A new interhemispheric 16-moment model of the plasmasphere-ionosphere system: IPIM

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Aurélie Marchaudon, P-L Blelly

To cite this version:

Aurélie Marchaudon, P-L Blelly. A new interhemispheric 16-moment model of the plasmasphere-

ionosphere system: IPIM. Journal of Geophysical Research Space Physics, American Geophysical

Union/Wiley, 2015, 120, pp.5728 - 5745. �10.1002/2015ja021193�. �hal-03166291�

A new interhemispheric 16-moment model of the plasmasphere-ionosphere

system: IPIM

A. Marchaudon ^1,2 and P.-L. Blelly ^1,2

1 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, Toulouse, France, ² Centre National de la Recherche Scientifique UMR 5277, Toulouse, France

Abstract We present a new interhemispheric numerical model: the IRAP plasmasphere-ionosphere model (IPIM). This model describes the transport of the multispecies ionospheric plasma from one hemisphere to the other along convecting and corotating magnetic field lines, taking into account source processes at low altitude such as photoproduction, chemistry, and energization through the coupling with a kinetic code solving the transport of suprathermal electron along the field line. Among the new developments, a 16-moment-based approach is used for the transport equations in order to allow development of strong temperature anisotropy at high altitude and we consider important but often neglected effects, such as inertial acceleration (centrifugal and Coriolis). In this paper, after presenting in detail the principle of the model, we focus on preliminary results showing the original contribution of this new model. For these first runs, we simulate the convection and corotation transport of closed flux tubes in the plasmasphere for tilted/eccentric dipolar magnetic field configuration in solstice and equinox conditions. We follow different flux tubes between 1.2 and 6 Earth Radii (R _E ) and demonstrate the capability of the model to describe a wide range of density (above 15 orders of magnitude). The relevance of the mathematical approach used is highlighted, as anisotropies can develop above 3000 km in the plasmasphere as a result of the mirroring effect related to the anisotropic pressure tensor. Moreover, we show that the addition of inertial acceleration may become critical to describe plasma interhemispheric transport above 4R _E . The ability of the model to describe the external plasmasphere is demonstrated, and innovative studies are foreseen, regarding the dynamics of the plasma along the magnetic field lines (in particular interhemispheric exchanges and “opening”/“closure” of a flux tube).

1. Introduction

A lot of effort has been done in the development of fluid models able to describe the dynamics of the magnetosphere-ionosphere-atmosphere system. However, the complexity of the coupling, e.g., the ion chem- istry in the lower part of the ionosphere, the electrodynamics, and the associated ion outflow in the upper part as well as the interhemispheric exchanges, is such that the ionosphere and the processes controlling it are more or less well described.

Those models are generally based on a five-moment multifluid approach [Blelly and Schunk, 1993]. In detail, they solve, as a function of time, systems of transport equations including continuity, momentum, and energy balance for the different species; when accounted for, the heat flux is expressed through a Fourier’s law. Depending on the models, the neutrals are either solved or obtained from an empirical model, but all models consider the main ionospheric ions. The mathematical approach for these models is either an Eulerian representation (generally when the model focuses on the dynamics of the atmosphere) or a mixed Eulerian-Lagrangian approach with resolution along a magnetic field line capable to follow convection effect (when the model focuses on the field-aligned dynamics of the ionospheric plasma). We can separate these models in three main classes: global, interhemispheric, and high-latitude models. These models were initiated more than 20 years ago, but they are still used and regularly updated.

Global models aim to reproduce and ideally forecast global variations of the thermosphere-ionosphere sys- tem, with respect to local time, season, magnetic activity, or tidal activity, over a limited range of altitudes.

Three main models belong to this class: the Thermosphere-Ionosphere-Electrodynamic General Circulation Model (TIEGCM) [Richmond et al., 1992; Solomon et al., 2012], the Coupled Thermosphere-Ionosphere Model

RESEARCH ARTICLE

10.1002/2015JA021193

Key Points:

• Interhemispheric exchanges of particles and energy

• Development of temperature anisotropy in relation with mirror force

• Effect of centrifugal acceleration on plasmasphere structure

Correspondence to:

A. Marchaudon,

[email protected]

Citation:

Marchaudon, A., and P.-L. Blelly (2015), A new interhemispheric 16-moment model of the plasmasphere-ionosphere system:

IPIM, J. Geophys. Res. Space Physics, 120, doi:10.1002/2015JA021193.

Received 6 MAR 2015 Accepted 14 MAY 2015

Accepted article online 20 MAY 2015

©2015. American Geophysical Union.

All Rights Reserved.

(CTIM) [Fuller-Rowell et al., 1996; Raeder et al., 2008], and to some less extent the Utah State University Time Dependent Ionospheric Model (USU-TDIM) [Schunk and Sojka, 1996; Sojka et al., 2014]. For such models, the coupling between the ionosphere and the magnetosphere is simplified. They are limited in altitude to collision-dominated regions (below 500 km), and no interhemispheric transfer is considered.

Interhemispheric models are mainly dedicated to the description of the ionospheric plasma transport along closed magnetic field lines, and they are designed to take into account energy and mass exchange between both hemispheres. The Sheffield University Plasmasphere-Ionosphere Model (SUPIM) [Bailey and Balan, 1996;

Balan et al., 2013] and the Field Line Interhemispheric Plasma model (FLIP) [Richards and Torr, 1996; Richards, 2013] use an empirical model for the neutral atmosphere, while the Coupled Thermosphere Ionosphere Plas- masphere model (CTIP) [Millward et al., 1996; Fuller-Rowell et al., 2011] solves the neutral atmosphere and specifically the electrodynamics in the equatorial region. Finally, one of the models which has proved the most appropriate to describe the plasma transport from the ionosphere to the plasmasphere is presently the SAMI3 model [Huba et al., 2008; Huba and Krall, 2013].

The major weaknesses of all these models are first the representation which does not allow for temper- ature anisotropies and thus is not appropriate for high-altitude/-latitude regions; then, the heat flux is only considered through thermal conduction, strongly limiting the energy transfer to collision-dominated regions; and finally, when considered, the energy exchange with suprathermal electron uses a very simplified analytic model.

High-latitude models are dedicated to the study of the direct influence of the interplanetary medium on the ionosphere through the exchange of momentum, energy, and mass along open magnetic field lines. In this class, we can mention the UAF Eulerian Parallel Polar Ionosphere Model (EPPIM) [Maurits and Watkins, 1996;

Maurits et al., 2008], based on an Eulerian approach and a five-moment resolution, and the TRANSCAR model family, using presently a 13-moment approach [Blelly et al., 1996, 2005]. This latter model has been devel- oped to describe properly the heat flux and temperature anisotropies which may develop from ion drag or from magnetic field divergence. It is based on a mixed Eulerian-Lagrangian approach: Eulerian description for field-aligned transport and Lagrangian description for flux tube convection over the polar cap. The TRAN- SCAR model is also coupled to a kinetic model describing the field-aligned transport of suprathermal electron.

