The ionosphere of the Earth

Transcription

1 Chapter 3 The ionosphere of the Earth The ionosphere of the Earth consists of weakly ionised gas. No more than about a fraction of the order of 10 3 of the atmospheric molecules are ionised. This means that the mass of neutrals surpasses overwhelmingly that of ions and electrons, and the result is that neutrals also have a major effect on the motion of the ionised species, whereas the effect of the latter ones on neutrals is weak or slow. As a result of ion chemistry, the ionosphere consists of ions of several species, and relative abundances of different ion species varies with altitude. A simplified picture of ionospheric plasma dynamics is obtained by considering that the atmosphere consists of gas of three particle species occupying the same volume. The three species consist of neutral molecules, ions and electrons. 3.1 Ionospheric regions The atmospheric regions discussed in the previous chapter contain ionized species as follows. Thermosphere: positive ions and electrons Mesosphere: positive and negative ions of small amount Stratosphere and below: no free electrons The region of main ionization exist thus in thermosphere. Historically the ionosphere has been divided into D, E and F layers, but nowadays it is more common to speak about D, E and F regions (Fig. 3.1). The lower part of the D region is in the mesosphere. The approximate altitude limits for these regions and typical daytime electron densities are: D region: km, n e = m 3 E region: km, n e = m 3 F region: km, n e = m 3. 39

2 onospheric variability iations h in upper F er of 40 CHAPTER 3. THE IONOSPHERE OF THE EARTH variations tion s of magnitude nd D regions nd space from seconds to [from Richmond, 1987] Figure 3.1: Typical ionospheric electron density profiles. The ionospheric layers were originally observed from radio wave reflections at different altitudes. It was then considered that the reflections take place at separate layers. Later, however, it was noticed that the reflections were from bulges in a continuous electron density profile, and therefore we nowadays rather speak about regions. The nomenclature D, E and F has turned out to be useful since the regions are physically different because of their different ion chemistry. A bulge called the F 1 layer forms in the bottomside F region electron density profile during daytime in solar maximum conditions. The D region disappears during night. At high latitudes the E region electron densities may increase drastically in the nighttime exceeding even the F region peak during auroral electron precipitation. Ionosphere has great variability: Solar cycle variations: one order of magnitude in upper F region Day-night variation: several orders of magnitude in lower F, E and D regions Space weather effects based on short-term solar variability (from seconds to weeks): even several orders of magnitude in lower F, E and D regions

3 3.2. BASICS OF PLASMA DYNAMICS 41 Figure 3.2: The most typical ions species and the corresponding electron density profile. The ion composition changes in accordance with the change of molecular constituents in the neutral atmosphere. As illustrated in Fig. 3.2, NO + and O + 2 are the dominant ions below 150 km. At greater heights O + ions are more abundant and the number of H + ions starts to increase rapidly above 300 km. D-region (not shown) contains negative and positive ions and ion clusters. The distribution of ions depends on solar and magnetospheric activity. The temperature of the ionosphere is controlled by the absorption of solar UV radiation in thermosphere. Since this radiation is able to ionize molecules and atoms, free electrons as well as ions are formed which carry some excess energy with them. Electrons have a larger heat conductivity so the electron temperature becomes usually higher than the ion temperature. Ions are heavier and interact by collisions more strongly with the neutral gas. Therefore much of the ion energy is transferred to the neutrals and the ion temperature is less increased. Thus, typically T e > T i > T n. However, in the high-latitude E region the Joule heating of ions may result in T i > T e. Electrons may also be strongly heated by plasma waves. An example of ion, electron and neutral temperature profiles is shown in Fig Basics of plasma dynamics The density of neutral molecules is denoted by n, their mass by m, pressure by p, temperature by T and velocity by u. Correspondingly, these quantities for ions and electrons are n i, n e, m i, m e, p i, p e, T i, T e, v i and v e. When the plasma consists of a single positive ion species with a charge e, the plasma neutrality implies that n i = n e.

4 42 CHAPTER 3. THE IONOSPHERE OF THE EARTH 500 T n T i Altitude / km T e Temperature / K Figure 3.3: An example of neutral, ion and electron temperature profiles Continuity equations Separate continuity equations are valid for all three gas species. Since only a very small fraction of neutrals is ionised, the continuity equation of the neutral molecules is not altered by ionisation, and therefore eq. (2.5) is valid for them. Since ions and electrons are continuously created by ionisation and they also disappear in recombination producing neutral molecules, the continuity equation for charged particles is n i,e + (n i,e v i,e ) = q l, (3.1) t where q is the production rate per unit volume and l the loss rate per unit volume. The unit of these quantities is particles per second per unit volume, i.e. m 3 s 1. Note that eq. (3.1) actually contains two equations, one for ions and one for electrons. Since we assume that all ions are singly ionised, the production and loss rates for ions and electrons are the same. The production term is determined by the density and properties of the neutral molecules as well as the ionising radiation and the detailed ionisation process. The loss rate depends on the electron/ion densities and the rate coefficients of the recombination process. In the true ionosphere, more than a single ion species are present. Then a separate continuity equation is valid for each species and production and loss terms have contribution from the ionospheric ion chemistry. This is because some ion species disappear in chemical reactions producing ions of other kind. The ion chemistry produces links between the different continuity equations, which greatly complicates the solving of the ion densities.

5 3.2. BASICS OF PLASMA DYNAMICS Momentum equations and mobility For the neutral gas, the momentum equation (2.11) is valid. In most cases, however, we do not solve it but simply assume a neutral wind velocity u, which remains unaltered. This is a reasonable approximation, because collisions with the dilute ion and electron gas can change the neutral velocity only very slowly. Ions and electrons experience Lorentz force density ±en i,e (E + v i,e B), which adds a new term in the momentum equations n i m i ( t + v i n e m e ( t + v e ) ) v i = n i m i g + en i (E + v i B) p i n i m i ν i (v i u) (3.2) v e = n e m e g en e (E + v e B) p e n e m e ν e (v e u). (3.3) Here E is electric field, B is magnetic induction, p i and p e are the pressures of the ion and electron gas, and the ion-neutral and electron-neutral collision frequencies are denoted by ν i and ν e, respectively. Because force densities due to collisions between ions and electrons are necessarily much smaller than the force densities due to collisions with neutrals, these terms are dropped out from the very beginning. The momentum equations are greatly simplified, if the plasma is in a stationary and homogeneous state and gravitational field together with p can be neglected. Then the momentum equation for the ion gas is To solve the ion velocity, we use a notation Then eq. (3.4) can be written as A cross product with B gives and a dot product with B gives e(e + v i B) m i ν i (v i u) = 0. (3.4) F = ee + m i ν i u. (3.5) v i = 1 m i ν i (ev i B + F). (3.6) v i B = 1 m i ν i [e(v i B)B eb 2 v i + F B] (3.7) v i B = F B m i ν i. (3.8) By inserting this in eq. (3.7) we first obtain v i B = 1 [ ] e (F B)B eb 2 v i + F B m i ν i m i ν i (3.9)

6 44 CHAPTER 3. THE IONOSPHERE OF THE EARTH and then, inserting this in eq. (3.6), v i = 1 { [ ] } e e (F B)B eb 2 v i + F B + F. (3.10) m i ν i m i ν i m i ν i This can now be solved for v i. If e B is a unit vector in the direction B, i.e. B = Be B, [ ( ) eb 2 ] 1 + v i = 1 [ ( ) eb 2 ( ) ] eb (F e B )e B + F e B + F. (3.11) m i ν i m i ν i m i ν i m i ν i Here ω i = eb m i (3.12) is the angular gyro frequency, also called angular cyclotron frequency, of the ion. Using this notation and noticing that 1/(m i ν i ) = 1/(eB) ω i /ν i, we get v i = 1 eb ω i ν i ω 2 i + ν 2 i [ (ωi ν i ) 2 (F eb )e B + ( ωi ν i ) ] F e B + F. (3.13) By dividing F into two components F = F + F this can also be written as v i = 1 eb ω i ν i ω 2 i + ν 2 i { F + ( ωi ν i ) [ F e B It is conventional to present this equation in matrix form v i = k ip k ih 0 k ih k ip k i F x F y F z ( ) ] } ωi 2 F ν i. (3.14), (3.15) where k ip = 1 eb k ih = 1 eb ω i ν i ω 2 i + ν 2 i ω 2 i ω 2 i + ν 2 i (3.16) (3.17) k i = 1 eb ωi ν i = 1 m i ν i (3.18) and the z axis points in the direction of B. This result is obtained by writing eq. (3.14) in component form with F = F x e x + F y e y and F = F z e z. Eq. (3.15) can be written briefly as v i = k i F = k i (ee + m i ν i u). (3.19) Here k i is the ion mobility tensor. The tensor presentation is a convenient mathematical description of the result in which the velocity points in a direction different

