2. Plasma dynamics

In this article, we consider the plasma dynamics in an accretion disk near a slowly rotating compact object such as a black hole or neutron star. The slowly rotating object can be characterized by its mass $M$ and spin parameter $a=J/Mc \ll 1$ (in the units of half-Schwarzschild-radius $r_g=GM/c^2$), in which $G$ is the gravitational constant, $c$ is the speed of light and $J$ is the angular momentum (Misner, Thorne & Wheeler Reference Misner, Thorne and Wheeler1973; Thorne et al. Reference Thorne, Price and MacDonald1986; Hobson et al. Reference Hobson, Efstathiou and Lasenby2006). Moreover, the plasmas we consider here are (i) located at larger radii from the Schwarzschild radius ($r\gg 2r_g=GM/c^2)$ and (ii) moving with a bulk velocity much lower than the speed of light ($v\ll c)$.

The accretion disk considered here is assumed to be unmagnetized initially, implying the magnetorotational instability is not operational yet. It has been shown in the pure hydrodynamical set-up that angular momentum transport can still be sustained in these types of disks (Fragile & Anninos Reference Fragile and Anninos2005; Paoletti et al. Reference Paoletti, van Gils, Dubrulle, Sun, Lohse and Lathrop2012; Ghosh & Mukhopadhyay Reference Ghosh and Mukhopadhyay2021). Instead of investigating the plasma dynamics of the accretion disk which has settled to its equilibrium structure, we concentrate on the time scale when seed magnetic fields are being generated in the disk by mechanisms such as Biermann battery, spacetime curvature drive, gravitomagnetic drive, etc. (Mahajan & Yoshida Reference Mahajan and Yoshida2011; Bhattacharjee, Das & Mahajan Reference Bhattacharjee, Das and Mahajan2015; Bhattacharjee & Stark Reference Bhattacharjee and Stark2021). Therefore, our analysis of wave modes will be limited to the range of seed magnetic field values in the accretion disk or where there are relatively unmagnetized regions of a larger structure.

Under these approximations, the plasma dynamics of individual species (labelled $s$) can be represented by the equation of motion and the continuity equation, respectively (Thorne et al. Reference Thorne, Price and MacDonald1986; Bhattacharjee & Stark Reference Bhattacharjee and Stark2021),

(2.1)\begin{align} m_sn_s\left(\frac{\partial}{\partial t}+\boldsymbol{v}_s\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{v}_i & ={-}\boldsymbol{\nabla}p_s+q_sn_s\boldsymbol{E}+\frac{q_sn_s}{c}\left(\boldsymbol{v}_s +\boldsymbol{\mathcal{N}}\right)\times\boldsymbol{B} +\frac{m_sn_s}{c}(\boldsymbol{v}_s\times\boldsymbol{B}_g) \nonumber\\ & \quad +m_sn_s\boldsymbol{g}, \end{align}

(2.2)\begin{align} \frac{\partial n_s}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(n_s\boldsymbol{v}_s)& =0,\end{align}

where $m_s$ is the species mass, $n_s$ is the plasma number density, $\boldsymbol {v}_s$ is the velocity, $p_s$ is the pressure, $\boldsymbol {E}$ is the electric field, $\boldsymbol {B}$ is the magnetic field and $\boldsymbol {g}=-\boldsymbol {\nabla }\varPhi _G$ is the Newtonian gravitational acceleration. It should be noted here that the two new terms containing $(\boldsymbol {\mathcal {N}}\times \boldsymbol {B})$ and $(\boldsymbol {v}_s\times \boldsymbol {B}_g)$ on the right-hand side of (2.1) are corrections due to rotating spacetime with $\boldsymbol {B}_g$ representing the ‘magnetic’-type gravitomagnetic field and its corresponding vector potential $\boldsymbol {A}_g=c \, \boldsymbol {\mathcal {N}}$. The shift vector is divergence-free $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\mathcal {N}}=0$ in the Boyer–Lindquist coordinate (Thorne et al. Reference Thorne, Price and MacDonald1986) and can be interpreted as the angular velocity of zero angular momentum observers (also known as fiducial observers) in a stationary background spacetime. It should be noted here that the Lorentz force term in (2.1) has a coupling between the magnetic field and gravity through the term $\boldsymbol {\mathcal {N}}\times \boldsymbol {B}$. The $m_sn_s(\boldsymbol {v}_s/c\times \boldsymbol {B}_g)$ term on the right-hand side of (2.1) is a new force that can be attributed to the frame-dragging caused by the spacetime's intrinsic rotation. The consequence of this new force has been studied quite extensively in the context of fluid dynamics in black hole accretion disks and is known as the Bardeen–Petterson effect (Bardeen & Petterson Reference Bardeen and Petterson1975; Nelson & Papaloizou Reference Nelson and Papaloizou2000).

