Interfacial Numerics For The Mathematical Modeling of Cloud Edge Dynamics

Transcription

1 Interfacial Numerics For The Mathematical Modeling of Cloud Edge Dynamics by R. P. Walsh B.Sc. (Hons), Memorial University of Newfoundland, 2014 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Faculty of Science c R. P. Walsh 2016 SIMON FRASER UNIVERSITY Fall 2016 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, education, satire, parody, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

2 Approval Name: Degree: Title: R. P. Walsh Master of Science (Mathematics) Interfacial Numerics For The Mathematical Modeling of Cloud Edge Dynamics Examining Committee: Chair: Ralf Wittenberg Associate Professor David Muraki Senior Supervisor Professor John Stockie Co-Supervisor Professor Nilima Nigam Internal Examiner Professor Date Defended: 8 December 2016 ii

3 Abstract In this thesis we present a two-dimensional thermodynamic model for cloud edge dynamics. We use the incompressible Euler equations to describe the atmospheric fluid dynamics and link these to the theory of moist thermodynamics through a constitutive law. This leads to a free boundary model for the interface separating clear and cloudy air. The model is further specialized to linearized disturbance equations about conditions that are critically saturated with zero liquid cloud water. Due to the presence of discontinuous derivatives induced by the clear/cloudy interface we adapt the immersed interface method (IIM) for computing the pressure in the system to second order accuracy. In addition, we investigate the unforeseen second order convergence of a naive method without the IIM. We conduct an analysis of local and global errors for the naive method leveraging off our analysis of the IIM and present numerical verification of the results when necessary. Keywords: Cloud edge dynamics; Cloud physics; Atmospheric fluid dynamics; Free boundary problem; Discontinuous Poisson equation; Immersed interface method iii

4 To my family, friends, and loved ones We couldn t have even done this if it wasn t for you and you and you and you The Hold Steady iv

5 Acknowledgements Funding for this work was, in part, provided by the Natural Sciences and Engineering Research Council of Canada through their Canada Graduate Scholarships-Master s program. I give special thanks to my senior supervisor Dr. David Muraki for the countless hours of discussion, coaching, mentoring, and guidance. His attention and care for students is unique and for this I am eternally grateful. I would also like to thank my co supervisor Dr. John Stockie for his continual support and insight throughout this project. Finally, I give special thanks to Caroline for her patience and care throughout this entire process. Words cannot describe the contributions and support she has given leading up to and throughout this work. v

6 Table of Contents Approval Abstract Dedication Acknowledgements Table of Contents List of Figures ii iii iv v vi viii 1 Introduction 1 2 Mathematical Model The Model Equations Constitutive Law For Moist Unsaturated Flow Constitutive Law For Moist Saturated Flow Time Dependence Of Entropy And Total Water Content The Complete Model The Marginal Cloud Limit Background Atmosphere For The Marginal Cloud Limit Nonlinear Equations For The Marginal Cloud Limit Linear Equations For The Marginal Cloud Limit Nondimensionalization Of The Linear Equations For The Marginal Cloud Limit The Marginal Cloud Model Pressure Equation Derivation of the Jump Condition Numerical Method The Immersed Interface Method in One Dimension A Numerical Example in One Dimension vi

7 3.2 The Immersed Interface Method: Pressure Equation Extending The IIM: Differentiating The Source Discretizing The Pressure Poisson Equation Calculating The Displaced Point Numerical Examples In Two Dimensions Open Contour Closed Contour Verification Of The Second Order Convergence For The Extended IIM An Alternate Discretization Discretizing The Pressure Poisson Equation Verification Of The Second Order Convergence For The Naive Method The Naive Method For 2 P = f Verification Of The Reduced Order Convergence For The Naive Method Applied To 2 P = f Quantifying The Global Error For The Naive Method Global Error Of The Naive Method For 2 P = ρ z Global Error Of The Naive Method For 2 P = f Further Analysis Functional Convergence Discrete Fourier Analysis Concluding Remarks 66 Bibliography 69 vii

8 List of Figures Figure 1.1 Holepunch cloud appearing over Simon Fraser University in Burnaby, British Columbia Figure 2.1 Diagram of cloud geometry Figure 2.2 Diagram of irregular nodes Figure 3.1 Diagram of irregular node in 1D Figure 3.2 Convergence for problem (3.1) Figure 3.3 Diagram of two-dimensional local coordinates Figure 3.4 Diagram of two-dimensional irregular node Figure 3.5 Plot of the functions for the open contour test problem (3.27) Figure 3.6 Plot of the functions for the closed contour test problem (3.28).. 47 Figure 3.7 Convergence of the LTE and global error for the open and closed contour examples (3.27) and (3.28) using the full IIM discretization (3.25) Figure 4.1 Convergence of the LTE and global error for the open and closed contour examples using the naive discretization (4.1) Figure 4.2 Convergence of LTE and global error for the open and closed contour examples using the IIM discretization (4.4) and the naive discretization (4.5) Figure 4.3 Convergence of the global error terms for the naive discretization (4.1) in (4.9) for solving (3.28) Figure 4.4 Convergence of the global error terms for the naive discretization (4.5) in (4.10) for solving (3.28) Figure 4.5 Contour plots demonstrating the potential functional convergence of the principal error in (4.9) Figure 4.6 Convergence plots for the functional convergence of the normalized principal error Figure 4.7 Convergence of the first 20 eigenbasis coefficients of C and C Figure 4.8 Plot of the eigenbasis coefficients of C and C as a function of wave number Figure 4.9 Plot of the normalized energy for both C and C viii

9 Figure 4.10 Plot of the eigenbasis coefficients of ( 2 h) 1 C and ( 2 h ) 1 C as a function of wave number ix

10 Chapter 1 Introduction Clouds play a major role in the everyday lives of most people whether you realize it or not. They contribute significantly to the global weather patterns, climate, and water cycle [3]. In addition to this, more locally, the presence of low clouds or fog in coastal cities and communities has a significant impact on visibility conditions and not to mention the rain and snow they can carry. Before we can define precisely what a cloud is however, we must discuss the contents of the atmosphere. The atmosphere is primarily composed of a mixture of dry air, water vapour, and liquid water droplets [1]. There are various other components such as solid ice particles and pollutants, but for our purposes we will not concern ourselves with this. When a parcel of the atmosphere contains both dry air and water vapour we describe this as being moist unsaturated air. Once the amount of water vapour, in a parcel, reaches a certain point, called the saturation point, any excess water vapour condenses to form liquid water droplets. This air is then referred to as being moist saturated. In the physical sciences we then define a cloud to be an aggregate of liquid water suspended in a mixture of dry air and water vapour [11]. A cloud, or cloudy air, can then be characterized as a region of moist saturated air typically surrounded by moist unsaturated, or clear, air. As you may already be aware there are many different types of clouds. Without getting too technical, clouds are categorized in stages the first of which we call an étage corresponding to the typical height of the base of the cloud [3]. The étages are named low, middle, and high with typical heights varying with latitude. Clouds are then further categorized into ten different genera such as cumulus, stratus, and cirrus to name a few [3]. This classification of clouds is entirely based on visual properties and does not always correlate well to classifying clouds based on mechanical or physical properties. For example, the physical processes considered in this work have been observed in stratocumulus, altocumulus, and cirrocumulus clouds. Stratocumulus clouds fall into the low étage with a typical base height below 2 km at all latitudes. Altocumulus clouds lie in the middle étage with a base height of 2-7 km in temperate regions and cirrocumulus clouds are in the high étage with a base height of 5-13 km. In this context it is more relevant to focus on the physical properties we discuss later 1

11 keeping in mind that these physical properties can apply to different types of clouds such as the ones just listed. A common physical characteristic of the clouds considered in this work is that they live in what we like to call the marginal cloud limit. The term marginal cloud limit refers to the cloud itself being marginal or near zero liquid water content. A sample altocumulus cloud of the type considered here can be seen in figure 1.1. As mentioned earlier, a cloud forms when an air parcel in the atmosphere reaches its carrying capacity, or saturation point, with respect to water vapour [1]. The saturation point of a parcel of air is characterized in terms of the partial pressure of water vapour, or vapour pressure. That is, water droplets exist when the amount of water vapour in clear air would cause the vapour pressure to exceed the saturation vapour pressure the additional water condenses to liquid and thus, produces a cloud. The edge of a cloud is then defined to be the interface where liquid water ceases to exist. There are at least two factors that can affect the location of a cloud edge. Background atmospheric winds can move and deform clouds in a strictly fluid mechanical process. Additionally, phase changes can also induce motion of the cloud edge. Phase changes can be strictly a thermodynamic process or a combination of both thermodynamic and fluid mechanical processes. Movement of moist air can induce phase changes through the advection of thermodynamic quantities, leading to changes in pressure, for example. As we will see later, in chapter 2, the theory of moist thermodynamics tells us that the saturation vapour pressure is a function of temperature. Thus, the liquid water content, and by association the edge, of a cloud is dictated by both the vapour pressure and the temperature. Part of the objective of this work is to model the edge dynamics of clouds accounting for both the fluid mechanical and thermodynamic processes. As we will see later, in chapter 2, the theory of moist thermodynamics can be linked to the theory of fluid mechanics through a coupling of the variables total density and total pressure in both clear and cloudy air. The atmosphere, on the scale of cloud dynamics, can be modeled as an incompressible inviscid fluid and can therefore be modeled through the incompressible Euler equations [3]. The state of a thermodynamic system can be completely specified by a set of state variables. State variables are a set of thermodynamic quantities which have the ability to completely specify all other state dependent thermodynamic quantities. A particularly convenient set of state variables for our model of cloud edge dynamics turn out to be the entropy of the system, pressure, and total amount of water. We will see in chapter 2 that their temporal dependence is easily characterized. The entropy of the system is an abstract thermodynamic variable and is defined as the first integral of the first law of thermodynamics. As long as we have our state variables specified this specifies the thermodynamic variable density which links directly to the Euler equations. In addition, since all other thermodynamic variables are determined the amount of liquid water, and thus the cloud edge, is also determined. Modeling the thermodynamic effects is then reduced to keeping track of how thermodynamic state variables change in time. To model 2

