10.2.1. The transport equation
In a selected
V1 volume
of the fluid, mass conservation of the component described with
c concentration can be expressed as:
Where
is the wind vector,
Sc is
the source term and
Dc is the
diffusion coefficient. Equation (10.1) represents the change of the total mass
of the material within volume
V1 as
the sum of the advective flux through the borders of the volume, source terms
inside the volume, and the diffusive flux. In dispersion models, wind field and
other meteorological data is obtained from measurement or a numerical weather
prognostic (NWP) model, thus the only unknown term in equation (10.1) is the
c concentration field. We can transform equation (10.1)
into a differential form using Gauss’ formula and generalizing the integrates to
any
V1
volumes:
This is the dispersion equation that describes advection, source and molecular
diffusion processes. Dry and wet deposition, chemical or radioactive decay is
part of the
Sc
term, while gravitational settling can be added as an extra advection component.
Turbulent diffusion, however, is not represented in equation (2).
Turbulence is usually taken into account with Reynold’s theory that splits the
wind and concentration field into time-averaged and turbulent perturbation
values:
From equations (10.2) and (10.3, 10.4) we can construct a dispersion equation
that represents both time-averaged and perturbation components:
Time averaging equation (10.5) will eliminate all the components that contain
a single perturbation term, as the Reynolds model is based on the assumption
that turbulent perturbations’ time average is zero. However, not all
perturbations will disappear as the covariance term’s time average is not
necessarily 0:
We can here conclude that dispersion equation (10.5) for turbulent flows can
be written in the same form as equation (10.2) with the addition of three eddy
covariance terms. Writing the turbulent components explicitly and assuming an
isotropic molecular diffusion, equation (10.6) can be rewritten in a form that
is widely used in atmospheric dispersion modelling:
The right side of equation (10.7) describes advection, source terms, molecular
diffusion and horizontal and vertical turbulent fluxes. In the atmosphere,
turbulent mixing is magnitudes more efficient than molecular diffusion thus the
third component of equation (10.6) can usually be neglected. However, in the
laminar layer within 0.1 − 3 cm of the ground, turbulence is very weak,
therefore molecular diffusion gets a large importance in the investigation of
soil-atmosphere fluxes and deposition processes, often treated as a resistance
term.
Equation (10.7) involves four new variables in the equation. There are two
ways for the closure of the turbulent dispersion equation: either we construct
new transport equations for the turbulent fluxes or we use parameterization to
express the turbulent fluxes with time-averaged concentration and wind values.
The former approach leads to the Reynolds Stress Models (RSM), while the latter
is the widely used gradient transport theory, or K-theory. These are presented
in details in Stull (1988), here we provide a brief outline of their results.
As an analogy of Fick’s law for molecular diffusion, gradient transport theory
is based on the assumption that the
x directional turbulent
flux is proportional to the first component of the gradient of the concentration
field:
where K
x is the x directional turbulent diffusion
coefficient or eddy diffusivity. Using this approach equation (10.7) results in
a form where turbulent fluxes are expressed as an additional diffusion
term:
where
K is a diagonal matrix of the
K
x, K
y,
K
z eddy diffusivities. Due to the different
atmospheric turbulent processes in horizontal and vertical direction,
K cannot be assumed to be
isotropic. Furthermore, while
Dc is a
property of the chemical species,
K is a property of the flow, thus
it varies in both space and time. Assuming an incompressible fluid and isotropic
horizontal turbulence and neglecting the molecular diffusion, the dispersion
equation can be written in a form:
where
is the horizontal divergence operator,
Kh is the
horizontal and
Kz is the
vertical eddy diffusivity. These two parameters need to be estimated at each
grid point and timestep through various parameterizations.
Reynolds Stress Models (RSM) construct new transport equations for the
turbulent fluxes based on the turbulent kinetic energy and dissipation values
(Launder et al., 1975). This approach has larger computational cost than
gradient transport models, however, its results proved to be more accurate in
several microscale cases (Rossi and Iaccarino, 2009, Chen, 1996). While RSMs
have been successfully applied in CFD-based microscale atmospheric models
(Riddle et al., 2004), on meso- and macroscale, computationally more efficient
eddy diffusivity approach is used (Draxler and Hess, 1998, Lagzi et al., 2009).
