3.2.1 Neutrino-driven buoyancy

In addition to the prompt convection associated with the deceleration of the shock as it stalls, the continued heating of the gas by neutrino absorption maintains a radial entropy gradient oriented in the same direction as gravity from the shock to the gain radius (Herant, Benz, & Colgate Reference Herant, Benz and Colgate1992, Janka & Müller Reference Janka and Müller1996). Some gravitational energy can be gained if the flow is able to interchange high and low entropy layers. The Brunt–Väisälä frequency ω_BV characterises the timescale of the fastest motions fed by buoyancy forces. Defining the entropy S in a dimensionless manner, the Brunt–Väisälä frequency is expressed as follows:

(5) \begin{eqnarray} S\equiv {1\over \gamma -1}\log {P/P_0\over (\rho /\rho _0)^{\gamma} }, \end{eqnarray}

(6) \begin{eqnarray} \omega _{\rm BV}^2\sim -{\gamma -1\over \gamma }\nabla S\cdot \nabla \Phi . \end{eqnarray}

This interchange can feed turbulent motions and push the shock further out, increasing the size of the gain region and diminishing the energy losses due to dissociation. Numerical simulations performed in the early 2000’s were disappointing though (Buras et al. Reference Buras, Rampp, Janka and Kifonidis2003), both because the duration of the simulation was limited to a few hundred milliseconds and as some equatorial symmetry was assumed due to limited computational resources. The equatorial symmetry precluded the growth of global l = 1 modes. Foglizzo, Scheck, & Janka (Reference Foglizzo, Scheck and Janka2006) pointed out that the negative sign of the entropy gradient is not a sufficient criterion for the convective instability in the gain region because the interchange has to be fast enough to take place before the advected gas reaches the gain radius. A criterion for linear stability compares the advection timescale to the buoyancy timescale estimated from the Brunt–Väisälä growth rate:

(7) \begin{eqnarray} \chi \equiv \int _{\rm gain}^{\rm shock} |\omega _{\rm BV}|{{\rm d} r\over |v_r|}<3. \end{eqnarray}

From this criterion, one can anticipate that the strength and consequences of neutrino-driven buoyancy may vary from one progenitor to another, depending on the radius of the stalled shock.

3.2.2 Instability of the stationary shock: SASI

Another source of symmetry breaking was discovered by Blondin, Mezzacappa, & DeMarino (Reference Blondin, Mezzacappa and DeMarino2003) who studied a simplified setup where neutrino heating was neglected in order to avoid any possible confusion with neutrino-driven convection. The standing accretion shock instability (SASI) corresponds to a global oscillatory motion of the shock surface with a period comparable to the advection time from the shock to the neutron star surface. This global (l = 1) and oscillatory character are distinct features of the linear regime when perturbations grow exponentially with time. In contrast, the convective instability is non-oscillatory and dominated by smaller azimuthal scales (typically l = 5 − 6) comparable to the radial size of the gain region.

The mechanism responsible for SASI relies on the interaction of acoustic perturbations and advected ones, such as entropy and vorticity perturbations (Galletti & Foglizzo Reference Galletti, Foglizzo, Casoli, Contini, Hameury and Pagani2005, Ohnishi, Kotake, & Yamada Reference Ohnishi, Kotake and Yamada2006, Foglizzo et al. Reference Foglizzo, Galletti, Scheck and Janka2007, Scheck et al. Reference Scheck, Janka, Foglizzo and Kifonidis2008, Guilet & Foglizzo Reference Guilet and Foglizzo2012). As illustrated in Figure 2, a perturbation of the shock surface produces entropy and vorticity perturbations which are advected towards the proto-neutron star surface. Their deceleration in the hot region close to the neutron star produces an acoustic feedback which propagates towards the shock surface, pushes it and regenerates new advected perturbations with a larger amplitude than the first ones. This advective-acoustic cycle has been described analytically in a Cartesian geometry (Foglizzo Reference Foglizzo2009, Sato, Foglizzo, & Fromang Reference Sato, Foglizzo and Fromang2009). The same coupling mechanism is responsible for the instability of the bow shock in Bondi–Hoyle–Lyttleton accretion (Foglizzo Reference Foglizzo2001, Reference Foglizzo2002, Foglizzo, Galletti, & Ruffert Reference Foglizzo, Galletti and Ruffert2005).

Figure 2.

SASI mechanism based on the coupling between acoustic waves (wavy arrows) and advected perturbations (circular arrows) between the stalled shock and the proto-neutron star.

4.2.1 The mystery of pulsar kicks potentially explained

The Garching team was first to recognise the potential of global l = 1 deformations to explain pulsar velocities of several hundreds of kilometers per second (Scheck et al. Reference Scheck, Plewa, Janka, Kifonidis and Müller2004). The global conservation of linear momentum implies that the final momentum of the neutron star is equal in magnitude and opposed in direction to the total momentum of the ejecta. The asymmetric distribution of ejected matter has been studied in a series of axisymmetric simulations where the neutrino luminosity was adjusted to trigger an explosion. They showed that the gravitational interaction pulls the neutron star in the direction of the closest dense regions of post-shock gas, over a timescale of several seconds which can exceed the timescale of direct advection of momentum from the accreted gas. Their statistical study produced a shape of the distribution of pulsar velocities in agreement with observational constraints (Scheck et al. Reference Scheck, Kifonidis, Janka and Müller2006). The magnitude of the kick is a stochastic variable, while the explosion energies and timescales were found to be deterministic in this study. The magnitude of the kicks obtained has been subsequently confirmed with a different axisymmetric code by Nordhaus et al. (Reference Nordhaus, Brandt, Burrows, Livne and Ott2010a, Reference Nordhaus, Brandt, Burrows and Almgren2012) which directly followed the motion of the neutron star. This kick mechanism has been confirmed in 3D simulations by Wongwathanarat, Janka, & Müller (Reference Wongwathanarat, Janka and Müller2010, Reference Wongwathanarat, Janka and Müller2013) who used an axis-free Yin–Yang grid rather than spherical coordinates.