12 the thermodynamic effects of both clear and cloudy air we will treat them as two separate systems as the thermodynamic theory for each is slightly different. This type of cloud edge modeling has been less explored by atmospheric scientists as most research has been focused on single isolated cumulus clouds where it is possible to model the cloud as primarily independent of its surroundings [13]. A large part of the reason the priority of past cloud modeling has been cumulus clouds is due to their convective instability and their contributions to global weather patterns and precipitation in the form of cumulonimbus clouds. After all, a large portion of cloud modeling is done with the intent of accurate weather forecasting. The processes we consider on the other hand contribute nothing in terms of precipitation and are convectively stable. Furthermore, the focus of cloud modeling is almost never that of resolving edge dynamics. The typical computational scale for modeling cumulus clouds and weather forecasting is on the order of kilometers which is far too coarse for the resolution of edge dynamics. Recently however, the work of Muraki and Rotunno [9] and Muraki et al. [10] has shed some light on the modeling of edge dynamics in convectively stable clouds. For both works the authors focused on the marginal cloud limit, mentioned earlier, which allowed them to use a nearly linear model. The thermodynamic and fluid mechanical changes are then strongly coupled and small thermodynamic changes can affect the presence of liquid water i.e. the cloud edge. The only nonlinearity present in their model was that of the density or buoyancy in their notation. The model contained a different characterization of the density based on whether the air was clear or cloudy due to the varying thermodynamic effects of each system. In the first paper [9] Muraki and Rotunno give a compelling explanation for a long unexplained phenomena observed in the full physics model. That is, simulations of a marginal cloud flow over an idealized two-dimensional mountain ridge gives rise to an upwind traveling front clearing the saturated air i.e. cloudy air transitioning to clear air. Through their nearly linear model, taking into account both fluid dynamic effects and moist thermodynamics, they were able to determine that the key factor in the motion of the cloud edge is the density variation mentioned earlier. They were then able to characterize, by a Rankine-Hugoniot condition, the speed of the edge motion thwarting previous attempts of explaining this phenomena. In a second paper Muraki et al. [10] used a similar approach to additionally explain the phenomena of a holepunch or fallstreak cloud. A holepunch cloud, depicted in figure 1.1, is a near circular region of clear air contained within a marginal thin cloud layer. The formation of such a hole is initiated by a disturbance in the cloud layer and is known to occur when ascending or descending aircraft penetrate a thin cloud layer. An elongated version of the holepunch cloud, called a canal cloud, is formed when an aircraft flies laterally through a weakly saturated, or marginal, thin cloud layer. Once the process has been initiated the hole, or canal, can expand as large as several kilometers in radius/width over a time scale of hours [10]. This phenomena has been known to scientists for many years with some 3

13 Figure 1.1: Ground view photograph of a holepunch cloud over Simon Fraser University taken atop Burnaby Mountain in Burnaby, British Columbia. Photo credit: Dr. Catherine Black of Simon Fraser University published observations occurring as early as the 1940 s [12] but yet only recently has anyone managed to give a theoretical explanation for the time scale upon which the process occurs [10]. The ability to model the motion of a cloud edge through a nearly linear model is somewhat remarkable and begs the question: What other cloud structures can be modeled and understood through this kind of nearly linear model? Building off the work in [9, 10] the objective of this work is to develop a general numerical framework for modeling the edge dynamics of convectively stable, marginally saturated, clouds with no precipitation. A computational cloud lab if you will. We will work with a free boundary model where the nonlinearity is induced by the free boundary. This will allow for exploration into other cloud structures with edge dynamics dominated by fluid mechanical and thermodynamic processes in the marginal cloud limit. As we mentioned previously the thermodynamic effects of clear and cloudy air are slightly different. The net effect of this on our model then is an induced derivative discontinuity in the density field. A result mirrored in both [9, 10]. Due to this irregularity special care must be taken when developing a numerical framework for our model. To handle the derivative discontinuities we employ the immersed interface method (IIM) [8] developed by Zhilin Li through the 1990 s. The IIM is a finite difference method designed to handle discontinuous sources and terms as well as singular sources while maintaining second order global accuracy as measured in the -norm. This method is particularly convenient in that it often results in only a slight correction to our standard finite difference methods to correct for discontinuities. 4

14 Currently, the primary use of the IIM in our numerical solution is in solving for the pressure. As we will see, when we present the model in chapter 2, determining the initial pressure is a challenge. To overcome this challenge we can derive a Poisson equation for the pressure term which has a sourcing term that depends on the vertical derivative of density i.e. a source containing a jump discontinuity. The IIM tells us precisely how to solve this problem when the sourcing term is known exactly. However, for our model the known quantity is density and we must therefore additionally approximate the derivative of the density i.e. we must additionally approximate the source. In chapter 3 we discuss the IIM as well as extending the IIM to additionally approximating the discontinuous sourcing term while maintaining second order global accuracy in the -norm. In addition to extending the IIM we also have the difficulty that the location of the interface, or our free boundary, and the Poisson pressure solve are coupled. That is, in order to determine the pressure we must know the location of the free boundary but to determine the location of the free boundary we require the pressure. The procedure we propose to overcome this coupling involves taking an initial guess for either the pressure or the interface and then iterating between determining the pressure and the interface until convergence is observed. While extending the IIM we discovered that the corrections suggested by the IIM methodology are not necessary to resolve the solution to second order accuracy, again in the -norm. This is a very curious result because, as we will see, without the IIM modifications the local error of the method is order one i.e. the method has an order one consistency error. Nevertheless, we consistently observe second order global accuracy in the solutions. We thus propose this method as a second discretization and the scope of this thesis is then limited by our ability to explain the observed order of accuracy. From then on we will focus on understanding why this additional method also generates second order global accuracy. To gain an understanding of why the global error is second order accurate ignoring the IIM corrections we will first compare to the analogous method when the exact source term is known. From this we will see that we can only ignore the corrections in the extended IIM and not when the source term is known exactly. We will then proceed to characterize the global error and what we will discover is that when ignoring the IIM corrections the global error has a component that depends directly on the ignored corrections. We then proceed to work with this error characterization gaining the most insight through a discrete Fourier analysis. In the end we will not be able to fully explain why ignoring the corrections generates a second order globally accurate solution but we will better understand how the order one local error can give rise to second order global accuracy. The remainder of this thesis will proceed as follows: In Chapter 2 we begin with a complete mathematical presentation of the full moist thermodynamic model for both the clear and cloudy air. We then proceed to derive the disturbance equations about conditions that are hydrostatic with zero liquid cloud water. Using the disturbance equations we can then produce a nearly linear model similar to [9, 10]. In addition, we provide a brief summary 5

15 of the model equations at the end of chapter 2 before deriving the Poisson solve for pressure. In Chapter 3 we will introduce the ideas of the IIM and demonstrate its performance for a one dimensional model problem. In addition we introduce the two dimensional IIM for our pressure solve as well as extend the IIM to approximating the source term. We then demonstrate the extended IIM s performance on two model problems we have developed. In Chapter 4 we introduce an additional discretization for our pressure solve, ignoring the IIM corrections, and investigate the global error of this discretization. Finally, in Chapter 5 we provide some concluding remarks as well as describe our plans for future work. 6

16 Chapter 2 Mathematical Model In this chapter we present a mathematical model for studying cloud edge motion. Physically, what distinguishes a cloud from the rest of the atmosphere is the presence of liquid water. That is, the fluffy white clouds we are all so accustomed to seeing are nothing more than suspended liquid water droplets. The more dense the cloud is the more liquid water is present. To be more specific, a cloudy region is characterized by a mixture of dry air, water vapour, and liquid water also referred to as moist saturated air. In the immediate area surrounding the cloud there is a mixture of dry air and water vapour also referred to as moist unsaturated air. Our mathematical model will use the incompressible Euler equations to describe the background atmospheric flow. We will link these to the theory of moist thermodynamics coupled through a constitutive law relating the total density and total pressure of the system. As we will see, the thermodynamic equations lead to a free boundary problem whereby on one side we have moist unsaturated air (clear) and on the other we have moist saturated air (cloudy). The free boundary separating clear and cloudy air, characterized by the presence of liquid water, will be referred to as an interface to distinguish this from the actual boundary of the problem domain. Our primary interest here is understanding the dynamics of convectively stable cloud layers in the marginal cloud limit. By convectively stable, as mentioned in the introduction chapter, we mean that we are assuming convective motion within the cloud does not generate unstable vertical motion, a typical characteristic of certain clouds. Furthermore, we will ignore convective dynamics entirely. For our model equations, we begin with presenting a very general model for the interaction of fluid motion and moist thermodynamics and specialize this to the marginal cloud limit. This is done by deriving a set of linearized disturbance equations about an atmosphere that is hydrostatic and critically saturated with zero liquid cloud water. We then discuss a model for determining the pressure in our system leading us into the numerical methodology for computing. 7

17 2.1 The Model Equations Before we jump into the model equations we should clarify that the geometry considered is that demonstrated in figure 2.1. Essentially, the two dimensional domain for our problem is split into two regions, unsaturated (clear) and saturated (cloudy), via the interface Γ. The interface is not necessarily a closed curve as depicted in figure 2.1. The only requirement of the interface is that it divides the domain into two sub-regions. Unsaturated z Saturated Γ Figure 2.1: Diagram demonstrating the cloud geometry of our model. The domain is split into two regions, unsaturated (clear) and saturated (cloudy), via the interface Γ. The interface Γ need not be closed but merely splits the domain into two regions. The horizontal coordinate is labeled x and the vertical labeled z. x As we mentioned in the introduction our starting point for the model equations is the incompressible Euler equations used to describe the fluid dynamical aspect of our model D u Dt = 1 ρ P gẑ (2.1) u = 0. (2.2) In the above Euler equations we have a system of four unknowns, the horizontal and vertical velocities u and w, the total pressure P, and the total density ρ. Our objective now is to link these to the theory of moist thermodynamics through a constitutive law to produce a closed system and obtain a model for the amount of liquid water in the atmosphere. As we will see, the key to obtaining a constitutive law linking the above equations to the theory of moist thermodynamics is the thermodynamic state variables entropy, total pressure, and total amount of water. The presence of liquid in a thermodynamic system causes a slight change in the governing physics and, as we will see, generates two separate constitutive laws: one for clear air and one for cloudy air. We have thus broken the derivation of the constitutive laws up into subsections for the moist unsaturated flow (clear) and for the moist saturated flow (cloudy). Once we have derived the two necessary constitutive laws it will 8

18 be clear that we require two additional differential equations to form a closed system. We will present these equations in the following section once the constitutive laws have been derived Constitutive Law For Moist Unsaturated Flow We begin our discussion of moist thermodynamics by introducing the concept of entropy. As a starting point we have the first law of thermodynamics which states that the rate of change of internal energy in a system U is the sum of the rate of heating Q and the rate of work done on the system W [1] du dt = dq dt + dw dt. We will model both dry air and water vapour as a composite system of ideal gases allowing us to rewrite the above first law as dq dt = C dt v dt + P dv dt where dw dt = P dv dt and du dt = C v dt dt. We also have that C v is the heat capacity at constant volume for the gas, T is the temperature, P is the pressure, and V is the volume [1]. For dry air and water vapour, more generally a simple ideal gas, the heat capacities at constant volume and pressure are constant [4]. The heat capacities at constant pressure will come up in a moment. We can apply the ideal gas law P = ρr s T, where R s represents the specific gas constant, to the above expression and divide by the temperature T and total mass to get ( ) ds dt = 1 dq T dt = c v dt T dt + ρr d 1 ρ s. (2.3) dt In the above expression q is the mass normalized heat Q, c v is the specific heat capacity or mass normalized heat capacity at constant volume, and s is the specific entropy. Looking carefully at equation (2.3) it is a perfect derivative with respect to time. That is, for an ideal gas the first law of thermodynamics has a first integral with the reciprocal of temperature as an integrating factor. The specific entropy s is then defined as the first integral of the first law of thermodynamics normalized with respect to mass with integrating factor 1 T. It can be shown in general that the reciprocal of temperature is an integrating factor for the first law and that a first integral always exists with the added knowledge that general heat capacities are functions of temperature alone [1]. Integrating equation (2.3) in time and 9

