Linear and nonlinear stability of a quasigeostrophic mixing layer subject to a uniform background shear

The aim of this work is to shed light by revisiting, from the kernel-wave (KW) perspective, the breakdown of a quasigeostrophic (QG) mixing layer (or vortex strip or filament) in atmosphere under the influence of a background shear. The QG mixing layer is modeled with a family of quasi-Rayleigh velocity profiles in which the potential vorticity (PV) is constant in patches. From the KW perspective, a counterpropagating Rossby wave (CRW) is created at each PV edge, i.e., the edge where a PV jump is located. The important parameters of our study are (i) the vorticity of the uniform shear $m$ and (ii) the Rossby deformation radius $L_d$, which indicates how far the pressure perturbations can vertically propagate. While an adverse shear ($m<0$) stabilizes the system, a favorable shear ($m>0$) strengthens the instability. This is due to how the background shear affects the two uncoupled CRWs by shifting the optimal phase difference towards large (small) wave number when $m<0$ ($m>0$). As a finite $L_d$ is introduced, a general weakening of the instability is noticed, particularly for $m>0$. This is mainly due to the reduced interaction between the two CRWs when $L_d$ is finite. Furthermore, nonlinear pseudospectral simulations in the nominally infinite-Reynolds-number limit were conducted using as the initial base flow the same quasi-Rayleigh profiles analyzed in the linear analysis. The growth of the mixing layer is obstructed by introducing a background shear, especially if adverse, since the vortex pairing, which is the main growth mechanism in mixing layers, is strongly hindered. Interestingly, the most energetic configuration is for $m=0$, which differs from the linear analyses for which the largest growth rates were found for a positive $m$. In the absence of a background shear additional modes are subharmonically triggered by the initial disturbance enhancing the turbulent character of the flow. We also confirm energy spectrum trends for broken-down mixing layers reported in the literature. We interpret the character of mixing-layer breakdown as being a phenomenological hybrid of Kraichnan's [R. H. Kraichnan, Phys. Fluids 10, 1417 (1967)] direct enstrophy cascade picture and the picture of self-similar vortex production.

Figure 1

Juno image of Jupiter's cloud tops near northern latitude ${48.9}^{\circ }$. The horizontal scale of the image is approximately 95 000 km (9.3 km/pixel). Being in the quasigeostrophic regime, these regions exhibit filamentary structure as well as large-scale coherent vortices. The Rossby deformation radius at these latitudes is approximately 20 000 km. (Image credit: NASA, JPL-Caltech, SwRI, MSSS, Gerald Eichstadt, and Sean Doran.)

Figure 2

From (a) the planetary scale in spherical coordinates to the synoptic scale in Cartesian coordinate. (e) Velocity profile for $L_d=\infty$, i.e., (c) the Rayleigh profile plus (d) the uniform shear. Two CRWs are formed at the two vorticity edges: NCRW at the northern edge and SCRW at the southern edge.

Figure 3

(a) Growth rate vs wave number $k$ and (b) the phase speed of the two CRWs taken in isolation for different values of the uniform shear $m$ and $L_d=\infty$. In (b) dashed lines depict the NCRW, while solid lines depict the SCRW. A NCRW (SCRW) is counterpropagating if its phase speed is below (above) the gray line, which depicts the average base flow velocity ($U_{av}=0$).

Figure 4

(a) Growth rate vs wave number $k$ and (b) the phase speed of the two CRWs taken in isolation for $m=0$ and different values of $L_d$. In (b) dashed lines depict the NCRW, while solid lines depict the SCRW. A NCRW (SCRW) is counterpropagating if its phase speed is below (above) the gray line, which depicts the average base flow velocity ($U_{av}=0$).

Figure 5

Growth rate in the plane $k\text{−}m$ for (a) $L_d=\infty$, (b) $L_d=10$, (c) $L_d=2$, and (d) $L_d=1$.

Figure 6

(a) Phase difference $\frac{\mathrm{\Delta }\epsilon }{\pi }$ and (b) interaction coefficient for $L_d=\infty$ and different values of $m$. In (a) the dashed line represents the phase difference corresponding to the maximum growth rate $\mathrm{\Delta }\epsilon =0.65$ for $m=0$.

Figure 7

(a) Phase difference $\frac{\mathrm{\Delta }\epsilon }{\pi }$ and (b) the interaction coefficient $\gamma$ vs wave number for different values of $L_d$ and $m=0$. The dashed line shows the value of the optimal phase difference $0.65\pi$ for the Rayleigh model (so $m=0$ and $L_d=\infty$) [33].

Figure 8

Growth rate ${\omega }_i$ in the plane $k\text{−}m$ for (a) $L_d=\infty$, (b) $L_d=10$, (c) $L_d=2$, and (d) $L_d=1$. The solid line represents $k_{\max }\left(m\right)$, the dashed line represents $k_{0.65\pi }\left(m\right)$, and the dotted line represents $k_{\mathrm{equal}}\left(m\right)$.

Figure 9

Phase difference corresponding to the maximum growth rate $\frac{\mathrm{\Delta }{\epsilon }_{\max }}{\pi }$ vs wave number for $m=0$ and different values of $L_d$. The black line represents the limit between the helping and the hindering configuration.

Figure 10

Potential vorticity fields for $m=0$ and $L_d=\infty$ at different times (a) $t=10$, (b) $t=24$, (c) $t=34$, (d) $t=60$, (e) $t=120$, and (f) $t=200$.

Figure 11

Potential vorticity fields for $m=-0.2$ and $L_d=\infty$ at different times (a) $t=10$, (b) $t=30$, (c) $t=50$, and (d) $t=140$. Note that at late times ($t=140$) small-scale filaments appear to undergo a secondary transition into coherent vortices.

Figure 12

Potential vorticity fields for $m=0$ and $L_d=2$ at different times (a) $t=10$, (b) $t=50$, (c) $t=100$, and (d) $t=200$. Similar to Fig. 11, torn shreds of vorticity appear to roll into smaller satellite coherent vortices at late times.

Figure 13

Temporal evolution of the momentum thickness $\theta$ for (a) $L_d=\infty$ and several values of $m$ and (b) $m=0$ and several values of $L_d$.

Figure 14

Spatially averaged perturbation kinetic energy for (a) $L_d=\infty$ and several values of $m$ and (b) $m=0$ and several values of $L_d$. The strength of the turbulence is growing with increasing $m$ and diminishing $L_d$. Note that the domain sizes are $L_x=49.2928$ and $L_y=60$ or $L_y=30$ (when $L_d\le 2$).

Figure 15

Fourier transform of the potential vorticity $\widehat{q}\left(\overline{k}\right)$ with $k$ equal to $k=0.1275$ (yellow), $k=0.2549$ (red), $k=0.3824$ (dark red), and $k=0.5099$ (black) for $L_d=\infty$ and (a) $m=0.5$ and (b) $m=0$.

Figure 16

Time-averaged kinetic energy spectra for (a) $L_d=\infty$ and several values of $m$ and (b) $m=0$ and several values of $L_d$.

Figure 17

Temporal behavior of the spatially averaged perturbation kinetic energy $K$ for (a) $L_d=\infty$ and different values of $m$ and (b) $m=0$ and different values of $L_d$. Note that the domain sizes are $L_x=\frac{2\pi }{k_{\max }}$ (see Table 1 for the values of $k_{\max }$) and the domain height is $L_y=60$ for all the configurations but for $L_d\le 2$ in which $L_y=30$.