Experimental and 2D fluid simulation of a streamer discharge in air over a water surface

The high reactivity and attractive properties of streamer discharges make them useful in many applications based on plasma-surface interactions. Therefore, understanding the mechanisms governing the propagation of a streamer discharge as well as its properties is an essential task. This paper presents the development and application of a 2D fluid model to the simulation of discharges triggered at the air-water interface by a pulsed nanosecond high voltage. Experimental characterization using 1 ns-time-resolved imaging reveals rapid transitions from a homogeneous disc to a ring and finally to dots during the discharge process. The simulation enables the determination of the spatio-temporal dynamics of the E-field and electron density, highlighting that the discharge reaches the liquid surface in less than 1 ns, triggering a radial surface discharge. As the discharge propagates along/over the water surface, a sheath forms behind its head. Furthermore, the simulation elucidates the transitions from disc to ring and from ring to dots. The former transition arises from the ionization front’s propagation speed, where an initial disc-like feature changes to a ring due to the decreasing E-field strength. The ring-to-dots transition results from the destabilization caused by radial electron avalanches as the discharge head reaches a radius of ∼1.5 mm. The simulation is further utilized to estimate a charge number and a charge content in the discharge head. This work contributes to a better understanding of discharge propagation in air near a dielectric surface, with the agreement between simulation and experiment validating the model in its present version.

In air at atmospheric pressure, the application of an Efield higher than the breakdown field of air (∼30 KV cm −1 ) could ionize the medium.If the generated electron avalanche locally produces charges larger than 10 8 (Meek's criterium [2]), the avalanche transits to a streamer.This transition is mainly due to the high space charge E-field that results from charge (electrons and positive ions) separation.With the aid of photoionization, this space charge field induces further electron avalanches at the streamer head, ensuring thus its propagation [6].
The interaction between a streamer discharge and a surface is encountered in the development of applications.In such a context, two steps of propagation occur: propagation in the gap and at the surface of medium that can be a solid or a liquid.The propagation in the gap is rather well understood and is mainly based on the space charge field at the streamer head and photoionization.However, the propagation at/over the surface is less understood due to the presence of a complex air-plasmamedium interface characterized by discontinuities of several parameters, such as density, chemical composition, dielectric permittivity, electrical conductivity, and others.Studies reporting such discharges, also known as surface discharges [7], are numerous [8][9][10].However, those addressing the transition from propagation in the gap to the surface are scarcer.Zhang et al [11] investigated the propagation of a discharge between a pin and a dielectric surface (ε r ∼ 8) located at 4 mm from the pin.They reported that when the discharge head approaches the dielectric surface, the production of photoelectrons at the discharge front decreases, which makes vertical propagation to cease.Meanwhile, charges accumulate on the dielectric surface, which induces a radial E-field and can trigger the radial propagation of discharges guided by the surface.
Many studies have reported the behavior of discharge propagation at the surface of dielectric materials.In the case of liquid surface, various structures (emitted light) on the surface were observed, such as dots, rings, discs, etc [12][13][14][15].The structures are sensitive to the experimental conditions (solution electrical conductivity and acidity, voltage polarity, discharge current, gas composition, etc.) as well as to the sensitivity/resolution of the diagnostic technique, such as the time exposure of the camera.For instance, Bruggeman et al [16] showed that a DC discharge on water anode appears as a homogeneous disc at an exposure time of 20 ms while moving dots appear at an exposure time of 100 µs.This behavior was also observed at shorter timescale in the case of 500 ns-pulsed DC single discharges.Indeed, in the case of a water anode, the discharge appears as a homogeneous disc for 1 µs-or 1 ns-integrated images [12].In the case of a water cathode, the 1 µs-integrated discharge show organized or unorganized filaments, depending on the voltage amplitude.
However, the 1 ns-integrated images show the propagation of highly organized plasma dots (streamers' head) whose speed, number, and charge depend on the discharge conditions, such as voltage amplitude, gap distance, and water electrical conductivity [17,18].
In the context of applications, it is crucial to quantify some fundamental parameters of the discharge, such as the charge deposited at the surface and the E-field at the streamer's head.For instance, we recently reported that increasing the gap distance by 200 µm results in the decrease in the number of streamers (i.e.plasma dot) by one streamer [17].Also, we observed that the increase of water electrical conductivity results in the reduction of the propagation length, among other effects [18].Allegraud and Rousseau [8] performed electrical measurements and measured the number and the charge of streamers propagating on a dielectric surface (ε r ∼ 5).They found a charge per filament of ∼ 1 nC.In a recent study, we utilized electrical measurement and high-speed imaging to determine the charge per streamer propagating at water surface.Depending on the voltage magnitude, we found a charge per plasma dot (i.e. a streamer head) of ∼ 3 − 5 nC.We also determined that the gap distance modifies the number of plasma dots propagating at water surface but not the charge per plasma dot.Kumada et al [19] employed a technique based on the Pockels effect to experimentally measure a 2D potential on a planar insulator (ε r = 2.7), hence allowing a precise evaluation of the charge in the head of the streamer.They found a charge per streamer of ∼0.01-0.02nC.This value is 100times smaller than that estimated from the techniques based on electrical measurements [8,17].
Many studies, including simulations, were performed to understand the fundamentals of the interactions between nanosecond discharge and a dielectric surface, particularly the transition from a volumetric discharge in the gap to a surface discharge.Currently, it is accepted that during discharge propagation in the gap, charges accumulate at the surface [11], resulting in a radial E-field sufficient to allow ionization [10].Depending on the experimental conditions, electron avalanches may transit to streamers, and the emission of the discharge may change drastically.In the case of positive nanosecond discharges over water cathode, a transition from discto-ring-to-dots was observed in less than ∼10 ns [13,17,18].However, in the case of the water anode, the emission was rather ring-like, and a transition from ring-to-dots was observed when the cathode pin was placed parallel to the water surface [20].In fact, the transition seems to be governed by different factors, mainly the E-field magnitude and direction, photoionization, and surface properties.Xiong and Kushner [21] have highlighted that such a ring-to-dot transition is probably due to the destabilization of the ring by random photoionization events.Moreover, Darny et al [9] investigated the influence of the surface properties on the emission behavior by discharging over SiO 2 -Si and glass plates.In the first case, the emission consisted of a homogeneous ring, while in the second case the emission consisted of filaments.The authors correlated these behaviors to the photoemissive properties of Si-SiO 2 , as the absorption of photons with energy higher than ∼ 1.1 eV may generate electron-hole pairs and influence the discharge propagation.
Motivated by the experimental investigation of nanosecond discharge in air over water [17,18], we aim here to simulate such a discharge by developing a simplified 2D fluid model.The model consists of solving the fluid equations for the density of electrons, positive ions, and negative ions.Such a simulation allows the determination of some fundamental properties of the discharge hardly accessible experimentally such as the spatio-temporal evolution of the species density and of the E-field.The emission of the discharge acquired during its propagation is compared to the source term of electronic impact avalanches.The results are further processed to discuss the disc-to-ring-to-dots transitions.

