a
A pa icle- ackingme hod o simula ing he
dispe sion o non-conse a i e adionuclides in
coas al wa e s
R. Pe i!aa*nnez
a,
*, A.J. Ellio
b
Depa amen o Fı´sica Aplicada I, Uni e si y o Se illa, EU Ingenie ı´aTe
´cnica Ag ı´cola, C a. U e a km 1,
41013 Se illa, Spain
b
Cen e o Applied Oceanog aphy, Uni e si y o Wales, Bango , Ma ine Science Labo a o ies,
Menai B idge, Anglesey LL59 5EY, UK
Abs ac
A pa icle- acking me hod has been used o simula e he dispe sion o non-conse a i e
adionuclides in he sea. Th ee dimensional u bulen diffusion and he in e ac ions be ween
wa e , suspended ma e and bo om sedimen s a e simula ed using a s ochas ic me hod.
Kine ic ans e coefficien s, as in fini e diffe ence models, a e used o desc ibe he ans e s
be ween he liquid and solid phases. Deposi ion o suspended ma e and e osion o sedimen
a e also included in he model. The me hod has been applied o simula e he dispe sion o
137
Cs and
239,240
Pu in he English Channel and he esul s ha e been compa ed wi h hose o a
fini e diffe ence model. The esul s om bo h echniques a e, in gene al, in good ag eemen .
Keywo ds: Model; Dispe sion; Pa icle- acking; Fini e diffe ence; Radionuclides; English Channel
1. In oduc ion
The s a e-o - he-a models used o simula e he dispe sion o adioac i i y in he
sea consis o fini e elemen s o , mainly, fini e diffe ence models ha sol e he
hyd odynamic equa ions oge he wi h he ad ec ion–diffusion dispe sion equa ion,
in ei he a wo dimensional o a h ee dimensional o m (Ha ms, 1997; Pe i!
aa*
nnez &
*Co esponding au ho . Tel.: +34-95-4233669; ax: +34-95-4232644.
E-mail add esses: [email p o ec ed] (R. Pe i!
aa*
nnez), [email p o ec ed] (A.J. Ellio )
Regue a, 1999). I he model is applied o simula e he dispe sion o non-
conse a i e adionuclides, hen he in e ac ions wi h he solid phases (suspended
ma e and bo om sedimen s) mus also be conside ed. This implies ha he
suspended ma e equa ion, including he se ling o pa icles, deposi ion and e osion
o he sedimen , mus be sol ed oo (Ald idge, 1998; Ma g elash ily, Made ich, &
Zheleznyak, 1997; Piasecki, 1998; Pe i!
aa*
nnez, 1999, 2000). Abso p ion/deso p ion
eac ions a e usually desc ibed using kine ic ans e coefficien s ins ead o he less
app op ia e equilib ium dis ibu ion coefficien s, kd. These models a e compu a-
ionally expensi e due o he ac ha a la ge numbe o equa ions mus be sol ed.
Also, since fini e diffe ence models wo k wi h adionuclide concen a ions, he whole
compu a ional g id mus be swep each ime s ep. Finally, he CFL condi ion
(Kowalick & Mu y, 1993) limi s he size o he ime s ep ha can be used i an
explici scheme is used o sol e he hyd odynamic equa ions. In consequence, la ge
CPU imes may be equi ed e en using ecen supe compu e s (Pe i!
aa*
nnez, 1999).
