Magnetohydrodynamics (MHD) I

Transcription

1 Magnetohydodynamics (MHD) I Yong-Su Na National Fusion Reseach Cente POSTECH, Koea, 8-10 May, 006

2 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations 4. MHD Equilibium Concept of Beta Equilibium in the z-pinch Equilibium in the Tokamak

3 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations 4. MHD Equilibium Concept of Beta Equilibium in the z-pinch Equilibium in the Tokamak

4 Confinement

5 Oigin of the Sta Enegy

6 Oigin of the Sta Enegy Themonuclea fusion

7 How to Confine the Sun on the Eath? Magnetic field ion

8 How to confine the sun on the eath? Review: Single paticle motion of the plasma Magnetic field fee motion along magnetic field lines gyation aound magnetic field lines ion

9 Magnetically Confined Plasmas Stong magnetic field: Lamo << L Magnetic field lines tace out magnetic sufaces, to paticles stay on these sufaces.

10 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations 4. MHD Equilibium Concept of Beta Equilibium in the z-pinch Equilibium in the Tokamak

11 Plasmas as Fluids The single paticle appoach gets to be complicated. A moe statistical appoach can be used because we cannot follow each paticle sepaately. Now intoduce the concept of an electically chaged cuent-caying fluid. Magnetohydodynamic (magnetic fluid dynamic) equations

12 Fluid Equations Continuity equation n t + ( nu) = S S : a volume souce ate of paticles Momentum balance equation du u mn = mn[ + ( u ) u] = F = nq( E + u B) dt t F : foce density When themal motions ae taken into account, u mn[ + ( u ) u] = nq( E + u B) P P : pessue tenso t

13 Fluid Equations Momentum balance equation u mn[ + ( u ) u] = nq( E + u B) P t ρ[ u + ( u ) u] = p + ρν u t Navie-Stokes equation This momentum density consevation equation fo species esembles in pats the one of conventional hydodynamics, the Navie-Stokes equation. Yet, in a plasma fo each species the Loentz foce appeas in addition, coupling the plasma motion (via cuent and chage densities) to Maxwell s equation and also the vaious components (electons and ions) among themselves. Equation of state p = γ Cn C, γ : constant

14 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations 4. MHD Equilibium Concept of Beta Equilibium in the z-pinch Equilibium in the Tokamak

15 Single-fluid MHD mass density chage density ρ = ni M + n e m σ = ( ni ne ) e cuent density j = e n u n u ) ne( u u ) mass velocity ( i i e e i e u = ( n Mu + n mue) / ρ i i e Mass and chage continuity equation ni, e + ( ni, eui, e) = 0 t The individual continuity equations subtacted fom one anothe σ + t j = 0 ρ t Multiply by the ion and electon mass and add togethe + ( ρu) = 0

16 Single-fluid MHD Momentum balance equation ui Mn i[ + ( ui ) ui ] = + eni ( E + ui B) pi+ Rie t ue mn e[ + ( ue ) ue] = ene ( E + ue B) pe+ R t Rαβ mα nα < ναβ > ( uα uβ ): = Momentum of species α tansfeed by collisions to species β add togethe ei u ρ [ + u u] t = σe + j B p

17 Single-fluid MHD genealised Ohm s law ue mn e[ + ( ue ) ue] = ene ( E + ue B) pe+ R t R = m n < ν > u u ) ei e e ei ( e i Assuming that the electons ae homogeneous and theefoe neglecting the electon pessue and velocity gadients along B E = j = en u u ) 0 en ee + Rei e ei = ue ui ) R ei e( e i m < ν > me < ν ei > ( = j = ηj e nee = η n e ( u u ) = ηn ej e E + ue B = ηj pe/ ne i E + u B = ηj + ( j B pe ) / ne e e ei u e u j / ne

18 The set of MHD Equations ρ σ + ( ρu) = 0 + t t j = 0 u ρ [ + u u] t = σe + j B p E + u B = ηj Simple Ohm s law small Lamo adius appoximation B = μ0 j E B = t B = 0

19 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations 4. MHD Equilibium Concept of Beta Equilibium in the z-pinch Equilibium in the Tokamak

20 Equilibium

21 MHD Equilibium As moe and moe chaged paticles ae added to a plasma, the cuents that flow along the magnetic field can become lage enough to modify the extenally ceated magnetic field. The plasma equilibium must then be detemined self-consistently: the pesence of the plasma itself modifies the magnetic field configuation. Fo a steady-state solution of the MHD equations fo the special case with u=0, E=0, η=0 and isotopic pessue.

