1. Introduction

Much attention has been given recently to the study of the global atmospheric electric circuit and, in particular, to its modeling (Rycroft et al. 2008; Tinsley 2008; Williams 2009; Rycroft and Harrison 2012; Williams and Mareev 2014). The most important aspects of modeling the global circuit are the description of its generators and the parameterization of the conductivity distribution.

The hypothesis originally proposed by Wilson (1924) that thunderstorms drive the global circuit is now widely recognized, with the alteration that electrified shower clouds are also taken into account. Historically, two different approaches have been used to express this idea quantitatively that differ as to whether thunderclouds should be regarded as current sources (Volland 1984) or voltage sources (Markson 1978). Willett (1979) treated thunderstorms as current sources, at the same time emphasizing that which type of source is most appropriate depends on the mechanism of the charge separation in thunderclouds.

One of the most comprehensive models of the global electric circuit was suggested by Hays and Roble (1979). However, generators in this model, as well as in many others (e.g., Holzer and Saxon 1952; Willett 1979), were restricted to a finite number of point current sources. Continuous spatial distribution of the external current density, which seems to describe generators more adequately, was used by Davydenko et al. (2004).

A number of models of atmospheric electricity focus on the role of the conductivity distribution in the global circuit (Makino and Ogawa 1985; Tinsley and Zhou 2006; Zhou and Tinsley 2010; Baumgaertner et al. 2013). It is a well-known fact that the conductivity of the air increases nearly exponentially with altitude, and that is why many models deal with a universal exponential vertical profile of the conductivity. However, large-scale conductivity inhomogeneities caused by various natural and anthropogenic factors can significantly affect the structure of the global circuit and, in particular, the ionospheric potential.

Markson (1978) analyzed the impact of ionizing radiation due to solar flares on the global circuit through an increase in conductivity in the upper atmosphere. The change of the atmospheric electric field at ground level was observed after the release of radioactive material from nuclear power plants (Takeda et al. 2011) and after nuclear explosions (Pierce 1972). It was established that during the period of nuclear testing in 1960s the ionospheric potential increased substantially, and its value lagged 1 year was highly correlated with nuclear radiation; Markson (2007) suggested that nuclear radiation affects the ionospheric potential by changing the conductivity of the air.

The concept of regarding thunderclouds as the generators of the global circuit is closely associated with another type of conductivity inhomogeneities. It is often assumed that the conductivity is reduced inside thunderclouds owing to the attachment of ions to hydrometeors (Rycroft et al. 2008; Zhou and Tinsley 2010; Odzimek et al. 2010); experimental evidence for this assumption may be found, for example, in observations by Rust and Moore (1974).

Here we develop an approach that enables us to estimate the influence of various large-scale conductivity inhomogeneities on the ionospheric potential and generalizes the equivalent-circuit models used by Markson (1978), Willett (1979), Volland (1984), and Odzimek et al. (2010). One of the most crucial advantages of our approach is that it is based on a well-posed problem formulation for a steady-state model of the global electric circuit, as discussed by Kalinin et al. (2014), in which thunderstorm generators are treated as a continuous distribution of the external current and the ionospheric potential can always be uniquely determined from the solution. Under several simplifying assumptions about the conductivity distribution and thunderclouds, we obtain an approximate solution to the model equations and, whenever possible, compare it with the results of direct numerical simulations. However, here we do not take into account the variability of several important factors including seasonal and diurnal variations of the global circuit (e.g., Mach et al. 2011; Hutchins et al. 2014) and restrict ourselves to simple model problems.

2. Problem formulation

Let us consider a simple steady-state model of atmospheric electricity in which the atmosphere is bounded by Earth’s surface Σ₁ and a surface Σ₂ representing the lower limit of the ionosphere—the former being encompassed by the latter. Setting both the dielectric permittivity and the magnetic permeability of the atmosphere equal to 1, stationary Maxwell’s equations read as follows (we use the Gaussian unit system):

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where E is the electric field, H is the magnetic field, J is the current density, ρ is the charge density, and c stands for the speed of light. The question of to what extent it is possible to describe the global circuit by stationary equations is beyond the scope of this study; several aspects of this problem were discussed by Boström and Fahleson (1976), Holzworth (1995), and Kalinin et al. (2014). Equations (1)–(4) must be supplemented with Ohm’s law:

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where σ stands for the conductivity and the second summand represents the external current density¹ as well as with the boundary conditions for E at Σ₁ and Σ₂. Since the air conductivity increases nearly exponentially with altitude, whereas its value near Earth’s surface is much less than the ground conductivity, both surfaces Σ₁ and Σ₂ can be regarded as perfect conductors, which yields the boundary conditions

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where the index τ indicates the tangential component. Substituting relationship (5) into (1) gives

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Given σ and J^ext, this relationship together with (2) and (6) form a system of equations in E and curlH. If curlH and E are found, one can obtain the magnetic field itself and the space charge density from (3) and (4), provided that necessary boundary conditions for H are established. Therefore, it is sufficient to find E for solving the entire system of (1)–(6).

