Ecology of Flows and Drift Wave Turbulence: Reduced Models and Applications

Transcription

1 1 Ecology of Flows ad Drift Wae Turbulece: Reduced Models ad Applicatios PhD Dissertatio Defese by Rima Hajjar PhD Adisor: P. H. Diamod

2 Backgroud. Publicatios + Pla of Dissertatio Chapter : The Ecology of Flows ad Drift Wae Turbulece i CSDX: a Model. Physics of Plasmas, 018. Chapter 3: Modelig the Ehacemet i Drift Wae Turbulece. Physics of Plasmas, 017. Chapter 4: Zoal Shear Layer Collapse i the Hydrodyamic Electro Limit. Physics of Plasmas, 018 (i preparatio) Coclusios ad Future Work O the side: Modelig of Alumium Impurity Etraimet i the PISCES-A He + Plasma. Joural of Nuclear Material, 015.

3 3 Fusio 101 Icreasig eed for sustaiable ad clea eergy. Nuclear fusio releases high outputs of eergy that ca be coerted ito electric power. The fusio reactio with the highest cross-sectio is: H H He 18MeV Challege: Igitio (E out > E i ) Cofiemet ad Lawso criterio: τ E T > kev. s. m 3 τ E = W plasma P loss Use eterally imposed magetic field lies to cofie the plasma i toroidal or liear deices. Turbulet trasport of particles ad eergy (maily due to istabilities) destroys cofiemet. = 3TV P i Not to scale

4 4 Drift Waes ad Zoal Flows DWs: plasma fluctuatios caused by radial desity gradiets. propagate i the electro directio at De k * m De 1 ks 1 ks Parallel resistiity is oe mechaism that ca destabilie DWs by itroducig a phase shift betwee ad φ, thus creatig a DW istability. De Te eb d l ( ) d yˆ Fortuately, oe mechaism that regulates these fluctuatios is the self geeratio ad amplificatio of Zoal Flows by turbulet stresses. Zoal Flows: Large scale sheared E B layers Decorrelate the turbulet eddies by shearig. Reduce turbulece ad trasport. Diamod et al, 005, PoP

5 5 Drift Waes/Flows = Predator/Prey Free eergy Drie Drift Waes Drie by turbulet stress Drie by turbulet stress Regulate by Shearig Suppress Aial flows Zoal flows Suppress PSFI + Collisioal Dampig + Noliear Dampig

6 6 D Naier-Stokes equatios PV 0 Hasegawa-Mima equatio resistiity Models to study DW turbulece Hasegawa-Wakatai equatios (3D) Simplicity Compleity Parallel e- respose time > period of ustable mode Parallel e- respose time < period of ustable mode HW i hydrodyamic limit Neglect aial flow HW i adiabatic limit Treat aial flow PV is cosered: Appropriate fluctuatio field=<(- ) > PV is ot cosered: Appropriate fluctuatio field=< +( ) + >

7 7 CSDX: a promisig testbed for eplorig DW turbulece models oer compressed rages of scales. Cui et al, 016, PoP a = plasma radius L = desity scale legth ρ = modified io Larmor Radius l corr = turbulece correlatio legth Models ad Results obtaied from CSDX ca be etrapolated to larger scale deices

8 8 What am I doig? Eplore the status of flows ad fluctuatios ecology. Iestigate the relatioship betwee microscopic DW turbulece ad macroscopic flows i magetically cofied plasmas. I particular, study the couplig relatio betwee parallel ad perpedicular flow dyamics i the plasma of CSDX. Model the eolutio of plasma mea profiles ad fluctuatios i CSDX, as the magitude of the magetic field B icreases. Aalytically cofirm the trasport bifurcatio pheomeo reported i CSDX as B is raised. Eamie the Drift Wae/Zoal Flow relatio i the hydrodyamic electro limit Releace to desity limit eperimet. 8

9 9 Why do I care? Mea flow structures, icludig both Zoal ad Aial Flows, play a importat role i regulatig turbulece (L-H trasitios, ITB formatio) uderstadig the mechaism of formatio of these flows is crucial i achieig better cofiemet i ITER. Eplai ad uderstad the physics behid the collapse of ZFs ad the ehacemet of turbulece i the hydrodyamic electro limit which is a importat ad uder-eplored problem iterpret the desity limit eperimets usig a simple robust mechaism of DW turbulece.

