Atmospheric Satellite Observations: Variation Assimilation and Quality Assurance

By Xiaolei Zou

Atmospheric Satellite Observations: Variation Assimilation and Quality Assurance provides an invaluable reference for satellite data assimilation. Topics covered include linear algebra, frequently used statistical methods, the interpolation role of function fitting, filtering when dealing with real observations, minimization in data assimilation systems, 3D-Var and the inverse problem it solves, 4D-Var and adjoint techniques, and much more. The book concludes with satellite observation of hurricanes.

• Contains mathematical concepts from several branches of study, including calculus, linear algebra, probability theory, functional analysis, and minimization
• Illustrates quality assurance for satellite observations using real data examples
• Includes a dedicated chapter on how different satellite instruments see hurricanes
• Reviews theory, system development, and the numerical experiments of three- and four-dimensional variational data assimilation (3D-Var/4D-Var)

• Language: English
• Publisher: Academic Press
• Release date: Mar 5, 2020
• ISBN: 9780128209530

Xiaolei Zou

Xiaolei Zou’s research interest is in atmospheric data assimilation. She developed a full-physics global four-dimensional variational data assimilation (4D-Var) system for the NCEP medium-range forecast model, a full-physics regional 4D-Var system for the PSU/NCAR mesoscale model, and another regional 4D-Var system for the coupled ocean/atmosphere mesoscale prediction system. Dr. Zou has pioneered and conducted research in the variational assimilation and quality assurance of space-based Global Positioning System radio occultation, satellite microwave and infrared brightness temperatures, multisensor surface rainfall, total ozone mapping spectrometer ozone, and airborne Doppler radar data for hurricanes. She has published over 180 papers in scientific journals and has been a fellow of the American Meteorological Society since 2008. She was a scientist at NCAR, United States (1993–97), a professor at the Florida State University, United States (1997–2014), a research professor at the University of Maryland, United States (2014–19), and is currently a distinguished visiting professor at Nanjing University of Information and Science and Technology (NUIST), China.

Book preview

Atmospheric Satellite Observations

Variation Assimilation and Quality Assurance

Xiaolei Zou

Earth System Science Interdisciplinary Center (ESSIC), University of Maryland (UMD), MD, United States

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1. Introduction to data assimilation

