A Comparative Study of CART and PTM for Modelling Water Age

J. Ocean Univ. China (Oceanic and Coastal Sea Research)DOI 10.1007/s1 1802-015-2393-7ISSN 1672-5182, 2015 14(1): 47-58http://www.ouc. edu.cn/xbywb/E-mail:xbywb@ouc.edu.cnA Comparative Study of CART and PTM forModelling W ater AgeWANG Haiyan', GUO Xinyu2), LIU Zhe'), *, and GAO Huiwang')1) Key Laboratory of Marine Environment and Ecology, Ministry of Education, Ocean University of China,Qingdao 266100, P. R. China2) Center for Marine Environmental Studies, Ehime University, Matsuyama 790-8577, Japan(Received May 9, 2013; revised May 30, 2013; accepted November 6, 2014)@ Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2015Abstract CART (Constituent-oriented age and residence time theory) and PTM (Particle-tracking method) are two widely usednumerical methods to calculate water age. These two methods are essentially equivalent in theory but their results may be different inpractice. The difference of the two methods was evaluated by applying them to calculate water age in an idealized one-dimensionaldomain. The model results by the two methods are consistent with each other in the case with either spatially uniform flow field orspatially uniform diffusion coefficient. If we allow the spatial variation in horizontal diffusion, a term called pseudo displacementarising from the spatial variation of diffusion coefficient likely plays an important role for the PTM to obtain accurate water age. Inparticular, if the water particle is released at a place where the diffusion is not the weakest, the water age calculated by the PTMwithout pseudo displacement is much larger than that by the CART. This suggests that the pseudo displacement cannot be neglectedin the PTM to calculate water age in a realistic ocean. As an example, we present its potential importance in the Bohai Sea where thediffusion coefficient varies spatially and greatly.Key words CART; PTM; pseudo displacement; water ageLiu et al, 2012). The mean water age is defined as themass-weighted arithmetic average of the ages of all of the1 Introductionwater particles within a target domain.Advection and diffusion are two important processes inAmong the aforementioned studies on mean water age,coastal material transport. The material transport time-constituent-oriented age and residence time theory (CART,scales play an important role in the algal bloom (Hilton www.climate.be/cart) (Deleersnijder et al, 2001) and par-et al, 1998). Because of the complex spatiotemporal ticle-tracking method (PTM) (Zhang, 1995) are two widelystructure of coastal currents, it is helpful to define auxil- used methods. For instance, Wang et al. (2010) studiediary variables, such as water age, to understand the mate- the mean age of Changjiang River water and de Brye et al.rial transport processes in coastal zone of oceans (Zim- (2013) studied the mean age of canal and dock water bymerman, 1976; Takeoka, 1984; Delersnijder et al, 2001; the CART; Chen (2007) studied the mean age of AlafiaMonsen et al, 2002; Delhez et al, 2004). Water age is River water and Liu et al. (2011) studied the mean age ofdefined as the time elapsed since the departure of a water the Tahan Stream, Hsintien Stream, and Keelung Riverparticle from an area, where its age is prescribed to be water by the PTM; Liu et al. (2012) used both the CARTzero, to its arrival at a water body of interest (Bolin and and PTM to investigate the mean age of Yellow RiverRodhe, 1973; Takeoka, 1984).water in the Bohai Sea.Numerical simulation is one of the major methods forThe CART obtains mean、studying water age. Compared with other methods (ie, Eulerian equations. As a Lagrangian method, the PTMfield observations and theoretical study), numerical traces water particles along their pathways and recordssimulation can consider both advection and diffusiontheir ages as time passes. These two methods are essen-processes in a realistic ocean with complex topography tially equivalent in theory (Liu et al, 2012). However, theand forcing conditions. Therefore, numerical simulation mean water age calculated by the CART and PTM may beis widely used in calculating mean water age (Chen, 2007; different in practice (Liu et al. 2012). In order to propose .Wang et al, 2010; Liu et al, 2011; de Brye et al, 2013; some useful su中国煤化工1 water age in arealistic oceanlHCNMH Gfference of the* Corresponding author. Tel: 0086-532-66786568two methods wds evaruatea y appiying unem to an ideal-E-mail: zliu@ouc.edu.cnized one-dimensional channel in this study. .