On Hamiltonian and Action Principle formulations of plasma fluid models ICTS Seminar Manasvi Lingam Institute for Theory and Computation Harvard University May 23, 2018 Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 1 / 41
Outline Why use Hamiltonian and Lagrangian methods? 1 On the ideal MHD Lagrangian 2 On the ideal MHD Hamiltonian 3 A unified action for extended MHD models 4 A unified Hamiltonian formulation for extended MHD models 5 Conclusions 6 Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 2 / 41
Why use Hamiltonian and Lagrangian methods? A reason for studying Hamiltonian and Action Principle formulations Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 3 / 41
Why use Hamiltonian and Lagrangian methods? A reason for studying Hamiltonian and Action Principle formulations Because we can . . . (to paraphrase Mallory’s famous comment about climbing Mt. Everest: “Because it’s there”) Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 3 / 41
Why use Hamiltonian and Lagrangian methods? Advantages of Hamiltonian and Action Principle formulations for plasmas Action principles: The construction of reduced models from a “parent” model - introduce new terms/impose orderings directly. Eliminate ‘fake’ dissipation, and associated spurious instabilities. Many plasma models do not even conserve energy. The action can be suitably discretized to construct variational integrators. Hamiltonian formulations: Well-suited for calculating plasma equilibria as well as analyzing their stability via the Energy-Casimir method. Can be used to establish underlying connections between outwardly different models. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 4 / 41
Why use Hamiltonian and Lagrangian methods? Advantages of Hamiltonian and Action Principle formulations for plasmas Credit: Kraus & Maj (2017); arXiv:1707.03227 Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 5 / 41
On the ideal MHD Lagrangian Why use Hamiltonian and Lagrangian methods? 1 On the ideal MHD Lagrangian 2 On the ideal MHD Hamiltonian 3 A unified action for extended MHD models 4 A unified Hamiltonian formulation for extended MHD models 5 Conclusions 6 Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 6 / 41
On the ideal MHD Lagrangian Eulerian and Lagrangian “viewpoints” for magnetofluids Lagrangian viewpoint: Fluid - continuum of particles; each ‘particle’ described by the coordinate q ( a , t ), where ‘ a ’ is the label. Attributes: Properties attached to the particle before it commences its trajectory; depend only on the label a and denoted by subscript ‘0’. Examples: ρ 0 ( a ), s 0 ( a ), etc. Eulerian viewpoint: Fluid variables are functions of r and t . These serve as observables since they can tracked. Examples: ρ ( r , t ), s ( r , t ), etc. Require a means of transitioning between these two viewpoints. This is accomplished via Lagrange-Euler maps. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 7 / 41
On the ideal MHD Lagrangian Lagrange - Euler maps The position and velocity in the two descriptions are identical, i.e. q ( a , t ); RHS is evaluated at a = q − 1 ( r , t ). r = q ( a , t ) and v ( r , t ) = ˙ Relations between attributes and their corresponding observables determined via imposition of physical laws. Locally, mass conservation dictates ρ 0 ( a ) d 3 a = ρ ( r , t ) d 3 r (1) and this leads to ρ = ρ 0 / J , where J is the Jacobian. Upon using the fact that D J / Dt = ( ∇ · v ) J , and simplifying further, the continuity equation ∂ρ ∂ t + ∇ · ( ρ v ) = 0 , (2) can be obtained. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 8 / 41
On the ideal MHD Lagrangian Lagrange - Euler maps (contd) Specific entropy (entropy per unit mass) follows via s ( r , t ) = s 0 , i.e. advection along streamlines. This leads to ∂ s ∂ t + v · ∇ s = 0 , (3) In the case of ideal MHD, the magnetic flux is assumed to be ‘frozen-in’: B 0 ( a ) · d 2 a = B · d 2 r (4) which leads to B i = ∂ q i /∂ a j B j 0 / J and this leads to the ideal MHD induction equation ∂ B ∂ t − ∇ × ( v × B ) = 0 , (5) after using the ∇ · B = 0 condition. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 9 / 41
