PRESENTATION BY BRODY FUCHS AND VANDANA JHA
ATS 780PROF STEVE RUTLEDGE
A computational study of the relationships linking lightning frequency and other thundercloud parameters
By MARCIA B. BAKER, HUGH J. CHRISTIAN and JOHN LATHAM
Introduction A simple computational model of thundercloud electrification has been developed. Relationships between lightning frequency f and cloud parameters such as radar
reflectivity Z, precipitation rate P , updraught speed w, cloud radius R, ice-crystal concentration Ni and graupel-pellet concentration Ng are measured.
Electric field-growth is assumed to occur via the non-inductive charging mechanism, for both Fletcher and Hallett-Mossop types of glaciation mechanisms.
A simple criterion is used to distinguish between cloud-to-ground and intracloud lightning discharges. f is found to be especially sensitive to w in situations where, as updraught speed increases, the temperature at balance level, Tbal, of the upper boundary of the charging zone falls. When Ni and the sizes of the ice hydrometeors are increased, the effectiveness of charge transfer is enhanced.
f is found to be roughly proportional to the first power of the parameters R, Ni, Ng and Z and at least the sixth power of w in some.
The relationship between f and P depends critically on whether or not w and Tbal are strongly linked.
Hallett-Mossop glaciation is capable of producing inverted-polarity lightning from thunderclouds; Fletcher glaciation is not.
Strong correlations between lightning flash-rate f and precipitation flux, updraught speed, peak radar reflectivity, total ice and water content of the clouds and CAPE.
A goal of the work described in this paper is to investigate the sensitivity of f to particular cloud properties and characteristics. These include updraught speed, cloud-top temperature, precipitation rate, radar reflectivity, cloud-width, ice-crystal concentration, and the prevailing glaciation mechanism
The average time interval t between lightning flashes depend upon: dρQ/dt, the charging rate per unit volume of cloud; Ecrit, the critical electric field for electrical breakdown to occur; R, the cloud radius The dimensionless parameter involving these four variables is
Eq. (1)
Hence, the lightning frequency can be calculated as:
Eq. (2)
Where ϕ is a function of the other non dimensional variables in the system.
Assumption: Charging is exclusively a consequence of the non-inductive mechanism (Reynolds et al. 1957; Takahashi 1978).
It involves rebounding collisions between graupel pellets of diameter Dg present in number concentration Ng(m-3),
Ice crystals of surface area a (m2) present in number concentration Ni (m-3).
Under this non- inductive process Eq. (1) may be rewritten as
where Z is the radar reflectivity and c is the volume of ice (in the form of crystals) in the cloud.
Assumption: The charge transfer is primarily located just below the balance level, where the altitude, pressure and temperature are Zbal, Pbal and & Tbal respectively, and the updraught of strength w balances the fall-speed V, of the graupel pellets
Prediction: Lightning frequency f is roughly proportional to R, Ni, Ng, vi, Z and w6.
1-D model of thunderstorm charge generation, electric field- growth and lightning activity, from which the dependence of f on the above-mentioned parameters can be estimated.
THE MODEL: QUALITATIVE DESCRIPTION Cloud structure
An updraught of cylindrical cross-section, constant speed w, and radius R is moving continuously through an environment whose meteorological sounding is fixed throughout the period covered by each cloud-simulation experiment.
This ascent creates a vertical cylindrical cloud whose base is at a pressure pcB , altitude zCB and temperature TcB (> 0 C).
Idealized version of the 19 July 1981 sounding used for modelling the evolution of lightning by Norville et al. (1991) and Helsdon et aZ. (1992).
Above zCB, the cloud contains droplets ice crystals of number concentration Ni and area a with linear dimension di
area a = (di)2. These solid and liquid particles ascend with
speed w until they reach cloud-top. q1is assumed constant above 0 C.
Figure 1. The meteorological sounding pressure p (mb)
and temperature T (K) used in the computations.
Ice crystals
Growth of ice crystals in the ascending air is calculated from the classical diffusion equation. Primary nucleation: ( Fletcher 1969), ice crystals are formed on nuclei existing within the cloud
(either in the air or in or on supercooled droplets), their number concentration Ni increasing exponentially as the temperature falls.
