2.1. The model and its 1.5D reduction

The equations of the DMRSW model in the rotating $(x,\,y)$ plane read (Dellar Reference Dellar2003):

(2.1)$$\begin{gather} D_{t} \boldsymbol{v} + f \hat{\boldsymbol{z}} \wedge \boldsymbol{v}+ g \boldsymbol{\nabla} h = \frac{1}{h} \boldsymbol{\nabla} ( h \boldsymbol{b} \otimes \boldsymbol{b} ) + \frac{1}{3h} \boldsymbol{\nabla} [-h^{2} D^{2}_{t}h + h^{3} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b})^{2} - h^{3} \boldsymbol{b}\boldsymbol{\cdot} \boldsymbol{\nabla} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b})], \end{gather}$$

(2.2)$$\begin{gather}D_{t}h + h {\boldsymbol{\nabla}} \boldsymbol{\cdot} \boldsymbol{v} = 0, \end{gather}$$

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

(2.4)$$\begin{gather}D_{t} \boldsymbol{b} = \frac{1}{h} \boldsymbol{\nabla} ( h \boldsymbol{v} \otimes \boldsymbol{b} ) , \end{gather}$$

where $D_{t} =\partial _{t}\boldsymbol {v} + \boldsymbol {v}\boldsymbol{\cdot}\boldsymbol {\nabla }$ is the material (Lagrangian) derivative, $f$ is the Coriolis parameter and $g$ is the gravity acceleration. (We used (2.2) to replace $h {\boldsymbol {\nabla }} \boldsymbol{\cdot} \boldsymbol {v}$ by $-D_{t}h$ in (2.1).) The dynamical variables of the model are the thickness of the layer $h$, the velocity $\boldsymbol {v} = u\hat {\boldsymbol {x}} + v \hat {\boldsymbol {y}}$ and the magnetic field $\boldsymbol {b}= a\hat {\boldsymbol {x}} + b \hat {\boldsymbol {y}}$ in the plane, where $(\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}, \hat {\boldsymbol {z}})$ are unit vectors along the respective axes, $\boldsymbol {\nabla } = \hat {\boldsymbol {x}} \partial _{x}+ \hat {\boldsymbol {y}} \partial _{y}$. The notation $\boldsymbol {\nabla } ( \boldsymbol{\mathsf{A}} \otimes \boldsymbol{\mathsf{B}} )$ is a shorthand for tensor notation: the $i$th component of such an expression is given by $\partial _{j} A_i B_j$, with summation over repeated indices from 1 to 3. We will work in the $f$-plane approximation, where Coriolis parameter $f$ is constant. According to the standard in geophysics conventions, $x$ and $y$ coordinates will be called zonal and meridional, respectively, and the components of the vector fields will be called correspondingly. The difference between DMRSW and (magneto)hydrostatic MRSW models resides in the last term on the right-hand side of (2.1), which is absent in the latter model.

The 1.5D reduction of this system consists in eliminating all dependence on one of the spatial coordinates, which we choose to be the zonal one, in a way that $\partial _{x}(\ldots ) \equiv 0$ for any quantity. It is easy to see that, as follows from (2.3) and (2.4), under this hypothesis, the meridional component of the magnetic field ceases to be independent:

(2.5)\begin{equation} hb = B H = \text{const.}\ \Rightarrow\ b = \frac{B H}{h}, \end{equation}

where $H$ is the mean thickness of the layer and $B$ is a mean meridional magnetic field. In fact, as magnetic fields in the DMRSW system, as it is written in (2.1)–(2.4), have dimensions of velocity, $B$ is at the same time the Alfvén wave speed $c_{a}$. Using (2.5) , the 1.5D DMRSW is written, in components, as

(2.6)\begin{align} \left.\begin{array}{c@{}} D_{t} u - f v =\displaystyle \dfrac{B}{h} a_{y},\\ D_{t}v + f u + g h_{y}= \displaystyle \dfrac{B}{h} \left(\displaystyle \dfrac{B}{h} \right)_{y} + \displaystyle \dfrac{1}{3 h} \left[ - h^{2}D^{2}_{t} h + h \left( \left[\left(\displaystyle \dfrac{B}{h} \right)_{y}\right]^{2} - \displaystyle \dfrac{B}{h} \left(\displaystyle \dfrac{B}{h} \right)_{yy} \right)\right]_{y},\\ D_{t} a = \displaystyle\dfrac{B}{h} u_{y},\\ D_{t}h + (v h)_{y} = 0, \end{array} \right\} \end{align}

where we used the shorthand notation $\partial _{y}(\ldots ) \equiv (\ldots )_{y}$, and from now on, $D_{t}= \partial _{t} + v \partial _{y}$. An analogous system, with the only difference that variables were not depending on $y$ instead of $x$, was used by Dellar (Reference Dellar2003) for a study of the linear wave spectrum. We should stress that in the $f$-plane approximation we are using, the orientation of the coordinate axes is arbitrary, so $y$ could also be considered as a zonal coordinate. In this case, $v$ becomes the zonal velocity, $-u$ becomes meridional velocity, and $b$ and $-a$ are zonal and meridional components of the magnetic field. Then, in the absence of a magnetic field, the resulting system becomes the one considered by Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003). Let us also mention that the $f$-plane approximation in (2.6) can be relaxed and the Coriolis parameter considered as a linear function of $y$: $f = f_{0} + \beta y$. The beta-plane approximation is appropriate for large-scale motions, which are sensible to the planet/star curvature. Non-hydrostatic effects are unimportant in such a case, so maintaining them in the equations of motion is not justified. As our goal is to study the simultaneous influence of non-hydrostatic and rotation effects, and in addition rotation will be considered weak in § 4, we will stick to the $f$-plane in what follows.