This coupling allows for a good description of the ion production and electron heating resulting from the interaction of energetic electron with thermal populations (neutrals and charged species).

All these models have proved their ability to describe the dynamics of the ionospheric plasma. However, they all suffer at some level from limitations in the approach; mainly, the higher moments are not consid- ered, which may be critical at high altitude. The goal of this paper is thus to introduce the new model, IRAP plasmasphere-ionosphere model (IPIM), that we developed to better describe the dynamics of the plasmasphere-ionosphere system. In the next sections (sections 2–4), we present the new features of this model, which will then be highlighted through a first application in section 5. Section 6 will be devoted to a discussion on the next studies and the evolution of the model.

2. The Model

We propose a new multifluid model whose mathematical approach combines the advantages of the three types of model described in section 1 and thus allows for a three-dimensional (3-D) description of the nearby Earth environment with complete coverage in altitudes and latitudes. The main feature of this new model is its ability to cover both hemispheres by solving the transport either along a closed magnetic field line, in the interhemispheric mode, or along an open field line, in the high-latitude mode. Among the new develop- ment for this model, a 16-moment approach is used in order to allow development of strong temperature anisotropies at high altitude, in region where the 13-moment is known to fail [Blelly and Schunk, 1993]. The resolution is based on a mixed Eulerian-Lagrangian approach, allowing for a 3-D coverage, as it was already the case for the 13-moment high-latitude TRANSCAR model [Blelly et al., 2005]. In this paper, we will focus on the interhemispheric mode, as this mode required the most critical developments: in particular, we need to solve the interhemispheric transport of the ionized species over a large range of altitudes (between about 90 km and 6 to 7 R _E ).

The model uses a bi-Maxwellian distribution functions for each species [Demars and Schunk, 1979; Blelly and

Schunk, 1993], which intrinsically introduces a nonisotropic diagonal pressure tensor. This model is based on

a center guide approximation, which allows to use a local frame based on the magnetic field geometry to write the transport equations. Basically, the motion of the plasma is organized by the magnetic field: flowing along the magnetic field lines and convecting perpendicularly to them. Thus, we define the parallel compo- nent (label ∥) as being the one aligned to the local magnetic field and the perpendicular components (label ⊥ ) those in the direction perpendicular to the local magnetic field. Furthermore, we assume that scalar parame- ters are functions of the sole field-aligned curvilinear abscissa. For each species 𝛼 (ions labeled i and thermal electron labeled e), we describe the transport of the density (n _𝛼 ); the field-aligned velocity (u _𝛼 ); the two diag- onal components of the pressure tensor, the parallel and perpendicular temperatures (T _𝛼 ^∥ and T _𝛼 ^⊥ ); and the field-aligned components of the heat flows for the parallel energy (q ^∥ _𝛼 ) and perpendicular energy (q ^⊥ _𝛼 ) (charac- teristic of the 16-moment versus the 13-moment). In the particular case of thermal electron, we do not solve for n _e and u _e and use instead the following electrodynamics relationships

⎧ ⎪

⎨ ⎪

⎩

n _e = ∑

i

n _i n _e u _e = ∑

i

n _i u _i − ^J

e − 𝜙 es

(1)

where J _∥ is the field-aligned current imposed to the model, e is the elementary charge, and 𝜙 es is the flux of the suprathermal electron, which population is obtained by a kinetic module coupled to the fluid model.

The next sections discuss successively the geometry of the model, the transport equations and collisional processes (chemistry and collisions), the numerical approach, and finally the inputs to the model.

2.1. The Geometry

The model is based on a joint geographic/geomagnetic representation of every flux tube. All the footprints are taken at an altitude of 90 km in geographic frame, and the curvilinear abscissa is measured along the field line, in geomagnetic frame, from the footprint. In case of closed field line, the footprint in the Southern Hemisphere is used as reference.

2.1.1. Metrics: Local Coordinate System

The metrics is essentially based on the structure of the magnetic field → − B = B⃗ b, where b ⃗ is a unit vector. Thus, the local frame used to express the transport equation has the curvilinear abscissa 𝜎 as first component. If we assume that the magnetic field has two contributions, internal due to the core of the Earth and external due to the presence of a current → − J , we may express the Maxwell laws as

⎧ ⎪

⎨ ⎪

⎩

→ −

∇ ⋅ B ⃗ = 0

→ − ∇ × B ⃗ = 𝜇 0 ⃗ J

⇒

⎧ ⎪

⎨ ⎪

⎩

→ − ∇ ⋅ b ⃗ = − ^{→ − ∇B}

B ⋅ b ⃗

→ −

∇ × ⃗ b = −

→ − ∇B B × b ⃗ + ^𝜇

B ⃗ J

(2)

from which we can derive the fundamental relationship for the metrics

→ − ∇B

B = ( → − ∇ × b ⃗ )

× b ⃗ − ( → − ∇ ⋅ b ⃗ ) b ⃗ + 𝜇 0

B b ⃗ × ⃗ J = 1

𝜌 ⃗𝜈 − → − ∇ ⋅ ⃗ b + 𝜇 0

B ⃗ b × ⃗ J (3)

In this equation, we have introduced 𝜌 ≥ 0 the local curvature radius of the field line, a second vector ⃗𝜈 per- pendicular to b, such that ⃗

( ⃗ b, ⃗𝜈 )

defines the local osculating plane of the field line. Furthermore, we introduce a parameter A defined by

𝜕 (AB)

𝜕𝜎 = 0 (4)

𝜌 and A are the two basic parameters of this metrics. We can then write

→ −

∇ ⋅ ⃗ b = 1 A

𝜕 A

𝜕𝜎 (5)

To ease writing of transport equation, we introduce, at this point, two operators

⎧ ⎪

⎨ ⎪

⎩

∇ _∥ X = ^𝜕X

𝜕𝜎

∇ _∥ ⋅ X = → − ∇ ⋅ ( X b ⃗ )

= ∇ _∥ X + X → − ∇ ⋅ ⃗ b = ¹

A

𝜕AX

𝜕𝜎

(6)

In case we account for the convection, we need to introduce some contributions of the electric field E ⃗ = E _⊥ ⃗ e, where e ⃗ is a unit vector. Starting from the Maxwell equations, we can write, similarly to equation (2),

⎧ ⎪

⎨ ⎪

⎩

→ − ∇ ⋅ E ⃗ = 0

→ −

∇ × E ⃗ = 0

⇒

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

→ −

∇ ⋅ ⃗ e = −

→ − ∇E

E

⋅ ⃗ e

→ − ∇ × ⃗ e = −

→ −

∇E

E

× ⃗ e

(7)

from which we can derive

→ − ∇E _⊥ E _⊥ =

( → − ∇ × ⃗ e )

× e ⃗ − ( → − ∇ ⋅ ⃗ e

) e ⃗ = 1 R ⃗𝜏 −

( → − ∇ ⋅ ⃗ e

) ⃗ e (8)

where we have introduced R ≥ 0 the local curvature radius of the “electric field line”: it is the curve which has the electric field as tangent vector. This curve is perpendicular to the magnetic field line, as E ⃗ ⊥⃗ B, and ⃗𝜏 is a vector perpendicular to e ⃗ lying in the local osculating plane to this curve.

