3]{2}P}{Q} +2(\omega_\eta+\omega_\kappa+\omega_\nu)\right), \end{gather}

where

(2.25)\begin{gather} P = 3 (\omega_A^2+\omega_M^2+\omega_\eta\omega_\kappa +(\omega_\eta+\omega_\kappa)\omega_\nu )- (\omega_\eta+\omega_\kappa+\omega_\nu )^2, \end{gather}

(2.26)\begin{gather}Q = \sqrt[3]{\sqrt{4P^3+R^2}+R}, \end{gather}

(2.27)\begin{align} R &= 9 \omega_A^2 (2\omega_\eta-\omega_\kappa-\omega_\nu) -9 \omega_M^2 (\omega_\eta - 2\omega_\kappa+\omega_\nu) \nonumber\\ & \quad + (\omega_\eta+\omega_\kappa-2\omega_\nu)(2\omega_\eta-\omega_\kappa-\omega_\nu) (\omega_\eta-2\omega_\kappa+\omega_\nu). \end{align}

The solutions (2.24a)–(2.24b) for the frequency may also be obtained by taking the transform of (2.11) for $\xi$.

The general solutions for the transforms $\hat {\psi }$ and $\hat {\xi }$ are then given by

(2.28)\begin{equation} [\hat{\psi},\hat{\xi}] = \sum_{m=1}^{3}[A_m,B_m]\,\mbox{e}^{\textrm{i} \lambda_m t}.\end{equation}

The coefficients $A_m$ and $B_m$ are evaluated from the initial conditions for $\hat {\psi }$ and $\hat {\xi }$ and their time derivatives.

For a given time, the streamfunctions of the velocity (and induced magnetic field) may be recovered from (2.28) by the inverse Hankel–Fourier transform

(2.29)\begin{equation} f(s,z) = 4{\rm \pi} s\int_{0}^{\infty}\int_{0}^{\infty}\hat{f}(k_s,k_z) {J}_1(k_ss)\, \mbox{e}^{\textrm{i}k_zz} k_s\,\mbox{d}k_s\,\mbox{d}k_z.\end{equation}

2.1.1. Evaluation of spectral coefficients

From (2.28), the initial conditions for $\hat {\psi }$ and its time derivatives are given by

(2.30)\begin{equation} {\rm i}^n\sum_{m=1}^{3}A_m\lambda_m^n= \left(\frac{\partial^n{\hat{\psi}}}{\partial{t^n}}\right)_{ 0}=a_{n+1}, \quad n=0,1,2.\end{equation}

where the subscript ‘0’ refers to the initial state. While the Hankel–Fourier transform of initial condition (2.1) gives $\hat{\theta}_0$ (Abramowitz & Stegun Reference Abramowitz and Stegun1972), the conditions of zero initial velocity and induced magnetic field are necessary to evaluate the initial time derivatives of $\hat {\psi }$. Algebraic simplifications using the curl of (2.2) with (2.7) give the right-hand sides of (2.30) as follows:

(2.31)\begin{gather} a_1 = 0, \end{gather}

(2.32)\begin{gather}a_2 = \frac{ g \alpha k_s}{k^2} \hat{\theta}_0= \frac{g \alpha k_s\delta^3 \,\mbox{e}^{{-}k^2\delta^2/8}}{16\sqrt{2}{\rm \pi} ^{3/2}k^2}, \end{gather}

(2.33)\begin{gather}a_3 ={-} g \alpha k_s (\nu+\kappa)\hat{\theta}_0={-} \frac{g \alpha k_s(\nu+\kappa)\delta^3 \,\mbox{e}^{- k^2\delta^2/8}}{16\sqrt{2}{\rm \pi} ^{3/2}}. \end{gather}

Equations (2.31)–(2.33) give the coefficients of the velocity transform,

(2.34a–c)\begin{equation} A_1\!=\!{-}\frac{a_3-{\rm i}a_2(\lambda_2+\lambda_3)}{(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)}, \quad A_2 \!=\!{-}\frac{a_3-{\rm i}a_2(\lambda_1+\lambda_3)}{(\lambda_2-\lambda_3)(\lambda_2-\lambda_1)}, \quad A_3 \!=\!{-}\frac{a_3-{\rm i}a_2(\lambda_1+\lambda_2)}{(\lambda_3-\lambda_1)(\lambda_3-\lambda_2)}. \end{equation}

From (2.28), the initial conditions for $\hat {\xi }$ and its time derivatives are given by

(2.35)\begin{equation} {\rm i}^n\sum_{m=1}^{3}B_m\lambda_m^n= \left(\frac{\partial^n{\hat{\xi}}}{\partial{t^n}}\right)_{ 0}=b_{n+1}, \quad n=0,1,2.\end{equation}

The right-hand sides of (2.35) are obtained as follows:

(2.36)\begin{gather} b_1 = b_2 = 0, \end{gather}

(2.37)\begin{gather}b_3 = \textrm{i} \frac{g \alpha B k_s k_z \delta^3 \,\mbox{e}^{- k^2\delta^2/8}}{16\sqrt{2}{\rm \pi} ^{3/2} k^2}. \end{gather}

The coefficients of the magnetic field transform are then given by

(2.38a–c)\begin{equation} B_1 ={-}\frac{b_3-{\rm i}b_2(\lambda_2\!+\!\lambda_3)}{(\lambda_1-\lambda_2)(\lambda_1\!-\!\lambda_3)}, \quad B_2 ={-}\frac{b_3-{\rm i}b_2(\lambda_1\!+\!\lambda_3)}{(\lambda_2-\lambda_3)(\lambda_2\!-\!\lambda_1)}, \quad B_3 ={-}\frac{b_3-{\rm i}b_2(\lambda_1\!+\!\lambda_2)}{(\lambda_3-\lambda_1)(\lambda_3 \!-\!\lambda_2)}.\end{equation}

In the sections below, we present approximate analytical solutions in the limit of zero viscous and thermal diffusion that describe the long-time evolution of magnetically damped waves initiated by a density perturbation in a stably stratified fluid. Two limits are analysed, that of strong stratification and that of strong magnetic field. The theory is then compared with computations of the general solution using $\nu , \kappa \ll \eta$.

2.2. The case of strong stratification

In the limit of zero viscous and thermal diffusion ($\nu =\kappa =0$), the orders of magnitude of the fundamental frequencies for this case are given by $\omega _A \gg \omega _M \gg \omega _\eta$. Since the diffusion of disturbances along the magnetic field lines does not result in a reduction in $\omega _A$, a diffusion-dominated regime where $\omega _\eta \gg \omega _A$ is not anticipated.

When terms up to second order in $(\omega _M/\omega _A)$ are considered, the frequencies (2.24a)–(2.24b) are approximated by (Appendix B)

(2.39a)\begin{gather} \lambda_{1,2} \approx{\pm} \omega_A \left(1+\frac{{\omega_M}^2}{{2\omega_A}^2} \right) +\textrm{i} \frac{{\omega_M}^2 \omega_\eta}{2{\omega_A}^2} ={\pm} R_1 +\textrm{i} I_{11}, \end{gather}

(2.39b)\begin{gather}\lambda_3 \approx \textrm{i} \omega_\eta \left(1-\frac{{\omega_M}^2}{{\omega_A}^2}\right) =\textrm{i} I_{12}. \end{gather}

The spectral coefficients in (2.34a–c) then take the approximate form

(2.40a–c)\begin{equation} A_1 \approx \textrm{i}\frac{a_2}{2\omega_A}, \quad A_2 \approx{-}\textrm{i} \frac{a_2}{2\omega_A}, \quad A_3 \approx\frac{a_2}{2\omega_A} \left(\frac{2 \omega_M^2 \omega_\eta}{\omega_A^3}\right).\end{equation}

where $a_2$ is given by (2.32). The transform of the velocity streamfunction is then,

(2.41)\begin{equation} \hat{\psi}\approx A_1 \mathrm{e}^{-\textrm{i} R_1 t - I_{11} t} + A_2 \mathrm{e}^{\textrm{i} R_1 t - I_{11} t}, \end{equation}

since $A_3 \to 0$. Further simplification using (2.40a–c) yields

(2.42)\begin{equation} \hat{\psi}\approx \textrm{i} \frac{a_2}{2\omega_A}( \mbox{e}^{-\textrm{i} R_1 t } - \mbox{e}^{\textrm{i} R_1 t })\, \mbox{e}^{- I_{11}t} \approx \frac{a_2}{\omega_A} \, \mbox{e}^{{-}I_{11}t} \sin{(R_1t)}.\end{equation}

A similar approach gives the spectral coefficients in (2.38a–c) in the approximate form

