Ion temperature in the upper atmosphere

JOURNAL OFGEOPttYSlCAL RESEARCH VOL. 72, NO. 13 JULY 1, 1967 Ion Temperaturein the Upper Atmosphere PETER M. B•xs Department oi Applied Electrophysics University of Cali/ornia, San Diego La Jolla, California, and Institute •or Radiation Physicsand Aerodynamics The problem of ion temperaturein the ionosphereis analyzed in terms of an energy budgetthat includesheatingby the ambient electrongas, coolingby the different gasesof the neutral atmosphere,and thermal conductionalong the lines of geomagneticforce. With the introduction of the ion temperaturefractional separationX, the time-dependentequation for ion temperatureexcludingtransporteffectsis reducedto a form that removesthe explicit dependence upon the neutral gas temperatureand separatesthe effectsof the electronand neutral gases.Adopting the appropriateaeronomicconditions,it is shown with different models of the neutral atmospherethat the ion temperature has a transitional behavior, increasingfrom the neutral gas temperaturenear 250 km toward the electron temperature above 700 km. While the ion thermal transport does not significantly affect ion temperatures below about 500 km at midgeomagneticlatitudes, it plays an increasinglyimportant role above this height, acting to keep the high-altitudeion temperaturesignificantlyless than the electrontemperature.In addition,as a result of ion coolingin He and H, the ion temperaturecan decrease with altitudein the transitionatmospheric regionsbetweenO and H dominance.The time constant for changesin ion temperature resulting from changesin aeronomic parameters is evaluatedand foundto be lessthan 10 minutesfor altitudesbelow 700km; abovethis altitude,conduction coolingmustbe considered. Finally, the theoretical expressions derivedhereare compared with recentionospheric Thomsonscatterdata. thoroughlyinvestigated.A first analysisof the It is well establishedfrom numerousexperi- O* ion energybahnee equationwas made by Hanson [1963]. It was shownthat for altitudes mentsusinga variety of techniquesthat in the below 300 km the energy transferred to the regionsof the atmosphereabove 150 km the O* ion gasby the electrongascouldbe rapidly temperatureof the electrongasis consistently lost to the neutral atmosphere, and, consehigher than that of the neutral gases.To exquently,only small differences would exist beplain this state of temperatureinequality a tween the ion and neutral gas temperatures. number of theoretical studies of the electron Above about 500 km, however,Hansonfound heat balance equation have been made with that the neutral gas concentration becomes the assumptionthat the principal source of sufficientlylow to prevent the ion gas from electronheatingarisesfrom the energyreleased dissipatingthe thermal energy obtainedfrom in the photoionization of O, O•.,and Ns [Drukthe electron gas without there being a sub1. INTRODUCTION arev, 1946; Hansonand Johnson,1961; Han- stantial difference between the ion and neutral son,1963; Dalgarnoei al., 1963; Geislerand gastemperatures. I-Ieneeit was predictedthat Bowhill, 1965; Da Rosa, 1966]. The most recent of theseanalysesare in generalagreement with the measuredvalues,but, becausethe deetron temperature is very sensitiveto minor changes in the solarEUV flux, the electronconcentration,and the assumed electronenergyloss processes, a high degreeof correspondence cannot be expectedat the presenttime. In contrast to the problem of electrontem- the ion temperatureprofileshouldhave a char- acteristic transition from the neutral gas temperature at low altitudes to the electron temperature at altitudes near 1000 km. Experimental support for the transitional behavior of the ion temperature can be inferred from satellite measurements[Boyd and Raitt, 1965] and Thomson scatter radar data [Evans, peratures,the thermal behaviorof the ion 1965]. It must be pointed out, however, that gasesof the upper atmosphere has not been although the results of Hanson indicate near 3365 3366 PETER M. BANKS equality of the electron and ion temperatures at high altitudes, in fact, there appear to be significantdeviationsof the ion temperature toward valuesconsistentlylessthan the electron temperature, even in the high atmosphericregions where the ion energy lossesare very small. Further, at altitudes above 500-600 km the measuredion temperaturedoesnot appear entirely compatiblewith the theoretical analysis of the ion energy balance equation based only upon local production and loss of O' ion thermal energy. The purpose of this paper is to present the enoughto ensurethat the temperaturedifferencesbetweenthe ion species are small; separations up to severalhundreddegreesKelvin were calculated for altitudes between 300 and 650 km. It. is noted, however,that. the effect of the ion temperatureseparationsupon the total ion energy balanceis not large since, under most.conditions,the He* and H* ions are only minor ionic constituentsin the regionsof temperature inequality. For altitudes above 650 km the calculated ion temperature differencesare small when only electrongasheating is considered,and, by identifying the O* results of an investigation into the factors inion temperatureas the commonion temperafluencing ion temperature in the atmospheric ture T,, an adequate descriptioncan be made regions above 300 kin. To reconcile the differ- of the thermal structure of the combined ion ences between experiment and theory, an gases. analysis has been made of the ion heat balance The heatingof the ionospheric ion gasesby equation to include the effects of multiple ions elastic electron-ion collisions and the loss of ion (0*, He*, H*) and the influence of ion heat thermal energy in collisions with neutral gas conductionalong the lines of geomagneticforce. particles representfactors that tend to distort In a previouspaper [Banks, 1966a] it has been the ion velocity distributionfunctionsfrom the shown that the introduction of these factors Maxwellian form. In fact, above 300 km the significantlyaffects the high-altitude ion tem- effectivetimes for ion energytransferby both perature profiles. electronsand neutralparticlesare muchlarger In brief outline, section2 of this paper con- than the inter-ion collisiontime [Banks,,1967]. tains the development of the terms to be inFurther, for altitudesbelow2000km the probeluded in the ion energy balance equation, lem of ion escapefrom the earth'satmosphere includinga discussion of ion energy production, can be neglectedas a consequence of the large loss, and thermal conduction.Section 3 is de- ion-ion Co.ulombcollisioncrosssection[Hanvoted to steady-stateand time-dependentsolu- son and Oftenburger,1961]. In this analysis, tions of the energy balance equations both whichis primarily directedtowarda description includingand excludingthe effectsof heat con- of the processesoccurringin the altitude 300duction. It is shown that numerical solutions 2000 km, it is assumedthat the individual ion to both heat flow and diffusion-type parabolic velocity distribution functions are Maxwellian equations can be readily obtained by lineari- with a commontemperature T,. zation and application of the triple diagonal The set of equationsdescribingthe number mesh method. In section 4 the application of density, temperature,and diffusionvelocity of these equationsto typical aeronomieconditions single componention gas moving under ambiis made. A summary of the results of this polar conditionsin response to gravity, presstudy is madein section5. sure, and temperaturegradients,and ion produetion and loss follows from Chapman and Cowling [1952] as 2. IoN ENERGY BALANCE EQUATIONS Equations. A preliminary analysis of the Dni q-X7.nic• = q-- l individual energy balance equations for 0', Dt He*, and H* ions for the aeronomieconditions Dc• 1 cgkT• of the upper ionospherehas been made in a -- g qVp• = --• previous study [Banks, 1967] of the temperaDt n•m• rniDi,•• ture coupling of ions subjected to different 2 1 DTi heating and energylossrates. As shownthere, Dt the rate of transfer of thermal energy between the ion gasesis not always sufficiently large -- (•ni]c)-l[P- iLl (la) (lb) q-: n-•.kV.Q q-õT•V. (lc) ION TEMPERATURE IN THE UPPER ATMOSPHERE 3367 bility of thermal diffusion,and thermoelectric where effects. Din Many of the salient features of ionospheric I (2a) ion temperature profiles can be understood, •'= •"p__,. p,,-]-p,,Di---• however,by consideringthe ion energy balance n Q = --KyT, (2b) p,, -- n,k(T,-]- T•) (2c) p,, = n,,kT,, (2d) and n• .... = ion,electron,neutralnumberdensity. c• = ion ambipolardiffusionvelocity. q = ion productionrate. l = ion loss rate. g = gravitational acceleration. mi = ion mass. k = Boltzmann's equation alone and using given values for the ambient ion density and diffusion velocity. Thus in the present analysisthe couplingsimplied by equationsla-c are brokenby adopting models of ionic composition that appear consistent with ionosphericmeasurements.For the ionosphericregionsabove 300 km the principal ion gasesare consideredto be 0', Itc *, and It • (the presence of N* [reported by Istomin, 1966] is nearly equivalentto a greater 0 • density) in a configurationthat is consistenteither with a given electron temperature profile and the calculated ion temperature or with experimental constant. T,.