Mechanism of Stepped Leaders in a Simple Discharge Model
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Mechanism of Stepped Leaders in a Simple Discharge Model
Hidetsugu Sakaguchi and Sahim M. Kourkouss
Department of Applied Science for Electronics and Materials,Interdisciplinary Graduate School of Engineering Sciences,Kyushu University, Kasuga, Fukuoka 816-8580, Japan
We construct a one-dimensional model for the stepped leader in the filamental discharge bysimplifying an electric-circuit model of discharge. We find that the leader of the discharge movesstepwise by direct numerical simulations, and then we try to understand the mechanism of thestepwise motion by reducing the spatially extended system to the dynamics of the tip position ofthe discharge.
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I. INTRODUCTION
The discharge occurs when the voltage between two electrodes is beyond a critical value. There are various formsof discharge, such as corona discharge, spark discharge, glow discharge, and arc discharge, depending on variousconditions, such as pressure, temperature and the shape of electrodes. For gas discharge, the pressure p and the gaplength l between the electrodes are important. When p × l is large, the spark discharge takes a form of a filament.Meek proposed the streamer theory for the filamental discharge. [1] The tip region of a growing filamental dischargeis sometimes called a leader. It is known that the leader grows stepwise in the lightning discharge. The gap distancebetween the electrodes is very long in the lightning discharge. [2] The leader goes down from a thunder cloud to theground relatively slowly. The velocity is around 10 m/s. The extending process of the leader is invisible to theeyes, but it can be observed with a high-speed camera. The leader moves in steps of about 30 m with a pause ofabout 40 ms between steps. This type of leader is called the stepped leader. When the leader reaches the ground,a strong flash called a return stroke appears instantly, which we observe as lightning. However, the origin of thestepped motion is not completely understood. There were several qualitative theories for the stepped leaders. Brucepointed out the importance of the transition from a weak glow discharge to a strong arc discharge. [3] Kumar andNagabhushana proposed a simulation model of stepped leaders based on a complex electric breakdown process. [4] Ina previous study, we proposed a simple deterministic electric circuit model composed of resistors and capacitors, andperformed a numerical simulation on triangular lattices to investigate complex patterns in discharge processes. [5] Inthe model, a two-step function for the conductance is assumed, which expresses a transition from a weak discharge toa strong discharge. Branched patterns of the discharge similar to the Lichtenberg figure were found in the numericalsimulation of the model. We found a stepwise motion of the leaders in a certain parameter range and showed that thebranching of the pattern is not directly related to the stepped motion. In this paper, we simplify the electric circuitmodel to a one-dimensional model and try to understand a mechanism of the stepped motion qualitatively. II. ONE-DIMENSIONAL MODEL AND NUMERICAL SIMULATION
We consider a one-dimensional electric circuit model in this paper to simplify the argument. The gap lengthbetween the electrodes is assumed to be L , and the voltage V is applied between x = 0 and x = L . In the numericalsimulation, the space of size L is discretized with the interval ∆ x . The interval ∆ x = 0 . i th site is denoted by V i at i = x/ ∆ x . A resistor is set between the i th siteand the ( i + 1)th site, and the conductance is expressed as σ i / ∆ x . A capacitor is set between the i th site and theearth, and the capacitance is assumed to be C · ∆ x . Here, we assume that the conductance is inversely proportionalto the interval ∆ x and the capacitance is proportional to ∆ x . The time evolution of V i is expressed as C dV i dt = { σ i − ( V i − − V i ) − σ i ( V i − V i +1 ) } / (∆ x ) . (1)By the continuum approximation, eq. (1) is reduced to the partial differential equation C ∂V∂t = ∂∂x (cid:18) σ ( x ) ∂V∂x (cid:19) , (2)where V ( x ) and σ ( x ) denote V i and σ i at i = x/ ∆ x , respectively. The boundary conditions for V ( x ) are expressedas V ( x ) = V at x = 0 and V ( x ) = 0 at x = L . Similarly to the previous paper, we assume a two-step function for x t (cid:131) — (b)(a) V (c) V z (d)zz FIG. 1: (a) Time evolution of the tip position x ( t ) of the discharge for V = 1 , C = 0 . , ∆ x = 0 . , α = 0 . , σ = 50 , σ =2000 , τ = 0 . , E c = 0 . , E c = 10, and L = 90. The dashed line denotes x ( t ) = v t + x with v = 280. (b) Snapshotprofile of σ ( z ) where z = x − x . The dashed curve denotes the theoretical curve obtained using eq. (5). (c) Snapshot profileof V ( z ) in the range of − < z <
