ALKO2 + HO2 → ALKOOH 7.5e-13 × exp(700/T)
ALKOOH + OH → ALKO2 3.8e-12 × exp(200/T)
C5H8 + OH → ISOPO2 2.5e-11 × exp(410/T)
C5H8 + O3 → 0.4 MACR + 0.2 MVK + 0.07 C3H6 + 0.27 OH + 0.06 HO2 + 0.6 CH2O + 0.3 CO + 0.1 O3 + 0.2 MCO3 + 0.2 CH3COOH 1.1e-14 × exp(–2000/T)
C5H8 + NO3 → ISOPNO3 3.0e-12 × exp(–446/T)
ISOPO2 + NO → 0.08 ONITR + 0.92 NO2 + HO2 + 0.51 CH2O + 0.23 MACR + 0.32 MVK + 0.37 HYDRALD 4.4e-12 × exp(180/T)
ISOPO2 + NO3 → HO2 + NO2 + 0.6 CH2O + 0.25 MACR + 0.35 MVK + 0.4 HYDRALD 2.4e-12
ISOPO2 + HO2 → ISOPOOH 8.0e-13 × exp(700/T)
ISOPOOH + OH → 0.8 XO2 + 0.2 ISOPO2 1.5e-11 × exp(200/T)
ISOPO2 + CH3O2 → 0.25 CH3OH + HO2 + 1.2 CH2O + 0.19 MACR + 0.26 MVK + 0.3 HYDRALD 5.0e-13 × exp(400/T)
ISOPO2 + CH3CO3 → CH3O2 + HO2 + 0.6 CH2O + 0.25 MACR + 0.35 MVK + 0.4 HYDRALD 1.4e-11
ISOPNO3 + NO → 1.206 NO2 + 0.794 HO2 + 0.072 CH2O + 0.167 MACR + 0.039 MVK + 0.794 ONITR 2.7e-12 × exp(360/T)
ISOPNO3 + NO3 → 1.206 NO2 + 0.072 CH2O + 0.167 MACR + 0.039 MVK + 0.794 ONITR + 0.794 HO2 2.4e-12
ISOPNO3 + HO2 → XOOH + 0.206 NO2 + 0.794 HO2 + 0.008 CH2O + 0.167 MACR + 0.039 MVK + 0.794 ONITR 8.0e-13 × exp(700/T)
ONITR + OH → HYDRALD + 0.4 NO2 + HO2 4.5e-11
ONITR + NO3 → HO2 + NO2 + HYDRALD 1.4e-12 × exp(–1860/T)
HYDRALD + OH → XO2 1.9e-11 × exp(175/T)
XO2 + NO → NO2 + HO2 + 0.5 CO + 0.25 GLYOXAL + 0.25 HYAC + 0.25 CH3COCHO + 0.25 GLYALD 2.7e-12 × exp(360/T)
XO2 + NO3 → NO2 + HO2 + 0.5 CO + 0.25 HYAC + 0.25 GLYOXAL + 0.25 CH3COCHO + 0.25 GLYALD 2.4e-12
XO2 + HO2 → XOOH 8.0e-13 × exp(700/T)
XO2 + CH3O2 → 0.3 CH3OH + 0.8 HO2 + 0.7 CH2O + 0.2 CO + 0.1 HYAC + 0.1 GLYOXAL + 0.1 CH3COCHO + 0.1 GLYALD 5.0e-13 × exp(400/T)
XO2 + CH3CO3 → 0.5 CO + CH3O2 + HO2 + CO2 + 0.25 GLYOXAL + 0.25 HYAC + 0.25 CH3COCHO + 0.25 GLYALD 1.3e-12 × exp(640/T)
XOOH + OH → H2O + XO2 1.9e-12 × exp(190/T)
XOOH + OH → H2O + OH 7.7e-17 × T2 × exp(253/T)
MVK + OH → MACRO2 4.1e-12 × exp(452/T)
MVK + O3 → 0.8 CH2O + 0.95 CH3COCHO + 0.08 OH + 0.2 O3 + 0.06 HO2 + 0.05 CO + 0.04 CH3CHO 7.5e-16 × exp(–1521/T)
MEK + OH → MEKO2 2.3e-12 × exp(–170/T)
MEKO2 + NO → CH3CO3 + CH3CHO + NO2 4.2e-12 × exp(180/T)
MEKO2 + HO2 → MEKOOH 7.5e-13 × exp(700/T)
MEKOOH + OH → MEKO2 3.8e-12 × exp(200/T)
MACR + OH → 0.5 MACRO2 + 0.5 H2O + 0.5 MCO3 1.9e-11 × exp(175/T)
MACR + O3 → 0.8 CH3COCHO + 0.275 HO2 + 0.2 CO + 0.2 O3 + 0.7 CH2O + 0.215 OH 4.4e-15 × exp(–2500/T)
MACRO2 + NO → NO2 + 0.47 HO2 + 0.25 CH2O + 0.53 GLYALD + 0.25 CH3COCHO + 0.53 CH3CO3 + 0.22 HYAC + 0.22 CO 2.7e-12 × exp(360/T)
