[Paper Review] The distribution of shock waves in driven supersonic turbulence

M. D. Smith, Mordecai‐Mark Mac Low+1 more|arXiv (Cornell University)|Aug 8, 2000

Solar and Space Plasma Dynamics2 references3 citations

TL;DR

This paper investigates shock wave distributions in three-dimensional, uniformly driven supersonic turbulence using numerical simulations. It finds a power-law distribution of fast shocks with number inversely proportional to the square root of shock jump speed, contrasting with the exponential decay seen in decaying turbulence, and attributes this to nonlinear mapping of velocity fluctuations. The results reveal that energy dissipation is dominated by a small number of strong shocks, while magnetic fields enhance transverse shocks and self-gravity generates rare, highly dissipative accretion shocks.

ABSTRACT

Supersonic turbulence generates distributions of shock waves. Here, we analyse the shock waves in three-dimensional numerical simulations of uniformly driven supersonic turbulence, with and without magnetohydrodynamics and self-gravity. We can identify the nature of the turbulence by measuring the distribution of the shock strengths. We find that uniformly driven turbulence possesses a power law distribution of fast shocks with the number of shocks inversely proportional to the square root of the shock jump speed. A tail of high speed shocks steeper than Gaussian results from the random superposition of driving waves which decay rapidly. The energy is dissipated by a small range of fast shocks. These results contrast with the exponential distribution and slow shock dissipation associated with decaying turbulence. A strong magnetic field enhances the shock number transverse to the field direction at the expense of parallel shocks. A simulation with self-gravity demonstrates the development of a number of highly dissipative accretion shocks. Finally, we examine the dynamics to demonstrate how the power-law behaviour arises.

Motivation & Objective

• To link the nature of driven turbulence to the statistical distribution of shock waves in astrophysical environments.
• To determine how shock strength distributions differ between driven and decaying turbulence, particularly in energy dissipation mechanisms.
• To examine the effects of magnetic fields and self-gravity on shock number, strength, and spatial distribution.
• To provide a theoretical and numerical basis for interpreting observational spectroscopic features in molecular clouds.
• To establish a shock probability distribution function (PDF) for use in predicting observable spectral signatures.

Proposed method

• Numerical simulations using the ZEUS finite-difference hydrocode with periodic boundary conditions and uniform driving at Mach number M = 5.
• Shock detection via artificial viscosity dissipation in grid zones, with shock jump speed calculated from velocity gradients across dissipation regions.
• Shock number distribution function (PDF) derived by counting shocks per unit jump speed interval, with analytical fits to observed power-law behavior.
• Inclusion of magnetohydrodynamics (MHD) with Alfvén Mach numbers A = 5 and A = 1 to assess magnetic field effects on shock orientation and strength.
• Self-gravity included in a large-scale simulation to examine accretion shock formation and energy dissipation.
• Dynamical interpretation based on mapping closure theory, relating initial Gaussian velocity distributions to final shock speed PDF via v_j ∝ v² scaling.

Experimental results

Research questions

• RQ1 What is the statistical distribution of shock jump speeds in uniformly driven supersonic turbulence?
• RQ2 How does the shock distribution in driven turbulence differ from that in decaying turbulence, particularly in terms of energy dissipation?
• RQ3 How do magnetic fields alter the spatial distribution and number of shocks, especially in relation to field orientation?
• RQ4 What role do accretion shocks play in energy dissipation in self-gravitating turbulent systems?
• RQ5 What physical mechanism explains the observed power-law dependence of shock number on shock jump speed?

Key findings

• The number of fast shocks follows a power-law distribution with N ∝ v_j^(-1/2), indicating an 'inertial range' of shock strengths in driven turbulence.
• Energy dissipation is dominated by a small number of strong shocks, with 66.9% of injected energy radiated through shocks, consistent with simulation results.
• The high-velocity tail of the shock distribution is steeper than Gaussian due to rapid decay of high-amplitude driving waves.
• Magnetic fields enhance shock numbers transverse to the field direction while suppressing parallel shocks, though the overall shape of the PDF remains unchanged.
• Self-gravity leads to the formation of rare, highly dissipative accretion shocks that are not prominent in the number distribution but dominate energy loss in collapsing regions.
• The power-law behavior arises from nonlinear mapping of initial velocity fluctuations, where v_j ∝ v², leading to a flat distribution of absolute shock speeds and resulting in the observed v_j^(-1/2) dependence.