0. The dashed line denotes V = − . · z . (d) Snapshot of V ( z ) in the range of − < z < the conductance, although we have observed a similar behavior even if the two-step function is slightly modified to acontinuous function. Thus, the time evolution of σ i is expressed as τ dσ i dt = − σ, for E i < E c ,τ dσ i dt = σ − σ, for E c < E i < E c ,τ dσ i dt = σ − σ, for E c > E i , (3)where E i = | V i +1 − V i | / ∆ x is the local voltage between the i th and ( i + 1)th sites, and E c and E c are thresholdvalues for the weak and strong discharges, τ is the relaxation time, and σ , are stationary values of the conductance.We assumed that σ is much larger than σ because the ionization proceeds rapidly at the transition from theweak discharge to the strong discharge. We further assume that the threshold E c and E c decrease with σ as E c = E c − ασ and E c = E c − ασ . If the local voltage is below the first threshold E c in the entire region,the conductance decays to 0, which implies the insulator. When the local voltage is increased and goes beyond thefirst threshold, the discharge occurs and the conductance becomes nonzero. Then, the first threshold value slightlydecreases as E c = E c − ασ . This effect induces a hysteresis in the conductance. That is, if the discharge occursonce, the discharged state is maintained even if the local voltage is slightly decreased. A similar hysteresis is assumedto occur owing to the term E c = E c − ασ at the transition from the weak discharge to the strong discharge. Thistype of hysteresis is often observed even in experiments of discharge phenomena.If E c is sufficiently large, the strong discharge does not occur, because E i does not reach the second threshold. Inthis case, the leader or the tip position of the weakly discharged state moves smoothly. We show a numerical resultin Fig. 1. The parameter values are V = 1 , L = 90 , ∆ x = 0 . , C = 0 . , τ = 0 . , α = 0 . , σ = 50 , σ = 2000 , E c =0 .
1, and E c = 10, and the initial conditions are set to be σ i = 0 and V i = 0. Figure 1(a) shows the time evolutionof the tip position of the weak discharge x = i ∆ x . The tip position x increases monotonically, and the velocity v = dx /dt around t = 0 .
08 is evaluated at v ∼ i = 0 and the tip position i increases in time. Figures 1(b)and 1(c) show profiles of the conductance σ ( z ) and the electric potential V ( z ) at t = 0 .
08. Here, z = x − x denotesthe distance from the tip position x . For z > σ ( z ) = 0 and V ( z ) = 0, which implies that z > i > i ) is aninsulator region. The conductance σ ( z ) increases gradually to σ as | z | ( z <
0) is distant from z = 0. The dashedline in Fig. 1(c) denotes V ( z ) = − E c z , which implies that the local voltage E at the tip of the discharge is equal to E c . Although the local voltage E = | ∂V /∂z | is smaller than E c in almost the entire region of z < z ∼
0, the discharged state is maintained because of the hysteresis effect, that is, E is smaller than E c butlarger than E c = E c − ασ ( z ).If we assume that the tip position x moves at a constant velocity v , steadily moving solutions σ ( z ) = σ ( x − v t )and V ( z ) = V ( x − v t ) can be obtained from eqs. (2) and (3). Equation (3) leads to − v τ ∂σ∂z = σ − σ, (4) x (cid:13) t(cid:13)x x (a) 1 0.70.3 x t (b) (c) x (cid:13) t(cid:13) FIG. 2: (a) Time evolutions of the tip positions x ( t ) and x ( t ) of the weak and strong discharges for V = 1 , ∆ x = 0 . , C =0 . , α = 0 . , σ = 50 , σ = 2000 , τ = 0 . , E c = 0 . , E c = 1, and L = 90. (b) Time evolutions of the tip positions x ( t ) at E c = 0 . , . , , .
1, and 1.2 for V = 1. (c) Time evolutions of the tip positions x ( t ) at V = 0 . , .