2.2. Lagrangian formulation of the 1.5D DMRSW model and scaling

Let us recall that in the Lagrangian description of the 1.5D RSW model (Zeitlin et al. Reference Zeitlin, Medvedev and Plougonven2003), the motion of the fluid layer is described in terms of the trajectories of Lagrangian parcels – fluid columns of variable depth $h$ with current positions $Y(y_{0},\tau )$ depending on the initial position $y_{0}$, the Lagrangian label, and on time $\tau$. The fluid motion, thus, is a time-dependent mapping of the domain of the flow, the whole $y$-axis in the present case, onto itself. According to the principle of Euler–Lagrange duality in hydrodynamics, the Lagrangian velocity of the fluid parcels $\partial _{\tau }Y \equiv \dot {Y}$ is the Eulerian velocity $v$ measured at the point $y = Y$. From now on, we will be using the dot and the prime notation respectively for the derivatives of Lagrangian variables with respect to time and with respect to space labels. The mass conservation in Lagrangian terms is just the conservation of the volume of the parcel, i.e. its area in the 1.5D case:

(2.7)\begin{equation} h(Y)\,{\rm d}Y = h_{0} \,{\rm d}y_{0}, \end{equation}

where $h_{0}$ is an initial thickness distribution. In what follows, we identify the above-used Eulerian coordinate with $Y$ and reserve the notation $y$ for the so-called mass-weighted coordinate (label) which is obtained from $y_{0}$ by a change of variables ‘straightening’ the initial thickness: $h_{0} \rightarrow H = \textrm {const.}$, see Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003). Thus,

(2.8)\begin{equation} h(Y)\,{\rm d}Y = H\,{\rm d}\kern0.05em y,\quad \rightarrow\quad h(Y) = \frac{H}{J},\quad J := \frac{\partial Y}{\partial y} \equiv Y', \end{equation}

where we introduced the Jacobian $J$ of the Lagrangian mapping. Notice that, as follows from the definition (2.8),

(2.9)\begin{equation} \dot{Y} = v(y,\tau). \end{equation}

We are now ready to rewrite (2.6) in Lagrangian form. We should emphasize that a Lagrangian description of the full (non-dissipative) three-dimensional MHD is known, and had been used in the seminal paper of Newcomb (Reference Newcomb1962) to construct an action principle for MHD. We should also mention in passing that the DMRSW equations could be directly obtained from this action by using mass-weighted coordinates, restricting vertical integration to a layer of finite depth and imposing the hypothesis of columnar motion, similar to the procedure leading to rSGN equations from the action principle for non-dissipative hydrodynamics, cf. e.g. Dellar & Salmon (Reference Dellar and Salmon2005). We, however, do not need such full machinery in the present 1.5D situation. There is no Lagrangian motion in the transverse direction, so it is sufficient to make a change of variables $Y \rightarrow y$ in the zonal fields $u$ and $a$, similar to what was done in ‘pure’ 1.5D RSW (Zeitlin et al. Reference Zeitlin, Medvedev and Plougonven2003), while the Lagrangian expression of the meridional magnetic field is already given by (2.5) and (2.8). The Lagrangian equations of the DMRSW model thus read:

(2.10)\begin{equation} \left.\begin{array}{c@{}} \ddot{Y} + f u + \displaystyle\dfrac{g H}{Y'} \left( \dfrac{1}{Y'}\right)' - c^{2}_{a}Y'' = \dfrac{H^{2}}{3} \left[ - \dfrac{1}{Y'^{2}}\ddot{\left( \dfrac{1}{Y'}\right)} + \dfrac{c^{2}_{a}}{Y'^{2}} \left( \dfrac{1}{Y'}\right)''\right]',\\ \dot{u} - f \dot{Y} - c_{a} a'=0,\\ \dot{a} - c_{a} u' = 0. \end{array} \right\} \end{equation}

The Lagrangian equations of the MRSW model are the same, modulo the term on the right-hand side of the first equation which should be set to zero in this case.

Notice that if the $f$-plane approximation is relaxed as explained earlier, the substitution $f \rightarrow f_{0} + \beta Y$ should be made in (2.10), with $f_{0} \equiv 0$ on the equatorial beta plane.

It is useful for what follows to rewrite this system in terms of variables $u,\ v,\ J, a$:

(2.11)\begin{equation} \left.\begin{array}{c@{}} \dot{v} + f u - \displaystyle \dfrac{g H}{J^{3}} J' - c^{2}_{a}J' = \dfrac{H^{2}}{3} \left[ - \dfrac{1}{J^{2}}\ddot{\left( \dfrac{1}{J}\right)} + \dfrac{c^{2}_{a}}{J^{2}} \left( \dfrac{1}{J}\right)''\right]',\\ \dot{J} - v' = 0,\\ \dot{u} - f v - c_{a} a'=0,\\ \dot{a} - c_{a} u' = 0. \end{array} \right\} \end{equation}

Notice that if $c_{a}\equiv 0$ and non-hydrostatic effects (the right-hand side of the first equation of each system) are neglected, (2.10) and (2.11) become the Lagrangian equations for 1.5D RSW obtained by Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003) (only there, they were equations for zonal and not meridional motion). At the same time, setting $c_{a}\equiv 0$ and keeping the non-hydrostatic terms gives Lagrangian formulation of the rSGN model. A crucial difference of (2.10) with (2.11) is the non-conservation of the geostrophic momentum $M = u - f\kern0.07em Y$ which, in turn, means non-conservation of potential vorticity, a known fact for MRSW and DMRSW (Dellar Reference Dellar2003) which becomes obvious in Lagrangian description. The conservation of the geostrophic momentum allowed Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003) to express the transverse velocity in terms of the Lagrangian position and thus obtain, without any approximations, a closed nonlinear second-order in time equation for the latter, which contained the entire evolution of the system. In the present case, the extra variables can be also eliminated in favour of $Y$, but due to the non-conservation of the geostrophic momentum, the resulting equation is of the fourth order in time. Indeed, cross-differentiating the second and the third equations in (2.10) gives

(2.12)\begin{equation} \ddot{u} - c^{2}_{a}u'' = f \ddot{Y}, \end{equation}

and, therefore, $u$ can be eliminated from the equation for $Y$ by additional differentiations.

