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Partial ages: diagnosing transport processes by means of multiple clocks

TL;DR: In this paper, partial ages are defined as the time spent in the different subregions of the path followed by the tracer parcel, and they can be computed in an Eulerian framework in much the same way as the usual age by extending the Constituent oriented Age and Residence Time theory.
Abstract: The concept of age is widely used to quantify the transport rate of tracers - or pollutants - in the environment. The age focuses only on the time taken to reach a given location and disregards other aspects of the path followed by the tracer parcel. To keep track of the subregions visited by the tracer parcel along this path, partial ages are defined as the time spent in the different subregions. Partial ages can be computed in an Eulerian framework in much the same way as the usual age by extending the Constituent oriented Age and Residence Time theory (CART, www.?climate.?be/?CART). In addition to the derivation of theoretical results and properties of partial ages, applications to a 1D model with lateral/transient storage, to the 1D advection-diffusion equation and to the diagnosis of the ventilation of the deep ocean are provided. They demonstrate the versatility of the concept of partial age and the potential new insights that can be gained with it.

Summary (3 min read)

1 Introduction

  • One of the main motivations for the study of geophysical flows — in the deep ocean, coastal regions, or inland waters — is the search for an improved understanding of the transport routes and transport rates of dissolved and suspended substances in the environment.
  • Basically, the age is a Lagrangian concept.
  • In numerical models, different tracers and ages can be defined to focus on specific aspects of the dynamics.

2 The concept of partial age

  • As described in Section 1, the age of a particle is the total time elapsed since the birth of this particle, which generally coincides with the time at which the particle entered the domain of interest.
  • This explains why the concept of age can be used to quantify transport rates.
  • By doing this, much information is therefore lost about the trajectory of the particle.
  • Assuming that the release of the particles in the coastal waters is considered as the birth of these particles, the ages of the two particles at time t are identical and given by the time t − t0 elapsed since they entered the domain of interest through the river outlet.
  • The partial ages account for the time spent in the different subdomains ωi and provide therefore a decomposition of the the time spent in ω, i.e., the age.

3 Eulerian equations for partial ages

  • Partial ages can be easily computed in an Eulerian framework by a straightforward extension of the procedure defined in the Constituent-oriented Age and Residence Time theory (CART) (Deleersnijder et al. 2001).
  • To understand the origin of Eq. 7 and the modification required to describe partial ages, it is important to realize that the particles located at the same location at a given time have different histories and, hence different ages.
  • For the partial age concentration distribution function ci , ageing should happen only when the tracer is located in the control domain ωi .
  • Using the linearity of the advection-diffusion operator L, the differential Eq. 8 for the age concentration α can be recovered by summing the n Eq. 17 for the partial age concentrations.
  • This means that the two scenarios cannot be differentiated by their partial age distributions.

4 One-dimensional model with lateral storage

  • If the cross-sectional areas and the different parameters are assumed to be constant and uniform, the velocity U itself is a constant and the system evolves toward a steady solution characterized by a uniform concentration field.
  • In accordance with the additivity property of partial ages (4), the mean ages can be split into their partial age components as am = amm + ams (29) and as = asm + ass (30).
  • The exchange process can then be described in a Lagrangian way by considering that any particle located in the main channel has a given probability pm to move to the storage zone during a given time interval dt while a particle located in the storage zone has a probability ps of taking the reverse course in the same time interval.
  • It is quite remarkable that the concept of partial age helps to better understand the net downstream rate of the material and the interaction between the longitudinal advection in the main channel and the diffusion like exchange with the storage zone.

5 Partial ages in a 1D advection-diffusion model

  • In other words, tracer parcels, whether located upstream or downstream from the point source, show exactly the same mean age if they are located at the same distance from the source.
  • Unlike the age itself, partial ages are not symmetric with respect to the origin (or at least not in the same sense as the mean age, see below).
  • The partial age a2+ associated with this second subpath is the usual (total mean) age a0 of a tracer or a water parcel when the age is defined as the time elapsed since touching the origin for the last time.
  • This property, which generalizes (52), is also valid at any particular time, not only at steady state.

6 Diagnosis of the ventilation of the deep ocean

  • In the World Ocean, it is customary to have recourse to the concept of age to quantify the rate at which deep waters are replaced with water originating from the surface layer (e.g., England 1995; Holzer and Hall 2000; Primeau 2005).
  • By considering multiple subdomains ωi , the corresponding partial ages provide new insights into the paths of the water parcels from the surface to the ocean’s interior by keeping track of the time spent by water parcels in the different subdomains.
  • As the surface is approached again through the upwelling region ω3, the age decreases toward zero as a result of the diffusive transport of young water parcels from the surface into the ocean’s interior, i.e., against the mean flow.
  • Detailed descriptions of the model can be found in the above-mentioned references.
  • In Fig. 9b, the partial ages are normalized by the average age a,j of the water in the corresponding sub-domain.

