A three dimensional σ-coordinate model to simulate the dispersion of radionuclides in the marine environment: application to the Irish Sea

A h ee dimensional s-coo dina e model o simula e he dispe sion o adionuclides in he ma ine en i onmen : applica ion o he I ish Sea R. Pe ia´n˜ez * Depa amen o Fı´sica Aplicada, E.U. Ingenie ı´a Tecnica Ag ı´cola, Uni6e sidad de Se6illa, C a. u e a Km 1 , 41013 Se6illa, Spain Abs ac A h ee dimensional model o simula e he ide induced dispe sion o adionuclides in he sea has been de eloped. The model uses no malized s-coo dina es in he e ical so ha esolu ion is no educed in he shallowe egions. The hyd odynamic equa ions a e sol ed and, simul aneously, he h ee dimensional ad ec ion di usion dispe sion equa ion (also w i en in no malized coo dina es) is sol ed oo. An ins an aneous low and dep h dependen eddy iscosi y has been used. The model has been applied o s udy he dispe sion o l37 Cs in he eas e n I ish Sea, whe e a nuclea uel ep ocessing plan eleases adionuclides. The hyd odynamic pa o he model has been es ed by compa ing obse ed and compu ed alues o idal ele a ions, phases and cu en s. The model gi es, in gene al, a good ep esen a ion o he wa e ci cula ion in he sea. Also, i gi es esul s in ag eemen wi h obse a ions when measu ed and compu ed le els o l37 Cs a e compa ed. In o de o show how he model can be used o ob ain wa e quali y pa ame e s o in e es , i has been applied o ob ain he u n o e ime o a egion o he sea. I has also been applied o simula e he dispe sion o a pollu an a e an hypo he ical acciden al discha ge. Keywo ds : Tides; Cu en s; Ad ec ion; Di usion; 137 Cs 1. In oduc ion Radionuclides a e being eleased o he sea om nuclea uel ep ocessing plan s (McKay and Bax e , 1985; Gueguenia e al., 1994; He mann e al., 1995; Cook e al., 1997). The e has been an inc easing in e es in de eloping and imp o ing models o simula e he dispe sion o hese a- dionuclides in he ma ine en i onmen (P andle, 1984; B e on and Salomon, 1995) due o he ac ha use ul oceanog aphic in o ma ion (like lush- ing and ansi imes) can be ob ained by means * Tel.: +34 95 4233669; ax: +34 95 4232644. o he applica ion o he models (P andle, 1984; Salomon e al., 1995). On he o he hand, hese models can be used as p edic i e ins umen s which can be applied in he assessmen o a- dioac i e con amina ion ollowing an acciden al elease o adionuclides. Al hough hese models a e wo dimensional, 2D (in eg a ed in he e ical di ec ion), he e has been an inc easing use o h ee dimensional models (3D) o sol e oceanog aphic p oblems (Da ies and Law ence, 1994; P oc o and James, 1996; Da ies e al., 1997) due o he limi a ions o 2D models, which o ins ance, can no gi e in o ma ion abou he e ical p o ile o cu - en s. A 3D model can also be applied o simu- la e he dispe sion o pollu an s in he sea. Indeed, P andle e al. (1993) ha e shown ha e ical s uc u e can be o p ima y conce n when ypical simula ed imes a e in he o de o one mon h o less. Mo eo e , he e ical a i- abili y is impo an o such ypical imes, e en in shallow wa e s (dep h :50 m), i idal mix- ing keeps he e ical di usion coe icien smalle han 10 −3 m 2 /s (P andle e al., 1993) (a alue in he o de o 10 −3 m 2 /s can be consid- e ed ep esen a i e o a s ong idal ac ion). Thus, a 3D app