Chemical Mixing Ratios: Eulerian & Lagrangian Approaches, Operator Splitting, Slides of Chemistry

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GLOBAL MODELS OF ATMOSPHERIC COMPOSITION

HOW TO MODEL ATMOSPHERIC COMPOSITION?

Solve continuity equation for chemical mixing ratios

C

(x i

,^

t)

Fires

Landbiosphere

Humanactivity

Lightning

Ocean

Volcanoes

Transport

Eulerian form:

i

i^

i^

i

C

C

P

L

t







 







U

Lagrangian form:

i

i^

i

dC

P

L

dt





U

= wind vector

P

i^

=

local sourceof chemical

i

L

i^

= local sink

Chemistry

Aerosol microphysics

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OPERATOR SPLITTING IN EULERIAN MODELS

i^

i^

i

TRANSPORT

LOCAL

C

C

dC

t

t

dt

… and integrate each process separately over discrete time steps:

(Local)•(Transport)

i^

o

i^

o

C

t

t

C

t

Split the continuity equation into contributions from transport and local terms:

Transport

advection, convection:

Local

chemistry, emission, deposition, aerosol processes:

i

i

TRANSPORT

i

i

LOCAL

dC

C

dt

dC

P

dt

U

i

L

C

C

These operators can be split further: •

split transport into 1-D advective and turbulent transport for

x, y, z

(usually necessary)

split local into chemistry, emissions, deposition (usually not necessary)

Reduces dimensionality of problem

QUESTIONS

1.

The Eulerian form of the continuity equation is a first-order PDE in4 dimensions. What are suitable boundary conditions for each ofthese dimensions?

1.

Textbooks will often tell you that operator splitting (transport vs.local in the continuity equation) requires time steps that are muchsmaller than the time scales for change in the system, but itactually also works fine for species that are very short-livedrelative to the time step. Error is largest for species that havelifetimes of magnitude comparable to the splitting time step.Explain.

SOLVING THE EULERIANADVECTION EQUATION

Equation is

conservative:

need to avoid

diffusion or dispersion of features. Also needmass conservation, stability, positivity… •

All schemes involve finite difference approximation of derivatives : order ofapproximation

→

accuracy of solution

Classic schemes: leapfrog, Lax-Wendroff, Crank-Nicholson, upwind, moments… •

Stability requires Courant number

u

t/

x

< 1

… limits size of time step •^

Addressing other requirements (e.g., positivity) introduces non-linearity in advection scheme

i^

i

C

C

u

t

x





 





VERTICAL TURBULENT TRANSPORT (BUOYANCY)

Convective cloud(0.1-100 km)

Model grid scale

Modelverticallevels

updraft

entrainment

downdraft

detrainment

Wet convection issubgrid scale in globalmodels and must betreated as a verticalmass exchangeseparate from transportby grid-scale winds.Need info on convectivemass fluxes from themodel meteorologicaldriver.

generally dominates over mean vertical advection

K-diffusion OK for dry convection in boundary layer (small eddies)

Deeper (wet) convection requires non-local convective parameterization

SPECIFIC ISSUES FOR AEROSOL CONCENTRATIONS

A given aerosol particle is characterized by its size, shape, phases, and chemical composition – large number of variables! •

Measures of aerosol concentrations must be given in some integral form, by summing over all particles present in a given air volume thathave a certain property •

If evolution of the size distribution is not resolved, continuity equation for aerosol species can be applied in same way as for gases •

Simulating the evolution of the aerosol size distribution requires inclusion of nucleation/growth/coagulation terms in

P

i^

and

L

, and size i

characterization either through size bins or moments.

Typical aerosolsize distributionsby volume

nucleation

condensation

coagulation

LAGRANGIAN APPROACH: TRACK TRANSPORT OF

POINTS IN MODEL DOMAIN (NO GRID)

U



t

U’



t

Transport large number of points with trajectories from input meteorological data base (U) + randomturbulent component (U’) over time steps



t

Points have mass but no volume

Determine local concentrations as the number of points within a given volume •

Nonlinear chemistry requires Eulerian mapping at every time step (semi-Lagrangian)

