TURBULENT TRANSPORT THEORY


 Clemence Kelley
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1 ASDEX Upgrade MaxPlanckInstitut für Plasmaphysik TURBULENT TRANSPORT THEORY C. Angioni GYRO, J. Candy and R.E. Waltz, GA
2 The problem of Transport Transport is the physics subject which studies the physical processes by which particles, momentum and energy are moved (transported) in a domain in the real space The goal is to identify the relationship between thermodynamic fluxes and thermodynamic forces Thermodynamic fluxes are particle, momentum and energy (heat) fluxes In present tokamaks, these are mainly imposed by external means ( particle sources, torque and heating powers) Thermodynamic forces are the spatial gradients of the particle and momentum densities and of the pressures (temperatures) These describe how the plasma reacts to the imposed fluxes, namely the radial profiles of density, rotation and temperature
3 Transport in fusion plasmas Due to the different behaviour of charged particles along and across a magnetic field, transport in fusion plasmas is strongly anisotropic Almost infinite in the direction parallel to the magnetic field, Small in the direction perpendicular to the magnetic field Parallel transport is large enough to strongly limit the variation of density and pressure along the field lines (on the flux surfaces) How large is the perpendicular (radial) transport is what determines the confinement properties of the plasma Collisional ( classical / neoclassical ) transport mechanisms provide a minimum unavoidable level of radial transport, which is lower than the measured values by almost one order of magnitude for the ion heat transport by at least two orders of magnitude for the electron heat transport
4 Outline Time scales relevant for turbulent transport Microinstabilities Turbulence and transport Main turbulence stabilisation mechanisms
5 Time scales : beyond neoclassical transport Time [s] Electrons Ions confinement 1 confinement Neoclassics 1 2 collision frequency collision frequency orbit frequency orbit frequency Cyclotron frequency Microinstabilities turbulence Cyclotron frequency plasma frequency plasma frequency Quasineutrality
6 Time scales : beyond neoclassical transport Electrons confinement Time [s] Ions 1 confinement 1 2 Parallel time scales Perpendicular time scales k v Dj j*k v collision frequency orbit frequency Cyclotron frequency plasma frequency collision frequency orbit frequency 1 5 Cyclotron frequency plasma frequency 1 8 ω b i =k ω b e =k =v i =v e trapped & passing ions trapped & passing electrons ω * j ω Dj
7 MICROINSTABILITIES GYRO, J. Candy and R.E. Waltz, GA
8 Particle orbits Dominant motion is gyromotion (magnetized plasma) Any force F implies a drift of the particle in a direction perpendicular to both the field B and the force F parallel motion curvature drift ExB drift grad B drift } drifts F B Mirror force also leads to particle trapping (banana orbits) since the magnetic field increases towards the axis of the torus
9 Gyrokinetic equation General kinetic equation Gyrokinetic equation : gyroaverage, f = F FM Linear gyrokinetic equation in simplified geometry, B = B R / R b
10 Normalized logarithmic gradients Gyrokinetic equation : gyroaverage, f = F FM Linear gyrokinetic equation in simplified geometry, B = B R / R b The (radial) gradient of the Maxwellian = E where 1 L X = X X and LB = R
11 Fluid model in simplified geometry Continuity (zero moment) Parallel velocity moment Energy balance ( second order moment ) Linear gyrokinetic equation in simplified geometry, B = B R / R ey
12 Fluid model in simplified geometry Continuity (zero moment) Parallel velocity moment Energy balance ( second order moment ) Linear gyrokinetic equation in simplified geometry, B = B R / R ey
13 Fluid model in simplified geometry Continuity (zero moment) Parallel velocity moment Energy balance ( second order moment ) Linear gyrokinetic equation in simplified geometry, B = B R / R ey
14 Fluid model in simplified geometry Continuity (zero moment) Parallel velocity moment Energy balance ( second order moment ) Linear gyrokinetic equation in simplified geometry, B = B R / R ey
15 Fluid model in simplified geometry Continuity (zero moment) Parallel velocity moment Energy balance ( second order moment ) Linear gyrokinetic equation in simplified geometry, B = B R / R ey
16 Normalizations, parallel and perpendicular time scales We are considering harmonic fluctuations exp( ik x  iω t ) Perpendicular motion characterisitc frequency is the drift frequency = / Parallel motion : = 4 Ions and electrons: same mobility perpendicular to the field line, very different mobility along the field line Parallel velocity moment ( parallel force balance ) Ions : inertia is dominant zero order, negligible parallel motion Electrons : parallel streaming is dominant adiabatic response
