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A charged particle of mass m and charge q moving with velocity v in the presence of an electric field E and magnetic field B, experiences the Lorentz force

(2.1)

such that the momentum equation for individual ions and electrons is

(2.2)

where we consider singly charged positive ions for simplicity (mainly protons in the magnetosphere and heavier ions such as O⁺ and O₂ ⁺ in the ionosphere), we have used q = e and q = − e for ions and electrons, and the last terms on the RHS represent momentum loss due to collisions with a background of uncharged particles, such as the atmosphere, with collision rates (frequencies) ν_i and ν_e. We assume initially that ν_i = ν_e = 0.

In a situation with no electric field, E = 0, and uniform magnetic field B, the momentum equations can be solved to show that the magnetic force causes particles to move in a circle in a plane perpendicular to B (Fig. 2.4a), of radius r_g with angular frequency Ω, where

(2.3)

the gyration being in a right‐handed sense about the field direction if q < 0 and left‐handed if q > 0. In a given field strength B, the gyroradius depends on the speed of the particle perpendicular to the magnetic field, v_⊥, such that the gyrofrequency is the same for all particles of a particular species (that is, with a given charge to mass ratio ). Hotter plasma (a larger range of v_⊥) will tend to have larger gyroradii. In a uniform magnetic field, the particles experience no force along the magnetic field direction, their initial parallel speed v_‖ is unchanged, and, in general, they are highly mobile along magnetic field lines. Each particle forms a current loop, but in a uniform plasma, the currents of adjacent particles cancel out. However, where there are gradients in plasma density or temperature (and hence pressure), a net magnetization current can flow. If the magnetic field has a gradient along its length, as shown in Figure 2.4d, then qv × B has a component that modifies v_‖ and expels particles from high‐field regions. In this way, particles are kept away from the Earth by this “mirror force” (see converging B regions near the Earth in Fig. 2.2) and tend to be trapped within the magnetosphere. Gradients in field strength perpendicular to B, such as in the Earth's inner dipole, lead to variations in gyroradius that cause charge‐dependent “gradient drift” and hence an electric current to flow (Fig. 2.4e). Gradient drift is enhanced by the curvature of the dipole field. This combined “gradient‐curvature drift” is significant for hot plasma with large gyroradii and rapid field‐parallel motions.

Figure 2.4 Schematic of (a) gyrating particles, (b) E × B drifting particles, (c) E × B drifting particles in the presence of neutrals, (d) particles mirroring in a high field region, and (e) a gradient in B producing charge‐dependent drift.

If the magnetic field is uniform, and an electric field is introduced (Fig. 2.4b), charged particles initially at rest are accelerated by the electric force, caused to deviate by the magnetic force, and then are decelerated by the electric force, performing a half gyration before coming to a rest again. The cycle repeats, and the particles follow cycloid trajectories with an average bulk drift in the E × B direction with a speed E/B E/B, that is a velocity

(2.4) (2.4)

An observer moving with the plasma would just see circular gyrations (as in Fig. 2.4a), the trajectories that are expected in the presence of a magnetic field but no electric field. This demonstrates that the electric field is dependent on one's frame of reference, a consequence of the theory of special relativity, and in a frame in which a magnetized plasma is drifting with velocity V, a motional electric field E exists, where

(2.5)

