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Nonlinear numerical simulation of equatorial spread-F — Effects of winds and electric fields
Adv. Space Res. Vol. 12, No.6, pp. (6)227—(6)230, 1992 Printed in Great Britain. All rights reserved.
0273-1177/92 $15.00 Copyright © 1991 COSPAR
NONLINEAR NUMERICAL SIMULATION OF EQUATORIAL SPREAD-F EFFECTS OF WINDS AND ELECTRIC FIELDS -
R. Raghavarao,* R. Sekar** and R. Suhasini** *
Code 914, Planetary Atmospheres Br., GSFC, Greenbelt, MD 20771, U.S.A. Physical Research Laboratory, Ahmedabad380 009, India
**
ABSTRACT Recent investigations by means of linear theory revealed the importance of the vertical winds on Rayleigh—Taylor (R—T) instability in the bottomside F—region along with other driving agencies like the zona]. electric field and winds. In the present paper, the nonlinear effects of vertical winds and zonal electric fields on the R—T instability are investigated. The results reveal that in the presence of downward wind or/and eastward electric field the evolutionary process of R—T even beyond 350 km altitude where their effects are considered to be less important according to linear theory.
INTRODUCTION It is well known that the phenomenon of Equatorial Spread—F (ESF) is due to the generation of field aligned irregularities in electron and ion densities with scale sizes ranging from few hundred kilometers to few centimeters. The atmospheric gravity waves are thought to be responsible for the generation of irregularities with few hundred kilometer scale sizes/i I, through spatial resonance mechanism /2/. The irregularities of few tens of kilometers to few hundred meters are believed to be generated by plasma fluid type collisional Rayleigh—Taylor instability /3/, which operates under the influence of gravity, on the bottomside of the nighttime equatorial F—region. Steep plasma density gradients created by the primary Rayleigh—Taylor mode, induce various other plasma instabilities giving rise to smaller scale irregularities/3 ,41. Plasma instability theories have been developed by linear and nonlinear approach to explain the growth of these irregularities. The linear instability mechanism, wherein the plasma density gradient is assumed to remain constant during its growth time, can only describe the zero order conditions for the development of the instabilities. Since the plasma densities and the gradients in them are changing with the growth time of the instability, the evolution of ESF is a nonlinear process. Further, the instability process evolves in both zonal and vertical directions as the irregularities are magnetic field aligned. Therefore a two dimensional model describing the basic plasma fluid equations is needed to simulate the near realistic situation in the nonlinear development of ESF. Scannapieco and Ossakow/5/, showed for the first time that the long wavelength irregularities could appear on both sides of the F—region peak by solving the basic plasma fluid equations through nonlinear simulation studies. They had demonstrated how a few percent perturbation in the background plasma density at the bottom side of the F—region could evolve into plasma bubbles eventually moving up beyond the F—peak resulting in the generation of irregularities. The dependence of the altitude of the F peak and the plasma density scale length in the evolutionary process/6/ and the development of large scale bubbles and their dynamics/7/ have been investigated by others. Further, Zalesak et a]. .18/ had shown that the finite E—region conductivities slow down the bubble evolution and the altitude independent eastward wind would tilt the bubble eastward at lower altitudes and westward higher above. It is to be noted that in the above simulation studies the gravity is considered to be the primary driving agency while the rest being secondary. In recent times, the has been recognised.
importance of various other driving agencies apart from gravity The effect of zonal electric field or vertical plasma drift and
JASR 12:6-0
(6)227
(6)228
R. Raghavarao et al.
zonal wind with tilted ionosphere to assist or inhibit the growth of the Rayleigh—Taylor mode has been included in the so called generalised Rayleigh—Taylor instability/9/. The importance of the vertical wind of measured magnitudes has been considered/10/ and the growth rates due to various agencies on the basis of co—ordinated measurements of plasma density, winds and electric fields at the onset time of ESF have been evaluated/ill. - The collective importance of various destabilising agencies to explain the observed electron density irregularities in the altitude region of 240 to 290 km has been demonstrated. Hanson et al/12/ investigated the time dependent behaviour of the growth rate of Rayleigh—Taylor instability in one dimension. The effects of vertical drift was also included along with gravity in their analysis. They pointed out that the F region vertical drift increases the growth rate just after sunset by an order of magnitude in about an hour when the drift direction is upward. In this paper, the nonlinear effects of vertical winds and zonal electric fields as additional contributions to Rayleigh—Taylor instability are investigated using a two dimensional nonlinear simulation model. THEORY AND MODEL DESCRIPTION The equatorial ionosphere is considered in a slab geometry with the earth’s magnetic field (B) along z axis. The x, y and z axes are directed westward, upward and northward respectively. The plasma fluid equations describing the model are the equations of momentum and continuity for ions and electrons and the current conservation equation. Under certain valid assumptions, these equations are reduced to two differential equations: V~Vjn°VcI~i)
=
—
B g 3ri + B W~Vin c 3X
—
~T
a (nc
axr3y
~
)
~_a~in
Ex
~.
