Approche cartésienne pour le calcul du vent en terrain complexe avec application à la propagation des feux de forêt

Appro he artésienne pour le al ul du vent en

terrain omplexe ave appli ation à la propagation

des feux de forêt

par

Louis-Xavier Proulx

Départementdemathématiques etdestatistique

Fa ultédesartset dess ien es

Mémoire présentéà laFa ulté desétudes supérieures

en vuede l'obtentiondugrade de

Maître èss ien es (M.S .)

enmathématiques appliquées

Août 2010

Fa ulté des études supérieures

Ce mémoire intitulé

Appro he artésienne pour le al ul du vent en

terrain omplexe ave appli ation à la propagation

des feux de forêt

présentépar

Louis-Xavier Proulx

a été évalué par un jury omposédes personnessuivantes:

Robert G. Owens (président-rapporteur) Anne Bourlioux (dire teurdere her he) Mi hel Delfour (membredujury)

Mémoire a eptéle:

SOMMAIRE

La méthode de proje tion et l'appro he variationnelle de Sasaki sont deux

te hniques permettant d'obtenir un hamp ve toriel à divergen e nulle à partir

d'un hamp initial quel onque. Pour une vitesse d'un venten haute altitude, un

hamp de vitesse sur une grille dé alée est généré au-dessus d'une topographie

donnée par une fon tion analytique. L'appro he artésienne nommée Embedded

Boundary Method est utilisée pour résoudre une équation de Poisson dé oulant

delaproje tionsurun domaineirrégulier ave des onditions auxlimitesmixtes.

Lasolution obtenue permet de orrigerle hamp initial an d'obtenir un hamp

respe tantla loide onservation dela masseet prenantégalement en ompteles

eets dûs à la géométrie du terrain. Le hamp de vitesse ainsi généré permettra

depropager unfeudeforêtsurlatopographieàl'aidedelaméthodeiso-niveaux.

L'algorithme est dé rit pour le as en deux et trois dimensions et des tests de

onvergen e sontee tués.

Mots lés :Appro he artésienne, feux deforêt, propagation, méthode de

SUMMARY

The Proje tion method and Sasaki's variational te hnique are two methods

allowingone toextra t a divergen e-free ve tor eld fromany ve tor eld. From

ahigh altitudewind speed, avelo ity eld is generated on a staggered grid over

a topography given by an analyti al fun tion. The Cartesian grid Embedded

Boundary method isused for solving aPoisson equation,obtained from the

pro-je tion, on an irregular domain with mixed boundary onditions. The solution

of this equation gives the orre tion for the initial velo ity eld to make it su h

thatit satisesthe onservationof mass and takesinto a ountthe ee ts ofthe

terrain. The in ompressible velo ity eld will be used to spread a wildre over

the topography with the Level Set Method. The algorithm is des ribed for the

two and three dimensional ases and onvergen e tests are done.

Keywords :Embeddedboundarymethod,wildres,spread,proje tionmethod,

TABLE DES MATIÈRES

Sommaire... iii

Summary... iv

Liste des gures ... vi

Liste des tableaux ... vii

Liste des sigles et abréviations... viii

Remer iements... 1

Introdu tion... 2

Chapitre 1. A mass- onsistent model... 4

1.1. Formulationof the problem... 5

1.1.1. Conservation of mass and in ompressibility... 5

1.2. Sasaki's variational te hnique... 6

1.2.1. Ellipti equation... 7 1.2.2. Boundary onditions... 9 1.2.3. Stability parameters... 11 1.3. Proje tion method... 12 1.3.1. Hodge de omposition... 12 1.3.2. Proje tion operator... 13

1.3.3. Comparison with the variational method... 13

Chapitre 2. Solving the model in 2 dimensions... 17

2.1. FiniteVolume Method... 18

2.1.1. Dis retization of the domain... 18

2.1.2. Divergen e operator... 18

2.1.3. Marker-and-CellProje tion ... 20

2.1.4. Boundary uxes ... 21

2.1.5. Symmetri matrix ... 22

2.1.6. Corre ted ve tor eld... 23

2.2. Embedded Boundary Method... 24

2.2.1. Sparse matrix... 28

2.2.2. Corre ted ve tor eld... 29

2.3. Convergen e and error analysis... 29

2.3.1. Flat terrain ase... 33

2.3.2. Sinusoidal hills ase... 37

2.3.3. Exponential hill ase... 40

2.3.4. Half- ylinder hill ase... 44

2.4. Fire spreadby anin ompressible ow... 49

2.4.1. Wind ee t depending onits altitude... 49

Chapitre 3. Solving the model in 3 dimensions... 53

3.1. Embedded Boundary Method... 53

3.1.1. Dis retization of the domain... 54

3.1.2. Embedded Boundary representation... 55

3.1.3. Divergen e operator... 55

3.1.4. Flux interpolation... 56

3.2.1. Flat terrain ase... 61

3.2.2. Sinusoidal hills ase... 62

3.2.3. Exponential hill ase... 64

Con lusions... 67

Bibliographie... 69

Annexe A. Proje tions... A-i

Cell-Centered Proje tion... A-i

MACProje tion...A-ii

LISTE DES FIGURES

1.1 Representation ofthe omputational domain

Ω

in 2and 3 dimensions, with the topography givenby

H(x)

and the high altitude wind

v

g

.... 6

2.1 Dis retization of the two-dimensional domainfor FVM... 18

2.2 Fluxes oneveryedge of ontrol volume

V

i,j

... 19

2.3 Staggered grid with the ell- entered s alar eld

ϕ

and the edge- entered omponentsof the velo ity eld

(u, v)

... 20

2.4 Standard 5 pointsten il Lapla ian

D

M

_(G

M

_(ϕ))

i,j

... 21

2.5 Interpolation of the boundary ux

F

N

x

+

2

1

,j

for applying the Diri hlet

ondition using

ϕ

B

,

ϕ

N

x

,j

and

ϕ

N

x

−1,j

... 22

2.6 The 3 dierent sten ilsfor the Lapla ian operator... 22

2.7 Symmetri matrix of the linear system with

N

x

= 6

and

N

y

= 8

... 23

2.8 Fluxes in a ut ell (

0 < κ < 1

) and in a full ell (

κ

= 1

). The ux

F

B

i,j

≡ 0

(redarrow)sin eaNeumann onditionisappliedattheterrain surfa e boundary... 25

2.9 Interpolationofthe ux

F

i+

1

2

,j

ona ut elledge using

G

M

_(ϕ)

i+

1

2

,j

and

G

M

(ϕ)

_i+

1

2

,j+1

... 26

2.10 Interpolation of the ux

F

i,j+

1

2

on a ut ell edge when the terrain is

onthe right hand side orleft hand side... 27

2.11 Sparse matrix

A

for solving Poisson equation on an irregular domain with anembedded exponential hill with