7 3.2. BASICS OF PLASMA DYNAMICS Ion mobilities (blue) and electron mobilities (red) height (km) k ep k ip k ih Ik eh I k iii k eii k ih Figure 3.4: Ion and electron mobility tensor components calculated for Tromso, , 24 UT. s/kg from the direction of the effective driving force F. The mobility tensor is composed of three different mobilities, the Pedersen mobility k ip, the Hall mobility k ih and the parallel mobility k i (Fig. 3.4). It can be shown that the mobility tensor is invariant in rotation around the z axis. This means that the directions of the x and y axes can be freely chosen. In the frame of reference where the neutral air is at rest, the neutral wind velocity u = 0, the ion velcity v i = v i u and E = E + u B (Galilean transformation). Eq. (3.19) must be valid in all inertial frames of reference. Therefore so that and v i = k i e E (3.20) v i u = k i e(e + u B) (3.21) v i = k i e(e + u B) + u. (3.22) Eqs. (3.19) and (3.22) are alternative presentations of the ion velocity. The electron velocity can be calculated in a similar manner and therefore the result can be written directly. One must, however, notice that e must be replaced by e and the angular gyrofrequency for both electron and ion are taken as a positive

8 46 CHAPTER 3. THE IONOSPHERE OF THE EARTH quantity. Thus the components of the electron mobility tensor are where k ep = 1 eb ω e ν e ωe 2 + νe 2 (3.23) k eh = 1 eb ωe 2 ωe 2 + νe 2 (3.24) k e = 1 eb ωe = 1, ν e m e ν e (3.25) and the two expressions for the electron velocity are ω e = eb m e, (3.26) v e = k e ( ee + m e ν e u) (3.27) v e = k e e(e + u B) + u. (3.28) At altitudes where the electron collision frequency is much smaller than the electron gyrofrequency, k ep 0 and k eh 1/(eB). Then electrons move at a velocity v e = E B/B 2 in a plane perpendicular to B. This happens practically 500 Ion (blue) and electron (red) collision and gyro frequencies height ν in ν en ω ib ω eb Hz Figure 3.5: Ion and electron collision and gyro frequencies calculated for Tromso, , 24 UT. Here ν in = ν i, ν en = ν e, ω ib = ω i and ω eb = ω e.

9 3.2. BASICS OF PLASMA DYNAMICS 47 at all ionospheric heights above the D region (Fig. 3.5). A similar drift is also experienced by ions, but the ion collision frequency becomes smaller than the ion gyrofrequency only in the F region (Fig. 3.5). So, in the F region (at altitudes above about 200 km) the ions and electrons move together in a direction, which perpendicular both to the electric field E and magnetic field B, the so called E- cross-b drift v i,e = E B B 2 (3.29) In the E region, electrons and ions move in a different way under the action of the electric field. This leads to electric currents, which will be discussed in Chapter 4. The parallel electron mobility is very high in the ionosphere. This is also a consequence of the small electron mass. Thus electrons can easily move along the magnetic field and they try to short circuit field aligned electric fields, which leads to the fact that the electric field tends to be perpendicular to the magnetic field. When field-aligned electric field is zero, the field aligned ion velocity calculated from eq. (3.15) is v iz = k i F z = 1 m i ν i u z = u z. (3.30) m i ν i Thus the neutral wind drives ions in the direction of the magnetic field and the ion velocity in this direction will be adjusted to the value of the neutral wind component Ambipolar diffusion Diffusion of charged particles in weakly ionised plasma contains one new aspect which is not encountered in neutral gas. Due to the different masses of ions and electrons, these two particle species tend to diffuse at different rates. This, however, would lead to charge separation and generation of large electric fields. As a matter of fact, a weak electric field will develop, which will adjust itself to a value which forces ions and electrons to the same diffusion rate. Let us first consider unmagnetised plasma (B = 0). Due to their light mass, electrons are very mobile and they can quickly short circuit any external electric field. The small electron mass also implies that, in the momentum equation (3.3), n e m e ν e (v e u) as well as m e g can be put to zero. When the convective derivative is also put to zero (stationary state and weak spatial variation in electron velocity), the momentum equation for electrons is simply E D = 1 en e p e = k BT e en e n e. (3.31) Here it is assumed that the electron gas obeys the ideal gas law and the electron temperature has no spatial variation. The result shows that a stationary state implies the existence of an electric field, which depends on the gradient of electron density.

10 48 CHAPTER 3. THE IONOSPHERE OF THE EARTH When the neutral air is at rest, the ion momentum equation then reads ee D + m i g m i ν i v i 1 n i p i = 0. (3.32) Next we divide the ion velocity into two components v i = v ic +v id in such a manner that m i g m i ν i v ic = 0. (3.33) This gives the convection velocity The diffusion velocity v id then obeys the equation v ic = g ν i. (3.34) ee D m i ν i v id k BT i n i n i = 0. (3.35) Here T i is the ion temperature. By inserting the diffusion electric field from eq. (3.31) we then obtain the diffusion flux density of ions n i v id = k B(T i + T e ) m i ν i n i = D a n i (3.36) Because the plasma must keep charge neutrality, the diffusion flux densities of electrons and ions must be identical. Eq. (3.36) defines the ambipolar diffusion coefficient D a = k B(T i + T e ) m i ν i. (3.37) We notice that, if electrons and ions have the same temperatures, the ambipolar effect doubles the diffusion coefficient compared to the neutral gas. In magnetised palsma diffusion becomes anisotropic. By neglecting collisions and gravity the stationary electron momentum equation reads e(e + v e B) k BT e n e n e = 0. (3.38) We divide the electric field in two components E = E 0 + E D in such a manner that E 0 = v e B. (3.39) Obviously E 0 is perpendicular to B. Eq. (3.39) gives the relation between electric field and electron velocity in a homogeneous plasma ( n e = 0) with no electron collisions. Then the electron velocity, electric field and magnetic field are perpendicular to each other. Then, we get from eq. (3.38) ee D k BT e n e n e = 0. (3.40)

11 3.2. BASICS OF PLASMA DYNAMICS 49 This gives E D = k BT e en e n e. (3.41) Eq. (3.41) indicates that inhomogeneities in the electron density create and additional electric field E D. This is the same diffusion electric field, which is generated in unmagnetised plasma (eq. (3.31)). The difference is that, in unmagnetised plasma, diffusion electric field is the only electric field, whereas in magnetised plasma an electric field perpendicular to the magnetic field is possible. The stationary ion momentum equation can now be written as e(e 0 + E D + v i B) + m i g m i ν i (v i u) k BT i n i n i = 0. (3.42) As above, we use convection and diffusion velocities v i = v ic +v id in such a manner that e(e 0 + v ic B) + m i g m i ν i (v ic u) = 0. (3.43) This is mathematically similar to eq. (3.4) and therefore its solution can be written directly. One only has to replace u by u + g/ν i. Hence the convection velocity is v ic = k i (ee 0 + m i ν i u + m i g). (3.44) The diffusion velocity must obey the equation e(e D + v id B) m i ν i v id k BT i n i n i = 0. (3.45) By inserting the diffusion electric field from eq. (3.41) we obtain ev id B m i ν i v id k B(T i + T e ) n i n i = 0. (3.46) Again, this equation is mathematically similar to eq. (3.4) and the solution can be written when we replace ee + m i ν i u by k B (T i + T e ) n i /n i. Therefore, the ion diffusion velocity is v id = k B(T i + T e ) n i k i n i (3.47) and the diffusion flux density is where n i v id = D a n i, (3.48) D a = k B (T i + T e )k i (3.49) is the ambipolar diffusion tensor. This results shows the close connection between the ion mobility tensor and the ambipolar diffusion tensor. The diffusion velocity can now be calculated from eq. (3.47). In the ionosphere we can assume that n i is vertical, pointing either upward or downward, and the

12 50 CHAPTER 3. THE IONOSPHERE OF THE EARTH n i h x N B I y z Figure 3.6: Vertical density gradient and inclined magnetic field. N points to geomagnetic north in the horizontal plane, h is upwards, B is along the z axis, x axis is located in the magnetic meridional plane and y completes the right-handed coordinate system. geomagnetic field has an inclination angle I. When altitude is denoted by h (see Fig. 3.6), and v id = 1 n i D a n i = 1 n i dn i dh n i = dn i dh e h = dn i dh (cos Ie x sin Ie z ), (3.50) D ap D ah 0 D ah D ap D a cos I 0 sin I. (3.51) After matrix multiplication we get the diffusion velocity in the xyz coordinate system v id = 1 dn D ap cos I i D ah cos I. (3.52) n i dh D a sin I This indicates that, although vertical density gradient is assumed, the diffusion velocity has a horizontal component. The upward vertical component of diffusion velocity is given by v idh = v idx cos I v idz sin I = 1 n i dn i dh (D ap cos 2 I + D a sin 2 I). (3.53) Hence diffusion velocity is opposite to the direction of the density gradient. Both Pedersen and parallel diffusion coefficient affect vertical ambipolar diffusion. If the magnetic field is vertical (i.e. at geomagnetic poles), cos I = 0 and sin I = 1, so that no Pedersen effect affects the diffusion. Above the magnetic equator the opposite is true; cos I = 1 and sin I = 0, and vertical diffusion is solely due to the Pedersen effect.

