Robotic information gathering (RIG) techniques refer to methods where mobile robots are used to acquire data about the physical environment with a suite of sensors. Informative planning is an important part of RIG where the goal is to find sequences of actions or paths that maximize efficiency or the quality of information collected. Many existing solutions solve this problem by assuming that the environment is known in advance. However, real environments could be unknown or time-varying, and adaptive informative planning remains an active area of research. Adaptive planning and incremental online mapping are required for mapping initially unknown or varying spatial fields. Gaussian process (GP) regression is a widely used technique in RIG for mapping continuous spatial fields. However, it falls short in many applications as its real-time performance does not scale well to large datasets. To address these challenges, this paper proposes an efficient adaptive informative planning approach for mapping continuous scalar fields with GPs with streaming sparse GPs. Simulation experiments are performed with a synthetic dataset and compared against existing benchmarks. Finally, it is also verified with a real-world dataset to further validate the efficacy of the proposed method. Results show that our method achieves similar mapping accuracy to the baselines while reducing computational complexity for longer missions.