•.n = electron, ion, neutral gas temperature. P = ion energyproductionrate. L = ion energylossrate. Din = ion-neutral diffusion coefficient. Q• - ion heat flux vector. data. It is assumed that the earth's magneticfield effectivelyconstrainsall charged particle motionssuch that ion heat flow, current, and diffusionare restricted to directions along the earth's magneticfield lines and that thereare no appliedelectricfields. With these restrictions the ion energy balanceequationcan be rewritten as K• = ion thermal conductivity. In deriving equations la-c it has been assumedthat m, >> m., • T,/D,, > •, TJD.•, where D.n is the electron-neutral diffusion coefficient, no external forces other than gravity OT• OTi -•T sin at -]-•sin•'I'•-z •- 3 i •'I O__•.• Oz (•n.k) -• sin •'I • K, are acting, the electrons,ions, and neutral gases have separate Maxwelltan velocity distributions, and that the effects of electron-ion col-- (•n.k)-•[P- L] (3) lisions can be ignored in the derivation of the where I is the magnetic dip angle, • is the ion momentum equation lb. The completesolutionto the problem of ion diffusionvelocity,z is a vertical coordinate,and temperaturesin the upper atmospheremust be n. is the electron concentration. Solutions to basedupon an analysisthat takes into account this equation determine the basic ion temperathe couplings between the electron, ion, and ture profiles for the different aeronomiccondineutral gas temperatures, the charged and tions. Ion gas heating. Under the influence of neutral particle number densities,and the relative speedsof particle diffusion.Actually, equa- thermal conduction and the nonlocal heating effects of photoelectrons,it is generally found tions la-c represent an idealized situation for only a single ion gas, and more complexequa- [Geislerand Bowhill, 1965; Evans, 1965] that tions are neededif the completecouplingprob- the electron temperature in the regions above lem is to be resolved.At the present time the 300 km lies significantlyabove that of the neusimultaneous solution of these equations has tral atmosphere,thus providing a heat source not been done in a satisfactory fashion since for the ion gases.By means of elastic electronthis involves consideration of many complex ion collisions an electron gas of temperature factors relating to ionospheric-magnetospheric T. and density n. transfers energy to the ion relations,multi-ion relative diffusion,the possi- gasesof the ionosphereat the rates PETER 3368 M. mosphericand ionospheric parameterstypical of quiet solar conditions,observations of elec- P,(O+) -- (4.8 -t- 0.5) X 10-?n, ßn+(O)T,-S/•[T,- T,] tron and ion temperaturesand densitiescan be usedto establishupper limits for external rates --6 P,(He +) = (1.9 q- 0.2) X 10 n, of ion heatingsincesourcesof energyas small ßn+(tte)T,-a/2[T,- T•] (4) P•(H+) = (7.7 q- 0.8) X 10-øn, ßn+(H)T,-a/2[Te-- Ti] ev cm-a see-• which are in the ratios 1'4' 16. Thus, for identical temperatures and electron concentrations, the electron gas heating rates for I-Ie+ and H + ions are equal to the 0 + heating rate when n+(He)/n+(O) -- 0.25 and n+(H)/n+(O) -0.063, respectively,implying that electronenergy transfer to the minor concentration BANKS ions can be a major source of heating for the over-all ion gas in certain regions (see Figure 3). In fact, the data of Carlson and Gordon [1966] indicate that for the period of their measurements at Areeibo (summer and winter, solar minimum) the heating of H + dominated the ion energy budget for altitudes above 600 km at all times, and occasionally the region of equality between the 0 + and I-I+ ion heating rates was probably as low as 4150km. Clearly, it is not possibleto deduce accurate ion temperature profiles from a knowledge of 0 + ion densitiesalone; the effectsof I-Ie+ and I-I+ must be included. In addition to electronheating, other possible sourcesof ion thermal energy involving dissipation of wave energy, electric fields [see Meqill and Carleton, 1964], chemicalreactions,or energetic particle impact can be postulated.However, from observational evidence there is strong support for the choice of the electron gas as the primary sourceof ion heating above 300 km. Thus from the daytime Thomson seatter measurementsof Evans [19615],Doupnik and.Nisbel [1966], Farley [1966a, b], Carru el al. [1966], and others, it has been found that the ion temperatures are always equal to or less than the electron temperature. This result tends to support the assumptionthat the electron gas is the major heat source.If there were additional energy processes,these would be expectedto lead, at times, to ion temperatures larger than the electron temperatures. as 1-5 ev cm-3 see-• can be expectedto raise the ion temperaturesignificantly(see,for example,the energybalanceshownin Figure10). It shouldbe notedthat the nighttimemeasurements of Carrg et al. [1966] indicate a slight enhancement(30-60øK) of T, above Te at 200 km. This effect may indicate the presence of an ion heat source in addition to the electrongas but, becauseof the difficulties involved in determiningthe densitiesof atomic and molecularions that may contaminatethe measured radio spectra, the results are not certain. It has been suggested that a nighttimeenhancement of T, would result from a low-en- ergy flux of protonsdescending from the proionosphereinto the ionosphere.A flux of this type has been investigatedby Praq ei al. [1966] in connectionwith nighttime optical emissions and electron concentration variations. By consideringthe energy transfer rates for energetic proton-ion elastic collisions the net heatingrate Pt of ambientions due to a proton flux of intensity • particles cm-2 sec-• and energy• (ev) is P, = 1.2 X 10-1an•(eff)cb/• ev cm-• sec-• (5) wheren,(eff) = n*(O) + 4n*(He) + 16n*(tt). Evaluation of (5) with typical nighttime parameters and the 4-kev proton flux data of Prag indicatethat the ion heatingrate is generally smaller than 10-2 ev cm-• scc-• and does not play an important part (for 4-key protons) in the ion energybudget. In any case, for a proton flux to be important in heating the ion gasthe inequality cbn•(eff)/• > 10'2-10 'a (6) must be satisfied.Thus, for 100-ev protonsin I-I* ionswith n+(H) = 10• cm-•, a flux of lff ø10n protons cm-' see-• would be required to significantlyalter the ion temperature due to ion-ion elasticcollisions. It is certain, however, that a flux of this intensity would be responsiIn fact, the ion temperaturefor the ionospheric ble for many ionosphericeffectsand would not regionsabove1500km is very sensitiveto minor remain undetected. Ion gas energy loss rates, The rates at changesin the ion energybalance.For the at- ION TEMPERATURE IN THE UPPER ATMOSPHERE 3369 densities are known quantities that are not which the ionospheric ions lose energy by means of elastic collisionsin the neutral gases altered by changesin ion temperature or comof the atmosphere have been discussedpre- position. Ion thermal conductivity. In response to viously [Banks, 1966b]. Since the present analysis is primarily concernedwith the regions gradients in ion temperature, a flux Q of above 300 km, ion processesof chemicalreac- thermal energy flows along the lines of magtion leading to ion speciesother than 0 +, Hie+, netic force. For a single ion gas the heat flux is and I-I + have not been introduced. For mixtures given by equation 2b, but when mixtures of of ions in unlike gasesit is assumedthat the ions are involved additional terms enter into polarization interaction determinesthe collision the expressioncorrespondingto thermal difcross sections and the energy transfer rates. fusion and ordinary diffusioneffects.However, For O+ in H and I-I+ in 0, however, the polari- partial compensationfor the omissionof these contributionsto the heat flow in equation 2b zation interaction is less important than the process of accidentally resonant charge ex- is found in equation4 when explicit accountis change (O+ + H • H + + O) in determining taken of the heat flux brought about by the the 0 + and H + energy transfer rates. For ions transport motion o• of ion gases. Evaluation of conductionand difin their parent gases,the effect of resonance of the heatinginfluences chargeexchange(X + + X-• X + X +) is domi- fusion indicates that heat conduction