17 Parallel and perpendicular time scales, trapping Motion to B : Drift frequency Motion to B : = Passing particles Trapped particles (Mirror force due to B 1/R) & Ions: bounce time longer than the characteristic time zero order no difference between passing and trapped Electrons : many bounces in a characteristic time different behaviour between passing and trapped electrons
18 Passing electrons : adiabatic response Fast unconstrained motion parallel to B Parallel force balance for passing electrons Balance of dominant terms δn ep = or in nonnormalized form n ep e φ T e
19 Ions : continuity and energy balance Neglect parallel motion (strong inertia) = Continuity and energy balance ~ ~ Quasineutrality condition n = n Assuming (for the moment) that all electrons are passing ( more precisely adiabatic ), quasineutrality becomes we obtain a homogeneous system of three equations in three unknowns e i
20 Mechanism leading to instability We look for the eigenvalues of this homogeneous system Can be solved analytically (trivial) here we focus on the basic coupling mechanism leading to an instability
21 Mechanism leading to instability We look for the eigenvalues of this homogeneous system imaginary roots pure growing mode γ ^ =
22 Ion Temperature Gradient mode Initial Temperature perturbation e y ex B ^ T < Cold plasma T Hot plasma ^ T > ^ T < T Cold plasma
23 Ion Temperature Gradient mode curvature and B drift Initial Temperature perturbation Generates a density perturbation e y ex v D B Cold plasma T Hot plasma ^ T < ^ T > ^ T < ^ n > ^ n < ^ n > Cold plasma T
24 Ion Temperature Gradient mode curvature and B drift ExB flow advection Initial Temperature perturbation Generates a density perturbation Passing electrons neutralise the charge separation e y ex v D B Cold plasma T Hot plasma ^ T < ^ T > ^ T < ^ n > ^ n < ^ n > Cold plasma T  E +++
25 Ion Temperature Gradient mode curvature and B drift ExB flow advection Initial Temperature perturbation Generates a density perturbation Passing electrons neutralise the charge separation Parallel force balance implies an electrostatic potential, the ExB flow enhances the perturbation e y ex v D B Cold plasma T ^ T < v ExB v ExB n ^ >  ^ T > ^ E Hot n < ^ +++ T < plasma ^ n > Cold plasma T
26 Ion Temperature Gradient mode e y ex STABLE SIDE v ExB ^ T < ^ T > ^ T < Cold plasma T v D Hot plasma ^ T < B ^ T < ^ T > UNSTABLE SIDE T v ExB Cold plasma Low field side unstable ( bad curvature ) region : warm plasma moves in the warm regions High field side stable region (reversed temperature gradient) warm plasma moves in the cold regions
27 ITG : what we neglected up to now... Back to our simple fluid model By keeping all the terms, we would have found that the eigenvalues are not just imaginary, but complex numbers in which an imaginary part (an unstable mode) occurs provided that the normalized logarithmic temperature gradient R/LT exceeds a certain value (threshold) (otherwise only real roots)
28 ITG mode : the threshold The mode is stable for values of R/LT smaller than the threshold value 1 (R/LT critical ).8 The threshold value is not a universal number, but depends itself on plasma parameters γ [c s / R] In particular it increases with increasing T i / T e and for adiabatic electrons increases with increasing R/Ln ( η mode, η = L n / L T ) i i GS R/L Ti
29 ITG mode : the spectrum Simple (reduced) fluid model γ γ = ω D = k θ ρ s c s γ is an increasing linear function of the wave number γ [c s / R] ω r [c s / R] k θ ρ i GS2
30 ITG mode : the spectrum Simple (reduced) fluid model γ γ = ω D = k θ ρ s c s γ is an increasing linear function of the wave number Parallel dynamics and finite Larmor radius effects, which were neglected in the simple fluid model, modify significantly the spectrum γ [c s / R] ω r [c s / R] stabilized by parallel dynamics stabilized by FLR k θ ρ i GS2
31 Kinetic electrons Up to now we have considered only the ITG mode with adiabatic electrons The inclusion of the electron dynamics gives rise to other modes : the trapped electron mode ( TEM ) and the electron temperature gradient (ETG) mode For these modes similar simple models can be considered, as we made for the ITG, namely assuming now adiabatic ions In reality, all these modes can occur concurrently, although at different scales, and with different dependences on plasma parameters
32 TEM and ETG These are both electron instabilities ( driven by an electron temperature gradient above a critical threshold) The TEM instability can be considered as the analogous of the ITG instability, still at the ion Larmor radius scale, but where the slow (average) motion along the field line is caused by trapping rather than by inertia The ETG instability is the analogous of the ITG at the electron Larmor radius scale
