National Academies Press: OpenBook

The Earth's Electrical Environment (1986)

Chapter: MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY

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Suggested Citation:"MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
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Page 217
Suggested Citation:"MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
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Page 218
Suggested Citation:"MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
×
Page 219
Suggested Citation:"MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
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Page 220
Suggested Citation:"MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
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Page 221

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THE GLOBAL ATMOSPHERIC-ELECTRICAL CIRCUIT 217 Electric Fields The electric field at the ground is a readily measurable quantity that has been determined at various stations for nearly a century. The main characteristics of these measurements are summarized in detail by Israël (1973). In fair weather, the electric field is directed downward perpendicular to the ground and is typically 100-150 V/m. Field reversals are rare during undisturbed periods, but they are frequent during stormy weather and in conditions of dust, smoke, and fog. The electric field in densely populated areas is usually much larger than 150 V/m, whereas in small towns and far away from cities it can be smaller. The main reason for the variability of the electric field near the ground is that it is a complex quantity subject to universal time variations of the global circuit as well as the local influences of turbulence, weather, smoke, aerosols, and other anthropogenic factors. Israël (1973) summarized electric-field observations that define (a) a latitudinal variation; (b) altitude variations including effects of clouds and aerosol layers; (c) diurnal variation giving continental and oceanic types; (d) annual variation with maximum values during southern hemisphere summer; and (e) possible variation due to solar influences. In fair weather the ground electric field varies owing to changes in columnar resistance, ionospheric potential, and the local electrical conductivity at the ground. During disturbed periods it fluctuates rapidly due to the space-charge variation associated with turbulence, lightning, thunderstorm charge location, precipitation in the form of snow or rain, cloud passages, fog, blowing snow, dust or aerosols, and other meteorological properties. The relationship of the ground electric field to these processes has been studied extensively. More recently, the rapid variation of electric field observed at a number of ground stations has been used to derive the charge structure within clouds, as discussed by Krehbiel (Chapter 8, this volume). Electrical Relaxation Time A fundamental property of the global atmospheric electrical circuit is the electrical relaxation time at various altitudes, which is defined as the time the electric current takes to adjust to 1/e of its final value after an electric field is suddenly applied, assuming that the conductivity remains constant. At high altitudes, near 70 km, the relaxation time is about 10–4 sec, increasing with decreasing altitude to about 4 s near 18 km and to about 5-40 min near the Earth's surface. The electrical relaxation time of the land surface of the Earth is about 10–5 sec. The maximum value of about 40 min in the atmosphere near the Earth's surface is the characteristic time that the global circuit would take to discharge if all thunderstorm activity suddenly ceased. Measurements have never shown a complete absence of a fair- weather electric field for any length of time, thereby suggesting a continuous operation of thunderstorms and other generators in maintaining the currents flowing in the global circuit. For time variations longer than about 40 min a quasi-static approximation can be applied when one is modeling the electrical properties of the global circuit. MATHEMATICAL MODELS OF GLOBAL ATMOSPHERIC ELECTRICITY Only a few mathematical models of global atmospheric electricity have appeared over the years (Kasemir, 1963, 1977; Hill, 1971; Hays and Roble, 1979; Volland, 1982). Since it is difficult to obtain global measurements to deduce the instantaneous properties of the global circuit, these models provide a convenient means of examining, through numerical experiments, the various interacting processes operating in the global circuit. The overall success of the models is judged on how well they represent observed properties at any place and time within the circuit. Some of the elements that need to be considered in any global model of atmospheric electricity are schematically illustrated in Figure 15.8. Thunderstorms are extremely complex, and some simplifying assumptions must be made to represent their properties in a global model. The usual assumption is to consider thunderstorms as dipolar current sources, with a positive source in the cloud top and negative source in the cloud bottom. The storms are smaller than the grid scale for a Figure 15.8 Schematic of various electrical processes in the global electrical circuit.

