Importance: Convective storms are one of the most costly natural hazards in the current climate. These storms are expected to change with future climate warming and may bring more devastating socioeconomic costs. Our understanding of these storms has substantially improved with advancements in theoretical understanding, model physics representation, and computational power. However, combining insights from simplified frameworks with those from realistic representations of these storms at scale remains challenging. Large-scale metrics for convective storms, like Convective Available Potential Energy (CAPE), could help bridge the gap between these paths. This thesis aims: (1) to evaluate existing model and reanalyses representation of CAPE distributions, (2) to understand the primary drivers of CAPE changes between climate states, and (3) to connect weather and climate scale variations with a full scaling of CAPE. Approach: This work uses an array of observations and modeling datasets with varying levels of authenticity and complexity. We approach the scientific questions by combining these datasets with simple theoretical frameworks. In Chapter 2, we use a radiosonde observational dataset (IGRA), against which we evaluate reanalyses (ERA-Interim and ERA5) and a convection-permitting model (WRF) in the current climate. In Chapter 3, we explain the full distributional projection of CAPE from the convection-permitting WRF simulations with CAPE-MSE surplus dependence and synthetic profiles. In Chapter 4, we propose and evaluate the full scaling of CAPE with ERA5 reanalysis and 11 models from the CMIP6 inter-comparison project across space and different temporal scales. Key Findings: We find that warming increases the occurrence of high CAPE conditions substantially in all climate models. While CAPE distributions in coarse-resolution models are not accurate, CAPE in a high-resolution convection-permitting model largely matches observations other than in extreme CAPE conditions, whose occurrence is underestimated in the current climate. The low biases arise from an underprediction of hot and humid conditions. Future projections of midlatitudes CAPE exhibit distributional shifts, so they cannot be expressed as a simple mean change; they also cannot be sufficiently predicted by changes in surface conditions alone. We find that the distributional shift can be captured with three mean changes at both the surface (Ts, RHs) and mid-troposphere (Tm), highlighting the importance of a lapse rate adjustment in mid-latitude summertime under climate change. Furthermore, the minimal three state parameters can be reduced to a single parameter of "MSE surplus". CAPE dependence on MSE surplus remains consistent across climate states in both the nudged convection-permitting model (WRF) and in 11 free-running CMIP6 models. On shorter timescales than climatological shifts, predicting CAPE variations requires at least one additional input, the convective layer depth. We, therefore, derived a robust CAPE scaling from entropy and buoyancy forms that effectively captures CAPE variations across spatial and temporal scales, including diurnal, seasonal, and climatological variations. This scaling provides key physical insights into how much and why CAPE changes, with strong implications for societal impacts. It allows model biases to be diagnosed and may provide a practical tool for weather analysis.