Effects of Alfvénicity on the compressible energy cascade in magnetohydrodynamic solar-wind turbulence: Evidence from Parker Solar Probe, Solar Orbiter, and WIND observations

The compressible magnetohydrodynamic (MHD) energy cascade in the solar wind is shaped by Alfvénicity, plasma fluctuations, and proton temperature, yet how they regulate the relative contributions of compressible and incompressible energy transfer across heliocentric distances remains poorly understood. We aim to investigate how Alfvénicity, plasma fluctuation amplitudes, and proton temperature regulate the compressible and incompressible MHD energy cascade rates in the solar wind and how the two components of the compressible cascade, the Yaglom-like term and the purely compressible term, respond to changes in plasma conditions. We applied compressible and incompressible exact relations to a large statistical sample of two-hour intervals from WIND (1 au), Parker Solar Probe (PSP; $<0.25$ au), and Solar Orbiter (SolO; 0.3--1 au), classifying intervals as Alfvénic (|σc|>0.75) or non-Alfvénic (|σc|<0.25) based on the normalized cross helicity and further separating them by solar-wind speed. The velocity-fluctuation amplitude is found to be the strongest predictor of the compressible cascade rate, łangle|ɛ_c|̊angle, among the variables examined, followed by density and magnetic-field fluctuations. łangle|ɛ_c|̊angle and łangle|ɛ_i|̊angle are tightly correlated across all regimes, with their ratio organized by Alfvénicity and wind speed---non-Alfvénic and Alfvénic fast winds cluster along łangle|ɛ_c|̊angle = łangle|ɛ_i|̊angle---while Alfvénic slow wind shows a systematic excess consistent with internal energy contributions. Across all regimes, the intervals with the largest łangle|ɛ_c|̊angle are systematically the hotter ones, with increases of nearly 1.5 and two decades in Alfvénic slow and non-Alfvénic wind, respectively---a trend primarily carried by the purely compressible cascade---while the dominant Yaglom-like term appears to control the overall cascade magnitude. Our results indicate that Alfvénicity and wind speed govern the efficiency of turbulent energy transfer across heliocentric distances. The tight coupling between compressible and incompressible cascade rates points toward the need for compressible, multi-fluid descriptions to fully capture the turbulent dissipation budget in the solar wind.