19 applying the ideal gas law we obtain ( ) ( ) T P s(t, P ) s 0 = c p log R s log T0 P0 (2.4) where c p is the specific heat capacity at constant pressure. The quantities s 0, T 0, and P 0 represent reference states from the integration of (2.3). The quantity s is defined as the specific entropy or the mass normalized entropy not to be confused with the non normalized entropy S found in most textbooks. We have just showed that for a single ideal gas the entropy of the system is given by (2.4). However, moist unsaturated air is a composite system containing both dry air and water vapour. In order to calculate the total entropy of a composite system we can make use of Gibbs theorem. Gibbs theorem states that the total entropy of a composite system is the sum of the component, or partial, entropies defined as the entropies of each component as if they occupied the volume of space alone at the same temperature T [15]. Thus, we can express the total entropy of the moist unsaturated air as { ( ) s T = log R d log T0 ( Pd P 0 ) } { ( ) (cp ) + r v T v log R ( ) } v e log (2.5) T 0 e 0 where r v = ρv ρ d and the specific entropy has been normalized with respect to the mass of dry air only. The constants (c p ) v and represent the specific heat capacities at constant pressure for both water vapour and dry air respectively. The constants R v and R d are the specific gas constants for each along with e and P d representing their partial pressures. The values of T 0, P 0, and e 0 are somewhat arbitrary as they define a reference state with corresponding entropy s 0. We have also absorbed s 0 into s for simplicity so that we avoid carrying along the constant s 0. In developing our expression for the entropy we introduced five new variables s, r v, T, P d, and e in addition to our original u, ρ, and P. We thus require five more equations to close the system. Before we go on to presenting more equations however we should point out that P, s, and r v form a complete set of state variables. By this we mean all other thermodynamic state functions such as ρ, T, P d, and e are uniquely determined once P, s, and r v are specified. What enables P, s and r v to define the state of a system is the existence of the ideal gas law. Both the dry air and water vapour have their own respective ideal gas laws: P d = ρ d R d T e = ρ v R v T. We can add these equations to eliminate the partial pressures and partial densities. In order to get the total pressure we use Dalton s law of partial pressures which states that the total 10

20 pressure in a composite system is the sum of the partial pressures [1] Now, adding the two ideal gas laws we obtain P = P d + e. (2.6) ( ) Rd + r v R v P = ρ T. (2.7) 1 + r v We were able to eliminate the partial densities using the fact that the total density ρ is the sum of the two partial densities, ρ = ρ d + ρ v, as well as the definition of r v. We can also divide the two gas laws to obtain P d r v R v = er d. (2.8) Equations (2.5)-(2.8) thus form a system of four equations containing all seven thermodynamic quantities P, s, r v, P d, e, T, and ρ. The existence of these four equations validates the statement of P, s and r v being a complete set of state variables as given these quantities we can use (2.5)-(2.8) to solve for the remaining four. In particular, these four equations allow us to write the total density ρ as the required constitutive law. ρ = ρ clr (P, s, r v ) (2.9) Combining equation (2.9) with the incompressible Euler equations (2.1) and (2.2) will produce a closed system provided we have differential equations for the time-development of s and r v. We will present the derivation of these equations in section Constitutive Law For Moist Saturated Flow In the previous section we demonstrated the constitutive law for the moist unsaturated, or clear, region of our model i.e. a mixture of dry air and water vapour. In this section we will present the analogous constitutive law for the moist saturated, or cloudy, region of our model, i.e. a mixture of dry air, water vapour, and liquid water. We have already discussed the concept of entropy and have shown it as the first integral of the first law of thermodynamics with integrating factor 1 T. In addition to this we have obtained an expression for the specific entropy of an ideal gas as well as stated that the total entropy of a system is the sum of all the component or partial entropies. In order to obtain an expression for the entropy of the moist saturated air we need only obtain an expression for the entropy of the liquid water and add this to the expression for entropy in the unsaturated case. To obtain an expression for the entropy of liquid water consider a small parcel of saturated air. The density of liquid water in the parcel is defined to be the total mass of 11

21 liquid water divided by the total volume of the parcel. The total mixing ratio of the parcel is then defined as r T = r v + r l (2.10) where r v is defined as before and r l = ρ l ρ d. We also have that the total density of the parcel is now given by ρ = ρ d + ρ v + ρ l. To obtain an expression for the entropy of liquid water we must go back to our definition of entropy ds dt = 1 dq T dt = 1 T ( du dt + dw dt ). (2.11) The liquid water is assumed to have zero volume and thus there is no work to speak of giving dw dt = 0. The change in internal energy for a constant volume process can be expressed as du dt = C dt l dt where C l is the heat capacity of the liquid water at constant volume [1]. Substituting this expression into (2.11) we obtain the following expression for the specific entropy of liquid water ( ) T s s 0 = c l log. T0 Once again, invoking Gibbs theorem for partial entropies we obtain ( ) ( ) s (c p ) = 1 + r v c l T v + r l log R ( ) ( ) d Pd R v es (T ) log r v log T0 P 0 e s (T 0 ) (2.12) where s has been normalized with respect to the mass of dry air only. Notice that in the above expression for entropy we changed the pressure of water vapour e to e s (T ). The reason for this is that for water vapour and liquid water to coexist stably we must be in a state where the evaporation rate E equals the condensation rate C. What we mean by stability in this context is keeping all state variables constant there should be no net evaporation in time. The condensation rate in the system is a function of the amount of water vapour in the system [1] which we can characterize as C = C(ρ v ). The evaporation rate E is a function of the temperature [1] and we thus have C(ρ ve (T )) = E(T ) (2.13) as a condition for the stable coexistence of vapour and liquid. The variable ρ ve is a special density called the equilibrium density of water vapour and is implicitly a function of 12

22 temperature due to the evaporation rate being a function of temperature. What expression (2.13) says is that when liquid water and vapour coexist the density of water vapour is a fixed function of temperature. This can then be substituted into the ideal gas law for water vapour to obtain e s = ρ ve R v T indicating that the saturation vapour pressure is a function of temperature alone. explicit expression for the saturation vapour pressure can be obtained from the Clausius Clapeyron equation for which we will omit a discussion. For a more detailed discussion of the saturation vapour pressure and the Clausius Clapeyron equation we refer to [1, 15]. The net result of the Clausius Clapeyron equation is ( ) ( lv e s (T ) = e s0 exp exp l ) v R v T 0 R v T where e s0 is the saturation vapour pressure of some reference state and l v is the latent heat of vaporization for water. In the unsaturated case we used P, s and r v as our state variables. For the saturated case however using r v as a state variable is inconvenient for reasons that will be clear in section In the description of the moist saturated region two new variables were introduced compared to the unsaturated region, r T and r l, and the vapour pressure e was removed. Thus, we now have eight thermodynamic variables in total P, s, r T, ρ, T, P d, r v, and r l. Therefore, in addition to equations (2.10) and (2.12) three more equations are required to specify our constitutive law knowing that we will define differential equations for s and r T in section To determine the remaining variables both Dalton s law of partial pressures and the ideal gas laws for vapour and dry air can be used as in the unsaturated case. Taking care in how the total density is defined for the saturated region we have our additional three equations as An P = P d + e s (T ) (2.14) ( ) Rd + r v R v P = ρ T (2.15) 1 + r T P d r v R v = e s (T )R d. (2.16) We thus have equations (2.10), (2.12) and (2.14)-(2.16) allowing us to write ρ = ρ cld (P, s, r T ) (2.17) 13

23 and therefore defining our constitutive law for the cloudy region. Combining (2.17) with the incompressible Euler equations will produce a closed system provided we have differential equations for s and r T to be presented in the next section. In addition to writing the constitutive law we are now prepared to differentiate between clear and cloudy air in our model. constitutive law to use we can additionally write To secern clear from cloudy air and know which r l = r l (P, s, r T ) using the same reasoning as the constitutive law. For r l > 0 we have cloud and thus use the cloudy constitutive law and otherwise for clear we have r l = 0 and use the clear constitutive law. This gives rise to a free boundary model where the cloud edge is tracked by stepping P, s, and r T forward in time and evaluating r l. In the next section we will present the remaining equations for s and r T model. followed by a summary of the complete free boundary Time Dependence Of Entropy And Total Water Content Up to now we have successfully specified the two constitutive laws required for clear and cloudy air (2.9) and (2.17) respectively. We have also established that (2.1) and (2.2) describe the fluid mechanics for which the velocity and pressure time dependence can be obtained. Finally, we require two equations to tell us how our remaining two thermodynamic state variables develop in time. In order to demonstrate how the entropy variable for both clear and cloudy air develops in time we go back to our definition of entropy. The entropy of the system is defined to be the first integral of the first law of thermodynamics with integrating factor 1 T. This allows us to express the time derivative of the entropy as ds dt = 1 dq T dt. For our model however we will be assuming that all processes are adiabatic. That is, there are no sources/sinks of heat present in our system implying that dq dt = 0. We thus have that for a fixed parcel of moist saturated air that the entropy remains constant. If we now embed the parcel in a dynamic atmosphere allowing the parcel to move and deform as necessary we have that the entropy within that parcel must remain constant or the advective derivative is zero Ds Dt = 0. (2.18) Finally, we only require an equation for how the total amount of water changes in time because in the clear region we have r v = r T due to the clear region being characterized by r l = 0. Our model assumes no precipitation is present giving that the total amount of water 14

24 contained within a parcel for both clear and cloudy air must remain constant giving Dr T Dt = 0. (2.19) A this point we have completely specified the complete model linking the incompressible Euler equations to the theory of moist thermodynamics. In the next section we will present a summary of the complete model. Our next step will be to further specialize this complete model to the marginal cloud limit The Complete Model As we saw in sections and the key to linking the incompressible Euler equations to the theory of moist thermodynamics was the constitutive laws. In deriving the constitutive laws a number of intermediate variables were introduced allowing us to express the total density in the system as a function of the state variables P, s, and r T. We then presented equations for the time dependence of s and r T giving the complete closed system D u Dt = 1 ρ P gẑ u = 0 (2.20a) (2.20b) Ds Dt = 0 (2.20c) Dr T = 0 (2.20d) Dt ρ clr (P, s, r T ), r l (P, s, r T ) = 0 (clear) ρ = (2.20e) ρ cld (P, s, r T ), r l (P, s, r T ) > 0 (cloudy). For the clear region we have the constitutive law ρ clr (P, s, r T ) given by the solution to the following system in terms of the state variables { ( ) s T = log R d log T0 ( Pd P 0 ) } { ( ) (cp ) + r v T v log R ( ) } v e log (2.21a) T 0 e 0 P = P d + e ( ) Rd + r v R v P = ρ T 1 + r v P d r v R v = er d r v = r T. (2.21b) (2.21c) (2.21d) (2.21e) These equations are initially presented in section with labels (2.5)-(2.8) the exception being equation (2.21e) which is discussed in section For the cloudy air we have a second constitutive law ρ cld (P, s, r T ) due to the presence of liquid water. In addition to the 15