In case of convection, the drift velocity ⃗ v _⊥ is then given by

⃗

v _⊥ = E ⃗ × ⃗ B B ² = E _⊥

B ⃗ e × b ⃗ = E _⊥

B ⃗ t = v _⊥ ⃗ t (9)

where ⃗ t is a unit vector parallel to the direction of the convection, such as ( b ⃗ ,⃗ t , ⃗ e )

is a right-handed orthonormal system. The velocity vector then writes ⃗ v = u b ⃗ + ⃗ v _⊥ .

The curvature radius R _c of the convection curve is given by 1

R _c =

→ − ∇v _⊥ v _⊥ ⋅ b ⃗ =

( → − ∇E _⊥ E _⊥ −

→ − ∇B B

)

⋅ ⃗ b = ⃗ b ⋅ ⃗𝜏 R + → −

∇ ⋅ ⃗ b (10)

R _c is the parameter used in the model to express the acceleration associated with the convection.

A summary of this complex geometry is presented in Figure 1, where the three different curvature radii and the three different osculating planes are sketched.

2.1.2. Application to Transport Terms

Beside scalar parameters (like density or temperature), we need to express transport equation for parallel component X _∥ of some vector → − X . The general expression for the convective derivative is given by

DX _∥ Dt = D

Dt ( → − X ⋅ b ⃗ )

= b ⃗ ⋅ 𝜕 → − X

𝜕 t + v _∥ ∇ _∥ X _∥ + → − X ⋅ D ⃗ b

Dt (11)

The first two terms on the right-hand side of equation (11) correspond to the projection along b ⃗ of the convec- tive derivative of → − X . b ⃗ being a unit vector, ^D⃗ _Dt ^b is perpendicular to b, and thus, the third term in equation (11) will ⃗ contribute if → − X has a component perpendicular to ⃗ b, which only occurs for the velocity in case of convection [Northrop, 1963; Cladis, 1986].

Moreover, we need to express divergence of the pressor tensor: → − ∇ ⋅ P. If we consider the two components P _∥ and P _⊥ , then we can write its contribution to the parallel component of the momentum equation as

→ − ∇ ⋅ P || | ^∥ = ∇ _∥ P ^∥ + (

P ^∥ − P ^⊥ ) → − ∇ ⋅ b ⃗

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

mirror

(12)

The guiding center approach associated with first a metrics related to the magnetic field and then an

anisotropic pressor tensor allows to account for the contribution of the mirror force to the dynamics and

provides a clear and simple description of this contribution. The term labeled “mirror” in equation (12) corre-

sponds to the mirror force emerging from the trapping of the plasma along the field line of an inhomogeneous

Figure 1. From the orthonormal base ⁽ b ⃗ ,⃗ t , ⃗ e )

, three planes useful for the metric representation are defined: Π ₁ (white plane) is the local osculating plane of the field line, defined by the vectors ( b ⃗ and

⃗𝜈 ). _𝜌 represents the local curvature radius of the field line and C ₁ the center of the corresponding osculating circle. Π ₂ (light grey plane) is the local osculating plane of the electric field streamline, defined by the vectors ( _⃗ e and _⃗𝜏 ). R represents the local curvature radius of the electric field streamline and C ₂ the center of the corresponding osculating circle. Π ₃ (dark grey plane) is the local osculating plane of the convection trajectory, defined by the vectors ( _⃗ e and ⃗ t ). R _c represents the local curvature radius of the convection trajectory and C ₃ the center of the corresponding osculating circle. Through convection, an element of volume of a field line undergoes a series of transformation: in the plane Π ₃ , perpendicular to the magnetic field, the element of volume undergoes a rotation and a volume deformation (top part of the figure) and in the plane Π ₁ , the element of volume undergoes a rotation, responsible of a field line

acceleration (bottom part of the figure).

magnetic field [Comfort, 1988; Parker, 2002].

This term expresses the field-aligned con- tribution of a transfer of thermal energy between parallel and perpendicular com- ponents in a diverging flux tube: if P ^⊥ ≥ P ^∥ , the term contributes upward as we move away from the ground, meaning that energy is provided by the perpendic- ular component. A similar contribution of the mirror force is present in higher-order moment equations for the 16-moment approach, as will be discussed later.

The compressibility of the fluid is character- ized by the term → − ∇ ⋅ ⃗ v, which is the rate of expansion of an elementary fluid volume.

The equations presented in section 2.1.1 allow for deriving the following expression

→ − ∇ ⋅⃗ v = → − ∇ ⋅ ( u b ⃗ )

+ → − ∇ ⋅⃗ v _⊥ = ∇ _∥ ⋅ u+⃗ t ⋅ → − ∇v _⊥ +v _⊥ → − ∇ ⋅ ⃗ t (13) where

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

⃗ t ⋅ → − ∇v _⊥ = v _⊥ ⃗ t ⋅ ( _{→ −}

∇E

E

− ^{→ − ∇B}

B

)

v _⊥ → − ∇ ⋅ ⃗ t = v _⊥ (

b ⃗ ⋅ ( → − ∇ × ⃗ e )

− ⃗ e ⋅ ( → − ∇ × ⃗ b )) (14) After simplification, we obtain the follow- ing expression

→ −

∇ ⋅ ⃗ v =

∇

⋅u

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

∇ _∥ u

⏟⏟⏟

longitudinal

+ u → −

∇ ⋅ b ⃗

⏟⏟⏟

transverse

−

→ − ∇

⋅⃗ v

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

v _⊥ (

2 ⃗ t ⋅ ⃗𝜈 𝜌 + 𝜇 0

B ⃗ J ⋅ ⃗ e )

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

convection

(15)

Equation (15) shows three contributions to the rate of expansion of the fluid. The first right-hand side (RHS) term expresses the longitudinal variation associated with the variation of the field-aligned velocity. The sec- ond RHS term expresses the transverse expansion of the fluid flowing in a variable section “nozzle” as the plasma is trapped along the magnetic field. The third RHS, → − ∇ _⊥ ⋅ ⃗ v _⊥ , characterizes the fact that the plasma is frozen in the magnetic field and the motion of the plasma through the field lines may result in a variation of the volume. This last term is strictly related to the convection and appears as such in an Eulerian description of the fluid. However, as we use a Lagrangian description for the convection, this term will be replaced by an equivalent term representing the rate of volume variation. Section 2.1.4 describes the way this contribution is accounted for in the transport equations.

