# Consistent Boundary Conditions for Age Calculations

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## Abstract

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## 1. Introduction

## 2. The Age Distribution Function and Its First Two Moments

## 3. Consistent Insulating, Departure, and Arrival Boundary Conditions

#### 3.1. Insulating Boundary

#### 3.2. Departure Boundary

#### 3.3. Departure Boundary: An Alternative Approach

#### 3.4. Arrival Boundary

#### 3.5. Arrival Boundary: An Alternative Approach

#### 3.6. Gas Exchanges through the Water–air Interface

## 4. A Simple Ventilation Assessment Problem

## 5. Discussion and Conclusions

- Set out the reasons why the age, rather than other timescales (or diagnoses of another nature), is likely to be of use to help interpret the aquatic processes under consideration;
- Select the constituent whose (mean) age is to be evaluated and explain the rationale of this choice;
- Define the age, especially where and when the age of a particle of the constituent under study is to set or reset to zero, as well as where, when, and how this particle will cease to be taken into consideration;
- Build the boundary conditions for the age distribution function in accordance with the outcome of the previous three steps;
- Derive consistent boundary conditions for the concentration and age concentration using the methodology developed in this article (see also Appendix D).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## References

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**Figure 1.**Schematic representation of domain of interest $\Omega $ delimited by insulating (or impermeable) boundary $\Gamma $, whose outward unit normal is vector $\mathit{n}$, with $\left|\mathit{n}\right|=1$. The velocity and the diffusivity tensor are time- and position-dependent.

**Figure 2.**Schematic representation of a semi-infinite domain ($x\in [0,\infty [$) with a departure boundary at $x=0$. On this boundary, we can either prescribe that all particles of the tracer under study have zero age (Dirichlet boundary condition) or that the age of the incoming particles is zero, which leads to a Robin boundary condition, as seen in Section 3.3.

**Figure 3.**Illustration of age distribution functions (19) (solid line) and (34) (dashed line), which are obtained under Dirichlet and Robin boundary conditions at the inlet ($x=0$). Dimensionless variables are used. They are identified by asterisks and are defined as follows: ${t}^{\ast}={U}^{2}t/K$, ${x}^{\ast}=Ux/K$, ${\tau}^{\ast}={U}^{2}\tau /K$ and ${c}^{\ast}=Kc/{U}^{2}$ (see Appendix B). The dimensionless age distribution functions are plotted at ${x}^{\ast}=3$ as functions of the age at different instants.

**Figure 4.**Schematic representation of a finite-sized domain ($x\in [0,L]$) with a departure boundary at $x=0$ and an arrival one at $x=L$. Dirichlet boundary conditions are prescribed on the boundaries. An alternative treatment of the arrival boundary leads to the implementation of Neumann boundary conditions at $x=L$.

**Figure 5.**Illustration of (

**a**) concentration (45), (

**b**) age concentration (46), and (

**c**) the associated age. Panel (

**d**) depicts the inconsistent age that is obtained as the ratio of inconsistent age concentration (49) and correct concentration (49). Dimensionless variables are represented. They are identified by asterisks and are ${x}^{\ast}=x/L$, ${\alpha}^{\ast}=U\alpha /L$, ${a}^{\ast}=Ua/L$ and ${a}_{inc}^{\ast}=U{a}_{inc}^{\ast}/L$.

**Figure 6.**Illustration of the ages obtained from the solution of the one-dimensional problem outlined in Figure 4. Panel (

**a**) depicts the age ensuing from Dirichlet boundary conditions imposed at the departure boundary ($x=L$) and, hence, is the same age as that represented in panel (

**c**) of Figure 5. The age obtained by prescribing Neumann boundary conditions at $x=L$, that is, age (52), is illustrated in panel (

**b**). Dimensionless variables are represented. They are identified by asterisks and are defined as follows: ${x}^{\ast}=x/L$ and ${a}^{\ast}=aU/L$.

**Figure 7.**Illustration of the gas fluxes ($\mathrm{kg}\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$) involved in boundary condition (53). The net gas (mass) flux through the water–air interface, which is positive in the upward direction (that is, from water to air), is the difference between the outgoing flux, $\rho {\varphi}_{C}^{\uparrow}={\left[\rho \varpi C\right]}_{\mathit{x}\in \Gamma}$, and the incoming one, $\rho {\varphi}_{C}^{\downarrow}={\left[\rho \varpi {C}^{s}\right]}_{\mathit{x}\in \Gamma}$, where ${C}^{s}$ is the surface concentration in equilibrium with the atmosphere.