(2.43a–c)\begin{equation} B_1 \approx \frac{b_3}{2\omega_A^2}\left({-}1+ \textrm{i}\frac{\omega_\eta}{\omega_A}\right),\quad B_2 \approx \frac{b_3}{2\omega_A^2}\left({-}1-\textrm{i}\frac{\omega_\eta}{\omega_A}\right),\quad B_3 \approx \frac{b_3}{\omega_A^2},\end{equation}

where $b_3$ is given by (2.37). The transform of the induced field streamfunction is then,

(2.44)\begin{equation} \hat{\xi} \approx B_1 \, \mathrm{e}^{-\textrm{i} R_1 t - I_{11} t} + B_2 \,\mathrm{e}^{\textrm{i} R_1 t - I_{11} t} + B_3 \, \mathrm{e}^{{-}I_{12}t}. \end{equation}

Further simplification using (2.43a–c) yields

(2.45)\begin{equation} \hat{\xi} \approx \frac{b_3}{\omega_A^2}\left[-\cos{(R_1t)} \, \mbox{e}^{- I_{11}t} +\frac{\omega_\eta}{\omega_A}\sin{(R_1t)} \, \mbox{e}^{- I_{11}t} + \mbox{e}^{- I_{12}t}\right]. \end{equation}

2.2.1. Long-time evolution of kinetic energy

Since $\vert {\hat {u}}\vert ^2=k^2\vert {\hat {\psi }}\vert ^2$, (2.42) gives

(2.46)\begin{equation} \vert {\hat{u}}\vert ^2\approx\frac{\vert {a_2}\vert ^2k^2}{\omega_A^2}\, \textrm{e}^{{-}2I_{11}t}\sin^2{(R_1t)}.\end{equation}

Using (2.31) and the expressions for the frequencies $\omega _A, \omega _M$ and $\omega _\eta$ in (2.42), we obtain

(2.47)\begin{equation} \vert {\hat{u}}\vert ^2 \approx\exp{\left(- \frac{k^2\delta^2}{4}\right)} \left(\frac{g\alpha \delta^6}{512{\rm \pi} ^{3}\beta}\right) \exp{\left(-\frac{V_M^2\eta k^4k_z^2}{g\alpha\beta k_s^2}t\right)} \sin^2{\left(\sqrt{g\alpha \beta}\frac{k_s}{k}t\right)}.\end{equation}

The initial temperature perturbation (2.1) releases energy into the poloidal flow on the time scale

(2.48)\begin{equation} t_A=\frac1{\sqrt{g \alpha \beta}} \frac{k_0}{k_{s0}}, \end{equation}

where $k_0=\sqrt {6}/\delta$ is the initial wavenumber of the disturbance (Appendix C) and $k_{s0}=k_0/\sqrt {2}$. This flow subsequently undergoes magnetic damping on the time scale

(2.49)\begin{equation} t_1=\frac{g \alpha \beta}{V_M^2 \eta k_0^4}, \end{equation}

as $k_{z0}=k_{s0}$. Since the kinetic energy is given by Parseval's theorem as (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017)

(2.50)\begin{equation} E_{k} = 16 {\rm \pi}^4 \int_{0}^{\infty} \int_{0}^{\infty} |\hat{u}|^2 k_s \, \mbox{d}k_s \, \mbox{d}k_z,\end{equation}

for $t \gtrsim t_1$, the kinetic energy is expanded as

(2.51)\begin{equation} E_{k} \approx 16{\rm \pi} ^4\int_{0}^{\infty}\int_{0}^{\infty} \left(\frac{g\alpha \delta^6}{512{\rm \pi} ^{3}\beta}\right) \exp{\left( - \frac{k^2\delta^2}{4}\right)}\exp{\left(-\frac{V_M^2\eta k^4k_z^2} {g\alpha\beta k_s^2}t\right)} k_s\, \mathrm{d}k_s\,\mathrm{d}k_z.\end{equation}

Using the substitutions $k_z=k\cos \vartheta$ and $k_s=k \sin \vartheta$ (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017),

(2.52)\begin{equation} E_{k} \approx\frac{{\rm \pi} g\alpha \delta^6 }{32\beta} \int_{0}^{\infty}\int_{0}^{{\rm \pi} /2}k^2\exp{\left(- \frac{k^2\delta^2}{4}\right)} \exp{\left(-\frac{V_M^2\eta k^4\cot^2\vartheta}{g\alpha\beta}t\right)} \sin\vartheta\,\mbox{d}\vartheta\,\mbox{d}k.\end{equation}

Letting $x=\cot ^2\vartheta$, (2.52) may be rewritten as

(2.53)\begin{equation} E_k \approx \frac{{\rm \pi} g\alpha \delta^6 }{64\beta} \int_{0}^{\infty}\int_{0}^{\infty} k^2 \exp \left( - \frac{k^2\delta^2}{4}\right) \exp \left( -\frac{V_M^2\eta k^4 x}{g\alpha\beta}t \right) x^{{-}1/2}(1+x)^{{-}3/2}\,\mbox{d}x\,\mbox{d}k.\end{equation}

Using the relation (13.2.5 in Abramowitz & Stegun Reference Abramowitz and Stegun1972)

(2.54)\begin{equation} \varGamma(\textit{a}){U}(a,b,z)=\int_{0}^{\infty} \,\mbox{e}^{{-}z \tau} \tau^{a-1}(1+\tau)^{b-a-1}\,\mbox{d}\tau, \end{equation}

where $\varGamma$ is the gamma function, ${U}$ is the confluent hypergeometric function, $a=1/2$, $b=0$ and $z=V_M^2 \eta k^4 t/g \alpha \beta$, (2.53) simplifies to

(2.55)\begin{equation} E_k\approx\frac{{\rm \pi} g\alpha \delta^6 }{64\beta} \int_{0}^{\infty} k^2 \exp{\left(-\frac{k^2\delta^2}{4} \right)} \varGamma\left(\frac{1}{2}\right) {U}\left(\frac1{2},0, \frac{V_M^2\eta k^4}{g\alpha\beta} t\right)\,\mbox{d}k.\end{equation}

Since the asymptotic value of ${U}$ as $z\to \infty$ is given by (13.1.8 in Abramowitz & Stegun Reference Abramowitz and Stegun1972)

(2.56)\begin{equation} {U}(a,b,z)\sim z^{{-}a} [1 + O (\vert {z}\vert ^{{-}1} ) ],\end{equation}

for $t \gg t_1$, (2.55) further reduces to

(2.57)\begin{equation} E_k \approx \frac{{\rm \pi} ^{3/2} g\alpha \delta^6 }{64\beta} \int_{0}^{\infty} \exp \left( -\frac{k^2\delta^2}{4} \right) \left(\frac{t}{k_0^4 t_1} \right)^{{-}1/2}\, \mbox{d}k.\end{equation}

Finally, integration over the wavenumber $k$ yields

(2.58)\begin{equation} E_k\approx\frac{3 {\rm \pi}^{2} g\alpha \delta^3}{32 \beta}\left(\frac{t}{t_1}\right)^{{-}1/2}, \end{equation}

by using the relation $k_0 \delta =\sqrt {6}$ for the initial perturbation (2.1). Thus, for $t \gg t_1$, the kinetic energy decays as $(t/t_1)^{-1/2}$.

2.2.2. Long-time evolution of magnetic energy

Since $|\hat {b}|^2=k^2|\hat {\xi }|^2$, (2.44) gives

(2.59)\begin{align} |\hat{b}|^2&\approx k^2 \frac{\vert {b_3}\vert ^2}{\omega_A^4} \left[\mbox{e}^{{-}2I_{12}t}-2\mbox{e}^{-(I_{11}+I_{12})t} \cos{(R_1t)}+ \mbox{e}^{{-}2I_{11}t} \cos^2{(R_1t)}\vphantom{\frac{\omega_\eta^2}{\omega_A^2}}\right.\nonumber\\ &\quad + 2 \, \mbox{e}^{{-}I_{12}t} \frac{\omega_\eta}{\omega_A} \mbox{e}^{{-}I_{11}t} \sin{(R_1t)} - 2 \, \mbox{e}^{{-}2I_{11}t} \frac{\omega_\eta}{\omega_A}\cos{(R_1t)}\sin{(R_1t)}\nonumber\\ &\quad + \left.\mbox{e}^{{-}2I_{11}t}\frac{\omega_\eta^2}{\omega_A^2}\sin^2{(R_1t)}\right]. \end{align}

Noting that the inequality ${R_1}^{-1} \ll {I_{12}}^{-1} \ll {I_{11}}^{-1}$ is satisfied by the time scales, the long-time evolution of $|\hat {b}|^2$ is determined by the term that decays slowest,

(2.60)\begin{equation} |\hat{b}|^2 \approx k^2 \frac{\vert {b_3}\vert ^2}{\omega_A^4}\, \mbox{e}^{{-}2I_{11}t}.\end{equation}

