Chanteur, G.: Spatial Interpolation for Four Spacecraft: Theory, in: Analysis Methods for Multi-Spacecraft Data, edited by: Paschmann, G. and Daly, P. W., p. 349, ESA Publications Division, Noordwijk, the Netherlands, ISBN: 1608-280X, 1998.
a,
b,
c
Denton, R. E., Torbert, R. B., Hasegawa, H., Dors, I., Genestreti, K. J., Argall, M. R., Gershman, D., Contel, O. L., Burch, J. L., Russell, C. T., Strangeway, R. J., Giles, B. L., and Fischer, D.: Polynomial Reconstruction of the Reconnection Magnetic Field Observed by Multiple Spacecraft, J. Geophys. Res.-Space, 125, e2019JA027481,
https://doi.org/10.1029/2019JA027481, 2020.
a
Dunlop, M., Southwood, D., Glassmeier, K.-H., and Neubauer, F.: Analysis of multipoint magnetometer data, Adv. Space Res., 8, 273–277,
https://doi.org/10.1016/0273-1177(88)90141-X, 1988.
a
Dunlop, M. W., Yang, Y.-Y., Yang, J.-Y., Lühr, H., Shen, C., Olsen, N., Ritter, P., Zhang, Q.-H., Cao, J.-B., Fu, H.-S., and Haagmans, R.: Multispacecraft current estimates at swarm, J. Geophys. Res.-Space, 120, 8307–8316,
https://doi.org/10.1002/2015JA021707, 2015.
a
Dunlop, M. W., Haaland, S., Escoubet, P. C., and Dong, X.-C.: Commentary on accessing 3-D currents in space: Experiences from Cluster, J. Geophys. Res.-Space, 121, 7881–7886,
https://doi.org/10.1002/2016JA022668, 2016.
a
Dunlop, M. W., Haaland, S., Dong, X.-C., Middleton, H. R., Escoubet, C. P., Yang, Y.-Y., Zhang, Q.-H., Shi, J.-K., and Russell, C. T.: Multipoint Analysis of Electric Currents in Geospace Using the Curlometer Technique, in: Electric Currents in Geospace and Beyond, 67–80, John Wiley & Sons, Inc.,
https://doi.org/10.1002/9781119324522.ch4, 2018.
a
Escoubet, C. P., Fehringer, M., and Goldstein, M.: Introduction: The Cluster mission, Ann. Geophys., 19, 1197–1200,
https://doi.org/10.5194/angeo-19-1197-2001, 2001.
a
Fadanelli, S., Lavraud, B., Califano, F., Jacquey, C., Vernisse, Y., Kacem, I., Penou, E., Gershman, D. J., Dorelli, J., Pollock, C., Giles, B. L., Avanov, L. A., Burch, J., Chandler, M. O., Coffey, V. N., Eastwood, J. P., Ergun, R., Farrugia, C. J., Fuselier, S. A., Genot, V. N., Grigorenko, E., Hasegawa, H., Khotyaintsev, Y., Contel, O. L., Marchaudon, A., Moore, T. E., Nakamura, R., Paterson, W. R., Phan, T., Rager, A. C., Russell, C. T., Saito, Y., Sauvaud, J.-A., Schiff, C., Smith, S. E., Redondo, S. T., Torbert, R. B., Wang, S., and Yokota, S.: Four-Spacecraft Measurements of the Shape and Dimensionality of Magnetic Structures in the Near-Earth Plasma Environment, J. Geophys. Res.-Space, 124, 6850–6868,
https://doi.org/10.1029/2019JA026747, 2019.
a
Friis-Christensen, E., Lühr, H., and Hulot, G.: Swarm: A constellation to study the Earth's magnetic field, Earth Planet. Space, 58, 351–358,
https://doi.org/10.1186/BF03351933, 2006.
a
Fu, H. S., Vaivads, A., Khotyaintsev, Y. V., Olshevsky, V., André, M., Cao, J. B., Huang, S. Y., Retinò, A., and Lapenta, G.: How to find magnetic nulls and reconstruct field topology with MMS data?, J. Geophys. Res.-Space, 120, 3758–3782,
https://doi.org/10.1002/2015JA021082, 2015.
