Editing Apps/PapaloizouPringleTori (section)

==Supplemental Relations==
Following PP84, we supplement the above-specified set of governing equations with a polytropic equation of state, as defined in our overview of [[SR#Time-Independent_Problems|supplemental relations for time-independent problems]].  Specifically, we will assume that {{Math/VAR_Density01}} is related to {{Math/VAR_Enthalpy01}} through the relation, 

<div align="center">
{{Math/EQ_Polytrope02}} ,
</div>
or we can set,
<div align="center"><font size="+1">
<math> 
H = (n+1) \frac{P}{\rho} .
</math> 
</font></div>
Inverting the first of these two expressions allows us to immediately derive the density distribution <math>\rho(\varpi,z)</math> throughout the torus from the derived algebraic expression for <math>H(\varpi,z)</math>.  Different density distributions will be obtained for different selected polytropic indexes and/or for different selected functional forms of the centrifugal potential. 

Here, also following PP84, we impose a steady-state velocity flow-field that is described by a fluid with uniform specific angular momentum, <math>~j_0</math>.  Drawing from our table of example [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|''simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,
<div align="center"><font size="+1">
<math> 
\Psi(\varpi) = \frac{j_0^2}{2\varpi^2} .
</math> 
</font></div>