Experimental background
The discharges at the water surface that we aim to model in this study were recently investigated experimentally in our group [17].To properly introduce the discharge conditions, a brief description of the setup is made.A scheme of the setup is shown in figure 1(a).A pulsed nanosecond positive power supply (NSP 120-20-P-500-TG-H, Eagle Harbor Technologies) was used to ignite an electrical discharge in air in contact with distilled water.The magnitude (V a ) and the duration of the pulse were fixed at 14 kV and 100 ns, respectively.The anode was a tungsten rod (diameter of 1 mm), and its tip was mechanically polished to obtain an aperture angle of ∼ 30 • .The cathode is made of a stainless-steel rod (4 mm diameter) placed at the bottom of a cylindrical Teflon cell (diameter of 67 mm and height of 5.7 mm) filled with 20 ml of distilled water.The distance (d) between anode pin and water surface was fixed to 600 µm.
An ICCD camera (PIMAX-4: 1024 EMB, Princeton Instruments) was used to monitor the behavior of the plasma emission at the solution surface.The camera is equipped with an RB-type intensifier that covers the wavelength range of 200 − 850 nm with a quantum efficiency between 2 and 15% depending on the wavelength.The dimension of the captured zone was 10 mm × 10 mm.A delay generator (Quantum Composers Plus 9518 Pulse Generator) was used to adjust the delay between the ICCD camera and the voltage pulse.The discharge emission (20 ns-integrated ICCD image) in the gap and at the water surface are shown in figures 1(b) and (c), respectively.The last one highlights the radial propagation of highly organized streamers over the water surface.
The voltage and current waveforms of the discharges were measured using a high-voltage probe (P6015A, Tektronix) and a current monitor (6585, Pearson), respectively.The waveforms were visualized and recorded using an oscilloscope (MSO54, 2 GHz, 6.25 GS/s).Figure 2(a) shows typical voltage-current waveforms of a discharge produced under the abovementioned conditions, and figure 2(b) shows a zoom on the characteristics where the breakdown has occurred (around ∼ 30 ns).We note that the discharge is ignited during the rising period of the voltage, and it can be identified by the current peak of ∼ 15 A and by a breakdown voltage (V bk ) of ∼ 8.2 kV.
The oscillations in the current peak are probably related to an RLC component due to connections in the electrical circuit.Note that the electric field that corresponds to V bk is higher than the breakdown field of air at atmospheric pressure (∼35 kV cm −1 ), and this is due to the fast rising of the voltage.Experiment (not shown here) performed at lower rising rate showed a significant decrease of V bk .
The time-resolved emission of the discharge was investigated in detail in [17,18], and details on the synchronization and the determination of the temporal sequence can be found in [13].Figure 3 shows the temporal evolution of the discharge emission over the water surface (1 ns-integrated images) during the first 5 ns, where the various propagation steps can be identified.In fact, the streamer discharge reaches the liquid surface in less than 1 ns (i.e.unresolved here).Then, radial propagation takes place with different patterns.During the first nanosecond, the emission is disc-like, while 1 ns later, it expands radially and takes the form of a ring-like structure.Then, the ring continues to expand radially and, after reaching a critical radius (∼ 1.6 mm) at t = 2 ns, it breaks into multiple single plasma dots, i.e. ionization front of individual streamers.Finally, these plasma dots continue to expand radially until they fade away.Under the present conditions, the maximal spatial extent of the discharge was ∼4 mm (see figure 9); in fact, this parameter strongly depends on the experimental conditions such as the applied voltage, the gap distance, electrical conductivity, dielectric conductivity, etc.We believe that its extension is generally governed by the intensity of the radial electric field at the ionization front.