oach should be used o build models ha can be used as p edic i e ools in he assessmen o he nea ield adioac i e con- amina ion o idal wa e s, since he ime scale in ol ed is o he o de o weeks and e ical s uc u e can be impo an , as said abo e. Only in he case o a e y s ong idal ac ion a 2D app oach can gi e enough accu a e esul s. Howe e , ew 3D dispe sion models ha e been de eloped: he model o Nies e al. (1997), o ins ance, s udies he 3D anspo o adionu- clides in he A c ic Ocean, bu he au ho s could no compa e he model p edic ions wi h obse a ions. In Pe ia´n˜ez (1998) some p elimina y esul s on a 3D dispe sion model o adionuclides we e p esen ed. Howe e , his model wo ked wi h laye s (in he e ical) o he same hickness, which implies a loss o esolu ion in he e ical di ec ion in he shallowe egions. On he o he hand, he model used a e y simple pa ame iza- ion o he eddy iscosi y coe icien (i was con- side ed cons an in ime and wi h he same alue o e he whole compu a ional domain). This wo k ocused on he s udy o some quali- a i e aspec s o h ee dimensional idal dispe - sion. Now he model has been imp o ed o o e come he abo e men ioned p oblems. Thus, no malized scoo dina es a e used in he e ical di ec ion, in such a way ha esolu ion is no los in he shallowe egions. On he o he hand, eddy iscosi y has been o mula ed in such a way ha i depends on he ins an aneous cu - en and dep h. Thus, a mo e ealis ic model has been de eloped. The model sol es he 3D hyd odynamic equa ions and, simul aneously, he ad ec ion di usion dispe sion equa ion, which has also been w i en in scoo dina es. This way, he ide induced dispe sion o a- dionuclides is ob ained. An implici nume ical scheme has been adop ed o sol e he hyd ody- namic and dispe sion equa ions in he e ical di ec ion in o de o e ain s abili y. This is he i s ime, o he au ho ’s knowledge, ha sco- o dina es a e used o sol e he dispe sion equa- ion. The model has been applied o he I ish Sea since he e is enough oceanog aphic in o - ma ion o es he model esul s ( ide ampli- udes, cu en s and cu en p o iles), he e is a well known sou ce o adioac i i y (Sella ield nuclea uel ep ocessing plan ) and he e a e measu emen s o adionuclide concen a ions o e he sea. The hyd odynamic pa o he model has been es ed by compa ing obse ed and compu ed idal ampli udes, idal phases and magni udes and di ec ions o idal cu en s. Ob- se ed and compu ed 137 Cs concen a ions ha e also been compa ed. The u n o e ime o an es ua y o he I ish Sea has been calcula ed o show how he model can be applied o ob ain wa e quali y pa ame e s o in e es . Finally, a nume ical expe imen has been ca ied ou o show he po en ial p edic i e powe o he model in he assessmen o con amina ion ol- lowing a hypo he ical acciden . The model equa ions a e p esen ed in he nex sec ion. Nex he nume ical me hods used o sol e hem a e desc ibed b ie ly and inally model esul s a e p esen ed and discussed. 