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 Two algorithms (the fractal (FPA) and the solid (SPA) particle aggregation) simulating fine particle dynamics were developed and incorporated in the Dynamic Lake Model with Water Quality (DLM-WQ). The previous aggregation model in DLM-WQ required a value for the probability of aggregation independent of particle size distribution (PSD) and concentration. In this study, we developed an algorithm that estimates the probability of aggregation based on PSD, sediment concentration and algal concentration. DLM-WQ was calibrated and validated using measured event-based fine particle data from two Lake Tahoe monitoring sites (1999–2010) and estimated external nutrient and fine particle loadings. Secchi depths estimated using the two algorithms with new aggregation rates varied slightly although predictions were very close to each other and to measured Secchi depths. However, Secchi depths and fine particles estimated by the two algorithms with constant aggregation rates deviate largely from those of measured values. Results using measured data and fine particles predicted by the two algorithms showed that fine particles account for approximately 50% of the total light scattering for all sources making it the most significant contributor. Because approximately 70–75% of the scattering is due to fine particles between 0.5 and 8 µm, management efforts should be targeted to control the transport of fine particles from the watershed to the lake. The updated DLM-WQ can be used to simulate lake response in terms of Secchi depth clarity to different fine sediment and nutrients inputs, and inform development of lake management guidelines to facilitate TMDL implementation where water quality is a concern.
 Suspended particles in lakes, reservoirs, and the ocean absorb and scatter light, affecting such characteristics as transparency, the heat content of ambient water, biogeochemical reactions, and ecosystem dynamics [Ulloa, 1994; Davies-Colley and Smith, 2001]. A light beam impinging on a particle is attenuated by scattering due to the processes of refraction and reflection and by absorption due to pigments associated with the particle. The magnitude of light scattering and absorption vary according to the size, shape, and number of suspended particles. Davies-Colley and Smith  reported that light attenuation peaks (attenuation cross section 1.3 m2 g−1) at a particle size of 1.2 µm for quartz-composition and this peak is not markedly different from other minerals found in water. Light attenuation in lakes from particles of diameter less than 0.1 µm and greater than 20 µm is low. They also demonstrated that because organic materials have much lower density and refractive indices relative to water, light attenuation peaks (attenuation cross section 0.8 m2 g−1) at larger sizes (∼5 µm). Light attenuates to approximately 0 m2 g−1 for organic cell sizes lower than 0.1 µm and greater than 20 µm. Size dependence of light attenuation by particles explains why mineral particles in natural waters attenuate light more appreciably than much more numerous but smaller (<1 µm) bacterial cells (an observation supported by Swift et al.  in Lake Tahoe). In addition, the detection limit of some particle measuring instruments (e.g., LiQuilaz, http://www.pmeasuring.com/) is approximately 0.5 µm [Terpstra, 2005; Swift, 2004; Nover, 2012]. For these reasons, we focus on the analysis and estimation of the dynamics of lake particles in the size range 0.5–16 μm.
 Settling times for fine particles in the range 0.5–16 µm are extremely slow, unless aggregation of fine particles generates larger particle and faster settling rates. Particle aggregation and settling change the particle size distribution in lake water, and the interaction of inert inorganic particles with biologically active materials can influence microbial ecology [Paerl, 1975; Friedrich et al., 1999; Murrell et al., 1999; Gerbersdorf et al., 2011]. The concentration and size distribution of particles in lakes and reservoirs changes depending on external inputs (e.g., streams, unchannelized runoff, atmospheric deposition, and shoreline erosion) and internal lake dynamics including aggregation and settling. Understanding particle dynamics and estimating aggregation rate and settling velocities are therefore essential for understanding lake systems [Weilenmann et al., 1989; Kiørboe et al., 1994; Chanudet and Filella, 2007; Arnous et al., 2010; Fathi-Moghadam et al., 2011].
 In particle aggregation models, the crucial factors determining particle removal rates from ambient lake water include (1) collision probability rate [Valoulis and List, 1984; McAnally and Mehta, 2001], (2) aggregation probability, (3) physical processes of particles dynamics [Sahoo et al., 2010, 2013a], and (4) settling rate [Weilenmann et al., 1989; Jürg, 1996; Burd and Jackson, 2009; Chakraborti et al., 2009]. The governing equations of particle dynamics presented below describe the collision rate between two particles. An equally important factor is the probability that colliding particles will stick together to form a larger aggregate. The probability of aggregation is influenced by particle “stickiness,” which increases when algae and bacteria release extracellular polymeric substances (EPS) or transparent exopolymeric particles (TEP) into surrounding water [Passow, 2012]. Particle settling rate is most commonly modeled using Stoke's law [Bonalumi et al., 2011; Chung et al., 2009; Monte et al., 2009], although this is extremely simplistic when applied to natural aquatic systems [Krishnappan and Marsalek, 2002; Fathi-Moghadam et al., 2011]. Aggregated particles with complex morphology and varying densities will settle differently than predictions based on Stoke's law, which assumes uniform density and spherical shape. Although numerous studies have estimated aggregation rates and settling velocities of particles in aquatic waterbodies, few have yielded robust results because of the complexity of this problem [Li and Logan, 1997a, 1997b; Jackson, 2001; Krishnappan and Marsalek, 2002; Chung et al., 2009; Fathi-Moghadam et al., 2011; Effler and Peng, 2012].
 Light has been shown to attenuate in Lake Tahoe approximately 58%, 25%, and 17% due to inorganic particles, algae, and colored dissolved organic matter plus water molecules, respectively [Swift et al., 2006]. A significant relationship (p < 0.001, R2 = 0.57) between individual Secchi disk readings and fine particle concentration in the 0.5–16 µm size range has been reported previously [Reuter et al., 2009]. Fine particles in lake water are complex and depend on a balance between external loading, particle collision rate, particle aggregation, particle settling rate, and lake dynamics.
