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): a ac =W, DC + 2 (KBC) Ĉz cz 0=W₁ 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 ôt ax (2) 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: dxdydz = 2(Cp) dvol +U. +V ++ (w-ws) a(pCV) a(Cp) ôt ôt dz (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. (b) Reynold's average (2) (which is your result from (a)) to get (10 pts): ac ac ac +u. +V +(w-ws) ac dz (1) = ac côz (4) (c) Now reduce the global expression of SPM conservation in (4) to the form needed for Rouse' Law (1), specifying each necessary assumption (I've already told you most of what you need to know.); 10 pts d) Explain why horizontal mixing is neglected in (4); 5 pts. 1