22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations

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1 .615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f = C + s m a. f ( ) B E = 1 E B = μ0j+ c σ E = 0 v B = 0 b. The partice and eectromagnetic equations are couped by σ= qn = q fdv v J = qnu = q vfd c. The coision operators C arise from eastic coisions and satisfy a corresponding set of conservation reations.. Derivation of fuid equations tae moments as foows: a. mass df dt C s dv = 0 b. momentum df mv C s dv = 0 dt c. energy mv df dt C s dv = 0.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 1 of 10

2 3. Introduce macroscopic quantities n (r, t), u (r, t), p (r, t). 4. The moment equations become a set of couped, time dependent PDE s reating the various macroscopic quantities. 5. There initia Botzmann equation is a singe scaar equation in f : f = f (r, v, t). There are seven independent variabes. 6. The resuting moment equations contain six fuid variabes: n, u, T, p, a functions of (r, t). There are four independent variabes. 7. The fuid equations are far simper to sove. 8. At a basic mathematics eve the moment method appears to be i conceived. You cannot sove a partia differentia equation by integrating over severa independent variabes and then soving a reduced equation. 9. Exampe consider = ( r,, t) ψ ψ θ satisfying ψ 1 ψ 1 ψ = 0 r r r r θ θ a. = rf ( r, θ) + g ( r, θ) b. Integrated over θ assuming periodicity: define G ( 1 ) = π Gdθ 1 ψ ψ = r f r r r c. This equation is a correct expression, but not a usefu one since two different averages appear: ψ, fψ r. Integrating over θ eads to a singe, simper reduced equation, but with two unnowns. 10. This is a genera property of moment equations. When we tae moments of the Botzmann equation, we wi obtain a set of correct reations, but there wi be more unnowns than equations. 11. How do we resove this probem? We wi cose the set of equations by soving the Botzmann equation and then evauating some of the higher order, additiona unnowns. 1. This woud seem to mae the entire procedure circuar. If we are going to sove the Botzmann equation anyway why bother with the moment equations?.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page of 10

3 13. There is method to this madness. a. First, even if we now the soution to the Botzmann equation the moments represent more usefu information in that they describe the measurabe physica quantities in the system. b. econd, and equay important, we are not ust going to simpy sove the Botzmann equation for the extra unnowns. The fu equation is enormousy compicated to sove. c. Instead, we sha sove the Botzmann equation by means of various e.g. m m, r a, ωω, etc. expansions ( ) e i L d. Each order in the expansion is exponentiay more painfu to cacuate than the previous order. e. By having a carefuy defined set of moment equations, we can determine beforehand exacty how many terms are needed in the expansions. In addition we can rewrite the moment equations in such a way as to further minimize the number of terms required. f. These two reasons (physica variabes, minimum agebra) are strong motivation for using the moment procedure..615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 3 of 10

4 Moment Equations 1. Procedure to be foowed: a. Cacuate exact moments: mass, momentum, energy. b. Introduce random veocity v=u (r, t) + w. c. Define physica variabes n, u, T, p d. Do some agebraic booeeping. e. Arrive at a set of moment equations (with more unnowns than equations).. Now focus on the momentum equation from which we derive the MHD equiibrium equation. how that this equation is vaid for both MHD stabiity anaysis and transport phenomena. Motivation: amost no nowedge of higher order unnowns is required. 3. pend haf a semester investigating appications of the MHD equiibrium equation. 4. Derive in detai the coision operators C. 5. pend haf a semester investigating appications of the transport equations in cyindrica geometry. 6. Consider next the conservation of mass equation: we derive this carefuy, carrying out a the steps. f q d + f + + B vf C s = 0 m a. v v ( E v ) b. Define n = fdv partice number density nu = vfdv partice fux f n c. dv = dvf = d. d f d ( f) f v v. = v v v =0 = d v vf = ( nu ).615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 4 of 10

