Kinetic interpretation of the classical Rayleigh-Taylor instability

Rayleigh-Taylor (RT) instabilities are prevalent in many physical regimes ranging from astrophysical to laboratory plasmas and have primarily been studied using fluid models, the majority of which have been ideal fluid models. This work presents a five-dimensional (two spatial dimensions, three velocity space dimensions) simulation using the continuum-kinetic model to study the effect of the collisional mean free path and transport on the instability growth. The continuum-kinetic model provides noise-free access to the full particle distribution function permitting a detailed investigation of the role of kinetic physics in hydrodynamic phenomena such as the RT instability. For long mean free path, there is no RT instability growth, but as collisionality increases, particles relax towards the Maxwellian velocity distribution, and the kinetic simulations reproduce the fluid simulation results. An important and novel contribution of this work is in the intermediate collisional cases that are not accessible with traditional fluid models and require kinetic modeling. Simulations of intermediate collisional cases show that the RT instability evolution is significantly altered compared to the highly collisional fluidlike cases. Specifically, the growth rate of the intermediate collisionality RT instability is lower than the high collisionality case while also producing a significantly more diffused interface. The higher moments of the distribution function play a more significant role relative to inertial terms for intermediate collisionality during the evolution of the RT instability interface. Particle energy flux is calculated from moments of the distribution and shows that transport is significantly altered in the intermediate collisional case and deviates much more so from the high collisionality limit of the fluid regime.

Figure 1

Initial conditions in number density (left), bulk velocity (center), and square of thermal velocity (right).

Figure 2

Time evolution of number density for varying collisionality. Left to right is $\mathit{\text{Kn}}$ of 0.1 (a), 0.01 (b), and 0.001 (c). Top to bottom is time 0.0, $1.5{\tau }_{\text{RT}}$, and $3.0{\tau }_{\text{RT}}$. Note that the low collisionality case (left column) presents no RT instability growth, and the intermediate collisionality case (middle column) presents significantly altered RT instability growth compared to the high collisionality case (right column) which approaches the fluid limit.

Figure 3

Time evolution of temperature for varying collisionality. Left to right is $\mathit{\text{Kn}}$ of 0.1 (a), 0.01 (b), and 0.001 (c). Top to bottom is time 0.0, $1.5{\tau }_{\text{RT}}$ and $3.0{\tau }_{\text{RT}}$.

Figure 4

Logarithm of $h$, the difference between spike and bubble heights, as a function of time for $\mathit{\text{Kn}}=0.01\text{and}0.001$ and a fluid simulation using the Euler equations. Data points early in time are excluded from the fit due to dynamic diffusion of the interface for the kinetic cases and wave launching for the fluid case. Note as $\mathit{\text{Kn}}$ decreases, the RT instability growth rate approaches the fluid simulation result.

Figure 5

Density of nonideal distribution, according to Eq. (9), for $\mathit{\text{Kn}}$ of 0.1 (left), 0.01 (center), and 0.001 (right), normalized to number density. Note the varying color scale for each subplot.

Figure 6

Terms of the expanded particle energy flux (3.5) in the $y$ direction for $\mathit{\text{Kn}}$ of 0.01 (top) and 0.001 (bottom), normalized to $n_0{v}_{\text{th},c}^3$. Energy flux is calculated at normalized time $3.0{\tau }_{\mathit{\text{RT}}}$ for the 0.01 case and $2.1{\tau }_{\mathit{\text{RT}}}$ for the 0.001 case to have similar amplitudes. Note the varying color scale of each column.

Figure 7

Comparison of energy-flux nonideal terms, Eq. (12), calculated from distribution function and those calculated from a first-order Chapman-Enskog expansion of the collision operator. Top and bottom rows of each comparison are $\mathit{\text{Kn}}$ = 0.01 and 0.001, respectively. Note the similarities in spatial distribution and magnitude and that color scales are constant by term and row (collisionality). Stress terms show more discrepancy because they are calculated from several stress tensor elements, so errors compound.

Figure 8

Non-Maxwellian density, $n_N$, compared with gas-frame $y$-direction skewness, $q_y$, and excess kurtosis, $\delta K_y$. Note the presence of local extrema for all quantities around the RT instability interface.