SU(N)上快速-慢速随机欧拉-阿诺德系统中的余伴随平均和跨尺度通量
Coadjoint averaging and cross-scale fluxes in a fast-slow stochastic Euler-Arnold system on $SU(N)$
AI总结：
研究 Zeitlin 的 SU(N) 近似随机版中能量和涡度相关跨尺度通量，引入快速-慢速随机框架，表明余伴随对称性抑制平均非线性通量，其打破可选择跨尺度转移方向。
AI中文摘要：
我们研究了在 Zeitlin 的二维欧拉动力学的 SU(N) 近似的随机版本中与能量和涡度相关的（跨尺度）通量的出现。受欧拉-阿诺德公式的启发，我们将非线性输运解释为沿余伴随轨道的运动，反映了保持所有卡西米尔不变量的潜在对称性。我们引入了一个快速-慢速随机框架，其中快速混合由沿选定方向作用的结构化快速随机强迫建模。我们表明，当快速动力学保持由自然辛结构给出的完整余伴随轨道对称性时，平均系统没有非平凡通量。相反，当这种对称性被余伴随轨道上的切向非哈密顿向量场打破时，我们确定了产生由欧拉-阿诺德非线性携带的非零通量的条件。因此，余伴随对称性抑制平均非线性通量，而其打破可以选择跨尺度转移的首选方向。
英文摘要：
We study the emergence of (cross-scale) fluxes associated with energy and enstrophy in a stochastic version of Zeitlin's $SU(N)$ approximation of 2-d Euler dynamics. Motivated by the Euler-Arnold formulation, we interpret the nonlinear transport as motion along coadjoint orbits, reflecting an underlying symmetry that preserves all Casimir invariants. We introduce a fast-slow stochastic framework in which rapid mixing is modeled by a structured fast stochastic forcing acting along selected directions. We show that when the fast dynamics preserves the full coadjoint orbit symmetry given by the natural symplectic structure, the averaged system exhibits no nontrivial flux. In contrast, when this symmetry is broken by tangential non-Hamiltonian vector fields on the coadjoint orbit, we identify conditions that yield nonzero fluxes carried by the Euler-Arnold nonlinearity. Thus, coadjoint symmetry suppresses averaged nonlinear flux, whereas its breaking can select a preferred direction of cross-scale transfer.