N

x

= 6

and

N

y

= 8

... 28

2.12 The onvergen eof

u

d

onthered pointsofthe grid

N

= 2

isdone with the mean of the blue points of the grid

N

= 8

... 30

2.13 Theneto oarseaveragingofa ell- enteredquantityusinga

volume-weighted average of values of the ne grid

N

= 8

to the oarse grid

N

= 2

... 32 2.14 Streamlines and divergen e of initial and nal ve tor elds

(u, v)

f lat

overthe at terrainsurfa e

H

f lat

... 34

2.15 Contour plot of the nal velo ity eld omponents

(u

d

_{, v}

d

₎

f lat

overthe

at terrain surfa e

H

f lat

... 34

2.16 Norms of the staggered absolute error

e

N

(top) and volume-weighted norms of the ell- entered absolute error

e

N

(bottom) of

(u

d

_{, v}

d

₎

f lat

... 35

2.17 Streamlines and divergen e of initial and nal ve tor elds

(u, v)

sin

overthe sinusoidalhills

H

sin

... 38

2.18 Norms of the staggered absolute error

e

N

(top) and volume-weighted norms of the ell- entered absolute error

e

N

(bottom) of

(u

d

_{, v}

d

₎

sin

.... 39

2.19 Streamlinesanddivergen eof theinitialandnal ve torelds

(u, v)

exp

overexponential hill

H

exp

... 41

2.20 Finalve tor elds

(u, v)

exp

overthe exponential hill

H

exp

... 41

2.21 Norms of the staggered absolute error

e

N

(top) and volume-weighted norms of the ell- entered absolute error

e

N

(bottom) of

(u

d

_{, v}

d

₎

exp

... 42

2.22 Streamlinesand divergen eof the initialand nalve tor elds

(u, v)

cyl

overhalf- ylinder hill

H

cyl

... 46

2.23 Volume-weighted norms of the ell- entered absolute error

e

N

of

(u

d

_{, v}

d

₎

cyl

... 46

2.24 Contour plot of horizontal omponent of numeri al and exa t velo ity

elds overa half- ylinderhill

H

cyl

... 47

2.25 Numeri alandexa tvelo itymagnitudeat

y

= 2.34

overahalf- ylinder hill

H

cyl

... 48

2.26 Final ve tor eld

(u

d

_{, v}

d

₎

cyl

(red) and exa t ve tor eld (bla k) on half- ylinderhill givenby

H

cyl

... 48

2.27 Fire spreading over dierent terrain shapes (wind inow is onthe left

side)... 50

2.28 Fire position

x(t)

over sinusoidal hills with dierent heights of the wind... 51

2.29 Fire position

x(t)

overanexponential hill with dierent heightsof the wind... 52

3.1 Interpolationof the ux

F

i

+

1

2

e

1

atthe entroidof the ut fa e ell edge

using bilinear interpolation. The s heme here is for the ase where

d

= 2

and

d

′

_{= 3}

... 57

3.2 Interpolationoftheux

F

i

+

1

2

e

1

atthe entroidofthe ut ellfa eusing

linear interpolation when the distan e between the fa e entroid and

the fa e enter onlyvaries inthe tangential dire tion

e

3

... 58

3.3 Theneto oarseaveragingofa ell- enteredquantityusinga

volume-weighted average of values of the ne grid

N

= 8

to the oarse grid

N

= 2

... 59 3.4 Divergen e of the nal ve tor eld

(u, v, w)

f lat

over the at terrain

H

f lat

... 61 3.5 Volume-weighted norms of the ell- entered absolute error

e

N

of

(u

d

_{, v}

d

_{, w}

d

₎

f lat

... 62

3.6 Volume-weighted norms of the ell- entered absolute error

e

N

of

(u

d

_{, v}

d

_{, w}

d

₎

sin

... 63

3.7 Divergen e of the nal ve tor eld

(u, v, w)

exp

over exponential hill

H

exp

... 64 3.8 Contour plot of the omponent

u

of the nal ve tor eld. ... 65

3.9 Ve tor plot of the nal ve tor eld

(u, v, w)

exp

over exponential hill

H

exp

... 65 3.10 Volume-weighted norms of the ell- entered absolute error

e

N

over

exponential hill

H

exp

... 66

A.1 ExpandedLapla ian

D

o

_(G

o

_(ϕ))

i,j

...A-ii

A.2 Staggered grid with the ell- entered s alar eld

ϕ

and the edge- entered omponentsof the velo ity eld

(u, v)

...A-ii

LISTE DES TABLEAUX

2.1 Parameters used for test ases in2D... 33

2.2 Convergen e rate of staggered absolute error

e

N

for

(u

d

, v

d

)

f lat

... 36

2.3 Convergen e rate of ell- entered absolute error

e

N

for

(u

d

_{, v}

d

₎

f lat

... 36

2.4 Convergen e rate of staggered absolute error

e

N

for

(u

d

_{, v}

d

₎

sin

... 37

2.5 Convergen e rate of ell- entered absolute error

e

N

for

(u

d

_{, v}

d

₎

sin

... 37

2.6 Convergen e rate of staggered absolute error

e

N

for

(u

d

, v

d

)

_exp

... 42

2.7 Convergen e rate of ell- entered absolute error

e

N

for

(u

d

, v

d

)

exp

... 42

3.1 Parameters used for test ases in3D... 61

3.2 Convergen e rate of

(u, v, w)

f lat

indierentnorms... 62

3.3 Convergen e rate of

(u, v, w)

sin

indierent norms... 63

LISTE DES SIGLES ET ABRÉVIATIONS

ANAG: Applied Numeri alAlgorithms Group

CFD: Computational FluidDynami s

EBM : Embedded Boundary Method

FDM: Finite Dieren e Method

FEM: FiniteElementMethod

FVM: FiniteVolume Method

MAC : Marker-and-Cell

MCM: Mass-ConsistentModel

REMERCIEMENTS

Je suis sin èrement re onnaissant envers ma dire tri e de re her he, Anne

Bourlioux, pour m'avoir proposé de travailler sur e nouveau projet. J'aimerais

aussi la remer ier pour son support nan ier qui m'a permis de me onsa rer

pleinement à mes re her hes ainsi que pour m'avoir donné l'opportunité defaire

un stage dere her heà Berkeley.

J'aimeraisaussitransmettremes remer iementsàPhillipColellapour son

a - euil haleureux au Berkeley Lab et pour avoir mis à ma disposition les outils

né essaires pour poursuivre mes re her hes sur e projet. Je suis également

re-devableàplusieurs membres del'équipeduANAG,plusparti ulièrementàTerry

Ligo ki, Dan Graves, Hans Johansen et Jerey Johnson qui ont été d'un très

grand se ours lors de mon apprentissage de EBChombo et surtout lorsqu'il est

venuletemps detrouverlesbugs.Je remer ie leFonds québé oisdela re her he

surla nature et les te hnologies (FQRNT) et le Centre dere her hes

mathéma-tiques (CRM) pour avoir re onnu la valeur de eprojet et pour m'avoir a ordé

une bourse de stage international qui m'a assuré une sé urité nan ière lors de

mon séjour à Berkeley.

Unmer ispé ialàAlexandreDesfossés-Fou aultpourses onseilssurMatlab,

ses ommentaires et ses orre tions sur e mémoire ainsi que son agréable

om-pagnieaulabomat.Finalement,un gros mer i àtous euxqui m'ont supporté de

prèsou deloin esdeuxdernières années. Vosen ouragementset nos dis ussions

British-Columbia,AlbertaandCaliforniafa eea hyearimportantforestres

on their territories. Fires are essential for maintaining the diversity and health

of forest e osystems, but they also bring a lot of drawba ks. Many

mathemati- al models are urrently used to predi t the propagation of res. Morea urate

simulations ould prevent the negative impa ts of wildreson publi health and

safety of individuals, and ould also de rease the loss of property and natural

resour es.

Simulation models for the spread of wildres, su h as PROMETHEUS and

FARSITE, use the equations of uid me hani s. Dierentfa tors must be taken

into onsideration when simulating the spread. The weather, the ee ts of the

topography, the dierent type of fuelsand obsta les are a few examples.Adding

more parameters and variables to the model in reases its a ura y but also the

omputational ost, hen e hoi es must be made. Most urrent models tend to

use the law of onservation of mass and onservation of momentum, but those

models oftennegle t the ee ts of the terrain and of the re as a dilationsour e

term.

