Instability in stratified accretion flows under primary and secondary perturbations

We consider horizontal linear shear flow (shear rate denoted by $\Lambda$) under vertical uniform rotation (ambient rotation rate denoted by ${\Omega }_0$) and vertical stratification (buoyancy frequency denoted by $N$) in unbounded domain. We show that, under a primary vertical velocity perturbation and a radial density perturbation consisting of a one-dimensional standing wave with frequency $N$ and amplitude proportional to $w_{0}sin(\varepsilon Nx/w_0)\approx \varepsilon Nx(\ll 1),$ where $x$ denotes the radial coordinate and $\varepsilon$ a small parameter, a parametric instability can develop in the flow, provided $N^2>8{\Omega }_0(2{\Omega }_0-\Lambda ).$ For astrophysical accretion flows and under the shearing sheet approximation, this implies $N^2>8{\Omega }_0^2\left(2-q\right)$, where $q=\Lambda /{\Omega }_0$ is the local shear gradient. In the case of a stratified constant angular momentum disk, $q=2,$ there is a parametric instability with the maximal growth rate $({\sigma }_m/\varepsilon )=3\sqrt{3}/16$ for any positive value of the buoyancy frequency $N.$ In contrast, for a stratified Keplerian disk, $q=1.5,$ the parametric instability appears only for $N>2{\Omega }_0$ with a maximal growth rate that depends on the ratio ${\Omega }_0/N$ and approaches $(3\sqrt{3}/16)\varepsilon$ for large values of $N.$

Figure 1

A stratified constant angular momentum disk, $q=2$, under primary and secondary axisymmetric perturbations. Top left panel: The Floquet tongues of instability as a function of the parameter $\varepsilon$ (numerical results). Within the subharmonic tongue emanating from $\alpha ={cos}^{-1}(\pm 0.5)$, the dashed lines indicate the inclination a of largest growth rate (plotted in the bottom left panel) at a given $\varepsilon$. Top right panel: Comparison between computations and analytical results [see Eqs. (39) and (45)]. Bottom left panel: A plot of the maximum scaled growth rate against the parameter $\varepsilon$ (numerical results). The intercept is $3\sqrt{3}/16$, in agreement with the analytical result [see Eq. (52)]. Bottom right panel: Width of the instability tongues emanating (I.T.E.) from $\alpha ={cos}^{-1}(\pm 0.5)$ as a function of the parameter $\varepsilon$ [analytical results from Eq. (45)].

Figure 2

A stratified Keplerian disk, $q=1.5$, under primary and secondary axisymmetric perturbations. Top panels: The Floquet tongues of instability as a function of the parameter $\varepsilon$ (numerical results) for $\text{Ri}=2.0$ (left panel) and $\text{Ri}=3.0$ (right panel). Within the subharmonic tongue emanating from $\alpha ={cos}^{-1}[\pm 0.5\sqrt{(9\text{Ri}-16)/(9\text{Ri}-4)}]$, the dashed lines indicate the inclination a of largest growth rate (plotted in the bottom left panel) at a given $\varepsilon$. Bottom left panel: A plot of the maximum scaled growth rate against the parameter $\varepsilon$ (numerical results) for $\text{Ri}=2.5,3.0,10.0$. The intercept is $(3\sqrt{3}/16)\sqrt{1-16/\left(9\text{Ri}\right)}/\left[(1-4/(9\text{Ri})\right]$, in agreement with the analytical result [see Eq. (52)]. Bottom right panel: Comparison between computations and analytical results for $\text{Ri}=3.0$ [see Eqs. (39) and (45)].

Figure 3

A stratified Keplerian disk, $q=1.5$, under primary and secondary asymmetric perturbations, $k_{10}/k_2=100,k_3/k_2=0.5$. Left panel: Time evolution of the element $G_{11}$ of the fundamental matrix solution of system (54) for $\text{Ri}=0.5$ and $\varepsilon =0.0,0.1,0.2$. Right panel: Time evolution of the spectral density of total (kinetic+potential) energy normalized by its initial value under initial isotropic conditions with vanishing initial potential energy for $\text{Ri}=0.5$ and $\varepsilon =0.0,0.1,0.2$.