# Timescale Methods for Simplifying, Understanding and Modeling Biophysical and Water Quality Processes in Coastal Aquatic Ecosystems: A Review

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## Abstract

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## 1. Introduction

“Nature is pleased with simplicity. And nature is no dummy.”—Commonly attributed to Isaac Newton

#### 1.1. What Are Timescales?

^{2}/time), water discharge (length

^{3}/time), growth, decay or uptake (1/time), water column production (mass/(area-time)), ingestion (mass food/(mass tissue-time))). Timescales can be defined and quantified for each of these processes by a variety of methods to be detailed in later sections. Regardless of the approach for estimating values for timescales, the following holds when using them in the fifth sense above: A smaller (or “shorter”) timescale indicates a faster process, whereas a larger (or “longer”) timescale suggests that the process is slower [27].

#### 1.2. Some Fundamental Concepts and Definitions

#### 1.2.1. Constituents, Particles, Parcels, and Types

^{3}, mol/L, cells/L, etc.). Since there is a huge number of constituents, it may be convenient to focus on groups of constituents, i.e., aggregates, whose concentrations may be seen to obey equations similar to those pertaining to individual constituents [63,64]. This is why many use the word “constituent” (or a similar term) even if the substance under consideration is actually an aggregate (e.g., salt). The water in an aquatic ecosystem is itself an “aggregate,” consisting of all of its constituents. Pure water is by far its dominant constituent, making the density of the water mixture close to that of pure water. The water mixture density may be regarded as a constant in most terms of the equations to be dealt with (the “Boussinesq approximation”).

#### 1.2.2. Lagrangian and Eulerian Descriptions and Approaches

#### 1.3. Transport Timescales

- Residence time—Although the term “residence time” is frequently used to mean a variety of things [88,92,93,94], one of the most common definitions is the time taken by a particle to leave a water body or defined region of interest [92,95,96,97]. Because particles originating at different locations and times within a water body may require different amounts of time to exit, residence time (according to this definition) is a function of location and time [87,92,97]. A strict interpretation of this residence time definition is the time taken to leave a water body for the first time (see Figure 3), an important distinction in tidal systems where oscillatory transport can cause particles to exit and then re-enter the domain of interest one or more times [26,95,98,99]. Numerical simulations currently offer the best methods for estimating time- and position-dependent timescales in realistic domains [66,97,100,101]; however, other (field-based [102,103,104,105], analytic [22,59]) methods may also provide trustworthy estimates, though with less resolution or with additional simplifying assumptions. Other residence time definitions, which are not location- and time-specific, also exist and see wide application (see “flushing time” below).
- Age—Age is defined as the time elapsed since a particle entered a water body or defined region [88,94,96,106]. Because the time to reach a specific location after entering will vary across the water body and over time, age (like residence time, as per our preferred definition above) is also time- and location-specific (see Figure 3). Age is seen as the complement to the location- and time-specific residence time: while age is the time taken since entering to reach location
**x**within a water body, residence time is the time remaining within the water body after reaching location**x**[87,88,96,106]. Some authors have generalized the common definition for age above, arriving at the following: “the time elapsed since the parcel under consideration left the region in which its age is prescribed to be zero” [63,64]. - Transit time—Transit time has been defined as the total time for a particle to travel across an entire water body or defined region, from entrance to exit [93,96]. Therefore, transit time is the sum of the location- and time-specific age and residence time (see Figure 3). Some authors have taken advantage of the fact that transit time is equivalent to age computed at the downstream boundary or exit of a water body [28,107]. Travel time is similar to transit time, in that it usually references the time taken to travel between two defined points in space [28]. The transit time and location- and time-specific age and residence time are easily derived analytically for a plug flow situation (see Appendix A).
- Exposure time—Exposure time goes forward where the strict definition of residence time stops. While the strict, spatially and temporally variable residence time only accounts for time spent within a defined region until leaving it the first time, exposure time accounts for the total time spent within the domain of interest [87], including “all subsequent re-entries” [95] (see Figure 3). Thus, exposure time may be of particular relevance in systems with oscillatory tidal flows [108]. When computing exposure time with a numerical model, it is important that the computational domain be larger than the domain of interest [95], since transport processes outside the domain of interest control particle re-entry.
- Flushing time—“Flushing time is a bulk or integrative parameter describing the general exchange characteristics of a waterbody without identifying detailed underlying physical processes or their spatial distribution” ([27], adapted from [87]). There are numerous methods for defining and quantifying flushing times, many of them mathematically quite simple. For example, if advection is expected to dominate exchange between the domain of interest and an adjacent water body (as for a river reach), an advective flushing time may be estimated simply as V/Q, where V is the volume of the domain of interest, and Q is the rate of volumetric flow through it. For this situation, V/Q estimates the time for all water in the domain of interest to be replaced, whereas ½(V/Q) represents the mean time for replacement of the original water. Analogously, if we assume that an estuary behaves similarly to a “plug flow reactor”, i.e., with perfect cross-sectional mixing but zero streamwise mixing, V/Q would represent the time needed to replace all the water initially in the estuary by water entering through its upstream boundary (Figure 4). Some variations on this approach include: (A) substitution of V with freshwater volume V
_{fw}and of Q with freshwater inflow rate Q_{fw}, if one is interested in the time to replace freshwater [52,109] (this is often called the “freshwater fraction method” [58,110]); or (B) substitution of V and Q, respectively, with scalar mass M and scalar flux F (in units (mass/time)), if one is concerned with time for replacement of a scalar quantity [87]. (Incidentally, the V/Q [90,104], V_{fw}/Q_{fw}[109], and M/F [111] formulations are sometimes called “residence times”.) It should be noted that the V/Q estimate depends on the (sometimes arbitrary) size of the domain of interest [112]. - e-folding flushing time—Another construct for quantifying time for flushing is the e-folding time (τ
_{e-fold}). This approach capitalizes on the frequently observed exponential-like decrease of constituent mass within a water body over time as it is subjected to flushing. This roughly exponential decrease is often observed in the results of coastal transport simulations [87,100,101,112,113,114,115] (see Figure 5) and tracer experiments [116,117]. Mathematically, the exponential form results from assuming a constant flow rate through a perfectly well-mixed system of constant volume, as for a CSTR (continuously stirred tank reactor) [87]. The well-mixed assumption employed here (Figure 6) is in stark contrast to the plug flow assumption above (Figure 4) and thus may be the more appropriate assumption for estuaries subject to strong (e.g., tidal) dispersive mixing. τ_{e-fold}may be obtained as (A) the reciprocal of the specific decay rate calculated from an exponential best-fit to a concentration time series [87,100,112,113] or simply as (B) the time when mass falls to 1/e (37%) of its initial value [114,117]. If the CSTR assumptions are perfectly met, τ_{e-fold}= V/Q, but if they are not met (e.g., for basins with bidirectional, tidal exchange flow), V/Q may not accurately characterize the effective flushing time captured by methods (A) or (B) above [87]. Although the well-mixed assumption is almost never satisfied, the e-folding construct is nonetheless employed widely and can work well in representing the net effect of all flushing processes acting on a basin. It is important to note the quantitative difference between this flushing time approach (which characterizes flushing of only 63%, or 1-e^{−1}, of initial mass; Figure 6) and the simple advective V/Q, V_{fw}/Q_{fw}, and M/F approaches above, whose aim is to characterize 100% replacement of initial mass or volume (Figure 4). Indeed, any perfect CSTR would never truly experience 100% replacement of initial mass, as suggested by the exponential dependency of concentration on time. Even so, for an inert constituent in a well-mixed system, the concentration tends to zero as time tends to infinity, resulting in a finite domain-averaged residence time, which is equal to the e-folding time [88,94,113,118]. - Tidal prism flushing time—Another class of flushing time approaches for estuaries—tidal prism models—prominently acknowledges tides as a flushing agent [119,120]. The most basic form for the tidal prism flushing time is V∙T
_{tide}/V_{p}[58], where V is estuary volume, T_{tide}is the tidal period, and V_{p}is the tidal prism volume (i.e., estuary volume difference between high and low tides). Applications of this general approach may vary in the way V and V_{p}are defined or calculated [27,110]. Moreover, authors have employed a range of assumptions and adjustments for capturing the influence of freshwater inflow or return flow at the seaward boundary [27,58,119]. Like the e-folding time, the tidal prism flushing (or “turnover” [58,110]) time is based on the assumption of well-mixedness [87,119]. - Turnover time—The V/Q [58], V
_{fw}/Q_{fw}[110], M/F [88,94], e-folding [121], and other bulk approaches [110] are also sometimes called “turnover times.” A relatively new approach for estimating bulk estuary turnover timescales is based on the total exchange flow (TEF) through a cross section at the estuary mouth; TEF is calculated using an isohaline framework [122], and the TEF timescale τ_{TEF}may be thought of as “the ratio of the mass of salt in the estuary to the salt flux into the estuary” [110]. (τ_{TEF}is also called a “residence time” [122].) In addition to physical processes, the term “turnover time” is frequently applied to biological or geochemical processes as well [62,90,123,124,125]. - Retention time—The term “retention time” is frequently, though not exclusively, used to refer to how long constituents (e.g., nutrients, sediment, organisms) remain within a particular aquatic environment or sub-environment [14,126]. Mechanisms influencing constituent retention can include both hydrodynamic processes (e.g., pools, eddies, and dead zones [14]; stratification and mixing [127]), sedimentation [14], biogeochemical processing [14], and motility of organisms [127]. Hydraulic “retention time” is sometimes treated interchangeably with “residence time” [128] or with expressions described herein as “flushing times” [129].

