## 1 Introduction
### Particle Dynamics  
In this section, we focus on the bottom boundary layer (BBL; for a recent review of the fluid dynamics of the BBL see [36]). The particle assembly \(C\) is a sum of many types of particles \(C=\sum_{i}C_{i}\), each with a specific settling speed, composition, size and any other property that is relevant to either hydrodynamic or optical properties. The general conservation equation for particles can be written as:  
\[\frac{\partial C_{i}}{\partial t}+\
abla(C_{i}(U-w_{i}))=\
abla(K\
abla C_{i} )+\sum_{j}f(C_{i-j},C_{j})+S_{i} \tag{1}\]  
where \(C_{i}\) is the mass concentration of particles of type \(i\), \(t\) is time, \(U\) is the 3-D velocity field, \(w_{i}\) is the settling velocity of these particles (we use the convention that it is positive downward), \(K\) is a diffusion coefficient, \(f(C_{i-j},C_{j})\) represents aggregation and disaggregation dynamics creating and destroying \(C_{i}\)-type particles, and \(S_{i}\) represents other sources and sinks (e.g., resuspension/deposition or biological production/consumption of particles). To solve Equation (1) for a given flow field, one needs boundary conditions (e.g., flux or concentration of particles at the boundaries of the domain) and initial conditions (e.g., the state of the particles at time \(t=0\)). Even then, Equation (1) cannot be solved analytically except for very simple cases such as we will address next. Equation (1) can be simplified to represent only the vertical dimension (\(z\), positive upwards), assuming that horizontal advection of horizontal gradients is negligible relative to vertical processes. After time averaging (or assuming steady-state), the steady balance between downward settling, upward diffusion, exchange among particle types (aggregation/disaggregation), and sources/sink is:  
\[-w_{i}\frac{dC_{i}}{dz}=\frac{d(K_{eddy}dC_{i})}{dz^{2}}+\sum_{j}f(C_{i-j},C_ {i})+S_{i} \tag{2}\]  
where \(C_{i}\) is the time- (and horizontally) averaged concentration of particles of type \(i\), \(K_{eddy}\) is an eddy diffusion coefficient (representing the mixing by the small-scale turbulent field), and \(w_{i}\) is a constant settling velocity for each particle type \(i\). Large aggregation rates are associated with large particle concentrations, e.g., following a major resuspension event or during an algal bloom. Large disaggregation rates are associated with large fluid shears, e.g., in the wave boundary layer, which is a layer a few centimeters thick next to the bottom. If particle concentrations and fluid shears do not vary greatly throughout a boundary layer, then an equilibrium size distribution develops, and the aggregation and disaggregation terms cancel one another. An assumed equilibrium under certain circumstances is unlikely to emerge in bottom boundary layers, for example when near-bed wave-generated shears are much greater than current-generated shears higher in the boundary layer. The law-of-the-wall is often invoked in the BBL, which is consistent with a linear eddy diffusivity profile where \(K_{eddy}=\kappa u_{*}z\), and \(\kappa\) is [PERSON]'s constant (\(\sim\)0.4), and \(u_{*}\) is the friction velocity (a function of BBL turbulence, e.g., due to wave and current shear). Neglecting aggregation dynamics and sources/sinks, and integrating Equation (2) in the vertical and solving the resulting differential equation results in the Rouse equation [37; 38] for a homogenous population of particles of type \(i\):  
\[C_{i}(z)=C_{i}(z_{a})\left(\frac{z}{z_{a}}\right)^{-\frac{w_{i}}{z_{a}}} \tag{3}\]  
where \(z_{a}\) is a reference elevation where a particle concentration is assigned. This is one of several different analytical solutions for the balance of settling and turbulent mixing, which vary depending on assumptions about the eddy diffusivity profile [39; 40]. An alternative derivation of Equation (3) using probability theory has also been proposed [41].