7 Conclusion

  • The concept of age, defined as the time elapsed since a given origin, is widely used to diagnose transport processes in the environment.
  • The concept of partial age provides a valuable extension of the concept of age.
  • While the concepts and computation procedure look similar to the ones used for the usual age, partial ages provide independent diagnostics, which can be used to gain deeper insights into transport routes and transport rates.
  • The analysis of the 1D advection-diffusion equation showed that the particles found at some location downstream a source have spent the same amount of time upstream the source than the time spent downstream by the particles found upstream the source at the mirror location.
  • Acknowledgments Éric Deleersnijder and Éric J.M. Delhez are both honorary research associates with the Belgian Fund for Scientific Research (F.R.S.-FNRS).

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Ocean Dynamics (2016) 66:367–386
DOI 10.1007/s10236-016-0922-6
Partial ages: diagnosing transport processes by means
of multiple clocks
Anne Mouchet
1,2
· Fabien Cornaton
3
·
´
Eric Deleersnijder
4,5
·
´
Eric J. M. Delhez
6
Received: 4 September 2015 / Accepted: 12 January 2016 / Published online: 9 February 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract The concept of age is widely used to quantify the
transport rate of tracers - or pollutants - in the environment.
The age focuses only on the time taken to reach a given
location and disregards other aspects of the path followed
by the tracer parcel. To keep track of the subregions visited
by the tracer parcel along this path, partial ages are defined
as the time spent in the different subregions. Partial ages can
Responsible Editor: Tal Ezer
´
Eric J. M. Delhez
e.delhez@ulg.ac.be
1
Laboratoire des Sciences du Climat et de l’Environnement
(LSCE), CEA/CNRS/UVSQ/IPSL, B
ˆ
at. 701 Orme des
Merisiers, 91191 Gif-sur-Yvette Cedex, France
2
University of Liege (ULg), Astrophysics, Geophysics
and Oceanography Department, Sart-Tilman B5c,
B-4000 Li
`
ege, Belgium
3
Groundwater Modelling Centre DHI-WASY GmbH,
Vo l m e r s t r a βe 8, 12489 Berlin, Germany
4
Universit
´
e catholique de Louvain (UCL), Institute
of Mechanics, Materials and Civil Engineering (IMMC)
& Earth and Life Institute (ELI), 4 Avenue G. Lema
ˆ
ıtre, 1348
Louvain-la-Neuve, Belgium
5
Delft University of Technology, Delft Institute of Applied
Mathematics (DIAM), Mekelweg 4, 2628CD Delft,
The Netherlands
6
University of Liege (ULg), Department of Aerospace &
Mechanical Engineering, Mathematical Modeling & Methods,
Sart-Tilman B37, 4000 Li
`
ege, Belgium
be computed in an Eulerian framework in much the same
way as the usual age by extending the Constituent oriented
Age and Residence Time theory (CART, www.climate.be/
CART). In addition to the derivation of theoretical results
and properties of partial ages, applications to a 1D model
with lateral/transient storage, to the 1D advection-diffusion
equation and to the diagnosis of the ventilation of the deep
ocean are provided. They demonstrate the versatility of the
concept of partial age and the potential new insights that can
be gained with it.
Keywords Age · Advection-diffusion · Tracer methods
1 Introduction
One of the main motivations for the study of geophysi-
cal flows in the deep ocean, coastal regions, or inland
waters is the search for an improved understanding of
the transport routes and transport rates of dissolved and
suspended substances in the environment. Many differ-
ent physical processes contribute to this transport over a
wide range of spatial scales and time scales, which makes
it difficult to describe the dynamics in terms of circu-
lation patterns or, even time dependent, velocity fields.
In this context, tracer methods (e.g., Thiele and Sarmiento
1990; England 1995; England and Maier-Reimer 2001;
Deleersnijder et al. 2001) are particularly useful and effi-
cient because they directly address the resulting integrated
effects of the different processes at stake on the transport
of contaminants, nutrients, and other substances. In numer-
ical models, specifically designed artificial tracers can also
be defined to isolate and quantify the effects of specific
processes. These advantages, together with the (apparent)
simplicity of the messages that they convey, explain why