#### 1.4. What Are Timescales Good for?

- A more meaningful substitute for primitive variables and native process rates: Computed or measured primitive variables (e.g., velocity, pressure, temperature, concentration; also known as state variables) and native process rates (e.g., velocity, production, growth) are not always conducive to interpretation in their raw form [79,89]. (Here, we use the term “native process rate” to refer to the typical rate variable(s) used in connection with a particular process, e.g., velocity or discharge for water movement, or specific growth rate for biomass growth.) On the other hand, diagnostic timescales can incorporate valuable contextual information that native process rates and primitive variables do not. For that reason, timescales can serve as auxiliary variables that might better illuminate a scientific problem [79,89]. For example, the primitive variable “velocity” alone contains no additional problem-specific information that can aid the user in understanding the practical effect of that velocity: it is just a velocity. Whereas the advective timescale τ
_{adv}—the timescale counterpart to velocity—typically conveys the time needed for a particle to traverse a specified water body or distance (e.g., the time taken by a fisherman’s cooler to travel to the river mouth from the upstream location where it, sadly, fell overboard). Therefore, in comparison to a process rate or primitive variable, a timescale can in many cases take the user farther on an interpretive level by communicating what the process rate, materially, means in the context of the scientific question at hand. - A common currency for comparing speeds of processes: Timescales provide a common cross-disciplinary currency by which the speed of disparate processes can be compared [23]. For example, consider the observed reduction in the concentration of a decaying pollutant in a river over the first couple days after release. Relevant process rates (e.g., decay (1/time), river discharge (volume/time)) can be transformed into timescales (τ
_{decay}, τ_{flush}) that can then be directly compared. Therefore, if τ_{decay}is, for instance, 0.2 day and τ_{flush}is 30 days, the ~2 order-of-magnitude difference in timescales suggests that decay is a much faster process than river-driven flushing and is likely primarily responsible for any significant concentration reduction in the couple days following pollutant release. Since they all carry the same units, timescales can thus help bridge the gap between scientific disciplines and make quick, back-of-the-envelope assessments of dominant processes possible. Timescale ratios can represent the competition between processes; in some cases, such dimensionless numbers can serve as simple indicators of how an ecosystem might respond to a combination of different physical, biological, or geochemical processes [21,23,25,132,133,134,135,136]. - Distilling numerical model outputs [89,137]: The output files of numerical fluid flow models can be immense. Making sense of all those gigabytes, or even terabytes, of spatially and temporally detailed data is a non-trivial effort [79,137,138]. Timescales can extract the essence from such comprehensive datasets. In contrast to other analysis techniques that might provide spatially (temporally) detailed glimpses of the output at limited points in time (space), timescales can integrate across space and/or time and take advantage of most, if not all, of the results [79,138]. For this reason, timescales derived from the results of complex numerical models may be considered “holistic” [79,138]. Importantly, a model-derived timescale, such as the transit time for a particle through an estuary, may be considered holistic in a second sense: it takes into account all processes and forcings included in the model that influence the transport (e.g., river flow, tides, wind, density gradients, etc.) [139]. It is this second meaning that we refer to hereinafter.
- Comparing systems across space or time: An effective way of enhancing understanding of an aquatic system is through comparison with other systems or through assessing the functioning of a single system under different conditions over time. Timescales can help encapsulate the general physical or ecological state of aquatic systems across space or time, do so in a way that is relatively simple and intuitive, and allow for easy comparisons.
- Building simple(r) models: The partial differential equations (PDEs) governing hydrodynamics and scalar transport are complex, as they are composed of many terms describing multiple influences on momentum and mass balances. Because high-quality (i.e., stable and accurate) numerical solutions to the governing equations can be computationally costly, justifiable simplification of these PDEs is therefore a worthwhile activity. One simplification approach implements timescales of variability in combination with other (e.g., velocity, length, pressure, density) scales to estimate the relative magnitudes of individual terms in time-marching equations [2]; terms that “scale” much smaller than other terms may be justifiably neglected, with the equations reducing to the most essential terms and, hopefully, the numerical solution becoming more tractable and efficient. Another method of simplification involves quantifying the primary processes with timescales, creating dimensionless ratios with those timescales, and then substituting those ratios appropriately into a time- or space-dependent equation. The conversion of a mathematical relationship into dimensionless form can significantly reduce the complexity—and increase the solvability—of the equation [21,23]).
- Assessing connectivity: Transport timescales can contribute substantially to assessments of connectivity between different aquatic systems or subregions within a system [56,95,140,141,142]. In fact, transport timescales can form the basis for one important assessment tool—the “connectivity matrix” [95,140] (see Section 3.4).
- In conceptual models: Timescales are often invoked in conceptual models or qualitative descriptions of how systems work. Even if not quantified or clearly defined, well-known terms such as “residence time” capture a general meaning that a scientific or management audience can conceptually follow. Timescales are frequently used (in mental models, written descriptions, cartoons, schematics, etc.) to qualitatively explain ecological phenomena such as phytoplankton bloom development in coastal systems [6,143], legacy phosphorus across watersheds [14], coastal hypoxia [11], nutrient release from sediments in shallow lakes [144], and eutrophication in lakes [145] and coastal systems [146].

## 2. How Are Diagnostic Timescales Estimated?

#### 2.1. Combining Process Rates with Other Scales

_{growth}as the reciprocal of a typical specific net growth rate in the euphotic zone, (2) estimate the timescale for vertical mixing ${\tau}_{mix}^{vert}$ as the square of the water column depth divided by ${K}_{vert}$, a typical (e.g., mean or mid-depth [25]) turbulent diffusivity for the water column, and (3) compare the two timescales. (An argument could be made to use half of the water column depth as the characteristic length scale, but since these scaling exercises are meant to be approximate, it may not matter significantly.) If ${\tau}_{mix}^{vert}$ is significantly shorter (i.e., at least an order of magnitude smaller) than τ

_{growth}, then we would expect vertical mixing to be rapid enough to prevent an algal bloom in the euphotic layer. If, on the other hand, ${\tau}_{mix}^{vert}$ is significantly longer than τ

_{growth}, then we would not expect vertical mixing to be strong enough to single-handedly prevent a surface bloom. If we instead wish to understand whether longitudinal dispersion is fast enough to limit algal accumulation within a defined water body, then (1) an algal growth timescale might be more appropriately based on a typical (e.g., mean) net growth rate over the water column, especially if vertically well-mixed, and (2) the mixing timescale would be more appropriately estimated as the square of the water body length divided by K

_{long}, a longitudinal dispersion coefficient [42]. Furthermore, if transport through a water body is known to be governed primarily by advection induced by river flow as opposed to dispersive processes, then an advective timescale (e.g., water body volume V divided by river discharge Q) may be a more relevant transport timescale to compare with the algal growth timescale. Incidentally, the relative importance of advection versus dispersion (or diffusion) is a matter that itself can be illuminated using this sort of scaling approach: The well-known Peclet number (i.e., the ratio of a diffusive timescale to an advective timescale) is a dimensionless ratio implemented for this very purpose [22,59,88].