Since the magnetic energy is given by Parseval's theorem as

(2.61)\begin{equation} E_m = \frac{16 {\rm \pi}^4}{\mu \rho} \int_{0}^{\infty} \int_{0}^{\infty} {\vert {\hat{b}}\vert }^2 k_s \, \mbox{d}k_s \, \mbox{d}k_z, \end{equation}

using (2.37) and the expressions for the frequencies $\omega _A, \omega _M$ and $\omega _\eta$ in (2.60), we obtain the magnetic energy for times $t \gtrsim t_1$,

(2.62)\begin{equation} E_m \approx \frac{{\rm \pi} \delta^6 V_M^2}{32 \beta^2} \int_{0}^{\infty}\int_{0}^{\infty} \frac{k^2 k_z^2}{k_s^2} \exp{\left(- \frac{k^2\delta^2}{4}\right)} \exp{\left(-\frac{V_M^2\eta k^4k_z^2} {g\alpha\beta k_s^2}t\right)} k_s\,\mbox{d}k_s\,\mbox{d}k_z. \end{equation}

Using the substitutions $k_z=k\cos \vartheta$ and $k_s=k\sin \vartheta$,

(2.63)\begin{equation} E_m \approx \frac{{\rm \pi} \delta^6 V_M^2}{32\beta^2} \int_{0}^{\infty}\int_{0}^{{\rm \pi} /2} k^4\exp{\left(- \frac{k^2\delta^2}{4} - \frac{V_M^2\eta k^4\cot^2\vartheta}{g\alpha\beta}t\right)} \cot^2\vartheta \sin\vartheta \, \mbox{d}\vartheta \, \mbox{d}k. \end{equation}

The integral in (2.63), of the form

(2.64)\begin{equation} I_A=\int_{0}^{\infty} k^4\exp ({-}a k^2-b k^4) \ \textrm{d} k, \end{equation}

where $a=\delta ^2/2$ and $b=({V^2_M \eta \cot ^2 \vartheta }/{g \alpha \beta }) t$, has the following solution (see 3.469, Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007):

(2.65)\begin{equation} I_A = \frac{\sqrt{a} \exp ( a^2/8b )}{32 b^{5/2}} \left[(a^2 + 2 b) K_{1/4}\left(\frac{a^2}{8b}\right)- a^2 K_{3/4}\left(\frac{a^2}{8b}\right)\right], \end{equation}

where $K_{n}(x)$ is the modified Bessel function of the second kind. In the limit $b\rightarrow \infty$ ($t \gg t_1$), $I_A$ is approximated by (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017)

(2.66)\begin{equation} I_A\approx \frac{\varGamma(1/4)b^{{-}5/4}}{16}+O(b^{{-}7/4}). \end{equation}

Therefore, (2.63) simplifies to

(2.67)\begin{equation} E_m\approx \frac{36 \sqrt{6} {\rm \pi}\delta V_M^2}{512\beta^2} I_\vartheta \varGamma{\left(\frac{1}{4}\right)} \left(\frac{t}{t_1}\right)^{{-}5/4},\end{equation}

where the integral $I_\vartheta$ is related to the beta function by (3.621, Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007)

(2.68)\begin{align} I_\vartheta &= \int_{0}^{{\rm \pi} /2} (\cot^2\vartheta)^{{-}5/4} \cot^2\vartheta \sin\vartheta\, \mbox{d}\vartheta = \frac{1}{2} \mathtt{B}(5/4,1/4), \end{align}

(2.69)\begin{align} &\approx 1.854. \end{align}

Thus, for $t \gg t_1$, the magnetic energy decays as $(t/t_1)^{-5/4}$.

2.3. The case of strong magnetic field

In the limit of zero viscous and thermal diffusion ($\nu =\kappa =0$), the orders of magnitude of the fundamental frequencies for this case are given by $\omega _M \gg \omega _A \gg \omega _\eta$. When terms up to second order in $(\omega _A/\omega _M)$ are considered, the frequencies (2.24a)–(2.24b) are approximated by

(2.70a)\begin{gather} \lambda_{1,2} \approx{\pm} \omega_M \left(1+\frac{{\omega_A}^2}{{2\omega_M}^2} \right) + \textrm{i} \frac{\omega_\eta}{2} \left(1- \frac{{\omega_A}^2}{{\omega_M}^2} \right) ={\pm} R_{2} +\textrm{i} I_{21}, \end{gather}

(2.70b)\begin{gather}\lambda_3 \approx \textrm{i} \frac{{\omega_A}^2\omega_\eta}{{\omega_M}^2} =\textrm{i} I_{22}. \end{gather}

The spectral coefficients in (2.34a–c) then take the approximate form

(2.71a–c)\begin{equation} A_1 \approx \frac{a_2}{2\omega_M}\left(-\frac{\omega_\eta}{\omega_M} + \textrm{i} \right), \quad A_2 \approx \frac{a_2}{2\omega_M} \left(-\frac{\omega_\eta}{\omega_M} -\textrm{i} \right), \quad A_3 \approx\frac{a_2}{2\omega_M} \left( \frac{2 \omega_\eta}{\omega_M} \right),\end{equation}

where $a_2$ is given by (2.32). The transform of the velocity streamfunction is then,

(2.72)\begin{equation} \hat{\psi}\approx A_1 \, \mbox{e}^{(-\textrm{i}R_2-I_{21}) t}+ A_2 \, \mbox{e}^{(\textrm{i}R_2- I_{21})t} + A_3 \, \mbox{e}^{{-}I_{22}t}.\end{equation}

Further simplification using (2.71a–c) yields

(2.73)\begin{equation} \hat{\psi}\approx\frac{a_2}{\omega_M}\left[ \left(-\frac{\omega_\eta}{\omega_M} \right)\, \mbox{e}^{{-}I_{21} t} \cos(R_2 t ) + \mbox{e}^{{-}I_{21} t} \sin(R_2 t ) + \left( \frac{ \omega_\eta}{\omega_M} \right)\, \mbox{e}^{{-}I_{22} t}\right]. \end{equation}

A similar approach gives the spectral coefficients in (2.38a–c) in the approximate form

(2.74a–c)\begin{equation} B_1 \approx \frac{b_3}{2 \omega_M^2} \left({-}1 -\textrm{i} \frac{\omega_\eta}{2 \omega_M}\right),\quad B_2 \approx \frac{b_3}{2 \omega_M^2} \left({-}1 +\textrm{i} \frac{\omega_\eta}{2 \omega_M}\right),\quad B_3 \approx \frac{b_3}{\omega_M^2}, \end{equation}

where $b_3$ is given by (2.37). The transform of the induced field streamfunction is then,

(2.75)\begin{align} \hat{\xi} &\approx B_1 \, \mbox{e}^{(-\textrm{i}R_2-I_{21}) t} +B_2 \, \mbox{e}^{(\textrm{i}R_2- I_{21})t} + B_3 \, \mbox{e}^{{-}I_{22}t}, \end{align}

(2.76)\begin{align} & \approx \frac{b_3}{\omega_M^2} \left[-\mbox{e}^{{-}I_{21} t} \cos(R_2 t ) - \frac{ \omega_\eta}{2 \omega_M} \mbox{e}^{{-}I_{21} t} \sin(R_2 t ) + \mbox{e}^{{-}I_{22} t}\right]. \end{align}

Since $\vert {\hat {u}}\vert ^2=k^2\vert {\hat {\psi }}\vert ^2$ and ${\vert {\hat {b}}\vert }^2 = k^2 {\vert {\hat {\xi }}\vert }^2$, (2.73) and (2.76) give

(2.77)\begin{align} \vert {\hat{u}}\vert ^2 & \approx\frac{\vert {a_2}\vert ^2k^2}{\omega_M^2} \left[\left(\frac{\omega_\eta}{\omega_M}\right)^2 \, \mbox{e}^{{-}2I_{22}t}- 2 \, \mbox{e}^{-(I_{21}+I_{22})t} \left(\frac{\omega_\eta}{\omega_M}\right)^2 \cos{(R_2 t)}\right.\nonumber\\ &\quad + \mbox{e}^{{-}2I_{21}t} \left(\frac{\omega_\eta}{\omega_M}\right)^2 \cos^2{(R_2t)} +2 \, \mbox{e}^{-(I_{21}+I_{22}) t}\left(\frac{\omega_\eta}{\omega_M}\right)\sin{(R_{2}t)}\nonumber\\ &\quad \left.-2 \, \mbox{e}^{{-}2I_{21}t} \left(\frac{\omega_\eta}{\omega_M}\right)\cos{(R_2t)}\sin{(R_2t)}+ \mbox{e}^{{-}2I_{21}t}\sin^2{(R_2t)}\right], \end{align}

