2.1. Base flow

The solutions of system (2.1) are conveniently decomposed into a ‘basic flow’ $(\boldsymbol {U},P, \boldsymbol {B}, \varTheta )$ and a ‘disturbance’ $(\boldsymbol {u},p,\boldsymbol {b}, \vartheta ),$ but the latter needs to be small compared with the former,

(2.5a–d)\begin{equation} \tilde{\boldsymbol{u}}=\boldsymbol{U}+\boldsymbol{u},\quad \tilde{p}=P+p,\quad \tilde{\boldsymbol{b}}=\boldsymbol{B}+\boldsymbol{b},\quad {\tilde{\vartheta}}=\varTheta+\vartheta. \end{equation}

We consider the linear stability of a stratified vortical flow with elliptical streamlines and with uniform vertical magnetic field (see figure 1)

(2.6a)$$\begin{gather} \boldsymbol{U}={\boldsymbol{A}}\boldsymbol{\cdot} \boldsymbol{x},\quad {\boldsymbol{A}}=\varOmega \left (\begin{array}{@{}ccc@{}} 0 & -E & 0\\ E^{-1} & 0 & 0\\ 0 & 0 & 0 \end{array}\right) \end{gather}$$

(2.6b)$$\begin{gather}\boldsymbol{B}=B\boldsymbol{e}_3, \end{gather}$$

(2.6c)$$\begin{gather}\varTheta=N^2x_3 \end{gather}$$

where $\varOmega$ is a constant that is a measure of the intensity of the flow and $E\geq 1$ is a measure of elliptical deformation of the streamlines, and $N$ is the Brunt–Väisälä frequency such that

(2.7)\begin{equation} N^2=-\frac{g}{\rho_0}\frac{{\rm d}\varrho}{{\rm d}\kern 0.06em x_3}. \end{equation}

Circular streamlines correspond to the case where $E=1.$ The parameter

(2.8)\begin{equation} \varepsilon=\tfrac{1}{2}\left (E-E^{-1}\right) \end{equation}

represents the departure of the streamlines of the unperturbed flow from axial symmetry. We note that

(2.9)\begin{equation} 0<\delta=(E-E^{-1})/(E+E^{-1})<1 \end{equation}

for the flow to be elliptical. We also note that, an equivalent form for the unperturbed velocity field has been used in previous studies (Waleffe Reference Waleffe1990; Miyazaki & Fukumoto Reference Miyazaki and Fukumoto1992; Miyazaki Reference Miyazaki1993; Kerswell Reference Kerswell2002; Mizerski & Bajer Reference Mizerski and Bajer2009; Bajer & Mizerski Reference Bajer and Mizerski2013)

(2.10)\begin{equation} \boldsymbol{U}={\varGamma}\left [-\left (1+\delta\right)x_2\boldsymbol{e}_1+\left (1-\delta\right)x_1\boldsymbol{e}_2\right], \end{equation}

where $2{\varGamma }={\varOmega } (E+E^{-1} )$ and $-{\varGamma }\delta$ represents the strain rate, but the expression (2.6a) seems more suitable for analysing resonant destabilisation (Mizerski & Bajer Reference Mizerski and Bajer2009).

Figure 1. A schematic drawing of the basic state: planar flow with elliptical streamlines, $\varPsi =-(\varOmega /2) (E^{-1}x_1^2+Ex_2^2)$ being the stream function in the presence of an axial uniform magnetic field ($\boldsymbol {B}=B\boldsymbol {e}_3)$ and an axial stable stratification ($\varTheta =N^2x_3,$ $N^2=-(g/\rho _0)({\rm d}\varrho /{\rm d}\kern 0.06em x_3)$, $N$ being the Brunt–Väisälä frequency). The gravity vector is given by $\boldsymbol {g}=-g\boldsymbol {e}_3$ with $g > 0$.

The basic buoyancy scalar $\varTheta$ varies linearly with the axial coordinate $x_3$ (axial stratification) and is proportional to the gravitational acceleration, $g,$ and to a background density (or temperature) gradient. This linear profile admits a constant Brunt–Väisälä frequency $N$ throughout the entire fluid. More general exact solutions of the combined stratified fluid/magnetic equations exist in an unbounded domain (Craik Reference Craik1989); the case in hand (i.e. (2.6)) is probably the simplest of these. As indicated previously, Miyazaki & Fukumoto (Reference Miyazaki and Fukumoto1992) considered an unbounded strained-vortex flow with stable exponential stratification in the axial direction at small Froude number, ${Fr}={\varGamma }^2L_0/g\ll 1$ with $L_0$ a characteristic length scale. One may show that the resulting linear differential system for the disturbances superimposed on the base flow is the same considering either an exponential basic stratification or a linear (with respect to space coordinates) basic stratification. Both profiles (exponential or linear) admit a constant Brunt–Väisälä frequency throughout the entire fluid.

2.2. Perturbed system

In the following, we consider the case of a non-diffusive fluid. Diffusivity effects with the assumption that the diffusion coefficients are equal $({\nu }=\kappa =\eta )$ or, equivalently, the magnetic and thermal Prandtl numbers are equal to one are briefly discussed at the end of § 4.

2.2.1. Linearised system in physical space

We substitute the solutions (2.5a–d) into the system (2.1) and linearise. Linearisation is not readily justified by the fact that the flow disturbances are very small with respect to the base flow. Linearisation is briefly discussed in the conclusion. Thus, we expect our analysis to break down when the disturbances become so large that nonlinear effects become important. The resulting perturbed equations are

(2.11a)$$\begin{gather} D_t{\boldsymbol{u}}=-\boldsymbol{\nabla} p-\left (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{U}+ \left ({\boldsymbol{B}}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{b}+ \vartheta\boldsymbol{e}_3, \end{gather}$$

(2.11b)$$\begin{gather}D_t{\boldsymbol{b}}= \left ({\boldsymbol{b}}\boldsymbol{\cdot}\boldsymbol{\nabla}\right){\boldsymbol{U}}+\left ({\boldsymbol{B}}\boldsymbol{\cdot}\boldsymbol{\nabla}\right){\boldsymbol{u}}, \end{gather}$$

(2.11c)$$\begin{gather}D_t{\vartheta}=-N^2u_3, \end{gather}$$

(2.11d)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol{u}}=0, \end{gather}$$

(2.11e)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol{b}}=0. \end{gather}$$

As for the linear part of MIPS (see (2.4)), it takes the form

(2.12)\begin{equation} \varpi_m=B\frac{\partial \vartheta}{\partial x_3}+ N^2b_3, \end{equation}

where $D_t\varpi _m=0,$ so that $\varpi _m=\text {const}.$ As shown later, for the purposes of studying stability, we may set $\varpi _m=0$ (see also Benkacem et al. Reference Benkacem, Salhi, Khlifi, Nasraoui and Cambon2022).

2.2.2. Floquet system in wave space

The disturbances are expressed in terms of Lagrangian Fourier modes, as discussed in section 1. These modes were used for shear waves by Moffatt (Reference Moffatt2010), who was probably the first to call them ‘Kelvin modes’, in reference to a pioneering paper by Lord Kelvin Kelvin (Reference Kelvin1887) in the nineteenth century. They are advected by the mean flow as Lagrangian invariants (Cambon Reference Cambon1982; Sagaut & Cambon Reference Sagaut and Cambon2008), as plane waves for which the direction and the speed of propagation depend on time, via a time-dependent wave vector:

(2.13)\begin{equation} \left [\boldsymbol{u},p,\boldsymbol{b},\vartheta\right](\boldsymbol{x},t)= \left [\hat{\boldsymbol{u}},\hat{p}, \hat{\boldsymbol{b}}, \hat{\vartheta}\right] (\boldsymbol{k},t)\exp\left ({\mathrm{i}}\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{k}(t)\right), \end{equation}

where $\mathrm {i}^2=-1.$ Accordingly, the material derivative of the fluctuating velocity can be rewritten as

(2.14)\begin{equation} D_t{{\boldsymbol{u}}}=\left (\partial_t+U_j\frac{\partial }{\partial x_j}\right)\left [\hat{\boldsymbol{u}}(\boldsymbol{k},t)\exp\left ({\mathrm{i}}\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{k}(t)\right)\right] \end{equation}

with $U_j=A_{jm}x_m,$ so that

(2.15)\begin{equation} D_t{{\boldsymbol{u}}}=\left [\partial_t{\hat{\boldsymbol{u}}}+\mathrm{i} \left ((d_t k_j)x_j+A_{jm}k_jx_m\right)\hat{\boldsymbol{u}}\right ] \exp\left ({\mathrm{i}}\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{k}(t)\right), \end{equation}

or, equivalently,

(2.16)\begin{equation} D_t{{\boldsymbol{u}}}=\left (\partial_t{\hat{\boldsymbol{u}}}+\mathrm{i} \left [\left (d_t \boldsymbol{k}+A^T\boldsymbol{k}\right)\boldsymbol{\cdot}\boldsymbol{x}\right]\hat{\boldsymbol{u}}\right ) \exp\left ({\mathrm{i}}\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{k}(t)\right). \end{equation}

In order to remove the explicit dependence on $\boldsymbol {x}$ in the resulting equations for the Fourier amplitudes $\hat {\boldsymbol {u}},$ $\hat {\boldsymbol {b}},$ $\hat {p}$ and $\hat {\vartheta },$ one has to ensure that $\boldsymbol {k}(t)$ varies in time according to the eikonal equation

(2.17)\begin{equation} {d_t{\boldsymbol{k}}}=-{\boldsymbol{A}}^T\boldsymbol{\cdot}\boldsymbol{k}, \end{equation}

where $d_t(\boldsymbol {\cdot })\equiv {{\rm d}(\boldsymbol {\cdot })}/{{\rm d}t}$ and $T$ denotes transpose. Equation (2.17) can be solved to give

(2.18a–c)\begin{equation} k_1=k_{p}\cos(\tau-\phi),\quad k_2=Ek_{p}\sin (\tau -\phi),\quad k_3=k_{30}, \end{equation}

where $\tau =\varOmega t$ being a dimensionless time,

(2.19a,b)\begin{equation} k_p^2=k_1^2+E^{-2}k_2^2=k_{10}^2+E^{-2}k ^2_{20},\quad \tan \phi=-E^{-1}k_{20}/k_{10}, \end{equation}

with $k_{j0}$ ($\,j=1,2,3)$ the initial wave vector component. For purposes of studying stability, we may set $\phi =0$. This is easily seen by making the substitution $\varOmega t'=\varOmega t+\phi,$ which eliminates $\phi$ from the equation.

