Time scales for pollution assessment
Time scales for pollution assessment can be considered using three commonly used methods: flushing time, residence time and age. In this article the methodologies and tools to quantify each of these methods is explained as well as the applicability of each method.
Transport time scales are useful tools to quantify the importance of hydrodynamic processes in the transport and fate of pollutants in coastal and estuarine water systems. Indeed, the water quality of a system depends crucially on the retention of pollutants within the system and its ability to flush them out. Transport time scales are often compared with the pollutant source time scales or biogeochemical processes to evaluate the relative importance of physical and water quality processes.   They are also used to understand population dynamics  and to serve as indicators to classify and compare estuarine systems.
The first correlation between retention time scales and water quality was proposed in the classical empirical model of lake eutrophication of Vollenweider (1976), which describes algal biomass as a function of the phosphorus loading rate scaled by the hydraulic residence time. Many other applications followed, from heavy metals problems, mineralization rates of organic matter and primary production (the reader is refered to Monsen et al. 2002  for a detailed list of applications).
Many retention time scales can be found in the literature, often with distinct definitions for the same concept (see for instance Oliveira and Baptista, (1997) , Shen and Haas, (2004)  for more details). Here, three fundamentally different concepts are presented, following the classification presented in Monsen et al. (2002)  : flushing time, residence time and age.
Flushing time is the "time to replace the freshwater volume of the estuary by the total freshwater input flux (river, discharges, rainfall, etc.)" (Officer and Kester, 1991) or, in a more general way, "the ratio of the mass of a scalar in a reservoir to the rate of renewal of the scalar". his transport time scale is a whole-system indicator of the renewal capacity, but does not allow for the distinction between several forcing mechanisms (e.g., the influence of tidal motion to flush out the system) or the spatial and time variability of the renewal capacity.
Residence time is the "time it takes for any waterparcel of the sample to leave the lagoon through its outlet to the sea".  It can be used to analyse the spatial variation of flushing properties and the variation of renewal for different environmental conditions (effect of tidal amplitude and phase, relative importance of different forcings – waves, currents, etc.). These detailed properties makes residence times very useful for comparative analyses of the effects of several engineering interventions (dredging, hard-struture building), and also to characterize changes in the system’s contaminant inputs (changes in the nature, location and frequency of the sources of contaminants). The specific way to define "the time to leave the system" can also lead to different concepts of residence times which can be very important for water quality analyses. If residence times are defined as the "time for a water parcel to leave the system once" (once-through residence times, Oliveira and Baptista, 1997 ), the concept is very useful to characterize the flushing of pollutants that are significantly altered once outside the system (for instance due to strong variations in salinity and/or temperature). An opposite definition is "the time for a water parcel to leave the system without returning at a later tide" (denoted re-entrant residence times in Oliveira and Baptista, 1997 ), which is useful for the analysis of the retention of conservative tracers in a system. Finally, residence times can also be defined as the "time spent in the domain of interest" (denoted as exposure time in Deleersnjider and Delhez, 2005). This definition is particularly important in coastal pollution since it quantifies the time of exposure of a system to a specific contaminant.
Using a reservoir concept, the age of a contaminant can be referred to as "the time elapsed since it entered the system".  Zimmerman (1976)  proposed a definition that explicitly accounts for the spatial variability of this time scale: “age of a water parcel is the time elapsed since the particle departed the region where its age is zero”. A general theory for age, based on tracer concentrations, can be found in Deleersnijder et al. (2001). Age is the complement of residence times and can be used to understand the pathways of contaminants and organisms within the system, in particular in the influence of external sources/sinks of contaminants to coastal systems.
Methodologies and tools
Each transport time scale can be estimated by several methodologies based on models or field data. The choice of the methodology depends on the specific goals and on the avaiability of data to characterize the system.
The Flushing Time (FT) is the least demanding parameter. It can be calculated in several ways, using either data or numerical model results:
Freshwater faction method
where S0 – ocean salinity, s – average salinity in domain, Q – freshwater input This method assumes steady state conditions and can only be applied to systems with a significant freshwater input. For complex systems, it requires a considerable amount of salinity data or the use of a numerical model.
Tidal prism method
where V - water volume at high tide, T - tidal period, P - tidal prism of a representative flood tide. This method only requires basin geometry and tidal range information. It assumes that
- the system is well mixed,
- tidal flow is the dominant flushing mechanism,
- the system is at steady-state with a sinusoidal tidal signal, and
- the system is fully flushed in a single tidal cycle.
The last assumption can be alleviated by a modified formulation (using (1-b)*P in the denominator) to account for an incomplete mixing, where b is the return flow factor (0-1).This method tends to underestimate the flushing time due to the steady-state hypothesis.
Continuously stirred tank reactor (CSTR) method
where C is the concentration of a pollutant at time t, due to an instantaneous load at time t0 that leads to the initial concentration C0. This method assumes that
- no further mass is introduced in the system,
- flow and volume in the CSTR are constant in time and
- instantaneous and complete mixing of the pollutant in the system.
Residence times are typically evaluated with numerical model results. Applications have been done with models of increasing complexity, including box-models, particle models and concentration models.
- Box-models are the simplest to set-up, requiring little information on the system and small computational resources. The residence time of the pollutants in each box can be computed based on the knowledge of the geometry of the system, the hydrodynamics and the pollution loads (the reader is referred to Hagy et al. (2000)  for a detailed application of this approach). Calculations of residence times based on this method are limited by the capability of box models to solve the underlying hydrodynamics, which are generally very complex in coastal systems.
- Particle models are by far the most popular approach to compute residence time. Residence times are computed based on the release of large numbers of particles, scattered throughout the domain of interest, at several release times within the tidal cycle and for different tidal amplitudes. This approach generally accounts for both advection and diffusion processes, through the use of Euler or Runge-Kutta methods for the advective term and random walk methods for the diffusive term. The advantages of this approach are its ability to handle the spatial and temporal variability of residence times, and its applicability to several types of tracers and multiple sources of contamination. Limitations include the limited hability to solve complex water quality processes using particles and the need to use very large numbers of particles for comprehensive studies (e.g. O(500,000) in the Gulf of Maine, Bilgili et al. 2005) , which may require very large computational resources.
- Concentration models (transport models) have also been used for residence time calculations, based on two and three dimensional applications. For these models, residence time can be defined as the time necessary to reduce the initial pollutant concentration by 1/e  or to reduce total mass by 1/e . This approach can readily account for water quality processes, but the accuracy of the residence time can be limited by numerical errors (mass imbalances, numerical diffusion,..) and cannot be applied for several simultaneous sources of the same contaminant.
The methodologies to compute age are the same as those used for residence time. The readers are referred to Shen and Haas, (2004) and Kennedy et al. (2006) for details on the application of concentration and particle models for the calculation of age.
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