(2.78)\begin{align} \vert {\hat{b}}\vert ^2& \approx\frac{\vert {b_3}\vert ^2k^2}{\omega_M^4} \left[\mbox{e}^{{-}2I_{22}t}-2 \, \mbox{e}^{-(I_{21}+I_{22})t}\cos{(R_2 t)}+ \mbox{e}^{{-}2I_{21}t} \cos^2{(R_2t)} \vphantom{\frac{\omega_\eta^2}{4\omega_M^2}}\right.\nonumber\\ & \quad -\frac{\omega_\eta}{\omega_M} \mbox{e}^{-(I_{21}+I_{22}) t}\sin{(R_2t)} +\frac{\omega_\eta}{\omega_M} \mbox{e}^{{-}2I_{21}t}\cos{(R_2t)}\sin{(R_2t)}\nonumber\\ &\quad \left.+\frac{\omega_\eta^2}{4\omega_M^2}\, \mbox{e}^{{-}2I_{21}t} \sin^2{(R_2t)}\right]. \end{align}

We note that energy is released into the poloidal flow and field on the time scale

(2.79)\begin{equation} t_M= (V_M k_{z0})^{{-}1}. \end{equation}

Since the inequality ${R_2}^{-1} \ll {I_{21}}^{-1} \ll {I_{22}}^{-1}$ is satisfied by the time scales, for ${I_{21}}^{-1} < t < {I_{22}}^{-1}$,

(2.80a,b)\begin{equation} \vert {\hat{u}}\vert ^2 \approx \frac{\vert {a_2}\vert ^2k^2}{\omega_M^2}\, \mbox{e}^{{-}2I_{21} t}, \quad \vert {\hat{b}}\vert ^2 \approx \frac{\vert {b_3}\vert ^2k^2}{\omega_M^4}\, \mbox{e}^{{-}2I_{21}t}, \end{equation}

which gives

(2.81)\begin{equation} \vert {\hat{u}}\vert ^2= \frac1{\mu \rho}\vert {\hat{b}}\vert ^2 \approx \frac{g^2 \alpha^2 \delta^6}{512 {\rm \pi}^3 V_M^2} \left(\frac{k_s^2}{k_z^2 k^2}\right) \exp{\left[- k^2 \left( \frac{\delta^2}{4} + \eta t \right) \right]}. \end{equation}

The kinetic and magnetic energies evolve in equipartition in this Alfvénic regime.

The long-time decay of classical MHD turbulence would be diffusion dominated as the Alfvén wave frequency progressively decreases and falls below the diffusion frequency (Lehnert Reference Lehnert1955; Moffatt Reference Moffatt1967; Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017). However, as we see below, stable stratification limits the dominance of magnetic diffusion as the frequency of internal gravity waves remains large relative to that of diffusion.

2.3.1. Long-time evolution of energy

From the relative magnitudes of the time scales given by ${R_2}^{-1} \ll {I_{21}}^{-1} \ll {I_{22}}^{-1}$, the long-time evolution of $\vert {\hat {u}}\vert ^2$ is described by the term that decays slowest in (2.77),

(2.82)\begin{equation} \vert {\hat{u}}\vert ^2 \approx\frac{\vert {a_2}\vert ^2k^2}{\omega_M^2} \left(\frac{\omega_\eta}{\omega_M}\right)^2 \, \mbox{e}^{{-}2I_{22}t}. \end{equation}

In the strong-field limit of Lundquist number $S \to \infty$, $\omega _\eta /\omega _M \to 0$, so a finite kinetic energy via (2.82) necessitates a progressive decrease of $\omega _M$ until the two frequencies are of the same order. In classical MHD turbulence at $S \gg 1$, the onset of diffusive decay occurs when $\omega _M \sim \omega _\eta$, at time $t \sim S^{1/2} t_\eta$ (Moffatt Reference Moffatt1967). In a stratified MHD layer, however, the decrease of $\omega _M$ would lead to the regime $\omega _A \gg \omega _M \gg \omega _\eta$, in which the evolution of kinetic energy is described by (2.58) (see § 2.2.1). We shall return to this point in § 2.4.

The long-time evolution of $\vert {\hat {b}}\vert ^2$ is determined by the term that decays slowest in (2.78),

(2.83)\begin{equation} \vert {\hat{b}}\vert ^2\approx\frac{\vert {b_3}\vert ^2k^2}{\omega_M^4} \,\mbox{e}^{{-}2I_{22}t}, \end{equation}

which remains finite for $S \to \infty$. Using (2.37) and the expressions for the frequencies $\omega _A, \omega _M$ and $\omega _\eta$ in (2.83), we obtain

(2.84)\begin{equation} \frac1{\mu \rho} \vert {\hat{b}}\vert ^2 \approx \frac{g^2\alpha^2\delta^6}{512 {\rm \pi}^3 V_M^2} \left(\frac{k_s^2}{k_z^2 k^2}\right) \exp{\left(-\frac{k^2 \delta^2}{4}\right)} \exp{\left(-\frac{2g\alpha\beta \eta k_s^2}{V_M^2k_z^2}t\right)},\end{equation}

which indicates that the magnetic energy undergoes damping on the time scale

(2.85)\begin{equation} t_2 = \frac{V_M^2}{g \alpha \beta \eta}. \end{equation}

The transformation to polar coordinates $(k,\vartheta )$ gives the magnetic energy for times $t \gtrsim t_2$,

(2.86)\begin{equation} E_m\approx\frac{{\rm \pi} g^2\alpha^2\delta^6}{32V_M^2}\int_{0}^{\infty}\int_{0}^{{\rm \pi} /2} \exp{\left(-\frac{k^2 \delta^2}{4}\right)} \exp{\left(-\frac{2t}{t_2} \tan^2\vartheta \right)} \frac{\sin^3\vartheta}{\cos^2\vartheta} \,\mbox{d}\vartheta\,\mbox{d}k.\end{equation}

Letting $x = \tan ^2 \vartheta$, and using the identity (2.54) with $a=2$, $b=3/2$ and $z=2t/t_2$,

(2.87)\begin{equation} E_m\approx \frac{{\rm \pi} g^2\alpha^2\delta^6}{64V_M^2} \int_{0}^{\infty} \exp{\left(-\frac{k^2 \delta^2}{4}\right)} \varGamma(2) {U}\left(2,\frac{3}{2},\frac{2t}{t_2}\right)\,\mbox{d}k.\end{equation}

The asymptotic value of ${U}$ as $z \to \infty$ ($t \gg t_2$) is obtained from (2.56). Integration over the wavenumber $k$ in turn yields

(2.88)\begin{equation} E_m \approx \frac{{\rm \pi} ^{3/2} g^2\alpha^2\delta^5}{256 V_M^2} \left(\frac{t}{t_2}\right)^{{-}2}.\end{equation}

For $t \gg t_2$, the magnetic energy of the disturbances decays as $(t/t_2)^{-2}$.

2.3.2. Wave transition time scales

As the waves generated in the regime $\omega _M \gg \omega _A$ undergo progressive diffusion, $\omega _M$ decreases with time, and eventually, Alfvénic waves would be transformed into waves whose frequency is buoyancy dominant. The kinetic energy undergoes this transition at time $t \lesssim t_2$ (§ 2.3.1). The time scale for the magnetic energy transition is obtained from (2.88) and (2.67) as follows:

(2.89)\begin{equation} 0.02175 \frac{g^2\alpha^2\delta^5}{V_M^2} \left(\frac{t}{t_2}\right)^{{-}2} \approx 3.637 \frac{\delta V_M^2}{\beta^2} \left(\frac{t}{t_1}\right)^{{-}5/4}, \end{equation}

which gives

(2.90a,b)\begin{equation} t\approx 0.43 M^{4/3} t_2, \quad M=\frac{V_M}{\delta \sqrt{g \alpha \beta}}, \end{equation}

where $M$ measures the initial ratio of the Alfvén wave frequency to the internal gravity wave frequency. The magnetic energy decays as $t^{-2}$ in the regime of strong field until $t \sim M^{4/3} t_2$, and subsequently crosses over to the regime of strong stratification wherein it decays as $t^{-5/4}$.

2.4. Computations of the long-time evolution of an isolated disturbance

The general solution of the initial value problem, given by (2.28), is computed. The values of the transforms $\hat {\psi }$ and $\hat {\xi }$ in $(k_s,k_z)$ space and their respective energies are calculated at several time points. The kinetic and magnetic energies are calculated from the exact velocity and magnetic field transforms in (2.28) by Parseval's theorem (see e.g. (2.50) and (2.61)), where the upper limits of the integrals are the truncation values of $k_s$ and $k_z$. (The truncation value of the wavenumbers is set to $3/\delta$.) The computed energies are then compared with their asymptotic values derived from the leading-order solutions of the transforms in the limit of zero viscous and thermal diffusivities ($\nu =\kappa =0$).

