## Abstract

Scaling relations connecting storm electrical generator power (and hence lightning flash rate) to charge transport velocity and storm geometry were originally posed by Vonnegut in the 1960s. These were later simplified to yield simple parameterizations for lightning based upon cloud-top height, with separate parameterizations derived over land and ocean. It is demonstrated that the most recent ocean parameterization 1) yields predictions of storm updraft velocity, which appear inconsistent with observation, and 2) is formally inconsistent with Vonnegut's original theory. Revised formulations consistent with Vonnegut's original framework are presented. These demonstrate that Vonnegut's theory is, to first order, consistent with recent satellite observations. The implications of assuming that flash rate is set by the electrical generator power, rather than the electrical generator current, are examined. The two approaches yield significantly different predictions about the dependence of charge transfer per flash on storm dimensions, which should be empirically and numerically testable. The two approaches also differ significantly in their physical explanations of regional variability in lightning observations.

## 1. Introduction

Dramatic (2–3 order of magnitude) spatial variability is found in global observations of lightning activity (Orville and Henderson 1986; Christian et al. 1999a; Boccippio et al. 2000b). Explanation of this variability is contingent upon

understanding why the underlying deep convective spectra (updraft magnitude, ice content) vary regionally, and

understanding how these spectra map to lightning production.

Such understanding is a necessary precursor to physically based attempts to infer thunderstorm properties from lightning observations. This study explores one framework in which to pursue problem 2), specifically scaling representations of a hypothetical quasi-steady-state thunderstorm with simple geometry. The intent of the study is not to suggest that such representations are adequate substitutes for the full time-dependent problem, rather to examine the implications (and predictions) of such approaches when carried to their logical conclusions.

A fundamental scaling relationship between thunderstorm electrical generator power, generator current (charge transport velocity) and storm geometry was originally derived by Vonnegut (1963, hereafter V63). This assumes that a thunderstorm can be conceptualized as a quasi-steady-state electrical dipole, that the dipole charge centers are of comparable size, and that this size scales with storm dimensions. With an additional assumption of monotonic mapping between generator power and lightning flash rate, this scaling forms one possible theoretical basis for linking lightning to thunderstorm dynamics, microphysics, and geometry. Subsequent simplifications to the approach translate the original parameters to more readily observable parameters such as cloud-top height, and appear to be consistent with observation (Williams 1985, hereafter W85; Ushio et al. 2001). The simplified approach was used by Price and Rind (1992, hereafter PR92) and Price et al. (1997), who used continental-based calibrations of the relationship to derive oceanic parameterizations of the same. Subsequently, Michalon et al. (1999) attempted to modify these parameterizations to account for land–ocean cloud condensation nucleus (CCN) spectrum differences, and Anyamba et al. (2000) used the inferred highly nonlinear relationship between lightning and cloud height to test a cloud-based proxy for lightning against integrated global lightning measurements.

PR92 sought to proxy global lightning rates from observable parameters (cloud-top height), and were constrained by the limited validation data available at that time. The uncertainty in this approach was acknowledged in Price et al. (1997). More recently, the deployment of low-earth orbit lightning sensors with high detection efficiency and little bias [the Optical Transient Detector (OTD) and Lightning Imaging Sensor (LIS) sensors, (Christian et al. 1996; Christian et al. 1999b)] and possible future deployment of geostationary lightning mappers (Christian et al. 1989) lead to the possibility that the reverse calculation may be possible, that is, inference of storm properties from lightning observations. Recent numerical modeling advances [e.g., the inclusion of breakdown mechanisms in the South Dakota School of Mines and Technology Storm Electrification Model (SEM) by Helsdon et al. (1992) and in the National Severe Storms Laboratory cloud model by Mansell (2000)] also allow direct testing of lightning-convection relationships. Physically based theoretical relationships may provide *one framework* in which to interpret both the new empirical data and modeling results. It is thus appropriate to reexamine the most recent simplifications to Vonnegut's theory (W85) and an extrapolation of it (PR92) for consistency with both the original theory and latest observations.

Vonnegut's basic theory, assumptions (both implicit and explicit), and subsequent simplifications are presented in section 2. In section 3, it is demonstrated (using satellite-observed storm flash rate distributions) that inversion of the PR92 ocean parameterization yields predictions about marine storm updraft velocities that may be at odds with the limited in situ measurements available, and it is shown that the parameterization involves a change of variable that is inconsistent with Vonnegut's original theory. Section 4 reexamines some of the basic assumptions, and identifies those now amenable to direct testing. Section 5 explores the significant assumption that flash rate scales as generator power, rather than generator current; the two approaches yield significantly different physical explanations of observed *f*(*z*) variability, predict very different geometric weightings for flux/generator current variability, and yield testable predictions of the dependence of per-flash charge transfer on flash rate itself. Regardless of the “correct” flash rate mapping, investigation of these approaches clearly defines parameters of interest for interpretation of both observational data and numerical modeling studies.

## 2. Scaling history, assumptions, validation, and application

V63 formulated a scaling relation for the electrical power generated by a thunderstorm. The following assumptions were either implicit or explicit in this formulation.

A thunderstorm can be conceptualized as an electrical dipole.

The horizontal and vertical scales of the two dipole charge centers are comparable and vary with storm scale.

A storm generator current can be conceptualized as a net charge transport velocity, which maintains the dipoles.

Time variation of the generator current is small and the generator is (as required under continuity) balanced by other (unspecified) currents.

Assumption 1) is a conventional representation of storm electrical structure (Wilson 1920). While more recent empirical studies have suggested that a tripole representation is more accurate (Simpson and Scrase 1937; Williams 1989), or, in some cases, that a multipole structure appears to hold (Stolzenburg et al. 1998), these empirical studies also confirm that the main negative and upper positive charge centers in storms dominate in terms of charge density. Since the region at and between these charge centers appears to be the dominant location of microphysically induced local charge separation (Takahashi 1978; Saunders 1994), this appears to be a reasonable working assumption.

Assumption 2) has not been rigorously tested (at least in terms of horizontal scales), and there may be reason to believe that upper positive charge centers are larger in spatial extent due to cloud-top divergence. This is likely to be a factor-of-2 type inaccuracy, at best. The assumption of horizontal and vertical scale similarity (and covariance with storm size) places constraints on the applicability of this approach to storms with aspect ratio ∼1, although separation into horizontal and vertical scale components (i.e., a different charge geometry model) is not precluded under the basic theory.

Assumption 3) is a simple description of how net convergence or divergence of differentially charged particles (driven by fall speeds and storm updrafts) yields bulk charge separation. Assumption 4) implies that all quantities discussed below refer to appropriately time-averaged values.

Under these four assumptions, the electrical generator power is given as the product of generator current and the potential drop between the dipoles:

Assuming tangential, spherical charge centers with volumetric charge density ± *ρ* and radius *R,* maintained by a generator current density *J* = *ρV*_{Q} (charge density × charge transport velocity; generator current *I* ∼ *JR*^{2}),

Under this charge geometry, the dipole separation is given by *z*_{d} = 2*R,* and area *A* = *πR*^{2}; hence,

(recognizing that under this geometry, it is assumed that *R* grows as *z*_{d}). This formulation follows W85, but captures the essence of V63. It is important to note (as V63 did) that the generator power is dominated by geometric terms (width and depth of the dipole region), with a linear modulation by the generator current density. This places strong constraints on what terms may dominate when comparing two storms of comparable shape and size.