10 10 How to do it? Formulate reduced models that self-cosistetly relate ariatios i mea plasma fields to fluctuatio itesity (total eergy/potetial estrophy). Reduced models are the ecellet cadidate: 1. Low computatioal cost if compared to DNS or LES. Good cadidate to describe the physics of a multiscale plasma such as CSDX plasma. 3. Essetial to uderstad the feedback loops betwee mea profiles (macro) ad fluctuatios (micro). 4. Easily coupled to other PMI codes. 5. Failure i model reductio suggests a gap i uderstadig Need to update the codes

11 The Ecology of Flows ad Drift Wae Turbulece: a Model for CSDX 11

12 1 D Naier-Stokes equatios PV 0 Models to study DW turbulece Hasegawa-Mima equatio resistiity Hasegawa-Wakatai equatios Simplicity Compleity Parallel e- respose time > period of ustable mode Parallel e- respose time < period of ustable mode HW i hydrodyamic limit Neglect aial flow HW i adiabatic limit Treat aial flow PV is cosered: Appropriate fluctuatio field=<(- ) > PV is ot cosered: Appropriate fluctuatio field=< +( ) + >

13 13 Eperimetal results i CSDX - 1 As magitude of B icreases: 1. Deelopmet of radial elocity shear. Decrease i turbulece leel Trasitio to a state of ehaced eergy i the perpedicular plae (Aalogy to larger MFE deices) 3. Steepeig of desity profile Cui et al, 015 ad 016, PoP

14 14 Eperimetal results i CSDX - As magitude of B icreases: 1. Deelopmet of aial elocity shear. Icrease i parallel Reyolds force Reyolds Work= Reyolds force elocity Trasitio to a state of ehaced eergy i the parallel directio 3. Steepeig of desity profile Hog, Hajjar et al, 018, PoP (submitted)

15 Formulatio of the Model }, {. }, { ) ( ) (. } {, ) ( s E y i ei th E ei th E c dt d dt d D dt d 15 Hasegawa Wakatai + Parallel Compressio Parallel Compressio breaks parallel symmetry Breakig of PV coseratio Defie a ew cosered eergy: y L L dy d 0 0 ) ( ) (

16 16 The model (mea fields + turb. Fluctuatio) t t t y t ( y D c c // y c S i i d d S d 1 / y lmi ) y y S y d Diffusio Sources Dissipatio Mea/Fluctuatio couplig terms Sik d d l Hajjar et al, 018, PoP 3/ mi P

17 Usig QL theory ad turbulet miig cocepts 1) Particle Flu: d D d f d ˆ d The electro parallel diffusio rate: α = k th ν ei ω. (Near adiabatic electros) The factor f represets the fractio of total eergy allocated for kietic eergy i the radial directio: ks f k (1 ) ( 1 ) ms kcs i k s lmi <k m k > Adiabatic electro without aial flow shear f k s k 1 c k s s lmi Pure DWs f k s 1 ks 17 17

18 18 ) Perpedicular Reyolds Stress 3) Reyolds Power rate d l y mi ci y lmi. d y d y d y [ l mi d d y l mi ci. ] Taylor s ID Diffusie Stress relaes the flow Residual stress dries the flow ia desity gradiet Drie Zoal Flows Regulate Drift Waes Predator-Prey Relatio

19 19 4) Parallel Reyolds Stress l mi d d k m k c s 3 s l [ mi s k ] Difficult to measure eperimetally Empirically, i aalogy with turbulece i pipe flows (à la Pradtl): l mi d d Tcs c L // ( l mi d ) d Measures parallel to perpedicular couplig d c l VT s mi. d L Turbulet diffusiity from Pradtl theory Eergy source proportioal to desity gradiet that accelerates the parallel flow Hajjar et al, 018, PoP

