Appendix CInternational Reference Atmosphere
Appendix DChemical MechanismD.1Chemical Species and Definitions of Symbols
D.2Photolysis
D.3Gas-Phase Reactions
D.4Heterogeneous Reactions
Appendix EBrief Mathematical ReviewE.1Mathematical Functions
E.2Scalars and Vectors
E.3Matrices
E.4Vector Operators
E.5Differential Equations
E.6Transforms
E.7Probability and Statistics
Further Reading
Index
Preface
Modern science dealing with complex dynamical systems increasingly makes use of mathematical models to formalize the description of interactive processes and predict responses to perturbations. Models have become fundamental tools in many disciplines of natural sciences, engineering, and social sciences. They describe the essential aspects of a system using mathematical concepts and languages and they can in this manner provide powerful approximations of reality. They are used to analyze observations, understand relationships, test hypotheses, and project future evolution. Disagreements between models and observations often lead to important advances in theoretical understanding. Models also play a critical role in the development of policy options and in decision-making.
In atmospheric science, mathematical models have long been central tools for weather prediction and climate research. They are now also used extensively to describe the chemistry of the atmosphere. The corresponding model equations describe the factors controlling atmospheric concentrations of chemical species as a function of emissions, transport, chemistry, and deposition. Chemical species are often coupled through intricate mechanisms, and the corresponding differential equations are then also coupled. Simulation of aerosol particles needs to account in addition for microphysical processes governing particle size and composition, as well as interactions with the hydrological cycle through cloud formation. The difficulty of modeling atmospheric composition is compounded by the need to resolve a continuum of temporal and spatial scales stretching over many orders of magnitude from microseconds to many years, from local to global, and involving coupling of transport and chemistry on all scales.
Mathematical modeling of atmospheric chemistry is thus a formidable scientific and computational challenge. It integrates elements of meteorology, radiative transfer, physical chemistry, and biogeochemistry. Solving the large systems of coupled nonlinear partial differential equations that characterize the atmospheric evolution of chemical species requires advanced numerical algorithms and pushes the limits of supercomputing resources.
The purpose of this book is to provide insight into the methods used in models of atmospheric chemistry. The book is designed for graduate students and professionals in atmospheric chemistry, but also more broadly for researchers interested in atmospheric models, numerical methods, and optimization theory.
The book is divided into three parts. The first part presents background material. Chapter 1 introduces the reader to the concept of model and provides a historical perspective on the development of atmospheric and climate models, leading to the development of atmospheric chemistry models. It reviews the different types of atmospheric chemistry models and highlights their role as components of observing systems.
Fundamentals of atmospheric dynamics and chemistry are presented in Chapters 2 and 3. Chapter 2 describes the vertical structure of the atmosphere, defines key parameters that characterize the dry and the wet atmosphere, and introduces the concept of static stability and geostrophic balance. It goes on to describe the general circulation of the atmosphere. Chapter 3 provides a summary survey of the chemical processes relevant to the atmosphere as well as the microphysical processes controlling the evolution of aerosol particles. Chapter 4 presents the fundamental mathematical equations on which atmospheric models are based and gives an introduction to the numerical methods used to solve these equations.
The second part of this book focuses on the formulation of model processes and reviews the numerical algorithms used to solve the model equations. Chapter 5 covers the formulation of radiative transfer, chemical kinetics, and aerosol microphysics. Chapter 6 reviews numerical methods to solve the stiff systems of nonlinear ordinary differential equations that describe atmospheric chemistry mechanisms. Chapter 7 presents numerical algorithms used to solve the advection equation describing transport by resolved winds. The formulation of small-scale (parameterized) transport processes including turbulent mixing, organized convection, plumes, and boundary layer dynamics is addressed in Chapter 8. Chapter 9 reviews formulations of emissions to the atmosphere, deposition to the surface, and two-way coupling between the atmosphere and surface reservoirs.
The third part of this book deals with the role of models as components of the atmospheric observing system. Chapter 10 focuses on model evaluation and presents different metrics for this purpose. It illustrates the importance of models for the interpretation of observational data. Chapter 11 covers fundamental concepts of inverse modeling and data assimilation. It shows how chemical transport models can be integrated with atmospheric observations through optimization theory to provide best estimates of the chemical state of the system and of the driving variables.