MACRO2 + NO → 0.8 ONITR 1.3e-13 × exp(360/T)
MACRO2 + NO3 → NO2 + 0.47 HO2 + 0.25 CH2O + 0.25 CH3COCHO + 0.22 CO + 0.53 GLYALD + 0.22 HYAC + 0.53 CH3CO3 2.4e-12
MACRO2 + HO2 → MACROOH 8.0e-13 × exp(700/T)
MACRO2 + CH3O2 → 0.73 HO2 + 0.88 CH2O + 0.11 CO + 0.24 CH3COCHO + 0.26 GLYALD + 0.26 CH3CO3 + 0.25 CH3OH + 0.23 HYAC 5.0e-13 × exp(400/T)
MACRO2 + CH3CO3 → 0.25 CH3COCHO + CH3O2 + 0.22 CO + 0.47 HO2 + 0.53 GLYALD + 0.22 HYAC + 0.25 CH2O + 0.53 CH3CO3 1.4e-11
MACROOH + OH → 0.5 MCO3 + 0.2 MACRO2 + 0.1 OH + 0.2 HO2 2.3e-11 × exp(200/T)
MCO3 + NO → NO2 + CH2O + CH3CO3 5.3e-12 × exp(360/T)
MCO3 + NO3 → NO2 + CH2O + CH3CO3 5.0e-12
MCO3 + HO2 → 0.25 O3 + 0.25 CH3COOH + 0.75 CH3COOOH + 0.75 O2 4.3e-13 × exp(1040/T)
MCO3 + CH3O2 → 2 CH2O + HO2 + CO2 + CH3CO3 2.0e-12 × exp(500/T)
MCO3 + CH3CO3 → 2 CO2 + CH3O2 + CH2O + CH3CO3 4.6e-12 × exp(530/T)
MCO3 + MCO3 → 2 CO2 + 2 CH2O + 2 CH3CO3 2.3e-12 × exp(530/T)
MCO3 + NO2 + M → MPAN + M 1.1e-11 × (300/T)/[M]
MPAN + M → MCO3 +NO2 + M kMCO3+NO2 × 1.1e+28 × exp(–14000/T)
MPAN + OH + M → 0.5 HYAC + 0.5 NO3 + 0.5 CH2O + 0.5 HO2 + 0.5 CO2 + M k0 = 8.0e-27 × (T/300)–3.5
k∞ = 3.0e-11
f = 0.5
C-7 degradation (lumped aromatics represented by toluene C7H8) Rate constant
TOLUENE + OH → 0.25 CRESOL + 0.25 HO2 + 0.7 TOLO2 1.7e-12 × exp(352/T)
TOLO2 + NO → 0.45 GLYOXAL + 0.45 CH3COCHO + 0.9 BIGALD + 0.9 NO2 + 0.9 HO2 4.2e-12 × exp(180/T)
TOLO2 + HO2 → TOLOOH 7.5e-13 × exp(700/T)
TOLOOH + OH → TOLO2 3.8e-12 × exp(200/T)
CRESOL + OH → XOH 3.0e-12
XOH + NO2 → 0.7 NO2 + 0.7 BIGALD + 0.7 HO2 1.0e-11
C-10 degradation (terpenes lumped as α-pinene C10H16) Rate constant
TERPENE + OH → TERPO2 1.2e-11 × exp(444/T)
TERPENE + O3 → 0.7 OH + MVK + MACR + HO2 1.0e-15 × exp(–732/T)
TERPENE + NO3 → TERPO2 + NO2 1.2e-12 × exp(490/T)
TERPO2 + NO → 0.1 CH3COCH3 + HO2 + MVK + MACR + NO2 4.2e-12 × exp(180/T)
TERPO2 + HO2 → TERPOOH 7.5e-13 × exp(700/T)
TERPOOH + OH → TERPO2 3.8e-12 × exp(200/T)
D.3.3 Halogen Chemistry
O(1D) reactions with halogens Rate constant
O1D + CFCl3 → 3 Cl 1.7e-10
O1D + CF2Cl2 → 2 Cl 1.2e-10
O1D + CCl2FCClF2 → 3 Cl 1.5e-10
O1D + CHF2Cl → Cl 7.2e-11
O1D + CCl4 → 4 Cl 2.8e-10
O1D + CH3Br → Br 1.8e-10
O1D + CF2ClBr → Cl + Br 9.6e-11
O1D + CF3Br → Br 4.1e-11
Inorganic chlorine reactions Rate constant
Cl + O3 → ClO + O2 2.3e-11 × exp(–200/T)
Cl + H2 → HCl + H 3.1e-11 × exp(–2270/T)
Cl + H2O2 → HCl + HO2 1.1e-11 × exp(–980/T)
Cl + HO2 → HCl + O2 1.8e-11 × exp(170/T)