20 0 Measuremets i CSDX Res = σ VT c s l mi L σ VT is the couterpart of the correlator k m k. σ VT represets the degree of symmetry breakig i k m k, ad quatifies the efficiecy of i driig a aial flow: = σ VTc s τ c L σ VT couples parallel to perpedicular flow dyamics ia: Hog, Hajjar et al, 018, PoP (submitted) d d y res = ω cil σ VT c Π y s τ c res Π

21 1 5) The miig legth l mi : The miig legth ehibitis turbulece suppressio ia aial ad aimuthal flow shear: l mi 1 l l 0 ( f l 0 0 y L // ) Shear Feedback Loop l l0 ( mi y ) 1 ε l mi I CSDX, the miig scale for turbulece l 0 i the absece of shear (ρ * =ρ/l ): l s L No Aial Shear f I betwee Bohm ad gyro-bohm Diffusio DCSDX D B *

22 The Big Picture Free eergy Drift Waes So May Loops: Feedback loop 1: DW+ZF Feedback loop : DW+AF Feedback loop 3: DW+ZF+AF = σ VTc s τ c L d( ) d = ω ci Limits Aial flows Parallel Symmetry Breakig: σ VT Zoal flows Regulate PSFI d d y = ω cil σ VT c s τ c + Collisioal Dampig + Noliear Dampig

23 Modelig Ehaced Cofiemet i Drift Wae Turbulece 3

24 4 D Naier-Stokes equatios PV 0 Models to study DW turbulece Hasegawa-Mima equatio resistiity Hasegawa-Wakatai equatios Parallel e- respose time > period of ustable mode Parallel e- respose time < period of ustable mode Simplicity Compleity HW i hydrodyamic limit Neglect aial flow HW i adiabatic limit Treat aial flow PV is cosered: Appropriate fluctuatio field=<(- ) > PV is ot cosered: Appropriate fluctuatio field=< +( ) + >

25 5 Free eergy Drie Drift Waes Whe parallel Reyolds power is egligible, ad whe eergy echage occurs maily betwee DWs ad ZFs, aial flow is treated as parasitic. Regulate by shearig Drie ia y Back to the predator/prey relatio betwee DWs ad ZFs Zoal flows Suppress + Collisioal Dampig + Noliear Dampig

26 Formulatio of the Model 6 Basic form of Hasegawa Wakatai HW equatios locally cosere the total Potetial Vorticity q = φ potetial estrophy ε is also cosered: y L L dy d 0 0 ) ( ) ( }, {. }, { ) ( ) (. } {, ) ( s E y i ei th E ei th E c dt d dt d D dt d

27 The Model 7 Diffusio Sources Dissipatio Mea/Fluctuatio couplig terms ε(u 0 ε) Hajjar et al, 017, PoP 3/ ) )( ( ) / ( P u u l t S u u t u S D t mi u c c

28 8 Closure usig QL theory ad miig legth cocepts 1) Particle Flu: d D d fl ˆ ) Vorticity Flu: (Taylor ID) d d mi s where f 1 ks k y u ˆ fl mi c u u du d Diffusie Stress d d fl ˆ mi Residual Stress d d The coefficiet c u reflects the shearig feedback o the mea profiles

29 9 3) The miig legth: D turbulece, the Rhies scale emerges as a coeiet miig legth for turbulece. Choose a hybrid miig legth: l mi l0 1 ( l / l + Weak PV miig l mi l 0 + Strog PV miig l mi l Rh Shear 0 Rh ) l Rh 1 l / ( u) 0 l0 ( ( u)) / Feedback Loop ε l mi

30 30 Recoery of eperimetal treds i CSDX Δ( 1 L ) L i = 1 L f 1 L f = 1 L i 0.70 umerically 0.55 eperimetally Δ( 1 L ) L i = 1 L f 1 L f = 1 L i 0.73 umerically 0.57 eperimetally Hajjar et al, 017, PoP

