- Email: [email protected]
Transient thermospheric heating and movement caused by an auroral electric field
P&met. Space Sci., Vol. 24, pp.
727
to
730. Pergamon
Press,
1976.
Printed
in Northern
Maad
TRANSIENT THERMOSPHERIC HEATING AND MOVEMENT CAUSED BY AN AURORAL ELECTRIC FIELD S. T. WU@ ad
K. D. COLE
Division of Theoretical and Space Physics, La Robe University, Bundoora, Victoria, 3083, Australia (Receioed 23 January 1976) Ab&ract--Two closed form solutions for the velocity distribution of the upper thermosphere were found using ma~etohydr~~amic fo~alism. One corresponds to a constant altitude, is timedependent, and has non-moving boundaries. This case asympotically approaches the steady solution obtained by Cole (1971). The other solution corresponds to a time- and attitude-dependence case with free boundaries. Solutions of electrodynamic (joule) and viscous heating for both cases are given. Some numerical results corresponding to the latter case are presented. It is clearly demonstrated that joule heating is dominant within the electric field region, and that viscous heating becomes important in the neighbourhood of the electric field region. It is also shown that the induced movement extends
beyond the electric field region as far as four times the original width of the electric field region.
1. U’ITRODUCTION
for the altitudes concerned (3160 km). With this in mind, the equation of motion governing this plasma flow can be written as follows negligible
When an electric field E, orthogonal to the geomagnetic field B, is applied in the ionosphere, the thermosphere is accelerated. The time constant required to reach a steady state is given by p/(olB2) where ur is the Pedersen conductivity and p the thermospheric density (see Cole, 1971). One specia1 case for analysis occurs when the characteristic time for existence of the electric field tE m p/(crlB2). This occurs often in the aurora1 zone and on the polar cap where electric fields can last sufficiently long to accelerate the thermosphere above altitudes of about 160 km. Above this altitude Pedersen currents dominate over Hall currents which may be neglected in an approximate solution of the problem. Consideration of the effects of electric fields below this altitude must take into account Hall currents. It is assumed in this paper that we are dealing with the thermosphere above about 160 km altitude which corresponds to Case II of Cole (1971).
*+J,B,=o
aV
P
at-'ax2
where u is the thermospheric flow velocity along y-direction, p is the mass density, CLis the molecular viscosity, B, is the magnetic induction in zdirection and 1, represents the Pedersen currents which can be given as 3, = a,(E + uB)
(2)
with CT,being the Pedersen conductivity and assuming it to be constant, and the independent variables are x and f. Further, we consider a small magnetic Reynofds number (i.e. p*ouI,) flow, so that the effect of changing the magnetic field in the plasma can be ignored. By substituting (2) in (l), we obtain 2
2-p~+o,Bo2u
II. THEORETICALANtiYSEs Let us suppose the electric field has the configuration as shown in Fig. 1. It is assumed that the plasma was bounded in the region of (-L, +L) and the motion is along the y-direction only. Pedersen current flows in the x-direction and a potential drop exists across the region -L 6 x s +I_., and Hall current fIows in y-direction but this is assumed
The initial specified as
(3)
and
boundary
conditions
can
be
u(x,O)=O u(-L, t) = U(L, t) = 0.
(4)
These conditions are the same as used by Cole (1971) to obtain this steady state solution. Later we shall discuss a rather general case, i.e. [u = u(x, z, t) and u(-L, t) = u(L, t) # 01.
* Permanent address: Dept. of Mechanical Engineering, The University of Alabama in Huntsville, Huntsville, AL 35807, U.S.A. 2
= -a,EB.
%t
727
s. T.
728
t
Wu
and K. D.
COLE
J it
wsh (MI) Y
-L
FIG. 1.