^{14}C uptake experiments. Timescales for algal losses to bivalve grazing have been calculated from water depth and grazing rates based on benthic biomass samples, published temperature-dependent pumping rate relationships, and laboratory-based expressions incorporating the food-limiting effect of concentration boundary layers [23,62]. Lopez et al. [148] estimated the specific loss rate of phytoplankton to zooplankton grazing based on tow net sampling, analyses to obtain carbon weight and community grazing rate, and measurements of phytoplankton biomass; that specific loss rate was then combined with benthic grazing losses to then obtain a collective timescale for loss [23]. Shen et al. [21] estimated the timescale for biochemical oxygen consumption based on temperature, surface dissolved oxygen concentration, and net oxygen consumption rate, which was taken as the sum of sediment oxygen demand and net water column respiration and based on previously published measurements and modeling constants. Crump et al. [91] calculated estuarine bacterial community doubling times from bacterial production (based on leucine incorporation) and bacterial cell counts. A timescale for contaminant depuration was calculated as the biological half-life of trace elements in mussels fed radiolabeled diatoms in a laboratory [150]. The timescale for 50% survival for larvae of broadcast spawning corals was quantified in laboratory experiments starting with gametes collected in the field (Nozawa and Okubo 2011); these “T

_{50}” values were ultimately compared with model-computed residence times to gain insight into ecological connectivity and the potential for self-seeding [135,136].

#### 2.2. Transport Timescales Based on Observational Data

#### 2.2.1. Drifter-Based Experiments

#### 2.2.2. Tracer-Based Experiments

#### 2.3. Transport Timescales Based on Numerical Models

#### 2.3.1. Forward Methods

#### 2.3.2. Backward Methods

_{1}–ω

_{6}; Figure 11A), and their novel extension of the adjoint approach permitted them to compute PRTs for particles initialized at specific points in space (numbered stars in Figure 11A). For each of those seven release locations, Figure 11B shows the PRTs representing time spent in subregions ω

_{1}–ω

_{6}before exiting the control region. For a given release location, the sum of all six PRTs (shaded portions of each bar in Figure 11B) equals the total residence time, i.e., the total time taken to leave the bay (top height of each bar). For pollutants discharged from a specific point location, this sort of information can quantify for resource managers how much time the pollutants spend in defined subregions on their way out of the bay [141], thereby highlighting areas potentially most impacted. PRTs are also displayed for each subregion ω

_{i}as time spent in ω

_{i}for particles released at every location in the domain (Figure 11C–H). These maps highlight the portions of the domain contributing particles spending the most time in a specific subregion and could, for example, provide insight into the major nutrient sources to a subregion and how much time those nutrients spend in the subregion before getting flushed out.

## 3. Timescale Applications for Explaining Ecosystem Processes and Variability in Water Quality

#### 3.1. Timescales in Conceptual Models

#### 3.2. Implementing Timescales in Building Simple Models

#### 3.2.1. Simple Models of the Physical Environment

^{2}= 0.74 and 0.95) regression models of Kärnä and Baptista [194] relating system-wide “renewing water age” (computed by a detailed 3D model) to observed river discharge and tidal range for the lower Columbia River Estuary (USA), thus allowing easy, quick estimates of water renewal timescales when 3D model simulations are not available; (2) the use by Mouchet and Deleersnijder [49] and [211] of mean ages and age distributions as a metric for evaluating the fidelity of the one-dimensional (1D) “leaky funnel” model to 3D models of ocean ventilation; (3) the derivation by Deleersnijder et al. [59] of simple estimates for mean residence time of sinking particles in the surface mixed layer; and (4) the development by Palazzoli et al. [179] of a simple polynomial relationship for the flushing-induced tracer decay coefficient (reciprocal of e-folding flushing time), as a function of wind speed and direction for the Virginia Coast Reserve, a complex system of interconnected shallow coastal bays and inlets on the United States east coast. Yet more examples are to be found in [22,212,213].

#### 3.2.2. Simple Ecological Models Using Physical Timescales

_{fw}). He started with an annual mass balance equation for total mass of biologically active, water-column nitrogen (m

_{N}) in an estuary,

_{N}. After making a number of simplifying assumptions (e.g., steady state, negligible nitrogen contribution from the ocean), Dettmann [214] arrived at the following dimensionless expression for F

_{E(l)}, the annual net export (export to the sea minus input from the sea) expressed as a fraction of upland loading:

^{2}= 0.94 with α = 0.3 month

^{−1}; Equation (3): r

^{2}= 0.85 with α = 0.3 month

^{−1}, ε = 0.69; Figure 13A,B). Moreover, the relationships make intuitive sense: the fraction of nitrogen input that is exported (denitrified) decreases (increases) as the transport timescale increases. This is logical because the longer nitrogen spends within an estuary, the more opportunity for it to incur denitrification and other loss processes, leaving less for export.

_{hab}is habitat averaged algal biomass concentration, B

_{in}is algal biomass concentration flowing into the habitat, μ

_{eff}is the effective phytoplankton growth rate (accounting for depth-averaged algal growth, respiration, zooplankton grazing, and clam grazing), and τ

_{tran}is transport time. Operative assumptions included a vertically well-mixed water column and steady-state conditions. A similar equation was derived also for habitat averaged phytoplankton net productivity. Results from the simple models (Figure 13C,D) showed clearly that the hypothesis does not always hold: Hydrodynamically “slower” habitats can be less productive than “faster” ones if benthic grazing is strong enough to render the effective phytoplankton growth rate negative. Further, it was evident that the range of possible outcomes broadens with longer transport times. Therefore, since it is difficult to predict the response of non-native bivalves to restoration, the ultimate functioning of created habitats—especially those with long transport times—is highly uncertain. This simple model was able to clearly demonstrate that widely held intuitive, management-relevant conceptual models of phytoplankton dynamics do not always hold—and can, in fact, be reversed—in the presence of strong benthic grazing. This same lesson could have been demonstrated with more complex 1D, two-dimensional (2D) or 3D models, but the ultra-simple timescale-based form of Equation (4) isolated the salient processes and conveyed the message more effectively than more complex approaches might have.

^{2}= 0.56). To aid in their global-scale estimates of denitrification, those authors then used this simple empirical model (Equation (5)) to estimate denitrification in lakes and reservoirs, and developed a similar estuary-specific relationship (% N removed = 16.1 (Water Residence Time)

^{0.30}, r

^{2}= 0.62). In this case, “water residence time” was likely defined and calculated in more than one way, given the large number of sources contributing to the dataset [226]. Regardless, and in spite of the gross simplification of complicated and site-specific transport processes by the single parameter “water residence time”, strong and useful relationships were obtained. Like Dettmann’s [214] relationship (Equation (3) above), the empirical models of Seitzinger et al. [54] are also consistent with intuition: as time spent by imported nitrogen within a water body increases, the longer the time available for processing and biogeochemical removal of that nitrogen.

#### 3.2.3. Simple Ecological Models Using Physical and Biogeochemical Timescales

_{in}is the phytoplankton biomass concentration entering a water body at the upstream boundary; B(x) is phytoplankton biomass at distance x downstream from the inlet (if the length of the domain is x, then B(x) is the same as B

_{out}, the concentration exiting the domain at the downstream boundary); μ

_{growth}and μ

_{loss}, respectively, are the algal specific growth and combined in situ loss (e.g., grazing, senescence, sedimentation) rates (1/time); and u is the transport velocity (length/time). Substituting in timescales for advective transport (τ

_{tran}=x/u), growth (τ

_{growth}=1/μ

_{growth}), and loss (τ

_{loss}=1/μ

_{loss}), and combining timescales into ratios, they arrived at the following dimensionless relationship:

_{out}) is a function of five parameters and variables; whereas the dependent variable in Equation (7) is a function of only two, allowing the relationship to be plotted (and, importantly, visualized) on a 2D surface (Figure 15 herein). Equation (7) and Figure 15 provide a simple tool for explaining why phytoplankton biomass can have a variety of relationships with transport time: biomass (${B}_{out}^{\ast}$) increases with time spent in a water body (i.e., moving rightward in Figure 15) if growth is faster than in situ loss (${\tau}_{loss}^{\ast}>1$), but decreases with transport time (${\tau}_{tran}^{\ast}$) if loss is faster than growth (${\tau}_{loss}^{\ast}<1$). If growth and aggregate loss rates are similar (${\tau}_{loss}^{\ast}\approx 1$), biomass does not change much while inside the water body (${B}_{out}^{\ast}\approx 1$), regardless of the transport time. In summary (and contrary to the intuition of some), transport time does not determine whether phytoplankton biomass increases or decreases within an aquatic system; rather, the growth-loss balance (represented by ${\tau}_{loss}^{\ast}$) does [23]. The reader is referred to a recent publication by Wang et al. [24], who developed an analytical model for downstream phytoplankton concentration in a 1D advective system, going beyond the model in Equations (6) and (7) by incorporating a non-linear reaction term (e.g., to incorporate the effects of self-shading or phytoplankton-dependent grazing). Reducing to Equation (6) above under simplified conditions, that model has two primary components—water age and accumulative growth—and agrees well with observations in the James River.

