I'm using the community land model (CLM) to do some university work and I'm trying to understand some of the equations that go into the model, one example is:
$$ \kappa{z}\frac{\partial{U}}{\partial{z}} = \left( 1 - 16\zeta\right)^{-1/4} $$
where $\zeta = \frac{z}{L}$ and and by integrating from $z$ to $z_{0}$ gives:
$$\partial{U} = \frac{u_{*}}{\kappa}\left[ ln \frac{z}{z_{0}} + \psi_{m}\left(\zeta\right)\right]$$
where
$$ \psi_{m} = 2 ln \left(\frac{1+\chi}{2}\right) + ln \left( \frac{1 + \chi^{2}}{2} \right) - 2tan^{-1}\chi + \frac{\pi}{2}$$
and $$ \chi = \left(1 - 16\zeta\right)^{1/4} $$
Could anyone give me some pointers on where to start understanding these equations?Eventually I would like to be able to write the entire solution in a step by step manner i.e. go from step 1 to step 2 above.