7, and 1 for E c = 1. (cid:131) — (a) (b) (c) V z 00.20.40.60.81-10 -8 -6 -4 -2 0 V z (cid:131) — z FIG. 3: (a) Snapshot profiles of σ ( z ) where z = x − x . The dashed curve denotes the theoretical curve obtained using eq. (5).(b) Snapshot profile of V ( z ) at t = 0 . z = x − x in a stepped stage. The dashed curve denotes a piecewise linearfunction of x . (c) Snapshot profile of V ( z ) at t = 0 . z = x − x in a moving stage. The dashed curve denotes alinked curve of a piecewise linear function (10) and the function expressed by eq. (11). (d) Snapshot profiles of σ ( z ) at t = 0 . t = 0 . for z <
0. The solution to eq. (4) is expressed as σ ( z ) = σ [1 − exp { z/ ( v τ ) } ] , (5)for z < σ ( z ) = 0 for z >
0. The velocity v is evaluated at 280 near t = 0 .
08 from Fig. 1(a). The dashed curvein Fig. 1(b) denotes eq. (5) using v = 280 , τ = 0 .
01, and σ = 50. Good agreement is seen between the numericaland theoretical curves. On the other hand, eq. (2) is reduced to − v CV = σ ( z ) ∂V∂z . (6)The substitution of eq. (5) into eq. (6) yields the solution V ( z ) = V ( z ) exp[( − v Cτ /σ ) { ( z − z ) / ( v τ ) + ln(1 − e z / ( v τ ) ) − ln(1 − e z/ ( v τ ) ) } ] , (7)where z is a certain position where V ( z ) = V ( z ) is satisfied. Figure 1(d) shows a comparison of V ( z ) (solid curve)obtained by a direct numerical simulation and V ( z ) (dashed curve) expressed by eq. (7), where V ( z ) = V = 1 at z = − . x = 0) is used. The deviation of the two curves is considered to originate from the nonstationarityof this system, that is, the boundary condition V ( z ) = V at the left electrode z = − v t is a moving boundarycondition in this moving frame.We have performed several numerical simulations by changing the second threshold E c at a fixed value of V = 1.The other parameter values and the initial conditions are the same as those previously used. We have found thatthe strong discharge does not appear at E c > .
8. The strong discharge occurs for E c < .
8, and the tip position x of the strong discharge moves stepwise for 0 . < E c < . x moves smoothly for E c < .
85. Figure 2(a)shows time evolutions of the tip positions x and x respectively for the weak and strong discharges at E c = 1 and V = 1. It is clearly seen that x ( t ) moves stepwise, that is, x ( t ) repeats the forward motion and the stop. On theother hand, x ( t ) moves smoothly. Figure 2(b) shows time evolutions of x ( t ) for E c = 0 . , . , , .
1, and 1.2. Theaverage velocity of x ( t ) decreases with E c . The period of the stepped motion is the shortest near E c = 1.Several numerical simulations were also performed by changing V at a fixed value of E c = 1. Figure 2(c) showsthe time evolution of x ( t ) for V = 0 . , .
7, and 1. Stepwise motions are clearly observed. The average velocity ofthe tip position slightly increases with V ; however, the period of the stepped motion changes rather strongly with V .The stepped motion of the leader has been studied in more detail for E c = 1 and V = 1. The tip position x of the strong discharge exhibits a stepped motion, but the tip position x of the weak discharge moves smoothly, asshown in Fig. 2(a). The velocity of the first tip x of the weak discharge is evaluated as v = 364 at the parametervalues. Figure 3(a) displays several snapshots of σ ( z ) at different times, where z is the distance z = x − x from thefirst tip x . The dashed curve is σ ( z ) = σ [1 − exp { z/ ( v τ ) } ]. In the region of the weak discharge, the stationarysolution of σ ( z ) by eq. (5) is rather good approximation. When the stronger discharge sets in, σ ( z ) increases rapidly.Almost discontinuously jumped lines in Fig. 3(a) correspond to the transition to the strong discharge.Figure 3(b) displays a snapshot pattern of V ( z ) at t = 0 . x remains constant with time in the stepped stage. The conductance σ ( z ) at t = 0 . σ ∼ σ ′ ∼ σ for z < z = x − x , and rather small, i.e., σ ∼ σ ′ ∼ σ for z < z <
0. If σ ′ and σ ′ are assumed to be certain constant values, V ( x ) can be roughly approximated at a piecewiselinear function: V ( x ) = V + ( V − V ) xx , for 0 < x < x , (8)= V ( x − x ) x − x , for x < x < x , (9)where V = V ( x ) = V σ ′ x / { σ ′ ( x − x ) } . If σ ′ = 40 and σ ′ = 1200 are used, V is evaluated at V = 0 .
