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

Abstract

Longitudinal and transverse structure functions, $D_{ll}=\langle {\it\delta}u_{l}{\it\delta}u_{l}\rangle$ and $D_{tt}=\langle {\it\delta}u_{t}{\it\delta}u_{t}\rangle$, can be calculated from aircraft data. Here, ${\it\delta}$ denotes the increment between two points separated by a distance $r$, $u_{l}$ and $u_{t}$ the velocity components parallel and perpendicular to the aircraft track respectively and $\langle \,\rangle$ an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, $\boldsymbol{u}=\boldsymbol{u}_{r}+\boldsymbol{u}_{d}$, where $\boldsymbol{u}_{r}$ is the rotational and $\boldsymbol{u}_{d}$ the divergent component of the velocity, we derive expressions relating the structure functions $D_{rr}=\langle {\it\delta}\boldsymbol{u}_{r}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{r}\rangle$ and $D_{dd}=\langle {\it\delta}\boldsymbol{u}_{d}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{d}\rangle$ to $D_{ll}$ and $D_{tt}$. Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, $D_{rr}$ and $D_{dd}$ both show a nice $r^{2/3}$-dependence for $r\in [2,20]\ \text{km}$. 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, $D_{rr}$ and $D_{dd}$ show no clean $r^{2/3}$-dependence, although the overall slope of $D_{dd}$ is close to $2/3$ for $r\in [2,400]\ \text{km}$. The ratio between rotational and divergent energy is approximately three for $r<100\ \text{km}$, 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.