Substituting the plane waves solution (2.13) into the system (2.11) and taking into account the eikonal equation (2.17), we obtain

(2.20a)$$\begin{gather} d_t{\hat{\boldsymbol{u}}}=-\mathrm{i} \hat{p}\boldsymbol{k}-{\boldsymbol{A}}\boldsymbol{\cdot} {\hat{\boldsymbol{u}}}+\mathrm{i} \left (k_3B\right)\hat{\boldsymbol{b}}+ \hat{\vartheta}\boldsymbol{e}_3, \end{gather}$$

(2.20b)$$\begin{gather}d_t{\hat{\boldsymbol{b}}}={\boldsymbol{A}}\boldsymbol{\cdot}\hat{\boldsymbol{b}}+\mathrm{i} \left (k_3B\right)\hat{\boldsymbol{u}}, \end{gather}$$

(2.20c)$$\begin{gather}d_t{\hat{\vartheta}}=-N^2\hat u_3, \end{gather}$$

together with $\boldsymbol {k}\boldsymbol {\cdot }\hat {{\boldsymbol {u}}}=0$ and $\boldsymbol {k}\boldsymbol {\cdot } \hat {{\boldsymbol {b}}}=0.$ The use of the latter conditions allows one to eliminate the Fourier amplitude of fluctuating pressure,

(2.21a,b)\begin{equation} \hat p(\boldsymbol{k},t)=-\mathrm{i} k^{-2}\left (2\varOmega^2\hat{c}_1+k_3\hat{\vartheta}\right),\quad \varOmega\hat{c}_1= Ek_1\hat u_2-E^{-1}k_2\hat u_1, \end{equation}

and to reduce the above seven-component Floquet system to a five-component version. Here, $k = \sqrt {k_1^2+k_2^2 + k_3^2}$ is the modulus of the wave vector. This reduction of components results from the fact that both Fourier modes for fluctuating velocity and for fluctuating magnetic field are two-component in the plane normal to the wave vector. Such projection can be done using the orthonormal Craya–Herring frame of reference, as used in several articles (e.g.from Salhi & Cambon Reference Salhi and Cambon1997).

We note that the case where the wave vector is vertical, so that $k_1 = k_2 = 0,$ $k_p = 0$ and $k_3 = \pm k,$ characterises a special class of disturbances, called horizontal perturbations, in which the vertical components $\hat {u}_3$ and $\hat {b}_3$ identically vanish (Bajer & Mizerski Reference Bajer and Mizerski2013). In that case, axial stratification has no effect on the horizontal perturbations, and then there is no instability without the Coriolis force.

2.2.3. Change of variables

At $k_3=0,$ the solution of the resulting Floquet system (2.20) is stable. Accordingly, we henceforth consider only perturbations with vertical wave number $k_3\ne 0.$ As in the studies by LZ04 and by Mizerski & Bajer (Reference Mizerski and Bajer2009), we transform the resulting Floquet system in terms of the following variables to facilitate subsequent calculations

(2.22a–d)\begin{align} \varOmega\hat{c}_2=-k_3\hat{u}_3,\quad \varOmega\hat{c}_3=Ek_1\hat{b}_2-E^{-1}k_2\hat{b}_1,\quad \varOmega \hat{c}_4=-k_3\hat{b}_3,\quad \varOmega^2\hat{c}_5=-k_3\hat{\vartheta}. \end{align}

Given (2.21a,b) we transform the system (2.20) according to these new variables,

(2.23a)$$\begin{gather} d_\tau{\hat{c}}_1=-4\varepsilon \frac{k_1k_2}{k^2}\hat{c}_1-2\hat{c}_2+\mathrm{i} \frac{(k_3B)}{\varOmega}\hat{c}_3+2\varepsilon \frac{k_1k_2}{k^2}\hat{c}_5, \end{gather}$$

(2.23b)$$\begin{gather}d_\tau{\hat{c}}_2=2\frac{k_3^2}{k^2}\hat{c}_1+\mathrm{i} \frac{(k_3B)}{\varOmega} \hat{c}_4+\frac{k_\perp^2}{k^2}\hat{c}_5, \end{gather}$$

(2.23c)$$\begin{gather}d_\tau{\hat{c}}_3=\mathrm{i} \frac{(k_3B)}{\varOmega}\hat{c}_1, \end{gather}$$

(2.23d)$$\begin{gather}d_\tau{\hat{c}}_4=\mathrm{i} \frac{(k_3B)}{\varOmega}\hat{c}_2, \end{gather}$$

(2.23e)$$\begin{gather}d_\tau{\hat{c}}_5=-\frac{N^2}{\varOmega^2}\hat{c}_2, \end{gather}$$

where $d_\tau {\hat {c}}_1= \varOmega ^{-1}d_t\hat {c}_1$ and $k_\perp =\sqrt {k_1^2+k_2^2}.$ Combining (2.23d) and (2.23e), we deduce the following relation:

(2.24)\begin{equation} -\frac{\varOmega}{k_3}\left [\mathrm{i}\varOmega\left (k_3B\right)\hat{c}_5+N^2\hat{c}_4\right]=\mathrm{i}\left (k_3B\right)\hat{\vartheta}+N^2\hat{b}_3=\hat{\varpi}_m=\text{const.}, \end{equation}

which represents the spectral counterpart of the MIPS (see (2.4)). Accordingly, system (2.23) can be further reduced to a fourth-order inhomogeneous Floquet system,

(2.25)\begin{equation} d_\tau{\hat{\boldsymbol{c}}}={\boldsymbol{D}}(\tau)\boldsymbol{\cdot}{\hat{\boldsymbol{c}}}+\hat{\boldsymbol{\varphi}}, \end{equation}

where the only non-zero elements of ${\boldsymbol {D}}(\tau )$ are

(2.26a)$$\begin{gather} D_{11}=-4\varepsilon \frac{k_1k_2}{k^2},\quad D_{12}=-2,\quad D_{21}=2\frac{k_3^2}{k^2}, \end{gather}$$

(2.26b)$$\begin{gather}D_{13}=D_{31}=D_{42}=\mathrm{i} \frac{\left (k_3B\right)}{\varOmega}, \end{gather}$$

(2.26c)$$\begin{gather}D_{14}=2\mathrm{i} \varepsilon \frac{N^2}{\varOmega (k_3B)}\frac{k_1k_2}{k^2}, \end{gather}$$

(2.26d)$$\begin{gather}D_{24}=\mathrm{i} \left (\frac{\left (k_3B\right)}{\varOmega}+ \frac{N^2}{\varOmega \left (k_3B\right)}\frac{k_\perp^2}{k^2}\right). \end{gather}$$

The non-zero components of the inhomogeneous term in (2.25) take the form

(2.27a)$$\begin{gather} \hat{\varphi}_{1}=2\mathrm{i} \varepsilon\frac{k_1k_2}{k^2}\frac{\hat{\varpi}_m}{(\varOmega^2 B)}, \end{gather}$$

(2.27b)$$\begin{gather}\hat{\varphi}_{2}=\mathrm{i} \frac{k_\perp^2}{k^2} \frac{\hat{\varpi}_m}{(\varOmega^2 B)} \end{gather}$$

and it can be seen as a time-varying forcing excitation.

Note that in the non-magnetised stratified case, one can use the fact that the PV is a Lagrangian invariant for a non-diffusive fluid (Pedlosky Reference Pedlosky2013) to derive a non-homogeneous two-component Floquet system in terms of the variables $\hat {c}_1$ and $\hat {c}_2.$

2.3. Homogeneous Floquet system

The linear system (2.25) has the properties ${\boldsymbol {D}}(\tau +T)= {\boldsymbol {D}}(\tau )$ and $\hat {\boldsymbol {\varphi }}(\tau +T)= \hat {\boldsymbol {\varphi }}(\tau )$ where $T=2{\rm \pi}$ is the period common to both the matrix ${\boldsymbol {D}}$ and the vector $\hat {\boldsymbol {\varphi }}$. Floquet theory does not address stability of the inhomogeneous system described by (2.25) where the ‘forcing excitation’ $\hat {\boldsymbol {\varphi }}(\tau )$ is present. However, the $T$-periodic nature of $\hat {\boldsymbol {\varphi }}(\tau )$ allows an extension to the theory (Slane & Tragesser Reference Slane and Tragesser2011). Following the study of Slane & Tragesser (Reference Slane and Tragesser2011), it is shown that the basic behaviour of the homogeneous system

(2.28)\begin{equation} d_\tau\hat{\boldsymbol{c}}={\boldsymbol{D}}\boldsymbol{\cdot}\hat{\boldsymbol{c}} \end{equation}

does not change with the addition of the term $\hat {\boldsymbol {\varphi }}(\tau ).$ Here, $\hat {\boldsymbol {c}}=(\hat {c}_1,\hat {c}_2,\hat {c}_3,\hat {c}_4)^{\rm T}$ in the canonical basis of $\mathbb {C}^4.$ In other words, for purposes of studying stability, one may set $\hat {\varpi }_\kappa =0,$ so that $\hat {\boldsymbol {\varphi }}=\boldsymbol {0}$ (Benkacem et al. Reference Benkacem, Salhi, Khlifi, Nasraoui and Cambon2022).

