 Our collaboration, Chimera, is the code that we're using to simulate core collapse supernovae. I'm going to give a brief background on neutrino driven core collapse supernovae and focus on fluid instabilities that develop in these explosions. Talk about supernova simulations and the input physics that go into them. Talk a little bit about the Chimera code and then talk about simulations of neutrino driven explosions in two dimensions and three dimensions. And I'll point out some key differences between the two and three dimensional simulations. And some preliminary results on numerical resolution study employing full physics simulations of core collapse supernovae and then conclude with the summary and some remarks. So core collapse supernovae is basically the transition from a massive star to a compact object like a neutron star or a black hole. So a massive star evolves on tens of millions of years until it dies in a supernovae in about one second and leaves behind a compact object. So at the end of its evolutionary stage the massive star is composed of it has an iron core in the center surrounded by lighter and lighter elements. These layers have been constructed by burning to heavier and heavier elements until it reaches iron. Iron cannot fuse to heavier elements and produce energy so nuclear fusion stops in the core. And the iron core is growing by burning of silicon to iron. Until the core reaches the Chandra Sekar mass where so this iron core is held up by pressure from degenerate electrons. At 1.4 solar masses the Chandra Sekar mass the electrons are no longer able to withstand the pull of gravity and the core which is a few thousand kilometers collapses down to tens of kilometers. At that point density is on the order of 10 to the 14 grams per cubic centimeter. At that point the equation of state's difference and holds collapse and a pressure perturbation is initiated and propagates radially. And this pressure perturbation transitions into a shockwave in very high densities. When this shockwave is produced the temperature increases and the iron is broken up into free nucleons, neutrons and protons. And neutrinos are produced by electron capture on neutrons and protons. As this shock propagates radially to lower densities the neutrinos become or the material becomes transparent to neutrinos and there's a burst of neutrinos as the shock propagates through this lower density region. The shock propagates further out but it loses energy due to neutrino emission and also due to dissociation of iron that is falling through the shock. And the shock stalls, sits there for a while, it is heated so neutrinos are pouring out of what's going to become the neutron star and it deposits energy in this region. It excites multidimensional flows, instabilities and eventually the shock is revived and the star explodes in a core collapse supernova and so it disrupts the entire star and distributes heavy elements into the galaxy. So why do we care about core collapse supernovae? They are a dominant source of heavy elements in the universe. They are laboratories for fundamental physics. You can look at core collapse supernovae to probe our knowledge of gravitational physics, neutrino physics and nuclear physics and they're targeted by ground based and space born instruments covering the entire electromagnetic spectrum and they're also targeted by neutrino detectors and gravitational wave detectors. So to aid in understanding this wealth of observational data simulations are typically used to understand and unravel the explosion mechanism which is still being debated and also gravitational wave detectors use templates to look for gravitational wave signals in their observations and they use simulations to help look for expected signals. When we compute gravitational wave signals there is a response in the gravitational wave signals due to fluid instabilities happening in the core. So if we are lucky to detect galactic supernova with gravitational wave observatories and simultaneously with neutrino observatory because both signals travel unhindered through the star bringing information about the dynamics at the center of the exploding star. So modeling core collapse supernovae is a multi physics problem. It involves gravity and gravity is relativistic at this scale. Hydrodynamics, neutrino transport and nuclear and particle physics inputs the equation of state and neutrino pasties are needed to get the dynamics correctly and in particular neutrino transport is very important. 99% of the gravitational energy that's released in a core collapse is radiated away by neutrino emission. It's about only 1% of the energy is ejected as kinetic energy in the matter and the source is neutrino deposition so the challenge is to compute the energy and momentum transfer between the radiation field and the fluid. So as I mentioned the shock wave bounces off this neutron star, propagates radially it loses energy due to neutrino emission and dissociation. It stalls at around 100 kilometers and we have this setup where we have like a stalled shock at roughly 100 kilometers then we have a neutron star between the proton neutron star and the shock is the gain radius. So below the gain radius you have net cooling by neutrinos and above the gain radius you have net heating by neutrinos and neutrinos capture on neutrons or protons and produce and deposit energy into the fluid. So the goal of a simulation is to study and compute this energy and momentum transfer and actually lepton transfer as well and to study how this energy, so you have inflow material there's a ramp pressure on the shock and this ramp pressure must be overcome by some pressure or stresses from below that will push the shock out and it's not so easy. So neutrinos are not in equilibrium with the fluid. So here is a plot of showing the contours of the neutrino mean free path versus radius and neutrino energy and this is the RMS energy of neutrinos in this region. This region here is the heating region which is separated, which is between the gain radius and the shock and this is the critical region where we compute the energy and momentum transfer. And here the mean free path is on the order of from 10 kilometers down to 1000 kilometers. So the mean free path is much larger than typical length scales in the system. So a kinetic description is warranted to study this exchange. So we have the talk just before mine about the Boltzmann equation which considers the phase space density which gives the number of particles per position space volume and momentum space volume. It is governed by the Boltzmann equation which is a balance between basically just advection, phase space advection and collisions. So in this high energy density environment space