Abstract

1.1 What does data assimilation do?

1.2 Thermodynamic variables and the equation of state

1.3 Atmospheric governing equations for numerical weather prediction

Chapter 2. Linear algebra and statistics

Abstract

2.1 Introduction

2.2 Inner product and adjoint of linear mapping

2.3 Least-squares fit

2.4 Gaussian distribution

2.5 Maximum likelihood estimate

2.6 Error of the fit

2.7 Linear regression

2.8 Lagrangian multiplier

2.9 Minimum variance estimate

2.10 Concluding remarks

Chapter 3. Interpolation

Abstract

3.1 Introduction

3.2 Polynomial function fitting

3.3 Local fitting

3.4 Regional fitting

3.5 Adding background fields

3.6 Successive correction

3.7 Optimal interpolation

3.8 Background error covariance of geopotential field

3.9 Concluding remarks

Chapter 4. Filtering

Abstract

4.1 Introduction

4.2 Fourier transform

4.3 Variance contributions from different scales

4.4 Aliasing

4.5 Nonrecursive filters

4.6 Filter design

4.7 Recursive filters

4.8 Empirical ensemble mode decomposition

4.9 Concluding remarks

Chapter 5. Minimization

Abstract

5.1 Introduction

5.2 Extrema of functional

5.3 Scalar function, gradient, and Hessian matrix

5.4 Iterative minimization, line search, and rate of convergence

5.5 Steepest descent and Newton’s methods

5.6 Conjugate gradient method

5.7 Rank-one update formula for search direction

5.8 Rank-two update formula for search direction

5.9 The L-BFGS method

5.10 Method of line search

5.11 Concluding remarks

Chapter 6. Adjoint model

Abstract

6.1 Introduction

6.2 Adjoint in ordinary differential equation

6.3 Nonlinear model

6.4 Tangent linear model

6.5 Adjoint model

6.6 Equivalence between adjoint variables and Lagrangian multipliers

6.7 Analytic adjoint equations

6.8 Computer programming of adjoint models

6.9 Adjoint sensitivity and relative sensitivity

6.10 Concluding remarks

Chapter 7. Microwave temperature sounding observations

Abstract

7.1 Introduction

7.2 Polar-Orbiting Operational Environmental Satellites missions carrying microwave temperature sounders

7.3 Polar-Orbiting Operational Environmental Satellites orbital characteristics

7.4 Absorption and emission of radiation

7.5 Two-point calibration equation and observation errors

7.6 Absorption and emission weighting functions

7.7 A fast radiative transfer model

7.8 Channel characteristics, scan pattern, field-of-view, and weighting function

7.9 Advanced Technology Microwave Sounder striping noise analysis and mitigation

7.10 Fengyun-3 microwave temperature sounders

7.11 Relative sensitivity of Advanced Microwave Sounding Unit-A brightness temperatures

7.12 Concluding remarks

Chapter 8. Three-dimensional variational data assimilation

Abstract

8.1 Introduction

8.2 Deterministic mathematical formulation

8.3 Statistic formulation

8.4 The National Meteorological Center method for constructing B matrix

8.5 Recursive filters for constructing B matrix

8.6 Comparison between 3D-Var and the Kalman filter

8.7 AMSU-A cloud liquid water path retrieval and cloud detection

8.8 Bias estimate and bias correction

8.9 Impacts of AMSU-A data assimilation on quantitative precipitation forecasts

8.10 Additional remarks

Chapter 9. Four-dimensional variational data assimilation

Abstract

9.1 Introduction

9.2 Four-dimensional variational formulation and gradient calculation using adjoint model

9.3 Penalty method for controlling gravity-wave oscillations

9.4 Adjoints of physical parameterization schemes with on–off switches

9.5 Development of a full-physics global four-dimensional variational system

9.6 Development of two regional adjoint modeling systems

9.7 Parameter estimation

9.8 Incremental four-dimensional variational and its equivalence to the Kalman filter

9.9 Comparison of four-dimensional variational with extended and ensemble Kalman filters

9.10 Additional remarks

Chapter 10. Global positioning system radio occultation observations

Abstract

10.1 Introduction

10.2 GPS and LEO satellite orbital features and GPS RO missions

10.3 Excess phase delay and excess Doppler shift

10.4 Bending angle and impact parameter

10.5 Refractivity retrieval

10.6 Two local observation operators

10.7 A ray-tracing observation operator of bending angle

10.8 A tangent-link observation operator of excess phase

10.9 Multipath occurrence and detection

10.10 Observation error sources

10.11 Impacts of liquid and ice clouds

10.12 Temperature, pressure, and water vapor retrievals

10.13 Postlaunch calibration of satellite microwave temperature sounders data

10.14 Concluding remarks

Chapter 11. Geostationary Operational Environmental Satellite imagers

Abstract

11.1 Introduction

11.2 Geostationary Operational Environmental Satellite satellite altitude

11.3 GOES missions and IGFOV characteristics

11.4 Advanced Himawari Imager and Advanced Baseline Imager instrument characteristics

11.5 Cloud detection

11.6 Advanced Baseline Imager bias characterization

11.7 Geostationary Operational Environmental Satellite imager data assimilation

11.8 Simultaneous assimilation of GOES and POES sensors measurements

11.9 Concluding remarks

Chapter 12. Satellite observations for tropical cyclones

Abstract

12.1 Introduction