2. Model equa ions I a ixed ini e di e ence g id in he e ical is used, he numbe o e ical g id boxes dec eases in he shallow egions. This has he e ec o educing he e ical esolu ion in he shallow wa e a eas. This p oblem can be sol ed ans- o ming he 3D hyd odynamic equa ions in o dep h ollowing scoo dina es. This way a con- s an numbe o g id boxes is used in he e ical a each ho izon al g id poin . The ans o ma ion o scoo dina es is (see Da ies, 1985a, o in- s ance): s=z+z h+z(1) whe e his he undis u bed (mean) dep h o wa e , zis sea su ace displacemen om he mean le el due o idal oscilla ions and zcoo dina e is mea- su ed om he mean sea le el o he sea bo om. Thus, he hyd odynamic equa ions a e ans- o med om he in e al −z5z5hin o he cons an in e al 05s51. The ans o med equa ions o an incomp essible low and o a homogeneous sea a e (Da ies, 1985a): (z ( +( (x(h+z)& 0 1 udsn +( (y(h+z)& 0 1 ndsn=0 (2) (u ( +g(z (x−V6=1 (h+z) 2 ( (sN(u (s(3) (6 ( +g(z (y+Vu=1 (h+z) 2 ( (sN(6 (s(4) w*=1 h+z(z ( (1−s)n +1 h+z ( (x(h+z)& s 1 udsn +1 h+z ( (y(h+z)& s 1 6dsn(5) whe e u,6and w* a e he componen s o he wa e eloci y along he di ec ions o x,yand s axis, espec i ely. gis g a i y, Vis he Co iolis pa ame e , V=2 sin ;( being he ea h o a- ional angula eloci y and he la i ude), and N is he coe icien o eddy iscosi y. The non linea ad ec i e e ms ha e been emo ed om hese equa ions since dimensional analysis has shown (Cha nock and C ease, 1957) ha hey a e impo - an only when zis compa able wi h he mean dep h, and his is no he case. These non linea e ms gene a e a esidual cu en ha may a ec he adionuclide anspo when s udying long- e m dispe sion ( ime scale o he o de o yea s). Howe e , his weak cu en can be neglec ed i ime scale o in e es is in he o de o weeks, as is he case. Mo eo e , P andle (1984) excludes he ad ec i e e ms in his long- e m dispe sion model o he Eu opean shel seas since hei e ec is o add addi ional s uc u e o esidual dis ibu ions, bu his s uc u e is o en exagge a ed due o poo opog aphic esolu ion (P andle, 1984). To sol e he equa ions, su ace and sea bed bounda y condi ions mus be speci ied. The su - ace bounda y condi ions a e: N(u (s s=0 =−(h+z)F s (6) N(6 (s s=0 =−(h+z)G s (7) whe e F s and G s deno e he componen s o wind s ess ac ing on he wa e su ace along he xand ydi ec ions, which can be w i en as in Pugh (1987) and Pe ia´n˜ez e al. (1994), and is he wa e densi y. Simila ly, a he sea bed: N(u (s s=1 =−(h+z)F b (8) N(6 (s s=1 =−(h+z)G b (9) whe e F b and G b a e he componen s o bo om s ess. Assuming a linea law o bo om ic ion: F b = ku b (10) G b = k6 b (11) whe e kis a ic ion coe icien and u b and 6 b a e he componen s o wa e eloci y a a gi en heigh abo e he bo om, which usually is 1 m (Da ies and S ephens, 1983). Al hough a quad a ic law o bo om ic ion is now mo e ex ended han a linea o mula ion, he linea law has been adop ed since a as e con e gence o he equa- ions is ob ained. Indeed, a linea law is mo e app op ia e in linea models (Da ies, 1985b). The 3D ad ec ion-di usion dispe sion equa ion o dissol ed adionuclides, which has been w i - en in scoo dina es, is: (C ( +u(C (x+6(C (y+w* (C (s =( (xK x (C (x+( (yK y (C (y +1 (h+z) 2 ( (sK 6 (C (s−lC(12) whe e Cis he adionuclide concen a ion, l he adioac i e decay cons an and K x ,K y and K 6 a e he di usion coe icien s along he x,yand e i- cal di ec ions, espec i ely. The ex e nal sou ce o adionuclides, whe e i exis s, should be included in his equa ion. A low dependen eddy iscosi y has been used in he model. This o mula ion has been used p e iously and has gi en good esul s o idal low s udies (Da ies and Law ence, 1994; Jones and