THE GLOBAL ATMOSPHERIC-ELECTRICAL CIRCUIT 218 global model that has a representative grid size of 5° in latitude and longitude. The storms provide current to the global circuit, which flows upward toward the ionosphere. The rapid increase in electrical conductivity with height confines the current to a vertical column that flows from the storm to the ionosphere where it then is rapidly distributed over the globe at ionospheric heights. Part of the upward-directed current also flows along the Earth's geomagnetic-field line into the magnetic conjugate ionosphere, where it is also redistributed globally. The dipolar magnetic-field-line configuration is important for current closure within the global model. From the ionosphere the current flows downward toward the Earth with a magnitude that is governed by the potential difference between the ionosphere and the Earth's surface and by the local columnar resistance. The Earth's orography is important because of the decreased columnar resistance and consequentially higher current flows over mountains than at sea level. Clouds, fog, aerosols, and other meteorological phenomena must be considered because of their influence on electrical conductivity, columnar and global resistance, and local generation processes. Perhaps the most difficult region to model is the planetary boundary layer whose electrical characteristics are discussed by Hoppel et al. (Chapter 11, this volume). The electric current flows freely along the highly conducting Earth's surface to the region underneath a thunderstorm, where it then flows upward into the storm, thus closing the circuit. The upward current flow into the thunderstorm is extremely complicated, consisting of lightning, precipitation, corona, conduction, convection, and displacement currents as discussed in a previous section. Simplifying assumptions are usually used to make the mathematical problem of modeling the global circuit more tractable. The model of Hays and Roble (1979) used the quasi-static approximation to simplify the mathematical procedure. They assumed that thunderstorms, on a global scale, can be represented as dipolar current-generator point sources that are randomly distributed in preferred storm regions around the surface of the Earth. In fair-weather regions far away from the storm centers, the problem of the distribution of the electrostatic potential is determined by the current return from the sources to the Earth's surface. Hays and Roble used the basic Green's function for the problem of the global electric circuit to solve for the electrostatic potential considering only conduction currents flowing in the global circuit. The model included orography in geomagnetic coordinates as shown in Figure 15.9 but neglects clouds, fog, pollution, and aerosols. The electric potential is determined by solving the electrostatic equation in spherical coordinates. Variables are separated and the horizontal solution is determined by an expansion in tesseral harmonics. Thirty-seven spherical harmonic components were used, giving an effective 5° grid in latitude and longitude. The model of Hays and Roble (1979) included an exponentially increasing conductivity variation with altitude and a latitudinal distribution that follows the latitudinal distribution of cosmic-ray ion production. It also included the coupling of thunderstorm currents flowing along geomagnetic-field lines into magnetic conjugate hemispheres. The model is on geomagnetic coordinates to account for the cosmic-ray variation and geomagnetic-field-line coupling. Figure 15.9 Perspective illustration of the Earth's orography in geomagnetic coordinates that is used in the global model of atmospheric electricity. We use the model of Hays and Roble (1979) and Roble and Hays (1979) to present a model simulation that illustrates various properties of the global circuit. In Figure 15.10 the boxed-in areas show regions assumed to have enhanced thunderstorm activity. The distribution is taken from the maps of thunderstorm frequency generated by Crichlow et al. (1971) and representing northern hemisphere summer conditions at 1900 UT. It is assumed that 500 point dipolar current sources are randomly distributed in the Gulf of Mexico area (region 3), 300 in Africa (region 1), 100 off the coast of Argentina (region 4), and 50 in Southeast Asia (region 2). The storms are randomly distributed in latitude, longitude, and altitude, and some fall over land areas, plains, and mountains and some over oceans. The combined current output from the ensemble of thunderstorms generates an ionospheric potential of 280 kV. The calculated global resistance including orogra

