Abstract
Abstract. We show how the rotating shallow-water MHD model, which was proposed in the solar tachocline context, may be systematically derived by vertical averaging of the full MHD equations for the rotating magneto fluid under the influence of gravity. The procedure highlights the main approximations and the domain of validity of the model, and allows for multi-layer generalizations and, hence, inclusion of the baroclinic effects. A quasi-geostrophic version of the model, both in barotropic and in baroclinic cases, is derived in the limit of strong rotation. The basic properties of the model(s) are sketched, including the stabilizing role of magnetic fields in the baroclinic version.
Highlights
The shallow-water magnetohydrodynamics model was introduced on heuristic grounds in Gilman (2000) in the context of the solar tachocline. (We will call it mRSW in what follows; with respect to the originally used acronym sMHD, this one reflects better the nature of the model; see below.) Its applications were further discussed in Dikpati and Gilman (2001a) and Dikpati and Gilman (2001b)
It is important to keep in mind the limits of applicability of the mRSW related to these hypotheses, the example of non-magnetic RSW and its applications in geophysical fluid dynamics show that such models remain useful far beyond their formal validity range
The added value, with respect to the pioneering work of Gilman (2000), is that we show that the mRSW equations arise universally from the vertical averaging of basic MHD and independently of the details of stratification and compressibility
Summary
The (rotating) shallow-water magnetohydrodynamics model was introduced on heuristic grounds in Gilman (2000) in the context of the solar tachocline. (We will call it mRSW in what follows; with respect to the originally used acronym sMHD, this one reflects better the nature of the model; see below.) Its applications were further discussed in Dikpati and Gilman (2001a) and Dikpati and Gilman (2001b). The primary purpose of the present paper is systematic derivation of the mRSW model from the full MHD equations by vertical averaging, which will (1) clarify the basic hypothesis underlying the model, and (2) immediately give multi-layer generalizations. These generalizations allow for incorporation of the baroclinic effects in the model. Another purpose is to establish the quasigeostrophic (QG) approximation of the (multi-layer) mRSW, arising in the limit of strong rotation, as is traditionally done in geophysical applications of RSW (cf., e.g., Pedlosky, 1982; Zeitlin, 2007). Shallow-water and QG approaches being standard working tools in the atmosphere–ocean community, the present work might help in establishing a common language with the astrophysical community
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