E-mail address: sportiss@cereve.enpc.fr (B. Sportisse).

Atmospheric Environment 35 (2001) 173}178

Short Communication

Box models versus Eulerian models in air pollution modeling

Bruno Sportisse

Centre d'Enseignement et de Recherche en Eau, Ville et Environnement, Ecole Nationale des Ponts et Chausse& es (ENPC-CEREVE),rue Blaise Pascal, 77455 Champs sur Marne, France

Received 25 June 1999; accepted 11 May 2000

Abstract

Box models are widely used in air pollution modeling. They allow the use of simple computational tools instead of thesimulation of 3D Eulerian grid models, given by a large set of partial di!erential equations. We investigate here thetheoretical justi"cation of such box models. The key point is the comparison with the underlying Eulerian modeldescribing the dispersion of pollutants in the atmosphere. We restrict the study to a vertical monodimensional case formore clarity. The main result is that the nonlinearity of the chemical kinetics, which is a characteristic feature ofchemistry, induces the loss of accuracy. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Box models; Reaction}di!usion; Chemical kinetics; Air pollution modelling

1. Introduction

Air pollution models describe the time and spaceevolution of certain trace gases in the atmosphere.A large number of phenomena have to be taken intoaccount such as emissions of primary pollutants, atmo-spheric chemical transformations, advection by the wind"eld, turbulent di!usion and dry deposition. The keyassumption is that the mean #ow "elds (density, temper-ature and wind velocity) may be computed independentlywith meteorological models (Seinfeld, 1985). These "eldsare then given o!-line for the equations describing thereactive dispersion of the chemical species.

For a predictive use or a sensitivity analysis there isnow a stringent need for models easy to simulate due tothe CPU requirements. Some simpli"ed models, such asbox-models, are therefore often used. They are based onthe assumption of a perfectly stirred reactor (PSR) andthey induce a loss of accuracy.

We will "rst present the Eulerian models and thecommonly used box-models. The theoretical justi"cationof box models is investigated in Section 2. Some numer-ical tests are reported in the last section.

2. Some models

2.1. Eulerian models

Let us recall that the atmospheric dispersion ofchemical species is described by a set of advection}reaction}di!usion partial di!erential equations subjectto appropriate boundary conditions (Seinfeld, 1985;McRae et al., 1982). For instance, for the species i,

Lci

Lt#div(<(x, t)c

i)"div(K(x, t)+c

i)#s

i(c,¹(x, t), t)

#Si(x, t), (1)

where x and t denote respectively the space and timecoordinates, c is the vector of chemical concentrations(whose index is i). <(x, t) is the "eld of wind velocity.K(x, t) is the di!usive coe$cient describing atmosphericturbulence given by a gradient model. In a "rst approxi-mation the turbulent di!usion can be mainly restricted tothe vertical direction (Seinfeld, 1985) and the moleculardi!usion can be neglected outside a "ne layer above theground. ¹(x, t) is the temperature "eld. S

i(x, t) is the

source term for species i, which describes the emissionsby industrial plants.

sistands for the chemical production of species i due

to chemical transformations in the atmosphere. In our

1352-2310/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 1 3 5 2 - 2 3 1 0 ( 0 0 ) 0 0 2 6 5 - X

simpli"ed model we only take into account gas-phasechemistry, by excluding aqueous-phase chemistry andmass transfer between both phases.

Some appropriate boundary conditions have now tobe speci"ed. A classical assumption is to consider thatwind advection is essentially a horizontal phenomenonwhile di!usion occurs in the vertical direction (convectivedi!usion of Rayleigh}BeH nard type (Lesieur, 1990)). Lat-eral boundary conditions are then those associated withusual linear hyperbolic systems (along the positive com-ponents of the wind "eld). Vertical boundary conditionsare the following ones (z stands in the sequel for thealtitude):

(1) At the ground (z"0),

!K(x, t)Lc

iLz

"Ei(x, t)!vi

$%1ci, (2)

where Ei(x, t) and vi

$%1are respectively the emission term

due to tra$c cars and the dry deposition velocity forspecies i.