'Fletcher'-type nucleation ->production of ice crystals at all levels above the 0 C isotherm, -> results in a population of crystals at those levels of a range of sizes, the smallest and most numerous having just been nucleated.
Secondary nucleation. Based on the laboratory experiments of Hallett and Mossop (1974) (H-M) , within a restricted temperature band (-3 to-8 C ) the freezing of supercooled water droplets accreted onto the surfaces of growing graupel or hail particles may be accompanied by the ejection of ice splinters.
In model, this process is represented by generating ice crystals at a constant rate within the (H-M) temperature band. These crystals grow as they are subsequently swept upwards through the cloud.
In computations a factor F F is used by which the values of Ni determined from the Fletcher equation can be multiplied for two reasons:
(1) to find the sensitivity of the predicted lightning activity to the ice-crystal concentration, without changing the shape of the size distributions generated by this glaciation mechanism; and (2) to produce ice-crystal concentrations which match field observations.
Graupel pellets
The cloud is inoculated at an altitude zi at a specified rate HF (m-2s-l) with embryonic spherical graupel pellets of diameter Do and mass density ρi.
These grow (at assumed constant ρi = 500 kg m-3) by the accretion of supercooled water which freezes onto their surfaces.
The pellets have a size-dependent terminal fall-speed Vg , and thus rise (relative to the ground) at a speed (w- Vg) until they reach the balance level (pbal, zbal, & Tbal), where (w - Vg) = 0; at which point they begin to descend through cloud towards earth with a continually increasing velocity (Vg- w).
Pellets cease growing when they fall through the 0 C isotherm at z = zF. Thus the pellets are continually growing throughout their period in the
supercooled regions of the cloud. They are monodisperse in size at all levels (except that between zi and
zbal there are two sizes, one for the upward-moving particles, and one for the descending ones).
Charge transfer
The ice crystals, pellets and supercooled water coexist within the thundercloud at all levels between zbal and the isotherms located at either 0 C or -3 C
Required for the effective operation of the non-inductive ice-ice collisional process of charge transfer.
All charge separation in the model takes place in this portion of the cloud, referred as the 'charging zone', of depth (in the case of Fletcher nucleation)
Lcz = Zbal – ZF The charge δQ separated when a rebounding collision occurs between a
graupel pellet and an ice crystal depends sensitively on the crystal area a and the relative fall-speed Vrel= Vg , of the interacting hydrometeors.
The quantitative relationship linking δQ, a and Vg is abstracted from the laboratory experiments of Saunders.
At temperatures colder than a reversal value Trev, the pellet is charged negatively by rebounding collisions with crystals, while at temperatures warmer than Trev the pellets become positively charged.
Lightning
The vertical electric field on the cylinder axis, E , increases at all levels until, at some altitude Zzap it reaches a value Ecrit at which lightning is initiated.
Ecrit is ~300 kV/m (Latham1981), and it is first achieved just below the top of the charging zone.
To determine whether a lightning flash is of the IC or C-G category, the gradient of the vertical electric field E immediately above and below the breakdown level Zzap (where E = Ecrit) is examined.
If the gradient is smaller in the upward direction, the lightning flash is of the IC form.
A C-G flash occurs if the gradient is smaller in the downward direction. The rationale for adopting this criterion is that the corona streamers
leading to lightning will grow preferentially in stronger ambient fields from which they can derive more energy in order to facilitate their propagation.
THE MODEL GOVERNING EQUATIONS AND PARAMETER VALUES
Assuming the liquid-water content is constant with altitude, we estimate tg,
Assume the rate of generation of graupel pellets, S(z, t) = HFδ (z-zi), constant in time, where HF(m2s)-1 is the pellet flux. This flux remains constant as the particles move in the vertical.
The number concentration of pellets rising through z is
THE MODEL GOVERNING EQUATIONS AND PARAMETER VALUES
Crystal growth is given by the usual diffusion equation, neglecting surface latent heating and shape effects:
Dv (m2/s) is the diffusion constant for vapor in air,(T) the density of vapor saturated with respect to ice, S(z) the supersaturation of the cloudy air with respect to ice (calculated from the prescribed vertical profiles of temperature and total water), pi (kg m-3) the density of ice, taken to be 920 kg m-3. The crystals grow at this rate as they rise in the updraught until they reach the anvil. After
this point they cease growing and are mixed evenly throughout the anvil depth ~1 km.