The dispersion equation (10.10) can be solved numerically with spatial
discretization of variables on a grid, which is often referred to as the
Eulerian approach. Under some assumptions, (10.10) can also be solved
analytically and provides a Gaussian distribution that is widely used in
Gaussian dispersion models. A stochastic solution also exists where instead of
solving the partial differential equation (PDE), equation (10.10), the
concentration field is given as a superposition of a large number of drifting
particles (Lagrangian approach).
10.2.2. Turbulence parameterization
Turbulence is a key process in dispersion simulations: while downwind
dispersion is usually dominated by advection, crosswind or even upwind turbulent
mixing is magnitudes more efficient than molecular diffusion. Turbulence is
treated on different scales: macroscale turbulence, with a scale larger than the
numerical weather prediction (NWP) model’s grid resolution, is computed
explicitly within the NWP thus it is taken into account in the advection term.
Subgrid-scale turbulence causes a velocity and concentration fluctuation, often
referred to as turbulent diffusion. We note that subgrid-scale velocity
fluctuation also has an effect on the large scale flow through turbulent
viscosity. This effect is treated with various turbulence parameterizations in
the NWP models.
Atmospheric dispersion can be regarded as a sum of two main effects: the
mechanical turbulence caused by wind shear, and the thermal turbulence caused by
buoyancy. Their characteristics and dependence on measurable variables is very
different (Table 10.1).
Mechanical turbulence estimates rely on 3D wind field measurement data to
obtain wind shear values. Surface roughness also has an important role in
generating mechanical turbulence through friction in the ground layer. Surface
roughness is usually measured with the
z0
roughness length, a characteristic length of surface obstacles. Typical
roughness lengths of different surfaces are shown in table 10.2. The models make
the difference between complex terrain and surface roughness upon the scale of
the problem: while the flow around large scale geometry covered with multiple
grid points is computed explicitly in the model, sub-grid scale geometry is
treated as roughness and parameterized through its effect on turbulence.
Table 10.1: Parameters that affect turbulence patterns in the planetary
boundary layer (PBL)
Table 10.2: Typical roughness length and albedo values of different
surfaces
Thermal turbulence depends largely on the atmospheric stability and the
surface’s radiation budget. Under stable conditions, thermal turbulence is low,
and mixing is determined by the wind shear strength. On the other hand, in a
convective boundary layer (CBL), thermal turbulence has a significant role in
dispersion processes. Surface parameters, like albedo (Table 10.2) or potential
evapotranspiration have a large importance in the estimation of sensible and
latent heat fluxes that determine the thermal turbulence intensity.
The strength of the turbulence can be described using various measures. One of
them is eddy diffusivities
(m
2s
-1) that represent
a diffusion coefficient in the dispersion equation (10.10). Another approach is
to estimate the
u’,
v’,
w’ wind fluctuation components (equation 10.3), and
calculate the turbulent kinetic energy (
TKE,
Jkg
-1) that describe the time-averaged energy of
subgrid-scale turbulent eddies (Stull, 1988):
While eddy diffusivities and turbulent kinetic energy explicitly describe
turbulence intensity, there are dispersion-oriented characteristics that try to
estimate the mixing efficiency instead of describing the turbulence itself.
Mixing efficiency is often treated as the deviation of a Gaussian plume, which
is widely used in Gaussian and puff models. Lagrangian models use a stochastic
random-walk simulation for mixing.
As both wind and temperature changes in space and time, atmospheric turbulence
is a non-homogenous, non-stationary field. Furthermore, thermal turbulence is
highly anisotropic, which means a difficult challenge for turbulence models
especially in a convective boundary layer and leads to the separated treatment
of horizontal and vertical turbulence in equation (R10.10). While horizontal
dispersion is usually dominated by advection, vertical mixing of the planetary
boundary layer (PBL) is caused by turbulence because of the large vertical wind
shear and temperature gradients, together with the fairly low vertical wind
speeds. This means that vertical turbulence is a key process in atmospheric
dispersion simulations, and requires sophisticated methods to estimate its
strength.
