J. Ocean Univ. China (Oceanic and Coastal Sea Research)
DOI 10.1007/s11802-015-2393-7
ISSN 1672-5182, 2015 14 (1): 47-58
http://www.ouc.edu.cn/xbywb/
E-mail:xbywb@ouc.edu.cn

A Comparative Study of CART and PTM for
Modelling Water Age
WANG Haiyan1), GUO Xinyu2), LIU Zhe1), *, and GAO Huiwang1)
1) Key Laboratory of Marine Environment and Ecology, Ministry of Education, Ocean University of China,
Qingdao 266100, P. R. China
2) 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 2015
Abstract 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-dimensional
domain. The model results by the two methods are consistent with each other in the case with either spatially uniform flow field or
spatially uniform diffusion coefficient. If we allow the spatial variation in horizontal diffusion, a term called pseudo displacement
arising from the spatial variation of diffusion coefficient likely plays an important role for the PTM to obtain accurate water age. In
particular, if the water particle is released at a place where the diffusion is not the weakest, the water age calculated by the PTM
without 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 age

1 Introduction
Advection and diffusion are two important processes incoastal material transport. The material transport timescales play an important role in the algal bloom (Hilton
et al., 1998). Because of the complex spatiotemporal
structure of coastal currents, it is helpful to define auxiliary variables, such as water age, to understand the material transport processes in coastal zone of oceans (Zimmerman, 1976; Takeoka, 1984; Deleersnijder et al., 2001;
Monsen et al., 2002; Delhez et al., 2004). Water age isdefined as the time elapsed since the departure of a waterparticle from an area, where its age is prescribed to bezero, to its arrival at a water body of interest (Bolin andRodhe, 1973; Takeoka, 1984).
Numerical simulation is one of the major methods forstudying water age. Compared with other methods (i.e.,field observations and theoretical study), numericalsimulation can consider both advection and diffusion
processes in a realistic ocean with complex topography
and forcing conditions. Therefore, numerical simulation
is widely used in calculating mean water age (Chen, 2007;
Wang et al., 2010; Liu et al., 2011; de Brye et al., 2013;* Corresponding author. Tel: 0086-532-66786568
E-mail: zliu@ouc.edu.cn

Liu et al., 2012). The mean water age is defined as the
mass-weighted arithmetic average of the ages of all of thewater particles within a target domain.
Among the aforementioned studies on mean water age,constituent-oriented age and residence time theory (CART,
www.climate.be/cart) (Deleersnijder et al., 2001) and particle-tracking method (PTM) (Zhang, 1995) are two widelyused methods. For instance, Wang et al. (2010) studied
the mean age of Changjiang River water and de Brye et al.
(2013) studied the mean age of canal and dock water by
the CART; Chen (2007) studied the mean age of Alafia
River water and Liu et al. (2011) studied the mean age of
the Tahan Stream, Hsintien Stream, and Keelung River
water by the PTM; Liu et al. (2012) used both the CART
and PTM to investigate the mean age of Yellow River
water in the Bohai Sea.
The CART obtains mean water age by solving two
Eulerian equations. As a Lagrangian method, the PTM
traces water particles along their pathways and records
their ages as time passes. These two methods are essentially equivalent in theory (Liu et al., 2012). However, themean water age calculated by the CART and PTM may bedifferent in practice (Liu et al., 2012). In order to propose
some useful suggestions for studying mean water age in arealistic ocean with two methods, the difference of thetwo methods was evaluated by applying them to an idealized one-dimensional channel in this study.

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2 Model Description
2.1 CART Model
To calculate mean water age a(t, x, y, z) using the
CART (Deleersnijder et al., 2001), two equations need tobe solved for the concentration C(t, x, y, z) and age concentration β(t, x, y, z) of the targeted water particles, respectively. The concentration C(t, x, y, z) is controlled by
Eq. (1).
  (uC )  (vC )  ( wC ) 
C
 



t
y
z 
 x


C 
C 
C 
  (KH) (KH)  ( KV
) , (1)
x y
y z
z 
 x
where t is time; x, y and z are three coordinates in space; u,v and w are velocities in x, y and z directions, respectively;
KH and KV are horizontal and vertical diffusion coefficients, respectively.
The age concentration β(t, x, y, z) is calculated by Eq.
(2).
  (u  )  (v )  ( w ) 

C 



t
y
z 
 x




 
 
  (KH) (KH)  ( KV
) . (2)
x y
y z
z 
 x
After solving Eqs. (1) and (2), the mean water age a(t, x,y, z) is calculated as the ratio of β(t, x, y, z) to C(t, x, y, z):a /C .