(1)
C
+~(nc ~y B
~
V~n
(2)
~
Equation (1) is the differential equation for the perturbation potential (-~ ) in a genera— lised form, wherein the effects of vertical winds (W ) and zonal eiecQric field (E are included apart from the Rayleigh—Taylor term duJ to the gravity (g). Here t?~e notations ~ ., n and c represent the ion—neutral collision frequency, plasma density and the velocity of light respectively. Equation (2) represents the plasma continuity equation wherein the dominant transport term due to perturbation potential (~ i~ and the recombination effects due to chemistry are included. The production is assumed to be zero as the case under study is a nighttime phenomenon. Here ~ represents the reco— mbination rate. The model calculations have been performed in the magnetic equatorial plane and in the altitude region of 252—532 km in steps of 2 km and a zonal extent of ± 100 km with grid size of 5 km. The boundary conditions are periodic in zonal direction both for and n and their first order derivatives
are taken as zero in the vertical direction.
The altitude profiles of ion—neutral collision frequency, recombination rate and plasma density at the onset time of the instability are used as basic inputs to the model and they are identical to those corresponding profiles that are adopted by Zalesak and Ossakow/7 I. Similar to their calculations, a sinusoidal perturbation in n, with a wavelength equal to —75 km and an amplitude 5% of the ambient density is imposed. With these inputs, the coupled differential equations (1) and (2) are solved simultaneously using successive over relaxation method maintaining skew symmetry in the zonal direction and by explicit finite difference method (Lax—Friedricks and Lax—Wandroff schemes) with flux corrected transport technique/l3/ respectively. The time evolution of the perturbation potential and the plasma densities are thus obtained. The model calculations
are performed in two parts;
1.
the effects of gravity
alone as the driving agency.
2.
the effects of vertically downward wind of magnitude 20 gravity as the driving agency. Winds of such magnitudes present in the F—region/14/.
m/s. along with are known to be
Equatorial Spread-F
(6)229
RESULTS AND DISCUSSION Figure 1 depicts the isoelectron density contours at 1100 sec after the onset of the instability for the cases a) the effects of gravity alone (No wind) and b) with a downward wind of magnitude 20 rn/s. along with gravity. The dotted profiles correspond to the background electron density at the onset time of the instability whose scale 51s hown at the top of the diagram. The contour corresponding to the value 1.5 x 10 cm (in the bottomside) is located at about 420 km at zero zonal distance in the “No wind” case (a), while in the “wind” case (b) it is located above 450 km which is well in the topside of the F region. It is also found that the effects of eastward electric field of magnitude 0.76 mv/m in the nonlinear simulation are identical to the effects of the downward wind of 20 m/s. Electric fields of such magnitudes have been observed at the evening time by different techniques/11,15,16/. The gain in time of the bubble development can be noted in the “wind” case as, the bubble reached the topside at 1100 sec while in the “No wind” case it took around 1400 sec for a similar spatial development. The detailed theory and the evolutionary patterns will be presented in a separate communication.
5 Cm3
No X IO~Cm3
No X10
~:~ ‘~
:‘~~
ZONAL DISTANCE
1Km)
ZONAL DISTANCE 1Km)
Fig. 1. Isoelectron density contours in zonal and vertical plane at 1100 sec after the onset time of the instability for the cases a) effects of gravity alone and b) with a downward wind of 20 m/s along with gravity.
The important conclusion arrived at, from the above exercise is that the effects of vertical winds and eastward electric fields are significant even beyond 350 km where they are considered to be less important according to linear theory. The growth rate due to gravitational term depends inversely on ion—neutral collision frequency (v in~ while the other terms do not depend on \).~. Since V decreases exponentially with altitude and less than 1 above 300 km, the ‘gravitational ~erm becomes dominant at higher altitudes. However, it is clear from the present exercise, that the evolutionary process of Rayleigh—Taylor instability is accelerated even beyond 350 km, when downward wind of 20 m/s or/and an eastward electric field of 0.76 mv/rn is introduced. ACKNOWLEDGEMENTS We thank Miss Ranna Patel for her assistance in preparing the contour diagrams. work was supported by the Department of Space, Government of India.