We denote by $\boldsymbol {\varPhi }(\tau )$ any fundamental matrix solution of the homogeneous system (2.28) where $\boldsymbol {\varPhi }(0)={\boldsymbol {I}}_4$. According to Floquet–Lyapunov theorem, $\boldsymbol {\varPhi }$ is expressible in the form (Kuchment Reference Kuchment1993),

(2.29)\begin{equation} \boldsymbol{\varPhi}(\tau)={\boldsymbol{F}}(\tau)\exp\left ({\boldsymbol{K}}\tau\right), \end{equation}

where ${\boldsymbol {F}}(\tau )$ is a non-singular continuous $2{\rm \pi}$-periodic $4\times 4$ matrix-function (whose derivative is an integrable piecewise-continuous function) and ${\boldsymbol {K}}$ is a constant matrix. The determinant of $\boldsymbol {\varPhi }$ is unity at $\tau =2{\rm \pi},$ $\vert {\boldsymbol {\varPhi } (2{\rm \pi} )}\vert =1$ because

(2.30)\begin{equation} \text{trace}\,{\boldsymbol{D}}=\sum_{j=1}^4D_{jj}=-4\varepsilon \frac{k_1k_2}{k^2}=-\frac{1}{k^2}{d_\tau k^2}. \end{equation}

It follows that whenever $\lambda$ is an eigenvalue of the monodromy matrix, ${\boldsymbol {M}}=\boldsymbol {\varPhi }(2{\rm \pi} ),$ so too are its inverse $\lambda ^{-1}$ and its complex conjugate $\lambda ^*$ (see LZ04). Consequently, in the stable case, eigenvalues of ${\boldsymbol {M}}$ lie on the unit circle. If any eigenvalue $\lambda$ of ${\boldsymbol {M}}$ has modulus exceeding one, this implies that there is indeed an exponentially growing solution. The growth rates are then given by

(2.31)\begin{equation} \sigma=\frac{1}{2{\rm \pi}}\log \left (\lambda\right). \end{equation}

In the Floquet system (2.28) figure four dimensionless parameters, namely

(2.32a–d)\begin{equation} \varepsilon=\tfrac{1}{2}\left (E-E^{-1}\right),\quad \mu={k_3}/{k_0},\quad {\mathcal{N}}=N/\varOmega,\quad {\mathcal{B}}=(k_0B)/\varOmega, \end{equation}

where $k_0=\sqrt {k_p^2+k_3^2}$ represents the modulus of the initial wave vector for $\varepsilon =0.$ The parameters $k_0B$, $N$ and $2\varOmega$ can be seen as the maximal frequencies of Alfvén, gravity and inertial waves, respectively (we return to this later).

For the stability analysis of system (2.28), we perform an asymptotic analysis to leading order in $\varepsilon$ to determine the maximal growth rates of instability (if it exists). In addition, we integrate numerically (using the fourth-order Runge–Kutta–Gill method) system (2.28) from $\tau = 0$ to $\tau =2{\rm \pi}$ and we determine the eigenvalues of the solution matrix numerically (using the double QR method).

3.1. Dispersion relation of MIG waves

In this section, we establish the dispersion relation of the MIG waves propagating in a non-diffusive unbounded fluid. The cases of inertia-gravity waves and magneto-inertia waves are briefly addressed. We denote by ${\boldsymbol {D}}_0$ the matrix ${\boldsymbol {D}}$ for $E=1$ (i.e. circular streamlines)

(3.1)\begin{equation} {\boldsymbol{D}}_0=\varOmega^{-1} \left (\begin{array}{@{}cccc@{}} 0 & -2\varOmega & \mathrm{i} \omega_a & 0\\ (2\varOmega)^{-1}\omega_r^2 & 0 & 0 & \mathrm{i} \left (\omega_a^2+\omega_g^2\right){\omega_a^{-1}}\\ \mathrm{i} \omega_a & 0 & 0 & 0\\ 0 & \mathrm{i} \omega_a & 0 & 0 \end{array}\right), \end{equation}

where

(3.2a)$$\begin{gather} \omega_r=2\boldsymbol{\varOmega}\boldsymbol{\cdot} (\boldsymbol{k}/k)= 2\varOmega \mu, \end{gather}$$

(3.2b)$$\begin{gather}\omega_a=\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{k}=Bk_0\mu=\varOmega {\mathcal{B}}\mu, \end{gather}$$

(3.2c)$$\begin{gather}\omega_g=N{\vert \boldsymbol{g}\times\boldsymbol{k}\vert}/(gk)=N\sqrt{1-\mu^2}=\varOmega {\mathcal{N}}\sqrt{1-\mu^2} \end{gather}$$

are the frequencies of inertial, Alfvèn and gravity waves, respectively. The eigenvalues $\sigma _j$ ($\,j=1,2,3,4)$ of the constant matrix ${\boldsymbol {D}}_0$ take the form (Salhi et al. Reference Salhi, Baklouti, Godeferd, Lehner and Cambon2017),

(3.3a)$$\begin{gather} -\varOmega^2\sigma_{1,2}^2=\omega^2_{1,2}=\tfrac{1}{2}\left [2\omega_a^2+\omega_r^2+\omega_g^2+\sqrt{\left (\omega_r^2+\omega_g^2\right)^2+4\omega_r^2\omega_a^2}\right], \end{gather}$$

(3.3b)$$\begin{gather}-\varOmega^2\sigma_{3,4}^2=\omega_{3,4}^2=\tfrac{1}{2}\left [2\omega_a^2+\omega_r^2+\omega_g^2-\sqrt{\left (\omega_r^2+\omega_g^2\right)^2+4\omega_r^2\omega_a^2}\right], \end{gather}$$

or, equivalently,

(3.4a)$$\begin{gather} \omega_1=-\omega_{2}=\frac{\varOmega}{\sqrt{2}}\sqrt{\left (4+2{\mathcal{B}}^2-{\mathcal{N}}^2\right)\mu^2+{\mathcal{N}}^2+\sqrt{\left [\left (4-{\mathcal{N}}^2\right)\mu^2+{\mathcal{N}}^2\right]^2+16{\mathcal{B}}^2\mu^4}}, \end{gather}$$

(3.4b)$$\begin{gather}\omega_3=-\omega_{4}=\frac{\varOmega}{\sqrt{2}}\sqrt{\left (4+2{\mathcal{B}}^2-{\mathcal{N}}^2\right)\mu^2+{\mathcal{N}}^2-\sqrt{\left [\left (4-{\mathcal{N}}^2\right)\mu^2+{\mathcal{N}}^2\right]^2+16{\mathcal{B}}^2\mu^4}}. \end{gather}$$

These are distinct and non-zero as long as

(3.5)\begin{equation} {\mathcal{B}}\ne 0\quad \text{and}\quad 0<\mu^2<1. \end{equation}

In the non-stratified case (${\mathcal {N}}=0),$ the frequencies $\omega _{1,2,3,4}$ are linear with respect to the variable $\mu$ (see (3.8)); in the presence of stratification, they are not. This has the consequence of making the asymptotic analysis calculations much more complex (see Appendix A) than in the cases without stratification which were studied by LZ04 (the magneto-elliptical instability) and Mizerski & Bajer (Reference Mizerski and Bajer2009) (the magneto-elliptical instability of rotating systems).

On the other hand, we note that (3.3), which can be rewritten as follows (Salhi et al. Reference Salhi, Baklouti, Godeferd, Lehner and Cambon2017)

(3.6)\begin{equation} \left (\omega^2-\omega_g^2\right)\left (\omega^2-\omega_a^2-\omega_g^2\right)-\omega_r^2\omega^2=0, \end{equation}

represents the dispersion relation for MIG waves in a non-diffusive fluid. As it was discussed in Salhi et al. (Reference Salhi, Baklouti, Godeferd, Lehner and Cambon2017), the dispersion relation (3.6) remains similar to that of $f$-plane MHD (Schecter, Boyd & Gilman Reference Schecter, Boyd and Gilman2001).

We can gain insight into the analysis of resonant cases of MIG waves by examining some limiting cases.

3.1.1. Inertia-gravity waves

In the absence of unperturbed magnetic field ($B=0,$ so that, $\omega _a=0$), the frequency $\omega _{3,4}$ given by (3.3) vanishes, whereas the frequency $\omega _{1,2}$ reduces to

(3.7)\begin{equation} \omega_{1,2}={\pm}\sqrt{{\omega_g^2+\omega_r^2}}=\varOmega\sqrt{(4-{\mathcal{N}}^2)\mu^2+{\mathcal{N}}^2}. \end{equation}

In that case, there are inertia-gravity waves where the simultaneous presence of the solid-body rotation and stable vertical stratification produces higher-frequency waves because $\omega _r^2\leq \omega _{1,2}^2$ and $\omega _g^2\leq \omega _{1,2}^2.$

3.1.2. Magneto-inertia waves

In the non-stratified case, $N=0,$ the frequency of gravity waves vanishes, $\omega _g=0.$ In that case, there are fast and slow magneto-inertia waves with frequency

(3.8a)$$\begin{gather} \omega_{1,2}={\pm}\tfrac{1}{2}\left (\sqrt{\omega_r^2+4\omega_a^2}+ \omega_r\right)={\pm}\varOmega\left (\sqrt{1+{\mathcal{B}}^2}+1\right)\mu, \end{gather}$$

(3.8b)$$\begin{gather}\omega_{3,4}={\pm}\tfrac{1}{2}\left (\sqrt{\omega_r^2+4\omega_a^2}- \omega_r\right)={\pm}\varOmega\left (\sqrt{1+{\mathcal{B}}^2}-1\right)\mu, \end{gather}$$

respectively. Note that, in the study by LZ04, the fast magneto-inertia waves are called ‘hydrodynamic modes’ and the slow magneto-inertia waves are called ‘magnetic modes’ because $\omega _{3,4}=0$ at vanishing unperturbed magnetic field.

3.2. Resonant cases of MIG waves

In this section, we establish the condition of resonances of order $n$ between two fast or between two slow modes or between a fast mode and a slow mode.

The resonant cases of MIG waves are those parameter values $(\mu, {\mathcal {N}},{\mathcal {B}},)$ such that

(3.9)\begin{equation} \omega_i-\omega_j= n\varOmega,\quad (i\ne j), \end{equation}

where $n$ is an integer. For the elliptical flow, the only resonant cases that can lead to instability are those for which the integer $n$ is even.

