VOLUME 88, NUMBER 24 PHYSICAL REVIEW LETTERS 17 JUNE 2002 Asymptotic Scaling Laws for Imploding Thin Fluid Shells M.M. Basko* and J. Meyer-ter-Vehn Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany (Received 5 February 2002; published 30 May 2002) Scaling laws governing implosions of thin shells in converging flows are established by analyzing the implosion trajectories in the (A, M) parametric plane, where A is the in-flight aspect ratio, and M is the implosion Mach number. Three asymptotic branches, corresponding to three implosion phases, are identi- fied for each trajectory in the limit of A, M ¿ 1. It is shown that there exists a critical value g cr  1 1 2n (n  1, 2 for, respectively, cylindrical and spherical flows) of theadiabatic index g, which separates two qualitatively different patterns of the density buildup in the last phase of implosion. The scaling of the stagnation density r s and pressure P s with the peak value M 0 of the Mach number is obtained. DOI: 10.1103/PhysRevLett.88.244502 PACS numbers: 47.40. –x, 52.57.Bc Compression of matter to very high densities is an in- teresting fundamental problem in itself, and a crucial issue in certain applications as, for example, inertial confine- ment fusion (ICF) [1]. In practical terms, it is usually a problem of reaching as high as possible matter density by applying a given limited external pressure p # P 0 . When applied quasistatically, a given pressure P 0 can produce only a limited maximum density r 0 , which corresponds to a practical minimum of the specific entropy in the com- pressed sample. The way to unlimited (at least in principle) compression of matter is opened by symmetrical implosions of thin fluid shells in a converging (cylindrical or spherical) geometry. So long as the hydrodynamic instabilities and drive asym- metries are kept under control, arbitrarily high matter den- sities (although for a brief instant only) can be reached behind the return shock after a sufficiently thin shell, ac- celerated by a fixed external pressure P 0 , converges to the center of symmetry. Here we address a pure problem of ideal one-dimensional (1D) hydrodynamics, namely, we establish how pressure and density (and the corresponding specific entropy) at stagnation scale with the shell in-flight aspect ratio A and its Mach number M in the asymptotic limit of A, M ¿ 1. Knowing this scaling allows one to answer such questions as, for example, how much thinner a shell should be taken in order to achieve a desired en- hancement in the degree of compression. The reason why this basic problem has not been solved before lies in the difficulty of describing nonlinear con- verging flows and, in particular, of determining the en- tropy generated in the stagnating fluid behind the shock rebounding from the center. Kemp et al. [2] made use of a self-similar solution for a spherical shell imploding with a Mach number M 0 [3] and obtained an approximate scal- ing P s P 0 ~ M 3.0 0 (g  53) for an intermediate range of Mach numbers 2 & M 0 & 20. However, it remained un- clear what happens in the limit of M 0 ! `, and how this self-similar solution could be related to properties of the accelerating pressure pulse. Here we approach the prob- lem from a different side, not requiring self-similarity but restricting ourselves to asymptotic values of A, M ¿ 1. Analytically derived scaling laws are validated by numeri- cal simulations. Once a specific implosion strategy is chosen and fixed, the entire multitude of all possible states of imploding shells may be considered as a five-parameter family of snapshot spatial profiles for the density rt , r , velocity ut , r , and pressure pt , r . We choose the peak density D  Dt   max r rt , r  rt , R, the corresponding pressure Pt   pt , R, the implosion velocity Ut   2ut , R, the shell radius Rt , and its effective thickness h  ht   1 D Z ` 0 rt , r  dr (1) to be such parameters. Optimal implosion strategy for the highest degree of final compression, as described in detail in Ref. [4], is accomplished by (i) setting a cold shell in motion by a carefully tailored pressure pulse, which generates a mini- mum amount (if any) of entropy, followed by (ii) the phase of adiabatic acceleration to a maximum implosion veloc- ity by the peak boundary pressure P 0 . Here we omit the first phase — which is absolutely insignificant for the scal- ing laws in question — and start with an isentropic density profile across a motionless shell, r0, r   D 0 μ r 2 r 0 R 0 2 r 0 ∂ 1g21 , (2) which corresponds to a uniform acceleration of all fluid elements (a UA, i.e., uniformly accelerated, profile) by a fixed pressure P 0 at the outer boundary; g is the adiabatic index. The interior of the shell at 0 , r , r 0 is void. The initial effective shell thickness is h 0  1 2g 21 R 0 2 r 0 . Under such drive conditions no shocks pass through the shell until the very moment of void closure, and the entropy parameter a  Pr g  a 0  P 0 D g 0 remains constant in space and time. From the five-dimensional parameters P, D, U, R, and h, which define the state of a shell in flight, only two dimensionless combinations can be constructed, namely, 244502-1 0031-9007 02 88(24) 244502(4)$20.00 © 2002 The American Physical Society 244502-1