2.1.3. Inertial Effects

As we solve the transport equation along a flux tube in a local frame, we must account for the field-aligned component of the inertial accelerations. First, our frame is corotating with the Earth around the pole axis ⃗ k (speed → −

Ω _E = Ω _E ⃗ k), and thus, we have the classical centrifugal and Coriolis accelerations

⃗𝛾 = − Ω ² _E ( k ⃗ × (

k ⃗ × ⃗ r ))

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

centrifugal

− 2Ω _E (

⃗ k × ⃗ v )

⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

Coriolis

(16)

Obviously, a contribution of Coriolis acceleration along the field line is only possible in the case of convec- tion, because the velocity has then a component perpendicular to the field line. However, it is not the only contribution associated with the convection; indeed, as we mentioned above (equation (11)), the convective derivative ^D _Dt ^{⃗ b} will contribute to the momentum equation through the component aligned with ⃗ t

⃗ t ⋅ D⃗ b Dt = ⃗ t ⋅ [(

⃗ v ⋅ → − ∇ )

⃗ b ]

= u 𝜌

( ⃗ t ⋅ ⃗𝜈 ) + v _⊥

R _c (17)

The contribution in equation (17) represents the inertial parallel acceleration induced by the convection.

At some point of geographic latitude 𝜆 g , we introduce the dip angle I between b ⃗ and the local horizontal plane (I > 𝜋 meaning b ⃗ is earthward). Then, the total inertial acceleration 𝛾 i∥ that we include in our model expresses as

𝛾 i∥ = −Ω ² _E r ((

k ⃗ ⋅ ⃗ b )

sin 𝜆 g − sin I )

−v _⊥ (

−2Ω _E (

⃗ k ⋅ ⃗ e ) + u _i

𝜌 ( ⃗ t ⋅ ⃗𝜈 )

+ v _⊥ R _c

)

(18) As shown in equation (18), the centrifugal acceleration is increasing with altitude while Coriolis acceleration is a function of the curvatures and the convection velocity which is expected to increase with magnetospheric activity and latitude. The influence of the curvatures on the field-aligned dynamics has already been studied:

the term with 𝜌 has been introduced by Cladis [1986], and the one with R _c has been introduced by Horwitz et al. [1994a, 1994b]; they were also proved to be important at high latitude.

2.1.4. Divergence Effects

The Lagrangian motion consists in computing the new location of the center of each element of volume as a result of the displacement in the direction perpendicular to ⃗ B. The expansion/contraction of the volume implies that any intensive quantity, as the density, should be affected. If n is the density of a species, constant in a volume V, N = nV is the number of particles in this volume, which remains unaffected by the transformation.

Thus, we can write

dN

dt = 0 (19)

So the effect of the transformation on n is given by dn

dt = d dt

( N V )

= 1 V

dN dt − N

V ² dV

dt = −n d log V

dt (20)

The right-hand side term in equation (20) appears to be a sink/source term for the Eulerian description of the field-aligned dynamics, and thus, we must add it to the continuity equation in order to account for the effect of the convection. As mentioned before, this term is equivalent to the term → −

∇ _⊥ ⋅ (

⃗ v _⊥ )

which would appear in a full Eulerian description of the continuity equation, with a fluid moving at speed v ⃗ _⊥ in the perpendicular direction.

2.2. The Transport Equation

Fluid transport equations are solved for thermal electron and main ions (O ⁺ , H ⁺ , N ⁺ , NO ⁺ , O ⁺ ₂ , and N ⁺ ₂ ), assuming for simplicity that molecular ions have the same velocity, temperature, and heat flux. The transport equations used in this model are presented in Blelly and Schunk [1993], and we present only the equations differing because of additional terms. For instance, the continuity equation, including the divergence effect, writes

𝜕 n _i

𝜕 t + ∇ _∥ ⋅ ( n _i u _i )

+ d log(V)

dt n _i = P _i − L _i n _i (21)

where P _i is the ion production rate and L _i n _i is the ion loss rate, proportional to the ion density. The produc- tion rate is a combination of photoionization and chemistry, while the loss rate only corresponds to chemical processes. The photoionization source is described in section 2.2.1, and the list of the chemical reactions considered in the model is given in Table 1.

The momentum equation, including inertial forces, writes

𝜕u i

𝜕 t + u _i ∇ _∥ u _i + k _b m _i n _i ∇ _∥

( n _i T _i ^∥ )

+ k _b m _i

(

T _i ^∥ − T _i ^⊥ ) → −

∇ ⋅ ⃗ b

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

mirror

−G _∥ − 𝛾 i∥ − e _i

m _i E _∥ = 𝛿u i

𝛿 t (22)

Table 1. Chemical Processes Taken Into Account in This Work, Including Photoionization

Process Reference

N ₂ + h 𝜈 → N ⁺ ₂ + e ⁻ Fennelly and Torr [1992] and Stolte et al. [1998]

O ₂ + h 𝜈 → O ⁺ ₂ + e ⁻ Fennelly and Torr [1992] and Stolte et al. [1998]

O + h 𝜈 → O ⁺ + e ⁻ Fennelly and Torr [1992] and Stolte et al. [1998]

N ₂ + h 𝜈 → N ⁺ + N Fennelly and Torr [1992] and Stolte et al. [1998]

O ₂ + h 𝜈 → O + O ⁺ + e ⁻ Fennelly and Torr [1992] and Stolte et al. [1998]

H + h 𝜈 → H ⁺ + e ⁻ Verner et al. [1996]

O ⁺ + H _→ H ⁺ + O Stancil et al. [1999]

O ⁺ + N _{2 →} NO ⁺ + N St.-Maurice and Laneville [1998]

O ⁺ + O _{2 →} O ⁺ ₂ + O St.-Maurice and Laneville [1998]

H ⁺ + O _→ O ⁺ + H Stancil et al. [1999]

N ⁺ + O _→ O ⁺ + N Rees [1989]

