1. Rouse' Law and the SPM profile equation (40pts): The Rouse' Law approach to suspended

particulate matter (SPM) conservation assumes that SPM concentration is conserved locally, in

each vertical column of water; i.e., SPM concentration may move up and down in the water col-

umn (depending on flow conditions), but horizontal transport of SPM has no effect on the con-

centration profile, even though individual particles will travel more or less with the flow and the

concentration profile (once known) is used to calculate horizontal transport. This Rouse' Law ap-

proach is the basis of most practical calculations of suspended load, except in some 3-D numeri-

cal models, where the total derivative (local time variations plus horizontal advection) is consid-

ered, also. The specific form of SPM conservation used in Rouse' law is (after Reynold's averag-

ing and simplification using physical reasoning):

ac

Ws OC + D (Kc DC)

dz

dz

Here, C is SPM concentration, and Kc is a vertical diffusivity for SPM.

(a) Derive the law for SPM conservation (15 pts):

ac ac ac

ac

+v. -(w-ws)

vs) = 0

Ət ex dy

dz

+U

0=Ws

a(pCV)_a(Cp)

=

Ət

where no Reynolds' averaging has been done so far.

There is a summary of the derivation in the first file on the equations of motion, after the discus-

sion of mass conservation. You can follow the approach mentioned there, by considering the flux

of suspended sediment. That is, define SPM as a volume concentration C (e.g., µ-liter SPM/liter

water), and then define fluxes in and out of a control volume dVol = dx dy dz. Note that the flux

you are defining is Cxpxvelocity and you are looking at the time-change of Cxp in volume dVol,

so you are going to use a control volume approach to define:

Ət

dxdydz =

(1)

a(Cp) dvol

dt

(2)

(3)

To do this, you will have to assume that the sediment velocity vector Us = {u, v, w - Ws}, where

{u, v, w} is the water velocity. The SPM settling velocity Ws is NOT a function of {x,y,z,t} and

does not have turbulent fluctuations - it is a constant.