Streaming instability: saturation in turbulent protoplanetary disks. (arXiv:1906.05371v1 [astro-ph.EP])

Streaming instability: saturation in turbulent protoplanetary disks. (arXiv:1906.05371v1 [astro-ph.EP])
<a href="http://arxiv.org/find/astro-ph/1/au:+Umurhan_O/0/1/0/all/0/1">Orkan. M. Umurhan</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Estrada_P/0/1/0/all/0/1">Paul. R. Estrada</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Cuzzi_J/0/1/0/all/0/1">Jeffrey N. Cuzzi</a>

The streaming instability for solid particles in protoplanetary disks is
re-examined assuming the familiar alpha ($alpha$) model for isotropic
turbulence. Turbulence always reduces the growth rates of the streaming
instability relative to values calculated for globally laminar disks. While for
small values of the turbulence parameter, $alpha < 10^{-5}$, the wavelengths of the fastest-growing disturbances are small fractions of the local gas vertical scale height $H$, we find that for moderate values of the turbulence parameter, i.e., $alpha sim 10^{-5}-10^{-3}$, the lengthscales of maximally growing disturbances shift toward larger scales, approaching $H$. At these moderate turbulent intensities and for particle to gas mass density ratios $epsilon < 0.5$, the vertical scales of the most unstable modes begin to exceed the corresponding radial scales so that the instability appears in the form of vertically oriented sheets. We find that for hydrodynamical turbulent instability models reported in the literature, leading to $alpha = 4times 10^{-5} - {10^{-4}}$, the streaming instability is present in principle for a narrow range of Stokes numbers, $sim 0.01

The streaming instability for solid particles in protoplanetary disks is
re-examined assuming the familiar alpha ($alpha$) model for isotropic
turbulence. Turbulence always reduces the growth rates of the streaming
instability relative to values calculated for globally laminar disks. While for
small values of the turbulence parameter, $alpha < 10^{-5}$, the wavelengths
of the fastest-growing disturbances are small fractions of the local gas
vertical scale height $H$, we find that for moderate values of the turbulence
parameter, i.e., $alpha sim 10^{-5}-10^{-3}$, the lengthscales of maximally
growing disturbances shift toward larger scales, approaching $H$. At these
moderate turbulent intensities and for particle to gas mass density ratios
$epsilon < 0.5$, the vertical scales of the most unstable modes begin to
exceed the corresponding radial scales so that the instability appears in the
form of vertically oriented sheets. We find that for hydrodynamical turbulent
instability models reported in the literature, leading to $alpha = 4times
10^{-5} – {10^{-4}}$, the streaming instability is present in principle for a
narrow range of Stokes numbers, $sim 0.01<tau_s< 0.05$ ($tau_s$ is the ratio
of the particle gas drag stopping time to the local orbit time). However, with
these levels of $alpha$ and canonical solids-to-gas abundances, we find that
the streaming instability stalls and saturates as growing modes approach
$epsilon = 1$ from smaller values, resulting in only modest particle
overdensities of factors of 4-20 at best. Our results are consistent with, and
place in context, published numerical studies of streaming instabilities.

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