In an axisymmetric stationary spacetime near a weakly rotating object, the gravitomagnetic field takes the following form:

(2.3)\begin{equation} \boldsymbol{B}_g=\frac{2G}{c}\left(\frac{\boldsymbol{J}}{R^3}-\frac{3(\boldsymbol{J}\boldsymbol{\cdot} \boldsymbol{r})\boldsymbol{r}}{R^5}\right) \end{equation}

with its corresponding vector potential

(2.4)\begin{equation} \boldsymbol{A}_g=\frac{2G}{c}\frac{\boldsymbol{J}\times \boldsymbol{r}}{R^3}. \end{equation}

Here, $R=\sqrt {r^2+z^2}$ is the distance to the plasma element from the central mass, and $\boldsymbol {r}$ is associated with the cylindrical coordinate system $(r,\phi, z)$. If we take the angular momentum of the object $\boldsymbol {J}$ along the $z$-axis and assume the disk to be thin ($z\ll r$), then (2.3) becomes

(2.5)\begin{equation} \boldsymbol{B}_g=\frac{2G}{c} \frac{\boldsymbol{J}}{r^3}, \end{equation}

which can be expressed in the following normalized form:

(2.6)\begin{equation} \boldsymbol{B}_g=\frac{2 \tilde{a}c^2}{r_g}\frac{1}{\tilde{r}^3}\hat{z}. \end{equation}

Here we have used the following normalizations: spin parameter $\tilde a=J/Mcr_g$ and radial distance $\tilde {r}=r/ r_g$. The spin parameter $\tilde a$ can have a maximum value of $1$. However, in order for the gravitomagnetic approximation to be valid, we consider cases where spin parameter $|\tilde {a}|<0.3$. This specific range has also been confirmed to produce the frame-dragging effect when modelling the accretion disk near a rotating object by means of Newtonian physics (Chakrabarti & Khanna Reference Chakrabarti and Khanna1992).

3. Electrostatic wave modes in electron–ion plasma

To obtain the electrostatic wave modes in an incompressible electron–ion plasma, we need (2.1), (2.2) and Poisson's equation,

(3.1)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}=\sum_s n_sq_s. \end{equation}

First, we assume the background electric field $\boldsymbol {E}_0=0$ and a uniform background plasma density, i.e. $\boldsymbol {\nabla }n_{0s}=0$. Next, we look for wave modes in the electron–ion plasma with wavevector $\boldsymbol {k}=k\hat {x}$ in a background magnetic and gravitomagnetic fields ($\boldsymbol {B}_0,\boldsymbol {B}_g$) $|| \hat {z}$. Then, using the following perturbations: $n=n_{0s}+\delta n_s$, $\boldsymbol {E}=\delta \boldsymbol {E}|| \boldsymbol {k}$ and $\boldsymbol {v}_s=\boldsymbol {v}_{0s}+\delta \boldsymbol {v}_s$, we linearize (2.1), (2.2) and (3.1) and solve for the dispersion relation of the system, yielding