4.2.2 Enhanced mixing triggered by asymmetric shock in SN1987A

The light curve of SN1987A, together with the early emergence of X-ray and gamma rays suggested that significant mixing disrupted the onion like structure of the star (McCray Reference McCray1993, Utrobin Reference Utrobin2004, Fassia & Meikle Reference Fassia and Meikle1999), probably triggered by the interaction of the shock with composition interfaces. The efficiency of this mixing is enhanced if the shape of the shock is non-spherical, as shown by Kifonidis et al. (Reference Kifonidis, Plewa, Scheck, Janka and Müller2006). This study considered shock deformations dominated by the modes l = 1, 2 as naturally produced by SASI during the first second after shock bounce. The subsequent hydrodynamical evolution of the shock across the stellar envelope up to the surface showed final iron group velocities up to 3300 km s^{− 1}, strong mixing at the H/He interface, and hydrogen mixed down to velocities of 500 km s^{− 1}. These features are closer to observations than those obtained by the same team from a spherical shock dominated by the smaller scale convective instability (Kifonidis et al. Reference Kifonidis, Plewa, Janka and Müller2003). The viability of this process in 3D was later confirmed by Hammer, Janka & Müller (Reference Hammer, Janka and Müller2010), who followed the evolution of the shock calculated in 3D by Scheck (Reference Scheck2007), across the stellar envelope. They found that the development of mixing instabilities is more efficient in 3D and allows for higher clump velocities due to the different action of drag forces. The effect of the progenitor structure was recently explored by Wongwathanarat, Müller, & Janka (Reference Wongwathanarat, Müller and Janka2014) with a series of red and blue supergiants of 15 and 20M _sol.

5.3.1 The parameter χ and the relative roles of neutrino-driven convection versus SASI

The numerical simulations performed at Garching and Oak Ridge confirmed the diversity of hydrodynamical processes ruling the multidimensional evolution during the last hundreds of milliseconds before the explosion. Neutrino-driven buoyancy seems to govern the post-shock dynamics of the progenitor of 11.2 M _sol. The global oscillations of SASI are clearly dominant for the 27 M _sol progenitor (Müller et al. Reference Müller, Janka and Heger2012b), while both instabilities seem to be entangled in the evolution the 15 M _sol progenitor. Such differences can be understood from the comparison of the advection time and the buoyancy times in these systems. A short advection time is both favourable to SASI (Foglizzo et al. Reference Foglizzo, Galletti, Scheck and Janka2007, Scheck et al. Reference Scheck, Janka, Foglizzo and Kifonidis2008), and stabilising for neutrino-driven convection (Foglizzo et al. Reference Foglizzo, Masset, Guilet and Durand2006). By adjusting the neutrino luminosity and the rate of dissociation at the shock, Fernandez et al. (Reference Fernandez, Müller, Foglizzo and Janka2014) were able to characterise each instability separately and compare the properties of the buoyant bubbles and turbulence induced before the explosion.

Besides a few examples in the parameter space, we do not have a clear picture of the dependence of the χ parameter on the main sequence mass yet. It should be noted that the development of SASI is not a sufficient criterion for a successful explosion. The axisymmetric simulation of a 25M _sol progenitor is a clear example of strong SASI activity without an explosion (Hanke et al. Reference Hanke, Müller, Wongwathanarat, Marek and Janka2013).

5.3.2 The compactness parameter ξ_2.5 and black hole versus neutron star formation

A systematic study performed by Ugliano et al. (Reference Ugliano, Janka, Marek and Arcones2012) used some educated prescription to account for multidimensional evolution using 1D simulations over a wide range of progenitor masses from 10 to 40 solar masses. The outcome of the core collapse can either be a supernova explosion with the formation of a neutron star, a failed supernova with the formation of a black hole, or a supernova with the delayed formation of a black hole through the fall back of enough ejecta. The diversity of mass loss processes during the life of a star results in a non-monotonous relation between its mass at the moment of core-collapse and its mass on the main sequence. In addition to stressing this labelling confusion, the study of Ugliano et al. (Reference Ugliano, Janka, Marek and Arcones2012) showed that the production of a neutron star or a black is also very variable for progenitors more massive than 15 M _sol. An interesting indicator is based on the compactness parameter ξ_2.5 defined by O’Connor & Ott (Reference O’Connor and Ott2011), measuring the radius of the innermost 2.5 M _sol:

(8) \begin{eqnarray} \xi _{2.5}\equiv {M/M_{\rm sol}\over R(M)/1000\ {\rm km}}. \end{eqnarray}

Although this indicator is not strictly deterministic, a threshold ξ_2.5 > 0.3 seems very favourable to black hole formation in the calculations of Ugliano et al. (Reference Ugliano, Janka, Marek and Arcones2012). From a series of 101 axisymmetric simulations from 10.8 to 75 M _sol using the isotropic diffusion source approximation of neutrino transport, Nakamura et al. (Reference Nakamura, Takiwaki, Kuroda and Kotake2014a) found that a high compactness parameter corresponds to a later explosion time, a stronger neutrino luminosity and explosion energy, and a higher Nickel yield.

5.3.3 The possible influence of pre-collapse convective asymmetries

The structure of the progenitor may also influence the asymmetry of the explosion mechanism through the convective inhomogeneities associated with thermonuclear burning in the Silicon and Oxygen shells (Arnett & Meakin Reference Arnett and Meakin2011). The quantitative impact of this process has been estimated by Couch & Ott (Reference Ott2013), who showed an example where these asymmetries had an effect comparable to a 2% increase of the neutrino luminosity and could be enough to turn a failed explosion into a successful one. Elaborating in this direction, the recent study of Müller & Janka (Reference Müller and Janka2014b) pointed out that the most efficient effect of pre-collapse asymmetries comes from inhomogeneities with the largest angular scale l = 1, 2.