 Lake Tahoe is renowned for its deep-water clarity. However, analysis of long-term lake clarity, expressed as Secchi depth, show that its clarity has declined by 10 m between1968 and 2009 at ∼0.22 m/year (R2 = 0.88, p < 0.001) [Sahoo et al., 2010]. As public concern for the clarity of Lake Tahoe is high, local, state, federal agencies, and policy-makers proposed development of a science-based restoration plan known as the Lake Tahoe total maximum daily load (TMDL). In essence, a TMDL is a water quality restoration plan required by the United States Federal Clean Water Act to ensure the achievement of water quality standards. The Lake Tahoe TMDL was developed and adopted in 2011 by California and Nevada to define fine sediment and nutrient load reductions required to achieve historic lake clarity. The TMDL research questions were to (1) identify pollutants causing Lake Tahoe clarity loss, (2) assess loadings of each pollutant type, and (3) define levels of load reduction needed to meet the water quality standards for Lake Tahoe's optical properties. The Dynamic Lake Model with Water Quality (DLM-WQ) [Sahoo et al., 2010] was developed to provide guidelines (defining nutrient and fine sediment load reduction) for Lake Tahoe TMDL establishment.
 It has been shown that the temporal and depth distribution of fine inorganic particle distributions in ultraoligotrophic Lake Tahoe plays an important role in lake clarity loss, a key beneficial use for this unique waterbody [Swift et al., 2006; Sahoo et al., 2010; Nover, 2012]. The DLM-WQ simulates light penetration and transparency in Lake Tahoe focusing on phytoplankton and inorganic particles dynamics in 0.5–1, 1–2, 2–4, 4–8, 8–16, 16–32, and 32–63 µm size categories. Swift et al.  showed that in Lake Tahoe, approximately 98% of particle light scattering was due to particles <16 µm in diameter.
 The DLM-WQ (Figure 1) had an aggregation model based on algorithms described by O'Melia  and Casamitjana and Schladow  with an optical model developed by Swift et al. . These algorithms represent the two particles after collision as one solid and spherical particle (Figure 2a) (the solid particle aggregation model (SPA)) and require a value for the probability of aggregation that is independent of particle size distribution (PSD) and concentration. In reality, the particle size and shape (e.g., nonspherical (Figure 2b) [Lee et al., 2000; Jackson, 2001] after aggregation depend on the orientation of the colliding particles (i.e., fractal particle aggregation (FPA)). Smaller particles primarily coalesce around larger particles although the solid particle aggregation model (SPA) assumes a resultant compact sphere. Both SPA and FPA conserve mass, although the area available for collision is greater for FPA [Lee et al., 2000; Burd and Jackson, 2009]. The objectives of this study are to (1) develop a fractal-based particle aggregation model (FPA) to predict particle aggregation and settling accurately by capturing the large variation in particle structure that exists in lake particles; (2) formulate a theoretical aggregation rate that is a function of PSD, sediment concentration, and algal concentration; (3) incorporate the new aggregation models in the DLM-WQ model and calibrate the updated models (SPA and FPA) using measured in-lake particle concentration between 1999 and 2008; and (4) generate guidelines for lake management and TMDL development.
2. Particle Dynamics Model
 The FPA modeling approach is appropriate for aggregation of particles of different densities (e.g., inorganic particle in collision with algal cells or bacteria) and orientation of collision. Jackson  proposed a particle aggregation model that looks at two different particle parameters when calculating the changes in particle size distribution. When two particles collide and stick together, the resulting particle has a mass that is equal to the sum of the two colliding particles. Particles in aquatic ecosystems have irregular shapes. Depending on the orientation of the colliding particles, the resulting new particle could have very different lengths. The SPA model only uses the mass to track changes in particle populations. Each mass category is associated with a single particle diameter. This approach provides insufficient flexibility for describing particle aggregation. Particles in the water column can be small and heavy (clay particles), or large and light (aggregated bacteria). To capture the wide range of particle types, it is therefore useful to use two parameters to describe the particles.
 In addition to mass, Jackson  proposed particle fractal length as another parameter that is conserved when two particles coagulate. Fractal length is defined as follows:
where λ is the fractal length, L is the particle length, and superscript D is the particle fractal dimension. When two particles with different fractal lengths collide, the interaction can be described as follows:
2.1. Particle Collision Rate
 All aggregation models (FPA and SPA) use three physical processes that cause particles to bump into one another. The processes are Brownian motion, shear, and differential settling. An aggregation kernel is calculated for each physical process [Jackson, 2001]. This kernel is a rate constant that is used when calculating the rate at which new particles are formed from the collision of smaller particles. The rate of formation is then a sum of the three different kernels multiplied by the number of particles in each bin that can interact
where i and j represent two bins of different particle size ranges, Ri,j is the rate of formation of new particles with mass (mi + mj) and fractal length λi,j from the collision of two smaller particles of masses mi, mj and fractal length λi, λj, ci[c(mi, λi)], and cj[c(mj, λj)] are the particle number concentrations of the colliding particles with associated mass and fractal length; βbr, βsh, and βds are the aggregation kernels associated with Brownian motion, fluid shear, and differential settling, respectively. Jackson  describes several ways of calculating the differential settling and shear aggregation kernels. There are two common formulations for calculating these kernels: (1) rectilinear and (2) curvilinear. Each formulation has different assumptions about the hydrodynamic behavior of the particles in an aqueous environment. The rectilinear formulation of these kernels does not take into account how the particle affects the surrounding water. Thus, a falling particle is not considered to be “pushing” water away from it as it is descending. The curvilinear formulation takes into account the effects that the larger particle has on the flow field, but neglects the effect of the smaller particle on the surrounding waters. To account for the porous “fractal” character of particles in water, Jackson  used results from Li and Logan [1997a, 1997b] to calculate the aggregation kernels for different circumstances. Li and Logan [1997a, 1997b] measured the interaction rates between particles of different sizes and found that their results fall somewhere between what the curvilinear and rectilinear formulation would predict. The classic rectilinear formulations are as follows:
 For differential sedimentation
 For shear
 For Brownian motion
where , , ε is the energy dissipation rate, T is the absolute temperature (K), K is the Boltzmann constant, ν is the kinematic viscosity, μ is the dynamic viscosity, and wi, settling velocity, is estimated assuming a balance between gravitational and drag forces as: .