This
(6)230
R. Raghavarao et al.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8 9. 10. 11.
12. 13. 14. 15. 16.
J. Rottger, J. Atmos. Terr. Phys., 43, 453 (1981). T. Beer, Planet. Space Sci.., 21, 297 (1973). G. Haerendel, Theory of Equatorial Spread—F, unpublished report, Max—Planck— Institute—fur Physik and Astrophysik, Garching, Garching, F.R.G. (1974). S.L. Ossakow, J. Atmos. Terr. Phys., 43, 436 (1981). A. J. Scannapieco and S. L. Ossakow, Geophys. Res. Lett., 3, 451 (1976). S. L. Ossakow, S. T. Zalesak, B. E. McDonald and P. K. Chaturvedi, J. Geophys. Res., 83, 2066 (1979). S. T. Zalesak and S. L. Ossakow, J. Geophys. Res., 85, 2131 (1980). S. T. Zalesak, S. L Ossakow and P. K. Chaturvedi, J. Geophys. Res., 87, 151 (1982). M. C. Kelley, J. Atmos. Terr. Phys., 47, 745 (1985). R. Sekar and R. Raghavarao, J. Atrnos. Terr. Phys., 49, 981 (1987). R. Raghavarao, S. P. Gupta, R. Sekar, R. Narayanan, J. N. Desai, R. Sridharan, V. V. Babu and V. Sudhakar, J. Atmos. Terr. Phys., 49, 485 (1987). W. B. Hanson, B. L. Cragin and A. Dennis, J. Atmos. Terr. Phys., 48, 205 (1986). 5. T. Zalesak, J. Comp. Phys., 31, 335 (1979). M. A. Biondi and D. P. Sipler, Planet. Space Sci., 33, 817 (1985). R. Raghavarao, J. N. Desai, B. G. Anandarao, R. Narayanan, R. Sekar, Ranjan Gupta, V.V. Babu and V. Sudhakar, J. Atmos. Terr. Phys., 46, 355 (1984). B. G. Fejer, J. Atmos. Terr. Phys., 43, 377 (1981).
0273-1177/92 $15.00 Copyright © 1991 COSPAR
NONLINEAR NUMERICAL SIMULATION OF EQUATORIAL SPREAD-F EFFECTS OF WINDS AND ELECTRIC FIELDS -
R. Raghavarao,* R. Sekar** and R. Suhasini** *
Code 914, Planetary Atmospheres Br., GSFC, Greenbelt, MD 20771, U.S.A. Physical Research Laboratory, Ahmedabad380 009, India
**
ABSTRACT Recent investigations by means of linear theory revealed the importance of the vertical winds on Rayleigh—Taylor (R—T) instability in the bottomside F—region along with other driving agencies like the zona]. electric field and winds. In the present paper, the nonlinear effects of vertical winds and zonal electric fields on the R—T instability are investigated. The results reveal that in the presence of downward wind or/and eastward electric field the evolutionary process of R—T even beyond 350 km altitude where their effects are considered to be less important according to linear theory.
INTRODUCTION It is well known that the phenomenon of Equatorial Spread—F (ESF) is due to the generation of field aligned irregularities in electron and ion densities with scale sizes ranging from few hundred kilometers to few centimeters. The atmospheric gravity waves are thought to be responsible for the generation of irregularities with few hundred kilometer scale sizes/i I, through spatial resonance mechanism /2/. The irregularities of few tens of kilometers to few hundred meters are believed to be generated by plasma fluid type collisional Rayleigh—Taylor instability /3/, which operates under the influence of gravity, on the bottomside of the nighttime equatorial F—region. Steep plasma density gradients created by the primary Rayleigh—Taylor mode, induce various other plasma instabilities giving rise to smaller scale irregularities/3 ,41. Plasma instability theories have been developed by linear and nonlinear approach to explain the growth of these irregularities. The linear instability mechanism, wherein the plasma density gradient is assumed to remain constant during its growth time, can only describe the zero order conditions for the development of the instabilities. Since the plasma densities and the gradients in them are changing with the growth time of the instability, the evolution of ESF is a nonlinear process. Further, the instability process evolves in both zonal and vertical directions as the irregularities are magnetic field aligned. Therefore a two dimensional model describing the basic plasma fluid equations is needed to simulate the near realistic situation in the nonlinear development of ESF. Scannapieco and Ossakow/5/, showed for the first time that the long wavelength irregularities could appear on both sides of the F—region peak by solving the basic plasma fluid equations through nonlinear simulation studies. They had demonstrated how a few percent perturbation in the background plasma density at the bottom side of the F—region could evolve into plasma bubbles eventually moving up beyond the F—peak resulting in the generation of irregularities. The dependence of the altitude of the F peak and the plasma density scale length in the evolutionary process/6/ and the development of large scale bubbles and their dynamics/7/ have been investigated by others. Further, Zalesak et a]. .18/ had shown that the finite E—region conductivities slow down the bubble evolution and the altitude independent eastward wind would tilt the bubble eastward at lower altitudes and westward higher above. It is to be noted that in the above simulation studies the gravity is considered to be the primary driving agency while the rest being secondary. In recent times, the has been recognised.
