Chapter 5. Currents in the ionosphere. 5.1 Conductivity tensor

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1 Chapter 5 Currents in the ionosphere 5.1 Conductivity tensor Since both ions and electrons can move in the ionosphere, they both can also carry electric currents and the total current is the sum of the currents carried by these charged particle species. As discussed in Section 3.1, the driving forces causing the motion of charged particles are the electric field and the neutral wind and, when these remain constant, a stationary situation may be reached in which the charged particles move at constant velocities. Due to the geomagnetic field, the mobility of each particle species will be a tensor. When electrons move at a velocity v e they cause a current density j e = n e ev e. When a single ion species is assumed, the current density due to ions is j i = n e ev i, and the total current density is the sum of these two current densities. Using eqs. (3.22) and (3.28), we readily obtain j = j i + j e = n e e(v i v e ) = n e e[k i e(e + u B) + u k e ( e)(e + u B) u] = n e e 2 (k i + k e )(E + u B). (5.1) By denoting σ = n e e 2 (k i + k e ), where σ the conductivity tensor, we can write j = σ (E + u B) (5.2) In a coordinate system with the geomagnetic field pointing in the direction of the z axis, the component presentation of the conductivity tensor is (see eq. (3.15)) σ = n e e 2 k ip + k ep k ih + k eh 0 k ih k eh k ip + k ep k i + k e = σ P σ H 0 σ H σ P σ. (5.3) Here σ P, σ H and σ are the Pedersen, Hall and field-aligned conductivities and they are defined to be positive. The sings in the conductivity tensor appear because k eh is defined to be negative in eq. (3.24) and, due to the great angular electron 95

2 96 CHAPTER 5. CURRENTS IN THE IONOSPHERE gyrofrequency, k eh > k ih. Then a positive Hall conductivity is obtained with a definition σ H = n e e 2 (k ih + k eh ). The components of the conductivity tensor can now be calculated using the expressions of the mobilities in eqs. (3.16) (3.18) and (3.23) (3.25). The results are σ P = n ee B σ H = n ee B ( ωi ν in ωi 2 + νin 2 ( ω2 i + ω ) eν en ωe 2 + νen 2 ω 2 e + ωi 2 + νin 2 ωe 2 + νen 2 σ = n e e 2 ( 1 m i ν in + 1 m e ν en ) (5.4) (5.5) ). (5.6) Just like earlier in the case of mobility, collisions between ions and electrons are neglected here in comparison with collisions with neutrals. At very great altitudes, where the neutral density is small, ion-electron collisions become important in calculating σ. Because of the light electron mass, these collisions would mainly affect the electron motion and it is sufficient to replace ν en by ν en +ν ei in eq. (5.6). Ion-electron collisions can still be neglected at great altitudes in calculating σ P and σ H. This is true, because k ip 0, k ep 0, k ih 1/(eB) and k eh 1/(eB) when ν in 0 and ν en 0. Then both ions and electrons drift at a same velocity E B/B 2, and the ion-electron collisions do not affect their motion. The conductivity tensor is, like the mobility tensor, is invariant in the rotation around B. When neglecting the neutral wind, eq. (5.2) can be written as j = σ P E σ H E ˆb + σ E ˆb, (5.7) where ˆb is the unit vector in the direction of B. The first term on the right hand side is the Pedersen current, which flows in the direction of the electric field and the second term is the Hall current, which flows opposite to the E B direction. The last term is the parallel current. By choosing E B and an x axis pointing in the direction of E, we obtain j x = j P = σ P E x (5.8) j y = j H = σ H E x. (5.9) These are the Pedersen and Hall current densities.the corresponding ion and electron velocities are and v ix = v ip = ek ip E x (5.10) v iy = v ih = ek ih E x (5.11) v ex = v ip = ek ep E x (5.12) v ey = v ih = ek eh E x. (5.13)

3 5.1. CONDUCTIVITY TENSOR ALTITUDE / km σ / σ P σ H σ / σ P 100 σ H Oulu, June at noon Oulu, January at noon CONDUCTIVITY / 10-4 (Ωm) -1 Figure 5.1: Typical ionospheric conductivity profiles. Since both k ip and k ep are positive, this means that the Pedersen current is carried by ions and electrons moving in opposite directions. The case is different for the Hall current. Since k ih is positive and and k eh is negative, the Hall motion of both ions and electrons is in the E B direction. Hence, in the case of the Hall current, current carriers of both species move in the same direction which is opposite to the direction of the current! This happens, because k ih < k eh so that ions lag behind the electrons, and the current direction depends only on the relative speed of ions and electrons, not on the direction of the two motions. Fig. 5.1 presents model calculations of conductivity profiles at Oulu at noon in June and January. This is based on IRI-90 ionospheric model and CIRA 72 neutral atmospheric model, and a sunspot number 100 is assumed. Conductivity is caused mainly by photoionisation. Pedersen and Hall conductivity have peaks in the E region and the field-aligned conductivity increases with altitude. This can be understood in terms of eqs. (5.4) (5.6). When collision frequency is allowed to approach infinity, all conductivities approach zero. When the collision frequency approaches zero, the field-aligned conductivity approaches infinity. This basic behaviour of the expressions in brackets in eqs. (5.4) (5.6) is modified by the electron density profile. The conductivities are scaled by the altitude-dependent electron density. The conductivities are weaker in winter, because the electron density is smaller due to the weaker solar illumination. The maximum value of Hall conductivity is typically larger than that of Pedersen conductivity. It also lies at a lower altitude (close to 100 km) than the peak of Pedersen conductivity (above 120 km in height). The latter occurs approximately at an altitude where ω i = ν in. The maximum in the