Da ies, 1996; Da ies e al., 1997) N=C N u¯ 2 +6¯ 2 h(13) whe e C N =0.0025 is a dimensionless expe imen- ally measu ed coe icien and u¯ and 6¯a e dep h mean cu en s: u¯ =& 0 1 uds(14) 6¯=& 0 1 6ds(15) Eddy iscosi y dec eases in he egion close o he sea bed (Da ies e al., 1997). Howe e , i has been aken cons an in he e ical since no a - emp has been made o sol e he high shea egion close o he sea bed. This app oxima ion has also been used by Da ies and Law ence (1994). The e ical di usion coe icien can be w i en as a unc ion o he eddy iscosi y (Kowalick and Mu y, 1993): K 6 =oN(16) whe e he non-dimensional numbe o anges om 0.1 o 0.5. 3. Nume ical solu ion All he equa ions a e sol ed using ini e di e - ences. A s agge ed g id is used in he ho izon al, wi h uni o m spacing Dxand Dy, and a g id in s coo dina es, wi h spacing Ds, is employed in he e ical di ec ion. Time s ep is ixed as D . The model is s a ed om es . I he hyd odynamic equa ions a e in eg a ed using an explici me hod besides he CFL c i e- ion, he e is ano he s abili y condi ion imposed by he e ical di usion e m, which is ela ed o he magni ude o eddy iscosi y and he wa e dep h: D B(hDs) 2 2N(17) Thus, in shallow wa e , his condi ion can lead o a ime s ep which is smalle han ha equi ed by he CFL c i e ion. This p oblem can be a oided using an implici me hod o ea he eddy iscosi y e m. In his wo k, he me hod de eloped by Saul’e (1957) has been used. In his me hod an al e na ing di ec ion sweep is em- ployed a al e na e ime s eps. De ails can be seen, o ins ance, in Da ies (1985b). Some bounda y condi ions a e also equi ed. Fo closed bo de s, a no lux condi ion is im- posed: q=0 (18) whe e qis he cu en componen which is no mal o he bounda y. Along open bounda ies, wa e ele a ions a e speci ied om obse a ions and he no mal componen o he su ace wa e cu en , q, is ob ained om a adia ion condi ion (Kowalick and Mu y, 1993; Glo ioso and Da ies, 1995): q=c hz(19) whe e c=gh. Some bounda y condi ions a e also equi ed by he dispe sion equa ion. The e is no lux o a- dionuclides h ough a closed bounda y, hus: (C (x i =0 (20) whe e x i is he no mal di ec ion o he bounda y. Along open bounda ies, he condi ion desc ibed in Pe ia´n˜ez e al. (1994) was applied: C i =aC i−1 (21) whe e C i is he concen a ion in he open bounda y and C i−1 ep esen s he concen a ion jus inside he compu a ional domain. The non dimensional numbe ais ob ained om a calib a ion exe cise. To sol e he dispe sion equa ion, a cen ed scheme is used o he ho izon al di usion e ms, upwind di e ences a e used o he ad ec i e e ms and he Saul’e me hod (Saul’e , 1957) is again employed o sol e he e ical di usion e m. I is well known ha upwind di e ences in oduce nu- me ical di usion. I has been shown (P andle, 1984) ha he magni ude o nume ical di usion is equi alen o inc easing he di usion coe icien K i by K% i , whe e: K% i =1 2(u i Dx i −u i 2 D ) (22) whe e he subindex i ep esen s he h ee di ec ions in space. Nume ical di usion in he e ical di ec- ion can be neglec ed due o small alues o he e ical eloci y (maximum alue is o he o de o 10 −4 m/s). In he ho izon al di ec ions, nume ical di usion has been educed by sub ac ing he ins an aneous alue o K% h o K h , whe e he index h ep esen s xo y(F ench, 1988). 