A new metric for sunlight exposure in rivers, lakes, and oceans

Vertical motion is an important driver of sunlight exposure in aquatic environments, shaping the growth and fate of materials and organisms. We derive a simple model accounting for turbulent depth fluctuations of particles to predict the depth that contributes the most sunlight exposure (effective depth) as well as the single depth that, if measured at one place over time, produces the same total sunlight exposure as a moving particle (functional depth). Field measurements of light and depth in rivers using neutrally buoyant drifters and buoys validate our model. Effective depth varied from 0.1 to 1.5 m below the water surface and was ~ 30% of the overall water depth on average. Functional depth varied from 0.67 to 2.3 m and was ~ 50% of the overall water depth on average. Functional and effective depth are physically based concepts incorporating turbulent motion, spatial variability, and water clarity offering new approaches to characterize light exposure in aquatic environments.

Life in motion experiences the environment differently than life not in motion (Nathan et al. 2008;Doblin and Van Sebille 2016).For example, aquatic macrophytes experience a different thermal and light regime than phytoplankton drifting down a river (Gardner et al. 2020).However, there is no framework that combines the frequency and magnitude of environmental conditions experienced by organisms in motion to predict the most important, or effective, condition.
In aquatic environments, motion controls sunlight exposure, hereafter referred to as "light."Phytoplankton and other suspended materials are turbulently advected, exposed to highly variable light at different depths (Gardner et al. 2020).Rates of photoreactive process such as photosynthesis (by photosynthetically active radiation [PAR]) and photodegradation of organic matter and contaminants (by ultraviolet light [UV]) are influenced by turbulent depth fluctuations (Cory and Kling 2018;Köhler et al. 2018).However, measurements and models of light exposure are often approximated as the depth-integrated or mean light (Sellers and Bukaveckas 2003;Ochs et al. 2013) under the assumption of instantaneous and complete mixing rather than light weighted by the distribution of depths experienced by particles moving downstream.Furthermore, classic models of bloom formation in oceanography that incorporate vertical fluctuations assume phytoplankton spend equal time at all depths in the turbulent layer (Sverdrup 1953;Franks 2015).
While tracking the light exposure of individuals is challenging, it is possible to make approximations by combining a frequency distribution of a stochastic variable linked to a predictable process (Wolman and Miller 1960;Doyle et al. 2005).In the case of light exposure for moving particles, combining the frequency distribution of vertical depths with the lightdepth decay curve (Kirk 1994), we can predict the effective and functional depths (Fig. 1).Effective depth (z eff ) is the depth that contributes the most to the total light exposure over the water column over a given time period.Effective depth is a weighted distribution found by multiplying the light intensity at a given depth by the time spent at that depth.Functional depth (z fun ) is the depth at which a stationary measurement at a fixed depth over time would have the same total light exposure as a moving particle over the same time period.For simplicity, we use the term total light exposure instead of radiant exposure or cumulative energy incident on a particle over a given time period.
To derive analytical models for z eff and z fun , we adapted the concept of effective discharge, or the level of streamflow that moves the most sediment and shapes rivers in the long-term (Wolman and Miller 1960).Effectiveness has also been used to find the effective discharge for carbon loading, nutrient uptake, and habitat diversity in rivers (Doyle et al. 2005;Doyle and Shields 2008).
The concepts and models presented here apply to neutrally buoyant particles in aquatic environments with a reflective, noflux bottom boundary (Tennekes and Lumley 1972).Conceptualizing particle motion as a function of turbulent perturbations of flow (Tennekes and Lumley 1972;Kundu et al. 2016) or heterogeneous turbulent diffusivity (Thomson 1984;Visser 1997) both show turbulent stress and diffusivity are highest along the reflective bottom boundary and cause upward ejection of particles.Hydrodynamic, random walk models, and field measurements in marine systems show the concentration of neutrally buoyant particles with depth is approximately lognormal due to vertical gradients in eddy diffusivity and reflection of particles at a bottom boundary, or mixed layer boundary (Visser 1997;Martins et al. 2010).We expect depth distributions of neutrally buoyant particles in rivers to be more skewed than those in oceans because the riverbed is shallower and variable in topography, pushing particles up into the water column more frequently.Therefore, the lognormal distribution is a useful first approximation for the depth distribution of a neutrally buoyant particle in our derivation of z eff and z fun .
Here, the depth (z) of a moving particle is the stochastic variable approximated as a lognormal distribution with two fitted parameters; the location (μ) and shape (σ) parameters (Eq.1).
Light or downwelling irradiance (E d ), is the predictable process linked to the stochastic depth variable (Eq.2), where E 0 is downwelling irradiance just below the water surface, K d is the diffuse attenuation coefficient for downwelling irradiance (m À1 ), and z is depth (m) (Davies-Colley et al. 1984).
Multiplying the frequency distribution of depth by the lightdepth exponential decay curve, the peak of the resulting curve indicates the depth that makes the greatest contribution to the total light exposure of particles (Eq.3).We analytically solved for z eff by taking the derivative of E(z)f(z) with respect to z and setting it to zero (Eq.4).
The z eff model has three parameters μ, σ, and K d , and ω is the wright-omega function.The μ and σ parameters are dimensionless (Matta et al. 2010) and do not represent the depths, but rather the location and shape of the depth distribution, respectively.We can perhaps interpret μ as a proxy for the depth of central tendency and σ as a proxy for depth variability of neutrally buoyant particles and are likely influenced by the overall depth of the waterbody and the amount of turbulence (Brown et al. 2009).
The functional depth (z fun ) considers the full depth distribution sampled along a river.If a sensor is placed at z fun and light is measured over a day, theoretically, the total daily light measured by that sensor should be the same as the total light exposure of particles moving downstream.Measurements at a single location can therefore represent the total light exposure of moving particles, thereby linking fixed-site measurements with a Lagrangian perspective (Doyle and Ensign 2009;Ensign et al. 2017).To calculate z fun , Eq. 3 is numerically integrated over the depth distribution (Eq.5) referred to as L. L is then substituted into Eq. 1 (Eq.6) and z fun can be estimated with the same three parameters, K d , μ, and σ (Eq.7).z fun does not vary with E 0 .The z eff and z fun models assume parameters are constant over some transit time (e.g., 1 d) and some representative length of river and are inherently spatial, while other metrics in limnology, such as photic depth or compensation depth, only represent one point.