Equation (2) makes it possible to introduce the electric potential in the atmosphere, a function ϕ such that E = −gradϕ, and to reformulate other equations in terms of this function. Conditions (6) mean that ϕ does not vary over each of the two boundary surfaces (we do not take into account magnetospheric and ionospheric generators of the global circuit, which might result in a variation of the potential across Σ₂), and thus if it is set equal to zero at Σ₁, its value at Σ₂ represents the ionospheric potential V_i. To eliminate the magnetic field from (7), let us point out that for any vector field X defined in a region Ω, the following two statements are equivalent (Girault and Raviart 1986):

1. There exists a vector field Y such that X = curlY within a region Ω.

2. divX is equal to zero, as is the flux of X through each connected component of the boundary of Ω.

Applying this general result to the right-hand side of (7) and substituting −gradϕ for E, we arrive at the equations

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Note that we have omitted the counterpart of (9) for Σ₂, which is redundant, being a consequence of (8) and (9). If a field E satisfies (7), its potential must satisfy (8) and (9); conversely, if the potential of E satisfies (8) and (9), there exists a field H such that (7) holds, although this H is not yet guaranteed to meet the requirement (3) and (unspecified here) boundary conditions.

Adding to (8) and (10) the boundary conditions

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we arrive at the system of equations describing the potential distribution in the atmosphere. It can be shown that if ϕ⁽¹⁾ and ϕ⁽²⁾ are two solutions to (8)–(10) with respective ionospheric potentials and , then ϕ⁽¹⁾ = ϕ⁽²⁾ and (Kalinin et al. 2014). Therefore, V_i can always be inferred from the solution of (8)–(10) and should not be specified explicitly. Furthermore, it can be proved that if some restrictions are imposed on the conductivity distribution and boundary surfaces, this problem is well posed; namely, there exists a solution to (8)–(10), which is unique in a certain class of functions [for more details, see Kalinin et al. (2014) and references therein]. This also enables us to establish the connection between the equations for ϕ [(8)–(10)] and the original equations [(1)–(6)] in terms of the fields. Indeed, if we suppose that the original equations have a solution E, H, then the potential of E satisfies (8)–(10), and now that the solution of the latter system is unique, one can say that every its solution corresponds to a solution of the original equations (in addition, from this it follows that the original equations uniquely determine E).

It is crucial to emphasize that the value of the constant V_i is not explicitly specified, but is inferred from the solution of (8)–(10). If we omitted (9) and set V_i equal to a given constant, the solution obtained might not correspond to the original Maxwell’s equations. Of course, if this constant were chosen to be close to the real value of V_i, the obtained potential distribution would also be close to the real one, but such an approach is inconsequent and does not correspond to any well-posed formulation of the problem.

3. Explicit formulas for the ionospheric potential and numerical modeling

Henceforth we confine ourselves to a specific problem geometry. Let Σ₁ and Σ₂ be concentric spheres, the radii thereof being equal to r_min and r_max, respectively, and let (r, θ, ψ) be spherical coordinates whose origin coincides with the common center of the spheres Σ₁ and Σ₂. Then, within the framework of the model described above, it is possible to deduce an explicit formula for the ionospheric potential, supposing that the conductivity is a function of r alone. First of all, it follows from (8) and (9) that for all h ∈ [r_min, r_max] we have

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Σ_h being the surface {(r, θ, ψ): r = h}. By means of the assumption imposed on the conductivity, one can rewrite this relationship as

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from which, by canceling equal factors which do not depend on θ and ψ, we obtain

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Hence, by integrating with respect to r from r_min to r_max and using (10), we get the required formula:

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Using a similar argument, one can derive a more general formula for the ionospheric potential in case the conductivity is of the form σ(r, θ, ψ) = a(r) · b(θ, ψ):

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However, this technique does not yield V_i in the most general case, when no assumptions are made about the conductivity distribution.

Although some explicit formulas can be derived for V_i, the most general problem is to find the distribution of the potential in the atmosphere and particularly the value of the ionospheric potential, making no additional assumptions about σ and J^ext. This problem can be surmounted by numerically solving (8)–(10) employing the Galerkin method, and we implemented this idea to construct a numerical model that calculates the spatial distribution of the potential, provided that σ(r, θ, ψ) and J^ext(r, θ, ψ) are axisymmetric (i.e., do not actually depend on ψ); axial symmetry was assumed only so as to speed up computations.