13 3.3. ABSORPTION OF IONIZING RADIATION 51 z z+dz z dz χ χ I+dI I ds = -secχ dz s Figure 3.7: Decrease of radiation intensity I in a slab of thickness ds. The ratio of parallel and Pedersen ambipolar diffusion coefficients is D a D ap = k i k ip = ω i ν i ω2 i + ν 2 i ω i ν i = ω2 i + ν 2 i ν 2 i = ω2 i ν 2 i + 1. (3.54) This approaches infinity when ν i approaches zero and unity when ν i approaches infinity. Hence, at intermediate inclination angles I, field-aligned ambipolar diffusion dominates at low collision frequencies, but at high collision frequencies field-aligned and Pedersen diffusion become equally important. 3.3 Absorption of ionizing radiation Let the intensity of the solar photon radiation at wavelength λ be I(λ, z) in units of photons m 2 s 1 at an altitude z. Assume a slab with an area A and thickness ds, perpendicular to the incident radiation (Fig. 3.7). The number of neutral molecules in the slab is nads, where n is the number density of the neutral molecules (single species assumed). If the absorption cross section of a single molecule in units of m 2 is σ, the the total absorption cross section of the slab is σnads. The number of photons incident to the slab per unit time is IA, and the number of absorbed photons per unit time is IσnAds. The decrease of intensity is given by the number of absorbed photons per unit time divided by the area A. Therefore di = nσids and di = nσi. (3.55) ds Absorption cross section σ is a measure for the probability of an absorption process. Generally, the absorption cross section depends on the wavelength for a given molecule or atom. If the ionization enters the atmosphere at a zenith angle χ, ds = dz cos χ = sec χdz. (3.56)

14 52 CHAPTER 3. THE IONOSPHERE OF THE EARTH Then eq. (3.55) gives di = σnds = σn sec χdz, (3.57) I where sec χ = 1/ cos χ. Using the notation I for the intensity at the top of the ionosphere (z = ) we then obtain I I di z I = σ sec χ n(z)dz. (3.58) The integration gives I(z) = I exp σ sec χ z n(z)dz = I exp( σ sec χ N T ) = I exp( τ), (3.59) where N T is the total number of neutral particles from height z to infinity per unit area and the optical depth τ is defined by τ(z) = σ sec χn T (z). (3.60) Each wavelength will have its own optical depth. Note also, that optical depth is not a constant, but a parameter replacing altitude in eq. (3.59): it takes into account absorption produced by all the electrons the radiation has passed when entering altitude z. In the case of an isothermal atmosphere (with a single molecular species as we have assumed so far) I(z) = I exp σ sec χ n 0 exp( z/h)dz The optical depth is now z = I exp[ σ sec χ Hn 0 exp( z/h)]. (3.61) τ(z) = σ sec χ n(z)h. (3.62) The real atmosphere is composed of different neutral particles, and each particle species has an own density and wavelength-dependent absorption cross section. Therefore, in a multicomponent atmosphere, eq. (3.58) can be written as I I di I = sec χ j z σ j n j (z)dz. (3.63) Here n j and σ j are the particle density and absorption cross section of gas species j, respectively. This gives I(z) = I exp sec χ j σ j z n j (z)dz = I exp( τ), (3.64)

15 3.3. ABSORPTION OF IONIZING RADIATION 53 Figure 3.8: Absorption cross sections for N 2 (top) and O (bottom) in the wavelength region from 0 to 100 nm and in values from to cm 2.

16 54 CHAPTER 3. THE IONOSPHERE OF THE EARTH Figure 3.9: Absorption cross section for O 2 in the wavelength region from 0 to 200 nm and in values from to cm 2. where the optical depth is τ(λ, z) = sec χ j σ j (λ) n j (z )dz. (3.65) z Figures 3.8 and 3.9 display the absorption cross sections for typical neutral constituents as a function of wavelength. Absorption takes place at UV and X ray wavelengths and is highly structured for molecular species. 3.4 Ionization rate The classical theory of ionospheric ionization by means of photoionization was presented by Chapman. To obtain the classical Chapman formula for the ion production rate q, this theory assumes that the atmosphere is isothermal and obeys the hydrostatic equation so that the scale height H is independent of altitude. We also assume that the atmosphere is composed of a single neutral species and the absorption cross section is constant. Constant cross section is equivalent to assuming monochromatic radiation. It was concluded above that the total number of photons absorbed per unit time in a slab with an area A and thickness s is IσnAds. Hence the total number of photons absorbed per unit volume per unit time is nσi. However, not all their energy goes to the ionization process. The ionization efficiency η is the (statistical) number of photoelectrons produced per photon absorbed. Then the number of photo

17 3.4. IONIZATION RATE 55 electrons produced per unit volume per unit time, the ionization rate q (called also the ion production function) can be expressed as q = η di ds = ηnσi = ηnσi exp( τ). (3.66) According to eq. (2.59), we write the neutral density profile as ( n = n m,0 exp z z m,0 H ) = n m,0 e h, (3.67) where n m,0 is the molecular density at the reference height z m,0 and h = (z z m,0 )/H is the distance from the reference height, scaled by H. Then, according to eq. (3.62), and the ionization rate is τ = σ sec χ Hn(z) = σ sec χ Hn m,0 e h (3.68) q(χ, h) = ηn(h)σi e τ(h) = ηn m,0 σi exp [ h sec χ σhn m,0 e h]. (3.69) The expression is still simplified if we now fix the reference height so that σhn m,0 = 1. (3.70) Then where q(χ, h) = ηi [ H exp h sec χe h] = ηi [ He exp 1 h sec χe h] [ = q m,0 exp 1 h sec χe h], (3.71) q m,0 = η H I e = ησn I m,0 e. (3.72) If we still choose a height variable h = h ln sec χ, we obtain [ q(χ, h) = q m,0 cos χ exp 1 h e h ]. (3.73) In terms of variable h, the ionization rate can be presented for all zenith angles χ with a single shape, which only has a scaling factor cos χ for different zenith angles. The expressions in eqs. (3.71) or (3.73) are called the Chapman production function. Since n increases and I decreases as we go down in altitude, the ion production rate q has necessarily a maximum at some altitude. From eq. (3.73) we can show that the maximum value is encountered at an altitude h m = 0 h m = ln sec χ. (3.74)

18 56 CHAPTER 3. THE IONOSPHERE OF THE EARTH Since on the other hand h m = (z m z m,0 )/H, we get z m = z m,0 + H ln sec χ. (3.75) This equation relates the altitude of maximum ionization rate z m to the reference height z m,0. For zero zenith angle, χ = 0, z m is the same as the selected reference height z m,0. The notation selected in eq. (3.67) becomes now clear: in subscript z m,0, m refers to the maximum ionization rate and 0 refers to the zero zenith angle. Thus, the maximum value of ionization rate is q m = q m,0 cos χ. (3.76) This shows that the maximum production is encountered at χ = 0, i.e. when the sun is shining overhead. Then also the maximum is encountered at lowest altitudes. When the zenith angle increases, the maximum rises to higher altitudes and the maximum value decreases (Fig. 3.10). It is obvious that, when the zenith angle is high enough, the curvature of the Earth and the atmosphere should be taken into account and theory should be modified. This is because, if the atmosphere would be flat, the ray path in the atmosphere would approach to infinity when the elevation angle would approach zero. At very large values of h (z z m,0 ) the profile in eq. (3.71) takes the form ( q = q m,0 exp z ) H (3.77) Figure 3.10: Chapman production profiles for different solar zenith angles.

19 3.4. IONIZATION RATE 57 Figure 3.11: The altitude of unit optical depth calculated for zero zenith angle. Ionization threshold wavelengths for several molecules and atoms are marked by arrows (as well as the Ly-α wavelength). so that well above the maximum the ionization rate profile decays with height in the same way as the atmospheric density. This holds for h > 2, i.e. at distances more than two scale heights above z m,0. This relation arises because at those heights the intensity of the radiation is only weakly reduced, and the rate of production is essentially proportional to the gas density. Eq. (3.68) can be used to show that at the altitude of maximum ionization rate z m, the optical depth τ m = 1. At the very same height the intensity of radiation has a value I m = I /e according to eq. (3.59). Figure 3.11 shows the altitude of unit optical depth at wavelengths below 300 nm. Some spectral wavelengths are absorbed very strongly, but the hydrogen Lyman-α at nm penetrates to the D region, where it ionizes NO. The calculation of ion production rate in a real atmosphere with an incident solar illumination with its true spectrum is complicated. The upper atmosphere is composed of different gases with different scale heights and the solar radiation spectrum consists of a myriad of lines and bands with different intensities (which vary with solar activity). The different constituents in the atmosphere have different ionization and dissociation threshold energies. Table 3.1 displays these energies in ev for some of the most common neutral particles and the corresponding wavelength (V p = hν = hc/λ) which fall mostly in the EUV regime. Photons with energies larger than V p or wavelengths shorter than λ can ionize or dissociate the given species. Dissociative ionization threshold wavelengths for O 2 and N 2 are 662 Å and 510 Å, respectively (not shown in Table 3.1).

