Hydrodynamic slope length factor for the USLE-M in grid cells

The Desmet and Govers (1966) slope length factor is given by

		( Ai,j-in + D^2)^m+1 - Ai,j-in^m+1
L i,j =	————————————.................................................................(10)
			D^m+2 xi,j^m (22.13)^m

where A_i,j-in is the catchment area the top of grid cell i,j, D is the size of the cell, and m is the slope length power normally used in the USLE. This equation results from subtracting the soil lost from the area upslope of the grid cell from the area including the grid cell and dividing the result by the area of the grid cell.

Originally, it was proposed that if this approach is applied using the USLE-M rather than the USLE to determine the soil losses, the result is

		QCe.i,j ( Ai,j-in + D^2)^m+1 - QCe.i,j-in Ai,j-in^m+1
L i,j = F	———————————————————...................................(11)
			QRe.i,j-cell D^m+2 xi,j^m (22.13)^m

where

			(1 - QCei,j-in)
F = QCe.i,j-in +	————————.....................................................................(12)
		(1 + Ai,j-in /D^2)^m

Details of the derivation of these equations can be found in Kinnell, PIA (2001), Slope length factor for applying the USLE-M to erosion in grid cells. Soil and Tillage Research 58, 11-17.

Eqs. 11 and 12 have now been replaced by

Q_Ce.i,j-eff (A_i,j-in.eff + D²)^m+1 - Q_Ce.i,j-eff A_i,j-in.eff^m+1
L_UMe.i,j = ———————————————————
Q_Re.i,j-cell D^m+2 x_i,j^m (22.13)^m

where

Q_Ce.i,j-eff = [(Q_Ce.i,j-cell D²) + (Q_Ce.i,j-in A_i,j-in.)] / (A_i,j-in.eff + D²)

.

and

Ai,j-in.eff = Ai.j-in QCe.i,j-in / QCe.i,j-all

The original approach developed by Desmet and Govers (1996) for determining the L factor for grid cells applies where runoff is produced uniformly over the whole area contributing runoff to the cell so that the runoff coefficient for the cell (Q_Ce.i,j-cell) equals that of the upslope area (QCe.i,j-in). When runoff is zero, the upslope area is effectively zero. In the context of determining the slope length factor for a cell, the approach given above considers the upslope area to vary depending on the ratio of the runoff ratio of the upslope area to that of the whole area (QCe.i,j-all). Details of the approach are subject to a paper to be published in the Soil Science Society of America Journal and the approach is used in the current software applying the USLE-M within AGNPS (AGNPS-UM) available from this WWW site.

It should be noted that these equations apply only to the USLE-M and cannot be used with the USLE

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