Time evolution of the isotropic electron distribution function while heating for the nonlinear kinetic equation with the Landau-Fokker-Planck collisional integral is studied. The considered heating sources are mono kinetic distribution, hot ions, and a quasi-linear diffusion operator. The investigation is mainly concentrated on the formation of the distribution function and tail acceleration. The time-dependent solutions allowing the solutions in self-similar variables are examined. Also presented are analytical asymptotic solutions and comparison them with numerical results

Krasheninnikov, S. I.

Potapenko, I. F.

Soboleva, T. K.

English

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DOI 10.1393/ncc/i2010-10581-5Colloquia: ICTT2009IL NUOVO CIMENTO Vol. 33 C, N. 1 Gennaio-Febbraio 2010Electron heating and acceleration for the nonlinearkinetic equationI. F. Potapenko(1)(∗), T. K. Soboleva(2) and S. I. Krasheninnikov(3)(1) Keldysh Institute for Applied Mathematics, RAS - Miusskaya Pl. 4, 125047 MoscowRussian Federation(2) Universidad Nacional Autonoma de Mexico - Mexico, D.F., Mexico(3) University of California at San Diego - La Jolla, California, USA(ricevuto il 9 Novembre 2009; approvato l’11 Febbraio 2010; pubblicato online il 16 Marzo 2010)Summary. — Time evolution of the isotropic electron distribution function whileheating for the nonlinear kinetic equation with the Landau-Fokker-Planck collisionalintegral is studied. The considered heating sources are mono kinetic distribution,hot ions, and a quasi-linear diffusion operator. The investigation is mainly con-centrated on the formation of the distribution function and tail acceleration. Thetime-dependent solutions allowing the solutions in self-similar variables are exam-ined. Also presented are analytical asymptotic solutions and comparison them withnumerical results.PACS 51.10.+y – Kinetic and transport theory of gases.PACS 52.50.-b – Plasma production and heating.PACS 52.65.Ff – Fokker-Planck and Vlasov equation.1. – PreliminariesIn many important cases one should treat plasma transport kinetically. The solutionsof the Landau-Fokker-Planck (LFP) equation, which is one of the key ingredients ofplasma kinetic equation, have much broader interest ranging from plasma physics tostellar dynamics (e.g., see [1-3] and the references therein).We consider the isotropic electron distribution function fe(v, t) and study the electronheating with a simple model:(1)∂fe∂t= Γ · I(fe) + H(fe), 0 ≤ v <∞, t ≥ 0,(∗) E-mail: firena@yandex.ruc© Societa` Italiana di Fisica 199200 I. F. POTAPENKO, T. K. SOBOLEVA and S. I. KRASHENINNIKOVwhere I(fe) is the LFP collisional integral for charged particles and H(fe) is the heatingsource. We focus on the interplay of the LFP collisional operator with the differentheating terms: monokinetic distribution, hot ions fi(v, t) (two component plasma), anda quasi-linear diffusion operator. Our interest is mainly concentrated on the evolutionand formation of the distribution function tails for t→∞, as v →∞. At the beginningwe shortly review some selected results of our works on the subject (see [4, 5] and thereferences therein) with a heating source H(fe) localized in the velocity space. Then abroader class of the heating terms resulting in enhancement of the tail of the distributionfunction is analytically analyzed. All asymptotic results are confirmed by the numericalcomputing