Cl + HO2 → OH + ClO 4.1e-11 × exp(–450/T)
Cl + CH2O → HCl + HO2 + CO 8.1e-11 × exp(–30/T)
Cl + CH4 → CH3O2 + HCl 7.3e-12 × exp(–1280/T)
ClO + O → Cl + O2 2.8e-11 × exp(85/T)
ClO + OH → Cl + HO2 7.4e-12 × exp(270/T)
ClO + OH → HCl + O2 6.0e-13 × exp(230/T)
ClO + HO2 → O2 + HOCl 2.7e-12 × exp(220/T)
ClO + NO → NO2 + Cl 6.4e-12 × exp(290/T)
ClO + ClO → 2Cl + O2 3.0e-11 × exp(–2450/T)
ClO + ClO → Cl2 + O2 1.0e-12 × exp(–1590/T)
ClO + ClO → Cl + OClO 3.5e-13 × exp(–1370/T)
HCl + OH → H2O + Cl 2.6e-12 × exp(–350/T)
HCl + O → Cl + OH 1.0e-11 × exp(–3300/T)
HOCl + O → ClO + OH 1.7e-13
HOCl + Cl → HCl + ClO 2.5e-12 × exp(–130/T)
HOCl + OH → H2O + ClO 3.0e-12 × exp(–500/T)
ClONO2 + O → ClO + NO3 2.9e-12 × exp(–800/T)
ClONO2 + OH → HOCl + NO3 1.2e-12 × exp(–330/T)
ClONO2 + Cl → Cl2 + NO3 6.5e-12 × exp(135/T)
Three-body and reverse reactions
ClO + ClO + M → Cl2O2 + M k0 = 1.6e-32 × (T/300)–4.5
k∞ = 2.0e-12 × (T/300)–2.4
Cl2O2 + M → ClO + ClO + M kClO+ClO × 5.8e26 × exp(–8649/T)
ClO + NO2 + M → ClONO2 + M k0 = 1.8e-31 × (T/300)–3.4
k∞ = 1.5e-11 × (T/300)–1.9
Inorganic bromine reactions Rate constant
Two-Body Reactions
Br + O3 → BrO + O2 1.7e-11 × exp(–800/T)
Br + HO2 → HBr + O2 4.8e-12 × exp(–310/T)
Br + CH2O → HBr + HO2 + CO 1.7e-11 × exp(–800/T)
BrO + O → Br + O2 1.9e-11 × exp(230/T)
BrO + OH → Br + HO2 1.7e-11 × exp(250/T)
BrO + HO2 → HOBr + O2 4.5e-12 × exp(460/T)
BrO + NO → Br + NO2 8.8e-12 × exp(260/T)
BrO + ClO → Br + OClO 9.5e-13 × exp(550/T)
BrO + ClO → Br + Cl + O2 2.3e-12 × exp(260/T)
BrO + ClO → BrCl + O2 4.1e-13 ×
exp(290/T)
BrO + BrO → 2Br + O2 1.5e-12 × exp(230/T)
HBr + OH → Br + H2O 5.5e-12 × exp(200/T)
HBr + O → Br + OH 5.8e-12 × exp(–1500/T)
HOBr + O → BrO + OH 1.2e-10 × exp(–430/T)
BrONO2 + O → BrO + NO3 1.9e-11 × exp(215/T)
Three-body reactions
BrO + NO2 + M → BrONO2 + M k0 = 5.2e-31 × (T/300)–3.2
k∞ = 6.9e-12 × (T/300)–2.9
Organic halogen reactions with Cl, OH Rate constant
CH3Cl + Cl → HO2 +CO +2 HCl 2.2e-11 × exp(–1130/T)
CH3Cl + OH → Cl +H2O + HO2 2.4e-12 × exp(–1250/T)
CH3CCl3 + OH → H2O + 3 Cl 1.6e-12 × exp(–1520/T)
CHF2Cl + OH → Cl +H2O + CF2O 1.1e-12 × exp(–1600/T)
CH3Br + OH → Br +H2O + HO2 2.4e-12 × exp(–1300/T)
D.4 Heterogeneous Reactions
The following table lists the dimensionless reactive uptake probabilities γ of different heterogeneous reactions. The symbol OC stands for organic carbon, SO4 for sulfate, NH4NO3 for ammonium nitrate and SOA for secondary organic aerosols. r denotes the particle radius.