Streamer model equations
The numerical simulation is performed using a 2D axisymmetric cylindrical fluid model in air at atmospheric pressure.The temporal evolution of the density of electrons (n e ), positive ions (n i ), and negative ions (n n ) are determined by solving the following equations: Starting with the second term in the left side of equations ( 1)-(3), i.e. the drift-diffusion terms, the velocities are introduced using the mobility coefficient (µ) and the electric field ( ⃗ E).In the case of electrons, positive ions, and negative ions, the velocities are written as   diffusion coefficient (defined in A.1.4); the diffusion of the ions is not considered here.
The right side of equations ( 1)-( 3) represents the various gains and losses of each species.S ph represents the rate of electron-ion pair production by photoionization.The production rate of electron and positive ion from electron-neutral collisions is considering with S α [22]: where ⃗ Γ e = n e ⃗ v e − D e ⃗ ∇n e and α is the first Townsend's coefficient defined in A.1.5.
In the same way, the production rate of negative ions by electron attachment to neutrals is considered with S η [22]: where η is the attachment coefficient defined in A.1.6.Secondary emission from the liquid surface ( Sγ ) represents the production of electrons under the flux of positive ions towards the liquid surface and is written as [23]: where γ = 0.1 is the secondary emission coefficient taken from [24].To solve the electrical potential V and to compute the electric field ⃗ E in the domain, the fluid equations are coupled with Poisson's equation: ) with ϵ = ϵ r ϵ 0 is the permittivity of the medium (ϵ r is the relative permittivity and ϵ 0 = 8.85 × 10 −12 F • m −1 ).The source term of Poisson's equation is composed of the volume charge density ρ (where q = 1.602 × 10 −19 C) and of the surface charge density σ; the latter is implemented with a geometric factor δ s to consider the accumulation of charges on the dielectric surface (δ s = 1 ∆z , ∆z being the finest mesh resolution on the liquid surface).The surface charge is calculated as follows: where Γe , Γn , and Γi are the fluxes of electrons, negative ions, and positive ions toward the liquid respectively.If they are not directed toward the liquid, they are not considered (i.e.equal to 0).Finally, S ph is computed by solving Helmholtz's equations [25] with the coefficients utilized here being similar to those introduced by Bagheri et al [25] as defined in the annex A.2:

Computational domain and boundary conditions
The stochastic characteristic of the discharges studied here was highlighted in [17].Indeed, for given experimental conditions and an applied voltage of 14 kV, breakdown can occur In water (ϵ r = 80), all densities as well as photoionization were set to zero.The electrical conductivity of water was not considered here.Experimentally, this parameter has significant impact on the discharge behavior [18] and needs to be considered in a more complete model.The ground is placed at the bottom of the liquid, and the Dirichlet boundary condition for Poisson's equation is used.As for the anode pin, the Dirichlet boundary condition is used for the voltage (8 kV), photoionization (S ph = 0), and positive ion density (n i = 0), while the Neumann boundary condition is used for n e and n n .In air (ϵ r = 1) above the liquid surface, the Neumann boundary condition is used to compute the flux when it is directed towards the liquid; otherwise, it is set to 0. To ignite the discharge, we first run the simulation with a uniform density (10 8 m −3 ) of ions and electrons.We remarked that the first stage of the simulation will produce a region close to the anode that has ions and electrons density of 10 19 m −3 with a Gaussian-like distribution.In the following and to gain computational time, Gaussian distributions of ions and electrons (n e,i ) near the electrode head are introduced (equation ( 13)), while the background density (n bgr ) is fixed at 10 8 m −3 , and the initial density of n n to 0: + n bgr (13) with n 0 = 10 19 m −3 , Z 0 = 4.59 mm, and σ = 50 µm.
To optimize the computational time, a non-uniform grid was used, with high resolution near the regions of interest such as the liquid surface [26].The grids of low resolution are used far from the regions of interest; more resolved grids in such regions do not change the results but elongate the computational time.The simulation is performed in a rectangular domain (figure 4) that has a size of 10.5 × 7 mm 2 with a grid size of 1047 × 1167.
Along the axial z-direction, the following dimensions were used: • From z = 0 to 3.9 mm, uniform grid with ∆z = 20 µm.
In the radial direction from r = 0 to 7 mm, a uniform grid with ∆r = 6 µm was used.A Finite Volume Method was used to solve the equations; it is fully described in Annex A.3.Electron fluxes are computed using a Scharfetter-Gummel scheme (A.3.4),because such a scheme is numerically stable for strong electric fields [26].The positive and negative ions fluxes are computed using 1st order upwind scheme (A.3.5),as their diffusion is not considered; the Scharfetter-Gummel scheme is in fact equivalent to a 1st order upwind scheme [27].We used a direct solver in python to solve Poisson's and Helmholtz's equations, and Ghost Fluid Method was used to capture the geometry of the pin with more accuracy (description of the utilized discretization is added in the annex A.4).The integration of the fluid equations was conducted using a 2nd order Runge-Kutta (annex 5, equations (A.5.1)-(A.5.3)).The time step was selected based on the CFL (Courant-Friedrichs-Lewy) conditions, detailed in equations (A.5.4) and (A.5.5)[28].Moreover, a correction of the electron flux is implemented to get rid of the restriction of the dielectric relaxation time to optimize the computational time (A.5.6 and A.5.7).The discretization of the equations (introduced above) is described in A.5.

Numerical results
Figure 5 shows the spatio-temporal evolution of E and n e .At t = 0, the initial conditions are shown with a maximal electron density of n e = 10 19 m −3 and an intense E-field at the tip of the anode with a maximal value of E = 332 kV • cm −1 .At t = 0.25 ns, the discharge is ignited and propagates vertically towards water surface with a maximal electric field E = 331 kV • cm −1 localized at the ionization front and where the density of electrons is also the highest with a maximal value of n e = 9 × 10 19 m −3 .The discharge propagates in the gap and approaches the water surface.Because of the discontinuity in dielectric permittivity between air and water, the E-field is enhanced, hence the electron density.The simulation shows that the discharge reaches the liquid surface within 0.4 ns.At this stage, positive charges accumulate on the water surface where a sheath is generated.For instance, at 0.4 ns, the highest E and n e at the water surface become 1267 kV • cm −1 and 3 × 10 21 m −3 , respectively.When the discharge reaches the liquid surface, its vertical propagation stops, and positive charge accumulates at the surface.Such an accumulation enhances the E-field, particularly its radial component, until radial propagation is ignited at t = 1.03 ns; at this moment, the highest E and n e at water surface reach 3842 kV • cm −1 and 5 × 10 21 m −3 , respectively.After neutralization of the accumulated charge at the water surface, a surface ionization wave starts to propagate followed by the formation of a sheath in its tail.In this time, we notice the decrease of both E and n e .For instance, one reads E = 792 kV • cm −1 and n e = 6 × 10 21 m −3 at t = 7.27 ns.At this point, it is important to compare the simulated values of E and n e with data available in the literature, where most of the studies are conducted at low permittivity.For instance, Ren et al [29] simulated the propagation of a discharge in air with an anode placed in contact with a dielectric surface with ε r = 4.2 (voltage magnitude and pulse were 10 kV and 300 ns, respectively).The authors reported E ∼ 240 kV•cm −1 in the sheath over the dielectric surface behind the streamer head and n e of 6.7 × 10 20 m −3 .Celestin [23] also simulated the propagation of discharge (voltage of 13 kV) over a dielectric surface (ε r = 5) in an air gap of ∼ 4 mm.They reported discharge dynamics similar to our finding with E ∼ 300 kV•cm −1 in the sheath behind the streamer head and n e ∼ 1.7 × 10 21 m −3 .Yoshiad et al [30] simulated the discharge in air between an anode pin and a dielectric (ε r = 10) surface located at 0.1 mm, and they found a maximum E-field in the sheath of ∼844 kV•cm −1 .
At this stage, the sheath formed at the water surface should be differentiated from the discharge head.The latter propagates slightly above the water surface and corresponds to the region (ionization front) detected experimentally.Figure 6 shows the temporal evolution of the E-field profiles (performed along the green dotted line shown in figure 5 att = 7.03 ns and at z = 4.06 mm) right above the water; the presented profiles corresponds to t = 0.4, 1.03, 2.44, and 7.27 ns (i.e. after ignition of the radial propagation).As the discharge propagates at the water surface and away from the anode, the E-field at the discharge head decreases significantly.For instance, at t = 0.4 ns, E (peak value) is ∼ 560 kV • cm −1 and decreases to ∼ 240 kV • cm −1 at t = 7.27 ns.These values are comparable to those reported by Hua and Fukagata [26] in their simulation of a discharge (at 12 kV) in air over a dielectric surface (ε r = 16).

