An idealized 1½-layer isentropic model with convection and precipitation for satellite data assimilation research. Part II: model derivation

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.

An idealized 1½-layer isentropic model with convection and precipitation for satellite data assimilation research. Part I: model dynamics

An isentropic 1½-layer model based on modified shallow water equations is presented, including terms mimicking convection and precipitation. This model is an updated version of the isopycnal single-layer modified shallow water model presented in Kent et al. (2017). The clearer link between fluid temperature and model variables together with a double-layer structure make this revised, isentropic model a more suitable tool to achieve our future goal: to conduct idealized experiments for investigating satellite data assimilation. The numerical model implementation is verified against an analytical solution for stationary waves in a rotating fluid, based on Shrira’s methodology for the isopycnal case. Recovery of the equivalent isopycnal model is also verified, both analytically and numerically. With convection and precipitation added, we show how complex model dynamics can be achieved exploiting rotation and relaxation to a meridional jet in a periodic domain. This solution represents a useful reference simulation or “truth” in conducting future (satellite) data-assimilation experiments, with additional atmospheric conditions and data. A formal analytical derivation of the isentropic 1½-layer model from an isentropic 2-layer model without convection and precipitation is shown in a companion paper (Part II).