We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, $h/r$ , as well as three parameters characterising the background flow, the strain rate, $\gamma$ , the ratio of the background rotation rate to the strain, $\beta$ , and the angle from which the flow is applied, $\theta$ . However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, $f/N$ ( $f$ and $N$ are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, $\gamma _c$ , beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain $\gamma _m$ at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of $\gamma _m$ occur as $\beta \to 1$ . For large values of $f/N$ , changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.

中文翻译：

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有限罗斯比数下背景剪切流中的平衡椭球涡平衡

我们考虑一个势涡度 (PV) 的均匀椭球体，我们利用罗斯比数的二阶平衡模型得出的解析解，这是准地转 (QG) 理论的下一个阶，即所谓的 QG+1模型。我们在存在外部背景剪切流的情况下考虑这个涡流，作为外部涡流影响的代理。对于 QG 模型，系统取决于四个参数，涡流的高宽比， $h/r$ ，以及表征背景流动的三个参数，应变率， $\伽马$ , 背景旋转速率与应变的比值, $\beta$ ，以及应用流动的角度， $\θ$ . 然而，QG+1 模型也取决于 PV，以及普朗特比率， $f/N$ ( $f$ 和 $N$ 分别是科里奥利频率和浮力频率）。对于 QG 和 QG+1，我们确定不同背景流动参数值的平衡，以将施加的应变率值增加到临界应变率， $\伽玛_c$ ，超出该平衡不存在。我们还计算了这个涡旋对二阶模式的线性稳定性，确定了边际应变 $\伽马_m$ 椭圆体不稳定性爆发的地方。结果表明，对于QG+1，最具弹性的气旋椭球体略长，而反气旋椭球体往往更扁。的最高值 $\伽马_m$ 发生为 $\beta \to 1$ . 对于较大的值 $f/N$ ，边际应变率发生变化，使反气旋椭球稳定，同时使气旋椭球不稳定。