SCHEMATIC
-CL 1
0 REPRESENTATION
OF AN AURORAL
+$
-x
OF A CROSS-SECTION
ELECTROJET
REGION. X
Equation (3) together with initial boundary conditions of Equation (4) form a complete mathematical problem which can be solved by Laplace trans; formation, its solution being v(x,
t)=f0
l-
cc
cash (x~~(~~/~)*‘=) cash (LB,(~,/~)“2) exp (
+16
_ c = “=“(2n+l)
x sin
@/&2n+ 1)2rr2t 4L2 >
---$(2nf1)‘~‘+4]
(2n + l)T(X + L 2L !I 7
(5)
where LY= alBo2/p. ‘Ibis solution recovers the steady state solution_ given by Cole (1971) immediately by setting tsm thus VW steady =
0 (x, m)
cash (M&L) cash WI)
with Ml being the Hartmann’s number (i.e. Mi = (uJ~)~‘~B,L). In the meantime, we notice that the equilibrium characteristic time t, = (Y-I = ~/u,B,~ which is identical with those obtained by Piddington (1954) and Cole (1971). As soon as the velocity solution was found, the joule heating and viscous heating can be computed as follows:
X sin
[
(-1)“+‘(2n+
exp [
1)
1
fi(2n + lf2?T2t 4L2p
--$2n+1)%r2+4 x sin (2n -+-l)n(x + L) (8) 2L * I As we may note, the expressions of equations cl,) and (8) for joule and viscous heating always consist of two parts, the first part representing the steady state and the second part representing the contribution from the transient heating. In the previous model we consider the plasma flow which is confined in the electric field region, and no momentum is allowed to be transferred to the neighbou~ng region because we have set the plasma flow velocity to be zero at (--L, L). Now we shall consider a more realistic case, namely that momentum. can be allowed to be transferred to the neighbouring layer (Ix/> L) through the action of viscosity. Thus we shall leave the boundary free (i.e. u(-L, 0) = V(L, b)# 0) and also include the altitude dependence. Thus the equation of motion becomes d2v a% P~+apv+___= al&E (9) 0
P
CL
v(x,
2,
ax2+2
0) = 0.
(10)
with E = 0 and or = 0 for Ix/> L. Here the independent variables are x, z and t. Using the method of the Fourier transform, the solution for Equation (9) together with the initial condition Equation (10) can be expressed in the following integral form
xexpr_(x-5)2-(z-n)ld5dl)
exp
“(2n+l)
z
n 0
(11)
--&(2n+l)%r2+4]
(2n+l)?r(x+L) 2L
1 2
(7)
Since the currents only flow within the region of the electric field, the expression of equation (10) can be approximated by evaluation of the integral within a finite area which corresponds to
729
Heating and movement due to aurora1 electric field the electric
field range,
such that
VELOCITV L
p‘
kmlsec
J
.’
As soon as the velocity is found, the joule heating and viscous heating can be calculated by Qi = al(E + IB~)~ and Q, =
(13)
pu($+$).
(14)
RESULTS
r....,..,..
The solution of the first case is an extension of the steady state solution given by Cole (1971), which shows that the equilibrium time constant (Y-’ is identical to those obtained earlier (Piddington, 1954; Cole, 1971). Thus, we shall not discuss it here in detail. However, the second case whose physics was discussed by Cole (1971) is more general because we have removed the restriction that the velocity be zero on the boundary, and included altitude dependence in the present investigation. In order to show more physical significance which leads to movement outside the electric field region, we have computed the velocity and heating for a specified aurora1 electric field. Viz., we considered an electric field region with a typical width of the movements of neighbouring layers of the electrojet, and also the asympotic behaviour of the 200
0
02.04
E'IG.3. VELOCITY
06
06
10
12
14
16
16 20 175
V DIMENSIONLESS TIME AT Z= 200 km.
AND
10 km and height of an order of 100 km with E-20mV M-’ and &-OS G. On Fig. 2 we plot the velocity profile within the aurora1 thermosphere at an altitude of 175 km for a different time. In this figure, we clearly observed velocity profile, the velocity reached 80% of its steady state value at t = T = 60 min (characteristic time), and the induced movements can be extended up to a factor of four of the original width of the electrojet. This agrees well with the early estimation given by Cole (1971) (his equation 30). In Fig. 3 we have plotted the velocity of the upper thermosphere at different altitudes versus characteristic time. It clearly indicates that the velocity has reached 75-90% of its steady state value, which shows that Cole’s steady state solution in these altitudes is a reasonably good approximation. In Fig. 4 we show the thermosphere velocity at mot 180
t = 1333
160 140 lx)
,: : , !
V (m/wc)100 -
100
0
1 DIMENSIO?&S
3
LENGTH
4
5
XI Lo
FIG. 2. VELOCITY PROFILE OF THE ELECTROJET AT DIFFERENT DIMENSIONLESS TIMES CALCULATED ACCORDING TO EQUATION (12) AT Z= 175 km (ALTITUDE), Lo BEING THE HALFWIDTHOFTHEAURORALELE‘TROJET.