a
Hamrin, M., Rönnmark, K., Börlin, N., Vedin, J., and Vaivads, A.: GALS-Gradient Analysis by Least Squares, Ann. Geophys., 26, 3491–3499,
https://doi.org/10.5194/angeo-26-3491-2008, 2008.
a,
b
Harvey, C. C.: Spatial gradients and the volumetric tensor, in: Analysis Methods for Multi-Spacecraft Data, edited by: Paschmann, G. and Daly, P. W., p. 307, ESA Publications Division, Noordwijk, the Netherlands, ISBN: 1608-280X, 1998.
a,
b,
c
Hasegawa, H., Nakamura, T. K. M., and Denton, R. E.: Reconstruction of the Electron Diffusion Region With Inertia and Compressibility Effects, J. Geophys. Res.-Space, 126, e2021JA029841,
https://doi.org/10.1029/2021JA029841, 2021.
a
Kincaid, D. and Cheney, W.: Numerical Analysis Mathematics of Scientific Computing, American Mathematical Society, 3rd Edn., ISBN: 9780821847886, 2002. a
Liu, Y. Y., Fu, H. S., Olshevsky, V., Pontin, D. I., Liu, C. M., Wang, Z., Chen, G., Dai, L., and Retino, A.: SOTE: A Nonlinear Method for Magnetic Topology Reconstruction in Space Plasmas, Astrophys. J. Suppl. Ser., 244, p. 31,
https://doi.org/10.3847/1538-4365/ab391a, 2019.
a,
b
Liu, Y. Y., Wang, Z., Chen, G., Yu, Y., Guo, Z. Z., and Xiong, X.: Testing the Linearity of Vector Fields in Cold and Dense Space Plasmas, Astrophys. J., 929, 155,
https://doi.org/10.3847/1538-4357/ac5d4b, 2022.
a
Liu, Z. X., Escoubet, C. P., Pu, Z., Laakso, H., Shi, J. K., Shen, C., and Hapgood, M.: The Double Star mission, Ann. Geophys., 23, 2707–2712,
https://doi.org/10.5194/angeo-23-2707-2005, 2005.
a
Maruca, B. A., Rueda, J. A. A., Bandyopadhyay, R., Bianco, F. B., Chasapis, A., Chhiber, R., DeWeese, H., Matthaeus, W. H., Miles, D. M., Qudsi, R. A., Richardson, M. J., Servidio, S., Shay, M. A., Sundkvist, D., Verscharen, D., Vines, S. K., Westlake, J. H., and Wicks, R. T.: MagneToRE: Mapping the 3-D Magnetic Structure of the Solar Wind Using a Large Constellation of Nanosatellites, Front. Astron. Space Sci., 8, 665885,
https://doi.org/10.3389/fspas.2021.665885, 2021.
a
Ogilvie, K. W., von Rosenvinge, T., and Durney, A. C.: International Sun-Earth Explorer: A Three-Spacecraft Program, Science, 198, 131–138,
https://doi.org/10.1126/science.198.4313.131, 1977.
a
Pollock, C., Moore, T., Jacques, A., Burch, J., Gliese, U., Saito, Y., Omoto, T., Avanov, L., Barrie, A., Coffey, V., Dorelli, J., Gershman, D., Giles, B., Rosnack, T., Salo, C., Yokota, S., Adrian, M., Aoustin, C., Auletti, C., Aung, S., Bigio, V., Cao, N., Chandler, M., Chornay, D., Christian, K., Clark, G., Collinson, G., Corris, T., Santos, A. D. L., Devlin, R., Diaz, T., Dickerson, T., Dickson, C., Diekmann, A., Diggs, F., Duncan, C., Figueroa-Vinas, A., Firman, C., Freeman, M., Galassi, N., Garcia, K., Goodhart, G., Guererro, D., Hageman, J., Hanley, J., Hemminger, E., Holland, M., Hutchins, M., James, T., Jones, W., Kreisler, S., Kujawski, J., Lavu, V., Lobell, J., LeCompte, E., Lukemire, A., MacDonald, E., Mariano, A., Mukai, T., Narayanan, K., Nguyan, Q., Onizuka, M., Paterson, W., Persyn, S., Piepgrass, B., Cheney, F., Rager, A., Raghuram, T., Ramil, A., Reichenthal, L., Rodriguez, H., Rouzaud, J., Rucker, A., Saito, Y., Samara, M., Sauvaud, J.-A., Schuster, D., Shappirio, M., Shelton, K., Sher, D., Smith, D., Smith, K., Smith, S., Steinfeld, D., Szymkiewicz, R., Tanimoto, K., Taylor, J., Tucker, C., Tull, K., Uhl, A., Vloet, J., Walpole, P., Weidner, S., White, D., Winkert, G., Yeh, P.-S., and Zeuch, M.: Fast Plasma Investigation for Magnetospheric Multiscale, Space Sci. Rev., 199, 331–406,
https://doi.org/10.1007/s11214-016-0245-4, 2016.