(3)

2.2 PTM Model
The PTM module (Zhang, 1995) used in this study isfrom estuarine and coastal ocean model coupled with asediment transport module (ECOMSED) (Blumberg,
2002). The three coordinates (x, y, z) of a particle in thismodule was controlled by Eq. (4).
K H
t  u t x(t  t )  x(t )   2 K H t 
x

K H

t  vt , y (t  t )  y (t )   2 K H t 
y


K
 z (t  t )  z (t )   2 KV t  V t  wt
z


(4)

where ∆t is time step; ξ is random number with zero mean
and unit variance.
The second term on right hand side of Eq. (4) represents pseudo displacement arising from spatial variationin diffusion.
For a particle released at time t0, its position is given by
Eq. (4) and its age is t−t0. The mean water age (t, x, y, z)
is the average of all the particles’ age at location (x, y, z)
at time t.

2.3 Model Configuration
We considered a one-dimensional finite domain (signed
as x direction) with a length L of 20 km. In the PTM and
CART, we used the same grid interval ∆x (=200 m) andtime step ∆t (=10 s).
In the one-dimensional domain, the governing Eqs. (1)–(3) in the CART can be simplified toC (uC ) 
C

 (K
),
t
x
x
x

(5)


 (u  ) 

C 
 (K
),
t
x
x
x

(6)

 (t , x),
C (t , x)(7)a(t , x) 

where K is the diffusion coefficient.
The initial values of concentration C(t, x) and age concentration β(t, x) were both set to 0 in the whole domain.
At the releasing point (x=xr), the concentration C(t, x) was
always set to 1, that is, the water particle was released
continuously at x=xr. The age concentration at x=xr was
set to 0, resulting in a zero age of water particle at x=xr
(Bolin and Rodhe, 1973; Takeoka, 1984). At the two ends
(x=0 and x=L), the concentration C(t, x) and age concentrationβ(t, x) both were set to 0, indicating that the waterparticle could not re-enter the model domain.
In the PTM, the Eq. (4) can be simplified as:x(t  t )  x(t )   2 K t Kt  u t .
x

(8)

To better understand the movement of a particle, theKt and u∆t were defined as charterms 2K t ,xacteristic diffusion displacement ∆xDif, pseudo displacement ∆xPse, and advection displacement ∆xAdv, respectively.
In the PTM, the initial and boundary conditions were
the same as those in the CART. In the experiments for
temporally varied flow field (Section 3.2.2), 1 particle
was released at x=xr at each time step within the first period of temporally varied velocity (T). In other experiments, a total of 1000 particles were released at x=xr at
the first time step. When a particle was released at x=xr,
its age was set to be 0. Subsequently, the particle age was
updated at every time step until it reached the end of thedomain (x=0 or x=L), where the particle was excluded
from the model.
We recorded the positions of all the particles releasedin the first period of temporally varied velocity during thetotal time of calculation. It should be pointed out that theperiod in the numerical experiments for constant velocity(in all Sections except Section 3.2.2) could be consideredas ∆t. For calculating the mean water age in a steady state,
we need not only the pathways of the particles released in
the first period, but also those in the second period andWANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

succeeding periods. Based on the fact that the velocity
field and diffusivity coefficients used for the PTM calculation were repeated at the same time in every period, we
assumed that the particles released in the second and succeeding periods have the same pathways as those releasedin the first period. The only difference is in the ages of theparticles. In this manner, we obtained the pathways of theparticles released after the second period without additional PTM calculations. This counting method for meanwater age has been used for the mean age of Yellow River
water in the Bohai Sea (Liu et al., 2012).
We stopped the calculation when the mean water age
did not change with time in the CART and PTM and therefore obtained the mean water age results in a steady state.
To better understand the mean water age distribution ina steady state, the age frequency distribution function was
calculated based on the results of the PTM. The age frequency distribution function φ(τ, x) is defined by Eq. (9)(Bolin and Rodhe, 1973), i.e., ( , x) 

1 dM ( , x),
M 0 ( x)d(9)where, M0(x) is the total number of particles at x; M(τ, x)
is the total number of the particles whose age is smaller
than or equal to an age τ at x.
To represent the relative mass of water particles between location x and releasing location xr in a steady state,
R is defined by Eq. (10):R ( x )  M * ( x) / M * ( xr ) .

(10)

In the CART, M*(x) and M*(xr) are the concentration atx and xr, respectively; in the PTM, M*(x) and M*(xr) are
the particle number at x and xr, respectively.

3 Results
Because advection and diffusion are two important
processes controlling material transport in coastal water,
we examine the mean water age distribution controlled bythem. In previous studies on mean water age by the PTM,
the displacement of a particle usually contains only ∆xDifand ∆xAdv (Chen, 2007; Liu et al., 2011) but does notcontain ∆xPse. Hence, in addition to the comparison of the
CART and PTM, we also pay some attention to the difference between the mean water ages calculated by thePTM with and without ∆xPse, respectively.