Comparison of the model output with experimental results
Experimentally, the discharge is identified by the emission captured by the ICCD camera.The radial position of the discharge head was previously reported as a function of time [17].To compare experimental and simulated predictions, we consider that the source term of electronic impact avalanches (S α ) is a suitable parameter for comparison with the emission [31,32], as the emission of photons was not simulated.S α represents the number of electrons generated by collision between the energetic electrons and the particles in the medium.Considering that these electrons have enough energy to ionize, they also have enough energy to excite atoms and molecules in the gas [31,32].Reporting the position of the maximum of S α , it is possible to describe the temporal evolution of the radial propagation over the liquid surface.Also, the shortest ICCD integration time of our equipment is 1 ns, while the model provides instantaneous data on S α .To compare data over the same period, several instantaneous profiles of S α are  successively superimposed during a period of 1 ns. Figure 7(a) presents a superimposition of five instantaneous S α -profiles from 3 to 4 ns.∆r Dots ∼ 0.5 mm as shown on the experimental image at t = 3 ns in figure 7(c).Despite the little difference between the experimental and the numerical values, we believe that the agreement is quite acceptable.Therefore, the experimental emission of the plasma dots can be correlated to the propagation of the ionization front observed numerically during 1 ns.It is worth noting that, due to the decrease of the E-field, S α (peak value) decreases from 1.8 × 10 32 to 1.3 × 10 32 m −3 • s −1 as time goes from 3 to 4 ns.
Furthermore, we utilized 20 ns-integrated images (e.g.figure 8(a)) to compare the discharge emission profile along plasma filaments with S α .Figure 8(b) shows such a comparison.The normalized experimental data shows a slight change of the slope at r ∼ 1.6 mm.In fact, this position corresponds to the edge of the disc, and the smoother slope (at r > 1.6 mm) corresponds to the filaments.On the other hand, the normalized S α data obtained by simulation shows a very similar behavior with a slope change at ∼1.4 mm.Such a great correlation between discharge emission and S α further validates that this quantity provides a convenient representation of the discharge emission.The continuous decrease of the intensity along the filament is expected, as it is strongly related to the E-field; this latter decreases continuously as the ionization front propagates radially.As for the change of slope, more details are provided in the discussion section.
Using this approach, it becomes feasible to compare the discharge position and propagation velocity predicted by the model with the experimental results (figures 9(a) and (b)).In figure 9(a), the uncertainty associated with each radius comes from their spatial extension ∆r Dots as defined in figure 7(c).These values are derived from the numerous 1 ns-integrated ICCD images (∼ 500 images).The agreement between the discharge position derived from simulation and experiment is excellent.Figure 9(a) shows the initial fast increase of the radius (it increases to 2.5 mm during the first 5 ns) followed by a slower increase (up to 3.35 mm after 12 ns).The propagation velocity was determined by fitting the experimental evolution of the discharge radius using a mathematical function r (t) = A + Be −t/τ .Then, the time derivative of the obtained function yields the velocity shown in figure 9(b) together with the uncertainty (gray shaded area).Experimentally, we measure an initial velocity of ∼ 0.5 against ∼ 0.9 mm•ns −1 numerically.This difference may be explained by the fast development of the discharge which is not resolved with the 1 ns-integrated images.Hence, the simulation results allow a precise estimation of the initial velocity.A few nanoseconds after discharge initiation, the experimental and numerical velocities become comparable.The decrease of the velocity during propagation is related to the decrease of the electric field at the streamer head, as discussed earlier (figure 6).

Discussion: pattern formation and properties of the plasma dots
The aim of this discussion is to deepen the understand of the radial propagation of the discharge, including the different steps observed experimentally, i.e. the uniform emission during the 1st ns, the transition to a ring at t = 2 ns, followed by its decomposition into several identical plasma dots.We also aim to finely report the properties of the plasma dots, namely its charge.