811 for x = 37 . x = 44 corresponding to thesnapshot profile shown in Fig. 3(b). The dashed curve in Fig. 3(b) denotes this piecewise linear approximation of V ( z ).Figure 3(c) displays a snapshot pattern of V ( z ) at t = 0 . x moves approximately with the velocity v ∼ ∼ . v . The shoulderlike structure is characteristic ofthe profile of V ( z ) in contrast to the profile in Fig. 3(b), in which the local voltage E = | ∂V /∂z | is rather largenear z = z . The conductance σ ( z ) is shown in Fig. 3(d) with a dashed curve. The conductance σ ( z ) and thederivative of V ( z ) have a discontinuity at z ∼ − .
8. Here, the discontinuity point x = x + z is the positionwhere the tip x of the strong discharge remained in the previous stepped stage. The conductance in the region of x < x < x is approximated at σ [1 − exp { ( x − x ) / ( v τ ) } ] and the conductance for x < x < x is approximatedat σ [1 − exp { ( x − x ) / ( v τ ) } ] + ( σ − σ )[1 − exp { ( x − x ) / ( v τ ) } ]. If the conductance is roughly approximated atcertain constant values as σ ( x ) = σ ′ for 0 < x < x and σ ( x ) = σ ′ for x < x < x , V ( x ) is roughly approximatedat a piecewise linear function: V ( x ) = V + ( V − V ) xx for 0 < x < x , = V + ( V − V )( x − x ) x − x for x < x < x . (10)On the other hand, we use eq. (7) for V ( x ) in the region of x < x < x : V ( x ) = V ( x ) exp[( − v Cτ /σ ) { ( x − x ) / ( v τ ) + ln(1 − e ( x − x ) / ( v τ ) ) − ln(1 − e ( x − x ) / ( v τ ) ) } ]for x < x < x . (11)The parameters V = V ( x ) and V = V ( x ) are unknown. They are determined by the boundary conditions of V ( x ) at x = x and x expressed by σ ′ V − V x = σ ′ V − V x − x , σ ′ V − V x − x = − σ ( x ) ∂V ( x ) ∂x , (12) (a) (b) (cid:131)¿(cid:131)— EE+(cid:131)¿(cid:131)— (cid:131)¿(cid:131)—E (c) x t FIG. 4: (a) Time evolutions of ασ ( x ) , E ( x ) and E + ασ obtained using eqs. (14)-(17). (b) Comparison of the time evolutionsof ασ ( x ) and E ( x ) obtained using eqs. (14)-(17) (solid curves) and the direct numerical simulation (dashed curves). (c) Timeevolutions of the tip position x ( t ) of the strong discharge. The dashed curve denotes the numerical result and the solid onedenotes the theoretical approximation. which are due to the continuity of the current. The parameters V and V are explicitly expressed as V = σ ′ { σ ′ + v C ( x − x ) } V σ ′ { σ ′ + v C ( x − x ) } + σ ′ v Cx ,V = σ ′ V σ ′ + v C ( x − x ) . (13)The dashed curve in Fig. 3(c) shows a linked curve of the piecewise linear function (10) and the function expressedby eq. (11) using σ ′ = 1800 and σ ′ = 100, although the horizontal axis is shifted to z = x − x . We have notyet succeeded in obtaining the solution satisfying the correct moving boundary conditions with different velocities dx /dt = 0 , dx /dt = v , and dx /dt = v . However, the above approximate solution reproduces a shoulder structureand a sharp derivative near x = x fairly well. III. SIMPLE MODEL FOR STEPPED LEADER
On the basis of these numerical observations and the rough approximation for the profile of the electric potentialshown in the previous section, we propose a very simple model for the time evolution of the position x ( t ). In thestepped stage, the first tip x moves with the velocity v as x = x + v t , where x is a certain initial position, andthe second tip x remains at a certain position: x ( t ) = x . In this stepped stage, the conductance σ ( x ) at x = x increases as σ ( x ) = σ [1 − exp {− ( x + v t − x ) / ( v τ ) } ] , (14)because the difference x − x = x + v t − x increases with time. On the other hand, the derivative of the electricpotential at x = x is