time, there's curvature to space time. So that gives rise to some additional complexity on the left hand side and on the right hand side we consider scattering of neutrinos on alpha particles and heavy nuclei, scattering on electrons and positrons, neutrons and protons. This must be treated inelastically. So we have an electron capture, so that's emission and absorption on free neutrons and protons and heavy elements and then we also have some pair processes. Now simulations in also core collapse supernova simulations in spherical symmetry have been carried out in with full physics solving the Boltzmann transport equation, employing general relativity and all the neutrino-vider interactions that we think are important. And this plot shows the shock radius versus time after shock formation in the fully relativistic case and in an approximate Newtonian non-relativistic case. And in both cases the shock reaches a maximum and then starts to recede and there is no explosion in general. This is a general, so for the lowest mass presentors you can achieve explosions but in general for higher mass stars you don't achieve any explosion when you simulate in spherical symmetry. So multi-dimensional effects are important. So simulations must be done in multi-D. So here this region between the... So this heating region between the shock and the proton-neutral star is... So it's continually heating in this layer and heats up the material and it becomes connectively unstable, so neutrino-driven convection sets in. In addition there is an instability in the shock wave itself that leads to large-scale sloshing motions of the shock and both of these instabilities leads to both... or simultaneously simultaneous up-flow and down-flows which is different from what you can get in spherical symmetry. If you would get to a situation where you would have enough energy to push the shock out so the neutrino radiation field has two components. It has a diffusive component and also an accretion component. So accretion sustains the neutrino luminosity but if the shock is propagating outward you shock off the neutrino luminosity due to accretion and reduce and there is like a self-regulating effect. But in multi-D you can have these down-drafts that are fairly energetic that feed the neutrino radiation field continuously and you can have a build-up of explosion more efficiently. So the standing accretion shock instability called short SASI and neutrino-driven convection typically develop in different regimes and it depends on basically the structure of the progenitor. If you consider the advection time as the time a parcel spends between the shock and the gain radius and the buoyancy time, the time for convective bubbles to develop. If this advection time is fast, perturbations will be just swept into the gain layer and in that case the standing accretion shock instability develops. So simulations have shown that when this ratio is less than 3 then standing accretion shock instability will dominate. Typically that's for more massive stars and for this parameter larger than 3 then the driven convection will develop. And both of these instabilities lead to non-radial mass motions and turbulence below the shock which are becoming, we're realizing that this is becoming important to understand the explosion mechanism. One effect is that these instabilities pushes the shock to larger radii and this leads to an increased residence time of fluid elements in the heating region and a more efficient energy transfer. Another effect is just renal stresses due to turbulent motions or non-radial motions. In this study by Couch and Ott, they kept the neutrino heating rate at the same rate but perturbed the material ahead of the shock in different ways to excite weaker and stronger turbulence. And they found that the turbulence was contributing significantly to the stresses below the shock. So more energetic multi-dimensional flows were favorable to explosion. And this is a study by Hanke et al. that looked at two-dimensional and three-dimensional simulations and they computed the spectra of the kinetic energy in two-dimensional simulations and in three-dimensional simulations and the first talk in this session talked about the inverse cascade of in two-dimensional turbulence and this seems to be an important effect and a detrimental effect for 2D simulations because typically both the SASI and the neutrino-driven convection inject energy at the scale here and in two-dimensional simulations this inverse cascade leads to accumulation on large scales. In 3D you have the correct direction of the cascade and if you compare 2D and 3D simulations, the 2D simulations are typically characterized by large-scale plumes whereas in 3D the forward cascade leads to more of a fragmentation of large-scale structures during the explosion. So these effects, both the neutrino-driven convection and the standing increase shock instability, the stresses as well as this effect between 2D and 3D have been studied using quite simplified parameterized models. In particular the neutrino physics has been simplified a lot but they're very important because you can run a lot more simulations and very parameters in a controlled way to gain this nice insight. So in the Camero code we're running multi-dimensional supernova simulations with spectral neutrino transport. So Camero consists of three heads. It has hydrodynamics, nuclear network and radiation transport. The hydrodynamics is a short capturing PPM implementation. It uses a moving grid to follow the concentration of mass during collapse and in multi-D solving the Boltzmann equation is very expensive. It has not been done in full 3D yet and we're solving for angular moments of the distribution function in a flux-limited diffusion approach in Camero. But it uses a modern opacity set. It includes relativistic corrections that are necessary. It also approximates the transport in a ray-by-ray approximation which is it solves in the spherically polar coordinate system. It solves spherically symmetric transport problems along each radial ray. This is done for computational expediency but it's an approximation that we're working to get rid of. So for hydro there's some tests showing decent capturing of shock and entropy wave interactions comparison with the Boltzmann code. This is the shock radius versus time after bounds. So some neutrino quantities. So this approximation is reasonable. In 2D simulations run simulations in axial symmetry across a range of progenitors and we see neutrino-driven explosions in all these models and the neutrino-driven explosions are initiated here comparing 1D shock trajectories and 2D shock trajectories. This is about the time when fluid instabilities become nonlinear