Electron pressure effects on driven auroral Alfvén waves

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2004ja010610, 2005 Electron pressure effects on driven auroral Alfvén waves Jörgen Vedin and Kjell Rönnmark Department of Physics, Umeå University, Umeå, Sweden Received 3 June 2004; revised 1 October 2004; accepted 8 November 2004; published 20 January [1] Fluid models for the auroral electron acceleration processes have almost exclusively been derived by assuming cold or isothermal electrons. The consequences of these assumptions have never been thoroughly analyzed. In this study we compare results from an isothermal simulation with those obtained when the pressure is calculated from a double adiabatic approximation and from stationary kinetic theory. We find that the reflection of shear Alfvén waves, as well as the current-voltage relation, is very sensitive to the description of the electron pressure variations. Using pressures calculated from steady-state kinetic theory, we find that driven shear Alfvén waves can build up auroral currents and fields that are consistent with a linear current-voltage relation. Citation: Vedin, J., and K. Rönnmark (2005), Electron pressure effects on driven auroral Alfvén waves, J. Geophys. Res., 110,, doi: /2004ja Introduction [2] Discrete auroras are created by precipitating electrons that have a rather well defined peak in their energy spectrum. This indicates that the electrons have been accelerated by a field-aligned electric field on their way toward the ionosphere. From rocket and satellite observations it is known that the auroral electrons often are accelerated through kv potential drops [e.g., Hoffman, 1993; Evans, 1968, 1974; Mizera and Fennel, 1977], and kinetic models predict that large potential drops are required to drive the currents associated with discrete auroras [Knight, 1973; Chiu and Schulz, 1978]. However, collisionless fluid models describing the auroral electron acceleration have not been able to explain how potential drops of that magnitude can be maintained. [3] During the past decades, several different fluid models have been developed to describe the auroral electron acceleration process. In most of these models the electrons have been assumed to be cold [Lysak and Dum, 1983; Lysak, 1985; Kletzing, 1994] or isothermal [Goertz and Boswell, 1979; Streltsov et al., 1998; Streltsov and Lotko, 1999; Rönnmark and Hamrin, 2000]. However, some recent studies indicate that kinetic effects and pressure gradients play a decisive role in the auroral flux tube dynamics, both according to theory [Janhunen, 1999; Vedin and Rönnmark, 2004] and observations [Hull et al., 2003a, 2003b]. The combination of a field-aligned electric field and converging magnetic field lines in auroral flux tubes strongly affects the parallel and perpendicular electron pressures, and the motion of the electron fluid is to a large extent determined by the balance between these pressure variations and the electric field. [4] In this study we present results from dynamic fluid simulations of the auroral electron acceleration process. The simulation model is based on Rönnmark and Hamrin [2000] Copyright 2005 by the American Geophysical Union /05/2004JA010610$09.00 but with some important differences. We will describe the electrons as a single fluid, while they used two electron fluids. Furthermore, their study was isothermal, but here we will let the temperatures vary dynamically according to different closures of the fluid equations. The purpose is to stress the importance of a correct equation of state when considering auroral flux tube dynamics. [5] A local equation of state relating density and pressure variations is usually assumed in collision-dominated gases and plasmas. Similar local equations of state can also be used to describe magnetized collisionless plasmas with anisotropic pressures, as long as the field-aligned particle motion can be ignored. However, in an auroral flux tube with high fluxes of accelerated field-aligned electrons the assumption of a local equation of state cannot be justified. In this case a kinetic model is needed to describe how electrons, once accelerated in one place, will later affect the pressures all along the field line. A few recent studies [Tikhonchuk and Rankin, 2002; Lysak and Song, 2003] have included a fully kinetic description of the electrons within a linearized theory. Such linear models can be