N ⁺ + H _→ H ⁺ + N Rees [1989]

N ⁺ + O _{2 →} O ⁺ + NO Rees [1989]

N ⁺ + O _{2 →} NO ⁺ + O Rees [1989]

N ⁺ + O _{2 →} O ⁺ ₂ + N Rees [1989]

N ⁺ ₂ + O _→ O ⁺ + N ₂ McFarland et al. [1974]

N ⁺ ₂ + O _→ NO ⁺ + N McFarland et al. [1974]

N ⁺ ₂ + O _{2 →} O ⁺ ₂ + N ₂ Rees [1989]

N ⁺ ₂ + e ⁻ _→ N + N Sheehan and St.-Maurice [2004]

NO ⁺ + e ⁻ _→ O + N Sheehan and St.-Maurice [2004]

O ⁺ ₂ + N _→ NO ⁺ + O Anicich [1993]

O ⁺ ₂ + N _{2 →} NO ⁺ + NO Rees [1989]

O ⁺ ₂ + e ⁻ _→ O + O Sheehan and St.-Maurice [2004]

where k _b stands for the Boltzmann’s constant and G _∥ is the field-aligned component of the gravity accelera- tion. The polarization electric field E _∥ comes from the ambipolar condition (equation (1)) and is given by the electron momentum equation

eE _∥ = − k _b n _e ∇ _∥ (

n _e T _e ^∥ )

− k _b (

T _e ^∥ − T _e ^⊥ ) → −

∇ ⋅ b ⃗

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

mirror

+∇ _∥ ⋅ 𝜖 s + m _e 𝛿 u _e

𝛿 t (23)

where ∇ _∥ ⋅ 𝜖 s is the contribution of the suprathermal electron with energy 𝜖 s .

The last two equations present two contributions of the mirror force to the momentum equation. If T _𝛼 ^⊥ ≥ T _𝛼 ^∥ , the term contributes to accelerate upward as we move away from the ground, which means that energy is provided by the perpendicular component, and thus, we see that an anisotropy on electron is likely to affect the upward acceleration of the ions.

The parallel energy equation for electron is given by

𝜕 T _e ^∥

𝜕 t + u _e ∇ _∥ T _e ^∥ + 2T _e ^∥

longitudinal

⏞⏞⏞

∇ _∥ u _e + 1

n _e k _b ∇ _∥ ⋅ q ^∥ _e − 2 k _b

q ^⊥ _e n _e

→ − ∇ ⋅ ⃗ b

⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

mirror

= 1 n _e k _b

( Θ _e − Λ _e ) + 𝛿 T _e ^∥

𝛿 t (24)

and the perpendicular energy equation writes

𝜕 T _e ^⊥

𝜕 t + u _e ∇ _∥ T _e ^⊥ + T _e ^⊥

transverse

⏞⏞⏞

u _e → − ∇ ⋅ b ⃗ + 1

n _e k _b ∇ _∥ ⋅ q ^⊥ _e + 1 k _b

q ^⊥ _e n _e

→ − ∇ ⋅ b ⃗

⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

mirror

= 1 n _e k _b

( Θ _e − Λ _e ) + 𝛿 T _e ^⊥

𝛿 t (25)

In these two equations, the terms labeled mirror again refer to the contribution of the mirror force and express the energy transfer between parallel and perpendicular components through the heat flow for the perpen- dicular energy. The two contributions are such that when computing the mean energy, they do not contribute to the energy balance, and consequently, there is no energization of the plasma due to this transfer. There is no dependency on the heat flow for the parallel component, as the mirror force contribution is in the par- allel direction and corresponds to a transfer from the perpendicular component to the parallel component.

That is why the mirror force contribution only appears in the transport equation for q ^⊥ _𝛼 through the term

k

m

n _𝛼 T _𝛼 ^⊥ (

T _𝛼 ^∥ − T _𝛼 ^⊥ ) → −

∇ ⋅ ⃗ b (equation not written [e.g., see Blelly and Schunk, 1993, equation 58]). If T _𝛼 ^⊥ ≥ T _𝛼 ^∥ , the term contributes to increase the heat flow upward as we move away from the ground.

As discussed for equation (15), the term labeled “longitudinal” is the longitudinal rate of variation of the fluid element, while the term “transverse” is the transverse rate of variation of the fluid element, both expressing the expansion/contraction of the fluid as it is moving along the flux tube and contribute to the cooling/heating of parallel and perpendicular components of the energy.

In the different transport equations, _𝛿t ^𝛿⋅ in the right-hand side stands for the expression of the elastic collision terms between one species and the others (charged and neutral particles) present in the medium. In the elec- tron energy equations, the thermal electron heating rate Θ _e and the thermal electron cooling rate Λ _e describe nonelastic energy transfer rates and are equally distributed between parallel and perpendicular energy.

Demars and Schunk [1979] give a thorough description of the collision terms for the 16-moment and different power law interaction, and we refer to this paper to get the expressions used in our model.

2.2.1. Kinetic Coupling

Source terms like the ion production rates P _i and thermal electron heating rate Θ _e are major sources in the fluid transport equations and result from a coupling with a kinetic code solving the field-aligned transport of suprathermal electron issued from photoionization and precipitation. This kinetic module is the one used for the TRANSCAR family [Lilensten et al., 1989; Lummerzheim and Lilensten, 1994]; it provides the distribu- tion of suprathermal electron along the magnetic field line as a steady state solution of the balance between transport and interactions with the background. The byproduct of this balance is thermal ion and electron production (photoionization and precipitation) and thermal electron heating (collisions).

2.2.2. Cooling Processes

Beside the source terms, there are some energy losses Λ _e for thermal electron. These sinks of energy cor- respond to the cooling by inelastic collisions on neutrals that lead to excited states. Mainly, the energy loss results from vibrational and rotational excitation of N ₂ and O ₂ , electronic excitation of O, and fine-structure excitation of O. The expressions for these rates are given by Schunk and Nagy [1978, 2000].