12.2 A general description of tropical cyclones

12.3 Typhoon Maria (2018) observed by Advanced Himawari Imager

12.4 Warm-core retrieval from AMSU-A and ATMS data

12.5 Conical-scanning microwave radiometers AMSR2 and MWRI

12.6 Microwave humidity sensors MHS, ATMS, MWHS, and MWHS2

12.7 Satellite total column ozone data from TOMS and OMPS

12.8 Hyperspectral infrared sounders AIRS, IASI, and CrIS

12.9 Vortex initialization

12.10 Additional remarks

Bibliography

Index

Copyright

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Preface

Atmospheric data assimilation extracts useful information about the atmosphere from physical measurements using a set of inverse methods. Introductory in nature, this book deals with several selected methods and describes only three types of satellite data in great detail. By adhering to these limitations, data assimilation can be presented at a level that is accessible to many college seniors and most first-year graduate students. The only mathematical prerequisite is a working knowledge of calculus, linear algebra, probability theory, functional analysis, and minimization. It is hoped that students will have less trouble understanding other data assimilation material not included in this book and be able to conduct more difficult tasks in data assimilation after they read this book.

The book consists of 12 chapters. Chapter 1, Introduction to data assimilation, gives a general introduction to data assimilation as well as atmospheric governing equations for numerical weather prediction. Important mathematical concepts and theories in linear algebra and statistics that will be encountered in later chapters are selectively reviewed. Chapter 3, Interpolation, focuses on interpolation by function-fitting methods, which was the earliest atmospheric data analysis methods used before the digital computer was invented, and two objective analysis methods, namely, successive corrections and optimal interpolation. Chapter 4, Filtering, introduces the concept and role of filtering. Minimization algorithms that are used for obtaining solutions of variational data assimilation problems are provided in Chapter 5, Minimization. Chapter 6, Adjoint model, extensively discusses adjoint techniques. Satellite orbital characteristics and quality assurances of nadir-looking microwave temperature sounding observations from polar-orbiting operational environmental satellites are provided in Chapter 7, Microwave temperature sounding observations. The 3D-Var and 4D-Var approaches are described in Chapter 8, Three-dimensional variational data assimilation, and Chapter 9, Four-dimensional variational data assimilation, respectively. Chapter 10, Global positioning system radio occultation observations, describes data processing procedures and challenges for assimilation of limb-viewing global positioning system (GPS) radio occultation observations from low-earth-orbiting satellite receivers. Data characteristics and assimilation of imager observations from geostationary operational environmental satellites are covered in Chapter 11, Geostationary operational environmental satellite imagers. The last chapter of the book—Chapter 12, Satellite observations for tropical cyclones—describes various features of tropical cyclones and their environments observed by the three satellite observations described in Chapters 7, 10, and 11 and three other satellite instruments and explains how they could be used for hurricane vortex initialization and data assimilation. Most of the material in this book is presented in a self-contained manner with sufficiently detailed mathematical derivations, figure illustrations, and careful interpretations.

Many people have helped me write this book. I am very grateful to Prof. J. Stephen whose excellent lecture notes on objective analysis inspired me to teach data assimilation at a level accessible to undergraduate and first-year graduate students and to appreciate the study of atmospheric data analysis from the early 1950s; to Drs. M. Navon, J. Derber, J. Sela, E. Kalnay, and F. LeDimet for our pioneering work from 1989–93 in developing the four-dimensional variational data assimilation system of the NCEP global medium-range forecast model; to Drs. R. Anthes, Y.-H. Kuo, S. Ware, M. E. Gorbonov, S. Sokolovskiy, M. Exner, and J. Hajj for my early involvement in GPS radio occultation data assimilation, creating a field of study that I have not left since 1993; to Profs. P. Ray, A. Barcilon, T.N. Krishnamurti, J. Obrien, and R.L. Pfeffer for having strongly supported me during the 17 years (1997–2014) of my tenure as professor at Florida State University; and to my graduate students and postdoctoral fellows Z. Qin, S. Yang, X. Zhuge, L. Lin, Y. Ma, X. Tian, F. Tang, H. Dong, Y. Han, M. Yin, X. Xu, and Z. Niu for their infinite patience in producing high-quality artistic figures that have given me enormous satisfaction. Special thanks go to my husband, Jordan Yao, for always believing in me, as I constantly push myself to do better than before; and to my two bright children, Yimei Laura Yao and Yige Noah Yao, who have made me a very happy mom and who remain a constant comfort to me when I am faced with difficulty.