368 Ocean Dynamics (2016) 66:367–386
tracer methods have been increasingly used to analyze
natural flows and unravel the underlying dynamics.
Tracer methods often proceed with the introduction of
characteristic timescales to quantify the transport rates of
tracers or of water itself. The seminal papers by Bolin and
Rodhe (1973), Zimmerman (1976), and Takeoka (1984)
provide clear definitions of many of such timescales includ-
ing the age, residence time, transit time, and turn-over
time.
The concept of age is particularly appealing because of
its clear definition (in contrast with other timescales that are
often defined in slightly different ways by different authors),
its versatility, and the ease with which it can be computed.
The age of a water/tracer particle is defined as the time
elapsed since a user-defined origin which is hence consid-
ered as the “birth” of the particle. In most applications, the
origin is chosen as the time at which the particle enters the
domain of interest or touches one of its boundaries. Accord-
ingly, the age at any interior point of the domain measures
the time taken by the water/tracer particle to reach that point
and represents therefore a valuable diagnostic to quantify
transport rates.
Basically, the age is a Lagrangian concept. Conceptually,
each particle can be considered to be equipped with a clock
that starts ticking at the instant of its birth or is reset when
a specific event takes place (e.g., when the particle hits the
surface of the ocean) and hence records the time elapsed
since this origin. The computation of the age is therefore
easily implemented in any Lagrangian model (e.g., Chen
2007; Liu et al. 2012; Villa et al. 2015)
Fortunately, the age can also be computed in an Eule-
rian framework if one takes into account the fact that the
water/tracer parcels to be considered according to the under-
lying continuum hypothesis are made of a collection of
water/tracer particles carrying different ages. The full Eule-
rian description of the age field requires the introduction of
a distribution of the ages of these particles. The appropri-
ate framework is therefore five-dimensional, i.e., space ×
time × age (Bolin and Rodhe 1973; Hall and Plumb 1994;
Delhez et al. 1999;Ginn1999; Deleersnijder et al. 2001;
Haine and Hall 2002). While special techniques can be
used to work in this five-dimensional space (Delhez and
Deleersnijder 2002; Cornaton 2012), this complexity is
avoided in most Eulerian studies by resorting to the steady
state hypothesis (e.g., Holzer and Hall 2000; Khatiwala et al.
2009) or by considering only the mean value of the ages of
the particles in a water parcel.
Implicitly following the second approach, Thiele and
Sarmiento (1990) and, later, England (1995) and Goode
(1996) proposed a differential equation for the mean age of
the water and used it to compute ventilation timescales in
the global ocean.
In the framework of their Constituent-oriented Age and
Residence Time theory (CART, www.climate.be/CART),
Delhez et al. (1999) and Deleersnijder et al. (2001) derived
a system of two equations allowing for the straightforward
computation of the mean age of every seawater constituent,
including substances involved in bio-geochemical reactions.
The first equation is just the basic transport equation for
the constituent, including appropriate reaction terms when
necessary. The second is a reaction-transport equation for
the so-called age concentration, which is the product of the
concentration of the constituent by its mean age. The age
concentration is an extensive variable; it satisfies a bud-
get equation that can be used to compute the mean age of
the constituent. This approach has been applied in a large
number of studies (e.g., Gourgue et al. 2007; Meier 2007;
Plus et al. 2009;
Bendtsen et al. 2009;deBryeetal.2012;
Bendtsen and Hansen 2013; Ren et al. 2014;DuandShen
2015).
The concept of age is a very flexible one. In numerical
models, different tracers and ages can be defined to focus
on specific aspects of the dynamics. For instance, in their
study of the dynamics of sediments in the Belgian coastal
zone, Mercier and Delhez (2007) define two different ages
providing a focus, respectively, on the rate of horizontal
transport of the suspended matter and on the time spent by
sediment flocs in the water column, i.e., on the frequency of
deposition-resuspension events. In de Brye et al. (2012), two
different tracers tagging the freshwater entering from the
upstream boundary and the water entering from the coastal
area are used to compute two water renewal timescales in
the Scheldt Estuary.
In the last two applications, different ages are associated
with different tracers to study different aspects of a single
system. The aim of this paper is to show that different ages
can also be attached to a single tracer to provide deeper
insights into the dynamics. In the following sections, the
concept of partial age will first be introduced and Eulerian
equations for partial ages will be derived. Then, applica-
tions to a 1D model with lateral/transient storage, to the
1D advection-diffusion equation and to the diagnosis of the
ventilation of the deep ocean will be considered.
2 The concept of partial age
As described in Section 1, the age of a particle is the total
time elapsed since the birth of this particle, which gener-
ally coincides with the time at which the particle entered
the domain of interest. Accordingly, the age a of a particle
that is located at x at time t measures the time taken by this
particle to reach x from the place where it was born at time
t a. This explains why the concept of age can be used to
quantify transport rates.