is genof low nant in determining the rates of ion energy erally dominant but for circumstances loss. Table I summarizesthe energy loss rates ion temperaturesand large ion densitiesdiffusion heating or coolingdue to ion transport usedin this analysis. The effectivenessof the neutral atmosphere can be important. The calculation of the ion thermal conducacting as an ion heat sink dependsupon the extent to which the neutral gas temperature is tivity K• for a single ion gas of atomic mass determined by ion-neutral energy transfer. At follows from the work of Chapman [1954] Since3/2 n(n) kT•/L• >> 1, wheren(n) is the neutral gas number density and L• is the ionneutral energy transfer rate, the effectivetime for ions to heat the neutral atmosphereis large when comparedwith the typical conduction cooling times of the neutral atmosphere [see Nicolet, 1960]. Thus in this analysisit is assumed that the neutral gas temperature and as ß ß5/2ev Ki -- 4.6 X 104A,-X/2T, cm -x sec -x oK- x where the correctionfactor appropriate for the second approximation has been applied. The analysisof the thermal conductivity of a fully ionized gas [Spitzer and HSrm, 1953] shows TABLE 1. Ion Energy Loss Rates Ion Mixture 0 + - 02 0+ 0 + 0 + - He 0 H He + - N2 He + - 02 He + He + He + - 0 H He H + - N2 H + - Os H + -- He H +-H H+ - 0 Energy Loss Rate (10-14 ev cm-a sec-•) 6.6 n+(O)n(N2) (Ti - T•) 5.8 n+(O)n(02) (Ti -- T•) 2.8 n+(O)n(He) (Ti - T,•) 0.21 n+(O)n(O)(Ti + T,•)in(Ti - T,•) 0.36 n+(H)n(O) Ti•n(Ti - T,•) 5.3 n+(I-Ie)n(N•) (Ti -- Tn) 4.5 n+(I-Ie)n(O•) (Ti -- T,•) 5.8 n+(He)n(O) (Ti -- T,•) 10.0 n+(He)n(H) (Ti -- T,•) 0.4 n+(He)n(He)(Ti -+- T,•)in(Ti -- T,•) 3.1 n+(I-I)n(N•) (Ti - T,•) 2.8 n+(I-I)n(O•.) (Ti - T•) 5.5 n+(H)n(He) (Ti - T,•) 1.4 n+(H)n(H)(Ti -+- T,•)•/•(Ti - T,•) 0.40 n+(O)n(H)T,•Xn(Ti - T,•) Rates taken from Banks [1966b]. Comment Polarization Polarization Polarization interaction interaction interaction Charge exchange Charge exchange Polarization Polarization Polarization Polarization interaction interaction interaction interaction Charge exchange Polarization Polarization Polarization interaction interaction interaction Charge exchange Charge exchange PETER 3370 M. BANKS that thermoelectriceffectsare important. Thus the stipulation that there is no net electron current flow due to temperature gradientsleads to the establishment of an electric field within the ing to explicit solutionsfor the steady-stateand time-dependent ion'temperatures. Solutionsneglectingtransport effect s. For equationsare introduced,it is possibleto re- low geomagnetic latitudeswherevertical motions of thermal ions and electronsare restricted by the geomagneticfield, and for seronomicconditions leading to large rates of energy production in the ion gases, the ion thermal conductivity is not of dominating importance in determiningthe profile of ion temperature. quire a condition of no net electric current When plasma that opposesthe electron current flow associatedwith the gradient in electron temperature and reducesthe electron thermal conductivity by a factor of 2.3. When the simultaneous ion and electron heat and current flow thermal conduction and diffusion are through the arebipolarhypothesis.In this case neglected,the time-dependention temperature the electric field establishedby the electrons equation becomesa function of only the local tends to increase the effective ion thermal conparameters of the neutral and ionized conductivity. It is found, however, that this en- stituentsand can be written from (3) as hancement is generally small and can be negOT• lectedin the ionosphere. - an,(efOT.-3/•(T.-- T,) -]- (]n.k)-•Px Ot For the ion gasesof the ionosphere,an extension of (7) must be made to include the effects of differing conductivitiesfor each ion i • species.An exact expressionis difficult to obtain and for the present analysis an approxi- Here, e• = 3.7 X 10-•, n,(eff) =, [n+(O) + mate, density weighted,conductivity has been 4n+ (He) + 16 n+(H)] is the effectiveion density for electrongas heating of the ion gases, usedin the form, P, is the rate of ion gas heating arising from = x + sourcesother than the electron gas; /(l) = -]- 4n+(I-I)]T, 5/"/n.ev cm-1 sec-• øK-1 (8) n+(l)/n, is the fractional abundancefor the/th ion species;fi•,, the ion energylosscoefficient which leads to errors of less than several per for an/th-type ion in the jth neutral gas,is the cent in binary mixtures. The effects of ion- numerical coefficient for ion energy loss dineutral collisions,which tend to reduce the vided by 1.4 X 10-'; and the indicated sumthermal conductivity below the value given by mations extend over all neutral (j) and ion (8), can be neglectedabove300 km. (l) speciespresent.Values of fi•, for the principal ionospheric constituents are listed in 3. SOLIJTIONS TO TI-IE ION ENERGY Table 2 with (T, + 3T,) '/' = 70. [BALANCEEQUATION The solutions to (3) using the appropriate seronomicvaluesof particle densitiesand temperatures must be obtained by numerical methods when the effects of ion heat conduction and diffusion are included in the ion energy balance. Thus the calculatedion temperatures depend directly upon the chosenmodel atmosphere, the ionic constituents,and the electron temperatures,making it difficult to reproduce a set of general ion temperature profiles applicable for all possiblevariations in seronomic conditions. In many situations, however, ion transport mechanismsare not of dominating importance in determining the values of ion temperature, and, when the ion energy budget is restricted to local heat production and loss, an important simplificationcan be made lead- Equation8 may be reducedin formby noting TABLE 2. Ion Energy Loss Coefficients Ion Mixture •il( X 10-lø) O + -- No. O + - O•. 4.6 4.2 O+ O+ - 2.0 7.7 He O He + - N2 He + - O2 He + He + He + - O H He H + - N•. H + - O2 H+ H+ - He H 3.8 3.2 4.2 7.2 14.0 2.2 2.0 3.9 50.0 ION TEMPERATURE IN THE from Table 2 that all of the adopted ion en- UPPER 3371 ATMOSPHERE heat source other than the local electron gas, ergylosscoefficients are independent of T•. The linearizationapproximationfor the square-root I (16) factorsdoesnot greatly affect the magnitudes of the charge exchangeenergy loss rates and with no explicit referenceto the neutral gas permits(9) to be expressed as temperature.Equation 16 showsthe simplere- X - 1-Jr(S,•Toa/•/S,) lation that exists between the ion temperature OT• Ot fractional separation,the ion and neutral particle weighteddensities,and the electrontem- _ [SoTo -•/2 q- P:'] -- IS.To-a/2q- S•]T, of ex(10) perature.As shownlater, the conversion whereP.' = (3/2 n.k) -• P. and the factorsS. and S• are givenby S. = 3.7 X 10-an•(eff) (11a) A usefultransformationof (10) can be made through the introductionof the ion temperature fractionalseparationX definedas X = (•"- Y2/(L- Y3 perimentaldata to plots of X versusaltitude permitsdirectreferencebetweenmeasuredvaluesand theory for differentstatesof the neu- tral and electron gases. Becauseequation 10 is linear in T{, the time solutions for the ion temperature can be obtained directly once the form of OT,/Ot and aT&/at are known. For aT,/at = aT&/at = o, where the ion temperature is initially at some value T{•, the time dependentsolutionfor T• is T,---- T,oe-t/'-+- T,•(1- e-t/') (1•) which is a measureof the separationof the ion and neutral gas temperaturesin terms of the electron-neutraltemperature difference.At low (17) where -•/2 qT• = SeT, S,T. -•/•SnTn -4-+ Px' (18a) altitudes where the ion to neutral energy loss rates are large for small valuesof (T• -- T•), (18b) T• • T•, and X • 0, while at high altitudes, Thus the relaxation from an initial temperature where the coolingeffect of the neutral atmosstate dependsupon the effectiveion density, the phereis not strong,T• • T• and X • 1. In electron temperature,and the densitiesof the terms of X, equation 10 can be rewritten as neutral atmosphericconstituents.When T, and T• have time variations comparable with or smaller than r, the solution to the time-dependention temperatureis more difficult to ob- ot - &L x tain. Ion temperature profiles including heat con- (13). duction. Solutions for the ion energy balance which, •ith 0T•0t = OT,/Ot= O,becomes OX ot equation, which includes heat conduction through the ion gases,must be obtained by numerical _ + - [SoTo -a/• q- S•]X (14) The steady-statesolutionsfor the ion temperatureseparation are obtainedfrom (14) by takingOX/Ot = O,giving For the calculation of steady-state ion temperature profiles, and for more general parabolic differential equations such as the electron continuity equation, it has been found that the implicit integration method of Diaz [1958] used previously [Banks, 1966a] can be greatly improved upon through appropriate linearization and application of the triple diagonalmeshmethod. For the problemof ion temperaturethe basic X = I + [P='T.a/•/S.