Chapter 1

Introduction to data assimilation

Abstract

This chapter provides a general introduction to data assimilation by answering questions such as what is data assimilation, why is data assimilation needed, and what does data assimilation do. Definitions of and relationships among atmospheric thermodynamic variables are then given, along with various forms of the equation of state and the first law of thermodynamics. Finally, the physical laws that govern the atmosphere and its motion are provided, including the fundamental principles of conservation of momentum, mass, energy, and water vapor. When formulated mathematically, these physical laws form a set of atmospheric governing equations for numerical weather prediction.

Keywords

Data assimilation; atmospheric governing equations; thermodynamic variables

1.1 What does data assimilation do?

I assume people reading this book are interested in the following questions: What is data assimilation?, Why is data assimilation needed?, What are the general concerns of data assimilation?, What is the theoretical basis of data assimilation?, How is data assimilation done?, What are important details of data assimilation? How can these details of data assimilation be taken care of with sufficient knowledge of both data and assimilation methods?, How can one method of data assimilation do better than other methods?, and What are the challenges for data assimilation?

One major purpose of atmospheric data assimilation is to provide initial conditions for numerical weather prediction (NWP) models. The atmospheric governing equations used in NWP models can be symbolically written as

(1.1)

is an initial condition vector consisting of the atmospheric state of 3D wind, potential temperature, density, potential temperature, and specific humidity (e.g., u, v, w, ρ, θ, q) at an initial time to, x(texcept at time t ), and F(x(t.

Taking the Euler scheme at the first time step and the leap-frog scheme at the follow-on time steps to discretize the time tendency term on the left-hand side of (1.1) (Haltiner and Williams, 1980), (1.1) can be written as

(1.2)

) is to be generated by the model. This is called a forward problem in mathematics. Therefore NWP solves forward problems.

, n=0, 1, 2, …, N) and an operator that links a model state to the data

(1.3)

) satisfies , n=0, 1, 2, …, N). Here, statistical methods of estimate define the optimality—the maximum likelihood estimate and the minimum variance estimate (see Chapter 2: Linear algebra and statistics).

Most inverse problems are substantially harder to solve than their corresponding forward problems. The mathematical inverse theory is a theory for solving inverse problems (Tarantola, 1987). It provides information about unknown input parameters that are required by a model, a means for assessing the correctness of a model, a means for discriminating between several models, a tool in experimental design for decision making, and a tool for obtaining key information in data. It must be emphasized that inverse theories and inverse methods provide insights about and improvements to an existing model but cannot provide a model itself.

We may use three examples to highlight why data assimilation is desirable and needed. ) using Fujita’s formula (Fujita, 1952) and is shown in Fig. 1.1B. Assimilation of the bogus SLP (Fig. 1.1B) using physics-included four-dimensional variational data assimilation (4D-Var) systems produces better track and intensity forecasts (Zou and Xiao, 2000; Xiao et al., 2000; Park and Zou, 2004; Tian and Zou, 2019b).

Figure 1.1 Sea-level pressure distributions near Hurricane Florence (A) from the ERA5 reanalysis at 0000 UTC September 6, 2018 and (B) after a 4D-Var bogus vortex initialization. The storm center position at this time from best track records is indicated by the hurricane symbol. 4D-Var, Four-dimensional variational data assimilation; UTC, coordinated universal time.