 The curvilinear kernel for differential settling motion for solid spheres and without any chemical attractive or repulsive forces is smaller [Pruppacher and Klett, 1980]
where . The effect of making this correlation is greatest for small particles impacting large ones.
 The curvilinear kernel for shear motion [Hill, 1992] is given by
where and .
 The curvilinear kernel for Brownian motion is the same as in the case of rectilinear Brownian motion. Jackson  used Li and Logan [1997a, 1997b] formulations for cases where the colliding particles are of different sizes. When the particles are of the same size, Jackson  proposed using the rectilinear formulation. The resulting equations for the differential settling kernel are as follows:
where and . For shear, the resulting equation is
2.2. Particle Dynamics
 The following equation is used to track changes in particle size distribution [Jackson, 2001]
where α is the probability that two colliding particles will stick together and n is the particle number spectrum. The equation is multiplied by 0.5 when the colliding particles are exactly the same, i.e., they have the same mass and fractal length. For all other conditions, the equation is not multiplied by 0.5. This equation is applied to each layer in the water column. The last term in the above equation is a sinking term that removes particles from the layer.
3.1. Modified Aggregation Model
 In our model, we included an additional term that adds particles from the layer above. To apply equation (11) to a multilayer environment, we add an additional index beyond the mass and fractal length. The additional index denotes the layer that the particles occupy. The transformed equation becomes
where li denotes the layer of interest. Our model considers seven arrays of fractal length and corresponding masses. To solve the above equation numerically, we turned the integrals in the above equation into sums, in the fashion proposed by Jackson . If two colliding particles formed a new particle that was big enough to move into a larger size category (either in fractal length, mass, or both) then the number of particles in the new class was augmented. If the resulting particle was not big enough to fit into a larger size bin, then the number of particles in the smaller size category of the two colliding particles was augmented. Moreover, if smaller particle(s) are “absorbed” by a larger aggregated particle, the number of particles in the larger bin does not change.
 The mass and fractal length of the smaller “absorbed” particle is stored in a “bank” and at the end of each time step, this excess mass and fractal length are spread evenly among all larger particles of the same class. Although the model can estimate mass and size of aggregated particles based on colliding particles of various sizes and types, it is difficult to keep track of the exact size and mass records of individual particles because of the enormous number of organic particles present in the lake. The external average annual fine inorganic particle loading to the lake in the 0.5–16 μm size range is approximately 4.5 × 1020 per year [Sahoo et al., 2013b].
 Sensitivity analysis was performed to determine the values of λ and mass of each particle size group compared with in situ lake particle data [Jassby, 2006]. The reported range of the fractal dimension (D) for various aquatic aggregates is broad, between 1.0 and 3.0 [Jiang and Logan, 1991; Li and Ganczarczyk, 1989; Logan and Wilkinson, 1991; Wiesner, 1992]. The fractal dimension represents the compactness of the aggregate with higher fractal dimension (e.g., 3) indicating denser structure and lower fractal dimensions (e.g., 1) indicating looser structure. For fractal aggregates, density is also inversely related to aggregate size. A D-value of 3.0 represents a uniformly distributed structure or more commonly, a compact structure of particle(s) that conserve total particle(s) volume when they aggregate; D-values near 1.75 are representative of cluster-cluster aggregation (combination of water, organic matter, and inorganic particles) [Jackson, 1998]. In our study, we used fractal dimension (D) values of 3.0 for particle size ranges 0.5–1, 1–2, and 2–4 µm because of their small size; however, we used a D-value of 2.7 for particles in the size range of 4–8 µm, a D-value of 2.4 for particle size range 8–16 µm, a D-value of 2.0 for particle size range 16–32 µm, and a D-value of 2.0 for particle size range 32 to <63 µm because these values fit the Tahoe data set well [Jassby, 2006]. Because small particles are compact, we assumed that the mass of particle size ranges 0.5–4 µm is 2650 kg/m3 while the mass of particle size ranges 4–<63 µm is 2100 kg/m3.
3.2. Probability of Aggregation Rate
 The collision rate between two particles is only part of the equation that governs particle dynamics. An equally important factor is the probability (α) that colliding particles will stick together to form a larger aggregate. Aggregates in aqueous environments are composed of a variety of source particles including clays and other inorganic matter, algae, bacteria, detritus, and extracellular polymeric substances (EPS) also known as transparent exopolymeric particles (TEP), and water filled pores [Droppo, 2001]. TEP is a branched, chain-like, algal or bacterial secretion that is largely produced from sugars, although proteins can also be substantial components. Algae and bacteria release TEP into the surrounding water while in a planktonic phase the same TEP is secreted by these cells when they are found in aggregates [Passow, 2012]. Thus, source particles in an aggregate are embedded in a TEP matrix [Droppo, 2001]. Studies with diatom cultures suggest that the generation of TEP varies by species and the physiological state of cells [Passow, 2012].