(2) At the top of the spatial domain to be studied,

!K(x, t)Lc

iLz

"0. (3)

Such a condition is associated with the de"nition of theatmospheric boundary layer, whose top is usually de"nedby KK0.

Such models will be referenced in the sequel as `Euler-ian modelsa. Their numerical discretization by "nite dif-ferences techniques (or by "nite elements algorithms)gives rise to a set of di!erential equations coupledthrough chemical phenomena. Numerous numericaltechniques are advocated for the solution of such ordi-nary di!erential equations (ODE) comprising solvers forsti! problems (Sandu et al., 1997; Verwer et al., 1996) andoperator splitting methods (Sportisse, 2000; Sportisseet al., 1998). We refer to Verwer et al. (1998) for an up-to-date overview of these techniques.

2.2. Box models

Alternative models are however often used in practicein order to deal with easier models de"ned on a smallset of grid cells. This is the case for the so-called`box-modelsa which mainly rely on an assumption of aperfectly stirred reactor (PSR). The equation to be integ-rated is then typically

dc6i

dt"s

i(c6 ,¹(t), t)#S

i(t)#

1

H(t)(E

i(t)!vi

$%1c6i) (4)

with the lateral boundary conditions eventually paramet-rizing lateral #uxes. H(t) stands here for the so-calledmixing height (the top of the boundary layer).

For a model including two layers (the upper box isdenoted by 1 while the lower box is indexed by 2) one has

dc6 1i

dt"s

i(c6 1, ¹1(t), t)#S1

i(t)#Exc1

i(c6 1

i, c6 2

i, t), (5)

dc6 2i

dt"s

i(c6 2, ¹2(t), t)#S2

i(t)#

1

H(t)(E

i(t)!vi

$%1c6 2i)

#Exc2i(c6 1

i, c6 2

i, t), (6)

where Exc1i(c6 1

i, c6 2

i, t) and Exc2

i(c6 1

i, c6 2

i, t) describe ex-

change terms between both layers. They are a function ofthe growth of the atmospheric boundary layer, whichexplains the direct dependence on the time t (Stull, 1988;Van Loon, 1996; Zannetti, 1990). If the mixing heightgrows, pollutants present in the upper layer are injectedinto the lower layer. An application of such simpli"edmodels to inverse modelling can be found for instance inVautard et al. (1996).

One may of course wonder if such models are indeedfounded. This is the question we investigate in this article.We will more particularly focus on the case of one-boxmodels given by Eq. (4).

3. From Eulerian models to box models

3.1. General result

For more clarity, we consider a monodimensionalEulerian model by neglecting wind velocity. Let us noticethat this is the practical situation for peaks of pollution(no wind, no transport far from the sources). We dealthen with a vector of species c(z, t) where z is the verticalcoordinate. Let n be the number of chemical species.

We assume as well that the mixing height H and thedi!usive coe$cient K(z) are constant with respect totime.

The Eulerian model (1) can then be written straightfor-wardly as

Lci

Lt"

LLz CK(z)

Lci

Lz D#si(c) (7)

with the following boundary conditions:

KLc

iLz

"0 at z"H, KLc

iLz

"vi$%1

ci!E

i(t)

at z"0. (8)

Let us de"ne the integrated concentration along a verti-cal column as

SciT(t) $%&"

1

HPH

0

ci(z, t) dz. (9)

Note that we have intentionally kept the usual notationof turbulence averaging.

174 B. Sportisse / Atmospheric Environment 35 (2001) 173}178

A straightforward integration gives

dSciT

dt"s

i(ScT)#

1

H(E

i(t)!vi

$%1Sc

iT)#R

i(c, t), (10)

where the residual term R is given by

Ri(c, t)"

vi$%1H

(SciT(t)!c

i(0, t))#

1

HPH

0

(s(c)!s(ScT)) dz.