22 MHD Equilibium ρ + ( ρu) = 0 t σ + t j = 0 u ρ [ + u u] t = σe + j B p E + u B = ηj B = μ0 j E B = t B = 0

23 MHD Equilibium ρ + ( ρu) = 0 t u ρ [ + u u] = σe t + σ + j = 0 t j B p edundant E + u B = ηj B = μ0 j E B = t B = 0

24 MHD Equilibium p = j B B = μ 0 B = 0 j Foce balance Ampee s law Closed magnetic field lines Two Maxwell s equations (well known) One (seemingly) simple foce balance: kinetic pessue balanced by j B foce

25 MHD Equilibium p = j B B = μ 0 B = 0 j Foce balance Ampee s law Closed magnetic field lines Two Maxwell s equations (well known) One (seemingly) simple foce balance: kinetic pessue balanced by j B foce p j B

26 Concept of Beta p = j B B = μ 0 B = 0 j Foce balance Ampee s law Closed magnetic field lines p = ( B) B / μ0 = [( B ) B ( B / )]/ μ ( p + B / μ ) = ( B ) B / μ 0 Assuming the field lines ae staight and paallel p + B / μ0 = const. 0 0 β = μ B 0 p /

27 Concept of Beta β = μ B 0 p / The atio of the plasma pessue to the magnetic field pessue A measue of the degee to which the magnetic field is holding a non-unifom plasma in equilibium Main Objectives of Fusion Devices Stable configuation High β = kinetic pessue 0 = μ magnetic pessue B High enegy confinement time p fusion powe = cost of decive stoed enegy τ E = applied heating powe

28 Simplest Case: The Linea Pinch (z-pinch) Cylindical co-odinates: Specify cuent pofile jz jz, Bθ, dp / d : = j /( 0 = I P πa ) fo < a

29 Magnetic Field Pofile in the z-pinch 1 Ampee: ( B ) j θ = μ 0

30 a I I d j B P 1 ) ( π μ μ π μ θ = = = j B 0 ) ( 1 θ = μ Ampee: Magnetic Field Pofile in the z-pinch

31 a I I d j B P 1 ) ( π μ μ π μ θ = = = a I B P 0 0 π μ = fo a < I B P π μ θ 0 = fo a > Magnetic Field Pofile in the z-pinch j B 0 ) ( 1 θ = μ Ampee:

32 Radial Foce Balance and β in the z-pinch With j z and B θ : compute foce balance: dp d = j IP μ I 0 πa π a B P z θ =

33 Radial Foce Balance and β in the z-pinch With j z and B θ : compute foce balance: dp d = j IP μ I 0 πa π a B P z θ = Bounday condition p(a)=0: μ I ( ) 0 p = P (1 ) 4 π a a

34 Radial Foce Balance and β in the z-pinch With j z and B θ : compute foce balance: dp d = j IP μ I 0 πa π a B P z θ = Bounday condition p(a)=0: μ I ( ) 0 p = P (1 ) 4 π a a

35 Radial Foce Balance and β in the z-pinch With j z and B θ : compute foce balance: dp d = j IP μ I 0 πa π a B P z θ = Bounday condition p(a)=0: μ I ( ) 0 p = P (1 ) 4 π a a Calculate β p μ0 < p > =, we find β p = 1 B ( a) θ geneal esult fo the z-pinch, not dependent on pofiles.