_{e}for longitudinal transport driven by gravitational circulation, τ

_{v}for vertical exchange, and τ

_{b}for consumption), creating timescale ratios, and substituting those ratios into their 1D equation, Shen et al. [21] arrived at the following predictor of bottom layer DO concentration, c:

_{s}is surface DO concentration, ${\tau}_{b}^{\ast}=\frac{{\tau}_{b}}{{\tau}_{v}}$, and ${\tau}_{e}^{\ast}=\frac{{\tau}_{e}}{{\tau}_{v}}$. (Equation (8) also incorporated the assumption that bottom and surface DO were equal at the estuary mouth.) ${\tau}_{b}^{\ast}$ (${\tau}_{e}^{\ast}$) represents the competition between consumption (gravitational circulation) and vertical exchange processes. Equation (8) succeeded in reducing the expression for c to a problem with only three independent variables. The relationship governing dimensionless bottom DO (c/c

_{s}) could thus be plotted in two dimensions, and the influence of the governing processes on the development (or avoidance) of hypoxia could be visualized (Figure 16). Notwithstanding the simplicity of Equation (8), estimates of bottom DO from this model compared well with observations (Figure 17), demonstrating how a complex hydrodynamic-biogeochemical problem could be broken down to a quantitatively accurate and illustrative algebraic relationship involving three timescales.

#### 3.3. Assessing Relative Speeds or Dominance of Processes

_{hyp}to water residence time. Akin to the DO consumption timescale τ

_{b}of Shen et al. [21], τ

_{hyp}was calculated as the ratio of an initial oxygen concentration to a volumetric oxygen consumption rate and represents the biogeochemically driven time to hypoxia occurrence. Residence time was taken to represent the time of restricted oxygen supply (i.e., how long biogeochemical consumption can operate uncountered by supply). The authors stated that γ “relates the two factors contributing to hypoxia generation—net biochemical oxygen consumption and restricted supply of oxygen, which is related to water residence time” [132]. They hypothesized that γ must be less than 1 for hypoxia to occur because, however slow oxygen consumption may be, hypoxia may still develop if hydrodynamically driven oxygen supply is impeded for an adequately long period of time. On the other hand, if oxygen consumption is rapid, hypoxia may be prevented if residence times are very short and oxygen is thus supplied on a frequent basis. Fennel and Testa [132] tested their hypothesis by estimating τ

_{hyp}and residence time for nine hypoxic estuary and shelf systems (see Figure 18 herein), finding that indeed γ < 1 (biogeochemical depletion is faster than replenishment) for the majority of hypoxic systems studied. (The non-conformance of two systems—the Gulf of St. Lawrence and the Namibian shelf—was explained by an assumed, uniformly applied initial oxygen concentration that was likely too high for those two environments due to the importance of low-oxygen source waters.) The implementation of timescales thus allowed the authors to capture a great deal of the physical-biogeochemical complexity surrounding hypoxia development and distill it down to a simple ratio that performs well in describing hypoxia occurrence.

- Estuarine nitrogen processing: In their studies covering several European estuaries, Middelburg and Nieuwenhuize compared water “residence time” estimates to turnover times for particulate nitrogen, nitrate, ammonium [90], and amino acids [123], providing insight into which nutrient forms may become limiting [90] and whether individual forms will be significantly modified during transport through an estuary [90,123].
- Hypoxia development in a tidal river: In their study of the effect of water diversion structures on water quality in a complex, heavily managed tidal environment, Monsen et al. [230] compared 2D model-computed e-folding flushing times to half-lives for biological oxygen demand (BOD) [231]. They found that when a physical barrier was installed on a branch of the San Joaquin River (CA, USA), consequently forcing all flow through the mainstem, flushing times on the mainstem could decrease enough (relative to BOD half-life) to prevent the development of hypoxia, a frequent occurrence in a deep portion of the mainstem San Joaquin.
- Nutrient processing on shelves and export to the open ocean: Sharples et al. [210] compared their global-scale, latitudinally varying estimates of continental shelf residence times (Figure 14A herein) with nutrient processing times (assumed independent of latitude) in a discussion of which shelf regions would be expected to experience more (middle to high latitudes) or less (low latitudes) nitrate removal before exchange with the open ocean occurs.
- Development of a unique estuarine bacterial community: In their study of the Parker River Estuary and Plum Island Sound (MA, USA), Crump et al. [91] studied the conditions for the development of a unique community of estuarine bacterioplankton, as opposed to the advected populations of riverine or marine origin that were prevalent in the estuary. They compared water residence times and bacterial doubling times across seasons and the salinity gradient, finding that a local estuarine community developed at intermediate salinity only in the summer and fall, when water residence time was much longer than average doubling time, thus allowing the local community ample time to develop. In contrast, no local bacterial community developed in spring, when residence time was similar to average doubling time—apparently short enough to prevent the development of new estuarine bacterioplankton populations [91].
- Benthic control of phytoplankton biomass: Several authors have compared benthic grazing timescales to transport and/or phytoplankton growth timescales to understand controls on estuarine aquaculture potential [134] or phytoplankton biomass [25,61,124,232,233,234]. Extending the conceptual model of Dame [233] (who expanded that of Smaal and Prins [234]), Strayer et al. [232] presented a graphical conceptual model (Figure 19A) of phytoplankton regulation as a function of hydrologic residence time on the horizontal axis and bivalve clearance time (i.e., time for a bivalve population to clear the overlying water column of phytoplankton through their pumping) on the vertical axis. They described three regimes within that 2D timescale space, each associated with a different control on phytoplankton biomass (advective loss, bivalve grazing, or phytoplankton growth), stating that the regime boundaries would vary as a function of phytoplankton net growth rate. The Strayer et al. [232] conceptual model (Figure 19A) was used to show how bivalve clearance rates changed as a function of bivalve invasion or population decline. The Strayer et al. [232] conceptual model was later extended through (1) the generalization of the benthic grazing timescale to include potentially any in situ loss process and (2) normalization of the loss and transport timescales by the algal growth timescale (Figure 19B) [23]. The latter model was derived from the simple, dimensionless expression in Equation (7), was consistent with the Strayer model control domains, and showed that the regime boundaries are in fact defined by two timescale ratios, i.e., at ${\tau}_{loss}^{\ast}$ = 1, ${\tau}_{tran}^{\ast}$= 1, and ${\tau}_{loss}^{\ast}$ = ${\tau}_{tran}^{\ast}$ (see description in Section 3.2.3). These conceptual models, together, demonstrate the utility of timescales (and their ratios) in understanding and delineating the conditions under which an ecosystem response (e.g., algal biomass accumulation) is controlled by one of several processes.

#### 3.4. Evaluating Connectivity

**x**within a water body. Age can be conceptualized with a clock attached to a particle, the clock beginning to tick when the particle enters the water body (or at the moment of the particle’s birth [140]); the age is the time noted at the instant the particle arrives at location

**x**. With partial age, on the other hand, every water particle has several clocks (one for each subregion) rather than one, and only one clock is ticking at a time, depending on the subdomain in which the particle is located [140]. Unlike the traditional concept of age, which provides only time spent in the system generally before reaching

**x**, partial age provides information on the histories of particles and “some knowledge of the paths followed by the particles to reach a given region” [140]. The authors applied this approach to the problem of ventilation of the world’s deep oceans by water parcels after they touch the surface. Those authors defined subregions of the world ocean (Figure 21A) and developed connectivity matrices based on simulations with a global ocean circulation model (Figure 21B). Manning et al. [103] developed a similar connectivity matrix for the Gulf of Maine based on the analysis of real drifter tracks. The reader is also referred to the work of Lin and Liu [141], who provided a method for computing “partial residence times” (i.e., the amounts of time spent by a particle in different subregions before leaving a water body; see Figure 11 and Section 2.3.2). Other studies employing timescales in the investigation of connectivity include:

- Exposure of marine protected areas (MPAs) to shipping-related pollution: Delpeche-Ellmann et al. [56] analyzed the paths of GPS-tracked surface drifters released in the Gulf of Finland’s main shipping fairway, providing insight into which MPAs on the edges of the Gulf are most likely to be affected by pollutants originating in the fairway, as well as timescales for transport to the MPAs. The transport timescales provide information for environmental managers regarding the time available to respond to pollutant spills and contain them before they reach MPAs.
- “Material connectivity”: Oldham et al. [229] noted that, in the field of hydrology, there have been numerous efforts at characterizing hydrological or hydraulic connectivity between landscapes; whereas, to their knowledge, there had been no attempts to “characterise connectivity in terms of the ‘effectiveness’ of transferring material,” a notion which those authors termed “material connectivity.” They argued that material connectivity must account for both physical transport and biological or chemical processing, since two environments may have strong hydrological connectivity between them but, if material carried by the water undergoes significant removal during transit, the material connectivity may be poor. The ratio of a transport timescale τ
_{tran}to a reaction or “material processing” timescale τ_{rxn}—termed the Damköhler number (Da) in the chemical engineering literature and generalized by Oldham et al. [229]—was proposed to capture the conditions under which material connectivity is strong or weak. For example, when reactions remove a constituent during transit and Da = τ_{tran}/τ_{rxn}>> 1, transport is very slow compared to in situ loss processes; the constituent material will be substantially lost during transport, resulting in material disconnectivity even under conditions of hydraulic connectivity. On the other hand, if Da << 1, transport is very fast compared to processing, the material behaves essentially conservatively, and material connectivity is therefore strong. Relatedly, Brodie et al. [237] estimated residence times for freshwater and several water quality constituents exported to the Great Barrier Reef and made the case that residence times of pollutants in that system are potentially much greater than those of the water itself, contrary to common assumptions. - Harmful algal bloom (HAB) initiation in geometrically complex estuaries: Qin and Shen [199] performed both theoretical analyses and 3D numerical modeling to understand the roles of estuary geometry and hydrodynamic connectivity between estuary subregions in determining where HABs are first observed to begin. (For their species of interest, a density of 1000 cells/mL was defined as the HAB threshold). Their idealized analytical model (in which residence time was a key parameter) predicted that the location of first HAB occurrence in a hydraulically interconnected system of two water bodies (e.g., the mainstem of a tidal river and its tributary) is determined by the relative ratios of residence time to volume (τ
_{r}/V) for the two water bodies. A HAB was predicted to be observed first in the water body with the larger τ_{r}/V ratio, i.e., the longer residence time and/or smaller volume. Results from numerical experiments with a 3D transport-reaction model of the lower James River (Figure 22A) were consistent with the theoretical model, demonstrating that—regardless of the initial source location of cells—flushing (represented by model-computed τ_{r}) and subregion volume V are indeed dominant factors determining where a HAB is first observed. Specifically, their 3D simulations were initiated with a non-zero algal concentration in the bottom layer of the lower James River mainstem (see Figure 22B), to represent cyst release in that region; initial algal concentrations were zero elsewhere, including in the tributaries. Nonetheless, only a few days were needed for concentrations in the tributaries to be higher than in the mainstem, initiated by cell transport from the mainstem driven by estuarine circulation. Simulated bloom-level densities ultimately developed first in the tributaries (Figure 22D), as predicted by the theoretical model. Both numerical and analytical results are consistent with, and help explain, first occurrences of toxic algal blooms in that system, which are frequently observed in the Lafayette River, a relatively small tributary to the James with a long residence time.

#### 3.5. Comparing Systems across Space or Time

- Ecosystem responses to management actions: To understand changes in hydrodynamics, water quality, and ecosystem processes induced by the installation of a temporary physical salinity-intrusion barrier in the Sacramento-San Joaquin Delta (CA, USA), Kimmerer et al. [62] employed high-speed boat-based isotope mapping (same approach as in [173]) to produce spatial patterns of water age with and without the barrier. Benthic grazing turnover time (i.e., time for benthic bivalve population to filter through the entire overlying water column) was also estimated as one measure of ecosystem response to related changes in salinity.
- Variability and drivers of estuarine flushing: In order to investigate the sensitivity of flushing in Mobile Bay (AL, USA) to river flow, wind, and baroclinic forcing, Du et al. [243] estimated both bulk (e-folding flushing time) and spatially variable (freshwater age) transport timescale metrics using a 3D numerical model. Deriving a simple empirical flushing time–discharge relationship based on a set of sensitivity runs and comparing to previous estimates based on a 2D depth-integrated model [244], they concluded that baroclinic processes reduce flushing times by approximately half. The spatial and temporal transport time patterns produced in these analyses (Figure 24 herein) could serve as valuable information toward interpreting variability in water quality and ecosystem processes.
- Retention of harmful algal cells: Ralston et al. [127] employed a 3D coupled hydrodynamic-biological model of the Nauset Estuary (MA, USA) to explore the physical and biological processes controlling recurrent blooms of the toxic alga Alexandrium fundyense. Implementing an e-folding approach to calculate A. fundyense residence times under a range of conditions, they explored the influence of swimming behavior, spring-neap tidal phase, wind, and stratification on retention of cells in one of the estuary’s salt ponds, concluding that all four processes are major factors determining retention. Although growth and mortality were turned off in these simulations, the computed residence times are particularly holistic, in that they not only include 3D hydrodynamic processes but also organism behavior (see Figure 25 herein).
- Ecosystem transformations by bivalves: The graphical timescale-based conceptual model of Strayer et al. [232] (see Figure 19A and Section 3.3 above) describes the evolution of five aquatic ecosystems in response to major changes in bivalve grazer populations. The process controlling phytoplankton was shown to be capable of shifting between advection, grazing, and algal growth as a function of either bivalve invasion or population decline.
- Hydrologic influence on zooplankton communities: Augmenting an 18-year field dataset with calculated water residence times, Burdis and Hirsch [33] explored several potential environmental drivers of zooplankton community structure in a natural riverine lake. As hypothesized, they found that water residence time was the most important driver of zooplankton abundance and community structure. Similar to Peierls et al. [57] and Hall et al. [238], use of a transport timescale allowed these authors to collapse spatial location and temporally variable hydrology into a single variable associated with each sample.

## 4. Discussion

#### 4.1. The Timescale “Tower of Babel”

#### 4.2. Holism of Timescales

^{2}/K, respectively) tend to be relatively atomistic (see filled circle in Figure 26). Consequently, classical timescales have proven useful in estimating the relative magnitudes of the terms in the governing equations of eco-hydrodynamics [139] or in comparing the speeds of different processes operating in an aquatic system (Section 3.3). It should be noted that while these classical algebraic timescale expressions may have the advantage of being mathematically simple, the methods to quantify the necessary parameters can be non-trivial.

^{2}/K allow for the direct comparison of the two processes. Whereas, residence time derived from a realistic 3D transport model will likely incorporate both processes into it, communicating their combined effect; this is something a classical timescale usually cannot achieve, unless one process is far more dominant than all others. Thus, atomistic timescales may bear little quantitative resemblance to holistic timescales [87,113], since they exclude the subtle and complex interplay between multiple processes operating in real systems and captured by realistic models [139]. Process attribution is perhaps less easy with a numerical PDE-based method than with simple algebraic expressions, but it is not impossible. It simply requires a different approach, such as sensitivity analyses that turn individual processes on or off, or coefficients up or down (e.g., [101,127]; triangles and five-pointed stars in Figure 26).

_{R}is the volumetric flow rate. Andutta et al. [22] also derived similar closed-form relationships for location-specific residence time and exposure time and for the water renewal time as well (not shown). With these expressions (see open circle in Figure 26), one can buy two processes for barely more than the calculational price of one!

_{decay}, where μ

_{decay}is the specific decay rate and is assumed positive), and ${\tau}_{hydro}$ is the time that would be taken by a conservative particle to leave the domain under hydrodynamic forcing only. These timescales satisfy [118]:

^{∗}= 1/${\tau}_{res}^{\ast}$ is an effective loss rate resulting from the combination of hydrodynamic transport processes and non-transport decay processes. It is possible also to express the combined effect of decay and oscillatory transport between a domain and its adjacent environment (as captured by the exposure time) with a simple expression similar to Equation (11) above [118]. Other moderately complex mathematical methods for estimating timescales could involve simpler numerical models, such as 1D models (e.g., [160,200]; filled square in Figure 26), a 2D depth-averaged model incorporating tides, water diversions, and river flow but not wind or stratification (e.g., [87]; open square in Figure 26), or a 1D physical-biological model (e.g., [189]; “$\times $“ in Figure 26).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Note

## Appendix A

**Figure A1.**Depiction of plug flow (velocity U is uniform over the flow cross section; longitudinal diffusion K

_{x}is zero) in an idealized channel, for which cross-sectional area A is longitudinally uniform, channel length is L, and volumetric flow rate Q (and therefore U) is constant and positive.