4 98 CHAPTER 5. CURRENTS IN THE IONOSPHERE Figure 5.2: Measured ionospheric electron density profiles and calculated conductivity profiles (Brekke et al., 1974). Hall conductivity is more closely related to the peak of the electron density profile. The field-aligned conductivity increases with altitude and it is much larger than the perpendicular conductivities (notice the scaling factor). Since recombination is fast in E region, electron density drops at night and therefore the conductivities are also much smaller at night than in the daytime. This is shown in Fig. 5.2 presents some electron density and conductivity profiles measured by an incoherent scatter radar at Chatanika, Alaska. In the auroral region electron densities may be quite variable in the E layer, because precipitation of charged particles form the magnetosphere may greatly increase the E region electron density. Such precipitation often occur in the eveningmidnight-morning sector. During auroral particle precipitation events the enhanced electron densities can increase the conductivities much above the quiet time values. The night profiles in Fig. 5.2 show that the altitude of the peak in the Hall conductivity can also greatly vary at night. This happens because the Hall conductivity is sensitive to the high-energy auroral particle precipitation which most strongly enhances the ionization below 125 km. The enhanced peak below 100 km is caused by precipitating particles which are energetic enough to reach these heights and cause ionisation there. 5.2 Currents and particle motion The behaviour of electric currents at different altitudes can be understood in terms of the motion of the current carriers, i.e. ions and electrons. This motion is determined

5 5.2. CURRENTS AND PARTICLE MOTION 99 a) E b) E c) E v i ExB θ i v i ek ip E B ek ih E ExB θ i ek ih E ekip E B v i ExB ek ih E θ i ek ip E B Figure 5.3: Ion velocity vector v i at (a) lower E region (b) upper E region (c) lower F region. by the driving forces (electric fields and neutral wind) and the mobility tensor of each particle species. Fig. 5.3 demonstrates the ion motion at different altitudes in the plane perpendicular to the geomagnetic field. The ions are driven by an electric field E and the ion velocity has a component both in the direction of E and E B. The former component is determined by the Pedersen and the latter one by the Hall mobility, i.e. v ip = k ip E and v ih = k ih E. The angle between v i and E is given by tan θ i = v ih v ip = k ih k ip = ω i ν in (5.14) and the ion speed by v i = ee m i (ω 2 i + ν 2 in) 1/2 = E B sin θ i. (5.15) Here the expressions of Pedersen and Hall mobilities in eqs. (3.16) and (3.17) are used. Since ν in decreases strongly with height and ω i has only a weak height dependence, θ i increases with altitude. Low down the ion velocity is nearly parallel to the electric field and high up in the F region nearly perpendicular to it. At high values of the ion-neutral collision frequency the ion speed approaches zero and at small values to E/B. Hence the ion velocity behaves as shown in Fig Panel a) refers to low altitudes; the ion velocity is small and nearly along E. Panel b) portrays a greater height; the angle between the electric field and the ion velocity is increased and ions move faster. Finally, in panel c) the ions have nearly reached their highest speed and move in a direction close to the E B vector. The electron velocity has a similar behaviour. The difference, however, is that the electron gyrofrequency is much higher than the ion gyrofrequency. Therefore the electron velocity turns towards the E B direction more quickly with altitude; actually the electrons travel nearly in the E B direction already at E region altitudes. Ion end electron velocities at different heights are shown schematically in Fig. 5.4, where electrons are moving at all altitudes in the E B direction (this is true, when altitudes considered are above about 90 km). Panel a) corresponds

6 100 CHAPTER 5. CURRENTS IN THE IONOSPHERE a) E b) E c) E ExB v i -v e v e v i B ExB v i -v e v e v i B ExB v i v e B Figure 5.4: Ion and electron velocity vector and their difference related to current density at (a) lower E region (b) upper E region (c) F region. to low altitudes, where ions move nearly parallel to the eletric field and hence the main part of the current is Hall current carried by electrons. When moving towards greater heights (b), ion velocity increases and the direction starts to rotate from the E to the E B direction. Then the current density has both the Hall and Pedersen parts, with the Pedersen current carried solely by ions. At F region altitudes (say, above 200 km, shown in c) both the ions and electrons are in the E B drift motion and no current flows. 5.3 Height-integrated currents As seen above, the most intense ionospheric current densities occur in the E region. Fore some purposes, especially when considering the magnetic effect of ionospheric currents on the ground or the conductive ionosphere from the point of view of the magnetosphere, the ionosphere between altitudes z 1 70 km and z km can be treated as a thin conductive slab (Fig. 5.5). This slab carries horizontal currents. If we assume that the geomagnetic field is vertical, then the parallel current must be zero at the lower boundary, j z (z 1 ) = 0. Let us denote the value at the upper boundary by j z (z 2 ) = j. In a stationary case the charge density does not change, and therefore ρ e / t is zero in the continuity equation ρ e / t + j = 0. Hence j = (j + j z ) = 0, (5.16) where j = σ E (5.17) z 2 z j z = j z 1 j z = 0 Figure 5.5: Field-aligned current density at two altitudes in the ionosphere.