包Springer48WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-582.3 Model Configuration2 Model DescriptionWe considered a one-dimensional finite domain (signed2.1 CART Modelas x direction) with a length L of 20km. In the PTM and .To calculate mean water age a(t, x, y, 2) using the CART, we used the same grid interval Or (=200m) andCART (Deleersnijder et al, 2001), two equations need to time step△t (=10s).be solved for the concentration C(t, x,y, z) and age con- In the one-dimensional domain, the goverming Eqs. (1)-centration B(t, x, y, z) of the targeted water particles, re- (3) in the CART can be simplified tospectively. The concentration C(t, x, y, z) is controlled byEq. (1).H(uC), d(5)d)x adxaC__ A(uC) 8(vC) . a(wC)dt d)ydzβ =C_ X(uB), 王以亟(6)didx dx^ax”’ac、p(,x)_axH dx'' 子y’dz0z'a(t,x)=(7)C(t,x)where t is time; x, y and Z are three coordinates in space; u,where K is the diffusion coefficient.v and w are velocities in x, y and z directions, respectively;The initial values of concentration C(t, x) and age con-KH and Kv are horizontal and vertical diffusion coffi-centration B(t, x) were both set to 0 in the whole domain.cients, respectively.The age concentration B(t, x, y, z) is calculated by Eq.At the releasing point (x=x), the concentration C(t, x) wasalways set to 1, that is, the water particle was released(2).continuously at x=Xr The age concentration at x=x, was0β c「H(uB) , d(vB), 8(wB)set to 0, resulting in a zero age of water particle at x=x,(Bolin and Rodhe, 1973; Takeoka, 1984). At the two endsz(x=0 and x=L), the concentration C(t, x) and age concen-.| d(K dB)+2(K. dB+2(k, dgtrationB(t, x) both were set to 0, indicating that the water| dx)y" dz2).(2)particle could not re-enter the model domain.In the PTM, the Eq. (4) can be simplified as:After solving Eqs. (1) and (2), the mean water age a(t, x,aKy, z) is calculated as the ratio of B(t,x,y,z) to C(t,x, y,z):x(t+Ot)-x(1)= ξV2KOt+gx Ot + uOt .(8))xa=βIC .3)To better understand the movement of a particle, the2.2 PTM Modelterms√2K△t ，Ot and uQt were defined as char-The PTM module (Zhang, 1995) used in this study isacteristic diffusion displacement OxrDif, pseudo displace-from estuarine and coastal ocean model coupled with ament△rpse, and advection displacement OrAdv， respec-sediment transport module (ECOMSED) (Blumberg,tively.2002). The three coordinates (x, y, z2) of a particle in thisIn the PTM, the initial and boundary conditions weremodule was controlled by Eq. (4).the same as those in the CART. In the experiments fortemporally varied flow field (Section 3.2.2), 1 particlex(1+ Ot)- x(1)= ξ√2KHOt +9出Ot +uOtwas released at x=x, at each time step within the first pe-riod of temporally varied velocity (T). In other experi-y(1+S)-v(0)=5N2KqAt+HNOIHOr+vOr，(4)ments, a total of 1000 particles were released at x=x, atthe first time step. When a particle was released at x=xr,its age was set to be 0. Subsequently, the particle age wasz(t +Ot)-z()= 5V2KvOt +2r Ot + wOtupdated at every time step until it reached the end of thedomain (x=0 or x=L), where the particle was excludedwhere△t is time step; ζ is random number with zero mean from the model.and unit variance.We recorded the positions of all the particles releasedThe second term on right hand side of Eq. (4) repre- in the first period of temporally varied velocity during thesents pseudo displacement arising from spatial variationtotal time of calculation. It should be pointed out that thein diffusion.period in the numerical experiments for constant velocityFor a particle released at time to, its position is given by (in all Sections ex中国煤化工considered .Eq. (4) and its age ist -to. The mean water age (t,x,y,z) as Ot. For calculatiMHcNMHGreleasedinsteady state,is the average of all the particles' age at location (x, y, z) we need not only tat time t.the first period, but also those in the second period and自SpringerWANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-5849succeeding periods. Based on the fact that the velocity where at x20.25L, x*=x-0.25L, l=0.75L; at x<0.25L, x*=field and diffusivity coefficients used for the PTM calcu- 0.25L-x, l=0.25L.lation were repeated at the same time in every period, weIn the case of K= =20m2s-', Fig.1a (black line) showsassumed that the particles released in the second and suc- that mean water age is zero at x, and increases as a para-ceeding periods have the same pathways as those released bolic function to the distance away from