The next assumption is that the net power dissipation by lightning varies monotonically with generator power, and in absence of an empirically confirmed functional form, linear mapping is implicitly assumed:

(*C* is a scaling constant and is dimensionless). A secondary assumption (by W85, but not by V63) relates the flash rate linearly with lightning power dissipation, hence storm cell flash rate *f*_{c} is assumed to vary linearly with *P.* Hence,

where *γ*^{–1} must have units of energy (work) for dimensional consistency. The implicit assumption is thus that each flash is responsible for a constant electrostatic energy. W85 notes that limited field data may support the assumption that *f*_{c} ∼ *P*; when flash rate is compared with gravitational power due to falling precipitation (not necessarily generator power), evidence of a constant *γ* is found. It is important to note that the linear *f*_{c} ∼ *P* mapping may reasonably be expected to fail in low flash rate storms where other dissipation mechanisms (conduction currents, precipitation currents) might dominate over net lightning current.

Further simplifications are required to make Eq. (5) empirically testable or operationally useful. A common assumption is that *z*_{d} (or *R*) varies as cloud-top height *z*_{t}, although other measures (e.g., the altitude of a fixed radar reflectivity contour) could also be reasonable proxies. The *z*_{t} approximation could be justified by observations that in many storms the lower negative charge region remains relatively constant in height,^{1} and that most upper positive charge is carried on small ice crystals with negligible terminal velocity (hence dipole separation can be linearly approximated by cloud-top height). Hereafter, *z*_{t} will thus be denoted simply as *z.*

These simplifications yield the basic relationship,

W85 further simplified this by invoking a scale similarity between horizontal and vertical dimensions (necessary under V63's original charge model, and thus reducing the *A* and *z*^{2} terms to a single length scale dependence of *L*^{4}):

A further significant simplification replaces *V*_{Q} with storm updraft velocity *w.* Here, *w* is the most appropriate velocity in the vertical profile to describe charge separation between the two charge centers, if such a velocity uniquely exists. Uniqueness or nonuniqueness of the relationship between *V*_{Q} and an identifiable *w* may be a significant flaw in this simplification. Thus,

The final implicit assumption is that variability in the charge density of the dominant dipole regions is small (and can thus be treated as a prescribed constant), an idea loosely corroborated by electric field soundings through MCSs (Stolzenburg et al. 1998; in such soundings, charge density in these regions does not appear to vary by orders of magnitude between storms). This has, at best, been only partially verified in continental storms.^{2}

It is again noted that the potential *direct* contribution (under these simplifications) from variability in storm updraft velocity is at best linear, and geometric terms from the size and separation of the dipole centers dominate. V63 hypothesized that updraft velocity would be positively correlated with cloud-top height, and W85 gave empirical evidence from continental, midlatitude radar-based observations to support a linear relationship between *w* and *z. *PR92 refined this empirical relationship over land to a power law with exponent 1.09, and presented a similar power-law relationship over oceans with exponent 0.38. Due to the limited maritime data available to PR92, their ocean *w*(*z*) fit is almost completely dominated by several outlier data points corresponding to cloud-top heights of 2 km.^{3} When only data points corresponding to cloud tops above the freezing level are considered, almost any *w*(*z*) relationship could be fitted (i.e., *w* and *z* appear uncorrelated in their small sample), although oceanic *w* are consistently lower than continental *w* for similar *z.*

The near-linear continental *w*(*z*) relationship presented by W85 and PR92 yields the familiar *f*_{c} ∼ *z*^{5} representation, where again, the highly nonlinear dependency is causally driven by geometric terms. This carries a single (measurable) parameter (cloud height), and was applied by PR92 to estimate global lightning from satellite [International Satellite Cloud Climatology Program (ISCCP)] cloud-top measurements. It is important to note that linear *w*(*z*) mapping carries with it the necessary corollary that *f*_{c} ∼ *w*^{5}, although this strong nonlinearity completely arises from the geometric contribution, rather than direct influence of *w* on charge separation.

A fifth-power *f*_{c}(*z*) continental relationship is consistent with data presented by W85, as well as by Shackford (1960), Jacobson and Krider (1976), and Livingston and Krider (1978). Observational matching of a fifth-power univariate dependency does not necessarily validate the sequence of simplifying assumptions, it only demonstrates that they are not inconsistent with observation. While large scatter is present in the storm-by-storm data, the fifth-power dependency appears to roughly hold when observations are averaged in altitude. Ushio et al. (2001), using global storm observations from the Tropical Rainfall Measuring Mission (TRMM) LIS and precipitation radar (PR), also found that a fifth-power relationship was consistent with (though not demanded by) the continental storm observations. Perhaps more importantly, these authors found an upper limit in *f*_{c} attainable by storms of a given *z,* strongly suggesting that scaling limitations of some sort (either through direct or indirect coupling to *z*) are at work in nature. PR92 compared their parameterizations over both land and ocean with regional flash rates (bulk production) observed by the Defense Meteorological Satellite Program (DMSP)–Optical Line Scan (OLS) satellite sensor. This test did demonstrate a net empirical usefulness of their parameterizations. However, this was (necessarily) not a strict test of the theory, which holds only for individual storm flash rates. Williams et al. (2000) and Boccippio et al. (2000b) have recently demonstrated that spatial variability in regional bulk flash production is dominated by variability in the frequency of storm occurrence rather than by variability in storm flash rates. Since a parameterization may accrue “skill” simply by activating whenever a deep cloud is present, validation against bulk lightning production is not necessarily a test of the fidelity of its underlying physics. In a similar integrated approach, a fifth-power weighting was used by Anyamba et al. (2000), who applied it to Television and Infrared Observation Satellite (TIROS) Observational Vertical Sounds (TOVS) cloud data and found some coherence of the weighted time series with Schumann Resonance (SR) global lightning measurements at Madden–Julian Oscillation (MJO) timescales.

It is again emphasized that any empirical tests of the univariate scaling relations are, of course, only tests of the observational consistency of the complete chain of assumptions leading to the particular formulation being tested [i.e., they cannot yet test the basic validity of Eq. (5) for lack of observations of the fundamental parameters; other constructions could conceivably yield similar univariate relationships]. Alternatively, numerical cloud modeling results could be used to investigate Vonnegut's original hypothesis [Eqs. (3), (5)] without the need for additional assumptions.

## 3. Testing Vonnegut's theory and later simplifications

### a. Implications of PR92 ocean parameterization

PR92 derived an ocean parameterization for *f*_{c}(*z*) based upon an assumption that the same nonlinear *f*_{c}(*w*) [derived from the *w*(*z*) dependency over land] holds over both land and ocean. Since large-scale observations of oceanic storm flash rate are now available from the LIS sensor, the necessary implications of this assumption (specifically, predictions about the distribution of oceanic updrafts) can be directly tested.

Two years of LIS data from December 1997 to November 1999 are used in this study. Flashes are identified using the LIS version 4 (v4) “flash” data product, which clusters contiguous optical lightning pulses observed from cloud top by the sensor. Preliminary validation of this product following techniques in Boccippio et al. (2000a) suggests that nominal LIS v4 flashes may represent fragmentations of true contiguous channel structures approximately 10%–20% of the time. Storms are identified using the LIS v4 “area” data product, which clusters flashes using a fixed spatial cutoff parameter. The median and mean diameters of LIS v4 optical areas are 26 and 30 km, although the “cores” of LIS areas are typically less than 10 km wide; much of the total diameter appears to derive from multiple scattering of light within clouds.