5.4.1 3D explosions are not easier than 2D

Using a simplified model of neutrino interactions with cooling and heating functions, Nordhaus et al. (Reference Nordhaus, Burrows, Almgren and Bell2010b) had claimed that the neutrino luminosity needed to obtain an explosion was lower in 3D than in 2D, in a comparable proportion as measured by Murphy & Burrows (Reference Murphy and Burrows2008) between 2D and 1D. This raised the hope that the weak explosions obtained in 2D axisymmetric simulations could become more robust once computer resources would allow first principle calculations including neutrino transport in 3D. A careful check by Hanke et al. (Reference Hanke, Marek, Müller and Janka2012) and Couch (Reference Couch2013) showed that the neutrino luminosity threshold in the idealised setup studied by Nordhaus et al. (Reference Nordhaus, Burrows, Almgren and Bell2010b) is actually very similar in 3D and 2D. The misleading results of Nordhaus et al. (Reference Nordhaus, Burrows, Almgren and Bell2010b) happened to be due to some numerical errors in the treatment of gravity in their CASTRO code, later corrected and acknowledged by Burrows, Dolence, & Murphy (Reference Burrows, Dolence and Murphy2012). Despite these corrections, the convergence between the different codes is not yet fully satisfactory (Dolence et al. Reference Dolence, Burrows, Murphy and Nordhaus2013).

Using the IDSA approximation of neutrino transport, Takiwaki, Kotake, & Suwa (Reference Takiwaki, Kotake and Suwa2014) studied the evolution of a 11.2 M _sol progenitor and observed that the explosion time is shorter and the explosion is more vigorous in 2D than in 3D. Similar conclusions were reached by Couch & O’Connor (Reference Couch and O’Connor2014) with a multi species leakage scheme for progenitors of 15 and 27 M _sol. Let us note that the stochastic nature of the dynamical evolution and the prohibitive cost of 3D simulations make it difficult to achieve numerical convergence. Beside the possible influence of numerical artifacts (e.g. Ott et al. Reference Ott2013 compared to Abdikamalov et al. Reference Abdikamalov2014), part of the difference between 2D and 3D simulations could be related to a more efficient energy cascade from large scales to small scales in 3D than in 2D (Hanke et al. Reference Hanke, Marek, Müller and Janka2012, Couch Reference Couch2013, Couch & O’Connor Reference Couch and O’Connor2014).

5.4.2 SASI does exist in 3D models of core collapse

According to the mechanism proposed by Guilet, Sato, & Foglizzo (Reference Guilet, Sato and Foglizzo2010) and checked on the axisymmetric simulations of Fernandez & Thompson (Reference Fernandez and Thompson2009), the saturation amplitude of SASI oscillations is not expected to be very different in 2D and 3D. The waves of entropy and vorticity created by the oscillations of the shock are expected to lose their large scale l = 1 coherence when disrupted by the parasitic growth of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. This description, however, ignores the interaction of SASI with small scale turbulence, whose properties may differ in 2D and 3D. Even though the first 3D simulations performed by Blondin & Mezzacappa (Reference Blondin and Mezzacappa2007) in an adiabatic approximation showed well-defined spiral shock patterns on a large scale, subsequent 3D simulations with less idealised setups suggested weaker amplitudes than in 2D (Iwakami et al. Reference Iwakami, Kotake, Ohnishi, Yamada and Sawada2008, Burrows et al. Reference Burrows, Dolence and Murphy2012). The clear dominance of non-oscillatory convective motions in some simulations even led Burrows et al. (Reference Burrows, Dolence and Murphy2012) to draw general conclusions such that SASI would be an artefact of simplified physics which does not exist in realistic 2D simulations, and even less in 3D. This reasoning missed the diversity of explosion paths later explored in a parametric manner by Fernandez et al. (Reference Fernandez, Müller, Foglizzo and Janka2014) in 2D or Iwakami, Nagakura, & Yamada (Reference Iwakami, Nagakura and Yamada2014a) in 3D, and exemplified by the axisymmetric simulation by Müller et al. (2012) of the 27 M _sol progenitor. The evolution of this progenitor was computed in 3D from first principles by Hanke et al. (Reference Hanke, Müller, Wongwathanarat, Marek and Janka2013) who confirmed the presence of the SASI mode with an amplitude which can exceed even the amplitude observed in 2D. Nevertheless, the explosion did not take place during the first 400 ms of evolution, whereas it did explode in 2D on this timescale. It is interesting to note the sensitivity of the dynamics to small differences in the equation of state and possibly general relativistic effects between the axisymmetric results of Hanke et al. (Reference Hanke, Müller, Wongwathanarat, Marek and Janka2013) and those of Müller et al. (2012), where less than 200 ms were sufficient for an explosion.

5.5.1 SASI effect on the pulsar spin

The spectacular conclusions regarding the pulsar spin obtained by Blondin & Mezzacappa (Reference Blondin and Mezzacappa2007) with a 3D adiabatic setup were naturally followed by more realistic 3D simulations. Fernandez (Reference Fernandez2010) was able to confirm the angular momentum budget when the progenitor does not rotate, with a setup similar to Blondin & Shaw (Reference Blondin and Shaw2007) but in 3D. By contrast, the spiral mode did not appear easily in the non-rotating simulation of Iwakami et al. (Reference Iwakami, Kotake, Ohnishi, Yamada and Sawada2009) who considered a stationary accretion flow incorporating the buoyancy effects due to neutrino heating and a more realistic equation of state. Significantly slower pulsar spin periods of the order of 500–1000 ms were estimated by Wongwathanarat et al. (Reference Wongwathanarat, Janka and Müller2010) who considered the collapse of a 15 M _sol progenitor with neutrino cooling and heating, self-gravity and general relativistic corrections of the monopole, and a Yin–Yang grid. Similar skeptical conclusions were reached by Rantsiou et al. (Reference Rantsiou, Burrows, Nordhaus and Almgren2011), who pointed out that the mass cut during the explosion process might not coincide with the separation of positive and negative angular momentum produced by SASI. A larger sample of non-rotating progenitors was considered by Wongwathanarat et al. (Reference Wongwathanarat, Janka and Müller2013), including a 20 M _sol progenitor, and measured final spin periods of 100–8 000 ms. They noted that the kick and spin do not show any obvious correlation regarding their magnitude or direction. Their analysis clarified an important difference between the kick and spin mechanisms: the spin up takes place during the phase of accretion onto the neutron star surface because it is dominated by the direct accretion of angular momentum. By contrast, the kick can grow on a longer time scale, even when accretion has stopped, through the action of the gravitational force. Using analytic arguments, Guilet & Fernandez (Reference Guilet and Fernandez2014) estimated the maximum spin of a neutron star born from a non-rotating progenitor by measuring the amount of angular momentum which can be stored in the saturated spiral mode of SASI. They obtained spin periods compatible to those measured in the simulations of Wongwathanarat et al. (Reference Wongwathanarat, Janka and Müller2013).

