Geometric features in oceanic mesoscale eddies such as tilt and anisotropy can influence the properties of the Reynolds stress that provides feedback between the eddies and the background flow. By regarding an eddy as a wave, previous studies have parameterized the Reynolds stress based on the equivalence in the tilt angle between the phase of the eddy stream functions and the variance ellipse for the Reynolds stress (RS-ellipse). However, the wave assumption cannot predict the anisotropy of the RS-ellipse, and also largely simplifies the eddy geometry, which would naturally be an ellipsoid rather than a wave. The present study explores the shape relation between elliptical eddies and the RS-ellipse, by mathematically reformulating the Reynolds stress based on the eddy shape. The new formula reveals that the shape relation is regulated by the horizontal extent of the occurrence probability distribution (PDF) of the eddy, and that the shape of the eddy and RS-ellipse are identical at the place of maximum PDF when the horizontal scale of the PDF is sufficiently larger than the size of the eddy. A similar tendency is found in eddies detected by satellite altimetry in the Kuroshio Extension jet region. A detailed analysis of the PDF in this region shows that the tilts of the eddies are likely to be consistent with the destabilization effect on the jet, suggesting a strong relation between the eddy geometry and the jet's stability in this region. These findings may open a path toward a new method to parameterize the Reynolds stress with the background state, exploiting the shape equivalence between the eddies and the RS-ellipse.

海洋中尺度涡旋的几何特征（例如倾斜和各向异性）会影响雷诺应力的属性，该应力在涡旋和背景流之间提供反馈。通过将涡旋视为波浪，以前的研究已经基于涡流函数的相位与雷诺应力的方差椭圆（RS椭圆）之间的等倾角的等价关系来对雷诺应力进行参数化。但是，波浪假设不能预测RS椭圆的各向异性，并且还大大简化了涡流几何形状，这自然是椭圆形而不是波浪。本研究通过基于涡流形状的数学公式重新形成雷诺应力，探索了椭圆形涡流与RS椭圆之间的形状关系。新公式表明，形状关系受涡的发生概率分布（PDF）的水平范围控制，并且当水平方向的比例在最大PDF位置时，涡和RS椭圆的形状相同。 PDF足够大于涡流的大小。在黑潮扩展喷气机地区的卫星测高仪检测到的涡流中也发现了类似的趋势。对这一区域的PDF的详细分析表明，涡流的倾斜可能与射流的失稳效应相一致，这表明该区域的涡流几何形状和射流的稳定性之间存在很强的关系。这些发现可能会开辟一条新方法，将雷诺应力与背景状态参数化，