**Figure 8.**Schematic representation of the vertical, one-dimensional domain of interest dealt with in Section 3.6 ($-h\le z\le 0$) (panel (

**a**)), and age (60) of the dissolved gas originating from the atmosphere for various values of dimensionless parameter $\u03f5$ (panel (

**b**)). The lower boundary ($z=-h$) is impermeable. Robin boundary conditions, related to gas exchanges, are prescribed at the water–air interface ($z=0$). Dimensionless variables are represented in panel (

**b**). They are identified by asterisks and are defined as follows: ${z}^{\ast}=z/h$ and ${a}^{\ast}=K(a-{a}^{a})/{h}^{2}$.

**Figure 9.**Schematic representation of the two-dimensional, “horizontal-vertical” domain of interest for the simple ventilation study dealt with in Section 4. The nature of the boundaries and the related boundary conditions are also indicated.

**Figure 10.**Illustration of the concentration, age concentration, and age from the solution of the partial differential problem (63) and (64) for ($P{e}_{x}$, $P{e}_{z})=(10,10)$ and ($P{e}_{x}$, $P{e}_{z})=(100,10)$. Dimensionless variables are represented. They are identified by asterisks and are defined as follows: ${x}^{\ast}=x/L$, ${z}^{\ast}=z/h$, ${\alpha}^{\ast}=U\alpha /L$, and ${a}^{\ast}=Ua/L$. The non-consistent age ensuing from inappropriate boundary conditions for the age concentration on the departure boundary is represented in the lowermost row of the graph.

**Figure 11.**Difference between the non-consistent age and the correct one for ($P{e}_{x}$, $P{e}_{z})=(10,10)$. Dimensionless variables similar to those of Figure 10 are represented. On the departure boundary, the age difference is infinite, but the colour is saturated at a value of 10.

**Figure 12.**Dimensionless width (${\mathsf{\Lambda}}^{\ast}=\mathsf{\Lambda}/L$) of the region where the error due to the inconsistent boundary conditions on the departure boundary is significant as a function of the horizontal Peclet number ($P{e}_{x}$) for various values of the vertical Peclet number, that is, $P{e}_{z}=110,100$.

**Table 1.**Outgoing and incoming specific fluxes (ratio of a flux to the water density) at the water–air interface for the age distribution function (${\varphi}_{c}^{\uparrow ,\downarrow}$), the concentration (${\varphi}_{C}^{\uparrow ,\downarrow}$), and the age concentration (${\varphi}_{\alpha}^{\uparrow ,\downarrow}$). As for the downward flux, the general expression and two simplified ones are taken into consideration, which consists in assuming that all the incoming gas particles have the same age, ${a}^{a}$ (fourth column), and that this age is zero (fifth column).

Variable | Outgoing (Upward) | Incoming (Downward) Specific Flux | ||
---|---|---|---|---|

Specific Flux | General Expression | ${\mathit{c}}^{\mathit{a}}={\mathit{C}}^{\mathit{s}}\mathit{\delta}(\mathit{\tau}-{\mathit{a}}^{\mathit{a}})$ | ${\mathit{c}}^{\mathit{a}}={\mathit{C}}^{\mathit{s}}\mathit{\delta}\left(\mathit{\tau}\right)$ | |

age distribution function | ${\varphi}_{c}^{\uparrow}=\varpi c$ | ${\varphi}_{c}^{\downarrow}=\varpi {c}^{a}$ | ${\varphi}_{c}^{\downarrow}=\varpi {C}^{s}\delta (\tau -{a}^{a})$ | ${\varphi}_{c}^{\downarrow}=\varpi {C}^{s}\delta \left(\tau \right)$ |

concentration | ${\varphi}_{C}^{\uparrow}=\varpi {C}^{s}$ | ${\varphi}_{C}^{\downarrow}=\varpi {C}^{s}$ | ${\varphi}_{C}^{\downarrow}=\varpi {C}^{s}$ | ${\varphi}_{C}^{\downarrow}=\varpi {C}^{s}$ |

age concentration | ${\varphi}_{\alpha}^{\uparrow}=\varpi \alpha $ | ${\varphi}_{\alpha}^{\downarrow}=\varpi {\alpha}^{a}$ | ${\varphi}_{\alpha}^{\downarrow}=\varpi {C}^{s}{a}^{a}$ | 0 |

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**MDPI and ACS Style**

Deleersnijder, E.; Draoui, I.; Lambrechts, J.; Legat, V.; Mouchet, A. Consistent Boundary Conditions for Age Calculations. *Water* **2020**, *12*, 1274.
https://doi.org/10.3390/w12051274

**AMA Style**

Deleersnijder E, Draoui I, Lambrechts J, Legat V, Mouchet A. Consistent Boundary Conditions for Age Calculations. *Water*. 2020; 12(5):1274.
https://doi.org/10.3390/w12051274

**Chicago/Turabian Style**

Deleersnijder, Eric, Insaf Draoui, Jonathan Lambrechts, Vincent Legat, and Anne Mouchet. 2020. "Consistent Boundary Conditions for Age Calculations" *Water* 12, no. 5: 1274.
https://doi.org/10.3390/w12051274