The aim of this thesis is to develop the rst part of a new model to predi t

e ientlyanda uratelythespreadofforestres.Theapproa h hosenheretakes

intoa ountthetopographi ee tsofthe terrainonthe winddire tion, whi his

oftennegle tedin urrentmodels. Thetangential wind tothe surfa eis non-zero

sin e the ee ts of vis osity, more spe i ally the fri tion of the wind eld with

the terrain, are negle ted. The nal velo ity ve tor eld must only satisfy the

ve tor eld into anin ompressible ve tor eld.After a shortdis ussion on

mass- onsistentmodels, hapteronepresentsSasaki'svariationalte hniquefollowedby

theproje tionmethod.Themodel willalso onstru t theinitial ve tor eldfrom

a high altitude wind. The dierent possibilities are dis ussed at the end of the

rst hapter.

Inorder toapply the onstraint of in ompressibility on the windve tor eld,

a Poisson equation is solved on an irregular domain. In fa t, the shape of the

topographyisembedded inthe omputationaldomainusingaCartesiangrid.This

methodis alledthe EmbeddedBoundary Methodand was developedby Phillip

Colella and his team at the Lawren e Berkeley National Laboratory (LBNL).

Thisapproa hisbased onthe FiniteVolumeMethodand isexplainedin hapter

twoforthe two-dimensional ase.Convergen eofthesolutionsanderror analysis

are also studied.

In hapterthree,themodeland thealgorithmare generalizedtothree

dimen-sions. The implementation of the model was done with EBChombo, a software

developedby theApplied Numeri alAlgorithmsGroup (ANAG)attheBerkeley

Lab.Thetest asesusedinthetwo-dimensional asearegeneralizedforadomain

A MASS-CONSISTENT MODEL

Models for simulating wind ow fall into two main ategories : prognosti

modelsand diagnosti models. The rstkind onsiders time-dependent

hydrody-nami equations su hasNavier-Stokes tofore ast howthe wind owwill evolve.

Thesemodels alsoin ludemanyfa torssu hasturbulen e,moisture, momentum

and heat. Elaborated models require pre ise data in order to deliver a urate

predi tions and su h data is not always available. On the other hand,

diagnos-ti models generate wind elds that satisfy spe i physi al onstraints. Models

that assurethe onservation of mass are typi ally alledmass- onsistent models.

These are simpler than prognosti models, they require less data and have the

big advantage of havinga low omputational ost.

The goal of this thesis is to onstru t a predi tive mass- onsistent model for

wildrepropagation. The model will generate a velo ity eld that will take into

a ounttheee tsofthetopographyonthewindow.Amorea uratewindeld

will lead to better predi tions of the re front spread. Sin e only the onstraint

of onservation of mass is onsidered, the simulationswill be fasterthan models

using also the momentum equation and hen e be useful for predi tions.

Following a des ription of the problem, two methods whi h generate

mass- onsistentowarepresented.Then,ashortdis ussion ontheinitializationofthe

1.1. Formulation of the problem

The model is explained for the two-dimensional ase, but it an be easily

generalizedtothreedimensions.Consider are tangularregionin

R

2

oflength

L

x

and height

L

y

. The topographi al height above mean sea level is expressed by a fun tion

H : R → R

dependingonthehorizontal oordinate

x

.The omputational domain

Ω

is the region ontained between the fun tion

H(x)

and the top of the re tangularregionasshowningure1.1.Theonlyinformationgivenforthewind

is the magnitude of a high altitude horizontal wind

v

g

= (u

g

_{, 0)}

from whi h an

initialvelo ityve toreld

v

= (u, v)

mustbe onstru ted.Thisinitialve toreld might not be divergen e-free, but sin e the wind is assumed to be invis id (no

vis osity)itmustbetangentialtotheterrainsurfa e,whi hmeansthatitsatises

the slip ondition

v

· n = 0

, where

n

is the outward normal at the topography surfa e. We then look for a orre tion that will transform the ve tor eld

v

in an in ompressible ve tor eld

v

d

whi h also fulls the slip ondition

v

d

· n = 0

. Boundary onditionsfor thesides and top ofthe domainhaveto bedenedsu h

that the ow will be allowedtogo through.

1.1.1. Conservation of mass and in ompressibility

In order toget anin ompressible ve tor eld

v

d

,wemustapply a onstraint to the ve tor eld. In uid dynami s, the onservation of mass is expressed by

the ontinuity equation

∂ρ

∂t

+ ∇ · (ρv) = 0

(1.1.1)

wherethe operator

∇ = (

∂

∂x

,

∂

∂y

,

∂

∂z

)

,

ρ = ρ(x, y, z)

is the density of the uid and

v

_{= (u(x, y, z), v(x, y, z), w(x, y, z))}

where

u

,

v

and

w

are the velo ity of the ow

in the

x

,

y

and

z

dire tions. Noti e that in our model, the variables

ρ

and

v

do notdependontime,sin eweareinterestedinastationaryow,hen e

∂ρ/∂t = 0

. The density of the uid isassumedto be onstant everywhere onthe domain, so

equation(1.1.1) be omes

0

2

4

6

8

10

0

10

20

30

40

50

60

PSfrag repla ements

x

y

Ω

v

_g

H

(x)

0

5

10

0

5

10

0

50

100

150

PSfrag repla ements

x

y

z

Fig. 1.1. Representationof the omputational domain

Ω

in2 and 3 dimensions, with the topography given by

H(x)

and the high altitude wind

v

g

.

where equation (1.1.2) is the in ompressibility onstraint on the ow. This

on-dition expresses and guarantees the onservation of mass for our model.

Mathe-mati ally, italso means that the ve tor eld is divergen e-free.

1.2. Sasaki's variational te hnique

There are dierent approa hes to apply the onstraint of mass onservation

on a given ve tor eld. A qui k look at the literature in atmospheri s ien es

showsthat mostmodels arebasedonavariational al ulusmethoddevelopedby

Sasaki [22 ℄. Ratto et al. [20℄ have reviewed these models whi h are adapted to

1.2.1. Ellipti equation

Sasaki's variational te hnique is a natural approa h in atmospheri s ien es.

Meteorologists onstru t their models with a variety of empiri al data, su h as

wind velo ity. The variational method allows them to nd, for a set of observed

wind data, a minimal orre tion to adjust the wind su h that it will be ome

divergen e-free.

Infa t,thismethodminimizesthevariationbetweenthe adjustedvalues

v

d

=

(u

d

_{, v}

d

_{, w}

d

₎

and the initial values

v

= (u, v, w)

in a generalized least squares formulation:

I(u

d

, v

d

, w

d

) =

Z

Ω



β

₁

2

(u

d

− u)

2

+ β

₂

2

(v

d

− v)

2

+ β

₃

2

(w

d

− w)

2



dV

(1.2.1)

where

β

i

(i = 1, 2, 3)

are the Gauss pre ision moduli. These weightsare used for the alibration of the adjustments of the wind eld

v

d

with the observedvalues

v

.They will be explainedin more detailsin se tion1.2.3.