On the ideal MHD Lagrangian Formulating the ideal MHD action Pick the domain and the choice of observables/attributes at first. Build each term in the action by appealing to physical reasoning. All terms must satisfy the Eulerian Closure Principle (ECP) - the action should be entirely expressible in Eulerian variables after using the Lagrange-Euler maps. The kinetic energy density has the property T [ q ] = 1 � q | 2 ↔ 1 � d 3 a ρ 0 | ˙ d 3 r ρ | v | 2 (6) 2 2 D D Construct the action, vary with respect to q and Eulerianize the equation(s) of motion. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 10 / 41
On the ideal MHD Lagrangian Formulating the ideal MHD action The potential energy term is given by , k B j q i , j q i 0 B k � 0 d 3 a ρ 0 U ( ρ 0 / J , s 0 ) + V [ q ] = 2 J D d 3 r ρ U ( ρ, s ) + | B | 2 � = (7) 2 D The action for ideal MHD is therefore given by � t 1 S = ( T [ q ] − V [ q ]) dt (8) t 0 The dynamical equation(s) found from δ S = 0. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 11 / 41
On the ideal MHD Lagrangian The ideal MHD equations We have seen earlier that the Eulerian equations for the density, entropy and the magnetic field follow automatically as a result of imposing local mass conservation, invariance and flux conservation respectively. Taking the variation of S leads to the Euler-Lagrange equation: � ρ 2 ∂ ∂ U � q i + A j 0 ρ 0 ¨ + · · · = 0 , (9) i J 2 ∂ a j ∂ρ and upon using the Euler-Lagrange maps and several identities (e.g. Morrison 2009), the MHD dynamical equation for the velocity is obtained: � ∂ v � ρ ∂ t + v · ∇ v = −∇ p + J × B . (10) Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 12 / 41
On the ideal MHD Lagrangian A brief comment on gyroviscosity In addition to the term quadratic in v , one can also introduce a term linear in v in the action, i.e. of the form � d 3 r v · M ⋆ , S = (11) D where M ⋆ can be viewed as an intrinsic momentum density. In the specific scenario where M ⋆ is expressible as ∇ × L ⋆ , choosing a suitable ansatz for L ⋆ leads to the inclusion of gyroviscosity - an important plasma term. In a simplified 2D limit, the gyroviscosity is given by � M j � π ls = N sjlk β∂ k ρ m N sjlk = 2 e ( δ sk ǫ jl − δ jl ǫ sk ) . (12) Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 13 / 41
On the ideal MHD Hamiltonian Why use Hamiltonian and Lagrangian methods? 1 On the ideal MHD Lagrangian 2 On the ideal MHD Hamiltonian 3 A unified action for extended MHD models 4 A unified Hamiltonian formulation for extended MHD models 5 Conclusions 6 Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 14 / 41
On the ideal MHD Hamiltonian Towards the Hamiltonian formulation Compute the canonical momentum Π = ∂ L /∂ ˙ q . Carry out a Legendre transform of the Lagrangian to obtain the Hamiltonian (in Lagrangian variables), and express it in terms of Eulerian variables. The Hamiltonian of ideal MHD is given by � ρ v 2 + ρ U ( ρ, s ) + B 2 � � d 3 r H = (13) 2 2 D These terms represent the kinetic, internal and magnetic energy densities respectively. It is more advantageous sometimes to work with σ = ρ s and M = ρ v . Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 15 / 41
On the ideal MHD Hamiltonian Towards the Hamiltonian formulation (contd) As noted earlier, physical functionals ought to be equally expressible in terms of Eulerian and Lagrangian variables: d 3 a δ ¯ δ Π · δ Π + δ ¯ � F F δ ¯ F ≡ δ q · δ q D � d 3 r δ F δρ δρ + δ F δσ δσ + δ F δ M · δ M + δ F = δ F ≡ δ B · δ B . (14) D Compute Lagrangian functional derivatives in terms of Eulerian ones, and map to find the Eulerian Poisson bracket. For example, the Euler-Lagrange map for the density can be written in integral form: � d 3 a ρ 0 δ ( r − q ) . ρ = (15) D Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 16 / 41
On the ideal MHD Hamiltonian Towards the Hamiltonian formulation (contd) The variation in the density is therefore given by � d 3 a ρ 0 ∇ δ ( r − q ) δ q δρ = − (16) D d 3 a δ ¯ � F � d 3 r δ F � d 3 a ρ 0 ∇ δ ( r − q ) δ q + . . . δ q · δ q + · · · = − δρ δρ D D D (17) Interchanging the order of integration and equating the coefficients of δ q and δ Π enables us to determine δ F /δ q and δ F /δ Π in terms of Eulerian functional derivatives such as δ F /δρ . . . . We plug in δ F /δ q and δ F /δ Π into the canonical Lagrangian-variable Poisson bracket to obtain the Poisson bracket of ideal MHD in Eulerian variables. Manasvi Lingam (Harvard University) HAP formulations for plasmas May 23, 2018 17 / 41
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