Brøns, Morten

Dam, Magnus

Juul Rasmussen, Jens

Naulin, Volker

Xu, Guosheng

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Bifurcation Analysis and Dimension Reduction of a Predator-Prey Model for the L-HTransitionDam, Magnus; Brøns, Morten; Rasmussen, Jens Juul; Naulin, Volker; Xu, GuoshengPublication date:2014Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Dam, M., Brøns, M., Juul Rasmussen, J., Naulin, V., & Xu, G. (2014). Bifurcation Analysis and DimensionReduction of a Predator-Prey Model for the L-H Transition. Poster session presented at 51st Culham PlasmaPhysics Summer School, Abingdon, Oxfordshire, United Kingdom.Bifurcation Analysis and Dimension Reduction of aPredator-Prey Model for the L-H TransitionMagnus Dam1, Morten Brøns1, Jens Juul Rasmussen2, Volker Naulin2, and Guosheng Xu31DTU Compute, Technical University of Denmark, Kgs. Lyngby, Denmark; 2Association Euratom-DTU, DTU Physics, Technical University of Denmark, Kgs. Lyngby, Denmark;3Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China. Email: magnusd@dtu.dkThe L-H transition denotes a shift to an improved confinement state of a toroidal plasma in a fusion reactor. A model of the L-H transitionis required to simulate the time dependence of tokamak discharges that include the L-H transition. A 3-ODE predator-prey type model ofthe L-H transition is investigated with bifurcation theory of dynamical systems. The model is recognized as a slow-fast system.INTRODUCTIONThe L- and H-modes1 are confinement statesof a toroidal plasma, referring to states of lowand high confinement, respectively. The tran-sition from the L- to the H-mode is called theL-H transition. The T-mode is a transient, in-termediate mode between the L- and H-mode,characterized by an oscillatory behavior. TheL-H transition still lacks a first principle ex-planation. Modeling the L-H transition mightcontribute to a better understanding of theunderlying mechanisms.0 0.2 0.4 0.6 0.8 1PedestalNormalized radius r/aPlasmapressure  H-mode profileL-mode profileFigure: Sketch of theradial pressure profiles inL- and H-mode.THE 3-ODE L-H TRANSITION MODELWe consider the minimal 3-ODE predator-prey type L-H transitionmodel suggested by Kim and Diamond [2, 3]. The model ignoresspatial dependencies. The dependent variables in the model are:I the drift wave turbulence level E ,I the shear of the zonal flow Vzf, andI the gradient of the ion pressure N .The model can be formulated asddtE = E (N − a1E − a2c23N 4 − a3V 2zf) ,ddtVzf = Vzf(b1E1 + b2c23N 4− b3),ddtN = Q(t)−N (c1E + c2) ,where ai, bi, ci, i = 1, 2, 3 are parameters and Q is the heating power.Introducing new variables and time,u = a1a1/32 c2/33 E , v = a1/32 a3c2/33 V 2zf,w = a1/32 c2/33 N , s = a−1/32 c−2/33 t,results in the non-dimensionalized systemu˙ = u(w − u− v − w4)v˙ = µ1v(u1 + µ4w4− µ2)w˙ = µ5 (σ − w(1 + µ3u))Here, µi, i = 1, . . . , 5 are new parameters and σ is the rescaledheating power.BIFURCATION ANALYSISThe nullclines areNu = {u = 0} ∪{u = w(1− w3)− v} ,Nv = {v = 0} ∪{u = µ2(1 + µ4w4)},Nw = {w(1 + µ3u) = σ} .Stability of the equilibrium points:I L is a stable node when it is below Nv.I H is always a saddle (unstable).I T is a focus point. Stability depends onthe value of σ.I QH is stable for σ > 1.0 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.5LTHQHwu  NuNvNwFigure: Nullclines andequilibrium pointsprojected onto theuw-plane.THREE TRANSITION TYPESThe bifurcation diagram structure depends on µi, i = 1, . . . , 5. Byvarying µ2 and µ3 the three different transition types are observed.0 0.5 1 1.5 200.10.20.30.40.5SNAHTC3TC2TC1(a)σ  0 0.5 1 1.5 200.10.20.30.40.5SNTC2(b)σ  0 0.5 1 1.5 200.10.20.30.40.5PF(c)σ  u∗v∗w∗/4u∗v∗w∗/4u∗v∗w∗/4Figure: Bifurcation diagrams showing the three types of transitions: (a) anoscillating transtion, (b) a sharp transition allowing hysteresis, and (c) a smoothtransition. Stable equilibria are shown as solid curves and unstable equilibria asdashed curves.0 0.5 1 1.5 200.20.40.60.81(a)σ  uvw/40 0.5 1 1.5 200.10.20.30.40.5(b)σ  uvw/40 0.5 1 1.5 200.10.20.30.40.5(c)σ  uvw/4Figure: Numerical solutions showing the three types of transitions: (a) anoscillating, (b) a sharp, and (c) a smooth transition.DIMENSION REDUCTION WITH GSPTPut µ5 = 1ε, where 0 < ε 1.I u and v are slow variables,I w is a fast variable.For ε > 0, but sufficiently small,solutions converge to the slowmanifold,Mε =M0 + εM1 + ε2M2 + · · ·The reduced system of the flow isfound by taking the limit ε→ 0:u˙ = u(w − u− v − w4)v˙ = µ1v(u1 + µ4w4− µ2)w =σ1 + µ3uThe reduced system contains thesame dynamics as the full system.QHTLwvu0 0.1 0.2 0.3 0.400.40.81.200.20.40.60.8Figure: The critical manifold,M0 = {w = σ1+µ3u} and anumerical solution.CONCLUSIONKim and Diamond’s 3-ODE L-H transition model was investigatedwith bifurcation theory.I The model contains three types of transitions: an oscillating, asharp with hysteresis, and a smooth transition.I The system can be reduced to a 2-ODE systemA spatio-temporal L-H transition model has been proposed by Miki etal. [4].References:1F. Wagner, G. Becker, K. Behringer, et al., Phys. Rev. Lett. 49, 1408 (1982).2E.-J. Kim and P. H. Diamond, Phys. Plasmas 10, 1698 (2003).3E.-J. Kim and P. H. Diamond, Phys. Rev. Lett. 90, 185006 (2003).4K. Miki, P. H. Diamond, O. D. Guercan, et al., Phys. Plasmas 19, 092306 (2012).

Bifurcation Analysis and Dimension Reduction of a Predator-Prey Model for the L-H Transition

Magnus Dam

Morten Brøns

Jens Juul Rasmussen

Volker Naulin

Guosheng Xu

Crossref

Physics of Plasmas

Bifurcation analysis and dimension reduction of a predator-prey model for the L-H transition

The L-H transition denotes a shift to an improved confinement state of a toroidal plasma in a fusion reactor. A model of the L-H transition is required to simulate the time dependence of tokamak discharges that include the L-H transition. A 3-ODE predator-prey type model of the L-H transition is investigated with bifurcation theory of dynamical systems. The analysis shows that the model contains three types of transitions: an oscillating transition, a sharp transition with hysteresis, and a smooth transition. The model is recognized as a slow-fast system. A reduced 2-ODE model consisting of the full model restricted to the flow on the critical manifold is found to contain all the same dynamics as the full model. This means that all the dynamics in the system is essentially 2-dimensional, and a minimal model of the L-H transition could be a 2-ODE model

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Bifurcation analysis and dimension reduction of a predator-prey model for the L-H transition

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Online Research Database In Technology