To simplify analysis, only LIS areas observed between 80–90 s are considered here. These represent the vast majority of LIS-observed areas; storms observed for less than this duration occur during periods of sensor data buffer overflow or near the edge of the LIS field-of-view (FOV), and account for less than 15% of all areas. The LIS flash detection efficiency ℱ is conservatively estimated here to be 75% based on *preliminary* cross-sensor comparison studies (e.g., Thomas et al. 2000).

The limited duration of LIS observation places a lower bound on observable per-cell flash rate *f*_{c}. While this cutoff is fundamentally probabilistic [observations of one flash (designated f1) in 80 s may occur from a spectrum of true instantaneous flash rates], for simplicity it is considered here to be discrete. The LIS minimum detectable instantaneous flash rate can be approximated by

where *δt* ∼ 80 s, ℱ ∼ 0.75, hence *f*_{cmin} ∼ 1 fl min^{–1}. In the context of PR92's *f*_{c}(*w*) relationship,

(*w* in m s^{–1}, *f*_{c} in fl min^{–1}), this corresponds to a minimum inferrable updraft of approximately 15 m s^{–1}.

The observed LIS storm instantaneous flash rate probability distributions Pr(*f*_{c}) over land and ocean (within the ± 35° orbit of the TRMM satellite) are shown in Fig. 1. Here, the truncated spectra are normalized using observations by Nesbitt et al. (2000) of the population of deep (ice-scattering) precipitation features identified using the TRMM microwave imager (TMI) and PR. These authors found that LIS observed no flashes in 50% of land features, and 98% of ocean features. This population of features exists below the LIS 1 fl min^{–1} cutoff and consists of an unknown mix of nonflashing and weakly flashing cells; the normalized distributions in Fig. 1 simply place these fractions below 1 fl min^{–1}, so the “complete” probability distribution functions (PDFs) integrate to unity. While there is a definitional mismatch between the precipitation features of Nesbitt et al. (2000) and the LIS areas used here, this suffices as a crude estimate (and will not greatly affect the inferences below).

It is obvious from Fig. 1 that the shapes of the observed distributions do not differ dramatically; that is, *when* storms with flash rates greater than about 1 fl min^{–1} occur over land and ocean, their flash rates are similar [their means differ by a factor of 2, as reported by Boccippio et al. (2000b)]. This immediately suggests that PR92-predicted land and ocean updraft distributions will also not differ dramatically (in shape).

Once an *f*_{c}(*w*) relationship has been either empirically or theoretically derived, the predicted probability of updraft occurrence Pr(*w*) can thus be calculated through a direct change-of-variable in the probability distribution

for the domain (*f*_{c} > *f*_{cmin}, *w* > *w*_{min}).

The conditional Pr(*w*), predicted by Fig. 1 and Eqs. (10) and (11), is shown in Fig. 2a (these distributions do not integrate to one because of the truncated observed flash rate spectra). Again, the shapes of the distributions do not differ dramatically, and the mean predicted updrafts for the population of >15 m s^{–1} updrafts in flashing storms are 23 m s^{–1} over land, and 20 m s^{–1} over ocean. Prior empirical observations of continental and oceanic updraft spectra have not been presented in ways amenable to direct intercomparison, although Lucas et al. (1994) and Zipser (1994) observe that the means of the upper 10% of *low altitude* aircraft-observed updrafts over land and ocean appear to differ by factors of 2–3, significantly larger than the ratio of 1.15 predicted here. Again, direct intercomparison is not yet formally possible, but the suggestion is that strongly nonlinear *f*_{c}(*w*) relationships, equal over both land and ocean (the core of the PR92 ocean parameterization assumption) might only hold if the population of low or zero flash rate (below the LIS truncation) storms is considered. Since this population may begin to violate underlying assumptions of the scaling law simplifications, this suggests that a global application of Eq. (10) is inconsistent with observation and may be inconsistent with theory.

The quantity Pr(*z*) can also be predicted, using the approach of Eq. (11). Again using PR92's parameterization, the predicted (truncated) cloud-top spectra are shown in Fig. 2b. Clearly, the PR92 ocean parameterization yields nonphysical cloud-height predictions upon inversion (70-km cloud tops for 1 fl min^{–1} storms). Note that these are not extrapolations, simply remappings. Noninvertibility, even of a purely empirical parameterization, suggests that key physical variability is not being correctly resolved.

### b. A self-consistent approach

A significant formal inconsistency also exists in the derivation of the ocean parameterization of PR92. To understand this (subtle) departure from theory, it is necessary to retrace this derivation. The derivation is presented in generalized functional form, to mitigate confusion by specific empirical fits and parameters.

PR92 began with three functional representations, *w*_{l}(*z*_{l}), *w*_{o}(*z*_{o}), and *f*_{cl}(*z*_{l}). The subscripts *l* and *o* denote land and ocean, respectively. Representation *w*_{l}(*z*_{l}) is a near-linear empirical power-law relationship (also found by W85). Relationship *w*_{o}(*z*_{o}) is a power-law relationship with exponent 0.38 (with caveats on its robustness as described above). Relationship *f*_{cl}(*z*_{l}) is a fifth-power relationship empirically observed and, under V63, *predicted* by Eq. (8) and linear *w*_{l}(*z*_{l}). PR92 ,*derive *Eq. (10) as *w*_{l}(*f*_{cl}) = *w*_{l}[*z*_{l}(*f*_{cl})], then assume *w*_{o}(*z*_{o}) = *w*_{l}[*z*_{l}(*f*_{cl})] and solve for *f*_{c}(*z*). However, it is obvious from this assumption that the resultant expression is for *f*_{cl}(*z*_{o}), a definitional inconsistency. Also, PR92 cite V63 and W85, yet derive a 5th-power *z* dependence of *work,* not power (as they claim), without inclusion of a generator current term *V*_{Q} (i.e., *w,* under W85's simplifications). Their implicit assumption is thus that flash rate scales linearly with instantaneous storm electric potential energy, which is restored after each discharge in an unspecified manner. Neglect of the generator current term indirectly leads to the inconsistencies with V63.

The *f*_{cl}(*z*_{o}) inconsistency can be more intuitively understood by recalling that, under V63, the strong nonlinearity in Eq. (10) for *f*_{c}(*w*) *only derives* from the fourth-power weighting of the geometric scaling law terms and the near-linear *w*(*z*) dependency over land (PR92's derivation differs). Yet this relationship is assumed to be regionally invariant, and provides the functional core for the remainder of the ocean parameterization. This inconsistency yields the PR92 *f*_{co}(*z*_{o}) parameterization

in which, under V63 and W85, the fundamental fourth-power geometric weighting given by Eq. (8) has been diminished, implying that over oceans,

This conclusion is both likely nonphysical (inverse relationship between updrafts and cloud heights) and inconsistent with the *w*_{o}(*z*_{o}) presented by PR92. As demonstrated in the prior section, Eq. (12) also predicts nonphysical cloud-top heights upon inversion. The correct implementation of variable *w*(*z*) relationships between land and ocean (should they actually occur) in the context of Vonnegut's theory [as simplified to Eq. (8)] should be,

with appropriate substitutions for land or ocean *w*(*z*) relationships (theoretical or empirical) and their inverses. Here, the parameter *k*_{1} is *invariant,* as the scaling should yield the same dependency regardless of the functional form of the parameters chosen. This approach is thus completely capable of being calibrated by land-only observations. If different land/ocean *w*(*z*) are stipulated, Vonnegut's theory demands differing univariate relationships (both in *w* and *z*) over land and ocean, in contrast to PR92's assumption of equivalent *f*_{c}(*w*). Further, all regional (land/ocean) variability in flash rates must, in the context of the theory, be fundamentally driven by variability in either true Pr(*w*), Pr(*z*), or *w*(*z*).