An al e na i e app oach is o make use o pa icle- acking echniques. In his
kind o model, he impac o he CFL c i e ion can be educed by making he
hyd odynamic compu a ions off-line. Also, ad ec ion can be simula ed o a high
deg ee o accu acy since nume ical dispe sion (which appea s when he ad ec ion
equa ion is sol ed wi h fini e diffe ences) is no in oduced. Pa icle- acking models
ha e al eady been used o simula e he dispe sion o conse a i e ace s and oils
spills (Hun e , 1987; Ellio , Dale, & P oc o , 1992; P oc o , Ellio , & Fla he ,
1994a, b). In such applica ions, a elease o adioac i i y o he sea is modelled as a
numbe o disc e e pa icles, each pa icle being equi alen o a numbe o uni s (Bq,
moles, a oms, e c.). Then he pa h ollowed by each indi idual pa icle is compu ed,
u bulen diffusion being modelled as a h ee dimensional andom walk (Mon e
Ca lo) p ocess. The densi y o pa icles is calcula ed o ob ain he adioac i i y
concen a ions a he end o he simula ion. The main difficul y ha appea s in he
simula ion o he dispe sion o non-conse a i e adionuclides is he ea men o
abso p ion and deso p ion: how o decide i each pa icle is fixed o suspended
ma e o bo om sedimen s (i ini ially dissol ed) o i i is edissol ed (i ini ially
p esen in he suspended ma e o he bo om sedimen ). The main con ibu ion o
his pape consis s o a new me hod de eloped o sol e his p oblem: a o mula ion
(sui able o a pa icle- acking model) o desc ibe he ans e s o adionuclides
be ween wa e , suspended ma e and bo om sedimen s, based upon kine ic ans e
coefficien s and a s ochas ic me hod, is p esen ed. I is impo an o poin ou ha
exac ly he same physical pa ame e s as in he equi alen fini e diffe ence models a e
used.
The pa icle- acking modelling echnique is well sui ed o p oblems in which high
con aminan g adien s a e in ol ed, since nume ical diffusion is no in oduced.
This, oge he wi h he ac ha i can gi e e y as answe s, e en in a PC, allows he
echnique o be conside ed as a e y use ul p edic i e ool in he assessmen o
con amina ion ollowing acciden al o delibe a e eleases o adionuclides.
The pa icle- acking model is p esen ed in he nex sec ion, hen an applica ion o
he English Channel is shown. The model has been used o simula e he dispe sion o
an ins an aneous hypo he ical elease o adionuclides om La Hague nuclea uel
ep ocessing plan , loca ed on he F ench sho e o he English Channel. Simula ions
ha e been ca ied ou o wo adionuclides wi h e y diffe en geochemical
beha iou s: he ela i ely conse a i e
137
Cs and he high eac i e
239,240
Pu. A fini e
diffe ence model o he Channel has been p e iously de eloped and alida ed
h ough he s udy o he dispe sion o hese adionuclides, compa ing obse ed and
compu ed concen a ions in wa e , suspended ma e and bo om sedimen s
(Pe i!
aa*
nnez and Regue a 1999; Pe i!
aa*
nnez, 2000). Thus esul s o he pa icle- acking
model will be compa ed wi h he ou pu o he fini e diffe ence model in equi alen
simula ions and he ela i e ad an ages o each echnique will be assessed.
2. The pa icle- acking model
2.1. Ad ec i e anspo
The posi ion ec o o a gi en pa icle, þD ðÞ, a ime þD is compu ed om
þD ðÞ ðÞ
D ¼q ðÞ;ð1Þ
whe e D is he ime s ep used in he model and q he cu en ec o o componen s u
and along he xand yaxes, espec i ely. Cu en s a e ob ained by unning a
hyd odynamic model in ad ance. S anda d idal analysis is used o de e mine he
idal cons an s ( ide ampli ude and phase) o each g id cell o he hyd odynamic
model. These cons an s a e e alua ed o bo h componen s o he flow and can be
de i ed o as many idal cons i uen s as desi ed. In his wo k, only he wo main
semidiu nal ides, M2and S2, will be conside ed. Once he idal cons an s a e known,
compu a ion o he flow ec o , q, jus in ol es he calcula ion and addi ion o a ew
cosine e ms. As a consequence, he e alua ion o he idal ad ec i e anspo o
pa icles is e y as and is no limi ed by he CFL c i e ion.
Fo eal applica ions, fi s o de accu acy in he pa icle- acking scheme is
adequa e. In simula ions o he mo emen o d ogues in an es ua ine en i onmen ,
Ellio and Cla ke (1998) ound no imp o emen in he esul s when a second o de
accu acy scheme was used o simula e he mo emen o su ace d i e s by he
pa icle- acking echnique. Mo eo e , in ocean dispe sion p oblems, he effec s o
u bulence will mask any small e o s in he ad ec ion scheme.
