2.3a as blue arrows. A typical ionospheric drift speed of 500 m s−1 equates to an electric field strength of 25 mV m−1.
In the dipolar inner magnetosphere and the ionosphere, where the magnetic field strength is high, the magnetic pressure is such that, on timescales longer than a few seconds, the field lines cannot be compressed (magnetic pressure waves can propagate, but are of little importance for convection) and only interchange motions are possible (∇ ∙ V = 0, or the flow is incompressible). A consequence of the rigidity of the field is that
as the curl of the gradient of a scalar field has the useful mathematical property that ∇ × ∇ Φ ≡ 0 for any Φ, satisfying ∇ × E = 0. The “zero” of Φ is chosen to be equatorward of the low latitude boundary of the convection, and Φ(A) at a position A is the integral of the electric field from the zero boundary to A:
(2.11)
where the value of the integral is independent of the path taken (a requirement for a conservative force that can be represented by a potential). The convection electric field is usually presented as a contour plot of the equipotentials of Φ, as for instance in Figure 2.1. The expression for E × B drift can be written
(2.12)
which indicates that the ionospheric flow is perpendicular to B (i.e., roughly horizontal) and perpendicular to the gradient in Φ, or in other words that equipotentials of Φ are equivalent to streamlines of the flow.
From equation (2.5), it is apparent that E is perpendicular to B, which can be written as B ∙ E = 0 or B ∙ ∇ Φ = 0. This indicates that there is no gradient of Φ along the magnetic field direction, that is, field lines are also equipotentials of Φ. This is the same as saying that the component of the electric field parallel to the magnetic field is zero, E‖ ≈ 0, which occurs because electrons are highly mobile along the magnetic field and can arrange themselves to nullify any field line voltage. If changes in the magnetospheric B field are assumed to be slow, the electric field distribution represented by Φ can be “mapped” upward along magnetic field lines to infer the electric field throughout the magnetosphere. In early models of convection, it was assumed that this electric field originated in the solar wind flow and mapped down along open magnetic field lines to drive E × B drift in the ionosphere (e.g., Stern, 1973; Lyons, 1985; Toffoletto & Hill, 1989). However, when temporal changes in magnetosphere/ionosphere convection are considered, and induced electric fields become important, it is clear that this view is too simplistic. Instead, the physics of stress balance, outlined above, comes to the fore (e.g., Parker, 1996; Vasyliunas, 2005).
The potential difference (or voltage) between two positions A and B in the convection pattern (in the ionosphere or anywhere in the magnetosphere) is
(2.13)
This is a measure of the amount of magnetic flux frozen into the flowing plasma that is carried across any path between A and B in unit time, that is ∆Φ represents the rate of transport of magnetic flux in the convection. The “cross‐polar cap potential” (CPCP) or “transpolar voltage,” usually written ΦPC, is defined as the voltage between the dawn and dusk sides of the polar cap, a measure of the antisunward magnetic flux transport in the Dungey cycle. If the polar cap is 2,000 km across and the antisunward flow speed is 500 km s−1 (an electric field of 25 mV m−1), then ΦPC is 50 kV.
2.3.2 Convection, Corotation, and Dawn‐Dusk Asymmetries
The twin‐cell convection pattern does not extend to the equator but is confined to high latitudes, which can be understood by considering the competition between the reconnection‐driven Dungey cycle and “corotation” (that is, rotation with the planet), as described by Wolf (1970) and Volland (1973). The magnetospheric ends of the field lines move under the influence of the Dungey cycle flow discussed previously, whereas the ionospheric ends experience ion‐neutral collisions with the corotating atmosphere. In the equatorial plane, the Dungey flow consists of sunward magnetic flux transport (e.g., Fig. 2.2b), which can be represented by a voltage between the dawn and dusk flanks of the magnetosphere (equal to ΦPC), corresponding to some constant electric field E0 (Fig. 2.5a). Corotation exerts a force to make the plasma follow circular trajectories at the Earth's rotational angular frequency (Fig. 2.5b). The combined convection and corotation potential is shown in Figure 2.5c. A teardrop‐shaped inner core of the magnetosphere corotates with the planet, avoided by the Dungey flow outside; a point of flow stagnation exists along the dusk meridian where the Dungey cycle and corotation forces cancel. Ionospheric plasma can accumulate in the inner region to form the plasmasphere, whereas the outer region is constantly replenished by plasma of solar wind origin from the magnetotail. The boundary between these two regions marks the equatorward edge of the high‐latitude convection pattern in the ionosphere. As the convection voltage increases, the stagnation point moves toward the Earth and the ionospheric convection pattern expands equatorward.
Figure 2.5 Plasma flow in the equatorial plane of the magnetosphere. (a) The Dungey cycle contribution to cold plasma flow, associated with a dawn‐to‐dusk convection electric field E0. Flow streamlines are also contours of the convection electrostatic potential Φ, where Φ1 < Φ2 < Φ3, etc. (b) The corotation contribution to cold plasma flow. (c) The resultant flow, giving rise to Dungey cycle flow in the outer magnetosphere and corotation in the inner magnetosphere, where the plasmasphere forms (shaded). A flow stagnation point exists along the dusk meridian. (d) Convection and gradient‐curvature drift of hot electrons. (e) Convection and gradient‐curvature drift of hot protons. (f) The effect of displacing the plasma sheet (shaded) sunward by convection: where gradient‐curvature drift paths (dotted circles) intersect the inner edge of the plasma sheet, divergence of the partial ring current leads to the formation of field‐aligned currents that form the region 2 FAC system.
This picture is appropriate for cold plasma, that is the bulk of the plasma sheet particles with gyroradii that are small with respect to the radial magnetic field gradient