3. Numerical Approach

3.1. Fluid Model

As described in Robineau et al. [1996], we use a Flux-corrected Algorithm for Solving Generalized Continuity Equations (LCPFCT) [Boris and Book, 1976; Boris et al., 1993] to solve the system of equations. This finite volume method is based on a conservative second-order Godunov explicit scheme which solves the following transport equation

𝜕𝜌

𝜕 t + ∇ _∥ ⋅ ( 𝜌 v) = S (26)

where 𝜌 is the variable solved, v the transport velocity, and S the other source terms. The scheme is noncen- tered and stabilized up to fourth order thanks to a numerical antidiffusive stage, but in case of steep gradients, the scheme is degraded to first-order accuracy through a stabilization procedure characterized by a diffusive numerical flux. It is able to solve conservative equations including transport terms or local source terms. The temporal dependency is solved using a fourth-order Runge-Kutta algorithm. In case the transport equation is not in a conservative form but rather in an advective form, we modify the equation to fulfill the conservation constrain. Thus, based, for instance, on equations (24) and (25), we will write

𝜕𝜌

𝜕 t + v∇ _∥ 𝜌 + 𝛼𝜌 ∇ _∥ v = 𝜕𝜌

𝜕 t + ∇ _∥ ⋅ ( 𝜌 v) + ( 𝛼 − 1) 𝜌 ∇ _∥ v − 𝜌 v∇ _∥ ⋅ ⃗ b (27)

where 𝛼 is a real parameter, generally associated with the degrees of freedom.

Figure 2. Synopsis of IPIM model: atmosphere and magnetosphere models are parametrized by ap and F _10.7 indices and provide inputs for the fluid model (atmosphere, convection electric field, and parallel current) and for the kinetic model (atmosphere and precipitation pattern). Beside this, the magnetosphere model provides the magnetic geometry used both in the fluid and kinetic models, and F _10.7 index is used to parametrize the solar flux which is required for the production of photoelectrons. There is a two-side coupling between fluid and kinetic models: thermal electron temperature and density are provided to the kinetic model, and ion production rate and thermal electron heating rate are provided to the fluid model.

3.2. Kinetic Model

The solver for the kinetic model is based on a discrete ordinary transfer algorithm (DISORT) [Stamnes and Swanson, 1981; Stamnes et al., 1988] and was originally developed for radiative transfer problem. This model has been already described in other articles [Blelly et al., 1996] where it was used to solve the transport of suprathermal electron along straight field lines. The initial version was not well suited for interhemispheric transport, and thus, some modifications have been made in order to describe the transport all along the flux tube, especially in the collisionless regions. However, some processes like mirror force are still missing, and this model will soon be replaced by a new improved model which will account for these processes.

4. Model Inputs

The IPIM model needs several inputs similar to the high-latitude TRANSCAR model [Blelly et al., 2005]. Figure 2

presents a flowchart of IPIM model, showing where the inputs parameters are included and the connections

between the different modules. As proxies of the Earth’s magnetic activity and of the global solar activity, we

use the ap (equivalent of Kp in physical units) and F _10.7 indices.The atmosphere and magnetosphere models,

parametrized by these indices, provide inputs for the fluid model (atmosphere, convection electric field, and

Table 2. The Characteristics of the Eccentric Tilted Magnetic Dipole ^a

M _E R _m X Y Z Longitude Latitude

7 . 75 × 10 ²² 6371.2 287.8 81. −400 . 333. 215.

a

M _E is the magnetic moment (in A m ² ) of the Earth dipole. R _m , X , Y , and Z are in km and refer respectively to the magnetic radius and the position in geocentric coordinates of the center of the dipole with respect to the center of the planet, where X is directed toward longitude 0, Y along longitude 90 ∘ in the equatorial plane, and Z is along the rotation axis. The geographic coordinates (longitude and latitude) of the magnetic dipole axis (also North Pole coordinates) are given oriented eastward.

parallel current) and for the kinetic model (atmosphere and precipitation pattern). It has to be mentioned that the electrodynamics are optional inputs, but the magnetosphere model provides the magnetic geometry used both in the fluid and kinetic models. F _10.7 index is also used to parametrize the solar flux which is required for the production of photoelectrons.

4.1. Environmental Inputs

IPIM being an ionosphere model, the main parameters of the coexisting neutral atmosphere are introduced as inputs from two empirical “time dependent” models covering the entire terrestrial globe from upper thermo- sphere (upper limit = 800 km) down to the ground; above 800 km, these atmospheric models are superseded by a simplified exospheric model using a Chamberlain approach [Chamberlain, 1963]. Neutral temperature, exospheric temperature, and number densities of the main constituents of the atmosphere, N ₂ , O ₂ , H, O, and N, are provided by the Mass Spectrometer Incoherent Scatter model (MSISE-90) [Hedin, 1987, 1991], with an extension to include a hot oxygen population contribution in the exosphere [Picone et al., 2002]. Meridional and zonal components of the atmospheric winds are provided by the Horizontal Wind Model (HWM-93) [Hedin et al., 1991, 1996].

Temperature and density of neutrals are used as inputs to the IPIM model in the fluid module for ion-neutral collisions and chemistry and in the kinetic module for energy degradation. Horizontal neutral winds are used in the fluid module for dynamical ion-neutral coupling in the momentum equation as a drag effect and in the thermal equations as frictional heating.

In order to compute the photoionization rates in the kinetic module of IPIM, we presently use the EUVAC model [Richards et al., 1994] or the Flare Irradiance Spectral Model (FISM) [Chamberlin et al., 2007]. This model describes the EUV solar spectral flux in the range 5–105 nm, through linear functions of the F _10.7 solar index.

4.2. Magnetic Geometry

For the validation of the model, we choose to use an eccentric tilted dipolar magnetic field. This representation is interesting as it allows for an analytical derivation of the metrics required to solve the transport but still is representative of the inner magnetosphere and gives access to the major source of asymmetries (season, local time LT) related to the magnetic field. In this case, we only have the core contribution, and therefore, we set

→ − J = 0 in the expressions given in section 2.

Using the eight first coefficients given by International Geomagnetic Reference Field (IGRF) [Finlay et al., 2010], we can derive the main parameters of the equivalent eccentric tilted dipolar magnetic field [Millward et al., 1996]. Mainly, the model is characterized by the location of the center of the dipole with respect to the center of the Earth, magnetic radius R _m of the planet, and the magnetic moment vector: orientation (geographic longitude and latitude) and dipole moment M _E . Table 2 gives the values used for the dipole in our model.

A magnetic field line is characterized by its McIlwain parameter L, where LR _m represents the apex of the field line in the magnetic equatorial plane. At a point of the field line, with coordinates (r _m , 𝜑 m , and 𝜆 m ) in spherical representation, the expression for the arc length 𝓁 from the apex is given by

𝓁 ( 𝜆 m ) = L R _m [ sin 𝜆 m

2

√

1 + 3sin ² 𝜆 m +

√ 3 6 log (√

3 sin 𝜆 m +

√

1 + 3sin ² 𝜆 m

)]

(28) from which we can derive the curvilinear abscissa 𝜎

𝜎 ( 𝜆 m ) = 𝓁 ( 𝜆 m ) − 𝓁 ( 𝜆 o ) (29)

where 𝜆 o is the latitude of the reference footprint.