Sin eweare looking foranadjustedve toreld

v

d

whi hfullsthe onserva-tion of mass, we add the onstraintgivenby equation(1.1.2) :

G

(u

d

_x

, v

_y

d

, w

_z

d

) =

∂u

d

∂x

+

∂v

d

∂y

+

∂w

d

∂z

= 0

(1.2.2) tothe fun tional

I

:

J

=

Z

Ω

EdV

= I +

Z

Ω

λGdV

(1.2.3)

where

λ

= λ(x, y, z)

is a Lagrange multiplier. Writing expli itly the fun tional

J

inequation (1.2.3)wehave :

J

(u

d

, v

d

, w

d

, λ

) =

Z

Ω



β

₁

2

(u

d

− u)

2

_{+ β}

2

2

(v

d

− v)

2

+ β

3

2

(w

d

− w)

2

+λ

 ∂u

d

∂x

+

∂v

d

∂y

+

∂w

d

∂z



dV

(1.2.4)

We now want to minimize the fun tional

J

under the strong onstraint of onservation of mass. To a hieve this goal, we must look at the rst variation

one knows that the solutions whi h minimize the fun tional

J

satisfy the Euler-Lagrange equations :

∂E

∂f

−

3

X

i=1

∂

∂x

i

∂E

∂(∂f /∂x

i

)

= 0

(1.2.5)

where

E

isthe integrandof thefun tional

J

,

f = (u

d

_{, v}

d

_{, w}

d

_{, λ)}

and

(x

1

, x

2

, x

3

) =

(x, y, z)

. Solving with the Euler-Lagrangeequations (1.2.5), wend

u

d

= u +

1

2β

2

1

∂λ

∂x

(1.2.6)

v

d

_{= v +}

1

2β

2

2

∂λ

∂y

(1.2.7)

w

d

= w +

1

2β

2

3

∂λ

∂z

(1.2.8)

∂u

d

∂x

+

∂v

d

∂y

+

∂w

d

∂z

= 0.

(1.2.9)

Notethatsolution(1.2.9) orresponds tothe in ompressibility onstraint(1.1.2).

Ratto et al. [20℄ noti ed that all mass- onsistent models assume the Gauss

pre- ision moduli tobe equal in the horizontal plane of the eu lidean spa e :

β

1

= β

2

.

(1.2.10)

In fa t, important distin tions are seen between the horizontal omponents and

the verti al omponent of the adjusted ve tor eld with the initial ve tor eld.

There is no su h big dieren e between the horizontal dire tions of the two

ve-lo ity elds.

Dierentiating equations (1.2.6), (1.2.7), (1.2.8) with respe t to

x

,

y

,

z

re-spe tively, and substituting the results in equation (1.2.9), this ellipti equation

isobtained :

∂

2

_λ

∂x

2

+

∂

2

_λ

∂y

2

+

 β

1

β

3



2

∂

2

_λ

∂z

2

= −2β

2

1

 ∂u

∂x

+

∂v

∂y

+

∂w

∂z



.

(1.2.11)

Using the solution

λ(x, y, z)

of equation (1.2.11), we an orre t the initial ve tor eld

v

with equations (1.2.6)-(1.2.8) and nd the divergen e-free ve tor eld

v

d

. In order to solve the ellipti equation, we need to spe ify the boundary

1.2.2. Boundary onditions

Forthevariationalproblemdes ribedabove,Ishikawa[13 ℄givestheasso iated

boundary onditions as:

I

∂Ω

δf

"

₃

X

i=1

∂E

∂(∂f /∂x

i

)

#

· ndA = 0

(1.2.12)

where

n

is the outward normal ve tor to the domain boundary

∂Ω

and

δf

an arbitraryrstvariationof

f

.Solvingthepartbetweenthebra ketsalwaysleadsto

λ

.Theintegralwillbezeroiftheintegrandiszero,hen etheboundary onditions be ome

λδv

d

· n = 0

on

∂Ω

(1.2.13)

where

δv

d

denotesthe rst variationof the velo ity.

Fromequation(1.2.13), eitherthe multiplier

λ

orthenormalvelo ity ompo-nentvariation

δv

d

·n

mustbezeroataboundary.Sherman[24℄ laimedthatonly oneofthosetwo onditionsmustbeimposedatatime,otherwiseitwould

overde-terminethe problem and the solution would not be unique. In fa t, Núñez etal.

[19℄haveshown thatthesole propertiesof

J

guaranteethe existen eand unique-nessofaeld

v

d

thatminimizes globally

J

.Mostauthorsofatmospheri s ien es arti les have followed the hoi e adopted by Sherman and her interpretation of

the boundary onditions.

For ow-through or open boundaries, the appropriate boundary ondition is

the homogeneousDiri hlet ondition,

λ = 0.

(1.2.14)

Forthis ondition,thenormalderivativeof

λ

mightnotbeequaltozero,whi his generallythe ase.Hen e,anon-zeroadjustmentoftheinitialvelo ity omponent

normaltotheboundarymighto ur.Moreover,a onstantvalueof

λ

atanopen boundary also implies that no orre tion ismade for the velo ity omponentsin

the non-normal dire tion, sin e the non-normal derivatives of

λ

are zero. This property is useful sin e we want to onserve the magnitude of the wind at the

Forno-ow-through or losedboundaries,thehomogeneousNeumann

bound-ary ondition,

∂λ

∂n

= 0

(1.2.15)

is hosen.Itimpliesthatthereisnoadjustmentinthenormalvelo ity omponent,

hen ethe variationof normal velo ity iszero :

δv

d

· n = 0.

(1.2.16)

Ifthe initial normal velo ity omponentthrough the boundary iszero :

v

_{· n = 0}

(1.2.17)

the adjusted owof mass a ross the boundary is also zero whi h means thatthe

orre ted ve tor eld will satisfy the slip ondition. The Neumann ondition is

thenappropriatefortheterrainsurfa eboundary.A ording toNúñezetal.[19℄,

amore appropriate boundary ondition for the terrainsurfa e would be

n

i

1

β

2

i

∂λ

∂x

i

= −v · n.

(1.2.18)

Noti e that this boundary ondition only oin ides with the homogeneous

Neu-mannboundary onditionintwo ases. Firstwhenthetopography isatandthe

initialvelo ityeldisparalleltotheterrainboundary:

v

· n = 0

.These ond ase is when all

β

i

= 1

and again the initial velo ity eld is parallelto the boundary. They also mentioned that the homogeneous Neumann ondition is in onsistent

whenusingtheFiniteElementMethod,hen eFDMshouldbeusedwiththis

on-dition. From equation (1.2.18), we understand why Barnard et al. [2 ℄ and Ross

et al. [21℄ mentioned that applying the losed boundary ondition requires that

the initialvelo ity eld mustrespe t the slip ondition atthe surfa e in orderto

satisfythe impenetrability onstraint whensolving the ellipti equation withthe

FiniteDieren e Method.

Forour model, the following boundary onditions are sele ted:

∂λ

∂n

= 0

forthe terrainsurfa e boundary (1.2.19)

1.2.3. Stability parameters

We nowdis ussthe parametrization of the Gaussianpre ision moduli.It was

noted that the oe ients

β

1

= β

2

and

β

3

play an important role in the orre -tionofthevelo ityeld.Theseweights analtertheowpatternandtheresidual

divergen e of the adjusted ow eld. Determining the right values for these

pa-rameters remains a major problem for wind models in atmospheri s ien es. It

hasbeennoted throughsimulationsof mass- onsistentmodelsthat the odesare

not dire tly sensitive tothe values of

β

1

and

β

3

but to their ratio. This is why a new parameter wasintrodu ed :

β =

β

1

β

3

.

(1.2.21)

We re all the fun tional

I

in equation (1.2.1)now with

β

1

= β

2

:

I(u

d

, v

d

, w

d

) =

Z

Ω



β

₁

2

(u

d

− u)

2

_{+ β}

2

1

(v

d

− v)

2

+ β

3

2

(w

d

− w)

2



dV.