The vertical profile of vertical eddy diffusivity is presented in Figure 10.1
after Kumar and Sharan (2012). It can be seen that far from the ground,
turbulence intensity decreases with height thus the
h
planetary boundary layer height can be defined as the elevation where turbulence
becomes neglectable. On the other hand, near-ground turbulence intensity fast
increases with height, and reaches its maximum value at the elevation
approximately 30–40 % of the PBL height. Near-zero eddy diffusivity at the top
of the PBL means that there is no vertical mixing upward, thus pollutants
released from the surface will stay in the boundary layer. The PBL height and
the advection speed together determine the volume in which the pollutant can
dilute within a specified time, thus they have a very significant effect on
concentration estimates.
Table 10.3: Typical PBL height values
PBL height has a strong diurnal and annual variability: it extends over 2000 m
on convective summer days, however, it can shrink to a few 10 meters on clear
nights (Table 10.3). A key process is the collapse of the mixed layer
approximately one hour before sunset, when heat fluxes from the surface are
stopped and thermal turbulence is ceased (Figure 10.2). The night time stable
boundary layer keeps the surface-based pollutants close to the ground, while the
residual layer is detached from the surface until approximately one hour after
sunrise.
10.2.2.1. Stability classes
Atmospheric turbulence is determined by wind shear and thermal
stratification. In the early years of dispersion modelling, it was an
obvious solution to define categories based on easily measurable wind and
radiation characteristics like albedo, cloud cover and sun elevation. The
most popular stability classification is Pasquill’s method that defines six
categories: from the very unstable A to the very stable F class (Table
10.4). Pasquill took into account wind speed, sun elevation and cloud cover
data to determine the stability class that provided pre-defined mixing
efficiency values. Despite the fact that this classification can handle only
six discrete values of turbulence intensity, Pasquill’s method was used for
scientific and regulatory purposes for many decades (Turner, 1997, Sriram et
al., 2006).
Table 10.4: Pasquill’s stability classes from the very unstable (A) to the
very stable (F) atmosphere. Note: neutral (D) class has to be used for
overcast conditions and within one hour after sunrise / before sunset.
[Source:
http://ready.arl.noaa.gov/READYpgclass.php]
10.2.2.2. Richardson number
The difference between thermal and mechanical turbulence can be regarded
as a separation of the subgrid-scale turbulent energy to potential (buoyant)
and kinetic energy. Their ratio is described with the gradient Richardson
number:
where
is density and
g is the gravitational
acceleration. The numerator of equation (10.12) describes buoyancy, while
the denominator is the wind shear term. It can be observed that the buoyancy
can take both positive and negative values, representing both the unstable
stratification that generates turbulence, and the stable stratification that
destroys turbulence. The wind shear term, however, can take only positive
values as it is always a generator of turbulence. The buoyancy term is the
square of the Brunt-Vaisala-frequency, and is often written with more
practical potential temperature gradients instead of density (Stull, 1988).
Richardson number gives a first-guess estimate on atmospheric stability
and turbulence intensity. In the
Ri < 0 case,
unstable stratification develops thermal turbulence, which describes a
convective boundary layer (CBL). Zero Richardson number means a neutral
stratification, thus only mechanical turbulence is present. If the
Richardson number is positive, but less than a critical
Ric
value (
0 < Ri <
Ric),
mechanical turbulence overrides stable stratification that leads to a
mechanic turbulence dominated boundary layer. For Richardson numbers larger
than
Ric,
atmospheric stability destroys mechanical turbulence and results in a stable
boundary layer (SBL). Estimates for the critical Richardson number are in a
range 0.15-0.5, but it was also showed that taking qualitative difference
between turbulence patterns based on a pre-defined critical value can lead
to unrealistic results. However, in many cases, when surface characteristics
and fluxes are not available to perform more sophisticated simulations, the
Richardson number is used to estimate the turbulence intensity and/or PBL
height.