The ne esidual cu en in he modelled a ea mus be added o qsince a esidual
anspo canno be gene a ed wi h he pu e ha monic idal cu en s ha a e used in
pa icle- acking calcula ions. The esidual flow ec o s can also be ob ained om
he (p e iously un) hyd odynamic model. Wind-induced anspo can be included
in he model by assuming ha he su ace wind-induced cu en is a pe cen age o
he wind speed, gene ally 2–3% (P oc o e al., 1994b). This cu en dec eases
loga i hmically below a dep h z1( he hickness o he wind-d i en su ace laye ) o
ze o a a dep h z2. Eckman heo y (Pugh, 1987) p edic s ha he su ace wind-
induced cu en due o a s eady wind blowing o e deep wa e should be deflec ed o
he igh o he wind di ec ion (in he no he n hemisphe e). Howe e , obse a ional
e idence sugges s ha his deflec ion angle can be neglec ed (P oc o e al., 1994b) in
shallow coas al wa e s.
2.2. Tu bulen diffusion
Th ee dimensional diffusion is simula ed using a andom walk me hod. I has been
shown (P oc o e al., 1994b; Hun e , 1987) ha i is a simula o o Fickian diffusion
p o ided ha he maximum size o he ho izon al s ep gi en by he pa icle, Dh,is
Dh¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12KhD
pð2Þ
in he di ec ion 2pRAN, whe e RAN is a andom numbe be ween 0 and 1. This
equa ion gi es he maximum size o he s ep. In p ac ice, i is mul iplied by RAN o
ob ain he eal size a a gi en ime and o a gi en pa icle. Simila ly, he size o he
e ical s ep is
D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
2K D
pð3Þ
gi en ei he owa ds he sea su ace o he sea bo om. Khand K a e he ho izon al
and e ical diffusion coefficien s, espec i ely.
2.3. Radioac i e decay
Conside he adioac i e decay equa ion:
qC
q ¼lC;ð4Þ
whe e lis he adioac i e decay cons an . This equa ion can be ea ed using a
s ochas ic me hod i i is assumed ha he p obabili y po emo al o a pa icle a
each ime s ep is (Hun e , 1987; P oc o e al., 1994b)
p¼1elD :ð5Þ
In p ac ice, a andom numbe is gene a ed o each pa icle on each ime s ep. I
RAN4p hen he pa icle is emo ed om he compu a ion.
2.4. T ans e s be ween wa e , suspended ma e and bo om sedimen s
Conside a wo phase sys em. I he ans e s o adionuclides be ween he wo
phases a e desc ibed h ough he kine ic ans e coefficien s k1and k2, he equa ions
ha gi e he ime e olu ion o ac i i y in he wo phases a e
qA1
q ¼k1A1þk2A2;ð6Þ
qA2
q ¼k1A1k2A2:
These equa ions a e easily sol ed using fini e diffe ences. In pa icle acking, a label
is gi en o each pa icle o diffe en ia e i i is in phase l o 2. I he pa icle is in phase
1, he p obabili y p1 ha he pa icle goes o phase 2 in each ime s ep is
p1¼1ek1D :ð7Þ
Simila ly, i he pa icle is in phase 2, he p obabili y p2 ha i goes o phase 1 each
ime s ep is
p2¼1ek2D :ð8Þ
Thus, in pa icle acking, he exchanges be ween wo phases can be modelled as wo
decay p ocesses wi h p obabili ies p1and p2. These p ocesses a e ea ed as he
adioac i e decay p ocess desc ibed abo e. I a gi en pa icle goes om one phase o
he o he , i s label is changed and he new co esponding decay p ocess is conside ed
a he nex ime s ep.