At the end of the volume, the reader will find several appendices with numerical values of physical constants and other quantities, unit conversions, and a list of important chemical reactions with corresponding rate constants. Some basic mathematical definitions and relations are also provided.
Over the years, both of us have benefited from numerous discussions with our colleagues, students, and postdoctoral fellows. Several of them have contributed to this book by reviewing chapters, making suggestions, and providing scientific material. We are deeply indebted to them. We would like to thank in particular Helen Amos, Alexander Archibald, Jerome Barre, Mary Barth, Cathy Clerbaux, Jim Crawford, Louisa Emmons, Rolando Garcia, Paul Ginoux, Claire Granier, Alex Guenther, Colette Heald, Jan Kazil, Patrick Kim, Douglas Kinnison, Monika Kopacz, Jean-François Lamarque, Peter Lauritzen, Sasha Madronich, Daniel Marsh, Iain Murray, Vincent-Henri Peuch, Philip Rasch, Brian Ridley, Anne Smith, Piotr Smolarkiewicz, Alex Turner, Xuexi Tie, Stacy Walters, Kevin Wecht, Christine Wiedinmyer, Lin Zhang, and Peter Zoogman. We would like also to acknowledge Sebastian Eastham, Emilie Ehretsmann, Natasha Goss, Lu Hu, Rajesh Kumar, Eloise Marais, Jost Müsse, Elke Lord, Barbara Petruzzi, Jianxiong Sheng, and Natalia Sudarchikova for their technical assistance during the preparation of the manuscript. A substantial fraction of this volume was written by one of us (G. P. B) at the National Center for Atmospheric Research, which is sponsored by the US National Science Foundation.
Symbols
The symbols used in the different chapters of this book are listed below with their corresponding units in the MKSA system. When no units are given, the quantity is either dimensionless or has no intrinsic dimensions. Appendix B gives further information on units, prefixes, and conversion factors. In some cases, when no confusion exists, the same symbols are used to characterize different variables. Scalars are represented as italics (alphabet letters) or as regular font (Greek and other symbols). Vectors and matrices are represented by lowercase and uppercase bold fonts, respectively.
A
a
Earth’s radius [m]
A
Surface area density of atmospheric particles [m2 m–3]
A
Averaging kernel matrix
B
B
Blackbody radiative emission flux [W m–2]
Bλ
Spectral density of blackbody emission flux (Planck function) [W m–2 nm–1]
C
c
One-dimensional constant flow velocity [m s–1]
c
Speed of light in vacuum [m s–1]
c*
Phase velocity of a wave [m s–1]
cg*
Group velocity of a wave [m s�
��1]
cp
Specific heat at constant pressure [J K–1 kg–1]
cv
Specific heat at constant volume [J K–1 kg–1]
Cc
Slip correction factor
CD
Drag coefficient
Ci
Mole fraction or molar mixing ratio of species i
CRMSE
Centered root-mean-square-error
D
d
Displacement height [m]
D
Divergence of the flow [s–1]
Da
Damköhler number
Dd
Detrainment rate associated with downdrafts in convective systems [kg m–3 s–1]
Di
Molecular diffusion coefficient for species i [m2 s–1]
Dp
Particle diameter [m]
Du
Detrainment rate associated with updrafts in convective systems [kg m–3 s–1]
DOFS
Degrees of freedom for signal
E
e
Water vapor partial pressure [Pa]
es
Saturation water vapor pressure [Pa]
e
Eigenvector
E
Emission flux [kg m–2 s–1]
E
Eliassen–Palm Flux [components Eφ and Ez in kg s–2]
E
Matrix of eigenvectors arranged by columns
E(k)
Spectral distribution of turbulent energy for a given wavenumber k [m3 s–2]
Ea
Activation energy [J mol–1]
Ed
Entrainment rate associated with downdraft in convective systems [kg m–3 s–1]
Eu
Entrainment rate associated with updraft in convective systems [kg m–3 s–1]
F
f
Coriolis factor [s–1]
fA
