  The current work presents a tensor formulation of the Lumley Decomposition
(LD), introduced in its original form by Lumley (1967b), allowing
decompositions of turbulent flow fields in curvilinear coordinates. The LD in
his form is shown to enable semi-analytical decompositions of self-similar
turbulent flows in general coordinate systems. The decomposition is applied to
the far-field region of the fully developed turbulent axi-symmetric jet, which
is expressed in stretched spherical coordinates in order to exploit the
self-similar nature of the flow while ensuring the self-adjointness of the LD
integral. From the LD integral it is deduced that the optimal eigenfunctions in
the streamwise direction are stretched amplitude-decaying Fourier modes
(SADFM). The SADFM are obtained from the LD integral upon the introduction of a
streamwise-decaying weight function in the vector space definition. The
wavelength of the Fourier modes is linearly increasing in the streamwise
direction with an amplitude which decays with the -3/2 power of distance from
the virtual origin. The streamwise evolution of the SADFM re-sembles reversed
wave shoaling known from surface waves. The energy- and cross-spectra obtained
from these SADFM exhibit a -5/3- and a -7/3-slope region, respectively, as
would be expected for regular Fourier modes in homogeneous and constant shear
flows. The approach introduced in this work can be extended to other flows
which admit to equilibrium similarity, such that a Fourier-based decomposition
along inhomogeneous flow directions can be performed.