This me hod has been compa ed wi h he fini e diffe ence solu ion o he sys em o
Eq. (6). I has been conside ed ha all adionuclides a e, a ¼0, in phase 1. Thus,
he solu ion gi en by each me hod e e s o he pe cen ages o adionuclides ha a e
in phase 1 a each ollowing ime s ep. Resul s ob ained by bo h me hods a e
p esen ed in Fig. 1, using 200 and 10,000 pa icles in he s ochas ic simula ion. High
fluc ua ions occu wi h 200 pa icles, bu he fini e diffe ence solu ion is well
modelled i 10,000 pa icles a e used in he pa icle- acking calcula ion. The mean
alue and s anda d de ia ion o he diffe ence be ween fini e diffe ence and
s ochas ic solu ions a e p esen ed in Table 1 o diffe en numbe s o pa icles in
Fig. 1. Time e olu ion o he pe cen age o adionuclides in phase 1 gi en by he solu ion o he
diffe en ial equa ions using fini e diffe ences and he pa icle- acking me hod, wi h 200 and 10,000
pa icles.

he s ochas ic me hod. I can be seen ha bo h he mean alue and s anda d
de ia ion o he diffe ence dec ease as he numbe o pa icles conside ed in he
s ochas ic simula ion inc eases. Accep able esul s a e ob ained o numbe s o
pa icles o he o de o 10,000.
The s ochas ic me hod can be ex ended o he case in which he e a e h ee
diffe en phases: wa e , suspended ma e and ac i e bo om sedimen s (pa icles
wi h a diame e 562.4 mm). I is conside ed ha he ans e o adionuclides om
he solid phases o wa e is go e ned by a kine ic ans e coefficien k2, he ans e
om wa e o suspended ma e by k1m and he ans e om wa e o he sedimen
by k1s. The abso p ion o adionuclides depends on he su ace o pa icles pe wa e
olume uni . Thus he exchange su ace and exchange eloci y concep s ha e been
used (Pe i!
aa*
nnez, Ab il, & Ga c!
ııa-Le!
oon, 1996; Pe i!
aa*
nnez & Ma !
ıınez-Agui e, 1997;
Pe i!
aa*
nnez, 1999, 2000). Following hese pape s:
k1m ¼w1
3m
R;ð9Þ
k1s ¼w1
3L
RH ;ð10Þ
whe e w1is he exchange eloci y, mis he suspended ma e concen a ion, and R
a e he densi y and mean adius o suspended ma e pa icles, Lis he a e age
mixing dep h ( he dis ance o which he dissol ed phase pene a es he sedimen ),
is a co ec ion ac o ha akes in o accoun ha no all he mass o he sedimen is
in con ac wi h wa e and Hgi es he hickness o he wa e laye abo e he sea
bo om ha in e ac s wi h he sedimen . In a wo dimensional dep h-a e aged
model, His equal o he wa e dep h. Since pa icle acking is h ee dimensional, H
is le as a ee pa ame e o be calib a ed.
The decay equa ions ha a e equi alen o he diffe en ial equa ions ha desc ibe
ans e s be ween he h ee phases, p esen ed o ins ance in Pe i!
aa*
nnez (2000), a e
qCd
q ¼k1mCdk1sCd;ð11Þ
Table 1
Mean alue and s anda d de ia ion o he diffe ence be ween he pe cen age o adionuclides in phase 1
gi en by fini e diffe ences and he s ochas ic me hod
a
NP hDis
100 0.654 2.861
200 0.637 2.220
500 0.296 1.557
1000 0.335 1.021
10,000 1.56 10
2
0.309
100,000 6.17 10
2
0.114
a
NP is he numbe o pa icles used in he simula ion.