5 q q f f f e. dv E Vf = E dv ex + ey + ez = m m vx vy vz 0 q q = f + = 0 m m vx f. dv v B vf dv ( vybz vzby) g. dv C = dvc = 0 (conservation of partices) h. dvs n (source of density) 7. Combine terms: note one equation, four unnowns n, u n ( nu ) + = n 8. imiar procedure for the momentum equation yieds t ( ) ( ) ( ) nmu + nm vv qn E + u B = mvc dv + p 9. a. Here denotes coisions) (due to conservation of momentum in ie partice b. p dv mvs = 0 (source of momentum, zero for practica appications) c. Q = Qf dv n 10. imiar procedure for energy equation yieds t mv + v E= v + E 1 1 mn v mn v qnu C d 11. a. Here E dv( m v )s (sources of energy, say due to rf). b. Aso, = vanishes from coision term because of conservation of energy in ie partice coisions.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 5 of 10

6 Pasma Booeeping 1. The moment equations can be written in more physica terms by introducing the random veocity and defining various physica quantities in addition to n and u.. The random veocity w: this is a change of independent variabes from v to w defined by 3. By definition dv=dw and w = 0 4. Then vv = uu + uw + wu + ww = ww + u u =0 =0 5. Define where p 1 = n m ω 3.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 6 of 10

7 1 Π = nm ww ω Ι 3 6. imiary 3 = + w u + ω = + v u u =0 p nm ( u ) ( ) v v = + w u + ω u + u + w u + ω w where =0 =0 3pu u P h = uu nm nm mn h 1 nm ω w is the heat fux, the fux of heat due to random motion. 7. Now define ( ) m u + w C dv = m wc dw R =0 ` m + + ω = + m ω ( u u w ) C d v u R w C d =0 u R +Q where R is the average momentum transferred due to unie coisions and Q is the heat generated due to unie coisions. 8. As they now stand, the moment equations can be written as n ( n u ) + = s n nm nm qn R ( u ) ( u u ) P ( E u B) =.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 7 of 10

8 1 ( 3 p 1 ) 3 nmu + + nmuu + pu + u P + h qnu E = ( u R + Q ) + s E Agebraic simpification 1. Define the convective derivative, moving with the fuid dq dt Q = + u Q. Define the temperature T = p n 3. The mass equation is OK as is 4. Momentum equation simpifications: ( nm ) + ( u nmu u ) u n = mn + mu + mu ( nu ) + mnu u du = mn + mu dt n 5. The momentum equation becomes du mn P qn ( E u B) R mu dt + + = n 6. Energy equation simpifications a. + u = u + + nu + u u nmu nmu mn mu n mu mn d mu u mn = + n dt b. 3 p 3 T n p u n T T nu nu T + = t 3 dt = n + T dt n.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 8 of 10

9 c. Energy equation becomes mn d 3 dt u + n + ( u P + h ) q n u E= dt dt ( u R ) m 3 + Q + E u + T n d. Note that u (momentum equation) is equa to mn d u + u P q n u E= u R m u n dt e. Now subtract (d) from (e) 3 dt mu 3 n + ( u P + h) u P = Q + E + T dt f. Note the foowing identity (recaing that by definition P i = P i ) ( u P) u P ( up i i) u = i P i x x n = P i x u i = P: u g. The energy equation becomes 3 dt mu 3 n + P : u + h = Q + E + T dt n 7. ummary of fuid moments dn n u dt + = n du mn P qn ( E u B) R mu dt + + = n 3 dt 3 mu n + P : u + h = + + Q E T dt n.615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 9 of 10

10 8. Assuming the coisiona terms are nown, the fuid unnowns are n,u,p, T, and h. (17 unnowns, 5 equations) 9. Even with a scaar pressure, there are sti 9 unnowns. 10. The moment equations above are exact, if not particuary usefu. They do, however, accuratey describe both MHD and transport phenomena..615, MHD Theory of Fusion ystems Lecture Prof. Freidberg Page 10 of 10