Although in this paper we primarily deal with steady-state problems, it is interesting to note that the corresponding nonstationary problems can be analyzed in a similar fashion within the framework of the quasi-stationary approximation, in which we assume that (2) holds despite H being time dependent. As in the steady-state case, one can introduce the electric potential, obtain a system of equations for it [from which the ionospheric potential is uniquely determined as a function of time V_i(t)], prove that this problem is well posed, derive some explicit formulas for V_i(t), and construct a numerical model calculating the distribution of the potential (Kalinin et al. 2014).

4. Influence of large-scale conductivity inhomogeneities

Let us now consider another special case, where weaker restrictions are imposed on the conductivity but the external current density is not an arbitrary function. The region occupied by the atmosphere being denoted by Ω, we can write Ω = Γ × [r_min, r_max], where Γ is the unit sphere and the multiplication sign indicates the Cartesian product. Suppose that

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and Γ_i ∩ Γ_j = Ø for i ≠ j. This partition of Γ induces a corresponding partition of Ω, namely

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where Ω_j = Γ_j × [r_min, r_max]. An example of such a partition is shown schematically in Fig. 1a. Using the spherical coordinates introduced in the previous section, let us suppose that σ(r, θ, ψ) = σ^(j)(r) and within Ω_j, whereas in Ω. In other words, the full solid angle is divided into several regions ω₁, ω₂, …, ω_n (ω_j being subtended by Γ_j) such that

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and

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(a) Partition of the atmosphere and (b) an equivalent electric circuit.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0001.1

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In case these conditions are satisfied, a fruitful approach to solving the problem (8)–(10) is based on the fact that as long as, for all j, the characteristic “horizontal” scale L_j of Ω_j is much greater than the characteristic vertical scale R = r_max − r_min, the derivatives with respect to θ and ψ in (8) can be neglected. To show this, let us rewrite (8) within Ω_j as

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where Δ_⊥ stands for the nonradial summands of the Laplace operator. Our ultimate goal is to compare the two terms in the left-hand side of (16) by means of nondimensionalization.

We use R = r_max − r_min and Φ, the maximum absolute value of the potential in Ω_j, to introduce the dimensionless radial distance, , and the dimensionless potential, . Taking the simplest assumption that the conductivity increases exponentially with height—namely,

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and we can transform the first term in (16) into

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which, by nondimensionalizing, can be rewritten as

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where we kept the conductivity dimensional, for it is a common factor in the two terms that we want to compare. As H is about 6 km (e.g., Ogawa 1985) and R is about 70 km in the real atmosphere,² one can assume that H ≪ R ≪ r_min, which allows us to approximate the last expression as

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Since the angles θ and ψ are already dimensionless and

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we cannot nondimensionalize the second term in (16) as straightforwardly as we did for the first one. However, let us note that if R ≪ L_j ≪ r_min, then the spherical geometry of the problem can be locally described by the Cartesian coordinates within the framework of the flat-Earth approximation, and, in consequence, it is possible to write approximately

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where is the dimensionless counterpart of the “flat” Δ_⊥ (i.e., of the horizontal terms of the Laplace operator in local Cartesian coordinates). Therefore, the second term in (16) can be estimated as

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It is reasonable to assume that and are of the same order. Then, comparing (17) and (18), one can see that the latter is much less than the former, provided that R ≪ L_j.

Even though the arguments given above do not involve rigorous comparison of the two terms in (16), they still constitute the motivation to neglect the latter of them. In other words, this means that we can neglect the horizontal component of the current in the atmosphere except for surface currents at Σ₁ and Σ₂. Such a “one-dimensional” approximation is convenient for the studies of atmospheric electricity [e.g., Stolzenburg and Marshall (1994) used similar approximation for estimating the charge density from electric field observations by means of (4)]. Applying it to (16), within Ω_j we get

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whence by integration we arrive at

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where C_j(θ, ψ) is a certain function. It immediately follows from the boundary conditions (10) that

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and therefore

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Substitution of (19) into the integrand yields the relationship

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from which it follows that C_j is actually a constant, independent of θ and ψ. With j in the range 1 ≤ j ≤ n, it gives n equations in n + 1 variables C₁, C₂, …, C_n and V_i. The equation that closes the system is obtained by substitution of (19) into (9), which gives

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where γ_j is the solid angle subtended by Γ_j (and thus the sum of γ_j is equal to 4π). Eliminating all C_j from (21) and (22), one can obtain the following formula expressing the ionospheric potential:

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It is easy to see that for n = 1 the approximate equation (23) coincides with the exact expression (11). Furthermore, in case the conductivity is of the form σ(r, θ, ψ) = a(r)b(θ, ψ), this formula turns into (12), provided that the regions Ω_j are taken to be infinitesimally small. To show this, let us point out that if b(θ, ψ) is a piecewise constant function, being equal to b_j within Ω_j, then (23) can be simplified to

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which turns into (12) as γ_j → 0, for this allows us to replace the summation over j by integration of continuous functions over θ and ψ. Thus, the approximate equation (23) serves as a generalization of the exact formulas derived in section 3.