applied when the electron distribution function is slightly perturbed from its ground state, but the fieldaligned electric fields responsible for auroral electron acceleration are strong enough to completely reorganize the electron distribution. There is still no kinetic theory that can be applied when a large fraction of the electrons is accelerated to suprathermal energies. [6] To compare with previous studies and illustrate the electron pressure effects on Alfvén waves, we will present results from isothermal and double adiabatic closures of the fluid equations. However, as mentioned earlier, we cannot expect such local equations of state to properly describe the auroral acceleration region. Therefore we will also present results where we include pressures calculated from stationary kinetic theory into our dynamic simulation. [7] In our model, the dynamics is driven by a generator in the equatorial magnetosphere. Field line resonances and 1of11

2 the reflections of monochromatic Alfvén waves are interesting in several ways and have been considered in numerous studies [e.g., Cheng et al., 1993; Streltsov et al., 1998; Streltsov and Lotko, 1999; Tikhonchuk and Rankin, 2000]. To allow comparisons with these studies, we first use a generator that creates monochromatic shear Alfvén waves. However, focusing only on periodic waves, we would obtain little information about the very important role played by driven Alfvén waves during transitions between different quasi-stationary states. Hence we also use a second generator to present simulations where the auroral flux tube goes through a transition from a currentfree state to a quasi-stationary state with a substantial, steady current. 2. Theory and Simulation Model [8] The geometry of the auroral current circuit and the generator region in the equatorial magnetosphere is sketched in Figure 1. We introduce a coordinate system based on the flux tube geometry, with z along the magnetic field lines, x increasing to the Earth, and y to the west. The origin of this coordinate system is at a point where an auroral field line crosses the equatorial plane. Since x and y are constant along the converging field lines, a distance dl is related to these coordinates by the metric dl 2 ¼ dx 2 þ dy 2 B 0 þ dz 2 ; where (z) is the geomagnetic field and B 0 is the field strength at z = 0 in the equatorial plane. We neglect the curvature of the field lines and assume homogeneity in the y direction Maxwell s Equations [9] The current density j = j x^x + j z^z will, according to Maxwell s equations, couple to the electric field components E x, E z, and the magnetic field B y. Ampere s and Faraday s laws for these fields are in our coordinate system expressed as t E x ¼ c z B y B y þ m 2B 0 j x ; ð2þ z t E z ¼ c x B y m 0 j z ; ð3þ B 0 t B y x E z E x þ E x ; 2 B 0 where c =(e 0 m 0 ) 1/2 is the speed of light. [10] The ion motion along the field lines is neglected, and we assume that the ion velocity U y in the y direction is given by the E B drift. We will consider the ions to be cold, which implies that the ion kinetic effects on Alfvén wave propagation are neglected, although they may be significant ð4þ Figure 1. The geometry of the auroral current circuit and the generator region in the equatorial magnetosphere. The curvature of the magnetic field lines is neglected in our model equations. at altitudes above a few R E. Including a prescribed mechanical force F y that drives the generator, the y component of the equation of motion for the ions is d t U y ¼ e m i U x þ F y n i m i ; where e is the proton charge, n i is the ion density, and m i is the ion mass. Using U y = E x / in this equation allows us to calculate the ion velocity U x and the current j x, which essentially is the ion polarization current. Inserting this current in Ampere s law gives [Rönnmark and Hamrin, 2000] t E x ¼ A 2 c z B y B y 1 A 2 F y ; 2 n i m i qffiffiffiffi 1 mi A 2 ¼ VA 2 eb 2 z B x E x qffiffiffiffi ; c 2 þ VA 2 mi B 1 x E x pffiffiffiffiffiffiffiffiffiffiffiffiffi and V A = / m 0 n i m i is the Alfvén velocity Fluid Equations [11] The evolution of the phase space density f is determined by the Vlasov equation eb 2 z B 0 ð5þ ð6þ d t f t f þ v rf e m ðe þ v BÞ@ vf ¼ 0; ð7þ where m is the electron mass. We consider one spatial dimension z and two velocity dimensions v z = _z and v?, implying that the phase space density is a function f = f(v z, v?, z, t) and the Vlasov equation can be rewritten t f þ v z f ee z m þ vz f þ v?v v? f ¼ 0: ð8þ 2 2 2of11