In the calculations, both $\nu$ and $\kappa$ are taken to be small relative to magnetic diffusivity $\eta$. The magnetic Prandtl number, $Pm=\nu /\eta = 10^{-8}$ and the Roberts number $q=\kappa /\eta =10^{-7}$. The strong-field regime is studied using the parameters $S=V_M \delta /\eta =10^4$ and $M=V_M/\delta \sqrt {g \alpha \beta } =316$. With small but finite magnetic diffusion, the frequencies in general change with time because the wavenumbers on which they depend are time varying. To compute the frequencies, the mean values of the wavenumbers are first calculated through ratios of $L^2$ norms; e.g.

(2.91a,b)\begin{equation} \bar{k}_z = \frac{\|k_z \hat{\psi} k\|}{\| \hat{\psi} k\|}, \quad \bar{k} = \frac{\| \hat{\psi} k^2\|}{\| \hat{\psi} k \|},\end{equation}

which are based on the velocity field. As we anticipate that the frequencies based on the induced magnetic field would evolve differently, the mean wavenumbers based on $\hat {\xi }$ are also evaluated following the definitions in (2.91a,b).

In figure 2(a,b), the imaginary parts of the frequencies $\lambda _1$ and $\lambda _3$ in (2.24a,b) are compared with the respective frequency approximations. The parameters are those for the case of strong field, $S=10^4$ and $M=316$. The frequencies are calculated using the mean wavenumbers based on the induced magnetic field, so that a transition to the regime of dominant stratification may be observed, as predicted by theory. Here, $\textrm {Im}(\lambda _1)$ agrees well with $\omega _\eta /2 (1 - \omega _A^2/\omega _M^2 )$ (see (2.70a)) until time $t \sim 10^3 t_2$, and with $\omega _M^2 \omega _\eta /\omega _A^2$ (see (2.39a)) for $t > 10^3 t_2$. As the time scale for the strong-field regime to cross over to the regime of strong stratification is $t \approx 928 t_2$ (from (2.90a,b), shown by the vertical dashed-dotted lines in figure 2), it is apparent that the frequencies obey this transition. Figure 3(a) compares the computed kinetic and magnetic energies with their asymptotic solutions for the case of strong magnetic field. The energies are normalized by the reference magnetic energy, given by (2.88). In the early phase that lasts until $t \sim t_2$, the energies evolve in approximate equipartition, in line with (2.81). The $\sim t^{-1/2}$ decay of $E_k$ and $E_m$ with stratification is much slower than that in the absence of stratification, where the energies would have decayed as $\sim (t/t_\eta )^{-5/2}$ (Moffatt Reference Moffatt1967). The contours of the streamfunction in this early phase (figure 4), obtained from the inverse Hankel–Fourier transform, reveal the predominantly Alfvénic character of the wave since the front propagates as $z/\delta \approx t/t_M$. (Here, $t_M$ is the Alfvén wave travel time.) The evolution of energies is consistent with the time variation of the fundamental frequencies $\omega _M$, $\omega _A$ and $\omega _\eta$ calculated using the mean wavenumbers. For $t > t_2$, $E_k$ continues on its $\sim t^{-1/2}$ decay as the frequencies calculated based on the velocity cross-over to the regime of strong stratification, where $\omega _A > \omega _M$ (figure 3c). The triangles in figure 3(a) represent the asymptotic kinetic energy for strong stratification, given by (2.58). For a larger Lundquist number $S=10^5$, the transition to the regime of dominant $\omega _A$ occurs at $t < t_2$ (figure 3d), so it is evident that the evolution of the velocity as damped Alfvén waves is confined to this early period. The induced magnetic field, on the other hand, evolves as Alfvén waves until $t \approx 0.43 M^{4/3} t_2$ (figure 3b), which explains why $E_m$ follows the $t^{-2}$ decay law given by (2.88) for the period $t_2 < t < 0.43 M^{4/3} t_2$. For $t \gg M^{4/3} t_2$, $\omega _A \gg \omega _M$, which explains why $E_m$ follows the $t^{-5/4}$ law given by (2.67).

Figure 2.

A comparison of the computed imaginary parts of the frequencies $\lambda _1$ and $\lambda _3$ (solid lines) with their approximations (symbols) for the case of strong magnetic field. The frequencies are calculated using the mean wavenumbers $\bar {k}_z$, $\bar {k}_s$ and $\bar {k}$ based on the induced magnetic field. The horizontal axis is normalized with respect to the time scale $t_2$, defined by (2.85). The vertical dash-dotted lines represent the theoretical time scale (2.90a,b) for cross-over of the magnetic energy from the regime of strong magnetic field to that of strong stratification.

Figure 3.

(a) Computed kinetic and magnetic energies (solid and dashed lines respectively) compared with asymptotic solutions (symbols) for the case of strong magnetic field. The triangles are obtained from (2.58), and the circles are obtained from (2.88) and (2.67). (b) Fundamental frequencies calculated using the mean wavenumbers based on the induced magnetic field. (c) and (d) Fundamental frequencies based on the velocity field. The vertical dotted lines in (a) and (b) represent the theoretical time scale (2.90a,b) for cross-over of the magnetic energy from the regime of strong magnetic field to that of strong stratification. The parameters used are $S=10^4$, $M=316$ for (a)–(c) and $S=10^5$, $M=316$ for (d).

Figure 4.

Contour plots of the streamfunction $\psi$ for (a) 10 and (b) 100 Alfvén wave times in a strong-field calculation using the parameters $S=10^4$ and $M=316$.

In summary, stable stratification produces an important effect that is absent in the classical evolution of disturbances in a strong magnetic field – the dominance of diffusion over wave motion is considerably delayed as the regime $\omega _M \gg \omega _A \gg \omega _\eta$ crosses over to the regime $\omega _A \gg \omega _M \gg \omega _\eta$, which persists for long times. As in freely decaying MHD turbulence at low $Rm$ (Moffatt Reference Moffatt1967), the induced magnetic field propagates as damped Alfvén waves for a much longer time than the velocity. The strong-field case is a useful first step in the analysis of the problem with added background rotation, presented in the next section.

3.1. Evaluation of spectral coefficients

From (3.11), the initial conditions for $\hat {\psi }$ and its time derivatives are given by

(3.12)\begin{equation} {\rm i}^n\sum_{m=1}^{5}D_m\lambda_m^n=\left(\frac{\partial^n{\hat{\psi}}}{\partial{t^n}}\right)_0 =d_{n+1}, \quad n=0,1,2,3,4.\end{equation}

As the initial conditions of the velocity and induced magnetic field are the same as in the case without rotation, algebraic simplifications give the right-hand sides of (3.12) in the limit of $\nu =\kappa =0$, as follows:

(3.13)\begin{gather} d_1 = d_3= 0, \end{gather}

(3.14)\begin{gather}d_2 = \frac{g \alpha k_s}{k^2}\hat{\theta}_0, \end{gather}

(3.15)\begin{gather}d_4 = \frac{g \alpha k_s}{k^2}\left({-}V_M^2k_z^2-\frac{g\alpha\beta k_s^2}{k^2}- \frac{4\varOmega^2 k_z^2}{k^2}\right)\hat{\theta}_0, \end{gather}

(3.16)\begin{gather}d_5 = g \alpha \eta V_M ^2k_z^2k_s\hat{\theta}_0, \end{gather}

where $\hat {\theta }_0$ is the transform of the initial temperature perturbation. The quantity within brackets in (3.15) gives the sum of the squares of the MAC frequencies, $-(\omega _M^2+\omega _A^2+\omega _C^2)$.

The coefficients of the fast and slow wave components of $\hat {\psi }$ in (3.11) may now be obtained using the roots of the characteristic (3.10). For example,

(3.17)\begin{gather} D_1 = \frac{d_5-{\rm i}d_4(\lambda_2+\lambda_3+\lambda_4+\lambda_5) +{\rm 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)}, \end{gather}

(3.18)\begin{gather}D_3 = \frac{d_5-{\rm i}d_4(\lambda_1+\lambda_2+\lambda_4+\lambda_5) +{\rm 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)}. \end{gather}

From (3.11), the initial conditions for $\hat {u}_\phi$ and its time derivatives are given by

(3.19)\begin{equation} {\rm i}^n \sum_{m=1}^{5}G_m\lambda_m^n=\left(\frac{\partial^n {\hat{u}_\phi}}{\partial{t^n}}\right)_0 =g_{n+1}, \quad n=0,1,2,3,4.\end{equation}

Algebraic simplifications give the right-hand sides of (3.19) in the limit of $\nu =\kappa =0$, as follows:

(3.20)\begin{gather} g_1 = g_2 = g_4=0, \end{gather}

(3.21)\begin{gather}g_3 = 2\,\textrm{i}\varOmega k_z \left(\frac{g \alpha k_s}{k^2}\right) \hat{\theta}_0, \end{gather}

(3.22)\begin{gather}g_5 ={-}2\,\textrm{i}\varOmega k_z\left(\frac{g \alpha k_s}{k^2}\right) (2\omega_M^2+\omega_C^2+\omega_A^2)\hat{\theta}_0. \end{gather}

