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

TL;DR

This work introduces an Integration-Free Method (IFM) to compute local spatial gradients, including curl and divergence, from multi-spacecraft measurements without relying on vector integration theorems. By constructing a local orthonormal frame anchored to the spacecraft, IFM uses finite-difference relations to obtain point-wise derivatives at each spacecraft, enabling applicability to both four-spacecraft tetrahedra and subsets with three spacecraft. Across 107 MMS intervals, IFM curls show excellent agreement with Curlometer results (average $R\approx0.99$), while providing a direct estimate of point-wise derivatives and higher-order quantities such as current density and vorticity. The method also includes a rigorously defined quality metric and links curl accuracy to tetrahedron shape, supporting its use for turbulent heating analyses and future multi-spacecraft missions beyond four spacecraft, such as HelioSwarm.

Abstract

The knowledge of local spatial gradients (curl, divergence etc.) is crucial to examine the three-dimensional variation of flow 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 a single volume-averaged estimate of the curl. Using our proposed method over 107 intervals of MMS (NASA) data, on the other hand, we successfully calculate the spatial derivatives at the position of each spacecraft of the tetrahedron and a three-spacecraft (non-collinear) subset of the same. The average value of all the curls calculated for a given tetrahedron shows an excellent agreement (correlation coeffcient ~ 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 mission, including those involving more than four spacecraft.

Figures (7)

• Figure 1: Schematic diagram of a tetrahedral multi-spacecraft system with the spacecraft being located at the vertices (with positions denoted by $\bm{R}$). The thick arrows denote the direction of the surfaces common to MMS 1 (located at $\bm{R}_1$).
• Figure 2: One possible orthonormal coordinate system with MMS 1 at centre and $\hat{x} = \hat{\bm{R}}_{21}$, $\hat{z} = (\mathbf{R}_{21} \times \mathbf{R}_{31})/\left |\mathbf{R}_{21} \times \mathbf{R}_{31}\right |$, and $\hat{y} = \hat{z}\,\times\,\hat{x}$.
• Figure 3: The magnitudes of $\bm{J}$ and $\bm{\omega}_p$ along with their curls and divergences calculated using IFM at $\bm{R}_1$ for 1 hour interval on September 14, 2017.
• Figure 4: Histograms of correlation coefficients between quantities calculated using IFM (3 s/c) and IFM (4 s/c) at $\bm{R}_1$ for 107 datasets. The three-spacecraft system is obtained by suppressing MMS 4.
• Figure 5: Comparison between averaged-IFM and Curlometer calculation of (a) $\bm{J}$ and (b) $\bm{\omega}_p$ in terms of correlation coefficient $R$ for 1 hour interval on September 8, 2015.