Methods
We evaluated z eff and z fun using modeled sensitivity analysis and field measurements of light and depth using neutrally buoyant drifters (HydroSphere, Planktos Instruments, LLC) and moored buoys.Data were collected on the Upper Mississippi River in Wisconsin, and the Neuse River in North Carolina between 2014 and 2016.Light was measured as illuminance, the light visible to humans in units of lux, as a proxy for more biologically relevant spectra such as UV or PAR.The drifters are 0.4 m diameter spheres responding only to turbulent eddies with characteristic length scales ≥ 0.4 m (Rutherford 1994;D'Asaro et al. 1996), and are not a perfect analog for small particles or solutes.In rivers, maximum eddy size is typically limited by depth (Rutherford 1994;Nadaoka and Yagi 1998;Jirka 2001) which was 1-10 m in our study reaches.Therefore, the largest turbulent eddies were 2-20 times greater than the drifter size suggesting our drifters can respond to multiple scales of turbulence.Our previous work showed drifters were an effective proxy for light exposure of moving particles.For more drifter details, see Supporting Information Fig. S2, Ensign et al. (2017), andGardner et al. (2020).
z eff And z fun were calculated for each drifter deployment according to Eqs. 4, 7 using parameters μ, σ, and K d .K d was estimated using the drifter measured light (lux) profile during each deployment, where K d is the slope of the linear regression between log(E d ) vs. z (Gardner et al. 2020).Parameters μ and σ were estimated by fitting the depth distribution from each deployment to a lognormal distribution using the maximum likelihood estimator.Drifter data confirmed depth distributions were best described as lognormal (Supporting Information Fig. S1).The μ and σ parameters were negatively correlated (R 2 = 0.84) (Supporting Information Fig. S3).
We modeled the sensitivity and potential range of z eff and z fun with different combinations of parameters μ, σ, and K d .z fun And z eff were calculated for 1820 permutations over parameter ranges informed by field measurements: μ from À1 to 2, σ from 0.2 to 2, and K d from 0.4 to 7 m À1 .The parameter combinations represent a wide range of conditions in water clarity and depth distribution parameters to be representative of conditions across rivers, reservoirs, and lakes.
For validation of z eff , we compare the predicted z eff using Eq. 4 and measured parameters μ, σ, and K d with the measured z eff from actual light observations at different depths from drifter deployments.When multiple drifters were deployed, they showed similar empirical z eff , suggesting z eff is not contingent on individual drifters or their specific flowpath (Supporting Information Fig. S4).Validating z fun requires comparing two independent measurements of total light exposure which should be equal; the total light measured by drifters and the total light measured at a fixed buoy at $ z fun in the same river over the same time period.We did not know z fun a priori to inform what depth to place sensors, therefore light (lux) sensors (HOBO Pendant) were fixed at different depths (0, 0.5, 1, 1.25, 1.5 m) from the water surface.The sensor that was closest to z fun estimated from drifter deployments was used to calculate total light exposure.All analyses were performed in R statistical software (R 3.3.4).Code is provided for z eff and z fun .