25 constitutive law we also have the amount of liquid water as a function of the state variables r l = (P, s, r T ). Both the constitutive law and the amount of liquid water are given by the solution to the analogous system for cloudy air ( ) ( ) s (c p ) = 1 + r v c l T v + r l log R ( ) ( ) d Pd R v es (T ) log r v log T0 P 0 P 0 (2.22a) P = P d + e s (T ) ( ) Rd + r v R v P = ρ T 1 + r T P d r v R v = e s (T )R d r T = r v + r l. (2.22b) (2.22c) (2.22d) (2.22e) These equations initially appear in section with the corresponding labels (2.12), (2.14)- (2.16), and (2.10) respectively. In the next section we will further specialize the model summarized here to the marginal cloud limit. 2.2 The Marginal Cloud Limit In this section we further specialize the model summarized in the previous section to the marginal cloud limit. To do this we will derive a set of disturbance equations about a background atmosphere that is hydrostatic and critically saturated with zero liquid cloud water. We begin with deriving a system for our leading order background atmosphere based on the assumptions just stated. We then use this system along with (2.20), (2.21), and (2.22) to derive equations for our disturbance quantities Background Atmosphere For The Marginal Cloud Limit The fundamental assumptions for the background atmosphere are that pressure, density, entropy, and the total mixing ratio are all functions of the vertical z-coordinate alone (P = P (z), ρ = ρ(z), s = s(z), and r T = r T (z)). Additionally it is assumed that velocity is constant in time horizontally and zero vertically ( u = (ū, 0)). The assumptions for the marginal cloud limit are that we are everywhere critically saturated with zero liquid cloud water. These assumptions can be substituted into the model equations (2.20) and (2.21) giving a set of equations defining our background atmosphere. From the vertical component of (2.20a) we obtain the relation 1 ρ P z = g (2.23) 16

26 known as hydrostatic balance. The horizontal component of (2.20a) along with (2.20b)- (2.20d) are all trivially satisfied by the assumptions. For the thermodynamic quantities we use system (2.21) putting us in the zero liquid cloud water regime. This gives ( ) ( ) s (c p ) = 1 + r v T v log R ( ) ( ) d Pd R v ē log r v log (2.24) T 0 P 0 P 0 ( ) P = ρ T Rd + r v R v (2.25) 1 + r v P d r v R v = ēr d (2.26) P = P d + ē (2.27) r T = r v (2.28) where ē is defined as e s ( T ) making us everywhere critically saturated. Equations (2.23)- (2.28) therefore describe the background atmosphere Nonlinear Equations For The Marginal Cloud Limit Now in order to derive the governing equations in the marginal cloud limit we look at small disturbances, or perturbations, from the background atmosphere. We begin by considering additive disturbances in u = ū + u, P = P + P, s = s + s, r v = r v + r v, r T = r T + r T, and r l = r l along with relative disturbances in ρ = ρ(1 + ρ), T = T (1 + T ), P d = P d (1 + P d ), and e = ē(1 + ẽ). Note that we have chosen to express the relative disturbances with a rather than a this was done so that it is clear that the relative disturbances are dimensionless. This notation is consistent with our notation for nondimensionalization as we will see later. Each of these can now be substituted the above complete model (2.20), (2.21), and (2.22). The remainder of this subsection is broken up into three parts for clarity. We will first derive the disturbance equations for the governing PDEs then the unsaturated constitutive law relations followed by the saturated constitutive law relations. Nonlinear Differential Equations For The Marginal Cloud Limit Substituting into the horizontal portion of equation (2.20a) we have where the on the D Dt u [ t + (ū + u ) (ū + u ) x D u Dt + ū u x = 1 ρ(1 + ρ) + w (ū + ] u ) 1 = z ρ(1 + ρ) P x ( P + P ) x indicates advection with respect to perturbed velocity D Dt = t + ( u ). 17

27 As for the vertical portion of (2.20a) we have w [ t + (ū + u ) w x D w Dt + ū w x = 1 ρ(1 + ρ) ] w 1 ( + w = P + P ) g z ρ(1 + ρ) z P ( ) ρ z g 1 + ρ where (2.23) has been used to remove the leading order P dependence. The horizontal and vertical components can be combined back into a single vector equation to obtain Substituting into equation (2.20b) we have D u ( ) Dt + ū u x = 1 P ρ(1 + ρ) ρ g ẑ. (2.29) 1 + ρ Once again, substituting into equation (2.20c) we have ū + u + w x z = 0 u = 0. (2.30) s + s ] + [(ū + u s + s s + s ) + w = 0 t x z D s Dt + ū s x + s z w = 0. (2.31) Precisely the same result can be obtained for (2.20d) D r T Dt + ū r T x + r T z w = 0. (2.32) Equations (2.29)-(2.32) form the nonlinear governing differential equations for the marginal cloud limit. Our next step will be to linearize these once we have derived the constitutive law relations for the marginal cloud limit. Moist Unsaturated Nonlinear Constitutive Law For The Marginal Cloud Limit For the moist unsaturated constitutive law relations (2.21) we can substitute the disturbances and cancel the leading order background dependence using (2.24) (2.27). Doing this 18

28 we obtain s ( ) ( = r (c p ) v T v log ( r v + r T v) (c ) p) v 0 ( r v R v ē log P = ρ(1 + ρ) P 0 ( Rd + ( r v + r v)r ) v 1 + ( r v + r v) ( log 1 + T ) ) ( r v + r v) R v log (1 + ẽ) ( ) T (1 + T Rd + r v R v ) ρ T 1 + r v R d log (1 + (c p ) P ) d d (2.33a) (2.33b) P d ( r v + P d ( rv + r v) ) R v = ēẽr d (2.33c) P = P d Pd + ēẽ r T = r v. (2.33d) (2.33e) Equations (2.33a) (2.33e) are therefore the fully nonlinear constitutive law relations for the unsaturated region. This, once again, allows us to write ρ = ρ clr (P, s, r T ). Moist Saturated Nonlinear Constitutive Law For The Marginal Cloud Limit For the saturated constitutive law relations (2.22) we again substitute our disturbances and cancel the leading order background to yield s ( ) ( ) ( = r v (c p ) v + r c l T l log ( r v + r T v) (c ) p) v + r c l l 0 R d log (1 + (c p ) P ) d ( r v + r v) R ( v es ( log T (1 + T ) )) d P 0 ( log 1 + T ) + r v R v log (2.34a) ( P Rd + ( r v + r ) ( ) = ρ(1 + ρ) v)r v 1 + ( r T + r T ) T (1 + T Rd + r v R v ) ρ T (2.34b) 1 + r T ( P d r v + P d ( rv + r v) ) = (e s ( T (1 + T )) ē)r d (2.34c) ( ē P 0 ) P = P d Pd + e s ( T (1 + T )) ē r T = r v + r l. (2.34d) (2.34e) We thus have system (2.34) as the fully nonlinear saturated constitutive law relations. In the next subsection we will linearize the PDEs (2.29) (2.32) and both sets of constitutive law relations (2.33) and (2.34) Linear Equations For The Marginal Cloud Limit In this section we present a linearization of the nonlinear equations for the marginal cloud limit presented in the previous subsection. Once again we have broken this section up 19

29 into three parts for clarity: the PDEs, the unsaturated constitutive law, and the saturated constitutive law. Linearized Differential Equations For The Marginal Cloud Limit Consider first (2.29) where u ( ) t + ( u ) u + ū u x = 1 P ρ(1 + ρ) ρ g ẑ. 1 + ρ Ignoring the inherently nonlinear advection term and noting that for small ρ we obtain ρ 1 ρ u t + ū u x = 1 ρ P g ρẑ (2.35) the linearized version of (2.29). No work is required for (2.30) as this equation is already linear. In order to linearize equations (2.31) and (2.32) the nonlinear advection term can be dropped to obtain s t + ū s x + s z w = 0 (2.36) r T t + ū r T x + r T z w = 0. (2.37) Equations (2.35) (2.37) as well as (2.30) form the linearized differential equations for the marginal cloud limit. Next we will consider both sets of constitutive law relations. Moist Unsaturated Linearized Constitutive Law For The Marginal Cloud Limit The nonlinear constitutive law equation (2.33a) can be linearized by noting that log(1 + x) x for small x and dropping the r v T and r vẽ terms to obtain s ( ) ( ) = r v (c p ) v T (c p ) log r v v T R d Pd T 0 ( ) r v R v ē R v log r v ẽ. P 0 (2.38) 20

30 Linearizing (2.33b) will require a bit more effort however, using ( r v + r v) 1 ( ) 1 r v 1 + r v 1 + r v and dropping any remaining nonlinear terms we can obtain P = ρ T 1 + r v [ ( r vr v + (R d + r v R v ) ρ + T r v 1 + r v )]. (2.39) Finally, to linearize (2.33c) the P d r v term can be dropped to obtain P d ( r v + P d r v ) R v = ēẽr d. (2.40) There is no need to linearize (2.33d) or (2.33e) as these equation are already linear. We thus have equations (2.38) (2.40), (2.33d), and (2.33e) as the linearized constitutive law relations for the unsaturated region. Moist Saturated Linearized Constitutive Law For The Marginal Cloud Limit Looking first at equation (2.34a) the added step in the linearization from the unsaturated case is that ( es ( log T (1 + T ) ( )) es ( log T ) ) P 0 P 0 + e s( T ) e s ( T T T. ) Substituting this into equation (2.34a) and linearizing as before we obtain s ( ) ( ) ( ) = r v (c p ) v + r c l T (c p ) l log r v v T T 0 R ( d Pd r R v es ( v log T ) ) R v e r s( T ) v e s ( T T T. ) P 0 (2.41) Next, equation (2.34b) can be linearized exactly as was done in the unsaturated case to give P = ρ T 1 + r T [ ( )] r vr v + (R d + r v R v ) ρ + T r T. (2.42) 1 + r T To linearize equation (2.34c) the additional approximation e s ( T (1 + T )) e s T + e s ( T ) T T is required. Substituting this into (2.34c) and once again linearizing as before we obtain P d ( r v + P d r v ) R v = e s( T ) T T R d. (2.43) 21

31 Finally, (2.34d) is linearized using the same linearization of e s ( T (1 + T )) to obtain P = P d Pd + e s( T ) T T. (2.44) There is no need to linearize (2.34e) as this equation is already linear. We thus have equations (2.41) (2.44) and (2.34e) as the linearized constitutive law relations for the saturated region. In the following subsection we will nondimensionalize the linear equations for the marginal cloud limit giving us the model we will actually compute with Nondimensionalization Of The Linear Equations For The Marginal Cloud Limit For the nondimensionalization process we introduce the following dimensionless variables dividing by characteristic scales x = x L, z = z H, t = t T, ũ = u w, w = U W ρ = ρ R, P = P P, Pd = P d P ē = ē s, s = P E, s = s E, T T = Θ. Once again we have this section broken up into three separate parts for clarity. Nondimensionalization Of The Linear Differential Equations For The Marginal Cloud Limit The dimensionless variables can be substituted into the above linear differential equations (2.35)-(2.37) and (2.30) to obtain ũ t + UT L ũ ū x = T P 1 P URL ρ x w ū x = T P 1 P WRH ρ z T W g ρ w z = 0 w t + UT L U ũ L x + W H s t + UT s ū L x + WT H r T t + UT L ū r T x + WT H s z w = 0 r T z w = 0. (2.45e) To maintain the divergence free condition within our model we will set U L = W H equation (2.45c) to (2.45a) (2.45b) (2.45c) (2.45d) reducing ũ = 0. 22