Ocean Dynamics (2016) 66:367–386 369
With the concept of age, the only information that is
available about the path of the particle from its initial loca-
tion to the observation point x is the total duration of the
journey. By doing this, much information is therefore lost
about the trajectory of the particle. In particular, all the
information about the different subregions visited by the
particle during its journey are lost. The concept of par-
tial age that we introduce in this paper aims at making a
better use of this information to diagnose transport routes
and rates.
To ease the introduction of the concept of partial age,
consider the idealized system schematized in Fig. 1 where
tracer particles are released by a river discharging into a
coastal region with different bays and sub-basins (e.g., Liu
et al. 2012). The figure shows the trajectories of two par-
ticles released at the same time t
0
and reaching the same
offshore location at the observation time t
. Assuming that
the release of the particles in the coastal waters is considered
as the birth of these particles, the ages of the two parti-
cles at time t
are identical and given by the time t
t
0
elapsed since they entered the domain of interest through
the river outlet. Obviously, the particles have different his-
tories, a fact that is not reflected by the common value of
their ages. In order to address the fact that different regions
are visited by the two particles, one introduces a partition
of the domain of interest ω into a set of (biologically, geo-
graphically and/or physically significant) nonoverlapping
subregions ω
i
, i ∈{1,...,n}, i.e.,
ω =
n
i=1
ω
i
, i = j : ω
i
ω
j
=∅ (1)
and defines the partial age a
i
of a particle with respect to a
subregion ω
i
as the time spent by the particle in this partic-
ular subregion (Liu et al. 2012; Deleersnijder et al. 2014).
Accordingly, each particle can be characterized by a set of
n partial ages a
1
, a
2
, ..., a
n
with respect to the n different
subregions. In particular, the partial ages, at time t
,ofthe
particle that immediately turns left after its release (Fig. 1)
are given by
a
1
= (t
1
t
0
)+(t
3
t
2
), a
2
= (t
2
t
1
), a
3
= (t
4
t
3
), a
4
= (t
t
4
)
(2)
For the second particle, the partial ages are
a
1
= (t
1
t
0
), a
2
= 0,a
3
= 0,a
4
= (t
t
1
) (3)
Obviously, these partial ages can be used to highlight the
different histories of the two particles; the additional pieces
of information available upon using partial ages can be used
to better describe and understand the dynamics of the sys-
tem under study. In particular, some knowledge of the paths
t
0
t
1
t
2
t
3
t
4
t
1
t
ω
1
ω
2
ω
3
ω
4
Fig. 1 Schematized picture of a river discharging into a coastal region
and of the paths of two particles visiting different subregions. The
domain is split into four non overlapping subdomains
followed by the particles to reach a given region can be used
to address connectivity issues.
From a Lagrangian point of view, the determination of
partial ages amounts to equipping each particle with n
different watches that are all set to zero at the birth of
the particle but with the ith clock ticking only when the
particle is in the ith subdomain ω
i
. If the different subdo-
mains form a partition of the domain of interest according
to Eq. 1, only one clock is ticking at a time (since the subdo-
mains do not overlap) and there is always one clock ticking
at any time (since the subdomains form a cover of ω). There-
fore, the usual age of a particle is simply the sum of its
partial ages, i.e.,
a =
n
i=1
a
i
(4)
The partial ages account for the time spent in the different
subdomains ω
i
and provide therefore a decomposition of the
the total time spent in ω, i.e., the (total) age. This additivity
property justifies the name “partial age” coined for a
1
, a
2
,
...,a
n
.
3 Eulerian equations for partial ages
Partial ages can be easily computed in an Eulerian frame-
work by a straightforward extension of the procedure
defined in the Constituent-oriented Age and Residence
Time theory (CART) (Deleersnijder et al. 2001).
CART relies on the solution of two coupled equations to
compute the mean age of a constituent. The first equation is
the usual transport equation for the constituent, i.e.,
∂C
∂t
=
L(C) (5)

370 Ocean Dynamics (2016) 66:367–386
where
L(·) ≡∇·
[
K ·∇(·) v(·)
]
(6)
is the linear advection-diffusion operator, v is the velocity
vector and K is the (supposedly symmetric and positive def-
inite) diffusivity tensor. The second equation is a reaction-
transport equation for the so-called age concentration α that
can be defined as the product
α(t, x) = C(t,x)a(t, x) (7)
of the concentration of the constituent by its mean age a.
It can be shown that the age concentration is an extensive
variable that characterizes the “age content” of a water par-
cel and satisfies the budget equation (Delhez et al. 1999;
Deleersnijder et al. 2001)
∂α
∂t
= C +
L) (8)
The mean age can thus be computed from Eq. 7 using the
solutions of the two coupled Eqs. 5 and 8 with appropriate
initial and boundary conditions.
Note that, while the above Eqs. 5 and 8 are expressed here
for a passive constituent that is simply transported by the
flow, the theory can easily cope with constituents involved
in chemical reactions or biological interactions through the
introduction of appropriate source/sink terms in Eqs. 5 and
8 (e.g., Delhez et al. 2004).
To understand the origin of Eq. 7 and the modification
required to describe partial ages, it is important to realize
that the particles located at the same location at a given
time have different histories and, hence different ages. To
account for this variety, CART relies on the concentration
distribution function c(t, x)to describe the distribution of
the concentration along the age dimension. The concentra-
tion distribution function is such that c(t, x)dτ measures
the concentration of the material with an age in the range
[τ,τ + ] so that the (usual) concentration and the mean
age are related to c through the equations
C(t,x) =
0
c(t, x,τ)dτ (9)
and
α(t, x) =
0
τc(t,x)dτ (10)
AccordingtoEq.7, the mean age appears therefore as
the normalized first-order moment of the age concentration
distribution.
In the same way, a partial age concentration distribution
function c
i
can be defined such that c
i
(t, x)dτ measures
the concentration of the material with a partial age a
i
in
the range [τ,τ + ]. The partial age concentration distri-
bution functions describe the distributions of the particles
present at the same time and location along the partial age
dimension so that
C(t,x) =
0
c
i
(t, x)dτ i ∈{1,...,n} (11)
Equations 9 and 11 show that c and the partial age con-
centrations c
i
each correspond to different decompositions
of the concentration C. When using the age concentration
c, the different tracer parcels contributing to the concentra-
tion C are differentiated by their age. By contrast, the same
tracer parcels are sorted out with respect to the time spent in
the subdomain ω
i
, i.e., their partial age, to build the partial
age concentration c
i
.
Similarly to Eqs. 7 and 10, mean partial ages can be
defined as the normalized first-order moments of the corre-
sponding distributions, i.e.,
a
i
(t, x) =
α
i
(t, x)
C(t,x)
(12)
where the so-called partial age concentrations α
i
are given
by
α
i
(t, x) =
0
τc
i
(t, x)dτ (13)
For a passive tracer, the concentration distribution func-
tion satisfies the differential equation (Delhez et al. 1999;
Deleersnijder et al. 2001)
∂c
∂t
+
∂c
∂τ
=
L(c) (14)
The second term of the left-hand side, the ageing term, can
be seen as an advection term acting at a unit velocity along
the age dimension; as time goes by, the material tends to get
older at the rate of one time unit per elapsed time unit.
A differential equation similar to Eq. 14 can also be eas-
ily written. In Eq. 14, the ageing term is active at all time,
whatever the location of the particles. For the partial age
concentration distribution function c
i
, ageing should hap-
pen only when the tracer is located in the control domain
ω
i
. Therefore, the differential equation for the partial age
concentration distribution reads
∂c
i
∂t
+ δ
ω
i
∂c
i
∂τ
=
L(c
i
) (15)
where
δ
ω
i
(x) =
1ifx ω
i
0ifx ω
i
(16)
is the characteristic function of the subdomain ω
i
.
Delhez et al. (1999) and Deleersnijder et al. (2001)
showed how the differential Eqs. 5 and 8 for the concen-
tration and for the age concentration can be recovered by
integrating (14) over the age dimension. The application