Heterogeneous reactions on tropospheric aerosols Reactive uptake probability γ
N2O5 → 2 HNO3 0.1 on OC, SO4, NH4NO3, SOA
NO3 → HNO3 0.001 on OC, SO4, NH4NO3, SOA
NO2 → 0.5 OH + 0.5 NO + 0.5 HNO3 0.0001 on OC, SO4, NH4NO3, SOA
HO2 → 0.5 H2O2 0.2 on OC, SO4, NH4NO3, SOA
Stratospheric sulfate aerosol reactions Reactive uptake probability γ
N2O5 → 2 HNO3 0.04
ClONO2 → HOCl + HNO3 f(sulfuric acid wt%)
BrONO2 → HOBr + HNO3 f(T, p, [HCl], [H2O], r)
ClONO2 + HCl → Cl2 + HNO3 f(T, p, [H2O], r)
HOCl + HCl → Cl2 + H2O f(T, p, [HCl], [H2O], r)
HOBr + HCl → BrCl + H2O f(T, p, [HCl], [HOBr], [H2O], r)
Nitric acid trihydrate reactions Reactive uptake probability γ
N2O5 → 2 HNO3 0.0004
ClONO2 → HOCl + HNO3 0.004
ClONO2 + HCl → Cl2 + HNO3 0.2
HOCl + HCl → Cl2 + H2O 0.1
BrONO2 → HOBr + HNO3 0.3
Ice aerosol reactions Reactive uptake probability γ
N2O5 → 2 HNO3 0.02
ClONO2 → HOCl + HNO3 0.3
BrONO2 → HOBr + HNO3 0.3
ClONO2 + HCl → Cl2 + HNO3 0.3
HOCl + HCl → Cl2 + H2O 0.2
HOBr + HCl → BrCl + H2O 0.3
References
Emmons L. K., Walters S., Hess P. G., et al. (2001) Description and evaluation of the Model for Ozone and Related Chemical Tracers, version 4 (MOZART-4), Geosci. Model Dev., 3, 43–67.
Lamarque J.-F., Emmons L. K., Hess P. G., et al. (2012) CAM-chem: Description and evaluation of interactive atmospheric chemistry in the Community Earth System Model, Geosci. Model Dev., 5, 369–411.
E
Brief Mathematical Review
E.1 Mathematical Functions
A function f from set S to set T is a rule that associates with each element x of set S a unique element y of set T. One writes
y = f(x)
(1)
where y is said to be a function of x, or to be the image of x under f. Function f maps therefore x on y. Element x is called the independent variable and element y the dependent variable. The concept can be extended to multiple independent variables:
y = f(x1, x2, .., xn)
A function written as (1) is said to be explicit. If expressed as
f(x, y) = 0
it is implicit.
Partial Derivatives
We define the partial derivative ∂f/∂x of a function f versus independent variable x as the variation of f relative to infinitesimal variation in x, while keeping all other independent variables constant.
Total Differential
The total differential of function f is defined as the variation in f for an infinitesimal perturbation of all independent variables. If f depends on variables xi (with i = 1, n), the total differential is thus
(2)
Total Derivative
By direct application of relation (2), the total derivative of function f versus any independent variable xk is
Time Derivative: Eulerian Versus Lagrangian
For a function f that depends, for example, on three spatial variables (x, y, z) and on time (t), the total derivative is
or
where u = dx/dt, v = dv/dt, and w = dz/dt are the velocity components. The total derivative on the left-hand side is called the Lagrangian time derivative because it expresses the change of function f following a moving parcel. The partial derivative ∂f/∂t on the right-hand side defines the change in the function at a given point of the domain in response to local sources and sinks. It is called the Eulerian time derivative. The remaining terms on the right-hand side represent the change in f versus time due to the advection of air parcels from other locations in the domain where the value of f is different.