33 Trapped electron mode (TEM) We have seen that the electron bounce time is small compared to the drift frequency Trapped particles (Mirror force due to B 1/R) & Bounce averaged parallel motion of trapped electrons is slow (similar to ions, for which the cause of slow parallel motion is inertia ) Trapped electrons therefore are not fast enough along the field line to ensure the adiabaticity condition A picture completely analogous to the ITG leads us to an instability whose drive is R/LTe
34 Trapped electron mode : density gradient drive This is not the end of the story When both fluctuations on the ion density and on the trapped electron density are present at the same time, another instability occurs in the presence of a density gradient, the R/Ln driven TEM ( ubiquitous mode ) With the simple fluid model, you can get a simple physical picture of this mode by considering the coupling mechanisms between the continuity equations of ion and trapped electron density fluctuations The instability arizes due to the fact that the vertical drift is in opposite directions for ions and electrons Also this instability involves a threshold, namely a critical logarithmic density gradient
35 Instability diagram, 1 or 2 modes can be unstable (Te = Ti, R/LTe = R/LTi) 1 5 R/L T ITG 1 root stable ITG & TEM 2 roots TEM 1 root R/L n Kinezero [ C. Bourdelle et al., NF 42, 892 (22) ] Typical values in experiments (no ITBs) at mid radius R/LT [4 7] in HMode, R/Ln [ 7], R/LTe [ 5 12] in Lmode In many exp. conditions, two modes are unstable
36 Instabilities at the different scales ITG / TEM & ETG Different instability domains and main drives Sources of free energy R/LTi, R/LTe, R/Ln instability drive k θρ i ITG destabilised by TEM R/L Ti R/L Te & pure ITG stabilised by R/L n R/L n stabilised by ei collisions {~ unaffected by collisions T / T e i ETG R/L Te stabilised by R/L n stabilised by ITG and TEM scale of the order of the ion Larmor radius ETG scale of the order of the electron Larmor radius T / T e i
37 TEMs are stabilised by collisions Since the TEM instability is due to the presence of trapped electrons, it is strongly affected by collisionality (mainly el.  ion collisions) Collisions produce trapping and detrapping processes, Coulomb diffusion in velocity space ( velocity pitchangle scattering ) Collisions are strongly stabilising, From simple model γ ω De ν ei ω De + and more effective on instabilities driven by temperature gradients γ [c i /R] R/L n = 6, R/L Te = 1 R/L = 1, n R/L = 1 Te GS2 τ = 2 ε =.16 R/L =.1 Ti ν [c ei i /R] ν ei /ωde
38 TURBULENCE and TRANSPORT 1 8 Q i [ GB ] GYRO time [a / c s]
39 From linear instabilities to turbulence Back to the gyrokinetic equation Up to now we neglected this term which describes the effect of the fluctuating potential on the perturbed distribution We were using a linear model ~ By including this term we introduce a nonlinearity, since φ depends on f through the Poisson equation, In addition we introduce toroidal mode coupling, since all toroidal modes are involved in this term This equation describes the development of a turbulent state Numerical codes (nonlinear fluid and kinetic codes) have been developed to compute this state ( movie GYRO shape.n16.mpg )
40 Turbulent transport Transport is produced by density and temperature fluctuations in the presence of a fluctuating electrostatic potential Γ This transport is usually called electrostatic or ExB transport. Electromagnetic effects can affect the electrostatic transport through effects on the fluctuating quantities Magnetic fluctuations (fluctuations of the magnetic field) can also produce transport, usually called magnetic flutter, due to the radial component of the fluctuating magnetic field. In many conditions, this contribution is small and we shall not discuss it here
41 Nonlinear spectrum vs linear spectrum Nonlinear transfer through mode coupling implies that the nonlinear spectrum is different from the linear one Scales which provide the largest contribution to transport are not those which correspond to the most unstable linear modes q i [ GB ] time [ a / c s ] fractional contribution per mode k θ ρ s linear phase nonlinear phase
42 Nonlinear spectrum vs linear spectrum Nonlinear transfer through mode coupling implies that the nonlinear spectrum is different from the linear one Scales which provide the largest contribution to transport are not those which correspond to the most unstable linear modes q i [ GB ] time [ a / c s ] fractional contribution per mode.25 1 linear phase nonlinear phase k θ ρ s
43 Fluxes in the velocity space Ion fluxes: both passing and trapped ions contribute to transport Electron fluxes: trapped electrons make almost all the transport 5 x 1 4 Ion heat flux 15 x 1 5 Electron heat flux trapped 1 5 trapped λ 1.5 passing E / T i 4 λ = λ V / V B / B 1 passing E / T e 4 5