20 58 CHAPTER 3. THE IONOSPHERE OF THE EARTH Table 3.1 The approach is to assume a neutral atmosphere model with height distribution of some of the major gases and the for finite number of wavelength steps to derive the individual production profiles. Two examples are shown Chemical in Fig. effects 3.12of and high Fig. energy particle precipita For atomic species, the ionization efficiency η is unity so that the whole energy of the ionizing photon goes into producing ion-electron pair(s). For molecules, a photon may produce dissociation or ionization, so the ionization efficiency η 1, Normal ionisation profile Daytime ionisation Figure 3.12: rates Calculated due to daytime solar photons, ionisation photoelectrons, rates due solar and photons, GCR photoelectrons, maximum and GCRequinox, during solar at latitude maximum 70 equinox, N, SZA at = 69.6 latitude (from 70P.Verronen, N, SZA = 2006) 69.6 (P. during solar Verronen, 2006). SGO, E.Turunen, NordAurOpt Workshop

21 3.4. IONIZATION RATE 59 Figure 3.13: Calculated ionisation rates (Hinteregger et al., 1965). and η depends on the wavelength of the photon. There is still another complexity associated with photoionization, and it caused by the energy of the photoelectron. In general, the energy of the ionizing radiation hν is much greater than the threshold energy V p, and the excess energy will then be left partly to the kinetic energy of the photoelectron and partly to the energy of the excited ion. The photoelectron may then in turn produce further ionization. Laboratory experiments have shown that the mean energy ɛ lost by a photoelectron per ionization is almost constant and greater than the ionization energy. Hence the energy of a photon consumed is hν = V p + (N e 1) ɛ, (3.78) where N e is the approximate number of electrons produced in the cascade of ioniza-

22 60 CHAPTER 3. THE IONOSPHERE OF THE EARTH tion processes. This gives N e = 1 + hν V p. (3.79) ɛ Table 3.2 gives values of ɛ for an ion-electron pair production by electron impact ionization for different neutral species. It is customary to set V p = 15 ev and ɛ = 34 ev in air. Table Recombination After ions and free electrons are produced in the ionization processes, they will take part in chemical reactions in which they may disappear or produce other types of ions. Some of the ionization will also be carried away by transport mechanisms such as diffusion, neutral winds and electric fields. This is controlled by the continuity eq. (3.1) containing both the transport, production and loss terms. In this section we will discuss the loss term which, together with production, determines the electron density in the absence of transport. When the convection term in the continuity equation is neglected and electron density is constant, production and loss are in balance, i.e. there is a photochemical equilibrium, q i = l i. (3.80) Electrons may recombine directly with positive ions to make neutral atoms or molecules in the reaction X + + e X + hν (3.81) which leads to a production of a photon. This is called radiative recombination. The probability of this reaction is low because of the demand put by simultaneous conservation of energy and momentum. Another possibility is dissociative recombination of a diatomic ions according to the reaction XY + + e X + Y. (3.82) The conservation laws are more easily satisfied by this reaction which produces two neutral atoms.

23 3.5. RECOMBINATION 61 The speeds of reactions like those in eqs. (3.80) and (3.81) are expressed in terms of reaction coefficients. The probability of a single electron to recombine with must be proportional to the ion density, since recombination implies that the electron must hit an ion. The number of electrons lost per unit volume and unit time must also be proportional to the number of electrons which can recombine, that is proportional to electron density. Since the electron and ion densities are identical, the loss term must be l i = αn 2 e, (3.83) where α is called the recombination coefficient or reaction coefficient. Magnitudes of the reaction coefficients for dissociative recombination are of the order of m 3 s 1 and for the radiative recombination m 3 s 1. These numbers indicate that dissociative recombination is much faster than radiative recombination. When the Chapman production function from eq. (3.71) is inserted in the photochemical equilibrium (3.80) and dissociative recombination is assumed, the result is [ q m,0 exp 1 h sec χ e h] = αn 2 e. (3.84) The electron density solved from this equation is [ q n e = α = qm,0 1 α exp 2 (1 h sec χ e h]. (3.85) When neglecting height variations of α (actually α does not depend directly on height but on temperature) we find that the electron density has a maximum when showing that the electron density maximum is given by n m (χ) = e h = cos χ. (3.86) qm,0 α cos χ = nm,0 cos χ. (3.87) We find that the maximum electron density is proportional to the square root of the solar zenith angle. The electron density profile in eq. (3.85) is called the Chapman α profile. It is representative of the ionospheric E region where the most common ions are O + 2 and NO +. The presence of NO + might be surprising at the first sight, since the neutral atmosphere contains only minute amounts of neutral NO. It turns out that NO + is produced by the E region ion chemistry. This will be discussed later in detail. If electrons are lost by attachment to a molecule M + e M, (3.88) the electron loss rate is only proportional to n e and can be expressed as l e = βn e, (3.89) where β is the attachment coefficient proportional to the density of neutral molecules, n M. In this case the plasma consists of electrons and positive and negative ions,

24 62 CHAPTER 3. THE IONOSPHERE OF THE EARTH and the positive ion density is the sum of electron and negative ion densities. By assuming the Chapman production function, in equilibrium the electron density profile is given by n e = q [ m,0 β exp 1 h sec χ e h]. (3.90) This is called the Chapman β profile. As we can see in the next section, any loss process resulting in a coefficient directly proportional to n e can be characterized as β type. If we again neglect height variations in β (which is not a good assumption since the density of neutral molecules decreases with height) we get for the maximum electron density n m = q m β cos χ = n m,0 cos χ. (3.91) In the E region, dissociative recombination with molecular ions is the most important loss mechanism for electrons. Production of negative ions takes place in D region, where ion chemistry is complicated. In the F2 region loss mechanisms associated with atomic oxygen ions dominate. Between the E and F2 regions lies the F1 region, where both dissociative recombination and a reaction containing atomic oxygen ions are active and one of them dominates, depending on the values of reaction rate coefficients. 3.6 O dominant ionosphere In the F-region (above 150 km) the dominant ions formed by solar irradiance are the O + ions. In the following we discuss a simplified model which assumes that these ions are the only species resulting from solar radiation and the other ions are produced as a result of chemical reactions. The O + ions can be lost by several reactions. One possibility would be radiative recombination with electron O + + e α r O + hν, (3.92) where α r is the radiative recombination coefficient (Table 3.3). This process is slow and is neglected in the following discussion. A more rapid loss occurs through two reactions, O + + N 2 k 1 NO + + N (3.93) followed by NO + + e α 1 N + O, (3.94) where k 1 is the rearrangement coefficient and α 1 is the dissociative recombination coefficient (Table 3.3). The rate of process (3.93) is k 1 [O + ][N 2 ] and the rate of process (3.94) is α 1 [NO + ]n e. Square brackets are used to indicate the concentration (number density) of each species.

25 3.6. O DOMINANT IONOSPHERE 63 O + 2 Another loss process for O + ions is through a similar chain of reactions with ions. First O + makes a charge exchange with an O 2 molecule O + + O 2 k 2 O + O + 2 (3.95) and then O + 2 recombines with an electron and dissociates according to O e α 2 O + O. (3.96) The rate of process (3.95) is k 2 [O + ][O 2 ] and the rate of process (3.96) is α 2 [O + 2 ]n e. In equilibrium conditions, and when radiative recombination is neglected, the production of atomic oxygen ions must be equal to their loss qiven in eqs.(3.93) and (3.95) q(o + ) = (k 1 [N 2 ] + k 2 [O 2 ])[O + ] = l(o + ). (3.97).

26 64 CHAPTER 3. THE IONOSPHERE OF THE EARTH Table 3.3: Typical reaction rates. Here T r = (T i + T n )/2 (Brekke, 1997). A similar equilibrium condition valid for molecular oxygen and nitric oxide ions. When their production by solar radiation is neglected, the production of O + 2 given by eq. (3.95) must be equal to loss given by eq. (3.96) q(o + 2 ) = k 2 [O 2 ][O + ] = α 2 [O + 2 ]n e = l(o + 2 ). (3.98) Production of NO + given by eq. (3.93) must be equal to loss given by eq. (3.94) Since charge neutrality must hold From eq. (3.99), and from eq. (3.98) q(no + ) = k 1 [N 2 ][O + ] = α 1 [NO + ]n e = l(no + ). (3.99) By inserting these in eq. (3.100) we obtain n e = [O + ] + [NO + ] + [O + 2 ]. (3.100) [NO + ] = k 1[N 2 ] α 1 n e [O + ] (3.101) [O + 2 ] = k 2[O 2 ] α 2 n e [O + ]. (3.102) n e = [O + ] + k 1 α 1 [N 2 ] n e [O + ] + k 2 α 2 [O 2 ] n e [O + ]. (3.103) The concentration of atomic oxygen ions, solved from this equation, is [O + ] = Inserting this in eq. (3.97) gives q(o + ) = n e 1 + k 1 [N 2 ]/(α 1 n e ) + k 2 [O 2 ]/(α 2 n e ). (3.104) k 1 [N 2 ] + k 2 [O 2 ] 1 + k 1 [N 2 ]/(α 1 n e ) + k 2 [O 2 ]/(α 2 n e ) n e. (3.105) Using a notation β for the coefficient of n e, this can also be written as q(o + ) = β n e. (3.106) The values of k 1 and k 2 are of the order of m 3 /s while α 1 and α 2 are of the order of m 3 /s. At altitudes above 250 km [N 2 ] < m 3, [O 2 ] m 3 and n e m 3, so α 1 n e and α 2 n e will be much larger than k 1 [N 2 ] and k 2 [O 2 ], respectively. Therefore, in the upper F region, β can be approximated by β, which is β = k 1 [N 2 ] + k 2 [O 2 ]. (3.107)