In the specific case where the land and ocean *w*(*z*) are considered to be power-law fits (as by W85 and PR92),

we thus have,

Column 2 of Table 1 shows the resultant power-law weightings using *w*(*z*) fits from PR92 and *k*_{1} derived from that study.^{4} Here, the height dependency is more consistently in the fourth- to fifth-power range, while the weak *w*_{o}(*z*_{o}) relationship reported by PR92 yields a much more nonlinear *f*_{co}(*w*_{o}) dependency.

Figure 3a shows Pr(*w*) predicted using the observed LIS Pr(*f*_{c}), Eq. (11), and the revised relations in column 2 of Table 1. Here, the means of the truncated updraft spectra differ by a factor of 3 (perhaps closer to observations), and the oceanic spectra are in general lower-valued and more tightly constrained. While this may be seen as an improvement, it is cautioned that the lower-valued, narrower oceanic spectra are primarily a result of the observed *w* underlying PR92's *w*_{o}(*z*_{o}) empirical fit, which are confined to values from about 5–12 m s^{–1}.

Figure 3b shows Pr(*z*) predicted using this approach. The new relations, when inverted, yield physically realistic cloud-top height values, and the truncated means in ocean storms are slightly (about 1.5 km) higher than in land storms (i.e., the same flash rate is predicted to occur in deeper storms over oceans than over land). Recall that this refers *only to the population of flashing storms*; it is cautioned that observational consistency may only be valid for the “reverse” problem, that is, prediction of *z* from *f*_{c}. No consideration has been given to the population of nonflashing storms, so the “forward” *f*_{c}(*z*) may easily be overpredicted using this formulation. This is also evident when Fig. 3b is compared against direct observations of oceanic Pr(*z*) spectra, for example, radar echo tops observed during the Coupled Ocean–Atmosphere Response Experiment (COARE) by Johnson et al. (1999), who find a significantly higher frequency of occurrence [e.g., Pr(*z* = 10 km) = 0.07 km^{–1}].^{5}

Equation (18) is not intended to supplant PR92's operational ocean parameterization for the forward problem, merely to illustrate the necessary consequences of deriving such a parameterization consistent with the underlying theory. Re-examination of the theory's underlying assumptions (including examination of critical velocities, as well as the appropriateness of the chosen observables, e.g., *z*_{t}) may be required to yield a theoretically consistent *and* operationally useful forward *f*_{c}(*z*) parameterization.

## 4. Understanding Vonnegut's theory

### a. Implications of the revised approach

The revised univariate scaling law implementations can be used to assess interdependencies even in absence of rigorous *w*_{o}(*z*_{o}) observations. First, in all likelihood, the parameters *a*_{l} and *a*_{o} are greater than 0 (cloud-top height is positively correlated with updraft speed). This has the necessary implication that flash rate–height relationships will always be at least fourth-power-weighted [if all assumptions leading up to Eq. (8) hold]. Equation (18) also demands that all variability in observed flash rates (e.g., between land and ocean) is determined by either: 1) differences in the underlying Pr(*z*), or 2) differences in the *w*(*z*) relationships, that is, *k*_{l}, *k*_{o}, *a*_{l}, and *a*_{o}.

Noting that *a*_{l} and *a*_{o} are likely >0, it is also evident that *f*_{c}(*w*) relationships will be at least linear in *w* (as given by V63). Furthermore, in the possible case that *a*_{o} is less than or equal to 1, this yields at least a fifth-power *f*_{c}(*w*) dependency, and possibly much higher [PR92's ,*w*_{o}(*z*_{o}) yields an 11.5 exponent, as in Table 1]. Also, the fact that Eq. (19)'s land and ocean variants are not necessarily equal means that a given observed flash rate (e.g., *f*_{cmin} for LIS) implies different (and likely lower) updrafts over oceans (contrasting PR92's assumption). This has the important implication that, under V63 and W85, *regionally invariant univariate relationships between lightning and storm properties are not necessarily predicted to exist.* Interestingly, the cause [regionally variant relationships between updrafts and realized cloud depth, *w*(*z*)] of this conclusion could be seen as implicating variability in, for example, vertical buoyancy profiles [“shape of the CAPE” (convective available potential energy)] already suspect as a factor in regional convective spectrum differences. V63 and W85 simply provide (geometrically based) energetic weighting on the potential impacts of such variability on flash production.

### b. Effects of truncated LIS observations

The LIS effective minimum observable flash rate of 1 fl min^{–1} places some restrictions on the use of these data to test updraft spectra predictions. In the case of PR92, the inferences were limited to updrafts greater than about 15 m s^{–1}; under the revised approach, the limit is about 7 m s^{–1}. The effects of including lower flash rate storms in the sample have been examined by testing candidates for the probability distribution of *f*_{c} < 1 fl min^{–1} storms.

Recalling section 3, Nesbitt et al. (2000) found that 98% of deep (ice-scattering) precipitation features over ocean were not observed by LIS to flash (50% over land) (i.e., they had *f*_{c} < *f*_{cmin}, possibly 0). Arbitrary hypothetical low flash rate extrapolations were constructed by assuming various percentages of these storms were truly flashing at a modest rate of 1 fl 3 min^{–1} (the remainder were assumed nonflashing and ignored, hence truncated updraft spectra were again predicted).

Inclusion of a population of low flash rate storms did little to alter the mean predicted updrafts over land. Inclusion of these storms over ocean did appreciably (and expectedly) alter the means. However, factor-of-2 differences in the predicted land–ocean updraft truncated means *only* occurred if a perhaps unreasonably large fraction (50%–90%) of all deep ocean cells were assumed to be flashing at rates greater than 0 but below the LIS *f*_{cmin}. This would be at odds with observations of oceanic lightning frequency reported by, for example, Zipser (1994), or observed by the author with an electric field mill during the Tropical Ocean and Global Atmosphere (TOGA) COARE. This formally reinforces the earlier conclusion of section 3, that a globally invariant *f*_{c}(*w*) assumption can only be reconciled with observation if a (perhaps unrealistically) large population of very low (but nonzero) flash rate storms is assumed to exist.

Using the revised parameterizations, factor-of-3 differences in the inferred truncated updraft means are maintained over a wide range of assumed low flash rate storm occurrence. Vonnegut's approach thus does not necessarily require invocation of a large ocean nonzero flash rate population to be consistent with observation.

### c. Ambiguity in w(z)

Since strict adherence to V63's theory demands that land–ocean flash rate differences be attributed either to differences in Pr(*w*), Pr(*z*), or *w*(*z*), it is necessary to examine the sensitivity of these results to ambiguity in *w*(*z*). Figure 4 shows the empirical (*w, **z*) data reported in PR92. As noted earlier, consideration of only data for *z* > 6 km leaves considerable ambiguity in the “true” *w*_{o}(*z*_{o}), should such exist. Sensitivity is tested by assuming a range of (*a*_{l}, *a*_{o}) (from 0.1–5.0), fitting appropriate (*k*_{l}, *k*_{o}) to the PR92 observed data, and repredicting updraft and cloud-top spectra.