xr. The meanin the first period. The only difference is in the ages of the water age calculated by the CART and PTM both agreesparticles. In this manner, we obtained the pathways of the well with the analytical solution (Fig.1a).particles released after the second period without addi-There is one major peak of frequency at approximatelytional PTM calculations. This counting method for mean 0 in the age frequency distribution function for the areawater age has been used for the mean age of Yellow River around x (i.e, x=0.3L) (Fig.1lb, red line; Fig.1d). Thiswater in the Bohai Sea (Liu et al, 2012). .peak corresponds to the young water particles thaWe stopped the calculation when the mean water age quickly spread into this area from Xr In addition to thesedid not change with time in the CART and PTM and there- young water particles, we can also identify the presencefore obtained the mean water age results in a steady state.of old water particles that have an age longer than 5dTo better understand the mean water age distribution in (Fig. Ib, red line), indicating that the water particles returna steady state, the age frequency distribution function was to x=0.3L from the area outside x=0.3L. Therefore, thecalculated based on the results of the PTM. The age fre- mean water age of about 3d at x=0.3L (Fig. 1a) resultsquency distribution function φ(T, x) is defined by Eq. (9) from the coexistence of newly released water particles(Bolin and Rodhe, 1973), i.e,from x, and the returned water particles from the areaoutside x=0.3L.1 dM(r,x)The age frequency distribution fiunction shows a more(,)=M6(x) dr(9)complex composition of mean water age in the region farwhere, M(x) is the total number of particles at x; M(r, x)away from x, (i.e., x=0.7L) (Fig.1c, red line) than that inis the total number of the particles whose age is smallerthe region around x, (ie, x=0.3L, Fig.1b, red line). Thereis one major peak of frequency at approximately 10d inthan or equal to an age tatx.the age frequency distribution function at x=0.7L (Fig.lc,To represent the relative mass of water particles bered line; Fig. 1d). This peak corresponds to the water par-tween location x and releasing location x, in a steady state,ticles that directly spread into this area from xXr. However,R is defined by Eq. (10):the frequency at approximately 5- -20d in the age fre-R(x)=M' (x)/M (x,).(10)quency distribution function at x= =0.7L is not muchsmaller than that at approximately 10d (Fig. lc, red line).In the CART, M(x) and M (x) are the concentration atAs a result of coexistence of such water particles, thx and xr, respectively; in the PTM, M (x) and M (x,) aremean water age at x=0.7L is about 18d (Fig.1a).Fig.le is the extension of results at x=0.3L and .x=0.7Lthe particle number at x and xr, respectively.to the whole model domain (i.e, x>0.25L). At any loca-tion x in the model domain, there are both water particles3 Resultswith small age and water particles with large age. TheBecause advection and diffusion are two importantmean water age at x is a result of coexistence of all theprocesses controlling material transport in coastal water,water particles at x (Fig.1a). Similar to the mean waterwe examine the mean water age distribution controlled byage, the age corresponds to the max. frequency in the agethem. In previous studies on mean water age by the PTM,frequency distribution function increases as a parabolicthe displacement of a particle usually contains only Orpirfunction to the distance away from the x, (Fig. 1d).In the above experiments, we also changed the valuesand ArAdv (Chen, 2007; Liu et al, 2011) but does notof K but the agreement of the mean water age between thecontain Oxpse: Hence, in addition to the comparison of theCART and PTM was kept. Apparently, the agreementCART and PTM, we also pay some attention to the dif-between two methods is independent of K.ference between the mean water ages calculated by thePTM with and without Arpse, respectively.3.1.2 Spatially varied diffusion coefficient3.1 DiffusionWe used a spatially varied diffusion coefficient con-trolled by Eq. (12) (Fig.2a):3.1.1 Constant and uniform diffusion coefficientIf the model only includes constant and uniform diffu-K(x)=Ko + Acos(2x/L) , x∈[0,L],(12)sion coefficient without advection, the analytical solutionfor mean water age can be found in Appendix A in Liu where, Ko=20m2s，A=15m's!et al. (2012). In this study, x,=0.25L; the mean water ageis given by Eq. (11):contains Orir中国煤化工se (Fig.2b, redline). Oxrpse is sC N M H G (Fig.2b). Arpsea(x)=(2部(11)is positive at x>0.5L while negative at x0.25L (Fig.3c). This is of water particles spend approximately 5 d spreading intoconsistent with the results reported by Visser (1997) that this area from xr. However, the frequency at approxi-the PTM with only Orpif causes the particles to gather in mately 5d in the age frequency distribution function atlow diffusion regions. As a result, it is difficult for the x=0.5L by the PTM without Orpse is smaller than that byparticles to leave the low diffusion region (x=0.5L) (Fig.3d, the PTM with Orpse (Fig.4a). On the other hand, the fre-blue line), and the mean water age at x>0.25L becomes quency at longer than 30d in the age frequency ditribu-much longer (Fig.3c, blue line) than that calculated by the tion function at x=0.5L by the PTM without Oxpse is largerCART. As Arpe is considered in the PTM, it helps the than that by the PTM with Arpse (Fig.4a). This indicateswater particles leave the low diffusion region and move again that in the calculation of PTM without Orpse, com-towards x=0 orx=L (Fig.3d, red line).pared with the calculation of PTM with Oxpse, it is diff-It must be noted that in the case of x,=0.5L, the PTMcult for the water particles to leave the low diffusion re-without Orpse also gather water particles in the low diffu- gion (x=0.5L) (Fig.3d, blue line) with a longer mean wa-sion region (x=0.5L) (Fig.3b, blue line). However, in this ter age at x>0.25L (Fig.3c).25.7 r.6-. 1:20aI (b)x=03L0.50三15.4).3= Frequency2 10PTM.2， Number4层.10.2L 0.4L.0.6L0.8L入Age (d).07 r0.060.05(C)x-0.7L10 t_(d.0.03此- Frequency3首曼6Number22NW.01 t40600.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L I1:0.08仓10.00.020.3L 0.4L 0.SL 0.6L 0.7L 0.8L 0.9L LFig.1 The diffusion coefficient is constant and uniform (20m2 s ") and water particle is released at 0.25L (x,=0.25L). (a)Comparison of mean water ages by CART (black line), PTM (red line), and analytical solution. The analytical solutionof Eq. (11) is overlapped by CART (black line). (b) Total number (black line, i.e.中国煤化工equencydistribution function (red line, i.e, o(, x) in Eq. (9)) at x=0.3L. The age range (7)requencyof particles with age longer than 60d is too small to be identified. (c) The same as:MHCNMHGTheagecorresponds to the maximum of frequency distribution function at x>0.25L. (e) The age Irequency a1Str1DuIn function中(t, x) (unit: (0.25d) ) at x>0.25L. The age range (t) is limited to 20d.包SpringerWANG et al. 1J. Ocean Univ. China (Oceanic and Coastal Sea Research) 201514: 47-58_5135 r280.0502”(a)b)T.03合25下全200.01 宜20亏16....-0.0器0t0.2L0.6L0.8L0.4L 0.6LL0.0150.15(d0.010日0.005-0.005-0.050.).4.0).20.4 0.60.81.0Time (d)Fig.2 (a) The distribution of spatially varied diffusion coefficient by Eq. (12). (b) Based on the diffusion coefficient shownin Fig.2(a), Orpir (black line) and Orpse (red line) calculated by Eq. (8). (c) The distribution of temporally varied velocity byEq. (13). (d) Based on the velocity shown in Fig.2(c), OxAdv calculated by Eq. (8). See Section 2.3 for the definitions ofAxpit, Oxrpses and AxAdv.6r(a)(b)一CART|心. CART .- PTMyPTM■PTMa02: PTMn50 r3.0 r0十.4 t(心) .仓30--CART.2 t-PTMy- PTMn0.4L).8LXFig.3 The spatially varied diffusion coefficient is controlled by Eq. (12). (a) Water particle is released at 0.5L (x,=0.5L).Comparison of mean water ages by CART (black line), PTM with△rpse (red line), and PTM without ATPse (blue line). (b)Water particle is released at 0.5L (x,=0.5L). Comparison of R (defined by Eq. (10)) by CART (black line), PTM with Arpse .(red line), and PTM without Oxpse (blue line). (c) The same as Fig.3(a), but for x,=0.25L. (d) The same as Fig.3(b), but forx,=0.25L.Figs.4c and 4d are the extension of results at x=0.5L to with Arpse (Fig.3c). Similar to mean water age, the agesthe whole model domain (i.e., x>0.25L) by the PTM with corresponding to the max. frequencies by the PTM withand without Orpse; respectively. At any location x, the and without Orpse both increase as a parabolic function tofrequency at small age by the PTM without Orpse isthe distance away from the x (Fig.4b). The age corre-smaller than that by the PTM with Orpse (Figs.4c and 4d)，sponding to the max. frequency by the PTM without Orpswhile the frequency at large age by the PTM without is slightly' 中国煤化工M with ArpseOrpse is larger than that by the PTM with Arpse (Fig.4c，(Fig.4b).Fig.4d). As a result, the mean water age by the PTMIn summaryYHC N M H Garied difusionwithout Arpse is significantly larger than that by the PTM coefficient, the PTM should include Arpse.