Residence and exposure times : when diffusion does not matter

Abstract

Under constant hydrodynamic conditions and assuming horizontal homogeneity, negatively buoyant particles released at the surface of the water column have a mean residence time in the surface mixed layer of h/w, where h is the thickness of the latter and w ( > 0) is the sinking velocity Deleersnijder (Environ Fluid Mech 6(6):541–547, 2006a). The residence time does not depend on the diffusivity and equals the settling timescale. We show that this behavior is a result of the particular boundary conditions of the problem and that it is related to a similar property of the exposure time in a one-dimensional infinite domain. In 1-D advection–diffusion problem with a constant and uniform velocity, the exposure time—which is a generalization of the residence time measuring the total time spent by a particle in a control domain allowing the particle to leave and reenter the control domain—is also equal to the advection timescale at the upstream boundary of the control domain. To explain this result, the concept of point exposure is introduced; the point exposure is the time integral of the concentration at a given location. It measures the integrated influence of a point release at a given location and is related to the concept of number of visits of the theory of random walks. We show that the point exposure takes a constant value downstream the point of release, even when the diffusivity varies in space. The analysis of this result reveals also that the integrated downstream transport of a passive tracer is only effected by advection. While the diffusion flux differs from zero at all times, its integrated value is strictly zero.

Appendix—Point exposure in a 1-D infinite or semi-infinite domain

Consider first an infinite channel whose cross-sectional area S(x) depends on the longitudinal coordinate x ∈ ] − ∞ , ∞ [. The velocity u(x) > 0 is assumed to be constant in time but varies spatially in such a way that the volumetric flow rate

$$ Q = S(x) u(x) $$

(26)

is a constant. When a spatially variable diffusivity κ(x) is taken into account, the concentration C(t, x) (or rather its average value over the cross section) obeys the equation

$$ \frac{\partial}{\partial t} \left[S(x)C \right] + \frac{\partial}{\partial_x} \left[Q C - \kappa(x) S(x)\frac{\partial C}{\partial x} \right] = 0 $$

(27)

where t denotes the time.

In order to evaluate the point exposure, a point discharge is introduced at the initial time at some location x ₀, i.e.

$$ C(0,x) = \dfrac 1 {S(x_0)} \delta(x-x_0). $$

(28)

In the context of one-dimensional channel with a variable cross section, the residence time and exposure time in a control domain ω can be computed from the solution of the initial value problem Eqs. 27–28 as

$$ \theta(x_0), \Theta(x_0) = {\int_0^\infty \int_\omega S(x) C(t,x) \textrm{d}x \textrm{d}t},$$

(29)

It is therefore appropriate to define the point exposure as

$$C^\Theta_{x_0}(x) = \int_0^\infty S(x)C(t,x) \textrm{d}t$$

(30)

which can be expressed as

$$C^\Theta_{x_0}(x) = S(x)\xi(x)$$

(31)

using the auxiliary variable

$$\xi(x) = \int_0^\infty C(t,x) \textrm{d}t.$$

(32)

A differential equation for this auxiliary variable can be obtained by integrating Eq. 27 for t ∈ ]0, + ∞ [. One gets

$$ \begin{array}{rll} && S(x)\left[\lim\limits_{t\to+\infty}C(t,x)- C(0,x)\right] \\ &&\quad + \dfrac{\textrm{d}\phantom x}{\textrm{d}x}\left(Q \xi - \kappa(x)S(x)\dfrac {\textrm{d}\xi} {\textrm{d}x} \right)=0. \end{array} $$

(33)

Using Eq. 28, assuming that the tracer is completely flushed out as t tends to infinity, one has:

$$\delta(x-x_0) = \dfrac{\textrm{d}\phantom x}{\textrm{d}x}\left(Q \xi - \kappa(x)S(x)\dfrac {\textrm{d}\xi} {\textrm{d}x} \right).$$

(34)

On the upstream side, assuming that the signal does not diffuse to − ∞, one gets

$$Q \xi - \kappa(x)S(x)\dfrac {\textrm{d}\xi} {\textrm{d}x} =0 $$

(35)

and

$$\xi (x) = \xi(x_0^-) \exp \left[-\int_x^{x_0} \dfrac Q {\kappa(x')S(x')} \textrm{d}x'\right], \qquad x<x_0.$$

(36)

Downstream, the solution takes the form

$$\xi(x) = \alpha + \beta \exp \left[\int_{x_0}^x \dfrac Q {\kappa(x')S(x')} \textrm{d}x'\right], \qquad x>x_0$$

(37)

where α and β are appropriate integration constants. To avoid the exponential growth as x tends to infinity, one must have β = 0 (unless κ vanishes at a sublinear rate), so that ξ(x) is a constant in this part of the domain.

The matching of the solutions in the upstream and downstream parts requires that

$$\lim\limits_{x\to x_0^+} \xi(x) = \lim\limits_{x\to x_0^-} \xi(x)$$

(38)

and

$$ \begin{array}{rll} &&\lim\limits_{x\to x_0^+} \left(Q \xi - \kappa(x)S(x)\dfrac {\textrm{d}\xi} {\textrm{d}x} \right)\\ &&\quad -\lim\limits_{x\to x_0^-} \left(Q \xi - \kappa(x)S(x)\dfrac {\textrm{d}\xi} {\textrm{d}x} \right)= 1 \end{array} $$

(39)

hence

$$\xi(x_0^-) = \alpha = \dfrac 1 Q.$$

(40)

Finally, one gets

$$C^\Theta_{x_0}(x) = \begin{cases}\dfrac {S(x)} Q \exp \left[-\displaystyle\int_x^{x_0} \dfrac Q {\kappa(x')S(x')} \textrm{d}x'\right] \qquad &x< x_0\\[12pt] \dfrac {S(x)} Q \qquad &x\geq x_0.\end{cases}$$

(41)

Obviously, the above-mentioned results also apply to a semi-infinite domain x ∈ ]a, + ∞ [ provided that the condition Eq. 35 is valid upstream the point of release. In particular, Eq. 41 applies also to any semi-infinite channel if the total tracer flux (advection + diffusion) vanishes at the upstream boundary.

Also, when the cross section of the channel is a constant, the velocity u is also constant, and Eq. 41 takes the simpler form Eq. 19.