Let us proceed with the scaling of the resulting equations. For our purposes, we use the ‘wave’ scaling based on the horizontal scale $L$, vertical scale $H$ and the velocity scale based on the maximal velocity of gravity waves:

(2.13a–c)\begin{equation} y \sim L,\quad u \sim \sqrt{g H}, \quad t \sim L/\sqrt{g H}. \end{equation}

Two characteristic non-dimensional parameters can be built from these scales and the Coriolis parameter $f$:

(2.14a,b)\begin{equation} \gamma = Bu^{-1/2} = \frac{f\kern0.06em L}{\sqrt{g H}}, \quad \delta= \frac{H}{3 L}, \end{equation}

the first being the non-dimensional Coriolis parameter, and at the same time the inverse square root of the Burger number, and the second being the effective shallowness of the layer. They control respectively the rotation and non-hydrostatic effects. It is also helpful, cf. Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003), to introduce the deviations $\phi (\tau, y)$ of the Lagrangian parcels from their initial positions: $Y = y + \phi$. The system of non-dimensional equations for $\phi$ and $u$ reads:

(2.15)$$\begin{gather} \ddot{\phi} + \gamma u - \frac{\phi''}{(1+ \phi')^{3}} - c^{2}_{a}\phi'' =-\delta^{2} \left\{\frac{1}{(1+ \phi')^{2}}\left[ \ddot{\left( \frac{1}{1+ \phi'}\right)} - c^{2}_{a} \left(\frac{1}{1+ \phi'}\right)''\right]\right\}', \end{gather}$$

(2.16)$$\begin{gather}\ddot{u} - c^{2}_{a}u'' = \gamma \ddot{\phi}, \end{gather}$$

and the passage between DMRSW and MRSW models is achieved by switching on/off the parameter $\delta$. The last step consists in eliminating $u$ by applying the wave operator $\partial ^{2}_{\tau \tau } - c^{2}_{a} \partial ^{2}_{y y}$ to (2.15), which gives the following closed equation for $\phi$:

(2.17)\begin{align} &(\partial^{2}_{\tau \tau} - c^{2}_{a} \partial^{2}_{y y}) \left( \ddot{\phi} - \frac{\phi''}{(1+ \phi')^{3}} - c^{2}_{a}\phi'' + \delta^{2} \left[ \frac{1}{(1+ \phi')^{2}} ( \partial^{2}_{\tau \tau} - c^{2}_{a} \partial^{2}_{y y} ) \left(\frac{1}{1+ \phi'}\right)\right]' \right)\nonumber\\ &\quad + \gamma^{2} \ddot{\phi} = 0, \end{align}

and we recall that $\dot {(\ldots )} \equiv \partial _{\tau } (\ldots ),\, (\ldots )' \equiv \partial _{y}$.

We should stress that no approximation was made in obtaining the ‘master’ equation (2.17), which is equivalent, up to zero modes of the wave operator, to the original system of equations and contains the full information on the evolution of the system starting from an arbitrary set of initial conditions for $Y$, and equivalently, for $\phi$. Let us mention that the initial value of $Y$ is $y$, by construction, the inverse of the initial value of $Y'$ gives initial $h$, and therefore initial $b$, initial value of $\dot {Y}$ gives initial $v$, and combinations of initial second derivatives of $Y$ allow to determine initial $u$ and $a$. For example, magneto-geostrophic adjustment of arbitrary initial perturbations, which was studied numerically in the framework of the standard MRSW by Zeitlin et al. (Reference Zeitlin, Lusso and Bouchut2015), that is, in the limit $\delta \rightarrow 0$ of the system (2.1)–(2.4), can be studied solely with (2.17), although such analysis is out of the scope of the present paper. We will use (2.17) both for development of the perturbation theory in wave amplitudes and for finding exact nonlinear wave solutions in § 5, but will start in the next section with finding conditions of shock formation in the non-dispersive limit.

4.1. Linear waves

We now proceed with analysis of small-amplitude wave motions in MRSW and DMRSW. For this, we introduce the parameter $\epsilon$ measuring the wave amplitude, by rescaling $\phi \rightarrow \epsilon \phi$ in (2.17). We are interested in the regimes of weak rotation, that is, $\gamma \rightarrow 0$, and weak non-hydrostaticity, that is, $\delta \rightarrow 0$, so we will consider that the parameters $\gamma ^{2}$ and $\delta ^{2}$ entering (2.17) are of the order of $\epsilon$ or smaller. Solutions of (2.17) are sought in the form

(4.1)\begin{equation} \phi = \epsilon \phi_{0} + \epsilon^{2} \phi_{1}. \end{equation}

Linear waves arise in the leading order in $\epsilon$ of (2.17):

(4.2)\begin{equation} \left( \frac{\partial^{2}}{\partial \tau^{2}} - c^{2}_{a} \frac{\partial^{2}}{\partial y^{2}}\right) \left( \frac{\partial^{2}\phi}{\partial \tau^{2}} - (1+ c^{2}_{a})\frac{\partial^{2}\phi}{\partial y^{2}} \right) = 0. \end{equation}

Although there are only two independent variables $\tau$ and $y$ in the system, it is convenient to introduce four of their linear combinations, the characteristic variables with corresponding characteristic velocities $c_{a}$ and $c = \sqrt {1 + c^{2}_{a}}$ for each kind of waves, as follows:

(4.3a,b)\begin{equation} \xi_{{\pm}} = y \pm c_{a} \tau, \quad \chi_{{\pm}} = y \pm c \tau. \end{equation}

Notice that the characteristic velocity $c$ is obtained from the eigenvalue $\mu ^{(1)}$ of the characteristic matrix of the previous section by linearization.