Figure 5a shows the *χ*^{2} of the empirical fits over land when only *z* > 6 km data are considered. Clearly a broad range of *w*_{l}(*z*_{l}) power-law exponents yield plausible fits, with optimal correlation corresponding to *a*_{l} = 2.39. Figures 5b and 5c show the predicted updraft and cloud-top height spectra, in which the logarithm of the PDFs are raster-plotted (i.e., each vertical “strip” of these figures corresponds to a predicted PDF using a given *a*). The overall *shapes* of the predicted spectra are fairly invariant over the broad range of low *χ*^{2} values, suggesting little sensitivity to the precise *w*_{l}(*z*_{l}) fit. Over ocean (Figs. 6a–c), the range of plausible *a*_{o} is even broader (consistent with the large scatter in the raw data), with an optimal fit for *a*_{o} = 0.92. Again, the sensitivity to the specific power law formulation is low. For updrafts, the ocean spectra are consistently narrow and low-valued (compared to land), regardless of the functional form of the fit. For all spectra except land updrafts, the truncated means (white overlay lines) are almost completely insensitive to the specific functional form of *w*(*z*); as a further illustration, Figs. 7a,b show the spectrum predictions using PR92's *w*(*z*) fits and the newly computed *w*(*z*) fits above; despite significant differences in *a,* the spectra do not differ dramatically.^{6} Univariate relations corresponding to the new fits are shown in the third column of Table 1.

This essentially indicates that the *z* dependence in *w*(*z*) is unable to significantly constrain the theory, and land–ocean differences are (here) driven by the small dynamic range of observed *w* in PR92's (*w*_{o}, *z*_{o}) data. These data effectively placed bounds on ocean Pr(*w*), which, when mapped through Vonnegut's theory for storm energetics, are consistent with observed flash rate spectra in ocean flashing storms. Succinctly, the (limited) available data on Pr(*f*_{c}), Pr(*w*), Pr(*z*), and *w*(*z*) over land and ocean are consistent with (or alternatively, cannot refute) Vonnegut's scaling theory. Equivalently, univariate relationships correctly derived from V63 and constrained by observed *w*(*z*) *must* yield internally self-consistent results.

### d. Re-examining the underlying assumptions

It is emphasized that observational consistency has only been demonstrated for the population of flashing storms, or the reverse problem of inferring storm properties from observed flash rates. There is good reason to expect that the forward prediction problem will fail under this approach, for lack of consideration of nonflashing deep storms [apparently dominant over ocean (Nesbitt et al. 2000)]. This suggests that assumptions in the simplifications to Eq. (3) must be reexamined, tested or revised. The most fundamental assumption, that flash rate is determined by generator power, is examined in section 5. The following are also likely candidates for reexamination:

*A*≠*A*(*w,**z*): in situ observations suggest that the area of updraft cores may be related to their intensity (LeMone and Zipser 1980; Zipser and LeMone 1980). This could introduce a slight nonlinearity, invalidating the strict horizontal–vertical scale similarity invoked by W85. This would not violate Eq. (3), which allows separability in the geometric terms, but would require fine-tuning of the univariate relationships. This is not a fundamental obstacle, and continued observation (and reporting) of updraft core areas through in situ or overflying aircraft observations will help facilitate such tuning.*z*_{t}∼*z*_{d}: this hypothesis is now indirectly testable through the current and planned deployment of very high-frequency (VHF) time-of-arrival 3D lightning channel mapping systems [such as the National Aeronautics and Space Administration (NASA) and New Mexico Institute of Mining and Technology Kennedy Space Center (KSC) Lightning Detection and Ranging System (LDAR) and Lightning Mapping Array (LMA) networks (Poehler and Lennon 1979; Rison et al. 1999)]. The distance between the upper and lower branches of intracloud channels should provide reasonable estimates of*z*_{d}, which can be directly compared against various measures of*z*_{t}. This again is thus not a fundamental limitation, as more optimal*z*_{d}proxies (either IR or radar-based) can presumably be identified.The assumption (under V63) that the volume of charge centers scales with storm size (rather than just the area) is perhaps suspect given more recent observations of the relative stratification of the main charge centers into thin layers, but this can easily be rectified by a more realistic charge model.

*w*∼*ρ*^{2}*V*_{Q}, or*w*∼*V*_{Q}and*ρ*^{2}is invariant: this is perhaps the most likely point of failure in simplifications to Vonnegut's theory. In addition to considerations such as critical velocities for lightning occurrence (Zipser 1994), which could conceivably be integrated into the theory, it could also be argued that updrafts themselves play only an indirect role in determining the net generator current density. Specifically, the large-scale charge separation can be interpreted as a net convergence (or divergence) of charged particles due to differential terminal velocities: Here,*D*is ice particle diameter [=*D*(*T,**w*)], we integrate over the ice particle spectrum, and*ρ*_{net}represents the net charge associated with each particle diameter in the spectrum. Parameter*V*_{t}is particle terminal velocity. The particle distribution is indirectly determined by*w*(available water supply, particle collision rates), as is*ρ*_{net}(*D*) (for the same reasons). In this representation, it is clear that the vertical profile of particle terminal velocities may be at least as important as*w*as a direct term in the generator current density, especially for low updraft storms (i.e.,*w*near*V*_{t}). The indirect influence of*w*(determination of the particle spectrum and charging rates) could easily be highly nonlinear. Any overall linear mapping that might exist between*w*and*J*(stipulated in V63's theory and W85's simplifications) would thus be fortuitous but not demanded.Such mapping [i.e., Eq. (20) above] is at least partially determinable from explicit numerical cloud models, which include local charge separation terms and track charge density on various hydrometeor categories. Thus, while

*w*∼*V*_{Q}may be a fundamental flaw in simplifications to V63, an appropriate mapping between the two may not be beyond our reach.

In summary, (1)–(3) above represent insufficiently tested assumptions in the simplifications to V63, which have at best limited and heuristic observational support. It is again emphasized that none are fatal to Eq. (3), and that all are testable using contemporary observational platforms and/or numerical models. Vonnegut's original theory (which even in simplified form appears not to be inconsistent with available observations) may thus yet hold promise as a physically based underpinning for the “inverse” problem of relating lightning flash rate observations to meteorological storm properties.^{7}

## 5. An alternate approach: Flash rate set by generator current

The idea that flash rate is set by generator power is a key underpinning of W85. Implicit in this assumption are that

lightning power dissipation scales linearly as generator power production, and

flash rate scales linearly as lightning power dissipation,

or alternatively, that unspecified mappings in 1) and 2) combine to yield a linear dependency *f*_{c} ∼ *γP.* However, it is also plausible to consider that net lightning current matches net generator current (or at least scales linearly with it); that is, *f*_{c} ∼ *I*_{l} , *I*_{l} ∼ *I,* and hence *f*_{c} ∼ *I.* This approach was, for example posited in a time-dependent model by Driscoll et al. (1992) and used to retrieve generator currents from observational data by Driscoll et al. (1994). One possible scaling derivation (again quasi-steady-state) from this alternate approach is presented below. A dipole geometry perhaps more consistent with recent observation (circular plates) is employed, although the specific charge configuration is not critical to the overall comparison with Vonnegut's hypothesis.

