^ecknical <v2ote 162

THERMAL BALANCE IN THE

F REGION OF THE ATMOSPHERE

DOUGLAS C. HUNT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical Ntote

162

SEPTEMBER 1962

THERMAL BALANCE IN THE F REGION

OF THE ATMOSPHERE

Douglas C. Hunt

NBS Boulder Laboratories

Boulder, Colorado

(Central Radio Propagation Laboratory

and the University of Wyoming)

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 50 cents
CONTENTS

PAGE

1. INTRODUCTION 1

1.1 Scope of Present Work 1

1.2 Review of Atmospheric Physics 1

1.3 Equilibrium Types in the Atmosphere 3

a. Mixing 3

b. Chemical 3

c. Diffusion k

• 2. THERMAX PROCESSES IN THE ATMOSPHERE k

2.1 The General Thermal Energy Continuity Equation ... k

2.2 Terms in the Heat Balance Equation 5

a. The Heat Conduction Term 5

b. The Net Heat Generation Term 6

c. The Heat Capacity Term 9

3. DETAILED DISCUSSION OF THERMAL PROCESSES IN THE ASSUMED

MODEL 10

3.1 The Gain Term, G(h,t) 10

3.2 The Conduction Term 22

a. Neutral Gases 22

b. Plasma 2k

c. Turbulent Conduction 25

3.3 The Loss Term, L(h,t) 25

3 A The Heat Capacity Term 26

111
Abstract

An investigation is made of thermal mechanisms and detailed properties of the atmosphere at F-region heights. The method used constructs

a thermal model of the atmospheric F region based on experimental data

combined with theoretical results and arguments. The heat balance equation for this model is simultaneously solved with the barometric equation. The resultant mass density profiles are then compared with observed mass density profiles to determine how much heating by some ex- ternal source is necessary in the F-re'gion.

Two cases are considered, one in which the time rate of change of

temperature is zero (quasi-steady state) and one in which it is not

zero (non-steady state). In the first case a complete solution is obtained. In the second case only a first approximation is carried out.

It is concluded that a heat flux of 1.13 ergs/cm -sec, originating in

solar electromagnetic radiation, can satisfy the thermal requirements

of the atmospheric F-region. The existence of other heat sources cannot

be excluded altogether but their necessity is ruled out. For a 1.13

ergs/cm -sec electromagnetic radiation heat flux, a model of height and

time variation of pertinent atmospheric parameters is constructed. These

parameters are the temperature, molecular weight, scale height, detailed

composition, optical depths, rate of heat generation due to absorption

of solar electromagnetic radiation, rate of radiative heat loss, and the

thermal conductivity. The effect of varying the parameters that determine the importance of the terms in the heat balance equation, and of

varying the boundary conditions is tested. It is concluded that the un- certainties in these quantities cannot seriously affect the results obtained .

11
PAGE

k. SOLUTIONS OF EQUATIONS 27

k.l Terms in Final Form of Heat Balance Equation .... 27

a. The Heat Capacity 27

b. The Gain 27

c. The Loss • 28

d. The Conduction 28

k.2 Boundary Conditions 29

a. The Height Range Considered 29

b. The Height h, and the Degree of

p

Dissociation . 29

c. The Temperature and the Total Mass Density at

130 km 30

d. The Conductive Flux at the Upper Boundary .... 30

^•3 Solutions of the Heat Balance Equation 30

a. Quasi-steady State Solution 30

b. Non-steady State Solution 31

5. RESULTS AND DISCUSSION 32

5.1 Results 32

a. Quasi-steady State 32

b. Non-steady State ^9

5.2 Boundary Conditions and Parameter Variation 60

5.3 Range of Validity of Quasi-steady State Model .... 70

5,h The Non-steady State 70

5.5 The Structure in the Observational Density and Scale

Height Profiles 7^

5.6 Conclusions 75

5.7 Future Studies Indicated . 75

6. BIBLIOGRAPHY, a 76

iv
FIGURES

FIGURE PAGE

1. dT/dt necessary to make C p dT/dt equal to the heat

generation rate G 11

2. Temperature model for daytime and nighttime 12

3. Solar electromagnetic flux outside the earth's atmosphere as a function of wavelength l6

k. Absorption cross sections of versus wave length ... 19

5. Absorption cross sections of N„ versus wave length ... 20

6. Photoionizat ion cross section of versus wave length. . 21

7a. Mass densities versus height for heat flux triplets

summing to 1.10 ergs/cm -sec 33

7b. Mass density versus height for heat flux triplets sump ming to 1.10 ergs/cm -sec 3^-

7c. Mass density versus height for heat flux triplets sumo ming to 1.10 ergs/cm -sec 35

8a. Scale heights versus height for heat flux triplets sump ming to 1.10 ergs/cm -sec 36

8b. Scale height versus height for heat flux triplets sump ming to 1.10 ergs/cm -sec 37

8c. Scale heights versus height for heat flux triplets sumP

ming to 1.10 ergs/cm -sec 38

9. Scale heights versus height for several times (x) . The

p

heat flux sum is 1.13 ergs/cm -sec 39

10. Mass densities versus height for several times (x)»

p The heat flux sum is 1.13 ergs/cm -sec kO

11. Temperature versus height for several times (x)« The

p heat flux sum is 1.13 ergs/cm -sec k2

\
FIGUEE PAGE

12. Mass densities versus time (x) for several heights.

o The heat flux sum is 1.13 ergs/cm -sec h-3

13. Temperature versus time (x) for several heights. The

heat flux sum is 1.13 ergs/cm -sec kk

Ik. Detailed composition versus height for a heat flux sum

of 1.13 ergs/cm -sec and x = 0° ^5

15. Detailed composition versus height for a heat flux sum

of 1.13 ergs/cm -sec and x

= 30° W>

16. Detailed composition versus height for a heat flux sum

p of 1.13 ergs/cm -sec and x = 60° • ^7

17. Detailed composition versus height for a heat flux sum

p of 1.13 ergs/cm -sec and x = 90° • ^-8

18. Optical depths versus height at several times (x) for

58^ A wave length range. The total heat flux is 1.13

p ergs/cm -sec 50

19. Optical depths versus height for 900 A wave length

range at several times (x)» The total heat flux is

o 1.13 ergs/cm -sec 51

20. Optical depths versus height for several times (x) in

the 304 A wave length range. The total heat flux is

p 1.13 ergs/cm -sec 52

21. Rate of heat generation versus height for several times

p

(x)« The total heat flux is 1.25 ergs/cm -sec 53

22. Radiative heat loss rate versus height for several times

p

(x) and a total heat flux of 1.25 ergs/cm -sec 5^

23. Molecular weight and thermal conductivity versus height

p for a total heat flux of 1.25 ergs/cm at x

= 0° .... .55

2k. Non-steady state mass densities versus height for several

p times (x) and a total heat flux of 1.0 ergs/cm -sec. . . 56

VI
FIGURE PAGE

25. Non-steady state temperatures versus height for several

p

tiroes (x)- The total heat flux is 1.00 erg /cm -sec, ... 57

26. Non-steady state mass densities versus time (x) for sev- eral heights and a total heat flux of 1.00 erg /cm -sec . . 58

27. Non-steady state temperatures versus time (x) for several

p heights and a total heat flux of 1.00 erg /cm -sec 59

28. Effect of varying H(l30) by varying T(l30) for a total

o heat flux of 1.10 ergs/cm -sec 6l

29. Effect of varying H(l30) "by varying M(l30) for a total

p heat flux of 1.25 ergs/cm -sec 62

30. Effect of varying "boundary mass density. The total heat

flux is 1.25 ergs/cm -sec and x is 0° 63

31. Effect on p(h) of varying cross sections for a total heat

flux of 1.25 ergs/cm -sec 6k

32. Effect of varying h for a total heat flux of 1.10

ergs/cm -sec at x = 0°. 66

33 • Effect of varying total heat flux 67

3^-. Effect of varying thermal conductivity for a total heat

flux of 1.10 ergs/cm -sec at x

= °° 68

35 • The effect of varying the conductive thermal flux at the

P

upper boundary. The total heat flux is 1.10 ergs/cm -sec

at a x of 0° 69

36. Effect of varying compositions on mass density profiles

p for a total heat flux of 1.10 ergs/cm -sec (x = 0°). . . . 71

37* Effect on composition and total mass density of Blamont,

et . al. boundary conditions for a total heat flux of 1.25

p

ergs/cm -sec and x = 0. Model curve is computed for

boundary conditions selected in paper "J2

Vll

THERMAL BALANCE IN THE F REGION OF THE ATMOSPHERE

Douglas C. Hunt

1. INTRODUCTION

1.1 The Scope of the Present Work

The primary properties of an atmosphere are its density, temperature, pressure, composition, and motion. With the advent of high altitude rockets and globe-circling satellites, these properties are becoming better known up to very great heights in the Earth's atmosphere.

However, there is still considerable uncertainty in these properties at

F-region and upper E-region heights (about 130-500 km). Mass densities

are here known to within a factor of two and from these pressures of

about the same accuracy may be computed. However, the detailed properties, the temperature, composition, and motion are poorly known. This

study makes use of the thermal balance in the F-region to investigate

the atmospheric temperature and density. First, the mechanisms which

contribute to the thermal balance are investigated both on the basis of

experimental data and theoretical arguments. Then a thermal model is

constructed (using photoionization by solar electromagnetic radiation

as the F-region' s primary heating source) which predicts the observed

F-region daytime mass density profiles very closely. Finally, a daytime

model of the detailed atmospheric properties, giving their height and

time variation, is derived.

1.2 Review of Atmospheric Physics

If there is no relative velocity between the motion of the center

of masses of the various constituents of a gas, no net motion of the

total center of mass of the gas, and if thermal diffusion can be neglected (conditions which usually prevail in the atmosphere), then the gas

is in hydrostatic equilibrium. In the atmosphere where the applied

force is primarily that of gravity, the equation of hydrostatic equilibrium may then be used as follows:

dp(h)/dh = -p(h)g(h), (l-l)

where h is the height relative to the surface of the earth, p(h) is the

hydrostatic pressure, g(h) is the acceleration of gravity, and p(h) is

the mass density of the gas. If the above equilibrium conditions do not

hold (which may happen for a short while around sunrise and sunset), the

continuity equation for momentum must be applied.