3 [12] When the phase space density f is known, we can calculate fluid quantities such as the electron density Z n ¼ fdv; ð9þ the field-aligned bulk flow velocity u z ¼ 1 Z v z fdv; n the parallel pressure Z P z ¼ ð10þ mv ð z u z Þ 2 fdv; ð11þ The hierarchy of fluid equations (15) (18) must be closed, either by assuming equations of state that determines the pressures or by finding equations for the evolution of the heat conductions q zz and q z? [Ramos, 2003]. In this study we present results from three different closures of the hierarchy of fluid equations. We start with the local isothermal and double adiabatic closures, and then we continue with a newly developed nonlocal closure of the fluid equations through stationary kinetic theory Model Equations [13] We now introduce new dimensionless independent variables through the perpendicular pressure the conduction of parallel heat Z mv 2 P? ¼? fdv; ð12þ 2 Z q zz ¼ and the conduction of perpendicular heat mv ð z u z Þ 3 fdv; ð13þ x ¼ W i c x; z ¼ W i c z; t ¼ W it; where W i = eb 0 /m i, and the dimensionless fields p E x ¼ E x = ffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 B 0 ; E z ¼ E z = ðcb 0 Þ; p B ¼ B y = ffiffiffiffiffiffiffiffiffi B 0 ; ð19þ q z? ¼ Z mvz ð u z Þv 2? fdv: 2 ð14þ From these definitions we can also construct the fieldaligned current density j z = enu z, the parallel temperature T z = P z /n, and the perpendicular temperature T? = P? /n. Taking moments of equation (8), we can derive the fluid equations relating the above defined fluid quantities to each other. Integrating the Vlasov equation over velocity space, we find the continuity equation for t n z ðnu z Þþnu z : ð15þ Multiplying the Vlasov equation by v z before integrating over velocity space, we find the momentum t j z ¼ e2 n m E z z e m P z þ j2 z en e ð m P z P? Þþ j2 : en ð16þ By multiplying the Vlasov equation by (v z u z ) 2 or v? 2 before integrating over velocity space, we will find the energy equations. These are evolution equations for the pressures defined in equations (11) and (12), and they are found to t P z z ðu z P z þ q zz Þ 2P z u z 2q z? and t P? z z ðu z P? þ q z? Þ: ð18þ n ¼ nm i = ðe 0 B 0 Þ; u z j ¼ n s c ¼ m 0c j z ; W i T ¼ T= ðmc 2 Þ; P ¼ nt; q ¼ 1 P c P q; F y F ¼ p ffiffiffiffiffiffiffiffiffiffiffiffi : n i m i cw i B 0 = ð20þ The governing equations (6), (3) (4), (15) (18) can then be rewritten in simulation t E x ¼ A z B 1 A 2 F; t E z ¼ ð@ x B þ jþ; ð21bþ B t j ¼ m i m ne t B x E z E x t n z j; P z þ j2 n ð21cþ ð21dþ z ; ð21eþ t P z z n P j z þ q zz 2P z n z z? ; t P? z j n P? þ q z? : ð21gþ 3of11

4 Figure 2. Plots from the isothermal simulation showing the fields related to the monochromatic Alfvén waves during one wave period, (a) the perpendicular electric field E x, and (b) the perpendicular magnetic field B y. The labels indicate the time in seconds, starting with label 0 at 41 s. The first half of the cycle is shown as full lines and the second half as dotted lines. [14] These equations describe the propagation of shear Alfvén waves but also high-frequency waves such as electrostatic oscillations at the plasma frequency. For a more comprehensive discussion about the physical processes described by equations (21) we refer to Rönnmark and Hamrin [2000]. They also describe the implicit algorithm used in the numerical integration of these equations Model Parameters [15] All results presented in this study are based on a single set of model parameters. The simulation plane has the dimensions 0 < z < L z, L x < x < L x, where L z = km (or 8.6 R E ) and L x is 2000 km at the generator and 78.6 km in the ionosphere due to the convergent magnetic field lines. In the results we will also use the height h = L z z above the ionospheric boundary. [16] The geomagnetic field strength is modeled by h ðþ¼ 0 exp ðz=l z Þ 2 ðlnðb I =B 0 Þ 0:6 1:8ðz=L z i Þ 2 þ2:4ðz=l z Þ 6 Þ ; ð22þ with an ionospheric field strength B I =56mTand B 0 = mt. This magnetic field model approximates the dipole field for the L = 7 shell. [17] As mentioned in section 1, we will present results using two different generators. When driving monochromatic Alfvén waves along the field lines, we will use the force 8 F y ðx; z; tþ ¼ F pz 0 cos exp x2 2L zg L 2 sin 2pt < z < L zg xg t : G 0 z L zg ; ð23þ where F 0 = Nm 3, L zg = V AG t G /4 with V AG as the Alfvén