Overall system energyEquation (1) Below follows a description of the partial energies introduced in equation (1). Bond Order and Bond EnergyA fundamental assumption of ReaxFF is that the bond order BO' ij between a pair of atoms can be obtained directly from the interatomic distance r ij as given in Equation (2). In calculating the bond orders, ReaxFF distinguishes between contributions from sigma bonds, pi-bonds and double pi bonds.

The nonlinear interaction between counterpropagating Alfvén waves is the physical mechanism underlying the cascade of energy to small scales in astrophysical plasma turbulence. Beginning with the equations for incompressible MHD, an asymptotic analytical solution for the nonlinear evolution of these Alfvén wave collisions is derived in the weakly nonlinear limit. The resulting qualitative picture of nonlinear energy transfer due to this mechanism involves two steps: first, the primary counterpropagating Alfvén waves interact to generate an inherently nonlinear, purely magnetic secondary fluctuation with no parallel variation; second, the two primary waves each interact with this secondary fluctuation to transfer energy secularly to two tertiary Alfvén waves. These tertiary modes are linear Alfvén waves with the same parallel wavenumber as the primary waves, indicating the lack of a parallel cascade. The amplitude of these tertiary modes increases linearly with time due to the coherent nature of the resonant four-wave interaction responsible for the nonlinear energy transfer. The implications of this analytical solution for turbulence in astrophysical plasmas is discussed. The solution presented here provides valuable intuition about the nonlinear interactions underlying magnetized plasma turbulence, in support of an experimental program to verify in the laboratory the nature of this fundamental building block of astrophysical plasma turbulence.a)

Turbulence is a ubiquitous phenomenon in space and astrophysical plasmas, driving a cascade of energy from large to small scales and strongly influencing the plasma heating resulting from the dissipation of the turbulence. Modern theories of plasma turbulence are based on the fundamental concept that the turbulent cascade of energy is caused by the nonlinear interaction between counterpropagating Alfvén waves, yet this interaction has never been observationally or experimentally verified. We present here the first experimental measurement in a laboratory plasma of the nonlinear interaction between counterpropagating Alfvén waves, the fundamental building block of astrophysical plasma turbulence. This measurement establishes a firm basis for the application of theoretical ideas developed in idealized models to turbulence in realistic space and astrophysical plasma systems.Introduction.-Turbulence profoundly affects many space and astrophysical plasma environments, playing a crucial role in the heating of the solar corona and acceleration of the solar wind [1], the dynamics of the interstellar medium [2][3][4], the regulation of star formation [5], the transport of heat in galaxy clusters [6], and the transport of mass and energy into the Earth's magnetosphere [7]. At the large length scales and low frequencies characteristic of the turbulence in these systems, the turbulent motions are governed by the physics of Alfvén waves [8], traveling disturbances of the plasma and magnetic field. Theories of Alfvénic turbulence based on idealized models, such as incompressible magnetohydrodynamics (MHD), suggest that the turbulent cascade of energy from large to small scales is driven by the nonlinear interaction between counterpropagating Alfvén waves [9][10][11][12]. However, the applicability of this key concept in the moderately to weakly collisional conditions relevant to astrophysical plasmas has not previously been observationally or experimentally demonstrated. Verification is important because the distinction between the two leading theories for strong MHD turbulence [11,12] arises from the detailed nature of this nonlinear interaction. Furthermore, verification is required to establish the applicability of turbulence theories, utilizing simplified fluid models such as incompressible MHD, to the weakly collisional conditions of diffuse astrophysical plasmas.

This paper presents the numerical verification of an asymptotic analytical solution for the nonlinear interaction between counterpropagating Alfvén waves, the fundamental building block of astrophysical plasma turbulence. The analytical solution, derived in the weak turbulence limit using the equations of incompressible MHD, is compared to a nonlinear gyrokinetic simulation of an Alfvén wave collision. The agreement between these methods signifies that the incompressible solution satisfactorily describes the essential dynamics of the nonlinear energy transfer, even under the weakly collisional plasma conditions relevant to many astrophysical environments.

Turbulence in space and astrophysical plasmas is governed by the nonlinear interactions between counterpropagating Alfvén waves. Here we present the theoretical considerations behind the design of the first laboratory measurement of an Alfvén wave collision, the fundamental interaction underlying Alfvénic turbulence. By interacting a relatively large-amplitude, low-frequency Alfvén wave with a counterpropagating, smaller-amplitude, higher-frequency Alfvén wave, the experiment accomplishes the secular nonlinear transfer of energy to a propagating daughter Alfvén wave. The predicted properties of the nonlinearly generated daughter Alfvén wave are outlined, providing a suite of tests that can be used to confirm the successful measurement of the nonlinear interaction between counterpropagating Alfvén waves in the laboratory.