32 The time scale will now be set by the horizontal advection requiring that UT L = 1 imposing that the time derivatives in the system are proportional to the background advection. This reduces equations (2.45d) and (2.45e) to s t + ū s x + s z w = 0 r T t + ū r T x + r T w = 0. z In addition to this we also maintain the relationship between the time derivative of the horizontal velocity and the pressure gradient by setting the scale for the pressure as P = URL T. Note that the scale of density R is set by the background density ρ which, under a Boussinesq approximation, can be treated as a constant. This reduces equation (2.45a) to ũ t + ū ũ x = 1 ρ P x. T P WRH = L2 As a result of our scaling choice for pressure we now have that. To maintain H 2 our gravitational dynamics we are now forced to have L2 = T H 2 W g setting our scale for W in terms of the ratio of L and H. This reduces equation (2.45b) to H 2 ( w L 2 t + ū w ) = 1 ρ P x z ρ. Now, if we assume the quantities s z and r T z are O(1) and take H2 = 1 then we have a L 2 completely nondimensionalized and scaled the linear differential equations such that they contain only O(1) terms. Note that s z, r T z and H2 need not necessarily be O(1) and we L 2 could still scale, by redefining the time scale, to obtain a comparable system. 23

33 Nondimensionalization Of The Linear Unsaturated Constitutive Law For The Marginal Cloud Limit We can substitute the dimensionless variables into the above linearized constitutive law equations (2.38) (2.40), (2.33d), and (2.33e) to obtain E s ( ) ( ) = r v (c p ) v T (c p ) log r v v T R d Pd T 0 ( ) r v R v ē R v log r v ẽ P 0 P = RΘR d ρ T [ ( ) ( )] r R v R v v r v ρ + P 1 + r v R d R T r v d 1 + r v P d (r v + P ) Rv d r v = ēẽ R d P = Pd Pd + ēẽ r T = r v. (2.46a) (2.46b) (2.46c) (2.46d) (2.46e) We will omit a discussion of the scaling for the constitutive law as it relies heavily on the set up of the background atmosphere as well as the values for each of the thermodynamic constants introduced and is not necessary for the remainder of this thesis. The main idea to take away from this section is that the above linear equations (2.46) allow us to write the density as ρ = c 1 P + c2 s + c 3 r T. The scaling of the above equations will then determine the relative sizes of c 1, c 2, and c 3 depending on the background atmosphere. Nondimensionalization Of The Linear Saturated Constitutive Law For The Marginal Cloud Limit Substituting the dimensionless variables into (2.41)-(2.44) and noting that equation (2.34e) is already dimensionless we obtain E s ( ) ( ) ( ) = r v (c p ) v + r c l T (c p ) l log r v v T T 0 R d Pd r R v (ẽs (Θ T ) ) R v ẽ v log r s(θ T ) v P 0 ẽ s (Θ T T T ) P = RΘR d ρ T [ ( ) ( )] r R v R v v r v ρ + P 1 + r T R d R T r T d 1 + r T P d (r v + P ) Rv d r v = ẽ s (Θ T ) T T R d P = Pd Pd + ẽ s (Θ T ) T T r T = r v + r l. (2.47a) (2.47b) (2.47c) (2.47d) (2.47e) 24

34 As with the unsaturated constitutive law we will omit a discussion of the scaling. The take home message remains the same only in this case we also obtain r l = k 1 P + k2 s + k 3 r T. 2.3 The Marginal Cloud Model In this section we present a brief summary of the contents of sections 2.1 and 2.2. Beginning in section 2.1 we presented a thermodynamic free boundary model for the amount of liquid water present in the atmosphere. The incompressible Euler equations were used for the fluid mechanical aspect and were linked to the theory of moist thermodynamics through a constitutive law relating the total density, pressure, specific entropy and total amount of water. The model is broken up into a two phase flow containing moist unsaturated air (clear) in one region, with the constitutive law defined by system (2.21), and moist saturated air (cloudy) in the other, with the constitutive law defined by system (2.22). Next we derived the equations specialized to the marginal cloud limit linearized about conditions that are hydrostatic with zero liquid cloud water. Then the system was nondimensionalized to arrive at the following model equations: u t + ū u x = 1 ρ P gρẑ u = 0 s t + ū s x + s z w = 0 r T t + ū r T x + r T z w = 0 ρ = c 1 P + c2 s + c 3 r T ρ =, r l(p, s, r T ) = 0 (clear) ρ = d 1 P + d2 s + d 3 r T, r l(p, s, r T ) > 0 (cloudy). (2.48a) (2.48b) (2.48c) (2.48d) (2.48e) where we have dropped both the s and the s for clarity. That is, the above equations are for the disturbance quantities even though we have dropped their indicators. For clarification on how these relate to the original variables of (2.20) please see section 2.2. The unscaled equations defining each constitutive law for the unsaturated and saturated regions are systems (2.46) and (2.47) respectively. 2.4 Pressure Equation In this section we begin thinking about computing solutions to the model summarized in the previous section. It can be seen from the differential equations (2.48) that we require initial conditions for u, s, and r T. In order to completely specify the thermodynamic state we additionally require the pressure which is then implied by our choice of initial conditions as well as the system. To compute solutions to our model we are going to need a way to 25

35 determine the pressure. One way to determine the pressure is by taking the divergence of the velocity equation to reveal 2 P = g ρ ρ z taking ρ as a constant. We thus have a Poisson equation for P. In order to use this method for P we require boundary conditions. With the intention of computing on a square domain on the top and bottom walls we would like to have no flow through and thus use homogeneous Neumann boundary conditions. On the left and right walls we impose periodic boundary conditions. There is one issue we have to deal with for the pressure Poisson solve: the right-hand side of this solve is a z-derivative of the density. Given that we are studying cloud physics the regularity of the right-hand side ρ and of P will be an issue. For the total density ρ it is quite clear from the definition, ρ = ρ d + ρ v in unsaturated air and ρ = ρ d + ρ v + ρ l in saturated air, that it must be continuous. For the z derivative however, recall that the total density is given by ρ = ρ clr in the unsaturated region and ρ = ρ cld in the saturated region. We therefore have that in the unsaturated region and ρ z = ρ clr z ρ z = ρ cld z in the saturated region. Thus, there is going to be a jump discontinuity in the z-derivative of ρ across the clear/cloudy interface. To present a well-posed Poisson solve for P we must also be sure of the regularity of P. Certainly the pressure across the interface of the cloud must be continuous. This however, is not enough for the Poisson solve to be well posed. We additionally require jump information about the normal derivative of P. In turns out that knowing P is smooth on either side of the interface, bounded in the whole domain, and that its Laplacian has only a jump discontinuity is enough to determine that the jump in the normal derivative of P must be zero. We will prove this in the next subsection. For now we can summarize our Poisson pressure solve as 2 P = P (0, y) = P (L x, y) Pˆn = 0; y = 0 Pˆn = 0; y = L y [P ] Γ = [Pˆn ] Γ = 0, dρ clr dz, r l = 0 (clear) dρ cld dz, r l > 0 (cloudy) (2.49) 26

36 where the domain Ω is assumed a square positioned from the origin with side lengths L x and L y. The curve Γ, referred to as the interface, is characterized by the location of the jump discontinuity in the right hand side ρ z and partitions the domain Ω into Ω + Ω and Ω = Ω\Ω + One comment to be made is regarding the ill-posedness of the of the boundary value problem (2.49). Since we are only interested in P the non-uniqueness of the solution P is not a problem. The null space of the above problem is the space of all constants and thus has no affect on P Derivation of the Jump Condition As mentioned in the last section knowing that 2 P = f, that P is smooth on either side of the interface, and bounded in the whole domain is enough to determine that the jump [Pˆn ] Γ = 0. Consider an arbitrary volume V within our domain Ω that contains some portion of the interface Γ. V V 1 V 2 Figure 2.2: Diagram demonstrating the control volume V containing an arbitrary portion of the interface Γ. Γ splits the volume V into two sub-volumes V 1 and V 2 for which our solution P is smooth. Γ We have a diagram demonstrating this construction in figure 2.2. Within this volume we know that P satisfies the following weak formulation given test functions φ C c (V ) V φ P dv = V φfdv. In addition it is also know that P is smooth on either side of the interface so Green s first identity can be applied in each of the sub volumes V 1 and V 2 as depicted in figure 2.2. Applying Green s first identity we obtain the expression φ 2 P dv + φ 2 P dv φ( ˆn)P ds φ( ˆn)P ds = φfdv V 1 V 2 V 1 V 2 V [ φ ( ] ˆn)P ds = 0. (2.50) Γ 27

37 No claim was made about which portion of the interface Γ the volume V contained and we therefore have that (2.50) holds for any portion of the interface Γ. This implies that [ ( ] ˆn)P = 0 everywhere along the interface Γ. This completes the derivation of the jump condition for the normal derivative of P. 28

38 Chapter 3 Numerical Method The numerical strategy employed to solve the previously stated marginal cloud model is the Immersed Interface Method (IIM) developed by Zhilin Li [7, 8]. The method itself is a finite difference method for handling PDEs with discontinuous coefficients and sources as well as singular sources. The method is straightforward in its implementation whereby the standard differencing techniques are used on all nodes away from the interface, referred to as regular nodes, and a corrected stencil is used on nodes near the interface, referred to as irregular nodes. The essential idea with this method is that we can derive the necessary corrections, involving the given jump information along the interface, to restore first order local accuracy to an otherwise inconsistent scheme (O(1) local truncation error). It then turns out that the elliptic inversion is enough to give second order global accuracy, measured in the -norm, even though the local truncation error near the interface is only first order [6]. This method was chosen due to the straightforward implementation of finite difference methods and its ability to achieve second order global accuracy in the -norm. In addition to this the method is also relatively new having been developed through the early to mid 90s for which our marginal cloud model appears to be a good application. The remainder of this thesis will focus primarily on the elliptic pressure solve (2.49). Whilst studying the elliptic pressure solve and the IIM it was observed that we are able to achieve second order global accuracy via a naive discretization of (2.49) provided the right-hand side is approximated appropriately. If the right side source is known exactly then second order accuracy requires an IIM correction to the finite difference stencil. We will be more precise by what we mean by this later. For now, this chapter will focus on developing the IIM for the elliptic pressure solve (2.49). 3.1 The Immersed Interface Method in One Dimension In this section we describe the IIM in a general framework, not the most general problem, to highlight the key aspects of the analysis and to help the reader visualize the method s 29

