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Many measurements have shown that the random temperature fields in the stably stratified atmosphere are not locally isotropic (LI). The local axial symmetry (LAS) hypothesis looks more appropriate under these conditions. The object of this paper consists in the development of a flexible tool for spectral studies of LAS scalar fields independently of their origin in stably stratified geophysical flows. A heuristic model of a 3D spatial spectrum is proposed in order to describe and study statistical properties of LAS temperature inhomogeneities from LI structures up to quasi-layered ones. To simplify the solution of this problem, a main assumption was accepted: the consideration is restricted to 3D spectra which may be given on a one-parametric family of surfaces of rotation. Such 3D spectra may be represented by a single function of one variable which is the parameter of the family. This approach allows one to introduce the generalized energy spectrum which describes an energy distribution according to inhomogeneity sizes. The family of surfaces determines the shape of inhomogeneities. The family of ellipsoids of rotation and power-law generalized energy spectrum is used as the simplest example of the model application in order to study the general properties of LAS-structure spectra. The behavior of vertical, horizontal, and oblique 1D spectra and coherency spectra is studied. The relationship between the suggested model and some existing models of temperature spectra is considered. The application of the model for the analysis of experimental data is shown for two sets of measurements. It is shown that the suggested model may quantitatively describe experimental 1D spectra and coherency spectra from a unique point of view. It is noted that the model may be used for both the planning of measurements and data processing. Possible generalizations of the model are considered for random fields with more degenerate symmetry and for space-temporal spectra.