自Springer52WANG et al. 1J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-580.070t(a)x=0.5Lb)...30 t.....日。? 0.03- PTMy!" .200.02-PTMn"V.PTMy,0一)230 4)60700.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L LAge (d)7070 00阿PTMTy504PTMh0.04;0仓40.0330y 300.01o10。.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L LFig.4 The spatially varied diffusion coefficient is controlled by Eq. (12) and water particle is released at 0.25L (x,=0.25L).(a) The age frequency distribution function q(t, x) at x= 0.5L by PTM with Arpse (red line) and by PTM without Arpe (blueline). (b) The age corresponds to the maximum of frequency distribution function at x>0.25L by PTM with Axrpse (red line)and by PTM without Arpse (blue line). (c) The age frequency distribution function o(t, x) (unit: (0.25d) |) for x>0.25L byPTM with Orpse. (d) The same as Fig.4(c), but for PTM without Arpse. The age range (I) in Figs.4(a), 4(C), and 4(d) is lim-ited to 70d.deals with each particle and is capable of considering the3.2 Advectioninternal information (e.g, uneven age of particles) insidea grid. The exchange of particles between neighboring3.2.1 Constant and uniform velocityIn the case of constant and uniform velocity (u) withoutgrids can be well described in the PTM. However, theuniformity of age of particles within each grid cannot bediffusion, the analytical solution for mean water age is x/u,considered in Eqs. (5)- (6) for CART.where x is the distance away from xr. Assuming u=0.005At t=105.5d, there is one major peak of frequency atms ! and x=0.25L, we show the analytical solution inapproximately 1.7 d in the age frequency distributionFig.Sa, in which the mean water age is zero at. x, and in- function for the area around x, (.e, x=0.32) (Fig.Sc, redcreases linearly with the distance away from the X. The ;line; Fig.5e). This peak corresponds to the young watermean water age obtained by the CART and PTM agreesparticles that spread quickly into this area from xr. In ad-well with the analytical solution (Fig.5a). From the agedition to these young water particles, we can also identifyfrequency distribution function, each particle' 's age equalsthe presence of old water particles that have an age longerto the mean age at any location x in the model domain.than 2.3 d (Fig.5c, red line), indicating that the water par-ticles can return to x=0.3L from the area outside x=0.3L3.2.2 Temporally varied velocitybecause of the negative velocity (Fig.2c). As the case ofIn this case, we assume a temporally varied velocity difusion, the mean water age (about 2d) at x=0.3Lgiven by Eq. (13) (Fig.2c):(Fig.5b) is a result of coexistence of newly released waterparticles from x, and the returned water particles from theu()=u0 + Bcos(2πt/T),(13) area outside x=0.3L.The age frequency distribution function shows a morewhere uo=0.005ms , B =0.01 ms，T=86400s.complex composition of mean water age in the region farAccording to Eq. (8), in the case of temporally varied away from x (i.e, x=0.7L) (Fig.5d, red line) than that invelocity without diffusion, the displacement of a particle the region around x (ie,妇=0.3L, Fig.5c, red line). Forcontains only Oradv (Fig.2d). Again, we released particles instance, there are two major peaks of frequency at ap-at x,=0.25L.proximately 20.6d and 20.9d in the age frequency distri-Fig. 5b shows that the mean water age by the PTM and bution (Fig.5d, red line). As a result of coexistence of theCART both are zero at x, at t=105.5d. The mean water water particles with different ages, the mean water age atage by the PTM increases linearly with the distance away x=0.7L is about 21中国煤化工from the x, with a small fluctuation. The mean water ageFig.5f is the extMHCNMHGand x=0.7ILby the CART also increases linearly with the distance to the whole modeSt any loca-away from the xp but with lttle fluctuation. The PTM tion x in the model domain, there are water particles with包SpringerWANG et al. 1J. Ocean Univ. China (Oceanic and Coastal Sea Ressearch) 201514: 47-5853both small and large ages. The mean water age at x is a responding to the max. frequency in the age frequencyresult of coexistence of all the water particles at x (Fig.5b). distribution function increases linearly with the distanceCompared with the age frequency distribution function away from the x, with a very small fluctuation (Fig.5e).under diffusion (Figs. le, 4c, and 4d), the age of frequencyIn summary, in the case of temporally varied velocity,distribution function under advection (Fig.5f) presents a the mean water age by the PTM and CART agrees wellsmaller range. Similar to the mean water age, the age cor- with each other.3555 [(b)2.....620- Theory .....。