By using (3.3) we deduce that the resonance condition between two fast modes ($\omega _1-\omega _2=n\varOmega )$ or between two slow modes ($\omega _3-\omega _4=n\varOmega )$ is described by the following algebraic equation,

(3.10)\begin{align} 4{\mathcal{B}}^2\left ({\mathcal{B}}^2-{\mathcal{N}}^2\right)\mu^4+\left (4{\mathcal{B}}^2{\mathcal{N}}^2-2n^2{\mathcal{B}}^2-4n^2+n^2{\mathcal{N}}^2\right)\mu^2+n^2\left (\frac{n^2}{4}-{\mathcal{N}}^2\right)=0, \end{align}

with $0<\mu ^2< 1.$ Because the replacement $\mu \rightarrow -\mu$ and/or $n\rightarrow -n$ in (3.10) result in the same condition, we may therefore assume without loss of generality that $\mu >0$ and $n>0$.

The condition (3.10) for resonance readily extends that by Bayly (Reference Bayly1986) for basic elliptical flow instability (${\mathcal {B}}=0$ and ${\mathcal {N}}=0).$ Note that ${\rm \pi} /\varOmega$ is the typical time (period) for a wave packet to run the closed elliptical streamline, in the limit of vanishing, but non-zero, $\varepsilon$; in the same limit, this period characterises the periodic alignment of the fluctuating vorticity with the mean (weak) strain, as $2{\rm \pi} /\omega$. The condition can be written $4 \varOmega \mu = n \varOmega$ from the seminal study (Bayly Reference Bayly1986), that immediately yielded $\mu = n/4=0.5$ (because only $n=2$ gives rise to $\mu <1),$ giving the origin of time-dependent instability tongues at vanishing $\varepsilon$. Even if a single-mode analysis is apparently sufficient, as emphasised by Craik & Criminale (Reference Craik and Criminale1986) without the need for nonlinearity (exact solution), a two-mode resonance is implied, as also suggested by the classical normal mode analysis. For both basic elliptical flow instability and precessional instability, we have to consider the modes with dispersion law $\omega _1 =+ \omega _r$ and $\omega _2 = - \omega _r$, and the subharmonic destabilising resonance is found for $\omega _1 - \omega _2 = 2\omega _r = \pm n \varOmega.$ Accordingly, the subharmonic order is $n=2$ for the basic elliptical flow instability, and $n=1$ for the precessional instability. Also in agreement with his triad instability principle, a detailed analysis of Waleffe (Reference Waleffe1992) shows that the elliptical instability corresponds to a forward (F)-interaction: the two modes with eigenfrequency $\omega _1$ and $\omega _2$ have opposite polarities and are coupled with the mean flow, which is associated to a zero frequency, the unstable modes are thereby two resonant inertial waves associated with the uniform background rotation.

In a similar manner, we determine from (3.3) the resonance condition between a fast mode and a slow mode ($\omega _1-\omega _3 = n\varOmega$ or $\omega _1-\omega _4 = n\varOmega$),

(3.11)\begin{align} \left [\left (4-{\mathcal{N}}^2\right)^2+16{\mathcal{B}}^2\right]\mu^4-2\left [2n^2{\mathcal{B}}^2+\left ({\mathcal{N}}^2-4\right)\left ({\mathcal{N}}^2-n^2\right)\right]\mu^2+\left ({\mathcal{N}}^2-n^2\right)^2=0. \end{align}

In the three following sections, we present asymptotic formulae for the maximal growth rates of the subharmonic instabilities (those corresponding to the destabilising resonances of order $n=2$). The asymptotic formulae are yielded by the asymptotic analysis at leading order in $\varepsilon$ of the Floquet system (2.28). Obviously, the instabilities related to higher-order resonances $(n = 4, 6, 8, \ldots )$ are excluded by the procedure leading to the asymptotic formulae. These instabilities (if they exist) can be captured by the numerical computations (see § 4).

3.3. Destabilising resonance between two fast modes

As shown by LZ04, the universal elliptic instability, which results from the resonances (of order $n=2)$ between two fast modes, persists in the presence of magnetic fields of arbitrary strength, although the growth rate decreases somewhat (LZ04). As a counterpart, in the presence of vertical (stable) stratification, it is completely suppressed when ${\mathcal {N}}$ reaches $1$ (Miyazaki & Fukumoto Reference Miyazaki and Fukumoto1992). In this section, we investigate the effects of the axial (stable) stratification and magnetic field when there are simultaneously present on this instability (referred to as IF instability).

A detailed analysis of the algebraic equation (3.10), knowing that $0\leq \mu ^2\leq 1,$ $0\leq {\mathcal {B}}<+\infty$ and $0\leq {\mathcal {N}}<+\infty,$ indicates that the resonant case of order $n = 2$ between two fast modes ($\omega _1-\omega _2=2\varOmega )$ exists for

(3.12)\begin{equation} \left ({\mathcal{B}},{\mathcal{N}}\right) \in \left ([0,+\infty{[} \times [0,1]\right)\cup {\mathcal{D}}_f, \end{equation}

which corresponds to the domains I, II and VI of the plane (${\mathcal {B}},{\mathcal {N}})$ shown by figure 2. Here, the domain ${\mathcal {D}}_f$ is defined as follows:

(3.13)\begin{equation} \forall \left ({\mathcal{B}},{\mathcal{N}}\right)\in {\mathcal{D}}_f\Leftrightarrow\ \text{for given}\ {\mathcal{N}}\in\, ]2,+\infty[\quad \implies \quad 1< f({\mathcal{N}})\leq {\mathcal{B}}<\sqrt{3}, \end{equation}

where $f :\, ]2, +\infty [\,\rightarrow\, ]1,\sqrt {3}[$ is a continuous decreasing function with reciprocal function $f^{-1}: \,]1,\sqrt {3}[ \,\rightarrow\, ]2, +\infty [,$

(3.14a)$$\begin{gather} \forall {\mathcal{N}}\in \,]2,+\infty[,\ f({\mathcal{N}})=\frac{1}{{\mathcal{N}}^2} \sqrt{{\mathcal{N}}^4+4{\mathcal{N}}^2-8+4\sqrt{({\mathcal{N}}^2-1)({\mathcal{N}}^4-4)}} \underset{{\mathcal{N}}\rightarrow +\infty}{\longrightarrow} 1, \end{gather}$$

(3.14b)$$\begin{gather}\forall {\mathcal{B}}\in \,]1,\sqrt{3}[,\ f^{-1}({\mathcal{B}})=\frac{2}{({\mathcal{B}}^2-1)} \sqrt{ 1+{\mathcal{B}}^2+{\mathcal{B}}\sqrt{4-({\mathcal{B}}^2-1)^2}}\underset{ {\mathcal{B}}\rightarrow 1^+}{\longrightarrow} +\infty. \end{gather}$$

For (${\mathcal {B}},{\mathcal {N}})$ belonging to the domain given by (3.12), the resonances (of order $n=2)$ between two fast modes occur when

(3.15a)$$\begin{gather} \mu^2=\frac{1-{\mathcal{N}}^2}{4-{\mathcal{N}}^2}\quad \text{if}\ {\mathcal{B}}=0, \end{gather}$$

(3.15b)$$\begin{gather}\mu^2=\frac{{\mathcal{B}}^2-1}{{\mathcal{B}}^4-{\mathcal{B}}^2-4}\quad \text{if}\ {\mathcal{B}}={\mathcal{N}}, \end{gather}$$

(3.15c)$$\begin{gather}\mu^2=\frac{(1+{\mathcal{B}}^2)(1-{\mathcal{N}}^2)+{\mathcal{B}}^2+3-\sqrt{\varDelta}} {2{\mathcal{B}}^2({\mathcal{B}}^2-{\mathcal{N}}^2)}\quad \text{if}\ {\mathcal{B}}\ne {\mathcal{N}}, \end{gather}$$

where

(3.16)\begin{equation} \varDelta({\mathcal{B}},{\mathcal{N}})=\left ({\mathcal{N}}^2\left ({\mathcal{B}}^2-1\right)-4\right )^2+16\left ({\mathcal{B}}^2-{\mathcal{N}}^2\right). \end{equation}

It immediately follows that

(3.17a,b)\begin{equation} \lim_{0\leq {\mathcal{N}}\leq 1, {\mathcal{B}}\rightarrow +\infty}{\mathcal{B}}^2\mu^2=1-{\mathcal{N}}^2,\quad \lim_{1< {\mathcal{B}}<\sqrt{3}, {\mathcal{N}}\rightarrow +\infty}{\mathcal{B}}^2\mu^2=1. \end{equation}

In the non-stratified magnetised case $({\mathcal {N}} = 0),$ the parameter $\mu = [1+\sqrt {1+{\mathcal {B}}^2}]^{-1}$ changes from $0.5$ (so that, $\theta = \widehat {(\boldsymbol {k}, \boldsymbol {W})} =\cos ^{-1}(\mu ) = {\rm \pi}/3)$ at ${\mathcal {B}}=0$ to zero (so that, $\theta ={\rm \pi} /2)$ as ${\mathcal {B}}\rightarrow +\infty$ (see also LZ04). Recall that $\boldsymbol {k}$ denotes the wave vector and $\boldsymbol {W}=\boldsymbol {\nabla }\times \boldsymbol {U}$ denotes the basic vorticity vector. In the non-magnetised stratified case (${\mathcal {B}}=0),$ the parameter $\mu =\sqrt {(1-{\mathcal {N}}^2)/(4-{\mathcal {N}}^2)}$ changes from $\mu =0.5$ at ${\mathcal {N}}=0$ to $\mu =0$ at ${\mathcal {N}}=1.$ For $({\mathcal {B}},{\mathcal {N}})\in [0,+\infty {[} \times [0,1]$ and when ${\mathcal {B}}$ is fixed, the parameter $\mu$ changes from $\mu = [1+\sqrt {1+{\mathcal {B}}^2}]^{-1}$ at ${\mathcal {N}}= 0$ to $\mu =0$ at ${\mathcal {N}} = 1.$ For (${\mathcal {B}},{\mathcal {N}})\in {\mathcal {D}}_f$ and when ${\mathcal {B}}$ is fixed, the parameter $\mu$ increases from $\mu ({\mathcal {B}},f({\mathcal {B}}))$ at ${\mathcal {N}}=f({\mathcal {B}})$ to $\mu =1$ as ${\mathcal {N}}\rightarrow +\infty.$ As an illustration, figure 3 shows the variation of $\mu$ versus ${\mathcal {N}}$ for three values of ${\mathcal {B}}=\sqrt {3}, 1.632$ and $1.532$.

