Reduced model of plasma evolution in hydrogen discharge capillary plasmas

A model describing the evolution of the average plasma temperature inside a discharge capillary device including Ohmic heating, heat loss to the capillary wall, and ionization and recombination effects is developed. Key to this approach is an analytic quasistatic description of the radial temperature variation which, under local thermal equilibrium conditions, allows the radial behavior of both the plasma temperature and the electron density to be specified directly from the average temperature evolution. In this way, the standard set of coupled partial differential equations for magnetohydrodynamic (MHD) simulations is replaced by a single ordinary differential equation, with a corresponding gain in simplicity and computational efficiency. The on-axis plasma temperature and electron density calculations are benchmarked against existing one-dimensional MHD simulations for hydrogen plasmas under a range of discharge conditions and initial gas pressures, and good agreement is demonstrated. The success of this simple model indicates that it can serve as a quick and easy tool for evaluating the plasma conditions in discharge capillary devices, particularly for computationally expensive applications such as simulating long-term plasma evolution, performing detailed input parameter scans, or for optimization using machine-learning techniques.

Figure 1

Radial temperature profiles $T\left(r\right)$, as defined in Eq. (5), for four average temperatures $\langle T\rangle$. The dashed vertical lines mark the boundary $R_b$ between the plasma-dominated region ($\kappa \propto T^{5/2}$) and the neutral-dominated region ($\kappa \propto T^{1/2}$) which occur at $T_b=0.9$ eV, represented by the dashed horizontal line.

Figure 2

Flowchart representation of the QUEST method algorithm. Flowchart symbols follow the ISO 5807 (1985) convention.

Figure 3

Comparison of QUEST simulation and 1D MHD simulation [8] results for electron temperature and electron density. The simulation conditions are given by “B” in Table 1. (a) Radial electron temperature profiles $T\left(r\right)$ for select times corresponding to the vertical lines in (d) and colored square markers in (f). (b) Radial electron density profiles $n_e\left(r\right)$ for select times corresponding to the vertical lines in (e). (c) Discharge current profile $I$. (d) On-axis electron temperature $T_0$. (e) On-axis electron density $n_{e0}$. (f) Variation of the on-axis electron temperature $T_0$ with current amplitude $I$. Solid lines indicate QUEST method results, while dotted lines indicate the (digitized) simulation results from Ref. [8] and blue crosses represent calculations using the simplified equilibrium model in Ref. [8]. The colored shaded bands in (d) and (e) represent the range of values between assuming the uniform regime and the quasistatic regime. The shaded grey region indicates the times at which the QUEST algorithm assumes uniform conditions, and marks the transition between the uniform and quasistatic regimes corresponding to the discrete jump in the electron temperature and density profiles.

Figure 4

Comparison of QUEST simulations and 1D MHD simulations [11] for on-axis electron temperature $T_0$ and electron density $n_{e0}$ as a function of time. The simulation conditions are given by G1–G6 in Table 1. Descriptions are the same as in Figs. 3. The electron density in G6 is increased by a factor of 10 to aid in visibility. The current discharge profiles are given in arbitrary units that are consistent across all plots.

Figure 5

Comparison of QUEST simulation results and those from Ref. [42] for average electron temperature $\langle T\rangle$ and electron density $\langle n_e\rangle$ as a function of time. The simulation conditions are given by “C” in Table 1. Solid lines indicate the QUEST method results, while dotted lines indicate the (digitized) simulation results from Ref. [42].

Figure 6

Variation of $\zeta (p,n)$ parameter, defined in Eq. (E1), with power $p$ for two temperature power laws $n$. The plasma limit corresponds to $n=5/2$, while the neutral limit corresponds to $n=1/2$.

Figure 7

(a) Comparison of the average temperature $\langle T\rangle$ evolution calculated by the zero-order Taylor series truncation (solid lines), and by including the full radial variation (dotted lines), for the simulation conditions G1–G4 in Table 1. The vertical lines indicate the transition time between uniform and quasistatic regimes. (b) Relative error $(\%)$ in average temperature $\langle T\rangle$ between the two methods in (a).