Since C_j is a constant, one can conclude from (19) and (20) that the potential is a function of r alone within each Ω_j; that is,

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Note that this solution satisfies (8) in each Ω_j, satisfies (9), and meets the boundary conditions (10) at Σ₁ and Σ₂. However, this solution, although assumed to be continuous on every Ω_j, is not necessarily continuous on entire Ω and may have discontinuities at the surfaces separating contiguous regions Ω_j. On the other hand, the existence and uniqueness theorem that can be proved for (8)–(10) implies that the potential must be an element of a certain class of continuous functions (which we do not specify here). From this, it is clear that we have actually found an exact solution to (8)–(10), which may not belong to . In other words, the method we have just used can be interpreted as an external approximation of the potential.

5. Equivalent circuit

The approximation described in section 4 turns out to be a generalization of classical multicolumn models of atmospheric electricity based on the concept of equivalent circuit. In such models the entire atmosphere is divided into two or more columns, some corresponding to thunderstorm regions, where the current flows upward, and others corresponding to fair-weather regions, where the current flows downward. Then, replacing different regions with equivalent resistors and current sources, it is possible to simulate the real atmospheric electric system by an equivalent circuit. Such a circuit, generalizing those used in models by Markson (1978), Willett (1979), Volland (1984), and Odzimek et al. (2010), is shown in Fig. 1b. It consists of n parallel vertical current paths whose lower ends are joined together, as are their upper ends. We suppose the jth vertical current path to be a series of infinitesimal elements consisting of a current source of strength and a resistor of resistance R_jk connected in parallel.

To establish the correspondence between this circuit and the model of the previous section, we say that n vertical current paths correspond to the regions Ω₁, Ω₂, …, Ω_n, the bottom and top of the circuit representing the boundary surfaces Σ₁ and Σ₂, respectively. We also demand that R_jk and the external currents be chosen according to the distributions σ^(j)(r) and defined in (13) and (14). More precisely, we suppose that the region Ω_j = Γ_j × [r_min, r_max] is partitioned into infinitesimally thin slabs Γ_j × [r_jk, r_j,k+1] with r_j,k+1 − r_jk = dr_jk, and the kth slab corresponds to the element with the resistance R_jk and the external current ; that is,

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Denoting the current in the jth path as I_j (which is defined to be positive if the current flows upward and negative otherwise), we get n equations of the form

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where V_i stands for the voltage between the top and bottom of the circuit, thus being an equivalent of the ionospheric potential. Then, since

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we arrive at the formula

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which is similar to that obtained by Odzimek et al. (2010).

Substituting the relationships (24) into (25) and replacing the sums over k with integrals over r, we arrive at (23) again. However, this is not surprising because the two approaches are actually equivalent, inasmuch as in either case we divide the atmosphere into one-dimensional columns and neglect the current flowing through their side surfaces.

The condition for the applicability of the approximation developed in section 4 is R ≫ L_j for all j, and since equivalent circuit models can be regarded as a special case of this approximation, the condition for their validity is the same. As characteristic horizontal dimensions of real thunderclouds are less than the height of the atmosphere, we conclude that, strictly speaking, this approximation cannot be directly applied to quantitative description of the potential distribution in the atmosphere (in other words, the horizontal component of the current in the vicinity of thunderclouds cannot be neglected). However, numerical simulations show that it is still useful for qualitative analysis of the global circuit, especially of how conductivity inhomogeneities affect the ionospheric potential. Several examples of this are given in the following sections.

