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A one-dimensional model of the atmospheric electric field near the Venusian surface
ICARUS 52, 3 4 6 - 3 5 3 (1982)
A One-Dimensional Model of the Atmospheric Electric Field near the Venusian Surface 1 I. T Z U R National Center for Atmospheric Research, 2 P.O. Box 3000, Boulder, Colorado 80307 AND
Z. L E V I N 3 NASA, Ames Research Center, Moffett Field, California 94035
Received November 13, 1981; revised July 19, 1982 A numerical model is used to simulate the buildup of an electric field from below the Venusian cloud layer to the surface. The steady-state profiles of the ion concentration, net space charge, diffusion and conduction current, and electric field are calculated. Two electric field sources are considered. The first is that produced by the higher diffusivity of positive ions relative to negative ions, which results in charging the surface with a net positive charge. The results show that the magnitude of the electric field and the net space charge developed near the surface are mainly dependent on the mixing conditions in the boundary layer. However, even in the case of relatively strong mixing, the maximum electric field is found to be 1.5 V m 1and it decays rapidly above 100 m. The second source of an electric field is assumed to be charge separation inside Venusian clouds. A steady-state conduction current in the region below the layer of clouds which represents the intensity of charge separation inside the clouds is used as a parameter. When this parameter is assumed to be 10 ~2A m-2, which is about the fair-weather conduction current in the atmosphere of Earth, an electric field of 5 kV m-I is developed near the surface. This electric field exists up to a few kilometers, decreases by an order of magnitude at about 20 km, and then decays rapidly.
INTRODUCTION T h e r e c e n t d i s c o v e r y that lightning m a y o c c u r o n V e n u s ( K s a n f o m a l i t i , 1979, 1980; T a y l o r et al., 1979; a n d S c a r f et al. 1980) raises the p o s s i b i l i t y o f e l e c t r i c a l a c t i v i t y in the a t m o s p h e r e of this p l a n e t . B o r u c k i et al. (1982) c a l c u l a t e d the electrical p a r a m e t e r s in the l o w e r V e n u s i a n atm o s p h e r e a n d s u g g e s t e d that the d o m i n a t i n g ions n e a r the p l a n e t ' s s u r f a c e are J Paper presented at "An International Conference on the Venus Environment," Palo Alto, California, November 1-6, 1981. 2 The National Center for Atmospheric Research is sponsored by the National Science Foundation. 3 On sabbatical leave in 1981 from Tel-Aviv University, Department of Geophysics and Planetary Science, Ramat Aviv, Israel 69978.
CO2+CO2 a n d SO2 CO2. T h u s the p o s i t i v e ions h a v e l o w e r m o l e c u l a r weight a n d h e n c e higher m o b i l i t y t h a n the n e g a t i v e ions. Consequently, a net positive diffusion c u r r e n t c a n be g e n e r a t e d t o w a r d the p l a n e t surface c r e a t i n g a p o s i t i v e surface c h a r g e a n d u p w a r d - d i r e c t e d electric field. A s a result a net s p a c e c h a r g e l a y e r is g e n e r a t e d n e a r the surface that is k n o w n as the electrode effect l a y e r (Israel, 1973). U n d e r s t e a d y - s t a t e c o n d i t i o n s , p o s i t i v e ion diffusion to the surface j u s t b a l a n c e s the c o n d u c t i o n a n d diffusion o f n e g a t i v e ions to the surface. I n a d d i t i o n , if charge s e p a r a t i o n o c c u r s in the c l o u d s o v e r h e a d ( - 5 0 k m altitude), a n electric field a n d c o n d u c t i o n c u r r e n t s will be d e v e l o p e d b e t w e e n the c l o u d l a y e r a n d 346
0019-1035/82/110346-08$02.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
VENUS' ATMOSPHERIC ELECTRIC FIELD the surface. This electric field may increase near the planet surface as a result of the low electrical conductivity in this region. In this paper a numerical model is used to study the development of an electric field below the layer of clouds. The above-mentioned sources of electric fields (e.g., the net diffusion of ions to the Venusian surface as a result of the differences in ion diffusivities and the electrical activity inside the clouds) are considered. A one-dimensional electrical model is used here to resolve the electrode effect layer and to derive the steady-state electric field from the ground to 40 km, as well as the profiles of conduction and diffusion currents, densities of positive and negative ions, and the net space charge.
347
Ions are produced by cosmic rays, and the sink is recombination of positive and negative ions. The altitude profiles of ion mobility, ion diffusion coefficient, ionization rate, and recombination rate are taken from Borucki e t al. (1982). The Venusian atmosphere is assumed to have weaker turbulent mixing than the terrestrial atmosphere. The eddy mixing coefficients are assumed to be linearly dependent on height as can be assumed in a surface layer under constant stability conditions (McBean e t a l . , 1979). The equations used in the model are described below. The continuity equation of small positive ions is +
++
+
+-
Nt = (DNz - NE)z + g -
aNN.
(2)
The continuity equation of small negative ions is
THE MODEL DESCRIPTION The present physical model involves the following principal assumptions: In the region under consideration there is neither cloud-to-ground lightning nor coronal currents from the ground to the cloud base; convective transport can be neglected; the aerosol sink of ions may be neglected below 40 km (see Borucki e t a l . , 1982); and the Venus topography does not screen the electric field near the surface. Even though it is known that high mountains and deep valleys exist on Venus, we will ignore them due to the fact that only a one-dimensional model is used here. The medium in which the ions flow (i.e., the Venusian atmosphere) can be considered as a collision-dominated partially ionized gas. Thus in the one-dimensional case the ion velocity may be written
Nt = (DNz + NE)z + g -
aNN.
(3)
To describe the time derivative of the electric field, the Poisson equation is differentiated with respect to time, which results in the equation + eEz.t = e ( N
- N)t.
(4)
Subtracting Eq. (3) from Eq. (2) and substituting the result for the right-hand side of Eq. (4) gives ++ ~Ez,t = e ( D N z ++ -DNz-
IxNE-
txNE)z.
