Introduction to atmospheric chemistry modelling

Continuity equation for chemical species i

ci = <ci> + ci' avec <ci'> = 0

'uuu

“Atmospheric Dispersion Equation” ( cf Seinfeld and Pandis )

Ri = terme de production/destruction chimique (réactions chimiques) (mol.m -3.s-1)Si = terme sources/puits (émissions, dépôts humides et secs, transports convectifs) (mol.m -3.s-1)

Neglect diffusion

+ first order closure for turbulence

+ Boussinesq approx.

Reynolds av.

Di = diff. Coeff. (m2.s-1, loi de Fick)

chemistry Sinks(deposition)

Dispersion equation generally written as :

Wind and turbulent parametters are suppose to be known ( e.g. calculated by an atmospheric model solving Navier-Stokes equations).

Analytical solutions of the dispersion equation (gaussian models).

If we consider a passive traceer , a constant wind and a stationary turbulence, the dispersion equation simplifies.

e.g. 1D propagation in the direction of the wind problem

We of course need initial conditions on c to solve the problem ….

e.g 1 . Explosive source : cx(x,0) = S1/3δ(x)

We can show that :

(try to draw the evolution of concentration at different time t=0, t1, t2 etc )

e.g2. Continuous punctual source ( chimney stack). Consider isotropic turbulence.

For which we can calculate analytical solution:

Draw the solution in the x,y plan

Practically this gaussian models can be use to calulate concentration in the vicinity (< 2km) of punctual sources like chimney). They can be complexified to include deposition effects. The dispersion (turbulent) parameters can be empirically estimated according to local atmospheric stability.

These models are simple and fast but show their limits as soon as the terrain becomes complex and the wind variable and are not adapted for regional transport studies.

Box models (OD model)

e.g consider the whole atmosphere as a 0D box , How would you write the atmospheric dispersion equation for a compound of mass m (or concentration c) in such a case ?

Life time of the species (cf also biogeochemical cycle class)

Coupled boxes ( coupling between two reservoirs)

)( iBidilk

Lagrangian approaches

Dilution terms depending on the chemical mixing ratios in the air parcel and in the background (account for the effect of turbulence)

Requires to determine air mass trajectories in a Eulerian framework.

Most of atmospheric chemistry model are Eulerian but can integrate Lagrangian capabilities.

Back trajectory models: Inverse the Lagrangian trajectories to locate the geographical origin of a compound measured in a location

Numerical solution of the dispersion Equation

A : Transport : solve the advection-diffusion equation for a tracer (considered passive for theses processes) : Cf numerical method classes + remember all the issues like numerical diffusion, stability (Courant criteria), explicit / implicit schemes etc

Domain and temporal discretisation (cf Numerical Method classes)

Operator splitting :

B: Chemistry : solve the sources and losses due to chemical reactions of compound i compounds j =/ i

+ …..

C : Other processes : use parameterizations to calculate production and losses due to dry deposition , wet deposition, convective transport, emissions in each grid cell at each time step

Solving Chemical Systems

Chemical System = ensemble of coumpounds reacting together ( couple system)

Chemical scheme : ensemble of reactions between compounds describing the coupled sustems

Ordinary differential equation system

Chemical solver = numerical algorithm to solve the ODE

The chemical tendency is often written as the contribution of production and losses

A priori the simplest method we could think of to integrate this system would be a forward Euler explicit scheme , where n is the relative to a given step of the temporal integration , and DT is the time step of the model

OK let’s try if that work with a simplified O3/NOx system …

Which we can iterate given initial conditions Ci0

Consider a time step of 0.01 s , which is already a very short time step for an atmospheric simulation

….. The numerical solution does not make sense, you obtain negative concentrations and concentrations will go fast to infinity …

In order to obtain stability you would need a time step consistent with faster kinetic constant of the system ,

Most of the chemical systems involves kinetics constant that span several order ( up to 12 !) of magnitude. Such systems are numerically “stiff”.

Characterization of the stiffness of a system

We can write generally our system in vectorial form :

If the system is linear

For this system we can calculate the Jacobian matrix

If the system is linear And J = A , matrix of coefficient independent from time and other system variables.

If you calculate the eigenvalues of the Jacobian matrix (λi) , the system will be ‘stiff’ if :

Practically , the complex systems can also be simplified through a number of hypothesis :

Quasi stationary approximation The very reactive species e.ge. OH , O, … are considered to reach equilibrium instantaneously ( compared to the typical time step of a model) . Their time derivative is 0 so you replace a differential equation by an algebraic equation ( you reduce the number of coupled equation in the ODE) . In the previous example we could apply that to [O] and assume that :

k1[O] = k3[NO2] … and reduce the previous ODE system to : (?)

