Pressure Profiles

As of now, we have put in a model of the pressure profiles by hand.  Jerry asked me to look through the various papers, and figure out if what we have done is sane.  Below is a collection of relevant quotes from the following sources

1) Garnier & Cherfils-Clerouin, Phys of Plasmas 15, 102702, 2008 Here
2) Sanz et al Phys. Plasmas  12, 112702 (2005) Here

A little summary of what I could find so far:

INSIDE HOT SPOT

From Reference 2:

 The flow inside the hot spot is subsonic, which allows us to adopt the subsonic flow ordering. To lowest order we get the flat pressure approximation $p(t,r)=p_h(t)$ ....  The momentum equation describes the fluctuations of the pressure and it can be  integrated a posteriori....

Finally, the fluctuations of the pressure field can be obtained from Eq.2: $p(t,r)=p_0(t,\frac{r}{R_h(t)})$...  This analysis shows that all hydrodynamic quantities and profiles can be computed in terms of the hot spot pressure $p_h(t)$.  This pressure will be determined by closing the system with a shell model for the region $r>R_h(t)$.

INSIDE SHOCKED SHELL

From Reference 2:

 The isobaric approximation is not valid in the shocked shell. Actually, the flow is neither subsonic as it is in the hot spot, nor supersonic as it is in the free-fall shell. However, heat flow and nuclear reaction can be safely neglected, so that the energy conservation acquires a simple form. We assume that the shocked shell is a polytropic gas of EOS $p=\gamma_s −1$  with an adiabatic exponent $s =7/4$. The choice of this EOS comes from observations by Herrmann13  and Saillard14  that the behavior of deuterium-tritium in the dense shell for the SESAME EOS15  is well described with the value $s =7/4$.

The above calculations can be pushed forward to obtain pieces of information about the spatial profiles of the density, velocity, and pressure inside the shocked shell. We have obtained the gradient of the velocity field at the shell inner surface:
$\partial_r u(t,r=R_h^+)=-\frac{\dot{p}_h(t)}{\gamma_s p_h(t)} - 2 \frac{\dot{R}_h}{R_h}$
Much more information can be obtained. For instance, by considering the momentum conservation equation in the vicinity of $r=R_h^+(t)$ , we get $\ddot{R}_h(t)= $..... which yields the pressure gradient
$\partial_r p(t,r=R_h^+)= -\rho_{sh}(t) \ddot{R}_h(t)$
 [Here $\rho_{sh}$ is the shell density, just outside the hot spot, nothing to do with the shock (although it has been shocked)].  By substituting typical values for the acceleration, shell density, and pressure (see the next paragraph), we can check that the isobaric approximation in the shell is not fulfilled in general.

IN FREEFALL REGION OUTSIDE THE SHOCK

From Reference 2:

Let us first consider the outer region $r>R_s(t)$ . The shell is in free-fall conditions with an evanescent pressure [It dies off smoothly??  Reference 1 has the same cryptic description of the pressure in this outer region] and the flow is supersonic. At time 0, the density profile is $\rho_0$ and the velocity profile is $u_0$. The density profile can be arbitrary, but we assume that the velocity profile is affine.  [I don't know what this word means in this context.  To me, this form  $u_0(r)=-V_i \left( 1 - \eta_0 \frac{r-R_0}{\Delta_0}\right)$ looks kinda arbitrary too, in the sense that it's not apparently related to the density, just linear in r with slope governed by $\eta_0$]

Reference two also had the output data from the acceleration phase of their code, including a pressure profile.  You can see they also assume (for the deceleration phase) that the pressure inside the hot spot is constant, but the just profile just outside seems to be proportional to the density.  This is the outer unshocked region because this initial condition begins when the shock crosses $R_h$.