(5)
Integrating Eq.-(5) with respect to z gives the electric field equation e ++ ---++ Et = -~ ( D N z - D N z - ~,,NE -
~NE)
On
V = IxE - D O---z'
(1)
where the first term on the right-hand side describes the conduction current and the second the diffusion current. Symbols are defined in the Appendix.
+ constant.
(6)
Under steady-state conditions, the constant of integration is the total steady-state current. A numerical model is used to simultaneously solve the three coupled partial dif-
348
TZUR AND LEVIN
ferential equations (2), (3), and (6) for assumed boundary conditions, initial conditions, and total steady-state current. This model uses the mathematical subroutine package PDECOL by Madsen and Sincovec (1979). The ground is assumed to absorb ions and the boundary condition for the ion concentration at this point is zero. Since the cloud layer is located above 45 km (Knollenberg and Hunten, 1980), the upper boundary is assumed to be at 40 km. At this height the conductivity is much higher with respect to its value at the lower boundary value and we can assume zero net space charge at this point. The concentration of positive and negative ions at the upper boundary is approximated by the balance between the ionization rate and the recombination of ions, -
+
N = N = ~/gla.
(7)
When a small net space charge was introduced at the upper boundary the same solution was obtained except in a distance of a few mesh points from the upper boundary. No boundary conditions are needed for the electric field because the electric field equation, Eq. (6), does not have derivatives of the electric field with respect to the spatial coordinate. At the boundaries the numerical code solves Eq. (6) by using boundary values of the other variables. The initial values of the concentration of ions are calculated from the source and sink balance, Eq. (7). The initial electric field is assumed to be zero. Near the surface, the molecular diffusion currents dominate. Large gradients in the concentration of ions may be developed, and small grid intervals are used to properly represent the diffusion currents in this region. A variable vertical grid is used which has small intervals near the lower boundary, 10 -7 cm, increasing with altitude to about 3 km at an altitude of 40 km.
RESULTS AND DISCUSSION The results are classified into two cases. In the first case, the electric field develops as a result of differential ion diffusion to the surface without any charge buildup in the clouds. In the second case, the electric field builds up as a result of both ion diffusion to the surface and currents resulting from charge separation in the clouds. In both cases, the steady-state profiles of the ion concentrations, net space charge, electrical currents, and electric fields are presented and discussed. These profiles are compared with the corresponding terrestrial profiles.
Case I
The higher diffusivity of the positive ions (Borucki et al., 1982) results in a net diffusion of positive charge toward the Venusian surface. As a positive surface charge accumulates on the surface, it generates an electric field that extends upward into the atmosphere. Thus as a steady-state develops, diffusion currents are balanced by conduction currents, and the total current is zero. Diffusion of ions can take place by both molecular and eddy diffusion. The eddy diffusion coefficient may be written as De = cz,
(8)
where c is a constant which depends on the surface roughness and the stability of the atmosphere. A typical profile of eddy mixing in the terrestrial boundary layer has a maximum of about 15 m 2 sec -I at approximately 600 m (McBean et al., 1979). For lack of data on the magnitude of the eddy mixing in the Venusian atmosphere we assumed three values of the coefficient c in Eq. (8), 0, 0. l, and 1 m sec -1. Seiff e/al. (1980)found the layer lying below the clouds to be stable in the Venusian atmosphere. Therefore, we assumed the depth of the boundary layer on this planet to be 10 m. The eddy diffusion coefficients are calculated according to Eq.
VENUS' ATMOSPHERIC ELECTRIC FIELD (8) up to 10 m, and assumed to be constant above it. When we increased the boundary layer depth (keeping the maximum eddy coefficient constant), we found no significant change in the electric field profile. Figure 1 shows the steady-state electric field developed for the various eddy mixing coefficients. When the eddy mixing is zero, the maximum electric field obtained is 10 mV m -l, and 10 cm above the surface it decreases rapidly to zero. As the eddy mixing increases, the maximum electric field and the altitude to which it extends is increased (Fig. 1). The increase in the value of the electric field is caused by the increase in positive ion concentration brought from above. This increase in ion concentration results in enhanced ion diffusion to the surface and further increase in the electric field. The three profiles shown in Fig. 1 exhibit only a very limited development of the electric field. The maximum electric field obtained is only 1.5 V m -I. It is produced near the surface and decays rapidly above 100 m to a value of 10 mV m -1 at about 1 km (Fig. 1). This electric field (1.5 V m -l) is more than two orders of magnitude smaller than the terrestrial fair-weather electric field. The conduction current (Fig. 2) is directed upward to balance the net positive ~0 5
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i0-~
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~-~
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ELECTRIC FIELD {V/m)
FIG. 1. The electric field as a function of height in the case of zero total steady-state current, Case 1. The eddy diffusion coefficients in the boundary layer, De, are given near each line.
105
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CONDUCTION CURRENT (A/m 2)
FIG. 2. As in Fig. 1, except for net conduction currents. diffusion current to the surface (negative ions move down and positive ones move up). The conduction current is a few orders of magnitude smaller than the diffusion current for each type of ion. The maximum conduction current predicted in this case is about 2 × 10 -16 A m -2 and is four orders of magnitude smaller than the fair-weather conduction current measured in the Earth's atmosphere. In Fig. 3, the diffusion current of the negative ions is shown (it differs from the positive diffusion current by less than two orders of magnitude). Without eddy mixing the diffusion currents are due to molecular diffusion, and very small currents are produced (Figs. 2 and 3). In the case of maximum mixing, the diffusion currents reach a value of about 10 -12 A m -2 near the ground and decay rapidly above 100 m (Fig. 3). The profiles of the concentration of the ions are shown in Fig. 4. Without the influence of the surface, the ion profile is determined by the balance between ionization rate and ion recombination, as in Eq. (7). In this case, the difference between the positive and negative ion concentrations is at least two orders of magnitude smaller than the ion concentration of each type of ion. Therefore, the profiles in Fig. 4 represent the positive or negative ion concentration. The dashed line in Fig. 4 corresponds to the ion concentration calculated from Eq. (7).
