Model order reduction approach to one-dimensional collisionless closure problem Camille Gillot, Guilhem Dif-Pradalier, Xavier Garbet, Philippe Ghendrih, Virginie Grandgirard, Yanick Sarazin To cite this version: Camille Gillot, Guilhem Dif-Pradalier, Xavier Garbet, Philippe Ghendrih, Virginie Grandgirard, et al.. Model order reduction approach to one-dimensional collisionless closure problem. 2021. hal-03099863 HAL Id: hal-03099863 https://hal.archives-ouvertes.fr/hal-03099863 Preprint submitted on 6 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Model order reduction approach to one-dimensional collisionless closure problem C. Gillot,1, 2, a) G. Dif-Pradalier,1, b) X. Garbet,1 P. Ghendrih,1 V. Grandgirard,1 and Y. Sarazin1 1)CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France 2)École des Ponts ParisTech, F-77455 Champs sur Marne, France (Dated: 18th October 2020) The problem of the fluid closure for the collisionless linear Vlasov system is investigated using a perspective from control theory and model order reduction. The balanced truncation method is applied to the 1D–1V Vlasov system. The first few reduction singular values are well-separated, indicating potentially low-dimensional dynamics. To avoid large- dimensional numerical work, a reduced model is formulated using rational interpolation, generalising the seminal work from Hammett and Perkins. The resulting models are found to outperform the state-of-the-art models for thermal phase velocities. Thanks to the versatility of this formulation, an application to toroidal gyrokinetic dynamics is discussed. I. INTRODUCTION observables are the fluid moments. Balanced truncation de- velops the distribution on components ranked by their dy- First-principle simulation of plasma turbulence faces a namical reachability and observability through moments. By double challenge of cost and complexity. The physical and construction, the balanced reduced model only depends on intellectual complexity stem from the non-linearity of the tur- the ground Vlasov equation and on the chosen observables. bulent dynamics. The cost is induced by the kinetic nature of Rational interpolation rather constructs a phenomenological the instability drive. Simulating the gyrokinetic description of model, matching the kinetic dispersion relation at well-chosen tokamak turbulence is now routine for numerous codes, but re- phase velocities. As such, it provides an effective generalisa- quires maintaining high-performance codes running smoothly tion of the Padé-based methods to an arbitrary set of expansion on supercomputers. While simulations get longer and longer, points. the amount of actually analysed information remains limited in In a first part, the two methods are reviewed and their con- proportion. A few fluid moments of the distribution function nection to the usual collisionless closures are discussed. In a (density, velocity, pressure. ) are extracted out of the tens second part, the two methods are applied numerically to the of grid points used to simulate dynamics in velocity space, 1D–1V Vlasov–Poisson problem. The precision of the recon- and the rest is discarded. Reaching more day-to-day investiga- struction and temporal evolution in the Vlasov problem are tion of turbulent transport requires lighter models for tractable compared. The influence of the model on the behaviour of simulations. However, the obvious fluid systems fail to sim- the Vlasov–Poisson problem is discussed, using the Landau ulate dynamics close to the marginal instability threshold1. damping and bump-on-tail instability as benchmarks. Finally, Collisionless closures are a way forward to avoid this short- the extensions to the toroidal gyrokinetic problem and to non- fall while retaining dynamics in a low-dimensional space2,3. linear simulations are discussed. Those so-called Landau-fluid models have been applied both to magnetohydrodynamics4–7 and to gyro-fluids8,9. The collisionless closure problem arises due to the non- II. REDUCTION OF THE KINETIC EQUATION determination of the fluid moment ℓ ¸ 1 in the ℓ-fluid reduc- tion. The goal is to find the best linear combination to recover We consider the linear one-dimensional electrostatic Vlasov relevant kinetic effects, to recover the kinetic response kin ¹Zº problem as a function of the phase velocity Z. Several methods have 10 0 been suggested in the literature . Asymptotic methods use a mC 5 ¸ 8:E 5 = 8:F q (1) Taylor2,11 or Padé12,13 expansion of the kinetic response func- : F 0¹Eº tion kin for Z 1 and Z 1 to constrain the free parameters. where is the spatial wave-number, is the derivative As such, they explicitly choose an asymptotic frequency range, of the equilibrium distribution