39 x + x j 1 x j x j+1 Figure 3.1: Diagram demonstrating the interface point Γ = x intersecting the second derivative centered finite difference stencil of both x j and x j+1. x j and x j+1 are thus irregular nodes while x j 1 is a regular node. The + and are a labeling to refer to particular sides of the interface. utility. For a comprehensive description of the IIM in one, two, and three dimensions we refer the reader to [8]. The ideas demonstrated in this section will be ubiquitous throughout the remainder of this thesis. As mentioned in the introduction of this chapter the IIM uses the standard stencil on all regular nodes and a corrected stencil on all irregular nodes. We say a node is irregular if the finite difference stencil used for that node is intersected by the interface, i.e. the interface intersects the axis connecting any two nodes in the stencil. In figure 3.1 we have a diagram demonstrating x j and x j+1 as irregular nodes since their stencil for the centered difference method for the second derivative is intersected by the interface Γ = x. All nodes are classified as being either regular or irregular and we note that in figure 3.1, x j 1 is a regular node. The hallmark of the analysis involved with the IIM is what we will refer to as a displaced point analysis (DPA) used to work around the discontinuities present in our domain. We will first demonstrate the process of deriving a corrected stencil using the DPA in one dimension to establish the general applicability and flavour of the method. Consider the sample problem P xx = f, x (0, 1) A 0 P (0) + B 0 P x (0) = 0 A 1 P (1) + B 1 P x (1) = 0 [P ] Γ = [P x ] Γ = 0. (3.1) where the source f has a jump discontinuity at the interface Γ. Problem (3.1) is meant to be an analogous one-dimensional version of (2.49) where we have stated generic boundary conditions; we will not worry about the derivative source for now. Note that for a onedimensional example the interface Γ is a single point in the domain as indicated in figure 3.1. As mentioned before to discretize this problem we use the standard scheme, center difference in this case, on all regular nodes ( d 2 ) P dx 2 = P i+1 2P i + P i 1 h 2 = f i (3.2) h 30

40 where h represents the spacing between nodes. It is well known that (3.2) is a locally second order accurate discretization for smooth P. This scheme preserves its local second order accuracy, T i = P i+1 2P i + P i 1 h 2 f i = d2 P dx 2 f i + O(h 2 ) xi = O(h 2 ), on all regular nodes as P and f are assumed to be smooth away from the interface. On the irregular nodes however, we do not expect (3.2) to be locally second order accurate as P C 4 (x j, x j+1 ) assuming x j and x j+1 are irregular as in figure 3.1. In fact (3.2) will have O(1) error on irregular nodes. In order to derive a first order scheme at an irregular node we use an undetermined coefficients approach. Let x j be an irregular node where the interface can lie on either side. We begin by assuming a discretization of the form γ j 1 P (x j 1 ) + γ j P (x j ) + γ j+1 P (x j+1 ) = f j + C j. (3.3) As with the derivation of any standard finite difference method we determine the γ k by minimizing the local truncation error T j = γ j 1 P (x j 1 ) + γ j P (x j ) + γ j+1 P (x j+1 ) f j C j. (3.4) Here is where the displaced point analysis comes into play. If we were dealing with smooth functions we could simply expand P (x j 1 ) and P (x j+1 ) about the center node x j. However, because P is only C 1 [x j 1, x j+1 ] the point that lies on the opposite side of the interface as x j cannot be expanded to the desired order at x j. Instead, we can expand each node about the interface point as P is piecewise smooth. We will refer to the point on the interface as the displaced point x. In one dimension there is, of course, only one point however in multiple dimensions we will have to pick a displaced point, x = (x, z ), somewhere on the interface. Note that in figure 3.1 we indicated that the interface is to the right of x j however, in the following derivation we will not make this assumption and the possibility of x being on the left of x j is not excluded. We will make the notational assumption that x j lies on the side of the interface as indicated in figure 3.1. In addition to this we will define a local coordinate system, labeled the ξ coordinate, such that the origin is x. Note that this is not necessary for the one-dimensional case, however in two and three dimensions it will be and the goal here is to present an easily generalized analysis. We can expand the 31

41 function P at each x k about the point x in the local coordinate in the following way P (x k ) = P (ξ k ) = P ± + ξ k P ± ξ + ξ2 k 2 P ± ξξ + O(h3 ) where the ± refers to P being on either the + or side of the interface. The function P appearing with no arguments is assumed to be evaluated at the origin of the local coordinate system. Substituting this into the truncation error (3.4) yields the expression T j = a 1 P + + a 2 P + a 3 P ξ + + a 4Pξ + a 5P ξξ + + a 6Pξξ f j C j + O(h) (3.5) where a 1 = γ k ; a 2 = γ k k K + k K a 3 = γ k ξ k ; a 4 = γ k ξ k k K + k K a 5 = γ ξ2 k k k K + 2 ; a 6 = In the above expressions for the a k the sets are defined as γ ξ2 k k k K 2. (3.6) K ± = {x i used in stencil x i is on the ± side of Γ}. Now, in order to make use of the continuous problem we must also expand f j about the interface point to give f j = f + O(h). This is somewhat unfortunate because we are now fated to be first order on irregular nodes unless we are able to approximate fξ, the next term in the Taylor series, to better approximate f j. Note that a displaced point analysis can be used to analyze finite difference schemes for smooth functions and the fact that we must also expand f j does not affect the order of accuracy you are able to show. So the O(h) error obtained from expanding f j for non-smooth functions is not a fault of the analysis but is inherit in the problem itself. The local O(h) error can also be numerically demonstrated once we have derived the scheme. Thankfully however, as mentioned earlier, being first order locally on the irregular nodes is sufficient to give second order global accuracy measured in the -norm. In determining the γ k s, unlike with standard finite difference methods, it is not sufficient to set the required a k coefficients to zero and one. This is going to result in far too many equations likely with no solution. Instead, by imposing the right relationships between the a k coefficients we can turn terms we do not know, such as P ξξ ±, into terms we do know, 32

42 namely, the jump conditions [P ξξ ]. Observe that by imposing the following a 1 + a 2 = 0 a 3 + a 4 = 0 a 5 + a 6 = 1 (3.7) we can write T j = a 1 [P ] + a 3 [P ξ ] + a 5 [P ξξ ] C j + (P ξξ f ) + O(h). In our case we have that [P ] = 0 and [P ξ ] = 0. We can derive [P ξξ ] from (3.1) by noting that P + xx f + = P xx f [P xx ] = [P ξξ ] = [f]. Therefore, by setting C j = a 5 [P ξξ ] we have that T j = O(h). If it were the case that [P ] 0 and/or [P ξ ] 0 we can simply modify C j to include the other jump terms to obtain the same order of accuracy. We thus have system (3.7) of three equations in three unknowns, recall the a k s depend on the γ k s, which uniquely determines the required coefficients. We can write down system (3.7) explicitly as p h p p + h γ = 0 (p h) 2 p 2 (p+h) where p is x j represented in the local coordinate ξ. A keen eye might notice that the system (3.7) is precisely same system that defines the coefficients for (3.2) and thus we have that γ j±1 = 1 h 2 and γ j = 2 h 2. Our scheme at the irregular node x j is therefore P j+1 2P j + P j 1 h 2 = f j + a 5 [P ξξ ]. (3.8) We have just shown that for a problem of the form (3.1) the standard center differencing scheme can be used with only a slight modification to the sourcing in the linear solve. Note that in the above discretization we require [f] as [P ξξ ] = [f]. For this we can use 33

43 log 10 (jj ~P! Pjj) log 10 (jj ~P! Pjj) log 10 (h) Numerical result x + ( ) (a) Standard discretization log 10 (h) Numerical result x + ( ) (b) IIM discretization Figure 3.2: Convergence plots for the numerical solution of (3.1) measured in the -norm. The data used in this plot was generated allowing the number of mesh nodes to vary from the difference of the gridded f values closest to the interface as this will give us an O(h) approximation to [f] A Numerical Example in One Dimension In this section we present a one-dimensional numerical example to demonstrate the results of section 3.1 and to validate the conducted analysis. Consider the problem 6(x x ) + 2 x x P xx = 6(x x ) + 4 x > x P (0) = P (1) = 9 64 [P ] Γ=x = [P x ] Γ=x = 0 (3.9) having exact solution (x x ) 3 + (x x ) 2 x x P =. (x x ) 3 + 2(x x ) 2 x > x Albeit, this is a relatively straightforward example, having a piecewise polynomial solution, nevertheless, (3.9) is sufficient to demonstrate the results presented above. For this particular example we take the point x to be 3 4, although, the choice of x is relatively arbitrary. Using the above methodology, we need only modify the scheme (3.2) at the two nodes closest to the interface Γ. The discretization at each follows (3.8) as these are our only irregular nodes. This, as mentioned before, results in a slightly modified sourcing term in our linear solve. 34

44 In figure 3.2 we can see convergence plots for both the standard discretization (3.2) applied on all nodes as well as the IIM discretization (3.8). As we can see, from the slope in figure 3.2a, the order of convergence for the standard discretization behaves as O(h). With this discretization we are making an O(1) local error on the irregular nodes which, due to the elliptic inversion, is sufficient to obtain a globally first order method. Note that the O(1) error made on the irregular nodes is precisely the correction a 5 [P ξξ ] subtracted from the right-hand side in the IIM discretization. For the IIM discretization, we can see from the slope in figure 3.2b, we have achieved the O(h 2 ) accuracy as desired. We will discuss the global error in more detail when we consider some two-dimensional examples in chapter 4. What we will show is that in certain circumstances we can make O(1) local errors at irregular nodes and still obtain the required second order accuracy globally measured in the -norm. 3.2 The Immersed Interface Method: Pressure Equation In this section we demonstrate the DPA applied in two dimensions to derive a scheme for problem (2.49). For now we will ignore the derivative on the right-hand side and assume the source is a known function, at least on the mesh nodes. As with the one-dimensional case we use the standard centered difference discretization 2 hp = P (x i 1, z j ) + P (x i+1, z j ) 4P (x i, z j ) + P (x i, z j 1 ) + P (x i, z j+1 ) h 2 = f i,j (3.10) on all regular nodes and a modified IIM discretization, to be determined, on the irregular nodes. We were able to show in one dimension that when P had a discontinuity in its second derivative the finite difference stencil for the irregular nodes remained the same and the scheme needed only to be modified via adding a constant to the source. For the twodimensional case we will use this as a starting point assuming the same happens in two dimensions. As such, we begin with a discretization of the form k K 5 γ k P k = f i,j + C i,j (3.11) where K 5 = {(i, j) x i,j the standard five point stencil}. In principle, because we already anticipate the γ k s to be the same as (3.10) we could substitute the Taylor series for each P k and solve for the required constant C i,j. However, it is difficult to do this without making assumptions about the orientation of the stencil with respect to the interface i.e. which nodes are on the + and sides of the interface. A more straightforward approach is to derive the necessary conditions for the γ k s and then verify that the γ k s from (3.10) satisfy these conditions. We will use a direct analysis in section 3.3 when we address differentiating the right-hand side in (2.49). Just as in the one-dimensional case to determine the constant 35