Fig. 1.2 gives another example where the two-dimensional distribution of space-based observations of ocean wind vectors (Ricciardulli and Wentz, 2015) for Hurricane Gordon at 0000 UTC September 17, 2016 (Fig. 1.2A) is compared with ocean wind vectors from the National Center for Environmental Prediction (NCEP) Final (FNL) large-scale analysis (NCEP, 2000), the ECMWF Interim reanalysis (Simmons et al., 2007), and the ECMWF ERA5 reanalysis. Hurricane Gordon was a category 1 hurricane at this time. The surface winds from the NCEP large-scale analysis are much weaker than the QuikSCAT observations. The horizontal resolutions are 0.25°×0.25° for QuikSCAT and the ECMWF Interim and ERA5 reanalysis, and 1.0°×1.0° for the NCEP FNL analysis. The QuikSCAT-observed surface winds within Hurricane Gordon at this time have an asymmetric distribution with stronger surface winds in the first and fourth quadrants than elsewhere and a tight structure with a radius of the maximum wind speed slightly less than 100 km (Fig. 1.2A). The center of the minimum ocean surface winds from QuikSCAT observations coincides with the best track record, while those from the NCEP FNL analysis (Fig. 1.2B) and the ECMWF ERA5 analysis (Fig. 1.2D) are misplaced, located to the west of the best track record. The radius of the maximum wind speed is about 200 km from the minimum center position of the NCEP FNL analysis (Fig. 1.2B), double that seen in the QuikSCAT observations. This is likely due to the too-coarse horizontal resolution of 1.0°×1.0°. Although having the same horizontal resolution as the QuickSCAT observations, the ECMWF Interim reanalysis (Fig. 1.2C) almost completely misses the cyclonic circulation of Hurricane Gordon. Although weaker, surface winds from the newer ECMWF reanalysis, the ERA5, have a cyclonic circulation that is closest to QuikSCAT in structure. The second maximum of surface winds located to the northeast, which is at a larger radial distance than the maximum closest to the center (Fig. 1.2A), is not revealed in all three large-scale analyses (Fig. 1.2B–D). It is therefore desirable to incorporate these satellite QuikSCAT surface wind observations into hurricane vortex initialization, data assimilation, and subsequent forecasts.

Figure 1.2 Spatial distributions of (A) QuikSCAT ocean surface wind observations, and (B–D) the 10-m winds from (B) the NCEP FNL analysis, (C) the ECMWF Interim reanalysis, and (D) the ECMWF ERA5 reanalysis for Hurricane Gordon at 0000 UTC September 17, 2000. The storm center position at this time from best track records is indicated by a hurricane symbol. The reference wind vector is 20 m s−1. The horizontal resolutions are 0.25°×0.25° for the QuikSCAT, ECMWF Interim and ERA5 reanalyses, and 1.0°×1.0° for the NCEP FNL analysis. ECMWF, European Centre for Medium-Range Weather Forecast; FNL, final; NCEP, National Center for Environmental Prediction; UTC, coordinated universal time.

The third example presents brightness temperature observations at 19 (Fig. 1.3A) and 85 GHz (Fig. 1.3B) at vertical polarization from the Special Sensor Microwave Imager (SSM/I, Raytheon, 2000) for Hurricane Bonnie at 1200 UTC August 23, 1998. The 19-GHz low-frequency brightness temperature observations have an asymmetric distribution. The highest brightness temperatures of more than 275K are located to the northeast of the hurricane center and are more than 55K higher than the surroundings. However, 85-GHz high-frequency brightness temperature observations have much lower values (less than 225K) in areas where the 19-GHz brightness temperature observations are larger than the surroundings. Such distributions of SSM/I observations are closely related to cloud distributions within Hurricane Bonnie. Fig. 1.3C shows the spatial distribution of the SSM/I liquid water path retrieval product at 2230 UTC August 24, 1998, and Fig. 1.3D presents the ice water path distribution from the Tropical Rainfall Measuring Mission Microwave Imager at 1700 UTC August 24, 1998 within Hurricane Bonnie. Liquid clouds contribute an additional amount of thermal emission measured at the 19-GHz frequency, and the ice scattering effect reduces the amount of thermal emission measured at the 89-GHz frequency by satellite microwave imagers. These structures seen in satellite microwave observations of brightness temperature within tropical cyclones instigated a series of studies to assimilate these structures into hurricane forecast models (Amerault and Zou, 2003, 2006; Amerault et al., 2008, 2009). Challenges for the assimilation of cloud-affected high-frequency brightness temperature observations involve an improved radiative transfer model in which particle sizes are made consistent with those in the explicit moist physics of a mesoscale model (Amerault and Zou, 2003), an estimate of background error covariances for hydrometeor variables (Amerault and Zou, 2006), development of an adjoint mesoscale model with explicit moist physics (Amerault et al., 2008), and an assimilation of SSM/I observations using the adjoint system of the Naval Research Laboratory Coupled Ocean/Atmosphere Mesoscale Prediction System (Amerault et al., 2009).

