||Yoshikawa A. and R. Fujii, Earth’s Ionosphere: Theory and Phenomenology of Cowling Channels, in Electric Currents in Geospace and Beyond, John Wiley & Sons, Inc, Hoboken, N.J., 10.1002/9781119324522.ch25, 2018.04, The Cowling channel is a generic name of a current system forming inside a high conductivity band, in which a secondary polarization electric field modifies the current flow. The polarization field is excited when a divergent part of Hall current driven by the primary electric field is prevented from flowing out to the magnetosphere as the field-aligned current (FAC).
The Cowling effect is now well known as enhancement of current flow in the direction of the primary electric field by the secondary Hall current [Chapman, 1956]. The Cowling effect was first investigated by Cowling  in connection with the solar atmosphere. The generation mechanism [Cowling and Boreger, 1948] was adopted to account for equatorial electrojet [Hirono, 1950, Untied, 1967] and auroral electrojet [Boström, 1964]. The Cowling effect has been investigated theoretically and observationally [e.g., Baujohann, 1983; Yasuhara et al., 1985; Haerendel, 2008, Amm et al., ; Amm and Fujii, 2008; Marghitu et al., 2011].
Figure 1 shows traditional picture of two-dimensional Cowling channel model elongated along east-west direction [e.g., Baumjohann, 1983], in which ionospheric Hall and Pedersen conductivity are height-integrated. The primary westward electric field (E1) drives northward Hall current and westward Pedersen current. The southward secondary field (E2) is generated so that the Pedersen current closes the primary Hall current between the conductivity gradients. The secondary Hall current flows in the same direction as the primary Pedersen current and forms the electrojet system.
Generally, it is difficult to specify polarization effects in the ionosphere from ground-based data alone. These data only allow to infer the resultant total electrodynamic fields, but cannot track back the chain of cause and consequence that led to the physical situation which then causes these observed total fields. Thus, using ground-based data alone in most cases we can only state whether an observed situation is consistent or not with the expectations from an “active” polarization effect.
To quantify the Cowling effect, we need to know the relative strength of the polarization electric field to total electric field and to what extent it cancels (closes) the primary Hall current. This problem is complementary to the question: How much curl-free Hall current flows out to the magnetosphere as FAC?
In order to reply to this problem provided by Amm et al., , modeling of Cowling channel has been further developed.
To describe the Cowling channel, Amm et al.,  and Fujii et al.,  introduce a parameter called the Cowling efficiency. It is defined as a ratio how much of Hall current is confined inside the ionosphere by the secondary Pedersen current excited by the polarization electric field. Definition of Cowling efficiency is practically important. It provides a general way to calculate quantitatively the polarization electric field, if the Cowling efficiency, the conductance, and either primary or the total electric fields are known [Amm et al., 2013].
It has been suggested that to identify the Cowling efficiency for specific phenomenon, one needs to know the impedance of the magnetospheric circuit, which completes the current circuit in the M-I system via FAC [e.g., Fujii et al., 2011]. However, it is questionable to assign a magnetospheric impedance for steady state because the M-I system is always changing dynamically.
The M-I coupling process via shear Alfven waves has been used to investigate the nonstationary FAC closure by ionospheric conducting current [e.g., Scholer, 1970]. Assuming specific electric field configurations of an incident wave, Glaβmeier  and Itonaga and Kitamura  have shown that a secondary polarization field due to gradients of Hall conductance can appear in the reflected wave. Actually, the Alfven wave approach can be used to describe not only local and dynamical phenomena but also more generally global quasi-static M-I coupling processes [Yoshikawa et al., 2010]. Therefore it is very important to understand the how the shear Alfven wave interacts with the Cowling channel.
Yoshikawa et al., [2011a] give a general theory about M-I coupling, independent of specific geometries or specific situations. This theory, based on the Alfven waves used in a way of a basis function for the M-I coupling process, is later applied in Yoshikawa et al., [2013a] and Yoshikawa et al., [2013b] specifically to a Cowling channel situation, but can be applied for any general case.
Most of Cowling channel models introduced so far rely on a thin-sheet ionosphere [e.g., Baujohann, 1983]. However, in a realistic ionospheric E-layer, a vertical distribution of the Pedersen conductivity and Hall conductivity has maximum peak around 125 km altitude and around 110 km altitude, respectively [e.g., Richmond and Thayer, 2000]. In order properly consider the ionospheric current closure, one also takes into account the ionospheric thickness [Amm et al., 2008]. One step in this direction is to assume that the Pedersen and Hall current flow thin layers at different altitudes [Fujii et al., 2011; Amm et al., 2011; Yoshikawa et al., 2011].
The classical picture illustrated in Figure 1 describes divergence-free approximation of auroral electrojet. However, a longitudinal boarder of Cowling channel is also important for considering finite aurora arc formation and Harang reversal region [Harang, 1947; Heppner, 1972; Marghitu et al., 2011], where the auroral electrojet is diverging.
Amm et al.,  give a review of the work available in the literature until 2008 regarding following aspects of ionospheric electrodynamics and Magnetosphere-Ionosphere (M-I) coupling:
-Polarization effect in the ionosphere (often referred to as “Cowling effect)”
-Inductive effect in the ionosphere
-The effect of the three-dimensional (3D) nature of the ionosphere for ionospheric electrodynamics
-The consequences of the above mentioned aspects to M-I coupling.
Marghitu,  provides an excellent review for auroral arc electrodynamics, by considering the 1D thin uniform arc, the 2D thick uniform arc, and the non-uniform arc. The various arc features are assembled together in a tentative 3D arc model.
The purpose of this chapter is to review the recent development of Cowling channel model after Amm et al.,  and Marghitu, . Recent work provide an extension of theoretical description of the classical Cowling channel with respect to the following aspects: 1) Taking into account the 3D nature of ionosphere by introducing two current layers at different altitudes, and 2) considering finite length of the Cowling channel by introducing a conductance boundary not only at the meridional borders of Cowling channel, but also at its zonal boundaries. Using this improved model, schematically illustrated in Figure 2 with Cowling efficiency description, we discuss current closure and their energy principle for evolution of Cowling channel. Energy flow inside the Cowling channel and impact of polarization effect on Joule dissipation in more general M-I coupling scheme are also provided. In addition, we also clarify how shear Alfven wave interacts to the Cowling channel and their application to the global magnetosphere-ionosphere coupling simulations..