Uniform-to-ring transition
Simulation and experimental results are first compared in the 1 ns-period where a homogeneous disc is observed.The simulated instantaneous S α -profiles are superimposed and summed during a period of 1ns to match the exposure period of the ICCD images.Figure 10(a) shows the integrated S α -profiles from 0 to 1 ns (red curve) as well as the experimental profile of the intensity (black curve) obtained from the image shown in figure 10(b) (plotted in polar coordinate); the area of the pin is also added as a gray rectangle to simulate the unlighted region.One ns after ignition, the simulation shows that S α extends up to r = 1.1 mm, i.e. the radial extension ∆r is 1.1 mm.The experimental emission profile shows uniform emission with a radial extension of 1.4 mm.In fact, the uniform emission observed experimentally corresponds to the 1 ns-integrated emission of the discharge from ignition (at the pin) to the beginning of radial propagation.The discharge propagates very rapidly during this period and, therefore, it appears homogeneous.Figure 10(c) shows S α from 1 to 2 ns (red curve) as well as the experimental emission profile at t = 1 ns (black curve).These data indicate that S α extends from r = 1.2 to 1.8 mm, i.e. a radial extension ∆r = 0.6 mm.Experimentally, the emission at t = 1 ns (figure 10(d)) is centered at r ∼ 1.7 mm with an extension of ∆r ∼ 0.6 mm.Despite a shift in the radial position of S α compared to emission, their radial widths are in great agreement.As mentioned earlier, the ring-like emission observed experimentally is also a superposition of ionization fronts that propagated during 1 ns.Therefore, the emission of the uniform disc and the ring differ by the propagation velocity of the ionization front that is much faster initially.This drastic decrease of the propagation velocity as a function of time is strongly correlated to the E-field at the streamer head: E ∼ 500 and 400 kV • cm −1 in the periods of 0-1 and 1-2 ns, respectively.

