This differential drift represents a current, j (A m−2), where j = e(niVi− neVe) and ni and ne 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, jH and jP, 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, νi and νe, 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 JH and JP 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
where
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:
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 ni ions and ne 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 ρq is its charge density. To a good approximation plasmas are quasi‐neutral (ρq ≈ 0), so the momentum equation becomes
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.
2.3 STEADY‐STATE MAGNETOSPHERIC/IONOSPHERIC CONVECTION
Maxwell stress together with pressure variations associated with plasma density and temperature are responsible for maintaining the magnetospheric shape presented in Figure 2.2a, in equilibrium with the solar wind flow and its associated ram pressure. If this equilibrium is disturbed, for instance dayside and nightside reconnection cause erosion of the magnetopause and inward collapse of the magnetotail, respectively, then large‐scale magnetospheric flows are excited to rebalance the system. As described in section 2.2.1, in a steady state this causes a continuous circulation of the magnetospheric flux and plasma, and this magnetospheric convection is coupled to the ionosphere by tension forces and gives rise to ionospheric convection. Antisunward motions of open field lines and sunward motions of closed field lines give rise to the general twin‐cell convection pattern shown in Figure 2.1. The nature of this steady‐state flow and the associated electrodynamics are discussed in the following sections.
2.3.1 Electrostatic Potential and Magnetic Flux Transport
Magnetospheric convection coupled to the ionosphere results in a twin‐cell convection pattern, sketched schematically in Figure 2.3a. In the nonrotating rest‐frame of the Earth, these plasma motions are equivalent to an electric field given by equation (2.5). At polar latitudes, the magnetic field is roughly vertical, with a magnitude B ≈ 50,000 nT, so the electric field is roughly horizontal. The