### a. Charge geometry

Two circular plates are stipulated with radius *R,* separation distance *z*_{d}, and equal and opposite areal charge densities *σ.* The potential difference between the plates (*z* = 0 is the lower plate, *z* = *z*_{d} is the upper plate) is

For convenience, call the geometric term above *G*_{1}(*R, **z*_{d}). This is not dimensionless; it has units of length. Parameter *G*_{1} is shown for a range of plausible (*R, **z*_{d}) values in Fig. 8a. For *R* > 2*z*_{d} , *G*_{1} ∼ *z*_{d} ; for *R* < 2*z*_{d} , *G*_{1} is nonlinear in *R.* The vertical electric field *E*_{z} (neglecting a constant offset) is

The field is maximum along the axis and at the plates:

For convenience, call the geometric term above *G*_{2}(*R, **z*_{d}). Note that *G*_{2} is dimensionless, and shown in Fig. 8b. For very large *R,* this correctly reduces to the infinite plate result (no dependence on *z*_{d}). For very small *R,* the *z*_{d} dependence also drops out. For intervening values, *G*_{2} varies nonlinearly but only takes on a small range of values (factor of 2 variation).

### b. Estimation of flash rate

To compute flash rate, assume that the interflash time is given by the time *δt* it takes the generator current to convey enough charge to bring the dipoles fully to breakdown field strength. An efficiency of lightning charge removal *η* is included:

where *J*_{gen} is the generator current density (prescribed by storm microphysics and dynamics, i.e., fixed here); *σ*_{crit} is the charge density transferred to the dipoles required to build a critical field for breakdown, *E*_{crit} (which varies with thermodynamic and microphysical parameters, but is assumed constant); and *z*_{d} is prescribed (assumed set by storm microphysics and dynamics). It is derived from Eq. (23):

(1/*G*_{2} is shown for reference in Fig. 8c). It is already implicit from Eq. (26) that this approach will predict at best a weak dependence of *f*_{c} on storm area, deriving completely from *G*_{2} in *σ*_{crit}. Flash rate is thus given by

As stipulated, the flash rate here balances the generator current, and hence *ρ*_{gen}*V*_{Q} is the dominant term; the geometric correction adds at best a factor of 2 variability. Hence *f*_{c} ∼ *I*_{gen} essentially implies *f*_{c} ∼ *J*_{gen}. Additional weightings by area *R*^{2} and *z*^{2}_{d} are not predicted as in V63 and W85. The prediction is thus that if *f*_{c} varies dramatically with storm geometry (as observations suggest, i.e., as *z*^{5}_{t}), it is because *ρ*_{gen}*V*_{Q} covaries with geometry, not because of energetically based geometric weighting. This is a highly important physical distinction between the explanations provided by the two approaches.

### c. Lightning power dissipation

Under this framework, power *P* again is a current run through a potential difference, and the lightning current is assumed to match the generator current. Hence,

This is dimensionally consistent in units of Watts. For *R* > *z*_{d} , *G*_{1}/*G*_{2} ∼ *z*_{d} /2 (Fig. 8d), hence,

(still dimensionally consistent). This is the expected result, since the potential is assumed given by a critical field occurring over a given dipole separation distance. The dimensional difference in *z*_{d} from V63 arises because of the assumption of 2D charge plates, in contrast to V63's spherical charge centers.

### d. Charge transfer per flash

The assumptions that flash rate scales with generator power or current carry different implications for the necessary charge transfer per flash *Q* (since both stipulate a steady state, should satisfy current continuity, and yield different predictions for *f*_{c}). Since W85 implies a fixed electrostatic energy dissipation per flash *γ*^{–1}, the charge transfer per flash will necessarily vary inversely with storm potential difference. Alternatively, in an approach which stipulates *f*_{c} ∼ *I*_{gen}, it might appear that a fixed charge transfer per flash is predicted. However, since *f*_{c} predicted under this approach essentially (to within a factor of 2) varies with *J*_{gen}, charge transfer per flash in this approach will necessarily vary directly with storm area (to maintain the stipulated lightning–generator current balance).

Under W85, since the lightning current is operating in the same Φ as the generator, lightning power dissipation is given by

Recalling the implicit assumptions in Eqs. (4) and (5),

The implication is thus that larger dipoles (deeper or wider storms, under scale similarity between the dipoles and storms, as assumed by V63 and W85) yield less charge transfer per flash (while supporting a higher *f*_{c }*because of their geometry*). This seems intuitively palatable for deep/high flash rate storms, but seems counterintuitive for large-scale, low flash rate clouds with large dipole moment change flashes (e.g., the stratiform regions of mesoscale convective systems, winter storms, and possibly oceanic storms). Geometry, in those cases and under *f*_{c} ∼ *γP,* seems to work in the wrong direction.

Note that this general relationship (*Q* varying inversely with storm scale) is insensitive to the particular dipolar charge configuration assumed. If V63 and W85 had, instead, stipulated thin charge plates (as above), *Q* would vary as

that is, *Q* ∼ *L*^{–1} (where, for *R* > 2*z*_{d}, *L* is essentially *z*_{d} and any *R* dependence drops out). This is illustrated in Fig. 9a, which shows 1/*G*_{1} for plausible scales. This general (inverse scaling) result derives because Eqs. (33) and (34) describe the implicit assumptions in W85 (fixed electrostatic energy dissipation per flash), not the formal geometry-dependent details.

Under the *f*_{c} ∼ *I*_{gen} approach derived here,

which essentially predicts that charge transfer per flash varies directly with storm area (for *R* ≫ *z*_{d}, the *z*_{d} dependence essentially drops out). This is again necessary as *f*_{c} ∼ *I*_{gen} is effectively equivalent to *f*_{c} ∼ *J*_{gen}, to within a factor of 2; hence, *Q* must vary with area to balance net current. That is,

This is quite different from the W85 prediction. Here, larger storms only yield slightly higher flash rates because of their geometry (and if the observations claim they yield significantly higher flash rates, it is because *ρ*_{gen}*V*_{Q} somehow covaries with *L*) and also yield more charge transfer per flash. Flash rates are more driven by generator current than storm geometry (by stipulation). Predictions of *Q* from Eq. (37), assuming *η* = 1, *E*_{crit} = 300 kV m^{–1}, are shown in Fig. 9b. Clearly, predicted *Q* are unrealistically high, suggesting *η* ≪ 1 (a value of 0.1 could yield plausible results).^{8}

Differences between an *L*^{–2} or *L*^{–1} dependence and an *L*^{2} dependence should be empirically testable based on measurements of *Q* and storm geometry across a spectrum of storm scales.

