The Velocity Diagnostic, revisited

CLICK THE PLOTS TO GET A BIGGER VERSION.

Anna had a few questions regarding the Be9 velocity diagnostic.  In particular, she was wondering what the reason was for the vastly differing values of the Be9/Al indicator, right near time T=0.  Here are some plots of the quantities involved:  First, the cumulative measure of Be9/Al:

You can see that there is an overall shift, by an order of magnitude in this ratio, right from the beginning.  If we look at the rate of this ratio, we see a similar feature:

I plotted each of the items in this ratio individually, to see what was driving the shift in the normalization.  It seem that our code predicts similar initial Al production near time T=0, but the initial Be9 production is quite different in each of the four runs.  Note that the tag for the ALuminum is BE9_RATE_CONTROL whereas the tag for the actual signal is BE9_RATE_SIGNAL.   So here are the rate plots, and the cumulative plots just below that.

Notice that the control starts at the same normalization at the initial time, but the signal doesn't.  The explanation for this is that the doppler shift is moving the neutron spectrum more and more into the relevant energies for Be9 production.  Anna was curious whether there is also a thermal effect, but it appears that the answer is no.  See below. 

Here it's interesting, the control and the signal seem to be rolling off at roughly the same times, which is weird because the Be9 ought to turn off well before the Al, at the turn around time.  I think what's going on is just that the neutron flux dies down pretty quickly after the ablator shell turns around.  I threw all the curves on one plot, just so you could get a visual.

I checked the temperature profiles at the 0th time step, the 1st time step, and the 10th time step, to see if the difference in the normalization might be due to thermal broadening, due to much higher hot spot temperatures right near the beginning of the simulation.  It does not look like that can be the explanation.  Below are the 0th, 1st, and 10th time steps, and the temperatures look roughly the same in the hot spot for all four cases.

That being said, there will be differences due to thermal broadening, because at the peak burn time, when most of the Be9 and Al are being produced, there are non-trivial temperature differences in the hotspot.  Here is a plot of the time evolution of the hotspot temperature for each of these cases.

But these are not as extreme as the difference between 4 and 15 kev, depicted in that cartoon plot we showed in the poster.  So I don't THINK the temperature effect is responsible fo rhte order of magnitude difference in the production of Be9.

Varying dE/dx model: part II

So my hunch was correct:  the parameter $A \log (\Lambda)$ that I varied yesterday was not actually impacting the RIF spectrum (other than through the yield / peak burn time) because in fact it corresponds to a parameter in the $\alpha$ particle deposition model in the Garnier-Cherfils paper, and has nothing whatever to do with Jerry's RIF calculation.  That's why yesterday's plots were so unexciting.

Just to remind myself later, what is this $A$?  Well, if we have

$\frac{dE}{dx} \propto \frac{1}{E} G(y) \ln (\Lambda(y))$

where

$y=\frac{E}{\Theta} \frac{m_e}{m_{\rm particle}}$

and

$G(y) = \frac{A y^{3/2}}{1 + A y^{3/2}}$

Then the parameter $A$ affects the normalization, and it affects the energy at which the function $\frac{dE}{dx}$ rolls over.

Yesterday I got Jerry to expose the $A$ parameter, so I could make the plot I was intending to make in the first place. I have to say, the results are somewhat unexciting (although totally what Jerry predicted off the cuff):

What we see, not surprisingly, is that changing the parameter $A$ changes the normalization.  What I had hoped is that, since changing $A$ affects the energy at which $\frac{dE}{dx}$ rolls over, that the measured SHAPE of the RIF spectra would change noticeably as we monkeyed with this parameter (this might give us some hope of constraining in in an observation of the RIF spectrum).  Sadly, it looks as if the shape of the measured RIFS is probably just dominated by the shape of the underlying 14s, and  the dependence on $A$ is weak at best. Jerry: Incidentally, I tried to run the Zimmerman model for $\frac{dE}{dx}$, to compare its normalization and shape to these, however I am getting NANs or INFs, so that seems to be broken for the time being.

Indeed, the shape is almost TOO insensitive for me to believe, so I made a plot where I re-normalized all the curves to the $A=0.5$ curve, and my hunch was correct, there seems to be NO dependence on the shape... this seems odd to me...

Could this be good news?  It's probably just wrong of course, but it might also be an indication that we can argue that differences in the shape of the RIF spectrum come from mix, and are not affected by uncertainties in the model of $\frac{dE}{dx}$ in the strongly coupled limit....

Before we decide this is right though, I'd really like to see the shape of $\frac{dE}{dx}$ as we shift $A$.  Maybe Jerry's "test_stopping" functions would print this out?  I should see what it does...


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Addendum:  Here are plots of $\frac{dE}{dx}$ for the various values of $A$.  The roll over is moving slightly, but not as much as I was expecting given Anna's chalkboard drawings.  Still, seems to be doing the right thing.  I was surprised that Zimmerman seems to agree with A=0.5 much better than A=1...

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Addendum 2:  Jerry found a bug that affects all RIF plots made before this date.  Here is an updated version of the first plot in this post.

This does more what we'd expect: the asymptotic behavior is the same in each case.  The zimmerman curve (not shown here) has the same asymptotic behavior, but more rif neutrons around 14.6 than any of these models.

Sensitivity to $A \log (\Lambda)$

I was speaking with Anna yesterday, and we agreed that the reason it's not so easy to write a RIF paper at the present time is because the RIF spectrum (i.e. the observable) is affected by two different unknown quantities: 1) the Mix and 2) uncertainty in the function $\frac{dE}{dx}$ in the strongly coupled limit where we reside.  Either we need to disentangle their effects using an observable that does not have degenerate dependence on the two, or we need to show that one of them is not so important, or we are more or less in trouble.

I know that for the most part, the calculations we have so far of the dependence of the RIF spectra on the form of $\frac{dE}{dx}$ have been done in the static limit.  I know that we have a transport parameter $A \log(\Lambda)$ in our code, which amounts to a dial on the amplitude and roll-over shape of the $\frac{dE}{dx}$ function (of energy) used in the hydro calculation.  I decided to see how dramatic an effect changes in this parameter have on the observable RIF spectrum.

Here is the (pretty undramatic) result:

By the time peak burn is achieved, the cumulative (in time I assume) RIF spectrum depends almost not at all on this $A \log(\Lambda)$ parameter.  Closer investigation shows that the blue and green curves identically coincide, as do the red, teal and magenta.  The difference seems to be entirely governed by the peak burn time, which is slightly earlier in the blue/green case than it is in the red/teal/magenta case.

Perhaps I assumed incorrectly that $\frac{dE}{dx}$ was being used in the computation of the RIF spectrum?  I'm a little puzzled by these results.

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 $R_h$.