Ocean Dynamics (2016) 66:367–386 371
of the same procedure to Eq. 15 leads to the differential
equation for α
i
, i.e.,
∂α
i
∂t
= δ
ω
i
C + L
i
) (17)
With (5), this equation forms a set of two coupled partial
differential equations whose solutions can be used to com-
pute the partial age a
i
from Eq. 12. The determination of the
full set of partial ages a
1
, a
2
, ..., a
n
requires the resolution
of (5) together with the n different versions of Eq. 17 corre-
sponding to the n nonoverlapping subdomains ω
i
included
in the partition of the domain of interest.
Using the linearity of the advection-diffusion operator
L,
the differential Eq. 8 for the age concentration α can be
recovered by summing the n Eq. 17 for the partial age con-
centrations. Since the Eqs. 8 and 17 are subjected to the
same auxiliary conditions, their solutions are such that
α =
n
i=1
α
i
(18)
i.e., the partial age concentrations α
i
form a decomposition
of the age concentration. The additivity property (4)ofthe
partial ages naturally follows from the definitions Eqs. 10
and 13.
It should be realized that, unlike a and α, the concen-
tration distribution function c cannot be expressed as the
sum of the corresponding partial concentration distribution
functions c
i
, i.e.,
c =
n
i=1
c
i
(19)
The distributions of age and partial ages are independent
of each other. A simple discrete example can be easily
conceived to show that c and the c
i
convey different infor-
mation. Consider two particles P
(1)
and P
(2)
of a given
tracer and their partial ages (a
(1)
+
,a
(1)
) and (a
(2)
+
,a
(2)
) mea-
suring the time spent in two sub-domains ω
+
and ω
forming a partition of the whole model domain ω.Fromthe
additivity property of partial ages, the (total) ages of the two
particles are given by
a
)
= a
)
+
+ a
)
∈{1, 2} (20)
Two hypothetical scenarios for these two particles are envis-
aged in Table 1.
The distributions of each partial age for the set P
(1)
P
(2)
of the two particles are identical in the two scenarios; in
both
A and B, the mixtures made on the two particles have
exactly one particle with an age a
+
= 1 (arbitrary unit), one
particle with an age a
+
= 2, one particle with an age a
=
2, and one particle with an age a
= 3. This means that the
two scenarios cannot be differentiated by their partial age
distributions. The (total) age distributions do however differ
(while sharing the same mean value); in the first scenario,
the ages of the particles are equal to 3 and 5, while they take
the common value of 4 in scenario
B. Since identical distri-
butions of the partial ages can lead to different distributions
of the age, one concludes that, in general, the concentration
distribution function c cannot be expressed as a function of
the partial concentration distribution function c
i
.
4 One-dimensional model with lateral storage
As a first illustration of the use of the concept of partial
ages, consider a one-dimensional flow connected with a lat-
eral storage zone as schematized in Fig. 2. Such a model
is commonly used in subsurface flow settings where water
and tracers can be stored in low-permeability subregions,
like aquitards, or to describe the transport of pollutants in
rivers when water and materials can be trapped in stag-
nant water zones like pools, gravel beds, adjacent wetland
areas and other hyporheic zones (e.g., Runkel and Chapra
1993; Wagner and Harvey 1997; Kumar and Dalal 2010).
In the so-called transient storage models (TSMs) or dead
zone models (DZMs), the river is divided into the main flow
zone, where downstream advection occurs, and lateral stor-
age zones, which stagnant waters (e.g., Bencala and Walters
1983). The two zones are coupled by diffusion.
In a 1D approach, the transport of a tracer in such a
system can be described by the set of coupled equations
(integrated over the cross-section)
∂(A
m
C
m
)
∂t
=−
∂(QC
m
)
∂x
+ A
m
β(C
s
C
m
) (21)
∂(A
s
C
s
)
∂t
=−A
m
β(C
s
C
m
) (22)
where C
m
and C
s
denote the concentration in the main
channel and in the lateral storage zone, A
m
and A
s
are the
cross-sections of two zones, Q = A
m
U is the flow rate
Table 1 Hypothetical age and
partial ages (arbitrary units) of
two individual particles P
(1)
and P
(2)
and mean values of
their mixture P
(1)
P
(2)
in
two scenarios
Scenario A a
+
a
a Scenario B a
+
a
a
P
(1)
123P
(1)
224
P
(2)
235P
(2)
134
P
(1)
P
(2)
1.5 2.5 4 P
(1)
P
(2)
1.5 2.5 4