Equation (4.2) and its general solution, modulo a constant which can be excluded by imposing decay boundary conditions on the $y$-axis, which we suppose hereafter, are

(4.4)\begin{equation} \frac{\partial^{4} \phi_{0}}{\partial \xi_+ \partial \xi_- \partial \chi_+ \partial \chi_-} = 0\ \Rightarrow\ \phi_{0} = f_+(\chi_+) + f_-(\chi_-) + g_+(\xi_+) + g_-(\xi_-), \end{equation}

where $f_{\pm }$ and $g_{\pm }$ are arbitrary (localized) functions of their arguments. The physical meaning of the thus obtained solution is obvious: it represents wave packets of non-dispersive Alfvén and magneto-gravity waves moving leftward or rightward along the $y$-axis with phase velocities $c_{a}$ and $c$, and with envelope functions $g_{\pm }$ and $f_{\pm }$, respectively.

4.2. Weakly nonlinear waves with dispersive corrections

Nonlinear and dispersive corrections to these linear wave solutions, as usual, lead to a modulation of their amplitudes. To take this effect into account and following the standard recipes, we introduce a slow-time dependence of the amplitude: $\phi _{0} = \phi _{0}(\tau, T, y)$, with $T = O(\epsilon ^{-1}\tau )$, and thus make a replacement

(4.5)\begin{equation} \partial_{\tau} \rightarrow \partial_{\tau} + \epsilon \partial_{T} \end{equation}

in (4.2). We recall that we supposed that $\gamma ^{2}, \delta ^{2} = O(\epsilon )$. For easy tracking of the terms due to rotation and non-hydrostaticity, respectively we put

(4.6a,b)\begin{equation} \gamma^{2} = \epsilon \bar{\gamma},\quad \delta^{2} = \epsilon \bar{\delta}, \end{equation}

with $\bar {\gamma }, \bar {\delta }$ being either one or zero. We rewrite (2.17) with these changes:

(4.7) \begin{align} &( ( \partial_{\tau} + \epsilon \partial_{T} ) ^{2} - c^{2}_{a} \partial^{2}_{y y} ) \left[ ( \partial_{\tau} + \epsilon \partial_{T} ) ^{2}(\epsilon\phi_{0} + \epsilon^{2} \phi_{1}) - \left( \frac{1}{(1+ (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})')^{3}} + c^{2}_{a}\right) \right.\nonumber\\ &\quad \,\times (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})''+\epsilon \bar{\delta} \left[ \frac{1}{(1+ (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})')^{2}}\left( ( \partial_{\tau} + \epsilon \partial_{T} ) {\left( \frac{( \partial_{\tau} + \epsilon \partial_{T} ) (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})'}{(1+ (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})')^{2}}\right)}\right. \right.\nonumber\\ &\quad \!\left. \left.\left. -\,c^{2}_{a} \left(\frac{( (\epsilon\phi_{0} + \epsilon^{2} \phi_{1} ) '}{(1+ (\epsilon\phi_{0} + \epsilon^{2} \phi_{1})')^{2}}\right)'\right)\right]' \right]' +\epsilon \bar{\gamma} ( \partial_{\tau} + \epsilon \partial_{T} ) ^{2 }(\epsilon\phi_{0} + \epsilon^{2} \phi_{1}) = 0, \end{align}

where we kept, for compactness, the prime notation for differentiation with respect to $y$ where it does not lead to confusion.

The terms of the first order in $\epsilon$ in (4.7) give linear wave equation (4.2) for $\phi _{0}$, with solutions (4.4). Collecting the terms of the second order in $\epsilon$ gives

(4.8) \begin{align} &\big( \partial^{2}_{ \tau^{2}} - c^{2}_{a} \partial^{2}_{y^{2}} \big) \big( \partial^{2}_{\tau^{2}} - \big(1+ c^{2}_{a}\big)\partial^{2}_{y^{2}} \big)\phi_{1} \nonumber\\ &\quad =- 2 \big[\big( \partial^{2}_{\tau^{2}} - c^{2}_{a}\partial^{2}_{y^{2}} \big) +\big( \partial^{2}_{\tau^{2}} - (1+ c^{2}_{a})\partial^{2}_{y^{2}} \big)\big]\partial^{2}_{\tau T}\phi_{0} - \big( \partial^{2}_{ \tau^{2}} - c^{2}_{a} \partial^{2}_{y^{2}} \big) 3 \phi'_{0}\phi''_{0}\nonumber\\ &\qquad -\bar{\delta} \big( \big( \partial^{2}_{\tau^{2}} - c^{2}_{a} \partial^{2}_{y^{2}} \big) \phi''_{0} \big) '' - \bar{\gamma} \partial^{2}_{\tau^{2}} \phi_{0}. \end{align}

