2. Basic model

We consider an electron–ion plasma in the presence of thermal radiation, magnetic and gravitational fields, which is dense enough to be treated as an incompressible fluid within the MHD model (Tsintsadze Reference Tsintsadze1998). The plasma is embedded in an ambient magnetic field directed along the $z$-axis, i.e. $\boldsymbol {B}=B_{0}\hat {z}$ (where $\hat {z}$ is the unit vector along the $z$-axis). Plasma evolution is limited to the $y$–$z$ plane, while vacuum is considered to extend in the direction normal to the plane i.e. along the $x$-axis. In order to address surface-wave phenomena at the interface between the plasma and vacuum, we assume that the temperature inside the plasma is kept constant, whereas small temperature fluctuations may occur at the interface, to be described as a function of the boundary coordinates. The MHD model equations, neglecting the electron inertia, read

(2.1)\begin{gather} -en_{e}\boldsymbol{E}-\frac{en_{e}}{c}\left( \boldsymbol{u}_{e}\times\boldsymbol{B} \right) -\boldsymbol{\nabla}(P_{g,e}+P_{r,e}) =0, \end{gather}

(2.2)\begin{gather}m_{i}n_{i}\frac{\textrm{d}\boldsymbol{u}_{i}}{\textrm{d}t} =en_{i}\boldsymbol{E}+\frac{en_{i}} {c}\left( \boldsymbol{u}_{i}\times\boldsymbol{B}\right) -\boldsymbol{\nabla}(P_{g,i}+P_{r,i}) -m_{i}n_{i}\boldsymbol{\nabla}\varPsi, \end{gather}

where $n_{e}$ and $n_{i}$ denote the number density for the electron and ion fluid component(s), $\boldsymbol {u}_{e}$ and $\boldsymbol {u}_{i}$ their respective fluid speed, $\varPsi$ is the gravitational potential, and $P_{g,e}$ and $P_{g,i}$ are the usual gas pressure terms for electrons and ions, respectively; similarly, $P_{r,e}$ and $P_{r,i}$ (defined in (1.3) above) denote the radiation pressure terms for the respective species. The basic set of equations involves Poisson's law(s) for the gravitational potential $\varPsi$ and for the electrostatic potential $\varPhi$, expressed respectively as

(2.3)\begin{gather} \nabla^{2}\varPsi =4{\rm \pi}\,G\,\rho_{i}, \end{gather}

(2.4)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E} ={-}\nabla^{2}\varPhi=4{\rm \pi}\,\sigma_{e}, \end{gather}

where $\rho _{i}=m_{i}n_{i}$ and $\sigma _{e}=en_{e}$ are the total ion mass density and the charge density respectively. Here, we have considered $n_{e}\simeq n_{e,0}+n_{e,1}$ and $n_{i}=n_{i,0}$, i.e. assuming density perturbations to be negligible (incompressibility assumption), yet retaining the electron density disturbance $n_{e,1}$ ($\ll n_{e,0}$) and neglecting their ionic counterpart. The continuity (mass conservation) equations read

(2.5)\begin{equation} \frac{\partial n_{e,i}}{\partial t}+\nabla.(n_{e,i}\boldsymbol{u}_{e,i})=0, \end{equation}

Maxwell's equations

(2.6)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{B} =0\,, \end{gather}

(2.7)\begin{gather}\boldsymbol{\nabla}\times\boldsymbol{B} =\frac{4{\rm \pi}}{c}\boldsymbol{J}\,, \end{gather}

(2.8)\begin{gather}\boldsymbol{\nabla}\times\boldsymbol{E} ={-}\frac{\partial\boldsymbol{B}}{\partial t} \end{gather}

and the MHD equation

(2.9)\begin{equation} \boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B}=\bar{\eta}\,\boldsymbol{J}, \end{equation}

where $\boldsymbol {v=}({m_{e}u_{e}+m_{i}u_{i}})/({m_{e}+m_{i}})$, while $\boldsymbol {J}=e( n_{i}\boldsymbol {u}_{i}-n_{e}\boldsymbol {u}_{e})$ is the charge current density and $\bar {\eta }$ is the thermal conductivity. Note here that gravity is significant only for the ions due to their large mass. Adding (2.1) and (2.2), we obtain

