Tapered Flow Model Series‏

This series of models is one of the most interesting and, in some ways, probably the most realistic. That is, purely cylindrical flows are an idealization. More likely the "cylindrical" flow is a bit sloppy along its lateral perimeter. The Gaussian taper is a way to develop a conical jet whose density and velocity roll off at the edges. As it turns out, the lobes take on quite a nice variety of morphologies depending on the ratios of wind and core densities. This may yet prove to be very useful for explaining he shapes of HST images of prePNe.

(Martin implemented the Gaussian tape in this way: one specifies a spherical flow (opening angle of 90 degrees) AND a flow taper function for the density and the velocity with angle, theta, from the flow axis such that both n_jet and v_jet fall off with elevation according to a Gaussian of angle tf = 1/e angle. I used tf=15 deg.)

The results are shown below. The window sizes are highly variable. The density and temperature are shown in the left panels. A zoom-in of the kinematics is shown to the right.

Bear in mind that observers NEVER see any emission from the jet. Rather, they observe shock-excited emission from the tip (speeds 100-200km/s), the gas along the lateral edges that is (or has been) shocked at the head (shock speeds 10-50 km/s), or both. That is, since the emissivity scales as density squared they observe the lobe tip and the relatively dense outer skin of the lobe.

The top panels show a light jet (nJet << namb(core)). A triangular lobe emerges. Its dense and fast leading edge is susceptible to thin-shell instabilities as it propagates into the ambient medium. The middle panels show an intermediate case where nJet = namb(core). A thin tapered lobe develops and maintains its shape after it reaches a distance where the ambient gas density (pressure) is low. The bottom panels show a heavy jet that simply blasts its way forward like a rounded piston. Its shape and propagation speed never change.

Look in the upper-right corner of the left panes where I show that average speed of the tip of the jet, <v_tip>. The top row (light jet emerging at 200 km/s) is significantly decelerated by the ambient medium at first. <v_tip> is just 90 km/s averaged over 400 y. The other two cases have <v_tip> ~ 200 km/s.

In the bottom row a hot sheath of gas (that was originally shock-heated near the leading tip) forms between the tapered flow and the lateral edges of the lobe. It's thermal pressure is significant, and the lobe edges expand nearly laterally and slowly. In the other cases the momentum of the wind suppresses the formation of a large and hot sheath. Nonetheless a warm sheath (at 10,000K) forms where the gas radiates its heat. Even so, a thin hot zone forms where the tapered wind rams into the sheath. I can't recall seeing this sort of behavior before.

One final note. The early hydro tapered-jet computations of Sahai & Lee (2003) at coarse resolution found that the walls of the jet are where wind streamlines slide and converge towards the tip of the lobe. That's the concept behind the Canto model developed originally to explain H-H outflows. Very smooth walls are essential if the streamlines are to move coherently towards the lobe tip. In our sims we see no trace of streamlines sliding up the lobe walls, nor do we see any evidence of flow convergence at the lobe tips.

THE SIGNIFICANCE OF R_jet: Part 1

In the past few days we've been musing about the relationship between Rjet and the shape of the developing flow. Muse no more. Rjet matters!!!! It is a shape- and size-controlling variable of paramount significance (forcing us to make some strategic choices in this project)!

Rjet really matters!!!! Rjet really matters!!!! Rjet really matters!!!! Rjet really matters!!!!

Here's the originating problem.data file for one of the runs (the only variable that changes is Rjet)

&ProblemData

! MODEL: jet 0deg (n=4e4, r=1000AU, T=100K) @ 200 km/s into 4e4 torus + AGB wind

! folder TapAGB45Rhn4e2v200namb4e4

!

! BACKGROUND or “AMBIENT” SECTION. Values apply to origin

tamb = 1d3 ! ambient temp, 1cu = 0.1K (100K=1000cu)

namb = 4e4 ! ambient central density cm^{-3. Usually 400 for 1/r}2 or tor$

stratified = t ! true = add a 1/r^{2 background 'AGB stellar wind'}

torus = f ! true - add torus to the background

torusalpha = 0.7 ! alpha and beta specify the geometry

torusbeta = 10d0 ! see Frank & Mellema, 1994ApJ…430..800F

rings = f ! true - add radial density modulations to AGB wind

!

! FLOW DESCRIPTION SECTION, values apply at origin at t=0

outflowType = 2 ! TYPE OF FLOW 1 cyl jet, 2 conical wind, 3 is clump

njet = 4d2 ! flow density at launch zone, 1cu = 1cm^-3

Rjet = 0.5, 1.0. 2.0 ! flow radius at launch zone, 1cu = 500AU (outflowType=1 only)

vjet = 2e7 ! flow velocity , 1cu = cm/s (100km/s=1e7cu)

tjet = 1d3 ! flow temp, 1cu = 0.1K (100K=1000cu)

tt = 0.0d0 ! flow accel time, 1cu = 8250y (0.02 = 165y)

open_angle = 90d0 ! conical flow open angle (deg)

tf = 45d0 ! conical flow Gaussian taper (deg) for njet and vjet; 0= disa$

sigma = 0d0 ! !toroidal.magnetic.energy / kinetic.energy, example 0.6

! OTHER PARAMETERS

lcooling = t ! radiative cooling?

buff = 8 ! central refinement of a grid with a resolution ½

Comments about the differences:

The primary reason to make Rjet large is to better resolve its surface (which is the lower boundary of the flow)

The primary reason to make Rjet small is to wipe out less of the background. This is surprisingly important.