3.1 Diffusion
3.1.1 Constant and uniform diffusion coefficient
If the model only includes constant and uniform diffusion coefficient without advection, the analytical solution
for mean water age can be found in Appendix A in Liu
et al. (2012). In this study, xr=0.25L; the mean water age
is given by Eq. (11):a( x) x*lx*
(2  ) ,
6K
l

(11)

49

where at x ≥ 0.25L, x*=x − 0.25L, l = 0.75L; at x<0.25L, x*=
0.25L−x, l = 0.25L.
In the case of K=20 m2 s−1, Fig.1a (black line) showsthat mean water age is zero at xr and increases as a parabolic function to the distance away from xr. The meanwater age calculated by the CART and PTM both agrees
well with the analytical solution (Fig.1a).
There is one major peak of frequency at approximately
0 in the age frequency distribution function for the areaaround xr (i.e., x=0.3L) (Fig.1b, red line; Fig.1d). This
peak corresponds to the young water particles thatquickly spread into this area from xr. In addition to theseyoung water particles, we can also identify the presenceof old water particles that have an age longer than 5 d(Fig.1b, red line), indicating that the water particles returnto x=0.3L from the area outside x=0.3L. Therefore, themean water age of about 3 d at x=0.3L (Fig.1a) resultsfrom the coexistence of newly released water particles
from xr and the returned water particles from the area
outside x=0.3L.
The age frequency distribution function shows a more
complex composition of mean water age in the region far
away from xr (i.e., x=0.7L) (Fig.1c, red line) than that in
the region around xr (i.e., x=0.3L, Fig.1b, red line). Thereis one major peak of frequency at approximately 10 d in
the age frequency distribution function at x=0.7L (Fig.1c,
red line; Fig.1d). This peak corresponds to the water particles that directly spread into this area from xr. However,the frequency at approximately 5–20 d in the age frequency distribution function at x=0.7L is not muchsmaller than that at approximately 10 d (Fig.1c, red line).
As a result of coexistence of such water particles, themean water age at x=0.7L is about 18 d (Fig.1a).
Fig.1e is the extension of results at x=0.3L and x=0.7L
to the whole model domain (i.e., x>0.25L). At any location x in the model domain, there are both water particles
with small age and water particles with large age. The
mean water age at x is a result of coexistence of all thewater particles at x (Fig.1a). Similar to the mean waterage, the age corresponds to the max. frequency in the age
frequency distribution function increases as a parabolic
function to the distance away from the xr (Fig.1d).
In the above experiments, we also changed the valuesof K but the agreement of the mean water age between the
CART and PTM was kept. Apparently, the agreement
between two methods is independent of K.

3.1.2 Spatially varied diffusion coefficient
We used a spatially varied diffusion coefficient controlled by Eq. (12) (Fig.2a):K ( x)  K 0  A cos(2πx / L) , x[0, L] ,(12)

where, K0=20 m2 s−1, A=15 m2 s−1.
According to Eq. (8), the displacement of a particle
contains ∆xDif (Fig. 2b, black line) and ∆xPse (Fig.2b, redline). ∆xPse is smaller than ∆xDif in this case (Fig.2b). ∆xPse
is positive at x>0.5L while negative at x<0.5L (Fig.2b, redline). Therefore, ∆xPse helps the particle move towards to50

WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

two ends of the channel.
According to Eq. (12), the minimum diffusion coefficient occurs at x=0.5L (Fig.2a). Fig.3a shows that themean water age by the PTM with ∆xPse is almost the sameas that by the PTM without ∆xPse when the releasing point
is 0.5L (xr=0.5L). Both of them agree well with the meanwater age by the CART. The mean water age is zero at xr
and increases as a parabolic function to the distance awayfrom xr. However, if the releasing point changes to 0.25L(xr=0.25L), the mean water age by the CART agrees wellwith that by the PTM with ∆xPse, but is much smaller thanthat by the PTM without ∆xPse at x>0.25L (Fig.3c). This isconsistent with the results reported by Visser (1997) that
the PTM with only ∆xDif causes the particles to gather inlow diffusion regions. As a result, it is difficult for theparticles to leave the low diffusion region (x=0.5L) (Fig.3d,
blue line), and the mean water age at x>0.25L becomesmuch longer (Fig.3c, blue line) than that calculated by the
CART. As ∆xPse is considered in the PTM, it helps thewater particles leave the low diffusion region and movetowards x=0 or x=L (Fig.3d, red line).
It must be noted that in the case of xr=0.5L, the PTM
without ∆xPse also gather water particles in the low diffusion region (x=0.5L) (Fig.3b, blue line). However, in thiscase ∆xPse is significantly smaller than ∆xDif (Fig.2b) and
it is ∆xDif that determines the displacement of water particles. In addition, since x=0.5L is also the releasing pointof water particles, the age there is always set to be zeroaccording to the boundary condition. Therefore, the meanwater ages by the PTM with and without ∆xPse are almost
the same (Fig.3a).
In the case of xr=0.25L, one major peak at approximately 5 d can be found in the age frequency distributionfunction at x=0.5L (Figs.4a, b). This peak indicates thatwhether the PTM contains ∆xPse or not, a certain numberof water particles spend approximately 5 d spreading into
this area from xr. However, the frequency at approximately 5 d in the age frequency distribution function at
x=0.5L by the PTM without ∆xPse is smaller than that by
the PTM with ∆xPse (Fig.4a). On the other hand, the frequency at longer than 30 d in the age frequency distribution function at x=0.5L by the PTM without ∆xPse is larger
than that by the PTM with ∆xPse (Fig.4a). This indicatesagain that in the calculation of PTM without ∆xPse, compared with the calculation of PTM with ∆xPse, it is difficult for the water particles to leave the low diffusion region (x=0.5L) (Fig.3d, blue line) with a longer mean water age at x>0.25L (Fig.3c).