350
TZUR AND LEVIN 105
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DIFFUSION CURRENT (A/m z)
--IO -15
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--10
J IIIII 13
NET SPACE CHARGE (C/m3)
FIG. 3. As in Fig. 1, except for diffusion currents of the negative ions.
FIG. 5. As in Fig. 1, except for net space charge.
The other two lines in Fig. 4 correspond to no mixing (solid curve) and the maximum mixing (dash-dot curve) assumed in Case 1, respectively. In the case of maximum mixing, the electrode layer effect penetrates up to about 1 km. Because the positive ions are more mobile, they flow faster out of the electrode layer and a net negative space charge is built up (Fig. 5). In the case of no mixing a net space charge develops only below 1 m, while for the assumed maximum mixing the net space charge penetrates above 100 m (Fig. 5). However, we emphasize that the net space charge shown in Fig. 5 is actually
very small as compared to that near the Earth's surface.
,o5~ ................., ........, ................., 'j..~
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'°'r ....................... ,,':i .............. ,',,,,,,1, io3
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105
ioe
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ION C,ONCENTRATION (ions/rn 3)
FIG. 4. As in Fig. 1, except for ion concentration. The dashed line represents the ion concentration without the influence of the surface charge.
Case 2
In the second case we assume that electric fields and conduction currents exist beneath the Venusian cloud layer. In the present calculations the electrification of the clouds is not simulated and the upper boundary of the model is taken well below the base of the clouds. It is also assumed that there are no ground-to-cloud lightning and no coronal currents from the ground to the cloud base. The steady-state conduction current which flows below the cloud layer is used as a parameter. This parameter depends either on the degree of charge separation inside the clouds or the intensity of some other electric field generator. It is assumed to vary from a minimum of 10 16A m -z, which is about the maximum predicted in Case 1, to an arbitrary maximum of 10 -12 A m -2. When oppositely directed steadystate currents were used (which corresponds to an opposite polarity of the electric field generator), the results were very similar and differed only in sign. In Case 2 the eddy mixing coefficients are calculated from Eq. (8), with c = 0.1 m sec-I and the depth of the boundary layer is assumed to be 10m. Figure 6 presents the profiles of the electric field for assumed steady-state conduc-
VENUS' ATMOSPHERIC ELECTRIC FIELD
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351
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102
0
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I08
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5xlO e
7xlO 8
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ION CONCENTRATION (ions/m 3)
F I E L D (Vim)
FIG. 6. The electric field as function of height for Case 2. The total steady-state currents, J t o t a l , a r e given near each line. The dashed line represents the fairweather terrestrial electric field.
FIG. 8. T h e ion concentration as function of height for Case 2. The total steady-state currents are given near each line.
tion currents of 10 -lz, 10 -14, and 10 -16 A m -z. Also shown in this figure is the terrestrial fair-weather electric field. The global thunderstorm activity on Earth is a s s u m e d to generate this electric field and produces a conduction current of about 10 -12 A m -2 in the fair-weather regions (Israel, 1973). The Venusian a t m o s p h e r e has a lower conductivity than that of the terrestrial a t m o s p h e r e (Borucki et al., 1982) and a conduction current of 10 -12 A m -2 is capable of producing an electric field of 5 kV m -~ near its surface (Fig. 6). This relatively high electric field penetrates up to about 10 km, where it is 103
V m -l, and then decreases very rapidly to about 1 V m - I at 40 km. The d e v e l o p m e n t of the electric field as a function of time is shown in Fig. 7. The two lines correspond to total currents of 10-14 A m -2 and zero, respectively. In both cases the eddy mixing coefficients are calculated from Eq. (8) with c = 0.1 m sec -l, and the depth of the b o u n d a r y layer is 10 m. As can be seen in Fig. 7, the time constant of the electric field d e v e l o p m e n t is about 105 sec in both cases. This long time constant, as c o m p a r e d to about 103 sec in the terrestrial case, is again a result of the lower conductivity in the lower Venus atmosphere. The ion concentration near the ground is shown in Fig. 8 for an assumed steady-state conduction current of 10 -~2 and 10 -14 A m -2, respectively. When a current density o f l 0 -14 A m -2 is assumed, an electric field of 50 V m -~ is produced near the s u r f a c e - too small to generate significant net space charge in the electrode effect region. H o w ever, an assumed current of 10 -12 A m -2 generates an electrode layer of 1 km in depth. The net space charge for the various assumed conduction currents is shown in Fig. 9. If local vertical convection currents or stronger mixing than that a s s u m e d in this model exist near the Venusian surface, they
0.6 ~
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TIME (s)
FIG. 7. The time d e v e l o p m e n t of the electric field at the surface. T h e left vertical axis is for zero total current, and the right one is for total current of l0 14 A m-Z,
352
TZUR AND LEVIN
-~,o, =,o-':¢A*,,~---_J7 ::I:
I
I
10-3
1@-5
10-7
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NET SPACE CHARGE
I
_io-lO
(C/m 3)
FIG. 9. As in Fig. 8, except for net space charge.
will carry a net space charge out of the electrode layer, resulting in more charge separation and enhancement of the electric field in the lower atmosphere of Venus. CONCLUSIONS
Calculations of the electric field profile in the lower Venusian atmosphere reveal that the smaller mass and higher mobility of the positive ions cause an electrode effect layer with excess negative ions to develop near the surface. As expected, inclusion of eddy mixing increases the height of the electrode effect layer. The ever-present clouds on Venus may have some charge separation in them. Currents produced by charge centers in the clouds or by some other source can affect the magnitude of the electric field below, all the way to the surface. Because the electrical conductivity is lower at the lowest levels of the atmosphere on Venus, a current even smaller than that observed in our terrestrial atmosphere can produce fields of similar magnitudes to those observed here. The presence of convection on Venus could also affect the magnitude of the field. However, measurements suggest that the atmosphere below the clouds is stable, so that if convection exists it is of a small scale and probably localized. In regions where it occurred, stronger fields and deeper electrode effects would be found.