We will now rewrite this equation using the characteristic variables introduced in (4.3) and the linear wave solution (4.4) for $\phi _{0}$, where the functions $f_{\pm }$ and $g_{\pm }$ are now supposed to depend on the slow time $T$, in addition to their characteristic arguments. We first replace the wave operators $( \partial ^{2}_{ \tau ^{2}} - c^{2}_{a} \partial ^{2}_{y^{2}} )$ and $( \partial ^{2}_{\tau ^{2}} - (1+ c^{2}_{a})\partial ^{2}_{y^{2}} )$ by their expressions in terms of $\chi _{\pm }$ and $\xi _{\pm }$ and get, not forgetting that these variables are not mutually independent,

(4.9) \begin{align} \frac{\partial^{4} \phi_{1}}{\partial \xi_+ \partial \xi_- \partial \chi_+ \partial \chi_-} &=-(\mathcal{R}_{1} + \mathcal{R}_{2} +\mathcal{R}_{3} + \mathcal{R}_{4})\nonumber\\ &\equiv- 2\partial^{2}_{\xi_+ \xi_-} \partial^{2}_{\tau T}[ \,f_+(\chi_+) + f_-(\chi_-)] - 2\partial^{2}_{\chi_+ \chi_-} \partial^{2}_{\tau T}[ g_+(\xi_+) + g_-(\xi_-)]\nonumber\\ &\quad -3 \partial^{2}_{\xi_+ \xi_-} [ \partial_{y}( \,f_+(\chi_+) + f_-(\chi_-) + g_+(\xi_+) + g_-(\xi_-) ) \partial^2_{y^{2}}( \,f_+(\chi_+)\notag\\ &\quad + f_-(\chi_-) + g_+(\xi_+) + g_-(\xi_-) ) ]\nonumber\\ &\quad - \bar{\delta}\, \partial^{2}_{y^{2}}[ \partial^{2}_{\xi_+ \xi_-} \partial^{2}_{y^{2}}( \,f_+(\chi_+) + f_-(\chi_-) + g_+(\xi_+) + g_-(\xi_-) ) ]\nonumber\\ &\quad -\bar{\gamma} \, \partial^{2}_{\tau^{2}}( \,f_+(\chi_+) + f_-(\chi_-) + g_+(\xi_+) + g_-(\xi_-) ) . \end{align}

Using the changes of variables

(4.10)\begin{equation} \left.\begin{array}{c@{}} \displaystyle y = \dfrac{1}{2} ( \chi_+ + \chi_- ) ,\quad \tau = \dfrac{1}{2 c} ( \chi_+ - \chi_- ) ,\\ \displaystyle y = \dfrac{1}{2} ( \xi_+ + \xi_- ) ,\quad \tau = \dfrac{1}{2 c_{a}} ( \xi_+ - \xi_- ) , \end{array}\right\} \end{equation}

we get

(4.11a,b)\begin{equation} \chi_{{\pm}} = \alpha_{{\pm}} \xi_+ + \alpha_{{\mp}} \xi_-, \quad \xi_{{\pm}} = \beta_{{\pm}} \chi_+ + \beta_{{\mp}} \chi_-, \end{equation}

where we introduced the notation

(4.12a,b)\begin{equation} \alpha_{{\pm}} = \frac{1 \pm \dfrac{c}{c_{a}}}{2}, \quad \beta_{{\pm}} = \frac{1 \pm \dfrac{c_{a}}{c}}{2}. \end{equation}

Notice that, as follows from the definition, $c = \sqrt {1 + c^{2}_{a}} > c_{a}$; therefore,

(4.13a–c)\begin{equation} \beta_{{\pm}} > 0, \quad \alpha_+ >0, \quad \alpha_-<0. \end{equation}

With the help of these formulae, all derivatives in (4.9) could be transformed into derivatives of the functions $f_{\pm },\ g_{\pm }$ with respect to their characteristic arguments. In this way, we get the following expressions for the terms $\mathcal {R}_{i},\ i = 1, 2,3,4$ on the right-hand side of (4.9):

(4.14) \begin{gather} \mathcal{R}_{1} = 2\partial_{T} [ c \alpha_+\alpha_- (\, f'''_+(\chi_+) - f'''_- (\chi_-) ) + c_{a} \beta_+\beta_- ( g'''_+(\xi_+) - g'''_- (\xi_-) ) ], \end{gather}

(4.15) \begin{align} \mathcal{R}_{2} &= 3\{\alpha_+\alpha_- (\, f'''_+(\chi_+) + f'''_- (\chi_-) ) ( g''_+(\xi_+) + g''_- (\xi_-) + f''_+(\chi_+) + f''_- (\chi_-) ) \nonumber\\ &\quad +\, ( g''_- (\xi_-) + \alpha_- f''_+(\chi_+) + \alpha_+ f''_- (\chi_-) ) ( g'''_+(\xi_+) + \alpha_+ f'''_+(\chi_+) + \alpha_- f'''_- (\chi_-) ) \nonumber\\ &\quad +\, ( g''_+ (\xi_+) + \alpha_{+ } f''_+(\chi_+) + \alpha_- f''_- (\chi_-) ) ( g'''_-(\xi_-) + \alpha_- f''_+(\chi_+) + \alpha_+ f'''_- (\chi_-) ) \nonumber\\ &\quad +\,\alpha_+\alpha_- ( g'_+(\xi_+) + g'_-(\xi_-) + f'_+(\chi_+) + f'_-(\chi_-) ) (\, f''''_+(\chi_+) + f''''_- (\chi_-) ) \}, \end{align}

(4.16) \begin{gather} \mathcal{R}_{3} = \bar{\delta}\, \alpha_+\alpha_- (\, f''''''_+(\xi_+) + f''''''_-(\xi_-) ) , \end{gather}

(4.17) \begin{gather}\mathcal{R}_{4} = \bar{\gamma} [ c^{2} (\, f''_+(\chi_+) + f''_- (\chi_-) ) + c^{2}_{a} ( g''_+(\xi_+) + g''_- (\xi_-) ) ], \end{gather}

where from now on, the prime denotes the derivative of the functions $f_{\pm }$ and $g_{\pm }$ with respect to their characteristic arguments.