importance of various other driving agencies apart from gravity The effect of zonal electric field or vertical plasma drift and
JASR 12:6-0
(6)227
(6)228
R. Raghavarao et al.
zonal wind with tilted ionosphere to assist or inhibit the growth of the Rayleigh—Taylor mode has been included in the so called generalised Rayleigh—Taylor instability/9/. The importance of the vertical wind of measured magnitudes has been considered/10/ and the growth rates due to various agencies on the basis of co—ordinated measurements of plasma density, winds and electric fields at the onset time of ESF have been evaluated/ill. - The collective importance of various destabilising agencies to explain the observed electron density irregularities in the altitude region of 240 to 290 km has been demonstrated. Hanson et al/12/ investigated the time dependent behaviour of the growth rate of Rayleigh—Taylor instability in one dimension. The effects of vertical drift was also included along with gravity in their analysis. They pointed out that the F region vertical drift increases the growth rate just after sunset by an order of magnitude in about an hour when the drift direction is upward. In this paper, the nonlinear effects of vertical winds and zonal electric fields as additional contributions to Rayleigh—Taylor instability are investigated using a two dimensional nonlinear simulation model. THEORY AND MODEL DESCRIPTION The equatorial ionosphere is considered in a slab geometry with the earth’s magnetic field (B) along z axis. The x, y and z axes are directed westward, upward and northward respectively. The plasma fluid equations describing the model are the equations of momentum and continuity for ions and electrons and the current conservation equation. Under certain valid assumptions, these equations are reduced to two differential equations: V~Vjn°VcI~i)
=
—
B g 3ri + B W~Vin c 3X
—
~T
a (nc
axr3y
~
)
~_a~in
Ex
~.
(1)
C
+~(nc ~y B
~
V~n
(2)
~
Equation (1) is the differential equation for the perturbation potential (-~ ) in a genera— lised form, wherein the effects of vertical winds (W ) and zonal eiecQric field (E are included apart from the Rayleigh—Taylor term duJ to the gravity (g). Here t?~e notations ~ ., n and c represent the ion—neutral collision frequency, plasma density and the velocity of light respectively. Equation (2) represents the plasma continuity equation wherein the dominant transport term due to perturbation potential (~ i~ and the recombination effects due to chemistry are included. The production is assumed to be zero as the case under study is a nighttime phenomenon. Here ~ represents the reco— mbination rate. The model calculations have been performed in the magnetic equatorial plane and in the altitude region of 252—532 km in steps of 2 km and a zonal extent of ± 100 km with grid size of 5 km. The boundary conditions are periodic in zonal direction both for and n and their first order derivatives
are taken as zero in the vertical direction.
The altitude profiles of ion—neutral collision frequency, recombination rate and plasma density at the onset time of the instability are used as basic inputs to the model and they are identical to those corresponding profiles that are adopted by Zalesak and Ossakow/7 I. Similar to their calculations, a sinusoidal perturbation in n, with a wavelength equal to —75 km and an amplitude 5% of the ambient density is imposed. With these inputs, the coupled differential equations (1) and (2) are solved simultaneously using successive over relaxation method maintaining skew symmetry in the zonal direction and by explicit finite difference method (Lax—Friedricks and Lax—Wandroff schemes) with flux corrected transport technique/l3/ respectively. The time evolution of the perturbation potential and the plasma densities are thus obtained. The model calculations
are performed in two parts;
1.
the effects of gravity
alone as the driving agency.