7 5.3. HEIGHT-INTEGRATED CURRENTS 101 is the horizontal (perpendicular) component of the current density, E = E + u B and ( ) σp σ σ = H (5.18) is the two-dimensional (perpendicular) conductivity tensor. This gives σ H σ P j x x + j y y = j z z. (5.19) By integrating the right hand side of eq. (5.19) over the height interval of the slab we get z2 j z z 1 z dz = j. (5.20) Similarly, by integrating the left hand side of eq. (5.19) we obtain z 2 z 2 j dz = (σ P E x σ H E x y)dz + (σ H E x + σ P E y y)dz z 1 z 1 z 1 = z 2 z 2 z 1 σ E dz. (5.21) If E and u are independent of height (the latter assumption is not very well valid), also E = E + u B is height-independent and we get z 2 z 1 j dz = (Σ E ), (5.22) where the height-integrated conductivity tensor Σ is defined by ( ) ΣP Σ Σ = H Σ H Σ P (5.23) and the height-integrated Pedersen and Hall conductivities, the Pedersen and Hall conductances are given by Σ P = Σ H = z 2 z 1 σ P dz (5.24) z 2 z 1 σ H dz (5.25) The unit of conductance is Siemens (S). The integration limit z 1 must be in the bottom of the ionosphere and z 2 at the top of the ionosphere. In conclusion, the height-integration of the current continuity equation (5.19) leads to j = J = [Σ (E + u B)], (5.26)

8 102 CHAPTER 5. CURRENTS IN THE IONOSPHERE a) b) c) j j j Figure 5.6: (a) Filed-aligned current density flowing towards the ionosphere along a magnetic field line and out of the ionosphere along the walls of the cylinder surrounding the field line. (b) Same situation as in (a) except that the radius of the cylinder is made infinitely large. (c) Filed-aligned current density flowing in and out of the ionosphere along magnetic field lines. where J = Σ E is the height-integrated current density, usually called only as height-integrated current. The unit of J is A/m. Assuming homogenous conductances and zero neutral wind velocity (u = 0), we obtain j = [(Σ P E x Σ H E y )e x + (Σ H E x + Σ P E y )e y ] ( Ex = Σ P x + E ) ( y Ex Σ H y y E ) y x = Σ P ( E) + Σ H ( E) z. (5.27) In a stationary situation E = 0, and then j = Σ P ( E) = (Σ P E) = J P. (5.28) This means that, with the present assumptions, Pedersen currents are connected to field-aligned currents. Therefore Hall currents must make closed loops in the ionosphere ( J H = 0). It turns out that the assumption of vertical geomagnetic field and horizontally homogeneous conductivity leads to the fact that the only magnetic disturbance on the ground level is due to the Hall currents. This can be seen as follows. Consider a case where a vertical downward line current enters the ionosphere and is connected by means of Pedersen currents to upward field-aligned currents on a cylindrical surface (Fig. 5.6 a). Then it is obvious from Ampère s law that B ds = 0 (5.29) C