Figure 1.3 Brightness temperature observations (A) of channel 1 (19 GHz, vertical polarization) and (B) of channel 6 (89 GHz, vertical polarization), (C) liquid water path retrievals at 2230 UTC August 24, 1998, from the SSM/I, and (D) ice water path retrievals from the Tropical Rainfall Measuring Mission’s Microwave Imager at 1700 UTC August 24, 1998, within Hurricane Bonnie whose center position is indicated by a black hurricane symbol. The dotted line in (D) shows the eastern SSM/I swath edge for convenience. SSM/I, Special Sensor Microwave Imager; UTC, coordinated universal time.

) includes: Global Positioning System (GPS) radio occultation bending angle and refractivity, microwave radiance from polar-orbiting operational environmental satellite (POES) sounders, infrared radiance from POES sounders, infrared radiance from geostationary operational environmental satellite imagers, ground-based GPS total precipitable water, satellite total column ozone, satellite surface wind over ocean, surface radar data, satellite water vapor wind vector retrieval, radiosonde data, surface station observations, air flight observations, airborne radar data within hurricanes, and data from special field experiments such as dropsonde data in tropical cyclones. To assimilate these observations in an available data assimilation system, data assimilation scientists still have to complete several tasks, including developing appropriate observation operators; quantifying error variances and biases of both models and data; developing physically sound algorithms for bias correction, cloud detection, data thinning, and quality control; ensuring minimization convergence; carrying out impact assessments; and seeking reasons for the improvement and/or degradation obtained by data assimilation.

) produced can approximately be written into a general form as follows

(1.4)

where xb is called the a posteriori weight. Therefore various data assimilation methods differ mostly in how the a posteriori weight W are much larger than 10⁶. The analyses are not directly derived from (1.4), and Eq. (1.4) is implicitly and approximately satisfied.

Data assimilation involves the following: (1) fitting of observations to within observation errors; (2) ensuring that the observation operators are physically consistent with what the data represent; (3) including past background information; (4) quantifying biases in both the model and observations and subtracting them from the data; (5) properly accounting for error statistics of both observations and background fields; (6) incorporating appropriate dynamic and physical constraints at the scales of interest; (7) ensuring that computational and data noise is suppressed, (8) the computational expense is affordable, and (9) analysis errors are quantified and delivered. In summary, atmospheric data assimilation is the process of incorporating various observational data into an NWP model to produce the best description of the atmospheric state at a desired resolution in an optimal sense statistically. It is more than a pure mathematical inverse problem. Physical understanding of the observables and what structures we are looking for is essential. Knowledge of the computational constraint is also important.

1.2 Thermodynamic variables and the equation of state

and thermodynamic variables. In this section, we provide definitions of the following thermodynamic variables: temperature (T), pressure (p), specific humidity (q), mixing ratio (w), internal energy (u), enthalpy (h). Various forms of the equation of state and the first law of thermodynamics reveal many constraining relationships among these thermodynamic variables in the atmosphere.

The atmosphere can be thought of as an ideal gas in which interactions among molecules are negligible. Although constantly moving in all directions, molecules in a volume of the atmosphere as a whole stay still. The kinetic energy of the molecular motion contributes to and determines the temperature of the atmosphere. The atmospheric temperature (T) is defined as follows (Bohren and Albrecht, 1998):

(1.5)

is the mass of a molecule (unit: kg), vi is the motion velocity of the ith molecule (i=1, …, N), k represents averaging over all molecules in the volume.

