We study the statistical properties of stationary, isotropic and homogeneous turbulence in two-dimensional (2D) flows, focusing on the direct cascade, that is on wave-numbers large compared to the integral scale, where both energy and enstrophy are provided to the fluid. Our starting point is the 2D Navier-Stokes equation in the presence of a stochastic forcing, or more precisely the associated field theory. We unveil two extended symmetries of the Navier-Stokes action which were not identified yet, one related to time-dependent (or time-gauged) shifts of the response fields and existing in both 2D and 3D, and the other to time-gauged rotations and specific to 2D flows. We derive the corresponding Ward identities, and exploit them within the non-perturbative renormalization group formalism, and the large wave-number expansion scheme developed in [Phys. Fluids {\bf 30}, 055102 (2018)]. We consider the flow equation for a generalized $n$-point correlation function, and calculate its leading order term in the large wave-number expansion. At this order, the resulting flow equation can be closed exactly. We solve the fixed point equation for the 2-point function, which yields its explicit time dependence, for both small and large time delays in the stationary turbulent state. On the other hand, at equal times, the leading order term vanishes, so we compute the next-to-leading order term. We find that the flow equations for simultaneous $n$-point correlation functions are not fully constrained by the set of extended symmetries, and discuss the consequences.
Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model
Reply to comment on ``Competing Interactions, the Renormalization Group and the Isotropic-Nematic Phase Transition''
我们工作的重点是确定各向同性竞争相互作用的通用系统的上下文中存在的各向同性-双相变的条件。评论0709.4205批评我们的结果，表明在2d的条纹相的低温波动在热力学极限线性发散。分析仅限于条纹相位，不适用于我们的信的中心结果。与评论中的建议相反，我们证明D. G. Barci和D. A.在PRL 98，200604（2007）中的Stariolo经历了在Kosterlitz无核普适类中的各向同性-无核相变。
Comment on: Competing Interactions, the Renormalization Group, and the Isotropic-Nematic Phase Transition, by D. Barci and D. Stariolo, Phys. Rev. Lett. 98, 200604 (2007)
Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation
协方差函数的时间排序过程，如前一篇论文（K。Kiyani和W. D. McComb Phys.Rev.E70，066303（2004））的方法进行了扩展，并用于表明与Kraichnan-Wyld扰动级数相关的二阶响应函数可以由局部（波数）能量平衡确定。这些时间排序程序也允许两次制定减少到时间无关的形式，通过指数近似，它被验证的响应方程没有红外发散在无限雷诺数。最后，单次马尔可夫封闭方程（在前面的论文中所述）的推导和证明是兼容的Kolmogorov分布，而不需要引入一个特设常数。
