Stability of a horizontal vortex in weakly stratified flow

A trailing vortex behind a wing moving through a stratified fluid will act to twist the density field into a pattern where the density profile continuously overturns along the axis of the vortex. This configuration is approximated here with a density field that is overturning periodically along the axis of the vortex. The matching velocity field is found approximately assuming weak stratification and constant axial vorticity at leading order and confining the flow to a fixed radius. This base flow is shown to be unstable to a wave triad consisting of two disturbance waves and a component of the base flow as the third “wave.” Three components of this base flow lead to instability: (1) the twirling component, (2) the streaming component, and (3) the rolling component. All three instabilities depend strongly on the axial length for the density field to overturn (the pitch). The twirling and rolling instabilities are important when the pitch is small and they also depend on the Froude number. The streaming instability is dominant when the pitch is large and is independent of the Froude number.

Figure 1

Schematic diagram of the geometry.

Figure 2

Contours of kinetic energy of the base flow at four axial positions (a) $z=0$, (b) $z=\frac{\pi }{2}$, (c) $z=\pi$, and (d) $z=\frac{3\pi }{2}$, with $\beta =0.5$ and $k=\text{Fr}=1$.

Figure 3

Contours of kinetic energy of the base flow at four axial positions (a) $z=0$, (b) $z=\frac{\pi }{2}$, (c) $z=\pi$, and (d) $z=\frac{3\pi }{2}$, with $\beta =0.5$, $k=5$, and $\text{Fr}=1$.

Figure 4

First five profiles of $\widehat{\sigma }$ for two azimuthal wave numbers (a) $m=1$ and (b) $m=-1$.

Figure 5

Dispersion profiles for $m_a=1$, $m_b=-1$, and $m_c=2$ for two example values of $k$: (a) $k=1$ and (b) $k=5$.

Figure 6

Frequency of disturbance waves with $m_c=1$: (a) $m_a=2$ and $m_b=1$, (b) $m_a=1$ and $m_b=0$, (c) $m_a=0$ and $m_b=-1$, and (d) $m_a=-1$ and $m_b=-2$.

Figure 7

Frequency of disturbance waves with $m_c=0$: (a) $m_a=m_b=2$, (b) $m_a=m_b=1$, (c) $m_a=m_b=0$, and (d) $m_a=m_b=-1$.

Figure 8

Frequency of disturbance waves with $m_c=2$: (a) $m_a=3$ and $m_b=1$, (b) $m_a=2$ and $m_b=0$, (c) $m_a=1$ and $m_b=-1$, and (d) $m_a=0$ and $m_b=-2$.

Figure 9

Values of ${\sigma }_1^2$ with $m_c=0$, $\text{Fr}=1$: (a) $m_a=m_b=2$, (b) $m_a=m_b=1$, (c) $m_a=m_b=0$, and (d) $m_a=m_b=-1$.

Figure 10

Growth rates (imaginary part of $\sigma$) for the most unstable modes with $m_c=0$ and $\text{Fr}=1$: (i) mode 2 with $m_a=m_b=-1$ and mode 3 with $m_a=m_b=1$, (ii) mode 3 with $m_a=m_b=0$, (iii) mode 0 with $m_a=m_b=1$ and mode 1 with $m_a=m_b=-1$, and (iv) mode 1 with $m_a=m_b=1$ and mode 0 with $m_a=m_b=-1$. To obtain dimensional values, multiply by $V/R$.

Figure 11

Values of ${\sigma }_1^2$ with $m_c=1$: (a) $m_a=2$ and $m_b=1$, (b) $m_a=1$ and $m_b=0$, (c) $m_a=0$ and $m_b=-1$, and (d) $m_a=-1$ and $m_b=-2$.

Figure 12

Growth rates (imaginary part of $\sigma$) for the most unstable modes with $m_c=1$: (i) mode 3 with $m_a=2$ and $m_b=1$ and (ii) mode 1 with $m_a=3$ and $m_b=2$. To obtain dimensional values, multiply by $V/R$.

Figure 13

Values of ${\sigma }_1^2$ with $m_c=2$, $\text{Fr}=1$: (a) $m_a=3$ and $m_b=1$, (b) $m_a=2$ and $m_b=0$, (c) $m_a=1$ and $m_b=-1$, and (d) $m_a=-0$ and $m_b=-2$.

Figure 14

Growth rates (imaginary part of $\sigma$) for the most unstable modes with $m_c=2$ and $\text{Fr}=1$: (i) mode 2 with $m_a=2$ and $m_b=0$, (ii) mode 0 with $m_a=3$ and $m_b=1$, (iii) mode 3 with $m_a=2$ and $m_b=0$, (iv) mode 0 with $m_a=0$ and $m_b=-2$, and (v) mode 3 with $m_a=2$ and $m_b=0$ and using the second dispersion mode for the $b$ disturbance. To obtain dimensional values, multiply by $V/R$.

Figure 15

Values of ${\sigma }_1^2$ for mode 3 with $m_c=2$, $m_a=2$, and $m_b=0$. The $a$ disturbance uses the first dispersion profile $\widehat{\sigma }\left(\widehat{k}\right)$, while the $b$ disturbance uses the (i) first, (ii) second, and (iii) third profiles. To obtain dimensional values, multiply by $V/R$.