Analytical edge power loss at the lower hybrid resonance: comparison with ANTITER IV and application to ICRH systems
UUnder consideration for publication in J. Plasma Phys. Analytical edge power loss at the lower hybridresonance: comparison with ANTITER IV andapplication to ICRH systems
V. Maquet † , A. Druart , A. Messiaen Laboratory for Plasma Physics - ERM/KMS, Avenue de la Renaissance 30, B-1000 Brussels. Université Libre de Bruxelles, B-1050 Brussels. International Solvay Institutes, CP 231, B-1050 Brussels.(Received xx; revised xx; accepted xx)
In non-inverted heating scenarios, a lower hybrid (LH) resonance can appear in theplasma edge of tokamaks. This resonance can lead to large edge power deposition whenheating in the ion cyclotron resonance frequency (ICRF) range. In this paper, the edgepower loss associated with this LH resonance is analytically computed for a cold plasmadescription using an asymptotic approach and analytical continuation. This power losscan be directly linked to the local radial electric field and is then compared to thecorresponding power loss computed with the semi-analytical code ANTITER IV. Thismethod offers the possibility to check the precision of the numerical integration made inANTITER IV and gives insights in the physics underlying the edge power absorption.Finally, solutions to minimize this edge power absorption are investigated and applied tothe case of ITER’s ion cyclotron resonance heating (ICRH) launcher. This study is alsoof direct relevance to DEMO.
Key words:
Plasma Heating, ICRH, lower hybrid resonance, power loss, edge modes.
1. Introduction
A potentially important power loss mechanism for ion cyclotron resonance heating(ICRH) arising in the presence of a lower hybrid (LH) resonance in the edge of a tokamakplasma has recently been discussed in Messiaen & Maquet (2020). The possibility of thispower loss was already pointed out in earlier work (Berro & Morales (1990); Lawson(1992)) and can be linked to a confluence between the fast and the slow wave at theLH resonance in non-inverted heating scenarios ( i.e. where ω > ω ci where ω ci is the thecyclotron frequency of the majority ions and ω is the driving angular frequency of theantenna) for toroidal wavenumber k z smaller than the propagation constant in vacuum k . The same paper provides simple rules to minimize the power loss at this LH resonanceconstraining the current distribution on the strap array in amplitude and phase.Edge power loss in tokamaks should be avoided as it can lead to a reduction of thedeposited heating power in the plasma core and to deleterious impurity release from thefirst wall of the device. A correlation between ICRH related impurity release and low | k z | < k present in the k z spectrum launched by ICRH antennas was investigated inMaquet & Messiaen (2020). This paper also proposes a new non-conventional antennastrap phasing minimizing power losses in the edge of the tokamak for effective heatingof the core plasma. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] J a n V. Maquet et al
In the edge of a tokamak, the plasma can be approximated by the cold plasmadispersion description. A recent upgrade of ANTITER II, called ANTITER IV, is nowdescribing the waves launched by an ICRH antenna in the cold plasma limit includingthe full description of the fast and slow waves confluence and the LH resonance aspects(Messiaen et al. (2021)). The present paper analytically derives the power loss at the LHresonance in section 2, compares the results with the power loss computed numericallyby ANTITER IV in section 3 and applies the results to relevant operational scenariosfor the ITER ICRH antenna in section 4.