5.5.2 Rotation effects on the shock dynamics

The effect of a moderate rotation on the dynamics of the collapse has received little attention so far, despite the fact that interesting correlations between the kick and spin directions would be expected in this regime. Indeed, the growth of a prograde spiral SASI mode is expected from the early numerical simulations of Blondin & Mezzacappa (Reference Blondin and Mezzacappa2007), and the linear stability analysis of Yamasaki & Foglizzo (Reference Yamasaki and Foglizzo2008) indicated that this effect of differential rotation can be significant even if the centrifugal force is weak. The 3D simulations of stationary accretion by Iwakami et al. (Reference Iwakami, Kotake, Ohnishi, Yamada and Sawada2009) confirmed that the amplitude of the prograde spiral mode of SASI is enhanced by rotation. Iwakami, Nagakura, & Yamada (Reference Iwakami, Nagakura and Yamada2014b) gradually increased the angular momentum of the infalling gas to measure the influence of rotation on the explosion threshold. According to the few points in their Figure 1, the change in the threshold of neutrino luminosity required to obtain an explosion seems to be a linear function of the angular momentum with a luminosity variation of ~ 10% for an angular momentum of the order of ~ 5 × 10¹⁵ cm ²s^{− 1}. More simulations would be necessary to check whether this effect really scales linearly with the rotation rate as expected. According to a tentative linear extrapolation, rotation effects with an angular momentum of ~ 10¹⁵ cm²s^{− 1} could induce a 2% shift of the neutrino luminosity, i.e. a similar order-of-magnitude effects as the pre-collapse asymmetries described by Couch & Ott (Reference Ott2013). For reference, 10¹⁵ cm² s^{− 1} is the equatorial angular momentum at the surface of a pulsar with a radius of 10 km and a spin period of 6 ms. The uncertain effect of mixing instabilities inside the neutron star remains to be evaluated. Compared to the models of stellar evolution by Heger et al. (Reference Heger, Woosley and Spruit2005), a value of 10¹⁵ cm² s^{− 1} is twice as large as the estimated angular momentum at 2 000 km with magnetic fields ( ~ 5 × 10¹⁴ cm² s^{− 1}), and 20 times smaller than without magnetic fields ( ~ 2 × 10¹⁶ cm² s^{− 1}).

Nakamura et al. (Reference Nakamura, Kuroda, Takiwaki and Kotake2014b) considered the collapse of the stellar core of a 15M _sol progenitor with a shell type rotation profile: Ω(r) = Ω₀/(1 + r ²/R ²₀) with R ₀ = 2 × 10⁸ cm and Ω₀ = 0, 0.1π, and 0.5π rad s^{− 1} corresponding to an initial angular momentum at 2 000 km of 0, 6 × 10¹⁵ cm² s^{− 1}, and 3 × 10¹⁶ cm² s^{− 1}. They measured a reduction of the threshold of neutrino luminosity of the order of 10% for Ω₀ = 0.1π, which is consistent with the results of Iwakami et al. (Reference Iwakami, Nagakura and Yamada2014b). They emphasised that the direction of the explosion is preferentially perpendicular to the spin axis but did not take into account the asymmetry of neutrino emission induced by rotation, which may favour the axial direction.

5.5.3 The growth of magnetic energy without differential rotation

Endeve et al. (Reference Endeve, Cardall, Budiardja and Mezzacappa2010, Reference Endeve, Cardall, Budiardja, Beck, Bejnood, Toedte, Mezzacappa and Blondin2012) used an adiabatic setup to show that even when the initial rotation of the core is neglected, the magnetic energy can grow from the turbulent motions induced by SASI to reach 10¹⁴G at the surface of the neutron star. The growth of this magnetic field, however, did not significantly affect the dynamics of the shock. Obergaulinger, Janka, & Aloy (Reference Obergaulinger, Janka and Aloy2014) addressed the same question in a more realistic setup including neutrino-driven convection and a M1 approximation of neutrino transport. They confirmed the lack of significant effects on the shock evolution except if the field is initially very strong. They found evidence for an accumulation of Alfven waves at the Alfven point as described by Guilet et al. (Reference Guilet, Foglizzo and Fromang2011), but did not find a significant contribution to the field amplification or heating.

5.5.4 Magnetic field amplification with fast rotation

Fast differential rotation provides a large energy reservoir for magnetic field amplification, such that magnetic effects cannot be neglected beyond a certain angular momentum. The shearing of a poloidal magnetic field into a toroidal one provides a linear amplification, which would be relevant only if the initial poloidal magnetic field were large enough. If the initial poloidal magnetic field is weak, the exponential growth of the magnetorotational instability (MRI) is a more promising mechanism of magnetic field amplification (Akiyama et al. Reference Akiyama, Wheeler, Meier and Lichtenstadt2003). A meaningful description of this process requires 3D simulations with a very high resolution, due to the very short wavelength of the MRI growing on a weak initial field. The prohibitive computing time required has led researchers to use axisymmetric simulations with artificial assumptions such as a strong initial poloidal field mimicking the outcome of some amplification processes (e.g. Moiseenko, Bisnovatyi-Kogan, & Ardeljan Reference Moiseenko, Bisnovatyi-Kogan and Ardeljan2006, Burrows et al. Reference Burrows, Dessart, Livne, Ott and Murphy2007, Takiwaki, Kotake, & Sato Reference Takiwaki, Kotake and Sato2009, Takiwaki & Kotake Reference Takiwaki and Kotake2011). In this framework, they obtained powerful magnetorotational explosions, with jets launched along the polar axis due to the winding of the initial poloidal magnetic into a very strong ( ≳ 10¹⁵G) toroidal field. The progress of computational power has opened new perspectives to address this problem in 3D. Winteler et al. (Reference Winteler, Käppeli, Perego, Arcones, Vasset, Nishimura, Liebendörfer and Thielemann2012) suggested that such magnetorotational explosions could be an important source of r-process elements. The 3D simulations of Mösta et al. (Reference Mösta2014) highlighted the role of a non-axisymmetric instability of the toroidal magnetic field which can break the jet structure and prevent such magnetorotational explosions, a non-runaway expansion of the shock being observed instead.

