## 4 Effect of Grounded Electrode Simulation Results
It is seen in Figure 8b that the ionization rate is four times higher with the GR substrate (curve 1) compared to the case without it (curve 2). The explanation of an increase of the ionization rate can be found from the analysis of the potential distribution. Figure 9 shows a zoom of the electric potential distribution near the liquid layer for two considered cases. The potential on the plasma-liquid interface \(\varphi_{w}\approx 1\) kV without the GR substrate beneath and \(\varphi_{w}=372\) V with the GR substrate. It is seen that without the GR substrate beneath the target (when the grounded substrate is 500 mm apart from the target) the electric potential on the plasma-liquid interface \(\varphi_{w}\) is more elevated since the potential drop between the liquid layer and the remote GR substrate is relatively large. The potential drop over the liquid layer is negligible since \(\epsilon_{w}=80\). The electric field strength is determined with the potential gradient according the Poisson equation for the electric potential \(\varphi\) and electric field \(\mathbf{E}\):  
\[\mathrm{div}\
abla\varphi=4\pi e(n_{e}-n_{i}),\mathbf{E}=-\
abla\varphi, \tag{1}\]  
where \(n_{e}\), \(n_{i}\) are electron and ion densities, \(e\) is the elementary charge. Thus, the elevation of \(\varphi_{w}\) on the plasma-water interface decreases the electric field near the surface and consequently the electron energy and the ionization rate. The white line in Figure 9a indicates a trace of the maximum of the ionization rate during streamer propagation from the discharge tube to the target. The potential profiles (over white line in Figure 9a) for different times during the streamer approaching the surface is shown in Figure 10 for the cases with and without the GR substrate. A higher electrical field near the liquid surface for the case with the GR substrate is provided with a steeper potential drop compared to the case without the GR substrate. Figure 9: Spatial distributions of electric potential (**a**) without the GR substrate, color palette is for 900–2100 V, and (**b**) with the GR substrate, color palette is for 240–1700 V, argon, \(U=4.9\) kV. The spatial distributions of the electron density, electron energy, normal component of electric field and ionization rate are shown in Figure 11 for enhanced plasma jet parameters with the GR substrate. Figure 10: Potential profile over z for different times, t = 381.1 ns (1) and 382.8 ns (2) for the case without the grounded substrate and t = 374.5 ns (3) and 375 ns (4) for the case with GR substrate, argon, \(U\) = 4.9 kV. The maximum of the electron density and energy are 2.5 \(\times\) 10\({}^{13}\), cm\({}^{-3}\) and 8.3 eV. The \(E_{z}\) and \(E_{r}\) are 56 kV/cm and 26 kV/cm, respectively. If the GR substrate is absent these values are smaller that leads to decrease of the rate of plasma enhanced chemical reactions. In conclusion of this section, in 2D fluid model simulations, the discharge ignition and streamer dynamics were studied for our experimental conditions. The streamer interaction with the target was calculated for the cases with and without grounded substrate beneath the target. It was shown that the electron density and energy, electric field strength and ionization rate considerably rise with the presence of the grounded substrate without increasing the discharge input power. This recommendation for the plasma jet enhancement was used in the cell-based experiments.