Vortex Dynamics and Scalar Transport in Rotating Stratified Turbulence-Construction of a ‘Vortex-Wave based Turbulence Model'-

9.1.采用矩约简方法建立了hamilton矩模型，其中每个涡均为均匀位涡椭球，其主轴和倾角在其他涡诱导的剪切作用下发生变化。为了评估该模型的有效性，并找到纳入耗散效应的线索，采用casl算法进行了直接数值模拟。研究了垂直高度稍有不同的两个同旋涡的相互作用。哈密顿矩模型能较好地预测临界合并距离。其次，研究了反向旋转涡旋对（偶极子）的相互作用。根据椭球矩模型，当两个细长涡旋初始处于临界距离内时，其垂直轴的倾角接近π/2，且其中一个椭球主轴随时间呈指数增长。casl计算表明，涡旋更能稳定地保持偶极子结构。它们通过从顶部和底部喷出细丝而存活下来，即使它们被放置得很近。对准地转线涡模型进行了改进。我们采用高斯积分法来计算相互能量积分，而不是以前的线模型中使用的矩近似。即使两个线涡放置得很近，积分精度也大大提高。改进后的模型可以避免先前的线矩模型所预测的对旋涡对的奇异行为。假设每个线涡是一个延长的球体，我们还引入了倾角与涡长细比参数之间的现实关系。后一种改进使我们能够更精确地模拟更大的涡。为了解析地证明椭球涡旋周围混沌运动的存在，我们将Melnikov方法应用于弱摄动椭球涡旋。沿异斜轨道和同斜轨道计算的Melnikov函数在任何情况下横向相交为零，表明存在流体粒子的混沌混合。事实上，数值计算的频闪庞加莱图即使在扰动非常弱的情况下也会在异斜轨道和同斜轨道附近散射。少

9.1.A Hamiltonian moment model is constructed following a moment reduction procedure, in which each vortex is modeled by an ellipsoid of uniform potential vorticity, whose principal axes and inclination angles change under the influence of shear induced by other vortices. Direct numerical simulations based on the CASL-algorithm are performed in order to assess the validity of the model and to find out a clue to incorporate dissipative effects. The interaction of two co-rotating vortices on slightly different vertical levels is studied. The Hamiltonian moment model can predict the critical merger distance fairly well. Next, the interaction of counter-rotating vortex pair (dipole) is investigated. According to the ellipsoidal moment model, the inclination from the vertical axis approaches to π/2 and one of the principal axes of the ellipsoid grows exponentially with time, if two slender vortices are placed within a critical distance initially. The CASL-computation tells that the vortices … More keep the dipole structure quite robustly. They survive from tilting down by ejecting thin filaments from their top and bottom, even if they are placed very close.9-2.Refinements on the quasi-geostrophic wire-vortex model are proposed. We adapt the Gaussian integration method in the evaluation of the mutual energy integrals, instead of the moment approximation used in the previous wire model. The accuracy of integration is improved drastically even if two wire-vortices are placed very close. The refined model can circumvent the singular behavior of a counter rotating vortex pair, predicted by the previous wire moment model. We also introduce a realistic relation between the inclination angle and the vortex-slenderness parameter, assuming that each wire-vortex is a prolate spheroid. The latter refinement enables us to model fatter vortices more accurately.9-3.In order to show the presence of the chaotic motion around an ellipsoidal vortex analytically, we apply Melnikov's method to a weakly perturbed spheroidal vortex. The Melnikov functions computed along the heteroclinic and homoclinic orbits intersect zero transversely in any cases, suggesting the occurrence of chaotic mixing of fluid particles. In fact, numerically computed stroboscopic Poincare plots scatter near the heteroclinic and homoclinic orbits even when the perturbations are very weak. Less