Equation (4.9) allows to determine the first correction to the linear Alfvén and magneto-gravity waves, unless its right-hand side contains resonant terms which produce a secular growth of the solution and thus a breakdown of the perturbation theory. This terms are, indeed present, but the above-introduced slow-time dependence allows to ‘kill’ them, according to the well-known procedure, and thus obtain the modulation equations for the amplitudes of the linear waves. In view of (4.11), (4.9) can be regarded either as a second-order partial differential equation with respect to the pair of independent variables $\chi _{\pm }$ or with respect to the pair $\xi _{\pm }$. It is easy to realize that in both cases, the only resonant terms are those depending only on one variable in the chosen pair of independent variables, as integration of both sides of the equation over the other variable produces a linear growth in the latter, in this case. Hence, a combination of such terms occurring in the right-hand side of (4.9), taking into account the expressions (4.14)–(4.17), should be equal to zero. Inspection of the expression (4.15) immediately shows an essential difference between the resonances in $\xi _{\pm }$ and in $\chi _{\pm }$, as there are neither nonlinear nor short-wave dispersive resonant terms in the former case. We should stress that the absence of the former is fully consistent with the results of § 3.

Elimination of the resonances in $\chi _{\pm }$ results in the following modulation equation for left- and right-moving magneto-gravity waves:

(4.18)\begin{equation} \pm\! 2 c \alpha_+\alpha_-\partial_{T} f'''_{{\pm}} + 3\alpha_-\alpha_+( 3 f''_{\pm}\, f'''_{{\pm}} + f'_{\pm}\, f''''_{{\pm}} ) + \bar{\gamma} c^{2} f''_{{\pm}} + \bar{\delta}\, \alpha_+\alpha_- f''''''_{{\pm}} = 0. \end{equation}

Similarly, elimination of the resonances in $\xi _{\pm }$ results in the modulation equation for left- and right-moving Alfvén waves:

(4.19)\begin{equation} \pm\! 2 c_{a} \beta_+\beta_-\partial_{T} g'''_{{\pm}} + \bar{\gamma} c^{2}_{a} g''_{{\pm}} = 0. \end{equation}

4.2.1. Magneto-gravity waves

Let us first recall that for small displacements of Lagrangian parcels, the non-dimensional thickness is

(4.20)\begin{equation} h = (1 + \epsilon \phi')^{-1} \approx 1 - \epsilon \phi', \end{equation}

where prime denotes the $y$-derivative, cf. (2.8). Therefore, up to a sign, $\phi '$, and hence each of $g'$ and $f'$ is a non-dimensional deviation $\eta (y, \tau )$ of the free surface from the state of rest.

Let us rewrite (4.18) in terms of $\eta _{\pm }$, the deviations of the surface produced by respectively left- and right-moving waves, dividing all terms by $\alpha _+\alpha _- <0$ (cf. (4.13)):

(4.21)\begin{equation} \mp\! 2 c \partial_{T} \eta''_{{\pm}} + 3( 3 \eta'_{{\pm}} \eta''_{{\pm}} + \eta_{{\pm}} \eta'''_{{\pm}} ) - \bar{\gamma} \frac{c^{2}}{\alpha_+\alpha_-} \eta'_{{\pm}} - \bar{\delta} \eta'''''_{{\pm}} = 0. \end{equation}

It can be easily checked that this equation can be rewritten as

(4.22)\begin{equation} ( ({\mp} c \partial_{T} \eta_{{\pm}} + 3 \eta_{{\pm}} \eta'_{{\pm}} - \bar{\delta}\, \eta'''_{{\pm}} ) ' + \varGamma \eta_{{\pm}} ) ' = 0, \end{equation}

where we rescaled the slow time $T\rightarrow T/2$ and introduced $\varGamma = \bar {\gamma } ({c^{2}}/{|\alpha _+\alpha _-|})>0$, which can be integrated once, giving for rapid decay boundary conditions:

(4.23)\begin{equation} ({\mp} c \partial_{T} \eta_{{\pm}} + 3 \eta_{{\pm}} \eta'_{{\pm}} - \bar{\delta}\, \eta'''_{{\pm}} ) ' + \varGamma \eta_{{\pm}} = 0. \end{equation}

This is the Ostrovsky equation (Ostrovsky Reference Ostrovsky1978), which is also called rotation-modified KdV equation. There exist an abundant literature on this equation, but dwelling into it is out of the scope of the present paper. We would only like to emphasize that, as already mentioned in § 1, unlike the famous KdV equation, this equation does not have soliton solutions, unless the coefficients $\bar {\delta }$ and $\varGamma$ have opposite signs (Galkin & Stepanyants Reference Galkin and Stepanyants1991), which is not the case here. The equation does not produce shock formation either.

