cap (ECPC) model of the Dungey cycle. If the dayside and nightside reconnection rates are unequal, the proportion of the magnetic flux associated with the Earth's dipole that is open, the open magnetic flux content of the magnetosphere, also known as the “polar cap flux,” FPC, will change with time. Global auroral imagery, used to estimate the size of the polar cap, shows that typically 0.5 GWb of the 8 GWb associated with the Earth's dipole is open, though this can vary between 0.2 and 1.2 GWb (e.g., Milan et al., 2007; Huang et al., 2009). Assuming a polar magnetic field strength of 50,000 nT, with 0.5 GWb of open flux the polar cap is approximately 1,800 km in radius.
The reconnection rate at the magnetopause is measured by a voltage, the amount of flux that is opened by reconnection in unit time, or equally the rate at which magnetic flux is transported into a region where its topology changes from closed to open. If the dayside and nightside reconnection rates, ΦD and ΦN, are equal, flux is closed as rapidly as it is opened, and FPC is constant. In such a situation, the rate of open and closing flux is also the rate at which flux is transported through the magnetosphere by the Dungey cycle, so ΦD = ΦN = ΦPC. On the other hand, if the rates are unequal, the rate of change of FPC is
Consider the situation in which ΦD > 0, ΦN = 0. Magnetic flux is opened at the dayside magnetopause and carried into the tail by the flow of the solar wind. The dayside magnetosphere is eroded, and the magnetotail flares outward, as indicated in Figure 2.9a, in which a limited portion of the subsolar magnetosphere has reconnected. The magnetopause is no longer everywhere in stress balance with the flow of the solar wind; there is a mismatch between the angle of attack of the solar wind on the magnetopause and the magnetic pressure within. As a consequence, the internal magnetic flux and plasma are redistributed by the pressure imbalance to re‐attain equilibrium: the eroded magnetopause moves outward, the magnetotail magnetopause moves inward, and there is a general sunward movement of flux in the closed magnetosphere. Once equilibrium is reached, the magnetosphere as a whole is blunter and the magnetotail larger in cross‐section, as flux has been removed from the dayside and added to the tail. Figure 2.9b (i–iv) shows the situation in the ionosphere: equatorward of the original dayside polar cap is a region of newly opened magnetic flux. The flows excited within the magnetosphere are communicated to the ionosphere, and ionospheric convection redistributes the flux to return the polar cap to a circular, but larger configuration, with a general antisunward motion in the polar cap and sunward motion at lower latitudes. A burst of nightside reconnection has a similar effect, closing open flux at the nightside of the polar cap, again leading to antisunward and sunward flow at higher and lower latitudes, respectively (Fig. 2.9c (i–iv)), reducing the width of the tail and returning closed flux to the dayside.
Figure 2.9 (a) Deformation of the magnetopause and resulting plasma flows following a burst of dayside reconnection. (b) (i)–(iv) The flows in the ionosphere excited by a burst of dayside reconnection. (c) Ionospheric flows excited by nightside reconnection. (d) Ionospheric flows excited due to dayside and nightside reconnection, in the limit that the redistribution of flux maintains a circular polar cap at all times
(from Cowley & Lockwood, 1992).
Figure 2.9d shows the convection pattern expected for dayside and nightside reconnection in the limit that the redistribution of flux in the magnetosphere is sufficiently rapid that the magnetopause and polar cap boundary are kept close to equilibrium at all times. The reconnecting portion of the OCB is shown dotted. Dayside and nightside reconnection frequently occur at the same time, and in this situation the convection is a superposition of these patterns, with the polar cap expanding or contracting at a rate given by equation (2.15). Figure 2.10 presents a sequence of simulated convection patterns in response to dayside and nightside reconnection (Milan, 2013). In panels (a) and (b) ΦD > 0, ΦN = 0, in (c) ΦD < ΦN, and in (d) ΦD = 0, ΦN > 0. Tracers in the flow show that where the boundary is not reconnecting, the ionosphere and boundary move together, for example, tracer 1 in panels (a) and (b): in this situation the OCB is said to be “adiaroic” (no flow across) (Siscoe & Huang, 1985). In the frame of reference of the moving boundary, where the flow is adiaroic, the electric field is zero, and elsewhere along the OCB it is equal to the reconnection electric field (the rate at which flux is being opened/closed along unit length of the boundary). The integral of the electric field around the boundary is then equal to the difference between ΦD and ΦN, the rate of change of flux in the polar cap, and equation (2.15) is seen to be an expression of Faraday's law. When the circular polar cap approximation is used, it is straightforward to calculate the electric field and hence Φ at all points around the OCB in the nonrotating Earth frame (this is the electric field in the boundary rest‐frame modified by the speed of the moving boundary). Then, assuming that j‖ is everywhere zero except at the OCB and the low latitude Φ = 0 boundary, and that conductance is uniform, Φ can be found as a solution of Laplace's equation, ∇2Φ = 0 (see equation (2.14)), subject to the boundary conditions (e.g., Siscoe & Huang, 1985; Freeman, 2003; Milan, 2013). This was the technique used to produce Figure 2.10.
Figure 2.10 The ionospheric convection excited in response to combined dayside and nightside reconnection
(from Milan et al., 2012; Reproduced with permission of John Wiley and Sons).
In the limit that the polar cap remains circular, the cross polar cap potential along the dawn‐dusk meridian is the average of the dayside and nightside reconnection rates (Lockwood, 1991),
The cross polar cap potential is sometimes defined as the difference between the maximum and minimum potentials in the convection pattern, ∆Φ = Φmax − Φmin, in which case, ∆Φ ≥ ΦPC.
Observational evidence for the evolution of the flows from day to night across the polar cap following the onset of dayside reconnection, as depicted in the top two panels of Figure 2.10, has been mixed. As a result, significant debate has ensued as to the validity and applicability of the Cowley and Lockwood (1992) paradigm of ionospheric flow excitation.