The equation that expresses the time rate of change of electron concentration, Ne, is the continuity equation:

where Ne is the electron density, IJN,,) is the loss rate, which is dependent upon the electron density, div stands for the vector divergence operator, and V is the electron drift velocity.

Figure 3-6: Rate of electron production as a function of reduced height (h-h„) and for selected values of the solar zenith angle. From Davies [1965].

The divergence of the vector in Equation 3.6 is the transport term, sometimes conveniently called the "movement" term. The continuity equation says that the time derivative of electron density within a unit volume is equal to the number of electrons that are generated within the volume (through photoionization processes), minus those that are lost (through chemical recombination or attachment processes), and finally adjusted for those electrons that exit or enter the volume (as expressed by the transport term). To first order, the only derivatives of importance in the divergence term are in the vertical direction, since horizontal Ne gradients are generally smaller than vertical ones. In addition, there is a tendency for horizontal velocities to be relatively small in comparison with vertical drift velocities. Consequently, we may replace div (Ne V) by d/dh (Ne Vh ) where Vh is the scalar velocity in the vertical direction. We rewrite Equation 3.6 as follows:

Now let us look at some special cases. If V/, = 0 (no movement), then the time variation in electron concentration is controlled by a competition between production q and loss L. At nighttime, we may take q - 0, and this results in

In principle, there are two mechanisms to explain electron loss: one defined by attachment of electrons to neutral atoms (in the upper ionosphere), and the second defined by recombination of electrons with positive ions (in the lower ionosphere). The attachment process is proportional to Ne alone, while recombination depends upon the product of Ne with Nj, where Nj is the number of ions. The process of attachment involves radiative processes and results in an extremely low cross section (probability of occurrence). We may ignore it in many practical situations and take recombination as the only major source for electron loss. Since Ne= Nj, the recombination process obeys the equation L = a Ne2, where a is the recombination coefficient. Recombination is very rapid in the D and E regions, the process being accomplished in the order of seconds to minutes. Attachment, the electron loss process for the upper ionosphere, has a time constant of the order of hours. This is the primary reason that the ionosphere does not entirely disappear overnight. Another reason is that there exists a second source of electrons associated with the plasmasphere, This reservoir of ionization is built up during the daytime through vertical drift, but "bleeds" into the ionosphere during nocturnal hours.

In the vicinity of local noon, dNe / dt = 0 and we may analyze the quasiequilibrium conditions suggested by Equation 3.7 when the left hand side of the equation = 0. The two main types of equilibrium processes are given in Table 3-2. The equilibrium processes identified in Table 3-2 are the dominant possibilities during daytime when photoionization is significant. During nocturnal hours, equilibrium is seldom achieved at F region heights, although it is approached in the period before sunrise.

Type |
Property |
Equation |

Photochemical |
Production balanced by loss |
L(Ne> » d/dh (NeVf) |

Drift |
Production balanced by drift |
L(Ne) « d/dh (NeVh) |

While the continuity equation appears quite simple, the generic terms (i.e., production, loss, and transport) represent a host of complex photochemical and electrodynamic processes, which exhibit global variations and are influenced by nonstationary boundary conditions within the atmosphere and the overlying magnetosphere. Notwithstanding these complications, the equation provides a remarkably clear view of the basic processes that account for ionospheric behavior. In fact, the relative contributions of terms in the continuity equation will account for the majority of the anomalous ionospheric properties; that is, those ionospheric variations which depart from a Chapman-like characteristic. This is especially true for the F2 layer within which the movement term attains paramount status. In the E and F1 regions where the movement term is small compared with production and loss (through recombination), photochemical equilibrium exists in the neighborhood of midday. All of this has had a significant bearing on the development of ionospheric models and prediction methods. Indeed, as it relates to the F region of the ionosphere, it may be said that the existence of a nonvarishing divergence term in the continuity equation has been the primary impetus for the development of statistical modeling approaches. Nevertheless, efforts to account for all terms in the continuity equation through physical modeling are ongoing.

The underlying assumptions used by Chapman in his theory of layer production are in substantial disagreement with observation. The Chapman layer was based upon an isothermal atmosphere, and it is well known that the atmosphere has a scale height, kT/mg, which varies with height. Moreover, the basic theory assumes a monochromatic source for photoionization and a single constituent gas. Corrections and extensions to the early Chapman theory have led to better agreement with observation and, to this day, the

Chapman layer provides a fundamental baseline for ionospheric profile modeling.

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