45 η ξ x θ Figure 3.3: Diagram demonstrating the two-dimensional local coordinate system of (ξ, η).the point x is the origin of the local coordinate system and θ is defined to be the angle between the normal direction and the horizontal axis. Γ C i,j we expand each P k about a displaced point. In two dimensions however, we have two problems, the first being how do we choose the displaced point as the interface is now a curve and not just a single point? The second problem is that we are only given the jump information about P and Pˆn. For the first problem, the displaced point can be any point along the interface that is nearby the irregular node we are discretizing. Typically the interface point is chosen as a projection, of the discretization node, onto the interface or where the interface intersects some axis. We will comment more on how to compute a projection in section 3.5. For now, just note that the choice of the displaced point x = (x, z ) is arbitrary and just needs to be close to the discretization node x i,j = (x i, z j ). To address the second problem we will have to translate the provided jump information into jump information about P in some coordinate system. It turns out that this is straightforward if we use the local coordinates at the chosen displaced point. The local coordinates at a displaced point are defined to be a translation and rotation of the Cartesian coordinate system. We translate the Cartesian system such that the origin is centered at the displaced point and we rotate such that one coordinate direction is normal to the curve and the other is tangential to the curve in the positive oriented direction. A diagram of the local coordinate system can be see in figure 3.3. The change of variables for the new system can be found in [8] and is as follows ξ = (x x ) cos(θ) + (z z ) sin(θ) η = (x x ) sin(θ) + (z z ) cos(θ) (3.12) where θ is the angle between the normal direction and the x-axis. To determine the jump information about P in the local coordinates we use the following theorem, the proof of which can also be found in [8]. Theorem 3.1. Let (x, z ) be a point on the interface Γ. Assume that Γ C 2 in a neighbourhood of (x, z ). Then from the jump conditions [P ] = 0 and [Pˆn ] = 0 and the PDE, 36

46 (x 2, z 2 ) (ξ 2, η 2 ) (x 3, z 3 ) (ξ 3, η 3 ) (x 0, z 0 ) (ξ 0, η 0 ) x (x 1, z 1 ) (ξ 1, η 1 ) (x 4, z 4 ) (ξ 4, η 4 ) Γ Figure 3.4: Diagram demonstrating the two-dimensional node numbering scheme at an irregular node. 2 P = f, we have the following relations: [P ] = 0, [P ξ ] = 0, [P η ] = 0, [P ξξ ] = [f], [P ηη ] = 0, [P ξη ] = 0. Now that we have jump information about P in the local coordinates along the interface it makes sense to proceed with our analysis using the local coordinates. When switching to the local coordinates we will have to be very careful with our node labeling scheme. We cannot simply label (x i, z j ) = (ξ i, η j ) as we have an issue when (x i, z j+1 ) (ξ i, η j+1 ). For this reason we adopt the local node numbering scheme as labeling the center node 0 and all others as 1, 2, 3, 4 starting with the right node and proceeding counter-clockwise with the labeling (x k, z k ) = (ξ k, η k ), k = 0,..., 4. In figure 3.4 we can see a diagram of this node numbering scheme. As was done in the one-dimensional case we now proceed by expanding each P k about the displaced point yielding P (x k, z k ) = P (ξ k, η k ) = P ± + ξ k P ± ξ + η kp ± ξ + ξ2 k 2 P ± ξξ + η2 k 2 P ± ηη + ξ kη k 2 P ± ξη + O(h3 ). (3.13) Again, we look to determine the constant C i,j by minimizing the truncation error T i,j = k K 5 γ k P k f i,j C i,j. (3.14) 37

47 Substituting (3.13) into the truncation error expression (3.14) we obtain where T i,j = a 1 P + + a 2 P + a 3 P + ξ + a 4P ξ + a 5P + η + a 6 P η + a 7 P + ξξ + a 8P ξξ + a 9 P + ηη + a 10 P ηη + a 11 P + ξη + a 12P ξη f C j + O(h) a 1 = γ k, a 2 = γ k, k K + k K a 3 = γ k ξ k, a 4 = γ k ξ k, k K + k K a 5 = γ k η k, a 6 = γ k η k, k K + k K a 7 = a 9 = γ ξ2 k k k K + γ η2 k k k K + a 11 = 2, a 8 = 2, a 10 = k K + γ k ξ k η k, a 12 = γ ξ2 k k k K 2, γ η2 k k k K 2, k K γ k ξ k η k. (3.15) Note the slight variation in the definition of the a k s in (3.15) compared to that of the one-dimensional case (3.6). Playing the same game as we did for the one-dimensional case it can be determined that the necessary conditions to achieve an O(h) truncation error are: a 1 + a 2 = 0 a 3 + a 4 = 0 a 5 + a 6 = 0 a 7 + a 8 = 1 a 9 + a 10 = 1 a 11 + a 12 = 0. (3.16) Provided the above conditions (3.16) are satisfied we have the following expression for the truncation error T i,j = a 1 [P ] + a 3 [P ξ ]+a 5 [P η ] + a 7 [P ξξ ] + a 9 [P ηη ] + a 11 [P ξη ] + a 10 [P ηη ] + a 12 [P ξη ] C j + ( 2 P f ) + O(h), which for our particular problem gives T i,j = a 7 [P ξξ ] C i,j + ( 2 P f ) + O(h). 38

48 It then follows that C i,j = a 7 [P ξξ ] resulting in T i,j = O(h) as desired. Note that to write 2 P we have used the fact that the Laplacian is invariant under our coordinate transformation, i.e. 2 P = P xx + P yy = P ξξ + P ηη. As we mentioned in the one-dimensional case, if we had more non-zero jump conditions in our problem we need only modify C i,j to incorporate the remaining non-zero jump conditions. Now that we have solved for the C i,j correction, the question of the γ k s still remains. It turns out, as we anticipated, that the solution to system (3.16), despite there being six equations, is the standard centered difference coefficients γ 0 = 4 h 2, and γ 1,2,3,4 = 1 h 2. A proof of this result can be found in [7]. Our newly derived scheme for an irregular node, x i,j, can then be written as P (x i 1, z j ) + P (x i+1, z j ) 4P (x i, z j ) + P (x i, z j 1 ) + P (x i, z j+1 ) h 2 = f i,j + a 7 [P ξξ ]. (3.17) As was the case with the one-dimensional example, having O(h) local error on the irregular nodes and O(h 2 ) on the regular nodes will give globally second order accuracy measured in the -norm. Once again, as with the one-dimensional case, we require [f] to determine [P ξξ ]. We can once again use the difference between the irregular node and a node on the opposite side of the interface to compute [f] as this will give us an O(h) approximation of [f]. We will save the numerical results for section Extending The IIM: Differentiating The Source Up to this point we have focused on discretizing the problem 2 P = f and have ignored the fact that system (2.49) has a derivative on the right-hand side. In the standard textbook presentation of the IIM the source is always assumed to be known exactly. We will now extend the idea of the IIM to additionally differentiating the source term. In our physical model, summarized in section 2.3, ρ represents the density of the atmosphere and as previously mentioned should be a continuous function. However, at the interface separating saturated and unsaturated air the derivative of the density has a jump discontinuity. Given that the function ρ is the known quantity we now need some scheme to differentiate ρ to at least first order on the irregular nodes. We saw that in the case of the one and twodimensional Laplacian that the standard finite difference techniques were sufficient with the addition of a constant. This constant depended on the jump in the solution across the interface, the layout of the stencil with respect to the interface, and the finite difference 39

49 coefficients. It seems reasonable, at this point, to expect that we can also differentiate the source term using our standard finite difference stencil modified by yet another constant. In order to derive this constant we will take a slightly different approach than what was done for the discrete Laplacian. Rather than determine the necessary conditions for first order accuracy we will examine the standard stencil directly to gain an understanding of what this computes in the presence of discontinuities. From there, as we will see, we can easily determine the required constant such that we obtain first order accuracy on the irregular nodes. Note that we could have also taken this approach for the discrete Laplacian however, the number of configurations to consider is large enough that it is actually easier to take the general approach. Here however, we have just two possible configurations which makes this a good example to demonstrate a direct analysis. Again, as mentioned many times now on all regular nodes we use the standard finite difference scheme, in this case centered difference. At all regular nodes we have ( z ) h ρ(x i, z j ) = ρ(x i, z j+1 ) ρ(x i, z j 1 ) 2h which we know to be second order accurate locally. (3.18) In the presence of discontinuities however, in particular a derivative discontinuity, this method becomes inconsistent and we must do a little more work to obtain a first order scheme. Before we analyze (3.18) directly when ρ contains discontinuities we will need to determine some jump information about ρ and its derivative. As we mentioned already ρ is a continuous function. It is easily shown by differentiating ρ along the interface that [ρ η ] = 0. Given that ρ z has a jump discontinuity and ρ η is continuous it must be the case that ρ ξ has a jump discontinuity. In order to apply the following method we will need to know either [ρ ξ ] or [ρ z ] since the two are related by sin(θ) [ρ z ] = [sin(θ)ρ ξ + cos(θ)ρ η ] = sin(θ)[ρ ξ ]. We will now proceed assuming we are able to determine at least one of these quantities to at least first order accuracy. To determine how (3.18) behaves when ρ has a jump in its z-derivative we once again use our DPA by expanding ρ about the displaced point x. As always we will assume the point x i,j lies on the side of the interface and initially we will assume x i,j+1 lies on the + side. In addition to this we will assume our grid is fine enough that we never have both x i,j+1 and x i,j 1 lying on the + side. We can now expand ρ about the displaced point, using the same node labeling as in the two-dimensional Laplacian seen in figure 3.4, as follows ρ(x i = x 2, z j+1 = z 2 ) = ρ(ξ 2, η 2 ) = ρ + ξ 2 ρ + ξ + η 2ρ η + O(h 2 ), ρ(x i = x 4, z j 1 = z 4 ) = ρ(ξ 4, η 4 ) = ρ + ξ 4 ρ ξ + η (3.19) 4ρ η + O(h 2 ). 40

50 From (3.12) it is easily obtained that ξ 2 = ξ 0 + h sin(θ) = ξ 0 + h 1, η 2 = η 0 + h cos(θ) = ξ 0 + h 2, ξ 4 = ξ 0 h 1, η 4 = η 0 h 2. Substituting these and (3.19) into (3.18) we obtain ( ) 2h ρ = (ξ 0 + h 1 )[ρ ξ ] + 2h 1 ρ ξ z + 2h 2ρ η + O(h 2 ) h substituting our expressions for h 1 and h 2 we see that ( z ) h ρ(x i, z j ) = ρ z + γ 2 ξ 2 [ρ ξ ] + O(h). Thus, in order to obtain a first order method we must subtract off the γ 2 ξ 2 [ρ ξ ] giving us the first order approximation ( z ) h ρ(x i, z j ) γ 2 ξ 2 [ρ ξ ] = ρ z (x i, z j ) + O(h). (3.20) We can apply the same process assuming now that x i,j 1 lies on the + side of the interface with the end result being ( z ) h ρ(x i, z j ) γ 4 ξ 4 [ρ ξ ] = ρ z (x i, z j ) + O(h). (3.21) Note that in the above two expressions (3.20) and (3.21) we have additionally expanded ρ z about the center node x i,j. Looking closely at both (3.20) and (3.21) notice that we can combine these two results into the more general expression ( z ) h ρ(x i, z j ) γ + ξ + [ρ ξ ] = ρ z (x i, z j ) + O(h) = ρ z + O(h) (3.22) where γ + and ξ + represent the finite difference coefficient and the local ξ-coordinate corresponding to the node on the + side of the interface. Equation (3.22) is now a first order method for differentiating the source in system (2.49). Note that (3.22) is precisely the result one would have guessed based on the results obtained for the one and two-dimensional Laplacian. 41