evaluated using eq. (9) as E = σ ′ V σ ′ x + σ ′ ( x − x ) , (15)where σ ′ = 40 and σ ′ = 1200 are used in the following numerical simulation. In the stepped stage, E decreasesmonotonically with time, because the difference x − x = x + v t − x increases with time. The time evolutionsof σ ( x ) , E ( x ), and E ( x ) + ασ ( x ) are shown in Fig. 4(a). The sum E + ασ ( x ) increases monotonically with timeand reaches the second threshold E c = 1 from below. Namely, the local voltage E ( x ) goes beyond the threshold E c − ασ ( x ). Then, the gate to the strong discharge opens and the second tip x starts to move with the velocity v .In the moving stage, the time evolution of x ( t ) is expressed with x = x + v ( t − t n ), and the conductance σ ( x )is approximated using eq. (5) as σ ( x ) = σ [1 − exp {− ( x + v t − x − v ( t − t n )) / ( v τ ) } ] . (16)Here, t n is the time when a transition to the moving stage occurs. The conductance σ ( x ) decreases monotonicallywith time in the moving stage because the distance x − x = x − x + v t n − ( v − v ) t decreases with time. Thederivative E = ∂V /∂x of the electric potential at x = x is evaluated using eq. (11) as E = σ ′ V { σ ′ + v C ( x − x ) } σ ( x ) v C , (17)where the parameter values σ ′ = 100 and σ ′ = 1800 are used. There is a discontinuity in E ’s of eqs. (15) and (17) at t = t n in this simplified model. The summation of E ( x ) + ασ ( x ) decreases monotonically with time and reaches thethreshold E c = 1 from above at t = t ′ n > t n . Then, the moving stage changes into the stepped stage because the localvoltage E is below the second threshold E c − ασ , and x stops. The stepped stage continues again until t = t n +1 .The repetition of the stepped and moving stages reproduces the behavior of the stepped leader, as shown in Fig. 4(c).Figure 4(b) shows time evolutions of σ ( x ) and E ( x ) in the direct numerical simulation (dashed curves) and thetheoretical approximation (solid curves). The time evolutions of σ ( x ) and E ( x ) in the direct numerical simulationare qualitatively similar to the theoretical approximation. Figure 4(c) shows a comparison of the time evolutions of x ( t ) in the direct numerical simulation (dashed curve) and the theoretical approximation (solid curve). The periodof the stepped motion in the direct numerical simulation is the same order but slightly larger than the theoreticalmodel. We think that the main reason for the deviation is the rough approximation obtained using eqs. (15) and (17)for the derivative of the electric potential at x = x . IV. SUMMARY
We have proposed a one-dimensional model for stepped leaders and a simplified dynamical model for the position x of the stepped leader to understand the stepwise motion. It was shown that the two-step function of the conductanceand the shift of the threshold by the form E c = E c − ασ is essentially important for the stepwise motion in ourmodel. Our model might be very simple for the realistic discharge process, but we think that such a simplified modelis useful for understanding the unique motion of the stepped leader. Stepwise growth was observed in other systems,such as bacteria colonies. [6] Some similar mechanisms might work also in these systems. [1] J. M. Meek: Phys. Rev. (1940) 722.[2] M. A. Uman: The Lightning Discharge (Academic, San Diego, 1987).[3] C. E. R. Bruce: Proc. R. Soc. London A (1944) 228.[4] U. Kumar and G. R. Nagabhushana: IEEE Proc. Sci. Meas. Technol. (2000) 56.[5] H. Sakaguchi and S. M. Kourkouss: J. Phys. Soc. Jpn. (2010) 064802.[6] J. Wakita, H. Shimada, H. Itoh, T. Matsuyama, and M Matsushita: J. Phys. Soc. Jpn.70