6. A simple model of a thundercloud

Even though the assumptions under which formulas (11) and (23) are valid do not enable us to deal with arbitrary distributions of the conductivity and the external current density, it is nevertheless possible to solve several model problems that reveal qualitatively certain mechanisms accounting for the variation of the ionospheric potential.³

Within the framework of the model described in section 2, it is the external current density J^ext that drives the global electric circuit and thereby determines the distribution of the potential in the atmosphere. In the real atmosphere the distribution of J^ext corresponds to the position of thunderclouds, which are widely recognized as the main generators of the global electric circuit. This correspondence is based on the fact that vertical charge separation in a thundercloud can be interpreted in terms of the external current. More precisely, let us consider a thundercloud that occupies the region

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in the atmosphere, where r_min ≤ r₁ < r₂ < r_max and is a certain region on the unit sphere parameterized by θ and ψ (for our purpose, it is convenient to restrict ourselves to simple geometric forms, as they make the calculations easier, while using a more complicated geometry does not seem to change the results significantly). To determine the external current density distribution corresponding to the thundercloud , we need to deal with a detailed microphysical description of the charging process (Mansell and Ziegler 2013). However, here we do not employ such an analysis and assume that this distribution is given by

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where Ω again stands for the region occupied by the entire atmosphere. The structure of J^ext is shown schematically in Fig. 2a.

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(a) The structure of the external current density distribution corresponding to a single thundercloud and (b) the structure of the global external current density distribution.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0001.1

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Recently it was recognized that electrified shower clouds also serve as generators of the global circuit and contribute significantly into the distribution of atmospheric electricity (Williams 2009; Liu et al. 2010; Mach et al. 2011). However, here we do not distinguish between the two types of generators, and the term “thunderclouds” refers here to both thunderstorms and electrified shower clouds.

We have already mentioned that the air conductivity increases nearly exponentially with altitude, which is an essential condition for our model to be valid. The simplest possible approximation for the conductivity is given by the formula

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where H is a scale height (equal to about 6 km). Given the external current density [(26) and (27)] and the conductivity [(28)], one can obtain the ionospheric potential by using (11):

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where is the solid angle subtended by (i.e., by the thundercloud). We have deliberately written the left-hand side of this formula as δV_i (instead of V_i), in as much as it expresses the ionospheric potential produced by a single thundercloud.

Since (8)–(10) are linear with respect to ϕ, the electric potential distributions satisfy the superposition principle. To be specific, let and be two external current density distributions and let ϕ⁽¹⁾ and ϕ⁽²⁾ be the solutions to (8)–(10) with and , respectively (the conductivity distribution is assumed to be the same in both cases). Then the electric potential distribution corresponding to is exactly ϕ⁽¹⁾ + ϕ⁽²⁾, and the ionospheric potential produced by is equal to the sum of the contributions from and calculated separately. This immediately follows from the linearity of the equations and the uniqueness of the solution: ϕ⁽¹⁾ + ϕ⁽²⁾, owing to the linearity of the equations, satisfies (8)–(10) with and therefore is the only solution.

The superposition principle makes it possible to regard δV_i given by (29) as the contribution to the ionospheric potential from a single thundercloud. The entire ionospheric potential is equal to the sum of such contributions from all the thunderclouds in the atmosphere. Therefore, as long as every thundercloud has geometric shape of the same type as and the corresponding external current density of the form (26) and (27) (with the same r₁, r₂, and J₀; see Fig. 2b), the ionospheric potential is given by the formula

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where γ stands for the total solid angle occupied by thunderclouds. Note that this formula can be derived both straightforwardly from (11) in a similar way to the calculation of δV_i above and by adding together the contributions given by (29).

It is noteworthy that (30) does not contain the height of the upper boundary of the atmosphere, and thus the ionospheric potential given by it does not depend on the choice of r_max in the model. In case the conductivity distribution is more complicated, V_i does depend on this choice, but this dependence is rather slow for sufficiently large values of r_max, as shown by our numerical experiments. Moreover, one can consider a problem with r_max = ∞ and ascertain that the solution to (8)–(10) with finite r_max approaches the solution to (8)–(10) with r_max = ∞ as r_max → ∞. According to our numerical simulation, the potential does not change much with height above 60 km, provided that thunderclouds (corresponding to J^ext) occur only in the troposphere (below 20 km) and the conductivity increases nearly exponentially with altitude with the scale height of about 6 km. Therefore, it seems reasonable to set the value of r_max − r_min to be about 70 km. Yet another argument to choose r_max − r_min close to 70 km is the fact that above this altitude the conductivity is essentially anisotropic and thus more complicated approximations are needed to include this region in our model (e.g., Park and Dejnakarintra 1973).

7. The impact of conductivity inhomogeneities inside thunderclouds

Although the approximate theory developed in section 4 and (23) have certain restrictions, they still allow us to solve rather complicated problems that involve large-scale conductivity inhomogeneities. This section is dedicated to the study of the variation of the ionospheric potential due to conductivity inhomogeneities inside thunderclouds. In models of atmospheric electricity the conductivity inside thunderclouds is often assumed to be reduced in order to allow for the attachment of ions to hydrometeors.

