Scale-by-scale two-point statistics in WRF Hybrid LES model

Atmospheric variability spans interacting regimes set by rotation, stratification, and diabatic forcing. One open question is that diagnosing scale-to-scale energy transfer remains challenging because observations rarely provide complete budget closure. We analyze the June 2019 European heatwave using the Weather Research and Forecasting (WRF) model with a hybrid, scale-adaptive LES closure and five nested domains, resolving horizontal separations from O(10²) m to O(10⁶)–O(10⁷) m.

Starting from the governing equations of motion in WRF hybrid vertical coordinate, we derive and appraise generalized two-point, Scale-by-Scale (SbS) budget equations for the second-order moments of horizontal velocity increments, reflecting the kinetic energy at each scale. Whilst equations are written for all scales and any point of the considered domains, their assessment against data is performed in a plane parallel to the ground. SbS energy budget equations account for the inhomogeneity, anisotropy, and all effects present in the first principles. We complement these diagnostics with height-dependent characteristic length scales (Kolmogorov, Taylor, Ozmidov, buoyancy, Rhines and Rossby deformation).
We show results for two cases:
i) In the free troposphere, where the SbS kinetic-energy budget is dominated by the advective term (reflecting non-linear interactions and energy transfer), which is balanced by the pressure-gradient contributions. Radial integration of the advective term reproduces the third-order structure function and exhibits a sign reversal near r ∼ 10⁵ m, reflecting transitions between downscale and upscale kinetic energy transfer across mesoscale–synoptic ranges.
ii) In the lower troposphere, we investigate daytime and nocturnal conditions. First, in daytime conditions, the boundary layer exhibits a classical behavior, in which energy is transferred across scales mainly by advective, non-linear effects. Second, for stable stratification during the night, the pressure contribution increases significantly, and the advective transfer adjusts to the pressure-imposed scale dependence, as already noted in the free atmosphere.

These results provide a physically interpretable framework for diagnosing atmospheric cascades across scales and motivate extending SbS budgets to include thermodynamic variables, such as the moist potential temperature and the water vapor content. The latter would allow us to quantify the contributions of radiative and diabatic forcings over short- and long-term timescales, relevant to climate variability.