velocity at the equatorial magnetosphere, and L xg = 100 km. The parameter L xg controls the transverse scale of the driven Alfvén waves. At the ionosphere this scale transforms to about 4 km, and in the acceleration region it corresponds to about twice the electron inertial length. Unless otherwise stated, the period t G of the generated wave will be 10 s. [18] When studying the transition from a current-free state to a quasi-stationary state, we will use a second force of the form F y ðx; z; tþ ¼ F 0 exp z2 L 2 x2 t 2 zg L 2 xg t 2 þ tg 2 ð24þ to generate a steady current. Here, L zg = 6000 km, while the other parameters have the same values as in the previous case. Notice that the characteristic time t G now is the time it takes the generator force to reach half its final value Initial and Boundary Conditions [19] The initial conditions for the simulation are found from stationary kinetic theory. The boundary conditions for the temperatures and densities in this initialization are T M = 1 kev and n M = m 3 for the magnetospheric electrons and T I = 1 ev and n I =110 9 m 3 for the ionospheric electrons. We apply an ambipolar electrostatic potential of the shape 0 z < 0:9L f amb ðþ¼ z z E amb ð0:9l z zþ z 0:9L z ; ð25þ where E amb is a constant. Assuming Maxwellian distributions at the boundaries, we use the algorithm presented by Vedin and Rönnmark [2004] to calculate the values of the fluid variables from kinetic theory. By adjusting the value of 4of11

5 Figure 3. Plots from the isothermal simulation showing the quasi-stationary state of (a) the field-aligned current density mapped to the ionosphere and (b) the potential. The snapshots are taken 100 s after the generator was started. E amb until j z = 0, we find an initial state for the densities and pressures. In all the simulations presented here, the ambipolar potential drop is Df amb 6.5 V. [20] The lower ionosphere is represented by a heightintegrated ionospheric Pedersen conductivity, which we take to be S P = 10 W 1. Integrating Ohm s law and Ampere s law across the ionospheric height leads to the condition E x ðl z Þ ¼ 1 B y ðl z Þ ð26þ m 0 S P at the ionospheric boundary. The perpendicular magnetic field B y, the current, and the pressures are extrapolated at the ionospheric boundary, while the density is kept constant. No ionospheric boundary condition is needed for the fieldaligned electric field E z. [21] Since the generator force is symmetric around the equator, we z E x z n z P z z P? = E z = B = j = 0 at the equatorial plane. An obvious consequence of these boundary conditions is that only modes with a symmetric perpendicular electric field will be included. At the x =±L x boundaries we use open boundary conditions allowing perturbations to flow out of the system without any reflection. 3. Isothermal Closure [22] For comparison we will first present results from an isothermal simulation of the auroral flux tube, i.e., we let the temperatures be constants in time. In this case the first four equations in (21) will be used together t j ¼ m i m ne z nt z þ j2 n where T z and T? are constants in time. z ; ð27þ [23] As discussed in section 1, the isothermal closure has in the past frequently been used in studies of auroral electron acceleration by Alfvén waves. This is the simplest alternative, and it has usually been chosen without explicit justification Monochromatic Results [24] Using the monochromatic generator (23), we can excite standing Alfvén waves. With a generator period t G = 10 s we are close to a resonance of the third harmonic mode. As illustrated in Figure 2, three quarters of the characteristic wavelength l = t G V A (z) fit nicely along the length L z of half the field line. The ionospheric conductivity S P is much higher than the Alfvén conductivity S A =1/m 0 V A, which means that the reflection coefficient for the electric field R ¼ S A S P S A þ S P ð28þ is close to 1, and the resulting E x is very small in the ionosphere. The current and the wave magnetic field are there amplified to twice their values in the incoming wave. It is not surprising that the results of our isothermal simulations mainly confirm the traditional picture of field line resonances [e.g., Cheng et al., 1993; Streltsov et al., 1998], since Alfvén waves in the magnetosphere usually have been studied within isothermal models Quasi-Stationary Results [25] When driving the system with generator (24), the fields converge to a quasi-stationary state after about a minute. In Figure 3 we present the quasi-stationary state of the field-aligned current density mapped to the ionosphere and the potential f defined by E z z f. The fieldaligned current builds up to 2 mam 2, which is a typical auroral current, and in the acceleration region the potential reaches about 50 V. However, there is almost no net potential drop between the ionosphere and the equatorial 5of11

