Editorial Type:
Article
Article Type:
Research Article

An Idealized 1½-Layer Isentropic Model with Convection and Precipitation for Satellite Data Assimilation Research. Part II: Model Derivation

Onno Bokhove aSchool of Mathematics, University of Leeds, Leeds, United Kingdom

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https://orcid.org/0000-0002-1005-8463
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Luca Cantarello aSchool of Mathematics, University of Leeds, Leeds, United Kingdom

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Steven Tobias aSchool of Mathematics, University of Leeds, Leeds, United Kingdom

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Online Publication:
16 Mar 2022
Print Publication:
01 Mar 2022
DOI:
https://doi.org/10.1175/JAS-D-21-0023.1
Page(s):
875–886
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Abstract

In this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Onno Bokhove, o.bokhove@leeds.ac.uk; Luca Cantarello, mmlca@leeds.ac.uk

Abstract

In this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Onno Bokhove, o.bokhove@leeds.ac.uk; Luca Cantarello, mmlca@leeds.ac.uk
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