In this study, estimates of inhomogeneous integral scales are derived from rapid distortion theory (RDT) for the case of wall-bounded high-Reynolds-number turbulence and from large-eddy simulation (LES) of a neutrally stratified atmospheric boundary layer (ABL). As for any inhomogeneous flow, integral scales in different directions are introduced. Downward integral scales are introduced since they differ from the usual vertical integral scales because of the presence of the wall. The study concentrates on the length scales based on the vertical velocity, which are the most affected by blocking by the wall, which is assumed to be horizontal.

It is shown from RDT that the asymptotic behaviour of the integral length scales for small heights depends crucially on the spectrum power law $-2p$. When $2p>1$ there is always one length scale which does not scale with the distance to the wall $z$. Only the downward integral scale is proportional to $z$ for any $2p$. These results show that the assumption, often made in studies of boundary layers, that all the lengths are proportional to $z$, is not compatible with the assumption of a spectrum decaying according to Kolmogorov's law, but rather with a spectrum following a $-1$ power law. It is an encouraging result since there is now widespread theoretical, experimental and numerical evidence that such a $-1$ power-law subrange exists in the spectra of high-Reynolds-number wall-bounded turbulence, for eddies larger than $z$. The RDT results allow an interpretation of the vertical profiles of the integral length scales computed from the LES outputs: above the third grid point, the vertical profiles of the integral length scales have a linear shape, as expected for high-Reynolds-number turbulence and $2p=1$. Very close to the surface, the upward integral length scales decreases with $z$ because of the fast decay of the spectrum ($2p>2$) from the LES subgrid model.

The longitudinal-to-transverse integral length scale ratio is computed using RDT and LES. This ratio is interpreted as the aspect ratio of elongated near-wall large eddies, which are ubiquitous features of LES of boundary layers in which shear plays an important role in the dynamics. The LES shows that the longitudinal-to-transverse integral length scale ratio is an increasing function of $z$, ranging between 1 and 3, which is of the order of magnitude of the published theoretical value of 3.5. From RDT, the evolution with $z$ of the longitudinal-to-transverse integral length scale ratio means either that the velocity shear $\beta$ decreases with $z$ and the spectral power law $2p$ varies in a non-trivial manner, or if both the RDT and LES are valid that the scale of the large eddies is proportional to $\beta z$ with $\beta$ varying from 1.3 to about 4.