4. Applica ion o he model As said abo e, he model has been applied o s udy he dispe sion o adionuclides in he I ish Sea. These adionuclides a e discha ged om a nuclea uel ep ocessing plan a Sella ield. The model has a ho izon al esolu ion Dx=Dy=5000 m. Ten laye s a e used in he e ical, hus, Ds= 0.1. Time s ep is ixed as D =60 s. S abili y condi ions a e sa is ied wi h his selec ion. The compu a ional domain is p esen ed in Fig. 1, whe e he loca ion o he nuclea uel plan is also shown. Wa e dep hs ha e been in oduced om ba hy- me ic maps and ange om 55 m in he wes o he compu a ional domain o a shallowe a ea a ound he B i ish coas . Wa e ele a ions a e speci ied along he open bounda y om obse a ions (Howa h, 1990). Only he main idal componen , M 2 , has been conside ed. The ic ion coe icien has been aken as k=0.0112, a simila alue o ha used by Jones and Da ies (1996). Wind e ec s ha e no been conside ed: F s =G s =0. The mean alue 0.3 has been aken o ein Eq. (16) and good esul s a e ob ained selec ing a=0.9 in Eq. (21). The ho izon al di usion coe icien s ha e been ixed as K x =K y =500 m 2 /s since esul s in ag eemen wi h obse a ions a e ob ained wi h hese alues. Indeed, Bowden (1950) sugges ed ha o he I ish Sea 500BK x ,K y B900 m 2 /s. Fig. 1. Map o he compu a ional domain. Le e s indica e he poin s whe e idal ampli udes and phases ha e been measu ed, numbe s indica e he poin s whe e idal cu en s ha e been measu ed and ci cles deno e he poin s whe e 137 Cs concen a- ions ha e been ob ained. The s a is Sella ield nuclea uel ep ocessing plan . Each uni in he xand yaxis is 5000 m (g id elemen numbe ). Table 1 Obse ed and compu ed ide ampli udes and phases o he poin s shown in Fig. 1 Poin Ampli ude (m) Phase (deg ees) Compu ed Obse edCompu ed Obse ed Di e ence Di e ence 285 300 −5.0a −3.91.98 2.06 311 317b 2.95 2.92 1.0 −1.9 312 323c 2.99 3.11 −3.8 −3.4 326328 0.6d 5.93.23 3.05 6.3 329 325 1.2e 3.35 3.15 331 331 3.10 3.08 0.6 0.0 332357 7.5g −0.042.75 2.73 339 8.0h 2.76 2.75 0.04 366 303 326I 1.90 2.30 −17.4 −7.0 324336 3.7j −4.62.51 2.63 −5.3 319 318k 0.32.48 2.62 7.8332m 2.50 2.55 −2.0 358 Di e ences be ween compu ed and obse ed alues a e gi en in % ela i e o he obse ed alue. 5. Resul s and discussion 5.1. Wa e ci cula ion Tidal ampli udes and phases, and magni ude and di ec ion o idal cu en s calcula ed wi h he model ha e been compa ed wi h he measu ed alues o a numbe o poin s inside he compu a- ional domain. These poin s a e shown in Fig. 1. Compu ed and obse ed idal ampli udes and phases a e p esen ed in Table 1. The di e ence be ween compu ed and obse ed alues in % ela- i e o he obse ed alue is also gi en. I can be seen ha he di e ence in ampli udes is B7% o all he poin s and only in he case o poin ‘i’a di e ence 10% is ob ained. In he case o idal phases, e o s a e B8% o all poin s. Obse ed and compu ed semi-majo axis mag- ni ude and o ien a ion o he M 2 idal cu en ellipse ha e also been compa ed o a numbe o loca ions and dep hs (Table 2). I can be seen ha he model gi es, in gene al, a good ep esen a ion o cu en magni ude and di ec ion in he sea. Compu ed and obse ed cu en p o iles a poin s 8 and 9 (see Fig. 1) a e p esen ed in Fig. 2. The shape o he p o ile is ep oduced by he model a bo h poin s and o bo h he uand 6 componen s o he wa e eloci y. Thus, i seems ha , in gene al, he model gi es a good ep esen- a ion o he wa e ci cula ion in he s udied a ea since good ag eemen be ween obse ed and com- pu ed idal ampli udes, phases and cu en s has been ob ained. Mo eo e , ou main objec i e is o s udy he dispe sion o adionuclides and he e- sul s o he hyd odynamic pa o he model seem good enough o allow an adequa e desc ip ion o he dispe sion p ocesses. 