2. Analytical derivation
The edge plasma is described by the cold dielectric plasma tensor and Maxwell’sequations expressed in the radial x direction for a slab geometry and Fourier analysisin the y, z directions where z represents the direction along the total steady magneticfield B . The plasma wave model considered leads to a system of 4 first order ordinarydifferential equations (ODEs) of the form Y ( x ) (cid:48) = A ( x ) Y ( x ) : dd x iωB z E y iωB y E z = 1 (cid:15) − k y (cid:15) k (cid:15) + ( (cid:15) k z − k (cid:15) ) (cid:15) k z − k y k z (cid:15) (cid:15) − k y k k y (cid:15) k y k z k k y k z (cid:15) (cid:15) ( k (cid:15) − k y ) − k y k z k k z (cid:15) k z k − (cid:15) iωB z E y iωB y E z . (2.1) Moreover, E x = − k y k (cid:15) ωB z + k z k (cid:15) ωB y − i (cid:15) (cid:15) E y , (2.2) ωB x = k y E z − k z E y . (2.3)In the expressions above, (cid:15) , (cid:15) and (cid:15) are the components of the cold dielectric plasmatensor (Swanson (2012)). The field components E y and B z can be associated to thefast wave components of interest for ICRH and E z and B y can be associated to theslow wave components. This system is singular at the location x where (cid:15) ( x ) = 0 which corresponds to the LH resonance in the cold plasma description. In what follows,the method used to derive the power loss at this LH resonance is similar to the one inFaulconer & Koch (1994) where the power loss at the Alfvén resonance was obtained froman asymptotic expansion of the system of differential equations and from an analyticalcontinuation around the singularity.2.1. Asymptotic expansion in the vicinity of the LH resonance
Choosing the position of the LH resonance at the origin x = 0 , one can expand (cid:15) ( x ) ina Taylor series as (cid:15) ( x ) = (cid:15) (cid:48) (0) x + O (cid:0) x (cid:1) where (cid:15) (cid:48) is the derivative of (cid:15) with respect to x . This leads to an asymptotic expression of A ( x ) of the system (2.1): A ( x ) = A (cid:15) (cid:48) x + O (cid:0) x (cid:1) (2.4)with (cid:15) (cid:48) ≡ (cid:15) (cid:48) (0) and where A ≡ − k y (cid:15) k (cid:15) k z (cid:15) − k y k k y (cid:15) k y k z k
00 0 0 0 − k y k z k k z (cid:15) k z k . (2.5) nalytic power loss at the lower hybrid resonance in ANTITER IV
3A straightforward computation shows that the matrix A satisfies A = 0 . (2.6)This property will be fundamental in the forthcoming computations. The matrix A canalso be expressed in a way that will become handy later: A = 1 k k (cid:15) k y k z (cid:0) − k y (cid:15) k k z (cid:1) . (2.7)2.2. Fields near the resonance
To derive expressions for the tangential ( y, z ) fields near the resonance we consider x asbeing a complex variable ( i.e. , we will consider the analytic continuation of our systemof equations over the complex plane). Close to x = 0 , one can truncate the expansion(2.4) as A ( x ) (cid:39) A (cid:15) (cid:48) x . (2.8)In this approximation, using the property (2.6), we observe that A ( x ) and its primitivecommute. This enable us to use the general theorem derived in Appendix A. The solutionto the system (2.1) reads iωB z E y iωB y E z = exp (cid:18)(cid:90) x A ( x (cid:48) )d x (cid:48) (cid:19) C = exp (cid:18) A (cid:15) (cid:48) log( x ) (cid:19) C = (cid:18) I + A (cid:15) (cid:48) log( x ) (cid:19) C , (2.9)where C ≡ (cid:0) C , C , C , C (cid:1) is a constant column vector depending on the initial con-ditions of the problem. Here, log denotes the principal value of the complex logarithm.The last equality in (2.9) is exact due to the property (2.6). More explicitly, our solutionis given by iωB z E y iωB y E z = C C C C + α ( x ) k (cid:15) k y k z (cid:0) − k y C (cid:15) k C k z C (cid:1) , (2.10)with α ( x ) ≡ log( x ) k (cid:15) (cid:48) and where we used relation (2.7).The expression above clearly shows that all fields except B y are singular at theresonance. Remarkably, one can nevertheless construct a particular combination of themwhich remains non-singular at x = 0 . Left multiplication of equation (2.10) by the linevector (cid:0) − k y (cid:15) k k (cid:1) leads to k (cid:15) E y − k y iωB z + k z iωB y = k (cid:15) C − k y C + k z C ≡ Υ. (2.11)There is a clear relationship between this combination of the fields and the radial field E x : recalling that the latter takes the form given in (2.2), Υ can be written as Υ = ik (cid:15) E x . (2.12) V. Maquet et al
We finally get from (2.10) iωB z = C + α ( x ) (cid:15) k Υ, (2.13) E y = C + α ( x ) k y Υ, (2.14) iωB y = C , (2.15) E z = C + α ( x ) k z Υ. (2.16)These are the expressions of the singular part ( i.e. up to O (cid:0) x (cid:1) corrections) of the y, z fields in the vicinity of the LH resonance (located at x = 0 ).2.3. Power loss at the LH resonance
Our next goal consists in computing the power loss at the LH resonance using theexpressions of the fields derived above. In our case, the power loss ∆P is given by thedifference of the Poynting flux S ( x ) after and before the resonance: ∆P ≡ lim κ → + S ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =+ κx = − κ . (2.17)The Poynting flux can be written as S ( x ) ≡ Re( E y H ∗ z − E z H ∗ y ) (2.18) = − ωµ Im[ E y ( iωB z ) ∗ − E z ( iωB y ) ∗ ] . (2.19)Defining a ≡ k y C , b ≡ k (cid:15) C and c ≡ k z C , one has Im [ E z ( iωB y ) ∗ ] = Im( C C ∗ ) + Im[ α ( x ) c ∗ Υ ] (2.20) Im [ E y ( iωB z ) ∗ ] = Im( C C ∗ ) + Im[ α ∗ ( x ) bΥ ∗ + α ( x ) a ∗ Υ ] . (2.21)Making use of these expressions and of the identity Υ = − a + b + c , the Poynting fluxcan be rewritten as S ( x ) = − ωµ { Im( C C ∗ − C C ∗ ) + Im[ α ∗ ( x ) bΥ ∗ + α ( x ) a ∗ Υ − α ( x ) c ∗ Υ ] } (2.22) = − ωµ Im( C C ∗ − C C ∗ ) + Im[ α ( x )] ωµ | Υ | . (2.23)We finally have ∆P = lim κ → + S ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =+ κx = − κ = | Υ | ωµ (cid:32) lim κ → + Im[ α ( x )] (cid:12)(cid:12)(cid:12)(cid:12) x =+ κx = − κ (cid:33) . (2.24)On the other hand, one has the identity Im[ α ( x )] (cid:12)(cid:12)(cid:12)(cid:12) x =+ κx = − κ (cid:39) − πk | (cid:15) (cid:48) | . (2.25)A proof of this equality is given in Appendix B. Plugging (2.25) into (2.24) leads to thefinal result: ∆P = − πk ωµ | (cid:15) (cid:48) | | Υ | (2.26) = − πk ωµ | (cid:15) (cid:48) | | (cid:15) E x | . (2.27) nalytic power loss at the lower hybrid resonance in ANTITER IV E z but on the local radial electric field E x . The powerloss also inversely depends on the derivative of the first dielectric tensor (cid:15) component: alarger density gradient in the edge will lead to lower power losses. Moreover, as relation(2.27) is exact, it provides an opportunity to verify the numerical integration done inANTITER IV when crossing this LH resonance.2.4. Parallelism with previous works
The result (2.27) reminds of the work of Faulconer & Koch (1994), where only the fastwave was taken into account in the plasma description. This leads to a system of 2 firstorder ODEs: dd x (cid:18) iωB z E y (cid:19) = 1 u (cid:18) − (cid:15) k k y − u + (cid:15) k u − k y (cid:15) k k y (cid:19)(cid:18) iωB z E y (cid:19) . (2.28)Here, u ≡ k (cid:15) − k z . (2.29)The authors found that the power lost at the Alfvén resonance ( u = 0 ) is proportionalto the square of | uE x | : ∆P = − πωµ | u (cid:48) | | uE x | . (2.30)A clear symmetry can be found between (2.30) and (2.27).
