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To provide a theoretical explanation for the emergence of zonally elongated structures from two-dimensional turbulence on a rotating sphere, a quasi-invariant of the system is obtained by a minimization process, which is a straightforward extension of a similar process proposed by a previous study on β-plane turbulence to the spherical geometry. The quasi-invariant is defined as a weighted sum of the energy density in the wavenumber space. The distribution of the weighting coefficient has airfoil-shaped contours, with which the anisotropic energy transfer that favors zonally elongated structures can be explained. Large number of numerical time-integrations of decaying two-dimensional turbulence on a rotating sphere are conducted to examine the conservation of the quasi-invariant. It is shown that the quasi-invariant is conserved well when the nonlinearity of the system is sufficiently weak; furthermore, energy is transferred in the wavenumber space apparently along the airfoil-shaped contours of the weighting coefficient for the quasi-invariant.

On a Quasi-Invariant Associated with the Emergence of Anisotropy in Two-Dimensional Turbulence on a Rotating Sphere