Thus, as the calculation proceeds, the ice-crystal concentrations below the anvil do not change in time but those in the anvil continually increase. The rate of precipitation due to graupel is
The charge transferred per rebounding collision between a graupel pellet of diameter D, and a crystal of area a
Rate of charging of a single graupel pellet at z is
The total rate of charge separation at any height where Q, (Dg, z) (C) is the average pellet charge at z.
The electric field at any point in space is that due to the charged discs and to their images in the earth's surface. The vertical field at z on the cloud axis is :
where the charge density ρQ (C m-3) is
The field gradients in the vicinity of the balance level, where lightning is usually initiated, to depend on whether the local charging rate is large or small compared with the rate of charge removal by vertical motions-that is, it will depend on the ratio
Model limitations
It has only skeletal dynamics, the sounding for the model cloud is prescribed and assumed to be the same for all model runs, and many of the cloud physical parameters are prescribed and invariant with time in each run.
One major quantitative limitation of the electrical component of the model is the way in which the charge redistribution associated with lightning is treated.
Arbitrariness exists in assumptions concerning charge storage in the ice-crystal anvil, and there is no account of the reported sensitivity of Trev to the value of q1.
Although a physically based rationale for the criterion for determining whether lightning flashes are internal or cloud-to-ground has been described, there is no direct evidence for its validity.
MODEL RESULTS
Table 1 presents typical values of N i at various levels within the charging zone forBoth the H-M and Fletcher types of glaciation process.
In the H-M case, for a specified value of w, Ni
= 0 at temperatures warmer than -3 C, increases steadily within the H-M temperature band (-3 to -8°C) then remains constant at all higher levels (colder temperatures) In the Fletcher case, we see a rapid increase in Ni with decreasing temperature and pressure within the zone.For the latter process the ice-crystal size distribution broadens continuously with altitude because of fresh activation of crystals.
On the other hand, since the H-M process creates no ice particles at temperatures colder than-8 C, the predicted size distributions narrow as ascent proceeds.
Table 2 presents values of P (mm h-') at the ground for a range of updraught speeds w and associated balance-pressure, Pbal.
P increases slowly with increasing w. Its values are lower than are normally associated with lightning-producing clouds because the pellet sizes are somewhat higher than might generally be the case.
Fig 3: As Fig. 2, but for the variation with pressure p (mb)
of net charge density, PQ (C km- 3)
*Immediately before the IC flash, E had achieved the break down value of 3KV/cm
at pbal and dropped off sharply on either side of this peak.
Fig 5: As Fig. 3, but for a cloud-to-ground lightning flash.Fig 4: As Fig. 2, but for a cloud-to-ground lightning flash.
Following breakdown at the balance level, positive charge was deposited just below pbal which reinforced that already existing, which was probably the legacy of the preceding flash. The location of the region of strongest electric field (about 85% of the breakdown value) was slightly lowered.
Table 3 presents characteristic computed values of various electrical parameters associated with successive lightning flashes from a single storm.
All flashes originate from the same level, Pbal. The intervals between consecutive flashes are seen to diminish steadily while the charge, Qflash, transferred by lightning increases.
The first seven flashes were IC and the final three, for which RQ is seen to exceed the threshold value, were C-G.
Figures 6 and 7 show, for H-M and Fletcher respectively, the sensitivities of t and t1 to w when , Pbal was kept constant by adjusting ql
Figure 6. The variations of (A) mean flash-interval, t(s),
and (B) time of occurrence of first lightning, t (s), with
updraught speed w (m s-l). Hallett-Mossop glaciation.
Solid lines, Pbal = 403 mb; squares and triangles, bal= 392 mb, t and t1 respectively. R = 1 km, Ni = 2 x l0 5m-3,
and HF= 0.01 m-2 s-l
Ti is independent of w in both cases, probably because as w increases, the increase in dρ
Q/dt associated with larger
pellets is roughly matched by a decreasing dρQ
/dt caused
by the smaller size of the ice crystals near Pbal.t1 decreases slowly as w increases, in both cases because the arrangement of charges producing the electric field is more rapidly established at higher values of w.