qCs
q ¼k2Cs;ð12Þ
qAs
q ¼k2 As:ð13Þ
whe e Cd,Csand Asa e adionuclide concen a ions in wa e , suspended ma e and
ac i e bo om sedimen s, espec i ely. A label is gi en o each pa icle o classi y in
which phase i is p esen . Depending on he label o he pa icle, he co esponding
decay equa ion is ea ed. I he pa icle is in suspended ma e , he p obabili y ha i
goes o he dissol ed phase, in each ime s ep, is
p¼1ek2D :ð14Þ
Simila ly, i he pa icle is in he bo om sedimen , he p obabili y ha i is
edissol ed is
p¼1ek2 D :ð15Þ
I he pa icle is ini ially dissol ed and i s dis ance o he sea bo om is smalle han
H, i can go o any o he wo solid phases wi h a p obabili y
p¼1ek1mþk1s
ðÞD :ð16Þ
A andom numbe is gene a ed o decide i he pa icle is effec i ely emo ed om
solu ion. I i is, he no malized p obabili y ha he pa icle goes o he sedimen is
calcula ed as
p¼ps
pmþps
;ð17Þ
whe e
pm¼1ek1mD ;ð18Þ
ps¼1ek1sD :ð19Þ
A second andom numbe is hen gene a ed. I RAN5p, hen he pa icle goes o
he sedimen . I RAN >p, hen i goes o he suspended ma e . O cou se, i he
dis ance o he pa icle o he sea bo om is la ge han H, only he decay o
suspended ma e is conside ed since such pa icles canno in e ac wi h he
sedimen .
A nume ical expe imen has been ca ied ou o es he me hod in which a olume
o wa e wi h a gi en suspended ma e concen a ion and ac i e sedimen on he
bo om is conside ed. A dissol ed adioac i e ace is added and he equa ions ha
gi e he ime e olu ion o ac i i ies in he h ee phases a e sol ed using fini e
diffe ences and he s ochas ic me hod. The ollowing ealis ic pa ame e s ha e been
used: w1¼2:1108m/s, k2¼1:2105s
1
,m¼10 ppm, R¼15 mm,
¼2600 kg/m
3
,L¼0:01 m, ¼1, ¼0:1, H¼0:2 m wi h 10,000 pa icles being
used in he pa icle- acking simula ion. The compa ison be ween bo h me hods is
p esen ed in Fig. 2, whe e he ime e olu ion o he ac ion o ace ha is
dissol ed, in suspended ma e and in he sedimen is p esen ed. The simula ion
shows ha he s ochas ic me hod solu ion is in e y good ag eemen wi h he fini e
diffe ence solu ion o he h ee phases. Indeed, solu ions co esponding o bo h
me hods canno be dis inguished in he case o wa e and sedimen .
2.5. Suspended ma e deposi ion and sedimen e osion
The suspended ma e concen a ions o e he model domain can be ob ained by
unning in ad ance a fini e diffe ence suspended ma e model. Resul s a e hen
analyzed in a simila way o cu en s, so ha he suspended ma e concen a ion a
each poin and o any ime can be ob ained as he simple calcula ion o cosine
unc ions.
Suspended ma e alls o he sea bo om wi h a se ling eloci y ws. I a pa icle (in
he pa icle- acking sense, no a suspended ma e pa icle) is fixed o suspended
ma e , i s posi ion abo e he bo om, z, a ime þD is ob ained om
z þD ðÞz ðÞ
D ¼ws:ð20Þ
I z þD ðÞ40, hen he pa icle is conside ed o fix o he sedimen and i s label is
app op ia ely changed.
A s anda d o mula o floccula ion has been used o ep esen he inc ease in he
se ling eloci y as he suspended ma e concen a ion inc eases (Cla ke & Ellio ,
1998; Eisma, 1993; Pe i!
aa*
nnez, 2000)
ws¼a1ma2;ð21Þ
whe e a1and a2a e ob ained om measu emen s o om model calib a ion. The
pa icle- acking model is h ee dimensional. I he fini e diffe ence suspended ma e
model is dep h-a e aged, i s ou pu is he dep h-a e aged suspended ma e
concen a ion. A Rouse p ofile is hen used o esol e he e ical s uc u e o
suspended ma e . This allows he calcula ion o he suspended ma e concen a ion
a heigh zabo e he bo om, mz, om he dep h-a e aged suspended ma e
concen a ion m(Cla ke & Ellio , 1998):
mz¼m1ws=bku*

h
z

ws=bku *;ð22Þ
whe e his wa e dep h, kis he on Ka man cons an (0.4), bis an a bi a y cons an
usually aken as 1 (Eisma, 1993; Cla ke & Ellio , 1998) and u*is he scala ic ion
eloci y
u*¼kq
jj
ln h=z0

1;ð23Þ
whe e z0is he bo om oughness. The co esponding alue o mzis used o calcula e
k1m a he posi ion o each pa icle om Eq. (9).