Fig.1 The diffusion coefficient is constant and uniform (20 m2 s−1) and water particle is released at 0.25L (xr=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., M(τ, x) in Eq. (9)) and age frequency
distribution function (red line, i.e., φ(τ, x) in Eq. (9)) at x=0.3L. The age range (τ) is limited to 60 d since the frequencyof particles with age longer than 60 d is too small to be identified. (c) The same as Fig.1(b), but for x=0.7L. (d) The age
corresponds to the maximum of frequency distribution function at x>0.25L. (e) The age frequency distribution function
φ(τ, x) (unit: (0.25 d)−1) at x>0.25L. The age range (τ) is limited to 20 d.

WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

51

Fig.2 (a) The distribution of spatially varied diffusion coefficient by Eq. (12). (b) Based on the diffusion coefficient shownin Fig.2(a), ∆xDif (black line) and ∆xPse (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), ∆xAdv calculated by Eq. (8). See Section 2.3 for the definitions of∆xDif, ∆xPse, and ∆xAdv.

Fig.3 The spatially varied diffusion coefficient is controlled by Eq. (12). (a) Water particle is released at 0.5L (xr=0.5L).
Comparison of mean water ages by CART (black line), PTM with ∆xPse (red line), and PTM without ∆xPse (blue line). (b)Water particle is released at 0.5L (xr=0.5L). Comparison of R (defined by Eq. (10)) by CART (black line), PTM with ∆xPse(red line), and PTM without ∆xPse (blue line). (c) The same as Fig.3(a), but for xr=0.25L. (d) The same as Fig.3(b), but for
xr=0.25L.

Figs.4c and 4d are the extension of results at x=0.5L to
the whole model domain (i.e., x>0.25L) by the PTM withand without ∆xPse, respectively. At any location x, thefrequency at small age by the PTM without ∆xPse issmaller than that by the PTM with ∆xPse (Figs.4c and 4d),while the frequency at large age by the PTM without∆xPse is larger than that by the PTM with ∆xPse (Fig.4c,
Fig.4d). As a result, the mean water age by the PTM
without ∆xPse is significantly larger than that by the PTM

with ∆xPse (Fig.3c). Similar to mean water age, the ages
corresponding to the max. frequencies by the PTM withand without ∆xPse both increase as a parabolic function to
the distance away from the xr (Fig.4b). The age corresponding to the max. frequency by the PTM without ∆xPse
is slightly longer than that by the PTM with ∆xPse(Fig.4b).
In summary, in the case of spatially varied diffusion
coefficient, the PTM should include ∆xPse.

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WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

Fig.4 The spatially varied diffusion coefficient is controlled by Eq. (12) and water particle is released at 0.25L (xr=0.25L).
(a) The age frequency distribution function φ(τ, x) at x=0.5L by PTM with ∆xPse (red line) and by PTM without ∆xPse (blueline). (b) The age corresponds to the maximum of frequency distribution function at x>0.25L by PTM with ∆xPse (red line)
and by PTM without ∆xPse (blue line). (c) The age frequency distribution function φ(τ, x) (unit: (0.25 d)−1) for x>0.25L by
PTM with ∆xPse. (d) The same as Fig.4(c), but for PTM without ∆xPse. The age range (τ) in Figs.4(a), 4(c), and 4(d) is limited to 70 d.

