The biogeochemical cycle that does not include the atmosphere is the phosphorus cycle.
Unlike other cycles such as the carbon and nitrogen cycles, which involve significant atmospheric components, the phosphorus cycle primarily operates through soil, water, and living organisms. Phosphorus is usually found in rocks and mineral deposits and does not exist in gaseous form like carbon dioxide or nitrogen gas.
In the phosphorus cycle, weathering processes help in the release of phosphate ions into the soil and water, which are then taken up by plants. Animals acquire phosphorus by consuming plants or other animals. When these organisms die or excrete waste, phosphorus returns to the soil, where it can again be utilized by plants, completing the cycle. Overall, the absence of an atmospheric component in the phosphorus cycle distinguishes it from the other major biogeochemical cycles.