CART曼15品1:. PTM0.2L.4L0.6L0.8L0.4L 0.6LL.0.6 r(c) 0.3L(d)x-0.7L2.5g42.09 82.4 F三3.5复.8 t宫2g1.50.6 t01.1.725 22.7520.420.620.821.021.221Age (d)35 n30-(030 1002:仓2015b 15100.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L LFig.5 Water particle is released at 0.25L (x,=0.25L). (a) The analytical solution for mean water age (x/u, where x is the dis-tance away from x) with a constant and uniform velocity (0.005 ms ). The mean water ages by CART and PTM overlapwith the analytical solution. (b) Comparison of mean water ages by CART (black line) and PTM (red line) with temporallyvaried velocity controlled by Eq. (13). (c) Total number (black line, M(, x)) in Eq. (9) of the particles with ages less thanor equal to an age (T) at x=0.3L; age frequency distribution function (red line, 0(, x)) in Eq. (9) at x=0.3L. The age range ()is from 1.5 to 3d. (d) The same as Fig.5(c), but for x=0.7L. The age range () is from 20.4 to 21.4d. (e) The age corre-sponds to the maximum of frequency distribution function at x>0.25L. (f) The age frequency distribution function ρ(T, x)(unit: (0.02d) ) at x>0.25L. The age range (7) is limited to 35 d. Same as Fig.5(b), the temporally varied velocity is con-trolled by Eq. (13) in Figs.5(c)- -5(f). Figs. 5(b)- -5([) are at time t=105.5d. .the PTM without Orpse collcts water particles in low dif-fusion region (Fig.3d).4 DiscussionFrom Eq. (12), the average magnitude of gradient of4.1 The Disappearance of the Mean Water Age Dif- spatially varied diffusion cofficient (ie, |0K/ax) is aboutference Between the CART and PTM Without 3x10~*ms-' at x>0.25L. In order to calculate the averageArpse in the Case of the Spatially Varied Diffusion of |aK/x| at x>0.25L, we first calculate the magnitude ofCoefficientgradient of spatially varied diffusion coefficient (]K/ax)The several experiments we discussed above show that at each location x at x>0.25L. The average of |K/ax\ atthe mean water age results by the CART and PTM gener-x>0.25L is the arithmetic average of all the |0K/ax| atally agree well with each other except for that in the case x>0.25L.of spatially varied diffusion coefficient (Section 3.1.2). InIt is of interests to know under what circumstances thethat case, if the water particle was not released at the phenomenon中国煤化Iy the CART isplace with weakest difusion, the mean water age by the smaller than tfYHCN MH Ghie at x>0.252CART is much smaller than that by the PTM without (Fig.3c) will vi. . ...n in |@K/ax|.Orpge (Fig.3c). The cause for this inconsistence is becauseWe used a spatially varied diffusion coefficient given包Springer54WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58by Eq. (14) to examine this problem.by the CART is smaller than that by the PTM withoutOrpse at x>0.25L will vanish, if the magnitude of gradientK(x)= K。+ Acos(2πx/L), x∈[0,L],(14)of spatially varied diffusion coefficient decreases to aBy changing value of A in Eq. (14) to 10, 8, 6.5, 5.5,certain value (i.e., the average of |0K/ax| not larger thanand 5m2s , we obtained the average of |0K/ax| as 2x10 3 ,1x10 'ms-).1.6x10-, 1.3x10-3, 1.1x10-, and 1x10-'ms-' at :x>0.25L,On the other hand, advection and diffusion coexist inthe realistic ocean. It is therefore necessary to examinerespectively.Again, we assumed x,=0.25L. When |0K/ax| decreases,the impact of velocity on the PTM without Orpse in spa-the mean water age difference between the CART andtially varied diffusion domain. In the next experiments,PTM without Orpse at x>0.25L decreases gradually (Fig.6).besides the diffusion coefficient given by Eq. (12), a con-When the average of |@K/ax| is 1x10 3 ms ', the meanstant and uniform velocity is added at x2>0.25L. In Figs.8a,water age by the CART is almost the same as that by the8b, 8c, and 8d, the velocity is set to 0.0015, 0.005, 0.015,PTM without Oxpse (Fig.6e). In this case, no matter theand 0.05 ms', respectively. Again, we assumed x,=0.25L.PTM includes Orpse or not, there is one major peak ofAccording to Eq. (8), the displacement of a particlefrequency at about 5d in the age frequency distributioncontains Oxpir, Orpse, and OrAdv at x20.25L. Here, to showfunction at x=0.5L (Fig.7a), being the same as in the casethe relative importance of Orpse and Axdv, a parameter