Citations
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Journal ArticleDOI
TL;DR: Lagrangian analysis is a powerful way to analyse the output of ocean circulation models and other ocean velocity data such as from altimetry as mentioned in this paper, where large sets of virtual particles are integrated within the 3D, time-evolving velocity fields.

309 citations


Cites methods from "Partial ages: diagnosing transport ..."

  • ...One method makes use of tracers, such as the multitude of age tracers described by Mouchet et al. (2016) and references therein....

    [...]

Journal ArticleDOI
29 Sep 2020-Water
TL;DR: The use of diagnostic timescales as simple tools for illuminating how aquatic ecosystems work, with a focus on coastal systems such as estuaries, lagoons, tidal rivers, reefs, deltas, gulfs, and continental shelves, is discussed in this paper.
Abstract: In this article, we describe the use of diagnostic timescales as simple tools for illuminating how aquatic ecosystems work, with a focus on coastal systems such as estuaries, lagoons, tidal rivers, reefs, deltas, gulfs, and continental shelves. Intending this as a tutorial as well as a review, we discuss relevant fundamental concepts (e.g., Lagrangian and Eulerian perspectives and methods, parcels, particles, and tracers), and describe many of the most commonly used diagnostic timescales and definitions. Citing field-based, model-based, and simple algebraic methods, we describe how physical timescales (e.g., residence time, flushing time, age, transit time) and biogeochemical timescales (e.g., for growth, decay, uptake, turnover, or consumption) are estimated and implemented (sometimes together) to illuminate coupled physical-biogeochemical systems. Multiple application examples are then provided to demonstrate how timescales have proven useful in simplifying, understanding, and modeling complex coastal aquatic systems. We discuss timescales from the perspective of “holism”, the degree of process richness incorporated into them, and the value of clarity in defining timescales used and in describing how they were estimated. Our objective is to provide context, new applications and methodological ideas and, for those new to timescale methods, a starting place for implementing them in their own work.

24 citations

Journal ArticleDOI
TL;DR: In this paper, partial residence times (PRTs) are defined as the amount of time a water parcel spends in different subregions until leaving the control region, which can provide detailed information and new insights into the water exchange process.
Abstract: Residence time (RT) is a diagnosis widely used to quantify the water exchange rate and mass transport timescale in semi-enclosed systems. The RT focuses on only the total time a water parcel spends in a system (i.e., control region). However, for a system that consists of several subregions (e.g., sub-bays or functional zones), the RT will not include information about the time spent in different subregions. To determine the RT compositions in different subregions, partial residence times (PRTs) are proposed and defined as the amount of time a water parcel spends in different subregions until leaving the control region. The equations for PRTs are derived using the adjoint method, which can quickly determine the variation of PRTs in time and space. To validate the PRT diagnostic equation and numerical model, a test is conducted in an idealized 1D channel with idealized tidal currents. The numerical results are in excellent agreement with the analytical solution. Finally, the PRT method is applied to tide-dominated Jiaozhou Bay. The PRTs in six functional subregions of Jiaozhou Bay are derived, and the detailed RT compositions of the different functional subregions are presented. The application indicates that PRTs could provide detailed information and new insights into the water exchange process.

17 citations

Journal ArticleDOI
TL;DR: In this article, the authors examined the water exchange between Chesapeake Bay and the adjacent coastal shelf, between different regions within the bay, as well as their relationships with river discharge, wind, and residence time.

12 citations

Journal ArticleDOI
TL;DR: In this paper, the authors combined two Lagrangian approaches and estimated a local diffusivity coefficient using the hydrodynamic model MARS 3D with a barycentric repositioning technique over a tidal period.
Abstract: Coastal waters are subject to great environmental and anthropogenic pressures. The diffusion and the transport of these waters are a key element for environmental, ecological and economic management. There are numerous indicators of hydrological characteristics based on theories of transport time scale. However, these indicators strongly depend on the geographical shape of the studied area and tend to give information after long integration time periods, generally on the order of weeks. Here, to qualify a coastal area’s dispersion more precisely, we combined two Lagrangian approaches and estimated a local diffusivity. This paper presents the numerical implementation and the results obtained over a tidally flushed, semi-enclosed water body located at mid-latitude. This new coefficient was estimated using the hydrodynamic model MARS 3D with a barycentric repositioning technique over a tidal period to ensure its reliability. We highlight the existing relationships between local diffusivity and both horizontal and vertical processes. Methodological aspects were analysed based on a reference case (number and distribution of particles, resolution, integration time period). The consistency and sensitivity of the coefficient were studied with different forcing conditions (hydrodynamical and meteorological regimes). In conclusion, our local diffusivity provides a new perspective for understanding the land–sea interface and coastal dispersion and holds potential for future studies of coastal marine ecosystems.