(1.2.22)

One an see that large values of

β

3

will imply minimal adjustments for

w

d

in

(w

d

_{− w)}

2

sin ethefun tionalisbeingminimized.Thesameargumentisvalidfor

smallvaluesof

β

3

thatwillenfor e biggeradjustmentsof

w

d

.Thesame reasoning

an be used with the parameter

β

. For

β ≫ 1

, ow adjustment in the verti al dire tionwillpredominate, sothat windismorelikelytogooveraterrainbarrier

rather than around it. For

β ≪ 1

, ow adjustment will o ur primarily in the horizontal plane, sothe wind is more likely to goarounda terrain barrierrather

than overit.

Noti ethatwhen

β → 0

,theadjustmentispurelyhorizontalandwhen

β → ∞

the adjustment is stri tly verti al.From a physi al perspe tive, this last remark

was used by the

WIN DS

software [11 ℄ so that

β

ould be dened as the at-mospheri stabilityparameter,wherethe strati ationobtained isunstable when

β ≫ 1

, stable when

β ≪ 1

and neutral when

β = 1

.

The mass- onsistentmodels usually adoptone of these two approa hes when

usingtheparameter

β

.Itiseither onstantintheentire domain

Ω

andthevalues dependonatmospheri stabilityorit anbeexpressedasafun tion

β = β(x, y, z)

1.3. Proje tion method

In the last se tion, we explained an approa h widely used in atmospheri

s ien estogetasolenoidalve toreldfromagivenve toreld.Here,weintrodu e

anothermethodforsolvingproblemsin omputationaluiddynami s.Chorin[4℄

has introdu edthis methodbased onthe Hodge de omposition whi hallows one

toextra t a divergen e-free ve tor eld fromany givenve tor eld.

1.3.1. Hodge de omposition

Theorem 1.3.1 (Hodge de omposition). Let

Ω

be a simply onne ted domain with smooth boundary

∂Ω

. Any ve tor eld

v

on

Ω

an be uniquely de omposed in the form

v

_{= v}

_d

_{+ ∇ϕ}

(1.3.1)

where

∇ · v

d

= 0

in

Ω

and

v

d

· n = 0

on

∂Ω

.

In other words, any ve tor eld

v

an be de omposed into two orthogonal omponents; one divergen e-free part

v

d

and a url-free part expressed as the gradient of a s alar eld

ϕ

. Applying the divergen e operator on ea h side of equation(1.3.1) we have

∇ · v = ∇ · v

d

| {z }

=0

+∇ · ∇ϕ

(1.3.2)

∇ · v = ∆ϕ

(1.3.3)

where

∆

isthe Lapla ian operator inCartesian oordinates. The solution ofthis ellipti equation given by

ϕ(x, y, z)

will give the orre tion to be added to the initialve tor eld

v

su hthat weget the solenoidal ve tor eld

v

d

:

v

_d

_{= v − ∇ϕ.}

(1.3.4)

The boundary onditions required for solving the ellipti equation an be found

by taking the normal omponent onboth sides of equation(1.3.1) :

v

_{· n = v}

_d

_{· n}

| {z }

=0

+∇ϕ · n

(1.3.5)

whi h is the Neumannboundary ondition :

∂ϕ

∂n

= v · n

on

∂Ω.

(1.3.7)

1.3.2. Proje tion operator

Sin e

v

d

is uniquely determined,we an dene a proje tion operator

P

, su h that

P(v) = v

d

.

(1.3.8)

Basedon the pro eduredes ribedabove, we an dene

P

as

P = I − ∇ (∇ · ∇)

−1

∇·

(1.3.9)

The operator

P

dened this way is idempotent,

P

2

_{= P}

, self-adjoint,

P = P

T

andthe normoftheoperatorislessthanorequaltoone,

kP(v)k

2

≤ kvk

2

.These properties are used toprovethat the operatoris stable.

1.3.3. Comparison with the variational method

On eagain,theproje tionmethodrequires thesolutionofthe following

Neu-mannproblem :

∆ϕ = ∇ · v

on

Ω

(1.3.10)

∂ϕ

∂n

= v · n

on

∂Ω.

(1.3.11)

Settingthe parameters

β

1

= β

2

= β

3

= 1

inequation (1.2.11)we have

∂

2

λ

∂x

2

+

∂

2

λ

∂y

2

+

∂

2

λ

∂z

2

= −2

 ∂u

∂x

+

∂v

∂y

+

∂w

∂z



(1.3.12) that is

∆λ = −2∇ · v

(1.3.13)

It an be noti ed that equation (1.3.13) isexa tly the same asequation (1.3.10)

up to the onstant -2. The solution of both ellipti equations is a s alar eld

fromwhi hthe gradient gives the orre tion tothe initial ve toreld (equations

There are two main dieren es between these methods. First, the boundary

onditions for solving the ellipti equations are not the same. In the proje tion

method,onlyNeumann onditionsareusedunlikethemixed onditions(Diri hlet

andNeumann)usedinthevariationalmethod.TheHodgede ompositionassumes

that the divergen e-free ve tor eld

v

d

will be parallel to the domain boundary

∂Ω

sin e

v

d

· n = 0

,whi hisdenitelynot the kindof ve toreld weare looking for.

Itis also important tonoti e that the proje tion

P

is well-denedsin e

v

d

is uniquely determinedasmentionedinthe Hodge de ompositiontheorem. Uni ity

of the proje tion is guaranteed by the Neumann boundary onditions. In the

urrentmodel,weareinterestedinusingbothDiri hletandNeumann onditions.

We will then lose the uni ity of the proje tion. Hen e, the orre ted eld

v

d

found with the proje tion

P

will be the losest ve tor eld to

v

su h that it is in ompressible.

Finally,the main dieren eresideinthe Gauss moduli.Sin e ourmodel does

not rely on experimental data, ex ept the high altitude wind speed, there is no

need for a alibration of the model with the weights

β

i

, hen e they will be set equal to one. In order to avoid onfusion with the notation, the s alar eld will

benotedby

ϕ

ratherthan

λ

andthe problemthatwill besolvedisthe following:























∆ϕ = −2∇ · v

on

Ω

∂ϕ

∂n

= 0

onterrain surfa e boundary

ϕ

= 0

onside and top boundaries

(1.3.14)

1.4. Ve tor field initialization

Thewaytheinitialve toreldisinitializedhasahugeimpa tonthestru ture

ofthe adjusted ve toreld. Remember thatthe orre ted ve toreld will bethe

losestve toreldtotheinitialve toreldthatsatisesthe onservationofmass.

Hen e the initializationpro ess isa veryimportantstep.

Therearedierenttypesofdatathatareavailableto onstru ttheinitialwind

of this data is usually poor and that its density is more than often insu ient

forresolvingvariations ofthe owabove omplextopography.Moreover,thedata

givesonlyinformationfortheverti alproleofthewind,neverforthehorizontal.

These reasonsen ourage aninitialization of the ve tor eld with a high altitude

wind,whi h is alled the geostrophi wind inthe literature.

Asin

WIN DS

[11 ℄,averti alprole anbe onstru tedfromthegeostrophi wind.MostofthemodelsstudiedbyRattoetal.[20℄splittheatmosphereintotwo

layers:thesurfa elayer(SL)andthePlanetaryBoundaryLayer(PBL).Overthe

PBL (1000m-2000m), the wind is assumed tobe onstant with height and given

by the geostrophi wind. The surfa e layer takes into a ount fri tion/vis ous

ee ts and is usually ontained in the rst 100m of the atmosphere. This layer

is not of great interest sin e the slip ondition is applied at the surfa e of the

terraininthismodel. Between the twolayers, dierentinterpolations hemes an

be used toget the verti al prole :linear, logarithmi orpowerlaw.