120
160
t = 0 6667
j
160
180
200
FIG. 4. AVERAGE VELOCITY vs ALTITUDE (Z) AT DIFFERENTCHARACTERISTICTIMES. values below 160 km are not physicallly valid, because the Hall currents become important in that part of the atmosphere; they are included simply for mathematical completeness. The
S. T. Wu and K. D. COLE
730
ALTITUDE (2)
175km
u 0.3 ;
TOTAL HEATING
w 0.2. s n go.>5 Y
JOULE HEATlNG 0
1
2
DIMENSIONLESS TIME
‘G
FIG. 5. %-HE AMOUNT OF JOULE HEATING AND VISCOUS HEATING OF AN ELECSFZOJET AT 175 km ALTITUDE vS DIMENSIONLESSTIME.
different times versus altitude. Physically this solution is only valid above 160 km altitude, because the Hall currents have been neglected in the present calculation. Finally, we plot the average joule heating, viscous heating and total heating through the disturbed volume of space versus dimensionless time at an altitude of 175 km in Fig. 5. These heating rates per unit volume are calculated in the following way. The volume is that of the disturbed atmosphere. Thus, the initial volume is the volume occupied by the electric field. Later as motion is induced, the volume of moving atmosphere will grow as time progresses until it approaches a steady state. From the results shown in Fig. 5 we notice that the joule
heating dominates in the initial stage and is concentrated in the electric field region. In the initial stage, there is no motion in the atmosphere, therefore the viscous heating is practically zero. Later, motion is set up due to the electric field, then viscous heating becomes important. We also plot the total average heating, which is simply the sum of the joule heating and viscous heating. This shows a minimum heating before approaching an asymptotic state. This is due to the fact that when the electric field is turned on, the joule heating occurs immediately, then as time progresses, the volume increases, thus the joule heating per unit volume decreases. In the meantime, viscous heating is increasing due to the movement of the atmosphere. Finally it reaches a steady value of heating. We conclude that this is the mechanism to sustain the motion in the upper atmosphere. This heating will correspond to a horizontal pressure gradient which has been discussed by Comfort et al. (1976). Ack~w~dge~~~n~ of the authors (STW) wishes to acknowledge the Australian-American Education Foundation which supported his visit’ to La Trobe University and MSFC/NASA support under a contract with OAH. REFERENCES Cole, K. D. (1971) El~odyna~~ heating and movement of the thermosphere. Pfanef. Space Sci. 19,5P-75. Comfort, R. H., Wu, S. T. and Swenson, G. R. (1976) An analysis of aurora1 E region neutral winds based on incoherent scatter radar observation at Chatanika. P&met.Suace Sci. 24, 541. ~d~n~on. f. -I%.[X954) The motion of ionized gas in comgineb magnetic electric and mechanical fieids of force. Mon. not, R. astr. Sot. 114,651-663.
727
to
730. Pergamon
Press,
1976.
Printed
in Northern
Maad
TRANSIENT THERMOSPHERIC HEATING AND MOVEMENT CAUSED BY AN AURORAL ELECTRIC FIELD S. T. WU@ ad
K. D. COLE
Division of Theoretical and Space Physics, La Robe University, Bundoora, Victoria, 3083, Australia (Receioed 23 January 1976) Ab&ract--Two closed form solutions for the velocity distribution of the upper thermosphere were found using ma~etohydr~~amic fo~alism. One corresponds to a constant altitude, is timedependent, and has non-moving boundaries. This case asympotically approaches the steady solution obtained by Cole (1971). The other solution corresponds to a time- and attitude-dependence case with free boundaries. Solutions of electrodynamic (joule) and viscous heating for both cases are given. Some numerical results corresponding to the latter case are presented. It is clearly demonstrated that joule heating is dominant within the electric field region, and that viscous heating becomes important in the neighbourhood of the electric field region. It is also shown that the induced movement extends
beyond the electric field region as far as four times the original width of the electric field region.
1. U’ITRODUCTION
for the altitudes concerned (3160 km). With this in mind, the equation of motion governing this plasma flow can be written as follows negligible
When an electric field E, orthogonal to the geomagnetic field B, is applied in the ionosphere, the thermosphere is accelerated. The time constant required to reach a steady state is given by p/(olB2) where ur is the Pedersen conductivity and p the thermospheric density (see Cole, 1971). One specia1 case for analysis occurs when the characteristic time for existence of the electric field tE m p/(crlB2). This occurs often in the aurora1 zone and on the polar cap where electric fields can last sufficiently long to accelerate the thermosphere above altitudes of about 160 km. Above this altitude Pedersen currents dominate over Hall currents which may be neglected in an approximate solution of the problem. Consideration of the effects of electric fields below this altitude must take into account Hall currents. It is assumed in this paper that we are dealing with the thermosphere above about 160 km altitude which corresponds to Case II of Cole (1971).