The mass density of the i atmospheric constituent, p ± (h), is re- lated to the number density of the i

l constituent, n^h), by:

p (h) = n.(h)M.(h)/N

o

. (1-2)
Here M. (h) is the gram molecular weight of the i constituent and N =

6.02 x 10 ^ particles/mole is Avogadro's number. If by analogy then

p(h) is given as follows:

p(h) = n(h)M(h)/N

Q

, (1-3)

where p(h) is the total mass density, n(h) is the total number density,

and M(h) is the mean gram molecular weight, then (l-2) and (l-3) yield:

p(h) = £ Pj. =

][ n

i (h)M

i/N

Q

= n(h)M(h)/N

Q (1-4)

giving

M(h) = £ n^h)]^ n

± (h) (1-5)

where

n(h) B^n± (h). (1-5-1)

In the upper atmosphere, because of the low pressure, the gas molecules behave in their interactions with each other almost exactly as

elastic rigid spheres. Thus the ideal gas law is applicable and is

written:

p(h) = n(h)kT(h). (1-6)

Here T(h) is the kinetic temperature and k = I.38 x 10 ergs/°K is

Boltzmann's constant. Dividing (l-l) through by p(h) and using (l-6)

yields

:

dlnp(h)/dh = -l/H(h). (1-7)

Here H(h) = RT(h)/M(h)g(h) and R = N k = 8.31^ x 10T ergs/ mole^K is the

gas constant. Integrating (l-7) from some low height, h , to h then

gives:
3

p(h) = P (h

o )exp(- f Mr). (1-8)

h ^ o Here H(h) is normally termed the "scale height" and physically is the

logarithmic decrement of the pressure. Then using (l-3) and (l-6) gives

what will be called the "barometric equation."

p(hjT(hjM(h) h

dh

p^ h) = M(h Mh) exp( - J sfhj)- (1_9)

h

o

1.3 Equilibrium Types in the Atmosphere

In order to know the detailed composition of the atmosphere one must know what process is dominant in its equilibrium behavior. The

following are important in the atmosphere.

a. Mixing

If there are strong instabilities or turbulences present, the atmosphere may be forced to maintain the same percentage composition over a height range. Except for some trace constituents, observations of composition indicate the atmosphere to be mixed up to the mesopause (about

85 km). The reasons for the instabilities leading to mixing are not well

understood above the tropopause (10-15 km). It is possible that they may

be caused by disturbances originating in the troposphere and propagated

upward to be dissipated above the tropopause.

b . Chemical

If a constituent is being produced or destroyed by some reaction,

this may be the controlling factor in the constituent ' s height distribution. The distribution of ozone, , in the stratosphere is an example

of this (Wulf and Deming, 1937)* Even though the atmosphere is generally

turbulent in the stratosphere, the chemical reactions involving CL are

rapid enough to produce an layer. Above the mesopause the dissociation of molecular oxygen, , starts to occur and soon becomes so rapid

that turbulence cannot keep up with it. Early theories of dissociation (Penndorf, 19^-9) assumed complete photochemical equilibrium, but

more recent work (Nicolet and Mange, 195*0 has shown that such an equilibrium cannot adequately explain this problem.
c. Diffusion

If the times for establishing turbulent or chemical equilibrium are

long compared with the diffusive time constant (the diffusive time constant is about 12 hours at 120 km, and decreases exponentially above

this height), then each constituent "will. tend to arrange itself according to its molecular weight. The heavier ones will fall and the lighter

ones will rise. Such an equilibrium is termed diffusive. In this case

the law of partial pressures may be applied and (l-l) and (l-6) may be

written for each constituent . They are then combined as before giving

the barometric equation for each constituent

.

p,(hjT(hJ h

dh

p i (h) =

X

T(h) exp[ " J iT[hT]

- (1" 10)

h

o

Here H.(h) = RT(h)/M.g(h).

The exact height above which diffusion predominates is not precisely known. Afternoon rocket observations (Meadows and Townsend, i960)

indicate a diffusive equilibrium between argon, A, and molecular nitrogen, N

p

, to start between 100 and 120 km, indicating that the total atmosphere should be diffusively controlled above these heights. Nicolet

(i960) asserts that, because of the increase of pressure when diffusion

replaces mixing, diffusion may be inhibited until 150 to 160 km. The

height of the start of diffusive equilibrium, h , will probably be a

function of time and will likely be highest during midday. This is because atmospheric chemistry and turbulence are related to the diurnal

variation of heat flux absorption.

2. THERMAL PROCESSES IN THE ATMOSPHERE

2.1 The General Thermal Energy Continuity Equation

The thermal balance in a gas is controlled by what will be called

the thermal energy continuity equation. In the thermal energy continuity

equation the rate at which the thermal energy of a unit volume of a gas

changes is due to the following causes:

a. Particle Divergence

This energy change is due to a net influx or outflow of molecules

.

This net flow may be due to vertical transport (mutual diffusion of one

constituent with respect to another in a several constituent gas ) or convection.
b

.

Energy Divergence

This energy change is due to different energies of the inflowing and

outflowing molecules rather than different numbers. This flow is that

due to ordinary thermal conduction.

c

.

Surface Pressures

This energy change is due to the work done by the surface pressures

on the unit volume of gas.

In the atmospheric F region, only the energy change due to vertical

thermal conduction is assumed to be important. Also, to include the effects of sources and sinks of thermal energy, a net rate of heat generation (per unit volume) term, G(h,t) = G, is included in the thermal en- ergy continuity equation. The thermal energy continuity equation is then

written in the following form, which will be referred to as the heat balance equation:

C p ar/dt = -V . q) + G. (2-1)

Here C is the specific heat of the gas at constant volume and V . q;

is the derivative with rjsspect to the vertical direction of the conductive heat flux vector, q.

The neglect of the other terms is done, of course, to make the final

equation more tractable and because such a neglect is a good approximation. However, any terms that do not add dimensions (time or space) to

the problem may be included if a model of the functions involved is available.

2.2 Terms in the Heat Balance Equation

-* -S

a. The Heat Conduction Term V . q)

z For a binary gas mixture, the heat flux vector, q, will consist of

three parts:

q

z = XBT/Sh + 5/2^(^0^ + n

2

C

2z )

+ knTk#lz -C

2z )
where X is the thermal conductivity. The first term is the ordinary

heat flow due to temperature inequalities in the gas. The second and

third terms occur because of the relative diffusion of the two gases.

The second term is relatively unimportant since it is the flow due to

the fact that the thermal flux is measured relative to the mass velocity

c and not to c (the mean molecular velocity). It may thus he transformed away. It this is done only the factor, knTk_,C , remains where

now C, is measured with respect to c. This is the actual heat flow due

to diffusion. This heat flow is a transient one which will disappear

when mutual diffusion has stopped. Also, calculations of the ordinary

conduction term compared with the diffusion term, based on average Fregion parameters (Champion, Minzner and Pond, 1959 ) Chapman and Cowling, 1952), indicate that a value of C, of about 100 m/sec is necessary

to make the diffusion term comparable in magnitude to the ordinary conduction term. Based on this and the previously postulated transport

model, the thermal flux is taken to be due only to ordinary conduction.

b. The Net Heat Generation Term

This term includes sources and sinks of heat in the F-region. It

will be taken to consist of three terms as follows:

G = G(h,t) - L(h,t) + G (h,t). (2-2)

The three terms in (2-2) will be discussed separately.

(1) The local loss term, L(h,t). This term includes whatever thermal energy is lost to the atmospheric particles due to "in situ" processes

such as radiation from collisionally de-excited particles or particles

excited due to chemical reactions

.

(2) The convective transfer term, G (h,t). This term includes the

net heat gain or loss due to the influx or outflow of molecules which

have been heated or cooled elsewhere. This term has been neglected primarily because there is insufficient information available about it to

include it intelligently. However, the data which do exist indicate

that horizontal pressure gradients are probably insufficient to cause

appreciable convective transfer. Johnson (i960) has suggested that convective transfer may be important in the vicinity of 200 km; this possibility is treated further in the discussion of results.

(3) The local gain term, G(h,t). This term includes all of the

possible "in site" methods of heat generation. Many have been suggested
by various workers at one time or another. Some can empirically or theoretically he argued to he small. A list of these with some argument as to their relative importance follows:

(a) Photonic heating: The possibility of heating the F-region by

solar electromagnetic radiation depends on several things. First, there

must be a certain solar wavelength range in which the probability of interaction with the F-region constituents is high. If there is such a wavelength range there must be enough photons in it to cause appreciable

interaction. lastly, there must be a way to convert efficiently the interaction energy into thermal energy of the molecules and atoms of the

F region. Such a wavelength range is in fact available in the far ultraviolet region of the solar spectrum. In the range from about 170-912

A the interation probability (the absorption cross section ) is large

-1 ry -I O O

(^ 10 -10 cm ). Also rocket measurements ("Violette and Rense, 1959;

Hinteregger, et al., i960) show the photon flux in this range to be on

the order of 5 ergs/cm -sec. A theoretical estimate of the amount of

absorbed energy turned into thermal energy (Hanson and Johnson, I961)

give a "thermal efficiency," e(h,x), for 3oi A of O.I5-O.3O. This would

o mean that from 0.75-1*50 ergs/cm -sec of thermal energy would be made

available to the F-region if all the F-region ionizing flux were in the

30^ A line. However, there is considerable flux in other interacting

wavelength ranges. Thus, since the amount of thermal energy available

per photon is roughly independent of photon energy (Hanson and Johnson,

I961), the amount of available thermal energy should be higher than the

indicated range.

(b

)

Dissipation of hydromagnetic waves: This mechanism as a source

of atmospheric heating was suggested by Dessler (1959 )• According to

Dessler, interaction of the so-called "solar wind" with the Earth's static magnetic field may cause instabilities which generate electromagnetic

waves. The magnetic component of these waves adds to the Earth's static

field giving a time-varying magnetic field which is called a hydromagnetic wave. Ions in the atmosphere tend to follow the hydromagnetic wave

and their extra kinetic energy may then collisionally heat the atmosphere

Dessler estimates the dissipation to peak at about 175 km with a maximum

dissipative value of 5 x 10 ergs/cm -sec. This would imply a total

heat flux of about 0.7 ergs/cm -sec. This must, however, be regarded as

a very rough estimate because the amplitudes of the hydromagnetic waves

are not well established. Dessler also notes that during magnetic storms

the amplitude of the waves may increase by a factor of 100, causing hydromagnetic waves to be a significant heat source. Some evidence for

the existence of hydromagnetic waves exists, (Sugiura, 1961), but it is

still generally thought to be a relatively minor heat source. Its height

profile of heat generation is quite similar to that of photonic heating

and so would be hard to distinguish from photonic heating.
8

(c) Corpuscular heating: This has been suggested (Krassovsky,

i960) as the primary F-region heating source. Krassovsky obtains, from

p Sputnik III data, electron energy fluxes of about 100 ergs/cm -sec at

p 1720-1880 km, and estimates that about 1 erg/cm -sec of this can reach

the F-region. Because of the relatively long mean free path and short

radius of gyration of these particles in the Earth's magnetic field at

these heights (> 1000 km), the chances of an energy transferring collision before the particle is thrown back out into space are quite small.

At the poles, however, the magnetic field cannot as effectively turn

back these particles and hence, the occurrence of auroras. Bright aup

roras may involve energy fluxes of as much as 2000 ergs/cm -sec (Mcllwain,

i960). Some resultant local heating may occur, but the infrequency of

strong auroras and the small fraction of the Earth's surface which they

cover, indicate global effects are quite small. If corpuscular heating

occurs, its profile should resemble that of photonic heating and -would

also be difficult to distinguish from the first two heating sources.