First of all, let us assume, as before, that all the thunderclouds occur at the same height and have geometric shapes of the same type as defined in section 6, as is shown in Fig. 2b. More precisely, we assume that thunderclouds occupy the region

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where r_min ≤ r₁ < r₂ < r_max and is a (not simply connected) region on the unit sphere parameterized by θ and ψ, and we assume that the external current density distribution is given by (26) and (27) with replaced by . Let us denote the total solid angle subtended by (i.e., by thunderclouds) by γ.

In models of atmospheric electricity, it is often assumed that the conductivity inside a thundercloud is lower than that of the surrounding air. The simplest method to parameterize this effect is to introduce a coefficient β, −1 < β ≤ 0, and to state that σ = (1 + β)σ⁰(r) inside thunderclouds and σ = σ⁰(r) outside them—that is to say,

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Here we cannot apply (11), but as the conductivity and the external current density are of the form (13)–(15), we can use the approximate theory of section 4.⁴ Then we immediately obtain from (23) the expression

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where we used the following notation:

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and . Let us emphasize that , being equal to the solid angle subtended by divided by 4π, represents the portion of Earth’s surface covered by thunderclouds.

According to (31), the ionospheric potential increases with decreasing β. Note that since 0 < K < 1 and 0 ≤ γ ≤ 4π, this formula gives a finite value of V_i for any β ∈ (−1, 0]. When β = 0, the ionospheric potential is equal to

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which coincides with the value given by the exact equation in (30), and as β → −1, V_i tends to the value

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Thus, the maximum relative change in the ionospheric potential due to taking account of the reduction of the conductivity in thunderclouds is presented by the equation

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Taking r_min = 6370 km, r_max = r_min + 70 km, r₁ = r_min + 5 km, r₂ = r_min + 10 km, and defining σ⁰(r) as in (28) with and H = 6 km, we obtain that K = 0.246. Then, taking J₀ = 3 × 10⁻⁹ A m⁻² and β₀ = −0.9, we require that V_i(β₀) = 240 kV, which yields that . Substituting the obtained values of K and into (32), we arrive at the estimate [V_i(−1) − V_i(0)]/V_i(0) = 3.07. Of course, the case β = −1 is not realistic, but for β₀ = −0.9 we have [V_i(β₀) − V_i(0)]/V_i(0) = 2.12 in a similar fashion. We see that taking account of the conductivity reduction results in a substantial increase in the ionospheric potential, and, in consequence, it is important to allow for it in models of atmospheric electricity.

The ionospheric potential given by (31) depends only on the total area covered by thunderclouds but not on their particular spatial distribution. Since the vertical potential profile changes most significantly in the vicinity of the boundaries of thunderclouds, it seems reasonable to assume that it is the total horizontal length L of these boundaries (i.e., the sum of the horizontal perimeter lengths of all the thunderclouds) that determines how much the value of V_i corresponding to (8)–(10) differs from the value obtained from (31). With this assumption made, let us estimate L, assuming that there are N thunderclouds in the atmosphere each of which covers a circle of radius δr on Earth’s surface. As the total area covered by thunderclouds is given by , we conclude that .

To use our numerical model described in section 3 to compare the approximate theoretical analysis given above with the direct numerical solution of (8)–(10), we need to demand that the spatial distribution of thunderclouds be axisymmetric. Using the same values of r_min, r_max, r₁, r₂, σ₀, H, and J₀ as before and demanding again that V_i(−0.9) must be close to 240 kV, we obtain that should be chosen close to 5.8 × 10⁻⁴. Besides, we require that the total length of the boundaries be chosen such that δr is about 5 km. The simplest axisymmetric arrangement of thunderclouds that satisfies these conditions is a number of narrow circular strips of the form

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(with certain θ₀ and δθ) situated at different latitudes θ₀. The solid line in Fig. 3a represents the dependence V_i(β) obtained from numerical simulations in case thunderclouds occupy two strips of width 0.04° at latitudes 30°N and 30°S, which corresponds to and δr = 4.45 km. Comparing the dependence obtained from numerical modeling with that given by the approximate theoretical analysis of section 4 (the dashed line in Fig. 3a), we observe that the former yields smaller values of the ionospheric potential than the latter. When β = 0, the two values are almost the same [because in that case the approximate equation (31) coincides with the exact equation (30)], and the larger the value of |β| is, the larger is the difference between them; for example, for β = −0.9, V_i determined by (31) is 24% larger than that inferred from the numerical solution. However, the two curves are qualitatively similar to each other.