(2.10)\begin{equation} \rho_{i}\frac{\textrm{d}\boldsymbol{u}_{i}}{\textrm{d}t}=e(n_{i}-n_{e})\boldsymbol{E+}\frac{1} {c}(\boldsymbol{J}\times\boldsymbol{B)-\nabla}(P_{g}^{(\textrm{total})}+P_{r} ^{(\textrm{total})}) -\rho_{i}\boldsymbol{\nabla}\varPsi, \end{equation}

where $P_{r}^{(\textrm {total})}=P_{r,e}+P_{r,i}$ and $P_{g}^{(\textrm {total})}=P_{g,e}+P_{g,i}$. It is necessary to point out here that both thermal and radiation pressure effects are negligible for the ions, compared with the electrons, due to the large mass disparity between them, viz. $P_{r,i}\ll P_{r,e}$ (recall that the electron inertia has been neglected, as mentioned above). Physically speaking, the electrons will move towards the interface more easily, thus charge separation may appear at the interface of the order of thermal energy, i.e. $e\delta \varPhi \sim k_{B}\delta T_{e}$. As a consequence, the separation of charges at the interface is normal to the surface of plasma, instead of the usual parallel component of electric field arising in the ordinary plasma wave description. The associated negative pressure gradient may be inferred from (2.4) as

(2.11)\begin{equation} \sigma_{e}\,\boldsymbol{E}=\frac{1}{4{\rm \pi}}\boldsymbol{\left( \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{E}\right) \boldsymbol{E}=}\frac{1}{8{\rm \pi}}\boldsymbol{\nabla}_{x}E^{2}, \end{equation}

where $E=-\boldsymbol {\nabla }\varPhi$, viz. $E=-({\partial \varPhi }/{\partial x})$. Using (2.7) and (2.11), (2.10) reduces to

(2.12)\begin{equation} \rho_{i}\frac{\textrm{d}\boldsymbol{u}_{i}}{\textrm{d}t}=\frac{1}{8{\rm \pi}}\boldsymbol{\nabla}_{x} E^{2}+\frac{1}{4{\rm \pi}}\boldsymbol{B}\left(\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\right) \boldsymbol{-\nabla}\left( P_{g}^{(\textrm{total})}+P_{r}^{(\textrm{total})}+\frac{B^{2}}{8{\rm \pi} }\right) -\rho_{i}\boldsymbol{\nabla}\varPsi. \end{equation}

Here $\rho _{i}=m_{i}n_{i0}$ is the ion mass density, the first term on the right-hand side represents the negative pressure gradient acting perpendicular to the surface between the vacuum and the plasma. Since the plasma under consideration is incompressible, the continuity equation becomes $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u=0}$, which allows us to express $u$ as the gradient of a scalar function, say $\varphi$, i.e. $\boldsymbol {u} =\boldsymbol {\nabla }\varphi$ (Landau & Lifshitz Reference Landau and Lifshitz1984), or

(2.13)\begin{equation} \nabla^{2}\varphi=0.\end{equation}

At equilibrium, the electrostatic field at the plasma–vacuum interface involves the surface charge density

(2.14)\begin{equation} E_{0} =4{\rm \pi}\int{}\sigma_{e}\,{\textrm{d}x} = 4{\rm \pi}\sigma_{se}, \end{equation}

where $\sigma _{se}=\sigma _{e}x$ is the equilibrium surface charge density of electrons. The resulting electrostatic potential $\varPhi$ at equilibrium is

(2.15)\begin{equation} \varPhi={-}4{\rm \pi}\sigma_{se}x. \end{equation}

Similarly, the gravitational potential $\varPsi$ at equilibrium can be obtained from (2.3) as

(2.16)\begin{equation} \varPsi=4{\rm \pi} G\rho_{si} x, \end{equation}

where $\rho _{si}=\rho _{i}x$ is the surface mass density of ions. For an oscillating plasma–vacuum interface, the oscillating electrostatic and gravitational potentials may be defined from (2.15) and (2.16) respectively as

(2.17)\begin{gather} \delta\varPhi=4{\rm \pi}\sigma_{se}\chi(y,z,t), \end{gather}

(2.18)\begin{gather} \delta\varPsi={-}4{\rm \pi} G\rho_{si}\chi(y,z,t), \end{gather}

where $\delta \varPhi$ and $\delta \varPsi$ are the perturbed electrostatic and gravitational potentials respectively, and $\chi (y,z,t)$ is the spatial displacement of the surface under the action of perturbations. (Cf. (12)–(13) in Tsintsadze (Reference Tsintsadze1998).) For surface waves with small amplitude, we may represent the small variations of the potential functions as $\{\delta \varPhi ,\delta \varPsi \}\sim \exp [\textrm {i}( k_{y}y+k_{z}z-\omega t) -kx]$, where $k>0$ is a positive quantity (representing an inverse characteristic decay length). (Note, for clarity, that $k$ is not the wavenumber norm $(k_{y}^{2}+k_{z}^{2})^{1/2}$ ($=k_{yz}$, say).) Note that both $\delta \varPhi$ and $\delta \varPsi$ decay exponentially in the $x-$direction and actually vanish for $x\rightarrow \infty$.