(T,T•)] (15) 1+ (S•T.a/•/S.) substitution or, as will be assumedin th• paper, for no methods. 0 = T{ TM can be used to reduce (3) to the form [seeNicolet, 1962] PETER 3372 d20 ds• - M. BANKS = o where f•(z) and f2(z) are functionsthat depend upon the neutral and charged particle densitiesand temperatures, the magnetic dip angle, and the numerical coefficientof the ion thermal conductivity. Although (19) may Be solved by trial and error using given boundary conditions,a more rapid method has Been developed using an iterative technique involving linearization and application of the triple diagonal mesh method. The advantage of this approach lies in the great flexibility permitted in the choice of appropriate boundary conditions. In the present work it has been found that at altitudes below 350 km thermal conduc- tion has only a slight effect upon the ion temperature. Hence the point boundary condition O(z = 0) = [/• (z = 0)//• (z = 0)] TM is used at the base altitude (usually 200 km). At high altitudes,usually for z > 1500 km, it is assumed that dO/dz is equal to some predetermined value that depends upon the assumed sourcesof ion heating above the altitude range of interest. To solve (19) numerically, 0 is expanded in terms of a function q(z), which is presumed to be known from a previousiteration or initial conditions, and a difference quantity A(z). Thus with (2o) the linearizedform of equation19 is d2/• .•/l(Z )g--,5/7' /• dz• Using a centraldifferencemethod, (21) can be reduced to the form (22) where Bi = --2 1 -}- --•/• gi --[gi+z- .J 2gi-Jl- g/-1] accordingto K•= D1 -- Ao 1 B• Di- L•-- B1 Ki-• (24) 1 Ki = Bi_ Li_l Li = Bi_ Li_l and then working downward from j -- j max to j -- 1 usingthe relation /•i--1 = Ki-1 -- Li-1 Ai (25) with the latter relation beingtrue only for dT•/dz -0 at the upper boundary. The iterative solutionsfor 0 (and T•) are found by taking the initial function !7 to be given by g• = [f• (j)/f, (j)lye, that is, the profile for 0 that neglects the conductioneffect. Using the individual values for the gj, the A• are calculated using equations23-26 and added to the previous values of the g•. The basic iteration processis then continueduntil all the Aj are essentiallyzero. The final function g is then equivalent to 0 and the steady-stateprofile of ion temperature. Using this method it has been found that the convergenceof the iterative procedureis very rapid with the ratio I•X•/g•[reaching10-• after four repetitionsin + /•i+l + Bi/•i + /•i--1 '--- Di from j -- 0 to a maximum value, j max, determined by the attitude step width • and the peak altitude desired.The boundaryconditions appropriateto (19) appear as Ao -- 0 and, for d T•/dz -- O,Ajma. , • -- Aj .... Since (22) is in the triple diagonalform, the solutionsfor the Aj are found by first calculating the coefficients Ks and L• for j -- 0, j max (23a) (23b) and the j's refer to integration steps extending typical examples. The time-dependention temperature equation that includes heat conduction can be solved through application of the implicit integration method describedby Diaz [1958]. Again, the substitution0 = T? simplifiesthe procedure,and the time relaxation calculations are carried forward in the manner describedby Da Rosa [1966] in an application to the problem of electron temperatures. It is noted, however, that in the implicit method care must be taken in the choiceof time integration elements for the problem of electron and ion temperatures to avoid instabilities in the numerical ION TEMPERATURE IN THE UPPER ATMOSPHERE solution. Essentially, this restriction for the ion temperatureproblem limits the time elements Al to valueslessthan ß given in (18b). For this reason the calculation of steady-state temperatureprofilesby the relaxationtechnique ,ooo 3373 Tn'IOOO'K A / B I T.' 2600'K •' is excessively timeconsuming andcanbetterbe _ doneby meansof the meshtechniquedescribed / •' • WITHOUTHEAT CONDUCTION above.It shouldbe notedagainthat mesh techniquecan be useddirectly to obtain rapid solutions to the parabolic electron-ion equations of continuity when both chemicalreactionsand particlediffusionare acting. ,oo 200 1000 4. APPLICATION TO IONOSPHERIC ION TEMPERATURES 1500 2000 2500 TEMPERATURE $000 ('K) Fig. la. Ion temperature profile calculated using the ionic composition data of Johnson [1966] with T• -- 1000øK, Te = 2600øK, and It has been shown in equation 16 that, neglecting ion heat conduction,the ion tempera- I -- 90 ø. Curve ture (or X) is determinedby the weighted ratios of the neutral gas and ion number densities.Becausethese ratios changerapidly with altitude it is found,as pointed out by Hanson [1963], that there shouldexist a transitional A includes the effect of ion heat conduction, while curve B has been calculated without conduction.The increasedthermal energy given to the ion gas above 600 for curve A is transported downward and deposited between 450 and 600 km, slightly raising the ion temperature above the values obtained when only local production and loss of energy is considered. A maximum gradient of 4.7øK km-• is found at 475 behavior for X and T•. Below 250 km the neu- tral gasdensityis usuallylarge enoughto in- km. sure that. S,,Tf/•/S6 >> 1 for the normal range of midlatitude electron concentrations giving X << 1 and T, ___7'• to within a few per cent. Near the ionospheric ?•-layer peak, the actual altitude dependingupon the particular distributions of neutral and chargedparticle densities and temperature, the rate of ion to neutral energytransferis not large enoughfor small valuesof (T, -- T•) to matchthe rate of heat input from the electrongas,and T, beginsto rise toward T6 (X --> 1). The conditionfor IOOO Ti = 2600øK I _ Te-T . ß' Te_T n 800 WITH HEAT CONDUCT• ,,' •...-'"'"'•' WITHOUT HEAT • ....,•.•....-'" CONDUCTION 600 _ lying midwaybetween7'• and 7'6is givenby (12) asX - % or,in termsof $6and&, 400 SnTe a/2 = 1 [X = «1 (27) The altitude variation of X in the lower transi- tion region(X < •/•) is complexevenwhen the perturbingeffectof heat conduction is neglected.Fortunately,the numericalsolutionsfor X(z) showthat the effectsof ion heatconduction are principallyfelt 'for X > % and that the ion temperatures calculatedfor X < are unchanged to within5% whenthe heatflux is included;a slightaltitudeuncertaintymay be introduced.This point is illustratedin Figures la and b for the electron and ion density profilesof Johnson [1966]matchedwith the 2000 ! 0!.?_ I 014 I 0!.6 i 018 ! 1.0 ION TEMPERATURE FRACTIONAL SEPARATIpN,'•' Fig. lb. Profile of the ion temperature fractional separation X using the data of Figure la. The use of the variable X ---- (T• -- T,,)/(T6 -T•) facilitates the organization and comparison of ion temperature data taken under different aeronomic conditions. Below X = % heat con- duction plays only a small part in determining the value of X (or T•), and calculations can be made assuming a local production and loss of ion thermal energy. Above X ---- % heat conduction can strongly• affect the ion temperature profile. 3374 PETER M. BANKS parameters of a 1000øK model neutral atmos- phere of Nicolet [1967] (see Table 3) and a constant electron temperature of 2600øK. For z < 600-km curve A, which includesthe effect of thermal conduction,differs from curve B, which omits conduction,by either 60øK or, equivalently, 10 km. Thus for X < • it is convenientto analyze the problem of ion temperaturesat midgeomagneticlatitudes by omitting thermal conduction;for X > x/• conduction must be included. This separation is not valid for all conditions,however, and the Thomsonscatter data of Evans [1965] (personalcommunications,1967) can be interpreted as indicatingthat the conditionX -- • may not be reached before heat conduction vvvvv vvvvv effects vvvvv are several times larger than the local ion-neutral gas coolingrates. Nevertheless,in the following sectionsit is convenient to use the X variable to separate those atmosphericregions where only local (X < •) and local plus transport (X > •) effectspredominate. Ion temperatures; X < x/•. In the real ionospherethe determinationof X is made complicated, even in the absenceof thermal conduction, by the relatively independentvariations of S• and Se. While the appropriate neutral gas densities needed for the evaluation of S• can be obtained from models of the neutral vvvvv atmos- phere, the derivation of model ionospheres consistent with the chosenneutral atmosphereis difficult and subject to considerableuncertainty. Hence a theoretical analysis cannot be relied upon to give accurate values of real ion temperaturesbut rather shouldbe used to indicate the important aeronomicprocessesthat determine them. In the