3.2 Advection
3.2.1 Constant and uniform velocity
In the case of constant and uniform velocity (u) without
diffusion, the analytical solution for mean water age is x/u,
where x is the distance away from xr. Assuming u=0.005
m s−1 and x=0.25L, we show the analytical solution in
Fig.5a, in which the mean water age is zero at xr and increases linearly with the distance away from the xr. The
mean water age obtained by the CART and PTM agrees
well with the analytical solution (Fig.5a). From the age
frequency distribution function, each particle’s age equals
to the mean age at any location x in the model domain.
3.2.2 Temporally varied velocity
In this case, we assume a temporally varied velocity
given by Eq. (13) (Fig.2c):u (t )  u0  B cos(2πt / T ) ,(13)

where u0=0.005 m s−1, B=0.01 m s−1, T=86400 s.
According to Eq. (8), in the case of temporally varied
velocity without diffusion, the displacement of a particle
contains only ∆xAdv (Fig.2d). Again, we released particles
at xr=0.25L.
Fig.5b shows that the mean water age by the PTM and
CART both are zero at xr at t=105.5 d. The mean water
age by the PTM increases linearly with the distance awayfrom the xr with a small fluctuation. The mean water age
by the CART also increases linearly with the distanceaway from the xr but with little fluctuation. The PTM

deals with each particle and is capable of considering theinternal information (e.g., uneven age of particles) inside
a grid. The exchange of particles between neighboring
grids can be well described in the PTM. However, theuniformity of age of particles within each grid cannot be
considered in Eqs. (5)–(6) for CART.
At t=105.5 d, there is one major peak of frequency at
approximately 1.7 d in the age frequency distributionfunction for the area around xr (i.e., x=0.3L) (Fig.5c, redline; Fig.5e). This peak corresponds to the young waterparticles that spread quickly into this area from xr. In addition to these young water particles, we can also identify
the presence of old water particles that have an age longer
than 2.3 d (Fig.5c, red line), indicating that the water particles can return to x=0.3L from the area outside x=0.3L
because of the negative velocity (Fig.2c). As the case ofdiffusion, the mean water age (about 2 d) at x=0.3L(Fig.5b) is a result of coexistence of newly released waterparticles from xr and the returned water particles from thearea outside x=0.3L.
The age frequency distribution function shows a more
complex composition of mean water age in the region far
away from xr (i.e., x=0.7L) (Fig.5d, red line) than that in
the region around xr (i.e., x=0.3L, Fig.5c, red line). For
instance, there are two major peaks of frequency at approximately 20.6 d and 20.9 d in the age frequency distribution (Fig.5d, red line). As a result of coexistence of thewater particles with different ages, the mean water age at
x=0.7L is about 21 d (Fig.5b).
Fig.5f is the extension of results at x=0.3L and x=0.7L
to the whole model domain (i.e., x>0.25L). At any location x in the model domain, there are water particles with

WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

both small and large ages. The mean water age at x is a
result of coexistence of all the water particles at x (Fig.5b).
Compared with the age frequency distribution function
under diffusion (Figs.1e, 4c, and 4d), the age of frequency
distribution function under advection (Fig.5f) presents asmaller range. Similar to the mean water age, the age cor-53

responding to the max. frequency in the age frequency
distribution function increases linearly with the distanceaway from the xr with a very small fluctuation (Fig.5e).
In summary, in the case of temporally varied velocity,
the mean water age by the PTM and CART agrees wellwith each other.

Fig.5 Water particle is released at 0.25L (xr=0.25L). (a) The analytical solution for mean water age (x/u, where x is the distance away from xr) with a constant and uniform velocity (0.005 m s−1). The mean water ages by CART and PTM overlap
with the analytical solution. (b) Comparison of mean water ages by CART (black line) and PTM (red line) with temporally
varied velocity controlled by Eq. (13). (c) Total number (black line, M(τ, x)) in Eq. (9) of the particles with ages less thanor equal to an age (τ) at x=0.3L; age frequency distribution function (red line, φ(τ, x)) in Eq. (9) at x=0.3L. The age range (τ)
is from 1.5 to 3 d. (d) The same as Fig.5(c), but for x=0.7L. The age range (τ) is from 20.4 to 21.4 d. (e) The age corresponds to the maximum of frequency distribution function at x>0.25L. (f) The age frequency distribution function φ(τ, x)(unit: (0.02 d)−1) at x>0.25L. The age range (τ) is limited to 35 d. Same as Fig.5(b), the temporally varied velocity is controlled by Eq. (13) in Figs.5(c)–5(f). Figs. 5(b)–5(f) are at time t=105.5 d.