In this work, we use the molecular weights based on the ionic composition suggested by Borucki et al. (1982). However, we believe that this ionic composition is not certain since even in the terrestrial atmosphere this parameter is not yet completely resolved. Therefore, the sensitivity of the results to the molecular weight of the ions is tested. Figure 10 shows three profiles of electric field produced by differences in the positive and negative ionic diffusion currents to the surface. Here the coefficient c in Eq. (8) is assumed to be 0. I, corresponding to stable atmospheric conditions. The lower electric field profile corresponds to the molecular weight ratio suggested by Borucki et al. (1982). Introduction of a molecular weight ratio of 10 produces a maximum electric field of 30 Vm-1. The effect of the ionization produced by natural radioactive substances emitted from the ground was also tested. This ionization source was assumed to be of the same magnitude as the terrestrial radioactive source (Israel, 1973). The effect of this ionization source on the results of Case 1 is very small since in this case charging of the surface depends mostly on the difference in the diffusivities of the positive and negative ions. In Case 2, where an electric field generator was assumed (i.e., charge separation 104
bo °
_~
= M/M 108/88 z
10-z
10-4
10-6
J
10--4
I
1013
J
10"2
i
~--I
-I J
IO0
I01
102
ELECTRIC FIELD (V/m)
FIG. 10. The electric field as function of height. T h e ratios of negative to positive ion molecular weight, M/ + M, are indicated at each line.
VENUS' ATMOSPHERIC ELECTRIC FIELD inside the cloud layer), the additional ionization rate increases the conductivity and results in a decrease in the electric field. This decrease is proportional to g~-0.5 and is limited to the region below 1 km. APPENDIX--SYMBOLS
Time derivative of the variable A Height derivative of the variable A
At Az +
-
D,D
De e
E g, gr n +
Diffusion coefficient (eddy mixing plus molecular diffusion) of the positive and negative ions, respectively Eddy diffusion coefficient Electron charge Electric field Ionization rate from cosmic and radioactive sources, respectively Concentration of ions
-
N,N
Concentration of positive and negative ions, respectively Time Height Positive-negative ion recombination Permittivity of free space
+ -
Electrical mobility of the positive and negative ions, respectively ACKNOWLEDGMENTS The authors are very grateful to Dr. John M. Gary for considerable mathematical advice and guidance in the solution of the model equations. This research was supported in part by NASA Grant W-15,028.
353
REFERENCES BORUCKI, W. J., Z. LEVIN, R. C. WHITTEN, R. G. KEESEE, L. A. CAPONE,O. B. TOON, AND J. DOBACK (1982). Predicted electrical conductivity between 0 and 80 km in the Venusian atmosphere. Icarus 51, 302-321. ISRAEL, H. (1973). Atmospheric Electricity, Vols. 1 and 2. Israel Program for Scientific Translations, Jerusalem. KNOLLENBERG, R. G., AND D. M. HUNTEN (1980). The microphysics of the clouds of Venus: Results of the Pioneer Venus particle size spectrometer experiment. J Geophys. Res. 85, 8039-8058. KSANFOMALITI, L. V. (1979). The lightning in the cloud layer of Venus. Kosm. Issled. 17, 747-762. KSANFOMALITI, L. V. (1980). Discovery of frequent lightning discharges in clouds on Venus. Nature 284, 244-246. MADSEN, N. K., AND R. F. SINCOVEC (1979). Algorithm 540 PDECOL, general collection software for partial differential equations ID3]. ACM Trans. Math. Software 5, 326-351. MCBEAN, G. A., K. BERNHARDT,S. BODIN, Z. LITYNSKA, A. P. VAN ULDEN AND J. C. WYNGAARD (1979). The Planetary Boundary Layer, Chap. 4, Numerical Modeling o f the Atmospheric Boundary Layer (G. A. McBean, Ed.). Technical Note No. 165, WMO-No. 530. Geneva, Switzerland. SCARF, F. L., W. W. L. TAYLOR, C. T. RUSSELL, AND L. H. BRACE (1980). Lightning on Venus: Orbiter detection of whistler signals. J. Geophys. Res. 85, 8158-8166. SEIFF, A., D. B. KIRK, R. B. YOUNG, R. C. BLANCHARD, J. T. FINDLEY, G. M. KELLY, AND S. C. SOMMER (1980). Measurements of thermal structure and thermal contrasts in the atmosphere of Venus and related dynamical observations: Results from the four Pioneer Venus probes. J. Geophys. Res. 85, 7903 -7933. TAYLOR, W. W. L., F. L. SCARF, C. T. RUSSELL, AND L. H. BRACE (1979). Evidence for lightning on Venus. Nature 279, 614-616.
A One-Dimensional Model of the Atmospheric Electric Field near the Venusian Surface 1 I. T Z U R National Center for Atmospheric Research, 2 P.O. Box 3000, Boulder, Colorado 80307 AND
Z. L E V I N 3 NASA, Ames Research Center, Moffett Field, California 94035
Received November 13, 1981; revised July 19, 1982 A numerical model is used to simulate the buildup of an electric field from below the Venusian cloud layer to the surface. The steady-state profiles of the ion concentration, net space charge, diffusion and conduction current, and electric field are calculated. Two electric field sources are considered. The first is that produced by the higher diffusivity of positive ions relative to negative ions, which results in charging the surface with a net positive charge. The results show that the magnitude of the electric field and the net space charge developed near the surface are mainly dependent on the mixing conditions in the boundary layer. However, even in the case of relatively strong mixing, the maximum electric field is found to be 1.5 V m 1and it decays rapidly above 100 m. The second source of an electric field is assumed to be charge separation inside Venusian clouds. A steady-state conduction current in the region below the layer of clouds which represents the intensity of charge separation inside the clouds is used as a parameter. When this parameter is assumed to be 10 ~2A m-2, which is about the fair-weather conduction current in the atmosphere of Earth, an electric field of 5 kV m-I is developed near the surface. This electric field exists up to a few kilometers, decreases by an order of magnitude at about 20 km, and then decays rapidly.
INTRODUCTION T h e r e c e n t d i s c o v e r y that lightning m a y o c c u r o n V e n u s ( K s a n f o m a l i t i , 1979, 1980; T a y l o r et al., 1979; a n d S c a r f et al. 1980) raises the p o s s i b i l i t y o f e l e c t r i c a l a c t i v i t y in the a t m o s p h e r e of this p l a n e t . B o r u c k i et al. (1982) c a l c u l a t e d the electrical p a r a m e t e r s in the l o w e r V e n u s i a n atm o s p h e r e a n d s u g g e s t e d that the d o m i n a t i n g ions n e a r the p l a n e t ' s s u r f a c e are J Paper presented at "An International Conference on the Venus Environment," Palo Alto, California, November 1-6, 1981. 2 The National Center for Atmospheric Research is sponsored by the National Science Foundation. 3 On sabbatical leave in 1981 from Tel-Aviv University, Department of Geophysics and Planetary Science, Ramat Aviv, Israel 69978.
CO2+CO2 a n d SO2 CO2. T h u s the p o s i t i v e ions h a v e l o w e r m o l e c u l a r weight a n d h e n c e higher m o b i l i t y t h a n the n e g a t i v e ions. Consequently, a net positive diffusion c u r r e n t c a n be g e n e r a t e d t o w a r d the p l a n e t surface c r e a t i n g a p o s i t i v e surface c h a r g e a n d u p w a r d - d i r e c t e d electric field. A s a result a net s p a c e c h a r g e l a y e r is g e n e r a t e d n e a r the surface that is k n o w n as the electrode effect l a y e r (Israel, 1973). U n d e r s t e a d y - s t a t e c o n d i t i o n s , p o s i t i v e ion diffusion to the surface j u s t b a l a n c e s the c o n d u c t i o n a n d diffusion o f n e g a t i v e ions to the surface. I n a d d i t i o n , if charge s e p a r a t i o n o c c u r s in the c l o u d s o v e r h e a d ( - 5 0 k m altitude), a n electric field a n d c o n d u c t i o n c u r r e n t s will be d e v e l o p e d b e t w e e n the c l o u d l a y e r a n d 346
0019-1035/82/110346-08$02.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
VENUS' ATMOSPHERIC ELECTRIC FIELD the surface. This electric field may increase near the planet surface as a result of the low electrical conductivity in this region. In this paper a numerical model is used to study the development of an electric field below the layer of clouds. The above-mentioned sources of electric fields (e.g., the net diffusion of ions to the Venusian surface as a result of the differences in ion diffusivities and the electrical activity inside the clouds) are considered. A one-dimensional electrical model is used here to resolve the electrode effect layer and to derive the steady-state electric field from the ground to 40 km, as well as the profiles of conduction and diffusion currents, densities of positive and negative ions, and the net space charge.
347
Ions are produced by cosmic rays, and the sink is recombination of positive and negative ions. The altitude profiles of ion mobility, ion diffusion coefficient, ionization rate, and recombination rate are taken from Borucki e t al. (1982). The Venusian atmosphere is assumed to have weaker turbulent mixing than the terrestrial atmosphere. The eddy mixing coefficients are assumed to be linearly dependent on height as can be assumed in a surface layer under constant stability conditions (McBean e t a l . , 1979). The equations used in the model are described below. The continuity equation of small positive ions is +
++
+
+-
Nt = (DNz - NE)z + g -
aNN.
(2)
The continuity equation of small negative ions is
THE MODEL DESCRIPTION The present physical model involves the following principal assumptions: In the region under consideration there is neither cloud-to-ground lightning nor coronal currents from the ground to the cloud base; convective transport can be neglected; the aerosol sink of ions may be neglected below 40 km (see Borucki e t a l . , 1982); and the Venus topography does not screen the electric field near the surface. Even though it is known that high mountains and deep valleys exist on Venus, we will ignore them due to the fact that only a one-dimensional model is used here. The medium in which the ions flow (i.e., the Venusian atmosphere) can be considered as a collision-dominated partially ionized gas. Thus in the one-dimensional case the ion velocity may be written
Nt = (DNz + NE)z + g -
aNN.
(3)
To describe the time derivative of the electric field, the Poisson equation is differentiated with respect to time, which results in the equation + eEz.t = e ( N
- N)t.
(4)
Subtracting Eq. (3) from Eq. (2) and substituting the result for the right-hand side of Eq. (4) gives ++ ~Ez,t = e ( D N z ++ -DNz-
IxNE-
txNE)z.
(5)
Integrating Eq.-(5) with respect to z gives the electric field equation e ++ ---++ Et = -~ ( D N z - D N z - ~,,NE -
~NE)
On
V = IxE - D O---z'
(1)
where the first term on the right-hand side describes the conduction current and the second the diffusion current. Symbols are defined in the Appendix.
+ constant.
(6)
Under steady-state conditions, the constant of integration is the total steady-state current. A numerical model is used to simultaneously solve the three coupled partial dif-
348
TZUR AND LEVIN
ferential equations (2), (3), and (6) for assumed boundary conditions, initial conditions, and total steady-state current. This model uses the mathematical subroutine package PDECOL by Madsen and Sincovec (1979). The ground is assumed to absorb ions and the boundary condition for the ion concentration at this point is zero. Since the cloud layer is located above 45 km (Knollenberg and Hunten, 1980), the upper boundary is assumed to be at 40 km. At this height the conductivity is much higher with respect to its value at the lower boundary value and we can assume zero net space charge at this point. The concentration of positive and negative ions at the upper boundary is approximated by the balance between the ionization rate and the recombination of ions, -
+
N = N = ~/gla.