Barnard et al. [2℄ and Ross et al. [21 ℄ have also noti ed that applying the

losedboundary ondition ontheterrain surfa erequiresthat the initialvelo ity

eld mustrespe t the slip ondition at the terrain surfa e inorder to satisfythe

impenetrability onstraint.Sin ethereisusuallynoinformationonthehorizontal

prole, i.e. on the verti al omponent of the wind eld, we will follow Barnard

and setthe verti al wind omponent tozero

w

= 0

.

1.4.1. Potential ow and onformal oordinates

Barnard etal. [2℄ have observed that la k of veri ation isa major di ulty

for mass- onsistent models. Many papers used real data to alibrate and verify

theresultsgivenbythemodel.Rossetal.[21℄foundoutthatif

β

1

= β

2

= β

3

and thattherighthandsideofequation(1.3.14)vanishes,then

ϕ

representsavelo ity potential. They also used simple terrainshapes (half- ylinder, hemisphere,

( onformal) oordinates tosolvethe Poisson equation :



























ξ

= x

η

= y

σ

(x, y) =

h

t

(x, y) − z

h

t

(x, y) − H(x, y)

(1.4.1)

where

h

t

(x, y)

istheheightofthedomainwhi hinour ase is onstantandwhere

H(x, y)

istheheightofthetopography.Thesurfa eoftheterrainisrea hedwhen

σ

= 1

and the top boundary when

σ

= 0

. They still solve the ellipti equation withCartesian oordinates but with adierentNeumann boundary ondition at

theterrainsurfa e inordertoget thepotential ow.They initializedtheir model

with a uniform ba kground wind in onformal oordinates and they found that

the generated owis ingood agreement with the analyti potential ow. Even if

the problem issolvedin Cartesian oordinates, onformal oordinates are useful

for onstru tingan initialvelo ity eldtangenttothe terrainand paralleltothe

SOLVING THE MODEL IN 2 DIMENSIONS

There exists many ways to solve the Poisson equation introdu ed in the last

hapter.Dieren es between previousmass- onsistent models omefrom

numer-i al algorithms for solving ellipti equations and from the hoi e of the values

used for the parameters. For instan e, Sherman [24℄ used the Finite Dieren e

Method (FDM) and dis retized the topography in a stair-step fashion. Ishikawa

[13℄solvedthe ellipti equationwith FDM as wellbut on astaggered grid. Ross

etal. [21 ℄ and Barnard et al.[2 ℄ used onformal oordinates to solve the ellipti

equation in order to get better integration of the terrain surfa e and ensure the

appli ationof the slip ondition.Forthofer [10℄ used the FiniteElementMethod

(FEM) tosolve the PDE.

This hapterwill onsiderthe Embedded Boundary Method(EBM) for

solv-ing the ellipti equation with the appropriate boundary onditions for the

two-dimensional ase.ThisisthersttimethatEBMisappliedtoanempiri almodel

usedtosimulatethe spreadofwildres. Theparti ularfeatureofthisapproa his

thatitembeds anirregularboundaryintoaCartesiangrid. Hen e,the generated

wind elds shouldbe in better agreement with the terrain surfa e.

Sin etheEBMisbasedontheFiniteVolumeMethod(FVM),thislastmethod

is rst introdu ed before moving on to EBM. The onvergen e of the algorithm

is tested for dierent terrain shapes and initial wind ve tor elds, and an error

analysis is also ondu ted. Finally, the divergen e-free ve tor elds are used to

2.1. Finite Volume Method

The FVM is a useful approa h for solving numeri ally partial dierential

equations (PDE). It allows the divergen e operator to be dis retized using the

divergen etheorem, su hasinthe Poisson equation.Thenite-volumeapproa h

hassome onsiderableadvantages su hasregularpredi tablememorya ess and

higher a ura y for less omputation. In this se tion, we explain how the FVM

an solveour ellipti equation without topography embedded in the re tangular

domain.

2.1.1. Dis retization of the domain

Considerare tangulardomain

Ω ⊂ R

2

.Thedomainhaslength

L

x

and height

L

y

andisdis retizedusingaCartesiangridwhosere tangular ontrolvolumesare denedas

Υ

i,j

= [(i −

1

2

)h

x

, (i +

1

2

)h

x

] × [(j −

1

2

)h

y

, (j +

1

2

)h

y

]

for

i = 1, . . . , N

x

and

j = 1, . . . , N

y

. The number of horizontal and verti al ells,

N

x

and

N

y

, are used todene the horizontal and verti al mesh spa ing

h

x

= L

x

/N

x

and

h

y

= L

y

/N

y

. The method uses ontrol volumes

V

i,j

= Υ

i,j

∩ Ω

.

PSfrag repla ements

j

i

h

x

h

y

L

x

L

y

V

i,j

Υ

i,j

Fig. 2.1. Dis retization of the two-dimensional domainfor FVM.

2.1.2. Divergen e operator

We are now looking to solve the following Poisson equation

where

v

= (u, v)

isthe initialve toreld and

ϕ

is as alareld. Equation(2.1.1) an be writtenin onservation form

∇ · F = −2∇ · v

(2.1.2)

wherethe ux

F

= ∇ϕ

is the onserved quantity.

To a hieve a dis retization of the divergen e operator, we rst re all the

di-vergen e theorem :

Theorem 2.1.1 (Divergen etheorem). Let

Ω

be a ompa t region in

R

n

with a

pie ewise smooth boundary

∂Ω

. If

F

is a ontinuously dierentiable ve tor eld dened on a neighbourhood of

Ω

, then we have

Z

Ω

∇ · FdV =

I

∂Ω

F

_{· ndA.}

(2.1.3)

Usingthe ellaveragevalueofthedivergen eof

F

andthedivergen etheorem in

R

2

we have;

∇ · F ≈

1

|V

i,j

|

Z

V

i

∇ · FdV =

1

|V

i,j

|

I

∂V

i

F

_{· ndA}

(2.1.4)

=

1

|V

i,j

|

h

h

y

F

i+

1

2

,j

+ h

x

F

i,j+

1

2

− h

y

F

i−

1

2

,j

− h

x

F

i,j−

1

2

i

(2.1.5)

where

n

istheoutwardunitnormaltothe ontrolvolume ell

V

i,j

and

|V

i,j

| = h

x

h

y

thevolumeof

V

i,j

.This orrespondstothemidpointruledis retizationofthe line integral.Hen e, the dis retized divergen eoperator

D

M

_(F)

i,j

is

D

M

(F)

i,j

=

F

_i+

1

2

,j

− F

i−

1

2

,j

h

x

+

F

i,j+

1

2

− F

i,j−

1

2

h

y

(2.1.6)

The uxes are illustrated ingure 2.2. In the Poisson equation, a dis retized

PSfrag repla ements

F

_i,j−

1

2

F

_i+

1

2

,j

F

_i,j+

1

2

F

_i−

1

2

,j

divergen e operator is also applied to the initial ve tor eld

v

. It would be de-sirable to use the same dis retization of the operator. We also need a onsistent

dis retizationforthegradientoperatorsin e

F

= ∇ϕ

.TheMACproje tionseems quiteappropriate forthis ase.

2.1.3. Marker-and-Cell Proje tion

Therearemanywaystodis retizemathemati aloperators.Thedis retization

mostly reliesonthe kind ofgrid onwhi hthe omputationis done.The

Marker-and-Cell(MAC)proje tionintrodu edbyHarlowandWel h[12℄usesastaggered

grid, where the omponents of the ve tor eld are dened on the edges of the

ontrol ells and the s alar eld is denedat the enter as shown inFigure 2.3.