Results
Sensitivity analysis over a range of conditions revealed important properties of z fun and z eff .First, z eff and z fun are constrained to a relatively narrow range of depths.The modal z eff is 0.08 m, median 0.20 m, and there is a 96% probability that z eff will be less than 2 m across all combinations of water clarity (K d ), μ, and σ (Fig. 2A).Similarly, the modal z fun is 0.47 m, median 0.75 m, and there is an 87% probability z fun lies between 0.2 and 2 m (Fig. 2B).Second, z fun will always be deeper in the water column than z eff (Supporting Information Fig. S6).Finally, z fun and z eff are highly sensitive to μ when holding all other parameters constant (Fig. 3).When σ is low and μ is high, z eff and z fun will be at their deepest in the water column.Overall, sensitivity analysis suggests channel morphology (i.e., overall depth) may play a larger role than water clarity or the amount of turbulence in determining the depth within the water column that contributes most to total light exposure.
z eff And z fun calculated from drifter measurements confirmed the properties revealed by sensitivity analysis.z fun was always deeper than z eff and are linearly correlated (R 2 = 0.92) (Supporting Information Fig. S7).Observed z eff and z fun were typically less than 2 m but shifted deeper compared to simulations.This can be explained by the wider range of conditions used in simulations and the study rivers were relatively deep (maximum 10 m depth from field measurements) with relatively low K d values (0.61-2.3 m À1 ) skewing z eff and z fun observations deeper.Dividing z fun and z eff by the overall water depth calculated as the average of 3-10 field measurements of water depth during each deployment along the Neuse River, we estimate that z fun lies at the 53% (AE 19%) depth, z eff at the 33% (AE 17%) depth, and the arithmetic mean depth of moving particles at 70% (AE 20%) depth from the water surface.
Analytical models predicted z eff and z fun reasonably well.Comparing drifter and buoy measured total light measured at z fun over the same time periods (Fig. 4A), symmetric mean absolute percentage error (SMAPE) was 26% and the linear regression had an R 2 = 0.98 with a slope of 1.21 (p = 0.0002).There was also good agreement between measured z eff at the actual depth where most light exposure occurred and predicted z eff derived from the depth distribution and its fitted parameters (Fig. 4B), with root mean square error of 27.5 cm, SMAPE of 33%, and a linear regression with an R 2 = 0.59 and slope of 0.75 (p = 0.0005).Despite the low precision (AE 0.25 m) that fixed measurements provided for measuring light at the exact z fun , as well as assumptions that μ, σ, and K d were constant over time and space, analytical models were validated by field data.

