Measuring the equations of state in a relaxed magnetohydrodynamic plasma

We report measurements of the equations of state of a fully relaxed magnetohydrodynamic (MHD) laboratory plasma. Parcels of magnetized plasma, called Taylor states, are formed in a coaxial magnetized plasma gun, and are allowed to relax and drift into a closed flux conserving volume. Density, ion temperature, and magnetic field are measured as a function of time as the Taylor states compress and heat. The theoretically predicted MHD and double adiabatic equations of state are compared to experimental measurements. We find that the MHD equation of state is inconsistent with our data.

Figure 1

A schematic of the experimental setup. A glass tube has been added in between the gun and the stagnation flux conserver (SFC) and is covered with a copper flux conserving shell. All three principal plasma diagnostics are located in the SFC at the location of the circle. $T_i$ is measured using ion Doppler spectroscopy along the vertical chord and $n$ is measured using HeNe laser interferometry along a horizontal chord. The long $\dot{B}$ probe array is aligned with the axis of the SFC.

Figure 2

A typical time trace of (a) plasma density, (b) ion temperature, and (c) magnetic field measured inside the stagnation flux conserver. The error bars for $T_i$ are indicated by the vertical bars at each time value whereas the uncertainty in $n$ and $B$ is $\sim 10\%$. The Taylor state enters the SFC at $44\mu \text{s}$ (indicated by the blue bar) and then compresses from 50 to 54 $\mu \text{s}$ against the closed end of the SFC, accompanied by a rise in $T_i$ (indicated by the pink bar).

Figure 3

(a) A time trace (for the same shot as Fig. 2) of the length of the Taylor state, (b) an associated increase in plasma pressure, and (c) an excursion of the thermodynamic state of the object in a $PV$ diagram for the compression process. Note that as the volume of the Taylor state decreases, the pressure shifts to higher isotherms.

Figure 4

Statistical variation of three equations of state for 192 compression events (length contraction ranges from 10% to 30%): (a) The magnetohydrodynamic equation of state for three-dimensional (3D) compression ($\gamma =5/3$), (b) perpendicular, and (c) parallel CGL equations of state as a function of time. The blue curve in each panel shows the standard error of the mean. Note that the MHD equation of state (a), and the perpendicular version of the CGL EOS (b) have a nonzero time derivative. However, the parallel version of the CGL EOS (c) has a nearly zero time derivative for most of the compression time.