of nonlinear LFP equation with high accuracy.The distribution functions are considered normalized to unity and the non-equilibriumelectron Te and ion Ti temperatures are as follows:(2) 4π∫ ∞0dvv2fe,i(v, t) = 1, Te,i(t) =4πme,i3∫ ∞0dxx4fe,i(x, t),where the first integral has a sense of the particle density. The system of the LFPcollisional integrals for such functions isI(fe) =1ρv2∂∂v[A∂fe∂v+ (Be + ρBi)fe],(3)I(fi) =ρv2∂∂v[A∂fi∂v+(1ρBe + Bi)fi].Here, the following notations for the integral operators A, B are used:(4) A =Ae + Ai3v, Ae =∫ v0dxx4fe(x, t)+v3∫ ∞vdxfe(x, t), Be =∫ v0dxx2fe(x, t),where Γ = 16π2e4LN/memi, L is the Coulomb logarithm and  = me/mi. In case ofheating, the local Maxwellian distribution depends on time fM ∼ vth−3 exp[−v2/vth2],where vth = ve,i(t) =√2Te,i(t)/me,i is the corresponding thermal velocity. We also usethe energetic variable ξ = v2/v2th. For v, ξ → ∞ the nonlinear integro-differential LFPintegral becomes linear,(5) I(f) =1v2∂∂v(2Tm1v∂f∂v+ f)=1ξ1/2∂∂ξ(∂f∂ξ+ f).The Coulomb diffusion influence is utmost in the cold region 0 ≤ v/vth < 1. In thehigh-velocity region v/vth  1 the LFP parabolic equation degenerates because of knownRutherford cross-section dependence on velocity and acquires more pronounced hyper-bolic type. This circumstance leads to inevitable retarding of the distribution tail for-mation.2. – Heating localized in the velocity spaceAt first, we consider the Cauchy problem for (1)-(4) without a heating term dealingwith one sort of particles ( = 1). The initial distribution function is located in theregion 0 ≤ v ≤ 1, the thermal velocity is a velocity scale unit. We are interestedELECTRON HEATING AND ACCELERATION FOR THE NONLINEAR KINETIC EQUATION 201in the relaxation of the distribution function tails in the high-energy region v → ∞.Substituting f(v, t) = G(v, t) · fM (v) in (5), we consider the period when the relaxationprocess is practically finished in the thermal velocity region, that is, the period whenGM (v) ≈ 1. Hence, we solve the problem for eq. (5) in the superthermal velocity regionwhere the second (transport) term prevails and the slow establishing of the equilibriumsolution GM = 1 occurs, and G → 0 as v  1, t  1. From asymptotic analysis thefollowing results are obtained. G is expressed in terms of the error function(6) G(v, t) = Φ{25vf[v − vf (t)vf (t)]5/2}, Φ(x) =2√π∫ ∞xdye−y2,and has a character of a propagating wave, whose front moves under the law vf (t) =(3 t)1/3 with the constant front width Δf (t) =√π. Then the distribution function forma-tion for t→∞, v →∞ actually means the distribution tail heating. For high energeticregion ξ  1 the rough approximation can be used G(ξ) = Φ(ξ5/2) ∼ exp[−ξ5/2]. Inother words, the electron distribution slowly tends to its equilibrium solution f(ξ) exp[−ξ] · exp[−ξ5/2] having an underheated tail.Further, we include into consideration the heating of the electrons by the sourceof the hot particles which is localized in a high energetic region ξ+  1. In a coldregion the distribution is supposed Maxwellian. Therefore for asymptotic analysis weagain use eq. (5). Let the heating source have small intensity σ  1, so the densityand the energy of the system practically do not undergo noticeable changes during theprocess under consideration. Then a stationary distribution is formed while the sourceis regarded as still located in a tail distribution region. Under the