27 3.6. O DOMINANT IONOSPHERE 65 Then, from eq. (3.106), n e = q(o+ ) β = q(o + ) k 1 [N 2 ] + k 2 [O 2 ]. (3.108) Since N 2 and O 2 have almost similar scale heights we have approximately β = β 0 ( exp z z 0 H ), (3.109) where β 0 is the loss coefficient at reference height z 0 and H is the scale height. In the lower F region, the reactions involving charge rearrangement between O + and O 2 and N 2 will become more abundant because of the increase of these neutral species. Therefore β must be used instead of β. According to eqs. (3.105) and (3.106) β = = k 1 [N 2 ] + k 2 [O 2 ] 1 + k 1 [N 2 ]/(α 1 n e ) + k 2 [O 2 ]/(α 2 n e ) β 1 + k 1 [N 2 ]/(α 1 n e ) + k 2 [O 2 ]/(α 2 n e ) (3.110) so that where Then eq. (3.106) gives 1 = 1 [1 + (k β 1 [N 2 ]/α 1 + k 2 [O 2 ]/α 2 ) 1 ] β ne = 1 β + 1, (3.111) α eff n e α eff = β k 1 [N 2 ]/α 1 + k 2 [O 2 ]/α 2. (3.112) ( n e = q(o+ ) 1 = q(o + ) β β + 1 ) α eff n e. (3.113) By studying the corresponding reaction coefficients (Table 3.3) and neutral densities it can be seen that below, say, 150 km the condition β α eff n e is easily valid. Then an approximate value for n e can be solved from eq. (3.108) and it is n e = q(o+ ) α eff. (3.114) Since β decreses with height and α eff n e increases with height below the F region maximum, there must be a transition height z t such that β t = α eff n et, (3.115)

28 66 CHAPTER 3. THE IONOSPHERE OF THE EARTH Figure 3.14: (a) Ionization rate q as a function of altitude, (b), (d)-(f) n e as a function of altitude assuming different recombination processes and (c) recombination coefficient β as a function of altitude. where n et is the electron density at z t. Above this altitude electron density varies as q/β (beta-type region) and below it as (q/α eff (alpha-type region). Fig displays schematically the formation of the F 1 region. Fig (a) shows the Chapman production function q. It has a maximum at an altitude z m. Fig (b) shows the electron density profile which would be generated if the ions were diatomic and would be recombined by means of dissociative recombination with a constant recombination coefficient α. This would produce a peak electron density at the same altitude z m as the production peak. Fig (c) shows the height variation of loss rate β, which decreases with altitude due to the decrease of neutral density. Fig (d) shows the electron density which would be produced if the ions were monoatomic and would decay by means of charge exchange (and the successive recombination would be quick). In the bottomside F region both decay processes are active. There is some transition altitude z t where the two recombination mechanisms are equally important; dissociative recombination dominates below this altitude and charge exchange dominates above. The shape of the resulting total electron density depends on whether the transition altitude lies above or below the production peak. If the transition altitude lies above the production peak (Fig (e)), a local maximum in the electron density profile is produced at z m. In the opposite case (Fig (f)) only a knee at z m will appear. This is roughly the mechanism of the generation of the F 1 region (see Fig. 3.1).

29 3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION 67 The condition in Fig (e), i.e. z t > z m can be fulfilled in the daytime under sunspot maximum conditions. The observation of an F 1 layer does not necessarily imply the existence of a maximum in the electron density profile; the F 1 layer is usually rather seen as a bulge in the profile. We have seen that, at distances larger than two scale heights above the ionization maximum, the ion production profile obeys eq. (3.77). Therefore we can write ( q(z) exp z ), (3.116) H O where H 0 is the scale height of oxygen atoms (which is assumed as the dominant neutral species at high altitudes). Since N 2 is more abundant than O 2, we can approximate eq. (3.107) to give ( β(z) exp z ), (3.117) H N2 where H N2 is the scale height of molecular nitrogen. According to eq. (3.108) we then obtain n e (z) = q(z) [( ) ] β(z) exp HO z 1. (3.118) H N2 H O Since H O /H N2 = 7/4, this leads to ( ) 3z n e (z) exp. (3.119) 4H O This is an unreasonable result, because it would indicate that the electron density increases with height at great altitudes. In the real ionosphere F region has a maximum at an altitude of several hundred kilometers after which the electron density decreases. Obviously the assumption of photochemical equilibrium (photochemical production is equal to chemical loss, eq. (3.79)) is not correct. This means that the transport term in the continuity equation cannot be neglected. The physical mechanism behind the non-negligible transport term is ambipolar diffusion in the topside ionosphere. 3.7 Multicomponent topside ionosphere and diffusion Diffusion without magnetic field When diffusion cannot be neglected, the continuity equation for ions in a horizontally stratified ionosphere is n i t + (n iv iz ) z = q i l i. (3.120)

30 68 CHAPTER 3. THE IONOSPHERE OF THE EARTH For simplicity, let us first neglect the geomagnetic field. Then, according to eqs. (3.34), (3.36) and (3.37), n i v i = n i v ic + n i v id = n ig ν i D a n i. (3.121) This is valid in geomagnetic poles where the magnetic field does not affect the vertical plasma motion. In a horizontally stratified ionosphere this equation can be written as ( ni n i v iz = D a z + n ) ig. (3.122) ν i D a We define the plasma scale height H p = k B(T i + T e ) m i g (3.123) and note that H p = ν i D a /g where the ambipolar diffusion coefficient D a is given by eq. (3.37). Then [ ni n i v iz = D a z + n ] i. (3.124) H p If production and loss approximately cancel in the continuity equation, a stationary state is obtained when v iz = 0, i.e. when the convection and diffusion velocities also cancel so that the ions have no vertical motion. Hence dn i dz + n i H p = 0. (3.125) This can be easily integrated and the result is n i = n i0 exp ( z z 0 H p ) (3.126) where n i0 is the ion density at a reference height z 0. Hence the equilibrium ion density profile is exponential. The decrease with altitude is determined by the plasma scale height. Notice that, assuming equal ion and electron temperatures, the scale height is twice as great as it would be without the ambipolar effect. Actually several ion species like O +, N +, H + and He + are present in the topside ionosphere. When this is taken into account, the basic theory of ambipolar diffusion must be modified. The approximations of the electron momentum equation are still the same as before, and therefore the diffusion electric field is given by eq. (3.41) E D = k BT e en e n e. (3.127) When we have k ion species with densities n j, j = 1, 2,... k, charge neutrality implies k n e = n j. (3.128) j=1

31 3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION 69 The convection velocity of each ion species is still (see eq. (3.34)) v jc = g ν j, (3.129) and the diffusion velocity is obtained from the equation (see eq. (3.35)) ee D m j ν j v jd k BT n j n j = 0, (3.130) where we have assumed that all ion species have the same temperature T i. If the total velocity for ion j is zero, v j = v jc + v jd = g ν j k BT i m j ν j n j n j + ee D m j ν j = 0. (3.131) By multiplying this by m j n j and calculating a sum over j we obtain k g m j n j k B T i n e + en e E D = 0, (3.132) j=1 where we have taken into account the fact that the sum of all ion densities is equal to the electron density. When the diffusion electric field is inserted here, the equation can be solved to give where 1 n e n e = g k B (T i + T e ) 1 n e k m j n j = j=1 m + g k B (T i + T e ), (3.133) m + = 1 k m j n j (3.134) n e j=1 is the mean ion mass. Because different ion species have different height distributions, these will affect the electron density via the height dependence of the mean ion mass. In a horizontally stratified situation eq. (3.133) can be written as 1 dn e n e dz = m + g k B (T i + T e ). (3.135) When inserting this in eq. (3.127) we get an alternative form for the diffusion electric field E z, m + g ee z = T e (T i + T e ) = m c +g 1 + c, (3.136) where c = T e (3.137) T i is the ratio of electron and ion temperatures. When eq. (3.135) is inserted in eq. (3.131), one can solve (note E D = E zˆk and g = gˆk) 1 n j n j z = m jg m + gc + = 1 ( 1 c ) k B T i (1 + c)k B T i H j 1 + c m+ = 1. (3.138) m j δ j