The coefficients of the fast and slow wave components of $\hat {u}_\phi$ in (3.11) may now be obtained using the roots of (3.10). For example,

(3.23)\begin{gather} G_1 = \frac{g_5-g_3(\lambda_4\lambda_5+\lambda_3\lambda_4+ \lambda_3\lambda_5+\lambda_2\lambda_3+\lambda_2\lambda_4+ \lambda_2\lambda_5)}{(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)(\lambda_1- \lambda_4)(\lambda_1-\lambda_5)}, \end{gather}

(3.24)\begin{gather}G_3 = \frac{g_5-g_3(\lambda_4\lambda_5+\lambda_2\lambda_4+ \lambda_2\lambda_5+\lambda_1\lambda_2+\lambda_1\lambda_4+ \lambda_1\lambda_5)}{(\lambda_3-\lambda_1)(\lambda_3- \lambda_2)(\lambda_3-\lambda_4)(\lambda_3-\lambda_5)}. \end{gather}

From (3.11), the initial conditions for $\hat {\xi }$ and its time derivatives are given by

(3.25)\begin{equation} {\rm i}^n\sum_{m=1}^{5}P_m\lambda_m^n= \left(\frac{\partial^n{\hat{\xi}}}{\partial{t^n}}\right)_0=p_{n+1}, \quad n=0,1,2,3,4. \end{equation}

The right-hand sides of (3.25) are obtained in the limit of $\nu =\kappa =0$, as follows:

(3.26)\begin{gather} p_1 = p_2=0, \end{gather}

(3.27)\begin{gather}p_3 = \textrm{i} B g \alpha \frac{k_sk_z}{k^2} \hat{\theta}_0, \end{gather}

(3.28)\begin{gather}p_4 ={-}\textrm{i} B g \alpha \eta k_s k_z \hat{\theta}_0, \end{gather}

(3.29)\begin{gather}p_5 = \textrm{i} B k_z d_4 - \eta k^2 p_4, \end{gather}

where $d_4$ is given by (3.15). The coefficients of the fast and slow wave components of $\hat {\xi }$ in (3.11) may now be obtained using the roots of (3.10). For example,

(3.30)\begin{gather} P_1 =\frac{p_5-{\rm i}p_4(\lambda_2+\lambda_3+\lambda_4+\lambda_5) -p_3(\lambda_4\lambda_5+\lambda_3\lambda_4+\lambda_3\lambda_5 +\lambda_2\lambda_3+\lambda_2\lambda_4+\lambda_2\lambda_5)}{(\lambda_1 -\lambda_2)(\lambda_1-\lambda_3)(\lambda_1-\lambda_4)(\lambda_1-\lambda_5)}, \end{gather}

(3.31)\begin{gather}P_3 = \frac{p_5-{\rm i}p_4(\lambda_1+\lambda_2+\lambda_4+\lambda_5)- p_3(\lambda_4\lambda_5+\lambda_2\lambda_4+\lambda_2\lambda_5+ \lambda_1\lambda_2+\lambda_1\lambda_4+\lambda_1\lambda_5)}{(\lambda_3- \lambda_1)(\lambda_3-\lambda_2)(\lambda_3-\lambda_4)(\lambda_3-\lambda_5)}. \end{gather}

From (3.11), the initial conditions for $\hat {b}_\phi$ and its time derivatives are given by

(3.32)\begin{equation} {\rm i}^n\sum_{m=1}^{5}Q_m\lambda_m^{n}=\left(\frac{\partial^n\hat{b}_\phi}{\partial t^n}\right)_0=q_{n+1}, \quad n=0,1,2,3,4. \end{equation}

The right-hand sides of (3.32) are obtained in the limit of $\nu =\kappa =0$, as follows:

(3.33)\begin{gather} q_1 = q_2 = q_3 =0, \end{gather}

(3.34)\begin{gather}q_4 ={-}2 B \varOmega g \alpha \frac{k_s k_z^2}{k^2} \hat{\theta}_0, \end{gather}

(3.35)\begin{gather}q_5 ={-}\eta k^2 q_4. \end{gather}

The coefficients of the fast and slow wave components of $\hat {b}_\phi$ in (3.11) may now be obtained using the roots of (3.10). For example,

(3.36)\begin{gather} Q_1 = \frac{q_5-{\rm i}q_4(\lambda_2+\lambda_3+\lambda_4+ \lambda_5)}{(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)(\lambda_1- \lambda_4)(\lambda_1-\lambda_5)}, \end{gather}

(3.37)\begin{gather}Q_3 = \frac{q_5-{\rm i}q_4(\lambda_1+\lambda_2+\lambda_4+ \lambda_5)}{(\lambda_3-\lambda_1)(\lambda_3-\lambda_2)(\lambda_3- \lambda_4)(\lambda_3-\lambda_5)}. \end{gather}

3.2. Long-time evolution of a disturbance in rapid rotation

We consider a rapidly rotating system where the fundamental frequencies satisfy the inequality $|\omega _C| \gg |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$. To obtain the roots of the characteristic equation (3.10) subject to the above inequality, we first consider the inertia-free limit ($\partial \boldsymbol {u}/\partial t = 0$) of the equation of motion (3.1), which, together with (2.3) and (2.4), yields the following equation for plane waves in the limit $\nu =\kappa =0$:

(3.38)\begin{equation} \lambda^3-2\,\textrm{i}\omega_\eta\lambda^2-\left(\omega_\eta^2+ \frac{\omega_A^2\omega_M^2}{\omega_C^2}+\frac{\omega_M^4}{\omega_C^2}\right) \lambda+\textrm{i}\left(\frac{\omega_A^2\omega_\eta\omega_M^2}{\omega_C^2}\right)=0. \end{equation}

Letting $x=\omega _A/\omega _C$, $y=\omega _\eta /\omega _C$ and $z=\omega _M/\omega _C$, we see that (3.38) is of the form

(3.39)\begin{equation} \lambda^3+a_2\lambda^2+a_1\lambda+a_0=0,\end{equation}

where

(3.40a–c)\begin{equation} a_2={-}2iy\omega_C,\quad a_1={-}(y^2+x^2z^2+z^4)\omega_C^2, \quad a_0=\textrm{i} x^2yz^2 \omega_C^3. \end{equation}

The solution of the cubic equation (3.39) subject to the inequality $z \gg x \gg y$ broadly follows the procedure outlined in Appendix B, and gives the slow wave frequencies

(3.41)\begin{equation} \lambda_{3,4}\approx{\pm} \frac{1}{2} (x^2+2z^2) \omega_C +\textrm{i} \left(y- \frac{x^2y}{2z^2}\right)\omega_C,\end{equation}

and the decay frequency

(3.42)\begin{equation} \lambda_5 \approx \textrm{i} \frac{x^2y}{z^2}\omega_C.\end{equation}

Now, polynomial long division using the roots in (3.41) of the characteristic equation (3.10), written in the form

(3.43)\begin{align} &\lambda^5 - 2 \textrm{i}\lambda^4y \omega_C - \lambda^3 (1+x^2 +y^2 +2 z^2) \omega_C^2 + 2 \textrm{i} y \lambda^2 (1 + x^2+ z^2) \omega_C^3\nonumber\\ &\quad + \lambda (y^2 + x^2 y^2 +x^2 z^2 +z^4) \omega_C^4- {\rm i}x^2 y z^2 \omega_C^5 =0, \end{align}

gives the cubic equation

(3.44)\begin{equation} \lambda^3 -\lambda^2 \textrm{i} \frac{x^2 y}{z^2} \omega_C+ \lambda ({-}1 - x^2 - 2 z^2)\omega_C^2 + \textrm{i} \frac{ x^2 y}{z^2} \omega_C^3- 2 \textrm{i} y ( z^2 - x^2 ) \omega_C^3 =0, \end{equation}

with the remainder approximately zero. The roots of (3.44) in approximate form are once again obtained through the cubic formula, as follows:

(3.45)\begin{gather} \lambda_{1,2} \approx{\pm} (1+z^2)\omega_C+\textrm{i} yz^2\omega_C, \end{gather}

(3.46)\begin{gather}\lambda_5 \approx \textrm{i} \frac{x^2y}{z^2}\omega_C. \end{gather}

Substituting for $x$, $y$ and $z$ gives

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

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

(3.49)\begin{gather}\lambda_{5} \approx \textrm{i} \frac{\omega_A^2\omega_\eta}{\omega_M^2}. \end{gather}

From (3.47)–(3.49), we note that the fast waves undergo damping on the time scale

(3.50)\begin{equation} t_3 = \left(\frac{\omega_C^2}{\omega_M^2 \omega_\eta} \right)_{0} =\frac{4 \varOmega^2}{\eta V_M^2 k_0^4}, \end{equation}

while the slow waves are damped on the time scale

(3.51)\begin{equation} t_\eta = (\omega_\eta)^{{-}1}_{0} = (\eta k_0^2)^{{-}1},\end{equation}

as for a freely decaying system (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017). The long-time decay the perturbations in a stratified fluid occurs on the time scale

(3.52)\begin{equation} t_2 = \left( \frac{\omega_M^2}{\omega_A^2\omega_\eta} \right)_{ 0} = \frac{V_M^2}{g \alpha \beta \eta},\end{equation}

which was obtained earlier for the strong-field case without rotation (see (2.85)).