### e. Prediction of updrafts and cloud-top heights

The *f*_{c} ∼ *I*_{gen} approach can also be used to examine predictions of univariate scaling relationships. Constraining this application are observations suggesting that *f*_{c} is highly nonlinearly dependent on *z*_{t}, while the predicted *f*_{c} has no (significant) direct *z* dependency. This has the necessary implication (under this approach), that *I* = *I*(*z*), or alternatively that *I* depends on a storm parameter, which itself is highly nonlinearly coupled with *z.* Ignoring the weak geometric dependency *G*_{2} (which, under scale similarity, is a constant anyway), we have, for land and ocean:

substituting (*κ*_{l}, *α*_{l}, *k*_{l}, *a*_{l}) or (*κ*_{o}, *α*_{o}, *k*_{o}, *a*_{o}) as appropriate for land or ocean. The (*κ, **α*) must be determined by empirical fits [e.g., PR92's empirical *f*_{cl} (*z*_{l})]. Leaving *η* as a free parameter, it is clear that

As with *f*_{c} ∼ *P*_{gen}, the existence of regionally variant *w*(*z*) requires regionally variant *f*_{c}(*w*). However, under *f*_{c} ∼ *P*_{gen}, *J* ∼ *w* is stipulated [regional microphysical variability in *J*(*w*) is disallowed]. Under *f*_{c} ∼ *I*_{gen}, no such restriction is imposed, and under empirical constraints, *J*(*w*) must vary regionally, presumably reflecting variability in local microphysics.

The univariate scaling relations using (*κ, **α*) from empirical *f*_{c}(*z*) fits by PR92 and Ushio et al. (2001) are shown in Table 2. Ushio et al. (2001) analyzed data for August 1998 from the TRMM LIS and PR, for spatial domains including Northern Hemisphere only, and the Tropics. Fits were made to raw (*f*_{c}, *z*) data and data averaged in altitude (as by W85). The tropical results (along with predictions from the *f*_{c} ∼ *P*_{gen} approach) are shown in Figs. 10 and 11. Since the *z* spectra shown in Fig. 11 are simply remappings of the climatological Pr(*f*_{c}) through purely empirical fits, they illustrate the disagreement between *f*_{c} ∼ *P*_{gen} predictions and observations. Specifically, the *f*_{c} ∼ *P*_{gen} cloud-top spectra appear to roll off too quickly, consistent with the earlier observation that they disagree with spectra reported by Johnson et al. (1999) [over oceans, the empirical *z* spectra are now consistent with Johnson et al. (1999)'s spectra]. This of course does not validate the *f*_{c} ∼ *I*_{gen} approach, since the empirical fits are stipulated.

Note that since deeper oceanic cloud tops correspond to the same flash rate as over land, *if* any degree of scale similarity between *z*_{d} and *z*_{t} holds, this approach predicts *Q*_{o} > *Q*_{l} (for the same flash rate), and if any degree of scale similarity between *R* and *z*_{t} holds, this approach *may* predict *Q*_{o} ≫ *Q*_{l}. Since the shapes of land and ocean *f*_{c} spectra are similar (Fig. 1) and differ primarily in frequency of occurrence, this prediction may be testable by examination of the relative frequency of occurrence of high dipole moment change flashes observed over tropical land and ocean by long-range extremely low-frequency (ELF) or SR techniques (at comparable ranges).

For updrafts (Fig. 10), the results are again dominated by the dynamic range in the constraining *w*(*z*) data. The *f*_{c} ∼ *I*_{gen} approach predicts higher mean updrafts over both land and ocean, and the land means are significantly higher than under *f*_{c} ∼ *P*_{gen} (30–40 m s^{–1} vs 23 m s^{–1}). The large scatter in the underlying (*f*_{c}, *z*) data (nonrobustness of the empirical fits against averaging or spatial domain) and the narrow dynamic range of the ocean spectra suggests that discrimination between the two theories will not be possible by comparison with observed ocean updraft spectra. Such discrimination might be possible over land. Overall, the land–ocean differences are again dominated by *w*(*z*), although for *different reasons* {microphysics, i.e., *J*[*z*(*w*)], rather than cloud geometry, i.e., Φ[*z*(*w*)]}.

### f. Summary

In summary, W85's (*f*_{c} ∼ *P*) approach has flash rate set, essentially, by storm geometry, with a small direct contribution from generator current density (which is stipulated to vary linearly with updrafts, i.e., *J* ∼ *w*). Under W85, charge transfer per flash varies inversely with storm scale, is insensitive to generator current, and is “what it needs to be” to have lightning power dissipation match generator power production under a prescribed potential difference (equivalently, each lightning flash transfers a constant electrostatic energy). Under this approach, *J* is predicted to be significantly smaller over ocean (for a given observed flash rate). Predicted cloud-top height spectra appear to be lower than observed.

In the alternative (*f*_{c} ∼ *I*) approach, flash rate is set directly by generator current density with slight modulation by storm geometry. Since flash rate observationally varies dramatically with storm scale, the implication is that generator current density must covary with scale (for unspecified reasons). Charge transfer per flash varies directly with storm scale (primarily area), and is what it needs to be to maintain near-critical breakdown fields under the inferred flash rate (equivalently, each lightning flash is responsible for a constant current). When constrained by empirical *w*(*z*) relationships, the implication is that *J* is nonlinearly dependent on *w,* and that this relationship is regionally variant, with exponent ∼2 over land, and ∼5–15 over ocean.^{9} Under this approach, *J* is stipulated to be identical over land and ocean (for a given observed flash rate). When driven by empirical *f*_{c}(*z*), the approach predicts higher valued updraft spectra than W85's approach.

Empirical testing of predictions of *Q* and its dependence on storm scale are needed to distinguish between the two approaches from observational data analysis. This ambiguity is nontrivial, as *J* is presumably the quantity of greatest interest in the pursuit of storm property inferences from flash rate observations (through its coupling to updrafts *w,* ice *q,* etc.).

The two approaches are shown schematically in Figs. 12a,b. “Open boxes” and dashed lines denote empirical fits or parameterizations. The sequencing of generator-driven electrical parameters is simply the order in which they might be computed under a prescribed *J*_{gen} and geometry in a fully time-and-space dependent calculation. The *f*_{c} ∼ *P*_{gen} approach [under empirical *w*(*z*) constraints] predicts regionally variant *f*_{c}(*w*) relationships; since microphysical contributions are disallowed under its *J* ∼ *w* assumption, it is clear that such regional variability in this approach must be driven by forcing and environmental effects on the relationship between updrafts and geometry. The *f*_{c} ∼ *I*_{gen} does not explicitly describe the generator current–meteorology connection and allows for microphysical contributions. Under empirical *w*(*z*) and *f*_{c}(*z*) constraints, it predicts regionally variant *f*_{c}(*w*) *and **J*(*w*). In this approach, regional variability (variability in forcing and environment) may manifest itself in *J*(*w*) by influencing local microphysics (although the exact influence is not explicitly described).

Since both approaches, when constrained by observed *w*(*z*), predict regionally variant *f*_{c}(*w*), the observation that Pr(*w*) varies regionally is *by itself insufficient to explain regional lightning variability.* A complete explanation must also consider why, for example, similar updrafts over land and oceans yield different cloud geometry (under *f*_{c} ∼ *P*_{gen}) or why similar updrafts over land and oceans yield different generator current density (under *f*_{c} ∼ *I*_{gen}).

Under either approach, geometry *G* and generator current density *J* are fundamental parameters of interest (especially in the context of understanding and/or parameterizing a lightning–microphysics or lightning–dynamics relationship). This places added emphasis on the importance of calculating or observing *G* and *J* during aircraft and ground-based electrification studies, and of reporting them during explicit modeling studies.

## 6. Conclusions

The history of scaling relationships between thunderstorm electrical energetics and thunderstorm dynamic and geometric properties has been reviewed, with particular care given to describing the (often implicit) assumptions made when simplifying the underlying theory. The particular implementation of these relationships derived by PR92 for application to oceanic domains has been shown to yield unrealistic predictions of oceanic updraft spectra and nonphysical predictions of cloud-top heights, and to contain a formal inconsistency with Vonnegut's original theory.

