The high-order surface quasigeostrophic $\left({\mathrm{SQG}}^{+1}\right)$ turbulence is investigated numerically in this work with a focus on how lateral divergence modifies cross-scale energy transfers. The ${\mathrm{SQG}}^{+1}$ model extends the classic SQG model by incorporating a velocity correction that introduces controlled divergence, whose strength is controlled by a characteristic Rossby number. We demonstrate that, at the minimal Rossby number, the system exhibits near-SQG behaviors. The potential energy cascades forward and converts to the inversely cascading kinetic energy through frontolysis, with negligible divergence effects. At higher Rossby numbers, however, divergence reorganizes the flow into cold cyclones embedded in expansive warm filaments and fundamentally alters the energy transfer dynamics. Our derived seven-term balance reveals that the compression of cold cyclones by large-scale background flows drives kinetic energy downscale, while the expansion of warm filaments sustains the upscale potential and kinetic energy transfers. These divergence-driven fluxes intensify with the increasing Rossby number, systematically energizing large-scale background flows. This work provides the first quantification of the divergence-driven energy fluxes in balanced submesoscale turbulence.

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• Geophysical fluid dynamics
• Two-dimensional turbulence