 TEP ranges in size from less than a micron to hundreds of microns long, and occurs at concentrations of up to thousands per milliliter [Alldredge et al., 1993]. During algae blooms, TEP occurs at abundances and sizes comparable to phytoplankton [Passow and Alldredge, 1995a, 1995b]. In a controlled experiment, Passow and Alldredge [1995a, 1995b] found that the fraction of phytoplankton enclosed in TEP increased from 30% to 70% in one day. TEP is also thought to trigger mass flocculation in aquatic systems [Passow and Alldredge, 1995a, 1995b] and to contribute greatly to particle stickiness [Engel, 2000, 2003]. It has been shown to be largely responsible for the formation of particle aggregates [Engel, 2000]. Logan et al.  showed that sedimentation rates in a freshwater lake increased eightfold when TEP concentrations were at a maximum. TEP's role in rapid sedimentation of phytoplankton is supported by large amounts of the material found in sediment trap studies. These results and others suggest that TEP is instrumental in increasing particles aggregation. Using a Couette Chamber to simulate shear, Engel  developed an equation to deduce the value of α from TEP concentration relative to Coulter Counter detectable particles. Particle size distribution measurements and microscopic observations show that there are many more organic particles than inorganic particles in Lake Tahoe [Jassby, 2006; Swift, 2004; Coker, 2000; Terpstra, 2005]. Since particle aggregates are made up of inorganic and organic material, and the organic fraction is often very large [Nover, 2012], one can postulate that TEP are responsible for particle aggregation in Lake Tahoe.
 Unfortunately, TEP measurements were not taken in Lake Tahoe. Because Lake Tahoe is deep and light does not reach the lake bottom, phytoplankton concentration (Chl a) is minimal below about 100 m [Tahoe: State of the Lake Report, 2011]. However, the lake turns over completely once in 3–4 years and so it is likely that TEP concentrations are greater than zero in deep parts of the lake. The probability of particle collision increases when more particles are present and although the concentration of very small particles is orders of magnitude higher than the concentration of larger particles, the increase in probability of aggregation is not several orders of magnitude. Due to aggregation, concentration of smaller particle reduces exponentially. In previous particle aggregation models [e.g., O'Melia, 1985; Casamitjana and Schladow, 1993; Jackson, 2001; Jassby, 2006; Sahoo et al., 2010], the probability of aggregation (α) was considered a constant irrespective of lake depth, particle size distribution, and particle concentration. In this study, we postulate that the probability of aggregation is a function of particle size distribution, particle and phytoplankton concentration (i.e., a function of TEP). The above reasoning suggests the following empirical function in which the value of α is directly proportional to the square of the logarithm of particle concentration (cp), inversely proportional to particle size (r) and is directly proportional to Chl a concentration with a minimum value equal to 1.0
where Ca is a constant determined by model calibration and varies with particle size i and j. Note that the above equation was only applied to Lake Tahoe and was not validated for other lakes/reservoirs.
3.3. Particle Sinking
 The model DLM-WQ determines the number of particles in each layer after aggregation and estimates how many particles in a specific size bin sink to a deeper layer. The model uses Stokes' Law to estimate the sinking rate
where V is the sinking rate, g is the gravitational acceleration due to gravity, r is the particle radius, b is the excess length stored in the “bank” for that particular size category at a particular depth (denoted by the index k), Dj is the fractal dimension in column j, ρp denotes aggregate particle density, ρw denotes lake water density, and µ is the dynamic viscosity of water. The minimum layer thickness considered for Lake Tahoe is 1 m. As an example, if V is estimated to be 0.5 m for a particular bin in a particular layer, then half of the particles in that bin will sink to the lower layer. This assumes that particles are evenly dispersed throughout the 1 m layer. Aggregation and sinking are repeated for each time step.
3.4. Required Data for DLM-WQ
 The meteorological data required for the DLM-WQ include precipitation (mm), air temperature (°C), net shortwave radiation (W m−2), net longwave radiation (W m−2) or surrogate of fraction of cloud coverage and air temperature (°C), relative humidity (%) or vapor pressure (mb), and wind speed (m s−1) measured at 10 m above the ground. The meteorological records from the Tahoe City SNOTEL (SNOwpack TELemetry) gages were precipitation, air temperature, shortwave radiation, dew point temperature, and wind speed. Vapor pressures were estimated using the measured records of dew temperature while longwave radiations are estimated using measured shortwave radiation, air temperature, and the aerodynamic formulae reported in Tennessee Valley Authority (TVA) .
 The required lake data for DLM-WQ includes lake outflow time series, lake bathymetry, the elevation of streams before entering the lake, and initial vertical profiles of the lake. The initial vertical lake profiles include chlorophyll a, particle number concentration in seven size bins, temperature, dissolved oxygen, all species of phosphorus and nitrogen, and silica collected at the midlake station at 0, 10, 50, 100, 150, 200, 250, 300, 350, 400, and 450 m. The elevations of each stream near the lake are estimated from GIS DEM and are used to estimate the depth and distance of stream plunging in the lake.
 The water quality parameters in the lake are collected at two stations: (1) an index station along the west shore (150 m deep) and (2) a midlake station in the deeper part of the lake (460 m deep). A comparison of lake particle data from the index and midlake stations shows that particle numbers were similar exhibiting similar patterns of variation, with the exception of a few sampling dates when a spike in particles was observed at both stations (Figure 3). Since water samples collected at the midlake station were collected to 460 m depth, covering nearly the full vertical profile, data from this station was used as representative of the average whole-lake conditions. Field data show much more variation along the vertical profile than on a horizontal gradient, justifying the use of the midlake station as the source of limnological input data.
 Fine particle profiles [Swift et al., 2006; Andrews et al., 2011a, 2011b; Nover, 2012], chlorophyll a concentrations profiles and Secchi depths [Tahoe: State of the Lake Report, 2011] collected at the midlake station during 1999–2010 were used for model calibration and validation (see Figure 1 for the modules' interactions and data used). The LiQuilaz LS-200 (Particle Measurement Systems Inc., Boulder, Colorado, USA) was used to count inorganic suspended particles in the lake for the period January 1999 to September 2003 [Swift et al., 2006] and the period March 2006 to September 2010 [Nover, 2012]. Fine particles in the lake are measured once per month at depths of 0, 10, 50,100, 150, 200, 250, 300, 350, 400, and 450 m from the water surface. It provides optically active suspended inorganic particles concentrations (#/ml) in 15 user selected size bins in the range 0.5–20 μm. The data of 15 bins were combined to produce the seven particle size classes (0.5–1, 1–2, 2–4, 4–8, 8–16, 16–32, and 32–63 μm) used in the DLM-WQ model. There were no particle data taken during 2004 and 2005 (Figure 3). Secchi depth was measured using a 25 cm diameter white disk once each month at the midlake station and approximately thrice each month at the index station. Secchi depth is the mean of the depths of disappearance and reappearance [Swift et al., 2006]. The depths of disappearance and reappearance vary between 0.5 m and maximum 4 m.