Exponential solution

)1(

)(

)()()( )(.())(.( tLt

i

itLtii

ii etL

tPetPttC

Here we assume that the production and loss are constant during the model time step ( this hypothesis holds if the time step is fine compared to the species lifetime). One of the problem is that it isnot mass conservative

•forward Euler if : Δt.Li < 0.01•exponential if : 0.01 < Δt.Li < 10•quasi-stat si: Δt.Li > 10.

Practically we can use these simple methods following the general criteria :

However for complex systems and even after simplifications, part of the ODE can remains stiff, or the solutions obtained not accurate enough. Implicit methods are then developed

Requiring to solve an algebraic system to calculate cn+1 from cn obtained the Jacobian of the ODE system needs to be inversed : numerical cost !

e.g. euler backward ( does that remind you something ?)

Developing chemical schemes

Inorganic chemistry (e.g. sulfur oxidation chemistry) : - Relatively well known- 20-30 species to consider - explicit methods used ( numerical cost is low)

Organic chemistry (VOC oxidationlatively well known-There are still some unknown mechanisms- 103 - 104 species for 104-105 reactions !- Is it necessary to treat VOC explicitly oxidation chains ?- What information should we conserve for application -in air quality and Earth system models ?

Estimation of kinetic constants for missing data :

link the reactivity to the structure of molecules

Chemical schemes reduction :

“Molecular families” grouping (based on reactivity with OH)

“Characteristic species” , species from the family are assumed to have the same reactivity as a given well known species

“Structural grouping “ Carbon bond mechanisms . Identifies reactive structures in molecules and group across molecules.

Validation of reduced schemes / comparison explicit vs reduced scheme

Other key processes (parameterizations)

EmissionsMany compounds (SO2,Nox, VOCs, HAPs, Heavy Metals etc …One compound can have several sources, anthropogenic or Natural.

Emission sectors :

heating Petro-chem

Forest

Ei s = EF i s × Ns ×Us (for a given chemical species and for a given sector )

EF : Emission factors represents the emission potential of a given emitter. The unit vary according to the process e.g : g.km-1 for car traffic sources; g.l-1 for solvant ; g.kg-1 for bioegenic emission ….

N : represent an “quantity of activity” on a given area (e.g a grid cell) - it scales up emission factor to give actual surface fluxes- , e.g. number of km made , quantity of biomass etc ..

U : a fonction modulating the emission typically in fonction of time (U(t)). Account for dirunal and seasonal variations of emissions.

e.g.Nox emissions / ParisThe integration of these different informations are typically done in SIG

Atmospheric models usesexternally precribed emissions inventory (except in the case of some natural emissions).

Interpolation (or aggregations) are necessary

Dry deposition

Absorption of chemicals by vegetation and soil

Analogy with the Ohm law : the vertical flux of the chemical compound is equivalent to electrical couant and the concentration difference is equivalent to electrical potential differences (tension)

X

tot

zXzXR

)()( 21

)(

1)()(

zV

zXzR

dXtot

cbatot RRRR

Ra : aerodynamic resistance , depends on wind velocity , surface roughness, turbulence close to the surface

Rb : quasi-laminar layer resistance, depends both on turbulence and molecular diffusivity of the species in the quasi laminar layer ( diffusivity will depends on local chemical gradients, thickness of the layer, temperature and the molecular nature of the species considered).

Rc : Canopy resistance . This resistance decomposes in different terms, stomatal resistance, cuticular resistance, soil resistance etc. Rc is connected to the plant physiology which depends on temperature light, soil humidity etc . Rc is often the largest resistance to dry deposition.

Wet deposition ( wet scavenging)

A very complex problem …

Washout : below cloud scavenging by hydrometeors ( rain, snow, ..)

Rainout : in cloud scavenging, suppose first that the compound dissolve in cloud water which is eventually rained out. Aqueous chemistry can happen . Release of compounds after cloud droplet evaporation can happen . Cmplex and species dependant ….

Usually parameterized as a first order process, where the kinetic constant representing washout will depend on rain intensity, hydrometeor type, and solubility of the gas ..

And the washout flux would be given by :

Convective transport of species …

Gets tricky because there is also a sub-grid wet scavenging associated with convective transport ….