(7)
When a small net space charge was introduced at the upper boundary the same solution was obtained except in a distance of a few mesh points from the upper boundary. No boundary conditions are needed for the electric field because the electric field equation, Eq. (6), does not have derivatives of the electric field with respect to the spatial coordinate. At the boundaries the numerical code solves Eq. (6) by using boundary values of the other variables. The initial values of the concentration of ions are calculated from the source and sink balance, Eq. (7). The initial electric field is assumed to be zero. Near the surface, the molecular diffusion currents dominate. Large gradients in the concentration of ions may be developed, and small grid intervals are used to properly represent the diffusion currents in this region. A variable vertical grid is used which has small intervals near the lower boundary, 10 -7 cm, increasing with altitude to about 3 km at an altitude of 40 km.
RESULTS AND DISCUSSION The results are classified into two cases. In the first case, the electric field develops as a result of differential ion diffusion to the surface without any charge buildup in the clouds. In the second case, the electric field builds up as a result of both ion diffusion to the surface and currents resulting from charge separation in the clouds. In both cases, the steady-state profiles of the ion concentrations, net space charge, electrical currents, and electric fields are presented and discussed. These profiles are compared with the corresponding terrestrial profiles.
Case I
The higher diffusivity of the positive ions (Borucki et al., 1982) results in a net diffusion of positive charge toward the Venusian surface. As a positive surface charge accumulates on the surface, it generates an electric field that extends upward into the atmosphere. Thus as a steady-state develops, diffusion currents are balanced by conduction currents, and the total current is zero. Diffusion of ions can take place by both molecular and eddy diffusion. The eddy diffusion coefficient may be written as De = cz,
(8)
where c is a constant which depends on the surface roughness and the stability of the atmosphere. A typical profile of eddy mixing in the terrestrial boundary layer has a maximum of about 15 m 2 sec -I at approximately 600 m (McBean et al., 1979). For lack of data on the magnitude of the eddy mixing in the Venusian atmosphere we assumed three values of the coefficient c in Eq. (8), 0, 0. l, and 1 m sec -1. Seiff e/al. (1980)found the layer lying below the clouds to be stable in the Venusian atmosphere. Therefore, we assumed the depth of the boundary layer on this planet to be 10 m. The eddy diffusion coefficients are calculated according to Eq.
VENUS' ATMOSPHERIC ELECTRIC FIELD (8) up to 10 m, and assumed to be constant above it. When we increased the boundary layer depth (keeping the maximum eddy coefficient constant), we found no significant change in the electric field profile. Figure 1 shows the steady-state electric field developed for the various eddy mixing coefficients. When the eddy mixing is zero, the maximum electric field obtained is 10 mV m -l, and 10 cm above the surface it decreases rapidly to zero. As the eddy mixing increases, the maximum electric field and the altitude to which it extends is increased (Fig. 1). The increase in the value of the electric field is caused by the increase in positive ion concentration brought from above. This increase in ion concentration results in enhanced ion diffusion to the surface and further increase in the electric field. The three profiles shown in Fig. 1 exhibit only a very limited development of the electric field. The maximum electric field obtained is only 1.5 V m -I. It is produced near the surface and decays rapidly above 100 m to a value of 10 mV m -1 at about 1 km (Fig. 1). This electric field (1.5 V m -l) is more than two orders of magnitude smaller than the terrestrial fair-weather electric field. The conduction current (Fig. 2) is directed upward to balance the net positive ~0 5
iiiii111
l 11111111
i lllllll]
1 11111111
I i iii11~
i ll,l,q
i ,fired
L lllil[d
IHI
l
D, =LOz mZ/s
D,=O.Iz 101
~
U"l
:It: i0-3
10-7 I 10-4
i0-~
i0~.
~-~
~0
ELECTRIC FIELD {V/m)
FIG. 1. The electric field as a function of height in the case of zero total steady-state current, Case 1. The eddy diffusion coefficients in the boundary layer, De, are given near each line.
105
i
i
111111
L 103 L_
I
349 ,
De=1,0z
I
1111111
I
,
'
[ I I11
I
I
Ltll
mZ/s
i0 I
:£ 10-3 I0~ 10-' t
I
IIIIIII
I 10"17
I
IIIIII1 10-16
CONDUCTION CURRENT (A/m 2)
FIG. 2. As in Fig. 1, except for net conduction currents. diffusion current to the surface (negative ions move down and positive ones move up). The conduction current is a few orders of magnitude smaller than the diffusion current for each type of ion. The maximum conduction current predicted in this case is about 2 × 10 -16 A m -2 and is four orders of magnitude smaller than the fair-weather conduction current measured in the Earth's atmosphere. In Fig. 3, the diffusion current of the negative ions is shown (it differs from the positive diffusion current by less than two orders of magnitude). Without eddy mixing the diffusion currents are due to molecular diffusion, and very small currents are produced (Figs. 2 and 3). In the case of maximum mixing, the diffusion currents reach a value of about 10 -12 A m -2 near the ground and decay rapidly above 100 m (Fig. 3). The profiles of the concentration of the ions are shown in Fig. 4. Without the influence of the surface, the ion profile is determined by the balance between ionization rate and ion recombination, as in Eq. (7). In this case, the difference between the positive and negative ion concentrations is at least two orders of magnitude smaller than the ion concentration of each type of ion. Therefore, the profiles in Fig. 4 represent the positive or negative ion concentration. The dashed line in Fig. 4 corresponds to the ion concentration calculated from Eq. (7).