 Detailed fine particle loading from all external sources were estimated in Lahontan and NDEP  and Sahoo et al. [2013b] and fine particles profiles in the lake were reported in Nover . In brief, the external nutrients and fine particles sources in the Tahoe basin include (1) 54 channelized streams, (2) 10 intervening zones those do not have channels and discharge directly into the lake, (3) 54 streams bank and channel erosion, (4) atmospheric deposition, (5) shoreline erosion, and (6) groundwater. All treated sewage effluents are exported out of the basin. The nutrients and fine particles from each source are numerically quantified for Lake Tahoe TMDL based on in situ measured data and modeling study. The watershed model that quantified nutrients' inputs to the lake from streams, intervening zones, and stream channel erosion, includes six major land-use types (1. single-family residential (SFR); 2. multifamily residential (MFR); 3. commercial/institutional/communications/utilities (CICU); 4. transportation; 5. vegetated; and 6. waterbody). The six major land-use types were refined to 21 subcategories in the watershed model [Lahontan and NDEP, 2010]. Fine particle loadings from streams, intervening zones, and stream channel erosion were calculated based on the watershed model estimated daily stream flow (cubic feet per second) and field particle flux (number of particles per second) data collected during 2002–2010 [Sahoo et al., 2013b].
 The atmospheric particles deposition on the lake is based on the inert soil-based particulate matter. Values for the atmospheric deposition were obtained from studies by a variety of investigators including the UC Davis-TERC, UC Davis-DELTA Group, the Desert Research Institute, and California Air Resources Board [e.g., Hackley et al., 2004, 2005; California Air Resources Board, 2006; Gertler et al., 2006; Lahontan and NDEP, 2010]. The values were estimated based on air samplings for deposition modeling and direct measurements using deposition buckets located on the lake and lakeshore. Data for the shoreline erosion were taken from the study by Adams and Minor  who estimated annual load to the lake using aerial photographs from 1938 to 1998. We used the annual values of groundwater nutrient loading to the lake reported by U.S. Army Corps of Engineers . The uncertainty associated with the fine particles and nutrient estimations were discussed in Sahoo et al. [2013b]. The atmospheric (seasonally averaged), shoreline erosion (annually averaged), and groundwater (annually averaged) loadings were assumed to be the same for all years while inputs from stream and intervening zones were daily loadings [Sahoo et al., 2010, 2013b].
3.5. Calibration and Validation
 The temporal and spatial process descriptions of the hydrodynamic module of the DLM-WQ are physically based and the equations are parameterized; therefore it is free from calibration [Hamilton and Schladow, 1997]. Thus, Sahoo et al.  calibrated and validated the water quality modules of the DLM-WQ using only five years of data collected during the period 2000–2004.
 The main features of this work include:
 1. Sahoo et al.  optimized the parameters associated with phytoplankton dynamics (e.g., maximum growth, respiratory, and mortality rate; light saturation value; temperature for optimum growth), nutrient utilization rates (e.g., phosphorus to chlorophyll ratio and nitrogen to chlorophyll ratio), bio-chemical reaction rates (e.g., half saturation constant for nitrogen (N), half saturation constant for phosphorus (P), half saturation constant for ammonia (NH4)), nutrient temperature multipliers, sediment fluxes, dissolved oxygen (DO) dynamics (e.g., DO-to-carbon ratio in respiration, DO per mass of nitrified, oxidation rate of chemical oxygen demand (COD)) and inorganic particles dynamics (e.g., constant coagulation rate). The equations for fine sediment, nutrient, dissolved oxygen, and chlorophyll dynamics are described in Sahoo et al. .
 2. The DLM-WQ estimates daily Secchi depth values. However, regulatory decisions are based on the annual average [Lahontan and NDEP, 2010]. The annual average DLM-WQ estimated and measured Secchi depths agree well with highest relative percent error 6%.
 3. DLM-WQ estimated concentrations of chlorophyll, nitrate, and dissolved oxygen followed the patterns of measured concentrations. In addition, simulated lake water temperature demonstrated a close match with those of measured values.
 4. A sensitivity analysis was performed to ensure that the model parameters were optimized. The sensitivity analysis illustrates that inorganic particles have greater effect on lake clarity than total nutrients. In addition, the DLM-WQ demonstrates the nonlinearity of Secchi depth reaction to load reductions.
 5. The DLM-WQ provided a firsthand scientific solution to managers that the historic clarity could be achieved if nutrients and inorganic particles loads would be reduced to approximately 55% from all sources (urban, nonurban, atmosphere, groundwater, and shoreline erosion) or approximately 75% from urban sources.
Sahoo et al. [2013a] reported that the heat (the evaporative heat losses and sensible heat exchanges) and water budgets were not accurately calibrated in Sahoo et al. . Incorporating the turbulent diffusion transfer module in the DLM-WQ and using modified precipitation inputs directly on the lake surface, Sahoo et al. [2013a] demonstrated that estimated and measured water surface temperatures and lake water level were in excellent agreement for the period 1994–2008 with R2 equal to 0.97 and 0.99, respectively. This indicates that the modified DLM-WQ simulates lake vertical dynamics adequately.
