# Unlikely existence of $k_x^{-1}$ spectral law in wall turbulence: an observation of the atmospheric surface layer

## TL;DR

This study investigates the presence of the $k_x^{-1}$ spectral law in wall turbulence within the atmospheric surface layer, finding no evidence for it and supporting a recent model correction instead.

## Contribution

It provides observational evidence against the $k_x^{-1}$ law in wall turbulence and supports a corrected model of energy-containing eddies attached to the wall.

## Key findings

- No support for the $k_x^{-1}$ spectral law in atmospheric surface layer data.
- Results align with a recent correction to the energy-containing eddy model.
- Mathematical discussion explains the discrepancy with the $k_x^{-1}$ law.

## Abstract

For wall turbulence, there has been predicted a range of streamwise wavenumbers $k_x$ such that the spectral density of streamwise velocity fluctuations is proportional to $k_x^{-1}$. The existence or nonexistence of this $k_x^{-1}$ law is examined here. We observe the atmospheric surface layer over several months, select suitable data, and use them to synthesize the energy spectrum that would represent wall turbulence at a very high Reynolds number. The result is not consistent with the $k_x^{-1}$ law. It is rather consistent with a recent correction to the prediction of a model of energy-containing eddies that are attached to the wall. The reason for these findings is discussed mathematically.