12 citations

References
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Journal ArticleDOI
TL;DR: In this article, a transient storage model was used to simulate solute transport in a very small (0.0125 m3s−1) mountain pool-and-riffle stream.
Abstract: The physical characteristics of mountain streams differ from the uniform and conceptually well- defined open channels for which the analysis of solute transport has been oriented in the past and is now well understood. These physical conditions significantly influence solute transport behavior, as demonstrated by a transient storage model simulation of solute transport in a very small (0.0125 m3s−1) mountain pool-and-riffle stream. The application is to a carefully controlled and intensively monitored chloride injection experiment. The data from the experiment are not explained by the standard convection-dispersion mechanisms alone. A transient storage model, which couples dead zones with the one-dimensional convection-dispersion equation, simulates the general characteristics of the solute transport behavior and a set of simulation parameters were determined that yield an adequate fit to the data. However, considerable uncertainty remains in determining physically realistic values of these parameters. The values of the simulation parameters used are compared to values used by other authors for other streams. The comparison supports, at least qualitatively, the determined parameter values.

795 citations


"Partial ages: diagnosing transport ..." refers background in this paper

  • ...In the so-called transient storage models (TSMs) or dead zone models (DZMs), the river is divided into the main flow zone, where downstream advection occurs, and lateral storage zones, which stagnant waters (e.g., Bencala and Walters 1983)....

    [...]

Journal ArticleDOI
19 Nov 2009-Nature
TL;DR: The results indicate that ocean uptake of anthropogenic CO2 has increased sharply since the 1950s, with a small decline in the rate of increase in the last few decades, and suggest that the terrestrial biosphere was a source of CO2 until the 1940s, subsequently turning into a sink.
Abstract: The release of fossil fuel CO(2) to the atmosphere by human activity has been implicated as the predominant cause of recent global climate change. The ocean plays a crucial role in mitigating the effects of this perturbation to the climate system, sequestering 20 to 35 per cent of anthropogenic CO(2) emissions. Although much progress has been made in recent years in understanding and quantifying this sink, considerable uncertainties remain as to the distribution of anthropogenic CO(2) in the ocean, its rate of uptake over the industrial era, and the relative roles of the ocean and terrestrial biosphere in anthropogenic CO(2) sequestration. Here we address these questions by presenting an observationally based reconstruction of the spatially resolved, time-dependent history of anthropogenic carbon in the ocean over the industrial era. Our approach is based on the recognition that the transport of tracers in the ocean can be described by a Green's function, which we estimate from tracer data using a maximum entropy deconvolution technique. Our results indicate that ocean uptake of anthropogenic CO(2) has increased sharply since the 1950s, with a small decline in the rate of increase in the last few decades. We estimate the inventory and uptake rate of anthropogenic CO(2) in 2008 at 140 +/- 25 Pg C and 2.3 +/- 0.6 Pg C yr(-1), respectively. We find that the Southern Ocean is the primary conduit by which this CO(2) enters the ocean (contributing over 40 per cent of the anthropogenic CO(2) inventory in the ocean in 2008). Our results also suggest that the terrestrial biosphere was a source of CO(2) until the 1940s, subsequently turning into a sink. Taken over the entire industrial period, and accounting for uncertainties, we estimate that the terrestrial biosphere has been anywhere from neutral to a net source of CO(2), contributing up to half as much CO(2) as has been taken up by the ocean over the same period.

522 citations


"Partial ages: diagnosing transport ..." refers background or methods in this paper

  • ...While special techniques can be used to work in this five-dimensional space (Delhez and Deleersnijder 2002; Cornaton 2012), this complexity is avoided in most Eulerian studies by resorting to the steady state hypothesis (e.g., Holzer and Hall 2000; Khatiwala et al. 2009) or by considering only the mean value of the ages of the particles in a water parcel....

    [...]

  • ...…space (Delhez and Deleersnijder 2002; Cornaton 2012), this complexity is avoided in most Eulerian studies by resorting to the steady state hypothesis (e.g., Holzer and Hall 2000; Khatiwala et al. 2009) or by considering only the mean value of the ages of the particles in a water parcel....

    [...]

  • ...In practice, the modeling results may, however, be strongly dependent on the thickness of the surface layer and other boundary conditions can be considered (e.g., Khatiwala et al. 2009; DeVries and Primeau 2010)....

    [...]

Journal ArticleDOI
01 Feb 1973-Tellus A
TL;DR: A brief review of the concepts age distribution, transit time distribution, turnover time, average age and average transit time (residence time) and their relations is given in this article.
Abstract: A brief review is given of the concepts age distribution, transit time distribution, turn-over time, average age and average transit time (residence time) and their relations The characteristics of natural reservoirs are discussed in terms of these concepts, and a classification is proposed based on whether the average age is larger, equal to or smaller than the average transit time Some examples illustrate the differences between these various cases DOI: 101111/j2153-34901973tb01594x

518 citations


"Partial ages: diagnosing transport ..." refers background in this paper

  • ...The seminal papers by Bolin and Rodhe (1973), Zimmerman (1976), and Takeoka (1984) provide clear definitions of many of such timescales including the age, residence time, transit time, and turn-over time....