In parallel, progress is being made in understanding the magnetic field amplification due to the MRI in the proto-neutron star. Buoyancy driven by radial gradients of entropy and lepton fraction has a strong impact on the dynamics, though the thermal and lepton number diffusion due to neutrinos allows for fast MRI growth, as shown by the linear analysis of Masada, Sano, & Takabe (Reference Masada, Sano and Takabe2006), Masada, Sano, & Shibata (Reference Masada, Sano and Shibata2007). The study of the non-linear phase of the MRI requires 3D simulations, which have been undertaken in local and semi-global models that describe a small fraction of the proto-neutron star. Taking into account global gradients responsible for buoyancy has, however, proven difficult in this framework because of boundary effects (Obergaulinger et al. Reference Obergaulinger, Cerdá-Durán, Müller and Aloy2009) or coarse radial resolution (Masada, Takiwaki, & Kotake Reference Masada, Takiwaki and Kotake2015). A local model in the Boussinesq approximation allowed Guilet & Müller (Reference Guilet and Müller2015) to estimate that a stable stratification decreases the efficiency of magnetic field amplification only slightly, but significantly impacts the structure of the magnetic field.

Neutrino radiation, neglected in most numerical simulations, can slow down the MRI growth through an effective neutrino viscosity deep inside the proto-neutron star or through a neutrino drag near its surface (Guilet, Müller, & Janka Reference Guilet, Müller and Janka2015). Further local simulations will be useful to study the MRI in this new growth regime, and ultimately global simulations (only achieved in 2D so far (Sawai, Yamada, & Suzuki Reference Sawai, Yamada and Suzuki2013, Sawai & Yamada Reference Sawai and Yamada2014)) will be needed to assess the magnetic field geometry and its impact on the explosion.

6.1.1 From gas to water dynamics: the shallow water analogy

In the simple fountain built by Foglizzo et al. (Reference Foglizzo, Masset, Guilet and Durand2012), the dynamics of water looks very similar to the gas dynamics induced by SASI in the equatorial plane of the star, on a scale one million times smaller and an oscillation period one hundred times slower than in the astrophysical flow. On the one hand, the fury of extreme gravity accelerating the stellar gas to one tenth of the speed of light in less than half a second. Iron nuclei broken apart upon compression by a shock wave which moves back and forth with a 30 ms period. On the other hand, water flowing at room pressure and temperature in a meter size fountain, giving rise to the back and forth motions of a surface wave with a 3 s period. The striking resemblance illustrated in Figure 3 is rooted in the similarity between the Euler equations describing an inviscid adiabatic gas and the Saint Venant equations describing shallow water. The conservation of mass, momentum and energy for a gas with an adiabatic index γ, density ρ, pressure P, entropy S, velocity v, and sound speed c ²_s = γP/ρ is written as follows:

(9) \begin{eqnarray} {\partial \rho \over \partial t}+\nabla \cdot (\rho v)=0, \end{eqnarray}

(10) \begin{eqnarray} {\partial v\over \partial t}+(\nabla \times v)\times v+\nabla \left[{v^2\over 2}+{c_{\rm s}^2\over \gamma -1}+\Phi \right]=c_{\rm s}^2\nabla S, \end{eqnarray}

(11) \begin{eqnarray} {\partial S\over \partial t}+v\cdot \nabla S=0. \end{eqnarray}

The pressure gradient ∇P in the Euler equation (10) has been decomposed into a enthalpy gradient ρ∇(c ²_s/(γ − 1)) and an entropy gradient ρc ²_s∇S.

Figure 3.

The shape of the rotating hydraulic jump observed in the non-linear regime in the SWASI experiment (right) is similar to the shape observed in the shallow water approximation (left) and the shape of the SASI in the numerical simulations of cylindrical gas accretion with local neutrino cooling (top).

The Saint Venant set of equations describing the shallow layer of water with depth H flowing with a velocity v over a surface $z=H_\Phi (r)$ is the following:

(12) \begin{eqnarray} {\partial H\over \partial t}+\nabla \cdot (H v)=0, \end{eqnarray}

(13) \begin{eqnarray} {\partial v\over \partial t}+(\nabla \times v)\times v+\nabla \left[{v^2\over 2}+c_{\rm w}^2+\Phi \right]=0. \end{eqnarray}

The local gravity g = 981cms^{− 2} in the laboratory infuences the speed c _w of surface waves in the experiment and the effective potential $\Phi \equiv gH_\Phi$, which mimics the Newtonian gravity of the central object with a hyperbolic shape:

(14) \begin{eqnarray} c_{\rm w}^2\equiv gH, \end{eqnarray}

(15) \begin{eqnarray} H_\Phi (r)\equiv -{(5.6\ {\rm cm})^2\over r}. \end{eqnarray}

The Froude number defined by Fr = |v|/c _w plays the same role as the Mach number ${\cal M}=|v|/c_{\rm s}$ in a gas. The pressure variation ΔP induced by the weight of water $\Delta P=\rho _{\rm w}g (z-H_\Phi (r))$ has a mean value noted P = ρ_wgH/2 related to the local speed of surface waves c _w:

(16) \begin{eqnarray} P=\rho _{\rm w} {c_{\rm w}^2\over 2}. \end{eqnarray}

This relation is analogous to the relation P = ρc ²_s/γ for a gas with an adiabatic index γ = 2.

6.1.2 The absence of buoyancy effects in shallow water

Besides the particular value of the adiabatic index, the main difference between the two sets of equations comes from the absence of buoyancy effects in shallow water. This limitation precludes the study of buoyancy driven instabilities such as neutrino-driven convection and their interesting interaction with SASI. This limitation also ignores the formation of buoyant entropy bubbles from shock oscillations driven by SASI (e.g. Fernandez et al. Reference Fernandez, Müller, Foglizzo and Janka2014), and the dominant role of buoyancy driven parasitic instabilities to saturate the amplitude of SASI (Guilet et al. Reference Guilet, Sato and Foglizzo2010). Even in the inviscid limit the dynamical system described by the shallow water equations can only describe the interaction of surface waves and vorticity perturbations, whereas the Euler equations describe the interaction of acoustic waves, entropy and vorticity perturbations. In a positive sense, the shallow water dynamical system offers an opportunity to assess the importance of buoyancy effects on the dynamics of the shock.