THE GLOBAL ATMOSPHERIC-ELECTRICAL CIRCUIT 219 phy is 275 Ω, and the total current flowing in the circuit is 1010 A. To illustrate regions of upward and downward current flow, the results are presented as differences from the ionospheric potential [φ(σ)-φ¥] in volts along constant- conductivity surfaces near 105, 50, 25, 8, 4, and 2 km as shown in Figure 15.11. In the middle atmosphere the model calculates the largest positive potential difference over the thunderstorm regions, indicating that currents are flowing upward toward the ionosphere. Negative potential regions indicate regions of downward current flow toward the Earth's surface. Figure 15.10 Regions of assumed thunderstorm distribution for model calculations. The five solid boxes indicate regions of thunderstorm occurrence for 1900 UT during northern hemisphere summer. In the equatorial region the geomagnetic-field lines are horizontal; therefore, any upward current flow into the equatorial ionosphere is redistributed only by horizontal conduction currents, and a single positive potential difference develops over the main thunderstorm region in equatorial Africa. The Gulf of Mexico thunderstorm region is off the geomagnetic equator, and upward-flowing currents can flow freely along the geomagnetic-field line into the conjugate hemisphere. Since the electrical conductivity along the geomagnetic-field line is large, the magnetic potential is symmetric about the equator with no potential differences existing in the ionosphere between the feet of conjugate geomagnetic-field lines. The calculated potential differences along the 105-km constant-conductivity surfaces shown in Figure 15.11 (a) are about 1.2 kV, which is relatively small compared with the calculated 280-kV constant ionospheric potential. The contours of the calculated potential difference along the constant-conductivity surface near 50 km are shown in Figure 15.11(b). The overall pattern is similar to that calculated at 105 km, indicating very little attenuation between the two altitudes. There is, however, a small attenuation in the magnetic conjugate region of the Gulf of Mexico thunderstorm region. Along the con

THE GLOBAL ATMOSPHERIC-ELECTRICAL CIRCUIT 220 stant-conductivity surface near 25 km, the potential difference pattern is different from that at 50 and 105 km, illustrating how the rapid increase of electrical conductivity alters the potential distribution in the upper atmosphere. Maximum potential variations occur over the main thunderstorm region, indicating a strong current flowing toward the ionosphere. In other regions the current flow is downward, with maximum values over the mountainous areas, primarily Tibet, Antarctica, and Greenland. The potential differences along constant-conductivity surfaces at 8, 4, and 2 km are shown in Figures 15.11(c), 15.11(d), and 15.11(e), respectively. In plotting these contours, the intense negative potentials generated under the thunderstorm areas are arbitrarily suppressed for contouring to illustrate the potential distortion due to orography. The hatched areas indicate these intensely disturbed regions. Also, the potential is zero whenever the height surface cuts a mountainous area that lies above it. These figures illustrate the effect of orography in altering the potential surfaces within the troposphere. These same potential surfaces are presented as perspective illustrations in Figure 15.12 for ease in visualizing the global circuit. Figure 15.11 Contours of calculated potential difference [φ(σ) -f¥] in kilovolts along a constant-conductivity surface s. Here, is the ionospheric potential, globally averagedat ionospheric heights (approximately 105 km). (a) Potential difference along the σm = 4.5 × 10–6 mho/m surface at about 105 km over the equator; (b) potential difference along the σm = 4.7 × 10–10 mho/m surface, which is 50 km over the equator; (c) potential difference along the σm = 7.3 × 10–12 mho/m surface, which is at an altitude of 25 km over the equator; (d) potential difference along the σm = 4.3 × 10–13 mho/m surface, which is at an altitude of 8 km over the equator; (e) and (f) give the potential difference along the constant-height surface at 4 km and 2 km, respectively. The calculated vertical ground potential gradient and current along the Earth's orographic surface are shown in Figures 15.13(a) and 15.13(b), respectively. The electric field in the fair-weather areas of the equatorial region is typically 130 V/m and increases with in

THE GLOBAL ATMOSPHERIC-ELECTRICAL CIRCUIT 221 Figure 15.12 Perspective illustration of the calculated potential difference along the same surfaces as specified in Figure 15.11.

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This latest addition to the Studies in Geophysics series explores in scientific detail the phenomenon of lightning, cloud, and thunderstorm electricity, and global and regional electrical processes. Consisting of 16 papers by outstanding experts in a number of fields, this volume compiles and reviews many recent advances in such research areas as meteorology, chemistry, electrical engineering, and physics and projects how new knowledge could be applied to benefit mankind.

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