6 Figure 4. Plots from the adiabatic simulation showing the fields related to the monochromatic Alfvén waves during one wave period, (a) the perpendicular electric field E x, and (b) the perpendicular magnetic field B y. The labels indicate the time in seconds, starting with label 0 at 36 s. The first half of the cycle is shown as full lines and the second half as dotted lines. magnetosphere. This clearly demonstrates the well-known inability of a collisionless isothermal model to support large field-aligned electric fields. Notice that when the magnetospheric and ionospheric electrons are represented by separate fluids [Rönnmark and Hamrin, 2000], this current corresponds to a potential drop of about 100 V, maintained by the inertia of magnetospheric electrons. 4. Double Adiabatic Closure [26] An adiabatic closure was used by Lu et al. [2003] in their study of field line resonances. They assumed isotropic pressures, but since it is clear that anisotropies are important we use the CGL [Chew et al., 1956] double adiabatic approach. In this model there is no heat flow relative to the bulk velocity of the electrons. This means that the heat conduction terms in the last two equations of (21) are set to zero and the energy equations are simplified t P z ¼ j j zp z 3P z ; n t P? z n P? þ j n z? : ð29þ [27] We emphasize that we do not think that the CGL model provides a good equation of state for auroral flux tubes, where precipitating energetic electrons represent a substantial heat flow through the ambient ionospheric and trapped electrons [Ramos, 2003]. These results are included only to indicate how sensitive the description of auroral Alfvén waves is to the assumed equation of state Monochromatic Results [28] When we determine the pressures from the CGL model, the properties of the Alfvén waves driven by a monochromatic generator are significantly different from the isotropic case. Examples of the fields in the adiabatic model are shown in Figure 4. First we notice that the generator period must be reduced to t G = 8 s in order to obtain a reasonably clean standing wave. The main reason for this is that the effective length of the field line has been shortened, since the third harmonic mode now mainly is reflected from the acceleration region at about 1 R E rather than from the ionosphere. There are also lower amplitude modes, which may be described as a fifth harmonic bounded at the acceleration region and a seventh harmonic going down to the ionosphere. Notice that the magnetic field that penetrates to the ionosphere is an order of magnitude smaller in the CGL model than in the isothermal case. In the CGL model, strong electron pressure gradients build up near the acceleration region and resist the flow of field-aligned current. This causes charge to accumulate and build up strong electric fields, while the magnetic field is reduced Quasi-Stationary Results [29] Also when we try to build up a steady current, the increasing pressure gradients in the CGL model efficiently limit the field-aligned current that can be closed in the ionosphere. The field-aligned current and the potential obtained 100 s after the generator was started are illustrated in Figure 5. Although the current shown in Figure 5a is time-independent, the flux tube is not in a steady state. The magnitude of the current is increasing with altitude. Part of 6of11