Sahoo et al. [2013b] updated the detailed fine particle and nutrient loadings from all external sources for the period 1999–2008. Since the Secchi depth of the lake depends on concentration of fine particles, algae and other water quality parameters, the water quality modules described in Sahoo et al.  and particle dynamics updated in this study are calibrated so that estimated values of dynamics, transparency, and water quality closely mirror measured values. Although the DLM-WQ was calibrated and validated using the data for the period 1999–2002 and 2002–2008, respectively, model results were presented for the entire simulation period 1999–2008. In addition, daily mid-day model results were presented for the entire simulation period while model time step was 30 min.
 The statistical efficiency criteria: the correlation coefficient (R), the Nash-Sutcliffe efficiency (NSE), the root-mean-square error (RMSE), and mean error (ME) were used to measure the predictive performances of the updated DLM-WQ. The mathematical expressions of R, NSE, RMSE, and ME are given as
where Pi and Oi are the predicted and observed values at event i, respectively; n is the total number of observations; and and are the mean of the predicted and observed values, respectively. Since these efficiency terms use error statistics relative to the observed values, they are statistically unbiased. The R estimates the direction and the strength of a linear relationship between observed and simulated values. The NSE is an index used for the predictive accuracy of the DLM-WQ model. The RMSE indicates the global discrepancy between the observed and simulated values. The ME measures the average of the total model errors. The ranges R, NSE, RMSE, and ME vary between −1 to 1, −∞ to 1, −∞ to ∞, and −∞ to ∞, respectively, while the model predictions are considered to be most precise if their values are close to 1, 1, 0, and 0, respectively.
4. Results and Discussions
4.1. Secchi Depth
Swift et al.  calibrated and validated the optical module using event-based measurements of chlorophyll and fine particles for the period 1999–2002. Using the same calibrated model coefficients of the optical module, this study extended the validation period up to 2008 using event-based measured data collected at the MLTP site (Figure 4a). The annual average relative percent differences between predicted and observed Secchi depths are approximately 10% (14%, 14%, 8%, −6%, and −11% for year 1999, 2000, 2001, 2002, and 2006, respectively) except those of 2007 (25%) and 2008 (24%) (Figure 4a), comparable to the mean relative percent difference reported by Swift et al. . Event-based estimated Secchi depths (Figure 4b) demonstrate strong agreement with measured Secchi depths with R = 0.46. The accuracy of model estimated annual average Secchi depth to observed value is above 80% although the model failed to capture some of the measured data points because of uncertainty in estimated particle concentrations (may be due to amplification error in measured data), interpolated particle and chlorophyll a input (note that profiles in the lake are measured once per month at depths of 0, 10, 50,100, 150, 200, 250, 300, 350, 400, and 450 m from the water surface) and optical model formulations and calibrated parameters which represent overall clarity trends for the lake.
 Although Secchi depths are measured approximately once per month, we report annual averages as regulatory decisions are based on annual average values [Lahontan and NDEP, 2010]. Simulated annual average Secchi depths are compared with measured values (Figure 5a) for cases: (1) SPA model with new α (described in section 3.2) referred to as SPA; (2) FPA model with new α (described in section 3.2) referred to as FPA; (3) SPA model with constant α [Sahoo et al., 2010] referred to as SPA constant; and (4) FPA model with constant α [Sahoo et al., 2010] referred to as FPA constant. For case SPA, the mean relative percent difference between estimated and measured is below 10%, in the same range as those reported by Swift et al. . The case SPA constant model simulated annual average Secchi depths close to measured records except those for years 1999, 2005, 2006, 2007, and 2008. The SPA constant model did not simulate higher fine particle loadings well (R equals only 0.03) during snow-melt and winter storm events (Figure 5b), because the α value is independent of PSD and fine sediment concentration. Sahoo et al.  used the SPA constant model to simulate years 2000–2004. Although this study used revised pollutant loadings [Sahoo et al., 2013b], the annual average Secchi depths (Figure 5a) estimated by the SPA constant model are identical to Sahoo et al.  for years 2000–2004. The annual average Secchi depths estimated by the FPA model closely match those of the SPA model. However, the Secchi depths estimated by the FPA constant model are very close to those of FPA and SPA models (Figure 5).
 The seasonal estimated Secchi depths (summer and winter) were compared to measured Secchi depths (Figure 6). The estimated winter Secchi depth trend follows measured Secchi depths closely. Estimated summer Secchi depths follow measured Secchi depths closely with the exception of 2007 and 2008 (Figure 5b). Variability is attributed to (1) seasonal data inputs from atmospheric deposition and groundwater loading, (2) insufficient measured data during storm events (lack of maximum concentration events), and (3) formulation and calibration of optical model [Swift et al., 2006] leading to overestimates during 2007 (25%) and 2008 (24%)(Figure 4a). Estimated annual average Secchi depth computed by FPA was found to be higher (0 to 7%) than those from SPA (Figure 5a), which is expected as the area available for collision in case of the FPA model is greater than that for the SPA model and the fine particle aggregation rate is correspondingly higher for FPA (Figures 7-12).
 Figure 5b shows a comparison between modeled daily Secchi depth time-series and measured data for all four cases. The modeled Secchi depth trends estimated by the SPA and FPA model closely track measured values. However, estimates by the SPA constant model deviate dramatically from measured values. Deviation between modeled Secchi depth by SPA and FPA model and field measurements is likely due to (1) seasonal atmospheric deposition and groundwater loading inputs and (2) uncertainty associated with the stream particles estimation [Sahoo et al., 2013b]. Fine particle concentrations at specific depths are affected by (1) stream plunge depth and associated nutrient and particle loadings, (2) loading from shoreline erosion, (3) atmospheric loading, (4) lake mixing and stratification dynamics, and (5) aggregation and settling rate. SPA and FPA algorithms estimate particle concentration somewhat differently for the same input because of model formulations and corresponding assumptions. As a result, estimated Secchi depths using the two algorithms vary slightly although they show similar trends. The new α generates major differences in particle dynamics (Figures 7-12) and consequently in Secchi depth estimation between SPA and SPA constant.
