Introduction
“What I cannot create, I do not understand” was once found written on Richard Feynman's blackboard. Albert Einstein stated that “You do not really understand something unless you can explain it to your grandmother.” The point made by these two great minds was that understanding something in physics means you come to the point when it seems obvious and you no longer need the equations to derive the result. Among the physical ideas that have been examined over the years, the stability/instability concept has been extremely fruitful, although its understanding in a given setting would not always win Feynman's or Einstein's approval. The stability of a ball inside a bowl is very intuitive, and its oscillation frequency when removed from its equilibrium position can be calculated just by looking at a sketch of the system. The instability of a pencil on its tip is equally obvious and frequently cited when introducing the concept of an unstable system. In plasma physics, Fried (Reference Fried1959) could analyze the filamentation instability of two counter-propagating particle beams from the very understanding of the physical mechanism at work. Yet, a similar intuitive derivation for the well-known two-stream instability is still lacking.
The Rayleigh-Taylor instability (RTI) plays a key role in many fields of physics, and its behavior in connection with inertial confinement fusion (ICF) (Pomraning, Reference Pomraning1990, Kawata et al., Reference Kawata, Sato, Teramoto, Bandoh, Masubichi and Takahashi1993), z-pinch physics (Douglas et al., Reference Douglas, De Groot and Spielman2001), or metallic hydrogen generation experiments (Piriz et al., Reference Piriz, Cortázar, López Cela and Tahir2006, Lopez Cela et al., Reference Lopez Cela, Piriz, Serna Moreno and Tahir2006), has been the topic of many recent works. The RTI occurs when a heavy fluids is accelerated against a light fluid (see Fig. 1). In ICF, a spherical Deuterium-Tritium target is compressed by a Laser. The laser ablates the target, creating a low density ablating plasma outside the pellet. During the early phase of the compression, the interface between the compressed target (the heavy fluid) and the low-density ablating plasma (the light fluid) accelerates (Priz et al., Reference Piriz, Sanz and Ibañez1997). An observer “sitting” on the interface would then feel a force pushing him from the heavy fluid to the light one, resulting in the RT unstable configuration pictured in Figure 1. In astrophysics, the RTI is frequently invoked to explain the filamentary structure of the Crab nebula for example (Hester et al., Reference Hester, Stone, Scowen, Jun, Gallagher, Norman, Ballester, Burrows, Casertano, Clarke, Crisp, Griffiths, Hoessel, Holtzman, Krist, Mould, Sankrit, Stapelfeldt, Trauger, Watson and Westphal1996). As the supernova remnant (the dense fluid) decelerates through the interstellar medium (the light fluid), the interface between both is again RT unstable as it experiences an acceleration from the heavy to the light medium.
Fig. 1. (color online) The Rayleigh-Taylor instability. At equilibrium, the pressure on both side of the interface is p 0. The pressure variation when moving the interface shows the perturbation is unstable if ρ 1 > ρ 2.
A Feynman/Einstein like heuristic approach to the RTI (Rayleigh, Reference Rayleigh1900, Taylor, Reference Taylor1950) was recently provided by Piriz et al. (Reference Piriz, Lopez Cela, Serena Moreno, Tahir and Hoffmann2006) for an incompressible fluid. In the usual, normal modes approach, where the fluid equations for both fluids are linearized, the linear growth rate is derived but the basic mechanisms at works remain hidden behind the equations (Chandrasekhar, Reference Chandrasekhar1961). In contrast, Piriz et al. proposed a direct derivation of the linear growth rate from the very description of the physics involved, “short-cutting” much of the equations.
Non-relativistic approach
Suppose the interface represented in Figure 1 is initially in equilibrium, both incompressible fluids exerting a pressure p 0 upon it. The system is accelerated downward with an intensity g m/s2. The interface is then displaced by z along a distance ~1/k, where k mimics here the wave-number introduced in the normal modes approach. The pressure of the upper-fluid at the interface increases by an amount ρ 1gz, while the pressure of the lower fluid also increases, but by a quantity ρ 2gz. The upper fluid now pushes the interface downward with the pressure p 0 + ρ 1gz, and the lower fluid pushes upward with the pressure p 0 + ρ 2gz. It is obvious that if p 0 + ρ 1gz > p 0 + ρ 2gz, i.e., ρ 1>ρ 2, the pressure balance amplifies the perturbation. Note that according to a similar analysis, moving the interface upward equally triggers an instability.