Vanishing rotation, vanishing short-wave dispersion or both give, under rapid decay boundary conditions, the following equations describing different dynamical regimes:

• (i) no rotation, no dispersion $\varGamma = 0,\,\bar {\delta } = 0$

  Hopf or inviscid Burgers equation:

  (4.24)\begin{equation} \pm\! c \partial_{T} \eta_{{\pm}} + 3 \eta_{{\pm}} \eta'_{{\pm}} = 0 \end{equation}
  describing wave-breaking and shock formation in finite time;

• (ii) no rotation, dispersion $\varGamma = 0,\ \bar {\delta } \neq 0$

  Fully integrable Korteweg–de Vries equation with solitary wave solutions:

  (4.25)\begin{equation} \mp\! c \partial_{T} \eta_{{\pm}} + 3 \eta_{{\pm}} \eta'_{{\pm}}- \bar{\delta} \eta'''_{{\pm}} = 0; \end{equation}

• (iii) rotation, no dispersion $\varGamma \neq 0,\ \bar {\delta } = 0$

  Reduced Ostrovsky (Ostrovsky–Hunter, Vakhnenko) equation:

  (4.26)\begin{equation} \pm\! c \partial_{T} \eta_{{\pm}} + 3 \eta_{{\pm}} \eta'_{{\pm}} + \varGamma \eta_{{\pm}} = 0, \end{equation}
  which is integrable in a range of parameters, and produces wave-breaking and shock formation in another range (Grimshaw et al. Reference Grimshaw, Helfrich and Johnson2012).

Thus, weakly nonlinear magneto-gravity waves undergo the modulation described by the Ostrovsky equation. In the limits of negligible rotation and/or dispersion, the modulation produces coherent structures, respectively solitons or shocks.

4.2.2. Alfvén waves

The analysis of the modulation equation for Alfvén waves is much simpler. Equation (4.19) under rapid decay boundary conditions can be integrated twice, giving the following second-order linear PDE:

(4.27)\begin{equation} \pm\! \partial_{T} g'_{{\pm}}(\xi_{{\pm}}) + \bar{\varGamma}\, g_{{\pm}}(\xi_{{\pm}}) = 0, \end{equation}

where we defined $\bar {\varGamma } = {\bar {\gamma } c_{a}}/{2 \beta _+\beta _-} >0$. By introducing ‘bi-characteristic’ variables $\rho _{\pm }=\xi \pm T$ for each $\xi _{\pm }$, this equation is transformed into the well-known and exhaustively studied Klein–Gordon equation:

(4.28)\begin{equation} \frac{\partial^{2} g_{{\pm}}}{\partial \rho^{2}_+} - \frac{\partial^{2} g_{{\pm}}}{\partial \rho^{2}_-} \pm \bar{\varGamma} g_{{\pm}} = 0, \end{equation}

and may be analysed as such. Thus, the modulation of weakly nonlinear Alfvén waves, unlike magneto-gravity waves, does not produce coherent structures, having a purely dispersive character which is due exclusively to rotation.

5.1. Non-dispersive rotating case $\delta = 0,\ \gamma \neq 0$

Equation (5.2) in this case becomes

(5.3)\begin{equation} ( V^{2} - c^{2}_{a} ) \left[ ( V^{2} - c^{2}_{a} ) J + \frac{1}{2 J^{2}} \right]'' + \gamma^{2} V^{2} J = A. \end{equation}

The method of treating it is the same as in Zeitlin et al. (Reference Zeitlin, Medvedev and Plougonven2003), where fully nonlinear wave solutions were obtained in Lagrangian RSW. First, the constant $A$ can be determined if solutions are sought in a form of periodic waves. By integrating (5.3) over one period of the wave, we get

(5.4)\begin{equation} A = \gamma^{2} V^{2}. \end{equation}

After substituting this result in (5.3) and multiplying by $[ ( V^{2} - c^{2}_{a} ) J + ({1}/{2 J^{2}}) ]'$, we can integrate the equation once more, and get

(5.5)\begin{align} &\frac{1}{2}( V^{2} - c^{2}_{a} ) \left(\left[ ( V^{2} - c^{2}_{a} ) J + \frac{1}{2 J^{2}} \right]'\right)^{2}\nonumber\\ &\quad + \gamma^{2} V^{2} \left[ ( V^{2} - c^{2}_{a} ) \left(\frac{J^{2}}{2} - J\right)+ \frac{1}{J} - \frac{1}{2 J^{2}} \right]= B = \text{const.} \end{align}

The first term of this equation can be rewritten as $( V^{2} - c^{2}_{a} ) [ ( V^{2} - c^{2}_{a} ) - {1}/{J^{3}} ]^{2} ({J'^{2}}/{2})$, and we thus arrive at the following ordinary differential equation for $J$:

(5.6)\begin{equation} \frac{J'^{2}}{2} + \frac{ \gamma^{2} V^{2} \left[ ( V^{2} - c^{2}_{a} ) \left( \dfrac{J^{2}}{2} -J\right)+ \dfrac{1}{J} - \dfrac{1}{2 J^{2}} \right] - B}{( V^{2} - c^{2}_{a} ) \left[ ( V^{2} - c^{2}_{a} ) - \dfrac{1}{J^{3}} \right]^{2} } = 0, \end{equation}

which is equivalent to a mechanical particle-in-a-well problem with a ‘particle’ with unit mass at position $J$, evolving at zero energy level in a ‘potential’ $\mathcal {V}(J)$ depending on parameters $B, C \equiv \gamma V, \varDelta \equiv V^{2} - c^{2}_{a}$:

(5.7)\begin{equation} \mathcal{V}(B,C, \varDelta; J) = \frac{C^{2} \left[ \varDelta \left( \dfrac{J^{2}}{2} -J\right)+ \dfrac{1}{J} - \dfrac{1}{2 J^{2}} \right] - B}{\varDelta \left[ \varDelta - \displaystyle \dfrac{1}{J^{3}} \right]^{2} } . \end{equation}

The constant $C$ may be absorbed in the independent variable: $y - V\tau \rightarrow C(\kern0.06em y - V\tau )$ and the constant $B$: $B \rightarrow \bar {B} = B/C^{2}$, which results in the potential depending effectively only on two parameters $\varDelta$ and $\bar {B}$. We will not present the analysis of the resulting particle-in-a-well problem, which follows that presented by Shecter et al. (Reference Shecter, Boyd and Gilman2001) and Zeitlin et al. (Reference Zeitlin, Lusso and Bouchut2015), although it was performed in these papers in the Eulerian framework. We should only recall that two types of solutions are possible when respectively $\varDelta >1$ and $\varDelta <0$: finite-amplitude magneto-gravity and Alfvén waves. We present the form of the potential well in (5.6) in these two cases in figure 1, demonstrating the existence of respective solutions.