4 Discussion
4.1 The Disappearance of the Mean Water Age Difference Between the CART and PTM Without∆xPse in the Case of the Spatially Varied Diffusion
Coefficient
The several experiments we discussed above show thatthe mean water age results by the CART and PTM generally agree well with each other except for that in the caseof spatially varied diffusion coefficient (Section 3.1.2). In
that case, if the water particle was not released at theplace with weakest diffusion, the mean water age by the
CART is much smaller than that by the PTM without∆xPse (Fig.3c). The cause for this inconsistence is because

the PTM without ∆xPse collects water particles in low diffusion region (Fig.3d).
From Eq. (12), the average magnitude of gradient of
spatially varied diffusion coefficient (i.e., |∂K/∂x|) is about
3×10−3 m s−1 at x>0.25L. In order to calculate the averageof |∂K/∂x| at x>0.25L, we first calculate the magnitude of
gradient of spatially varied diffusion coefficient (|∂K/∂x|)
at each location x at x>0.25L. The average of |∂K/∂x| atx>0.25L is the arithmetic average of all the |∂K/∂x| atx>0.25L.
It is of interests to know under what circumstances thephenomenon that the mean water age by the CART issmaller than that by the PTM without ∆xPse at x>0.25L(Fig.3c) will vanish along with the reduction in |∂K/∂x|.
We used a spatially varied diffusion coefficient given

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WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

by Eq. (14) to examine this problem.
K ( x)  K 0  A cos(2πx / L) , x  [0, L] ,(14)

By changing value of A in Eq. (14) to 10, 8, 6.5, 5.5,
and 5 m2 s−1, we obtained the average of |∂K/∂x| as 2×10−3,
1.6×10−3, 1.3×10−3, 1.1×10−3, and 1×10−3 m s−1 at x>0.25L,
respectively.
Again, we assumed xr=0.25L. When |∂K/∂x| decreases,
the mean water age difference between the CART andPTM without ∆xPse at x>0.25L decreases gradually (Fig.6).
When the average of |∂K/∂x| is 1×10−3 m s−1, the meanwater age by the CART is almost the same as that by thePTM without ∆xPse (Fig.6e). In this case, no matter the
PTM includes ∆xPse or not, there is one major peak offrequency at about 5 d in the age frequency distributionfunction at x=0.5L (Fig.7a), being the same as in the casewhere the average of |∂K/∂x| is 3×10−3 m s−1 (Fig.4a). The
frequency at about 5 d and longer than 30 d by the PTM
without ∆xPse at x=0.5L is almost the same as that by thePTM with ∆xPse (Fig.7a). In the whole domain, the age
corresponding to the maximum frequency (Fig.7b) andthe age frequency distribution function (Figs.7c, d) by thePTM without ∆xPse is almost the same as that by the PTM
with ∆xPse. These features are not found in the results
when the average of |∂K/∂x| is 3×10−3 m s−1 (Fig.4). Therefore, it is likely that the problem that the mean water age

by the CART is smaller than that by the PTM without∆xPse at x>0.25L will vanish, if the magnitude of gradientof spatially varied diffusion coefficient decreases to acertain value (i.e., the average of |∂K/∂x| not larger than
1×10−3 m s−1).
On the other hand, advection and diffusion coexist in
the realistic ocean. It is therefore necessary to examine
the impact of velocity on the PTM without ∆xPse in spatially varied diffusion domain. In the next experiments,
besides the diffusion coefficient given by Eq. (12), a constant and uniform velocity is added at x≥0.25L. In Figs.8a,8b, 8c, and 8d, the velocity is set to 0.0015, 0.005, 0.015,
and 0.05 m s−1, respectively. Again, we assumed xr=0.25L.
According to Eq. (8), the displacement of a particle
contains ∆xDif, ∆xPse, and ∆xAdv at x≥0.25L. Here, to show
the relative importance of ∆xPse and ∆xAdv, a parameter R*
is defined in Eq. (15):
R* 

xPse
K

.
xAdv
ux

(15)

The average of R* equals to 2, 0.67, 0.2, and 0.067 in
Figs.8a–8d, respectively. In order to calculate the averageof R* for x>0.25L, we first calculate R* at each location x
for x>0.25L. The average of R* for x>0.25L is the arithmetic average of all the R* for x>0.25L.

Fig.6 Water particle is released at 0.25L (xr=0.25L). (a) Comparison of mean water ages by CART (black line), PTM
with ∆xPse (red line), and PTM without ∆xPse (blue line) with spatially varied diffusion coefficient controlled by Eq. (14)when A is set to 10 m2 s−1. (b)–(e) The same as Fig.6(a), but for spatially varied diffusion coefficient controlled by Eq.
(14) when A is set to 8, 6.5, 5.5, and 5 m2 s−1, respectively.

WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

55

Fig.7 The same as Fig.4, but for spatially varied diffusion coefficient controlled by Eq. (14) when A is set to 5 m2 s−1.
Water particle is released at 0.25L (xr=0.25L).

