Simulations of a Conducting Sphere Moving through Magnetized Plasma: Alfvén Wings, Slow Magnetosonic Wings, and Drag Force

TL;DR

The study uses 3D MHD simulations to model a conducting sphere moving through a magnetized plasma, quantifying the drag via the dimensionless coefficient $C_A$ as a function of the Alfvén Mach number $M_ ext{A}$ and the plasma parameter $\beta$. It confirms that sub-Alfvénic drag is well described by the Alfvén-wing framework, while a slow magnetosonic wing provides a corrective, $\beta$-dependent contribution that becomes more pronounced as $M_ ext{A}$ increases toward unity. The results reveal that $C_A$ is roughly constant at low $M_ ext{A}$ but rises near $M_ ext{A}\approx1$, with higher $\beta$ amplifying the drag due to the slow-wing effect; in low-$\beta$ cases, the slow wing is strongly pressure-driven and more distinct from the Alfvén wing, whereas in high-$\beta$ cases the two wings align. These findings have implications for magnetic interactions in planet–star and binary compact-object systems and motivate extensions to include internal conductor fields and non-ideal MHD effects.

Abstract

Plasma-mediated interaction between astrophysical objects can play an important role and produce electromagnetic radiation in various binary systems, ranging from planet-moon and star-planet systems to binary compact objects. We perform 3D magnetohydrodynamic numerical simulations to study an ideal magnetized plasma flowing past an unmagnetized conducting sphere. Such flow generates magnetic disturbances and produces a drag force on the sphere, and we explore the corresponding drag coefficient as a function of the flow speed relative to Alfvén speed and the $β$ parameter of the background plasma. We find that the drag is generally well-described by the Alfvén wing model, but we also show that slow magnetosonic waves provide a correction through their own wing-like features. These give rise to a nontrivial dependence of the drag coefficient on the plasma $β$, as well as enhanced drag as the flow speed approaches the Alfvén speed.

Figures (11)

• Figure 1: A sketch of the simulated system while it is in a steady state. The black lines are the magnetic field lines in the $x$-$z$ plane, which asymptotically approach $\vb{B}=B_0\vu{x}$ for $r\gg R$, where $R$ is the radius of the conducting sphere. The blue lines are the velocity field lines in the $x=2R$ plane, which similarly approach an asymptotic value of $\vb{v}=v_0 \vu{z}$ for $r\gg R$. The orange sphere in the center represents the conductor, and the transparent orange cylinders emerging from it represent the theoretical locations of the Alfvén wing characteristics.
• Figure 2: Diagnostic parameters for our fiducial run, with $M_\mathrm{A} =0.3$, $\beta=2$, on the $x$-$z$ plane after the simulation has achieved a steady state. The conductor is shown as a gray sphere in the center of the domain. Vector quantities are plotted with field lines placed in front of a background which displays magnitude. The field lines initialize at $y=0.15$ when $r=10R$ and may vary in $y$ across the domain. From left-to-right, top-to-bottom, the quantities are (a) $\vb{B_\mathrm{bkg} }$, (b) $\vb{B}$, (c) $\vb{v}$, (d) $|\vb{\delta B}|$, (e) $\delta B_x$, and (f) $\delta B_z$. All evolved quantities show disturbances along the Alfvén wing characteristics. We note that the velocity field lines wrap around the wings since motion stops entirely along the characteristics.
• Figure 3: Density (a) and pressure (b) for our fiducial run, with $M_\mathrm{A} =0.3$, $\beta=2$, on the $x$-$z$ plane after the simulation has achieved a steady state.
• Figure 4: Diagnostic parameters for our fiducial run on the $x=-2R$ plane (seen from below) after the simulation has achieved a steady state. Shown are (a) $\vb{v}$ and (b) $\delta B_y$. The vector field lines for $\vb{v}$ initialize at $x=2.5R$ since there is a strong out-of-plane component close to the wing [see Figure \ref{['fid_visual']}(c)].
• Figure 5: Time evolution of the drag coefficient $C_\mathrm{A}$ (see Eq. \ref{['drag_coef']}) measured at different radii (see Eq. \ref{['energy_flux']}) in our fiducial run ($M_\mathrm{A} =0.3$ and $\beta=2$). For all simulations, the system is evolved until the value for $C_\mathrm{A}$ stabilizes, as shown in this case. $C_\mathrm{A}$ is evaluated at $r=\{1.25, 2.50, 3.75\} R$ (purple, magenta, and orange, respectively).