(d) Coronal conduction: This mechanism originally suggested by

Chapman (l95l)> should be closely related to corpuscular heating. It

is the conduction of heat to a cool speck (the Earth) imbedded in the

hot gas of the Sun's corona. Chapman estimates a heat flux of about

p 0.2 ergs/cm -sec may be brought into the atmosphere by this means.

Again, because of the Earth's magnetic field, the conducting particles

should enter the atmosphere mainly in the polar regions and give very

little energy to the temperate latitudes before being thrown back into

space. Also, as pointed out before, such a flux is unlikely to lead to

appreciable global effects. This heating mechanism should be distinguishable from the true "in situ" heat sources.

(e) Meteorites: Johnson (1958), in his paper on atmospheric heating, has estimated from rocket data that the heating rate by micrometeorites would be 0.1°K/day. This would be caused by meteors of average

—12

mass 10 gm and velocity of about 12 km/sec, statistics which Johnson

says fits current meteoric theories best. In general, meteorites are

thought to expend most of their energy in the E- and upper D-regions of

the atmosphere. Kaiser (1955) has estimates that sporadic meteorites

-k 1 3 should produce about 2.3 x 10 electrons/cm -sec in the E-region. If a thermal input of 6 e.v. /ionization is effected, that the total meteoric

_o p heat flux would be about 10 ergs/cm -sec . During meteor showers this

value may rise considerably. Lovell (1950) has estimated that the ionization rate may then be 3 x 10 electrons/cm -sec, leading to a meteoric

heat flux of about 0.05 ergs/cm -sec. Thus it is seen that meteorites

are almost certainly never an important F-region heat source.
(f) Joule heating: Krassovsky (1959) has proposed that atmospheric

heating may "be due to currents induced by magnetic fields imprisoned by

solar corpuscular streams incident upon the Earth. This theory has not

been developed in detail, but as Bates (1959) points out, one unattrac22 tive feature of it is the fact that magnetic field energy of 10 ergs

must be changed into upper atmospheric thermal energy each hour. This

is of the order of the field energy occurring during a magnetic storm.

This then suggests that, as is the case with hydromagnetic heating, such

a mechanism could be significant only during magnetic storms

.

(g) Dissipation of infrasonic waves: Daniels (1952) has proposed

a theory based on acoustical waves generated by the ocean. He estimates

o the average radiated acoustical energy to be 100 ergs/cm -sec. Under

isothermal conditions and if e is unity for this process, he concludes

all this energy would be absorbed between 130 and 155 km and could heat

the F-region by 300°K/hr. The absence of isothermality, the presence

of temperature inversions, and winds in the 0-100 km range render the

transport of such waves rather uncertain. Also, e for this process will

probably be considerably less than unity. These reasons, plus the relatively uncertain method by which the acoustical energy radiation rate was

derived, make this mechanism a doubtful heat source.

To summarize, while there are a number of ways in which the temperate latitude F-region may be heated, the only one which is well established both theoretically and experimentally as a strong thermal energy

source is heating by solar electromagnetic radiation. It will, therefore, be assumed that the F-region is heated entirely by a solar photon

flux.

c . The Heat Capacity Term c pdT/dt

This term adds the extra dimension, time, to the total differential

equation represented by the right hand side of the heat balance equation.

When this term is negligible compared to the terms on the right side of

the heat balance equation, the atmosphere will be said to be in the quasisteady state due to the fact that the G term may be time dependent. It

is appropriate, therefore, to investigate the conditions under which it

is possible to neglect the heat capacity term. Heating by solar electromagnetic radiation, which has been selected as the F-region' s heating

source, has a strong diurnal variation. Thus the general form of the

diurnal temperature variation is the following: in the morning there

will be a temperature rise. The rise will continue until sometime in

the afternoon and then will decline somewhat until sunset at the con- sidered height. A decline in the temperature will then occur, rather

rapidly at first, until a minimum is reached sometime early in the morning. Thus at two times, near sunrise and sometime during the day (pro-
10

bably in the early afternoon), dT/dt = T will "be zero, implying the

quasi- steady state to be valid for at least a short time around these

times and at the height in question. Now to obtain a rough quantitative

• idea of the importance of the heat capacity term, values of T necessary

to make this term equal to G are plotted in figure 1. These TVs are obtained by using the expression, |T| = |[G(h,t) - L(h,t)]|/c p(h,t). The

p(h,t) are taken from a daytime mass density profile (Kallman, 1961).

The daytime density profile gives densities near a time when T« at

all heights so that the values of G(h,t) - L(h,t) used to compute the

•T's are taken from computations made for noon assuming the quasi-steady

state to hold. Since the ratio of G to p(h,t) is roughly constant in

• time, these |t| 's will be representative of the entire day. Approximate

temperature profiles are plotted for noon and 0800 hours, assuming the

quasi-steady state to hold( figure 2). These are thought to approximate

the maximum daytime and minimum nighttime temperature profiles. Using

these curves, and assuming most of the day-to-night temperature changes

• occur in a six hour period, the following table of average |t| 's is computed:

h(km) 150 200 300 400 500

|T|(°K/hr) 26 53 80 87 91

where h is the height above the Earth's surface. These numbers should

now be compared with |t| 's (figure l).

From these considerations it is seen that, on the average, the heat

capacity term during the day will be only about ten percent of G(h,t

)

above approximately 170 km. It is concluded that, except at low Fregion heights, soon after sunrise, and around sunset, the daytime Fregion atmosphere is represented approximately by the quasi-steady state.

3. DETAILED DISCUSSION OF THERMAL PROCESSES

IN THE ASSUMED MODEL

3.1 The Gain Term, G(h,t)

The attenuation of a photon beam may be accomplished through photoionization, photodissociation, resonance or fluorescent scattering,

Compton scattering by free (modified line) or bound (unmodified line)

electrons, or through pair production. Since this absorption is a statistical process the beam attenuation will be proportional both to the

beam intensity and to the incremental path length traversed. Thus for

a monoenergetic beam we may write:
11

1000

X

o

HEIGHT h (km)

Figure l

dT/dt Necessary to Make C pST/dt

Equal to the Heat Generation Rate G
12

2000

HEIGHT h(km)

Figure 2

Temperature Model for Daytime and Nighttime
13

dl(x,h) = - Y

M. ± U,li)l(X,h)as (3-1)

where the negative sign implies absorption. Here l(X,h) is the beam intensity per unit wavelength, |j,.(X,h) is the linear absorption coefficient

tor the i atmospheric constituent, and ds is the path length traversed.

For purposes of atmospheric absorption a plane parallel atmosphere will

first be assumed in which ds = secxdh, where x is the solar zenith angle.

Also, Beer's law will be assumed to hold; it states that the proportionality constant between u,. and the absorber concentration is a constant

independent of temperature and pressure. Thus:

^(X,h) = A

i (x)p

i

(h). (3-2)

Here A.(x) is the absorption cross section for the i atmospheric con- stituent in cm /gm. Equations (3-l) and (3-2) then upon integration

give, using the ds definition:

l(X,h) = I

oo (x)exp[-T(X,h)secX ]. (3-3)

Here T(x,h) = ) | A.(x)p.(h)dh is the optical depth for wavelength X and

. h

l

I (x) is the beam intensity per unit wavelength outside the Earth's atmosphere .

The rate of ionization or the rate of heat generation is given by:

G(x,h) = e(X,h)dl(X,h)/ds = e(X,h)cosxdl(X,h)/dh. (3-M

Here e(X,h) is the quantum efficiency or number of electrons produced

per unit energy increment absorbed (in the case of ionization) or the

amount of thermal energy produced per unit energy increment absorbed

(in the case of heat generation). The latter meaning will always be

assumed in this paper.

The total heat generation rate due to all wavelengths is then given

by:

00 G(h, X ) = J G(X,h)dX s G(h,t). (3-5)
1k

Equations (3-3 )> (3~^-)> and (3-5) then give the total heat generation

rate, G(h,t), as:

00 , G(h,t) = J e(\,h)lj\) £ A

i (x)pi (h)exp[-T(x,h)secX]dX. (3-6)

o

i

The terms in (3-6) will now he considered separately.

N,

a. p.(h)

As mentioned "before, h, probahly lies in the upper E-region. Since

' is very stable against photodissociation, while

p may be dissociated,

it is likely that there is a considerable amount of N

p

and atomic oxygen,

0, present. In diffusive equilibrium the lighter constituents persist to

higher heights so that even though N

p

will predominate at L, will become dominant in the upper F-region. It is fairly certain that at h ,

,

p

will be considerably dissociated (Kupperian, et al., 1958) and so whether photochemical or diffusive equilibrium between and predominates, will be the major constituent. Even a small amount of N

p

dissociation can give rise to a large percentage of N at high heights due

to diffusive separation. However, N will probably never be a predominate species until possible the lower exosphere. At sea level, the at-

-k mosphere is about 5-2 x 10 percent helium, He. Thus about the same fraction of the atmosphere will be He at h, (ra 130 km). Then assuming

a diffusive equilibrium for He above h , it can be shown that He will

be a minor atmospheric constituent throughout the F-region. According

/ \ 9 to Kuiper (1952 ), about 10 atomic hydrogen, H, atoms are released per

2

second per cm column due to photodissociation of water vapor near 80

km. It must also escape at the top of the atmosphere at the same rate.

Also, He escapes there at about 10 /cm -sec (Mitra, 1952). Since H eso capes about 10 times more rapidly than He (due to its lighter mass),

-k its concentration at the exosphere base must be about 10 that of He.

Since He has already been argued to be a minor constituent, H must also

be only a trace constituent in the F-region.

b. I (\) 00 s '

The radiations which may interact with the major constituents as postulated in the preceding paragraph are now examined. The atmosphere
15

is known to be opaque to wavelengths from approximately 0-2900 A and

from about 2k microns to 1 cm. The second wavelength range does not

carry enough energy to be an appreciable energy source and, as will be

seen in the next section, has a small probability of being absorbed in

the F-region. The wavelengths in the 2k micron to 1 cm range are thought

to be emitted from the upper photosphere, and should be approximately

equivalent to the radiation from a 6000°K black body. Since the peak

energy density for a 6000°K distribution occurs at ^-9°0 A and falls off

approximately as \ , the wavelength range starting at 2k microns will

have little energy in it . The wavelengths between 2100-2900 A are almost exclusively absorbed by , so only the spectrum between and 2100

A is of interest. Farther into the ultraviolet the equivalent black

bocy temperature of the sun becomes less, so that to agree with spectrographs (Johnson, et al., 1951 ) and photoelectric measurements (Friedman,

1955 ) a temperature of about k^00°Yi is taken for the range of wavelengths

from 2100-13^4-0 A. From 1350-10^0 A occur' a number of emission lines and

particularly H Lyman a at 1216 A. H Lyman a is thought to account for

about 95 percent of the intensity in this wavelength range. Estimates

of the incident intensity in this range have been made from rocket observations by several people and have been tabulated (Friedman, 1960a).

In the range from HA-0-60 A the fluxes have been measured at 210 and 255

km (Hinteregger, i960) using photon counters. An extrapolation of these

measurements to outside the atmosphere has been done (Denison, I961) and

these are taken to be representative of the solar flux in this range.