9 5.3. HEIGHT-INTEGRATED CURRENTS 103 along any path C which encircles the whole current system. Assume, for a moment, that the ionosphere is isotropic so that Hall currents do not exist. The magnetic induction caused by each current element is perpendicular to the direction of the current. Therefore the magnetic induction caused by the vertical currents can only be horizontal. Due to cylindrical symmetry of the vertical currents, it can only have a cylindrical symmetry, and therefore the field direction can only be azimuthal. The horizontal Pedersen currents also produce horizontal field which only can be azimuthal. Then, if we choose the integration path C to be a horizontal circle with its centre on the line current, it is obvious that eq. (5.29) can be valid only if B = 0 at every point of the integration path. This is also valid, if C lies on the ground level. This means that the effects of the field-aligned current and the Pedersen current cancel on the ground. When the ionosphere is anisotropic, the Hall currents create additional magnetic effects. The Hall currents are circular loops and they produce magnetic induction which has both a vertical and a radial component. Hence, in the case of Fig. 5.6 a, the magnetic effects of the field-aligend and the Pedrsen currents cancel on the ground level and only the field caused by the Hall currents can be observed. The same arguments apply to any radius of the current cylinder, even if we let the radius grow towards infinity (Fig. 5.6 b). Then the magnetic effects of the field-aligned and Pedersen currents still cancel below the ionosphere, but not above the ionosphere (the space above the ionosphere is now inside the cylinder and there the integral of B is not zero). If we have several vertical line currents entering the ionosphere or leaving the ionosphere, the same arguments are valid separately for their magnetic effects and therefore the total magnetic effect below the ionosphere is caused by Hall currents even in this case. Then Pedersen currents of downward and upward current systems can be connected as in Fig. 5.6 c, but still the magnetic effects of field-aligned and Pedersen currents cancel on the ground. Any field-aligned current system can be built from line currents and therefore the result is also valid for any current system. If, however, the field aligned currents are not vertical or the ionosphere is not horizontally homogeneous, the above arguments are not valid and the magnetic induction caused by the field-aligend and Pedersen currents can be observed on the ground. The geomagnetic field is nearly vertical at high latitudes, and there the magnetic fields on the ground can be mostly due to Hall currents. Sometimes, however, the ionosphere is so inhomogeneous that this is not a good approximation. For a century, magnetic observations at high latitudes on the ground have been used to deduce the ionospheric current systems. The magnetic disturbances observed by a network of magnetometers can be used to calculate the ionospheric current, when it is assumed that no vertical currents exist. Since field-aligned currents (FAC) do exist, the assumption is not valid, and therefore the result of such an analysis is called the equivalent current system. By definition, ionospheric equivalent currents are divergence-free ( J = 0) horizontal sheet currents, that produce the same magnetic field below the ionosphere as the real (unknown) 3-dimensional current system. Some additional data, e.g. on ionospheric conductances, FAC or electric field is needed in order to determine the real 3-dimensional current system.

10 104 CHAPTER 5. CURRENTS IN THE IONOSPHERE In the KRM method (named after Kamide, Richmond and Matsushita) the ionospheric equivalent currents and ionospheric conductances are assumed to be known. If the ionospheric electric field is assumed to be given by a potential, the global solution can be found. One way of solving the ionospheric parameters is trial and error analysis, where distributions of conductivities, electric field etc. are varied, until a satisfactory agreement with observations is achieved. The AMIE procedure (Assimilative Mapping of Ionospheric Electrodynamics) is an objective way of doing the optimization in a least squares sense. In AMIE different types of measurements are assimilated, together with some statistical models. AMIE requires the ionospheric Pedersen and Hall conductance distributions as input. The method of characteristics (Inhester, 1985 and Amm, 1998) is used for local studies. In the method of characteristics the input quantities are the ionospheric electric field, equivalent currents and the conductance ratio Σ H /Σ P. As will be discussed later, in the auroral ionosphere (i.e. at high latitudes) large scale eastward or westward currents flow. Then, a rough estimate of the current density can be made as follows. Assume that we have an infinite current sheet where the height-integrated current is flowing in the eastward direction (Fig. 5.7). The magnetic perturbation on the ground can be calculated from the Ampère s law and it is B x = µ oj y 2 (5.30) The measured magnetic deviations are commonly given in a coordinate system where X is in the geographic north, Y in the geographic east and Z downward. Alternative coordinate system is H in the geomagnetic north, D the deviation toward east and Z downward. If the current has an arbitrary direction, then B = B xˆx + B y ŷ = µ o 2 (J yˆx J x ŷ). (5.31) x y J B z Figure 5.7: Infinite sheet current and magnetic perturbation below the sheet.

11 5.4. CURRENTS AT DIFFERENT LATITUDES Currents at different latitudes The field-aligned currents at high latitudes are connected to the magnetosphere of the Earth along the magnetic field lines. At low and mid latitudes no field-aligned currents exist. Also, the geomagnetic field is not even close to vertical there, and then the perpendicular current calculated from the electric field and conductivity tensor should have a large vertical component. Since the ionosphere is restricted vertically, at low and mid latitudes the ionosphere will be polarised in such a manner that the current will be horizontal. The conductivity tensor defined in eq. (5.3) is valid in the coordinate system xyz, where z points along the magnetic field, x is in the magnetic meridian plane and points in magnetic north and up and y points in magnetic east (Fig. 5.8). In order to calculate the horizontal and vertical currents we carry out a rotation to the coordiante system x y z, where z points downward towards the Earth s center in the magnetic meridian plane and x lies in the horizontal direction and points to magnetic north. The easiest way of doing this it to make a rotation by means of an orthogonal matrix The inverse of this matrix is U = U 1 = sin I 0 cos I cos I 0 sin I sin I 0 cos I cos I 0 sin I. (5.32). (5.33) According to (5.2), the presentation of Ohm s law is j = σ E in the xyz coordinates, when neutral wind is neglected. Correspondingly, in the x y z coordinates x x I B y I z z Figure 5.8: Coordinate system xyz where z is along B and the rotated system x y z where z is downwards.

