A thermodynamically consistent formulation of the pseudo-incompressible equations for atmospheric modelling with an extension to moist processes

Abstract

Atmospheric flows are composed of motions occurring on a large range of temporal and spatial scales. All of these motions can be captured by the compressible Navier-Stokes equations, however, as sound waves do not play an important role in these processes, it is theoretically appealing and can be numerically advantageous to remove the sound waves entirely from the governing equations. Many ``soundproof'' equation sets have been developed to tackle this issue. The most well known of these are the Boussinesq equations, the anelastic equations and and the pseudo-incompressible equations. These equation sets are derived by introducing different approximations into the governing equations. The Boussinesq approximation replaces the continuity equation by an incompressibility condition, the anelastic approximation disregards the time derivative of the density in the continuity equation to yield a divergence constraint and the pseudo-incompressible approximation ignores the effect of pressure perturbations on the density which also yields a divergence constraint. In this work we will be focusing on the pseudo- incompressible approximation and we aim to extend this model in two ways. The first is to numerically implement a ``thermodynamically-consistent'' formulation of the pseudo-incompressible equations and the second is to extend the model to include moisture, i.e. phase changes and diabatic terms. These two aspects are outlined in more detail in the following paragraphs. Typically, the pseudo-incompressible equations are written in ``\pi-\theta'' form i.e., with the Exner pressure \pi and the potential temperature \theta utilised in the momentum equation's pressure and gravity terms and not thermodynamic pressure p and the density \rho. An outline for the latter formulation is given in Klein & Pauluis 2011 and it requires that all terms up to first order in the pressure perturbation are retained in the momentum equation which is a condition which is automatically fulfilled in the ``\pi-\theta'' case. This complicates the numerics in two ways: 1) it adds a pressure perturbation dependent source term to the momentum equation and 2) it alters the form of the projection step used to enforce the divergence constraint. A method to resolve these issues will be illustrated in the first part of the present work. For the remainder we will examine the effect of moisture on our pseudo-incompressible model. So far only the anelastic equations have been utilised extensively as a basis for modelling moist flows. The Boussinesq equations are unsuitable for moist atmospheric applications due to the assumption of constant density and the pseudo-incompressible equations are rarely used in moist form due to the complications that latent heat causes in the divergence constraint. In Almgren et al 2008 a pseudo-incompressible model with source terms and compositional changes for supernovae is developed which overcomes these complications. Motivated by this work we have created a non-thermodynamically consistent pseudo-incompressible model for moist atmospheric flows. This entailed the derivation of a background state which varies in time as a result of the release of latent heat. The creation of such a background state solves the problems created by the heat source in the divergence constraint and allows the pseudo-incompressible model to incorporate diabatic terms. This model is valid for flows with large density and potential temperature variations and is therefore more generally applicable than the anelastic approximation which is only valid for small variations. In the analysis presented in Lipps & Helmer 1982, for example, they require the potential temperature to be a slowly varying function of the vertical coordinate for their moist anelastic model. If we focus on atmospheric motions then this restricts the validity of their model to motions in the troposphere and in this paper they state that ``for severe mid-latitude convection...the present analysis is expected to have limited validity''. However, the assumption of small density and potential temperature variations is valid for most atmospheric processes of interest and our model is advantageous only in specific cases. We will also develop the moist model to be thermodynamically consistent. This is carried out in much the same fashion as in the dry case except the background variables in the gravity term will now be functions of time as well as height. Moreover, we will examine the effect of the pressure choice used, full value or hydrostatic, for the condensation rate calculation. To implement our models numerically we have incorporated it into an in-house finite volume code for low Mach number flow. The models are then verified against a number of testcases which are used to examine various aspects of the interest, e.g. how the model performs under buoyancy driven flow. In summary, the thesis will have the following outline: we begin with a derivation, starting from the compressible equations, of the dry thermodynamically consistent set. This is followed by a derivation of the moist form of the thermodynamically consistent set. The numerics used for modelling each set of equations is then outlined and finally each model is tested and compared using several standard dry and moist testcases.

Die Atmosphärenströme definieren sich durch Prozesse auf unterschiedlichsten zeitlichen wie räumlichen Skalen. Diese Prozesse können mit den "kompressiblen'' Navier-Stokes Gleichungen modelliert werden. Es ist anzunehmen, dass eine Nichtberücksichtigung von Schallwellen zu numerischen Vereinfachungen führt. Dies ist auch von theoretischem Interesse. Speziell unter schallfreien Bedingungen wurde eine Reihe von Gleichungen entwickelt, sog. ``schallfreie Gleichungen'', welche das Problem der Schallwellefilterung lösen. Ein Lösungsweg beschreibt die ``pseudo-inkompressiblen Gleichungen'', welche der Fokus dieser Arbeit darstellt. Die pseudo-inkompressible Annäherung definiert die Dichte als eine Funktion von hydrostatischem Druck und Temperatur. Diese Annäherung resultiert in einer Divergenz-Gleichung, die, im Gegensatz zur Evolutions-Gleichung, Schalleffekte im Modell vernachlässigbar macht. Die Studie entwickelt die pseudo-inkompressible Annäherung in zwei Richtungen weiter: 1) Implementierung einer ``thermodynamisch konsistenten'' Form in die pseudo-inkompressiblen Gleichungen und 2) die Expandierung des Modellraums, um diabetische Terme und Phasenübergange modellieren zu können. Die numerische Implementierung der ``thermodynamisch konsistenten'' Form und der Modellexpandierung stellt eine ``in-house'' Entwicklung von pseudo- inkompressiblen Finite-Volumen-Routinen dar. Nach der Implementierung werden standardisierte Teststudien durchgeführt und die Ergebnisse mit anderen Modellen verglichen.