5.2. Radionuclide dispe sion The dispe sion o 137 Cs eleased om he nu- clea uel ep ocessing plan a Sella ield has been simula ed. As a i s app oach, l37 Cs was consid- e ed o be pe ec ly conse a i e, i.e. no ac ion is emo ed om he wa e column due o biological o geochemical p ocesses. This is usual in some models (P andle, 1984; Ab il and Ga cı´a-Leo´n, 1992). Indeed, he mean alue o he Cs dis ibu- ion coe icien , k d , in coas al wa e s is 3×10 3 l/kg (IAEA, 1985). Since ypical suspended ma e concen a ions in he eas e n I ish Sea a e o he o de o 1 ppm (Ke shaw and Young, 1988), i can be calcula ed ha only 0.3% o he o al Cs con en in a gi en wa e olume is ixed o solid pa icles. The majo sou ce o 137 Cs o he I ish Sea has been he discha ges om Sella ield. O he sou ces, such as nuclea weapon es allou con ibu ed Table 2 Obse ed and compu ed semi-majo axis and o ien a ion o he M 2 idal cu en ellipse a se e al dep hs and loca ions shown in Fig. 1 Compu ed aluesObse ed aluessh (m)Poin Di ec ion (deg ees) Axis (m/s)Axis (m/s) Di ec ion (deg ees) 9 1.05 0.11 1.1050 0.30 0.91 10 0.932 1.550 0.44 14 0.412 52 0.92 0.62 6.6 0.52−32 −1.53 0.4920 0.60 0.40 6 0.58 8.74 45 0.98 −5 0.995 45 0.50 0.79 −7.1 0.67−11 −12.36 0.5830 0.50 −14 0.587 20 0.76 −17.90.48 −9 0.898 55 0.42 0.86 2.5 0.52−6 6.78 0.6655 0.88 0.72 3 0.919 3.145 0.62 4.60.679 45 0.80 0.66 6 O ien a ion is gi en in deg ees measu ed an iclockwise om eas ( hus ange om −90 o 90°). B1% o he o al inpu (Je e ies and S eele, 1989). Obse ed and compu ed 137 Cs dis ibu ions ha e been compa ed o a numbe o yea s. The eal inpu om Sella ield (Je e ies and S eele, 1989) was in oduced in he model o each yea . This inpu was 2970 TBq/yea o 1980, which is equi alen o 5.6×10 9 Bq pe ime s ep. Howe e , he inpu has been aking place since he 1960s. Thus, ins ead o s a ing he model om ze o concen a ions, we ha e as- sumed an uni o m backg ound o 1900 Bq/m 3 . This backg ound ep esen s he e ec o p e i- ous discha ges. In Pe ia´n˜ez e al. (1994) i was shown ha model esul s do no depend upon he way he backg ound is c ea ed. Thus, he same esul s would be ob ained i a la ge dis- cha ge is pe o med and some ime is allowed o elapse so ha he discha ge is dis ibu ed o e he sea. To sa e CPU ime, he uni o m back- g ound op ion was chosen. Thus, discha ges om Sella ield a e ca ied ou o e his uni o m backg ound and esul s a e ob ained a e a sim- ula ion pe iod o 25 days. These esul s a e compa ed wi h obse a ions. Obse ed and com- pu ed 137 Cs concen a ions in su ace wa e s along he B i ish coas line (Fig. 1), no h and sou h om Sella ield, can be seen in Fig. 3. I can be seen ha he model gi es he gene al dis ibu ion pa e n o 137 Cs in he sea. An in- ense peak is ob ained a Sella ield (poin 0 in he xaxis) and concen a ions dec ease as we mo e sou h o no h om Sella ield. The ac i - i y le els measu ed in he sea ha e