12 106 CHAPTER 5. CURRENTS IN THE IONOSPHERE the presentation is j = σ E. The transformations of the current and field presentations are carried out by U and U 1, i.e. Since we get from eq. (5.34) Hence and by using eq. (5.32) and (5.33) we get σ = U σ U 1 = = sin I 0 cos I cos I 0 sin I j = U j = U σ E. (5.34) E = U E E = U 1 E (5.35) j = U σ U 1 E. (5.36) σ = U σ U 1 (5.37) σ P σ H 0 σ H σ P σ sin I 0 cos I cos I 0 sin I σ P sin 2 I + σ cos 2 I σ H sin I (σ σ P ) sin I cos I σ H sin I σ P σ H cos I (σ σ P ) sin I cos I σ H cos I σ P cos 2 I + σ sin 2 I Now we can calculate the current density by j = σ E j = = j x j y j z (σ P sin 2 I + σ cos 2 I)E x σ H sin IE y + (σ σ P ) sin I cos IE z σ H sin IE x + σ P E y σ H cos IE z (σ σ P ) sin I cos IE x + σ H cos IE y + (σ P cos 2 I + σ sin 2 I)E z. (5.38). (5.39) Expression (5.39) is valid at all latitudes. At high latitudes the inclination angle I 90 and we get eq. (5.3) from eq. (5.38). At middle and low latitudes the current must be horizontal as discussed in the beginning of the section. Also at high latitudes, the field-aligned currents may not be able to flow (e.g. due to magnetospheric conditions). Then the vertical current must be zero, j z = (σ σ P ) sin I cos IE x + σ H cos IE y + (σ P cos 2 I + σ sin 2 I)E z = 0. (5.40) This happens because the ionosphere will be polarized, i.e. there will be a charge accumulation on both sides of the conduction layer. The polarisation charges will modify the electric field to make the current horizontal. The polarization electric field E z is obtained from eq. (5.40) E z = (σ σ P ) sin I cos IE x + σ H cos IE y σ P cos 2 I + σ sin 2 I. (5.41)

13 5.4. CURRENTS AT DIFFERENT LATITUDES 107 When this is inserted in the expressions of j x and j y, the result is (the primes are neglected from now on) where ( jx j y ) = ( σxx σ xy σ yx σ yy ) ( Ex E y σ xx = σ σ P σ sin 2 I + σ P cos 2 I σ xy = σ σ H sin I σ yx = σ sin 2 I + σ P cos 2 I σ yy = σ P + ), (5.42) (5.43) (5.44) σ 2 H cos 2 I σ sin 2 I + σ P cos 2 I. (5.45) Hence the conductivity of the plasma sheet is determined by a two-dimensional conductivity tensor, which is determined by the components of the three-dimensional conductivity tensor and the inclination angle. One should notice that the twodimensional conductivity tensor is not rotation-invariant. This means that eq. (5.42) is only valid in a coordinate system where z points downward in the magnetic meridian plane, x is horizontal and points to magnetic north and y points to magnetic east. One should also notice that the conductivity tensor is height-dependent, because the Hall, Pedersen and field-aligend conductivities are height-dependent. Although twodimensional, this conductivity tensor is quite different from the height-integrated conductivity tensor in eq. (5.23). At all altitudes in the ionosphere where the currents flow, we have σ σ P and σ σ H. If the inclination is not too small (i.e. we are not too close to the magnetic equator), σ sin 2 I σ P cos 2 I and σ sin 2 I σ H cos 2 I so that we can use an approximation σ xx σ P (5.46) sin 2 I σ xy σ yx = σ H (5.47) sini σ yy σ P. (5.48) On the other hand, if we are close enough to the magnetic equator, σ sin 2 I σ P cos 2 I and σ sin 2 I σ H cos 2 I. The we have an approximation σ xx σ (5.49) σ xy σ yx = 0 (5.50) σ yy σ P + σ2 H σ P = σ C. (5.51) Here σ C is called the Cowling conductivity. Hence at low latitudes ( ) ( ) ( ) ( ) jx σ 0 Ex σ E = = x j y 0 σ C E z σ C E y (5.52)

14 108 CHAPTER 5. CURRENTS IN THE IONOSPHERE Figure 5.9: Model calculations of conductances as a function of magnetic latitude. The current density in the direction perpendicular to the geomagnetic field (in the geomagnetic east-west direction)at low latitudes is determined by the Cowling conductivity and plays a role in the equatorial electrojet, which will be discussed later in this chapter. Figure 5.9 shows the two dimensional conductivity tensor components by eq. (5.42) in the height-integrated form. At low latitudes, Σ xx is equal to Σ and Σ yy is the Cowling conductance. At high latitudes, Σ xx and Σ yy merge into Σ P, which is usually somewhat smaller than Σ xy representing the Hall conductance at high latitudes. 5.5 The atmospheric dynamo The ionospheric currents at middle and low latitudes are driven by electric fields generated by the atmospheric dynamo. The driving force of the dynamo is the moving neutral atmosphere, mainly the tides. At E region altitudes the neutral wind exerts a force on the charged particles via collisions, and the force is different on ions from the force on electrons; more specifically the force density is m i n e ν in (v i u) on ions and m e n e ν en (v e u) on electrons. The result is that ions and electrons move at different velocities and, due to the geomagnetic field, also in different directions so that electric currents are generated. These can be interpreted to be a result of an effective electric field u B. Let us assume that, originally, no electric field is present in the ionosphere. Then the current density according to (5.2) is j = σ (u B). (5.53)