Notations for Differentiation
If equation y = f(x) represents a mathematical relationship between a dependent variable y and an independent variable x, the first and second derivatives of y versus x can be expressed in different equivalent forms:
The notation can be generalized for higher-order derivatives. In this book we generally use Leibniz’s notation but adopt the Lagrangian notation in some cases. The “dot” notation of Newton is used in the fluid dynamics literature to express time derivatives.
For a function y = f(x, y) of two independent variables x and y, the first and second partial derivatives are often expressed using the following notations:
Differential operators for a scalar field φ and a vector field a are expressed as follows:
where ∇ is a differential operator expressed as a vector whose Cartesian components are .
Higher-Order Derivatives
The second-order derivatives of function f(x, y) versus independent variables x and y are expressed by
This can be generalized to all higher-order derivatives, for example
Taylor Expansion of f(x)
If f(x) is a function whose successive derivatives exist, it can be expressed by an infinite polynomial series
where f(x0) is the value of f estimated at a point x0, and [∂if/∂xi]xo represents the ith order derivative of function f evaluated at xo. In many applications, function f(x) is approximated in the vicinity of point x0 by a finite Taylor expansion limited to order n. Terms with higher order derivatives are neglected. The accuracy of the approximation increases with the value of n. Nonlinear functions can be linearized around x0 by limiting the Taylor expansion to the first order
Taylor expansions are the basis for many numerical methods.
E.2 Scalars and Vectors
Scalars
A scalar is a physical quantity that is completely defined by a single number. Examples are temperature or pressure at a given location and time. A scalar field associates a scalar value to every point in space. In many atmospheric applications, a scalar field such as the temperature or the concentration of a chemical species is provided by a real function f(x, y, z, t) expressed as a function of three independent spatial variables (x, y, z) and time (t).
Vectors
A vector x (noted by a bold symbol) of dimension n is an ordered collection of n elements called components:
The elements of a vector are usually numbers but can also be functions. A vector field associates a vector to every point of a Euclidean space. Vectors are used in physics to represent physical quantities that have both a magnitude and a direction such as force, velocity, acceleration, flux. Vectors are also basic tools of matrix algebra, used in atmospheric chemistry for statistical applications. In such applications, one may for example define a concentration vector x where the components xi represent the concentrations of the different species i.
Vector elements are usually arranged as a column, as shown in the example above; one sometimes refers t
o column vectors, but “column” is generally assumed by default. When vector elements are arranged in a row one refers to a row vector. A column vector is an (n × 1) matrix (n rows × 1 column). When column vectors are written out horizontally in text, we express the vector as its transpose (1 × n) to avoid ambiguity with row vectors. Thus we write x = (x1, x2, …xn)T for the column vector in the example above.
In a Cartesian coordinate frame, a vector a can be described by its orthogonal projections on the axes of the reference frame. In a 3-D space (x, y, z), for example, we write
a = axi + ayj + azk
where ax, ay, az are the three components of vector a and (i, j, k) are the unit vectors along the axes (x, y, z) of the coordinate frame. The norm of the vector is
and measures the length (or magnitude, or amplitude) of the vector. The sum of two vectors a and b is a resultant vector c represented by the diagonal of a parallelogram whose adjacent sides are the two vectors. The components of the resultant vector are
cx = ax + bx cy = ay + by cz = az + bz
Scalar Product
The scalar product (also known as the inner or dot product) of vectors a and b is a scalar whose value is
a · b = ‖a‖‖b‖ cos θ = axbx + ayby + azbz
Here θ denotes the angle between the two vectors. The following rules are satisfied by scalar products (m is a scalar):
Vector Product
The vector product (also known as the outer or cross product) of vectors a and b is a vector c = a × b directed perpendicularly to the plane defined by a and b. Stretch out the three fingers of your right hand so that your middle finger is perpendicular to the plane defined by your thumb and your index finger (the “right hand rule”). If your thumb is vector a, and your index finger is vector b, then c is oriented in the direction of your middle finger. The amplitude of c is
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