Multi-point Monin-Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

The Multi-point Monin-Obukhov similarity (MMO) was proposed to address the incomplete similarity in the original Monin-Obukhov similarity theory (MOST). However, similar to MOST, MMO was also proposed as hypotheses based on phenomenology. Here we derive analytically MMO for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations using the spectral forms of the Navier-Stokes and potential temperature equations. We show that for large-scale motions (wavenumber $k<1/z$) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e., as $z$ decreases from $z>-L$ to $z<-L$), where $z$ is the height from the surface and $L$ is the Obukhov length. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k < -1/L$ to $k > -1/L$, resulting in a singular perturbation problem, which we analyze using the method of matched asymptotic expansions. The scales and scaling ranges identified in MMO are derived. We also derive the corrections to the spectra due to finite ratios of the length scales. The analytical derivations for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.