10.2.2.3. Monin–Obukhov-theory
Monin and Obukhov developed a theory in 1954 for atmospheric turbulence
that fully described vertical profiles of turbulence intensities. Their
basic idea was to combine the vertical turbulent momentum and heat fluxes
into a single number with dimension of length, the so called
Monin–Obukhov-length
where
T’,
u’,
w’ are the temperature, horizontal and vertical wind
turbulent fluctuations,
T is the temperature and
is the von Karman constant, usually taken equal to 0.4
(Stull, 1988). Equation (10.12) is valid for dry air, however, for moist
air, virtual temperature or virtual potential temperature is used (Stull,
1988). Hardly measurable turbulent fluxes can be estimated with the
definition of two parameters: the
u*
friction velocity and the
q turbulent heat flux. Using
these new parameters, the Monin–Obukhov-length can be expressed as a
function of measurable variables (Foken, 2006)
where
cp
is the specific heat and
is the density of the air. There are different methods for
the determination of
u*
and
q using net radiation components and (potential)
temperature gradient, which largely differ in the convective and in the
stable boundary layer.
The Monin–Obukhov-length in (10.13) is a reference scale for boundary
layer turbulence, thus any
z height can be given with
the dimensionless
z/L parameter. The
z/L is not only a new vertical coordinate, but also
a stability parameter: at any given
z height,
turbulence intensity can be described with the
z/L
value (Figure 10.1).
Monin and Obukhov defined universal functions for neutral, stable and
unstable cases that describe vertical profiles of wind and temperature as a
function of the
z/L height (Foken, 2006). The
u*
friction velocity is often also computed as a function of
z/L using an iterative solution (Cimorelli et al.,
2005). The turbulent momentum and heat fluxes are given explicitly from the
u*
and
q values (as in equations 10.13-10.14), however,
dispersion models require eddy diffusivity profiles as well. These are
obtained through model-dependent parameterizations using the
u* value, the
vertical wind and temperature profile provided by the Monin–Obukhov-theory
and the universal functions of
z/L that describe
stability. The Monin–Obukhov-theory was validated against numerous
measurement and advanced computational fluid dynamics datasets, and although
some weaknesses were shown, it still means the most reliable basis for all
fields of atmospheric modelling (Foken, 2006).
10.2.3. Chemical reactions and radioactive decay
Chemical transformation of air pollutants can be diversified and complex
especially in the troposphere or stratosphere, where thermal as well as
photochemical reactions can occur. Photochemical reactions can produce radicals,
which have high affinity to react with other chemical compounds in the
atmosphere. This driving force and rich variety of atmospheric components can
really provide a complex network of chemical reactions. Formation of air
pollutants can be described by reaction mechanisms. A reaction mechanism is a
set of elementary reactions by which overall chemical change occurs. Variation
of concentrations of air pollutants in the atmosphere can be constructed from
the rate equations and can be calculated by a set of ordinary differential
equations (ODEs)
where
is the chemical rate constant for
n
th reaction and
is the order of reaction in respect for a given chemical
species. Usually, for a thermal reaction the rate constant is a “real” constant,
however, for a photochemical reaction the rate constant depends on light
intensity, and this dependence can be expressed through the annual and diurnal
variation of the solar zenith angle (
)
where
a and
b are parameters
depending on the given photolytic reaction.
Radioactive decay in environmental models can be interpreted as a first order
chemical reaction
Solution of this equation (10.17) is an exponential function in time.
Radioactive decay constant (
k) is an inherent property of
the radionuclides. Models can take into account consecutive decays with
different decay constants, which is mathematically not a complicated task.
Equations (10.16, 10.17) are added to the source term
(
Sc) of atmospheric transport
equations (10.10). Models using concentration field (e.g., Eulerian model) can
easily handle this mean field description of the concentrations described by
ODEs.
However, if we consider dispersion of air pollutants as a dispersion of
individual particles, using deposition, chemical and decay rate constants could
be meaningless, because these quantities represent “bulk” properties of the
system. Here we deal with individual particles, where macroscopic parameters can
hardly be used. We can overcome this problem using probabilistic nature for
first order reaction, radioactive decay and deposition. The probability that
individual particles during a time step
will transform, decay or deposit can be estimated by the
following relation
where
k is either the “macroscopic” radioactive decay
constant (first order rate constant) or wet or dry deposition constant. Thus we
can connect microscopic event to macroscopic properties and constants.
Copy of this source: http://elte.prompt.hu/sites/default/files/tananyagok/AtmosphericChemistry/ch10s02.html
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