R*where the average of |0K/ax| is 3x103ms-1 (Fig.4a). Theis defined in Eq. (15): .frequency at about 5d and longer than 30d by the PTMwithout Orpse at x=0.5L is almost the same as that by theR*Arpse|aK(15)PTM with Orpse (Fig.7a). In the whole domain, the ageArndv|~ |udx|corresponding to the maximum frequency (Fig.7b) andthe age frequency distribution function (Figs.7c, d) by theThe average of R* equals to 2, 0.67, 0.2, and 0.067 inPTM without Arpe is almost the same as that by the PTM Figs.8a- 8d, respectively. In order to calculate the averagewith Arps.: These features are not found in the results ofR" for x>0.25L, we first calculate R" at each location xwhen the average of |aK/2x| is 3>10-3 ms 1 (Fig.4). There- for x>0.25L. The average of R" for x>0.25L is the arith-fore, it is likely that the problem that the mean water age metic average of all the R* for x>0.25L.35o|(a)30b)仓三20当1- CART, 15个CART10-PTMy。PTMy5PTMp5 ..0.2L0.4L 0.6L0.8L0.4L 0.6L 0.8L30，(d2巨20三20...曼15”CART2 15CART-PTMy.1i PTMy.PTMn..。PTMn0.410.6L.8L35 rt (e)25仓20-CART0- PTMy5 ..........一PTMn0.4LXFig.6 Water particle is released at 0.25L (x,=0.25L). (a) Comparison of mean w中国煤化工), PTMwith Oxrpse (red line), and PTM without Oxpse (blue line) with spatially varied diffusEq. (14)whenA is set to 10m's'. (b)- (e) The same as Fig.6(a), but for spatially varied cYHCN MH G1by Eq.(14) whenA isset to 8, 6.5, 5.5, and 5m2 s，respectively.包SpringerWANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58550.1(@)x-0.5L0.082of 6); 0.06旦15官0.04- PTMy:曼10一PTMn0.02m PTMy. PTMn0” P IMnT10 2(50 6(0.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L LAge (d)0m70 m05@PTMyCPIMn0.0450。。三40仓40I 0.0330岛3002<20200.010面Fig.7 The same as Fig.4, but for spatially varied diffusion coefficient controlled by Eq. (14) whenA is set to 5 m2s '.Water particle is released at 0.25L (x,=0.25L).50 [a3(b)40.....252(30 t曼20- CART10. PTMy .- PTMy0上“” PTMn"i PTMn'0.3L 0.4L 0.5L 0.6L 0.7L 0.8L 0.9L0F3.d)巨2.06....2P 1.5r CART.0... PTMn .0.5- PTMn.Fig.8 Water particle is released at 0.25L (x,=0.25L). (a) Comparison of mean water ages between those by CART (blackline), PTM with△xpse (red line), and PTM without△xpse (blue line). The spatially varied diffusion coefficient is con-trolled by Eq. (12) and a constant and uniform velocity (0.0015ms ) is added for x20.25L. (b)- (d) The same asFig.8(a), but the velocities are set to 0.005, 0.015, and 0.05 ms , respectively.For R* values, the mean water age by the CART agreesthe average of R* decreases to 0.67 (Fig.8b) and 0.2well with those calculated by the PTM with OXxpse (Fig.8，(Fig.8c), the mean water age by the CART is still shorterred and black lines). Regarding the PTM without OXrpse, than that by the PTM without OXpse. However, comparedthe agreement is not kept for some values of R'*. For in- with Fig.8a, the mean water age difference between thestance, the mean water age by the CART is much shorter CART and PTM without Axpse greatly decreases. Ththan that by the PTM without Orpse when the average of mean water age by the CART agrees well with that by theR" is2 (Fig.8a). In this case, advection is too weak to PTM without Oxpse when the average of R”is 0.067cover up the mean water age difference caused by spa-(Fig.8d). In this case, advection is strong enough to covertially varied diffusion cofficient (Fig.3c). In the calcula- up the age difference caused by spatially varied diffusiontion of the PTM without Orpse, it is difficult for water par- coefficient.中国煤化工reases to a cer-ticles to leave the low diffusion region (x=0.5L) and tain extent (ifYHger than 0.067),therefore a lot of water particles gather there (Fig.3d).the phenomendCNMHGytheCARTis.In the case with both advection and diffusion, even as shorter than that by the PTM without Arpse at x>0.25L自Springer56WANG et al. 1J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58will vanish.石=|az(18)4.2 Application to the Realistic Ocean|aKrIn this section, we take the Bohai Sea as an example to的二0z(19) .study the application of CART and PTM to a realisticocean. The Bohai Sea is a semi-enclosed water body withWe interpolated Kv at equal distance in the verticalan average depth of 18m. It is divided into 5 subregions,direction before calculating Az and A4 by Eq. (18) and Eq.namely Laizhou Bay, Bohai Bay, Liaodong Bay, the cen-(19).tral Basin, and Bohai Strait. The major rivers that flowThe annual KH for the surface layer of the Bohai Seainto the Bohai Sea include the Yellow River, the Haihe(Fig.10a) shows that the Kx is higher in the coastal areaRiver, the Luanhe River, and the Liaohe River (Fig.9). .