Friedman (1960b) has tabulated the experimental measurements of hard x- rays in the regions 0-20 A and kk-60 A. These will be taken as representative of the 0-60 A range and the amount of energy in 20-M+- A range is

obtained by interpolation. (All this information is shown in figure 3«)

A smooth curve is drawn and the important emission lines, with an estimate as to their contained energy, is indicated. This will be taken as the rough spectrum of I (x) to be used for qualitative arguments and approximate quantitative comparisons . The graph refers mainly to sunspot

maximum, since most of the measurements were taken then.

c. A.(X)

1

The point at 10A in figure 3 represents the total x-ray flux from

0-20 A. Therefore, pair production as an attenuating mechanism may evidently be neglected. The question of the loss of protons by Compton

scattering will now be considered. As previously noted, an appreciable

photon energy flux only exists for wavelengths above about 30 A. With

regard to the Compton effect this is a classical energy. The well known

Klein-Nishina formula for the Compton scattering cross section per electron gives high cross sections for these energies (from about 10 "" -10

p

cm ). However, for such relatively low photon energies, the photon is

hardly degraded at all in a scattering process. Also, the scattering
16

10 i i i i

1

1

i i r i i i i i i | HLY.a \

<I o

Eo

v>

UJ

HEH rn

LY

iTx iOI

I .„50NT-\ r\HLY)9

fern

\HLY.r j10 10' WAVELENGTH X(A)

Figure 3

Solar Electromagnetic Flux Outside the Earth's Atmosphere
IT

angle is essentially 0° in such low energy interactions. Thus, most of

the photons in question will suffer negligible Compton scattering and

this may be neglected as an attenuation mechanism. The remaining important attenuation processes are the following:

1. Ionization k. Resonance absorption

2. Dissociation 5* Preionization

3. Dissociative ionization 6. Predissociation

The absorption due to the three important F-region constituents

will be discussed separately* In the case of N and , experimental

measurements are possible, whereas theoretical estimates are virtually

impossible. Thus experimental results will be exclusively used. In

the case of the situation is reversed and so theory must be used.

(1) A (\): Friedman (1960c) summarizes the recent work which has

Op t been done on the absorption cross section of the atmospheric constituents,

For 0„, starting at 2026 A, there are absorption bands with a small cross

-23 2 section of about 10 cm . At 1759 A absorption starts in the dissocia-

-19 2 tion continuum with a cross section of about 5 x 10 cm . The dissociation continues until the first ionization potential of at 1027 A

where it becomes small compared to ionization. The absorption cross

-17 2 section reaches a broad, flat peak of about 1.6 x 10 cm at 1^-50 A.

After this, absorption in bands is observed up to the ionization limit.

It is especially to be noted that H Lyman a has a low cross section of

-21 2 about 8.5 x 10 cm . From 1027-8^0 A an absorption continuum of super-

_-i o imposed bands is observed with a maximum cross section of about k x 10

2

cm . From 850-7^-0 A, the continuous absorption has resonance bands su- perimposed with the absorption cross section reaching a maximum of 2.2 x

-17 2

10 cm at 510 A. Most of the band absorption cross sections do not

-17 2

exceed k x 10 cm . Below 200 A the cross section is taken to be twice

that of because of the lack of experimental data in this region and

because this is a good approximation for high energy photons. A rough

graph of a„ (\) = An (x)M /n is shown in figure k.

Q O P

(2) \t '• Above 1220 A, N is virtually transparent to electromagnetic radiation. Between 1200-796 A band absorption is present. Between

the bands from 10^-0-910 A the absorption cross section is less than k x

-20 2

10 cm , which may be taken as roughly representative of the cross sec- tion from I2OO-796 A. At 79& A and 66l A there exist two ionization limits for N . This accounts for the absorption continuum below 800 A.
18

Again, below 200 A the absorption cross section is taken as twice that

for photoionizat ion of N. The absorption cross section for N

p

below

800 A is shown in figure 5 where a = A^. M_ /u .

(3) An

(\): As mentioned previously, only theoretical estimates

are available as to the absorption cross sections for 0. These are for

absorption by photoionization and take no account of possible preionization. Since preionization is thought to occur in

p

from 950-850 A it

would seem possible that some might occur in near this wavelength

range.

The values of o~„ = AJYL/n are taken from the work of Bates and

0' o Seaton (19^9) above k60 A, Dalgarno and Parkinson (i960) from k60-22.5

A, and Victoreen (19^-9) below 22.5 A. The photoionization cross section is shown plotted in figure 6. The various ionization limits appear as discontinuites on the cross section curve.

These cross section graphs are approximate and do not include all

of the details of the cross sections in the wavelength range presented.

A factor of error of about ± 1.5 to ± 2.0 is attached to all numbers

quoted

.

(k) e(\,h): The thermal efficiency will vary with height through

the different relative compositions of both the neutral and ionized

parts of the atmosphere at different heights. It may vary with X through

the different amounts of thermal energy which may be transferred to the

neutral gas by photons of different energies. The variation of e is a

radiative transfer problem of some complexity and is avoided in the work

of this paper by considering only the heat flux, i.e., the product

I (X)e(X,h). 00 The value of e may be roughly estimated by considering that the main

result of photonic interaction with the atmospheric F~region is photoionization and inquiring how these fast photoelectrons lose their energy.

This has been done by Hanson and Johnson (1961) and the following discussion will be based on their work.

The energetic photoelectrons have several interactions. They may

collide elastically with neutral particles, electrons, or positive ions,

thus sharing their kinetic energy. They may also make inelastic collisions with neutral particles or positive ions leading to dissociation,

ionization, or excitation. If the excitation energy is lost by radiation which is not locally absorbed, some of the prospective thermal en- ergy is lost. Also, if the dissociated or ionized particles do not re- combine locally, this internal energy is lost. The excitation energy

may also be lost by collisional deactivation, in which case it may still
WAVELENGTH X(A)

Figure k

Absorption Cross Section of Versus Wavelength
20

10"

i i ii | i i i i i i i

1 1 r

HE L

~ 10

CME

o

OUJ

if)

if)

if) O

CCO

-IT

10"

10

-19 I I I I I I I I I I I I I

HE US

10 I0

2 WAVELENGTH X(A)

1

I0

;

Figure 5

Absorption Cross Section of N

p

Versus Wavelength
21

10

-16

E 10 o

O

UJCO

CO

COo

cro

10"

10

-19

1 I I I I I II |

I I I I I I I I |1 1 1

1—I I I _

10 10 WAVELENGTH X(A)

HEI- HEMI I I I I I I I I / I I I / I I I I I I I I I I I I I I I

I0

J

Figure 6

Photoionization Cross Section of Versus Wavelength
22

be available for atmospheric heating. Dissociative recombination may

also be a heat source. Hanson and Johnson show that for most dissociative recombinations the maximum amount of fragmentation energy available

is about 6 ev/dissociation for N? and and about 2.7 ev/dissociation

for NO (fragmentation energy is the kinetic energy of the two formed

atoms). All the rest of the energy will be lost as radiation or imrecovered internal energy. Hanson and Johnson favor this heating mechanism and thus for 30^ A photons get an e of about 0.15 • On the basis of

this argument, about the same energy per photon would be available for

any photon energetic enough to photoionize. Thus for purposes of argument it will be assuned that e(h,\) = e(h ,\) = 6/[lic/x](ev), where h

is the height of maximum heat gain, ti is Planck's constant, and c is

the velocity of light. This then makes the heating effect proportional

to the photon flux and not the energy flux.

3.2 The Conduction Term

2 2

This term may be written l/r d/dr[r \(T)dT/dr] where X(t) is the

thermal conductivity and r is the distance from the Earth's center. If

a plane geometry were considered the term would be d/dh[X(T)dT/dh]

.

The value of X(T) will be taken to consist of two parts, the first

due to kinetic energy transfer in short range interactions (neutral gas

conductivity) and the second due to kinetic energy transfer in long range

Coulombic interactions (plasma conductivity). Since these conductivities

will be seen to be very weak functions of the neutral particle and the

plasma densities, they will be assumed to be independent of each other

and their effects additive. As with cross section calculations, the theoretical estimates are best for atomic species where few empirical values

are available. Conversely, experimental values for simple molecular

gases are available while theoretical estimates are not as reliable.

The conductivity for neutral and ionized gases will be considered separately.

a. Neutral Gases

(l) Simple gases: Using a method of successive approximations,

Chapman and Cowling ( 1952b) solved the Boltzmann equation for the velocity distribution function. The first approximation is taken to be the

Maxwellian distribution. Then, using the definition of thermal flux,

the second approximation to X(t) is calculated in terms of the particle

interaction. In the case of the spherically symmetric, rigid, elastic

particles the second approximation is given as:
23

6kcL (nm) '

Here m is the particle mass and d is the particle diameter. Chapman

and Cowling ( 1952c) shew that on this model the second approximation is

almost exact

.

Keyes (1952) has measured the thermal conductivity of several gases

in the temperature range 73-1273°K. All of the atoms, and the molecules

l/2 without too much structure, follow the T ' law very closely. The conductivity for N and as given by Keyes are used here and are (in ergs/

cm-sec- Kj:

X (T) = 191 T

1' 2

± kff> (3-12-1)

2

1/2

X (T) = 20^ T ' ± k$>. (3-12-2)

2

The two atoms measured hy Keyes are He and A. Their conductivities (in

ergs/cm-sec-°K) are:

x^Or) = 530 t

1/2

± 2% (3-12-3)

XA

(T) = 75.5 T

1/2

± 6$. (3-12-4)

If an interpolation "between the conductivities for these two atoms is

2 — T_/? made, assuming that the (dm) ' dependence shown in (3-ll) is true and

using values of d given by Brokaw (1961), a value of 3^1 is found for the

coefficient of X (t) for atomic oxygen. Thus the conductivity of is

taken:

X (T) = 361 T

1/2

± 10$. (3-12-5)

(2) Mixed gases: The problem of computing the conductivity in

this case is complicated. Approximate formulas for the conductivities

of mixtures are given, for example, by Chapman and Cowling (l952d) and

by Brokaw (1961). Chapman and Cowling give the formula for the first

approximation to the conductivity of a binary mixture. Then with the

parameters appropriate to a gas composed of rigid, elastic, spherically

symmetric molecules, the conductivity assumes the form:
2k

(n + k n )(n + k n ) / K o(T ) = 1 + v \( + v \ ^T1/2

. (3-13)

Here k.(i = 1 - 5) are rather complex functions of the molecular weights

and conductivity coefficients of the gases 1 and 2 involved. The magnitudes of the k. are such that formula (3-13) niay he approximately written:

kn-, + k'n , /p V(T) ° n[ + n

g

T

7 • C3-H-X)