51 3.4 Discretizing The Pressure Poisson Equation In section 3.2 we successfully derived a first order finite difference scheme for the Laplacian when P has a discontinuous second derivative and in section 3.3 for the first derivative when ρ has a discontinuous first derivative. As a result, we have also understood how the standard finite difference techniques behave under the stated continuity. Namely, ( z ) 2 hp (x i, z j ) = 2 P + [P ξξ ] h ρ(x i, z j ) = ρ z + [ρ ξ ] k K + 5 k K + 2 γ k ξ 2 k 2 + O(h), (3.23) µ k ξ k + O(h), (3.24) ( ) where 2 h represents the standard 5-point discrete Laplacian and z is the standard h centered difference derivative as derived in sections 3.2 and 3.3. It is also assumed that (x i, z j ) is an irregular node otherwise we have the standard result of O(h 2 ) local truncation error. Note that in expression (3.24) we have switched notation from expression (3.22). This was done to highlight the structural similarity in the discretization of the Laplacian and the z-derivative. The set K 2 + is the analogous set to K+ 5 and only contains two elements. Each node of the z-derivative stencil must occur on either side of the interface and thus the sum in (3.24) is really not a sum at all. We would also like to point out that 2 P and 2 P (x i, z j ) are equivalent up to O(h), and similarly for ρ z, via a first order Taylor expansion. That is, 2 P = 2 P (x i, z j ) + O(h) where we note that the O(h) error constant in (3.23) and (3.24) changes. From expressions (3.23) and (3.24) we can discretize (2.49) as ( ) 2 hp (x i, z j ) = z h ρ(x i, z j ) + [P ξξ ] k K + 5 γ k ξ 2 k 2 [ρ ξ ] k K + 2 µ k ξ k (3.25) at irregular nodes resulting in a first order LTE, T i,j truncation error for the above is given by ( ) T i,j = 2 hp (x i, z j ) z h = 2 P ρ z + O(h) = O(h). ρ(x i, z j ) [P ξξ ] k K + 5 = O(h). To see this we note the γ k ξ 2 k 2 [ρ ξ ] k K + 2 µ k ξ k (3.26) Thus, using the DPA we have successfully derived a locally first order discretization at irregular nodes. According to the conjecture in [6, 7] this is sufficient to generate a globally second order accurate solution as measured in the -norm. 42

52 3.5 Calculating The Displaced Point Before we can present any numerical examples we must take care of an outstanding issue. That is, how can we calculate the displaced point x. So far throughout this chapter we have used the DPA to derive a first order discrete Laplacian in both one and two-dimensions as well as a method for differentiating the source term. However, up to this point we have neglected to discuss how to compute this displaced point. The necessity to compute this displaced point comes from the dependence of the discretization on the local coordinates as well as the jump values at the displaced point. As mentioned before the displaced point can be any point along the interface that is close to the node we are discretizing. We are free to choose this point however we please and in this section we present one such strategy. This strategy is referred to as the orthogonal projection and is also outlined in [8]. Suppose we are looking for the orthogonal projection of the irregular node x i,j. Our interface Γ is defined by the zero level-set of our liquid water mixing ratio r l. Thus, to find a point on the interface we can move a distance α in the direction normal to the level curve at x i,j. Our displaced point can then be written as x = x i,j + α r l (x i,j ). We can then determine α by substituting x into r l to give r l (x ) = r l (x i,j + α r l (x i,j )). Then taking a Taylor series to second order we have the following quadratic equation r l (x i,j ) + ( r l (x i,j )) 2 α r l(x i,j )He(r l (x i,j )) r l (x i,j ) T α 2 = 0. where He(r l (x i,j )) represents the hessian of r l evaluated at x i,j. Note that the computed displaced point will have third order accuracy provided r l (x i,j ) and He(r l (x i,j )) are computed using the standard five point finite difference stencils [8]. In order to completely define our local coordinate system we will need the normal vector to the interface at the displaced point. This can easily be obtained by conducting a bilinear interpolant of both (r l ) x and (r l ) y in the mesh element containing the displaced point. This allows for evaluating both functions at the displaced point giving us our normal vector to second order accuracy as desired. The formula for computing the bilinear interpolant G(x, y) of a grid function G i,j can be written as follows G(x, y) = 1 1 G i+k,j+l x k ȳ j, 4 k=0,l=0 ( 2(x xi ) x k = 1 + (2k 1) h ( 2(y xj ) ȳ k = 1 + (2l 1) h ) 1, ) 1. where x i,j is assumed to be the lower left node of the mesh element [8]. 43

53 3.6 Numerical Examples In Two Dimensions In this section we present two two-dimensional test problems used for numerical verification and give a description of the design principles. These two test problems will be the foundation for all numerical experiments to follow and we will refer to them as the open and closed contour examples. In the first example we present a problem having an open contour with a solution that is periodic in x. In the second example we present a problem having a closed contour which we pose as a Dirichlet problem Open Contour The origin of this test problem stems from wanting a problem that resembles (2.49) as closely as possible. We thus choose a periodic interface given by the zero level set of φ(x, z) = z 1 2 ɛ sin (p(x s)) so that the solution P will contain a natural periodicity allowing us to impose periodic boundary conditions in x. After designing the interface we then proceeded to design the solution P allowing us to impose the required continuity namely, [P ] Γ = 0, [Pˆn ] Γ = 0, but [ 2 P ] Γ 0. One such design for P is as follows: φ 2 f(z), x Ω + P = φ 2, x Ω where Ω is the region below the interface and Ω + is the region above the interface. We can easily see that this design for P is continuous across our interface as φ = 0 along the interface. Further, we can see that P has a continuous gradient, and therefore continuous normal derivative, via 2φf(z) φ P + φ 2 f(z), x Ω + = 2φ φ, x Ω since, again, φ = 0 along the interface. The function f(z) was added to remove the ± symmetry on either side of the interface and was taken to be f(z) = 1 + δz a. In addition to 2π sin(2πx) the f(z) we also added the harmonic function 1000 to both parts of P so that P did not necessarily vanish on the interface. Putting all the pieces together we have our solution P satisfying our required continuity as follows: φ 2 (1 + δz a ) + e2πz sin(2πx) 1000, x Ω + P =. φ 2 + e2πz sin(2πx) 1000, x Ω 44

54 From the design of P we then obtain our sourcing term ρ z by taking the Laplacian of P to yield 2(1 + δz a )(1 + ɛ 2 p 2 cos 2 (p(x s)) + φɛp 2 sin(p(x s)))+ ρ z = 4φaδz a 1 + φ 2 a(a 1)δz a 2, x Ω + 2(1 + ɛ 2 p 2 cos 2 (p(x s)) + φɛp 2 sin(p(x s))), x Ω containing a jump discontinuity along the interface Γ. The complete problem can then be stated as follows 2 P = ρ z, (x, z) Ω + Ω, P (0, z) = P (1, z) Pˆn (x, 0) = 2φ(x, 0) + 2π sin(2πx) 1000 Pˆn (x, 1) = 2(1 + δ)φ(x, 1) + aδφ 2 (x, 1) + 2πe2π sin(2πx) 1000 [P ] Γ = [Pˆn ] Γ = 0 (3.27) where the top and bottom Neumann boundary conditions come directly from P. Finally, in order to use our IIM discretization (3.25) we need to derive a continuous ρ. To do this we simply integrate ρ z from 0 to z for (x, z) Ω and for (x, z) Ω + we first integrate from 0 to the interface Γ and then from the interface to z. Thus, giving us the continuous density ρ = where 2(1 + ɛ 2 p 2 cos 2 (p(x 2)))(I 1 (z) I 1 (d(x)))+ 2z 2ɛp 2 sin(p(x s))(i 2 (z) I 2 (d(x)))+ 4aδ(I 3 (z) I 3 (d(x))) + a(a 1)δ(I 4 (z) I 4 (d(x))), x Ω + ( ( )) 1 + ɛ 2 p 2 cos 2 (p(x s)) + ɛp 2 sin(p(x s)) z ɛ sin(p(x s)), x Ω I 1 (z) = z + δ za+1 a + 1 ( z I 2 (z) = z 2 1 ( z a+1 ɛ sin(p(x s)) + δ 2 a + 2 ( z I 3 (z) = z a a ) ɛ sin(p(x s)) 2a a I 4 (z) = z a 1 ( (2ɛ sin(p(x s)) + 1) 2 4(a 1) d(x) = 1 ɛ sin(p(x s)). 2 z a )) 2(a + 1) za ɛ sin(p(x s)) a + 1 ) 2ɛ sin(p(x s)) + 1 z + z2 a a + 1 The values for each of the constants used in the above problem are: 45

55 (a) ρ (b) ρ z (c) P Figure 3.5: In figure (a) is a plot of the continuous density ρ, (b) the jump-discontinuous z derivative of the density, and (c) a plot of the exact solution P for the open contour test problem. p s ɛ δ a 4π and in figure 3.5 we can see plots of the density ρ, the source ρ z, and the exact solution P Closed Contour To design our closed contour example we took a slightly different approach than for the open contour example. This time we began with a continuous density term that would yield a discontinuous z-derivative. The density was chosen to be ρ = A x 2 + z Bz + C which specifies the interface as the zero level-set of φ(x, z) = x 2 +z 2 1. The constants A, B and C allow us to control the density stratification of ρ to allow for a semi realistic density source. To design a complete problem we now need only solve for a P such that 2 P = ρ z with P being continuous. Recall that in section we showed P being smooth on either side of the interface as well as bounded in the domain is enough to guarantee that [Pˆn ] = 0 given that its Laplacian has a jump discontinuity. A continuous solution P can easily be 46

56 (a) ρ (b) ρ z (c) P Figure 3.6: In figure (a) is a plot of the continuous density ρ, (b) the jump-discontinuous z derivative of the density, and (c) a plot of the exact solution P for the closed contour test problem. derived in polar coordinates to be P = Br sin(θ) Br sin(θ) [ Ar (2 A) 4 ] [ Ar3 4 + r ] (r 2 +1) r + (2 3A) (r 2 1) 2 2r, x Ω., x Ω + We can then take Dirichlet boundary conditions from the above exact solution to state the complete problem as 2 P = ρ z, (x, z) Ω + Ω, P x= 1.5 = P ( 1.5, z) P x=1.5 = P (1.5, z) P z= 1.5 = P (x, 1.5) P z=1.5 = P (x, 1.5) [P ] Γ = [Pˆn ] Γ = 0 (3.28) where we label the exterior of the interface as Ω and the interior as Ω +. The values of A, B, and C are taken to be -1, -3, and 10 respectively. In figure 3.6 we can see plots of the density ρ, the discontinuous source ρ z, and the exact solution P. 47