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**Figure 1.**Cartoon depicting the relationships between water parcels, particles, and molecules, cells, etc., as defined herein. A water parcel is a mixture of particles, the most numerous of which are pure water particles. A particle is a material point at which many atoms, molecules, cells, etc., of an individual constituent or aggregate are concentrated. (Following Deleersnijder et al. [63]).

**Figure 2.**Cartoon depicting a water parcel as it is transported through an aquatic ecosystem between times t

_{1}and t

_{2}. The water parcel’s volume is constant, but its shape is not. Due to diffusion (magenta arrows), the particles contained within the parcel at t

_{2}are not the same as the particles contained in the parcel at t

_{1}. Each “particle” is composed of multiple molecules, atoms, or cells of a particular constituent or aggregate. (Following Deleersnijder et al. [63]).

**Figure 3.**Schematic depicting the relationships between space- and time-dependent age, (strict) residence time, transit time, and exposure time, following Zimmerman [96], Delhez [98], Shen and Haas [121], Viero and Defina [130], Andutta et al. [22], and others. The dots represent successive locations for a single particle following a trajectory passing through locations

**x**

_{i}at times t

_{i}.

**x**

_{0}and t

_{0}are the initial location and time.

**Figure 4.**Simplified depiction of advective, river-driven estuarine flushing, idealized as plug flow (perfect mixing over the flow cross section, zero mixing in the streamwise direction). Panels (

**A**–

**D**) follow a progression through time of river water gradually replacing estuarine water initially present at time t

_{0}. V is estuarine volume, and Q is river discharge. River water is depicted as magenta; original estuarine water is blue; water outside the estuary mouth is orange. Gray dashed lines represent upstream and downstream boundaries of the estuary.

**Figure 5.**Based on a series of 45-day numerical particle transport simulations of Galveston Bay (TX, USA) by Rayson et al. [100]: (

**a**) e-folding flushing times for particles initialized on each day for a period spanning mid-March to mid-July 2009. Triangles represent start times for simulations used for exponential fits shown in (

**b**), with the blue triangle representing a high discharge period and the red triangle representing a low discharge period. (

**b**) Example exponential fits for particle -tracking simulations with the three different start times indicated by the triangles in (

**a**). Blue (red) dots and and dashed lines represent the model output and curve fit, respectively, for high (low) discharge periods. (

**c**) RMSE (root mean square error) of the exponential best fit for all times modeled. (Reproduced with permission from M. Rayson, Journal of Geophysical Research: Oceans; published by Wiley, 2016.).

**Figure 6.**Simplified depiction of the e-folding flushing time, driven by river, tidal, and/or other flushing processes. Panels (

**A**–

**D**) follow a progression through time of C, the estuarine concentration of a tracer or other constituent. The e-folding mathematical construct is based on the assumption of perfect mixing within the water body of interest (in this case, the estuary). Dark gray dashed lines represent upstream and downstream boundaries of the estuary. Dark purple represents initial estuarine water. Light gray represents replacement water.

**Figure 7.**Results from the drifter field studies of Manning et al. [103] in the Gulf of Maine. Upper Panel: calculated residence times in days (italics), low frequency speed in cm/s, and direction in degrees True. Number of observations (“nobs”) is in parentheses. Lower Panel: tracks of drifters entering waters offshore Cutler Maine from the northeast and heading southwest in the Eastern Maine Coastal Current. Transit time (7.3 d) is the mean time for drifters to traverse the region outlined in purple. (Modified from Manning et al. [103], with permission from Elsevier).

**Figure 8.**From the drifter studies of Storlazzi et al. [102] in Faga’alu Bay (American Samoa): (

**A**) a deployed drifter; individual drifter tracks, with orange symbols representing drifter deployment locations and red circles representing drifter recovery locations for conditions of (

**B**) calm and (

**C**) strong winds. (Modified from Storlazzi et al. [102].) From the drifter field studies of Pawlowicz et al. [104]: tracks for drifters released in (

**D**) the northern Strait of Georgia (SoG) and (

**E**) Victoria Sill in the Salish Sea. Statistics in legends represent the number of tracks for each category; when two numbers are provided separated by a slash, the first is number of tracks, and the second is the number of unique drifter IDs [104]. “JdF” is “Juan de Fuca” Strait. (Modified from Pawlowicz et al. [104] and licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)).

**Figure 9.**(

**A**) Water age “τ”, (

**B**) chlorophyll a fluorescence, and (

**C**) nitrate, based on concurrent mapping by Downing et al. [173] aboard a high-speed boat in the Cache Slough Complex of the Sacramento-San Joaquin Delta (USA). Low (high) fCHLA generally corresponded with small (large) τ. Nitrate had roughly the opposite pattern relative to τ. For (

**D**) Prospect Slough and (

**E**) the Sacramento Deep Water Ship Channel (“DWSC”), fits to an exponential relationship between change-in-nitrate versus change-in-water-age along boat tracks, used to estimate total-ecosystem net nitrate uptake rate. Estimated uptake rates were 0.039 d

^{−1}in Prospect Slough and 0.006 d

^{−1}in the DWSC. (Adapted from Downing et al. [173] (https://pubs.acs.org/doi/10.1021/acs.est.6b05745), with permission from American Chemical Society. This is an unofficial adaptation of an article that appeared in an ACS publication. ACS has not endorsed the content of this adaptation or the context of its use. Further permissions related to the material excerpted should be directed to the ACS).

**Figure 10.**Spatially variable residence times computed by Defne and Ganju [101] with coupled 3D hydrodynamic and particle tracking models applied to Barnegat Bay-Little Egg Harbor (NJ, USA). The scenarios shown are (

**a**) tidal forcing only, (

**b**) tidal plus remote coastal forcing, (

**c**) like (

**b**) but with river flow added, (

**d**) like (

**c**) but with meteorological forcing added. Two inlets—Little Egg Inlet at the southern end and Barnegat Inlet near the center—connect the ocean and estuary (see Figure 1 in [101] for detailed site map). (Modified from Defne and Ganju [101]).

**Figure 11.**Lin and Liu’s [141] (

**A**) bathymetry map of Jiaozhou Bay (China), showing six subregions (ω

_{1}–ω

_{6}) in which partial residence times (PRTs) were calculated in (

**B**–

**H**), and seven release points (stars) for which PRTs in the subregions are shown in (

**B**). (

**B**) For particles initiated at each of seven locations, PRTs shown are time spent in each of six subregions before leaving the bay. For a given release location, the sum of the PRTs equals the total residence time within Jiaozhou Bay. (

**C**–

**H**) Spatial maps for each subregion representing time spent in the subregion for particles initiated at every location in the domain. Dashed lines represent the boundaries of each subregion. (Adapted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Ocean Dynamics, Partial residence times: determining residence time composition in different subregions, Lin and Liu, 2019. https://www.springer.com/journal/10236).

**Figure 12.**(

**A**) A global map of coastal residence times (CRTs) simulated by Liu et al. [207] using high-resolution, coupled global ocean-ice models and a novel variation on an age tracer approach; (

**B**) simple model of CRT as a function of Coriolis parameter f and a geometric parameter χ, which is the ratio of total coastal system volume to total open boundary area. (Modified with permission from Xiao Liu, Geophysical Research Letters; published by Wiley, 2019).

**Figure 13.**Dettmann’s [214] simple models for fraction of upland nitrogen loading to an estuary that is (

**A**) exported (Equation (2) herein) and (

**B**) denitrified (Equation (3) herein), expressed as functions of “freshwater residence time” and fit to data for several estuaries. “γ” in Dettmann’s [214] denitrification plot (

**B**) is referred to as “ε” in Equation (3) and the text herein. (Modified from Dettmann [214].) Calculations of habitat-averaged phytoplankton (

**C**) biomass and (

**D**) productivity based on Lucas and Thompson’s [215] simple models expressed as a function of transport time (Equation (4) herein for algal biomass). (Modified from Lucas and Thompson [215]).