Discussion
z eff And z fun may have practical and ecological implications for how to measure and understand photoreactive processes in aquatic environments.One practical implication of z fun is that the total light exposure of carbon, phytoplankton, and photo-reactive contaminants transported downstream can be approximated by simply measuring light at one fixed-depth over time.Our results suggest that z fun is most likely between  0.4 and 1.4 m from the water surface across a range of environmental conditions (Table 1) (see Supporting Information for heuristics to estimate z fun without a drifter).
The ecological relevance of z eff and z fun requires further exploration.In theory, reactions occurring at z eff should contribute the most to time-integrated rates of photoreactive processes, but this would be challenging to test in the field.Depths of ecological relevance in limnology and oceanography, such as photic depth, compensation depth, critical depth, are biological concepts linked to photosynthesis (Sverdrup 1953;Banse 2004;Franks 2015).However, z eff and z fun are physical concepts that link water clarity, spatial variability, and turbulent motion of particles along many kilometers of travel.z eff And z fun may apply to a wide range of photoreactive processes.Our results and models can apply to any wavelength(s) of light with corresponding measurements of diffuse attenuation coefficients (K d ).UV light may be of particular interest given its role in driving carbon photochemistry, degrading contaminants, and damaging organisms' cellular function (Cory et al. 2014;Häder et al. 2015).UV attenuates more rapidly with depth compared to longer wavelengths (Markager and Vincent 2000) and z fun and z eff for UV should be shallower and vary within a smaller range compared to lux or PAR.For life in motion, moderate depths ($ 1 /3 depth on average), rather than the water surface, make the greatest contribution to total light exposure.The concept of effectiveness approximates the dominant environmental condition, here the depth that contributed the most sunlight exposure, and may play a role in the fate of photoreactive organisms and materials in motion.
Table 1.Recommended depth from water surface to measure light to approximate the total light exposure of moving particles over a given time period.Depths were grouped into bins by water depth and water clarity and are based on the z fun model.Mean particle depth represents the arithmetic mean particle depth of the depth distribution (which is less than the overall depth of the waterbody).Numbers in parentheses represent standard deviation (SD) estimated from all simulated values within each bin.For a similar table of z eff , see Supporting Information Table S1.

Fig. 1 .
Fig. 1.Conceptual figure of effective and functional depths.(A) Effective depth is the peak of the frequency distribution of depths of a moving particle multiplied by the light attenuation curve of the waterbody.(B) Functional depth integrates the effective depth curve and defines the depth, which if held constant over time, produces the same total light exposure of a moving particle.

Fig. 2 .
Fig. 2. The potential range of (A) z eff and (B) z fun simulated over a wide range of conditions in water clarity, μ, and σ (n = 1820) compared with the range of z eff and z fun calculated from parameters measured by drifter deployments in the Neuse and Upper Mississippi Rivers in yellow (n = 20).Histograms are smoothed using the kernel density estimate in R.

Fig. 3 .
Fig. 3. Sensitivity of (A) z eff and (B) z fun to μ, σ, and water clarity (K d ) and interactions between these three parameters holding one parameter constant for visual clarity: in the left column K d was a constant 2 m À1 , in the middle column μ was a constant of 1, and in the right column σ was a constant of 1. Statistics reported in the text include all simulations, not just what is visualized here.μ is a fitted location parameter and negative values do not imply depth is negative.

Fig. 4 .
Fig. 4. (A) Validation of z fun by comparing the total light measured near z fun on a moored buoy with the total light measured by drifter in the same river and over the same time period during six field deployments, where SMAPE = 26%, percent bias = À29%, and R 2 = 0.98.Log-log plot is for visualization, and reported statistics are not log-transformed.(B) Validation of z eff by comparing the drifter depth of maximum light exposure with the predicted z eff from depth distribution parameters, where RMSE = 27.5 cm, SMAPE = 33%, percent bias = À18%, and R 2 = 0.59.