Figure 2. Domains of the $({\mathcal {B}}, {\mathcal {N}})$ plane for which the magneto-gravity-elliptic instability operates. The subharmonic instability resulting from a resonance (of order $n=2)$ between two fast modes (referred to as IF instability) exists for $({\mathcal {B}}, {\mathcal {N}})$ belonging to domains I, II and VI. The subharmonic instability resulting from a resonance between two slow modes (referred to as IS instability) exists for $({\mathcal {B}}, {\mathcal {N}})$ belonging to domains II, IV, V and VI. The subharmonic instability resulting from a resonance between a fast mode and a slow mode (referred to as IM instability) exists for $({\mathcal {B}}, {\mathcal {N}})$ belonging to domains I, II, III and IV. For $({\mathcal {B}}, {\mathcal {N}} )$ belonging to domain VII, there is no subharmonic instability, whereas instabilities related to higher order resonances $(n = 4, 6, 8,\ldots )$ can exist. Domain I: $({\mathcal {B}}, {\mathcal {N}})\in [0, \sqrt {3}]\times [0,1];$ domain II: $({\mathcal {B}}, {\mathcal {N}})\in [ \sqrt {3}, +\infty {[} \times [0,1];$ domain III: $({\mathcal {B}}, {\mathcal {N}})\in [0, \sqrt {3}]\times [1,2];$ domain IV: $({\mathcal {B}}, {\mathcal {N}})\in [\sqrt {3}, +\infty {[} \times [1,2];$ domain V: $({\mathcal {B}}, {\mathcal {N}})\in [\sqrt {3}, +\infty [ \,\times\, [2,+\infty [;$ domain VI: $({\mathcal {B}}, {\mathcal {N}})\in {\mathcal {D}}_f$ where ${\mathcal {D}}_f$ is defined by (3.13); domain VII: $({\mathcal {B}}, {\mathcal {N}})\in [0, \sqrt {3}[ \,\times\, [2,+\infty [\setminus {\mathcal {D}}_f$.

Figure 3. Resonant cases of order $n=2$ in the case where $({\mathcal {B}},{\mathcal {N}})\in {\mathcal {D}}_f$ (see (3.13)). Variation of $\mu =\cos (\theta )=k_3/k_0$ versus ${\mathcal {N}}=N/\varOmega$ for ${\mathcal {B}}=k_0B/\varOmega =\sqrt {3},1.632$ and ${\mathcal {B}}=1.532.$ Solid lines represent the resonances between two fast modes (fast, fast) and dotted lines represent the resonances between two slow modes (slow, slow).

According to our asymptotic analysis at leading order in $\varepsilon$ (see Appendix A), the maximal growth rate of the subharmonic instability (if it exists) resulting from the resonance between two fast modes is of the form

(3.18)\begin{equation} \frac{\sigma_{mf}}{\varepsilon}=\frac{\left (3-{\mathcal{B}}^2\mu^2\right)\left (1+{\mathcal{B}}^2\mu^2\right)}{8} \frac{\left [\left ({\mathcal{N}}^2-{\mathcal{B}}^2+2\right)\mu^2+\left (1-{\mathcal{N}}^2\right)\right]} {\left [\left ({\mathcal{N}}^2-2{\mathcal{B}}^2-4\right)\mu^2+\left (2-{\mathcal{N}}^2\right)\right]} \end{equation}

in which $\mu ^2$ is given by (3.15).

Some results reported in previous studies (Waleffe Reference Waleffe1990; Kerswell Reference Kerswell2002 and LZ04) can be recovered from (3.18) by using (3.15)

(3.19a)$$\begin{gather} \frac{\sigma_{mf}}{\varepsilon}=\frac{9}{16}\quad \text{for}\ {\mathcal{N}}=0,\ {\mathcal{B}}=0, \end{gather}$$

(3.19b)$$\begin{gather}\frac{\sigma_{mf}}{\varepsilon}=\frac{9}{4}\frac{(1-{\mathcal{N}}^2)}{(4-{\mathcal{N}}^2)}\quad \text{for} \ 0\leq {\mathcal{N}}\leq 1,\ {\mathcal{B}}=0, \end{gather}$$

(3.19c)$$\begin{gather}\frac{\sigma_{mf}}{\varepsilon}= \frac{1}{4}\left (1+\frac{1}{\sqrt{{\mathcal{B}}^2+1}+1}\right )^2 \quad \text{for}\ {\mathcal{N}}=0,\ 0\leq {\mathcal{B}}<+\infty. \end{gather}$$

Equation (3.19b) indicates that the maximal growth rate is zero at ${\mathcal {N}}=1$. Therefore, in the non-magnetised stratified case (${\mathcal {B}}=0),$ the subharmonic instability is completely suppressed by stratification when ${\mathcal {N}}$ reaches 1 (Miyazaki & Fukumoto Reference Miyazaki and Fukumoto1992; Kerswell Reference Kerswell2002). For the non-stratified magnetised case (${\mathcal {N}}=0),$ (3.19c) indicates that $\sigma _{mf}/\varepsilon$ decreases as ${\mathcal {B}}$ increases so as (see also LZ04)

(3.20)\begin{equation} \lim_{{\mathcal{N}}=0,\ {\mathcal{B}}\rightarrow +\infty} \frac{\sigma_{mf}}{\varepsilon}=\frac{1}{4}. \end{equation}

However, in the stratified magnetised case, the analysis of (3.18) is more subtle as shown in the following.

For $({\mathcal {B}},{\mathcal {N}})\in {]}0,+\infty {[} \times {]}0,1]$ and when ${\mathcal {N}}$ is fixed, $\sigma _{mf}/\varepsilon$ also decreases from $\sigma _{mf}/\varepsilon =[9(1-{\mathcal {N}}^2)]/[4(4-{\mathcal {N}}^2)]$ at ${\mathcal {B}}=0$ to zero (and not to $1/4)$ as ${\mathcal {B}}\rightarrow +\infty.$ Indeed, from (3.18), one easily deduces that

(3.21)\begin{equation} \lim_{0<{\mathcal{N}}\leq 1,\ {\mathcal{B}}\rightarrow +\infty} \frac{\sigma_{mf}}{\varepsilon}=0, \end{equation}

because $\lim _{{\mathcal {B}}\rightarrow +\infty }{\mathcal {B}}^2\mu ^2= 1-{\mathcal {N}}^2,$ as indicated previously. Therefore, the ${\mathcal {N}}\rightarrow 0$ limit is, in fact, singular (discontinuous). As an illustration, figure 4 shows $\sigma _{mf}/\varepsilon$ versus ${\mathcal {B}}$ for five values of ${\mathcal {N}} = 0.0, 0.2, 0.5, 0.7$ and $0.9.$ At ${\mathcal {N}}=1,$ one has $\sigma _{mf}=0$ independently of ${\mathcal {B}}.$

Figure 4. Maximal growth rate of the destabilising resonances of order n = 2 between two fast modes (see (3.18)). The figure shows $\sigma _{mf}/\varepsilon$ vs ${\mathcal {B}} = k_0B/\varOmega$ for ${\mathcal {N}}= N/\varOmega = 0$, 0.2, 0.5, 0.7 and ${\mathcal {N}} = 0.9$.

For $({\mathcal {B}},{\mathcal {N}})\in {\mathcal {D}}_f$ and for fixed ${\mathcal {N}},$ the maximal growth rate $\sigma _{mf}$ increases from $0$ at ${\mathcal {B}}=\sqrt {3}$ to $\sigma _{mf} (\,f ({\mathcal {N}}),N)$ at ${\mathcal {B}}=f ({\mathcal {N}})$ with

(3.22)\begin{equation} \lim_{{\mathcal{N}}\rightarrow +\infty}\frac{\sigma_{fm}\left (\,f\left ({\mathcal{N}}\right),N\right)}{\varepsilon}=\frac{1}{2}. \end{equation}

This is illustrated by figure 5 which displays the variation of $\sigma _{mf}/\varepsilon$ versus ${\mathcal {B}}$ ($1<{\mathcal {B}}<\sqrt {3})$ for ${\mathcal {N}}=10$ and ${\mathcal {N}}=50.$ Therefore, this subharmonic instability, which occurs when

(3.23a,b)\begin{equation} 2\varOmega < N<+\infty\quad \text{and}\quad \varOmega< k_0B=\varOmega f({\mathcal{N}})<\sqrt{3}\, \varOmega, \end{equation}

is the results of the simultaneous presence of axial (stable) stratification and magnetic field. On the other hand, the above analysis clearly shows that, for $({\mathcal {B}},{\mathcal {N}})\in [0,+\infty {[} \times [0,1],$ the IF instability is completely suppressed by the stratification when ${\mathcal {N}}$ reaches 1. When $0<{\mathcal {N}}<1,$ stratification acts to render the IF instability less efficient especially for large ${\mathcal {B}}$ because its maximal growth approaches zero as ${\mathcal {B}}\rightarrow +\infty$ (see (3.21)).

Figure 5. Maximal growth rates of subharmonic instabilities resulting from the resonances between two fast modes ($\sigma _{mf})$ or between two slow modes ($\sigma _{ms})$ in the case where $({\mathcal {B}},{\mathcal {N}})\in {\mathcal {D}}_f$ (see (3.13)). The variation of $\sigma _{mf}/\varepsilon$ and $\sigma _{ms}/\varepsilon$ versus ${\mathcal {B}}=k_0B/\varOmega$ for ${\mathcal {N}}=N/\varOmega =10$ and ${\mathcal {N}}=50$ is shown. Numerical results (see § 4) for $\varepsilon =0.05$ are represent by symbols.