**Figure 14.**Based on the global scale, simple mathematical modeling of Sharples et al. [210]: (

**A**) average residence time “T

_{res}” on the continental shelf; (

**B**) estimated proportion of riverine DIP (dissolved inorganic phosphorus) exported to the open ocean; (

**C**) estimated annual DIP mass export to the open ocean. The authors performed the same calculations for dissolved inorganic nitrogen and provided uncertainty estimates (not shown here). (Modified with permission from J. Sharples, Global Biogeochemical Cycles; published by Wiley, 2017).

**Figure 15.**Contours of ${B}_{out}^{\ast}$, the ratio of outgoing algal biomass concentration to incoming concentration, as a function of two dimensionless parameters, ${\tau}_{loss}^{\ast}$ (the ratio of the algal loss timescale to the growth timescale) and ${\tau}_{tran}^{\ast}$ (the ratio of the transport timescale to the algal growth timescale). Based on the simple, timescale-based mathematical model of [23], Equation (7) herein. (From Lucas et al. [23]).

**Figure 16.**Contours of c:c

_{s}(ratio of bottom layer dissolved oxygen concentration (DO) to surface layer DO), as a function of two dimensionless parameters, ${\tau}_{b}^{\ast}$ (the biochemical consumption timescale normalized by the vertical exchange timescale) and ${\tau}_{e}^{\ast}$ (the timescale for transport driven by gravitational circulation normalized by the vertical exchange timescale). Based on the simple, timescale-based mathematical model of Shen et al. [21] (Equation (8) herein). Rectangular regions delineate regimes associated with control of DO by particular processes and/or likelihood of hypoxia. (Reuse and minor adaptation from Shen et al. [21], with permission from Wiley. © 2013, by the Association for the Sciences of Limnology and Oceanography, Inc.)

**Figure 18.**Hypoxia timescale versus residence time for several hypoxic estuarine and shelf systems, as estimated by Fennel and Testa [132]. Systems falling below the diagonal 1:1 line are consistent with the authors’ hypothesis that γ, the ratio of the hypoxia timescale to the residence time, is less than unity for hypoxia to occur. Systems analyzed: (1) Pearl River Estuary (China); (2) East China Sea; (3) Northern Gulf of Mexico; (4) Long Island Sound (USA); (5) Chesapeake Bay (USA); (6) Northwestern Black Sea; (7) Baltic Sea; (8) Gulf of St. Lawrence (Canada); (9) Namibian Shelf. (Redrawn from Fennel and Testa [132] with the permission of K. Fennel.)

**Figure 19.**(

**A**) The conceptual model of Strayer et al. [232], which extended that of Dame [233] and described three domains of control of phytoplankton (i.e., by advection, bivalve grazing, or phytoplankton growth). Strayer et al. [232] explained that domain boundaries may be different from those shown, depending on phytoplankton net growth rates. Arrows describe how bivalve clearance times in five estuarine, river, and stream ecosystems changed over time as a result of bivalve invasion or population decline. Ecosystems are the following: HR, the Hudson River (NY, USA) after the Dreissena polymorpha (zebra mussel) invasion; SB, Suisun Bay (CA, USA) after invasion by Potamocorbula amurensis; CB, the Chesapeake Bay (USA) after the decline of oyster populations; ENAS, a typical eastern North American stream after unionid decline; and PR, the freshwater tidal Potomac River (MD, USA) after the Corbicula fluminea invasion. (Redrawn from Strayer et al. [232] with the permission of D. Strayer.) (

**B**) Reprise of Figure 15 with shaded areas added to describe domains of control on phytoplankton biomass [23], extending the conceptual model of Strayer et al. [232] in panel (

**A**). Contours represent values of ${B}_{out}^{\ast}$, the ratio of outgoing algal biomass concentration to incoming concentration. (From Lucas et al. [23].)

**Figure 20.**(

**A**) Zoom-in of the computational mesh of De Brauwere et al. [95], showing subregions of the Scheldt Estuary referred to in (

**B**). Subregions were based on the compartmentalization of [235]. (

**B**) Connectivity matrix based on computations of “subdomain exposure times” with a 2D tracer transport model. Colors represent the relative time spent in a particular subregion numbered on the horizontal axis by tracer initialized in a subregion on the vertical axis. (Modified from De Brauwere [95], with permission from Elsevier).

**Figure 21.**(

**A**) Horizontal partitioning of the world ocean by Mouchet et al. [140] for use with a global ocean circulation model to evaluate connectivity between 30 different subdomains (each horizontal partition is split into three boxes in the vertical dimension, denoted by “s” for surface, “i” for intermediate, or “d” for deep in (

**B**)). (

**B**) Connectivity matrix showing computed “partial age” (a

_{i,j}) for all subdomains, i.e., the mean time spent by particles in any subdomain i (vertical axis) before reaching the subdomain of interest j (horizontal axis). Partial age is normalized by the mean (total) water age in the corresponding sub-domain. (Adapted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Ocean Dynamics, Partial ages: diagnosing transport processes by means of multiple clocks, Mouchet et al., 2016. https://www.springer.com/journal/10236).

**Figure 22.**(

**A**) Map of the lower James River (USA) and its tributaries [199]. From a 3D model simulation performed by Qin and Shen [199], algal cell densities (

**B**) specified as the initial condition (non-zero cell densities initially only in the bottom layer of the lower James River mainstem), and computed cell densities (

**C**) after 0.75 d; (

**D**) after 24.29 d, when average surface density of the entire Lafayette River first reached bloom levels (1000 cells/mL); and (

**E**) after 33.54 d, when the average surface density of the mainstem first reached bloom levels. These results are consistent with observations and with a simple theoretical model indicating that a simple parameter—the ratio of subregion residence time to its volume—can predict where harmful algal blooms are first observed [199]. (Modified from Qin and Shen [199], with permission from Elsevier).

**Figure 23.**From Peierls et al. [57], maps of the (

**A**) New River Estuary (NewRE) and (

**C**) Neuse River Estuary (NRE); for the (

**B**) NewRE and (

**D**) NRE, observation based ln(chlorophyll a) versus flushing time estimated with the “date-specific freshwater replacement method” [239]. (Adapted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Estuaries and Coasts, Non-monotonic Responses of Phytoplankton Biomass Accumulation to Hydrologic Variability: A Comparison of Two Coastal Plain North Carolina Estuaries, Peierls et al., 2012. https://www.springer.com/journal/12237).

**Figure 24.**Maps of computed vertical mean freshwater age in Mobile Bay for (

**A**) the dry season and (

**B**) the wet season, based on the 3D numerical modeling of Du et al. [243]. Timeseries of (

**C**) river discharge, (

**D**) wind speed, and (

**E**) computed freshwater age averaged over the main bay. For the age timeseries, surface water is gray, bottom water is black, and the vertical age difference is cyan [243]. (Modified with permission from J. Du, Journal of Geophysical Research: Oceans; published by Wiley, 2018).

**Figure 25.**Computed residence times for the toxic alga A. fundyense based on the 3D model of Ralston et al. [127] for a pond within the Nauset Estuary (MA, USA). The different bars represent a variety of swimming and forcing cases under spring and neap tide conditions. “Swim”: diel vertical migration up to 1/k

_{w}depth, where k

_{w}is the light attenuation coefficient. “Don’t swim”: no vertical migration. “Swim to surface”: diel migration to the surface. “Swim + barotropic”: diel vertical migration to 1/k

_{w}with barotropic physics (uniform water density and thus no stratification). “Swim + barotropic + no wind”: diel vertical migration to 1/k

_{w}with barotropic physics and zero wind forcing. Horizontal lines: the residence time for tidal exchange assuming a well-mixed pond (volume of pond/tidal volume exchange), shown for reference. (Redrawn from Ralston et al. [127] with the permission of D. Ralston).

**Figure 26.**Schematic of diagnostic timescale holism as a function of the mathematical complexity of the calculation method (for computed timescales only). Timescales based on simple algebraic expressions tend to be less holistic, but potentially more useful for purposes of assessing dominant processes (Regime A). Complex numerical models have the potential to produce highly holistic timescales (Regime C) as well as timescale estimates at high spatial and temporal resolution. The effective level of holism depends on the process richness captured by the model simulation. More holistic timescales may be less useful for disentangling the relative speeds (and potential dominance) of individual processes (“process attribution”). Moderately holistic timescales may be derived from moderately complex numerical models or methods (Regime B) or from complex models that exclude some important processes (mid-region of Regime C). Examples: ●—τ

_{adv}= L/U or τ

_{diff}= L

^{2}/K. ○—$\overline{{\tau}_{res}}$ and $\overline{{\tau}_{exp}}$ from Andutta et al. [22], Equations (9) and (10) herein. Timescales derived from the 1D models of Delhez and Deleersnijder [200] or Vallino and Hopkinson [160] ($\u220e$); the 2D depth-averaged model of Monsen et al. [87] (□); the 1D hydrodynamic-biological model of Delhez et al. [189] ($\times $); the 3D hydrodynamic and transport model of Gross et al. [174] ($\ast $); the 3D hydrodynamic and particle tracking model described by Defne and Ganju [101], with progressively more physical processes included (starting with white triangle progressing upward to black triangle; also see Figure 10 herein); the 3D hydrodynamic-ecological model of Ralston et al. [127] with progressively more physical processes and dinoflagellate swimming behaviors (starting with white five-pointed star up to black five-pointed star; also see Figure 25 herein).