following conditions:t → ∞, σ → 0, ξ → ∞, such that Δt = σ · t · ln(1/σ) is finite, ξ+ = ξ/2T0 − ln(1/σ) isfinite, the new quasi–steady-state non-equilibrium distribution is established inside themomentum interval between the energy source ξ+ and the bulk distribution (or the sinkξ− < ξ+). If, in particular, the source and the sink are monokinetic distributed functionsH(f) = σ · ξ−1/2 · [δ(ξ − ξ+) − δ(ξ − ξ−)], with ξ− < ξ+, then f(ξ) = Ce−ξ + σ ·Δ(ξ),whereΔ(ξ) = η[ξ+ − ξ] + η[ξ − ξ+] exp [−(ξ − ξ+)]− η[ξ− − ξ] + η[ξ − ξ−] exp [−(ξ − ξ−)]and η[y] is a unit function. Numerical results show that the analytical asymptotic estima-tions obtained for the linear case are fit for a wider class of parameters. The functionaldependence of the steady-state non-equilibrium electron distribution is unsensitive tothe extent to which the source and sink are located in momentum space. Figure 1demonstrates the quasi–steady-state distribution function that has plateau essentiallyexceeding the equilibrium distribution. The size of plateau is shortening with time be-cause of normalization v+2/vth2(t). However, the distribution tail formation is describedby formulas (6), f(ξ) ∼ exp[−ξ5/2] in the region ξ  ξ+.Now let pass to the classical problem of electron and ion temperature relaxation:T 0e + T0i = Te(t) + Ti(t) = 2Teq. For (1)-(4) with H(fe) = 0 the heating source forelectrons are hot ions. For ρ → 0 the usual collision time ordering is the following.During time te electrons approach their equilibrium, next ions’ maxwellization occurswithin time ti  te. During temperature relaxation that takes time tei  ti  te,the distribution functions remain Maxwellians ∼ e−ξ with time-dependent electronand ion temperatures. Then the formula for temperatures can be obtained from the202 I. F. POTAPENKO, T. K. SOBOLEVA and S. I. KRASHENINNIKOV0 5 15 2510−2010−1510−1010−5           f(0,t)v2/vth2 ______   f(v,t)Fig. 1. – Formation of the quasi–steady-state non-equilibrium distribution function vs. v2/v2th(t)for different time moments and ξ+ = 4, ξ− = 0.5, σ = 10−5. The dashed-line curve is theMaxwell distribution.system (1)-(4): T 3/2e dTe/dt = C (Ti − Te). Such an approach imposes a severe restric-tion on the initial temperatures ρ1/3  Ti0/Te0  ρ−1/3.Let us setfe,i(v, t) =12πv−3e,i · f ′(v2/v2e,i, t′) , t = t0t′, t0 = 2πΓ (2Teq/me)3/2 , Te,i = TeqTe,i′,and the squared ratio of thermal speeds ε = v2i /v2e = ρTi/Te. We substitute theseexpressions in formulas (1)-(4) and return to the original notations. Then the equationfor the electron function reads(7) Te3/2∂f∂t=1ρ1√ξ∂∂ξ[(Ae + Ai)∂f∂ξ+ (Be + Bi) f + ρ√Te∂Te∂tξ3/2f].Here, coefficients Ae, Be correspond to the electron-electron collisions(8) Ae =23[∫ ξ0dy y3/2fe(y) + ξ3/2∫ ∞ξdy fe(y)], Be =∫ ξ0dy y1/2fe(y),and take a simple form Ae  2/3ξ3/2∫∞0dy fe(y), Be  2/3ξ3/2fe(0) for ξ  1 andAe = Be = 1 for ξ  1. For the electron-ion collisions(9) Ai =23ε[∫ ξ/ε0dyy3/2fi(y) +(ξε)3/2 ∫ ∞ξ/εdy fi(y)], Bi = ρ∫ ξ/ε0dy y1/2fi(y).ELECTRON HEATING AND ACCELERATION FOR THE NONLINEAR KINETIC EQUATION 203Table I. – Comparison of the electron-electron and electron-ion coefficients.