Turbulence closure remains a major source of uncertainty in atmospheric and oceanic models, particularly on coarse computational grid resolution, where subgrid scales strongly influence large-scale dynamics. While machine-learning-based closures have shown promise, most existing approaches rely on offline training and require large amounts of subgrid-scale data, raising concerns regarding generalization and stability in non-stationary geophysical flows. In this work, we present two online learning strategies for turbulence-closure modeling, in which model parameters are adaptively updated during large-eddy simulations (LES) using limited statistical information from high-fidelity reference data. Both approaches are demonstrated in idealized two-dimensional beta-plane geophysical turbulence, serving as a testbed for atmospheric and oceanic flows. The first approach employs Ensemble Kalman Inversion (EKI) to perform parametric online calibration of physics-based closures, including Smagorinsky-, Leith-, and backscatter-type models [1]. Closure parameters are iteratively updated to minimize mismatches between LES kinetic energy spectra and filtered direct numerical simulation (DNS) spectra. This derivative-free framework is data-efficient, requiring only a small number of DNS snapshots, and preserves physical interpretability while enabling uncertainty quantification of calibrated parameters. The second approach uses reinforcement learning (RL) to learn a state-dependent closure policy online [2]. Here, a neural network policy dynamically adjusts closure coefficients based on resolved-scale spectral information, with rewards defined through the agreement with DNS enstrophy spectra. This formulation allows for spatially and temporally varying closures, naturally incorporates backscatter, and improves the representation of extreme events. By contrasting these two online learning methods, we highlight trade-offs between interpretability, flexibility, and data requirements. Our results demonstrate that online learning offers a robust pathway for adaptive turbulence modeling in atmospheric and oceanic simulations, bridging physics-based closures and data-driven methods within a unified framework.