been ep o- duced by he model. A dis ibu ion map o e he sea is p esen ed as an example, in Fig. 4. This map is no signi ica i ely di e en om ha ob ained om obse a ions (Je e ies and S eele, 1989). The shape o he 1500 Bq/m 3 iso- line sugges s ha he e is an inpu o non-con- amina ed wa e be ween Anglesey and he Isle o Man. This is consis en wi h he ac ha s ong cu en s (o he o de o 1.5 m/s) a e obse ed in his a ea (Howa h, 1990) and hese cu en s p oduce a esidual low which en e s he eas e n I ish Sea be ween Anglesey and he Isle o Man. The inpu om Sella ield o 1982 was 2000 TBq/yea , which is equi alen o 3.8×10 9 Bq pe ime s ep. The uni o m backg ound was now selec ed as 1300 Bq/m 3 . As can be seen in Fig. 5, he gene al dis ibu ion o 137 Cs along he coas is again ep oduced by he model. Fig. 2. Cu en p o iles a poin s 8 (a) and 9 (b) o Fig. 1. The inpu o 1984 was 434 TBq/yea (8.2×10 8 Bq pe ime s ep) and he backg ound was aken as 1200 Bq/m 3 . In he case o yea 1985, he inpu was 325 TBq/yea (6.2×10 8 Bq pe ime s ep) and he backg ound was selec ed as 900 Bq/m 3 . Ob- se ed and compu ed dis ibu ions o 137 Cs along he coas o yea s 1984 and 1985 can be seen, espec i ely, in Figs. 6 and 7. The gene al be- ha iou o 137 Cs is again ep oduced by he model. Thus, i seems ha he model gi es a ealis ic ep esen a ion o he dispe sion p ocesses o a- dionuclides in he I ish Sea. 5.3. P edic i6e s udies Once he model has been es ed, i can be used as a p edic i e ool ha can be applied, o in- s ance, in he assessmen o con amina ion ollow- Fig. 4. Dis ibu ion map o 137 Cs concen a ions (Bq/m 3 )in su ace wa e o yea 1980. Each uni in he xand yaxis is 5000 m (g id elemen numbe ). Fig. 3. Obse ed and compu ed 137 Cs concen a ions (Bq/1) a se e al loca ions along he B i ish coas (see Fig. 1) o yea 1980 no h (posi i e dis ances) and sou h (nega i e dis ances) om Sella ield. Fig. 5. Same as Fig. 3 bu o yea 1982. Fig. 7. Same as Fig. 3 bu o yea 1985. ing an acciden al elease o adionuclides in any si e o he sea. I mus be no ed ha he model has been de eloped o adionuclides, bu can be used o o he dissol ed conse a i e pollu an s by se ing l=0. Also, he model can be applied o ob ain some wa e quali y pa ame e s. As an applica ion example, he u n-o e - ime o he Solway es ua y (Fig. 1) has been calcula ed. This pa ame e is de ined as he ime in which concen a ion inside a bounded egion dec eases by a ac o e −1 (P andle, 1994). I is compu ed assuming an a bi a y concen a ion in he egion o in e es and ob aining he ime e olu ion o his concen a ion. In ou calcula ion, an ini ial con- cen a ion o 10 uni s/m 3 was conside ed in he Solway. The ime e olu ion o his concen a ion is shown in Fig. 8. F om his ime e olu ion, he u n-o e - ime can be es ima ed as 2.6 days. I is in e es ing o no e ha some oscilla ions appea in concen a ions. They a e due o idal oscilla- ions and we e al eady obse ed in a simila ex- pe imen ca ied ou wi h a wo dimensional model (Pe ia´n˜ez e al., 1996a). I an acciden al discha ge o a dissol ed conse - a i e pollu an occu s in he Solway, he model has es ima ed ha concen a ion should dec ease by a ac o 0.37 in a ime o :2.6 days. Two Fig. 8. Time e olu ion o pollu an concen a ion (a bi a y uni s) in he Solway es ua y.Fig. 6. Same as Fig. 3 bu o yea 1984.