Comments about the flow size:

1. Increasing the surface area while holding njet fixed changes the mass and momentum of the flow. That is, njet is the density at the launching surface. Increasing Rjet increases the launch surface area and the total flow momentum.

2. Increasing Rjet means that the launch sphere covers more of the ambient medium. It is the innermost zone of the ambient medium where it density is largest, so flows starting at a large Rjet never see much of the density in the extrenal environment. This is especially important for tori (not used in the sims above).

Both of these conditions (1 and 2) mean that flows launched with a large Rjet will penetrate significantly further.

Comments about the flow shape:

1. About large-scale shapes: Much of the shaping of collimated flows occurs when they are very small. Launching them at Rjet = 0.5 or Rjet = 2 can have a profound impact on their overall shapes.

2. About features near the y axis: These change profoundly with the size of Rjet (that is, the details of the pixelated shape of the launch sphere), as we suspected that they would. Look at the attached figure near the y axis. Small Rjets are almost certain to produce dense artifacts near the y axis. However, there are still residual artifacts even when Rjet=2.

3. Look at the right panel of the figure and then at the others to its left. In the right panel you might imagine that the thin-shell instability has disturbed the surface. Not so fast. The shape of the wiggles along the boundary at 440 y is very clearly related to the wiggles at 110y — all of which are flow artifacts. Thus the ripples at 440y are born of "original" artifacts.

This problem becomes more severe when Rjet is smaller. But the artificial ripples in the large-scale flow are presisiten with any value of Rjet. And these artificial features are not confined to the region of the y axis. They're all along the leading edge of the flow!

Conclusions:

1. For many types of flows (but probably not dense cylindrical jets or bullets) Rjet should be at least 2.0. Even then, of wiggles appear at the leading edge of the flow near the y-axis at early times then you can't trust the resulting structures as the flow edge grows. Any wiggles or ripples will be hydrodynamically amplified in time. The only way to control these is to run at higher spatial resolution.

One way to remove some of the early wiggles that develop near the y axis is to launch the flow into the grid at 45 degrees.

[This may be easier than it sounds. Martin already has a way of modulating the flow density of conical outflows around the boundary with a gaussian taper that he applies to a spherical wind. The form of the taper is exp(-theta/user-half-width)^2. That algorithm might be changed to exp(-(theta-45deg)/user-half-width)^2. If you want to sharpen the edges (by suppressing the Gaussian wings) then the taper can be exp(-(theta-45deg)/user-half-width)^Xwhere X>2.

The lay-down of the external medium prior to the start of the flow MUST somehow adjust itself for the size of the luanch sphere. After a moment's though I suspect that this can be done by scaling njet so that the density of the initial medium at the launch surface is held fixed as Rjet is changed.

Note that njet ALWAYS describes the flow density at the launch surface. However, namb does not describe the density of the external medium at the launch surface.

THE SIGNIFICANCE OF R_jet: PART 2

The series of sims below is similar to the one that I sent yesterday with these exceptions: the opening angle of the Gaussian-tapered flow is 15 deg, not 45, and the structure of the ambient (environment) gas includes the torus in addition to the AGB wind. In this case the outcomes of each of the Rjet sims is similar in shape but not in size. Oddly, it is the Rjet=2 sim in which add behavior develops near the y axis.

Comments on size: The mass and momentum in the Rjet = 2 sim is 4x larger than that in the Rjet = 0.5 case, so the flow penetrates further as Rjet increases. However, the odd knot that develops near the y axis — combined with the fact that the flow barely encounters the torus on its lateral edge — mean that the head of the flow spreads out laterally and accretes more ambient with time. This slows it down.

Comments on shape: Although the general pattern of the flow stays about the same, the width-to-length ratio increases with Rjet. No surprise here, of course, Interestingly the flow width is set right at the earliest times by the size of the flow orifice. The width is not affected appreciably by the thermal pressure in the backflow.

Conclusions: in the narrow-flow sim (<~15-deg spread) the torus isn't a significant shaping factor. So for njet < namb the shape of the outflow is controlled largely by the initial conditions of the flow: vjet, jet mass, and opening angle. njet plays a role in determining the length of the jet. Rjet determine sits width. It appears safe to run models with a small Rjet since the issues that we saw in the email of yesterday (artificial structure forming near the y axis in the first 100 years) don't materialize for the narrower flows.

OVERALL CONCLUSIONS

It seems from this that the sims with small Rjet (that is, a high degree of flow-surface pixelation) work well other than the jet width is determined by initial conditions and not the flow hydro. For larger values of Rjet (that is, more diffuse momentum flux after launch) Rjet should be at least two (or the number of AMR refinement levels should be incremented by two) or else odd behaviors that develop at early times are amplified as the flow evolves will have a huge impact ont eh structure of the flow along its leading edge.