Here k and k' are new constants which may he identified with the simple

gas conductivities. This is equivalent to weighting the simple gas conductivities hy their number densities. Keyes (1952) also gives the conductivity of a mixture of N„ and CCL as a function of mole percent N at

l/2 temperatures 323-623°K. Although C0

p

does not follow a T ' law too

closely, it is still seen that a weighting of the simple gas conductivities by the number densities gives a good approximation to the mixture

conductivity. At 520°K, for example , a mixture of fifty percent N

p

and

fifty percent C0

p gives a conductivity exactly halfway between the conductivities of pure N

p

and C0

p

, just as (3-13-1) predicts. Thus the

defining equation for the mixture conductivity is taken to be:

\ (T) =7 n.X.(T)/ Y n. . (3-1*0

nr Z_j 1 i

v " Z_j 1

N

b. Plasma

In general, the ability of a gas to conduct heat is directly proportional to the value of the mean free path of a typical particle in

the gas. To see this, consider a plane in the gas across which thermal

energy is heing transported by the molecules on one side of the plane

to the molecules on the other side. The longer the mean free path of

the molecules on one side of the plane, the greater the chance that these

molecules will be ahle to cross this plane and deliver their kinetic energy to the molecules on the other side. A general kinetic theory result is that the mean free path of the particle in a gas is inversely

proportional to the interaction cross section. Thus the thermal conductivity is inversely proportional to the interaction cross section. In

neutral gases the interaction cross section is relatively small and of

the order of the physical cross sectional area of the particles. In

ionized gases however, where long range Coulomb forces are present, the

interaction cross section will be quite large. Therefore, the thermal
25

conductivity of a plasma is "usually quite small. Chapman and Cowling

(l952e) give the classical (at higher temperatures quantum statistics

are unnecessary) expression for the thermal conductivity of a simple gas

of particles having a Coulomb interaction. Here X(t) is proportional to

5/2 l/2 / k

T ' m ' /q log10n, where q is the particle charge. Calculations for a 1000°K electron gas show the conductivity to be about 1 erg/cm-sec-°K,

compared with a pure neutral particle conductivity of about 10 ergs/

cm-sec-°K. Only if the electron temperature is about lO,000°K will the

conductivity be comparable to that of the neutral gas . Hanson and Johnson (1961) estimate an electron temperature maximum of about l600°K in

the F-region, but point out their estimate may be low because of an overestimate of the excitation cross section of 0. A I0,000°K temperature

implies an average electron energy of about 6.5 e.v. Then considering

the thermalizing effect of a 1000°K electron gas on these fast electrons,

even in the vicinity of 200 km where the fast electron production is

greatest and the 1000°K electron gas density quite small, shows it will

take only about 0.1 sec. for thermalization.

The conduction of a heavy ion gas will be a factor of about ^0

greater than that of an electron gas at the same temperature, but the

ions will undoubtedly have almost the same temperature as the neutral

gas; thus their contribution can be neglected. On the basis of the previous arguments (and that at temperate latitudes, the Earth's magnetic

field impedes vertical conduction) the ionized particle conductivity is

neglected.

c. Turbulent Conduction

If turbulence is present in the atmosphere, the thermal conduction

will be much greater because of the increased chance for energy transferring collisions between the molecules. As previously mentioned, diffusive equilibrium is quite likely to prevail above about 120-130 km, rendering doubtful any pronounced turbulence (which would lead to mixing

)

above these heights. Also, Johnson (1958) has argued against the existence of turbulence which was postulated by Booker (1956) as being present as high as 170 km. This turbulent level is suspected since the

radio star scintillation level is a function of the inhomogeneities in

the electron-ion gas, and these inhomogeneities were taken to be too

large (according to Johnson) at 170 km by Booker. Thus it will be assumed

here that above 13 km turbulent conduction may be neglected.

3.3 The Loss Term, L(h,t)

Bates (l95l) has discussed the radiative cooling of the atmosphere

in some detail on the basis of transition probabilities and constituent
26

abundance . He concluded that the only significant radiative loss should

3 3 be from the P_ - P

p

transition in the atomic oxygen ground state. The

energy difference is 0.020 e.v. and therefore is of the order of F-region

kinetic temperatures. Thus the upper level may be populated and depopulated by thermal energy collisions giving rj.se to net radiative loss (if

the photons are not reabsorbed). Bates assumes and shows that the distribution of atoms between the two levels is probably characteristic of

the gas temperature, T, i.e., that thermal excitation of the upper level

is much faster than photon emission. Since the radiative temperature

of this transition is considerably less than the F-region kinetic temperature, T, it can be assumed that all the emitted photons escape the

F-region. On this basis, Bates gives the following expression for this

radiative loss:

L(h,t) = I.67 x 10"18

n (h)exp(-228/T)/[l + 0.6 exp(-228/T)

+ 0.2 exp(325.3/T)]ergs/cm -sec. (3-15)

This is assumed to be the entire loss term, L(h,t).

3.^- The Heat Capacity Term

In the solution of the equation with this term included (the nonsteady state ) the density and the specific heat change with both height

and time. It is to be noted, however, that p and c have no explicit

time dependence, but only an implicit dependence through the temperature.

This implicit time dependence is treated in the solution of the nonsteady state equation by assuming an equilibrium is attained around each

separate time the equation is solved. The specific heat of a mixture of

perfect gases is given, by analogy with (l-3) to (l-5) as the sum of the

specific heats of the individual gases weighted by the mass densities

of the individual gases. Thus c is given:

= Z

pi

cv/Iv ( 3 c -l6) V

1

Here c^ is the specific heat of the ith constituent. Thus the heat

i

balance equation (2-10-2) is now written:
27

( I

P i

Cv/ I P i

) ' P(^t)BT(h,t)/dt

1

1 l

= , X

o 5/Bh {(a + h)

2 V n X / V n [M?(h,t)/ah]}

(a + h) . .

+ J

e(\,h)lco (x)^A (X)p (h)e3q>[-T(\,h)secX]cLX

ionizing

wavelengths

-I.67 x 10~18

n (h)exp(-228/T)/[l + 0.6 exp(-228/T)

+ 0.2 exp(-325.3/T)]. (3-17)

where a is the average radius of the earth.

k. SOLUTIONS OF EQUATIONS

k.l Terms in Final Form of Heat Balance Equation

In order to attempt a solution of (3-17)? several further approximations must he made in accordance with the discussion of Chapter 3«

These assumptions will he discussed for each term in equation (3-17)*

a. Heat Capacity Term

As previously indicated, this term is quite likely negligible (^

ten percent of gain term) during most of the day and at most heights.

Therefore the solution of (3. 17) is attempted first for the quasi-steady

state and second for the non-steady state.

b . The Gain Term

In order to integrate this term over X, the integration is replaced

by a summation. Wavelength ranges are selected and step functions are

sbustituted for A.(x) and e(X,h)l (x). These ranges are selected according to the graphs of I (X) and A. (X) shown in the previous chapter. For

the non-steady state two wavelength ranges are selected, centered on the

two He lines at 30l+ A and 'pQk A. For the quasi-steady state an additional

range centered in the H Lyman continuum around 900 A is used. The approximate ranges of the intervals are:

1. He II range: 0-475 A ( 200-1+75 A)

2. He I range: 1+75-720 A (1+75-720 A)

3. H Lyman continuum range: 72O-I76O A (72O-9II A).
28

The numbers in parentheses are the actual approximate ranges considered to contribute to F-region heating. The flux quantity considered

is always the product, el (x) (the heat flux), now considered only as a function of wavelength.

The atmospheric composition for the non-steady state is taken to be

and KL in a diffusive equilibrium above the ' selected h, . For the quasisteady state is also taken as a constituent and is assumed to be in

diffusive equilibrium above the selected h and to have a fixed degree

of dissociation at h n

. No dissociation of HL is considered in accordance

d d

with the arguments of Chapter 3> first section. For large zenith angles

(x)> the path length traveled by a solar electromagnetic wave is not simply given by dh secx (as in a plane atmosphere), but by dh Ch(x,R) where

Ch(x,R) is called the Chapman function and R = (a + h)/H(h). Here "a"

again is the Earth's radius. This function takes account of the curvature of the Earth's atmosphere. Up to x

= 90° j Ch(x,R) ^ s a "weak function of height and so may be substituted directly into the gain term.

Thus the gain term may be written:

G(h,t) = Y

eIJj ^ A. iPi (h,t)exp[-T^(h,t)Ch(x,R)]. (4-l)

J i

Here el ) . is the total heat flux in the j wavelength range, A., is

°° ^ th th ^ 1

the absorption cross section for the i constituent in the j waveV

h

m

length range (taken from figures k to 6), and t. = ) J A..p.dh where

h is some high F-region height. . h

m i

c. The Loss Term

As seen in (3-15) the loss term is proportional to n and a function

of T. Computation shows that the T dependent coefficient varies from

about 7.1 to 8.6 from temperatures of 4oO-1500°K. Thus an average value

of 8.0 is taken and the loss term is then:

L(h,t) = 9.57 x 10~19n (h,t)(ergs/cm3 -sec). (k-2)

d. The Conduction Term

In the integration of the heat balance equation the planar approximation is adopted in the sense that integrals of the form (a + h) c

j (a + h)
29

2

°° (a + h) f(a + h)dh are taken to "be equal to f(a + h)dh. Then the

(a + h)

conduction term maintains its form as in (3-17)« The form of the heat

balance equation which is solved is

( Y

p±

cv y Y p±) p(ii,t)aT(h,t)/at

1

1 1

=

p V3h[(a + hf( V n X ./ V n . )OT(h,t)/dh]

(a + hf 11.