7 Figure 5. Plots from the adiabatic simulation showing the quasi-stationary state of (a) the field-aligned current density mapped to the ionosphere and (b) the potential. The snapshots are taken 100 s after the generator was started. this increase is related to charge accumulation, which is the source of increasing electric fields. The growing perpendicular electric field E x corresponds to a polarization current and allows part of the current to close above the acceleration region. In Figure 5b the potential has reached 5 kv and is steadily increasing. From Figure 6, which shows the relation between the maximum upward current from the ionosphere and the corresponding voltage at different times, we see that the current saturates after about 20 s but the potential drop continues to increase linearly with time for 3 min. The parallel and perpendicular pressures after 100 s are shown in Figure 7. Within the CGL model the pressures grow in proportion to the potential and constrain the current to an almost constant value. from the ionosphere determines the total potential drop [Knight, 1973], which for a wide parameter range may be approximated by the linear relation [Fridman and Lemaire, 1980] pffiffiffiffiffiffiffiffiffiffiffi Df s 2pT M ¼ n M m m i B I B 0 j I ; ð31þ but the shape of the stationary potential f s (z) that we would allow us to calculate the pressures is not determined by the current. However, from the previous step in the fluid simulation we can compute the potential f(z) = R z 0 E z(z 0 )dz 0 and the total potential drop Df. To obtain a f s (z) that is consistent with the required current density j I, we define 5. Nonlocal Kinetic Closure [30] In the third approach we will replace the energy equations (21f), (21g) with the simple model t P z ¼ g P z P s z t P? ¼ g P? P s ð30þ? : f s ðzþ ¼ Dfs Df fðþ: z ð32þ Here, the pressures with superscript s are computed from stationary kinetic theory and g is a constant that determines the timescale on which the fluid pressures (without superscript) will converge to the stationary pressures. In the results presented here we use g =1s 1, which is of the same order of magnitude as the time it takes an electron to travel from the generator to the acceleration region. [31] In our simulations the field-aligned current density is mainly determined by the z component of Ampere s law (21b). The electron momentum equation (21e) is then used to find the field-aligned electric field E z that is needed to drive the required current. Thus when solving the momentum equation, the pressures should in principle be regarded as functions of the given current density. Unfortunately, there is no way to compute the pressures directly from the current. In a stationary state the upward current density j I Figure 6. The adiabatic current-voltage relation sampled every tenth second in the simulation. 7of11

8 Figure 7. Plots from the adiabatic simulation showing the quasi-stationary state of (a) the parallel pressure, and (b) the perpendicular pressure. The snapshots are taken 100 s after the generator was started. On field lines with downward current we assume Df s = Df. The pressures P s z and P? s along the magnetic field lines are then computed from the potential f s (z). This is done by the method outlined by Vedin and Rönnmark [2004], taking particles trapped below a growing potential drop into account as described by Eliasson et al. [1979]. [32] By adjusting the pressures in the simulation to the pressures obtained from stationary kinetic theory, we will include some kinetic effects in the fluid model. However, the above equations for the evolution of the pressures will not describe the correct dynamics on timescales shorter than the time it takes electrons to pass through the region with significant field-aligned electric field. If the electric field varies little during a typical electron transit time (a few seconds), the electron distribution will be similar to that in a stationary field and the equations (30) can be justified. This is, at least approximately, the case when we use the generator (24) which approaches its full strength after about 20 s. However, when we use the generator (23) with a rise time of 2.5 s to drive monochromatic waves, this is more questionable, and therefore we only present quasi-stationary results in this section Quasi-Stationary Results [33] In Figure 8 we present the field-aligned current and the potential in the quasi-stationary state 100 s after the generator was turned on. We see from Figure 8a that the current, mapped to the ionosphere, is essentially constant at Figure 8. Plots from the nonlocal kinetic simulation showing the quasi-stationary state of (a) the fieldaligned current density mapped to the ionosphere and (b) the potential. The snapshots are taken 100 s after the generator was started. 8of11