31 31 Summary o umerical results As B icreases: + Steepeig of the desity profile with B. + Deelopmet of aimuthal elocity shear with B. + Icrease i the magitude of the Reyolds work, i.e., turbulece regulatio with B. These treds are qualitatiely isesitie to: + Magitude of the shearig coefficiet c u + Outer edge Boudary Coditio o orticity. + Magitude of l 0 + The presece of a residual stress

32 3 R T : a criterio for turbulece suppressio? Need to quatitatiely predict whe trasport barriers are formed. R T = y E B γ eff = local Reyolds power desity effectie icrease i turb. kietic eergy Whe R T >1 eergy trasfer to the shear flow eceeds the effectie icrease i turbulet eergy reductio of trasport ad formatio of a barrier. BUT, γ eff =? What does it really deped o? What about o-kietic turbulet eergy (such as iteral turbulet eergy): E = + ( φ? Ma et al, 011, NF

33 33 R DT : a better criterio for turbulece suppressio R DT Here 1 τ prod = Γ is the rate of turbulet estrophy productio due to relaatio of mea desity profile (relatio with γ eff i R T ). Ad 1 τ trasfer = u u is the rate of turbulet estrophy destructio ia couplig with the mea flow (relatio with Reyolds power desity i R T ia Taylor ID). R DT emerges aturally i this model from the turbulet estrophy equatio: prod trasfer d 3/ u ( u) P dt R DT ca be easily geeralied to comple models by epadig the compariso of sources ad siks for potetial estrophy. 33 u u

34 Zoal Flow Shear Layer Collapse i the Hydrodyamic Electro Limit 34

35 35 D Naier-Stokes equatios PV 0 Hasegawa-Mima equatio resistiity Hasegawa-Wakatai equatios Models to study DW turbulece Parallel e- respose time > period of ustable mode Parallel e- respose time < period of ustable mode Simplicity Compleity HW i hydrodyamic limit Neglect aial flow HW i adiabatic limit Treat aial flow PV is cosered: Appropriate fluctuatio field=<(- ) > PV is ot cosered: Appropriate fluctuatio field=< +( ) + >

36 36 Backgroud: Desity Limit Eperimets Eperimets show that as approaches G = I/πa, MHD actiity is triggered alog with strog disruptios, edge coolig, MARFE Recetly, a Ohmic L-mode discharge eperimet i HL-A showed that, as / G is raised: + Ehacemet of edge turbulece. + Edge coolig. + Drop i α = k th /(ν ei ω ) from 3 to Drop i edge shear. Note the low alues 0.01 < β < 0.0 i this eperimet. Hog et al, NF, 018)

37 Hydrodyamic Plasma Limit α = k th ν ei ω = Parallel Diffusio rate DW frequecy α 1 adiabatic plasma limit ad φ are strogly coupled α 1 hydrodyamic plasma limit ad φ ted to decouple Simulatios results show ehacemet of turbulece ad weakeig of edge shear layer as the plasma respose passes from the adiabatic to the hydrodyamic limit. Howeer, these results do ot eplai WHY turbulece is ehaced i the hydrodyamic limit Numata et al, 007 Hypothesis: Flow Productio drops i Hydrodyamic Limit 37 37

38 Eergy ad Mometum Flues Adiabatic regime ( k th ω ν ei 1): k k gr y k r k m k r m k 1 ks De d d < 0 ad gr >0 k r k m >0 Mometum flu <0 ad eergy flu>0 k Causality implies a couter flow spi-up eddy shearig ad ZF formatio De * * ks k i 1 k s Hydrodyamic regime ( k th ω ν ei 1 ): gr y k k gr is idepedet of k m r k r hydro * ks k r k m k k Coditio of outgoig wae eergy flu does ot costrai the mometum flu, as gr is idepedet of k m o implicatio for Reyolds stress r (1 i) k r hydro PV coseratio ca also be used to square PV miig with ZF formatio 38