followingsectionsthesemajor factors that influenceion temperatures in the regionX < • are discussed. For the regions near the F•-layer peak it is permissibleto ignore ions other than 0 + if n+(O)/n+(H) >> 16 and n+(O)/n+(He) >> 4. Further, if atomic oxygen is the dominant ion coolingagent,then X is givenby 1 X = I q-2.1X 10-7n(O)T.a/•'/n. (28) which, with (12), leads to a result similar to that of Brace et al. [1965]. Since both the neutral gas and electron concentrationsare sub- ject to variation throughout the courseof a day, it is clearthat X at a given altitudewill ß o o ION TEMPERATURE IN THE UPPER ATMOSPHERE also change.As an example, a rise in the neutral gas temperature from 800 to 1000øK would lead, usingthe modelsof Nicelet [1967], to a factor of 2.9 increasein n(O) and a consequent decreasein X. Countering this effect, however,are the diurnal variations in n, which, from the data of Evans [1965], appear to act in such a way as to give no the dominatingcontrol in the diurnal variations in T•. Using the simple model describedabove,the main features of the ion temperature transition, which begins near the F•-layer peak, can be explicitly obtainedfor the simple model of a singleion species(0 +) and neutral gas (0) in diffusive equilibrium and a constant electron temperature. In this ease So -- $o• exp (--z/ H•) and S• -- S,• exp (--z/H,,), whereH• and H, are the scale heightsof the ion and neutral gases,respectively,and z is measuredrelative to the referencepoint. a. Using equation 16 the altitude dependenceof X is, within the approximationsmade, X= 6OO _ 500 _ 3375 Te - •000 øK N• _ Tn=IOOOøK 800 ø 650 ø 400 500 200 I 0 I I 0.2 I 0.4 I I . I 0.6 I 0.8 I 1.0 ION TEMPERATUREFRACTIONAL SEPARATION, X Fig. 2. Ion temperature fractional separation X for an electron temperature of 2000øK, a magnetic dip angle of 90ø, and 650øI•, 800øI•, and 1000øI• model atmospheres of Nicelet [1967]. The ionic composition has been taken from Johnson [1966]. For increasing neutral atmosphere exospheric temperatures the ion temperature fractional separation at a given altitude decreases. 1 1+ • exp [--z{l/H. -- l/H,}] density data of Johnson [1966], and three model atmospheres(1000, 800, 650øK). For [X < •1 (29) decreasingexospherictemperaturesthe transiIf H• • 2H,, which is generally considered tion zone moves to progressivelylower altito be true for conditionsof diffusiveequilibrium tudes,and the maximum ion temperaturegradibetween the dominant O* ions and the neutral ent increases and moves downward atmosphere,the altitude variation of X must proceedat leastasrapidlyas km-• (530 km) to 4.5øK krn-• (460 km) to 1 X= 1+ • and the transition exp [--•/2H,] distance • valuesX• and X• is givenby between the from 2.8øK 6.6øK km-• (400 kin). It must also be noted that the Thomson scat- ter data of Evans [1965] (personalcommunications, 1967) generally show dTe/dz > 0 for z > 400 km, which by equation 28 implies an increasein the width of the ion temperature transition zone. However, this may be a particular feature of solar minimum and during solar maximum the work of Geisler and Bowhill [1965] would indicate dTe/dz <: 0 with a narrowingof the transitionzone. While S• generally decreaseswith altitude, Thus the transition distance between X•0.1 there occur changesin slope correspondingto and X• -- 0.5 is 4.4 H•, which, with the 1000øK the atmospherictransition zonesbetweenatomic modelatmosphereof Nicelet [1967] (seeTable oxygen, helium, and hydrogen dominance.For X < %, however, these changesare not im3) is about 270 •. A more detailed analysishas been made of portant in comparisonwith the uncertainty in the altitude dependence of X includingthe ef- So, which dependsdirectly upon n,(eff), that fectsof energylossesto the other neutral gases is, the weightedvalues of the 0 +, He+, and H + of the atmosphere.The resets, shownin Figure ion densities. For solar maximum the densities 2, were obtainedusingTo -- 2000øK, the ion of He + and H + are sufficiently low such that 3376 PETER M. BANKS n,(eff) --n*(O) for X < % [seeBauer, 1966]. For quiet and moderatesolar conditions,characterized by exospheric ternperatures lower than 1000øK, however,both Itc + and It + ions are presentin appreciablenurnbersand, while still minor in terms of density, can be the major recipientsof thermal energy from the electron gas. For the regionsof chemical reaction and also for diffusive equilibrium as a minor ion, the equilibrium ratio between It + and 0 + is given [Hanson and Oftenburger, profile of Johnson [1966] has been applied to a cool (650øK) neutral atmospheremodelwith It + density profiles calculatedfrom (32). The 1961] as stances. results,shown in Figure 3, indicate that the inclusion of H + lowers the altitude at which a given value of X is reachedand, in general,decreases the width of the X transition zone. In contrast,He+ ions, becausetheir abundanceis not great and their heating rate is a factor of 4 lessthan that of It +, can be neglectedin the ion energybalance (X < •) in most circumFinally, it shouldbe noted that sinceT{ _ - T,for z < 250 km it is possibleto use Thom- 8 son scatter data to obtain the diurnal varia- which, for low exospherietemperatures,implies tions of the neutral gas temperature, a quana large concentration of H +. Table 4, taken tity which has not been consistentlymeasured on a worldwidescale.Complicationsarise,howfrom the model atmospheresof Nicolet [1967] lists the equilibrium ratios at various altitudes ever, in choosinga proper altitude for measto show that during solar minimum the pres- urement; at. low altitudes the presence of ence of H + cannot be ignored. The horizontal molecular ions can make interpretation of the bars indicate the altitude transition between Thomson scatter spectra difficult, while at 0 + and H + dominance in terms of heat received higher altitudes there may exist diurnal variafrom the electron gas. Although experimental tions in T{ correspondingto increasesabove data are sparse,both the Electron 1 satellite T• due to enhanced daytime electron concendata of Istomin [1966] and the Thomson scat- trations. Further, since models of the neutral ter data of Carlson and Gordon [1966] taken atmosphereare characterizedby the nearly isoat Areeibo indicate that H + is an important thermal exospheric temperature, it is necesionic constituent in the 400-600-km regions sary to choose an altitude of measurement during solar minimum, but, as would be ex- where the gradient in the neutral gas temperapected on the basis of changesin the exo- ture is small. Analysis of the data of Evans spheric temperature, there are large fluctua- [1965], Doupnik and Nisbet [1966], and tions in the heights of constant n+(H)/n+(O) M. Petit and P. Waldteufel (personalcommunithroughout the day and for different seasons cation, 1966) indicatesthat z = 225 km may representan adequate compromisefor the cool [seealsoFarley, 1966b]. To show the effect of H + upon ion tempera- neutral atmospheresfound during solar miniture profiles for X < •, the 0 + ion density mum. During solar maximum it may not be TABLE 4. Altitude, km Hydrogen Ion ConcentrationRatios, N+(H)/N+(O) T• -- 600ø T. = 650ø T. = 700ø T = 750ø T. = 800ø 250 300 350 2.9 (-3)* 1.1 (-2) 4.2 (-2) 1.8 (-3) 6.2 (-3) 2.1 (-2) 1.0 (-3) 3.4 (-3) 1.1 (-2) 6.0 (-4) 1.8 (-3) 5.2 (-3) 3.5 (-4) 9.8 (-4) 2.7 (-3) 400 1.6 (-1) 7.2 (-2) 3.2 (-2) 1.5 (-2) 7.2 (-3) 450 5.8 (-1) 2.4 (-1) 9.9 (-2) 4.2 (-2) 1.9 (-2) •00 2.• (0) 7.8 (-•) 3.0 (-•) •.2 (-•) •.0 (-2) * 2.9 (-3) --- 2.9 X 10-8. Data taken from ModelsJ 1.53D [Nicolet,1967]. ION TEMPERATURE IN THE UPPER ATMOSPHERE 3377 siderable error. Here, typical profiles of ion temperature will be obtained using both theoretical and experimental[Taylor et al., 1963; Johnson,1966] modelsof the ionic composition. 500 The calculated electron and ion concentrations are based upon a model of O* and I-I* in dif• 400 fusive equilibriumabove a given referencealtitude a [see Bauer, 1966] where the electron concentrationis known and ne• --• n•*(O). For the ion temperatureproblem, He* and its rela300 H*IONS B = WITH H+IONS tion to O* and I-I* is not of dominating importance and will be neglected; the He* deetron heating rate is only four times greater 200 I 0.2] I 0.4I I 0.6I I O.[8 ] 1.0 0 than that of 0 +, and there is conflictingeviION TEMPERATURE FRACTIONALSEPARATION,• dence for n* (He)/n* (0) ratios at altitudes beFig. 3. Effects of H + ions upon an ion temlow 1000 km. While He* becomesan important perature profile for T• -- 650øK, Te -- 2000.