Figure 3. Convection + corotation potential contours in the equatorial plane with a stagnation point at L _max = 6 . The equipotential contours followed by the three flux tubes at L _max = 2 , 5, and 6 are represented by black solid lines.

Setting AB = 1, we can express A( 𝜆 m ) as A( 𝜆 m ) = (

L R _m ) 3 cos ⁶ 𝜆 m

√

1 + 3 sin ² 𝜆 m

(30)

and the expressions for the two curvature radii are

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

1 𝜌 = ³

LR

1+sin

𝜆

cos 𝜆

( ^{1+3 sin}

𝜆

)

1 R

= − ³

LR

sin 𝜆

cos

𝜆

( 1+3 sin

𝜆

)

(31)

4.3. Corotation and Convection Electric Fields

As we solve transport equation in the frame of a magnetic flux tube, we need to take into account corotation of the field line with the Earth, as well as convec- tion. From a topological point of view, the knowledge of the motion perpendicular to → − B of any point belonging to one field line allows to derive the motion of all the points on this field line.

Corotation can be locally interpreted as a meridional electric field given by

→ − E = − ( → − Ω _E × ⃗ r )

× → − B (32)

Thus, the corotation can be seen as deriving locally from an electric potential which can be added to the convection potential generally deduced from empirical models.

Similarly to Blelly et al. [2005], our new model has a module transporting the flux tube horizontally. For its val- idation, we choose to use a very simple analytical model [Schulz, 2007] combining corotation and convection electric fields. This model can be parameterized such that the location in the equatorial plane of the stag- nation point, where corotation and convection exactly compensate, can be chosen arbitrarily. The potential pattern in the equatorial plane used is presented in Figure 3 and shows a stagnation point located at 6.01 R _m and 15 MLT (magnetic local time), mimicking very quiet conditions. Such a large distance from Earth for the stagnation point is generally quite rare as the plasmasphere is very dynamic and undergoes regular plasma erosion, but it allows to push the model to its numerical limits by simulating closed plasmaspheric flux tubes with an apex as far as 6 R _m . In the following, to refer to a specific flux tube, we identify it from its L shell posi- tion at 15 MLT in the stagnation region. This value L _max is thus the highest L shell value in R _m unit that the tube will reach during its motion. In Figure 3, we have superimposed on the potential map the trajectory followed by three different flux tubes (black solid curves) whose results will be presented in the next section. The three tubes have their L _max at 2, 5, and 6, respectively. The closest tube from Earth (L _max = 2) is characterized by an almost circular motion totally dominated by corotation. For the tube with L _max at 5, convection effect starts to be important, giving an elliptical form to the trajectory and implying that the flux tube remains a longer time in the region around 15 MLT. Finally, for the tube farthest to Earth (L _max = 6), the tube almost attains the stagnation point and consequently spends a very long time in the region around 15 MLT, explaining the very angular shape of the trajectory. This flux tube undergoes a strong expansion before the stagnation point and a strong contraction after.

5. Applications and Discussion

For a validation of the model, we have run two plasmasphere simulations respectively for equinox and solstice

(summer in Northern Hemisphere) conditions and for very quiet conditions (ap = 4 and F _10.7 = 150). For

Figure 4. Density profiles with altitude of the main ions and thermal electron for the closed flux tube L _max = 6 . The profiles are plotted at the stagnation point ( L = 6 and 15 MLT) and for solstice conditions (summer in Northern Hemisphere). Each species is color coded with solid line corresponding to the Southern Hemisphere and dash-dotted line to the Northern Hemisphere (O ⁺ : blue, H ⁺ : red, e ⁻ : black, and molecular NO ⁺ +N ⁺ ₂ +O ⁺ ₂ ions: magenta).

these two simulations, we have run several flux tubes spanning the entire plasmasphere from 1.2 to 6 R _m and each flux tube is run for 30 days before the equinox/solstice date in order to eliminate the transients. A careful study of each 30 days run shows that although flux tube equilibrium is quickly reached for flux tubes below L _max = 2 . 5, it is not the case for tubes above this threshold, for which equilibrium will be reached after days and will be broken well before by storm/substorm cycle governing the magnetosphere time constant. A deep analysis of this observation is beyond the scope of this paper and requires a dedicated study.

In this section, we only focus on some very specific results to show the original contribution of this new model, and we only present results for the three different tubes shown in Figure 3. In the following, all the color panel plots will show modeled parameter along a flux tube displayed with respect to 24 MLT along the x axis and to altitude along the y axis with a logarithmic scale in order to emphasize the dynamics at all altitudes. The 24 MLT representation is chosen because it is more appropriate for plasmaspheric representation.

Figure 4 presents altitude profiles of the main ions and thermal electron obtained at the stagnation point along the tube corresponding to L _max = 6 for solstice conditions. The Northern Hemisphere part of the pro- files is plotted with dash-dotted lines, while the southern part is plotted with solid lines. The outstanding performances of IPIM are evident in this figure, as for such a flux tube the species dynamics cover more than 20 orders of magnitude in densities over more than 30,000 km in altitude range. This figure shows a stan- dard ion dynamics with heavier molecular ions dominating below 200 km, then O ⁺ ion dominating up to 2000–3000 km where they drop radically, replaced by lighter H ⁺ ion showing an almost vertical profile up to the apex of the flux tube, corresponding to an “hydrostatic” equilibrium (see discussion at the end of this section). The maximum electron density reached in the F ₂ layer (10 ¹¹ m ⁻³ ) is in agreement with observations.

On the other hand, very clear differences are observed between hemispheres caused by different solar illu-

mination during solstice conditions. Below 200 km, the Northern Hemisphere strongly more illuminated than

the southern one shows denser E region and F ₁ layer (higher densities of molecular ions and electron). On the

other hand, although O ⁺ ion is denser at almost all altitudes in the Northern Hemisphere, the F ₂ region peak

is slightly stronger and located at lower altitude in the Southern than in the Northern Hemisphere. Moreover,

H ⁺ ion density is higher in the Southern than in the Northern Hemisphere between 200 and 4000 km. We also

identify the upper transition height above which H ⁺ ion density dominates over O ⁺ ion density (z ₅₀ parame-

ter). We can see that this upper transition height is higher in the Northern than in the Southern Hemisphere by

almost 1000 km. All these results are in agreement with previous studies showing that H ⁺ /O ⁺ transition is sys-

tematically higher in the summer hemisphere and could be caused by transport of O ⁺ ion from the summer

to the winter hemisphere where they are converted in H ⁺ by charge exchange reaction [Titheridge, 1976].

