Watershed Scale Physically Based Water Flow, Sediment and Nutrient Dynamic Modeling System - Landscape Ecology for Sustainable Environment and Culture - page 144

@ C ss
dt

þ @ q tx

@ x þ @ q ty @ y ¼ J e J d þ W s ¼ J n

ð 8 : 4 Þ

where C ss = concentration of sediment particles in the flow [M/L 3 ], q tx ; q ty ¼
total sediment transport areal flux in the x- or y-direction [M/L 2 T], J e = sediment
erosion volumetric flux [M/L 3 T], J d = sediment deposition volumetric flux [M/
L 3 T], W s = sediment point source/sink volumetric flux [M/L 3 T], J n = net sedi-
ment transport volumetric flux [M/L 3 T].
The total sediment transport flux in any direction has three components,
advective, dispersive (mixing), and diffusive, and may be expressed as (Julien
1998 ):

q tx ¼ u x C ss ð R x þ D Þ @ C ss
@ x

ð 8 : 4a Þ

q ty ¼ u y C ss ð R y þ D Þ @ C ss
@ y

ð 8 : 4b Þ

where u x , u y = flow (advective) velocity in the x- or y-direction [L/T], R x ,
R y = dispersion (mixing) coefficient the x- or y-direction [L 2 /T], D = diffusion
coefficient [L 2 /T].
Note that both dispersion and diffusion are represented in forms that follow
Fick's Law. However, dispersion represents a relatively rapid turbulent mixing
process while diffusion represents a relatively slow a Brownian motion, random
walk process (Holley 1969 ). In turbulent flow, dispersive fluxes are typically
several orders of magnitude larger than diffusive fluxes. Further, flow conditions
for intense precipitation events are usually advectively dominated as dispersive
fluxes are typically one to two orders smaller than advective fluxes. As a result,
both the dispersive and diffusive terms may be neglected.
Similarly, the suspended sediment transport in channels is described by the 1-D
advection-diffusion equation that includes a source-sink term describing sedi-
mentation and resuspension rates and laterally distributed inflow of sediments. The
concentration of particles in flow is governed by the conservation of mass (Julien
1998 ):

@ C ss

@ t þ @ q tx @ x ¼ J e J d þ W s ¼ J n

ð 8 : 5 Þ

Individual terms for the channel advection-diffusion equation are identical to
those defined for the overland plane. Similarly, the diffusive flux term can be
neglected. The dispersive flux is expected to be larger than the corresponding term
for overland flow. However, the channel dispersive flux still may be neglected
relative to the channel advective flux during intense runoff events. The distributed
runoff inflow to the channel and the suspended sediment concentration in the
runoff are simulated by the overland component.