15 5.5. THE ATMOSPHERIC DYNAMO 109 If the conductivity or the u B field is not horizontally homogeneous, it may happen that [σ (u B)] is not zero. Then, according to the current continuity equation, ρ e / t + j = 0, the charge density must depend on time so that electric charges will be accumulated in the ionosphere. These create a polarisation electric field E, which can be expressed as a gradient of a electrostatic potential φ. E = φ. (5.54) If vertical currents cannot flow, the polarisation electric field will produce an electric current, which will adjust the total current to be horizontal. The total current density is j = σ L (u B φ) h, (5.55) where the subscript h means the horizontal component and σ L is the two-dimensional conductivity tensor in eq. (5.42). The polarisation electric field gets a value which leads to a stationary situation where j = j h = 0. In conclusion, the driving force causing the currents is the neutral wind and the currents generate electric fields which try to make the currents stationary. This is called the atmospheric dynamo. The currents flow in a resistive medium and therefore they heat (mainly) the ions and the neutral molecules. Hence, the result is that the mechanical energy in the neutral wind will be changed to into heat. The system is analogous with a generator producing currents which heat a resistor. If we assume that u, B and φ are independent of height in the E region where the currents flow, we can height-integrate eq. (5.55). This gives J = Σ L (u B φ) h, (5.56) where Σ L = σ L dz (5.57) is the height-integrated conductivity tensor and J is the height-integrated current. Since u is often height-dependent, this approximation is not necessarily good. Because the field-aligned conductivity is very high, the magnetic field lines within the E and F regions are nearly equipotentials. Then the electric field generated by the atmospheric dynamo in the E region will be mapped upwards to the F region altitudes. The effect of the neutral wind is small in F region where the plasma (both ions and electrons) move at a velocity v = E B B 2. (5.58) In the eq. above, the electric field is the polarisation electric field produced by the atmospheric dynamo. Because the ion life time is long in the F region, this motion can essentially affect the electron density profile in the F region. This is a mechanism by which the motion of neutral air in the E region is converted into plasma motion in the F region. Hence we have an E region dynamo which is running an F region motor.

16 110 CHAPTER 5. CURRENTS IN THE IONOSPHERE 5.6 Sq current system When the geomagnetic field is observed on the ground level, it is found that the field has a regular diurnal variation with occasionally appearing disturbances especially at high altitudes. An example of such regular magnetic variations at different latitudes is shown in Fig These variations have a seasonal variation and they are an indication of an ionospheric current with a regular daily variation. Since the variation is diurnal, it must be connected to the Sun. An obvious explanation is that the currents must be caused by the atmospheric dynamo and the ionospheric conductivity as explained in the previous section. The winds are controlled by the atmospheric tides, which are mainly due to the solar heating. Therefore this global pattern of currents is called the Sq (solar quiet) current system and the corresponding magnetic variations are called Sq variations. The seasonal changes in the Sq system are connected to the fact that the relative solar illumination of the two hemispheres varies with season. Since the Moon produces gravitational tides in the atmosphere and they can be separated from the solar tides, it is also possible to divide the magnetic variations and ionospheric currents to solar and lunar parts. The magnitude of the lunar variation is much smaller than the Sq variation and, because it is of gravitational origin, the lunar variation is semidiurnal. Fig shows an example of the Sq current system during sunspot minimum in equinox conditions. The left hand panel presents the dayside and the right hand Figure 5.10: Magnetic variations at low and middle latitudes due to the Sq current system (1 γ= 1 nt).

17 5.7. MAPPING OF THE ELECTRIC FIELD 111 Figure 5.11: Sq current systems during day (left) and night (right) at equinox. panel the nightside hemisphere. The current direction is shown by arrows. The numbers are in units 10 3 A, which means that the total current flowing between two neighbouring lines is 10 4 A. At equinox, both the northern and southern hemisphere is illuminated in a similar way and therefore the solar tides and the resulting currents are symmetric, as seen in the figure. Fig does not represent the total current system at high latitudes, where additional current systems appear. Also, at magnetic equator, the magnetospheric ring current may be visible. 5.7 Mapping of the electric field At high latitudes the horizontal ionospheric currents may be connected to fieldaligned currents. Such current systems are driven by magnetospheric electric fields. These electric fields are mapped to ionospheric altitudes where they are are superposed to the electric fields generated by the atmospheric dynamo. The mapping of the electric fields is associated with the fact that the magnetic field lines have a tendency of being equipotentials. The fact that the geomagnetic field lines are equipotentials can be easily understood in terms of an equivalent circuit in Fig Here R 0 represents the ionospheric horizontal resistance, R 1 is the field-aligned resistance and V is the voltage of the generator driving the current system. The current flowing in the circuit is I = V/(R 0 + 2R 1 ). Therefore the voltage in resistor R 0 is V 0 = V R 0 /(R 0 + 2R 1 ). If R 0 R 1, obviously V 0 V so that V 1 V 0, i.e. both ends of the resistors R 1 are nearly at the same potential. Because the field-aligned conductivity is very high, the field-aligend resistance between the magnetosphere and the ionosphere is very small, and therefore the magnetic field lines are nearly equipotentials. Consider the two field lines in Fig They are separated by a distance s m in the magnetic equatorial plane and their separation is s i in the ionosphere. The electric field lies in the magnetic meridional plane and is perpendicular to the field