350
TZUR AND LEVIN 105
ii1.. I
i ,lllm I
i i.l.. I
\
i iHlllq
10 5
, ,,,,,u I
i 11111111
i IlIHII I
i 11111111
i iiiiilli
I IIIIIId
i iiii1,
IO3 ~
De=O.Iz r n 2 / ~
103
Jlllul I
i0 ~
De=O ~ iO_i
I0 -I o_ =
10-3
i0-5
10-5
i0 -?
iO-r tllltu[
I innnnd
10-17
i nlllml
10-16
i illunl
10-15
i tnltnfi
10-14
~ i tlilllt
in~
I tlllllJI
--10 -16
10-13
DIFFUSION CURRENT (A/m z)
--IO -15
--10 -14
--10
J IIIII 13
NET SPACE CHARGE (C/m3)
FIG. 3. As in Fig. 1, except for diffusion currents of the negative ions.
FIG. 5. As in Fig. 1, except for net space charge.
The other two lines in Fig. 4 correspond to no mixing (solid curve) and the maximum mixing (dash-dot curve) assumed in Case 1, respectively. In the case of maximum mixing, the electrode layer effect penetrates up to about 1 km. Because the positive ions are more mobile, they flow faster out of the electrode layer and a net negative space charge is built up (Fig. 5). In the case of no mixing a net space charge develops only below 1 m, while for the assumed maximum mixing the net space charge penetrates above 100 m (Fig. 5). However, we emphasize that the net space charge shown in Fig. 5 is actually
very small as compared to that near the Earth's surface.
,o5~ ................., ........, ................., 'j..~
~,o-,~-
~.= ~
/
i ~
'°'r ....................... ,,':i .............. ,',,,,,,1, io3
io4
105
ioe
ioz
io8
io9
ION C,ONCENTRATION (ions/rn 3)
FIG. 4. As in Fig. 1, except for ion concentration. The dashed line represents the ion concentration without the influence of the surface charge.
Case 2
In the second case we assume that electric fields and conduction currents exist beneath the Venusian cloud layer. In the present calculations the electrification of the clouds is not simulated and the upper boundary of the model is taken well below the base of the clouds. It is also assumed that there are no ground-to-cloud lightning and no coronal currents from the ground to the cloud base. The steady-state conduction current which flows below the cloud layer is used as a parameter. This parameter depends either on the degree of charge separation inside the clouds or the intensity of some other electric field generator. It is assumed to vary from a minimum of 10 16A m -z, which is about the maximum predicted in Case 1, to an arbitrary maximum of 10 -12 A m -2. When oppositely directed steadystate currents were used (which corresponds to an opposite polarity of the electric field generator), the results were very similar and differed only in sign. In Case 2 the eddy mixing coefficients are calculated from Eq. (8), with c = 0.1 m sec-I and the depth of the boundary layer is assumed to be 10m. Figure 6 presents the profiles of the electric field for assumed steady-state conduc-
VENUS' ATMOSPHERIC ELECTRIC FIELD
~s~~... ~,_,~_~,~. -' I0 5
, ,,.,, , ,,
, ,,b,,q j.~
351
iO s
103
I01
j~l
,o'ff IL~ l,~I--~ ~, ~'J~,o,o-'~cA/~ ~)
=10-~4(A/m2)~
Terrestriol I0-t - Electric Field ~ . Jmo, =10-12 (A/mz)~
10-5
I0-'~
io-r
10-7 IIIIIIL I IIIULII I IIIIIU[ t IIIIIHL I Illllt| IO-a
iO-I
I00
ELECTRIC
I01
I lllll I IIIII
102
0
IOs
I08
3xlO a
5xlO e
7xlO 8
9xlO e
ION CONCENTRATION (ions/m 3)
F I E L D (Vim)
FIG. 6. The electric field as function of height for Case 2. The total steady-state currents, J t o t a l , a r e given near each line. The dashed line represents the fairweather terrestrial electric field.
FIG. 8. T h e ion concentration as function of height for Case 2. The total steady-state currents are given near each line.
tion currents of 10 -lz, 10 -14, and 10 -16 A m -z. Also shown in this figure is the terrestrial fair-weather electric field. The global thunderstorm activity on Earth is a s s u m e d to generate this electric field and produces a conduction current of about 10 -12 A m -2 in the fair-weather regions (Israel, 1973). The Venusian a t m o s p h e r e has a lower conductivity than that of the terrestrial a t m o s p h e r e (Borucki et al., 1982) and a conduction current of 10 -12 A m -2 is capable of producing an electric field of 5 kV m -~ near its surface (Fig. 6). This relatively high electric field penetrates up to about 10 km, where it is 103
V m -l, and then decreases very rapidly to about 1 V m - I at 40 km. The d e v e l o p m e n t of the electric field as a function of time is shown in Fig. 7. The two lines correspond to total currents of 10-14 A m -2 and zero, respectively. In both cases the eddy mixing coefficients are calculated from Eq. (8) with c = 0.1 m sec -l, and the depth of the b o u n d a r y layer is 10 m. As can be seen in Fig. 7, the time constant of the electric field d e v e l o p m e n t is about 105 sec in both cases. This long time constant, as c o m p a r e d to about 103 sec in the terrestrial case, is again a result of the lower conductivity in the lower Venus atmosphere. The ion concentration near the ground is shown in Fig. 8 for an assumed steady-state conduction current of 10 -~2 and 10 -14 A m -2, respectively. When a current density o f l 0 -14 A m -2 is assumed, an electric field of 50 V m -~ is produced near the s u r f a c e - too small to generate significant net space charge in the electrode effect region. H o w ever, an assumed current of 10 -12 A m -2 generates an electrode layer of 1 km in depth. The net space charge for the various assumed conduction currents is shown in Fig. 9. If local vertical convection currents or stronger mixing than that a s s u m e d in this model exist near the Venusian surface, they
0.6 ~
/
- 40
0.5 0.4
03
i
ao
02 I
0.1
/ /I/~Jto~l i01
i0 2
l0 s
I0 4
~
-FO
= IO-~'(A/m2)-_
IOs
TIME (s)
FIG. 7. The time d e v e l o p m e n t of the electric field at the surface. T h e left vertical axis is for zero total current, and the right one is for total current of l0 14 A m-Z,
352
TZUR AND LEVIN
-~,o, =,o-':¢A*,,~---_J7 ::I:
I
I
10-3
1@-5
10-7
iIiilUJ
i iiiiiid
_i0-16
I ILLlilL
I IIILIId
_i0-14
I ILbllld
L ILHI~[
_10-12
NET SPACE CHARGE
I
_io-lO
(C/m 3)
FIG. 9. As in Fig. 8, except for net space charge.
