Prezentace aplikace PowerPoint

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Atmospheric Dispersion (AD)
Matus Martini
Seinfeld Pandis Atmospheric Chemistry and
Physics Nov 29, 2007
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Outline

• Eulerian approach
• Lagrangian approach
• eqns for mean concentration, solutions for
  instantaneous and continuous source
• Gaussian plume eqn
• AD parameterizations (P-G curves), plume rise

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Air pollution dispersion models

• Box model air pollutants inside the box are
  homogeneously distributed
• Gaussian model is perhaps the oldest (circa 1936)
  
• Lagrangian model - statistics of the trajectories
  of a large number of the pollution plume parcels.
  
• Eulerian model - fixed three-dimensional
  Cartesian grid
• Dense gas model
• Hybrids (Plume in Grid model)

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Leonhard Euler (1707-1783) v. Joseph Louis
Lagrange (1736-1813)
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Lagrange v. Euler

• PROS over Eulerian models
• no Courant number restrictions
• no numerical diffusion/dispersion
• easily track air parcel histories
• invertible with respect to time
• CONS
• need very large points for statistics
• inhomogeneous representation of domain
• convection is poorly represented
• nonlinear chemistry is problematic
• Embedding Lagrangian plumes in Eulerian models
  (PinG model)
• Release puffs from point sources and transport
  them along
• trajectories, allowing them to gradually dilute
  by turbulent mixing
• (Gaussian plume) until they reach the Eulerian
  grid size at which
• point they mix into the gridbox

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Eulerian approach
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Eulerian approach

• If we assume that the presence of small
  concentration species does not affect the
  meteorology to any detectable measure, the
  continuty eqn can be solved independently of the
  coupled momentum and energy eqns
• 1. sufficient heat can be generated by chemical
  reactions to influence the temperature
• 2. absorption, reflection, and scattering of
  radiation by trace gases and particles could
  result in alterations of the fluid behavior
• The flow of interest is turbulent, the fluid
  velocities uj are random functions of space and
  time
• deterministic and stochastic component of
  velocity

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Eulerian approach

• since uj is random ci resulting from the
  solution must also be random -gt probability
  density function for a random process as complex
  as AD is almost never possible -gt mean of
  ensemble of realizations ltcigt
• convenient to express ci as ltcigt ci where by
  definition ltcigt 0
• If the single species decays by a 2nd - order
  reaction
• ltRgt - k ( ltcgt2 ltc2gt)
• closure problem (emergence of new dependent
  variables ltc2gt)
• Eulerian description of turbulent diffusion will
  not permit exact solution even for the mean
  concentration ltcgt

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Lagrangian approach

• behavior of representative fluid particles
• consider a single particle located at location x
  at time t in a turbulent fluid
• trajectory of the subsequent motion Xx,tt
  at any later time t
• probability density function
• integrated over all possible starting points x
• Q(x, t x, t) transition probability density
  the particle originally at x, t will undergo
  a displacement to x at t

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Lagrangian approach

• Ensemble mean concentration
• General formula for the mean concentration
• particles present at t0 added from
  sources between t0 and t
• We need complete knowledge of the turbulence
  properties -gt Q
• Except the simplest circumstances Q is
  unavailable!
• Integrals hold only when no undergoing chemical
  reactions, conservative species!

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• Eulerian statistics are readily measurable (fixed
  network of anemometers)
• can include detailed chemical mechanisms
• serious mathematical obstacle closure problem
• Lagrangian displacements of groups of particles
  released in the fluid
• difficulty of accurately determining the required
  particle statistics not directly applicable to
  problems involving nonlin chem reactions
• Exact solution for the mean concentration even of
  inert species in a turbulent fluid is not
  possible in general!
• Approximations

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Mean concentration K theory

• Eulerian approach approx AD eqn
• Molecular diffusion is negligible compared with
  turbulent diffusion
• linearization incompressible atmosphere
• Ri almost always nonlin, the most obvious approx

• reaction processes are slow compared w/turbulent
  transport
• distribution of sources is smooth (violated
  near strong isolated sources)

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Mean concentration statistical theory

• Lagrangian approach stationary, homogeneous
  Gausian flow
• In highly idealized example u(t) is a random
  variable depending only on time, and is
  stationary, Gaussian random proces, u(t) pdf
• stationarity implies that the statistical
  properties of u at two different times depend
  only on tt and not on t and t individually,
  transition probabilty density
• Then mean concentration is itself Gaussian (!)

