具有分数阶应力闭合的精确可解埃克曼层
An Exactly Solvable Ekman Layer with a Fractional-Order Stress Closure
AI总结：
本研究通过分数阶应力闭合放松局地性假设，在应力表述下用Mittag-Leffler函数精确求解埃克曼层，发现表面偏转角小于45度且输运仍垂直风向，远场呈幂律衰减。
AI中文摘要：
旋转海洋中风驱动表层流的经典理论通过局部通量-梯度定律闭合动量平衡，其中给定深度处的湍流应力与该深度处的剪切力成正比。该理论预测表层流偏离风向四十五度，这超过了大多数直接测量结果。本研究探讨当放宽局地性假设时会发生什么。从湍流应力与平均剪切力之间的精确积分关系出发，并要求记忆核不携带任何首选垂直尺度，我们获得了一个幂律核，从而得到一个分数阶应力定律。所得方程不能以速度作为变量来表述，因为有界剖面的分数阶导数在表面处为零，因此无法施加风应力，而导数的另一种定义则会使表层流无界。相反，以应力作为变量来表述时，该问题在零到一之间的每个阶数上都可以用Mittag-Leffler函数以封闭形式求解。此时表面偏转角仅取决于闭合阶数，且在整个范围内小于四十五度，而深度积分输运量则保持恰好垂直于风向，因为该约束源于动量平衡而非闭合关系。远场衰减遵循幂律而非指数律，其振幅在局地极限下消失，因此该极限是奇异的。在突然施加的应力下，表层瞬态呈代数衰减。三种独立算法结果高度一致，且未使用任何观测或模型数据。
英文摘要：
The classical theory of the wind-driven surface layer of a rotating ocean closes the momentum balance with a local flux-gradient law, in which the turbulent stress at a given depth is proportional to the shear at that depth. It predicts a surface current deflected forty-five degrees from the wind, which exceeds most direct measurements. This study asks what follows when locality is relaxed. Beginning from the exact integral relation between turbulent stress and mean shear, and requiring the memory kernel to carry no preferred vertical scale, we obtain a power-law kernel and therefore a stress law of fractional order. The resulting equation cannot be posed on the velocity, because the fractional derivative of a bounded profile vanishes at the surface, so the wind stress cannot be applied, while the alternative definition of the derivative leaves the surface current unbounded. Posed on the stress instead, the problem is solvable in closed form in Mittag-Leffler functions at every order between zero and one. The surface deflection then depends on the closure order alone and is smaller than forty-five degrees throughout, whereas the depth-integrated transport stays exactly normal to the wind, because that constraint follows from the momentum balance and not from the closure. The far field decays as a power law rather than exponentially, and its amplitude vanishes in the local limit, so that limit is singular. Under a suddenly applied stress the surface transient decays algebraically. Three independent algorithms agree closely, and no observational or model data are used.