øI<, ion for X > • during active solar conditions I -- 90 ø with the electron concentration data of characterized by elevated exospherietemperaJohnson [1966]. In curve A only O+ ions were used, while in curve B H + and O+ ions were intures,P,(He+)/P•(O +) -- 4, and P•(H+)/P•(O +) cluded in the ratio given by equation 32. The = 16, indicating that although the O*-I-Ie* influence of H + is strongest in the cooler models transition causessome changein the high-altiof the neutral atmosphere where n(H)/n(O) ratude ion heating, it is not as significantas the tios are enhanced. 0+-I-I * possibleto obtain accurate values of T• (exospheric) since the ion and neutral gas temperatures are equal only near 200-225 km, while the neutral gas temperature reachesits exospherievalue at somewhathigher altitudes. Ion temperatures; X > %. For X > x/• heat conduction along the lines of magnetic force can play an important part in keeping the ion gasescooler than the electron gas by transportingion thermal energy from high to low altitudes. While the introduction of this heat flux does not greatly alter the ion energy balancefor X < %, at the higher altitudesthe conductionterm can becomemany times larger than the ion-neutral energy transfer rate (see transition characteristic of low and medium solar activity. Further, in He* the energy production-thermal conduction ratio changesby only a factor of 2 from the O* ease; for H* a factor of 4 is involved. The model electron and ion concentrations used here have been calculated from the rela- tions, I + T,/T½ na (O)J -IFn•-•-•-• Lna(H) I (33a) ITn/Tg--1 1+ õLn(O) Lna(N) Figure 10). Nevertheless,it is not possibleto neglect completely the cooling action of the high-altitudeneutral gassincefor low ion densities the ion temperature profile may be de- n+(H) I termined by an energy budget where both lossesand conductionare important; that is, there may be a changein dT,/dz from positive (33c) to negativevalues. For the determination of ion temperatures which represent an extension of the work of at high altitudesthe densitiesof both the neu- Kockarts and Nicolet [1963] to the casewhere tral and chargedgasesmust be known. The T• -•= T{. Figure 4 showsa typical set of electheoreticalproblemof calculatingan ionosphere tron and ion concentrationprofiles calculated that is consistentwith a given neutral atmos- for T -- 800øK, ne• -- 2.0 X 10' cm-8, and n.- + 8[.(H) .(o) 1 phericmodelis difficultand can lead to con- various values of Te. L J 3378 PETER M. BANKS 1500 .. when T, = T,, showingthat H(e) • [1 + T,/T,] H(O) for all ion density ratios. Although ne decreasesmonotonically with altitude, the effective ion heating density, n•(eff) used in S, does not as a result of the rapid increasein n+(H) in the regionsbelow n+(H)/ n+(0) = 1. In terms of the ion effectiveheating scaleheight H•(eff), the altitude dependenceof n• (eft) is 1000 ,• t•••($00km)= 2x105cm-$ ..,o, .',H,.\ ni(eff) = n•a(eff)exp [--z/Hi(eff)] 500 (35a) where :500105 104 105 CHARGEDPARTICLE NUMBER DENSITY (crn-$) Fig. 4. Electron and ion density profiles for an 800øK neutral atmosphere with various electron temperatures and a 300-km electron concentration boundary condition of 2 X 10• cm-a. The ion densities apply only to an electron temperature of 1500øK but are similar in form at higher temperatures. In the regionsX ;> x/• there occur transitions in the major atmosphericand ionosphericconstituents that affect the density factors appearing in the ion heating and energy loss rates, leading, in certain cases,to changesin the sign of dT•/dz. In terms of equation 16, both & and Sehave a complexaltitude dependence,and if the equivalent scale height for Se is smaller than that for S• it follows that dT•/dz < O. In the region X < x/• where O+ and 0 are the predominateion and neutral gases,H(&) :> [1 + T,/T•]H(&)ensuring that dT•/dz > 0. Above X -- • there occur two important transitions; namely, from 0 to H and O+ to H +, the perturbing effects of He being neglected. [1 -Jr-15/(H+)]H(e) Hi(eff) ----/(It+)[1 -]- 15(T,/T,){i(It +) -- 1}] [(16/15) -- f(H+)] and ](H +) : n+(H)/ne. The altitude behavior of n•(eff) for a typical electron-ionmodel with T, = 2500øK, T = 800øK, and ne• =. 2 X 10= cm-a at 300 km is shown in Figure 5 indicating the increase with altitude in the density factors that enter into the ion gas heating rate for the atmospheric [0 -• H] and ionospheric [0 + -• H +] transition regions.Although there is an increasein n,(eff) with altitude, analysis of equations33 and 35 in terms of both S, and & indicates that in the transition regions the density factors in the ion-neutral lossesalso Tn'800øK N+(O)N+(H) Ne %- :,5OO'K •/ \ Ni (elf) The influence of these transitions upon the electron concentrationprofile can be calculated from equation33 and expressedin terms of the electronscaleheight H(e) and the atomic oxygenscaleheightH (0) as n, -- neaexp [--z/H(e)] (34a) where • • I I0$ [1 q- Te/Ti]H(O) 9(n)j =[I q-128 n(O)/ __ 1nu8 n(O) [1 q- T,/Ti]H(O) 1 I 104 105 CHARGEDPARTICLE NUMBERDENSITY (cm -$) (34h) Fig. 5. Electron, ion, and effective ion heating densities for a neutral atmosphere of 800øK, an electron temperature of 2500ø, and the boundary electron concentration of 2.0 X 10• cm-a. Although the electron concentration decreaseswith height, the effective ion density for heating, n• (eft) = n+ (O) q- 16 n* (H), increaseswith height in the O* -- 1:[*transition region,affectingstrongly the high-altitude ion temperatureprofiles. --I1-+-•6 1n+(H)l/I1 n+(H) -] n+(O)_+_ n+(O)l ION TEMPERATURE IN TIlE increaserapidly (the 0 + -- 0 energy loss rate is a factor of 7 smaller than the It + -- It rate, for equal densities). In fact, for particular ranges.of parametersthat can be derived from (33), (34), and (35), it has been found that $, decreasesless rapidly with altitude than Se, implying that the ion temperature will decrease with altitude [dT,/dz < 0] even without thermal conductionacting. This behavior cannot continue outside the 0 + -- It + and O -- It transition regions,however,since in the limit /(It +) -• I it is foundthat S,/Se cc exp (--z/ [1 q- T•/T,]It(I-I)) and X must againincrease UPPER ATMOSPHERE 3379 result for valuesof (To -- T,) larger than those associatedwith local energy production and loss. Examples of ion temperature profiles calculated for atmosphericmodelsspanninga wide range of atmospheric conditions for different electron temperatures and reference electron concentrationsare shown for I = 90ø in Figures 6-9. It is found that the conduction effect is generally large during conditions of low electron concentrationwhen n,(eff) falls below 10• cm-8 at approximately 500 kin. For larger values the transition X -• I occursrapidly in the 400-600-km region, preventing any substantial differencebetween To and T• at higher altitudes. In the present examples, n•(eff) is determined by the choice of the electron concentrationboundarycondition,the chargedparticle temperatures,and the n+(H)/n+(O) ratio implied by the particular atmosphericmodel. Thus the curves in Figures 6-8 with the larger with altitude giving dT•/dz > O. It shouldbe noted that in the case of the real ionosphere and neutral atmosphere there may be a considerable deviation of the n+(It) profiles from those correspondingto 0 + -- It + hydrostatic equilibrium.If, for example,the n+(I-I) density is smaller than that predicted for hydrostatic equilibrium, there may occur situationswhere values of n•, tend to have smaller values of 0 + -- It energy lossesoutweigh those due to (To -- T{) at high altitudes.Likewise,with the 0 + -- O, which leads to a more rapid increase cooler model neutral atmospheresthere is a in the ratio S,•/S• and decreasein X with alticonsiderableenhancementof n+(H) at the low tude than indicated for the case of hydrostatic altitudes and, for given values of n,, and Te, equilibrium.In fact, large decreases in X above there is again a decreasein (To -- T{). The X -- •/• for similar atmospherictemperatures part thermal conductionplaysin bringingabout and electronconcentrationsmay indicate a sub- an increased ion-electron temperature separastantial departure of the ion density profiles 1200 from the equilibriumconfigurationin terms of there beinga deficiencyof It +ions. Te = 2000OK Experimental evidencefor dT•/dz • 0 in I000 the transition region is not conclusive.The 1963 Thomsonscatter data of Evans [1965] for z > 600 km include several monthly averaged ion temperature profiles where lower values of .,• 800T• are seen at the higher altitudes. Because • 700there exists considerable experimental uncertainty in the data for these high-altitude • 600 regions, however, further work is needed to 500 05IXIO s establishthe true importance of high-altitude neutral gas cooling. For X > •/• thermal conduction acts to re300 ! ! 0 0.2 0.4 0,6 . 0.8 1.0 duce gradients in ion temperature and leads II00 - Tn =800øK I-' 900 - _ • 400 Ne(.