36 Equilibium in the Tokamak p = j B p j = p B = 0 : j and B lie in the sufaces p=const.

37 Equilibium in the Tokamak p = j B p j = p B = 0 : j and B lie in the sufaces p=const. Flux though any cuve on p=const. suface has same value: flux sufaces

38 Equilibium in the Tokamak p = j B p j = p B = 0 : j and B lie in the sufaces p=const. Flux though any cuve on p=const. suface has same value: flux sufaces Pessue is flux suface quantity. Tokamak consists of nested flux sufaces set up by B field lines

39 Equilibium in the Tokamak Two sots of cuves on the tous: winding aound poloidally: tooidal fluxes winding aound tooidally: poloidal fluxes

40 Equilibium in the Tokamak Two sots of cuves on the tous: - winding aound poloidally: tooidal fluxes - winding aound tooidally: poloidal fluxes Chose the poloidal fluxes ψ (magnetic flux) and I pol (cuent): B o/ B R B Z μ0i pol = πr 1 Ψ = πr Z 1 Ψ = π R R

41 Gad-Shafanov Equation e-wite foce balance in tems of fluxes (' denotes d/dψ): R 1 ψ ψ + R = p Δ* ψ = μ0(π ) + μ0 I pol I pol R R R Z Gad-Shafanov equation (GS-eqn) GS-eqn is nonlinea in ψ. To solve it, one may specify p(ψ) and I pol (ψ).

42 Equilibium in the Tokamak Poloidal field coils allow flexible shaping of coss-section

43 Equilibium in the Tokamak Plasma equilibium in ASDEX Upgade

44 Summay I 1. What is confinement? Why is single paticle motion appoach equied?. Fluid desciption of plasma Fluid equations 3. Single fluid equation 7 MHD equations 4. MHD equilibium Concept of beta Equilibium in the z-pinch Equilibium in the tokamak GS equation

45 The long fom of the name MHD means ``magnetic fluid dynamics''. MHD is a simplified model of a magnetised plasma in which the plasma is teated as a single fluid which can cay an electic cuent. By single fluid, we mean that thee is only one density (the mass density) and tempeatue, and thee is also only one velocity -- you don't have to teat the electons and ions sepaately. The basic fluid velocity is the ExB dift, and the diffeence in the velocities of the electons and ions appeas as the electic cuent. (Go back to fluid dift motion if any of these tems ae unfamilia.) The basic physics in the MHD model is the same as fo an odinay fluid, with the addition that the magnetic field can exet a foce. The fact that the Loenz foce is opposite fo ions and electons gives ise to a bulk ``magnetic foce'' wheneve the cuent flows acoss magnetic field lines. But since the velocity is the ExB velocity, the magnetic field is advected by the fluid plasma. This couples field and plasma togethe, giving the MHD system its ich chaacte. You can think of a tube of magnetic field lines confining a plasma, with the pessue pushing outwads and the magnetic field pushing inwads. This happens because a localised magnetic field always implies the existence of a cuent, and in this case the cuent flowing along the tube causes pat of the magnetic field to twist aound the tube. The sense of it is the ``ight-hand ule'' again: with the thumb pointing up and finges culed back towad the palm, the cuent flows along the thumb and the magnetic field culs in the diection of the finges. Now, the foce is given by the thid diection: J-coss-B, and this points inwad towads the axis of the cuent column, acting to confine the plasma. This is the vey basics of how MHD woks. The full ange of MHD phenomena extends as well to instabilities and tubulence, but this pictue of foce balance is what is behind the idea of a magnetically confined plasma, one example of which is the ``tokamak''.

46 Poloidal field coils maintain plasma shape and position: Vetical field compensates expansion(pessue, cuent) - 'Shafanov-shift' due to lage field on outside Note: A stellaato poduces all fields by extenal coils (no net tooidal cuent and no axisymmety)

47 Poloidal field coils maintain plasma shape and position: Vetical field compensates expansion(pessue, cuent) - 'Shafanov-shift' due to lage field on outside