E osion o he sedimen has been desc ibed in e ms o he e osion cons an
concep (Nicholson & O’Conno , 1986). Thus, he p obabili y ha a pa icle is
emo ed om he sedimen and inco po a ed o he wa e column as suspended
ma e is
p¼1eE jqjMD ;ð24Þ
whe e Eis he e osion cons an and Mis some powe o he wa e eloci y ypically
in he ange 2–5 (P andle, 1997). Thus, he e osion desc ip ion used in p e ious fini e
diffe ence models (Pe i!
aa*
nnez, 1999, 2000) has been con e ed in o a s ochas ic o m. I
is conside ed ha e osion can only ake place i he wa e eloci y is la ge han a
Fig. 2. Time e olu ion o he ac ion o adionuclides in wa e , suspended ma e and bo om sedimen s
gi en by fini e diffe ences and by he pa icle- acking me hod wi h 10,000 pa icles. Solid lines co espond
o he s ochas ic solu ion and dashed lines o he fini e diffe ence solu ion.
3.2. Discussion
Pa icle acking is a powe ul ool ha can be applied in he assessmen o
adioac i e con amina ion ollowing an acciden al elease in aqua ic en i onmen s in
gene al. Also, he me hod can be applied o bo h conse a i e and non-conse a i e
adionuclides, using he same concep ual app oach o he in e ac ions be ween
Fig. 6 (con inued).

Fig. 7.
239,240
Pu ac i i y concen a ions in wa e (mBq/l), suspended ma e (Bq/g) and ac i e sedimen
(Bq/g) gi en by he fini e diffe ence (a) and he pa icle- acking (b) models.
liquid and solid phases as used in fini e diffe ence models. The pa icle- acking
echnique, combined wi h idal analysis and mean flow da abases o he
hyd odynamics, p esen s wo clea ad an ages o e fini e diffe ences: speed o
Fig. 7 (con inued).
compu a ion and he ac ha i does no in oduce nume ical diffusion. The numbe
o pa icles used in simula ions depends on he speed o compu a ion and/o he
accu acy equi ed. Fo example, P oc o e al. (1994b) used 4000 pa icles o
simula e an oil spill du ing eal- ime o ecas ing o an inciden in he A abian Gul .
Howe e , i is possible ha , due o he geochemical beha iou o ce ain
adionuclides, he ac i i y concen a ions in some o he phases may no be
esol ed. This is he case o
137
Cs in suspended ma e . The numbe o pa icles used
in he simula ion should be o he o de o 10
6
o be able o calcula e, wi h
easonable accu acy, ac i i y concen a ions in suspended ma e bu hen he
compu ing ime would be simila o ha o he fini e diffe ence model. Howe e , i
we a e in e es ed in he assessmen o con amina ion ollowing an acciden , i is
p obably enough o calcula e ac i i ies in wa e and bo om sedimen s, which a e he
phases whe e almos all he adioac i i y is p esen . I he mos impo an aspec is
he speed o compu a ion, he numbe o pa icles can be educed. Fo ins ance, i
137
Cs dispe sion is simula ed using 20,000 pa icles, good esul s a e s ill ob ained in
wa e and bo om sedimen s and he compu a ion ime is educed o 2.6% o ha
equi ed by he fini e diffe ence model. In he case o
239,240
Pu, he numbe o
pa icles could be educed mo e and ac i i ies in he h ee phases may s ill be
calcula ed since Pu is mo e widely dis ibu ed be ween he h ee phases han Cs.
The pa icle- acking me hod is ully 3-D and akes accoun o ho izon al
ad ec ion plus u bulen diffusion in he x–y–zdi ec ions. In he p esen applica ion,
he idal and esidual flows ha e been compu ed by a dep h-a e aged 2-D
hyd odynamic model. A 2-D hyd odynamical model is adequa e due o he e y
s ong idal cu en s and he e ically well-mixed cha ac e o he coas al wa e s.