Fractional area of a model grid cell experiencing precipitation
fA
Fractional area of land suitable for saltation
fi,I
Fraction of soluble compound i partitioned in ice water
fi,L
Fraction of soluble compound i partitioned in liquid water
F
Mass flux [kg m–2 s–1]
F
Radiative flux [W m–2]
Air mass factor
F
Force vector with its three components Fx, Fy, and Fz [N]
F
Forward model
FD,i
Deposition flux of species i [kg m–2 s–1]
Fλ
Spectral density of the radiative flux [W m–2 nm–1]
G
g
Vector of gravitational acceleration [m s–2]
g
Amplitude of gravitational acceleration [m s–2]
g
Amplification function in numerical methods
g
Asymmetry factor
g
Gain factor
G
Green function
G
Gravity wave drag [m s–2]
G
Gain matrix
Gm
Grade of model m
H
h
Mixing depth [m]
H
Atmospheric scale height [m]
Effective (constant) scale height [m]
Hi
Dimensionless Henry’s law constant for species i
I
i
Unit vector in the zonal (x) direction
I
Light intensity [W m–2]
I
Identity matrix
IAB
Segregation ratio for chemical compounds A and B
Ii
Condensation growth rate of species i [m3 s–1]
J
j
Unit vector in the meridional (y) direction
j
Radiative source term [Wm–2 sr–1 nm–1 m–1]
J
Radiative source function [Wm–2 sr–1 nm–1]
J
Photodissociation (photolysis) frequency [s–1]
J
Cost function
J
Jacobian matrix
Ji,j
Coagulation rate between particles i and j [m–3 s–1]
J0
Nucleation rate [m–3 s–1]
K
k
Unit vector in the vertical (z) direction
k
Wavenumber [m–1]
k
Boltzmann’s constant (1.38 × 10–23 J K–1)
k
von Karman’s constant (0.35)
k
Chemical rate constant [first order: s–1; second order: cm3 s–1; third order: cm6 s–2]
kext
Mass extinction cross-section [m2 kg–1]
kG,i
Conductance for vertical transfer of species i in the gas phase [m s–1]
kW,i
Conductance for vertical transfer of species i in the water phase [m s–1]
K
Eddy diffusion coefficient [m2 s–1]
K
Equilibrium constant
K
Henry’s law constant [M atm–1]
K*
Effective Henry’s law constant [M atm–1]
K
Eddy diffusion tensor
K
Jacobian matrix (Chapter 11)
Ka
Acid dissociation constant
Km
Eddy viscosity coefficient [m2 s–1]
Kn
Knudsen number
Ki
Air–sea exchange velocity for species i [m s–1]
Kθ
Eddy diffusivity of heat [m2 s–1]
L
l
Mixing length [m]
Loss rate constant or loss coefficient of species i [s–1]
L
Characteristic length [m]
L
Liquid water content [kg water/kg air]
L
Monin–Obukhov length [m]
L
Lagrange function
Li
Loss rate of species i [m–3 s–1]
Lvap
Latent heat of vaporization of liquid water [J kg–1]
Lλ
Spectral density of the radiance at wavelength λ [W m–2 sr–1 nm–1]
M
m
Mean molecular mass of air (4.81 × 10–26 kg)
m
Refraction index
m
Wavenumber
Ma
Molar mass of air (28.97 × 10–3 kg mol–1)
Md
Mean vertical downdraft convective flux of air [kg m–2 s–1]
Me
Mean subsidence flux compensating for convective fluxes [kg m–2 s–1]
Mi
Molar mass of species i [kg mol–1]
Mk
Moment of order k for a given aerosol distribution
Mu
Mean vertical updraft convective flux of air [kg m–2 s–1]
Mw
Molar mass of water (18.01 × 10–3 kg mol–1)
MAD
Mean absolute deviation
MAE
Mean absolute error
MFB
Mean fractional bias
MFE
Mean fractional error
MNAE
Mean normalized absolute error
MNB
Mean normalized bias
N
n
Unit outward vector normal to a surface
na
Number density for air [m–3]
ni
Number density for species i [m–3]
nN
Particle number size distribution function [m–4]
nS
nV
Particle volume distribution function [m3 m–4]
Avogadro number (6.022 × 1023 molecules per mole)
NMB
Normalized mean bias
P
p
Pressure [Pa]
pd
Pressure of dry air [Pa]
pi
Production rate of species i [kg m–3 s–1]
ps
Surface pressure [Pa]
P
Phase function for scattered radiation
P
Ertel potential vorticity [m2 s–2 K kg–1]
Modeling of Atmospheric Chemistry Page 2