(3.2)\begin{align} & 1-\frac{\omega_{{\rm pe}}^2}{(\omega-\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{v}_{0e})^2-\gamma_e k_b T_ek^2/m_e-\tilde{\omega}_e^2/m_e^2}\nonumber\\ & \quad -\frac{\omega_{{\rm pi}}^2}{(\omega-\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{v}_{0i})^2-\gamma_i k_bT_ik^2/m_i-\tilde{\omega}_i^2/m_i^2}=0, \end{align}

where $T_e$, $T_i$ stand for electron and ion temperature, respectively; electron plasma frequency $\omega _{{\rm pe}}=\sqrt {4{\rm \pi} n_eq_e^2/m_e}$, ion plasma frequency $\omega _{{\rm pi}}=\sqrt {4{\rm \pi} n_iq_i^2/m_i}$, $\tilde {\omega }_s=(q_s/c B_0+m_s \omega _g)$ is the modified cyclotron frequency for both species, gravitomagnetic frequency $\omega _g=B_g/c$ and $\gamma$ is the adiabatic index for each species.

For simplicity in isolating the new physics, for the rest of this article we assume background flow $v_{0s}\approx 0$ for both species and write (3.2) as

(3.3)\begin{equation} 1-\frac{1}{\tilde{\omega}^2-\beta_e^2-\tilde{\omega}_{{\rm ce}}^2(1-\epsilon_e)^2}- \frac{1}{Z^2}\frac{\alpha}{{\tilde\omega}^2-\tilde{\gamma}\alpha\beta_e^2-\alpha^2 Z^2\tilde{\omega}_{{\rm ce}}^2\left(1+\dfrac{1}{\alpha Z}\epsilon_e\right)^2}=0. \end{equation}

In (3.3), we have used the following normalizations: $\tilde {\omega }=\omega /\omega _{{\rm pe}}$, $\tilde {\omega }_{{\rm ce}}=\omega _{{\rm ce}}/\omega _p$, ${\alpha =m_e/m_i}$, $\beta _e=\sqrt {\gamma _e}(k v_{{\rm th}(e)}/\omega _{{\rm pe}})$, $\tilde {\gamma }=\gamma _i/\gamma _e$, $q_i=Ze$, $\epsilon _e=\omega _g/|\omega _{{\rm ce}}|$, $\epsilon _i=1/Z\alpha \,\epsilon _e$ where thermal velocity for both species $v_{{\rm the}(s)}=\sqrt {k_bT_s/m_s}$, $Z$ is the atomic number, and electron cyclotron frequency $|\omega _{{\rm ce}}|=eB/m_ec$.

In (3.3), it should be noted that both the second and third terms on the left-hand side are modified by the factors $(1-\epsilon _e)^2$ and $(1+({1}/{\alpha Z})\epsilon _e)^2$, respectively. These two terms are explicit contributions from gravitomagnetism with the factor $\epsilon _e=\omega _g/|\omega _{{\rm ce}}|$ measuring the relative strength between the gravitomagnetic and magnetic field. In the absence of a gravitomagnetic field, (3.3) becomes the standard dispersion relation for electrostatic waves in a magnetized electron–ion plasma.

Figure 1(a) plots the dispersion relation for different values of $\epsilon _e$, where the traditional upper hybrid oscillation is represented by a black dotted line. For the rest of the article, we assume a hydrogen plasma for simplicity. For different values of $\epsilon _e$, we notice different vertical shifts in the frequency plot from the $\epsilon _e=0$ case (black line). As $|\epsilon _e|$ increases from 0, the frequency at $\beta _e=0$ shifts from the classical upper hybrid oscillation in a direction depending on the sign of $\epsilon _e$. For $\epsilon _e < 0$, the frequency is pushed above the classical value, whereas for $\epsilon _e > 0$, the frequency is pushed below it. In the latter case, we see that setting $\epsilon _e=1.0$ results in a starting frequency at $\omega _{{\rm pe}}$. In this case, there is an effective cancellation of the magnetic and gravitomagnetic fields, resulting in $\omega _s=0$. Here the wave behaves as a standard unmagnetized electron plasma wave. We note, however, that for $\epsilon _e>1.0$, the gravitomagnetic field will then no longer be cancelled by the magnetic field and the frequency will once again rise above the plasma frequency. At $\epsilon _e=2.0$, the mode will match the case of $\epsilon _e=0$, but now the mode is primarily influenced by the gravitomagnetic field as opposed to the magnetic field.