•00krn) =I jr; X•i I - ION TEMPERATURE FRACTIONALSEPARATION,Y to situations where substantial separations can exist between the high-altitude electron and ion temperatures. The effectivenessof conduction dependsstrongly upon the rate of ion energy Fig. 6. Ion temperature fractional separation X for an 800ø neutral atmosphere, an electron temperature of 2000øK, a dip angle of 90ø, and productionand, hence,the factor n•n, (eft). For tration boundary condition. Ion heat conduction large ion densitiesthe thermal conductivityis not adequateto carry downwardthe additional heat obtained from the electron gas that would various values is included. of the 300-kin electron concen- The lower electron concentrations en- hance the importance of heat conduction, leading to larger separationsbetween ion and electron temperaturesor, equivalently, lower values of X. PETER 3380 M ß BANKS 1200 T. - 2000øK / I I - 90 ø Ne(:500k) IXIO cm 2XIO5 5XIO5 be found at midgeomagneticlatitudes during high and low solar activity, the measured ionic distributions of Taylor et al. [1963] and Johnson [1966] have been used with 1394 and 800øK model atmospheresto obtain the respective examples of high and low solar activity shownin Figures 11a and b. A tentative conclusionis that the high-altitude separation between electron and ion temperatures may be smaller IXIO6 at solar minimum than at solar maximum even when the possibility of enhanced I-Ie* densities is considered. This con- clusion may be weakened if the true n*(It) densities $oo o 0 2 0 4 06 0.8 1.0 IONTEMPERATURE FRACTIONAL SEPARATION, •r Fig. 7. Ion temperature fractional separation X for a 1000øK neutral atmosphere,an electron temperature of 2000øK, a dip angle of 90ø, and various values of the 300-km electron concentra- tion boundary condition. Ion heat conduction is included. For higher exospheric temperatures the n*(II)n*(O) ratio is decreased limiting high- altitude energy production,enhancingheat conduction, and giving smaller Values of X in the isothermalregion. tion at high altitudesis seenin Figure 10 where the production,conduction,and ion energyloss terms are plotted as a function of altitude using the information obtained in deriving the profile shownin Figure 7. Above 520 km the conduction term is negative in the ion energy balance, correspondingto a downward transport of ion thermal energy. This energy reappears below 520 km where there is a net influx of ion thermal energyfrom above. In terms of ion temperature profiles to be expected during the course of the solar cycle it is seen that there are counteracting tendencies. For quiet solar conditions the electron concentrationtends to be low, but, for cool model atmospheric models and hydrostatic are reduced below those consistent with the experimental models used here [see, for example,Walker, 1967, and the implications of the Alouette data of Chan et al., 1966]. Thus during solar maximum generallysmaller values of X shouldbe seenin the regions700-1000 km in comparisonwith solar minimum conditions. The actual ratio of values depends strongly upon the highly variable H* distribution in the 500-1000-km range. Comparisonwith experiment. Measurements 1200 I100 I000 900 800 700 600 500 400 3000 I 0.2 I 0.4 I I 0.6 I I 0.8 X 10 I 1.0 ION TEMPERATUREFRACTIONALSEPARATION,X; equilibrium,the n+(It)/n+(O) ratio shouldbe Fig. 8. Ion temperature fractional separation large [see Carlsonand Gordon, 1966], leading X for a 1394øK neutral atmosphere, an electron to substantial values of n•(eff). Likewise, for temperature of 2000øK, a dip angle of 90ø, and active solar conditions the electron concentra- tion generally is large, but the n+(It)/n+(O) ratio is small. In both casesthe dependenceof To, which indirectly determines T• and the magnitude of the ion thermal conductivity, upon position in the solar cycle is not yet known. Nevertheless,to show ion temperature profiles,which may be indicative of those to various values of the 300-km electron concentra- tion boundary condition. Ion heat conduction is included. The density of H + ions is low in this model, permitting the energy balance heat conduction term to greatly exceed the high-altitude energy loss rate. He + ions, normally a major constituent of the high ionosphere for elevated exospheric temperatures, have been neglected but, as described in the text, their inclusion does not greatly alter the ion temperature profiles. ION TEMPERATURE IN THE UPPER ATMOSPHERE 1200 and observed ion temperatures are shown in Figures 12 and 13 for the September 1963 and November 1964 data of Evans [1965] (personal communications,1967) using the a,Te= 1500øK I100 b,Te = 2000OK c,Te = 2500OK d,Te "•,000*'K measured values of T, 900 ,.,, 800 • Tn= 1•94 øK a,,•?•00' /2/1 800* 700 600 500 400 :•oo o I.O ION TEMPERATURE FRACTIONAL SEPARATION,'•z' Fig. 9. Ion temperature fractional separation X for a 300-km electron concentration 3381 bound- ary condition of 2.0 X 10• cm-•, a dip angle of 90ø, and model neutral atmospheres with exospheric temperatures of 800ø, 1000ø, and 1394øK. Changes in the indicated electron temperatures are less important than changes in the neutral atmosphere gas densities and relative composition in determining the profiles of X. Changes in the H + ion distribution from hydrostatic equilibrium would also affect strongly the values of X for z 2> 700 km. of ion temperaturesin the ionospherehave been made by Nagy ei al. [1963], Evans and Lowenihal [1964], Boyd and Raiti [1965], Evans [1965] (personal communications, 1967), Doupnilcand Nisbei [1966], Farley [1966a, b], .Knudsonand ,Sharp [1966], Wali [1965], Carru ei al. [1966], and others. The comparison of the present theory with these results is difficult owingto the lack of simultaneousmeasurements of the appropriate atmosphericand ionospheric parameters. Presently, Thomson scatter experiments provide the best experimental verification for the ion temperature transitional behavior; satellite data, averaged over wide spatial regions for different aeronomic conditions,do not yield sufficientlyprecise information about the ion temperature profiles. By adopting a model of the neutral atmosphere consistentwith the measurements of T• in the 200-250-km range (where T• T•), and using the observedaltitude profilesof n, and T,, it is possibleto calculate profiles of and n, and neutral atmosphereshaving the indicated exospheric temperatures. While the agreement between experimentand theory appears adequatebelow 600 km, the effect of the ion thermal conductivity in lowering T• significantlybelow T, cannot be regardedas being fully establishedsince the experimental data terminate near the altitude where, at midgeomagnetic latitudes, there is an increasing separation between the theoretical profilesincludingand excludingconduction. Direct rocket measurements of electron and ion temperatures and electron concentrations between 180 and 365 km have been made by Nagy ei al. [1963] under conditionsthat indicated a neutral atmosphere characterized by T• -- 1500øK. It was observed that the ion temperature was essentiallyconstant,while the electron temperature varied between 2500øK (200 km) to greater than 3000øK (365 kin). Comparing these results with the somewhat cooler 1394øK atmosphericmodel used in Figure 8, it is found that the calculated(T• -- T,) -- 75øK at 365 km, a value lessthan the 175ø1( experimental uncertainty, and that the transitional behavior of the ion temperature should \\ Ne(500kin)2X105cm -$ E • Iooo _ CONDUCTION I0-I I00 I01 I0 z ION ENERGY BALANCE (eVcm-Ssec -I) Fig. 10. Altitude dependence of the different terms in the ion gas energy balance for a 1000øK model neutral atmosphere, an electron temperature of 2500øK, a dip angle of 90ø, and an electron concentration boundary condition of 2.0 X 10• cm• at 300 kin. The production and loss of ion thermal energy are equal near 300 km, but con- ion temperaturefor conditionsthat match duction becomesimportant at higher altitudes, preximatelythoseunder which the experimental acting as heat sink (-- sign) above 520 km and a data were obtained. Examples of calculated heat source (q- sign) below. 