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

position

t

o

position t

o

+



t

EMBEDDING LAGRANGIAN PLUMES IN EULERIAN MODELS

Release puffs from point sources and transport them along trajectories,allowing them to gradually dilute by turbulent mixing (“Gaussianplume”) until they reach the Eulerian grid size at which point they mixinto the gridbox

Advantages: resolve subgrid ‘hot spots’ and associated nonlinear processes (chemistry, aerosol growth) within plume •

Difference with Lagrangian approach is that (1) puff has volume as well as

mass, (2) turbulence is deterministic (Gaussian spread) rather than stochastic

S. California fire plumes,Oct. 25 2004

THE INVERSE MODELING PROBLEM

Optimize values of an ensemble of variables (

state vector

x

) using observations:

THREE MAIN APPLICATIONS FOR ATMOSPHERIC COMPOSITION:1.

Retrieve atmospheric concentrations (

x

) from observed atmospheric

radiances (

y

) using a radiative transfer model as forward model

2.

Invert sources (

x

) from observed atmospheric concentrations (

y

) using a

CTM as forward model

3.

Construct a continuous field of concentrations (

x

) by assimilation of sparse

observations (

y

) using a forecast model (initial-value CTM) as forward model

a priori estimate

xa

a

observation vector

y

forward model

y = F(x)

+

“MAP solution” “optimal estimate”

“retrieval”

ˆ

ˆ

x +

ε

Bayes’theorem

SIMPLE LINEAR INVERSE PROBLEM FOR A SCALAR

use single measurement used to optimize a single source

a priori

bottom-up estimate

xa

a

Monitoring sitemeasuresconcentration

y

Forward model gives

y = kx

“Observational error”





instrument

fwd model

y = kx







2

2

2

2

ln

|^

ln

|^

ln

a a

x

x

y

kx

P x

y

P y

x

P x



Max of

P

x|y

is given by minimum of cost function

2

2

2

2

a a

x

x

y

kx

J x



Solution:

a

a

x

x

g

y

kx

where

g

is a

gain factor

2

2

2

2

a

a

k

g

k









(^2)

2

2

1

a

k















Alternate expression of solution:

(

)

a

y

kx

x

ax

a x

g





















where

a = gk

is an

averaging kernel



solve for

dJ

dx

Assume random Gaussian errors, let

x

be the true value. Bayes’ theorem:

Variance of solution:

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GENERALIZATION:

CONSTRAINING

n

SOURCES WITH

m

OBSERVATIONS

1 n

j^

ij

i

i

y

k x



Linear forward model:

A cost function defined as

,^

1

1

2

2

1

1

,^

,

n

j^

ij

i

n

m

i^

a i

i

n

i^

j

a i

j

y

k x

x

x

J x

x













is generally not adequate because it does not account for correlation betweensources or between observations. Need vector-matrix formalism:

1

1

T

T

n

m

x

x

y

y

x

y

y

Kx

Jacobian matrix

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GAUSSIAN PDFs FOR VECTORS

A priori

pdf for

x

:

Scalar

x

Vector

2

2

exp[

]

a a

a

x

x

P x







1

,

1

,

,

1

,

,^

,

var(

cov(

cov(

var(

a

a

n

a n

a

n

a n

n

a n

x

x

x

x

x

x

x

x

x

x

x

x

a

S

1

2 ln

( )

(

)

(

)

T

P

c











a

a

a

x

x

x

S

x

x

1/ 2

(^1) / 2

1

( )

exp[

2

(

)

T

n

a

P





-

a

a

a

x

(x - x ) S

(x - x )]

S

1

(

,...

)

T

n

x

x



x

where

S

a

is the

a priori

error covariance matrix describing error statistics on (

x-x

a

In log space:

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OBSERVATIONAL ERROR COVARIANCE MATRIX

( )





i^

m

y

F x

ε

+

ε

observation

true value

instrument error

fwd model error

observational error

i^

m

ε

=

ε

+

ε

Observational error covariance matrix

1

1

1

var(

)

cov(

,^

)

cov(

,^

)

var(

n )

n

n

























S











is the sum of the instrument and fwd model error covariance matrices:

i^

m

ε

ε

ε

S

= S

+ S

How well can the observing system constrain the

true value

of

x

?

1

2

2 ln

(

)

(

)

(

)

T

P

c















y | x

y

Kx

S

y

Kx

Corresponding pdf, in log space: