 Solution

Write dimensional variables as their scales times dimensionless variables, with positive scales $S_0,Q_0,m_0,N_0,t_0$. For compactness defineA consistent explicit choice isThese identities give $t_0{m}_0/({\rho }_i{S}_0)=1$, $K{t}_0{N}_0^3=1$, $f{Q}_0^2/(\sin \alpha S_0^{8/3})=1$ and ${\rho }_{w}g{t}_0{Q}_0/(S_L{N}_0)=1$. Substitution therefore leaves the unit coefficients in all four balances, and the sole supply parameter is $\mu =M_L{S}_L/Q_0$. Expanding its scales yieldsAfter dropping scale hats, eliminate the instantaneous quantities to obtain the useful two-dimensional systemThese scales assume a positive slope, positive friction coefficient and nonzero lake area; they nondimensionalize exactly the hydraulic law printed in the source.