**A little of introduction**

Velocity of a fluid is the variation of particles position with time. However it is not velocity alone to determine fluxes. Fluxes in fact depend both on velocity of fluid particles and the section (the area) through which they flow. The area of the fluxes can vary: when there are waves. In the latter case, the section through which the flow passes can suddenly increase (or decrease). The first illustrating case is the kinematic wave from which, the variation of the fluxes is ruled by

$c := \frac{\partial Q}{\partial A}$

where $Q$ is discharge and $A$ is the cross-sectional area through which water moves.

In other cases, the variation of effective area is more subtle, as in unsaturated (vadose) porous media flow, where the flow that happens through pores is not visible, and flow can vary because pores of different size can be active or quiescent depending on the value of the global hydraulic head.

Jeff McDonnell (GS), and Keith Beven (GS) with their nose for interesting topics, actually wrote recently a paper where they discuss it (which I did not find very illuminating)

One aspect is that velocity of water itself vary (in time) and this variation, is, obviously, due to some acting force (stress) as the second law (Newton law) of mechanics asks. The acting force can be usually be thought as a consequence of a potential field that, in hydrological problems, we often recognise as being the hydraulic head (gradient of pressure terms, expressed in unit of length). Thus flow can actually vary as a consequence of local variations of hydraulic head pressures. These variations are communicated through the medium and the velocity to which they move is the pressure wave celerity (by definition).

To sum up, deformation in the pressures field travel, they result in variation of the flow area and of temporal variation of local (eulerian) velocity of a flow field.

One important point that Beven and McDonnell made (but this has been known from decades) is that transport of properties (passive tracers) depends on water molecules velocity and therefore between transported properties (like temperature, water isotopes, water age) and flow can be various discrepancies.

One indicator of transport processes is the mean travel time. There is no doubt that it must be:

$<T_r >_t := \int_0^\infty (t-\tau) p(t-\tau|t) d(t-\tau) $

where $<T_r >_t$ is the mean travel, $t$ is the actual time, and $\tau < t$ is the injection time and $p(t-\tau|t)$ is the travel time distribution conditional to the actual time. Implicitly, it is assumed that the probability is evaluated at the closure (boundary) of a control volume, and, for deformation, I think it is the outlet of a catchment. This distribution is time variant but often, in the past it was assumed invariant. Moreover, it usually differs from the residence time distribution, which actually refer to the distribution of ages inside the control volume. (Details in here). The distribution represent the ensable distribution of infinite random experiment that regards the motion of a single molecule or, after the hypothesis of ergodicity, the distribution of the travel time of a bunch of molecules, which are injected together in the system.

How $<T_r >_t$ is affected by celerity ? Let say that celerity has to do with fact that the probability is time varying. I cannot prove it at the moment, but I can give some heuristic.

**Back to the basics**

Let’s me re-discuss the topic from the scratch. When we have a velocity field, $u(\vec{x},t)$ in a fluid it can be either stationary or time varying. Assume it is stationary. Then:

$\frac{\partial u(\vec{x},t)}{\partial t} = 0$

which implies that $u(\vec{x},t) = u(\vec{x})$, i.e. it does not depends on space, and it is called stationary. Does it implies that forces does not act ? No, forces act, because still we can have gradient of velocity:

$\nabla u(\vec{x}) \equiv (\frac{\partial u(\vec{x})}{\partial x}, \frac{\partial u(\vec{x})}{\partial y}, \frac{\partial u(\vec{x})}{\partial z} ) \neq 0 $

However, forces they do not change in time and are subject to stationary resistances (dissipation) that do not allow for the overall system to accelerate (increase the total kinetic energy).

So, because variation of velocity means acceleration, we have forces there. When the we follow a first particle moving (the so called Lagrangian vie of the motion), it is locally accelerated, and it feels a force. Its travel time in the domain is completely determined by the velocity field (this is a statement always valid) and (that is true just for this case) does not vary in time. If a second particle is injected in the same place of the first one, it will have the same travel time. Particles (parcels, molecules) injected in different points will have different trajectories, all stationary, and the overall distribution of of travel times for a bunch of particles injected together (or, BTW, at a different time) is not time varying. The fluxes that comes out from such a velocity field is not constant but reflect the mass (volume) of particle injected in the fields. What is going in will go out in a certain composition of ages which however is easy to disentangle from the knowledge of the spatial field and the amount of injected matter.

In the above case no net energy is transferred and we do not have to invoke celerities besides energy.

Now, assume we have a time varying velocity field, so that $u(x,t)$. Therefore:

$ \frac{\partial u(\vec{x},t)}{\partial t} \neq 0$

So we have acceleration in (usually) many points. This acceleration can only be generated by a local force (which has to be time varying). The way to obtain it in a fluid is to generate a time-varying pressure gradient (yes, excluding time varying gravity, time varying electromagnetic fields, time varying applied work, time varying temperature or density gradient, ;-) etc. , all things we usually not deal with in our hydrological problems). Actually it can be shown that these pressure variations can travel from one place to another (in ways that are different in different phenomena: see Beven and Mcdonnell) so that this pressure variation is the cause of the varying velocity field, according to the thermodynamic fact that pressure is a generalized force. The velocity of these travelling pressures is the celerity. Conjointly with pressure field variations, also fluxes in previously unaffected area are activated.

So we obtained in full the phenomenology that we described at the beginning of this page. In this case distribution of particles’ travel times is affected in two ways we anticipated: by mobilising water (and/or enlarging the sectional area) and by increasing (locally) water velocity.

From the above analysis is also evident that the phrase "celerity (of pressure waves) affects flows" is true, but in an indirect way, since flow is affected by velocity fields and their spatial extension, both modified by pressure waves. Celerity of water is probably a meaningless or imprecise naming which does not describe properly the phenomenon. In some cases (finally) the pressure waves is identifiable as movements of the free surface, and this generates actually, a lot of misunderstanding in the visualisation of the general process.

From the above analysis is also evident that the phrase "celerity (of pressure waves) affects flows" is true, but in an indirect way, since flow is affected by velocity fields and their spatial extension, both modified by pressure waves. Celerity of water is probably a meaningless or imprecise naming which does not describe properly the phenomenon. In some cases (finally) the pressure waves is identifiable as movements of the free surface, and this generates actually, a lot of misunderstanding in the visualisation of the general process.

**Consequences on the mean travel time**

Distribution of the mean travel time can vary because the newly activate area can contain water of ages different from those activate previously, and because locally the velocity of water can increase (or decrease) at the passage of the pressure wave which travel with its own celerity. The latter effect can skew the distribution of travel time to the left (smaller travel times), but unless the variation is such that the whole the control section is affected (which can be the case) has not significant effects of the mean travel time, which is an integrated quantity. So it is probable that the set of particles close to the outlet passed through the control volume in antecedent times, and are not affected by a single perturbation.

Due to time variation of the velocity field, when we inject two water volumes in the same place at different times, the travel time of the two volumes is different, and inherently, the related transport.

**Well, the velocity field is not superimposed, but caused by the quantity of water itself.**

So far we discussed the cause of varying velocity fields which I suggested was due to pressure waves. However, is is interesting to know also what generates these wave pressures. In case of water flow is usually the water amount itself. The velocity field, in fact, is not something independent of the water quantity and distribution. On the contrary depends on it. So when we inject water, we cause pressure waves that propagates with a law dependent on the medium we are analysing: surface, vadose zone, or ground.