32 70 CHAPTER 3. THE IONOSPHERE OF THE EARTH Figure 3.15: Relative ion concentration profiles for typical topside ionosphere ions. Difusive equilibrium for electrons and ions in temperature of 1200 K is assumed (solid lines). Dashed lines correspond to photochemical equilibrium (Ratcliffe, 1972). where H j = k B T i /(m j g). This describes the vertical distribution of the jth ion species and we see that in diffusive equilibrium (v j = 0) the scale height δ j is different from H j. If the mass of the jth ion is the same as the mean mass m + and if the temperature ratio c is unity, the scale height for this ion species is the same as the plasma scale height given by eq. (3.123). The scale heights of heavier and lighter ions are smaller and greater, respectively. Especially, when m j < c/(1 + c)m +, the scale height is negative. This means that the number density of the lightest ions may increase with height. An example of the height distribution of some ions is given in Fig One should notice that exact results for electron and ion densities cannot be calculated by integrating eqs. (3.135) and (3.138). This is because both equations contain m + which is height-dependent. As a matter of fact, the ion momentum equations (3.130) are coupled by means of the diffusion electric field, which depends on electron density, i.e. on all ion densities. In order to find all ion densities, the whole set of coupled differential equations should be solved. Another point worth noticing is that, in the above theory, it was assumed that the vertical velocity of all ion species is zero. This is not exactly the case. In the topside ionosphere the diffusion electric field points upwards. This field accelerates light positive ions, which will have upward velocity. Inside the polar cap this phenomenon is known as polar wind. The classical theory of polar wind is based on diffusion electric field. The polar wind, however, also contains heavier ions, which means that there must be other acceleration mechanisms present generating this phenomenon.

33 3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION 71 Finally, one should notice that ambipolar diffusion cannot be expressed in terms of a single diffusion coefficient, when more than one ion species are present in the ionosphere. Mathematically this is associated with the fact that the ion momentum equations (3.130) are coupled, as discussed above Diffusion with magnetic field When we take into account the geomagnetic field and assume a single ion species in the topside F layer, we can start from eqs. (3.48), and (3.44), which give v id = D a ( 1 ) n i (3.139) ni and v ic = k i (m i g). (3.140) Here we have assumed that the electric field E 0 and the neutral wind u in eq. (3.44) are both zero. When we assume identical electron and ion temperatures (T e = T i = T ), the ambipolar diffusion tensor D a = 2k B T k i, and therefore the convection velocity can be written as ( ) mi v ic = D a 2k B T g. (3.141) According to eq. (3.53), the vertical component of the diffusion velocity is (here we denote altitude by z instead of h) ( v idz = (D ap cos 2 I + D a sin 2 I) 1 dn ) i. (3.142) n i dz Comaprison of eqs. (3.139) and (3.141) then shows that, obviously ( v icz = (D ap cos 2 I + D a sin 2 I) m ) i 2k B T g so that the total vertical ion velocity is v iz = v idz + v icz = (D ap cos 2 I + D a sin 2 I) In accordance with (3.54), D ap D a = k ip k i = 1 eb ω i ν i ω 2 i + ν 2 i (3.143) ( 1 dn i n i dz m ) i 2k B T g. (3.144) m i ν i = ν2 i. (3.145) ωi 2 + νi 2 When ν i is much smaller than ω i, this ratio is small. This means that, at sufficiently high altitudes, field-aligned motion dominates and Pedersen motion can be neglected. When this approximation is used, eq. (3.144) gives a vertical ion flux density ( n i v iz = D a sin 2 dni I dz + n ) ( im i 2k B T g = D a sin 2 dni I dz + n ) i. (3.146) H p

34 72 CHAPTER 3. THE IONOSPHERE OF THE EARTH This shows that, at geomagnetic poles, the flux density is equal to that with no magnetic field and, at the geomagnetic equator, the flux density will drop to zero. To be more precise, a small vertical flux density still exists at the geomagnetic equator. This is due to the Pedersen component of the ambipolar diffusion tensor. In a stationary situation n i / t = 0. If the loss term is l = βn i, the complete continuity equation reads d(n i v iz ) dz = q l = q βn i. (3.147) In order to obtain the differential equation for n i, the ion flux density in eq. (3.146) must be derivated and inserted here. For doing that, we have to know the hight dependence of D a. Since D a is inversely proportional to the ion-neutral collision frequency and the collision frequency is proportional to the neutral density, we can write ( ) z D a = D a 0 exp, (3.148) H where H is the scale heigth of the neutral density. (3.146) and the derivative is calculated, the result is d(n i v iz ) dz = D a sin 2 I When this is inserted in eq. [ ( 1 dni H dz + n ) i + d2 n i H p dz + 1 ] dn i. (3.149) 2 H p dz Then the continuity equation is [ d D a sin 2 2 ( n i 1 I dz + 2 H + 1 ) dni H p dz + n ] i = q βn i. (3.150) HH p This is a linear differential equation of second order. For solving it, the height dependence of the production should be known. (i) At lower altitudes, where ν i is large enough, we can use an approximation D a 0. Then also n i v iz 0 and production and loss are in balance, i.e. n i = q β. (3.151) Since both q and β are proportional to neutral density and q is also affected by intensity of solar radiation, q decreases more slowly with altitude than β does. Then the ion density must grow with altitude at heights where recombination is more important than diffusion (see Fig. 3.14). (ii) At higher altitudes, both q and β are small and, in addition, even the small production and loss terms tend to balance. In this case d(n i v iz )/dz 0. Then eq. (3.149) gives d 2 ( n i 1 dz + 2 H + 1 ) dni H p dz + n i HH p = 0. (3.152) This is a linear homogenous differential equation of second order with constant coefficients, and the standard method of solution is to use a trial function n i

35 3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION 73 exp(az). Then dn i /dz = an i and d 2 n i /dz 2 = a 2 n i. This leads to a characteristic equation ( 1 a 2 + H + 1 ) a + 1 = 0. (3.153) H p HH p Using the basic properties of the equation of second order we immediately see that a 1 + a 2 = 1/H 1/H p and a 1 a 2 = 1/(HH p ). Then, necessarily, a 1 = 1/H and a 2 = 1/H p, so that the general solution of eq. (3.152) is ( n i = C 1 exp z ) ) + C 2 exp ( zhp. (3.154) H This shows that the ion density goes to zero at great heights. Then, according to eq. (3.146), the ion flux density must do the same. This means that ( ) ( z dn i D a 0 exp H dz + n ) i = D a 0 e z/h H p ( C 1 H e z/h C 2 e z/h p + C 1 e z/h + C ) 2 e z/h p = H p H p H p ) D a 0 C 1 ( 1 H p 1 H (3.155) must approach zero. Since H and H p are constants, this is possible only if C 1 = 0. Therefore ) n i = C 2 exp ( zhp. (3.156) The result indicates that, at altitudes where production and loss are small or balance, ion density must decrease with increasing altitude. Since at lower altitudes ion density increases with altitudes according to eq. (3.151) and at higher altitudes ion density decreases, a maximum of ion density must be found at some altitude. At the density peak dn i /dz = 0. Then, according to eq. (3.146) n i v iz so that the ion velocity at the peak altitude is = D a sin 2 I n i H p, (3.157) v iz = D a sin 2 I H p. (3.158) In summary, we have found that by assuming β type loss process and by taking into account ambipolar diffusion, we can explain formation of the F region peak and the decrease in ion density above it. In general, the F region peak doesn t form at the altitude of maximum ionization rate z m.

36 74 CHAPTER 3. THE IONOSPHERE OF THE EARTH 3.8 Ionization and recombination in the E region The main ion species produced by solar radiation between 90 and 130 km are O + 2 and N + 2. We shall next study the photochemical equilibrium for these ions. The O + 2 ions will recombine with electrons by means of dissociative recombination O e α 2 O + O (3.159) or they can carry out charge exchanges with neutral constituents such as NO and N 2 in reactions or O NO k 3 NO + + O 2 (3.160) O N 2 k 4 NO + + NO. (3.161) with rate coefficients k 3 and k 4, thus forming NO + ions which are abundant in the E region. NO + ions can also be borne in reaction N O k 5 NO + + N. (3.162) Another loss mechanism for N + 2 is dissociative recombination N e α 3 N + N. (3.163) The NO + ions produced in reactions (3.159), (3.160) and (3.161) are also destroyed by means of dissociative recombination NO + + e α 1 N + O. (3.164) These reactions are the most important ones which make the E region ion chemistry. The E region electron density is stationary, if production and loss of each ion species are in balance. In the above model of ion chemistry, O + 2 is produced only by means of direct photoionization. The loss is by means of direct dissociative recombination and charge exchange with neutral N 2 and NO. Hence the balance equation between production and loss is q(o + 2 ) = α 2 [O + 2 ]n e + k 3 [O + 2 ][NO] + k 4 [O + 2 ][N 2 ]. (3.165) Similarly, N + 2 is produced by means of direct photoionisation and lost by means of direct dissociative recombination and in producing NO +. Thus q(n + 2 ) = α 3 [N + 2 ]n e + k 5 [N + 2 ][O]. (3.166) When direct photoionization of the minority neutral constituent NO is neglected, NO + is produced by the three reactions (3.159), (3.160) and (3.161). The production is q(no + ) = k 3 [O + 2 ][NO] + k 4 [O + 2 ][N 2 ] + k 5 [N + 2 ][O]. (3.167)