From (2.14), (2.15) and (2.17) we can write

(2.19)\begin{equation} \frac{E_{x}^{2}}{8{\rm \pi}}=2{\rm \pi}\sigma_{se}^{2}+k\sigma_{se}\delta\varPhi|_{x=0} =2{\rm \pi}\sigma_{se}^{2}+4{\rm \pi}\sigma_{se}^{2}k\chi\left( y,z,t\right), \end{equation}

where $\delta E_{x}=k\delta \varPhi$. For linear oscillations, the perpendicular component of the ion velocity $u_{i,x}$ (to the surface) is given as the time derivative of the surface displacement from the equilibrium position, viz. $u_{ix}={\partial \chi (y,z,t)}/{\partial t}$. Also, as discussed above (2.13), for an incompressible plasma $u=\nabla \varphi$ or, in our case, $u_{i,x}={\partial \varphi }/{\partial x}$. Comparing both definitions, we arrive at ${\partial \varphi }/{\partial x}|_{x=0}={\partial \chi }/{\partial t}$. The plasma–vacuum interface is subject to two competing (oppositely directed) pressure mechanisms, namely the negative pressure gradient due to the surface electrons i.e. $\nabla _{n}E^{2}$ acting upwards and the thermal pressure acting downwards to the surface. To describe this situation, we may follow Laplace's formula (Landau & Lifshitz Reference Landau and Lifshitz1984) for an incompressible radiative electron–ion plasma,

(2.20)\begin{equation} P-P_{0}={-}\epsilon\left( \frac{\partial^{2}\chi}{\partial y^{2}} +\frac{\partial^{2}\chi}{\partial z^{2}}\right), \end{equation}

where $P$ and $P_{0}$ denote the pressure terms due to the two distinct media (plasma and vacuum respectively) and $\epsilon$ is the surface tension coefficient. Furthermore, we may obtain an evolution equation for the plasma variation on the interface by taking into account (2.19)–(2.20) and integrating (2.12) – taking into account (1.3) – to find

(2.21)\begin{align} \left.\left[ \frac{\partial\varphi}{\partial t}-\frac{k\sigma_{se}}{\rho_{_{i}} }\delta\varPhi-\frac{B_{\boldsymbol{0}}}{4{\rm \pi}\rho_{i}}\frac{\partial}{\partial y} \int\delta B_{x}\,{\textrm{d}x}+\delta\varPsi+\left( \frac{\partial P_{r,e}}{\partial T_{e} }\right) \frac{\delta T_{e}}{\rho_{i}}-\frac{\epsilon}{\rho_{i}}\left( \frac{\partial^{2}\chi}{\partial y^{2}}+\frac{\partial^{2}\chi}{\partial z^{2}}\right) \right] \right|_{x=0}=0. \end{align}

Assuming $T_{e}\gg T_{i}$, we henceforth neglect radiation and thermal effects associated with the ion component; as for the electrons, we have introduced $\delta T_{e}$, which denotes temperature fluctuations on the interface as a function of the surface coordinates. (The subscript ‘$e$’ may later be omitted where obvious.) At the interface, the sharp boundary conditions can be obtained from (2.17)–(2.18) as

(2.22)\begin{gather} \frac{\partial(\delta\varPhi)}{\partial t}=4{\rm \pi}\sigma_{se}\frac{\partial\chi }{\partial t}=4{\rm \pi}\sigma_{se}\left(\frac{\partial\varphi}{\partial x}\right|_{x=0}=0, \end{gather}

(2.23)\begin{gather} \frac{\partial(\delta\varPsi)}{\partial t}={-}4{\rm \pi} G\rho_{si}\frac{\partial\chi }{\partial t}={-}4{\rm \pi} G\rho_{si}\left(\frac{\partial\varphi}{\partial x}\right|_{x=0}=0.\, \end{gather}

From (2.8)–(2.9) we obtain the following MHD equation:

(2.24)\begin{equation} \frac{\partial\boldsymbol{B}}{\partial t}=\boldsymbol{\nabla}\boldsymbol{\times}\left( \boldsymbol{u}_{i}\boldsymbol{\times} \boldsymbol{B}\right). \end{equation}

Linearizing (2.24) and writing down the $x-$component, we obtain

(2.25)\begin{equation} \frac{\partial B_{x}}{\partial t}=B_{0}\frac{\partial u_{i,x}}{\partial z}=B_{0}\frac{\partial^{2}\varphi}{\partial z\text{ }\partial x}. \end{equation}

Next, differentiating (2.21) with respect to $t$ and making use of (2.22)–(2.25), we get