A magnetically damped rotating system that initially lies in the regime $|\omega _C| \gg |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$ would eventually evolve into the regime $|\omega _A| \gg |\omega _\eta | \gg |\omega _C| \gg |\omega _M|$ because frequencies $\omega _M$ and $\omega _C$ of the fast wave progressively decrease with time (see figure 3 and also Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017). The approximate roots of the characteristic equation (3.10) in the evolved state are given by

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

(3.54)\begin{gather}\lambda_3 = \lambda_4 \approx \textrm{i}\omega_\eta, \end{gather}

(3.55)\begin{gather}\lambda_5 \approx \textrm{i}\frac{\omega_M^2}{\omega_\eta}. \end{gather}

Figure 5(a) shows the computed long-time evolution of the four fundamental frequencies based on the velocity field for a Lehnert number $Le = V_M/2 \varOmega \delta = 5.8 \times 10^{-3}$, which represents rapid rotation. (The above definition of $Le$ follows from $(\omega _M/\omega _C)_{ 0}$, introduced in § 1.) The dimensionless parameters $S$ and $M$ are set to $10^4$ and $316$ respectively, as for the non-rotating system considered in § 2.4. The progressive decrease of $\omega _M$ and $\omega _C$ due to the small but finite magnetic diffusion is evident. In figure 5(b)–(d), the computed real and imaginary parts of the frequency roots $\lambda _1$, $\lambda _3$ and $\lambda _5$ are compared with their respective approximations in the initial and final regimes, given by (3.47)–(3.49) and (3.53)–(3.55), respectively. The time is normalized by the time scale $t_3$, defined in (3.50). The approximations of the frequency roots in the initial and final phases of evolution closely match the actual (computed) frequencies. The long-time evolution should hence be characterized by damped internal gravity waves. In the final phase of decay, the general solution for the transform of $\psi$ is given by

(3.56)\begin{equation} \hat{\psi} = D_1 \,\mbox{e}^{\textrm{i}\lambda_1t}+ D_2 \,\mbox{e}^{\textrm{i}\lambda_2t}+ (D_3+D_4 t )\, \mbox{e}^{\textrm{i}\lambda_3t}+D_5 \,\mbox{e}^{\textrm{i}\lambda_5t}, \end{equation}

the coefficients $D_i$ of which are approximated by

(3.57)\begin{equation} \left. \begin{array}{c@{}} D_1 \approx d_2 \left(\dfrac{\omega_\eta \omega_M^2}{2\omega_A^4} -\dfrac{{\rm i}}{2\omega_A} \right), \quad D_2 \approx d_2 \left(\dfrac{\omega_\eta \omega_M^2}{2\omega_A^4} + \dfrac{{\rm i}}{2\omega_A} \right),\\ D_3 \approx d_2 \left(\dfrac{2 \omega_C^2}{\omega_A^2 \omega_\eta} + \dfrac{\omega_M^2}{\omega_A^2 \omega_\eta} - \dfrac{\omega_\eta \omega_M^2}{\omega_A^4} \right), \\ D_4 \approx d_2 \left(\dfrac{\omega_C^2}{\omega_A^2} + \dfrac{\omega_\eta^2 \omega_M^2}{\omega_A^4} \right), \\ D_5 \approx d_2 \left(-\dfrac{2 \omega_C^2}{\omega_A^2 \omega_\eta} - \dfrac{\omega_M^2}{\omega_A^2 \omega_\eta} \right). \end{array} \right\} \end{equation}

Noting that the real part of $D_1$ is much smaller than the imaginary part, we obtain

(3.58)\begin{gather} \hat{\psi} \approx \frac{{\rm i} d_2}{\omega_A}\sin(\omega_At) \exp{\left(-\frac{\omega_\eta\omega_M^2}{2\omega_A^2}t\right)}, \end{gather}

(3.59)\begin{gather}|\hat{u}|^2 \approx k^2\frac{d_2^2} {\omega_A^2}\sin^2(\omega_At) \exp{\left(-\frac{\omega_\eta\omega_M^2}{\omega_A^2}t\right)}. \end{gather}

Since the long-time decay of energy takes place on the time scale (see (2.49))

(3.60)\begin{equation} t_1 = \left(\frac{\omega_A^2}{\omega_\eta\omega_M^2}\right)_{ 0} =\frac{g \alpha \beta}{V_M^2 \eta k_0^4}, \end{equation}

for $t \gg t_1$,

(3.61)\begin{equation} E_k \approx 16{\rm \pi} ^4\int_{0}^{\infty}\int_{0}^{\infty}k^2 \frac{d_2^2}{\omega_A^2} \exp{\left(-\frac{\omega_\eta\omega_M^2}{\omega_A^2}t\right)}k_s\,\textrm{d}k_s\,\textrm{d}k_z, \end{equation}

which gives

(3.62)\begin{equation} E_k \approx\frac{3 {\rm \pi}^{2} g\alpha \delta^3}{32 \beta} \left(\frac{t}{t_1}\right)^{{-}1/2}, \end{equation}

in line with the analysis in § 2.2.1. Likewise, we expect the long-time evolution of magnetic energy to be similar to that in the absence of rotation (see (2.67) and figure 3a). Thus, the long-time evolution of disturbances in a rapidly rotating stratified medium would be that of damped internal gravity waves.

Figure 5.

(a) Long-time evolution of the magnitudes of the four fundamental frequencies in a rapidly rotating system with the dimensionless parameters $Le= 5.8 \times 10^{-3}$, $S= 10^4$ and $M=316$. (b), (c) and (d) Comparison of the numerical frequency roots $\lambda _1$, $\lambda _3$ and $\lambda _5$ of the characteristic equation (3.10) (solid and dashed lines) with their analytical approximations (symbols) for the initial and final regimes of evolution given by $|\omega _C| \gg |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$ and $|\omega _A| \gg |\omega _\eta | \gg |\omega _C| \gg |\omega _M|$ respectively. Here, $\mbox {Re}(\lambda _1)$ and $\mbox {Im}(\lambda _1)$ evolve in time from their approximations in (3.47) to those in (3.53); $\mbox {Re}(\lambda _3)$ and $\mbox {Im}(\lambda _3)$ evolve from their approximations in (3.48) to those in (3.54). The imaginary root $\lambda _5$ evolves from its approximation in (3.49) to that in (3.55).

3.3. The role of slow magnetostrophic waves in unstable stratification

The regime $|\omega _C|>|\omega _M|\gg |\omega _A| \gg |\omega _\eta |$, where $\omega _A^2 <0$, is of relevance to a large region of the parameter space where convection-driven dipole-dominated dynamos exist. Notably, here the local value of $|\omega _M / \omega _C|$ is not far less than unity. To obtain the approximate roots of the characteristic (3.10) in this regime, we consider the limit of zero magnetic diffusion ($\omega _\eta =0$), in which (3.10) reduces to

(3.63)\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 (3.63) are given by

(3.64)\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}

(3.65)\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}

Letting $\omega _M/\omega _C=z$ and $\omega _A/\omega _C=x$, we obtain,

(3.66)\begin{gather} \lambda_1=\frac{\omega_C}{\sqrt{2}}\sqrt{1+x^2+2z^2+\sqrt{1+2x^2+x^4+4z^2}}, \end{gather}

(3.67)\begin{gather}\lambda_3=\frac{\omega_C}{\sqrt{2}}\sqrt{1+x^2+2z^2-\sqrt{1+2x^2+x^4+4z^2}}, \end{gather}

for the forward-travelling fast and slow waves respectively. In the regime $|\omega _C|>|\omega _M|\gg |\omega _A|$ considered in this section, the frequencies in (3.66) and (3.67) are approximated by first expanding the inner square root,

(3.68)\begin{equation} f = \sqrt{1+2x^2+x^4+4z^2}, \end{equation}

as series expansions until second order in $x$ and sixth order in $z$, sequentially. Algebraic simplifications give

(3.69)\begin{equation} f\approx 1+x^2 - z^2(2x^2-2) -2 z^4 + 4 z^6,\end{equation}

by neglecting terms of $O(x^2 z^4)$ and $O(x^2 z^6)$. Therefore,

(3.70)\begin{gather} \lambda_1 \approx \omega_C\sqrt{1+x^2+2z^2-x^2z^2-z^4+2z^6}, \end{gather}