Fig.8 Water particle is released at 0.25L (xr=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 controlled by Eq. (12) and a constant and uniform velocity (0.0015 m s−1) is added for x≥0.25L. (b)–(d) The same as
Fig.8(a), but the velocities are set to 0.005, 0.015, and 0.05 m s−1, respectively.

For R* values, the mean water age by the CART agrees
well with those calculated by the PTM with ∆xPse (Fig.8,
red and black lines). Regarding the PTM without ∆xPse,
the agreement is not kept for some values of R*. For instance, the mean water age by the CART is much shorter
than that by the PTM without ∆xPse when the average of
R* is 2 (Fig.8a). In this case, advection is too weak tocover up the mean water age difference caused by spatially varied diffusion coefficient (Fig.3c). In the calculation of the PTM without ∆xPse, it is difficult for water particles to leave the low diffusion region (x=0.5L) andtherefore a lot of water particles gather there (Fig.3d).
In the case with both advection and diffusion, even as

the average of R* decreases to 0.67 (Fig.8b) and 0.2(Fig.8c), the mean water age by the CART is still shorter
than that by the PTM without ∆xPse. However, comparedwith Fig.8a, the mean water age difference between the
CART and PTM without ∆xPse greatly decreases. The
mean water age by the CART agrees well with that by thePTM without ∆xPse when the average of R* is 0.067(Fig.8d). In this case, advection is strong enough to coverup the age difference caused by spatially varied diffusion
coefficient. Therefore, if the velocity increases to a certain extent (i.e., the average of R* not larger than 0.067),
the phenomenon that the mean water age by the CART isshorter than that by the PTM without ∆xPse at x>0.25L

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WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

will vanish.

4.2 Application to the Realistic Ocean
In this section, we take the Bohai Sea as an example tostudy the application of CART and PTM to a realistic
ocean. The Bohai Sea is a semi-enclosed water body withan average depth of 18 m. It is divided into 5 subregions,
namely Laizhou Bay, Bohai Bay, Liaodong Bay, the central Basin, and Bohai Strait. The major rivers that flow
into the Bohai Sea include the Yellow River, the Haihe
River, the Luanhe River, and the Liaohe River (Fig.9).

Fig.9 Bathymetry of the Bohai Sea (unit: m). The Bohai
Sea is divided into 5 subregions: Laizhou Bay (LZB),
Bohai Bay (BHB), Liaodong Bay (LDB), the central basin (CB), and Bohai Strait (BS). The black dots represent
the locations of the river mouths of the Yellow River, theHaihe River, the Luanhe River, and the Liaohe River,respectively. Line AB denotes the section along which
vertical diffusion coefficient is given in Fig.10b.

The diffusion coefficient and velocity used here were
calculated by the model validated by Wang et al. (2008).
The horizontal resolution was 1/18 degree in both thezonal and meridional directions. In the vertical direction,
21 sigma levels were distributed (0.000, −0.002, −0.004,−0.010, −0.020, −0.040, −0.060, −0.080, −0.100, −0.120,−0.140, −0.170, −0.200, −0.300, −0.400, −0.500, −0.650,−0.800, −0.900, −0.950, and −1.000). The spatial variability of horizontal diffusion coefficient (λ1), the ratio of
the spatial variability of horizontal diffusion coefficient to
the horizontal velocity (λ2), the spatial variability of vertical diffusion coefficient (λ3), and the ratio of the spatial
variability of vertical diffusion coefficient to the verticalvelocity (λ4) are calculated by Eqs. (16)–(19), respectively:
K

2 K 

2
1   H    H  , x   y 
2



(16)



2

K
 K

2   H u    H v  ,x  y


(17)

3 

KV
,
z

(18)

4 

KV
w.
z

(19)