Formation of the plasma dots and estimation of its parameters
The ring-to-dots transition involves an important mechanism that is based on the destabilization of the ionization wave.We believe that the destabilization is induced by the ignition of streamers in the front of the ionization wave.Indeed, with the assistance of photoionization, the ionization front of the ring generates radial electron avalanches at its head.The electrons produced by the avalanches join the discharge region behind the ionization front, while the newly produced positive ions ahead of the front generate an intense space charge field that will ignite new electron avalanches, and so on.While propagating at the surface, the electric field at the ionization front can be disturbed due to avalanches close to its head.Under slight perturbations, the initial circular ionization front will deform and break into a number of individual streamers (plasma dots).This scenario was already proposed by Xiong and Kushner [21] for discharges in air.In the conditions of our study, we aim to identify the moment when destabilization of the ionization front becomes significant.In fact, we previously highlighted a change in the slope of the discharge emission profile along the filament; this behavior was also correlated to the ionization source term derived from the simulation.The position that corresponded to the slope change is correlated with disc-to-ring transition.We further report in figure 11 the variation of radial E-field (peak value) as a function of r, and a transition can be observed at r = 1.4 mm that corresponds to the discharge at t = 1.46 ns with radial E-field of 375 kV • cm −1 .In fact, this moment corresponds also to the establishment of the surface discharge after being disconnected from the volumetric one (details below).
To further understand the occurrence of radial propagation, figures 12(a) and (b) show the distribution of the radial Efield at two moments: at 0.49, and 1.46 ns that correspond to a moment before and after the establishment of the surface discharge.As figure 12(a) shows, the accumulation of charge on the surface generates an intense radial E-field on the liquid surface with a maximal value of E = 412 kV • cm −1 .It is interesting to note that in addition to the surface propagation, a volumetric-radial expansion of the plasma channel is also noticed in the gap.This volumetric plasma is explained by the high E-field in this region (∼ 90 kV • cm −1 ) that continues to produce electron avalanches.It is worth noting that at 0.49 ns, the surface and volumetric discharges are connected.At 1.46 ns, the E-field at the surface becomes much higher than that of the volumetric one, which results in a faster propagation at the surface that leads to a disconnection between the two zones.Figures 12(c ∇n e ); its magnitude is provided by the colormap and not by the arrow length.At 0.49ns, the flux of electrons is globally along z-direction, while at 1.46 ns, the radial flux is more significant.Such a difference in electron fluxes may be linked to instability developments, and the high  The destabilization of the surface ionization front by radial electron avalanches produces plasma dots, i.e. streamers.This assumption can be further supported by reporting some properties of the discharge head, in particular the number of charged species.This latter can be determined by knowing the volume of the discharge head as well as the space charge density.The simulation provides information on the spatial extension of the ionization front along the r-and z-axes (∆r and ∆z, respectively), as well as on the space charge density in the discharge head (n h ).Because the simulation was 2D, its extension over the surface (r∆θ) can be derived from the experimental results.Figure 13(a) shows the space charge density derived from simulation at 1.46 ns, i.e. when the surface discharge is developed.Figure 13(b) is a zoom on the discharge head that clearly shows its extension along z-axis (∆z = 70 µm) and raxis (∆r = 40 µm).The space charge density at the streamer head is n h ∼ 1.6 × 10 20 m −3 .The extension along the θ-axis of the plasma dots is determined from figure 13(c) (it illustrates the moment when experimentally the ring starts to be destabilized): r∆θ ∼ 0.5 mm.The volume of a plasma dot can be thus estimated as r∆θ∆r∆z as shown in figure 13(d), and the number of charged species in the discharge head can be finally estimated as: This value is representative of the Meek's criterion [2] that defines the moment when an electron avalanche transits to a streamer, i.e. when the charge number reaches 10 8 .We have performed similar calculation of N s at later times during propagation and found that the values decrease with time but remain close to that of Meek's criterion until the extinction observed at t = 13 ns.
Finally, we conducted a comparison between the numerical and the experimental estimations of the charge (Q Dot ) and electric field (E Dot ) of a plasma dot.Experimentally, the charge of a plasma dot is determined using time-integration of the discharge current (Q total ) and the number of plasma dots (N): Q Dot = Q total /N.In the present condition of voltage, conductivity, and gap distance, Q Dot−exp.∼ 4 nC [17].From simulation, the number of charged species in the streamer head at 1.46 ns are estimated previously: N s = 2.2 × 10 8 .Knowing the elementary charge (q = 1.6 × 10 −19 C), it is feasible to estimate the charge in the discharge's head, noted Q Dot−simul.= qN s = 0.034 nC.Clearly, there is a tremendous gap between the experimental and calculated values, which cannot be attributed to uncertainties.Experimental data are very scarce in literature and can be separated into two categories.The first category is based on electrical measurements, while the second one is based on Pockels effect.For instance, Allegraud and Rousseau [8] investigated the discharge propagation over a 2 mm-glass plate (ε r ∼ 5), and estimated a charge per streamer of ∼ 1 nC; this value is comparable to what we have measured previously [17,18].However, using Pockels effect, Kumada et al [19] measured a charge per streamer of 0.01-0.02nC in similar discharge conditions.On the other hand, Tanaka et al [33] measured a charge density of 0.5 nC • mm −2 , which corresponds to a charge per streamer of ∼0.04 nC.Therefore, the values determined by Pockels effect are closer to the value estimated from our simulation.The high values of charge per streamer derived from electrical measurements can be explained by the fact that the measured current considers the total charge produced in the discharge during its propagation instead of the sole charge produced at the ionization front.Such an overestimation of the charge explains therefore the experimental estimation of the Efield generated by a single plasma dot (∼ 10 4 kV • cm −1 ) [17] while simulation provides a lower value of ∼ 500 kV • cm −1 .Therefore, coupling experimental measurements to the simulation allows a more precise estimation of the charge contained in the head of a streamer propagating at/above the liquid surface.

Conclusion
In this paper, we developed a 2D fluid model to simulate the discharge in air and at water surface.Such a discharge produced by pulsed nanosecond high voltage was characterized by 1-ns-time-resolved imaging, and fast transitions from homogeneous disc to ring to dots are observed.
The simulation allowed us to determine the spatio-temporal dynamics of E-field and electron density.We found that the discharge reaches the liquid surface in less than 1 ns, and then a radial surface discharge is ignited due to the accumulation of charge at the surface.While propagating at the water surface, a sheath is formed behind the head of the discharge.The highest value of the E-field in the sheath and at the discharge head are ∼ 3842 kV • cm −1 at t = 1.02 ns and ∼560kV.cm−1 at t = 0.40 ns, respectively, while the highest electron density is ∼ 5 × 10 21 m −3 at t = 1.02 ns.All the values decreased significantly as the discharge propagated radially.
The simulation was further exploited to understand disc-toring-to-dots transitions which is due to the propagation speed of the ionization front.Indeed, initially, the propagation is too fast and the 1 ns-integrated images do not allow to resolve the propagation.Therefore, a disc-like is observed.However, 1 ns later, the propagation speed decreases due to a decrease of the E-field, and a ring is thus observed.The ring-to-dots transition can be explained by the destabilization of the ring by radial electron avalanches.The simulation show that this phenomenon becomes efficient at t = 1.46 ns when the discharge head reaches a radius of ∼ 1.45 mm; the experimental transition was observed at t = 2 ns for a radius of ∼1.6 mm.On the other hand, both simulation and experiment were used to estimate the volume of the discharge head, i.e. plasma dot.Knowing the charge density, a charge number of ∼ 2.2 × 10 8 was determined, which agrees with Meek's criterion for avalanche-to-streamer transition.Finally, the charge contained in the discharge head was estimated to be 0.034 nC which is too small as compared to the experimental estimation from electrical measurements (∼4 nC) but agrees well with measurements based on Pockels effect (∼0.01-0.04nC).The value provided by electrical measurements is overestimated as it accounts for all the produced charge and not just the charge in the discharge head.Overall, the results reported in this paper provide more advanced knowledge on discharge propagation in air in contact with a dielectric surface.The agreement between simulation and experimental results validates this first version of the model that will be further improved to include other parameters.

