# Partial ages: diagnosing transport processes by means of multiple clocks

## 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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##### Citations

^{1}, Utrecht University

^{2}, Geophysical Fluid Dynamics Laboratory

^{3}, Columbia University

^{4}, IFREMER

^{5}, Florida State University

^{6}, University of Miami

^{7}, Delft University of Technology

^{8}, Université catholique de Louvain

^{9}, Stockholm University

^{10}, Massachusetts Institute of Technology

^{11}, Princeton University

^{12}, University of Southampton

^{13}, British Antarctic Survey

^{14}, University of Oxford

^{15}, University of Kiel

^{16}, National Oceanography Centre

^{17}, Plymouth Marine Laboratory

^{18}, Sukkur Institute of Business Administration

^{19}, Yale University

^{20}, California Institute of Technology

^{21}, Los Alamos National Laboratory

^{22}, University of New South Wales

^{23}

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....

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24 citations

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##### References

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)....

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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)....

[...]

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....

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...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)....

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428 citations

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)....

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##### Frequently Asked Questions (2)

###### Q2. What are the future works in "Partial ages: diagnosing transport processes by means of multiple clocks" ?

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.