+ I

eI»5j ZA

j i

P i(

ll ^)exp[-T

j

(h,t)Ch(x,R)]

3 I

- 9-57 x 10" 19

n (h,t). (k-3)

k.2 Boundary Conditions

The boundary conditions will be considered separately and the effect of the variation of each will be shown in 5»

a. The Height Range Considered

The height range is taken as 130-500 km. This includes most of the

F-region. The upper boundary is taken such that the atmosphere is approximately isothermal above it . The lower boundary must be taken low

enough to insure the absorption of most of the incident radiation and

high enough to avoid complications due to E-layer absorption and turbulence. One-hundred thiry km is considered a good compromise.

b. The Height, h , and the Degree of Dissociation

The detailed composition is considerably affected and the total

density somewhat affected by the choice of this parameter. Since it is

thought to lie between 110 and 150 km, the choice of 130 km has been

made. The degree of dissociation is poorly known as a function of

height. The mass density and the total mass density are known moderately well from 100-200 km, but the degree of dissociation still depends on h . For the h of 130 km chosen, a dissociation degree of 80

percent has been taken as yielding the best fit to the observed data.
30

c . The Temperature and the Mass Densities at 130 km

Height amid time density profiles are observationally known in the

F-region from satellite and rocket measurements. From these, scale

heights may be derived knowing only a boundary pressure at some high

height. This is done by integration of the hydrostatic equation from

some high height to the (lower) height in question and use of the relation that H(h) = p(h)/p/(h)g(h). The boundary pressure will affect the

situation at low heights very little. Thus a scale height of 15 km is

taken from Kallmann's (i960) data at 130 km. This fixes the temperature

since the choice of h, and the dissociation degree of CL fixes the mole- d 2

cular weight. Thus a temperature of ^50°K is taken at 130 km for the

quasi-steady state and ^20°K at 130 km for the non-steady state. Recent

experimental work (Blamont, et al., I961), using the twilight flash from

rocket injected sodium, has measured the temperature at 130 km to be

about 350°K. This would suggest a lowering of h, and an increase in the

dissociation to maintain H(l30) at its observed value. Boundary conditions consistent with the temperatures of Blamont, et al., are discussed

in 5* Most of the density measurements have been made during sunspot mini

mum, during the winter, or at high latitudes. Thus the value of p(l30)

is taken to be 10 gm/cm , which is near the upper limit of the measure-

—12 ^ ments. This implies boundary densities of p (130) = I.56 x 10 gm/cm ,

p (130) = 7.73 x 10~13gm/cm3

, and p (130) = 7.67 x 10"12

gm/cm3 for

°2 2 -12 "3 the quasi-steady state and p (130) = 8A x 10 gm/cm and p = 1.6k

x 10 gm/cm for the non-steady state case.

d. The Conductive Flux at the Upper Boundary

This is the so called coronal conduction term and includes any thermal energy conducted into the atmosphere from above 500 km. Since the

conductivity at these heights is large, a small temperature gradient of

about l°K/km, which is suggested by scale height data, will give a conductive flux of about 1 erg/cm -sec. It is felt, however, that scale

height data at high heights is unreliable and an isothermal exosphere is

more likely. Thus the coronal conduction flux Fm

( ergs/sec ), is taken to

be zero. m

h.3 Solutions of the Heat Balance Equation

a. Quasi- steady State

In this case, the problem is a non- linear, second order total differential equation. If the p. and T. are known as analytic functions of

1 J
31

T, an analytic solution of (4-3) might be effected. In particular, if

a single constituent with constant scale height is considered, the equation may be solved in terms of exponential integrals. However, in general, numerical methods must be used because of the non-linearity of the

equation. The formal solution is:

[T(h)/T(h

o

)]3/2 . 1 -3 /V^

| £ elJi[1 . exp( .ch[x)R)Ti{h3)]

{ o

J o i

- J

m

9-57 x 10"19

n (h)dh + F /(a + h)

2 jdh (k-k)

h

th

Here X(h) = ) n.X./ / n - > ^« is ^Qe conductivity coefficient for the i

i i

constituent, and h and h are the upper and lower boundary heights.

The method of solution is to assume a temperature distribution, com- pute the p. 's using (1-10), and then compute a new temperature distribution using (h—k). The process is then iterated until the successive temperature and density distributions are consistent. In practice, about two

iterations are necessary to achieve convergence.

b. Non-steady State Solution

When the heat capacity term is included, the equation to be solved

is a non-linear, second order, partial differential equation in h and t.

For an analytical solution of such an equation, three boundary conditions

on the dependent variable are necessary. For example, T(h ,t), T(h ,t),

and T(h,t ) (where t is some initial time) may be specified. The standard method of analytical solution of conduction problems is according to

a transform theory, usually using Laplace transforms. This reduces the

partial differential equation to a total differential equation which is

then solved for the transform of the required function. Then if an inverse transform is tabulated or can be calculated, T(h,t) is obtained.

The non-steady state problem cannot, in general, be solved in closed form

except for situations which are not physically attainable. A list of con- duction geometries and boundary condition sets which may be solved is

available (Carslaw and Jaeger, 1959)' If; f°r example, we consider a two

dimensionally infinite atmosphere with all of the suggested boundary conl/2

ditions constant in height and time and with c p(z,t)/At ' being constant,

then the solution of the "nighttime" problem, with only the heat capacity

term and conductive loss, may be written:
32

[

T(h,t) = T(h,t

Q

) + lT(iio; t) - T(h,t

Q

)]

o (ho - h)/(2c

v

p/T

1//2A)l/2

t

1 - 2/n1/

2

j

exp(-t

2

)dtl. (k-5)

The method of solution attempted here is to use an iterative method

h

"based on the quasi-steady state calculations. The term J [-c (h)p(h,t)

h

V

5T(h,t )/dt}dh is added to the integrand in {h-k). Using values or this

term calculated from the quasi-steady state densities and temperatures,

new densities and temperatures iteratively consistent in height are computed. The process is then iterated to obtain time as well as height

convergence . An alternative method would he' to integrate once over time

and use the conduction term as the correction term.

5. RESULTS AM) DISCUSSION

5*1 Results

a. Quasi-steady State

The first and one of the most important results is to determine the

heat fluxes necessary to give a daytime mass density profile which agrees

with observation. One of the most recent daytime mass density profiles

is that of Kallmann (1961); she has made a careful compilation of moat

of the current observational data. Thus Kallmann's mass density profiles

(both for daytime and nighttime) are used and are shown by x's on figures 7a > T° > Tc an(i 10. Also, Kallmann's scale height profiles, derived

from her density profiles, are taken as the observational scale heights

and are shown as x's in figures 8a, 8b, 8c and 9-

In figures 'Ja, Jb, 7G and 8a, 8b, 8c are shown mass density versus

height and scale height versus height curves. There curves are drawn

o assuming a heat flux sum of 1.1 ergs/cm -sec (Hunt and VanZandt, I961)

and a zenith angle, x> °f zero degrees. This x i- s chosen in order to

get maximum densities that may then be compared to the observational daytime densities which are maximum daytime densities. Each curve refers

to different relative amounts of heat flux in the three wavelength

ranges. The purpose of these curves is to: (l) see how sensitive the

mass density and scale height curves are to changes in the relative

amounts of heat fluxes in the three wavelength ranges and (2) to determine which heat flux set gives mass density and scale height curves of

similar shape to the observational curves.

p

It is seen that a heat flux of 1.1 ergs/cm -sec gives densities

somewhat lower than the observational curves. This is because of Hunt
33

10-ii

10" 12

e

o

£

CL 10" l3

in

UJQ

10"

10 -15

100

X= OBSERVATIONAL VALUES _

eiooij ( ergs/cm -sec)

200 300

HEIGHT h(km)

400 500

Figure 7a Mass Densities Versus Height for Heat Flux

Triplets Summing to 1.10 ergs/cm- sec
3U

10"

10 -12

roE

E

o>

a. |0— ,3 >-

enz

UJQ

10 -14

IO-'5

100

X = OBSERVATIONAL VALUES _

eiooJj ( ergs /cm*- sec)

X. /O.I5, 0.80, 0.15

x y 0.20, 0.45, 0.45

0.45, 0.20, 0.45 -

200 300

HEIGHT h(km)

400 500

Figure Jb

Mass Densities Versus Height for Heat Flux

o Triplets Summing to 1.10 ergs/cm -sec
35

10 -ii

io-' 2

E

E

O. |0 -I3

CO

z.

Ixl Q

10 -14

IO-'5 100

X = OBSERVATIONAL VALUES

eicoh ( ergs/cm -sec )

200 300

HEIGHT h(km)

400 500

Figure 7C

Mass Densities Versus Height for Heat Flux

Triplets Summing to 1.10 ergs/cm -sec
36

100

80

-£ 60

\-

x

o

LUX

LU

_l<

oto 40

20

X = OBSERVATIONAL VALUES

eI oo'i (ergs /cm'1 - sec)

0.15, 0.80, 0.15

0.20, 0.45, 0.45

0.45, 0.20, 0.45

100 200 300 400

HEIGHT h(km)

Figure 8a

Scale Heights Versus Height for Heat Flux

o Triplets Summing to 1.10 ergs/cm -sec

500
37

100

80

E

60

I

CD

LU<O

CO 40

20

X= OBSERVATIONAL VALUES

>0 €Ioo)j ( ergs/cm -sec)

0.20, 0.30, 0.60

0.35, 0.35, 0.40

0.80, 0.15, 0.15

100 200 300 400

HEIGHT h (km)

Figure 8b

Scale Heights Versus Height for Heat Flux

P

Triplets Summing to 1.10 ergs/cm -sec

500
38

100

80 —

£ 60

X

o

LdX

UJ

_l<

o

CO 40 —

20 —

1 1 1 X•/

/

x

X= OBSERVATIONAL VALUES // /

S

DAYTIME s^X

/ . / ^^^

// ^r/~^^

X ^^ ^r

/ ^r ^y^ ^

/ ^r ^y ^y

/ /y/y^

/ yT >T yf

/ ^r S^ ^y^

/ y^ yS ylT

/ ^r ^y ^^

P< yT yT S^

S ^r ^r ^r ~ ///\S ~ / y/ /yT

/ /// €I coM ergs/cm2-sec)

1 X / S / S\ N 0.45, 0.45, 0.20

/ * s /y y \ / ^ /// X 0.20, 0.30, 0.60

/ //s \^

/ //y ^ 0.15, 0.15, 0.80

/ ///

x

/ 0/ ///

1 ///

1 ///

1 Jr

1 /

1 /

1 1

1 1

1 1 ~ ! ~

1

1

X'

1 1 1

100 200 300

HEIGHT h (km)

400 500

Figure 8c

Scale Heights Versus Height for Heat Flux

o Triplets Summing to 1.10 ergs/cm -sec
39

100

80 —

e

60

X

X

o

x

LlJ <

O

40

20

100 200 300 HEIGHT h (km)

400 500

Figure 9

Scale Heights Versus Height for Several Times (x).

p

The Heat Flux Sum is 1.13 ergs/cm -sec
40

ioI0"'2

10 — «3

£

£

>

COz

UJQ

| -I4

I0" 15

X = OBSERVATIONAL VALUES

300 HEIGHT h(km)

Figure 10

Mass Densities Versus Height for Several Times (x).

o The Heat Flux Sum is 1.13 ergs/cm -sec
in

and VanZandt's slightly different boundary conditions; it is also due

to their assumption of a pure oxygen atmosphere which increases thermal

conduction. It is also seen that the dependence of eI on X which is

00 chosen is not very great. Thus a heat flux spectrum is chosen using:

(l) figure 3 "to get the relative values of I in the three wave length

ranges, (2) the expression for e(\) that e(x) = 6/(hc/x )[e.v.

]

; and (3)

a total heat flux sum of 1.13 ergs/cm -sec. This heat flux sum seems to fit the observed daytime densities best when an el -versus-X profile

consistent with (l) and (2) is chosen, as is seen in figure 10. It is

to be noted that there is of the order of ± 25 percent error to be associated with the observational p(h) and H(h) curves. As can be estimated

by comparing figures "Ja, Jb, and 7C with figure 10, this would lead to

about a ± 10 percent uncertainty in the estimate of the heat flux sum.

o In figures 9 "to 23 a heat flux sum of 1.13 ergs/cm -sec is used. The

o individual fluxes chosen are el )„„, = 0.21 ergs/cm -sec, el ) r. 0) = 0.6l

2

co '301+ 2

°° 5o4

ergs/cm -sec, and el^j - 0.31 ergs/cm -sec. This choice implies I )qro, = 1.5^ ergs/cm -sec, I^)^ =2.75 ergs/cm -sec, and I^Lqq = °«89

ergs/cm -sec for a total F-region ionizing flux of 5 .18 ergs/cm -sec.