Figure 1 .
Figure 1.(a) Scheme of the experimental setup used to generate nanosecond discharges in air in contact with distilled water.(b) and (c) show the emission profiles along the gap and over water surface, respectively, of a typical discharge captured with 20-ns-integrated camera.

Figure 2 .
Figure 2. Current-voltage waveforms of a typical discharge produced at Va of 14 kV, pulse width of 100 ns, and a gap distance of 600 µm: (a) overall view and (b) zoom on the breakdown region.

Figure 3 .
Figure 3. Temporal evolution of the discharge emission over water surface observed from 1 ns-integrated ICCD images.The discharge was produced using a high voltage of amplitude Va = 14 kV and a pulse width of 100 ns.

Figure 4 .
Figure 4. Simulation domain showing parameters, boundary conditions, as well as the grid.Note that only 1/4 of the points are added for clarity.

Figure 5 .
Figure 5. Spatio-temporal evolution of the discharge determined by the simulation: (a) shows the E-field distribution in kV • cm −1 and (b) shows the distribution of ne in m −3 .

Figure 6 .
Figure 6.E-field radial profiles along the green dotted line (drawn in figure 5 at t = 7.03 ns) that show E-field values at the discharge head propagating over the liquid surface.

Figure 7 .
Figure 7. (a) Ionization term Sα from the numerical simulation of the discharge propagating at the surface of water at 3.0, 3.2, 3.4, 3.7, and 4.0 ns.(b) 1 ns-integrated ICCD image of discharge emission at t = 3 ns plotted in polar coordinates.(c) Comparison between the experimental z-integrated intensity (normalized) and numerical profiles for the period 3−4 ns.
Figure 7(b) shows a typical experimental image of plasma dots at t = 3 ns plotted in polar coordinates to facilitate the identification of their positions.
Figure 7(c) compares the intensity profile over a 1 ns-integrated image of plasma dots (acquired at 3 ns, i.e. from 3 to 4 ns) to the S α -profiles over the same period.The latter is obtained by summing the S α -profiles along the z-axis to reproduce an equivalent emission profile.At 3 ns, S α peaks at ∼ 2.0 mm, while it peaks at ∼ 2.4 mm at t = 4 ns.Experimentally, the mean position of the plasma dots was defined as the position of the maximum of the black curve in figure 7(c), i.e. at r Dots ∼ 2.3 mm.The length of an experimental plasma dots distribution is

Figure 8 .
Figure 8.(a) 20 ns-integrated ICCD image of the discharge over the water surface.(b) Comparison between the maximum of the normalized ionization term as a function of r (red curve) and the normalized profiles of the experimental emission performed on the filaments (black dots).

Figure 9 .
Figure 9.Comparison between the simulation (red curve) and the experimental results (black points) of the temporal evolution of (a) the radius position of the discharge over the liquid surface and (b) the propagation velocity; the gray zone refers to the uncertainty.The green star in (a) indicates the moment of ring-to-filament transition.
) and (d) show the map of the norm of the E-field at 0.49 and 1.46ns of the discharge head at the surface.The arrows in the figures indicate the flux of electron ( ⃗ Γ e = n e ⃗ v e − D e ⃗

Figure 10 .
Figure 10.Comparison between the profiles derived from experimental data (normalized) and simulation (Sα) at (a) 0-1 and (b) 1-2 ns.(c) and (d) show the 1 ns-integrated ICCD images of discharge emission plotted in polar coordinates.

Figure 12 .
Figure 12. 2D map of the radial electric field (kV • cm −1 ) at (a) t = 0.49 ns and (b) t = 1.46ns.(c) and (d) show zoom of the streamer head; the electron flux (m −2 s −1 ) is also shown, and its magnitude is given by the colormap.

Figure 13 .
Figure 13.(a) Space charge density at t = 1.46ns.(b) is a zoom of the streamer head.(c) Experimental emission of the discharge at t = 2 ns that captures the transition from ring-to-dot in 1 ns-integrated ICCD image.(d) is a schematic illustration of the plasma dot volume that shows their dimensions r∆θ, ∆r, and ∆z.