3382 PETER M. BANKS 1200 / I100 Tn I000 Ti Te 900 "• Ti (,JOHNSON • 800 ß • 700 6OO S(1965) 5O0 / 400 FOR 1200 .EST, ?EPT. 1965 ,• 500 800 I000 1500 2000 2500 Ne- 5.4X105 cm '3AT 500 km / :5000 TEMPERATURE ( eK ) Fig. 11a. Ion temperatureprofilesthat may be characteristicof high (Taylor) and low (John- 300 " 800 I IOOO I 15OO / I 2000 2500 2800 TEMPERATURE son) solar activity. For high solaractivity a neutral atmosphere with T• _-- 1394øK,T6 -- 2000øK, Fig. I2. Comparisonbetween computed and and the ionic compositionof Taylor [19,63] was experimentalion temperatureusing T,• _-- 1000øK chosen; for low solar activity a neutral atmos- and the Thomson scatter data of Evans [196,5] for phere with T• -- 800øK, T6 = 2500øK,and the 12{)0 EST, September 1963. H + ion den•ties were ioniccomposition of Johnson[1966]applied. computed for chemical equilibrium below 500 km and for dieusire equilibrium with O* •o• have begun at an altitude somewhatabove the rocketapogee. Sincethe conduction effectdependsuponthe magneticdip angle it would be expectedthat X -• 1 at the geomagnetic equatorbut that at progressivelyhigher geomagneticlatitudes for similar atmosphericand ionosphericconditions there would be increasinglylarger separations betweenT• and T•. Thus the data of Farley [1966a, b] taken at the geomagnetic equator, when comparedwith those of Doupnil• and ilOO IIOC ßDATA FROM EVANS (1967) LOOC FORDAYTIME, NOVEMBER, 1965 IOOO 90C 900 Ne(300km) ß2.9XlOScrn -$ WITH/ Tn=750øK - HEATTi WITHOUT HEATTI CONDUCTION/ CONDUCTION I 800 X (TAYLOR) 700 X ( dOHNSON ) • 70C- / •• 600- // J ....• _•.,.,.•'"'"Te(EVANS) "' 600 500 ,oo/ 400 •00 I 500 0 0.2. 0.4 0.6 0.8 ION TEMPERATUREFRACTIONALSEPARATION,•.. Fig. lib. Ion temperature fractional separation X for high and low solar activity usingthe data of Figure 11a. During solar maximum the I-I+ densitiesare greatly reduced,permitting heat conductionto carry downwardion thermal energy and the establishment of significant differencesbetween the high-altitude electronand ion temperatures. ¾ 700 i i000 / i 1500 i 2000 • 2 500 i :5000 3zoo TEMPERATURE (øK) Fig. 13. Comparisonbetween computed and experimentalion temperaturesusing T•- ?50øK and the Thomsonscatterdata of Evans (personal communication,1967) for daytime, November 1964. The effect of thermal conduction is seen to beginnear500km for the theoreticalresults,but the experimentaldata do not extendhigh enough to permit verificationof its effectiveness .in keep- ing the ion temperature below the electron temperature at high altitudes. ION TEMPERATURE IN THE UPPER 5. Nisbet [1966] from Areciboand Evans [1965] (personalcommunications, 1967) from MillstoneHill, showthat at low latitudesTs/T• approaches a valueof I morerapidlythan at high latitudes, where Te/T• -- 1.2-1.5 is not uncommonat 600 km and perhaps above. However, becausethe electron concentrationsare considerablyhigher at lower geomagnetic latitudes a direct comparisonis dillcult. Simultaneous satellite measurements of Te and T• wouldbe expectedto indicatedirectlythe effect of thermal conductionin maintaining T•/T• • 1 at high altitudes. Time-dependent ion temperatures. Below X -- • the time responseof the ion temperature to changesin the aeronomicparametersis determinedby the time constantß given by equation 18b; above this point conduction dominates. Table 5 lists ion temperature time constantsfor the ion temperature model shown in Figure 9 with T• -- 1000øK, T• -- 2000øK, andns (300km) -- 2 X 10• cm-•. The results indicatea rapid increasein ß with altitude,with there being a shift from ion-neutral (S,) to electron-ion (S•) control at the point where S•T2•/2 -- S• or X --• %. Changesin the aeronomic parametersthat occur in times shorter than ß will result in a time lag in the response of T•. Thus, during a solar eclipse,changesin Ts and ns may be rapid enoughto substantially upset T• from steady-state conditionsnear ATMOSPHERE 3383 CONCLUSIONS The principal resultsof the precedinganalysismay be summarizedas follows' 1. The use of the ion temperature fractional separation (X) facilitates the analysis of ion temperatureprofilesin terms of the relatively independentvariations in the electron and ionic composition, the electron temperature, and the neutral gasdensities.Further, it is found that the ion energy budget for X < % does not involve significant transport contributions; for X •, % sucheffectsmust be introduced. 2. For X < % there appears to be good agreementbetweenthe theory basedupon ion heating by the ionosphericelectron gas and experimentalresults when allowanceis made for the difficulties involved in evaluating the theoretical modelswith appropriate aeronomic parameters consistentwith the experimental conditions. 3. The heat received from the electron gas by H + and He+ is important in the over-all ion energy balance when (n+(It)/n+(O) >_ 0.03 and n+(He))/n +(0) >_ 0.12, respectively.Since there exists considerableuncertainty regarding the ionosphericHe+ densitiesfor X < %, and becauseP•(H +) -- 4P• (He+), it appears that I-I+ heating is of more importancethan I-Ie+ heatingin the ionosphere.Thus, during periods of low and medium solar activity characterized by low exospherietemperatures,I-I+ can play 500-km altitude. Above X -- %, ß becomeslarger than 10 an important part in determiningthe ion temminutes. However, solutionsfor the time-de- perature profile. 4. For X > % the transitional behavior of pendent ion temperature equation indicate that the ion thermal conductivity is adequate ion temperature is modified by the effect of to permit changesin X from 0.7 to 0.9 in less thermal conductionsuch that a significant difthan 10-15 minutes.Hence,there shouldbe no ference between T, and T• appears at high long-term coolingor heating effectsexcept at altitudes. Since thermal conduction is limited the geomagnetic equatorwheresubstantialtime to directionsparallel to the geomagneticfield lags may occurat high altitudesbecauseverti- lines, conductionis relatively ineffectiveat the geomagneticequator in altering ion temperacal thermal conductionis greatly retarded. ture profiles. Thus for equivalent aeronomie TABLE 5. Ion TemperatureTime Constants Altitude, km 300 400 500 600 700 Time Constant,sec 3.6 15. 72. 240. 540. conditions (Ts -- T•) should increase with geomagneticlatitude. The magnitude of the conduction effect depends strongly upon the effective ion density for heating, with large values leading to relatively small electron and ion temperatureseparation. 5. The cooling effect of the high-altitude IX > %] neutral gases,principally It and I-Ie, can bring about a changein sign of dT•/dz 3384 PETER M. BANKS and decreaseT, significantlyunder certain conditions in the regionswhere 0 + to I/e + to I-I transitionsoccur.Departures of I-I+ density profiles from hydrostatic equilibrium tend to enhance this effect, especiallyfor cooler neutral atmosphereswhere the 0 to I/transition occurs rapidly giving an effectiveion heating density scale height, which is less than that of the neutral gas. 6. The ion temperature time constant for changes in aeronomic conditions is less than 10 minutes for z < 600 kin, leading to the conclusion that for normal ionospheric behavior characterizedby slow density and temperature variations the ion temperature has a steadystate value. During periodswhen rapid changes occur there may be a substantialdelay before the ion temperature reaches its new steadystate value. Boyd, R. L. F., and W. $. Raitt, Positive ion temperatures above the F-layer maximum, Space Res., 5, 207-209, 1965. Brace, L. I-I., N. W. Spencer, and A. Dalgarno, Detailed behavior of the midlatitude ionosphere from the Explorer XVII satellite, Planetary Space Sci., 13, 647-666, 1965. Carlson, I-I. C., and W. E. Gordon, Radar spectrographic estimates of ionic compositionfrom 225 to 1400 kilometers for solar minimum winter and summer conditions, J. Geophys. Res., 71, 5573-5578, 1966. Carru, I-I., M. Petit, and P. Waldteufel, Measures de temperatures electroniques et ioniques par diffusion incoherent, paper presented at Interunion Symposium on Solar-Terrestrial Physics, Belgrade, 1966. Chan, K. L., L. Colin, and $. O. Thomas, Electron densities and scale heights in the topside ionosphere: Alouette I observationsover the American continents, NASA SP-3031, 1966. Chapman, S., The viscosity and thermal conductivity of a completely ionized gas, Astrophys. J., 120, 151-157, 1964. Chapman, S., and T. G. Cowling, The Mathematical Theory o] Nonuniform Gases, Cambridge University Press,Cambridge,1952. 7. Diffusion cooling, which was mentioned with respectto equation 3, has been evaluated for its effect upon the present ion temperature profiles.It is found that st 600 km a downward Dalgarno, A., M. B. McElroy, and R. J. Moffett, Electron temperatures in the ionosphere,Planediffusionvelocity greater than about 7 X 108 tary SpaceSci., 11,463-484, 1963. cm sec-• would be required to give a' heating Da Rosa, A. V., The theoretical time dependent effect equal to that of thermal conduction.If thermal behavior of the ionospheric electron gas, J. Geophys. Res., 71, 4107-4120, 1966. present,such a diffusionvelocity would yield a Doppler shift of nearly 200 cyclessec-• at a Diaz, $. B., Partial differential equations, in Handbook o] Automation, Computation, and Thomsonscatter radar frequencyof 430 Mhz; Control, edited by E. M. Grabbe, S. Ramo, and such values do not appear to have been exD. E. Wooldridge, John Wiley PublishingComperimentally observed. pany, New York, 1958. Acknowledgments. 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