(11)

Without loss of generality, we can assume that the kineticscheme only includes monomolecular or bimolecular re-actions (trimolecular elementary reactions do not occurvery often due to the low probability of collisions). Thechemical production s

iis then a sum of linear and quad-

ratic terms and can be written as

si(c)"(s@

i(t), c)

n#(c, sA

i(t)c)

n, (12)

where (. , .)n

stands for the usual scalar product in Rn.s@i(t) is a vector of Rn and sA

i(t) is an n]n matrix. They

are associated respectively with monomolecular and bi-molecular reactions. The eventual time dependence isdue to photolysis. These matrices are completely de"nedby stoichiometric coe$cients and kinetic rates. Let usnote that they may depend on altitude and time throughtemperature. We have neglected this dependence formore clarity.

We have then easily

si(c)!s

i(ScT)"(s@

i, c!ScT)

n#(c!ScT, sA

i(c!ScT))

n

#(c!ScT, chiAiScT)

n

#(ScT, sAi(c!ScT))

n. (13)

By using the fact that

PH

0

(c!ScT) dz"0, (14)

since ScT does not depend on z, we straightforwardlyobtain

PH

0

(s(c)!s(ScT)) dz"PH

0

(c!ScT, sAi(c!ScT))

ndz (15)

and "nally

Ri(c, t)"

vi$%1H

(SciT(t)!c

i(0, t))

#

1

HPH

0

(c!ScT, sAi(c!ScT))

ndz. (16)

If we omit the term associated with dry deposition theerror is then directly related to the non-linearity of chem-istry (sA

iO0). Needless to say that this is unfortunately

the key feature of any realistic kinetic scheme!

It is, however, much more interesting to derive an errorbound for the deviation e

i(t) associated with species i de-

"ned by

ei(t)"c6

i(t)!Sc

iT(t), (17)

where c6iand Sc

iT denote respectively the solution of the

box model (Eq. (4)) and the average of the Euleriansolution (Eq. (10)).

From now on we omit the dry deposition velocity sinceit does not have a strong in#uence (see the next sectionand the numerical results). We have then easily

dei

dt"s

i(c6 )!s

i(ScT)!

1

HPH

0

(si(c)!s

i(ScT)) dz. (18)

Therefore by using that c6 and ScT do not depend on z:

dei

dt"!

1

HPH

0

(si(c)!s

i(c6 )) dz. (19)

This equation is associated with the initial conditionei(0)"0. It is unfortunately rather di$cult to obtain

a general result further.

3.2. A simple example

The nonlinear case is relevant for chemical kineticsand we will consider the `cartoon examplea s

i(c)"!c2

i.

This is of course related to reactions under the formC#CP2. We have, then,

dei

dt"

1

HPH

0

(c2i!c6 2

i) dz. (20)

By introducing the #uctuation c@"c!ScT, we obtaineasily

dei

dt"Sc@

i2T#Sc2

iT!c6 2

i. (21)

We use c6i"e

i#Sc

iT and therefore,

dei

dt"Sc@

i2T!2e

iSc

iT!e2

i. (22)

We assume from now on without any loss of generalitythat Sc

iT and Sc@

i2T are strictly positive bounded func-

tions. Let us make some comments on Eq. (22):

f If ei"0 (e.g. at t"0), de

i/dt"Sc@

i2T'0. This proves

that for all "nite t, ei(t)'0. Let us note that e

i"0 is

an equilibrium point if and only if Sc@i2T"0.

f Let us search for an upper bound of ei(t). We have for

any "nite t:

dei

dt#e2

i(Sc@

i2T. (23)

B. Sportisse / Atmospheric Environment 35 (2001) 173}178 175

Fig. 1. Distributions K(z) (m2 s~1).

Fig. 2. Time evolution of the relative error ERR.