Collisions with the boundaries of an air parcel by these moving molecules contribute to the pressure, defined as the force acting on a unit area. The mathematical expression of pressure is thus

(1.6)

where V is the volume, N is the total number of molecules in Vis the number flux quantifying the total strikes of molecules on a unit area in unit time in the xrepresents the momentum change of a collision a molecule makes with the wall.

From (1.5) and (1.6), we obtain the equation of state of the atmosphere, that is,

(1.7)

), the equation of state (1.7) can be equivalently written as

(1.8)

is the gas constant:

(1.9)

(1.10)

and M is the total mass of molecules in V:

(1.11)

is defined as the mass of molecules in a unit volume, that is,

(1.12)

Substituting (1.10) into (1.8), we get the third expression for the equation of state:

(1.13)

grams of that gas. For examples, 1 mol of oxygen (O2) is 32 g, 1 mol of nitrogen (N2) is 28.01 g, 1 mol of water vapor (H2O) is 18 g, and 1 mol of dry air is 28.96 g. We may express the molecule weight (m) in the mole unit as follows:

(1.14a)

(1.14b)

) is called Avogadro’s number:

(1.15)

The universal gas constant can be defined as (see 1.10):

(1.16)

By applying the ideal gas law as the mass of a molecule (i.e., the total mass of molecules in V ), we obtain an equation of state of the following form:

(1.17)

Note that . We may also apply the ideal gas law (1.13) to any amount of an ideal gas (i.e., the total mass of molecules in V ), which leads to the following equation of state:

(1.18)

Substituting (1.9) into (1.18), we obtain

(1.19)

The equation of state (1.19) is valid for any amount of an ideal gas with molecule weight m (unit: mole). It is the most general form of the equation of state. Applying it to dry air,

(1.20)

):

(1.21)

Substituting (1.21) into (1.20), we finally obtain a commonly used equation of state for the dry atmosphere in NWP:

(1.22)

We may also apply (1.19), which is valid for any ideal gas, to water vapor to obtain the equation of state for water vapor:

(1.23)

) is defined as the gas constant for water vapor:

(1.24)

Substituting (1.24) into (1.23), we obtain another expression of the equation of state for water vapor:

(1.25)

). The following are some useful relationships among these five measures of water vapor content in the atmosphere:

(1.26a)

(1.26b)

and

(1.26c)

is a constant defined as

(1.27)

We may now derive the equation of state for moist air. Since pressure is additive, the total pressure (p) is the sum of dry air pressure (pd) and vapor pressure (e):

(1.28)

Substituting the equations of state (1.20) and (1.23) into (1.28), we obtain the equation of state for moist air by a step-by-step derivation:

(1.29)

) is introduced next:

(1.30)

The equation of state for moist air (1.29) then becomes

(1.31)

Comparing (1.31) and (1.22), we find that the dry air equation of state can be used for moist air by simply replacing T is the temperature that dry air would have if its pressure and density were equal to those of a given parcel of moist air.

). They are defined as

(1.32a)

and

(1.32b)

since all heat added to a system goes to increase temperature if the volume of the system does not change. For constant pressure processes, some of the added heat is used to do work on the surroundings through expansion so that less heat goes to increase temperature. The denominator (dT) in (1.32b) will be smaller than that in (1.32a) for the same amount of heat added (dq), which is the numerator of (1.32a) and (1.32b).