a
Riachy, S., Mboup, M., and Richard, J.-P.: Multivariate numerical differentiation, J. Comput. Appl. Math., 236, 1069–1089,
https://doi.org/10.1016/j.cam.2011.07.031, 2011.
a
Shao, P., Shen, C., Ma, Y., Rong, Z., Zhou, Y., Zhang, C., Dunlop, M., and Ji, Y.: Flapping motion configurations of geomagnetotail current sheet, J. Atmos. Sol.-Terr. Phys., 243, 106019,
https://doi.org/10.1016/j.jastp.2023.106019, 2023.
a
Shen, C., Li, X., Dunlop, M., Liu, Z. X., Balogh, A., Baker, D. N., Hapgood, M., and Wang, X.: Analyses on the geometrical structure of magnetic field in the current sheet based on cluster measurements, J. Geophys. Res.-Space, 108, 1168,
https://doi.org/10.1029/2002JA009612, 2003.
a
Shen, C., Li, X., Dunlop, M., Shi, Q. Q., Liu, Z. X., Lucek, E., and Chen, Z. Q.: Magnetic field rotation analysis and the applications, J. Geophys. Res.-Space, 112, A06211,
https://doi.org/10.1029/2005JA011584, 2007.
a
Shen, C., Rong, Z. J., Dunlop, M. W., Ma, Y. H., Li, X., Zeng, G., Yan, G. Q., Wan, W. X., Liu, Z. X., Carr, C. M., and Rème, H.: Spatial gradients from irregular, multiple-point spacecraft configurations, J. Geophys. Res.-Space, 117, A11207,
https://doi.org/10.1029/2012JA018075, 2012.
a,
b
Shen, C., Zeng, G., Zhang, C., Rong, Z., Dunlop, M., Russell, C. T., Escoubet, C. P., and Ren, N.: Determination of the Configurations of Boundaries in Space, J. Geophys. Res.-Space, 125, e2020JA028163,
https://doi.org/10.1029/2020JA028163, 2020.
a
Shen, C., Zhang, C., Rong, Z., Pu, Z., Dunlop, M. W., Escoubet, C. P., Russell, C. T., Zeng, G., Ren, N., Burch, J. L., and Zhou, Y.: Nonlinear Magnetic Gradients and Complete Magnetic Geometry From Multispacecraft Measurements, J. Geophys. Res.-Space, 126, e2020JA028846,
https://doi.org/10.1029/2020JA028846, 2021a.
a,
b,
c
Shen, C., Zhou, Y., Gao, L., Wang, X., Pu, Z., Escoubet, C. P., and Burch, J. L.: Measurements of the Net Charge Density of Space Plasmas, J. Geophys. Res.-Space, 126, e2021JA029511,
https://doi.org/10.1029/2021JA029511, 2021b.
a,
b,
c
Shen, C., Zhou, Y., Ma, Y., Wang, X., Pu, Z., and Dunlop, M.: A General Algorithm for the Linear and Quadratic Gradients of Physical Quantities Based on 10 or More Point Measurements, J. Geophys. Res.-Space, 126, e2021JA029121,
https://doi.org/10.1029/2021JA029121, 2021c.