44 Quasilinear estimates, phase relations What is relevant for transport is the phase relation between the fluctuation in the transported quantity and the fluctuation in the electrostatic potential e.g. for particle transport n ~ v ~ Γ E n ik yφ k k k B This quantity can be computed also within linear theory, by φ means of the (complex) linear relationship between n k and k n φ k Z k k Often in core transport the nonlinear state does not change significantly the linear phase relationships (ratio of transport channels often well estimated by linear theory ) The nonlinear saturation amplitude is more difficult to be properly described by a mixing length model, in any case... This allowed the successful development of quasilinear transport models for core transport, and many comparisons with experiments ( nonlinear runs are still very heavy in terms of computer time ) 2 φ k
45 GyroBohm units, scaling of transport The natural scaling of transport can be obtained by building a diffusivity given by a displacement by one Larmor radius in the time of one inverse growth rate ρ and γ ~ c s D ρ 2 γ s R s = c s Ω c D c s Ω c This scaling is called gyrobohm. It is the natural scaling of local transport, derived from the dimensionless form of the equations The name gyrobohm shows that it is given by the Bohm diffusion reduced by one normalized gyroradius Bohm diffusion is given by a displacement of one Larmor radius in a time of the order of one gyroperiod m 1/2 T 3/2 1 D B ρ s 2 Ωc = = T e B ρ D D s GB = B = a e B a where a is a characteristic length ( major / minor radius or a gradient length ) R
46 Scaling of transport, profile stiffness The gyrobohm transport increases strongly with temperature Q n = T χ T T Q n = χ^ χ GB ρ χ D s GB = B = a The dimensionless parameter S describes how steep the heat flux increases with increasing log. temperature gradient ( R R S T = R L T L Tcrit χ^ ( χ^ m 1/2 T 3/2 2 2 a e B Q n R Stiff 1 m 1/2 T 5/2 2 2 S 2 e B ( R L T R L Tcrit Nonstiff ( Due to the gyrobohm scaling, the hotter a plasma is, the stiffer it becomes ( in a reactor, plasma profiles will be very stiff ) R L T
47 MAIN STABILIZATION MECHANISMS GEOMETRY (magnetic shear) ROTATION (ExB velocity shear)
48 Magnetic shear Magnetic shear is due to the different winding numbers of the field lines ( safety factor profile is not constant ) s =.1 It has a strong effect since the instabilities are field aligned due to the fast parallel motion It leads to a deformation of the structures s = 1.
49 Negative shear provides strong stabilisation Negative shear tilts the eddies in such a way that the drive is reduced Maximum transport for s ~.5 s = s > s < χ GB R/LT = 9 D χ i χe χ i R/Ln = 1.5 R/Ln = 4.5 χ GB Turbulence stabilisation by negative shear is observed in exps to allow the development of transport barriers, namely of very large local values of R/LT (particularly in the electron channel) D .5. χ e GYRO, Kinsey PoP s
50 Turbulence stabilisation by sheared flows Various mechanisms can lead to the development of a profile of the radial electric field which is not constant along the minor radius (e.g. strong toroidal rotation profile due to beam torque in the core, fast ion losses in the edge, etc... ) Then an E x B rotation (mainly Er x Bt ) is produced which is not constant along the minor radius, namely a sheared rotation (sheared flow is a velocity field with a radial gradient, the radial gradient of a velocity has the dimension of a frequency, and is called shearing rate) By the presence of a sheared rotation, eddies get tilted and ripped apart, leading to turbulence suppression movie GYRO d3d.n16.2x_.6.mpg
51 General conclusions and outlook Theory of turbulent transport has made impressive progresses in the last decades and is able today to explain qualitatively ( sometimes quantitatively ) a big part of the phenomenology Some big challenges still remain Interaction among different scales ( electron Larmor radius, ion Larmor radius, and MHD mesoscales ) Global simulations without any assumption on the plasma equilibrium (full f models) Transport problem within a full gyrokinetic approach (prediction of plasma profiles from the imposed fluxes ) The phenomenology linked to the (edge) transport barrier formation ( in particular the L  H transition ) remains the most important observation still to be explained by theory Theory is the only way we have to build a reliable predictive capability
52 Review papers This is a short list of review papers and a book where you can find more informations and lists of references to more specific papers W. Horton, "Drift Waves and Transport" Review of Modern Physics 71, (1999) J. Weiland, "Collective Modes in Inhomogeneous Plasmas", Institute of Physics Pub. (1999), ISBN13: X. Garbet et al,"physics of transport in tokamaks" Plasma Phys. Control. Fusion 46, B557B574 (24) E.J. Doyle et al, "Plasma confinement and transport", in "Progress in the ITER Physics Basis" Nucl. Fusion 47, S18S127 (27) Movies presented in this talk come from the General Atomics GYRO code webpage:
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