For each reach, the sand transport component of the model proceeds as follows.

1. Mass of sand in each morphological environment (eddy, main-channel pool, channel margin) is initialized at some user-defined level of relative "fullness" (0-1).
   

2. Total monthly tributary inputs of fine and sand-size material for reaches beginning at the confluences of the Paria and LCR are added to the upstream boundary mass based on output from the USGS Paria model (Topping 1996) or LCR rating curves (D. Topping, USGS, Boulder, Colorado, USA, unpublished data). Tributary inputs for other reaches are currently assumed to be zero until ungaged tributary estimates become available.
   

3. Based on the maximum hourly value of the average on-peak diurnal water discharge for the current time step, potential storage in each environment is computed based on potential storage-discharge relations (Appendix 1).
   

4. Empirically derived bulk erosion rates (percentage of mass lost over the month) are applied to eddy (above-water, below-water), channel margin, and main-channel pool environments. All material lost through this process is added to the export mass from the reach.
   

5. Current mass in each morphological environment is compared to the potential storage under the discharge regime for the month. If potential storage is less than the existing mass within an environment, the mass difference is removed and added to the export mass.
   

6. Sediment inflow to the reach potentially available for deposition in each environment is based on user-defined proportions. If sufficient storage exists, all of the material destined for an environment is deposited. If the potential mass for deposition exceeds the potential storage, the difference is added to the export mass.
   

7. Some of the export mass may be retained in the reach if storage conditions permit. For example, during a period of increasing discharge, potential storage in the main-channel pool environment will decrease while potential storage in eddies will increase (Appendix 3). If sediment inflow to the reach is low, the eddy environment will not fill, but the extra material released from the main-channel environment will be available for deposition in the eddy environment within the same reach.
   

8. Based on these transfers, the mass of sand is updated in the main channel, and in each slice (Fig. 3) of eddy and margin environments. The mass of sediment exported from the reach is equal to the sum of sand inflow to the reach (from upstream reach and tributaries) and the amount eroded from the main-channel pools, eddies, and margins, less the amount stored. The export mass from reach xbecomes the inflow mass for reach x+1. The mass of sand in the main channel and in each discharge slice in the eddies and margins at the end of timestep x becomes the initial conditions at the start of timestep x+1.
   

Lagrange estimation of downstream concentration profiles-temperature model

One of the most difficult computational problems in river ecosystems is to represent gains and losses of materials and energy suspended in the water column, as water moves rapidly downstream (e.g., detritus, drifting insects, temperature). We use a Lagrange "sampling" method to deal with such variables (Fig. 4). We think of taking a representative sample parcel of water entering the system each month, and following how concentrations will change in that parcel due to gains (from atmosphere, river bottom, tributaries) and losses (sinking, decomposition, heat loss) as it moves downstream. Dynamic change along each Lagrange sample track is assumed to follow relatively simple linear dynamics (dx/dt = a -bx, where a is the input rate, b is the loss rate, and x is reset at each at the time when the sample parcel passes each tributary input point) for which we can obtain an analytical solution for downstream concentration changes.

The net result of processes effecting gains and losses to the sample parcel of water determines the output (concentration or temperature) at the downstream boundary of each reach in our simulation. The downstream concentration (C_i+1) is computed by the equation

Eq. A4.1 consists of three components:

Ceq :

the equilibrium concentration or temperature that would occur if the sample water parcel moved forever over an infinitely long reach at fixed gain and loss rates and no change in cross-sectional area;

Ci - Ceq :

the extent of the departure of the equilibrium concentration or temperature from the upstream boundary condition (Ci); and

exp{-vWiLi/Qi} :

a modifier on the equilibrium departure (a multiplying factor 0-1), which is an exponentially decreasing function positively related to the sinking or heat exchange rate, wetted width, reach length, and inversely related to discharge. For example, the longer the reach, the more closely the downstream boundary concentration or temperature will approach the equilibrium level.

Eq. A4.2 was derived by initially formulating the change in mass within a reach over time as a difference equation:

That is, the rate of change in mass or temperature over time (ΔM/Δt) within a reach will be a function of the difference between the gains due to resuspension (RW, a function of the resuspension rate and the surface it acts on, which is proportional to the wetted width) and the losses due to sinking (sWC, a function of the sinking rate, the surface on which this process occurs, and the concentration, C). Note that C is a function of mass divided by the volume of water, and Eq. A4.2 is therefore a differential equation. If we solve this equation over the period t that it takes the water, moving at a velocity v to pass through a reach of length L (t = L/v) and replace any reference to mass in Eq. A4.2 with its concentration equivalent,C/A, where A is cross-sectional area, which is equal to the volume for a 1 foot (0.3 m) thick section of water, we get the solution for the concentration at the bottom of the reach (C_i+1) given by Eq. A4.1.