Figure 7 as Figure 6 ,but for Fletcher glaciation
with R=1km, FF=1000, and HF =0.01m-2s-1
The difference between the f (and also the t l ) values at the two pressures is much greater for Fletcher than for H-M ice production.
This is probably because, in the former case, additional ice crystals are nucleated as cloudy air ascends between the two pressure levels; and these contribute significantly to dpQ/dt and dE/dt.
As R decreases, more remote charges contribute less to the electric field near Zbal (where the field is a maximum), which is thereby diminished; and thus the times required both to create and regenerate lightning are increased.
t starts to increase significantly from its large-R values for values of R in the region of 2 km, where 2R/Lcz = 1.
electrical characteristics of the cloud depend on the ratio 2R/Lcz
The function in E depends on this ratio
Figure 10: The variation with cloud radius R (km) of the mean
flash-interval,t(s). Hallett-Mossop glaciation.
w = 10 m s-l, P bal= 414 mb, N,= 2 x 105 m-3, and HF= 0.01
m-2s-1
Figure 11: The variationof mean flsh-interval,t (s), with
ice-crystal concentration N,(m-3 x l0-4).Hallett- Mossop
glaciation.w=10ms-l, Pbal =403mb, R=1km, and HF = 0.01m-'s-l
t is roughly linear with Ni for the higher ice-crystal concentrations and changes more slowly as Ni decreases.
For higher values of Ni the more effective charging causes a larger fraction of the total separated charge to be localized in the vicinity of Zbal (where the field is a maximum and breakdown occurs), than for lower values. This approximate linearity is predicted by the Eq.
Reversed-polarity clouds
The variation with cloud radius R (km) of the mean flash-
interval, (s). Hallett-Mossop glaciation. Inverted-polarity
cloud.w=7.1ms-l, ha1=488mb, Ni=2x lo6m-3, and
HF=0.01m-2s-1
In this situation, with the top of the charging zone at the level of the charge-reversal isotherm, the charge structure in the thundercloud is necessarily non-classical ( i s - /+).
Lightning is found to occur, in this situation, with values of f (and also t l ) several times greater than in the classical (+/-) case (Fig. 10).
Reversed-polarity clouds
Computations made for the Fletcher-type ice-formation process revealed that lightning could not be produced from reverse-polarity thunderclouds for realistic ice- crystal concentrations.
Values of Ni 5 or 6 orders of magnitude in excess of those given by the classical Fletcher equation were required before significant lightning activity occurred.
Such concentrations have never been observed in developing cumulonimbus, and if they did exist would cause almost instantaneous total glaciation.
It thus appears that the non-inductive charging process can produce reverse-polarity thunderclouds (and can thereby explain positive lightning flashes to ground) for H-M, but not Fletcher, types of ice formation
Figure16. The variation of mean flash-interval, t(s), with
the charging parameter CQ (m4x l0^18)
linearity between f (and also l/t1) and CQ
f(l/f) increases about ten times faster than CQ over the range of parameter values considered. This is probably because the contribution of freshly created ice crystals (not present in the H-M case) to the charge transfer and associated field-growth is appreciable.
Figure17. The variation of( A)mean flash-interval, i(s), and
(B) time of first lightning, Il(s),with the charging
parameter CQ (m4x 10^18) defined by Eq.(20). Fletcher
glaciation.
Their ‘model’ cannot be regarded as a quantitatively adequate thunderstorm electrification model.
It can be used as a vehicle enabling sensitivity tests to be conducted into the possible relationships between f.
Models have not computed this, because of the complexity and unknown features of thunderstorm characteristics following the first lightning stroke and the cloud parameters defined earlier.
The simplicity of this first model in which explicit cloud microphysics is retained, is helpful in identifying the separate sensitivities of f to the individual cloud parameters.
Baker et al 1999 The precise value of Trev remains under dispute, and Baker
et al. (1995) used a single, intermediate value Trev=-15C for all of their calculations. They took no account of reports that the value of Trev depends on that of the liquid-water-content L in the vicinity of the charging.
Trev from -8C to -24C Baker et al. (1995) used only one value 300 kV/m of critical
breakdown field Ecrit . Ecrit may depend upon the atmospheric pressure and hydrometeor characteristics in the region of breakdown. Hence, Ecrit was ranged from 200 to 400 kV/m.
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