Noether Currents for Eulerian Variational Principles in Non Barotropic Magnetohydrodynamics and Topological Conservations Laws

Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs

Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabelling symmetries are derived using Noether's first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noether's theorem, in conjunction with a fluid relabelling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same met…

Variational nonlinear WKB in the Eulerian frame

Nonlinear WKB is a multiscale technique for studying locally-plane-wave solutions of nonlinear partial differential equations (PDE). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small, and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the gener…

On Noether's theorem for the Euler-Poincaré equation on the diffeomorphism group with advected quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler-Poincaré theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied …

拉格朗日坐标下一维浅水磁流体力学方程的对称性与守恒律

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

本文研究一维浅水磁流体力学方程在拉格朗日坐标下的对称性分类与守恒律，利用Noether定理构造新守恒律，并给出不变解和部分不变解。

Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to…

Variational formulations of sound-proof models

We derive a family of ideal (nondissipative) 3D sound-proof fluid models that includes both the Lipps-Hemler anelastic approximation (AA) and the Durran pseudo-incompressible approximation (PIA). This family of models arises in the Euler-Poincaré framework involving a constrained Hamilton's principle expressed in the Eulerian fluid description. The derivation in this framework establishes the following properties of each member of the entire family: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, a Lie-Poisson Hamiltonian formulation possessing con…

变分声学

Variational Acoustics

本章从经典连续介质力学原始拉格朗日量系统推导声学控制方程，回顾拉格朗日框架下可压缩无耗散流体的作用量构造，通过摄动展开从变分原理恢复线性声学方程，得到弱非线性近似，理想气体结果为特例。