Hierarchical Physics

To find the physical state of the material within each zone, Cloudy attempts to solve a large, nonlinear system of equations coupling a wide range of physics, using iterative solutions of subsystems. Outer iterations for the pressure, temperature and electron density equilibrium surround a base set of solutions of ionization, iso-sequence and chemical balance. Originally, the ionization solvers could be treated as essentially independent. However, recent additions to the included physics (specifically chemistry and iso-sequence solutions) mean that the components of this base iteration are becoming more strongly coupled. As a result, the nature of the base iteration must be considered with care.

At a given stage of the base iteration, the system to be solved could be thought of as reducing to a large, sparse matrix system, for the densities ni of every species at the most accurate level of physics (i.e. iso-levels, ions, chemistry, dust -- all together):

Sumj Aij nj = Si

However, because of the sparseness of the matrix, the wide range of physical timescales (and hence poorly-conditioned system matrix), and the poor coupling between, say, the He-like lithium iso-sequence and CH3CH2OH, it makes sense to look at this as a system of reduced problems, such as

For i in B:

Sum{j in B} Aij nj = Si - Sum{j not in B} Aij nj

to solve iteratively. To put these together you need a coarse-grained solution which embeds these fine scale solutions (capitals stand for the index of the coarse grained vectors and matrices)

SumJ AIJ nJ = SI (*)

where the J index goes over a list of macro states {B,....}, and the rates AIJ and sources SI come from sums of the Aij, weighted by the relative abundances of the microstates in the fine-scale solutions.

At convergence, the nI will be consistent with the ni, but if they are not, then the after the coarse solution the ni have to be re-scaled, and the fine-scales re-solved, to improve the weighted matrix elements in (*).

In practice, it is also useful at the end of the fine solution to scale the results to be consistent with the current coarse-solution totals. In this way, the appropriate weightings are produced automatically for processes summed over the fine scale abundances. The re-scaling factor required here should tend to unity as a consistent solution is approached, and can therefore be used to monitor convergence.

Note that things are actually somewhat more complex than this, because the iso-levels are embedded in the ions, which are embedded in the molecular network which functions as the coarsest-grained solution. However the argument above extends naturally to this kind of hierarchy (it's essentially a  multigrid V-cycle). A uniform abstract representation of the state hierarchy would make it easier to ensure that the different levels were coupled in a consistent fashion.

It is not clear what is the best manner to embed the excited states of electronic iso-sequences within the coarser solutions. On kinematic grounds it would make more sense to group the levels with the parent ion, since they will typically vary together. However, external physics (such as pressure evaluation and chemical interactions) will more naturally be expressed in terms of densities of particles with a definite ionization, i.e. this would suggest that the excited states should be treated as constituents of the base ion.