4.2. Fine Particles
 As only fine particles in the 0–50 m depth range (i.e., Secchi depth ranges) can directly affect the Secchi depth estimation, we used output from the DLM-WQ simulation of daily particle concentrations at surface, 10 and 50 m depth. These results are compared with in situ measured values in Figures 7-12.
 Years 2005 and 2006 were wet years (1445 and 1067 mm precipitation, respectively) compared to other years when precipitation was low to moderate (2000: 795 mm, 2001:709 mm, 2002:750 mm, 2003: 767 mm, 2004: 808 mm, 2007: 640 mm, 2008: 874 mm, and 2009: 847 mm measured at Tahoe City meteorological station). Since precipitation drives stream runoff and stream particles are directly related to stream flow [Sahoo et al., 2013b], lake particle concentrations were high during 2005–2006 (see Figure 3b). The spikes in lake particle concentrations (Figures 7-12) indicate the amplification of error due partly to laboratory measurements (for example Figures 11a and 12a, 2 July 2001) and partly to abrupt changes in lake algae and particle concentration at the sampling time. In-lake measurements revealed numerous particle spikes in the range 0.5–1.0 µm during the period 1999–2003, and especially at depths of 10 and 50 m (Figure 7); however, particle concentrations in the range of 1.0–2.0 and 2.0–4.0 µm were relatively low between 1999 and 2003. Hydrologic loads are higher during spring due to snow-melt and rain-on-snow events [Sahoo et al., 2013b], and consequently, particle concentrations in the lake were observed to be higher during spring (Figure 3).
 Measured lake particles (Figures 7-12) show that lake particles decrease exponentially from bin size 1 (0.5–1.0 µm) to bin size 5 (8–16 µm). While DLM-WQ estimates follow measured trends at the surface and at 10 m depth, estimates are higher than measured values at 50 m depth. This may be due to the deep chlorophyll maximum (DCM) that is typically observed in Lake Tahoe during spring and summer. The DCM not only increases TEP concentration but also increases the number of organic particles and therefore coagulation rates resulting in larger, faster settling particles. Deviations between the DLM-WQ estimates and measurements appear because stream input estimates are based on stream particle models [Sahoo et al., 2013b], which are estimated based on the time averaged inputs instead of maximum concentration events (MCE) and seasonal atmospheric loading. The DLM-WQ estimated and predicted particle values for 0 to 50 m are in good agreement and change hydrologically over time for the cases of the SPA and FPA model; however, estimated and predicted particle values deviate largely for the cases of SPA constant (Figures 8, 9, 11, and 12) and FPA constant models (Figures 8 and 12). This deviation occurs because SPA constant and FPA constant do not aggregate particles at a higher rate even though in-lake fine particle concentrations increase significantly during snow melt and storm events. As a consequence, SPA constant and FPA constant estimate much higher particle concentrations compared to measurements resulting in accumulation of particles over time (Figures 8 and 9 for SPA constant and Figure 8 for FPA constant). Area available for aggregation in the case of the FPA model is greater than that in the SPA model and therefore in-lake fine particles predicted by FPA constant are lower than those of SPA constant (Figures 7-12). In overall, fine particles predicted by FPA constant are very close to those of FPA and SPA except particle concentrations in the range of 1.0–2.0 µm (Figure 8).
 We used the DLM-WQ to quantitatively evaluate the factors that contribute to Secchi depth over the period 1999–2008. Figure 13 presents the average light scattering and absorption contributed by chlorophyll a, fine particles, colored dissolved organic matter and water molecules. Estimated values presented in Figure 13a are based on the in situ lake measured profiles of chlorophyll and fine particles while estimated values presented in Figures 13b and 13c are based on the DLM-WQ simulated chlorophyll and fine particle concentrations. DLM-WQ simulated chlorophyll and fine particle concentrations from the internal and external nutrient and particle loadings [Sahoo et al., 2010]. Figure 13 demonstrates the agreement of light scattering and absorption percentages estimated by the DLM-WQ (updated SPA and FPA) using updated external loadings [Sahoo et al., 2013b] and those estimated by the optical model [Swift et al., 2006] using measured in-lake data.
 It is not possible for DLM-WQ to simulate each individual Secchi depth measurement with absolute accuracy because (1) the time averaged groundwater, atmospheric, and shoreline erosion inputs were assumed to be the same for all simulating years, (2) there is uncertainty associated with nutrient and particle estimation using the watershed model, (3) errors propagate from one module to the next (Figure 1), and (4) errors exist inherently in Secchi depth measurements [e.g., Preisendorfer, 1986]. With these limitations, the DLM-WQ (with updated SPA and FPA) was able to closely simulate the 1999–2008 averaged light scattering and absorption of each component.
 Light scattering is found to be the dominant attenuant, which is similar to the findings in other aquatic ecosystems [e.g., Kirk, 1994; Mobley, 1994; Effler et al., 2010; Peng and Effler, 2010, 2011; Effler and Peng, 2012]. Figure 13 demonstrates that fine particles accounted for approximately 50% of the total light scattering for all sources making it the most significant contributor. Both organic (chlorophyll a) and inorganic particles contribute to clarity loss. This is consistent with the findings of Peng and Effler , who found a strong linear relationship between inverse Secchi depth transparency and light scattering for the water column of Lake Ontario using scanning electron microscopy interfaced with automated image X-ray analysis (SAX) and an individual particle analysis (IPA) technique. Additionally, Effler and Peng  found a strong inverse relationship between Secchi depth and light scattering for Onondaga Lake as well, indicating that lake water transparency is largely attributable to variations in scattering associated with particles.