Figure 5. MLT/altitude color plots of thermal electron density over 24 MLT along closed flux tubes (south hemisphere at the bottom of the plot). Flux tubes L _max = 2 for (top left) equinox and (top right) solstice conditions (summer in Northern Hemisphere). Flux tubes reaching a L _max = 6 for (bottom left) equinox and (bottom right) solstice conditions.

For each color plot, the white solid curves indicate H ⁺ /O ⁺ upper transition height ( z ₅₀ ) in each hemisphere.

Figure 5 presents color panels of the diurnal and seasonal dynamics of the thermal electron for flux tubes at L _max = 2 and L _max = 6 and for equinox and solstices conditions. For all flux tubes, the sunrise and sunset are clearly identified by the strong increase or decrease of electron density in the E and F regions. As expected, the z ₅₀ height, represented by the white horizontal curves in both hemispheres, is always observed at higher altitude during dayside and for L _max = 6 than for L _max = 2. At L _max = 2, the diurnal electron dynamics is essen- tially observed below z ₅₀ irrespective of the season, while at L _max = 6, the diurnal electron dynamics remains important up to the apex. During equinox conditions, we can see an almost identical behavior in both hemi- spheres for both L _max , especially for sunrise and sunset, while the solstice conditions reveal a complex and strongly different behavior between hemispheres, especially at L = 6. For solstice conditions and both L, we can see that the typical dayside ionosphere lasts longer in the Northern summer hemisphere. We can also see that at L _max = 6, the ionosphere remains almost identical throughout the entire 24 MLT in the Northern Hemisphere. This specific behavior can be explained by the fact that the flux tube reaching very high alti- tude combined with its seasonal inclination allows the summer hemisphere to receive solar photons almost throughout the day. Finally, for L _max = 6, we can see the nonnegligible effect on the ionosphere caused by the stagnation point around 15 MLT. These results will be discussed in detail in further studies. Although it is out of the scope of this paper to make a precise comparison between these simulation results and exper- imental observations, globally, the modeled results show consistent behavior with observations, confirming the numerical validity of our new model. For instance, the magnitude and diurnal/seasonal pattern of elec- tron density are in agreement with incoherent scatter radar (ISR) observations at ionospheric altitudes, such as Millstone Hill for low latitude [Liu et al., 2007], EISCAT Tromsø for high latitude [Sedgemore et al., 1996; Zhang et al., 2005], and with satellite observations at plasmaspheric altitudes [Reinisch et al., 2009].

We have presented results for the extreme case L _max = 6 to show the ability of the model in such conditions.

In the following, we concentrate on an inner tube corresponding to L _max = 5, still sensitive to the competition between corotation and convection but without a stagnation point.

As discussed in section 2.1.3, because these terms may be larger than the field-aligned gravity acceleration

and thus may significantly contribute to the plasma dynamics along the flux tube, this new model accounts

for the centrifugal and Coriolis field-aligned accelerations (referenced hereafter as inertia). In our simulation,

Figure 6. (left) MLT/altitude color plot of the sum of accelerations (gravity, centrifugal, and Coriolis) applied to H ⁺ ion over 24 MLT. The plot is obtained for the closed flux tube L _max = 5 during equinox conditions. The vertical white solid line is the time of the flux tube cut shown in Figure 6 (right) (16 MLT). (right) Flux tube cut showing acceleration along the tube at 16 MLT.

as the convection is rather moderate, no real impact of the Coriolis acceleration is observed, and thus, we will not make any distinction between the two contributions to the inertia acceleration. Basically, the inertia is large when we are far from the Earth and when the motion is fast. Thus, we may expect to reach the largest value for the inertia close to the apex, when it is located at a high L shell value in a region where corotation and convection contribute to accelerate the transport of a tube, that is, in the 15–03 MLT sector (see Figure 3 for the track in the equatorial plane). Figure 6 presents the sum of inertia and gravity accelerations (positive upward and negative downward) in the format discussed above for the color panel plots, with a snapshot corresponding to a cut at 16 MLT (white vertical line in the color panel) plotted along the field line. Not sur- prisingly, gravity dominates at low altitude, and the total acceleration is downward up to about 5000 km in both hemispheres. However, above this altitude, depending on the position of the tube along its trajectory, the acceleration may become significantly upward and maximizes in a region close to the apex. For the rea- son presented before, the acceleration increases when the corotation/convection is enhanced after the flux tube crosses the stagnation region and results in a rapid decrease of the L shell from L = 5 at 15 MLT to L ≤ 4 around 20 MLT; this rapid motion leads to a maximum acceleration of about 4 m s ⁻² in the evening sector.

Due to the tilt of the magnetic field, the maximum of the inertia acceleration is located in the Southern Hemi- sphere; the snapshot at 16 MLT clearly shows the location of the maximum along the magnetic field line, close to the geographic equator and southward of the geomagnetic equator.

Figure 7 presents one impact of this motion on the energetics of the plasma. If we define the anisotropy as 𝜆 = ^T

T

−1, this figure shows the anisotropies (in %) for electron (top) and H ⁺ ion (bottom) in the format defined before. Focusing on the transition altitude between O ⁺ and H ⁺ ion in both hemispheres (white horizontal curves), it is clear that the largest anisotropies are obtained in this region, with maximum values between 20%

for H ⁺ ion and 30% for electron. However, they do not develop at the same time, because the mechanisms responsible for the anisotropy are different for the ions and the electrons. Similar to Blelly and Schunk [1993]

results in the case of electron, the anisotropy is always positive, meaning that the perpendicular temperature is always larger than the parallel temperature. Due to their mass, electrons are generally more sensitive to thermal flows than particle flows, and then heat fluxes contribute the most to the energy transfer. As discussed by these authors, we have q ^∥ _e ∕q ^⊥ _e ≈ 3 in the collision-dominated region, and the energy transfer associated with the mirror contributions in equations (24) and (25) helps to thermalize both components. As O ⁺ ion is the major species in the lower part of the tube, the electron density decreases rapidly with altitude, confining the collision-dominated region well below the transition region. Above this region the ratio decreases, and there is opposite contributions of the mirror terms, resulting in an increase of T _e ^⊥ and a decrease of T _e ^∥ . The effect maximizes where ^q _n

e

is maximum, just above the transition region.

Concerning H ⁺ ion, the impact of the field-aligned velocity is significant. When the tube is leaving the stag-

nation region and the plasma is compressed as L decreases, H ⁺ ion velocity is upward and maximizes below

the transition region, leading to a negative velocity gradient above the transition where H ⁺ , being the major

species, is more or less in hydrostatic equilibrium (see discussion below). In such conditions, the two contribu-

tions longitudinal and transverse to the rate of expansion of the fluid element in equations (24) and (25) have