Let us note that

M"supt

Sc@i2T (24)

and consider the following ODE:

dv

dt#v2"M, v(0)"0. (25)

We have directly

1

2JMlogA

v(t)#JM

v(t)!JMB"t, (26)

which proves that v(t)(JM for all t.Let us now compare e

iand v. The function

u"v!eiis such that

du

dt#v2!e2

i'M!Sc@

i2T, u(0)"0. (27)

Then

du

dt#u(t)(v(t)#e

i(t))'0, u(0)"0. (28)

This proves that u(t)*0 and then that v(t)*ei(t). As

v(t)(JM for all t we have proved that

ei(t))Jsup

tSc@

i2T (29)

which seems to be quite logical.

f Let us assume that the Eulerian solution converges forlarge t to a homogeneous solution (this is the case if weneglect boundary conditions given by emissions), thenc@itends to 0 and through (22) it is easy to prove that

ei(t) tends to 0 as well.

This simple example proves the dependence of the errorinduced by the box model on the #uctuation c@.

4. Some numerical tests

We have made some tests with a kinetic scheme givenin (Sportisse, 1999) comprising 16 species and 12 reac-tions. The mixing height is "xed to a value of 400 m. Theset of numerical parameters we use can be found in(Sportisse, 1999).

Fig. 1 gives the vertical distribution of four laws K(z)we have used. Distributions (2)}(4) have only distinctvalues for the boundary coe$cients K(0) and K(H), inorder to assess the sensitivity of Eulerian models to suchvalues.

We de"ne the relative error at a "xed time t in order tocompare the Eulerian model and the box model:

ERR(t)"1

n

n+i/1ASc

iT(t)!c6

i(t)

Sci(t)T B

2, (30)

where n is the number of species and c6 is computed withthe box model (R"0). The time evolution of the relativeerror (ERR) is given in Fig. 2. The length of computationis 3 h.

The errors associated with the use of a box model growwith time and they are rather huge (much bigger thanthe usual threshold of 1% advocated in air pollutionmodels). The PSR assumption is indeed not satis"ed inthis case (this is of course related to the rather low valueof K which actually characterizes pollution episodes!).The error due to dry deposition has no in#uence (see thepro"les with K

4with and without dry deposition in

Fig. 2).

176 B. Sportisse / Atmospheric Environment 35 (2001) 173}178

Fig. 3. Time evolution of the relative error for NO, NO2

andO

3with K

4(z).

Fig. 4. Sensitivity to K(0) and K(H).

The relative error for some of the most importantspecies such as NO, NO

2and O

3is plotted in Fig. 3 in

the peculiar case of K4(z). The loss of accuracy is parti-

cularly important for these species. The same kind oferrors occurs for the other pro"les.

Remark 1 (Sensitivity to the di+usion pro,le). We cannotice that Eulerian solutions are extremely sensitive tothe values of K at the top and at the ground, especially tothe cancelling (or not) of K. Fig. 4 gives the comparisonsrespectively between the Eulerian solutions computedwith the pro"les 1 and 2 (same extreme values but di!er-ent values in the domain), with the pro"les 2 and 3 (K(0)and K(H) range from 0 to 0.5) and with the pro"les 3 and 4(K(0) and K(H) range from 0.5 to 1). The unit of di!usionis m2 s~1.

We have kept the same de"nition as before for therelative error between Eulerian models by replacing in

(30) the solution issued from the box model by an Euler-ian solution.

Let us note that boundary conditions are not wellposed with K(0)"0 or K(H)"0 since Lc/Lz is then free.We rather advocate the use of low values such asK(0)"K(H)"0.5.

5. Conclusions

We have investigated the justi"cation of one-box mod-els in a monodimensional case. The key point is that higherrors may occur due to the nonlinearity of atmosphericchemistry. A complementary result concerns the de"nitionof well-posed boundary conditions for Eulerian models.

A further work may be the investigation of multi-boxmodels. The drawback is however that they are notalways well-de"ned since they include a parametrizationof the exchange terms. As the key assumption is theperfectly stirred reactor, the conclusions of the formerinvestigation give somehow an indication of the loss ofaccuracy associated with these models.

Acknowledgements

I thank my colleague Eric Cances for his help inderiving the error bound.

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