PSfrag repla ements

u

_i−

1

2

,j

u

i+

1

2

,j

v

_i,j−

1

2

v

_i,j+

1

2

ϕ

i,j

Fig. 2.3. Staggered grid with the ell- entered s alar eld

ϕ

and the edge- entered omponents of the velo ity eld

(u, v)

.

Let

D

M

and

G

M

be the dis rete divergen e and dis rete gradient operators

overthestaggeredgrid.Thedivergen eisdenedusingthedivergen etheoremon

the ontrol ell asbeforeand the gradient by entered dieren eoverthe edges :

D

M

(v)

i,j

=

u

_i+

1

2

,j

− u

i−

1

2

,j

h

x

+

v

i,j+

1

2

− v

i,j−

1

2

h

y

(2.1.7) and dene

G

M

(ϕ)

_i+

1

2

,j

=

ϕ

i+1,j

− ϕ

i,j

h

x

(2.1.8)

G

M

(ϕ)

_i,j+

1

2

=

ϕ

i,j+1

− ϕ

i,j

h

y

(2.1.9)

Rememberthat our methoduses a proje tion operator

P

denedinequation (1.3.9).Let

P

M

bethe dis retizationof this proje tion. It is then dened as

and it is easy to verify that

D

M

_(P

M

_{(v)) = 0}

. This means that the dis retized

proje tionoperatorisexa t.See AppendixAforadis ussionofthedis retization

of proje tions.

FortheMACproje tion,thedis retizationoftheLapla eoperator(

∆ = ∇·∇)

is dened as

D

M

_(G

M

_(ϕ))

i,j

whi h orresponds to the standard 5 point sten il Lapla ianfor interior ontrol volume ells(

i

= 2, ..., N

x

− 1

and

j

= 2, ..., N

y

− 1

) asshown in Figure2.4.

Fig. 2.4. Standard 5 point sten ilLapla ian

D

M

_(G

M

_(ϕ))

i,j

.

Despite the fa t that the omponents of the ve tor eld are not ollo ated,

the MACproje tion has the advantage that the no-owboundary ondition an

besetexpli itlyatwalls forre tangulardomainssin e theedges ofthe boundary

ells mat h the domain boundaries.

2.1.4. Boundary uxes

In order to use the divergen e operator on the ontrol volume ells on the

boundariesof thedomain, theuxes attheboundaries mustbespe ied.Forthe

bottom boundary, the problem uses a Neumann ondition

∂ϕ

∂n

= 0

. In this ase, the ux issimply zero,

F

i,

1

2

= 0

.

For the top and side boundaries, the Diri hlet ondition

ϕ

= 0

is used. This type of ondition does not pres ribe dire tly a parti ular value for the ux at

the boundary. Hen e, we follow Johansenand Colella[14 ℄ and use a three-point

gradient sten il inorder to get aspe i value of the ux on the boundary. The

gradientformula isgivenby

∂ϕ

∂n

=

1

d

2

− d

1

 d

2

d

1

ϕ

B

− ϕ

1

−

d

1

d

2

ϕ

B

− ϕ

2



(2.1.11)

where

ϕ

B

is the value of the fun tion

ϕ

on the boundary given by the Diri hlet ondition,

ϕ

1

and

ϕ

2

are the losest grid pointvaluesof

ϕ

to

ϕ

B

.Those two grid

pointsarerespe tivelyatdistan es

d

1

and

d

2

fromtheboundary.Thissten ilhas adis retization error of order

O(h

2

₎

. Figure2.5 showshow the uxon the right

boundary is al ulated by

∂ϕ

∂x

= F

N

x

+

1

2

,j

=

1

h

x



3 ϕ

B

− ϕ

N

x

,j

−

1

3

ϕ

B

_{− ϕ}

N

x

−1,j



(2.1.12) PSfrag repla ements

F

_N

_x

₊

1

2

,j

ϕ

N

x

−1,j

ϕ

N

x

,j

ϕ

B

Fig. 2.5. Interpolation of the boundary ux

F

N

x

+

1

₂

,j

for applying the Diri hlet ondition using

ϕ

B

,

ϕ

N

x

,j

and

ϕ

N

x

−1,j

.

Notethat this interpolation leads toan outward pointing gradient. Withthe

uxes at the boundary xed, it is easy to see that the Lapla ian operator will

have 3 dierent sten ils overthe domain aspi tured inFigure2.6.

Fig. 2.6. The 3 dierent sten ils for the Lapla ian operator.

2.1.5. Symmetri matrix

Solving the Poisson equation with the FVM redu es to nd a solution of a

toformthe ve tor

ϕ

ofthe

ϕ

i,j

dened atthe enter of the ontrol volume ells :

ϕ

= (ϕ

1,1

, . . . , ϕ

N

x

,1

, ϕ

1,2

, . . . , ϕ

N

x

,2

, . . . , ϕ

1,N

y

, . . . , ϕ

N

x

,N

y

)

(2.1.13)

The oe ientsin front of the unknown values of

ϕ

i,j

llup the matrix

A

of size

N

x

N

y

× N

x

N

y

.More pre isely, the maindiagonal ontains the oe ients of

ϕ

i,j

,thediagonalsaboveandbelowthemaindiagonalthe oe ientsof

ϕ

i±1,j

and thetwoextradiagonals,with

N

x

−2

zeroelementsbetweentheupper/lower diag-onal, the oe ients of

ϕ

i,j±1

. Hen e, the symmetri matrix

A

has the stru ture shown ingure 2.7 whi h isthe same asthat of the FDM.

0

10

20

30

40

0

5

10

15

20

25

30

35

40

45

i

j

Non−zero entries in Poisson Matrix

Fig. 2.7. Symmetri matrix ofthe linear system with

N

x

= 6

and

N

y

= 8

.

Allknown values su has

ϕ

B

and the RHSof the Poisson equation,

D

M

_(v)

i,j

, are put in the ve tor

b

. The linear system

Aϕ

= b

is solved using the solver

mldivide

in MATLAB whi h uses a dire t method.

2.1.6. Corre ted ve tor eld

On eweget thesolutiongivenbytheve tor

ϕ

,we an omputethe orre ted ve tor eld withequations(1.2.6)-(1.2.8) of hapter1.The orre ted eldonthe

staggered grid is easily al ulated with the MACgradient operators :

u

d

_i+

1

2

,j

= u

i+

1

2

,j

+

1

2

G

M

_(ϕ)

i+

1

2

,j

(2.1.14)

v

d

_i,j+

1

2

= v

i,j+

1

2

+

1

2

G

M

_(ϕ)

i,j+

1

2

(2.1.15)

Noti e that it is important to al ulate

G

M

_(ϕ)

i+

1

2

,j

and

G

M

_(ϕ)

i,j+

1

2

on the

boundaries with the same gradient sten il used for omputing the uxes on the

boundaries. Otherwise, the nal divergen ewill not be zero on the ells next to

the boundaries.

2.2. Embedded Boundary Method

We will now fo us on the Embedded Boundary Method for solving

numeri- allythe ellipti equation(2.1.1)onanirregulartwo-dimensionaldomain

Ω

.This approa h uses a nite-volume dis retization on the Cartesian grid on whi h the

boundary of the topography is embedded. In this hapter, we will fo us on the

simple2D ase whi h isdis ussed by Johansen and Colella[14℄.