*+J,B,=o
aV
P
at-'ax2
where u is the thermospheric flow velocity along y-direction, p is the mass density, CLis the molecular viscosity, B, is the magnetic induction in zdirection and 1, represents the Pedersen currents which can be given as 3, = a,(E + uB)
(2)
with CT,being the Pedersen conductivity and assuming it to be constant, and the independent variables are x and f. Further, we consider a small magnetic Reynofds number (i.e. p*ouI,) flow, so that the effect of changing the magnetic field in the plasma can be ignored. By substituting (2) in (l), we obtain 2
2-p~+o,Bo2u
II. THEORETICALANtiYSEs Let us suppose the electric field has the configuration as shown in Fig. 1. It is assumed that the plasma was bounded in the region of (-L, +L) and the motion is along the y-direction only. Pedersen current flows in the x-direction and a potential drop exists across the region -L 6 x s +I_., and Hall current fIows in y-direction but this is assumed
The initial specified as
(3)
and
boundary
conditions
can
be
u(x,O)=O u(-L, t) = U(L, t) = 0.
(4)
These conditions are the same as used by Cole (1971) to obtain this steady state solution. Later we shall discuss a rather general case, i.e. [u = u(x, z, t) and u(-L, t) = u(L, t) # 01.
* Permanent address: Dept. of Mechanical Engineering, The University of Alabama in Huntsville, Huntsville, AL 35807, U.S.A. 2
= -a,EB.
%t
727
s. T.
728
t
Wu
and K. D.
COLE
J it
wsh (MI) Y
-L
FIG. 1.
SCHEMATIC
-CL 1
0 REPRESENTATION
OF AN AURORAL
+$
-x
OF A CROSS-SECTION
ELECTROJET
REGION. X
Equation (3) together with initial boundary conditions of Equation (4) form a complete mathematical problem which can be solved by Laplace trans; formation, its solution being v(x,
t)=f0
l-
cc
cash (x~~(~~/~)*‘=) cash (LB,(~,/~)“2) exp (
+16
_ c = “=“(2n+l)
x sin
@/&2n+ 1)2rr2t 4L2 >
---$(2nf1)‘~‘+4]
(2n + l)T(X + L 2L !I 7
(5)
where LY= alBo2/p. ‘Ibis solution recovers the steady state solution_ given by Cole (1971) immediately by setting tsm thus VW steady =
0 (x, m)
cash (M&L) cash WI)
with Ml being the Hartmann’s number (i.e. Mi = (uJ~)~‘~B,L). In the meantime, we notice that the equilibrium characteristic time t, = (Y-I = ~/u,B,~ which is identical with those obtained by Piddington (1954) and Cole (1971). As soon as the velocity solution was found, the joule heating and viscous heating can be computed as follows:
X sin
[
(-1)“+‘(2n+
exp [
1)
1
fi(2n + lf2?T2t 4L2p
--$2n+1)%r2+4 x sin (2n -+-l)n(x + L) (8) 2L * I As we may note, the expressions of equations cl,) and (8) for joule and viscous heating always consist of two parts, the first part representing the steady state and the second part representing the contribution from the transient heating. In the previous model we consider the plasma flow which is confined in the electric field region, and no momentum is allowed to be transferred to the neighbou~ng region because we have set the plasma flow velocity to be zero at (--L, L). Now we shall consider a more realistic case, namely that momentum. can be allowed to be transferred to the neighbouring layer (Ix/> L) through the action of viscosity. Thus we shall leave the boundary free (i.e. u(-L, 0) = V(L, b)# 0) and also include the altitude dependence. Thus the equation of motion becomes d2v a% P~+apv+___= al&E (9) 0
P
CL
v(x,
2,
ax2+2
0) = 0.
(10)
with E = 0 and or = 0 for Ix/> L. Here the independent variables are x, z and t. Using the method of the Fourier transform, the solution for Equation (9) together with the initial condition Equation (10) can be expressed in the following integral form
xexpr_(x-5)2-(z-n)ld5dl)
exp
“(2n+l)
z
n 0
(11)
--&(2n+l)%r2+4]
(2n+l)?r(x+L) 2L
1 2
(7)
Since the currents only flow within the region of the electric field, the expression of equation (10) can be approximated by evaluation of the integral within a finite area which corresponds to
729
Heating and movement due to aurora1 electric field the electric
field range,
such that
VELOCITV L
p‘
kmlsec
J
.’
As soon as the velocity is found, the joule heating and viscous heating can be calculated by Qi = al(E + IB~)~ and Q, =
(13)
pu($+$).
(14)
RESULTS
r....,..,..