The first law of thermodynamics states that energy in a closed system is conserved. Therefore the change in the internal energy (u) of a system with unit mass is equal to the amount of heat (q) received by the system, subtracting the work (w) done by the system to its surroundings:

(1.33)

Eq. (1.33) can be rewritten as

(1.34)

Based on (1.34), the first law of thermodynamics can be stated as follows: When heat is added to a system, some amount contributes toward increasing the internal energy of the system; the remaining is used by the system to do work to its surroundings. For a system of gases the work done by the system to its surroundings is through expansion and evaluated by the change in volume:

(1.35)

Substituting (1.35) into (1.34), we obtain

(1.36)

), defined by

(1.37)

the right-hand side of (1.36) can be expressed in terms of enthalpy instead of internal energy:

(1.38)

For studying the atmosphere, which consists of ideal gases, the internal energy (u) does not change with specific volume, that is,

(1.39)

The amount of heat energy that converts to internal energy increases the temperature of the system:

(1.40)

Substituting (1.40) into (1.36), we obtain the third form of the first law of thermodynamics:

(1.41)

. The first law of thermodynamics (1.38) can then be expressed in terms of enthalpy [see (1.37)]:

(1.42)

Substituting (1.42) into (1.38), we obtain

(1.43)

Eqs. (1.41) and (1.43) are the two mathematical expressions of the first law of thermodynamics most commonly used by the atmospheric science community.

From ) can be expressed as partial derivatives of internal energy (u) and enthalpy (h) with respect to temperature (T):

(1.44a)

and

(1.44b)

Since u and h are also thermodynamic variables.

are related to each other. We may now derive this relationship. For ideal gases, there are only two independent thermodynamic variables. If we take temperature (T) and pressure (pare functions of T and p, that is,

(1.45a)

(1.45b)

and

(1.45c)

Taking partial derivatives of both sides of (1.45c), we have

(1.46)

Eq. (1.46) must be valid for all processes. Because T and p are independent variables, (1.46) is always true only if the coefficients of dT and dp , and (1.44a) and (1.44b) into the coefficient of dT :

(1.47)

are nearly constant, and their values are

(1.48)

. Substituting (1.22) into (1.43), we first obtain the following mathematical expression of the first law of thermodynamics on energy conservation:

(1.49)

in (1.49), we have

(1.50)

Therefore the temperature and pressure changes in an adiabatic process are constrained by (1.50). The ordinary differential Eq. (1.50) has the following analytical solution:

(1.51)

Assuming an air parcel with temperature T and pressure p can be derived from (1.51), that is,

(1.52)

is finally defined as the potential temperature of the air parcel with temperature T and pressure p:

(1.53)

).

1.3 Atmospheric governing equations for numerical weather prediction

Atmospheric governing equations are the mathematical equations expressing the physical laws of the atmosphere. NWP requires the atmospheric governing equations, initial or boundary conditions or both, and numerical methods for integrating the governing equations forward in time. In this section, we state these physical laws and describe how they are formulated mathematically.

The physical laws that control the atmospheric state consist of the fundamental principles of conservation of momentum, mass, energy, and water vapor. The conservation of momentum comes from Newton’s second law of motion, which gives three equations of motion. Conservation of mass leads to an equation of continuity. Conservation of energy gives a thermodynamic equation, which is the result of combining the first and second laws of thermodynamics. Finally, conservation of water gives an equation of advection of water vapor.

) relative to the Earth equals the sum of all forces it experiences, that is,

(1.54)

The acceleration of the motion of an air parcel relative to the Earth can be expressed as (Haltiner and Williams, 1980)

(1.55)

). The pressure force can be expressed as

(1.56)

where p is density. Substituting (1.55) and (1.56) into (1.54), we obtain

(1.57)

The second term on the left-hand side of ), that is,

(1.58)

). Due to gravity, the large-scale motions of the atmosphere are quasihorizontal with respect to the Earth’s surface.

Substituting (1.58) into (1.57), we obtain the governing equation on the momentum change of the atmosphere:

(1.59)

) in the unit volume, that is,

(1.60)

This is the governing equation on the change of density in the atmosphere.

) of the atmosphere:

(1.61)

) of an open system of ideal gases is equal to the heat added to the system divided by the temperature of the system:

(1.62)

Substituting (1.53) and (1.61) into (1.62), we obtain the governing equation on the change of potential temperature:

(1.63)

include phase changes