will carry a net space charge out of the electrode layer, resulting in more charge separation and enhancement of the electric field in the lower atmosphere of Venus. CONCLUSIONS
Calculations of the electric field profile in the lower Venusian atmosphere reveal that the smaller mass and higher mobility of the positive ions cause an electrode effect layer with excess negative ions to develop near the surface. As expected, inclusion of eddy mixing increases the height of the electrode effect layer. The ever-present clouds on Venus may have some charge separation in them. Currents produced by charge centers in the clouds or by some other source can affect the magnitude of the electric field below, all the way to the surface. Because the electrical conductivity is lower at the lowest levels of the atmosphere on Venus, a current even smaller than that observed in our terrestrial atmosphere can produce fields of similar magnitudes to those observed here. The presence of convection on Venus could also affect the magnitude of the field. However, measurements suggest that the atmosphere below the clouds is stable, so that if convection exists it is of a small scale and probably localized. In regions where it occurred, stronger fields and deeper electrode effects would be found.
In this work, we use the molecular weights based on the ionic composition suggested by Borucki et al. (1982). However, we believe that this ionic composition is not certain since even in the terrestrial atmosphere this parameter is not yet completely resolved. Therefore, the sensitivity of the results to the molecular weight of the ions is tested. Figure 10 shows three profiles of electric field produced by differences in the positive and negative ionic diffusion currents to the surface. Here the coefficient c in Eq. (8) is assumed to be 0. I, corresponding to stable atmospheric conditions. The lower electric field profile corresponds to the molecular weight ratio suggested by Borucki et al. (1982). Introduction of a molecular weight ratio of 10 produces a maximum electric field of 30 Vm-1. The effect of the ionization produced by natural radioactive substances emitted from the ground was also tested. This ionization source was assumed to be of the same magnitude as the terrestrial radioactive source (Israel, 1973). The effect of this ionization source on the results of Case 1 is very small since in this case charging of the surface depends mostly on the difference in the diffusivities of the positive and negative ions. In Case 2, where an electric field generator was assumed (i.e., charge separation 104
bo °
_~
= M/M 108/88 z
10-z
10-4
10-6
J
10--4
I
1013
J
10"2
i
~--I
-I J
IO0
I01
102
ELECTRIC FIELD (V/m)
FIG. 10. The electric field as function of height. T h e ratios of negative to positive ion molecular weight, M/ + M, are indicated at each line.
VENUS' ATMOSPHERIC ELECTRIC FIELD inside the cloud layer), the additional ionization rate increases the conductivity and results in a decrease in the electric field. This decrease is proportional to g~-0.5 and is limited to the region below 1 km. APPENDIX--SYMBOLS
Time derivative of the variable A Height derivative of the variable A
At Az +
-
D,D
De e
E g, gr n +
Diffusion coefficient (eddy mixing plus molecular diffusion) of the positive and negative ions, respectively Eddy diffusion coefficient Electron charge Electric field Ionization rate from cosmic and radioactive sources, respectively Concentration of ions
-
N,N
Concentration of positive and negative ions, respectively Time Height Positive-negative ion recombination Permittivity of free space
+ -
Electrical mobility of the positive and negative ions, respectively ACKNOWLEDGMENTS The authors are very grateful to Dr. John M. Gary for considerable mathematical advice and guidance in the solution of the model equations. This research was supported in part by NASA Grant W-15,028.
353
REFERENCES BORUCKI, W. J., Z. LEVIN, R. C. WHITTEN, R. G. KEESEE, L. A. CAPONE,O. B. TOON, AND J. DOBACK (1982). Predicted electrical conductivity between 0 and 80 km in the Venusian atmosphere. Icarus 51, 302-321. ISRAEL, H. (1973). Atmospheric Electricity, Vols. 1 and 2. Israel Program for Scientific Translations, Jerusalem. KNOLLENBERG, R. G., AND D. M. HUNTEN (1980). The microphysics of the clouds of Venus: Results of the Pioneer Venus particle size spectrometer experiment. J Geophys. Res. 85, 8039-8058. KSANFOMALITI, L. V. (1979). The lightning in the cloud layer of Venus. Kosm. Issled. 17, 747-762. KSANFOMALITI, L. V. (1980). Discovery of frequent lightning discharges in clouds on Venus. Nature 284, 244-246. MADSEN, N. K., AND R. F. SINCOVEC (1979). Algorithm 540 PDECOL, general collection software for partial differential equations ID3]. ACM Trans. Math. Software 5, 326-351. MCBEAN, G. A., K. BERNHARDT,S. BODIN, Z. LITYNSKA, A. P. VAN ULDEN AND J. C. WYNGAARD (1979). The Planetary Boundary Layer, Chap. 4, Numerical Modeling o f the Atmospheric Boundary Layer (G. A. McBean, Ed.). Technical Note No. 165, WMO-No. 530. Geneva, Switzerland. SCARF, F. L., W. W. L. TAYLOR, C. T. RUSSELL, AND L. H. BRACE (1980). Lightning on Venus: Orbiter detection of whistler signals. J. Geophys. Res. 85, 8158-8166. SEIFF, A., D. B. KIRK, R. B. YOUNG, R. C. BLANCHARD, J. T. FINDLEY, G. M. KELLY, AND S. C. SOMMER (1980). Measurements of thermal structure and thermal contrasts in the atmosphere of Venus and related dynamical observations: Results from the four Pioneer Venus probes. J. Geophys. Res. 85, 7903 -7933. TAYLOR, W. W. L., F. L. SCARF, C. T. RUSSELL, AND L. H. BRACE (1979). Evidence for lightning on Venus. Nature 279, 614-616.