a,
b,
c,
d,
e,
f
Shi, Q. Q., Shen, C., Pu, Z. Y., Dunlop, M. W., Zong, Q.-G., Zhang, H., Xiao, C. J., Liu, Z. X., and Balogh, A.: Dimensional analysis of observed structures using multipoint magnetic field measurements: Application to Cluster, Geophys. Res. Lett., 32, L12105,
https://doi.org/10.1029/2005GL022454, 2005.
a
Shi, Q. Q., Shen, C., Dunlop, M. W., Pu, Z. Y., Zong, Q.-G., Liu, Z. X., Lucek, E., and Balogh, A.: Motion of observed structures calculated from multi-point magnetic field measurements: Application to Cluster, Geophys. Res. Lett., 33, L08109,
https://doi.org/10.1029/2005GL025073, 2006.
a
Sonnerup, B. U. O. and Teh, W.-L.: Reconstruction of two-dimensional coherent MHD structures in a space plasma: The theory, J. Geophys. Res.-Space, 113, A05202,
https://doi.org/10.1029/2007JA012718, 2008.
a
Spence, H. E., Alexandrova, O., Arzamasskiy, L., Argall, M. R., Caprioli, D., Case, A. W., Chandran, B. D. G., Chen, L.-J., Dors, I., Eastwood, J. P., Forsyth, C., Galvin, A. B., Genot, V. N., Halekas, J. S., Hesse, M., Horbury, T. S., Jian, L., Kasper, J. C., Klein, K. G., Kretzschmar, M., Kunz, M. W., Lavraud, B., Le Contel, O., Mallet, A., Maruca, B., Matthaeus, W. H., Owen, C. J., Retino, A., Reynolds, C., Roberts, O. W., Schekochihin, A. A., Skoug, R. M., Smith, C. W., Smith, S. S., Steinberg, J. T., Stevens, M. L., Szabo, A., TenBarge, J. M., Torbert, R. B., Vasquez, B. J., Verscharen, D., Whittlesey, P. L., Zank, G. P., and Zweibel, E.: An Overview of HelioSwarm: A NASA MIDEX Mission to Reveal the Nature of Turbulence in Space Plasmas, in: AGU Fall Meeting 2022, AGU Fall Meeting 2022, Chicago, IL, United States, 12–16 December 2022, SH12E-1485, American Geophysical Union,
https://agu.confex.com/agu/fm22/meetingapp.cgi/Paper/1177250 (last access: 1 February 2024), 2022. a
Torbert, R. B., Dors, I., Argall, M. R., Genestreti, K. J., Burch, J. L., Farrugia, C. J., Forbes, T. G., Giles, B. L., and Strangeway, R. J.: A New Method of 3-D Magnetic Field Reconstruction, Geophys. Res. Lett., 47, e2019GL085542,
https://doi.org/10.1029/2019GL085542, 2020.
a,
b,
c
Vogt, J., Paschmann, G., and Chanteur, G.: Reciprocal Vectors, in: Multi-Spacecraft Analysis Methods Revisit, edited by: Paschmann, G. and Daly, P. W., Chap. 4, p. 33, ESA Communications, ISBN: 987-92-9221-937-6, 2008. a
Vogt, J., Albert, A., and Marghitu, O.: Analysis of three-spacecraft data using planar reciprocal vectors: methodological framework and spatial gradient estimation, Ann. Geophys., 27, 3249–3273,
https://doi.org/10.5194/angeo-27-3249-2009, 2009.
a,
b,
c,
d
Yang, J., Toffoletto, F. R., Wolf, R. A., Sazykin, S., Ontiveros, P. A., and Weygand, J. M.: Large-scale current systems and ground magnetic disturbance during deep substorm injections, J. Geophys. Res.-Space, 117, A0422,
https://doi.org/10.1029/2011ja017415, 2012.
a
Zhou, Y., Shen, C., and Ji, Y.: Undulated Shock Surface Formed After a Shock–Discontinuity Interaction, Geophys. Res. Lett., 50, e2023GL103848,
https://doi.org/10.1029/2023GL103848, 2023.
a