The calculation of the linear growth rate is straightforward from this stage. Let us consider a transverse direction, say y, to Figure 1 so as to account for the three-dimensional nature of the system. The surface of the interface over a depth D along the y axis is S ~ D/k. The force acting upon it thus reads,
As it moves, the interface also moves a layer of fluid on both sides. The volume of fluid displaced over the surface S should be proportional to S itself, and the height of the layer moved on both side is proportional to 1/k (which is why the normal mode calculation assumes the fluid thickness is much larger than 1/k). We can thus write the expression of the mass involved in the displacement,
Since this amount of fluid is displaced by a distance z, we can write Newton's law F = M a from Eqs. (1) and (2) as,
where,
and a proportionality coefficient equal to unity has been assumed for the mass in Eq. (2). Eq. (3) has exponentially growing solutions if δ 2 > 0, and δ is exactly the linear growth rate of the RTI (Rayleigh, Reference Rayleigh1900) where the Atwood number A t = (ρ 1 − ρ 2)/(ρ 1 + ρ 2) is immediately identified. It is clear now that the sum of the densities relates to the total amount of fluid involved in the motion, while the difference relates to the pressure shift generated by the displacement. Note that a mass factor different from unity in Eq. (2) would yield a slightly different denominator for the linear growth rate. Indeed, such intuitive calculations frequently yield the correct scalings with some pre-factor close to unity. In the present case, the result is exact.
Relativistic version
The relativistic version of the RTI is especially relevant to supernova and Gamma-Ray-Bursts physics (Waxman & Piran, Reference Waxman and Piran1994, Levinsona, Reference Levinsona2010), where ultra-relativistic inhomogeneous flows are involved. Adapting the normal modes method to such settings, Allen and Hugues (Reference Allen and Hughes1984) found the relativistic counterpart of Eq. (4),
Let us now analyze the problem from the intuitive standpoint explained above. Starting from Eq. (4), where exactly shall we have to introduce relativistic expressions? The displacement itself is not relativistic. The interface corrugation is a still, initial condition. The only relativistic modification will have to do with the inertia of the fluid displaced. A volume of fluid dV has the mass ρ dV in the non-relativistic limit. If the particles it is made of have relativistic motion, for example, a temperature T such as k BT ~ mc 2, the energy pdV adds up to the mass within an amount ∝ pdV/c 2. Relativistic fluid theory shows indeed that the correct factor is 3 so that the relativistic mass density is ρ + 3p/c 2 (Landau & Lifschitz, Reference Landau and Lifschitz1987b). The density term in Eq. (2) needs therefore to account for this extra inertia. While the interface has not been displaced, the pressure is the same on both side and the correction per unit of volume reads 3p 0/c 2 for both fluids. Updating the pressure here for the corrugated interface would introduce a second order term in z, which is neglected in the present linear regime. The relativistic counterpart of Eq. (2) is thus readily obtained replacing ρ 1 + ρ 2 by ρ 1 + 3p 0 /c 2 + ρ 2 + 3p 0/c 2, and the new linear growth rate is,
The calculation starting from the linearized relativistic fluid equations (Allen & Hughes, Reference Allen and Hughes1984) yields the same expression with a term 8p 0/c 2 instead of 6p 0/c 2.
Discussion
Eq. (6) thus give, up to a numerical factor, the correct value of the RTI linear growth rate. We now discuss the discrepancy between the factors 6 and 8. To do so, we can start from the relativistic Euler equation in the absence of gravitational field (Landau & Lifschitz, Reference Landau and Lifschitz1987a),
where p is the pressure, ɛ is the energy density, x i = (ct, r), and u i = (γ, γv/c). As previously said, our problem is relativistic in the sense that the energy density can be so, not by virtue of some relativistic velocity of the fluid elements. Setting thus γ = 1, one can check that the temporal component (i = 0) of the equation above yields v · ∇p = 0. The spatial part (i = 1, 2, 3) then gives
Neglecting v · ∇v as a second order quantity and adding the acceleration gives the premise of Allen & Hugues' Eq. (3),
Eqs. (7) and (9) show that that the correct relativistic inertia is not simply the energy density ɛ, but the energy density plus the pressure, ɛ + p. Setting then ɛ = ρ c 2 + 3p gives p + ɛ = ρ c 2 + 4p for both fluids, from which the factor 8 eventually arises.
ACKNOWLEDGEMENTS
The author acknowledges the financial support y Projects ENE2009-09276 of the Spanish Ministerio de Educación y Ciencia and PAI08-0182-3162 of the Consejería de Educación y Ciencia de la Junta de Comunidades de Castilla-La Mancha. Thanks are due to Roberto Piriz for fruitful discussions.