It is emphasized that the operational utility of PR92's ocean parameterization for prediction of bulk (long timescale) lightning production is not addressed in this study, merely its inconsistency with the underlying theory from which it purports to derive, and its (inferred) inconsistency with the latest oceanic instantaneous storm flash rate observations and limited knowledge of oceanic updraft spectra. The empirical and theoretical inconsistencies should strongly caution against further attempts to improve the PR92 ocean parameterization from a theoretical standpoint [e.g., inclusion of the effects of CCN spectra (Michalon et al. 1999)], although of course purely empirical fine-tuning is not precluded. The inconsistencies should further caution against use of these parameterizations for the reverse problem (inference of storm properties from lightning observations themselves), although the parameterizations were never intended for this purpose.

The revised univariate scaling relations [Eqs. (18), (19)] are consistent with Vonnegut's original scaling framework and provide significant constraints on what the true univariate scaling dependencies under this theory should be, even in absence of adequate observations over oceanic regions. Using these revised parameterizations, predictions of land and ocean updraft and cloud-top height spectra from the observed flash rate distributions yield physically plausible results. This is primarily a demonstration that univariate relationships correctly derived from Vonnegut's theory *must* yield internally self-consistent results. The utility of these univariate relationships for the application that motivated PR92 (forward prediction of flash rates from observed cloud-top heights) has *not* been demonstrated here. An important implication of the revised approach is that regionally invariant univariate relationships between lightning and storm properties are not necessarily expected to exist.

Three significant assumptions in the simplified scaling laws are identified for further testing. Two, better estimation of the dipole separation distance using proxy parameters, and identification of storm cell area/height interdependency (or lack thereof), are tractable using existing data or data currently being collected, and represent “fine-tuning” of the approach. The third, the mapping between updraft velocity and generator current density, is of greater importance, is relevant for any approach, and is in principle derivable from numerical cloud models with explicit microphysics and electrification. Scaling theory, even if not operationally applied, thus helps provide an interpretation and analysis framework (and motivation) for observational or modeled data.

An alternative scaling approach, assuming flash rate varies directly with generator current, was examined. This approach necessarily yields little direct (geometrically based) dependence of flash rates on storm scales. Observations of strong flash rate/scale dependencies thus imply that generator current density varies with storm scale, for unspecified (and not necessarily directly causal) reasons. Vonnegut's approach (under Williams' simplifications) implies that charge transfer per flash will vary inversely with storm scale (especially depth), while the generator current approach implies that charge transfer per flash varies directly with storm scale (primarily area). Vonnegut's approach stipulates that, over land, generator current density varies linearly with updrafts *w,* and in absence of claims to the contrary, this must be assumed to apply over oceans as well. The generator current approach, paired with empirical data, requires generator current density to vary as *w*^{2} over land, and as *w*^{5}–*w*^{15} over ocean. For a *given observed flash rate,* Vonnegut's approach predicts significantly smaller generator current density over oceans, while the generator current approach stipulates that it be the same over land and ocean. Neither approach predicts a regionally invariant flash rate/updraft relationship, which implies that observations of regional variability in updraft spectra are insufficient, by themselves, to fully explain regional variability in lightning. Vonnegut's approach suggests that flash rate observations should be normalized by storm area and depth to capture variability in generator current density, while the generator current approach does not. For physically based inference of storm properties from lightning flash rate observations, the distinction is important, as generator current density is the parameter most closely coupled to meteorological parameters of interest. Determination of whether flash rate is governed by power or current (perhaps through examination of the predictions of dependence of charge transfer per flash on storm geometry) is thus required to resolve the normalization issue.

## Acknowledgments

Thanks are given to Earle Williams, Kevin Driscoll, William Koshak, Walt Petersen, Steve Nesbitt, Stan Heckman, and Tomoo Ushio for extensive discussions and suggestions, and early access to data analyses in progress. This research was partially supported by NRA-99-OES-03, under the direction of Dr. Ramesh Kakar.

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## Footnotes

*Corresponding author address:* Dennis J. Boccippio, National Space Science and Technology Center, NASA/Marshall Space Flight Center SD-60, Marshall Space Flight Center, AL 35812. Email: Dennis.Boccippio@msfc.nasa.gov.

^{1}

This observation seems corroborated in isolated storms by a number of studies reviewed by W85, and by results of Krehbiel et al. (1984) and Heckman and Ushio (1999, personal communication) utilizing time-of-arrival channel and electric field-based charge center locations, but called into question in MCSs and supercells from electric field soundings by Stolzenburg et al. (1998).

^{2}

Alternatively, the assumption *w* ∼ *ρ*^{2}*V*_{Q} would yield the same result but not demand invariant charge densities.

^{3}

Within the context of the theory, *w*(*z*) is a statistical relationship between instantaneously observed separation velocities in the charging zone and dipole separation distance (not a vertical profile of *w*). Proxying this relationship by maximum updraft velocity and cloud-top height is, again, an observational convenience that may or may not be intrinsically meaningful.

^{4}

Parameter *k*_{1} may be estimated by equating Eq. (18) [using the PR92 ,*w*_{l}(*z*_{l}) relationship] to PR92's empirical land *f*_{c}(*z*) fit and solving. This yields an expression that scales as *z*^{–0.19}. This term varies negligibly over tropospheric cloud heights and is replaced with a constant value of 0.62, yielding *k*_{1} = 1.4314 × 10^{–5}. Alternatively, a fit of Eq. (18) may be made directly to land (*f*_{c}, *z*) data, fixing (*a*_{l}, *k*_{l}) and hence removing any *z* dependency from *k*_{1}.

^{5}

Note that the spectra reported by Johnson et al. (1999) also included warm precipitation features, which comprise at least 20%–30% of the overall spectrum. To enable direct comparison with predicted Pr(*z*) as normalized by the results of Nesbitt et al. (2000), this warm population should be excluded from consideration, yielding Pr(*z* = 10) ∼ 0.1.

^{6}

Note that under the new *w*(*z*) fits, the calibration constant *k*_{l} must be recomputed; following the same approach as earlier [matching to PR92's empirical land *f*(*z*) curve], *k*_{l} is now estimated at 1.7204 × 10^{–5}.

^{7}

At the very least, belief in Eq. (6) suggests that we should normalize flash rate observations by the geometric terms *A**z*^{2}_{d}, if such are available, if we wish to relate them to generator current density *V*_{Q} Belief in Eq. (3) further suggests that results from 1D or 1.5D cloud–electrification models that impose spatial scales upon the simulated updrafts [e.g., Solomon and Baker (1998)] should be interpreted with care.

^{8}

Driscoll et al. (1994) argued that the ratio of lightning to generator current (*η* here) might be 0.33–0.66, while instantaneous discharge charge removal efficiency (perhaps analagous to *η* here) has been reported as 0.3–0.5 in thundercloud discharges (Winn and Byerley 1975) and 0.2–0.4 in laboratory (polymethylmethacrylate) discharges (Williams et al. 1985).

^{9}

Interestingly, an empirically driven power-law *J*(*w*) prediction with very small coefficient and very large exponent might be interpreted as evidence of a critical velocity and an updraft spectrum that rarely strays very high above it.