9 Figure 9. The nonlocal kinetic current-voltage relation sampled every tenth second in the simulation compared with the linear Knight relation. about 1 mam 2 all the way from the generator to the ionosphere. In the upward current region there is a steep potential drop of magnitude close to 4 kv at altitudes between 5000 km and km, as seen in Figure 8b. The relation between the maximum upward current from the ionosphere and the corresponding voltage at different times is plotted in Figure 9. The crosses, which represent values taken from the simulation every 10 s, clearly gather around the line that shows the linear [Knight, 1973] relation. The distance between consecutive crosses also decreases with time, which indicates that the current and potential approach a stationary state. [34] The quasi-stationary states of the parallel and perpendicular pressures after 100 s can be seen in Figure 10. At this time the pressures calculated from the nonlocal kinetic model are comparable to those in the CGL model, but they are not growing at the same rate. The increase in the pressure gradients is in the nonlocal model limited by the heat conductions. In the CGL model the growing pressures keep the current constant, but here the slightly slower growth of the pressures allows the current to increase in proportion to the potential drop. In Figure 11, which illustrates the main forces in the momentum equation (16), it can be seen that there is a thin layer near the bottom of the acceleration region where the main contribution to the parallel electric field comes from the parallel density gradient, T z ln n. This is consistent with the observations of Hull et al. [2003a, 2003b], although our simulations are unable to resolve the >25 mv/m field-aligned electric fields they observe in very thin layers. At higher altitudes, where the density is rather constant, the electric field is supported mainly by the parallel temperature gradient. In general, the term involving the pressure anisotropy and magnetic field gradient gives a smaller but still significant contribution to the electric field. The term in the stationary electron momentum equation that depends on the current is much smaller than the large terms containing gradients of the potential and pressures. The small difference between these large terms determines the current, and this explains why a very accurate description of how the pressures depend on the potential is required in a fluid model to obtain a linear current-voltage relation. 6. Discussion and Conclusions [35] The mirror force is often invoked to provide a simple intuitive explanation for field-aligned auroral electric fields. Unfortunately, the mirror force, which is a single-particle concept, has no well-defined meaning in kinetic and fluid theories. Since the pressure calculations in this study are based on the conservation of energy and magnetic moment, the pressure variations may in some sense be viewed as Figure 10. Plots from the nonlocal kinetic simulation showing the quasi-stationary state of (a) the parallel pressure and (b) the perpendicular pressure. The snapshots are taken 100 s after the generator was started. 9of11

10 Figure 11. The main forces in the momentum equation 100 s after the generator was started in the nonlocal kinetic simulation. consequences of the mirror force. However, we emphasize that just as there is no pressure force in the equation of motion for a single particle, there is no mirror force in the fluid equations. The mirror force on all the particles is included in the force on a plasma volume, but it cannot be unambiguously separated from other contributions to the net force. [36] By treating the electrons as a single fluid, our model hides not only the mirror force but also other conspicuous aspects of auroral physics. In particular, the beams of accelerated magnetospheric kev electrons that precipitate to cause discrete auroras cannot be discerned. Reports of auroral electron measurements often present the cool ionospheric and hot magnetospheric components separately, but in our model they are always combined. Frequently, observational constraints do not allow densities and temperatures to be calculated by integrating over all of velocity space as assumed in our model. In some cases, this can make comparisons between fluid model predictions and observations difficult. For example, it may not be trivial to compare our temperatures to temperatures reported from observational studies of the auroral acceleration region. However, fluid models are convenient for large-scale numerical simulations, and a properly closed fluid model provides a simple and accurate