Figure 9 shows scale heights -versus -height for several values of

X« Here it is seen that relatively good agreement is obtained between

the observational scale heights and the x

= 0° theoretical profile.

Above about 350 km, the observational scale heights have a greater slope

than the theoretical ones (indicating a considerable M(h) gradient), but

this may be due to the inaccuracies in the observed scale heights at

these heights. Figure 11 gives the temperature -versus -height profiles

for several x' s * The sharp temperature gradients near 130 km are due

to the low conductivity there, necessitating high gradients to conduct

away the large energy fluxes caused by photonic heating in the region

around 200 km. At high heights there is little photonic heating or ra- diative loss, and so very little conduction is required. This results

in approximate isothermaiity above about 300 km. Figure 12 shows mass

density-versus-x at several heights. At 210 km Martin and Priester

(i960) give an observational curve and this is compared. The form and

degree of variation is quite similar up to about x

= 50° • Figure 13

shows the temperature as a function of time for several heights. Figures

lk-YJ show the atmospheric composition as a function of height for sev- eral times. The 0~ densities of Friedman (i960) are shown from 130-200

km and good agreement with these is obtained. The points from 160-200

km are densities obtained near x

= 90° and so may be slightly lower

than noontime values. In this case the calculated densities are somewhat low, indicating some mixing of may occur above 130 km. Near 130

Jon, the slopes of the p.(h) curves must be nearly the same since dT/dh is

J
U2

2000

1500

o

UJ

QC

UJ

Q.

UJI1000 —

500 —

1 1 1

_ ^

30° __

// 60°

A/ 90°

1 1 1

100 200 300 400

HEIGHT h (km)

Figure 11

Temperature Versus Height for Several Times (x)

The Heat Flux Sum is 1.13 ergs/cm -sec

500
U3

10"

1-12

X= OBSERVATIONAL VALUES

X—

~ in-13

h=i50 km —

—iX- 210 —

X

.210

20 40 60 80 100 ZENITH ANGLE X (DEGREES)

Figure 12

Mass Densities Versus Time (x) for Several Heights

o The Heat Flux Sum is 1.13 ergs/cm -sec
kh

2000

o

LU

I-<

luQLU

40 60 ZENITH ANGLE X (DEGREES)80

Figure 13

Temperature Versus Time (x) for Several Heights.

The Heat Flux Sum is 1.13 ergs/cm -sec
200 300 HEIGHT h (km)

Figure 1^

Detailed Composition Versus Height

o For a Heat Flux Sum of 1.13 ergs/cm -sec and x

= °

C

500
U6

10

10

i-l2

—

.

I0"

13 roE

o\

e

c»

'

Q-

>-

(-CO

z.

LUQ I0"

14

10

10"

15 _

100 200 300 HEIGHT h (km)

400 500

Figure 15

Detailed Composition Versus Height

o For a Heat Flux Sum of 1.13 ergs/cm -sec and x

= 30c
hi

10"

-^ 10

,J K)E

o\

E

en —

Q- >

H

coz

UJQ

I0"

14

icf'V

A" 16

100 200 300 400 HEIGHT h (km)

Figure 16

Detailed Composition Versus Height

For a Heat Flux Sum of 1.13 ergs/cm -sec and x

= 60c

500
300 HEIGHT h (km)

Figure 17

Detailed Composition Versus Height

o For a Heat Flux Sum of 1.13 ergs/cm -sec and x

= 90°

500
h9

large. As dT/dH gets smaller (above the peak of production) the slopes

of the p.(h) will "begin to he characteristic of their individual scale

J heights. This effect causes p and p not to cross until above 250 km.

Figures 18-20 show the optical depths for the three wavelength ranges as a function of height for several times. At the peak of production, the

optical depths are approximately 1 + 3H/dh. Thus the production peak is

at about 178 km for x

= 0° . At high heights the optical depth's slope

h

should become less as | p(h)dh approaches a pure exponential. The oph

tical depths have been made to not go to zero at 500 km by using an h

of 510 km in their calculation. This was felt to be more physically rea- sonable than having the optical depths go to zero at the upper boundary.

Figure 21 shows the total rate of heat generation-versus -height for sev- eral times. The x

= peak of production is at about 178 km (as predicted by the optical depth curves) and because of the increased absorption

path length, increases with x "to about 215 km at x

- 90° • Figure 22 shows

the radiative loss rate as a function of height for several times. It is

seen that the radiative loss rate becomes increasingly important at lower

heights compared to the gain rate (figure 21 ). At the lower heights,

however, other gains become important. Figure 23 shows the thermal con- ductivity-versus-height for x -

U and the molecular weight as a function

of height for x

= 0° . The thermal conductivity is seen to change by a factor of only about 2.k over the F-region and the molecular weight by

a factor of about 1.8.

b . Non- steady State

Here only the first approximation was computed (this was done for

the afternoon case when BT/dt is negative). Thus the results have more

qualitative than quantitative value. The heat fluxes used are el ) „i = O.67 ergs/cm -sec and el ) nv = 0.33 ergs/cm -sec. The quasi-steady

state is assumed to be valid at noon, which is necessarily true in the

first approximation; also the temperature at 130 km is taken to be ^20 K

and to remain constant in time. The diffusive height is taken as before

at 130 km and 100 percent dissociation of is assumed at h . In figures 2h and 25 the mass densities and temperatures are shown as a function of height for several times during the afternoon. A fairly good fit

to the observational densities is achieved for x ^ 20 . In figures 26

and 27 the variation with x of total mass density and temperature is shown

for several heights. The afternoon density maximum is seen to occur at

about x

- 50° in contrast to the observed densities (shown by dashed lines

in figures 26 and 27) which peak at about x

= 25°.
50

100 200 300 400 HEIGHT h(km)

Figure 18

Optical Depths Versus Height at Several

Times ( x ) for 5&^A Wavelength Range.

The Total Heat Flux is 1.13 ergs/cm2

- sec

500
X

tQ.UJo

<

o

a.o

300 HEIGHT h (km)

Figure 19

Optical Depths Versus Height at Several

Times (x) for 900A Wavelength Range.

o The Total Heat Flux is I.13 ergs/cm -sec

500
52

100 200 300 400 HEIGHT h(km)

Figure 20

Optical Depths Versus Height at Several

Times (x) for 30^-A Wavelength Range

o The Total Heat Flux is 1.13 ergs/cm -sec

500
100 200 300 400 HEIGHT h(km)

Figure 21

Rate of Heat Generation Versus

Height for Several Times (x)«

The Total Heat Flux is 1.25 ergs/cm -sec

500
o

<u

t/>

i toE

CO

Id

CE

IT) ino

100 200 300 400 HEIGHT h(km)

Figure 22

Radiative Heat Loss Rate Versus Height for

Several Times (x)» The Total Heat Flux

is 1.25 ergs/cm -sec

500
55

300 HEIGHT h(km)

Figure 23

Molecular Weight and Thermal Conductivity

Versus Height for a Total Heat Flux

of 1.25 ergs/cm -sec at x = 0°
56

10

rfii

iff*

I0

,J -

toE

e

a>O.

>-

(/)

111 Q -14

I0 n

X=0BSERVATI0NAL VALUES

DAYTIME'

NIGHTTIME'

10'

100 200 300 400 HEIGHT h (km)

Figure 2k

Non-steady State Mass Densities Versus Height

For Several Times (x) and a Total Heat

Flux of 1.00 erg /cm -sec

500
57

2500

2000

3

<

OS

300 HEIGHT h (km)

Figure 25

Non-steady State Temperatures Versus

Height for Several Times (x).

The Total Heat Flux is 1.00 erg /cm

2 -sec

500

L
58

X = OBSERVATIONAL VALUES

h = i50km

, nH2

X x x l'£_»x

210

20 40 60 80 ZENITH ANGLE X (DEGREES)

Figure 26

Non-steady State Mass Densities Versus

Time (x) for Several Heights and a

2 Total Heat Flux of 1.00 erg /cm -sec
59

2500

2000

o

UJ

or

enu

CL

UJ

40 60 ZENITH ANGLE X (DEGREES)

Figure 27

Non- steady State Temperatures Versus

Time (x) for Several Heights and a

o Total Heat Flux of 1.00 erg /cm -sec
60

5-2 Boundary Conditions and Parameter Variation

In this discussion all of the variations, unless otherwise stated,

refer to the chosen quasi-steady state model for x

= 0° • The following

variations are considered:

a. Scale Height at 130 km

The boundary scale height may be varied if the boundary molecular

weight (i.e., the h and degree of dissociation chosen) or the boundary

temperature is changed. It may also be varied by changing T and M together. Only the first two variations are treated here. The effect of

varying the boundary temperatures is shown in figure 29. This is a ra- ther serious variation. The effect of varying the molecular weight is

shown in figure 28 where a pure atmosphere with T(l30) = ^50°K is compared with the chosen model. The variation is seen not to be as serious

since a pure oxygen atmosphere fairly well approximates the thermal situation in the F-region. The variation in p(h), if H(l30) is increased to

22 km (probably a more physically realizable situation) instead of decreased to 8 km, will be of about the same magnitude and in the opposite

sense . The scale height at 130 km is perhaps uncertain by about ± 2.0

percent. Thus the resultant density curves may be uncertain as much as

± 30 percent at high heights due to these types of variation of H(l30).

b. Boundary Density

If the scale height at 130 km is kept fixed at 15 km and the boundary density varied, the resultant density variation is shown in figure

30. The boundary density has considerable effect on the resultant density profiles. The boundary density is uncertain by about ± 20 percent.

This makes the density profile uncertain by about the same amount.

c. Choice of Absorption Cross Sections

The effect of doubling and halving the absorption cross sections

used on the computed densities is shown in figure 31 • It is seen that

the density profiles are rather insensitive to this variation. This is

because near the lower boundary most of the incident electromagnetic ra- diation has been absorbed even if the A., are made considerably less.

Thus the temperature gradients near the lower boundary will be high since

h

they are determined by (G - L)dh. This means most of the temperature

h

rise will occur before the change in the A., becomes effective. Thus

the temperature profiles will not be significantly altered and the resultant density versus height curves will also be changed very little. It

is to be noted here that if all the A., and p.(l30) are changed in the
toE

>-

CO

LUQ

100 200 300 HEIGHT h (km)

Figure 28

Effect of Varying H(l30 by Varying T(l30) for

o a Total Heat Flux of 1.10 ergs/cm -sec at x

-

C
62

100 200 300 HEIGHT h (km)

400 500

Figure 29

Effect of Varying H(l30) by Varying M(l30)

for a Total Heat Flux of 1.25 ergs/cm -sec
HEIGHT h (km)

Figure 30

Effect of Varying Boundary Mass Density.

o The Total Heat Flux is 1.25 ergs/cm -sec and x 1S C
6U

10

io-V

E

o

E

c^ 10"

>-

CO

z.UJQ

200 300 HEIGHT h (km)

Figure 31

Effect on p(h) of Varying Cross Sections for

a Total Heat Flux of 1.25 ergs/cm -sec at y

O

c

400
65

same manner, such that the products A..p.(l30) in (^-4) are unchanged,

J

J. _L the resultant density profiles "will merely he multiplied hy the factor

hy which the p.(l30) were changed.

d. The Effect of h^d

The scale height and the degree of dissociation at 130 is kept

fixed to investigate this variation. The degree of dissociation of

at 130 km is kept fixed by picking the degree of dissociation at the h

in question such that an extrapolation to 130 km (assuming diffusive

equilibrium to hold between the chosen h, and 130 km and picking a suitable temperature model in this range) gives 80 percent dissociation there,

The total densities are plotted versus height in figure 32 for several

h, from 110-150 km. The effect of h, is seen to be small on the total

d d ' mass densities. The effect is also relatively small on the temperatures

and somewhat more in the scale heights because of the variation of M(h).