(i.e, estuaries, >40m2s~) than in the offshore area (ie,the central basin, <20 m2 s "). The distribution of Ku(Fig. l0a) induces a high个in the coastal area (i.e., estu-aries, >0.003 ms) in the surface layer of Bohai SeaB(Fig. l0c).2 is small in the offshore area (i.e, the centralbasin, about 0.002ms ') and it is even less than 0.001 ms-'Luanhe RiverLDBin some areas (Fig. 10c). Apparently, the magnitude of gra-.....dient of spatially varied horizontal diffusion coefficient isHaihe River39°smaller in the offshore area than in the coastal area. Ac-cording to the analysis for the spatially varied diffusionBHBI CB50coefficient in Section 3.1.2 and Section 4.1, the Sxpsecannot be neglected in the PTM to calculate mean water3850 age when the magnitude of gradient of spatially variedYellow RiverLZBdiffusion coefficient is larger than 0.001ms . The 2 forthe surface layer of the Bohai Sea (Fig. 10e) shows that A237°is larger in the coastal area (i.e., estuaries, >0.2) than in18°119°120*the offshore area (i.e, the central basin, <0.2). This indi-Fig.9 Bathymetry of the Bohai Sea (unit: m). The Bohaicates that the impact of horizontal advection comparedSea is divided into 5 subregions: Laizhou Bay (LZB),with horizontal diffusion is stronger in the offshore areaBohai Bay (BHB), Liaodong Bay (LDB), the central ba-than in the coastal area. According to the analysis for thesin (CB), and Bohai Strait (BS). The black dots representspatially varied diffusion coefficient with constant andthe locations of the river mouths of the Yellow River, theuniform velocity in Section 4.1, △xpse should be consid-Haihe River, the Luanhe River, and the Liaohe River,respectively. Line AB denotes the section along whichsion cofficient and velocity in one-dimensional domain)vertical diffusion coefficient is given in Fig. l0b.is not less than 0.2. If we consider the effects of bothhorizontal diffusion coefficient and horizontal velocity inThe diffusion coefficient and velocity used here were the Bohai Sea, the PTM should include Arpse at least forcalculated by the model validated by Wang et al. (2008). the coastal area (i.e, estuaries) of the Bohai Sea in theThe horizontal resolution was 1/18 degree in both the horizontal direction.zonal and meridional directions. In the vertical direction,The annual Kv along transect AB in the Bohai Sea21 sigma levels were distributed (0.000, -0.002, -0.004，(Fig. l0b, location being shown in Fig.9) shows that the-0.010, -0.020, -0.040, -0.060, -0.080, - -0.100, -0.120， Kr is smaller for the surface layer (i.e, shallower than 5m)-0.140, -0.170, -0.200, -0.300, -0.400, -0.500, -0.650，than in the middle and bottom layers (i.e, deeper than 10-0.800, -0.900, -0.950, and -1.000). The spatial vari-m). The distribution of Kr (Fig. 10b) induces a higherλ3 inability of horizontal diffusion coefficient (h1), the ratio of the surface layer (ie, shallower than 5 m, >0.003 ms |)the spatial variability of horizontal diffusion coefficient to than in the middle and bottom layers (i.e, deeper than 10the horizontal velocity (2), the spatial variability of ver- m, about 0.002 ms ) (Fig. 10d). In some areas, a3 is eventical diffusion coefficient (3), and the ratio of the spatial less than 0.001 ms 1 (Fig. 10d). This indicates that thevariability of vertical diffusion coefficient to the vertical magnitude of gradient of spatially varied vertical diffu-velocity (4) are calculated by Eqs. (16)-(19), respec- sion cofficient is smaller in the middle and bottom areatively: .than in the surface area. According to the analysis for thespatially varied diffusion coefficient in Section 3.1.2 and0KH )2 (aKH )丫Section 4.1, the Oxpse cannot be neglected in the PTM to(16)=ax( dycalculate mean water ame when the mamnitudoof gradientof spatially varied中国煤化工_larger than .0.001ms-'. Fig.10CHCNMHGargeforthe(KH( aKH(17)whole Bohai Sea (> 10-) because the vertical velocity isdyextremely small, indicating that the impact of vertical自SpringerWANG et al. 1J. Ocean Univ. China (Oceanic and Coastal Sea Research) 201514: 47-5857advection compared with vertical diffusion is very small whole Bohai Sea.in the whole Bohai Sea. Hence, it is necessary for theIn summary, the PTM must include Orpse for the realis-PTM to consider Oxpse in the vertical direction for the tic shallow waters (e.g, the Bohai Sea).m2 st>B 41°N0°N500.010.02540°40100.020 .宫39°300.015200.010 .38V.0250.00537030°