description of the electrodynamics, and in particular the current-voltage relation, in the auroral region. [37] The electron distribution function and the volumes of velocity space accessible to auroral electrons have been investigated in numerous studies [Knight, 1973; Whipple, 1977; Chiu and Schulz, 1978; Gurgiolo and Burch, 1988; Janhunen, 1999; Vedin and Rönnmark, 2004]. By inspection of how the populated volumes of velocity space vary with the potential in these studies, we see that there will be few electrons with energy less than the local potential. This indicates that the temperatures will at least be comparable to the potential, and an isothermal approximation is unreasonable unless the potential is small compared with the equatorial thermal energy. Furthermore, these studies show that the size of the loss cone increases with the total potential drop. The strong asymmetries caused by the loss cone, field aligned acceleration, and the mixing of magnetospheric and ionospheric electron populations with very different energies contribute to a substantial heat conduction that violates the adiabatic assumption. Since the electron distribution, and hence the pressure, at one altitude responds to electric fields all along the field line, these arguments speak not only against the isothermal and adiabatic approximations, but against any attempt to close the hierarchy of fluid equations by a relation that neglects nonlocal, kinetic effects. [38] The nonlocal kinetic closure is in this study achieved by equations (30), describing a simple relaxation of the pressures in the simulated electron fluid toward the pressures predicted by kinetic theory in a steady state with the same upward current. We stress that this should be considered as a first attempt to include some nonlocal, kinetic effects in a fluid simulation of auroral electron acceleration. This simple model can be justified only when the potential varies very slowly, and work is in progress to find energy equations that give a more realistic description of the dynamics also on timescales of the order 1 10 s. [39] Many studies involving auroral field line resonances of Alfvén waves have been based on the assumption of isothermal electrons. Our results show that at least if the amplitudes are large enough to provide significant auroral electron acceleration, the propagation and reflection of Alfvén waves is substantially influenced by temperature variations. This suggests that when considering auroral electron acceleration due to field line resonances, the results derived from isothermal models should be carefully revised. [40] In earlier studies of the reflection and damping of shear Alfvén waves at the acceleration region [Vogt and Haerendel, 1998; Fedorov et al., 2001; Pilipenko et al., 2002], it was necessary to assume an independently derived current-voltage relation. Here, we have shown that different closures of the fluid equations correspond to different current-voltage relations and in particular that the nonlocal kinetic closure leads to a linear relation. However, the mechanisms behind the reflection and damping of Alfvén waves by the potential drop at the acceleration region are essentially the same as in the earlier studies, and results regarding for example the dependence on the perpendicular scale length should apply also to our simulations. Hopefully, we can in future studies apply the nonlocal kinetic closure also to higher frequencies in order to confirm this. [41] It is widely accepted that the upward current carried by auroral electrons is linearly related to the potential drop in a steady state [Knight, 1973]. It is also well known that shear Alfvén waves with short perpendicular wavelength will produce a field-aligned electric field that can accelerate auroral electrons. Driven shear Alfvén waves should also be involved in transitions between different stationary states, but the connection between shear Alfvén waves and static potentials has remained unclear. In particular, it has not been possible to derive a linear current-voltage relation from the collisionless fluid models describing Alfvén waves. In this study we emphasize that the electron pressure variations caused by the auroral current must be properly included in a fluid model. By introducing pressures calculated from stationary kinetic theory into our dynamic fluid simulations, 10 of 11

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