 The relative contribution of fine particles of different sizes shown in Figure 14 shows that predicted light scattering percentages by DLM-WQ: SPA and DLM-WQ: FPA are very close to those estimated by the optical model [Swift et al., 2006] using measured data from Lake Tahoe. Figures 13 and 14 indicate that the DLM-WQ (updated for SPA and FPA) simulates particle dynamics adequately. As stated earlier, DLM-WQ (SPA and FPA) with constant coagulation rate could not simulate the particle dynamics adequately (Figures 7-12). Because the external average annual particle loading in the 0.5–16 μm size range was approximately 4.5 × 1020 per year [Sahoo et al., 2013b], a constant coagulation rate underestimated particle aggregation during snow melt and storm events. Although FPA algorithms are different than those of SPA, both FPA and SPA include the same physical processes (e.g., Brownian motion, fluid shear, and differential settling; and Stoke's law for settling velocity estimation) [Sahoo et al., 2010; this study] and consequently both FPA and SPA with new variable aggregation rate (equation (13)) produce similar results. Thus, the new variable aggregation rate (equation (13)) plays an important role in the Lake Tahoe particle dynamics simulation.
 Compared to SPA constant, FPA constant produced stable results in terms of estimation of particles (Figures 7-12 except Figure 8) and Secchi depths (Figure 5). Figure 14 shows that approximately 70–7% of scattering is due to fine particles between 0.5 and 8 µm, corresponding to silt and clay. Using the SAX and IPA technique, Peng and Effler  found similar results for the water column of Lake Erie with minerogenic scattering from clay mineral particles in the size range of 1–20 μm contributing more than 75% of total scattering. Given the importance of fine particles to the clarity of Lake Tahoe, the updated algorithms in DLM-WQ will provide a link between erosion control measures (i.e., load reduction scenarios) in the watershed and lake response.
 With few exceptions, the period of spring snowmelt (May to July) exhibits an increase (greater than 50% of the total attenuation) in fine particle scattering (Figure 15). Conversely, Figure 15 illustrates that chlorophyll scattering and absorption is relatively larger during winter months when stream flow is minimal and deep mixing occurs. It is evident from Figure 15 that winter mixing (e.g., during January to April 2008) increases the percentage of chlorophyll scattering and decreases the fine particle concentration (Figures 7-12) by moving particles to the lake bottom. While deep mixing transports particles toward the lake bottom, bio-stimulatory nutrients locked below the transition zone and photic zone (>100 m below the surface) move up to the photic zone resulting in increasing phytoplankton growth. The greatest Secchi depth clarity is observed during winter deep mixing (Figure 5). These findings echo the results of Jassby et al. , which was based on a 30 years' time series of data from Lake Tahoe.
 The dynamics of fine particles in the lake are complex and depend on a balance between external loading, particle aggregation rates, settling rates, and lake dynamics. The settling velocity of fine particles is extremely low (approximately 3 × 10−2 m/day, estimated using Stokes' law for the 0.5–1 µm size class), and they would remain suspended indefinitely if aggregation mechanisms and enhanced settling were not important. Two algorithms simulating particle aggregation dynamics were developed for incorporation into the lake clarity model (DLM-WQ): a fractal-based particle aggregation model (FPA) and a solid particle aggregation (SPA) model. Although both algorithms include the same physical processes (e.g., collision probability rate and Stoke's law for settling velocity estimation), FPA dynamics are different to those of SPA. The previous aggregation submodel in DLM-WQ required a value for the probability of aggregation independent of particle size distribution (PSD) and concentration. We introduce a new algorithm for probability of aggregation (variable aggregation rates) based on PSD, sediment, and algal concentrations (equation (13)). The updated aggregation models with variable aggregation rates enable estimation of fine particles that are close to measured data collected during 1999–2008. Both algorithms (FPA and SPA) predicted annual average Secchi depth close to measured data. In addition, FPA constant produced stable results in terms of estimation of particles and Secchi depth compared to those of SPA constant.
 The accuracy of model estimated annual average Secchi depths to observed values is above 80% although the model failed to capture some of the event-based data points because of the seasonal atmospheric particle data inputs and uncertainty in estimated stream particle data during storm events. The DLM-WQ elucidated the relative roles of organic and inorganic fine particles that have implications for Lake Tahoe clarity restoration planning. Both biological (e.g., phytoplankton and detritus) and inorganic fine particles are important contributors to clarity loss in Lake Tahoe. Fine particles accounted for approximately 50% of the total light scattering for all sources making it the most significant contributor. With few exceptions, the period of spring snowmelt (May to July) exhibit an increase (greater than 50% of the total attenuation) in fine particle scattering. The residence times of bio-stimulatory nutrients in Lake Tahoe are on the order of several decades [Jassby et al., 1995]; however, residence time of suspended inorganic particle may be shorter given that they aggregate to form larger particles and settle faster. Because approximately 70–75% of the scattering is due to fine particles between 0.5 and 8 µm, management efforts should be targeted to control the transport of fine particles from the watershed to the lake. The updated DLM-WQ can be used to simulate lake response in terms of Secchi depth clarity to different fine sediment and bio-stimulatory nutrients (nitrogen and phosphorus) input scenarios (i.e., load reduction scenarios), and inform development of lake management guidelines to facilitate Lake Tahoe TMDL implementation and other aquatic systems where water quality is a concern.
 This work was partially supported by an EPA/NSF-sponsored Water and Watersheds grant through the National Center of Environmental Research and Quality Assurance (R826282), the U.S. EPA sponsored (R819658 and R825433) Center for Ecological Health Research at UC Davis, grants from the Lahontan Regional Water Quality Control Board and the Tahoe Regional Planning Agency, and by the UC Davis Tahoe Environmental Research Center. The research described has not been subjected to any EPA review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. Field collection of water samples by Bob Richards and Brant Allen underpin the results, as do the particle size analyses by a cadre of students and technicians. The authors thank three anonymous reviewers and Associate Editor, for their helpful comments and suggestions for improvement of the paper.