Lagrangian techniques, such as the finite-time Lyapunov exponent (FTLE) and hyperbolic Lagrangian coherent structures, have become popular tools for analyzing unsteady fluid flows. These techniques identify regions where particles transported by a flow will converge to and diverge from over a finite-time interval, even in a divergence-free flow. Lagrangian analyses, however, are time consuming and computationally expensive, hence unsuitable for quickly assessing short-term material transport. A recently developed method called Objective Eulerian Coherent Structures (OECSs) (Serra and Haller in Chaos Interdiscip J Nonlinear Sci 26(5):053110, 2016) rigorously connected Eulerian quantities to short-term Lagrangian transport. This Eulerian method is faster and less expensive to compute than its Lagrangian counterparts, and needs only a single snapshot of a velocity field. Along the same line, here we define the instantaneous Lyapunov Exponent, the instantaneous counterpart of the FTLE, and connect the Taylor series expansion of the right Cauchy-Green deformation tensor to the infinitesimal integration time limit of the FTLE. We illustrate our results on geophysical fluid flows from numerical models as well as analytical flows, and demonstrate the efficacy of attracting and repelling instantaneous Lyapunov exponent structures in predicting short-term material transport.

拉格朗日技术（例如有限时间Lyapunov指数（FTLE）和双曲拉格朗日相干结构）已成为分析非恒定流体流动的流行工具。这些技术确定了在有限的时间间隔内，即使在无散度的流中，由流传输的粒子将汇聚和扩散的区域。然而，拉格朗日分析耗时且计算量大，因此不适合快速评估短期物料运输。最近开发的一种称为客观欧拉相干结构（OECS）的方法（Serra and Haller in Chaos Interdiscip J Nonlinear Sci 26（5）：053110，2016）严格地将欧拉量与短期拉格朗日运输联系起来。与拉格朗日对应方法相比，这种欧拉方法计算速度更快且成本更低，并且只需要一个速度场快照。沿着同一条线，我们在这里定义瞬时Lyapunov指数，即FTLE的瞬时对应物，并将右Cauchy-Green变形张量的Taylor级数展开与FTLE的无穷小积分时间限制联系起来。我们从数值模型和分析流中说明了我们在地球物理流体流方面的结果，并展示了在预测短期物质运输中吸引和排斥瞬时Lyapunov指数结构的功效。