3. ANTITER IV
The previous results can now be used to estimate the accuracy of the calculation inANTITER IV. ANTITER is a semi-analytic code describing an antenna in front of aplasma in plane geometry in the cold plasma limit. The code uses Fourier analysis in thepreviously defined ( y, z ) directions and numerical integration in the radial x one. An idealFaraday screen is assumed at the antenna mouth together with single-pass absorption inthe plasma (Messiaen et al. (2010)).The antenna is described by a set of boxes recessed into a metal wall containinginfinitely thin straps. The edge plasma electron density profile used for the study isthe ITER worst case plasma profile for IRCH (2010low – Carpentier & Pitts (2010)) as itis representative of the large SOL to be expected in large machines like ITER or DEMO.This electron density profile is presented in the figure 1 along with the antenna and theLH positions.ANTITER II is only describing the fast wave, based on the fact that the fast and slowwave component can be considered decoupled in first approximation in the ion cyclotronrange of frequencies for a large density range. This is no longer true at resonances.ANTITER IV extends the above description to include a detailed description of boththe slow and fast waves (Messiaen et al. (2021)). The LH resonance is handled by addinga small amount of collisions in the cold plasma tensor which corresponds to adding animaginary part to the dielectric tensor components. The plasma part is finally describedat the antenna position by four admittance matrices (cid:18) ωB z ωB y (cid:19) = (cid:18) ξ − ξ − ξ − ξ − (cid:19)(cid:18) E y E z (cid:19) (3.1) V. Maquet et al x (m) n e ( c m - ) ITER 2010 low density profile
LH resonance positionAntenna position
Figure 1.
ITER 2010 low electron density profile. The antenna and the LH resonanceposition are also displayed.
Figure 2.
The four plasma impedance matrices seen at the antenna position as a function of ( k y , k z ) . describing the relationship between the four tangential plasma components E y , E z , B z and B y for all the wavelets ( k y , k z ) of the Fourier expansion. The real part of the fourimpedance matrices found at the antenna position ( i.e. at the lower end of the electrondensity profile of figure 1) are presented in figure 2. These matrices are important forthe derivation of the active Poynting’s power flux. nalytic power loss at the lower hybrid resonance in ANTITER IV r (m) -6 -5 -4 -3 -2 / Collision frequencies considered
LH resonance V / m Fourier component (ky,kz)=(-3,0.64) (k
02 2 E y )(-k y i B z )(k z i B y )( )LH resonance Figure 3.
Illustration of the near singular behavior of individual components of the fields andtheir finite sum for a pair ( k y , k z ) around the LH resonance. Fields near the LH resonance with ANTITER IV
We first investigate equation (2.11). If ANTITER IV correctly describes the wavesat the LH resonance, relation (2.11) should lead to a non-singular behaviour. The fieldsultimately depend on the amount of collisions added to the dielectric tensor terms in orderto integrate the system (2.1), in the vicinity of the singularity, by analytical continuationin the complex plane. The collision coefficient in ANTITER IV does not bear a physicalmeaning and its sole purpose is to bypass the resonance. Three collisions profile differingby one order of magnitude are presented in figure 3a. Each component of the Fourier fields k (cid:15) E y , k y iωB z , k z iωB y and their sum are presented figure 3b for the three collisioncoefficient selected. As expected from relation (2.16), the fields E y and B z computed forlow collision coefficient show near singular behaviour at the lower hybrid resonance whiletheir sum Υ stays regular even for vanishing amounts of collisions. A smaller number ofcollisions leads to smaller integrating steps in ANTITER IV but improves the accuracy ofthe power loss calculation. Therefore, theoretical expressions like (2.11) and (2.27) givesan opportunity to assess the precision of the integration made in ANTITER IV. For therest of the computations, the second collisions coefficient profile presented in figure 3a isselected. 3.2. Power loss at the LH resonance with ANTITER IV
The power losses at the LH resonance using the Poynting flux (2.19) and the analyticalpower loss (2.27) computed with the fields of ANTITER IV are compared. This exerciseis performed for a pure fast wave excitation ( i.e. E y ( k y , k z ) = 1 at the plasma edgewhile ensuring E z ( k y , k z ) = 0 ). For this specific excitation, the power loss is limited tothe wavenumbers smaller than the wave propagation constant in vacuum | k z | < k asthey correspond to the fast wave undergoing a wave confluence with the slow wave. Thisfact is verified in figure 4. We also observe that the relative error between the analyticalPoynting flux given in (2.27) and the numerical integration of ANTITER IV is negligiblein the region of interest ( i.e. where the power loss is not negligible). The same test canbe performed for a pure slow wave excitation ( i.e. E z ( k y , k z ) = 1 and E z ( k y , k z ) = 0 ).Here we see a strong interaction which is no more limited to the region | k z | < k butextending to k < k z < m − . A negligible relative error between the two methods isagain observed.These results give further confidence in the ANTITER IV calculations. They hint at V. Maquet et al
Figure 4.
