Origin of hydrodynamic instability from noise: From laboratory flow to accretion disk

We attempt to address the old problem of plane shear flows: the origin of turbulence and hence transport of angular momentum in accretion flows as well as laboratory flows, such as plane Couette flow. We undertake the problem by introducing an extra force in Orr-Sommerfeld and Squire equations along with the Coriolis force mimicking the local region of the accretion disk. For plane Couette flow, the Coriolis term drops. Subsequently we solve the equations with the WKB approximation method. We investigate the dispersion relation for the Keplerian flow and plane Couette flow for all possible combinations of wave vectors. Due to the very presence of extra force, we show that both flows are unstable for a certain range of wave vectors. However, the nature of instability between the flows is different. We also study the Argand diagrams of the perturbation eigenmodes. This helps us to compare the different timescales corresponding to the perturbations as well as accretion. We ultimately conclude with this formalism that fluid gets enough time to be unstable and hence plausibly turbulent particularly in the local regime of the Keplerian accretion disks. Repetition of the analysis throughout the disk explains the transport of angular momentum and matter along outward and inward directions, respectively.

Figure 1

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=0$ for the Keplerian flow.

Figure 2

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=0$ for plane Couette flow.

Figure 3

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^2$ and $m/u_0=10$ for the Keplerian flow.

Figure 4

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=10$ for the Keplerian flow.

Figure 5

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^4$ and $m/u_0=10$ for the Keplerian flow.

Figure 6

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^4$ and $m/u_0={10}^2$ for the Keplerian flow.

Figure 7

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}=10$ and $m/u_0=10$ for the Keplerian flow. $\text{Im}{\left(\beta \right)}_{\max }=0.33$.

Figure 8

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^2$ and $m/u_0=10$ for the Keplerian flow. $\text{Im}{\left(\beta \right)}_{\max }=0.822$.

Figure 9

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^3$ and $m/u_0=10$ for the Keplerian flow. $\text{Im}{\left(\beta \right)}_{\max }=0.896$.

Figure 10

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^4$ and $m/u_0=10$ for the Keplerian flow. $\text{Im}{\left(\beta \right)}_{\max }=0.91$.

Figure 11

Variation of $\text{Im}{\left(\beta \right)}_{\max }$ as a function of Re for $m/u_0=10$ and $m/u_0=100$ for the Keplerian flow. The dashed and dotted lines represent $\text{Im}{\left(\beta \right)}_{\max }=0.91$ and 2.12, respectively.

Figure 12

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^2$ and $m/u_0=-{10}^{-2}$ for plane Couette flow. At ${\alpha }_1={\alpha }_3=0, \text{Im}\left(\beta \right)\to \infty$. It is indicated by a white point at the center of the plot.

Figure 13

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=-{10}^{-2}$ for plane Couette flow.

Figure 14

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^2$ and $m/u_0=-{10}^{-2}$ for plane Couette flow.

Figure 15

Variation of $\text{Im}\left(\beta \right)$ in three dimensions as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=-{10}^{-2}$ for plane Couette flow.

Figure 16

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^2$ and $m/u_0=-10$ for plane Couette flow.

Figure 17

Variation of $\text{Im}\left(\beta \right)$ as a function of ${\alpha }_1$ and ${\alpha }_3$ for $\text{Re}={10}^{10}$ and $m/u_0=-10$ for plane Couette flow.

Figure 18

Argand diagram for $\text{Re}=10$, 100 and 1000 for $m/u_0=10$ and ${\alpha }_1=1.0$ for the Keplerian flow. For each Re, ${\alpha }_3$ is varied from $-2000$ to 2000.

Figure 19

Argand diagram for $\text{Re}={10}^4, m/u_0=10$ and for ${\alpha }_1=1.0$, 5.0, and 10.0 for the Keplerian flow.

Figure 20

Variation of ${\left[\text{Re}\left(u\right)\right]}^2$ as a function of time, for $\text{Re}{\left(\beta \right)}_{\max }$ and $\text{Im}{\left(\beta \right)}_{\max }$ from Fig. 19 corresponding to ${\alpha }_1=1.0,5.0,\mathrm{and}10.0$.

Figure 21

Variation of $\text{Im}{\left(\beta \right)}_{\max }$ as a function of $m/u_0$, for ${\alpha }_1=1,5,10$ and $\text{Re}={10}^4$ for the Keplerian flow.

Figure 22

Argand diagram for $\text{Re}={10}^4, m/u_0={10}^2$ and for ${\alpha }_1=1.0$, 5.0, and 10.0 for the Keplerian flow.

Figure 23

Argand diagram for $\text{Re}={10}^{10}, m/u_0=10$ and for ${\alpha }_1=1.0$, 5.0, and 10.0 for the Keplerian flow.

Figure 24

Argand diagram for $\text{Re}={10}^{10}, m/u_0={10}^2$ and for ${\alpha }_1=1.0$, 5.0, and 10.0 for the Keplerian flow.