Geoscience Reference
In-Depth Information
The distribution of residence times
In a well-mixed system at steady state, the distribution of residence times (the histogram
of age groups) is exponential. Let us call
n
(
t
,
θ
) the number of particles that, at time
t
,
have resided in the system for a time
>θ
. Using the elementary rules of differentiation, the
quantity d
n
(
t
,
θ
)suchas:
n
t
,
θ
−
d
n
(
t
,
θ)
d
n
(
t
,
θ)
=
θ
+
d
n
(
t
,
θ)
=
d
θ
(6.7)
d
θ
is the number of particles which have resided in the system for a time comprised between
θ
and
θ
+
d
θ
. This is the number of individuals in the “class of age”
θ
.
the probability that a particle leaves the system in the unit time and
assume that it is independent of the time the particle stayed in the box (particles do not
age). The assumption that the system is well mixed therefore amounts to a constant
p
,
independent of
Let us call
p
=
1
/τ
. We can describe this problem as “Lagrangian” because we actually refer to
fluxes along the residence time coordinate that are not very different from the usual fluxes
along a directional spatial coordinate. Let us break down on a yearly base, the balance of
individual molecules of water in a lake with an inlet and an outlet. At time
t
,letussayon
January 1st, the fraction of water which resided between 10 and 11 years in the lake forms
bin 10 (
θ
10). After a year, the bin is flushed and replaced by all the water which had
resided between 9 and 10 years in the lake (the bin 9), while the water from bin 10 is either
promoted to bin 11 or is lost into the outlet. The number of water molecules n (t,
θ
=
θ
)inbin
θ
changes because water leaves the system and also because it ages. The rate of change is
given by:
n
t
,
θ
+
d
n
(
t
,
θ)
=−
pn
(
t
,
θ)
d
t
−
θ
+
d
n
(
t
,
θ)
(6.8)
n
t
,
θ
=
θ)
+
∂
n
(
t
,
θ)
θ
+
d
n
(
t
,
d
θ
(6.9)
∂
θ
Every year, the residence time of a water volume element that remains in the sys-
tem increases by one year and, therefore, d
θ
=d
t
. We can rearrange the previous
equation as:
∂
n
(
t
,
θ)
θ)
−
∂
n
(
t
,
θ)
=−
pn
(
t
,
(6.10)
∂
t
∂
θ
At steady state, the left-hand side of this equation vanishes and the right-hand side may be
integrated as:
e
−
p
θ
e
−
θ/τ
n
(
t
,
θ)
=
n
(
t
,0
)
=
n
(
t
,0
)
(6.11)
which demonstrates our second proposition. By integrating this expression over the entire
population, we would find that the mean residence time is simply
. Using proper statisti
cs
,
we would also discover that the standard deviation of the residence-time distribution is
√
τ
τ
.
The deceptively simple approach of residence-time distributions in a well-mixed reservoir
therefore packs a rich amount of information.