Figure 1. (a) Plot of $\omega$ versus $\beta$ for the upper hybrid wave with modifications due to different $\epsilon _e$ values. The classical upper hybrid oscillation is given in the dashed black line. (b) Plot of the cutoff frequency $\omega (\beta =0)$ in $\omega _{{\rm ce}}$, $\omega _g$ phase space. Contours are given in black and additional annotation shows the relative contributions of the gravitomagnetic and magnetic fields.

The interplay between the magnetic and gravitomagnetic fields can further be visualized by constructing a $\omega _{{\rm ce}}, \omega _g$ phase space plot showing the cutoff frequency $\omega (\beta _e=0)$ for each combination. Figure 1(b) gives $\omega _g$ and $\omega _{{\rm ce}}$ each in the range of $-1$ to 1 (still normalized to $\omega _{pe}$) and the colour coding is $\omega (\beta _e=0)$ using the same normalization. The diagonal blue region is where both fields are parallel and equal in magnitude in the plasma ($\epsilon _e=1)$; however, because the electron has a negative charge in the magnetic field and a positive effective charge (its mass) in a gravitomagnetic field, the effect of the fields on the electron will cancel. This implies that a purely magnetized plasma near a weakly rotating object can sustain a Langmuir wave perpendicular to the background field under the right conditions, which is possible because both magnetic and gravitomagnetic fields cancel in their impact on electrons.

In contrast, the yellow regions at the upper left-hand and lower right-hand corners indicate that magnetic and gravitomagnetic fields are colinear ($\epsilon _e=-1$) in the plasma. Here the force on an electron in the gravitomagnetic field will be in the same direction as that from the magnetic field, so the effective magnetic field or cyclotron frequency will increase. Contours of $\omega (\beta _e=0)$ are denoted in black. The remaining four regions in the phase space can be distinguished based on the relative magnitudes of the background magnetic and gravitomagnetic fields within the plasma. This implies unmagnetized or weakly magnetized plasmas near the object can sustain electrostatic wave modes mimicking those seen in magnetized plasma.

Figure 2(a) shows the lower hybrid wave dispersion for different values of $\epsilon _e$. In contrast to the high-frequency modes, we do not see any significant differences in the dispersion relations between $\epsilon _e$ values of the same magnitude but opposite sign. As the classical lower hybrid oscillation frequency depends on both the electron and ion cyclotron frequencies, there will always be one species where the gravitomagnetic field effectively increases the cyclotron frequency and one species where it effectively reduces this frequency. Under the assumption of $\epsilon _e \gg \alpha$, the lower hybrid oscillation (setting $\beta _e=0$) is

(3.4)\begin{equation} \omega_{{\rm LH}}^2 \sim \frac{\alpha \omega_{{\rm ce}}^2}{1+\omega_{{\rm ce}}^2}+\epsilon_e^2 \omega_{{\rm ce}}^2 \end{equation}

to lowest order in $\alpha$ and $\epsilon _e$. The dashed horizontal line shows the classical lower hybrid oscillation frequency, and all tested values of $\epsilon _e$ cause an increase from this value due to the $\epsilon _e^2$ dependence in the equation.

Figure 2. (a) Plot of $\omega$ versus $\beta$ for the lower hybrid wave with modifications due to different $\epsilon _e$ values. The classical lower hybrid oscillation is given in the dashed black line. (b) Plot of the cutoff frequency $\omega (\beta _e=0)$ in $\omega _{{\rm ce}}$, $\omega _g$ phase space. Contours are given in black.

In figure 2(b), we once again plot the phase space of the gravitomagnetic and cyclotron frequencies and colour code by the cutoff frequency $\omega (\beta _e=0)$, this time for the lower hybrid wave. When the magnitude of the magnetic field increases, the relative gravitomagnetic contribution diminishes. We note that no field cancellation is observed in this lower frequency mode in the plasma under this set of plasma conditions. The contour lines (black) form a unique island-type structure for lower values of both magnetic and gravitomagnetic fields, but the contours become mostly vertical in the case of $|\omega _{{\rm ce}}| \gg |\omega _g|$.