3.4. Destabilising resonances between two slow modes

As shown by LZ04, in the presence only of the magnetic field, there exist other subharmonic instabilities, which are due to the presence of the magnetic field, in addition to the universal elliptical instability. One of them is the subharmonic instability resulting from the resonances (of order $n=2$) between two slow modes (referred to as IS instability). In this section, we study the effects of the vertical (stable) stratification on the IS instability.

From the resonant condition described by relation (3.10), we deduce that the resonant cases of order $n = 2$ between two slow modes, so that $\omega _3-\omega _4=2\varOmega,$ exist for

(3.24)\begin{equation} \left ({\mathcal{B}},{\mathcal{N}}\right) \in \left ([\sqrt{3},+\infty{[} \times [0,+\infty[\right)\cup {\mathcal{D}}_f. \end{equation}

This corresponds to the domains II, IV, V and VI shown in figure 2.

For $({\mathcal {B}},{\mathcal {N}})$ belonging to these domains, the resonant cases of order $n=2$ between two slow modes occur at the following points of the $\mu$ axis

(3.25a)$$\begin{gather} \mu^2=\frac{{\mathcal{B}}^2-1}{{\mathcal{B}}^4-{\mathcal{B}}^2-4}\quad \text{if}\ {\mathcal{B}}={\mathcal{N}}, \end{gather}$$

(3.25b)$$\begin{gather}\mu^2=\frac{(1+{\mathcal{B}}^2)(1-{\mathcal{N}}^2)+{\mathcal{B}}^2+3+\sqrt{\varDelta}} {2{\mathcal{B}}^2({\mathcal{B}}^2-{\mathcal{N}}^2)}\quad \text{if}\ {\mathcal{B}}\ne {\mathcal{N}}. \end{gather}$$

It immediately follows that

(3.26a,b)\begin{equation} \lim_{{\mathcal{N}}\geq 0,\ {\mathcal{B}}\rightarrow +\infty} \mu^2{\mathcal{B}}^2=1,\quad \lim_{{\mathcal{B}}> 1,\ {\mathcal{N}}\rightarrow +\infty} \mu^2{\mathcal{B}}^2=1. \end{equation}

In the non-stratified magnetised case, (3.25) reduces to $\mu = [ \sqrt {{\mathcal {B}}^2+1}-1]^{-1},$ or equivalently, $\mu ^2{\mathcal {B}}^2= 1+2\mu.$ Thus, the parameter $\mu$ changes from $1$ at ${\mathcal {B}}=\sqrt {3}$ to $0$ as ${\mathcal {B}}\rightarrow +\infty$ (see also LZ04). In the stratified magnetised case, (3.25) indicates that, at fixed ${\mathcal {B}}$ such that (${\mathcal {B}},{\mathcal {N}})\in [\sqrt {3},+\infty {[} \times [0,+\infty [,$ the parameter $\mu$ changes from $\mu =[\sqrt {{\mathcal {B}}^2+1}-1]^{-1}$ at ${\mathcal {N}} = 0$ to $\mu =1/{\mathcal {B}}$ as ${\mathcal {N}}\rightarrow +\infty.$ For fixed ${\mathcal {B}}$ such that (${\mathcal {B}},{\mathcal {N}})\in {\mathcal {D}}_f,$ the parameter $\mu$ decreases from $\mu ({\mathcal {B}},f({\mathcal {B}}))$ at ${\mathcal {N}}=f({\mathcal {B}})$ to $\mu =1/{\mathcal {B}}$ as ${\mathcal {N}}\rightarrow +\infty$ (see figure 3).

As shown in Appendix A, the maximal growth rate, denoted by $\sigma _{ms},$ of the IS instability is also described by (3.18) (repeated here for the sake of clarity)

(3.27)\begin{equation} \frac{\sigma_{ms}}{\varepsilon}=\frac{\left (3-{\mathcal{B}}^2\mu^2\right)\left (1+{\mathcal{B}}^2\mu^2\right)}{8} \frac{\left [\left ({\mathcal{N}}^2-{\mathcal{B}}^2+2\right)\mu^2+\left (1-{\mathcal{N}}^2\right)\right]} {\left [\left ({\mathcal{N}}^2-2{\mathcal{B}}^2-4\right)\mu^2+\left (2-{\mathcal{N}}^2\right)\right]} \end{equation}

in which $\mu$ is now given by (3.25) and not by (3.15).

In the non-stratified magnetised case (${\mathcal {N}}=0)$, (3.27) reduces to

(3.28)\begin{equation} \frac{\sigma_{ms}}{\varepsilon}= \frac{1}{4}\left (1-\frac{1}{\sqrt{{\mathcal{B}}^2+1}-1}\right )^2\underset{{\mathcal{N}}=0,\ {\mathcal{B}}\rightarrow +\infty}{\longrightarrow}\frac{1}{4}, \end{equation}

where $\sqrt {3}< {\mathcal {B}}<+\infty,$ in agreement with the previous results by LZ04. In that case, ${\sigma _{ms}}/{\varepsilon }$ increases from $0$ at ${\mathcal {B}}=\sqrt {3}$ to $1/4$ as ${\mathcal {B}}\rightarrow +\infty.$

For $({\mathcal {B}},{\mathcal {N}}) \in ([\sqrt {3},+\infty {[} \times [0,+\infty [)\cup {\mathcal {D}}_f,$ we use (3.26a,b) to deduce from (3.27) the following limits

(3.29a,b)\begin{equation} \lim_{{\mathcal{N}}>0,\ {\mathcal{B}}\rightarrow +\infty} \frac{\sigma_{ms}}{\varepsilon}=\frac{1}{2}, \quad \lim_{{\mathcal{B}}>1,\ {\mathcal{N}}\rightarrow +\infty} \frac{\sigma_{ms}}{\varepsilon}=\frac{1}{2}. \end{equation}

Indeed, when ${\mathcal {B}}\gg 1$ and ${\mathcal {B}}\gg {\mathcal {N}}>0,$ an equivalent form for ${\sigma _{ms}}/{\varepsilon }$ can be written as

(3.30)\begin{equation} \frac{\sigma_{ms}}{\varepsilon}\underset{{\mathcal{B}}\rightarrow +\infty}{\sim}\frac{\left (3-{\mathcal{B}}^2\mu^2\right)\left (1+{\mathcal{B}}^2\mu^2\right)}{8} \frac{\left (1-{\mathcal{N}}^2-{\mathcal{B}}^2\mu^2\right)} {\left (2-{\mathcal{N}}^2-2{\mathcal{B}}^2\mu^2\right)} \underset{{\mathcal{B}}\rightarrow +\infty}{\longrightarrow}\frac{1}{2} \end{equation}

because $\lim _{{\mathcal {B}}\rightarrow +\infty }\mu ^2{\mathcal {B}}^2=1.$ When ${\mathcal {N}}\gg {\mathcal {B}}>1,$ an equivalent form for ${\sigma _{ms}}/{\varepsilon }$ can be written as

(3.31)\begin{equation} \frac{\sigma_{ms}}{\varepsilon}\underset{{\mathcal{N}}\rightarrow +\infty}{\sim}\frac{\left (3-{\mathcal{B}}^2\mu^2\right)\left (1+{\mathcal{B}}^2\mu^2\right)}{8} \underset{{\mathcal{N}}\rightarrow +\infty}{\longrightarrow}\frac{1}{2} \end{equation}

because $\lim _{{\mathcal {N}}\rightarrow +\infty }\mu ^2{\mathcal {B}}^2=1.$

It follows that, for $({\mathcal {B}},{\mathcal {N}}) \in [\sqrt {3},+\infty {[} \times [0,+2]$ and when ${\mathcal {N}}$ is fixed, ${\sigma _{ms}}/{\varepsilon }$ increases from $0$ at ${\mathcal {B}}=\sqrt {3}$ to $0.5$ as ${\mathcal {B}}\rightarrow +\infty.$ This is illustrated by figure 6 which shows ${\sigma _{ms}}/{\varepsilon }$ versus ${\mathcal {B}}$ for ${\mathcal {N}}=0, 0.5, 1$ and ${\mathcal {N}}= 2$. By comparing (3.28) and (3.31), we can remark that, in this case, the ${\mathcal {N}}\rightarrow 0$ limit is, in fact, singular (discontinuous). Consequently, we can conclude that in the presence of the stratification with ${\mathcal {N}}\leq 2$ the IS instability is reinforced because ${\sigma _{ms}}/{\varepsilon }$ tends towards $1/ 2$ like ${\mathcal {B}}\rightarrow +\infty,$ whereas without stratification $({\mathcal {N}} = 0),$ ${\sigma _{ms}}/{\varepsilon }$ approaches $1/4$.

On the other hand, for $({\mathcal {B}},{\mathcal {N}}) \in {\mathcal {D}}_f\cup ([\sqrt {3},+\infty {[} \,\times\, ]2,+\infty [)$ (this corresponds to the domains V and VI shown in figure 2) and when ${\mathcal {N}}$ is fixed, ${\sigma _{ms}}/{\varepsilon }$ also increases from $\sigma _{mf}(\,f({\mathcal {N}}),{\mathcal {N}})/\varepsilon$ at ${\mathcal {B}}=f({\mathcal {N}})$ to $0.5$ as ${\mathcal {B}}\rightarrow +\infty.$ Indeed, for $({\mathcal {B}}_1,{\mathcal {N}}) \in {\mathcal {D}}_f$ and $({\mathcal {B}}_2,{\mathcal {N}}) \in [\sqrt {3},+\infty {[} \,\times\, ]2,+\infty [,$ we have (see figure 5),

(3.32)\begin{equation} 0\leq\sigma_{mf}\left ({\mathcal{B}}_1,{\mathcal{N}}\right)\leq \sigma_{ms}\left ({\mathcal{B}}_1,{\mathcal{N}}\right)\leq \sigma_{ms}\left ({\mathcal{B}}_2,{\mathcal{N}}\right)<\frac{\varepsilon}{2}. \end{equation}

It clearly appears that in the simultaneous presence of the axial magnetic field and stratification with ${\mathcal {N}}>2,$ the subharmonic instability resulting from the resonances (of order $n=2)$ between two slow modes, which emerges beyond the threshold ${\mathcal {B}}_c=f({\mathcal {N}})<\sqrt {3},$ is dominant with a maximal growth rate approaching $\varepsilon /2$ for large ${\mathcal {B}}.$

Figure 6. Maximal growth rate of the destabilising resonances of order $n=2$ between two slow modes (see (3.27)). The figure shows $\sigma _{ms}/\varepsilon$ versus ${\mathcal {B}}$ for ${\mathcal {N}} = 0.0$, 0.5, 1.0, 2 and ${\mathcal {N}}=3$.