(2.26)\begin{align} \left.\left[ \frac{\partial^{2}\varphi}{\partial t^{2}}-V_{E}^{2} k\frac{\partial\varphi}{\partial x}-V_{A}^{2}\frac{\partial^{2}\varphi }{\partial z^{2}}-4{\rm \pi} G\rho_{si}\frac{\partial\varphi}{\partial x}+\frac {1}{\rho_{i}}\left( \frac{\partial P_{r,e}}{\partial T_{e}}\right) \frac{\partial\delta T_{e}}{\partial t}-\frac{\epsilon}{\rho_{i}}\left( \frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2} }\right) \frac{\partial\varphi}{\partial x}\right] \right|_{x=0}=0, \end{align}

where $V_{A}=B_{\boldsymbol {0}}/\sqrt {4{\rm \pi} \rho _{i}}$ is the standard (magnetic-field related) Alfvén speed and $V_{E}=E_{0}/\sqrt {4{\rm \pi} \rho _{i}}$ represents an analogous characteristic (electric field related) speed for the ions.

To evaluate the (non-relativistic) temperature oscillations $\delta T$, we shall make use of the energy equation (Tsintsadze Reference Tsintsadze1995)

(2.27)\begin{equation} \frac{\partial}{\partial t}\left( \frac{3}{2}k_{B}T_{e}+\frac{U_{r,e}} {n}\right) +\left( \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right) \left[ \frac {3}{2}k_{B}T_{e}+\frac{P_{g}}{n}+\frac{4}{3}\frac{U_{r,e}}{n}\right] +\boldsymbol{\nabla}\boldsymbol{\cdot}\left( \frac{\boldsymbol{S}_{e}}{n}\right) =0, \end{equation}

where $T_{e}$ is the (non-relativistic) electron temperature, and $n$ ($\simeq n_{e,0}=n_{i,0}$) in the latter expression is the incompressible fluid density (at quasi-equilibrium) and $U_{r,e}$ denotes the electron radiation energy density.

The Poynting vector $\boldsymbol {S}$ ($=\boldsymbol {S}_{e}$) is defined in terms of the electron heat flux through radiative heat conduction (Chandrasekhar Reference Chandrasekhar1984; Mihalas & Mihalas Reference Mihalas and Mihalas1984) as

(2.28)\begin{equation} \boldsymbol{S}_{e}\text{ }={-}\frac{\lambda c}{3}\,\boldsymbol{\nabla}U_{r,e} ={-}L_{0}\boldsymbol{\nabla}(k_{B}T_{e}), \end{equation}

where $L_{0}=4( \lambda c/3) \beta T_{e}^{3}$ is the coefficient of thermal radiation conductivity and $\lambda$ represents the Rosseland radiation mean free path $\lambda =A\,T_{e}^{\kappa }$, where $\kappa =1,2,3,\ldots$ (i.e. a positive integer) and $A$ is a positive quantity. (Note that Boltzmann's constant will henceforth be omitted for simplicity, hence the temperature symbol $T_{e}$ appearing in forthcoming relations will be expressed in energy units.)

Substituting the expressions for $U_{r,e}$ and for $\lambda$ into (2.28), we obtain

(2.29)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{S}_{e}={-}\frac{4\beta\,c\,A}{3}\frac{1}{\eta }\boldsymbol{\nabla}\boldsymbol{\cdot}\left( \boldsymbol{\nabla}T_{e}^{\eta}\right) ={-}\frac{4\beta cA}{3}[2(\boldsymbol{\nabla}T_{e,0}^{\eta -1})\,\boldsymbol{\nabla}(\delta T_{e})+(\boldsymbol{\nabla}^{2}T_{e,0}^{\eta -1})\delta T_{e}], \end{equation}

where $\eta =\kappa +4$. At the interface, we may write

(2.30)\begin{equation} \nabla_{x}S_{e,x}=\frac{\partial S_{e,x}}{\partial x}={-}\frac{4\beta cA} {3}\left(2k\frac{\textrm{d}T_{e,0}^{\eta-1}}{\textrm{d}x}+\frac{\textrm{d}^{2}T_{e,0}^{\eta-1}}{{\textrm{d}x}^{2} }\right)\delta T_{e}, \end{equation}

where $T_{e,0}$ denotes the electron temperature at equilibrium, which is a function of the surface coordinate(s), while $\delta T_{e}$ represents small temperature variations on the interface and is independent of the surface coordinates.