    [...]

  • ...The appropriate framework is therefore five-dimensional, i.e., space × time × age (Bolin and Rodhe 1973; Hall and Plumb 1994; Delhez et al. 1999; Ginn 1999; Deleersnijder et al. 2001; Haine and Hall 2002)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors evaluated the reliability of the stream tracer approach to characterize hyporheic exchange in St. Kevin Gulch, a Rocky Mountain stream in Colorado contaminated by acid mine drainage.
Abstract: Stream water was locally recharged into shallow groundwater flow paths that returned to the stream (hyporheic exchange) in St. Kevin Gulch, a Rocky Mountain stream in Colorado contaminated by acid mine drainage. Two approaches were used to characterize hyporheic exchange: sub-reach-scale measurement of hydraulic heads and hydraulic conductivity to compute streambed fluxes (hydrometric approach) and reachscale modeling of in-stream solute tracer injections to determine characteristic length and timescales of exchange with storage zones (stream tracer approach). Subsurface data were the standard of comparison used to evaluate the reliability of the stream tracer approach to characterize hyporheic exchange. The reach-averaged hyporheic exchange flux (1.5 mL s−1 m−1), determined by hydrometric methods, was largest when stream base flow was low (10 L s−1); hyporheic exchange persisted when base flow was 10-fold higher, decreasing by approximately 30%. Reliability of the stream tracer approach to detect hyporheic exchange was assessed using first-order uncertainty analysis that considered model parameter sensitivity. The stream tracer approach did not reliably characterize hyporheic exchange at high base flow: the model was apparently more sensitive to exchange with surface water storage zones than with the hyporheic zone. At low base flow the stream tracer approach reliably characterized exchange between the stream and gravel streambed (timescale of hours) but was relatively insensitive to slower exchange with deeper alluvium (timescale of tens of hours) that was detected by subsurface measurements. The stream tracer approach was therefore not equally sensitive to all timescales of hyporheic exchange. We conclude that while the stream tracer approach is an efficient means to characterize surface-subsurface exchange, future studies will need to more routinely consider decreasing sensitivities of tracer methods at higher base flow and a potential bias toward characterizing only a fast component of hyporheic exchange. Stream tracer models with multiple rate constants to consider both fast exchange with streambed gravel and slower exchange with deeper alluvium appear to be warranted.

428 citations

Journal ArticleDOI
TL;DR: In this paper, the authors employ a one-dimensional diffusive analog of stratospheric transport, and the general circulation model (GCM) of the Goddard Institute for Space Studies (GISS), to estimate the time over which tropospheric tracer concentrations must be approximately linear in order to determine stratosphere age unambiguously, if the growth time constant is greater than about 7 years.
Abstract: Estimates of stratospheric age from observations of long-lived trace gases with increasing tropospheric concentrations invoke the implicit assumption that an air parcel has been transported intact from the tropopical tropopause. However, because of rapid and irreversible mixing in the stratosphere, a particular air parcel cannot be identified with one that left the troposphere at some prior time. The parcel contains a mix of air with a range of transit times, and the mean value over this range is the most appropriate definition of age. The measured tracer concentration is also a mean over the parcel, but its value depends both on the transit time distribution and the past history of the tracer in the troposphere. In principle, only if the tropospheric concentration is increasing linearly can the age be directly inferred. We illustrate these points by employing both a one-dimensional diffusive analog of stratospheric transport, and the general circulation model (GCM) of the Goddard Institute for Space Studies (GISS). Within the limits of the GCM, we estimate the time over which tropospheric tracer concentrations must be approximately linear in order to determine stratospheric age unambiguously; the concentration of an exponentially increasing tracer is a function only of age if the growth time constant is greater than about 7 years, which is true for all the chlorofluorocarbons. More rapid source variations (for example, the annual cycle in CO2) have no such direct relationship with age.

406 citations


"Partial ages: diagnosing transport ..." refers background in this paper

  • ...The appropriate framework is therefore five-dimensional, i.e., space × time × age (Bolin and Rodhe 1973; Hall and Plumb 1994; Delhez et al. 1999; Ginn 1999; Deleersnijder et al. 2001; Haine and Hall 2002)....

    [...]

Frequently Asked Questions (2)
Q1. What are the contributions in "Partial ages: diagnosing transport processes by means of multiple clocks" ?

The age focuses only on the time taken to reach a given location and disregards other aspects of the path followed by the tracer parcel. In addition to the derivation of theoretical results and properties of partial ages, applications to a 1D model with lateral/transient storage, to the 1D advection-diffusion equation and to the diagnosis of the ventilation of the deep ocean are provided. They demonstrate the versatility of the concept of partial age and the potential new insights that can be gained with it. 

Detailed applications of the concept of partial age are deferred to further studies but the applications discussed above demonstrate the versatility of the concept age and the kind of information that can be gained with it.