**Table 1.**Processes operating in aquatic systems, associated native process rates and their units, and common mathematical expressions for their corresponding timescales. Scales combined with process rates to construct timescales include: L (length), L

_{z}(vertical length), V (volume), M (integrated mass within a water body), B

_{p}(phytoplankton biomass concentration), B

_{a}(areal biomass concentration), DO (dissolved oxygen concentration), η (nutrient concentration). Specific growth or decay rate μ may be positive (growth) or negative (decay). Decay rate μ

_{decay}is assumed positive. Unless specified otherwise, concentrations here are assumed volumetric. Timescale expressions shown here may be adjusted if available parameters or units are different from those shown.

Process | Native Process Rate | Units | Timescale | Relevant Citations |
---|---|---|---|---|

Diffusion/Dispersion/Mixing | Diffusion/Dispersion/Mixing Coefficient (K) | length^{2}/time | L^{2}/K | [42,49,52,59,147] |

Advection | Velocity (U) | length/time | L/U | [23,49,59] |

Flushing by river flow | Volumetric flow rate (Q) | length^{3}/time | V/Q | [87] |

Flushing by scalar flux | Mass flux rate (F) | mass/time | M/F | [42] |

^{1} Growth or decay | Specific growth or decay rate (μ) | 1/time | 1/|μ| | [25] |

^{2} Decay by one-half | Specific decay rate (μ_{decay}) | 1/time | ln(2)/μ_{decay} | [125] |

^{3} Growth by factor of 2 | Specific growth rate (μ_{growth}) | 1/time | ln(2)/μ_{growt}_{h} | [127] |

Sinking/settling | Sinking speed (w) | length/time | L_{z}/w | [59,60] |

^{4} Productivity | Areal Productivity (P_{a}) | biomass/(length^{2}-time) | B_{a}/P_{a} | [125] |

^{4} Benthic consumption | Grazing/Filtration/Clearance rate (BG) | length^{3}/(length^{2}-time) | L_{z}/BG | [23,25,62,124] |

Zooplankton grazing | Zooplankton community grazing rate (ZG) | biomass/(length^{3}-time) | B_{p}/ZG | [23,148] |

Oxygen consumption | Net oxygen consumption rate (C_{DO}) | mass O_{2}/(length^{3}-time) | DO/C_{DO} | [21,132] |

^{4} Nutrient uptake | Nutrient uptake rate (υ) | mass nutrient/(length^{3}-time) | η/υ | [90] |

^{1}This timescale is sometimes called an e-folding time or mean life [149] for decaying substances.

^{2}This timescale is typically called a “half-life.” If the decay rate carries a negative sign, then the applicable expression is ln(0.5)/μ

_{decay}.

^{3}This timescale is typically called a “doubling time.”

^{4}These timescales are sometimes referred to as “turnover” times.

**Table 2.**Compilation of transport timescales estimated in previous studies. Data is based on sources in “Author(s)” column. max(τ)/min(τ) is the ratio of the maximum timescale value to the minimum value for a water body and set of conditions. Q is volumetric flow rate. V is water body volume. M is total tracer mass. $\dot{M}$ is mass loading rate. L is length. U is mean velocity. U’ is average deviation from depth-mean velocity. A

_{o}is tidal amplitude. TEF is “total exchange flow” [122], an approach for estimating a salinity turnover time. “tc” is tidal cycles. Footnotes provide methodological information. Other specifics such as temporal or spatial averaging of parameters or timescales vary between authors; please see those publications for details.

Author(s) | Water Body | Time Period/Conditions | Timescale^{(approach)} | Value | $\frac{\mathit{m}\mathit{a}\mathit{x}\left(\mathit{\tau}\right)}{\mathit{m}\mathit{i}\mathit{n}\left(\mathit{\tau}\right)}$ |
---|---|---|---|---|---|

Jouon et al. [112] | SW lagoon of New Caledonia | Constant, moderate trade wind | V/Q ^{1} | 6.8 d | 2 |

Mean residence time ^{2} | 10.8 d | ||||

e-folding ^{3} | 11.4 d | ||||

Lemagie and Lerczak [110] | Yaquina Bay (USA) | Q = 10 m^{3}/s, A_{o} = 125 cm | TEF ^{1,4} | 3.96 tc | 26 |

Tidal prism ^{1,4} | 1.27 tc | ||||

Freshwater fraction ^{1,4} | 12.63 tc | ||||

Transit (e-folding) ^{2,4} | 32.6 tc | ||||

Flushing (e-folding) ^{2,4} | 5.16 tc | ||||

Monsen et al. [87] | Mildred Island (USA) | June 1999 (low flow) | V/Q ^{5} | 31-50 d | 17–28 |

e-folding ^{6} | 7.7 d | ||||

$M/\dot{M}$^{7} | 8.3–9.1 d | ||||

Mean age ^{6,8} | 1.8 d | ||||

Oveisy et al. [147] | Bay of Quinte (Canada) | Summer 2004 | e-folding ^{3,10} | 44 d | 14 |

V/Q ^{1,9,11} | 64 d | ||||

Residence time ^{2,10} | 52 d | ||||

Dispersion ^{3,10} | 1.7 y | ||||

Rayson et al. [100] | Galveston Bay (USA) | Mid–late April 2009 (peak flow) | Freshwater fraction ^{1} | ~10 d | 2 |

TEF ^{1,12} | ~20 d | ||||

Mean residence time ^{2,12} | ~20 d | ||||

e-folding ^{2,12} | ~20 d | ||||

Mean age ^{3,12} | ~20 d | ||||

Rayson et al. [100] | Galveston Bay (USA) | Late July 2009 (low flow) | Freshwater fraction ^{1,12} | ~200 d | 10 |

TEF ^{1,12} | ~20 d | ||||

Mean residence time ^{2,12} | ~25 d | ||||

e-folding ^{2,12} | ~50 d | ||||

Mean age ^{3,12} | ~30 d | ||||

Tartinville et al. [113] | Mururoa atoll Lagoon (French Polynesia) | Tides, wind, hoa inflow, stratification | L/U ^{1} | 8.3 d | 1113 |

L/U’ ^{1} | 5.3 d | ||||

Diffusion ^{13} | 5900 d | ||||

e-folding ^{2} | 114 d |

^{1}3D hydrodynamic model.

^{2}3D model with particle tracking.

^{3}3D model with tracer(s).

^{4}Based on power law regression of computed timescales as a function of discharge and tidal amplitude.

^{5}2D hydrodynamic model.

^{6}2D model with particle tracking.

^{7}2D model with tracer.

^{8}Mean of average ages for two locations and two time periods.

^{9}Observations.

^{10}Mean of timescales calculated for individual tributary inflows.

^{11}Based on total discharge from all main tributaries.

^{12}Estimated based on visual inspection of published figures.

^{13}Diffusivity based on Okubo [245].

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lucas, L.V.; Deleersnijder, E. Timescale Methods for Simplifying, Understanding and Modeling Biophysical and Water Quality Processes in Coastal Aquatic Ecosystems: A Review. *Water* **2020**, *12*, 2717.
https://doi.org/10.3390/w12102717

**AMA Style**

Lucas LV, Deleersnijder E. Timescale Methods for Simplifying, Understanding and Modeling Biophysical and Water Quality Processes in Coastal Aquatic Ecosystems: A Review. *Water*. 2020; 12(10):2717.
https://doi.org/10.3390/w12102717

**Chicago/Turabian Style**

Lucas, Lisa V., and Eric Deleersnijder. 2020. "Timescale Methods for Simplifying, Understanding and Modeling Biophysical and Water Quality Processes in Coastal Aquatic Ecosystems: A Review" *Water* 12, no. 10: 2717.
https://doi.org/10.3390/w12102717