ξ ∼ ε Ae, Be ∼ ε3/2 Ai ∼ ε, Bi ∼ ρε  ξ  ε2/3 Ae, Be  ε Ai ∼ ε, Bi ∼ ρξ ∼ ε2/3 Ae, Be ∼ ε Ai ∼ ε, Bi ∼ ρξ  ε2/3 Ae, Be  ε Ai ∼ ε, Bi ∼ ρThe normalization conditions can now be rewritten as follows:(10)∫ ∞0dx√x fe,i(x, t) = 1,∫ ∞0dxx3/2fe,i(x, t) =32and Te(t)+Ti(t) = 2. Now we replace eq. (7)-(9) by approximate equations for the smallmass ratio ρ  1. For fixed ρ, ε  1 and different magnitudes of the variable ξ, thecomparison by the order of coefficients (8) and (9) yields the result given in table I. Ascan be seen, the influence of the coefficients Ai, Bi around ξ ∼ 1 is small but it enhancessensibly in the vicinity of 0 ≤ ξ ≤ ε2/3. Thus in the vicinity of the point ξ  0, thereexists a composed boundary layer with a width of δ ∼ ε2/3. Within its internal domain0 ≤ ξ < ε coefficients Ai, Bi depend essentially on the ion distribution fi but for ξ  εthis dependence disappears. We obtain for ρ→ 0, ε→ 0 and ξ  ε from (7)-(9)(11) Te3/2∂fe∂t=1ξ1/2∂∂ξ[1ρ(Ae∂f∂ξ+ Bef)+(TiTe∂fe∂ξ+ fe)+ Te1/2∂Te∂tξ3/2fe],and we obtain the boundary condition and the temperature equation from (10), (11)(12)(TiTe∂fe∂ξ+ fe)ξ=0= 0, Te3/2dTedt=23[Tife(0)− Te∫ ∞0dξfe(ξ)].The main part ε  ξ ≤ ε2/3 of the boundary layer δ is correctly described by eq. (11).Examination of (7)-(9) shows that the solution can be replaced in the internal domainξ ∼ ε by the boundary condition (12) with the approximation error of O(ρ). From (10)-(12) for ρ = 0, we formally obtain the solution f (0)(ξ)  exp[−ξ], θ3/2θt  (T − θ) thatshould be “corrected” in the vicinity of ξ ≈ 0. To adjust (12) we assume(13) fe(ξ, t) =2√πe−ξ[1 +Te − TiTi· u(ξ, ε)],insert (13) in (10)-(12) and neglect the squared terms of u. After rather lengthycomputation one can obtain u(x)  δψ(ξ/δ) + O(ε), where δ = (3√πε/4)2/3 andψ(x) =∫∞xdy/(1 + y3/2). The applicability of the known formula for temperaturesε 1 is hundred times less rigorous than ρ 1 and the corrected formula reads(14) Te3/2dTedt 43√πTi − Te1 + 2.9ε2/3.204 I. F. POTAPENKO, T. K. SOBOLEVA and S. I. KRASHENINNIKOV0 1 3 500.250.50.751v/vth f(v,t)/fM(v,t)                 Fig. 2. – Temporal evolution of the electron distribution function tails for the quasi-linear dif-fusion operator (15) located in 0.05 ≥ v ≥ 0.75.The electron function achieves its maximum absolute deviation from its local Maxwelldistribution at ξ = 0: fe(ξ, t)  2/√π·e−ξ[1+2.9ε2/3(Te − Ti)/Ti]. The relative deviationof the electron distribution from equilibrium can be larger in the tail region than wherethe tail formation is described by (6). The same behavior of the distribution functiontails is valid for any localized heat source having time dependence as T ∼ t2/3.An example is the interaction of RF waves with a plasma. It is described by a LFPequation with an added quasi-linear term (usually 2D in velocity space)(15) H(f) =1ξ1/2∂∂ξ[ξ3/2Dql∂f∂ξ], Dql ={const, if ξ1 ≤ ξ ≤ ξ20, otherwise.For this case the numerical result demonstrates in fig. 2 the tail formation. It has acharacter of a propagating wave with slightly changing slope in time because of heating.3. – Self-similar solutions with accelerated tailsUnlike the cases considered above the situation changes drastically when in H(f)from (15) the coefficient of the quasi-linear diffusion Dql(ξ, t) increases with velocityincreasing. We consider a special class of functions, for which it is possible to constructa self-similar solution. Thus from (1), (15) instead of (11) we obtain(16) Te3/2∂f∂t=1ξ1/2∂∂ξ[A∂f∂ξ+ Bf + ξ3/2Te1/2D(ξ, t)∂f∂ξ+ Te1/2∂Te∂tξ3/2f],ELECTRON HEATING AND ACCELERATION FOR THE NONLINEAR KINETIC EQUATION 2050 20 40 60 8000.511.522.5F/ FMaxwv 2/eP=3/2 Fig. 3. – Temporal evolution of the electron distribution function tail