## 1 Introduction  
Atmospheric pressure (AP) gas-liquid discharge plasma has been actively investigated for its promising application prospects in water purification [1; 2; 3], biomedicine [4; 5; 6], and materials processing [7; 8; 9]. However, random micro discharge channels occur easily which can lead to non-directed energy distribution leading to serious issues and adversely affecting industrial applications. Hence a stable gas-liquid discharge at AP is expected. Over the last two decades, various factors have been studied for improving discharge stability, including the electrode structures [10; 11], the driving power supply [12; 13], the working gases characteristic [14; 15], and many other experimental parameters. In addition, the plasma characteristic could be different even under the same conditions [16], because it is influenced by electron transport. Thus, it is necessary to discuss the mechanism that leads to the change of the discharge mode, such as the glow to arc transition (GAT), which is a common result of thermal and electronic instability. Both of those instabilities are intensified by positive feedback via Joule heating [17]; more specifically, a slight increase in temperature can lead to a significant increase in vibrational-translational relaxation rate, which improves gas temperature further via their exponential relationship [18]. In addition, the heat release happens in a small volume that allows thermal instability to develop rapidly. Nanosecond pulse voltage can be used to generate stable non-thermal plasma, where the fast-rising time can prevent the GAT effectively. Since the rising time of nanosecond pulse voltage is shorter than the character time of gas heating (in microsecond scale) [19], the increase in gas temperature is not obvious. When the pulse voltage is applied, the rapidly rising voltage pulse induces a high electric field, which promotes the occurrence of non-equilibrium plasma [20], and the electrical energy delivered in the plasma discharge is mainly deposited in the energetic electrons instead of heating the heavy particles. Therefore, the nanosecond pulse voltage can be employed to generate a stable gas-liquid discharge with high electron density and low gas temperature [21; 22]. [PERSON] et al. [23] studied the influences of peak voltage and discharge gap distance on pulsed discharge behavior over the liquid surface. [PERSON] et al. [24] reported the physical process at the gas-liquid interface in a single pulse mode and high-frequency burst. [PERSON] et al. [25] compared the pulsed discharge plasma and sinusoidal discharge plasma, the results showing that nanosecond pulsed discharge plasma has lower gas temperature and higher efficiency of bacterial inactivation in liquid than that of the sinusoidal discharge plasma. [PERSON] et al. [26] used bipolar nanosecond pulse power to obtain a gas-liquid discharge plasma with electron density in the order of \(10^{15}\) cm\({}^{-3}\) and gas temperature of 350 K. They studied it further and reported in [27] that the electron density can be elevated to the order of \(10^{17}\) cm\({}^{-3}\) for the transient to spark discharge, with a gas temperature of 380 K.  
In addition, the floating electrode (FE), as a kind of independent structure between electrodes, shows the potential to improve the discharge plasma characteristics [28; 29]. As [PERSON] et al. [30] reported, the FE can influence the place of initial current density increase. [PERSON] et al. [31] compared electric field intensity distributions in a plasma jet with and without the inner-FE. The results show that the removal of FE improves plasma reactivity and lowers gas breakdown voltage. [PERSON] et al. [32] developed a diffuse-like discharge and studied the spatial electric field distribution when the FE is set. The report shows that the spatial electric field near the FE is advantageous for generating uniform-density plasma. But there are few studies on the gas-liquid discharge when FE is employed.