In 1941, Kolmogorov proposed the 4/5 law for three-dimensional isotropic turbulence, which links measurable third-order structure functions with energy transfer. It is only after around 60 years later that this theory is generalized to two-dimensional turbulence, then the theory is applied to quantify the direction and magnitude of energy transfer. However, the applicability of these theories is limited for several reasons: (i) they are derived for the inertial range, so they do not cover the forcing scales; (ii) they apply to scenarios with unidirectional energy transfer, while the geophysical turbulence is known to transfer energy to both large and small
scales simultaneously. To overcome these two obstacles, we derive a forcing-scale-resolving global theory that expresses different inertial ranges in one formula and captures the bidirectional energy transfer. Therefore it is well applicable to geophysical fluid data to obtain the information of energy transfer, and it applies to a broader range of the measured data by avoiding the artificial choice of inertial ranges. Also, the new theory brings about qualitative improvement by the ability to detect the scales of energy input, which was impossible before.
We justify the new global theory in classic two-dimensional turbulence and apply it to atmospheric and oceanic data to obtain the direction and magnitude of energy transfer and the energy input distribution. Thus, the new turbulence theory provides a more comprehensive understanding of energy transfer in geophysical flows. The new theory is also applicable to turbulence with more complexity such as magnetohydrodynamical turbulence and two-dimensional rotating stratified turbulence.
