I read reference (1)，but I am confused about how to introduce a distance vector in Lorentz force like the author does it in equation (11):
$$\rm J\times B = -J\cdot \nabla \left( B\times r\right)+\left(J\cdot \nabla B\right)\times r$$
$\rm J$ is a current, $\rm B$ is magnetic field, $\rm r$ is the distance vector
and equation (8)
$$\rm \omega\times u=u\cdot\nabla\left( \omega\times r\right)$$
$\omega$ is the rotating speed vector of the reference frame relative to the absolute inertial frame, $\rm r$ is the distance vector.
Then, how to introduce a distance vector when decompose a force?
The author says that $\rm -J\cdot\nabla \left(B\times r\right)$ is a globally conservative term while $\rm \left(J\cdot\nabla B\right)\times r$ is a locally conservative term, and he does not explain it.
How can we judge a term is a globally conservative term or a locally conservative term?
- Ni, Ming-Jiu, et al. "A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh." Journal of Computational Physics 227.1 (2007): 205-228.