There tangulardomain

Ω

uses the sameCartesian gridasmentionedearlier. Thetopographywhi hisgivenby thefun tion

H(x)

isrepresentedasapie ewise linear fun tion where the oordinate

x

= (i ±

1

2

)h

x

is dened on the ontrol volume ell edges. The fun tion

H(x)

orresponds to the lower boundary of

Ω

. The geometryof the irregular domain

Ω

isrepresented with its interse tion with the Cartesian grid. The method uses ontrol volumes

V

i,j

= Υ

i,j

∩ Ω

and their fa es

A

i±

1

2

,j

and

A

i,j±

1

2

whi haretheinterse tionof

∂V

i,j

withthe oordinatelines

{x = (i ±

1

2

)h

x

}

and

{y = (j ±

1

2

)h

y

}

. The interse tion of the boundary of the irregulardomainandtheCartesian ontrolvolumesarethefa es

A

B

i,j

= Υ

i,j

∩∂Ω

.

In order to onstru t an appropriate divergen e operator, areas and volumes

are writtenas nondimensional terms :

volume fra tions :

κ

i,j

= |V

i,j

|(h

x

h

y

)

−1

fa e apertures :

α

i±

1

2

,j

= |A

i±

1

2

,j

|h

−1

y

and

α

i,j±

1

2

= |A

i,j±

1

2

|h

−1

x

boundaryapertures :

α

B

i,j

.

Itisassumedthatthosevalues anbe al ulatedwithana ura yoforder

O(h

2

₎

.

Noti e that

α, κ

∈ [0, 1]

. When

κ

= 0

, the ontrol volume ell is ompletely ontained in the topography and when

κ

= 1

, the ell is full, meaning that the topography doesnot ut the ell.

Asbefore,weuse thefa tthatthe Lapla ianoperator

∆ϕ

inLHSofequation (2.1.1) an be written on a onservative form as

∇ · F

with

F

= ∇ϕ

. It is then

possible todis retize the divergen eoperator

∇·

using the divergen e theorem :

D

M

(F)

i,j

=

1

κ

i,j

α

_i+

1

2

,j

F

i+

1

2

,j

− α

i−

1

2

,j

F

i−

1

2

,j

h

x

+

α

i,j+

1

2

F

i,j+

1

2

− α

i,j−

1

2

F

i,j−

1

2

h

y

+ α

B

_i,j

F

_i,j

B

!

(2.2.1)

wheretheuxes

F

i±

1

2

,j

,

F

i,j±

1

2

andtheuxontheembeddedboundary

F

B

,dened

at the entroid of every ell fa es, are linear ombinations of

ϕ

i,j

and of the boundary values

ϕ

B

. This dis retization takes into a ount the stru ture of the

volume ell, parti ularly when we are al ulating the divergen e in a ut ell

(

0 < κ < 1

). The left pi ture in Figure 2.8 gives an example of uxes lo ated atthe fa e entroids ina ut ell, while the fa e entroids oin ide with the fa e

enters ina full ell.

Fig. 2.8. Fluxesina ut ell(

0 < κ < 1

)andinafull ell(

κ

= 1

). The ux

F

B

i,j

≡ 0

(redarrow)sin eaNeumann onditionisapplied at the terrain surfa e boundary.

Remember that a homogeneous Neumann ondition

∂ϕ

∂n

= 0

is applied on the terrain surfa e boundary whi h implies that

F

B

i,j

≡ 0

. Hen e, the divergen e operatorbe omes

D

M

(F)

i,j

=

1

κ

i,j

α

_i+

1

2

,j

F

i+

1

2

,j

− α

i−

1

2

,j

F

i−

1

2

,j

h

x

+

α

i,j+

1

2

F

i,j+

1

2

− α

i,j−

1

2

F

i,j−

1

2

h

y

!

(2.2.2)

Itisimportantthattheuxesbedenedatthe entroidofthefa es

A

i±

1

2

,j

and

A

_i,j±

1

2

inordertokeepagoodapproximationofthe dis retizationofthe integral.

For uxes dened on the verti al fa es

A

i±

1

2

,j

used with the above ell. For instan e, the ux

F

i+

1

2

,j

on the fa e

A

i+

1

2

,j

when

0 < α

_i+

1

2

,j

<

1

an be al ulated using the interpolationformula :

F

_i+

1

2

,j

=

 1 + α

i+

1

2

,j

2



G

M

(ϕ)

_i+

1

2

,j

+

 1 − α

i+

1

2

,j

2



G

M

(ϕ)

_i+

1

2

,j+1

(2.2.3)

This interpolation is illustrated in Figure 2.9. Note that for full fa es (without

embedded boundary), the fa e aperture

α

i+

1

2

,j

= 1

, hen e the regular entered

dieren eis used :

F

_i+

1

2

,j

= G

M

_(ϕ)

i+

1

2

,j

.

(2.2.4) PSfrag repla ements

G

M

(ϕ)

_{i+ 1}

2

,j

G

M

_(ϕ)

i+ 1

2

,j+1

F

_i+

1

2

,j

Fig. 2.9. Interpolation of the ux

F

i+

1

2

,j

on a ut ell edge using

G

M

(ϕ)

_i+

1

2

,j

and

G

M

_(ϕ)

i+

1

2

,j+1

.

For the uxes dened on the horizontal fa es

A

i,j±

1

2

, a linear interpolation

s hemeusing the uxineitherthe rightorleft ellis required.It isimportantto

usetheappropriateinterpolationsothe omputationmakessense.Theux

F

i,j+

1

2

onthe fa e

A

i,j+

1

2

when

0 < α

i,j+

1

2

<

1

an be al ulated using thisinterpolation

formula :

F

_i,j+

1

2

=

 1 + α

i,j+

1

2

2



G

M

(ϕ)

_i,j+

1

2

+

 1 − α

i,j+

1

2

2













G

M

(ϕ)

_i+1,j+

1

2

if terrain is onthe left side

G

M

_(ϕ)

i−1,j+

1

2

if terrain is onthe right side

(2.2.5)

The two ases are shown in Figure2.10.

When the divergen e is al ulated in a full ell (

κ

i,j

= 1

) for whi h all fa e apertures

α

= 1

, we re over the MAC divergen e operator given by equation

Fig. 2.10. Interpolationof the ux

F

i,j+

1

2

ona ut ell edge when

the terrainis onthe right hand side or lefthand side.

(2.1.6)for the FVM :

D

M

(F)

i,j

=

F

_i+

1

2

,j

− F

i−

1

2

,j

h

x

+

F

i,j+

1

2

− F

i,j−

1

2

h

y

(2.2.6)

A riti al feature of the EBM, as explained by Johansen and Colella [14℄, is

the assumptionthat the solution an be extendedsmoothlyoutsideof

Ω

. As an be seen in Figure 2.10, some grid values

ϕ

are overed by the terrain. Johansen and Colella assume that there are solution values for them that are su iently

smooth so that a trun ation error analysis based on Taylor expansions will be

valid.

Fortopandsideboundarieswhi hhaveDiri hlet onditions,thesamegradient

sten il is used as in the previous se tion. Johansen and Colella use one more

onstraint on the dis retization of the domain that is related to this gradient

formula : the interpolation sten il must not rea h into ells with zero volume

(

κ

= 0

), hen ethe Cartesian grid mustbe ne enough.

The divergen eon the right handside of equation(2.1.1) is

D

M

(v)

i,j

=

1

κ

i,j

 α

_i+

1

2

,j

u

i+

1

2

,j

− α

i−

1

2

,j

u

i−

1

2

,j

h

x

+

α

i,j+

1

2

v

i,j+

1

2

− α

i,j−

1

2

v

i,j−

1

2

h

y



(2.2.7)