18 112 CHAPTER 5. CURRENTS IN THE IONOSPHERE V 1 R 1 V R 0 V 0 R 1 V 1 Figure 5.12: Schematic figure of the ionosphere-magnetosphere current circuit. lines, because they are equipotentials. The potential difference between the field lines must be the same everywhere; in the magnetic equatorial plane it is s m E m and in the ionosphere s i E i. Hence s i E i = s m E m (5.59) so that E i = s m s i E m = r mi E m, (5.60) where r mi is the mapping factor determined by the relative distances of the magnetic field lines in the magnetosphere and in the ionosphere. For a dipole magnetic field, expressions for the mapping factor can be easily found. Since s m > s i, r mi > 1 and E i > E m. Hence, the magnetospheric electric fields are much smaller than those in the ionosphere. Although Fig is drawn in magnetic meridional plane, acorresponding factor is also obtained in the azimuthal direction because the magnetic field lines converge towards the ionosphere in that direction, too. Fig shows an example of the mapping the electric field from the magnetosphere to the ionosphere. The bottom curve is the eastward component of the equatorial electric field determined from whistlers (a certain type of audio frequency electromagnetic waves propagating along magnetic field lines from one hemisphere E i s i s m E m Figure 5.13: Mapping of electric field between the magnetosphere m and ionosphere i. The upper part of the magnetic flux tube is shown.

19 5.7. MAPPING OF THE ELECTRIC FIELD 113 Eastward electric field [mv/m] ionosphere magnetosphere 9 10 July, Universal time Figure 5.14: Electric field eastward component in the ionosphere by an IS radar (top) and in the magnetosphere (bottom) by whistler measurements. Note the different scales. to the other) at L values The top curve is the eastward electric field at the ionospheric F region measured by an incoherent scatter radar nearly at the same field line. The same pattern is repeated in both curves but the intensity of the ionospheric electric field is about ten times the intensity of the magnetospheric field. At ionospheric altitudes the field-aligned conductivity is lower than in the magnetosphere, which affects the mapping of the electric field. If we do not put the field-aligned conductivity to zero, the mapping can be treated as follows. Using an approximation of vertical geomagnetic field, we can write the current density in the form j = (σ P E x + σ H E y )e x + (σ P E y σ H E x )e y + σ E z e z. (5.61) Here the z axis is upwards in the northern hemisphere (antiparallel to B, notice the signs of σ H ) and x and y axes are horizontal. We assume a horizontally homogeneous ionosphere but take into account the vertical variation of the conductivities. In a stationary state j = 0 so that E x j = σ P x + σ E y H x + σ E y P y σ E x H y + (σ E z ) z = 0. (5.62) In steady state E = 0, and therefore E y / x = E x / y. Thus the equation above can be written as E x j = σ P x + σ E y P y + (σ E z ) z = 0. (5.63) We notice that the Hall conductivity has disappeared from the equation. This can be understood when we remember that, under the present assumptions, Hall

20 114 CHAPTER 5. CURRENTS IN THE IONOSPHERE 10 3 (σ /σ P ) 1/ Altitude / km Figure 5.15: Square root of parallel-to-pedersen conductivity ratio in the ionosphere. currents are purely horizontal and only Pedersen currents are connected to the fieldaligned currents. The static electric field is given by the gradient of the electrostatic potential φ, i.e. E = φ. (5.64) By inserting this in eq. (5.63) we obtain 2 φ x + 2 φ 2 y σ P ( ) φ σ = 0. (5.65) z z Next we make a coordinate transformation z σp = z (5.66) σ and use a notation σ m = σ P σ (5.67) and write eq. (5.65) in the form 2 φ x + 2 φ 2 y + 1 ( ) φ σ 2 σ m z m = 0. (5.68) z The difference between this equation and eq. (5.65) is that this equation contains only a single conductivity σ m. Fig shows an example of the mapping ratio (σ /σ P ) 1/2 at altitudes extending from D to F region, showing that (σ /σ P ) 1/2 strongly increases with height in the ionospheric altitudes. This is also obvious from eqs. (5.4) and (5.6) which indicate that both 1/σ P and σ increase with height. Thus the transformation in eq. (5.63) means a contraction of the z coordinate. Solving eq. (5.64) analytically implies an assumption on the z dependence of σ m. We notice that dσ m dz = dσ m dz dz dz. (5.69)

