An integration-free method for calculating curl and divergence in space plasmas using multi-spacecraft data

The knowledge of local spatial gradients (curl, divergence etc.) is crucial to examine the three-dimensional variation of fields including velocity and magnetic fields in space plasmas like the solar wind . Here we propose a simple method to calculate the same using the in-situ data of multi-spacecraft systems. Unlike the popular Curlometer method which depends on the vector integration theorems, our integration-free method is based on the construction of a local orthonormal coordinate system and the associated finite difference approximations. The Curlometer is applicable to a four spacecraft system arranged in a tetrahedron and yields an averaged estimate of the curl. Using our proposed method over 107 intervals of Magnetospheric Multiscale (MMS) Mission -- NASA data, we successfully calculate the spatial derivatives at the position of each spacecraft of the tetrahedron. The average value of all the curls calculated for a given tetrahedron shows an excellent agreement (correlation coefficient ∼ 0.99) with the curls calculated using Curlometer formula. The quality of the calculated curl (using our method) is found to improve if the spacecraft configuration approaches a regular tetrahedron. The current framework facilitates investigation of turbulent heating rates and the exploration of local flow features like Beltramization, existence of current sheets, etc., in present and future multi-spacecraft missions. black and magnetospheric plasmas