Petrov and Waters model

This aims to be a flexible model, based upon the physics of the leader channel described in section 3.3.2, which is adaptable for stroke polarity and terrain altitude.

The procedures which will be described here lead to the values of attraction radius for negative flashes at sea level that are shown for comparison in Table 3.5. A mast height of 190 m is calculated for 50 per cent probability of upward flashes (Table 3.8, section 3.5.4). This model agrees with values deduced from the electrogeometric approach for currents up to the median, but generic models predict significantly greater attraction radii for the larger peak currents.

(i) Upward leader criterion

A physical condition for the launch of a successful upward leader that will complete the junction with the down-coming negative leader [68] is not in this case based on a critical radius for leader inception, but upon a critical interaction between a putative upward discharge and the downward leader. In Figure 3.10, the leader channel is represented by a vertical linear charge of length L with a charge per unit length q and leader tip charge Q. The mast is shown as an ellipsoid of height h and half width b.

Q E L

q

h

b E_min

E^*_cr H

S

x₀

Figure 3.10 Analytical modelling of a downward leader above a ground mast [53]

At large distances S between the lightning channel and the mast, the range of the electric field intensification above the top of the mast may be insufficiently extensive to support a successful upward leader, although there may well be corona streamer activity and weak leader growth as the downward leader approaches. The streamer corona from the top of the mast propagates to a distance where the electric field falls to the minimum streamer gradient Es, at a distance x₀from the top of the mast. For standard sea level conditions, the electric field E is equal to about 5 kV/cm for a streamer of positive polarity and about 10 kV/cm for negative polarity. The criterion for the lightning strike to the mast is thus a critical upward streamer length so that an upward leader can be successfully developed.

Evaluation of the critical streamer length is made from long spark studies: if an upward positive leader of charge quper unit length grows simultaneously with a hemi-spherical upward leader corona region of radius Lu, the charge per unit radius within the leader corona will be equal to that in the leader. Thus the charge within the upward leader corona zone is given by q_uL_u. This charge produces at the hemispherical surface of the upward leader corona a field:

ES = q_u

2π ε0L_u (3.40)

The critical streamer length Luis associated with a critical value of qu. In the case of a positive upward leader, long spark studies show that the minimum charge for positive leader inception is about 20–40μC/m (section 3.3.1.1), so enabling the minimum L_uto be found. It is known also from optical and electronic measurements that the minimum length of the streamer zone of the positive leader in long air gaps is about 0.7 m [53]. This corresponds to a critical streamer charge q_uof 20μC/m.

(ii) Termination to a plane ground

For specific cases of gaps with simple geometry, analytical expressions for the poten-tial and electric field may be obtained. In particular, the axial electric field distribution

created by the vertical downward leader channel represented in Figure 3.10 at a height Habove an earth plane surface has the form:

E(x, H , L)= If the leader is sufficiently close to ground so that this field is equal to the critical streamer propagation field Es, then the leader will complete the strike to ground without any upward discharge growth.

Although the streamer is rarely visible in lightning photographs, it can be inferred from long gap studies to be essential for the propagation of the leader. For analytical purposes, we may represent the net charge in the streamer zone as the charge Q (Figure 3.10). Petrov and Waters [53] showed in the following way the correspon-dence between the streamer charge Q and the linear charge density q consequently established upon the leader channel. If the streamer zone of the downward leader is represented as a hemisphere, with a radius equal to the streamer zone length L_S, then we can write:

Q= qLS (3.43)

But the field at the head of the streamer zone is:

E_S = Q

2π ε₀L_S (3.44)

The important relationship between the charges Q and q is therefore from Equations (3.43) and (3.44):

Q= q²

2π ε₀E_S (3.45)

As a result, the charge Q of the leader head and the leader channel charge density q are both determined by the streamer zone length. For example, the charge q on the channel is 0.39 mC/m for a median current stroke (Table 3.3). Equation (3.45) gives the charge Q on the negative streamer system as 2.7 mC and the streamer zone length LS = 7 m. So the charge Q can be expressed in terms of q, and in the case of E_g = ES the height H represents the striking distance rs. From Equation (3.42):

E_S =

which yields a value of striking distance:

rs =

 q

4π ε₀E_S

 [1 +√

5] (3.47)

Using now the relationship between q and i₀ obtained from Equations (3.26) and (3.29a), and a value of ES = 10 kV/cm for the propagation field of the downward negative leaders, we get:

r_s = 1.16i₀^2/3 [m, kA] (3.48)

These striking distances shown in Figure 3.11a must be regarded as a lower limit for the striking distance to a plane ground, since no upward positive discharge has been assumed. In practice, such upward leaders are observed from local asperities on level terrain, and rs may consequently be larger than in Equation (3.48).

(iii) Strikes to a mast

When a vertical mast is approximated by a semiellipsoid of half width b as in Figure 3.10, the electric field between it and a vertical coaxial downward leader can also be represented analytically. The striking distance is then calculated by employing the criterion for a critical upward streamer length from (i) earlier. Petrov and Waters [68] showed that for a negative flash the striking distance is:

r_s = 0.8[(h + 15)i0]^2/3 [m, kA] (3.49)

the factor 0.8 (h+ 15)^2/3giving a convenient numerical representation (for b= 1 m) of the analytically calculated field enhancement in the leader-to-mast space for the attainment of the strike condition. Striking distances calculated from Equation (3.49) are shown in Figure 3.11b, together with those for the electrogeometric model. These give comparable values for a 20 m mast but the Petrov–Waters model suggests that an electrogeometric approach significantly underestimates the striking distance for a 60 m mast. Most importantly, both the electrogeometric and generic modelling show that the minimum rolling sphere radius of 20 m recommended in international standards is optimistically larger (for reliable protection) than the striking distance for low current strikes of 3 kA and below.