21 5.7. MAPPING OF THE ELECTRIC FIELD 115 Since the terms in the brackets in the Pedersen conductivity in eq. (5.4) are proportional to the collision frequencies and the terms in the field-aligned conductivity in eq. (5.6) are inversely proportional to the collision frequencies at high altitudes, σ m depends only weakly on z. Therefore dσ m /dz is proportional to dz/dz = 1/(σ /σ P ) 1/2. Fig indicates that (σ /σ P ) 1/2 increases with altitude even more steeply than exponentially. However, if assume an exponential growth, dz/dz decreases exponentially with height. Then ( ) σ m = σ 0 exp z, (5.70) α where σ 0 is a constant and α is the scale height of σ m. With this approximation, eq. (5.68) reads 2 φ x + 2 φ 2 y + 2 φ 2 z 1 φ = 0. (5.71) 2 α z The solution of this equation depends on the source field at some altitude z 0, and it tells how the field at the boundary is mapped to other heights in the ionosphere. We assume that the source potential is maintained by some generator. If z 0 is at the bottom of the ionosphere, the generator is the atmospheric dynamo. If, on the other hand, z 0 lies at the top of the ionosphere, the generator lies in the magnetosphere, and the field is mapped to z 0 as determined by the mapping ratio r mi. We assume that the potential has a sinusoidal variation at a boundary z 0 with wave numbers k x and k y in the x and y directions, respectively. This choice is reasonable, because eq. (5.71) is linear and any potential function at the boundary can be built from sinusoidal functions of different wave lengths. Hence the solution is φ(x, y, z ) = φ (z ) exp[i(k x x + k y y)], (5.72) which gives 2 φ x + 2 φ = (k 2 2 y 2 x + ky)φ 2 (z ) exp[i(k x x + k y y)] = k 2 φ (z ) exp[i(k x x + k y y)], (5.73) where k = (kx 2 + ky) 2 1/2 is the horizontal wave number. By inserting this in eq. (5.71) we obtain d 2 φ dz 1 dφ k 2 α dz φ 2 = 0. (5.74) This is a homogeneous linear differential equation of second order. The standard approach is to use a trial solution φ exp(γz ), which leads to a characteristic equation The solutions of the characteristic equation are γ 2 1 α γ k2 = 0. (5.75) γ 1,2 = 1 2α ± ( ) k 2 2α. (5.76)

22 116 CHAPTER 5. CURRENTS IN THE IONOSPHERE Hence the general solution of eq. (5.74) is φ (z ) = K 1 exp(γ 1 z ) + K 2 exp(γ 2 z ), (5.77) where K 1 and K 2 are integration constants. This leads to a solution φ(x, y, z ) = [K 1 exp(γ 1 z ) + K 2 exp(γ 2 z )] exp[i(k x x + k y y)] (5.78) for eq. (5.71). This can also be written as φ(x, y, z ) = [K 1 exp(γ 1 z 0) + K 2 exp(γ 2 z 0)] exp[i(k x x + k y y)] = {C 1 exp[γ 1 (z z 0)] + C 2 exp[γ 2 (z z 0)]} exp[i(k x x + k y y)], (5.79) where C 1 = K 1 exp( γ 1 z ) and C 2 = K 2 exp( γ 2 z ). This notation is adopted in order to expose the deviation from the reference level z 0 explicitly in the solution. We notice that γ 1 > 0 and γ 2 < 0 in eq. (5.76). Therefore γ 1 gives exponential growth with increasing height above the reference level. Correspondingly, γ 2 gives exponential growth with decreasing height below the reference level. Such solutions are not physically realistic, since the potential must be attenuated with increasing distance from the source region. Therefore we must have C 1 = 0 above and C 2 = 0 below z 0. In order to make the solution continuous at z 0 we must have C 2 = C when z > z 0, C 1 = C when z < z 0 and C 1 + C 2 = C when z = z 0. Hence, below the generator region, e.q. (5.79) can be written as φ(x, y, z ) = φ 0 exp[ γ 1 (z 0 z )] exp[i(k x x + k y y)]. (5.80) Altitude / km Altitude / km Damping ratio Damping ratio Figure 5.16: Mapping of potential from high to low altitudes during day (left) and night (right) conditions. Numbers indicate wavelengths.

23 5.7. MAPPING OF THE ELECTRIC FIELD 117 Figure 5.17: Damping of the dynamo electric field from 130 km to higher altitudes. Numbers indicate wavelengths. Now, γ1 determines the damping of the potential versus height. Note from eq. (5.76) that the damping factor γ1 depends on the scale height α of conductance σm and the horizontal wavenumber k = 2π/λ of the source potential. Fig shows results of electric field mapping from the magnetosphere down to the ionosphere for various horizontal wavelengths. The left hand panel indicates a daytime and the right hand panel a nighttime case. It is seen that long wave lengths (λ >100 km) km are not much attenuated, but the shorter the wave length is, the higher is also the attenuation. At 100 km, where most of the perpendicular current flow, structures at wavelengths shorter than 10 km during day and 5 km during night are heavily attenuated. Fig portrays the upward mapping of E region electric fields caused by the atmospheric dynamo. Also in this case, the shorter wave lengths are more heavily attenuated. The attenuation levels off at great altitudes. This is a result of the increase of the field-aligned conductivity with height.

24 118 CHAPTER 5. CURRENTS IN THE IONOSPHERE