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The dependence of V_i on the parameters α and β, as obtained from theoretical analysis (dashed lines) and numerical simulations (solid lines): (a) V_i(β) in the case where there are only inhomogeneities inside thunderclouds, (b) V_i(α) for β = 0 and β = −0.9 in case r₁ < r₂ < r₃ < r₄, (c) V_i(α) for β = 0 and β = −0.9 in case r₃ < r₁ < r₂ < r₄ and β′ = β, and (d) V_i(α) for β = 0 and β = −0.9 in case r₃ < r₁ < r₂ < r₄ and β′ = α + β + αβ. The parameters used are as follows: r_min = 6370 km, r_max = r_min + 70 km, r₁ = r_min + 5 km, r₂ = r_min + 10 km, J₀ = 3 × 10⁻⁹ A m⁻², σ⁰(r) = σ₀ exp[(r − r_min)/H], , H = 6 km, and thunderclouds occupy two strips of the width 0.04° at latitudes 30°N and 30°S; r₃ and r₄ are taken to be r₃ = r_min + 12 km, r₄ = r_min + 25 km for case (b) and r₃ = r_min + 3 km, r₄ = r_min + 12 km for cases (c),(d); the region in cases (b)–(d) covers the entire Northern Hemisphere. In case (b) the results of theoretical and numerical calculations for β = 0 differ by less than 0.5%, and that is why only the solid line is shown.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0001.1

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8. The impact of conductivity inhomogeneities outside thunderclouds

Conductivity inhomogeneities outside thunderclouds also occur in the atmosphere. Natural phenomena such as ionizing radiation due to solar flares can influence the conductivity, as well as anthropogenic factors, particularly nuclear radiation caused by weapons testing and accidents in power plants. Although it is usually difficult to determine precisely the effect of such events on the conductivity distribution, in many cases simple estimates can be made.

As before, we assume that thunderclouds occupy the region defined in the previous section, and the external current density distribution is given by expressions (26) and (27) with replaced by . Now we also assume that significant conductivity inhomogeneities occur inside the region

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where r_min ≤ r₃ < r₄ ≤ r_max and is a region on the unit sphere. Note that although and have similar geometric shapes, no assumptions are made about their positions relative to each other; two possible choices of these positions are shown schematically in Fig. 4. We assume that

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where α ≥ 0, −1 < β ≤ 0, β′ > −1. In other words, we assume that, outside the region , we have, as before, σ = (1 + β)σ⁰(r) inside thunderclouds and σ = σ⁰(r) outside them, whereas, inside the , we have σ = (1 + β′)σ⁰(r) inside thunderclouds and σ = (1 + α)σ⁰(r) outside them. We also assume that thunderclouds are distributed uniformly over Earth’s surface;⁵ if we denote the total solid angle subtended by a region Γ on the unit sphere by μ(Γ), we can rewrite this condition as

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The position of thunderclouds, the structure of the external current density distribution, and the region (hatched) in the cases (a) r₁ < r₂ < r₃ < r₄ and (b) r₃ < r₁ < r₂ < r₄.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0001.1

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Now we will analyze some particular cases of this problem depending on its geometry and the values of the coefficients α, β, and β′.

In case r₁ < r₂ < r₃ < r₄ or r₃ < r₄ < r₁ < r₂ (Fig. 4a)—that is to say, the segments [r₁, r₂] and [r₃, r₄] do not intersect—the entire region of disturbed conductivity lies above or below thunderclouds, and therefore nothing depends on β′ since . As the conductivity and the external current density are of the form (13)–(15), it is possible to apply (23), which yields the following formula for the ionospheric potential:

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where δV₀, K, and γ are the same as those of the previous section; P is defined by

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and are equal to α/(1 + α) and β/(1 + β), respectively; and stands for the solid angle subtended by divided by 4π, or, in other words, the portion of Earth’s surface covered by the region .

The dependence V_i(α, β) given by (34) is shown in Fig. 5a. The calculations show that V_i increases with decreasing β; it also increases with decreasing α, unless β = 0, in which case (34) yields the same value of V_i for all values of α.⁶ We observe that the rate of change of V_i with respect to β is much greater than its rate of change with respect to α, even if α ≠ 0.

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V_i as a function of α and β, as obtained from theoretical analysis, in cases (a) r₁ < r₂ < r₃ < r₄, (b) r₃ < r₁ < r₂ < r₄, β′ = β, and (c) r₃ < r₁ < r₂ < r₄, β′ = α + β + αβ. The parameters used are as follows: r_min = 6370 km, r_max = r_min + 70 km, r₁ = r_min + 5 km, r₂ = r_min + 10 km, J₀ = 3 × 10⁻⁹ A m⁻², σ⁰(r) = σ₀ exp[(r − r_min)/H], , H = 6 km, , and is chosen such that V_i(α = 0, β = −0.9) = 240 kV; r₃ and r₄ are taken to be r₃ = r_min + 12 km, r₄ = r_min + 25 km for case (a) and r₃ = r_min + 3 km, r₄ = r_min + 12 km for cases (b),(c).

