Building a weak shockwave from linear modes

TL;DR

This paper demonstrates that weak shockwaves in fluids can be constructed from linear acoustic modes, revealing insights into their propagation, conservation properties, and limitations in collisionless shocks.

Contribution

It introduces a novel approach to modeling weak shocks as a sum of linear modes, contrasting with traditional nonlinear shock theory.

Findings

Weak shock density profiles can be expressed as sums of linear acoustic modes.

The constructed shock propagates at the sound speed with exact matter conservation.

Collisionless shocks cannot be decomposed into linear modes due to dispersive effects.

Abstract

In shockwave theory, the density, velocity and pressure jumps are derived from the conservation equations. Here, we address the physics of a weak shock the other way around. We first show that the density profile of a weak shockwave in a fluid can be expressed as a sum of linear acoustic modes. The shock so built propagates at the speed of sound and matter is exactly conserved at the front crossing. Yet, momentum and energy are only conserved up to order 0 in powers of the shock amplitude. The density, velocity and pressure jumps are similar to those of a fluid shock, and an equivalent Mach number can be defined. A similar process is possible in magnetohydrodynamic. Yet, such a decomposition is found impossible for collisionless shocks due to the dispersive nature of ion acoustic waves. Weakly nonlinear corrections to their frequency do not solve the problem. Weak collisionless shocks could be inherently nonlinear, non-amenable to any linear superposition. Or they could be nonexistent, as hinted by recent works.