Power loss at the LH resonance for a pure E y ( k z , k y ) excitation. Red lines delimitsthe | k z | < k . Here k = 1 . m − . Figure 5.
Power loss at the LH resonance for a pure E z ( k z , k y ) excitation. Red lines delimitsthe | k z | < k . Here k = 1 . m − . possibilities to minimize the edge power absorption. For a field-aligned antenna with afield-aligned Faraday shield (FS), figure 4 shows that the power losses can be minimizedby avoiding | k z | < k in the power spectrum. The antenna power spectrum can be easilymodified by shaping the E y spectrum excited by the antenna ( i.e. by varying the currentamplitude and phase distribution over the straps). A FS that is not aligned with thebackground magnetic field will excite a spurious E z spectrum that can in turn lead tosignificant new losses for low k z above | k z | < k as shown in figure 5. These new lossescan be reduced by further depleting the low k z part of the power spectrum at the expenseof a reduction in the power coupled to the plasma core. We also see that for an equalexcitation of E y and E z , the losses due to E z are one order of magnitude larger than thelosses due to E y .Finally, one can also minimize power losses using gas puff (Zhang et al. (2019)) whichwill lead to larger density gradient near the lower hybrid location. This last method canlead to a substantial decrease in the edge power losses and at the same time increasesthe power coupling to the core plasma.
4. Application
The results of the previous sections indicate how to minimize power losses into thepresence of a LH resonance in the plasma edge for a given plasma profile. Here weuse ANTITER IV to minimize those losses for a given plasma density profile using amultidimensional minimization procedure. nalytic power loss at the lower hybrid resonance in ANTITER IV -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) Normalized Power Spectrum (0,2.9,3.8,0.4)(0 0)(0 0 ) -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) -3 Normalized Edge Power Spectrum (0,2.9,3.8,0.4)(0 0)(0 0 )
Figure 6. (a) Normalized k z power spectrum and (b) normalized k z edge power loss spectrumfor a current distribution on straps of constant amplitude and three different toroidal phasing.A poloidal phasing of π/ is imposed for load resilience. The toroidal phasing (0,2.9,3.8,0.4)minimize the edge LH power losses. ITER-like antenna
The ITER antenna is composed of 24 straps grouped into triplets (Lamalle et al. (2013)). For a fixed and even current amplitude on the straps, the remaining degree offreedom left to minimize the power losses is to change the phase distribution of the array.The spectrum minimizing the edge power losses found with ANTITER IV correspondsto the phasing (0,2.9,3.8,0.4) and is presented in figure 6a along with the conventionalphasings (0 ππ and (0 π π ) . Figure 6b presents the related edge power loss spectrum.The respective percentage of power lost P loss /P tot for each phasing is 0.25, 0.36 and 1.03%. While those numbers are small, the power coupled to the plasma is in the MW range(10 MW for one ITER launcher) which leads to power losses of about 10 kW (100 kW inITER).A misaligned antenna box, but with aligned FS, deforms the power loss spectrum butdoes not lead to direct spurious E z excitation and should only modestly change the resultabove. The same computation as in figure 6 but for a magnetic field tilted at an angleof 15 ◦ is presented in figure 7. It leads to the phasing (0,2.8,3.9,0.4) and a respectivepercentage of power loss P loss /P tot of 0.55, 0.95 and 1.61 %.While for the field aligned FS case the minimization only leads to a marginal decreaseof the power loss at the LH resonance, a non-aligned FS will create an undesirable E z excitation and will greatly increase the power loss at the LH resonance. The misalignmentof the FS with the magnetic field can be treated in ANTITER IV using the poloidalelectric field E y computed in the aligned case and rotating it by an angle of 15 ◦ . Theresult is presented in figure 8 and leads to the phasing (0,3.6,2.3,5.9) and a respectivepercentage of power loss P loss /P tot of 6.48, 7.98 and 7.12 %.One can finally verify that minimizing the power loss at the lower hybrid corresponds tothe minimization of the local radial electric field E x at this resonance. This is performedwith a FS and an aligned antenna box by toroidally varying the power ratio between thetwo inner straps and the total power coupled P central /P tot and by adding a phase ∆φ tothe best phasing (0 , . ∆φ, . ∆φ, . . The result, displayed in figure 9, correspondsindeed to a minimum of E x excitation.0 V. Maquet et al -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) Normalized Power Spectrum (0,2.8,3.9,0.4)(0 0)(0 0 ) -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) -3 Normalized Edge Power Spectrum (0,2.8,3.9,0.4)(0 0)(0 0 )