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0001.1

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Equation (34) is rather cumbersome; by expanding it to first order in α, we get a more simple formula:

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This equation expresses the dependence of V_i on α and β more clearly; however, it can be applied only for relatively small values of the parameter α; in the limit α → ∞ one can obtain a more appropriate formula by letting in (34).

In case r₃ < r₁ < r₂ < r₄ (Fig. 4b), we have [r₁, r₂] ⊂ [r₃, r₄] and the region of disturbed conductivity overlaps with the region of thunderclouds . Using the same argument as before, we obtain the following formula:

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where stands for β′/(1 + β′) and the other parameters are the same as before.

Let us assume, for the sake of simplicity, that the parameter β′ may be expressed as a function of α and β. More precisely, we consider two hypotheses about the value of β′:

1. β′ = β, hence 1 + β′ = 1 + β.

2. β′ = α + β + αβ, hence 1 + β′ = (1 + α)(1 + β).

The dependence V_i(α, β) given by (35) is shown in Fig. 5b for the case (i) and in Fig. 5c for the case (ii). In both these cases V_i increases with decreasing β or α, as shown by the calculations. We see that these dependences are similar to that shown in Fig. 5a, although here V_i changes more significantly with α.

Expanding (35) to first order in α, in the case (i), we have

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and in the case (ii),

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Note that in either case the coefficient of does not tend to zero as β → 0. In the limit α → ∞ these equations do not hold and one should use (35) with instead.

It is interesting to compare the functions V_i(α, β) given by (34) and (35) with corresponding functions obtained from numerical modeling. For β = 0 and β = −0.9, these functions corresponding to different parameters are shown in Figs. 3b–d. In the calculations the same distribution of thunderclouds over Earth’s surface was used as that described in the previous section, with two narrow circular strips of width 0.04° at latitudes 30°N and 30°S, and the region was taken to cover the entire Northern Hemisphere, in which case condition (33) holds. As before, we see that the values of the ionospheric potential obtained from numerical simulations are smaller than their counterparts calculated by the approximate theoretical analysis of section 4. If α = β = 0, there are no inhomogeneities, and the two values nearly coincide, which might have been expected since the approximate formulas become exact in this limit. In case the region of disturbed conductivity lies entirely above thunderclouds (Fig. 3a), the difference between the two curves changes only slightly as α increases; it is also remarkable that for β = 0 the results of the theoretical analysis nearly coincide with those given by numerical modeling (and that is why only one curve is shown in Fig. 3a); that is, conductivity inhomogeneities above (or below) thunderclouds do not significantly affect the ionospheric potential, unless we take the reduction of conductivity inside thunderclouds into account. In case the region of disturbed conductivity overlaps with the region of thunderclouds (Figs. 3b and 3c), the theoretical dependences V_i(α, β) given by (34) and (35) are qualitatively similar to those calculated directly from (8)–(10), but the relative difference between them increases substantially with increasing α. It is also clear that in either case the larger |β| is, the larger is the relative difference between the two values of the potential.

9. Conclusions

To estimate the influence of large-scale conductivity inhomogeneities in the atmosphere on the ionospheric potential, we developed an approximate theory on the basis of a well-posed problem formulation for the stationary model of the global electric circuit. Although this approximation is valid only when the spatial distributions of the conductivity and the external current density satisfy several constraints, it yields a method to estimate the impact of various natural and anthropogenic phenomena on the ionospheric potential and generalizes classical multicolumn models of atmospheric electricity formulated in terms of electric networks.

Comparing the results of our approximate theoretical analysis with those obtained from numerical simulations when the distributions of the conductivity and the external current density are axisymmetric, we conclude that the approximate theory describes the dependence of the ionospheric potential on the parameters of inhomogeneities qualitatively correctly, but quantitatively the values obtained from this theory may differ significantly from the more precise values obtained from numerical modeling, and the larger the conductivity reduction inside thunderclouds is, the larger is the difference. An immediate consequence of this is that the models of atmospheric electricity formulated within the framework of equivalent circuit can hardly serve as a reliable means of estimating the impact of various perturbations on the global electric circuit, and the reason for it is that, in such models, the horizontal component of the current is neglected, except for surface currents at Earth’s surface and the upper boundary of the atmosphere. However, such models qualitatively explain the results of the direct calculations and yield relatively simple formulas that can be used for parameterization of atmospheric electricity in high-resolution weather and climate models.