The above discussion is appropriate for the magnetosphere and F region ionosphere, where collisions can be discounted. Figure 2.4c shows how Figure 2.4b must be modified in the E and D region ionosphere where a dense background of neutrals exists, such that the collisional terms of equation (2.2) become significant. As the particles E × B drift, they are occasionally brought to a rest by collisions, before being re‐accelerated by the electric field. The particles' drift motion in the E × B direction is slowed by the collisions, and ions and electrons acquire an additional drift parallel and antiparallel to E, respectively, resulting in bulk drifts of V_i and V_e. This differential drift represents a current, j (A m⁻²), where j = e(n_iV_i− n_eV_e) and n_i and n_e are the number densities of ions and electrons. The current has components in the −E × B and +E directions, known as the Hall and Pedersen currents, j_H and j_P, respectively. In the polar ionosphere, where B is directed vertically, these currents flow horizontally. The magnitude of these currents depends on E, on the electron density, and on the ion‐neutral and electron‐neutral collision frequencies, ν_i and ν_e, which are altitude dependent. In the F region, where collisions are rare, the ionospheric plasma undergoes E × B drift, in the E region significant currents flow, and in the D region collisions are so prevalent that plasma motions and hence currents are negligible. Integrating in height through the ionosphere, total horizontal currents J_H and J_P have associated conductances ∑_H and ∑_P, which mainly depend on E region electron density, and hence are largest in the sunlit ionosphere and the auroral zones. The ionospheric (i.e., field‐perpendicular) current J_⊥ (A m⁻¹) driven in the presence of an electric field E is

(2.6)

where is the unit vector of B. Typical values of ∑_P at polar latitudes are 10 S or mho (ohm⁻¹) in daylight, 1 S in darkness, and 10 S in the auroral zone; ∑_H ≈ 2∑_P.

We now consider the plasma to be a collection of charged particles that can be described as a fluid, particles that attract and repel through the Coulomb force and whose relative motions give rise to magnetic fields that give structure to the fluid. Particles are free to move along the magnetic field lines, but cannot move across the field due to their gyratory motions. To first order, magnetized plasmas have the remarkable property that as an element of fluid moves (E × B drifts), it carries its internal magnetic field with it: the field is said to be “frozen‐in” to the fluid (Alfvén, 1942). If the fluid element expands or contracts in volume, the magnetic flux permeating it remains constant and the field strength within decreases and increases accordingly. If the element becomes distorted, the magnetic field within bends to acquire the new shape. This occurs as the motions of the plasma particles generate electric fields and currents, which in turn modify the existing magnetic field in such a manner as to give the appearance that the magnetic flux is frozen‐in. The frozen‐in flow approximation holds in regions where particle gyroradii are small with respect to gradients in the magnetic field, otherwise charge‐dependent flows are excited (see Fig. 2.4e).

To the momentum equations discussed above, we must add Maxwell's equations, which govern the evolution of magnetic and electric fields:

(2.7)

that is, the laws of Gauss for the electric and magnetic fields, and Ampère and Faraday, respectively, in a form appropriate for a plasma; displacement current is neglected from the Ampère‐Maxwell law as it is only significant for high‐frequency phenomena, which are not pertinent to this discussion. To understand the dynamics of the plasma, we consider the momentum equation of a unit volume of the fluid, containing n_i ions and n_e electrons, which is found by combining the ion and electron momentum equations (2.2) and including the effect of gas pressure (associated with random thermal motions of the particles):

(2.8)

where V is the velocity of the element (the mass‐weighted mean of the ion and electron velocities within the element), ρ is its mass density, P is its pressure, and ρ_q is its charge density. To a good approximation plasmas are quasi‐neutral (ρ_q ≈ 0), so the momentum equation becomes

(2.9)

where Ampère's law, equation (2.7), has been used to substitute for j. The forces that accelerate the plasma are gradients in pressure and two magnetic terms known as Maxwell stress. The first magnetic term indicates that in regions where the magnetic field is bent, the plasma element experiences a force so as to straighten the field. The second term indicates that where there are gradients in the field strength perpendicular to B, the plasma experiences a force that tries to smooth out that gradient. These terms are known as the magnetic tension force and magnetic pressure force, respectively. These magnetic forces have the effect of maintaining stress balance within the magnetosphere and coupling different regions within the magnetosphere‐ionosphere system: moving an element of plasma in one location exerts a magnetic tension and pressure on adjacent elements, and stress is transmitted both along and across the field lines. In equilibrium, the three terms on the RHS of equation (2.9) balance each other. If the magnetosphere‐ionosphere system is disturbed, then flows are excited to return the system to equilibrium. As discussed in the following sections, these forces are responsible for magnetospheric convection and its manifestation in the ionosphere.