Figure 1. Graphs of potential (5.7) as a function of $J$ at: (a) $\bar {B} = -0.97$, $\varDelta = 3$; (b) $\bar {B} = -0.97$, $\varDelta = -5$; and (c) a zoom of panel (b) showing that the left zero of the potential there is positive. Periodic finite-amplitude wave solutions with bounded $h \propto 1/J$ correspond to the intervals between successive positive zeros of the potential.

5.2. Dispersive non-rotating case $\delta \neq 0,\ \gamma = 0$

The double derivative in (5.1) should be lifted or, equivalently, (5.1) is integrated twice with integration constants equal to zero, as there is no need to differentiate twice (2.15) in this case because the original equations for $u$ and $v$ are uncoupled in the absence of rotation. Hence,

(5.8)\begin{equation} ( V^{2} - c^{2}_{a} ) \left[ ( V^{2} - c^{2}_{a} ) J + \frac{1}{2 J^{2}} \right]' + \delta \left[ \frac{V^{2} - c^{2}_{a}}{J^{2}} \left( \frac{1}{J}\right)'' \right]' = 0. \end{equation}

This equation can be integrated again, with an integration constant $A$. We thus have

(5.9)\begin{equation} ( V^{2} - c^{2}_{a} ) J + \frac{1}{2 J^{2}} + \delta \frac{V^{2} - c^{2}_{a}}{J^{2}} \left( \frac{1}{J}\right)'' = A. \end{equation}

Unlike the previous case, integration over the period of a supposed periodic wave solution does not allow to determine the constant $A$ because of the second term in (5.9), so we leave the equation as it is. It is natural to come back to the non-dimensional thickness variable $h = J^{-1}$ in (5.9) which gives, after the introduction of the above-defined $\varDelta$ and rescaling of $A$: $A \rightarrow \bar {A}= A/\varDelta$,

(5.10)\begin{equation} \delta h'' + \frac{1}{2 \varDelta} + \frac{1}{h^{3}} - \frac{\bar{A}}{h^{2}}=0. \end{equation}

This is, once more, a particle-in-a-well mechanical problem, where $\delta$ plays the role of particle mass and $h$ its position, and can be treated by standard techniques. (In fact, the equation can be directly integrated in terms of elliptic functions, but we will limit ourselves here by a qualitative analysis of solutions.) By multiplying the left-hand side by $h'$ and integrating once with an integration constant $E$, we get

(5.11)\begin{equation} \delta \frac{h'^{2}}{2} + \frac{h}{2 \varDelta} - \frac{1}{2~h^{2}} + \frac{\bar{A}}{h} = E. \end{equation}

Here, the first term is the ‘particle's kinetic energy and the rest of the left-hand side is a potential:

(5.12)\begin{equation} \mathcal{V}( \bar{A},\varDelta; h) = \frac{h}{2 \varDelta} - \frac{1}{2h^{2}} + \frac{\bar{A}}{h}. \end{equation}

As said, (5.11) can be explicitly integrated by the method of separation of variables, but existence of periodic and/or decaying solutions can be analysed qualitatively in a very simple way by looking for the existence of potential well(s) in $\mathcal {V}(h)$ at positive $h$. It easy to see that such a well can exist only at $\varDelta >0$, that is, if the velocity of the solution exceeds the Alfvén velocity. In figure 2, we give an example of such a well. As follows from the figure, the potential has a maximum at $h=h_{max}\approx 0.54$ and a minimum at $h = h_{min} \approx 1.68 > h_{max}$, growing monotonically at $h > h_{min}$. Following the standard mechanical interpretation, for $\mathcal {V}(h_{min}) < E < \mathcal {V}(h_{max})$, the ‘particle’ oscillates between two zeros of $E - \mathcal {V}(h)$ (closed contour in figure 2b), which corresponds to periodic wave solutions for $h(y - V \tau )$ in our MSW problem, and for $E = \mathcal {V}(h_{max})$, the ‘particle’ follows the separatrix trajectory (dashed contour), reaching the ‘top of the hill’ in infinite ‘time’, which corresponds to a solitary wave solution for $h$ which rises monotonically from its asymptotic value $h = h_{max}$ at $y - V \tau \rightarrow -\infty$ to the maximum value given by the non-trivial solution $h > h_{max}$ of the equation $E = \mathcal {V}(h_{max}) = \mathcal {V}(h)$, and then monotonically descending to the asymptotic value $h = h_{max}$ at $y - V \tau \rightarrow +\infty$.

Figure 2. (a) Graph of the potential (5.12) as a function of $h$ at $\bar {A} = 2$, $\varDelta = 1$, and (b) the corresponding phase portrait of the system (5.10) in the ($h,\ h'$) plane. Dashed, separatrix trajectory.

The existence of such solutions of both types for SGN equations is known, e.g. Dutykh & Ionescu-Kruse (Reference Dutykh and Ionescu-Kruse2016). We are not aware, to the best of our knowledge, of their derivation in the Lagrangian framework. The specificity of these solutions, which were already sketched by Dellar (Reference Dellar2003) in the present magnetized system, is that their speed of translation is limited from below by the Alfvén velocity.