Howe e , he pa icle- acking me hod is sui able o use wi h a 3-D hyd odynamic
model (e.g. Ha ms, Ka che , & De hleff, 2000) and he manne in which e ical
s abili y can inhibi e ical mixing can be pa ame e ised in such applica ions by
making he e ical s ep-size a unc ion o he s abili y o he wa e column.
4. Conclusions
The pu pose o his pape is o demons a e he iabili y o new pa icle- acking
algo i hms o he simula ion o p ocesses ha in ol e chemical specia ion. The
pa icle- acking me hod is commonly used o oil and chemical spill applica ions
whe e he e is a need o apid esponse simula ions. In such applica ions, a
significan inc ease in compu a ional speed is achie ed by compu ing he
hyd odynamics ‘off-line’ and hen using idal p edic ion and mean flow da abases
o econs uc he wa e mo emen . This a oids he CFL c i e ion du ing he apid
esponse applica ion, al hough he pa icle- acking ime-s ep mus be kep
easonably sho in o de o main ain he accu acy o he fi s o de ad ec ion
scheme ha is used o compu e he pa icle mo emen . While i would also be
easible o sol e fini e diffe ence specia ion equa ions using a p e-compu ed flow
field, he p oposed echnique has he ad an age ha he compu a ional ime will be
e y sho i a ela i ely small numbe o pa icles is used in he simula ion.
Mo eo e , he me hod will be mo e efficien i he con aminan pa ch co e s only a
small ac ion o he g id domain (Hun e , 1987) and he accu acy o he esul s can
be imp o ed by eleasing la ge numbe s o pa icles}al hough a he expense o an
inc eased simula ion ime. One o he main ad an ages o he pa icle- acking
me hod is he ad-hoc manne in which pa icle cha ac e is ics can be defined. Fo
example, non-Fickian dispe sion can be eadily simula ed by a aching an age o
each pa icle (equal o he ime since he elease o he pa icle in o he compu a ion)
and hen making he diffusi e s ep o he pa icle a unc ion o i s diffusion ime.
In he new model, he in e ac ions be ween he liquid and solid (suspended ma e
and bo om sedimen ) phases ha e been desc ibed in e ms o kine ic ans e
coefficien s, as in ac ual fini e diffe ence models. A s ochas ic me hod has been
de eloped o simula e such in e ac ions. Deposi ion o pa icles and e osion o he
sedimen a e also included in he model. A Rouse p ofile is also used o es ima e he
e ical s uc u e o suspended ma e concen a ions. As a demons a ion o he
me hod, he pa icle- acking model has been used o simula e he dispe sion o
adionuclides in he English Channel. A fini e diffe ence model was p e iously
de eloped and alida ed o he Channel. Thus he ou pu om bo h models has
been compa ed. Two adionuclides wi h diffe en geochemical beha iou s,
137
Cs and
239,240
Pu, ha e been used o he compa isons. The ag eemen be ween bo h models
is, in gene al, a he good. Howe e , i is possible ha , due o he pa icula
beha iou s o ce ain adionuclides, ac i i y concen a ions in one o he phases
canno be calcula ed.
Acknowledgemen s
R Pe i!
aa*
nnez was suppo ed by ENRESA and he EU unde con ac FIGE-
CT2000-00085. A J Ellio was unded by he Chie Scien is ’s G oup o MAFF
unde con ac AE1021 and by he P ocu emen Execu i e o he MoD unde
con ac NPC1B/2012.
Re e ences
Ald idge, J. N. (1998). CSERAM: A model o p edic ion o ma ine adionuclide anspo in bo h
pa icula e and dissol ed phases. Radia ion P o ec ion Dosime y,75, 99–103.
Cla ke, S., & Ellio , A. J. (1998). Modelling suspended sedimen concen a ions in he Fi h o Fo h.
Es ua ine, Coas al and Shel Science,47, 235–250.
Eisma, D. (1993). Suspended ma e in he aqua ic en i onmen . Be lin: Sp inge .
Ellio , A. J., & Cla ke, S. (1998). Shallow wa e ides in he Fi h o Fo h. Hyd og aphic Jou nal,87, 19–
24.
