Chemical transport by weakly nonlinear internal gravity waves in stars

TL;DR

This work provides the first rigorous asymptotic treatment of passive chemical transport by weakly nonlinear internal gravity waves in stars, using a 2D Boussinesq framework with a tracer. It shows that a coherent wave packet drives transport that scales as $a^4$, while random superpositions yield $a^8$, with the latter behaving diffusively across many packets as $D_w o rac{ au_c^3}{4} a^8 D_c^2 ilde{k}^2$. The analysis identifies frequency-resonant quartic wave interactions as the mechanism enabling net vertical transport, and validates the theory against Dedalus simulations for 2-, 3-, and 4-wave cases. The results offer a principled way to parameterize wave-driven chemical mixing in stellar evolution models and highlight the need to bridge theory and numerical simulations for accurate predictions in radiative zones.

Abstract

While it is well-known that internal gravity waves (IGWs) transport chemicals in the radiative zones of stars, there remains substantial uncertainty on the amount of, and physical mechanism behind, this transport. Most previous studies have relied on heuristic theories, or numerical simulations that may be hard to extrapolate to stellar parameters. In this work, we present the first rigorous asymptotic calculation of (passive) chemical transport by IGWs, in the limit of small wave amplitude. We find that the net transport by a coherent packet of waves scales like wave amplitude to the fourth power, and verify these analytic calculations with numerical simulations. Because the transport is equally likely to be positive as negative, the transport by a random superposition of waves is expected to scale as wave amplitude to the eighth power. These results show that closer comparisons between theoretical arguments and numerical calculations are essential for interpreting numerical simulations of chemical transport by IGWs, and making accurate predictions of this process for stellar evolution modeling.

Figures (9)

• Figure 1: Schematic illustration of the nonlinear processes contributing to the second-order slow tracer field $c_{2s}$. Solid lines indicate velocity fields driving the evolution of the tracer fields shown by dashed lines. Cases I, II, and III correspond to the three terms in equation \ref{['eq:c2s']}. In Case I, the second-order fast tracer $c_{2f}$ is generated through four interaction pathways listed in equation \ref{['eq:c2f']}. In Case II, the first-order slow tracer field $c_{1s}$ consists of the two terms given in equation \ref{['eq:c1s']}.
• Figure 2: Snapshots of $2-$wave interaction at an intermediate time, showing the vertical velocity $w$, passive tracer $c(x,z)$, and the horizontal mean of the passive tracer $\langle c(z) \rangle$. The analysis is confined to the region $(x,z)\in[0,2\pi]\times[0.8\pi,2\pi]$ delineated by the black dashed lines to avoid boundary artifacts.
• Figure 3: Transport $c(x,z,t)-c_0$ driven by 2--wave interaction at (a) $x_0=\pi$, (b) $x_0=\pi$, $z_0=0.998\pi$, and (c) $x_0=\pi$, $t=4500$. The black curves show the time-averaged fitting given in \ref{['eq:sim_fit']}, computed over the interval $t=30~T$ to $t=115~T$, where $T$ is the wave period. The red curve in (b) corresponds to the theoretical prediction from \ref{['eq:c1s']} at $x_0=\pi$, $z_0=0.998\pi$. The red curve in (c) shows the theory \ref{['eq:c1s']} at $x_0=\pi$ with the fitted transport time. The fitted and theoretical parameters are reported in Table \ref{['tab']}.
• Figure 4: Vertical transport $\langle c\rangle(z,t)-c_0$ induced by 2--wave interaction. The blue curves show simulation results for $c(z,t)$ averaged over $x$ at (a,b) $z_0=0.998\pi$ and (c) $t=4500$. (b) shows a zoomed-in view of (a). The black dashed curves in (a,b) are the time-averaged $\langle c \rangle$ fitted using the function in \ref{['eq:sim_fit']}. The red curves in (a-c) indicate the theoretical predictions. A comparison between the fitted parameters and theoretical values is provided in Table \ref{['tab']}.
• Figure 5: Transport induced by a 3--wave interaction with identical frequencies. Blue curves show simulation results of (a,b) $c(t)-c_0$ at $x_0=\pi$ and $z_0=0.998\pi$, (c) $c(t)-c_0$ at $x_0=\pi$ and $t=4500$, (d,e) $\langle c \rangle(z,t)-c_0$ at $z_0=0.998\pi$, and (f) $\langle c \rangle(z,t)-c_0$ at $t=4500$. The middle panels (b,e) are zoomed-in views of the left (a,d). The black curves in (a,b,d,e) are the time-averaged transport evolution. The red curves indicate theoretical predictions. A comparison between the fitted parameters and theoretical values is provided in Table \ref{['tab']}.