Radial foils are a simple and reliable platform to study high energy density plasmas (HEDP). This is an ideal setup to understand basic plasma processes or to impact the field of Laboratory Astrophysics. A metallic foil is stretched on the circular anode of the pulsed-power generator as shown to the right. The cathode is a simple hollow metal pin, contacting the foil at its geometrical center. Before delving into the mechanisms driving radial foil discharges, we present a simple electromagnetic model describing the basic forces involved in such experiments. the electron current flowing from the cathode to the anode establishes a toroidal field \(B_\theta\) around the pin. This field interacts with the current in the foil and generates an upward force, which lifts the foil. However this lift is not uniform. In fact, one can clearly see that the magnetic field decays as \(\frac{1}{r}\) where r is the distance from the center of the pin cathode, i.e.
\(B_\theta = -\frac{\mu_0 I}{2\pi r}\overrightarrow e_\theta\)
\(\overrightarrow e_\theta\) is the cylindrical coordinate system vector pointing in the toroidal horizontal direction. \(I\) is the total current.It can be shown that the force density \(f\) in \(N/m^3\)
\(f=\frac{\mu_0}{h}(\frac{I}{2\pi r})^2\), as shown in the figure below.
Here \(h\) is the foil thickness. This simple models shows that the force density is controlled by the plasma current but also the geometrical foil parameters. While changing the current on a pulsed-power generator is not easily achievable (and raising above its maximum value not recommended), changing the foil thickness or the pin radius \(r\) allow to cover a wide range of plasma parameters. As in wire arrays, radial foil dynamics goes through several phases, which we decided to highlight by using a series of schematic diagrams. It is possible to group these different phases into two distinct regimes. In the first regime, there are little instabilities and violent events are absent from the actual discharge. As instabilities appear in the second regime, they break the symmetry and trigger energetic plasma responses. The figure below summarizes the experimental evidence presented in the remainder of this paper. Starting from the initial plane configuration a, vertical JB forces slowly lift the foil plasma. The surface plasma produced above the foil is pushed upwards, expands quickly, and forms what we refer as the background plasma (\(n_e<<10^{18} cm^{-3}\)) However, most of the initial foil mass turns into a dense plasma \(n_e = 10^{19} cm^{−3}\).
The surface plasma formed below the foil accretes there as the dense foil plasma impedes its upward motion. This surface plasma plays an important role in connecting the dense plasma, moving upward, to the cathode pin, which is stationary At this stage most of the current flows in the dense foil plasma and the cathode, resulting in the strong toroidal magnetic field below the foil. A plasma jet forms on axis as most of the mass and force density peak there. The jet is surrounded by a low density plasma \(n_e=10^{18} cm^{−3}\). This is the front of the dense foil plasma, expanding faster due to lower density \(n_e = 10^{18} cm^{−3}\). This is the front of the dense foil plasma, expanding faster due to lower density.
This plasma carries a secondary current, which generates a weak toroidal field B above the dense foil plasma. As the foil continues its ascension a vertical plasma column forms above the cathode. This column is under radial compression i.e., pinch, unlike the rest of the foil plasma, which expands outward. As the column shrinks the jet becomes denser and its features are more defined. The continuous lift of the foil stretches the central plasma column vertically. At this stage, the \(J \times B\) force in the dense plasma has a radial component pointing outward and the plasma expands radially, forming a bubble-shaped cavity. As kink instabilities grow, a vertical magnetic field appears above the plasma foil, also measured by the Bdot probe. As the bubble grows, the outer dense plasma moves away from the central kinked plasma column. Due to the 1/ r magnetic field decay in the column periphery, the radial expansion of the bubble slows down.
On the other hand, the vertical plasma near the top of the bubble continues to accelerate steadily as the \(J \times B\) force does not weaken in this direction, resulting in the vertical elongation of the bubble. As kink instabilities grow, \(J\times B\) forces become asymmetric, shear forces appear, and the plasma sheet loses its integrity. Then the dense plasma sheet mixes with the low density and background plasmas. During this phase the current possibly reconnects at several locations. As the primary discharge current recedes from the dense plasma, the acceleration stops and the plasma becomes ballistic.
More information regarding the basic plasma dynamics of radial foils can be found in this publication. Instabilities studies have also been published. After the plasma properties were measured, and the dynamics analyzed we discovered that collimated plasma jets generated by radial foil plasmas can be used to understand the dynamics of celestial objects.
In particular, the Hall term has always regarded as negligible in most astrophysical plasmas where dynamo rules over most other plasma processes. The Hall effect generates electric fields as \(\frac{1}{en_e}\overrightarrow J\times \overrightarrow B\) while dynamo generates electric fields as \(\overrightarrow u \times \overrightarrow B\). The Hall effect is usually expressed when the ion inertial length \(\delta_i\) is on the order of or larger than the plasma scale length L. When resistivity is overcome by plasma dynamo, as it is often the case in space plasmas, then it is important to study the ratio of the electric field generated by the Hall effect \(E_H\) to the electric field generated by dynamo \(E_D\). We discover than the usual criteria
\(\frac{\delta_i}{L}\) should be replaced by \(\frac{\delta_i}{LM_A}\)
where \(M_A\) is the Alfvén Mach number. When looking at strongly collimated plasma jets with large \(Re\) and \(Re_M\), we discovered two puzzling effects:
- inside the jet, the usual Hall criterion was over estimating the impact of the Hall term by a factor of 2 to 5;
- despite the fact that the Hall term is weak in the jet, the jet was strongly influenced by the Hall effect.
These two contradictory statements can be easily reconciled. It is relatively easy to highlight the impact of the Hall effect since the plasma will behave differently when the electrical current flows in the opposite direction. When the electrical currents flow radially away from the plasma jet, that jet tends to be denser, taller and more stable than when the plasma current runs towards it. The figure to the left shows the palsma electron density for both current directions. The Hall effect impacts plasma jets because it is dominant in the low plasma density surrounding the jet itself. While it is weak inside the jet, the "rest of the universe" is dominated by Hall physics and the jets react to the Hall electric current surrounding it. This results was published in a PRL recently.