Ellio , A. J., Dale, A. C., & P oc o , R. (1992). Modelling he mo emen o pollu an s in he UK shel
seas. Ma ine Pollu ion Bulle in,24, 614–619.
Gueguenia , P., He mann, J., Ke shaw, P. J., Bailly du Bois, P., & Ba on, Y. (1996). A ificial
adioac i i y in he English Channel and he No h Sea. In P. Gueguenia , P. Ge main, & H. Me i ie
(Eds.), Radionuclides in he oceans (pp. 121–154). Les Ulis: Les Edi ions de Physique.
Ha ms, I. H. (1997). Modelling he dispe sion o
137
Cs and
239
Pu eleased om dumped was e in he Ka a
Sea. Jou nal o Ma ine Sys ems,13, 1–19.
Ha ms, I. H., Ka che , M. J., & De hleff, M. (2000). Modelling Sibe ian i e unoff}implica ions o
con aminan anspo in he A c ic Ocean. Jou nal o Ma ine Sys ems,27, 95–115.
Hun e , J. R. (1987). The applica ion o Lag angian pa icle acking echniques o modelling o
dispe sion in he sea. In J. Noye (Ed.), Nume ical modelling. Applica ions o ma ine sys ems (pp. 257–
269). No h-Holland: Else ie .
Kowalick, Z., & Mu y, T. S. (1993). Nume ical modellingo ocean dynamics. Singapo e: Wo ld Scien ific.
Ma g elash ily, N., Made ich, V., & Zheleznyak, M. (1997). Th ee ox: A compu e code o simula e h ee
dimensional dispe sion o adionuclides in s a ified wa e bodies. Radia ion P o ec ion Dosime y,73,
177–180.
Nicholson, J., & O’Conno , B. A. (1986). Cohesi e sedimen anspo model. Jou nal o Hyd aulic
Enginee ing,112, 621–640.
Pe i!
aa*
nnez, R. (1999). Th ee dimensional modelling o he idal dispe sion o non conse a i e adionuclides
in he ma ine en i onmen . Applica ion o
239,240
Pu dispe sion in he eas e n I ish Sea. Jou nal o
Ma ine Sys ems,22, 37–51.
Pe i!
aa*
nnez, R. (2000). Modelling he idal dispe sion o
137
Cs and
239,240
Pu in he English Channel. Jou nal o
En i onmen al Radioac i i y,49, 259–277.
Pe i!
aa*
nnez, R., Ab il, J. M., & Ga c!
ııa-Le!
oon, M. (1996). Modelling he dispe sion o non conse a i e
adionuclides in idal wa e s. Pa 1: Concep ual and ma hema ical model. Jou nal o En i onmen al
Radioac i i y,31, 127–141.
Pe i!
aa*
nnez, R., & Ma !
ıınez-Agui e, A. (1997). U and Th concen a ions in an es ua y affec ed by phospha e
e ilize p ocessing: Expe imen al esul s and a modelling s udy. Jou nal o En i onmen al
Radioac i i y,35, 281–304.
Pe i!
aa*
nnez, R., & Regue a, J. (1999). A nume ical model o simula e he idal dispe sion o adionuclides in
he English Channel. Jou nal o En i onmen al Radioac i i y,43, 51–64.
Piasecki, M. (1998). T anspo o adionuclides inco po a ing cohesi e/non cohesi e sedimen s. Jou nal o
Ma ine En i onmen al Enginee ing,4, 331–365.
P andle, D. (1997). Tidal cha ac e is ics o suspended sedimen concen a ions. Jou nal o Hyd aulic
Enginee ing,123, 341–350.
P oc o , R., Fla he , R. A., & Ellio , A. J. (1994a). Fo ecas and hindcas simula ions o he B ae oil
spill. Ma ine Pollu ion Bulle in,28, 219–229.
P oc o , R., Fla he , R. A., & Ellio , A. J. (1994b). Modelling ides and su ace d i in he A abian Gul :
applica ion o he Gul oil spill. Con inen al Shel Resea ch,14, 531–545.
Pugh, D. T. (1987). Tides, su ges and mean sea le el. Wiley: Chiches e (p. 15).