The effect on the detailed composition is, as previously mentioned, quite

large. A lower h n implies more 0_ and and less N_. d 2 2

e. The Variation of the Total Heat Flux

For heat fluxes of 3«3> 1*7; 1*1; an^ 0.3 ergs/cm -sec, total mass

densities versus heights are plotted in figure 33 • ^he mass density profile is seen to be very sensitive to this parameter since the total rate

of heat generation, G(r,t), is directly proportional to Eel .

f

.

The Variation of Thermal Conductivity

The effect of increasing the conductivity by 50 percent is shown in

figure 3^- compared with the total mass density profile for the chosen

thermal conductivity. The density profiles are quite sensitive to this

quantity since the conductive flux is directly proportional to thermal

conductivity

.

g. The Effect of F

m

The density profiles for several values of F are shown in figure

IT

35* The value of F of 0.575 x 10 ergs/ sec is just that required to

maintain the nighttime scale height at 500 km. The temperature and thus

the scale height is relatively insensitive to F except at high heights

where the photonic input is small compared to any coronal conduction.

Thus figure 35 shows the mass density profile to be relatively insensitive to F . m
E

E

>-

CO

300 400 HEIGHT h (km)

Figure 32

Effect of Varying h for a

2 Total Heat Flux of 1.10 ergs/cm -sec at x

500

C
67

HEIGHT h (km)

Figure 33

Effect of Varying Total Heat Flux for x

=

C
68

10

io"

^ 10 E

E

o>

>-

CO

H2

io

13

io

14 -

10

NORMAL THERMAL CONDUCTIVITY

THERMAL CONDUCTIVITY

INCREASED BY 50%

100 200 300 HEIGHT h (km)

400 500

Figure 3^

Effect of Varying Thermal Conductivity

o For a Total Heat Flux of 1.1 ergs/cm -sec at x

=
1=

E

en

>

en

UJQ

HEIGHT h (km)

Figure 35

The Effect of Varying the Conductive

Thermal Flux at the Upper Boundary.

o The Total Heat Flux is 1.10 ergs/cm -sec at a x of C
70

h. Composition

Figure 36 shows the total mass density profiles for pure 0, p

, an~ N

p atmospheres. The temperature is adjusted so that the "boundary scale

height is always 15 km. It is seen "by comparing the observational density profile in figure 10 that atomic oxygen is the best approximation

to the thermal situation in the F-region.

i. Blamont, et al. (1961) Boundary Conditions

If the previously mentioned boundary temperatures of 350°K is taken,

a diffusive height of about 110 km along with 100 percent dissociation

of is needed to keep the boundary scale height at 15 km. This is a

case of varying T/M together so that their quotient remains the same. The total density and detailed composition are plotted in figure 37* Such

low h., and such a high degree of dissociation indicate that a temperature of 350°K is probably somewhat low. Thus the detailed composition

according to this boundary temperature is taken as an extreme model with

the correct detailed composition closer to the model selected in this paper. If however, the upper range of the Blamont, et al. temperature at

130 is used (395°K) and if an h of 100 km is taken, a composition model

may be made which has a sufficient amount of in it.

5.3 Range of Validity of Quasi- steady State Model

On this model the afternoon and morning densities and temperatures

at a fixed height are symmetric about noon (x = 0°). However, satellite

observations (Martin and Priester, i960) indicate the diurnal mass variation to be symmetric about noon at low heights and about x

= 30° at high

heights. Thus if the zero of x "were shifted to about + 20°, the agreement

with observation would be quite good to about x

= ±50° (see figure 11 ).

Beyond this, the quasi-steady state drops considerably below the observed

densities due to the increasing importance of the heat capacity term.

5.^ The Non-steady State

As seen in figure 26, there is rough agreement between the form of

the observational density-versus-x and the computed form. The computed

curve has too large an amplitude and the amplitude maxima are generally

20°-30° too late in the afternoon. However, this discrepancy is due in

large part to the assumption that the quasi-steady state holds at noon

and not at x

= 20°, as the observations indicate.

The complete solution to the heat balance equation, including the

heat capacity term, can probably closely approximate the entire diurnal

density and temperature variation as the following qualitative discussion
^ 10 —

E

£

CD

UJQ

200 300 400 HEIGHT h (km)

Figure 36

Effect of Varying Compositions on Mass Density Profiles

For a Total Heat Flux of 1.10 ergs/cm2

- sec at X

= 0°

500
roE

E

o>

>

HEIGHT h (km)

Figure 37

Effect on Composition and Total Mass Density of

Blamont et . al. Boundary Conditions for a Total Heat

o Flux of 1.25 ergs/cm -sec and x

= 0° . Model Curve is

Computed for Boundary Conditions Selected in Paper.
73

indicates. The heat capacity term is a measure of the sluggishness of

the atmosphere in responding to thermal influences. It tends to oppose

any atmospheric temperature changes. Thus the morning temperatures and

densities will not be able to rise as rapidly as they would according to

the quasi-steady state and will peak later than noon, as is observed.

Also the afternoon temperature drop will be inhibited giving rise to high- er densities at large x' s than those predicted by the quasi-steady state.

The nighttime temperature drop will be likewise lessened by the heat ca- pacity although isothermality will extend probably as low as 200-250 km

by the night's end (Nicolet, 1961). If the total heat capacity of the

atmosphere and the average conductive flux is estimated, it may be seen

that the nighttime F-region atmosphere cannot be cooled on the average

by more than about 800 K in twelve hours through conductive loss . This

is seen by using the equation

T_/p_

At = pc HAT/KT ' T'. (5-1)

Here T' = dT/dh and the bars mean averages of the indicated quantities

over the height ranges in which they are important. The assumed values

are:

— — 1 ? "3 p = 10 gms/cm

c = 8.3 x 10 ergs/gm-°K

H = k0 km

T

7" = 5°K/km

KT ' = 6 x 10 ergs/sec-cm-°K.

According to (5-l) it will take about 25 hours for the atmosphere to cool

800°K. Taking average radiative loss values increases the denominator of

(5-l) by about 10 percent and so decreases the cooling time by about the

same amount. Observations (figure 9) show the nighttime scale heights to

be about 60 percent of the daytime scale heights from lj-00 to 500 km. A

decrease from about 1700°K (predicted by the quasi-steady state model) to

about 900°K would indicate a similar percentage decrease in the scale

height

.
7^

5.5 The Structure in the Observational Density

and Scale Height Profiles

An anomaly is seen in the observational mass density and scale

height curves (figures 9 and 10 ) at about 200 km. This anomaly shows

up as a pair of inflection points in the mass density curve and as an

inversion in the scale height curve. A scale height inversion will almost certainly imply a negative T". For this to occur (2-10-2) shows

that:

h . h

|

m

(c

v

pT + L)dh >

J

m

Gdh. (5-2)

On the theory as used here it is difficult to get an inversion in the

temperature except possibly near sunrise or late in the night. However,

( 5-2 ) could not be satisfied all day and all night as the observations

indicate.

There are several possible explanations. Johnson (i960) has pointed

out the possibility of time dependent heat convection from the polar regions at about 200 km. This would cause a sharp temperature rise. Then

at slightly higher heights , where the convective flux stops, the values

• of c pT and L could be high enough to satisfy (5-2). Such a convective

flow could lead to sharp gradients in c , which have been neglected in

the heat balance equation. These would have the effect (if positive)

of increasing the left hand side of (5-2). Another possibility is that

the charged particle conduction could be high here due to a high electron temperature (caused by low electron densities and the input of many

high energy electrons near 200 km). Thus a cool spot could be created

which would account for the observed inversion. It is, however, unlikely

that electron temperatures high enough for this to occur (^ 10,000°K)

could actually exist in the F-region. It can be asserted that whatever

the mechanism it is not due to differential absorption of ultraviolet

flux.

5.6 Conclusions

The primary conclusion is that ultraviolet photons are a major Fregion heating source. The necessary sunspot maximum heat flux is found

to be 1.13 ergs/cm -sec, giving an estimated photon energy flux of 5*18

ergs/cm -sec. The estimated error (due mainly to the uncertainty in the

T/M ratio at 130 km) is about ± 30 percent. This compares favorably with

the extrapolation of Hinteregger ' s (i960) measured ultraviolet flux values made by Denison (1961), who estimates 5.66 ergs/cm -sec in the Fregion ionizing flux.
75

The non-steady state heat balance equation can qualitatively ex- plain the diurnal variation of the F-region mass densities and temperatures during the afternoon. The quasi-steady state quantitatively ex- plains the F-region diurnal variation of mass densities and temperature

for x < 50° and up to about 300 km. The detailed composition models and

ultraviolet flux absorption parameter models are valid over the same ranges (x < 50°, h < 300 km).

The uncertainties in the parameters and boundary conditions necessary in the theory are insufficiently large to invalidate the above con- clusions .

5.f Future Studies Indicated

The complete solution of the non-steady state for daytime and nighttime is the most important extension of the present work. The method of

solution attempted in the present study c6uld be improved by a more ac-

• curate numerical calculation of T and better boundary conditions . When

the non-steady state problem is completely solved, seasonal and latitudinal variations may be investigated through the transformation of T to

ar/ax.

The use of the heat balance equation (with appropriate extra gain

terms ) to explain the critical points in the density and scale height

curves near 200 km is a study which should be done. The magnitude of

the density inflection and the scale height inversion is a measure of

the strength of the necessary gain terms.

The heat balance equation may also be used to study atmospheres of

other planets. For example, the Martian atmosphere is thought to be

mostly Mp. Also, the force of gravity on Mars is less than on Earth and

the solar flux is less at Mars than at Earth. Appropriate boundary con- ditions are not known but may be roughly estimated from the previous

facts. This is all the information necessary to apply the heat balance

equation.
76

ACKNOWLEDGMENTS

The author wishes to express his gratitude to his wife for aid and

encouragement, to Dr. T. E. VanZandt for many valuable consultations and

helpful criticisms, and to Mr. D. L. Obitts for technical advice on computer programming. Also, the author is grateful to both the National

Science Foundation and to the National Bureau of Standards for partial

support of this work.
77

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