function with respect to the and are limited to it. Other authors have introduced the excita- velocity, 5 ¹Eº the fluctuation of the distribution function, q tion frequency as a parameter14,15. This renders the simulation the electrostatic potential. The velocity is normalised to the of the closed model very difficult, because strongly non-linear. thermal velocity, and the potential to the thermal potential. These methods have an even larger number of free parameters, Without loss of generality, we will only consider one value of and rely heavily on physical choices and orderings11. :¡ 0, the other cases can be deduced by symmetry. In this paper, we will discuss two different methods, namely We consider a choice of observable quantities, for instance balanced truncation16–18 and rational interpolation19–21. The a few fluid moments like the density =, the velocity D, the balanced truncation method considers the Vlasov equation as pressure ?. The electric potential q is considered as an input, a control system. The forcing is the electric potential, and unconstrained by Poisson equation, and forces the dynamics of the state 5 . The mapping from the electric potential q to these moments is a linear time-invariant dynamical system. This mapping corresponds to an open-loop control system. The a)Electronic mail: [email protected]. eventual Poisson equation provides a closed-loop condition, b)Electronic mail: [email protected]. feeding back the observable = into the input q. Control theory 2 and model order reduction theory have developed useful tools This transfer function is real, all its poles are on the real axis, for analysing the structure of such systems, and for constructing so the model is conservative: it is not able to reproduce a reduced-order models matching such an input–output relation damping mechanism as in the kinetic case. It should be noted q 7! ¹=, D, ?º, without solving for the full state space, here the that even- and odd-dimensional models in this family behave distribution function 5 22. differently. Even-dimensional models correctly reproduce the The frequency response of the density from the Vlasov dy- adiabatic response = = −q for l ! 0. Odd-dimensional namics can be computed analytically as models are more precise at high frequency, at the expense of an incorrect adiabatic response. For instance, the 5-dimensional ¹ ¸1 F 0¹Eº =¹Zº = q¹Zº dE (2) fluid model writes E − Z − 8 f −∞ j: j ¹ ¹E5 − 10E3 ¸ 15Eº 5 dE ≈ 0 (8) with Z = l/j:j the phase velocity of the perturbation, nor- malised to the thermal velocity. We have introduced a small Z 2 − 7 f Fluid 5 ¹Zº = (9) damping rate . The integral on the right-hand-side is sin- Z 4 − 10Z 2 ¸ 15 gular if f = 0. The so-called Landau prescription enables to evaluate it for positive damping rate f. This corresponds to It correctly decays asymptotically as 1/Z 2, but the adiabatic the Laplace transform formalism, and enforces causality. The response is ¹Z = 0º = −7/15 instead of the correct −1. resulting =/q function is holomorphic, and can be extended to the negative complex plane. When the equilibrium distribution function is a Maxwellian, IV. REVIEW OF THE BALANCED TRUNCATION METHOD the frequency response if expressed using the Fried and Conte function Z23–25. Balanced truncation allows approaching the model reduc- = Z Z tion problem as a linear optimisation problem18. Instead of a kin ¹Zº = = −1 − p Z p (3) q 2 2 term-by-term matching of some expansion, it can be thought as a uniformly-weighted matching. This method is systematic, and only depends on (i) the original model equation, (ii) an III. THE FLUID HIERARCHY TRUNCATION energy functional on the input variables and (iii) a quadratic functional on the output variables. The reduced model is con- In order to reduce the simulation cost, we need a reduction structed by removing hard-to-reach and hard-to-observe states step. The traditional method involves the formulation of the from the dynamics. fluid hierarchy. It consists in the projection of the Vlasov equation (1) against graded polynomials. For instance, the fluid moments <ℓ verify A. Notions of reachability and observability ¹ ℓ <ℓ = E 5 dE (4) Given a potential perturbation q¹Cº, the response of the ¹ distribution function is given by ℓ−1 mC <ℓ ¸ 8:<ℓ¸1 = −8:qℓ E F dE (5) ¹ 5 ¹C, Eº = 6g ¹Eº8:q¹C − gºdg (10) where < is the density =, < is the momentum # D, < is 0 1 eq 2 −8:E g−fg 0 the pressure ?. The equation for each moment <ℓ requires the 6g ¹Eº = e F ¹Eº (11) one for the next moment, ad infinitum. Practical applications require to truncate this hierarchy somehow. 6 defines an infinite family of distribution functions, indexed The last relevant moment has to be expressed using the by g.