At equilibrium, (2.27) reduces to $\boldsymbol {\nabla }\boldsymbol {\cdot }( {\boldsymbol {S}_{e}}/{n}) =0$, hence following the procedure of Tsintsadze et al. (Reference Tsintsadze, Rozina, Shah and Murtaza2007) – cf. (1.3)–(2.2) therein – one obtains the relation

(2.31)\begin{equation} \frac{\textrm{d}^{2}T_{e,0}^{\eta}}{{\textrm{d}x}^{2}}=0. \end{equation}

Integrating the above equation twice, with the boundary condition $T_{e,0}=0$ at $x=x_{s},$ we get

(2.32)\begin{equation} T_{e,0}=\theta|x_{s}-x|^{1/\eta}, \end{equation}

where $x_{s}$ is the surface coordinate and $\theta$ is the constant temperature inside the plasma. Equation (2.32) indicates that at equilibrium, temperature is a function of surface coordinate $x$. Furthermore, it shows that at $x=x_{s}$ the temperature diminishes, representing the sharp variation on the boundary, i.e. across the plasma–vacuum interface. Substituting (2.29) into the linearized $x$-dimensional version of (2.27) leads to

(2.33)\begin{gather} \left[ \left( \frac{3}{2}+\frac{1}{n}\frac{\partial U_{r,e}}{\partial T_{e} }\right) \frac{\partial\delta T_{e}}{\partial t}+\frac{\partial\delta\varphi }{\partial x}\frac{\partial}{\partial x}\left( \frac{3}{2}T_{e,0}+\frac {P_{g}}{n}+\frac{4}{3}\frac{U_{r,e}}{n}\right) \right. \nonumber\\ \left.\left. -\frac{4\,\beta\,c\,A}{3n}\left( 2k\frac{\partial T_{e,0}^{\eta-1} }{\partial x}+\Delta T_{e,0}^{\eta-1}\right) \delta T_{e}\right] \right|_{x=0} =0. \end{gather}

Recall that the density $n( \simeq n_{e,0}=n_{i,0})$ is constant, i.e. time- and space-invariant. Applying a Fourier transformation, viz. $\delta {\varphi }\sim \exp [\textrm {i}( k_{y}y+k_{z}z-\omega t) +kx]$ on the above equation, we obtain

(2.34)\begin{equation} \delta T_{e}=\frac{\mu}{\digamma\left( \textrm{i}\omega+\nu\right) }, \end{equation}

where

(2.35)\begin{equation} \left. \begin{aligned} & \mu =\frac{\partial\varphi}{\partial x}\frac{\partial}{\partial x}\left( \frac{3}{2}T_{e,0}+\frac{P_{g}}{n}+\frac{4}{3}\frac{U_{r,e}}{n}\right),\\ & \digamma =\frac{3}{2}+\frac{1}{n}\frac{\partial U_{r,e}}{\partial T_{e} },\\ \textrm{and}\qquad & \nu =\frac{4\beta cA}{3n\left( \dfrac{3}{2}+\dfrac{1} {n}\dfrac{\partial U_{r,e}}{\partial T_{e}}\right) }\left( 2k\dfrac{\partial }{\partial x}T_{e,0}^{\eta-1}+\dfrac{\partial^{2}}{\partial x^{2}}T_{e,0} ^{\eta-1}\right). \end{aligned} \right\} \end{equation}

Substituting (2.34) in (2.26), we obtain the following dispersion relation:

(2.36)\begin{equation} \omega^{2}-k^{2}V_{A}^{2}\cos^{2}\theta-\frac{b^{2}k^{3}}{2}-k\left( \gamma_{g}-i\gamma_{g}^{{\prime}}\frac{\left( \nu/\omega\right) }{1+\left( \nu^{2}/\omega^{2}\right) }\right) +k\left( kV_{E}^{2}+4{\rm \pi} G\rho _{si}\right) =0, \end{equation}

where

(2.37)\begin{align} \gamma_{g}=\dfrac{4\beta T_{e}^{3}}{3\rho_{i}}\dfrac{\left( \dfrac{3}{2} +\dfrac{1}{n}\dfrac{\partial P_{g}}{\partial T_{e,0}}+\dfrac{4}{3n}\dfrac{\partial U_{r,e}}{\partial T_{e,0}}\right) \dfrac{\partial T_{e,0}}{\partial x} }{\left( \dfrac{3}{2}+\dfrac{1}{n}\dfrac{\partial U_{r,e}}{\partial T_{e} }\right) \left( 1+\dfrac{v^{2}}{\omega^{2}}\right) }=\dfrac{4\beta T_{e}^{3} }{3\rho_{i}}\dfrac{\left( \dfrac{3}{2}+\dfrac{1}{n}\dfrac{\partial P_{g} }{\partial T_{e,0}}+\dfrac{16}{3}\dfrac{1}{n}\beta T_{e}^{3}\right) \dfrac{\partial T_{e,0}}{\partial x}}{\left( \dfrac{3}{2}+\dfrac{4}{n} \beta_{r,e}T_{e}^{3}\right) \left( 1+\dfrac{v^{2}}{\omega^{2}}\right) } \end{align}

