A Helmholtz decomposition of structure functions and spectra calculated from aircraft data

Longitudinal and transverse structure functions, Dₗₗ= δuₗδuₗ and Dₜₜ= δuₜδuₜ , can be calculated from aircraft data. Here, δ denotes the increment between two points separated by a distance r , uₗ and uₜ the velocity components parallel and perpendicular to the aircraft track respectively and \, an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, u=uᵣ+ud , where uᵣ is the rotational and ud the divergent component of the velocity, we derive expressions relating the structure functions Dᵣᵣ= δuᵣ· δuᵣ and Ddd= δud· δud to Dₗₗ and Dₜₜ . Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, Dᵣᵣ and Ddd both show a nice r2/3 -dependence for r∈ [2,20]\ km . In this range, the ratio between rotational and divergent energy is a little larger than unity, excluding gravity waves as the principal agent behind the observations. In the upper troposphere, Dᵣᵣ and Ddd show no clean r2/3 -dependence, although the overall slope of Ddd is close to $2/3$ for r∈ [2,400]\ km . The ratio between rotational and divergent energy is approximately three for r<100\ km , excluding gravity waves also in this case. We argue that the possible errors in the decomposition at scales of the order of 10 km are marginal.