The solution of the first case is an extension of the steady state solution given by Cole (1971), which shows that the equilibrium time constant (Y-’ is identical to those obtained earlier (Piddington, 1954; Cole, 1971). Thus, we shall not discuss it here in detail. However, the second case whose physics was discussed by Cole (1971) is more general because we have removed the restriction that the velocity be zero on the boundary, and included altitude dependence in the present investigation. In order to show more physical significance which leads to movement outside the electric field region, we have computed the velocity and heating for a specified aurora1 electric field. Viz., we considered an electric field region with a typical width of the movements of neighbouring layers of the electrojet, and also the asympotic behaviour of the 200
0
02.04
E'IG.3. VELOCITY
06
06
10
12
14
16
16 20 175
V DIMENSIONLESS TIME AT Z= 200 km.
AND
10 km and height of an order of 100 km with E-20mV M-’ and &-OS G. On Fig. 2 we plot the velocity profile within the aurora1 thermosphere at an altitude of 175 km for a different time. In this figure, we clearly observed velocity profile, the velocity reached 80% of its steady state value at t = T = 60 min (characteristic time), and the induced movements can be extended up to a factor of four of the original width of the electrojet. This agrees well with the early estimation given by Cole (1971) (his equation 30). In Fig. 3 we have plotted the velocity of the upper thermosphere at different altitudes versus characteristic time. It clearly indicates that the velocity has reached 75-90% of its steady state value, which shows that Cole’s steady state solution in these altitudes is a reasonably good approximation. In Fig. 4 we show the thermosphere velocity at mot 180
t = 1333
160 140 lx)
,: : , !
V (m/wc)100 -
100
0
1 DIMENSIO?&S
3
LENGTH
4
5
XI Lo
FIG. 2. VELOCITY PROFILE OF THE ELECTROJET AT DIFFERENT DIMENSIONLESS TIMES CALCULATED ACCORDING TO EQUATION (12) AT Z= 175 km (ALTITUDE), Lo BEING THE HALFWIDTHOFTHEAURORALELE‘TROJET.
120
160
t = 0 6667
j
160
180
200
FIG. 4. AVERAGE VELOCITY vs ALTITUDE (Z) AT DIFFERENTCHARACTERISTICTIMES. values below 160 km are not physicallly valid, because the Hall currents become important in that part of the atmosphere; they are included simply for mathematical completeness. The
S. T. Wu and K. D. COLE
730
ALTITUDE (2)
175km
u 0.3 ;
TOTAL HEATING
w 0.2. s n go.>5 Y
JOULE HEATlNG 0
1
2
DIMENSIONLESS TIME
‘G
FIG. 5. %-HE AMOUNT OF JOULE HEATING AND VISCOUS HEATING OF AN ELECSFZOJET AT 175 km ALTITUDE vS DIMENSIONLESSTIME.
different times versus altitude. Physically this solution is only valid above 160 km altitude, because the Hall currents have been neglected in the present calculation. Finally, we plot the average joule heating, viscous heating and total heating through the disturbed volume of space versus dimensionless time at an altitude of 175 km in Fig. 5. These heating rates per unit volume are calculated in the following way. The volume is that of the disturbed atmosphere. Thus, the initial volume is the volume occupied by the electric field. Later as motion is induced, the volume of moving atmosphere will grow as time progresses until it approaches a steady state. From the results shown in Fig. 5 we notice that the joule
heating dominates in the initial stage and is concentrated in the electric field region. In the initial stage, there is no motion in the atmosphere, therefore the viscous heating is practically zero. Later, motion is set up due to the electric field, then viscous heating becomes important. We also plot the total average heating, which is simply the sum of the joule heating and viscous heating. This shows a minimum heating before approaching an asymptotic state. This is due to the fact that when the electric field is turned on, the joule heating occurs immediately, then as time progresses, the volume increases, thus the joule heating per unit volume decreases. In the meantime, viscous heating is increasing due to the movement of the atmosphere. Finally it reaches a steady value of heating. We conclude that this is the mechanism to sustain the motion in the upper atmosphere. This heating will correspond to a horizontal pressure gradient which has been discussed by Comfort et al. (1976). Ack~w~dge~~~n~ of the authors (STW) wishes to acknowledge the Australian-American Education Foundation which supported his visit’ to La Trobe University and MSFC/NASA support under a contract with OAH. REFERENCES Cole, K. D. (1971) El~odyna~~ heating and movement of the thermosphere. Pfanef. Space Sci. 19,5P-75. Comfort, R. H., Wu, S. T. and Swenson, G. R. (1976) An analysis of aurora1 E region neutral winds based on incoherent scatter radar observation at Chatanika. P&met.Suace Sci. 24, 541. ~d~n~on. f. -I%.[X954) The motion of ionized gas in comgineb magnetic electric and mechanical fieids of force. Mon. not, R. astr. Sot. 114,651-663.