Excitation of magnetohydrodynamic modes with gravitational waves: A testbed for numerical codes

We consider a gravitational wave oscillating in an initially homogeneous, magnetized fluid. The fluid is perfectly conducting and isentropic, and the magnetic field is initially uniform. We find analytic solutions for the case in which the gravitational wave is linear and unaffected by the background fluid and field. Our solutions show how gravitational waves can excite three magnetohydrodynamic (MHD) modes in the fluid: Alfvén waves, and both fast and slow magnetosonic waves. Our analytic solutions are particularly useful for testing numerical codes designed to treat general relativistic MHD in dynamical spacetimes, as we demonstrate in a companion paper.

Figure 1

Perturbation of the magnetic field $\delta \mathit{B}=(\delta B_m+\delta B_p){\widehat{\mathit{B}}}_0$ induced by a standing gravitational wave when $\mathit{k}·{\mathit{v}}_A=0$. The unperturbed magnetic field ${\mathit{B}}_0$ is perpendicular to the wave vector of the gravitational wave $\mathit{k}=k\widehat{\mathit{z}}$. Both stationary modes $\delta B_p$ and $\delta B_m$ have the same $\sin kz$ spatial dependence, but they oscillate with different amplitudes and at different frequencies.

Figure 2

Gravitational-wave induced Alfvén mode $\delta B_A{\widehat{\mathit{b}}}_A$ when $\mathit{k}·{\mathit{v}}_A\ne 0$. The direction of perturbation ${\widehat{\mathit{b}}}_A$ is perpendicular to both ${\widehat{\mathit{B}}}_0$ and $\widehat{\mathit{z}}$.

Figure 3

Gravitational-wave induced magnetosonic modes ($\delta B_{m1}{\widehat{\mathit{b}}}_{m1}$ and $\delta B_{m2}{\widehat{\mathit{b}}}_{m2}$) and the particular solution ($\delta B_p{\widehat{\mathit{b}}}_p$) when $\mathit{k}·{\mathit{v}}_A\ne 0$. All three unit vectors ${\widehat{\mathit{b}}}_{m1}$, ${\widehat{\mathit{b}}}_{m2}$ and ${\widehat{\mathit{b}}}_p$ are perpendicular to $\widehat{\mathit{z}}$. All the standing modes have $\sin kz$ spatial dependence, but they oscillate with different amplitudes and at different frequencies.