3.5. Destabilising resonances between a fast mode and a slow mode

In this section, we study the effects of the vertical (stable) stratification on the subharmonic instability resulting from the resonances (of order $n=2$) between a fast mode and a slow mode (hereinafter, referred to as IM instability). Without stratification, the IM instability occurs for all magnetic field strengths where its maximal growth rate approaches $\varepsilon /4$ as ${\mathcal {B}}\rightarrow +\infty$ (see LZ04).

The resonance of order $n=2$ between a fast mode and a slow mode, i.e. $\omega _1-\omega _3=2\varOmega$ or $\omega _1-\omega _4=2\varOmega,$ exists for

(3.33)\begin{equation} ({\mathcal{B}} , {\mathcal{N}}) \in {]}0, +\infty{[} \times [0, +\infty[. \end{equation}

In the case where $\omega _1-\omega _4=2\varOmega,$ one has $\mu =1,$ but this resonant case does not induce any instability because the Floquet system (2.28) is stable at $\mu =1,$ as already indicated.

In the case where $\omega _1-\omega _3=2\varOmega,$ we deduce from the algebraic equation (3.11) that the points of the $\mu -$axis characterising the resonances between a fast mode and a slow mode are given by

(3.34)\begin{equation} \mu^2=\frac{\left (4-{\mathcal{N}}^2\right)^2}{ \left (4-{\mathcal{N}}^2\right)^2+16{\mathcal{B}}^2}. \end{equation}

The latter expression implies that, for fixed ${\mathcal {N}}\geq 0,$ the parameter $\mu$ decreases from $1$ at ${\mathcal {B}}=0$ to zero as ${\mathcal {B}}\rightarrow +\infty.$ Inversely, for fixed ${\mathcal {B}}> 0,$ the parameter $\mu$ increases from $\mu =1/\sqrt {1+{\mathcal {B}}^2}$ at ${\mathcal {N}} = 0$ to $1$ as ${\mathcal {N}}\rightarrow +\infty.$

However, according to our asymptotic analysis (see Appendix A), only the resonant cases for which the couple $({\mathcal {B}},{\mathcal {N}})$ belongs to the domain $]0,+\infty {[} \times [0,2[$ (this corresponds to the domains I–IV shown in figure 2), are destabilising. In that case, the maximal growth rate of the IM instability, denoted by $\sigma _{mm},$ is found as (see Appendix A.3.3)

(3.35)\begin{equation} \frac{\sigma_{mm}}{\varepsilon}=\frac{(1-\mu^2)}{16} \frac{\left [(1-\mu^2)(4-{\mathcal{N}}^2)+4{\mathcal{N}}^2\right]}{\sqrt{4+{\mathcal{N}}^2}} \sqrt{\frac{(4-{\mathcal{N}}^2)^3}{(4-{\mathcal{N}}^2)^2+ {\mathcal{B}}^2{\mathcal{N}}^4}}. \end{equation}

It immediately follows that $\sigma _{mm}=0$ for ${\mathcal {N}}=2,$ independently of the magnetic field strength.

In the non-stratified case (${\mathcal {N}}=0),$ (3.35) reduces to

(3.36)\begin{equation} \frac{\sigma_{mm}}{\varepsilon}=\frac{1}{4}\left (1-\mu^2\right)^2= \frac{{\mathcal{B}}^4}{4\left (1+{\mathcal{B}}^2\right)^2} \underset{{\mathcal{B}}\rightarrow +\infty}{\longrightarrow}\frac{1}{4}, \end{equation}

in agreement with the study by LZ04. Therefore, ${\sigma _{mm}}/{\varepsilon }$ increases from $0$ at ${\mathcal {B}}=0$ to approach $1/4$ as ${\mathcal {B}}\rightarrow +\infty.$

However, in the simultaneous presence of the vertical (stable) stratification and the magnetic field and for fixed ${\mathcal {N}}\leq 2,$ ${\sigma _{mm}}/{\varepsilon }$ increases from $0$ to reach its maximum value (which is less than $1/4)$, then it decreases tending towards zero when ${\mathcal {B}}\rightarrow +\infty.$ Also in this case, the ${\mathcal {N}}\rightarrow 0$ limit is, in fact, singular (discontinuous). As an illustration, figure 7 shows ${\sigma _{mm}}/{\varepsilon }$ vs ${\mathcal {B}}$ for ${\mathcal {N}}= 0$, 0.5, 1, 1.5 and ${\mathcal {N}}= 2.$

Figure 7. Maximal growth rate of the destabilising resonances of order $n=2$ between a fast and a slow mode (see (3.35)). The figure shows $\sigma _{mm}/\varepsilon$ versus ${\mathcal {B}}$ for ${\mathcal {N}} = 0.0, 0.5, 1.0$ and ${\mathcal {N}}=1.5$.

We can therefore conclude that the effect of (stable) stratification on the IM instability is to suppress it if ${\mathcal {N}}$ exceeds $2,$ or to make it less efficient otherwise $(0<{\mathcal {N}}<2)$ because ${\sigma _{mm}}$ approaches zero for large ${\mathcal {B}}.$

4.1. Identification of instabilities

For the identification of the instabilities picked up by the numerical procedure we use the resonance conditions described by (3.10) and (3.11) as explained as follows.

On the one hand, for a given value of the couple $({\mathcal {B}},{\mathcal {N}})$, we use (3.15), (3.25) and (3.34) and determine the values of $\mu$, say $\mu _0$, characterising the resonant cases of order $n=2.$ On the other hand, for the same value of the couple $({\mathcal {B}},{\mathcal {N}})$, we consider several values of $\varepsilon$ uniformly distributed in the interval $[0, 1]$ and, for each of these values of $\varepsilon,$ we integrate numerically the Floquet system (2.28) and determine the growth rate $\sigma$ for 2000 values of $\mu$ (evenly distributed) in $]0, 1[.$ Obviously, $\sigma (\mu ) = 0$ if there is no instability. Thus, in the plane $(\mu, \sigma +\varepsilon ),$ the region of instability that emanates from the point of abscissa $\mu _0$ characterises a subharmonic instability. As an illustration, figure 8 shows $\sigma +\varepsilon$ versus $\mu$ for ${\mathcal {B}}=4$ and ${\mathcal {N}}=0.5$ and $100$ values of $\varepsilon$ evenly distributed in the interval $[0,0.8].$ In that case, there are three subharmonic instabilities IF, IM and IS which correspond to the regions of instability emanating from the points $(\mu _0 = 0.1755, 0)$, $(\mu _0 = 0.2282, 0)$ and $(\mu _0 = 3074, 0),$ respectively. The other instabilities appearing in figure 5 are related to higher-order resonances $(n = 4, 6, 8,\ldots ),$ but as they are dominated by the subharmonic instabilities, we do not seek to identify them one by one.

Figure 8. Magneto-gravity elliptic instabilities. The figure shows $\sigma +\varepsilon$ versus $\mu$ for ${\mathcal {B}}=4$ and ${\mathcal {N}}=0.5$ and $100$ values of $\varepsilon$ evenly distributed in the interval $[0,0.8].$ The regions of instability labelled by $f$, $m$ and $s$ denote the IF, IM and IS instabilities that emanate from the points of the $\mu$-axis of abscissa $0.1775, 0.2282$ and $0.3074,$ respectively. The other instabilities picked up by the numerical procedure are related to higher-order resonances $(n = 4, 6, 8,\ldots )$: for $n=4$, $\mu =0.3836$ (fast, fast), $\mu =0.4792$ (fast, slow), $\mu =0.6359$ (slow, slow); for $n=6,$ $\mu =0.5821$ (fast, fast), $\mu =0.7252$ (fast, slow).

4.2. Comparison between the asymptotic formulae and the numerical results

We recall that the present asymptotic results (at leading order in $\varepsilon )$ clearly shows that subharmonic instability exists when

(4.1)\begin{equation} ({\mathcal{B}} , {\mathcal{N}}) \in \left ([0, +\infty{[} \times [0, 2]\right)\cup ([\sqrt{3}+\infty{[} \times [2,+\infty[)\cup {\mathcal{D}}_f. \end{equation}

Obviously, the procedure leading to the asymptotic formulae excludes the instabilities related to higher-order resonances $(n = 4, 6, 8,\ldots ).$ Numerical calculations indicate that, when $({\mathcal {B}} , {\mathcal {N}})$ belongs to domains I–VI (see figure 2) deprived of a very narrow band which are specified below, one of the subharmonic instabilities listed in § 3 is dominant.

In figure 9, we show the continuous variation of the maximal growth rate $\sigma _m$ (maximum $\sigma$ over $0 \leq \mu \leq 1$) of the dominant instability normalised by $\sigma _0= 9/16$ plotted as a function of $0\leq {\mathcal {B}}\leq 4$ and $0\leq {\mathcal {N}}\leq 4.$ Figure 9(a) displays the numerical results for $\varepsilon = 0.1,$ where the grid consists of 201 points evenly distributed in each one of the ranges $0\leq {\mathcal {B}}\leq 4$ and $0\leq {\mathcal {N}}\leq 4.$ Figure 9(a) shows the analytical results for $\sigma _m/\sigma _0$ where

(4.2)\begin{equation} \sigma_m=\max(\sigma_{mf},\sigma_{ms},\sigma_{mm}). \end{equation}

Recall that $\sigma _{mf},$ $\sigma _{ms}$ and $\sigma _{mm}$ are given by equations (3.18), (3.27) and (3.35), respectively.