2.
the effects of vertically downward wind of magnitude 20 gravity as the driving agency. Winds of such magnitudes present in the F—region/14/.
m/s. along with are known to be
Equatorial Spread-F
(6)229
RESULTS AND DISCUSSION Figure 1 depicts the isoelectron density contours at 1100 sec after the onset of the instability for the cases a) the effects of gravity alone (No wind) and b) with a downward wind of magnitude 20 rn/s. along with gravity. The dotted profiles correspond to the background electron density at the onset time of the instability whose scale 51s hown at the top of the diagram. The contour corresponding to the value 1.5 x 10 cm (in the bottomside) is located at about 420 km at zero zonal distance in the “No wind” case (a), while in the “wind” case (b) it is located above 450 km which is well in the topside of the F region. It is also found that the effects of eastward electric field of magnitude 0.76 mv/m in the nonlinear simulation are identical to the effects of the downward wind of 20 m/s. Electric fields of such magnitudes have been observed at the evening time by different techniques/11,15,16/. The gain in time of the bubble development can be noted in the “wind” case as, the bubble reached the topside at 1100 sec while in the “No wind” case it took around 1400 sec for a similar spatial development. The detailed theory and the evolutionary patterns will be presented in a separate communication.
5 Cm3
No X IO~Cm3
No X10
~:~ ‘~
:‘~~
ZONAL DISTANCE
1Km)
ZONAL DISTANCE 1Km)
Fig. 1. Isoelectron density contours in zonal and vertical plane at 1100 sec after the onset time of the instability for the cases a) effects of gravity alone and b) with a downward wind of 20 m/s along with gravity.
The important conclusion arrived at, from the above exercise is that the effects of vertical winds and eastward electric fields are significant even beyond 350 km where they are considered to be less important according to linear theory. The growth rate due to gravitational term depends inversely on ion—neutral collision frequency (v in~ while the other terms do not depend on \).~. Since V decreases exponentially with altitude and less than 1 above 300 km, the ‘gravitational ~erm becomes dominant at higher altitudes. However, it is clear from the present exercise, that the evolutionary process of Rayleigh—Taylor instability is accelerated even beyond 350 km, when downward wind of 20 m/s or/and an eastward electric field of 0.76 mv/rn is introduced. ACKNOWLEDGEMENTS We thank Miss Ranna Patel for her assistance in preparing the contour diagrams. work was supported by the Department of Space, Government of India.
This
(6)230
R. Raghavarao et al.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8 9. 10. 11.
12. 13. 14. 15. 16.
J. Rottger, J. Atmos. Terr. Phys., 43, 453 (1981). T. Beer, Planet. Space Sci.., 21, 297 (1973). G. Haerendel, Theory of Equatorial Spread—F, unpublished report, Max—Planck— Institute—fur Physik and Astrophysik, Garching, Garching, F.R.G. (1974). S.L. Ossakow, J. Atmos. Terr. Phys., 43, 436 (1981). A. J. Scannapieco and S. L. Ossakow, Geophys. Res. Lett., 3, 451 (1976). S. L. Ossakow, S. T. Zalesak, B. E. McDonald and P. K. Chaturvedi, J. Geophys. Res., 83, 2066 (1979). S. T. Zalesak and S. L. Ossakow, J. Geophys. Res., 85, 2131 (1980). S. T. Zalesak, S. L Ossakow and P. K. Chaturvedi, J. Geophys. Res., 87, 151 (1982). M. C. Kelley, J. Atmos. Terr. Phys., 47, 745 (1985). R. Sekar and R. Raghavarao, J. Atrnos. Terr. Phys., 49, 981 (1987). R. Raghavarao, S. P. Gupta, R. Sekar, R. Narayanan, J. N. Desai, R. Sridharan, V. V. Babu and V. Sudhakar, J. Atmos. Terr. Phys., 49, 485 (1987). W. B. Hanson, B. L. Cragin and A. Dennis, J. Atmos. Terr. Phys., 48, 205 (1986). 5. T. Zalesak, J. Comp. Phys., 31, 335 (1979). M. A. Biondi and D. P. Sipler, Planet. Space Sci., 33, 817 (1985). R. Raghavarao, J. N. Desai, B. G. Anandarao, R. Narayanan, R. Sekar, Ranjan Gupta, V.V. Babu and V. Sudhakar, J. Atmos. Terr. Phys., 46, 355 (1984). B. G. Fejer, J. Atmos. Terr. Phys., 43, 377 (1981).