and

(2.38)\begin{align} \gamma_{g}^{{\prime}}=\dfrac{4\beta T_{e}^{3}}{3\rho_{i}}\dfrac{\left( \dfrac{3}{2}+\dfrac{1}{n}\dfrac{\partial P_{g}}{\partial T_{e,0}}+\dfrac{4} {3n}\dfrac{\partial U_{r,e}}{\partial T_{e,0}}\right) \dfrac{\partial T_{e,0} }{\partial x}}{\left( \dfrac{3}{2}+\dfrac{1}{n}\dfrac{\partial U_{e,r}}{\partial T_{e}}\right) }=\dfrac{4\beta T_{e}^{3}}{3\rho_{i}}\dfrac{\left( \dfrac{3} {2}+\dfrac{1}{n}\dfrac{\partial P_{g}}{\partial T_{e,0}}+\dfrac{16}{3}\dfrac{1} {n}\beta T_{e}^{3}\right) \dfrac{\partial T_{e,0}}{\partial x}}{\left( \dfrac{3}{2}+\dfrac{4}{n}\beta T_{e}^{3}\right) }, \end{align}

where $b^{2}=2\epsilon /\rho _{i}$ is the capillarity constant. (Note that the equation of state (1.3) was used in the last step, to show the dependence of the coefficients on the electron temperature $T_{e}$.) Equation (2.36) represents the propagation of capillary–gravity surface waves at the interface between a (magnetized radiative) plasma and vacuum.

Let us now explore the dispersion equation (2.36) in some special cases.

2.1. Capillary radiative Jeans surface instability

First, let us assume that the imaginary part of (2.36) is absent, hence that the dissipative term is absent in (2.27) (viz. $S_{e}=\boldsymbol {0}$), or $\nu =0,$ to obtain the dispersion relation describing the magnetocapillary radiative Jeans surface instability of an inhomogeneous incompressible electron–ion plasma:

(2.39)\begin{equation} \omega^{2}=k^{2}V_{A}^{2}\cos^{2}\theta+\frac{b^{2}k^{3}}{2}+k\gamma _{g}-k(kV_{E}^{2}+4{\rm \pi} G\rho_{si}). \end{equation}

Note, for comparison, that the dispersion relation of the Jeans instability derived by Tsintsadze et al. (Reference Tsintsadze, Rozina, Shah and Murtaza2008) was for a homogeneous compressible dusty (i.e. three-component) plasma. The associated growth rate of the Jeans surface instability reads

(2.40)\begin{equation} \operatorname{Im}\omega=\sqrt{k(kV_{E}^{2}+4{\rm \pi} G\rho_{si}) -\left( k\gamma_{g}+k^{2}V_{A}^{2}\cos^{2}\theta+\frac{b^{2}k^{3}}{2}\right) }. \end{equation}

Equation (2.40) governs the fragmentation of astrophysical objects via their surface oscillations when the electron surface charge density and the ion surface mass density couple together to enhance the oscillation rate of these objects, whereas gravity radiation, magnetic field and surface tension effects stabilize the Jeans surface instability.

2.2. Gravitational radiation

We may now assume that the plasma is unmagnetized and also neglect the surface tension. We thus obtain the simpler dispersion relation

(2.41)\begin{equation} \omega^{2}=k\gamma_{g}-k(4{\rm \pi} G\rho_{si}+kV_{E}^{2}). \end{equation}

We see that radiation does not modify the longitudinal dispersion relation, but it only appears as gravitational radiation on the interface, and actually plays a stabilizing role on the Jeans surface instability. Equation (2.41) provides us with a new definition of the Jeans wavelength, i.e. the Jeans wavevector

(2.42)\begin{equation} k_{J}=\frac{\gamma_{g}-4{\rm \pi} G\rho_{si}}{V_{E}^{2}} \end{equation}

incorporates the effect of gravitational radiation. If we substitute the numerical values, shown below in § 3, in (2.42), we may eventually obtain the threshold value of the Jeans wavevector, $k_{J}=5.53\times 10^{-6}\ \textrm {cm}^{-1}$. It may be noted here that for any length scale larger than $k_{J}(=5.53\times 10^{-6}\ \textrm {cm}^{-1}),$ the electromagnetic field can overcome the Jeans instability as is evident from (2.41). Moreover, in the absence of surface charge and mass density, pure radiative gravity waves will propagate on the interface as

(2.43)\begin{equation} \omega^{2}=k\gamma_{g} \end{equation}

so that the group velocity of gravity radiation becomes, in terms of the wavenumber $k$ (and thus the wavelength $\lambda =2{\rm \pi} /k$)