6.1.3 Inner boundary condition

The inner boundary condition in the experiment faces the same difficulty as the adiabatic simulation of Blondin & Mezzacappa (Reference Blondin and Mezzacappa2007): without any non-adiabatic process to shrink the volume of the accreted fluid on the surface of the neutron star, the fluid has to be continuously extracted through the inner boundary. This is done in the experiment by allowing water to spill over the upper edge of the inner cylinder (Figure 4). As long as the radius R _ns of the cylinder is large enough to allow the free fall of water inside it, this inner boundary condition can be translated into the condition that the radial component of the flow velocity |v _ns| reaches the local wave velocity c _w(R _ns) associated to the depth H _edge of water above the upper edge of the inner cylinder. Denoting by Q the flow rate,

(17) \begin{eqnarray} 2\pi R_{\rm ns} H_{\rm edge}|v_{\rm ns}| = Q, \end{eqnarray}

(18) \begin{eqnarray} v_{\rm ns}^2=gH_{\rm edge}. \end{eqnarray}

Figure 4.

Water is injected inward from a circular slit, in a uniform and stationary manner. A circular hydraulic jump is produced by the vertical surface of the cylinder at the centre. Water is evacuated by spilling over the upper edge of this cylinder.

6.1.4 SWASI, a shallow water analogue of SASI

In view of the similarity of the equations, the formalism of advected-acoustic cycles studied by Foglizzo (Reference Foglizzo2001, Reference Foglizzo2002, Reference Foglizzo2009) can be translated into a linear coupling process between surface gravity waves and vorticity perturbations, to produce a cycle between the hydraulic jump and the inner boundary. How would the absence of feedback from the advection of entropy perturbation affect the stability of this cycle in the shallow water analogue? The perturbative study and numerical simulations of the shallow water equations performed by Foglizzo et al. (Reference Foglizzo, Masset, Guilet and Durand2012) confirmed the existence of an instability very similar to SASI, named SWASI as a shallow water analogue of a shock instability. Despite the simplicity of the shallow water approximation, an excellent agreement was found between the oscillation period measured in the experiment and the one predicted by the shallow water equations.

6.2.1 Input parameters

The variable parameters of the experiment are the flow rate 0 < Q < 4Ls^{− 1}, the thickness of the injection slit 0.5 < H _inn ⩽ 2 mm, the height − 5.6cm < z _ns < 0 and diameter 8 ⩽ D _ns ⩽ 16 cm of the inner cylinder. The radius of injection is set to 33 cm. The minimum size of the injection slit is limited by its building accuracy. The diameter of the inner cylinder sets the maximum flow rate above which the inner boundary should be modelled as a drowned boundary rather than a critical point. A central spinner, made of a light object in free rotation along a central vertical axis, is in passive contact with the surface of the water in the vicinity of the inner cylinder. Its rotating motion helps visualise the angular momentum in the vicinity of the neutron star. In the new version described in Section 6.3.4, the rotation of the full experiment is motorised and can be adjusted electronically.

6.2.2 Hydraulic jumps versus shock waves

The hydraulic jump in the experiment is a reverse version of the familiar circular hydraulic jump observed in kitchen sinks. The water flowing inward faster than the speed of waves (Fr > 1) is abruptly decelerated to slower velocities (Fr < 1) into a thicker layer of water. Mass flux is exactly conserved across this transition, and momentum flux is approximately conserved due to the minor drag on the fountain surface. Jump conditions similar to the Rankine–Hugoniot conditions are deduced from these two conservation laws. The conservation of the energy flux is not expected since the radial flow of water is described by only two variables, v and H, without any equivalent of the entropy variable in a gas. The same is true for isothermal shocks in astrophysics, where the excess energy is usually assumed to be radiated away. In the hydraulic jump with a moderate Froude number, excess energy is viscously dissipated within the thickness of the jump by one or several horizontal stationary rollers. Other processes such as wave emission, the formation of bubbles or splashes can participate in the evacuation of the excess energy at a higher Froude number (e.g. Chanson Reference Chanson2009). The 2D formulation of the shallow water equations does not resolve the internal structure of the hydraulic jump and simply assumes that energy is instantly dissipated across an idealised discontinuity.

The description of hydraulic jump as a discontinuity is also unable to describe the limit of Froude numbers approaching unity 1 < Fr < 2, where the radial extent of the hydraulic jump increases with a succession of oscillations referred to as an undular hydraulic jump. According to the classification of Chow (Reference Chow1973), the range of Froude numbers most favourable to the simplest shallow water description of the experiment is 4 < Fr < 9.

6.2.3 Viscosity effects

The strongest effect of water viscosity takes place ahead of the hydraulic jump, in the shallowest region where the vertical shear is responsible for a significant viscous drag. In the experiment presented in Foglizzo et al. (Reference Foglizzo, Masset, Guilet and Durand2012), this viscous drag has been measured and modelled as a laminar drag in Equation (13):

(19) \begin{eqnarray} {\partial v\over \partial t}+(\nabla \times v)\times v+\nabla \left[{v^2\over 2}+c^2+\Phi \right]= \alpha \nu _{\rm w} {v\over H^2}, \end{eqnarray}

where ν_w ~ 0.01cm²s^{− 1} is the kinematic viscosity of water and α ~ 3 is a dimensionless number measured in the experiment.

Fortunately this viscous drag is very small in the deeper region after the hydraulic jump and does not significantly affect the dynamics of the interaction between surface waves and vorticity perturbations. In order to compensate for the viscous drag ahead of the hydraulic jump, the water can be injected from the outer boundary with a higher velocity than the local free fall velocity, so that the velocity immediately before the jump R _jp ~ 20 cm coincides with the local inviscid free fall velocity $v_{\rm ff}(R_{\rm jp})\equiv (2gH_\Phi (R_{\rm jp}))^{1/2}$.

Increasing the injection flow rate increases the development of turbulence in the fountain, with an expected transition for a Reynolds number of the order of Re ~ 2000. The viscosity of water seems too weak to affect the dynamics of SASI in the laminar regime, but a turbulent viscosity could significantly affect the efficiency of the cycle between surface waves and vorticity perturbations.