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Instantaneous point source

• Eulerian approach
• eddy diffusivities Kxx, Kyy, Kzz const

Solution
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Instantaneous point source

• If we define
• the two expressions are identical
• Evidently, there is a connection between Eulerian
  and Lagrangian approaches

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Continuous source, steady state

• Lagrangian approach
• Source began emitting at t 0, mean
  concentration achieves a steady state
  (independent of time), and source strength q in
  g s-1

slender plume approx advection dominates plume
dispersion (neglecting diffusion in the
direction of the mean flow)
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Continuous point source, steady state

• Eulerian approach

Solution where
slender plume approx interest only in the plume
centerline
Lagrangian and Eulerian expressions are identical
if
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Recapitulation

• Lagrangian and Eulerian solutions are identical
  if
• Instantaneous point source
• Continuous point source
• In most applications of the Lagrangian formulas,
  the dependence of sy2 and sz2 on x are
  determined empirically
• Relationship between K theory and the Gaussian
  formulas

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Summary of AD theories, Lagrange / Euler

• So far only physical processes responsible for
  the dispersion of a cloud or a plume due to only
  velocity fluctuations (instantaneous or
  continuous source in idealized stationary,
  homogeneous turbulence)
• Because of the inherently random character of
  atmospheric motions, one can never predict with
  certainity the distribution of concentration of
  marked particles emitted from a source. Although
  the basic equations describing turbulent
  diffusion are available, there does not exist a
  single mathematical model that can be used as a
  practical means of computing atmospheric
  concentrations over all ranges of conditions.
• The deciding factor in judging the validity of a
  theory for atmospheric diffusion is the
  comparison of its predictions with experimental
  data. Theory gives ensemble mean concentration
  ltcgt, whereas a single experimental observation
  constitues only one sample from the
  hypothetically infinite ensemble of observations.
  (Its practically impossible to repeat an
  experiment more than a few times under identical
  conditions in the atmosphere.)

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Gaussian spreading in 2D have a binormal
distribution
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Plume rise Dh
H effective stack height
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Gaussian Plume Equation

• Lagrangian approach under certain idealized
  conditions (stationary, homogeneous turbulence),
  the mean conc. of species emitted from a point
  source has a Gaussian distribution

• in the slender plume case sx -gt 0

f - crosswind dispersion g vertical dispersion
g1 no reflections, g2 reflection from the
ground, g3 - reflection from an inversion aloft
q emission rate, H effective stack height
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Gaussian Plume Equation

• Eulerian approach It can be shown (use of Green
  function) that we can get to the same result by
  solving the AD eqn (but with const eddy
  diffusivities)!

Johann Carl Friedrich Gauss (1777-1855)
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Dispersion parameters in Gaussian models

• Derived from concentrations measured in actual
  atmospheric diffusion experiments
• where sv , sw are standard deviations of the
  wind velocity fluctuations
• Fy , Fz characterize PBL
• friction velocity u, convective velocity scale
  w
• Monin-Obukhov length L
• Coriolis parameter f
• mixed layer depth zi (upper boundary, the height
  of an elevated layer impermeable to diffusion)
• surface roughness z0
• height of pollutant release above the ground H

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AD Parameterizations

• From two standard deviations more is known about
  sy , since most experiments are ground-level.
  Vertical concentration distributions are needed
  to determine sz
• Ground-leveled releases are not exactly gaussian
  in vertical.
• For complete parameterization we need all these
  variables
• not always available!
• Pasquill stability categories A F (1961)
• Surface windspeed
• Daytime incoming solar radiation
• Nighttime cloud cover
• Correlations for sigmas based on readily
  available ambient data!

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AD Parameterizations
Pasquill stability classes
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Pasquill-Gifford (P-G) curves
Horizontal and
vertical dispersion coeff
Distance from source m
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Behavior of a plume

• initial source conditions exit velocity,Tplume
  Tair
• stratification
• wind speed
• gases are usually released at T hotter than the
  ambient air and are emitted with considerable
  initial momentum
• Buoyant plume Initial buoyancy gtgt initial
  momentum
• Forced plume Initial buoyancy initial
  momentum
• Jet Initial buoyancy ltlt initial momentum

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Analytical properties of Gaussian Plume Eqn

• along the centerline (y0) at the ground
  (z0)
• we need effective stack height H !!
• Maximum ground-level concentration
• derivative w.r.t x 0
• critical downwind distance xc , critical wind
  speed uc

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Critical downwind distance as a function of
source height and a stability class(plume that
has reached its final height)
no xc for stable stratification!
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Summary 3

• Gaussian expressions
• fail near the surface, since no vertical shear
  is present
• no chemical reactions, either
• Eulerian approach
• AD eqn provides more general approach (special
  cases uniform wind speed and constant eddy
  diffusivities), key problem is to choose the
  functional forms of the wind speeds and the eddy
  diffusivities
• Generally, exact solution for the mean
  concentration even of inert species in a
  turbulent fluid is not possible!
• Therefore approximations, K-theory,
  linearizations
• in stationary, homogeneous Gausian flow the
  solution for ltcgt is itself Gaussian!
• Instantaneous and continuous point source
  stationary, homogeneous turbulence, and const
  eddy diffusivities -gt Gaussian plume eqn
  (Lagrange agrees with Euler)
• Experimental data gt parameterizations, P-G
  curves convenient for determining sy , sz
• We saw why the stack height and PBL meteorology
  matter.