37 3.8. IONIZATION AND RECOMBINATION IN THE E REGION 75 Figure 3.16: Daytime solar minimum ion profiles (Johnson, 1966). The loss is only due to dissociative recombination, i.e. l(no + ) = α 1 [NO + ]n e. (3.168) The total production of ions due to photoionization (O + 2 and N + 2 ) must be equal to the production of electrons. Therefore q(e) = q(o + 2 ) + q(n + 2 ) = (α 2 [O + 2 ] + α 3 [N + 2 ])n e + (k 3 [NO] + k 4 [N 2 ])[O + 2 ] + k 5 [N + 2 ][O]. (3.169) From eqs. (3.167) and (3.168) we notice that the last terms in eq. (3.168) are Hence (k 3 [NO] + k 4 [N 2 ])[O + 2 ] + k 5 [N + 2 ][O] = α 1 [NO + ]n e. (3.170) Since charge neutrality must always hold, q(e) = (α 1 [NO + ] + α 2 [O + 2 ] + α 3 [N + 2 ])n e. (3.171) [NO + ] + [O + 2 ] + [N + 2 ] = n e (3.172) By introducing the relative abundance ratios of the respective ions r 1 = [NO + ]/n e r 2 = [O 2 + ]/n e (3.173) r 3 = [N + 2 ]/n e

38 76 CHAPTER 3. THE IONOSPHERE OF THE EARTH we get r 1 + r 2 + r 3 = 1. (3.174) Then, from eq. (3.171) (and using n e instead of e) q(n e ) = (α 1 r 1 + α 2 r 2 + α 3 r 3 )n 2 e. (3.175) By introducing the effective recombination coefficient α eff α eff = α 1 r 1 + α 2 r 2 + α 3 r 3 (3.176) we can write q(n e ) = α eff n 2 e. (3.177) Formally, this suggests that the shape of the electron density profile in the E region is an α function. One should, however, notice that the relative abundances are height dependent and therefore the height dependence of the electron density profile is not only due to the height dependence of the production. Fig shows some rocket observations of various ion density profiles. It indicates that the O + and N + 2 densities decrease very rapidly below 150 km. Hence, we may neglect the N + 2 density in E region and put r 3 to zero so that and the mean ion mass is r 1 + r 2 = 1 (3.178) m + = r 1 m NO + + r 2 m O + 2. (3.179) It is a common practice to use a mean mass of 30.5 u at these altitudes, which leads to This corresponds to the assumption 30 r r 2 = (3.180) r 1 = 3/4 and r 2 = 1/4 (3.181) in eq. (3.180). Hence NO + is assumed to be more abundant than O + 2 in the E region. As seen in Fig. 3.16, this is not true at all altitudes. If the assumption by eq. (3.181) is used, the effective recombination coefficient is where α 1 and α 2 are given in Table 3.3. α eff = 0.75 α α 2, (3.182)

39 3.9. RECOMBINATION TIME CONSTANT Recombination time constant F region Let us study the continuity equation (3.23) and neglect the convection term as above, but leave the time derivative of electron density. Then the continuity equation reads dn e dt = q e l e, (3.183) where q e and l e are the electron production and loss rates, respectively. In the F region, where O + dominates, the loss is proportional to electron density. Then dn e dt = q e βn e. (3.184) If q e is constant, this can be solved by separating the variables. Hence which gives so that n e n e0 Assuming zero electron density at t = 0, where t dn e = q e βn e 0 dt, (3.185) ( ) qe βn e ln = βt, (3.186) q e βn e0 n e = q e β q e βn e0 exp( βt). (3.187) β n e = n e [1 exp( βt)], (3.188) n e = q e β. (3.189) If, on the other hand, production is switched off at t = 0, electron density will decay exponentially according to n e = n e0 exp( βt). (3.190) In these cases the electron density will change with a time constant τ = 1 β. (3.191) Now, eq. (3.190) can be written as n e = n e0 exp( t/τ) and τ is the recombination time constant in the F region. In the case of eq. (3.188), electron density will approach to a value q e /β and in the case of eq. (3.190) to zero.

40 78 CHAPTER 3. THE IONOSPHERE OF THE EARTH When q e (t) is not constant in eq. (3.185), we have a complete linear differential equation of first order. A standard way of solving it is to use the method of integrating factors. For a linear differential equation y + p(x)y = r(x) (3.192) the integrating factor is [ exp ] p(x) dx. (3.193) In the case of eq. (3.184) the integrating factor is ( exp ) β dt = exp(βt) (3.194) By multiplying eq. (3.184) by this integrating factor we obtain dn e dt exp(βt) + βn e exp(βt) = q e (t) exp(βt). (3.195) The equation can be written in a more simple form to give This can now be integrated and the result is so that d dt [n e exp(βt)] = q e (t) exp(βt). (3.196) n e exp(βt) + C = t 0 q e (t) exp(βt) dt, (3.197) t n e = exp( βt) q e (t) exp(βt) dt + C. (3.198) 0 By applying an initial condition n e = n e0 when t = 0 we see that C = n e0, and finally we obtain n e = exp( βt) n e0 + t 0 q e (t) exp(βt) dt. (3.199) Thus the time variation of n e depends on how the production varies with time. Notice also that putting q e to a constant in eq. (3.199) (and n e0 as zero) leads to eq. (3.188) E region In the E region the loss rate is α eff n 2 e, so that the continuity equation is dn e dt = q e α eff n 2 e. (3.200)

41 3.9. RECOMBINATION TIME CONSTANT 79 Again, if q e is constant, the variables can be separated and the resulting equation can be integrated. The resulting function is hyperbolic tangent (this is left as an exercise). If we assume that there has been some production causing an electron density n e0 and the production is switched off at t = 0, we obtain n e n e0 dn e n 2 e = α eff t 0 dt = α eff t. (3.201) The integration on the left hand side gives which can be solved for n e. The result is n e n e0 n e n e0 = α eff t, (3.202) n e = n e0 n e0 α eff t + 1. (3.203) Hence there is no exponential decay in this case. The electron density is halved after a time t 1/2 = 1 α eff n e0, (3.204) or, equivalently, the decay time corresponding to eq. (3.191) (time when electron density has decreased to 1/eth fraction) τ = e 1 n e0 α eff (3.205) Hence the speed of decay depends on the initial electron density. If the changes in the production are small or if we only consider small time intervals, we only have to consider small changes of electron density. Then we can write n e = n e0 + n e (3.206) where n e0 is the background density and n e a small variation. Thus the continuity equation is dn e dt = dn e dt = q e α eff (n e0 + n e) 2 = q e α eff n 2 e0 2α eff n e0 n e, (3.207) where the small second order term α eff n e2 is neglected. Hence we have a linear equation for n e, mathematically similar to eq. (3.184). The difference is that q e in eq. (3.184) is replaced by q e α eff n 2 e0 in eq. (3.207) and β by 2α eff n e0. Therefore, following eq. (3.191), the time constant for the variations n e is τ = 1 2α eff n e0. (3.208) This is often called the E region recombination time constant.

42 80 CHAPTER 3. THE IONOSPHERE OF THE EARTH 3.10 Ionization and recombination in the D region Figure 3.17: D region ionization by diferent sources (G.C.R:= galactic cosmic rays) (Whitten and Popoff, 1971). The lower part of the ionosphere below 90 km and down to about 60 km is called the D region. In the D region the atmospheric density is about 10 6 times the typical F region density and collisions are dominant. The photochemistry is very complex and not fully understood. Five important ionization sources the D region can be identified. Their contribution to the ion production is illustrated in Fig The Lyman-α line at nm penetrates down to the D region and ionizes nitric oxcide NO, which is a minor constituent in the neutral atmosphere. This happens because the ionization limit of NO is 134 nm. 2. EUV radiation between wavelengths and nm ionizes excited oxygen molecules O Hard X rays between 0.2 and 0.8 nm ionizes all constituents, thereby acting mainly on the abundant molecules N 2 and O Galactic cosmic rays ionize all constituents. 5. Relativistic electrons (100 kev - 1 MeV) and protons (1-100 MeV) may occasionally penetrate deep into the D region and produce ionization.

