# Vector Calculator

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### Contents:

- Overview
- Inputs
- Outputs
- Curlometer
- Minimum Variance of current density
- History
- Bugs/Caveats
- Other MPE/ISSI PlugIns

### Overview

The Cluster vector calculator returns the curl, gradient, divergence and barycentric average of a vector quantity. The vector quantity along with the spacecraft position from all four Cluster spacecraft must be supplied. The vector quantities are based on a linear approximation of the spatial gradients, and should be used with care if the four spacecraft are located in very different plasma regimes. A particular useful application may be the Curlometer to determine electric current density from Amperes law.

### Input data

Required inputs are a vector quantity (e.g. magnetic field, electric field, plasma velocity etc.) and the spacecraft sepration vector. In addition, a time interval and a normalization factor can be given. The inputs are automatically joined to the timetags of the input vector from Spacecraft C3

### Outputs

The outputs, curl, gradient, divergence and barycentric average of the input vectors are returned to the worklist. Additionally, the curl is also returned in an a MVA system (curl rotated by the rotations matrix from a minimum variance of the curl time series). When used as a curlometer, this MVA rotated curl represents the current in a current sheet aligned system.

### Curlometer

A special application of the PlugIn is to calculate current density according to the so0called curlometer technique; Current density can be expressed from Ampere's law as

      J = curl (B) / mu_0

If the input vector quantity is a magnetic field, and the CurlNormalize slot contains mu0 (magnetic permeability = see QSAS constants), the curl output represents a current density and has A m^-2 as unit.

### Minimum Variance analysis of current density

When used as a Curlometer, a minimum variance of the current density is performed. A set of eigenvalues and eigenvectors from the minimum variance of the curl quantity as well as a velocity are also printed in the output field of the PlugIn. Under certain conditions, (e.g. 1D current sheet appproximation) the eigenvector corresponding to the lowest eigenvalue can be used as an estimate of the current sheet normal. An approximate velocity of the current sheet motion is also given.

### Bugs/Caveats

None known.

### History

11 May 2005 : Current direction flipped
27 May 2005 : QSAS 2.1.0 version + fixed units for div, grad

### Other MPE/ISSI PlugIns

- Generic residue analysis of conservation laws
- Vector calculator/ Curlometer
- DeHoffmann-Teller analysis
- Solar wind utility. Calculates IMF clock angle, energy input, dynamic pressure etc.
- EDI filter utility
- Separate plasma velocity into paralell and perpendicular components
- Plasma calculator. Calculates plasma beta, magnetic pressure, resistivity inertial lengths etc.
- Modify, mask and manipulate components of a vector
- Construct a matrix time series from scalar time series

SEH, 12 Jan, 2004