Accurate forecasts of the global ocean state are essential for marine operations, hazard assessment, and scientific research. Recently, advances in artificial intelligence have enabled the development of data-driven, end-to-end neural models that learn ocean dynamics directly from multi-decadal historical datasets. However, existing methods often struggle to capture the inherent structure and multi-scale interactions within oceanic data, which leads to poor representation of key dynamical features and reduced forecast accuracy. To alleviate the issues, we propose a structure-aware network (SAN) with multi-scale interactions. First, we design variable-aware and depth-aware modules that incorporate learnable embeddings for multiple variables (e.g., temperature and salinity) and depths respectively, enabling explicit modeling of oceanic data structure. Second, we design a multi-scale interaction module comprising multi-scale feature selection and adaptive feature aggregation to extract improved feature representations across spatial scales. We conducted extensive experiments on the global ocean reanalysis dataset and the results demonstrate that the proposed method achieves promising performance.