43 3.10. IONIZATION AND RECOMBINATION IN THE D REGION 81 Main ions are in the upper D-region NO +, O + 2 in the lower D-region cluster ions H + (H 2 O) n, (NO) + (H 2 O) n negative ions like O 2, negative cluster ions Figure 3.18: Concentration of positive ions detected by a mass spectrometer in the D region (Narcisi and Bailey, 1965). The positive ions formed directly by solar radiation in D region are O + 2, N + 2 and NO +. The N + 2 ions rapidly experience charge exchanges with O 2 to produce an O + 2 ions in the reaction N 2 + k + O 6 2 N 2 + O 2 + (3.209) The reaction coefficient k 6 is given in Table 3.3. The resulting O + 2 will quickly participate in charge transfer in reactions O NO k 3 NO + + O 2 (3.210) and O N 2 k 4 NO + + NO, (3.211)

44 82 CHAPTER 3. THE IONOSPHERE OF THE EARTH which both produce NO +. The latter of these reactions produces neutral NO, which, however, is not an abundant constituent. Still, NO + is an important ion species in D region, since it is produced in reaction (3.210). This production is large because the production of the primary ions O + 2 is effective (due to the large concentration of neutral O 2 ) and the concentration of the target molecules N + 2 is also great. The concentrations of a few important D region ion species are shown in Fig. 3.18, which displays results of rocket-borne mass spectrometer observations. The figure indicates that NO + (mass number 30) and O + 2 (mass number 32) are major ion species above about 85 km. Below that, hydrated ions (molecular mass numbers 19-37) such as H + H 2 O, H 3 O + H 2 O are found. These hydrates occur when water vapour concentration exceeds about m 3. Heavy ions with molecular masses above 45 are observed below 75 km. These are believed to be metallic ions, probably debris from meteor showers. An important feature of the D region is the formation of negative ions such as O 2. These are produced by three-body reactions O 2 + M + e β 1, β 2 O 2 + M, (3.212) where M is one of the abundant neutral molecules O 2 or N 2. The reaction coefficients β 1 and β 2 are given in Table 3.3. In such reactions the role of M is to remove excess kinetic energy from the reactants. The electron affinity of O2 is small, only 0.45 ev. Therefore the electron might be rapidly detached again by a low-energy quantum, such as infrared or visible light. This reaction is O 2 + hν ρ O 2 + e (3.213) with a reaction coefficient ρ. Negative ions may also be destroyed in collisions with neutrals. Such reactions are and O 2 + M γ 1, γ 2 O 2 + M + e (3.214) O 2 + O γ 3 O 3 + e. (3.215) The rate coefficients are given in Table 3.4; γ 1 is valid if M is N 2 and γ 2 if M is O 2. Other possible loss processes are ion-ion recombination in reactions O 2 + O + 2 α i1 O 2 + O 2 (3.216) and O 2 + O M α i2 O 2 + O 2 + M. (3.217) These equations illustrate the major chemical reactions taking part in the D region. Observations of ions in the D region have shown that the amount of negative ions increases by decreasing height (Fig. 3.19). While there is less than one negative ion per 100 electrons at 90 km at daytime, there may be more than 100 negative

45 3.10. IONIZATION AND RECOMBINATION IN THE D REGION 83 Figure 3.19: Negative ion-electron concentration ratio λ below 90 km. The model is based on molecular oxygen ions only (Risbeth and Garriot, 1969). ions per electron at 50 km. At night even larger concentrations of negative ions are possible. Since the ion chemistry in D region is very complicated, we will make a simplified model in order to understand the main features of D region. Let us denote the densities of electrons, positive ions and negative ions by n e, n + and n, respectively. Positive ions are destroyed in recombinations with electrons (effective recombination coefficient α D ) and in recombinations with negative ions (effective recombination coefficient α i ). In addition, negative ions are destroyed in collisions with neutrals (detachment coefficient γ) and by photons (detachment coefficient ρ). Radiative recombination is neglected. When the convection term is neglected, the electron continuity equation reads dn e dt = q α D n + n e βn e n n + γn n n + ρn, (3.218) where q is the electron production rate due to photoionization and n n is the neutral number density. The continuity equation for positive ions is dn + dt = q α D n + n e α i n n +. (3.219) Since electrons and positive ions are created in the same processes, they have equal production rates. Finally, the continuity equation for negative ions is dn dt Because of charge neutrality in plasma, we have = α i n n + + βn e n n γn n n ρn. (3.220) n + = n e + n. (3.221) By introducing the ratio between negative ions and electrons λ λ = n n e (3.222)

46 84 CHAPTER 3. THE IONOSPHERE OF THE EARTH we have n + = (1 + λ)n e. (3.223) With these notations α D n + n e = α D (1 + λ)n 2 e and α i n n + = α i λ(1 + λ)n 2 e. Then eq. (3.219) can be written as so that (1 + λ) dn e dt + n dλ e dt = q (α D + α i λ)(1 + λ)n 2 e, (3.224) dn e dt = q 1 + λ (α D + α i λ)n 2 e n e dλ 1 + λ dt. (3.225) In eq. (3.218), γn n n = γλn n n e, and ρn = ρλn e. Therefore eq. (3.217) gives dn e dt These two equations lead to = q α D (1 + λ)n 2 e βn e n n + γλn n n e + ρλn e. (3.226) [ 1 dλ 1 + λ dt = βn n λ ρ + γn n + (α i α D )n e + q (1 + λ)n e ]. (3.227) Under most conditions λ is fairly constant, one value at night and the other in the daytime. The electron attachment and detachment processes often occur so rapidly that they nearly balance each other. This means that recombination with positive ions becomes unimportant. Then, also in stationary situation the production and recombination of positive ions are in balance. In this approximation we can put dλ/dt to zero in eq. (3.227) and neglect the two last terms inside the brackets. Then which leads to βn n λ(ρ + γn n ) = 0, (3.228) λ = βn n ρ + γn n. (3.229) At night the detachment rate due to chemical reactions dominates over detachment rate due to solar radiation so that γn n > ρ and In the daytime, however, the opposite is true and λ = β γ. (3.230) λ = β ρ n n. (3.231) We see from Table 3.3 that β is proportional to [O 2 ] or [N 2 ] and λ will therefore increase by decreasing height in agreement with Fig This increase, however, will be stronger in the daytime than at night.

47 3.10. IONIZATION AND RECOMBINATION IN THE D REGION 85 Figure 3.20: Schematic electron density profiles in the D region for quiet and active Sun. If we assume that attachment and detachment are not in balance but λ is still constant, eq. (3.224) gives dn e dt = q 1 + λ (α D + α i λ)n 2 e. (3.232) This equation is mathematically similar to eq. (3.199), and therefore the electron density obeys a similar profile with an effective recombination coefficient and a production profile α eff = α D + λα i (3.233) q = q 1 + λ. (3.234) In a stationary case the electron density has a shape of an α profile n e = q α eff q = (1 + λ)(α D + λα i ). (3.235) Fig illustrates some typical electron density profiles observed in the D region for active and quiet solar conditions. The electron density decreases usually from m 3 at 90 km to less than 10 7 m 3 below 60 km. A ledge in the profile is often observed between 80 and 90 km where the density can decrease by one magnitude. This ledge is believed to occur due to the presence of hydrated ions in this height region. Since the hydrated ions are very effective in recombining with electrons, they will rapidly destroy the electrons as soon as they are created.

48 86 CHAPTER 3. THE IONOSPHERE OF THE EARTH Figure 3.21: Schematic figure of electron concentration in the lower D region with time. R and S represent sunrise and sunset, respectively (Ratcliffe, 1972). From radio wave propagation studies it has been observed that the electron density in the D-region is during quiet conditions fairly constant at day and night but changes rather rapidly at sunset and sunrise (see Fig. 3.21). It has been recently found that ion chemistry is coupled to the neutral chemistry in the D region and may lead to ozone loss (Fig. 3.22). This is because when protons or electrons precipitate into the atmosphere, ions and secondary electrons are produced. Ions and electrons react chemically and produce odd hydrogen (e.g. H or OH), odd nitrogen (e.g. N, NO and NO 2 ) and negative ions. This trio then affects ozone O 3 via catalytic reaction chains like and NO + O 3 NO 2 + O 2 (3.236) NO 2 + O 3 NO + 2O 2 (3.237) OH + O H + O 2 (3.238) H + O 3 OH + O 2 (3.239) 3.11 Plasmasphere In the lower ionosphere heavy ions such as NO + and O + 2 dominate, but in the upper ionosphere the lighter ions such as O +, He + and H + become more abundant. In the daytime, plasma is produced and lost by different processes in the E and F region. Above the F layer electron density peak we would expect plasma to be in diffusive equilibrium. This equilibrium is, in general, established between electrons and O + ions which, together, create the diffusion electric field pointing upwards. The lighter ions such as He + and H + experience an upward force in this electric field and are therefore accelerated upwards.

49 3.11. PLASMASPHERE 87 Figure 3.22: Schematic figure of effects of particle precipitation in the D region (Turunen, 2007). Figure 3.23: Earth s plasmaphere. At some height the ionosphere dominated by O + gives way to the protonosphere, which is dominated by H +. Protonosphere is also known as geocorona. Actually the term geocorona refers to the solar far ultraviolet light that is reflected off the cloud of neutral hydrogen atoms that surrounds the Earth. At these altitudes charge exchange processes H + + O H + O + (3.240) and O + + H O + H + (3.241) are extremely rapid. They tend to establish a chemical equilibrium in which concentrations at the transition altitude are related by [H + ][O] = 9 8 [H][O+ ], (3.242)