for p = 3/2, e = 3/2 · v2th.The normalization conditions (10) lead formally to the following equation for T (t):Tt =dTdt=∫ ∞0dyf(y)∂∂y[y3/2D(y, t)].The only case, when eq. (16) formally admits stationary solutions, ∂f/∂t = 0, corre-sponds to Dql(ξ, t) = D(ξ)/√T (t). Note, we have the same law T (t)  t2/3. TakingD(ξ)ξ3/2 = D0ξp−3/2, where D0 is the normalization constant and 5/2 > p > 0 is anadjustable parameter, we find the distribution function in the high-velocity region(17) f(ξ →∞) ∝ exp[− ξ5/2−p5/2− p], 5/2 > p ≥ 0.For the case p = 5/2 we obtain a power-law tail f(ξ → ∞) ∝ ξ−5/2. For p = 3/2 thesolution of (1), (15) with Dql(ξ, t) = D0 is Maxwellian. Figure 3 shows the distribu-tion function tail formation for different time moments. At the beginning the tail has aCoulombian character and starting from the time τ ∼ 1/D0 it spreads into superthermalvelocity region following the character of diffusion action. The formation of the solu-tion (17) is presented for p = 2 in fig. 4. Numerical results also show that in the regionof 0 ≤ v ∼ 4vth the distribution is mostly close to Maxwellian and display the transitionregion between the Maxwellian part and the enhanced tail.4. – ConclusionsObtained results are consistent with the preceding results. The distribution functionis close to Maxwellian in the thermal velocity region. The perturbation of the electrondistribution function in case of heavy hot ions, which has a character of a boundarylayer, gives small correction in the temperature exchange but left the distribution tail206 I. F. POTAPENKO, T. K. SOBOLEVA and S. I. KRASHENINNIKOV0 50 100 15010−30010−20010−100P=2ξ1/2 ____    f(0,t)f(v,t) Fig. 4. – Temporal evolution of the electron distribution function for p = 2. The dashed-linecurve is the corresponding Maxwellian distribution function. In agreement with analytic results,the tail of the distribution function follows expression (17) (dotted line).underheated as well as any source localized in the velocity space. A broader class ofheating terms resulting in an enhancement of the distribution function tail is analyzedanalytically. Numerical simulation of nonlinear LFP equation confirms asymptotic resultswith high accuracy and reveal additional details that can help to test and benchmarkcomplex kinetic codes.∗ ∗ ∗IFP thanks Dr. I. Gamba for fruitful discussions and the hospitality at ICES UT inAustin where the manuscript was elaborated. Valuable and helpful comments of refereeare gratefully acknowledged. This work was partially supported by the PFI DMS N3 ofRussian Academy of Sciences.REFERENCES[1] Batishchev O. V., Batishcheva A. A., Catto P. J., Krasheninnikov S. I.,LaBombard B., Lipschultz B. and Sigmar D. J., J. Nucl. Mater., 241-243 (1997)374.[2] Takizuka T. and Hosokawa M., Contrib. Plasma Phys., 46 (2006) 698.[3] Risken H., The Fokker-Planck Equation-Methods of Solution and Applications (Springer,Berlin) 1989.[4] Batishcheva A. A., Batishchev O. V., Shoucri M. M., Krasheninnikov S. I., CattoP. J., Shkarofsky I. P. and Sigmar D. J., Phys. Plasmas, 3 (1996) 1634.[5] Potapenko I. F., Bobylev A. V. and Mossberg E., Transp. Theory Stat. Phys., 37(2008) 113.

Electron heating and acceleration for the nonlinear

kinetic equation

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Electron heating and acceleration for the nonlinear kinetic equation

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