PROLA: Principal-Orthogonal Low-rank Adaptation for Predictive Spatiotemporal Weather Downscaling

High-resolution weather prediction is important for resolving local atmospheric patterns missed by coarse global forecasts. Large pretrained weather backbones have improved global forecasting, but adapting them to produce future high-resolution fields from recent coarse states remains challenging and costly. We formulate this setting as Predictive Spatiotemporal Weather Downscaling (\textbf{PSWD}), where the target is a future high-resolution trajectory rather than a same-time refined field. To make this structure explicit, we propose a framework that decomposes prediction into spatial and temporal stages using a shared pretrained backbone, fixed numerical scaffolds, and lightweight residual heads. For backbone adaptation, we introduce \textbf{PROLA} (PRincipal-Orthogonal Low-rank Adaptation), which splits a fixed low-rank budget between pretrained principal directions and their orthogonal complement. PROLA further rescales rank-1 optimizer updates using gradient signals, improving adaptation without increasing trainable rank. Experiments in the PSWD setting show that PROLA outperforms representative low-rank adaptation baselines under matched trainable-parameter budgets. These results support PSWD as a challenging setting for weather backbone adaptation and PROLA as an effective method for this setting.