6.2.4 Uncertain vertical profile of velocity

The inviscid formulation of shallow water equations corresponds to an idealised situation where the velocity v is uniform in the vertical direction. The non-uniform vertical velocity profile viscously induced by the no-slip condition introduces a modification of the quadratic terms such as (∇ × v) × v and ∇v ²/2 in the shallow water equations, since the vertically averaged value of v ² is no longer equal to the square of the vertical average of v. Correction coefficients known as the Coriolis and Boussinesq coefficients could account for this effect if the vertical profile of velocity were known, for example in the idealised cases of a laminar flow or in fully developed turbulence.

6.2.5 A 2D view of the equatorial plane of the stellar core

The shallow water experiment is intrinsically limited to 2D since the radial attraction of the central neutron star is mimicked from a projection of the vertical gravity of the laboratory onto the inclined surface of the fountain. The shallow water equations could, however, be solved in 3D if necessary, for example to assess the importance of buoyancy by comparison with 3D simulations of SASI.

6.2.6 From shallow water equations to the experiment and back

From the point of view of understanding the essence of the physics of SASI using analogies, the idealised set of inviscid shallow water equations is the simplest formulation with the most direct astrophysical connections. These equations can be studied analytically and solved numerically without using the experimental fountain. The water experiment is more complex because of viscosity effects which are not trivially captured in models and have no direct astrophysical interpretation such as the development of a boundary layer, the viscous drag, Coriolis and Boussinesq coefficients, the radial extension of the hydraulic jump and its connection to bubble entrainment, splashes or unsteady behaviour unrelated to SASI. A careful interpretation of experimental results is needed to identify the results which are governed by the simplest shallow water equations. The experiment is complementary to simulations because its limitations are different from the numerical ones, such as the difficulty of first order numerical convergence due to the presence of a shock (Sato et al. Reference Sato, Foglizzo and Fromang2009). This difficulty is inherent to any shock capturing method. The simplicity of use of the experiment makes it very convenient to rapidly explore a large parameter space and identify new phenomena. From an astrophysical perspective, the immediate next step consists in plugging the experimental parameters of this new phenomenon in a numerical simulation of the simplest shallow water equations with a viscous drag. If the phenomenon is confirmed numerically, its sensitivity to the viscosity is tested by performing new shallow water simulations in the inviscid limit. The next step could be a 2D equatorial simulation of adiabatic gas dynamics to incorporate buoyancy effects, and a series of simulations to successively incorporate a hard inner boundary with a cooling function, neutrino-driven convection with a heating function, a third dimension, an equation of state for nuclear matter, neutrino transport and interactions, relativistic corrections, a progenitor profile for the initial conditions and an inner boundary condition which allows the contraction of the proto-neutron star. It is important to bear in mind the extreme simplicity of the shallow water setup compared to the most advanced simulations such as Hanke et al. (Reference Hanke, Müller, Wongwathanarat, Marek and Janka2013). It is equally important to bear in mind that even the simplest setup like the shallow water formulation contains interesting hydrodynamical puzzles and surprises.

6.3.1 SASI is not familiar yet

Part of the apparent complexity in the physical problem of core collapse comes from hydrodynamical instabilities such as neutrino-driven convection and SASI, which break the spherical symmetry. Most physicists are familiar with the convective instability, even though its interaction with a shock wave and an advection flow is far from trivial. By contrast, SASI is much less familiar to astrophysicists. Its advective-acoustic mechanism relates to aeroacoustic instabilities in ramjets (Abouseif, Keklak, & Toon Reference Abouseif, Keklak and Toon1984), rocket motors (Mettenleiter, Haile, & Candel Reference Mettenleiter, Haile and Candel2000), and can be traced back to the whistling of a kettle (Chanaud & Powel Reference Chanaud and Powell1965). In astrophysics the instability of Bondi–Hoyle–Lyttleton accretion also relates to this category of unstable advective-acoustic cycles (Foglizzo et al. Reference Foglizzo, Galletti and Ruffert2005), but the conical geometry of the shock make it less intuitive than SASI and this problem has been much less studied than stellar core collapse. Characterising the fundamental properties of SASI in their simplest formulation makes it possible to enlighten more complex simulations (e.g. Blondin & Mezzacappa Reference Blondin and Mezzacappa2007).

6.3.2 The linear coupling between vorticity perturbations and waves

The translation of gas dynamics into shallow water dynamics is in many respects a fruitful exercise, which can produce physical pictures sometimes more intuitive with water than with a gas although they are absolutely equivalent in the end. Let us describe the coupling between vorticity waves and surface gravity waves in an inhomogeneous flow. A vortical perturbation can be advected with the flow at uniform velocity without ever affecting the pressure equilibrium. Let us imagine that this perturbation is slowly advected in a direction where the depth of the shallow layer of fluid increases, as illustrated in Figure 5. Because we have an intuitive representation that the surface of slow water (i.e. Fr ≪ 1) remains horizontal even when the depth increases abruptly (e.g. $\partial H_\Phi /\partial r \gg 1$), we are able to guess that the vortical motion must perturb this flat surface by transporting some water from the deep regions to shallower regions and conversely. We also have an intuitive understanding that the perturbed water surface will return to horizontal through the propagation of surface waves. The same intuitive reasoning can be translated into an isentropic or isothermal gas decelerated by some external potential. A similar approach was used by Foglizzo & Tagger (Reference Foglizzo and Tagger2000) to explain the feedback from the advection of an entropy perturbation across the adiabatic compression produced by an external potential. It should be noted that these two feedback processes are fundamentally different but since entropy and vorticity perturbations are advected with the same velocity, the feedback they produce adds up in a coherent advective-acoustic cycle studied analytically by Foglizzo (Reference Foglizzo2009) and numerically by Sato et al. (Reference Sato, Foglizzo and Fromang2009). If a magnetic field were present, vorticity and entropy perturbations would no longer propagate at the same velocity and the advective-acoustic cycle would be transformed into five distinct MHD cycles (Guilet & Foglizzo Reference Guilet and Foglizzo2010).

Figure 5.

The coupling between vorticity perturbations and surface gravity waves in shallow water can help understand the coupling between vorticity perturbations and acoustic waves in a gas. The vorticity perturbation is shown in green at three successive positions in the upper illustration, viewed from above. The change in water elevation ± δH produced by the vortical motion over the gradient of depth is viewed horizontally in the lower illustration. It is a source of surface gravity waves.