(2.44)\begin{equation} \frac{\textrm{d}\omega}{\textrm{d}k}=V_{g}=\sqrt{\frac{\gamma_{g}}{4k}}=\sqrt{\frac{\gamma_{g} }{8{\rm \pi}}\lambda}. \end{equation}

2.3. Dissipative instability

Finally, in the presence of a dissipative surface effect, i.e. $\nu \neq 0$, there may exist, in (2.36), values of propagation vector

\begin{align*} k_{\pm }=({1}/{b^{2}})(V_{E}^{2}-V_{A}^{2}\cos ^{2} \theta )\pm ({1}/{b^{2}})\sqrt {(V_{E}^{2}-V_{A}^{2}\cos ^{2}\theta )^{2} -(b^{2})^{2}( \gamma _{g}^{{\prime }}-4{\rm \pi} G\rho _{si}) },\end{align*}

for which (2.36) reduces to

(2.45)\begin{equation} \omega^{2}+\frac{\textrm{i}\gamma_{g}^{{\prime}}\left( \nu/\omega\right) }{1+(\nu^{2}/\omega^{2})}=0, \end{equation}

or, assuming $|\nu |^{2}\ll \omega ^{2}$,

(2.46)\begin{equation} \omega^{3}+\textrm{i}k_{{\pm}}\gamma_{g}^{{\prime}}|\nu|\approx 0, \end{equation}

where $k_{\pm }=({1}/{b^{2}})(V_{E}^{2}-V_{A}^{2}\cos ^{2}\theta )\pm ({1}/{b^{2}})\sqrt {(V_{E}^{2}-V_{A}^{2}\cos ^{2}\theta )^{2}-(b^{2})^{2}( \gamma _{g}^{{\prime }}-4{\rm \pi} G\rho _{si}) }$. Among the three possible complex roots of the above equation, one root leads to instability of surface waves due to electromagnetic (thermal) radiation in the plasma medium with a growth rate, say $\hat {\gamma }$, given by

(2.47)\begin{equation} \hat{\gamma}=\frac{\sqrt{3}}{2}\left( k_{{\pm}} \gamma_{g}^{{\prime} }\mid\nu\mid\right) ^{{1}/{3}}, \end{equation}

provided that radiation acts for a short time, say $\tau \ll \omega ^{-2}$ (viz. $|\nu |^{2}\tau ^{2}\ll 1)$. Equation (2.47) shows that the growth rate of the radiative dissipative instability depends on the temperature inhomogeneity (due to radiative heat flux) as well as the thermal radiation pressure.

3. Numerical analysis

We have investigated the dispersion relation, as well as the growth rate of unstable modes, relying on (2.39), (2.41), (2.44) and (2.47) in their respective regions of validity, both numerically and graphically, to explore the impacts of the surface charge and mass densities, as well as the radiation energy flux and thermal radiation pressure on the growth rate of the surface Jeans instability in an incompressible, self-gravitating, radiative electron–ion plasma. For this purpose, we have chosen a set of representative values for the low-density and high-temperature plasmas (Post et al. Reference Post, Jensen, Tarter, Grasberger and Lokke1977) as $n_{i0}=n_{e0}=n_{0}\approx 10^{12}\ \textrm {cm}^{-3}$, $T_{e,0}=( 2\times 10^{4}-8\times 10^{4})\ \textrm {K}$ and $B_{0}=5.88\ G$. By substituting these parameters and taking the surface coordinate $x=0.12\ \textrm {cm}$ and $T_{e,0} =2\times 10^{4}\ \textrm {K}$ in (2.41), one can easily calculate various physical parameters, such as the electric Alfvén velocity $V_{E}(={E_{0}^{2}}/{4{\rm \pi} \rho _{i}}={4{\rm \pi} \sigma _{se}^{2} }/{\rho _{i}}) =1.28\times 10^{6}\ \textrm {cm}\,\textrm {s}^{-1}$ and the gravitational radiation acceleration $\gamma _{g}=2.70\times 10^{9}\ \textrm {cm}\,\textrm {s}^{2}$; the imaginary part associated with the surface Jeans instability turns out to be $\textrm {Im}(\omega )=6.05\times 10^{3}\ \textrm {s}^{-1}.$ Furthermore, if $\gamma _{g}$ increases to a higher value of $9.10\times 10^{9}\ \textrm {cm}\,\textrm {s}^{2}$ (for $T_{e,0}=3\times 10^{4}\ \textrm {K}$), while keeping all other parameters fixed, the magnitude of the imaginary part reduces to $\textrm {Im}(\omega )=8.92\times 10^{2}\ \textrm {s}^{-1}$. This analysis clearly depicts that inhomogeneous gravity radiations tend to stabilize gravitational collapse as shown in (2.36), (2.39) and (2.41).