(3.71)\begin{gather}\lambda_3 \approx \omega_C\sqrt{x^2z^2+z^4-2z^6}. \end{gather}

Further approximation of (3.71) using series expansions until second order in $x$ and fourth order in $z$ gives

(3.72)\begin{equation} \lambda_3\approx \omega_C\left(z^2+\frac{x^2}{2}-z^4+\frac{x^2z^2}{2}+\frac{3x^2z^4}{4}\right). \end{equation}

Neglecting the terms of $O(x^2z^2)$ and $O(x^2z^4)$ and substituting for $z$ and $x$, we obtain

(3.73)\begin{equation} \lambda_3\approx \frac{\omega_M^2}{\omega_C}+ \frac{\omega_A^2}{2\omega_C}-\frac{\omega_M^4}{\omega_C^3}. \end{equation}

Following a similar approach, (3.70) is approximated by

(3.74)\begin{equation} \lambda_1 \approx \omega_C+\frac{\omega_M^2}{\omega_C}+ \frac{\omega_A^2}{2\omega_C}-\frac{\omega_M^4}{\omega_C^3}. \end{equation}

For small but finite magnetic diffusion, the imaginary parts of the frequency roots $\lambda _1$ and $\lambda _3$ are taken from (3.47) and (3.48), respectively. The root $\lambda _5$ remains the same as in (3.49).

Figure 6 compares the approximations for $\lambda _1$ and $\lambda _3$ with their numerical values for $k_s=k_z$ for the fixed parameters $E_\eta = 10^{-5}$ and $F_\eta =0.1$, defined by

(3.75a,b)\begin{equation} E_\eta = \frac{\eta}{2 \varOmega \delta^2}, \quad F_\eta=\frac{\eta}{|\sqrt{g \alpha \beta} | \delta^2},\end{equation}

measuring the initial ratios of the diffusion frequency to the rotation frequency and buoyancy frequency, respectively. A good agreement between the computed and approximate roots is obtained until $\omega _M/\omega _C \approx 0.3$.

Figure 6.

Comparison of the computed real and imaginary parts of the fast and slow MAC wave frequencies (solid and dashed lines) with their approximations (symbols) for wavenumbers $k_s=k_z=k_0/\sqrt {2}$. The fast and slow wave frequencies are given by $\lambda _1$ and $\lambda _3$, respectively. The fixed parameters are $E_\eta =10^{-5}$ and $F_\eta =0.1$, defined in (3.75a,b).

Since the general solutions for the velocity and magnetic field transforms (3.11) consist of linear superpositions of the fast and slow MAC wave parts, the energy carried by each part can be evaluated separately. (See Sreenivasan & Narasimhan (Reference Sreenivasan and Narasimhan2017) for a similar separation of fast and slow MC wave solutions.) For the frequency inequality considered here, the exponential increase of the perturbations occurs on a much larger time scale than magnetic diffusion. For times $t \lesssim t_\eta$, the poloidal and toroidal kinetic energies,

(3.76a,b)\begin{equation} E_{k,p} = 16 {\rm \pi}^4 \int_{k_s} \int_{k_z} (k \hat{\psi})^2 k_s \,\mbox{d}k_s \,\mbox{d}k_z, \quad E_{k,t} = 16 {\rm \pi}^4 \int_{k_s} \int_{k_z} \hat{u}_\phi^2 k_s\, \mbox{d}k_s\, \mbox{d}k_z, \end{equation}

may be obtained from the fast and slow wave parts of $\hat {\psi }$ and $\hat {u}_\phi$, respectively. Similar integrals for the magnetic energy are calculated based on $\hat {\xi }$ and $\hat {b}_\phi$.

The total kinetic and magnetic energies carried by the fast and slow MAC wave parts of the solution are examined separately in figures 7–9. The ratio $F_\eta$ is set to 0.1. Because the times considered are much smaller than the time scale $t_2$ (3.52), the values of the energy for unstable stratification are nearly the same as for stable stratification. For ${\boldsymbol B}$ aligned with ${\boldsymbol \varOmega }$ (as in this study), the instantaneous value of $\omega _M/\omega _C$ is nearly the same as its initial value, of $O(Le)$. For a spherical shell dynamo, we anticipate that $\omega _M/\omega _C$ would be higher than $Le$ as the azimuthal wavenumber is much greater than the axial wavenumber in fully developed columnar convection. In figures 7–9, we present the fast and slow wave energies at different values of $\omega _M/\omega _C$. For $E_\eta =10^{-5}$ (figure 7), the fast wave kinetic energy is dominant with weak magnetic fields while the slow wave energy matches the fast wave energy with stronger fields ($\omega _M/\omega _C \approx 0.25$). The equivalence of the slow and fast wave kinetic energy persists at higher $\omega _M/\omega _C$. The magnetic energy carried by the slow MAC wave is consistently greater than that of the fast wave (figure 7c). Because the $z$ vorticity makes the dominant contribution to the enstrophy, we evaluate

(3.77)\begin{equation} W_{z} = 16 {\rm \pi}^4 \int_{k_s} \int_{k_z} (k_s \hat{u}_\phi)^2 k_s\, \mbox{d}k_s\, \mbox{d}k_z, \end{equation}

for the slow and fast waves. The $z$ enstrophy of the slow waves matches that of the fast waves for $\omega _M/\omega _C = 0.24$ (figure 7d). For the case $E_\eta =4 \times 10^{-5}$, which represents perturbations of smaller length scale $\delta$, the parity between the slow and fast wave kinetic energy for $\omega _M/\omega _C =0.24$ approximately holds (figure 8b), although the slow wave energy would fall much below that of the fast wave for higher values of $E_\eta$ ($\sim 10^{-3}$) representing still smaller scales. The progressive increase of the $z$ enstrophy of the slow wave with $\omega _M/\omega _C$ (figure 8c,d) is a significant result. The generation of $z$ vorticity by the Lorentz force – an essential process in the generation of helicity $\boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {\zeta }$ in rapidly rotating dynamos (Sreenivasan & Jones Reference Sreenivasan and Jones2011) – may be interpreted through the enhanced vorticity of the slow MAC waves. The Elsasser number $\varLambda = V_M^2/2 \varOmega \eta$, which follows from $(\omega _M^2 / (\omega _C \omega _\eta ) )_{ 0}$, takes values of $\approx 22$ and 250 for the two cases considered in figure 8. For parity between the intensities of the fast and slow wave motions, the leading-order slow wave frequency $\omega _M^2/\omega _C$ can be $O(10^2)$ times the diffusion frequency $\omega _\eta$ on the scale of the buoyancy perturbations.

Figure 7.

Computed evolution of the total kinetic energy ($E_k$) magnetic energy ($E_m$) and the $z$ enstrophy ($W_z$) carried by the fast and slow MAC waves. The ratio of Alfvén to inertial wave frequencies ($\omega _M/\omega _C$) is given above each panel. The fixed parameters in the calculations are $E_\eta =10^{-5}$ and $F_\eta =0.1$, defined in (3.75a,b). The energy is normalized by the steady state value of the fast wave. The symbols represent the energy calculated from the frequency approximations in figure 6.

Figure 8.

Computed evolution of the kinetic energy ($E_k$) and $z$ enstrophy ($W_z$) carried by the fast and slow MAC waves for two values of $\omega _M/\omega _C$. The fixed parameters in the calculations are $E_\eta = 4\times 10^{-5}$ and $F_\eta =0.1$. The energy is normalized by the steady state value of the fast wave.

Figure 9 indicates that the progressive increase of $\omega _M/\omega _C$ brings the fast wave magnetic energy closer to that of the slow wave, which is consistent with the result that the energies of both waves would merge and tend to the Alfvén wave energy at much stronger fields given by the regime $\omega _M \gg \omega _C$ (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017).

Figure 9.

Evolution of the magnetic energy ($E_m$) carried by the fast and slow MAC waves for progressively increasing $\omega _M/\omega _C$. The fixed parameters in the calculations are $E_\eta =4 \times 10^{-5}$ and $F_\eta =0.1$. The energy is normalized by the steady state value of the fast wave.

The analysis of isolated disturbances in a forced damped system presents an interesting comparison with that in an unforced freely decaying system; here, the kinetic energy of the slow MC waves is much smaller than that of the fast MC waves while the respective magnetic energies are approximately equal (Sreenivasan & Narasimhan Reference Sreenivasan and Narasimhan2017). With added buoyancy, the kinetic energies of the slow and fast MAC waves are equal while the magnetic energy of the slow waves surpasses that of the fast waves. Therefore, the generation of helicity in rapidly rotating dynamos can occur through slow MAC wave motions at horizontal length scales much smaller than the width of the fluid layer. Furthermore, the generation of the induced magnetic field may occur predominantly through the slow waves.