Figure 9. Magneto-gravity elliptic instabilities. Maximal growth rate of dominant instability normalised by $\sigma _0 = 9/16$ plotted as a function of $0\leq {\mathcal {B}}\leq 4$ and $0\leq {\mathcal {N}}\leq 4.$: (a) numerical results for $\varepsilon = 0.1$; (b) asymptotic analysis results.

As can be seen, the agreement between the numerical results and the asymptotic formulae is quite good except for $({\mathcal {B}} , {\mathcal {N}})$ belonging to a narrow band around ${\mathcal {N}}= 3$ and ${{\mathcal {B}}\succsim f({\mathcal {N}} = 3) = 1.6035}.$ This can also be observed more quantitatively from figure 10 which shows $\sigma _m/\varepsilon$ versus ${\mathcal {N}}$ for some selected values of ${\mathcal {B}} = 1$ (figure 10a), ${\mathcal {B}} = 2$ (figure 10b), ${\mathcal {B}} = 3$ (figure 10c) and ${\mathcal {B}} = 4$ (figure 10d).

Figure 10. Magneto-gravity elliptic instabilities. Maximal growth rate of the dominant instability normalised by $\varepsilon$ versus ${\mathcal {N}}$ for selected values of ${\mathcal {B}} = 1$ (figure 10a), ${\mathcal {B}} = 2$ (figure 10b), ${\mathcal {B}} = 3$ (figure 10c) and ${\mathcal {B}} = 4$ (figure 10d). The figure compares the asymptotic formulae with the numerical results at $\varepsilon = 0.1$. The subharmonic instability resulting from a resonance between two fast (respectively, slow) modes is labelled by IF (respectively, IS), whereas that resulting from a resonance between a fast and a slow mode is labelled by IM. From figure 10(a), we observe that an instability related to higher-order resonances (IHOR) is present for ${\mathcal {N}}> 3.7$.

A closer examination of the regions of instability in the plane $(\mu, \sigma +\varepsilon )$ for a given $({\mathcal {B}}, {\mathcal {N}})$ belonging to the narrow band around ${\mathcal {N}} = 3$ reveals that beyond $\varepsilon \approx 0.1$ the subharmonic instability resulting from the resonances between two slow modes disappears and it is the instability related to resonances of order $n=4$ between a fast mode and a slow mode which becomes dominant. This is illustrated by figure 11 which shows $\sigma +\varepsilon$ vs $\mu$ for ${\mathcal {N}} = 3.1$ and ${\mathcal {B}} =\sqrt {3}.$ It can be observed that, before disappearing, the subharmonic instability coalesces with the instability related to the resonances of order $n=6$ between two slow modes. Recall that for the identification of the different regions of instability we use (3.10) and (3.11).

Figure 11. Magneto-gravity elliptic instabilities. The figure shows $\sigma +\varepsilon$ versus $\mu$ for ${\mathcal {N}} = 3.1$ and ${\mathcal {B}} =\sqrt {3},$ and 100 values of $\varepsilon$ regularly distributed in the interval $0 \leq \varepsilon \leq 0.15$. The subharmonic instability IS emanates from the point $(0.6589, 0)$ and disappears beyond $\varepsilon \approx 0.1.$ Before disappearing, the IS instability coalesces with the instability associated with a resonance of order $n = 6$ between two slow modes which emanates from the point $(0.5262, 0).$ The region of instability emanating from the point $(0.7168, 0)$ corresponds to the instability related to resonances of order $n = 4$ between a fast mode and a slow mode.

Similar conclusions are drawn from the analysis of the numerical results for $0.1 < \varepsilon$ in the sense that, for $({\mathcal {B}} , {\mathcal {N}})$ belonging to one of the domains I–VI, the dominant instability corresponds to one of the subharmonic instabilities listed in § 3. As for the agreement between the asymptotic formulae and the numerical results at $\varepsilon >0.1,$ it is not as satisfactory as in the case of weak ellipticity $(\varepsilon \precsim 0.1).$ Indeed, the difference between the numerical results and the asymptotic formulae increases as $\varepsilon$ increases especially for $({\mathcal {B}} , {\mathcal {N}})$ belonging to the domains V and VI. This is illustrated by figure 12 that displays $\sigma _m/\varepsilon$ versus ${\mathcal {N}}$ for ${\mathcal {B}} = 4$ and five values of $\varepsilon = 0.1$, 0.2, 0.3, 0.4 and $\varepsilon = 0.5.$

Figure 12. Magneto-gravity elliptic instabilities. Maximal growth rate of dominant instability normalised by $\varepsilon$ versus ${\mathcal {N}}$ for $\varepsilon = 0.1$, 0.2, 0.3, 0.4, 0.5 and ${\mathcal {B}} = 4$.

We point out that the present numerical computations, as well as the asymptotic analysis, do not detect any instabilities for $2 < {\mathcal {N}} < 3$ and $0 \leq {\mathcal {B}} \leq f({\mathcal {N}}=3) = 1.6035.$ As an illustration, figure 13 shows the continuous variation of the maximal growth rate $\sigma _m$ of the dominant instability normalised by $\sigma _0=9/16$ plotted as a function of $0\leq {\mathcal {B}}\leq 4$ and $0\leq {\mathcal {N}}\leq 4$ for $\varepsilon = 0.5$ (figure 13a) and $\varepsilon =1$ (figure 13b).

Figure 13. Magneto-gravity elliptic instabilities. Maximal growth rate of dominant instability normalised by $\sigma _0 = 9/16$ plotted as a function of $0\leq {\mathcal {B}}\leq 4$ and $0\leq {\mathcal {N}}\leq 4$: (a) $\varepsilon = 0.5$; (b) $\varepsilon = 1$.

4.3. Accounting for diffusivity, in the simplest case

As indicated in the introduction, Singh & Mathur (Reference Singh and Mathur2019) investigated the effects of differential diffusion between momentum and density ($Sc=\kappa /{\nu })$ in their local stability analysis for an elliptical vortex with a uniform stable stratification along the vorticity axis. They showed that in the case where $Sc=\kappa /{\nu }=1$ viscous effects are purely suppressive, whereas for sufficiently small $Sc < 1,$ there is an oscillatory instability the signature of which is nevertheless present with zero growth rate in the inviscid limit.

The study of the effects of differential diffusion in the case of an elliptical vortex with a uniform stable stratification and a uniform magnetic field involves three dimensionless numbers, namely, the Reynolds number and the thermal and magnetic Prandtl numbers, in addition to the parameters $\varepsilon,$ ${\mathcal {B}}$ and ${\mathcal {N}}.$ This requires a detailed study and is beyond the scope of the present work. For instance, we consider the very special case where the diffusion coefficients are equal, ${\nu }=\kappa =\eta.$

In that case, the dispersion relation of the MIG waves is obtained by replacing $\omega$ in (3.6) by $(\omega -{\nu } k^2),$ and the Fourier amplitudes of the velocity, magnetic field and buoyancy scalar perturbations may be associated with those in the inviscid limit $\hat {\boldsymbol {u}}, \hat {\boldsymbol {b}}$ and $\hat {\vartheta }$ by the substitution (Cambon et al. Reference Cambon, Teissedre and Jeandel1985; Landman & Saffman Reference Landman and Saffman1987)

(4.3)\begin{equation} \left (\hat{\boldsymbol{u}}^{(v)}, \hat{\boldsymbol{b}}^{(v)}, \hat{\vartheta}^{(v)}\right)=\left (\hat{\boldsymbol{u}}, \hat{\boldsymbol{b}}, \hat{\vartheta}\right)\exp\left (-{\nu}\int_0^tk^2(s)\,{\rm d}s\right). \end{equation}

Accordingly, the maximal growth rate of the dominant instability, if attainable, is

(4.4)\begin{equation} \sigma_m^{(v)}=\sigma_m-Re^{-1}k_0^2L_0^2\left (1+\frac{(E^2-1)(1-\mu_m^2)}{2}\right), \end{equation}

where $Re=\varOmega L_0^2/{\nu }$ is the Reynolds number, $L_0$ is a characteristic length scale and $\mu _m$ is the value of $\mu$ where $\sigma _m$ occurs. At $L_0 k_0 \sim 1,$ the dominant instability survives the diffusive effects if $Re > Re_c,$ where

(4.5)\begin{equation} Re_c=\frac{2+(E^2-1)(1-\mu_m^2)}{2\sigma_m}. \end{equation}

Figure 14 shows the variation of $Re_c$ vs ${\mathcal {N}}$ for $E = 1.5$ (so that $\varepsilon = 0.41667)$ and ${\mathcal {B}} = 1, \sqrt {3}, 3$ and ${\mathcal {B}} = 4.$ It appears that in the case where ${\mathcal {B}}\geq \sqrt {3},$ the dominant instability survives the diffusion effects for $Re_c\sim 50$ and $({\mathcal {B}}, {\mathcal {N}} )\in [\sqrt {3},+\infty {[} \times [0,+\infty [.$ As there are no instabilities when $2 < {\mathcal {N}} < 3$ and $0 < {\mathcal {B}} < f({\mathcal {N}})=1.6035,$ the critical Reynolds number $Re_c$ takes large values for ${\mathcal {N}}$ in the left (respectively, right) neighbourhood of ${\mathcal {N}} = 2$ (respectively, ${\mathcal {N}} = 3).$ This is the case for ${\mathcal {B}}=1$ that we observe in figure 14.

Figure 14. Case where the diffusivity coefficients (kinematic, magnetic and thermal) are equal. Variation of the critical Reynolds number $Re_c$ versus ${\mathcal {N}}$ for $E = 1.5$ (therefore $\varepsilon = 0.41667)=$ and ${\mathcal {B}} = 1$, $\sqrt {3}$, $3$ and ${\mathcal {B}} = 4.$ Dominant instability can survive the effects of diffusion, provided $Re > Re_c$.