39 Scalig of trasport flues with α Plasma Respose Adiabatic (α >>1) Hydrodyamic (α <<1) Particle Flu Γ Γ adia 1 α Γ hydro 1 Turbulet Viscosity χ χ adia 1 α Residual stress Π res Π res adia 1 α Π res χ = ω α ci ( ω )0 α χ hydro 1 α Π res hydro- α α ω 1 Mea orticity gradiet d( y) d = Πres χ - which represets the productio of ZF - decreases ad becomes proportioal to α 1 i the hydrodyamic limit. Weak ZF formatio for α 1 weak regulatio of turbulece ad ehacemet of trasport

40 40 Oe step backward: Releace to the Desity Limit Eperimets αν 1 ei 1 whe icreases, α decreases, the ZF productio weakes ad turbulece is ehaced. No appeal to: 1) ZF dampig effects associated with plasma collisioality, charge echage (murky, case sesitie). ) The deelopmet of other istabilities, such as resistie ballooig modes which are ot releat i this eperimet because of the low β alues.

41 41 All Roads Lead to MHD istabilities Edge Fuelig icreases Iward turbulece spreadig α decreases Adj. T steepes ZF productio decreases, Turbulece ad trasport icrease Resistiity icreases J decreases Plasma Coolig Adj. J steepes MHD Actiity

42 4 What did I lear while pursuig a PhD? Reduced models are a powerful tool to describe comple turbulet systems. They describe feedback loops ad allow the study of plasma profiles across timescales ragig from a few turbulet correlatio times up to system equilibrium time scales. Reduced models distill what is leared from simulatios, basic theory ad eperimets. Capacity of drift wae turbulece to accelerate both oal ad aial flows ia the Reyolds stresses i both parallel ad perpedicular directios. Importace of parallel symmetry breakig i determiig the eergy brachig i the system as well as the stregth of the parallel to perpedicular flow couplig. Relatio betwee wae eergy flu, Reyolds stress ad PV miig is essetial i regulatig turbulece i both adiabatic ad hydrodyamic plasma limits, where predators feed o the prey i the former case, or are simply ot produced i the latter. Mechaism for oset of turbulece whe k th 1 is the collapse of the ZF regulatio ω ν ei

43 43 Recommedatios for future work Numerical simulatios of a slow eolutio plasma trasitio form the adiabatic to the hydrodyamic plasma limit. Addig charge-echage effects, ad io-eutral collisios to the model, so to umerically study the role of collisioal ZF dampig Geeralie the model to iclude a iestigatio of both flows ad fluctuatios i H-mode hydrodyamic plasma limit (Need to add temperature equatios for both ios ad electros, EM effects).

44 44 You made a differece. THANK YOU Pat, George for your patiece ad immese kowledge. My family. Luch group people. Awesome Sa Diegas. Aram Dalto.

45 Backup 45

46 46 Scalig of l 0 from eperimetal results k r is calculated form desity fluctuatios.

47 47 Additioal Numerical Results Steepeig of desity profile for differet amplitudes of the desity source S

48 Numerical Results without residual orticity flu Same treds as with a residual stress 48

49 49 Variatios of the shearig factor c u c u =6 c u =600

50 50 Results for Neuma Boudary coditios for orticity For Low B Steepeig of desity. Icrease i Reyolds work (magitude) Deelopmet of elocity shear.

51 51

52 How does ZF collapse square with PV Miig Rossby waes: PV = φ + βy is cosered betwee θ 1 ad θ. Quatitatiely The PV flu Γ q = h ρ s φ Total orticity Ω + ω is froe i Chage i mea orticity Ω leads to a chage i local orticity ω Flow geeratio, ia Taylor s ID. Drift waes: I HW, the q = l φ = l 0 + h + φ φ is cosered alog the lie of desity gradiet. Chage i desity from positio 1 to positio chage i orticity Flow geeratio ia Taylor ID Adiabatic limit α 1: +Particle flu ad orticity flu are tightly coupled (both are prop. to 1/α) Hydrodyamic limit α 1 : +Particle flu is proportioal to 1/ α. +Residual orticity flu is proportioal to α. PV miig is still possible without ZF formatio Particles carry PV flu 5