6.3.3 Open questions accessible to the SWASI fountain

The SWASI fountain is currently being used as a complementary tool to numerical simulations to address the following questions:

1. (i) Comparing the growth rate of SWASI in the experiment to the growth rate deduced from the shallow water equations with various prescriptions for the viscous drag and the turbulent viscosity can teach us about the sensitivity of the SWASI mechanism to experimental limitations.

2. (ii) Understanding the interaction of SWASI with the turbulence produced by the vertical shear may shed light on the physics of SASI and its interaction with the turbulence induced by parasitic instabilities or by neutrino-driven convection.

3. (iii) Measuring the experimental conditions for the symmetry breaking between the clockwise and anti-clockwise modes of SWASI can help us understand the nature of this non-linear process.

4. (iv) Understanding the saturation amplitude of SASI is essential to predict which type of progenitors could lead to the strongest SASI oscillations. Is the growth of parasitic instabilities proposed by Guilet et al. (Reference Guilet, Sato and Foglizzo2010) always the dominant saturation mechanism? Testing the saturation mechanism of SWASI in the shallow water setup may be simpler than SASI in a gas because of the action of a single parasitic instability (Kelvin–Helmholtz) associated with the fragmentation of vorticity waves.

6.3.4 A new experiment including the angular momentum of the stellar core

A new experiment has been built at CEA Saclay with similar dimensions as the one published by Foglizzo et al. (Reference Foglizzo, Masset, Guilet and Durand2012). The main difference is the capability to motorise the rotation of the full experiment so that water can be injected with a non-zero angular momentum. This device should be able to experimentally address the effect of rotation on the growth of the prograde spiral mode (Yamasaki & Foglizzo Reference Yamasaki and Foglizzo2008) and its saturation amplitude. Based on the adverse rotation of the central spinner observed by Foglizzo et al. (Reference Foglizzo, Masset, Guilet and Durand2012) in a flow without angular momentum, one can expect to experimentally demonstrate the competition between the angular momentum of the background flow and the adverse angular momentum in the region close to the neutron star (Blondin & Mezzacappa Reference Blondin and Mezzacappa2007).

6.4.1 From bouncing balls to fluid mechanics

As an introduction to the fluid experiment, the public was invited to comment on a side experiment were two bouncing balls fall vertically, both guided by a wire attached between the ceiling and the ground. The diameter of the lower bouncing ball is bigger ( ~ 8 cm) than the upper one ( ~ 5 cm). Used with a single ball, this experiment first illustrates the question of elastic versus inelastic bounce. When the two balls are dropped simultaneously the spectacular transmission of energy and momentum from the lower ball to the upper one is always a surprise to the public witnessing the ejection of the upper ball to the ceiling. This experiment is a good introduction to the simplest and naive solution to the problem of core-collapse. The SWASI experiment takes the public two steps further, both in the direction of fluid mechanics and with the additional degree of freedom allowed by transverse motions. The radial flow of water in the experiment is presented as a view of the dynamics in the equatorial plane of the stellar core.

6.4.2 Familiarity with water and kitchen sink hydraulic jumps

One key aspect of the public success of the SWASI fountain is the simplicity of the setup, and the familiarity of the public with water. The main action on the fountain was a change in flow rate from 0 to 1 L s^{− 1} which showed the radial injection of water, and the formation of a circular hydraulic jump at a radius of ~ 20 cm. The comparison with the circular hydraulic jump visible in every kitchen sink helps illustrate the analogy with shocks, and identify the subsonic and supersonic parts of the flow. The changes in composition are also introduced during this phase of stationary accretion: while the supersonic gas is made of iron nuclei, the post-shock gas is dissociated into protons, neutrons and electrons, and the neutron star is essentially made of neutrons with neutrinos diffusing out. It is important to introduce the neutrinos as a key actor of the supernova mechanism early enough. This is one of the difficulties of the presentations since this particle is generally unknown to the public, and absent from the analogous experiment. The post-shock gas is described as dense enough to capture a fraction of the outgoing neutrino flux, but insufficiently to revive the shock when distributed spherically. Unlike the stable jump in a kitchen sink, the hydraulic jump in the supernova fountain moves in an irresistible manner. Its motion is as inevitable as the fall of a pen in vertical equilibrium on a table. Like the falling pen the initial symmetry of the equilibrium is broken in a random direction. The breaking of the circular symmetry appears with the development of m = 1 sloshing oscillations along a random direction with a 3 s period. As the oscillation of the hydraulic jump reaches larger amplitude, a second symmetry breaking leads to the rotation of the spiral wave in a random direction, clockwise, or anti-clockwise. The central spinner raises the curiosity of the public by rotating erratically in the direction opposite to the hydraulic jump.

6.4.3 Capturing astrophysical processes accessible on a human scale

The notions of time-scale and length-scale are introduced with the fountain, applying the factor one million to scale down the neutron star radius, the shock radius, the size of the iron core and the size of a supergiant. The factor ~ hundred in timescale is applied to speed up the observed dynamics of the hydraulic jump. As the neutron star is identified with the surface of the inner cylinder, the extreme properties of these compact stars are stressed: nuclear densities, extreme spin rates, mysterious velocities up to 1 000 km s^{− 1}. The scaling factors enable the public to capture representations of extreme astrophysical processes on a human scale.

6.4.4 Public access to the physics of explosion threshold, kick, and spin

The connection between the dynamics of the fountain and stellar explosions is explained along three observations:

1. (i) the explosion threshold is understood by observing the radial extent of the post-shock material which can intercept neutrinos in the direction of largest shock displacement. Neutrino absorption will be enhanced in this direction compared to the spherically symmetric case.

2. (ii) the spectacular pulsar kicks up to 1 000 km s^{− 1} can be reached if the explosion takes place in an asymmetric manner: this is a consequence of the conservation of linear momentum, which can easily be experienced by feeling the recoil when throwing a heavy object

3. (iii) the significant spin up of the neutron star is illustrated by the motion of the central spinner. The conservation of angular momentum is invoked to accept that the spinner and the hydraulic jump rotate in opposite directions, noting that the injected flow is purely radial.