Figure 1(a) depicts the onset of instability of Jeans surface waves under the influence of gravitational radiation acceleration, as described by (2.39). For $\gamma _{g}=9.13\times 10^{6}\ \textrm {cm}\,\textrm {s}^{2}$, the threshold value of the Jeans wavevector is $k_{J}=3.25\ \textrm {cm}^{-1}.$ Increasing $k$ beyond $k_{J}$ tends to suppress the growth rate of the instability due to the competition between the surface charge and mass densities and the gravitational acceleration along with magnetocapillary effects in the magnetized plasma under consideration; the growth rate at the specific value of $k( =2.5\ \textrm {cm}^{-1})$ is $8.71\,\times 10^{4}\ \textrm {s}^{-1}$ and if the wavevector increases further, this results in a decrease in Im$\omega =6.06\times 10^{4}\ \textrm {s}^{-1}$. Furthermore, by increasing the gravity radiation ($\gamma _{g}$) through $T_{e,0}$, the threshold value of $k_{J}$ decreases significantly to reduce the frequency spectrum of magnetocapillary gravity surface waves, while the stability rate increases quickly as a function of $\gamma _{g},$ pointing out that gravitational collapse may be stabilized through gravitational radiation acceleration. Moreover, the variation of Jeans surface wave instability as a function of electron temperature ($T_{e,0}$) and wavenumber ($k$) can be inspected from figure 1(b). Similar results can be visualized in figure 2, which shows a comparison of the growth rate of the surface Jeans instability with and without the effect of gravity radiation.

Figure 1. (a) The angular frequency ($\omega$) of magnetocapillary gravity surface waves (as given in (2.39)) is shown as a function of the wavenumber ($k$) for different values of the gravitational radiation acceleration parameter, namely $\gamma _{g}=9.13\times 10^{6}\ \textrm {cm}\,\textrm {s}^{2}$ (blue curve) and $\gamma _{g}=5.82\times 10^{8}\ \textrm {cm}\,\textrm {s}^{2}$ (red curve). The solid curves are for the imaginary part Im($\omega$) (occurring below a certain wavenumber cutoff; cf. panel b), while the dashed curves are for Re$(\omega )$. (b) The variation of the instability in Jeans surface waves is plotted against the electron temperature ($T_{e,0}$) and the wavenumber ($k$).

Figure 2. Comparison of the growth rate of the surface Jeans instability without gravity radiation (blue curve) and with gravity radiation (red curve). The solid curves are for $\operatorname {Im}(\omega )$, while the dashed curves are for Re$(\omega )$.

Next, in order to visualize the impact of surface charge on the fragmentation of astrophysical objects, (2.39) is plotted in figure 3 in the absence of $\gamma _{g}$. This shows an increase in the value of the Alfvén velocity $( V_{E})$, through equilibrium surface charge density of electrons $( \sigma _{se})$, results in the increase of the amplitude of surface oscillation. Furthermore, the group velocity $( V_{g}=\sqrt {\gamma _{g}/4k})$ of pure gravity radiations (shown in (2.41)) propagating at the plasma–vacuum interface is plotted as a function of wavenumber $(k)$ in figure 4 at fixed gravity radiations $\gamma _{g}=9.13\times 10^{6}\ \textrm {cm}\,\textrm {s}^{2}$, to depict that the group velocity becomes a function of wavelength $( \lambda =2{\rm \pi} /k)$. Finally, in order to visualize the influence of temperature inhomogeneity on the growth rate of radiative dissipative instability of self-gravitating electron–ion plasma, (2.47) is plotted in figure 5. Figure 5 suggests that the growth rate increases with an increase of the thermal radiation energy density and the radiation heat flux via $T_{e,0}$ via $\gamma _{g}^{{\prime }}.$

Figure 3. The frequency $(\omega )$ of magnetocapillary gravity surface waves in the absence of gravitational radiation (as described by (2.39)) is shown as a function of the wavenumber $(k)$ for different values of electric Alfvén velocity of ion grains, $V_{E}=1.285\times 10^{6}\ \textrm {cm}\,\textrm {s}^{-1}$ (red curve) and $V_{E}= 1.284\times 10^{6}\ \textrm {cm}\,\textrm {s}^{-1}$ (blue curve). The solid curves are for $\textrm {Im}(\omega )$, while the dashed curves are for Re$(\omega )$.

Figure 4. Group velocity $( V_{g})$ (as shown in (2.44)) as a function of the wavenumber $(k)$ at fixed values of $\gamma _{g}.$

Figure 5. The growth rate of gravitational radiation instability $( \hat {\gamma })$ (as given by (2.47)) is plotted against the wavenumber ($k$).