pyrfu.pyrf.pid_4sc

pyrfu.pyrf.pid_4sc(r_mms, v_mms, p_mms)

Compute Pi-D term using definition of [1] as :

\[Pi-D = - \Pi_{ij}D_{ij}\]

with \(\Pi_{ij}\) the deviatoric part of the pressure tensor :

\[ \begin{align}\begin{aligned}\Pi_{ij} = P_{ij} - p\delta_{ij}\\p = \frac{1}{3}P_{ii}\end{aligned}\end{align} \]

and \(D_{ij}\) the deviatoric part of the strain tensor :

\[ \begin{align}\begin{aligned}D_{ij} = \frac{1}{2}\left ( \partial_i u_j + \partial_j u_i \right ) - \frac{1}{3}\theta\delta_{ij}\\\theta = \nabla . u\end{aligned}\end{align} \]

The pressure-strain interaction is then \(-(P \cdot \nabla) \cdot u = -p\theta + Pi-D\).

Parameters:

• r_mms (list of xarray.DataArray) – Time series of the position of the 4 spacecraft [km].

• v_mms (list of xarray.DataArray) – Time series of the bulk velocities of the 4 spacecraft [km/s].

• p_mms (list of xarray.DataArray) – Time series of the pressure tensor of the 4 spacecraft [nPa].

Returns:

• d_xyz (xarray.DataArray) – Time series of the deviatoric part of the strain tensor \(D_{ij}\) [s^-1].

• pi_xyz (xarray.DataArray) – Time series of the deviatoric part of the pressure tensor \(\Pi_{ij}\) [nPa].

• ptheta (xarray.DataArray) – Time series of the pressure dilatation \(p\theta\) [nW/m^3].

• pid (xarray.DataArray) – Time series of the Pi-D [nW/m^3].

Notes

The units are given for positions in km, velocities in km/s and pressures in nPa. No conversion is applied: other input units carry through.

References

[1]

Yang, Y., Matthaeus, W. H., Parashar, T. N., Wu, P., Wan, M., Shi, Y., et al. (2017). Energy transfer channels and turbulence cascade in Vlasov-Maxwell turbulence. Physical Review E, 95, 061201. doi : https://doi.org/10.1103/PhysRevE.95.061201