The Hamiltonian description begins with a canonically conjugate set of fluid el- ement variables ... Lagrangian equilibria are characterized by and. Stability ... MHD equilibria with flow, i.e. stationary Eulerian equilibria, clearly do not correspond ...
Lagrangian, Eulerian, and Dynamically Accessible Stability of MHD flows T.
1 Andreussi ,
1
LAGRANGIAN STABILITY
STABILITY COMPARISON
We consider what happens to the Hamiltonian formalism in terms of (Q, Π) when an expansion about a given reference trajectory is performed as follows:
Upon comparing the set of first order perturbations of the Eulerian variable, we see that, in general, the following is true
where η and πη are the displacements in the linear energy principles. Following Ref. [5], the second variation of the Hamiltonian in terms of the canonically conjugate variables (η,πη) results
Moreover, it can be shown that the three corresponding energy expressions are equivalent upon considering the same perturbations. Thus,
P. J. Morrison2, F. Pegoraro3
Alta S.p.A., Pisa, Italy 2 Phys. Dept. and IFS, University of Texas, Austin, USA 3 Phys. Dept., Pisa University, Pisa, Italy
RELABELING TRANSFORMATION The Hamiltonian description begins with a canonically conjugate set of fluid element variables (q(a, t), π(a, t)) where a labels a fluid element. In terms of these variables the Poisson bracket is canonical and has the form
where the quadratic form δWfr is that obtained by Frieman and Rotenberg [2]. A sufficient condition for stability is given by δWfr>0. The first order Eulerian perturbations that are induced by the Lagrangian variation are given as follows:
This Poisson bracket, together with the Hamiltonian
where the subscript 0 indicates the label configuration and , gives the MHD equations. Lagrangian equilibria are characterized by and Stability of these static configurations is assured if the equilibrium position qe is a local minimum of the potential energy (the so-called "δW energy principle" [1]) MHD equilibria with flow, i.e. stationary Eulerian equilibria, clearly do not correspond to Lagrangian equilibria.
EULERIAN STABILITY: SYMMETRIC CASE As shown in Ref. [4][7], symmetric configurations possess families of Casimir invariants Ci and the resulting energy-Casimir functional .
can be exploited to obtain MHD equilibrium equations (first variation) and stability conditions (second variation).
All the quadratic forms are in fact the same physical energy contained in a perturbation away from an equilibrium state, but how much of that energy can be tapped depends on the constraints embodied in the perturbations.
AN EXAMPLE: STABILITY AGAINST CONVECTION As shown in Ref. [3], in the presence of an external gravity force, a fluid in mechanical equilibrium (with dρ/dy0. The same problem can be investigated with the energy-Casimir approach and with the Lagrangian and Dynamically Accessible formulations (which, for static equilibria, are identical). Energy-Casimir Hydrodynamic equilibria with translational symmetry are extrema of
We consider v=0 and, after some manipulations, the second variation results
The perturbed variables are arbitrary and completely independent of each other, except we have built in For typical fluids ∂ρ/∂s|p0 (see Ref. [6]).
REFERENCES [1] [2] [3] [4] [5] [6] [7]
I.B. Bernstein, E.A. Frieman, M.D. Kruskal, R.M. Kulsurd, Proc. R. Soc. Lond. A 244 17 (1958), E. Frieman, M. Rotenberg, Rev. Mod. Phys. 32 898 (1960), L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, 1987), T. Andreussi, P.J. Morrison, F. Pegoraro, Phys. Plasmas 19 052102 (2012), P.J. Morrison, Rev. Mod. Phys. 70 467 (1998), E. Hameiri, Phys. Plasmas 10 2643 (2003), T. Andreussi, P.J. Morrison, F. Pegoraro, Phys. Plasmas 20 092104 (2013).