We interpolated KV at equal distance in the verticaldirection before calculating λ3 and λ4 by Eq. (18) and Eq.
(19).
The annual KH for the surface layer of the Bohai Sea(Fig.10a) shows that the KH is higher in the coastal area
(i.e., estuaries, >40 m2 s−1) than in the offshore area (i.e.,the central basin, <20 m2 s−1). The distribution of KH(Fig.10a) induces a high λ1 in the coastal area (i.e., estuaries, >0.003 m s−1) in the surface layer of Bohai Sea(Fig.10c). λ1 is small in the offshore area (i.e., the central
basin, about 0.002 m s−1) and it is even less than 0.001 m s−1
in some areas (Fig.10c). Apparently, the magnitude of gradient of spatially varied horizontal diffusion coefficient issmaller in the offshore area than in the coastal area. According to the analysis for the spatially varied diffusion
coefficient in Section 3.1.2 and Section 4.1, the ∆xPse
cannot be neglected in the PTM to calculate mean water
age when the magnitude of gradient of spatially varied
diffusion coefficient is larger than 0.001 m s−1. The λ2 forthe surface layer of the Bohai Sea (Fig.10e) shows that λ2
is larger in the coastal area (i.e., estuaries, >0.2) than in
the offshore area (i.e., the central basin, <0.2). This indicates that the impact of horizontal advection comparedwith horizontal diffusion is stronger in the offshore area
than in the coastal area. According to the analysis for the
spatially varied diffusion coefficient with constant anduniform velocity in Section 4.1, ∆xPse should be considered in the PTM when R* (indicating the effect of diffusion coefficient and velocity in one-dimensional domain)
is not less than 0.2. If we consider the effects of bothhorizontal diffusion coefficient and horizontal velocity inthe Bohai Sea, the PTM should include ∆xPse at least forthe coastal area (i.e., estuaries) of the Bohai Sea in thehorizontal direction.
The annual KV along transect AB in the Bohai Sea(Fig.10b, location being shown in Fig.9) shows that theKV is smaller for the surface layer (i.e., shallower than 5 m)than in the middle and bottom layers (i.e., deeper than 10m). The distribution of KV (Fig.10b) induces a higher λ3 inthe surface layer (i.e., shallower than 5 m, >0.003 m s−1)
than in the middle and bottom layers (i.e., deeper than 10m, about 0.002 m s−1) (Fig.10d). In some areas, λ3 is evenless than 0.001 m s−1 (Fig.10d). This indicates that themagnitude of gradient of spatially varied vertical diffusion coefficient is smaller in the middle and bottom area
than in the surface area. According to the analysis for the
spatially varied diffusion coefficient in Section 3.1.2 andSection 4.1, the ∆xPse cannot be neglected in the PTM tocalculate mean water age when the magnitude of gradientof spatially varied diffusion coefficient is larger than
0.001 m s−1. Fig.10f shows that the λ4 is very large for thewhole Bohai Sea (>102) because the vertical velocity isextremely small, indicating that the impact of verticalWANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

advection compared with vertical diffusion is very small
in the whole Bohai Sea. Hence, it is necessary for the
PTM to consider ∆xPse in the vertical direction for the57

whole Bohai Sea.
In summary, the PTM must include ∆xPse for the realistic shallow waters (e.g., the Bohai Sea).

Fig.10 (a) The annual horizontal diffusion coefficient for the surface layer (1 m) of the Bohai Sea. The contour interval is
10 m2 s−1. (b) The annual vertical diffusion coefficient along transect AB (location is shown in Fig.9). The contour interval
is 0.005 m2 s−1. (c) Based on the horizontal diffusion coefficient shown in Fig.10(a), the spatial variability of horizontal
diffusion coefficient (λ1) calculated by Eq. (16). The contour interval is 0.001 m s−1. (d) Based on the vertical diffusion coefficient shown in Fig.10(b), the spatial variability of vertical diffusion coefficient (λ3) is calculated by Eq. (18). The contour interval is 0.001 m s−1. (e) Based on the horizontal diffusion coefficient shown in Fig.10(a), the ratio of the spatial
variability of horizontal diffusion coefficient to the horizontal velocity (λ2) is calculated by Eq. (17). The contour interval
is 0.1. (f) Based on the vertical diffusion coefficient shown in Fig.10(b), the ratio of the spatial variability of vertical diffusion coefficient to the vertical velocity (λ4) is calculated by Eq. (19). The contour interval is 0.5. Colors represent thecommon logarithm of λ4.

5 SummaryThe difference in mean water age given by the CART
and PTM was studied in an idealized one-dimensional
computation domain. The model results by the two
methods are consistent with each other in the case witheither spatially uniform flow field or spatially uniform
diffusion coefficient. In the case with a spatially varied
diffusion coefficient, the mean water ages given by the

CART and PTM with ∆xPse agree well with each other. If
the water particle is released where the diffusion is weakest, the mean water ages given by the CART and PTM
without ∆xPse also agree well with each other. If the waterparticle is released at other places, the mean water age by
the CART is much shorter than that by the PTM without∆xPse. If the magnitude of gradient of spatially varied diffusion coefficient decreases to a certain extent, this difference decreases. The difference also decreases alongwith the increasing of velocity. As a general conclusion,

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WANG et al. / J. Ocean Univ. China (Oceanic and Coastal Sea Research) 2015 14: 47-58

we recommend that the PTM should include the pseudodisplacement caused by the spatial variation in the horizontal and vertical diffusion in a realistic sea area (suchas Bohai Sea), especially in the place where the diffusion
coefficient varies greatly in space.

Acknowledgements
This work was funded by the National Natural Science
Foundation of China (Nos. 41176007 and 40706007). It
was carried out while H. Wang was visiting the Center for
Marine Environmental Studies at Ehime University, Japan.
She thanks the China Scholarship Council (CSC) for
supporting her stay in Japan.

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