and this aid in the development of explosions. We have done simulations in 3D as well. Here, so just going back, this shows the fluid entropy at the end of a simulation which is characterized by a few large expanding hot bubbles and between these hot bubbles there is downflows that continue to feed the neutrino radiation by the accretion component. Now comparing the shock trajectories for 1D, 2D and 3D we see that the 3D shock trajectories grow a little bit slower than 2D and this is similar to the observation that was seen in the parametrized models. So we believe that the fundamental difference between 2D and 3D turbulence is contributing to this. Initially this showed entropy distributions in 3D and in 2D. The instability is set in at about the same time but it grows more forcefully in 2D. But in both cases the explosion is associated with large bubbles. Now looking at the neutrino heating efficiency comparing 2D and 3D. So neutrino heating efficiency is the net energy deposition by neutrinos divided by in this gain region divided by how much is coming into the gain region from below. And we see that as you increase the heating efficiency you get closer to explosion and the 2D models have a higher overall higher neutrino heating efficiency. At the same time the advection time through the gain layer over the heating time scale gives an indication on how long a fluid parcel is subject to neutrino radiation and when this ratio is approaching and increases one that's when you typically see a runaway explosion and you see in 2D simulations this is also higher. Now looking at the turbulent kinetic energy which we just compute as anything that's not part of the mean radial flow we see that in 2D it is much higher than in 3D. So this will contribute to a higher Reynolds stress in 2D simulations. So looking at the explosion energy we also see that in 2D the explosion energy is much higher at this point than in the 3D simulation. Now we've learned from running 2D simulations that we need to run to very long times to get the final explosion energy so this is not any indication of what the final explosion energy is but we see that the 2D is growing much faster so I think it's reasonable to... it's a good guess that the 3D will be lower. So to try to understand the impact of resolution we have started a numerical resolution study these are very preliminary results but I want to show them anyway our objective is to understand the energy and momentum transport in Supernova as it develops and to look at models with different resolutions and we want to employ full physics simulations So this is like a follow-up study to this very nice paper that looked at the same picture or the same question using very parametrized models so they used parametrized neutrino heating and cooling and fixed inner boundary and fixed accretion rate Now we know from self-consistent models that the accretion down onto the proton neutron star is also regulating the neutrino output so we want to include that in the model so we've run models in a reduced 3D domain using a variety of angular resolutions and this resolution here is what we used in this 3D simulation that I just showed So as we increase the resolution we see that the average shock radius is decreasing and this leads to a higher heating efficiency So if we look also at the kinetic energy on average the low resolution model has higher kinetic energy than the higher resolution model and if we look here at the Reynolds stresses at different times during the simulation So this is the Reynolds stress divided by the pressure so it gives like a relative contribution to the stress compared to the thermal pressure and it becomes significant when this fluid flow is highly nonlinear and you can also see that in 3D in the low resolution model the stresses are on average higher than in the higher resolution models So now if we look at compute energy spectra over a temporal interval here we see that this is not resolved and also for the low resolution models there is an accumulation of power at large scales So we increase the resolution the power is less distributed to smaller scales and this was also seen in this paper and discussed there in the context of more parametrized models and looking at the distribution of entropy we see that for the lowest resolution we have clearly formed these large scale bubbles but as we increase the resolution they seem to be shredded by multi-D they are more turbulent Okay so I don't think I'm not sure how much time do I have It's okay, I'm out of time Oh okay This is the last slide before I just wanted to show gravitational wave signals This is from 2D explosion simulations though So you want to keep that in mind but this shows the gravitational wave amplitude would be observed on Earth or if the supernova was 10 kPa away and you can see dynamics from the supernova or you can see the supernova dynamics imprinted in the gravitational wave signals Initially there is a shock expands into the stellar mantle it sets up a configuration that is unstable to convection this is a prompt convection that dies out quickly and you can see this in the signal then there is a quiescent phase until the neutrino driven convection or the standing accretion shock instability induce, so there is like flows that impinge on the proton neutron star and induce oscillations in the proton neutron star and they lead to the strongest part of the signal then as the explosion sets in there is an offset at tail which is due to asymmetry in the mass distribution and then as the explosion is really going and these downflows onto the proton neutron star sees this stochastic nature on the signal disappears and you have this smooth curve so one can see responses to supernova dynamics in the gravitational wave signal and that is interesting ok so this is my summary I have shown multi-D simulations of neutrino driven explosions non-radial flows tend to improve conditions for explosions compared to 1D simulations due to neutrino driven convection and the standing accretion shock instability the fluidity experiences longer exposure to neutrino driven... neutrino radiation due to these effects and also renal stresses seem to help push the shock out the ultimate driver is the gravitational energy that is in the collapsing star so for a non-radial shock it can effectively transform radial kinetic energy into non-radial kinetic energy and also it's also what's driving the neutrino radiation now the inverse cascade into these seem to result in over energetic explosions and we need to do 3D simulations and the explosion dynamics provides a response in the gravitational wave signal so I just want to thank collaborators and funding agencies and all of you for your attention