Figure 7. (a) Normalized k z power spectrum and (b) normalized k z edge power loss spectrumfor a current distribution on straps of constant amplitude and three different toroidal phasing.A poloidal phasing of π/ is imposed for load resilience. The toroidal phasing (0,2.8,3.9,0.4)minimize the edge LH power losses. -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) Normalized Power Spectrum (0,3.6,2.3,5.9)(0 0)(0 0 ) -20 -10 0 10 20 k z (m -1 ) P o w e r ( W m ) Normalized Edge Power Spectrum (0,3.6,2.3,5.9)(0 0)(0 0 )
Figure 8. (a) Normalized k z power spectrum and (b) normalized k z edge power loss spectrumfor a current distribution on straps of constant amplitude and three different toroidal phasing.A poloidal phasing of π/ is imposed for load resilience. The toroidal phasing (0,3.6,2.3,5.9)minimize the edge LH power losses. Figure 9.
Map showing respectively: (a) the % of power loss in the edge and (b) The maximumelectric field at the LH resonance as a function of the power ratio P central /P tot and a centralphase deviation ∆φ from the phasing minimizing the edge power loss (0 , . ∆φ, . ∆φ, . . nalytic power loss at the lower hybrid resonance in ANTITER IV
5. Conclusion
The paper presents an analytical derivation of the power loss that arises in the presenceof a lower hybrid resonance in the plasma edge of a fusion machine. To do so we useda slab geometry and a cold plasma model. The power loss found is linked to the localradial electric field E x at the LH resonance position and is inversely proportional to theslope along the radial direction x of the first cold dielectric tensor component (cid:15) . Theanalytical formula of the power loss is then used to verify the accuracy of the numericalintegration performed by the semi-analytical code ANTITER IV using the same slabdescription and cold plasma model. Good agreement is found between the two. Finally,we explore possible scenarios that could minimize the ITER ICRH power losses:(i) In a screen-aligned scenario, one should avoid the excitation of the lower | k z | < k part of the antenna power spectrum.(ii) In case of direct E z excitation due to a misalignment of the FS with the backgroundmagnetic field, the lower k z region to be avoided in the power spectrum is enlarged above k .The fact that the power loss is proportional to the derivative of (cid:15) along x provides amethod to directly influence the power loss at the LH resonance by shaping the plasmadensity profile near the antenna. An easy way to do so would be to use gas puff (Zhang et al. (2019)) and will be explored in a future paper.While the results are in line with earlier work ( e.g. Berro & Morales (1990); Lawson(1992)), the limits of the model should also be emphasized. The model neglects finitetemperature effects preventing the detailed description of the wave conversion at the LHresonance to new electrostatic waves ( e.g. ion Berstein waves). The model does not takeinto account the poloidal and toroidal inhomogeneity of the plasma density profile. It alsoneglects possible non-linear effects ( e.g. ponderomotive force) arising in the presence ofstrong fields excited by the antenna. The model also uses plane geometry and an antennarecessed into the wall of the machine.
Acknowledgements
This work has been carried out within the framework of the EUROfusion Consortiumand has received funding from the Euratom research and training programme 2014-2018and 2019-2020 under grant agreement No 633053. The views and opinions expressedherein do not necessarily reflect those of the European Commission.
Declaration of interests
The authors report no conflict of interest.
Appendix A. A theorem about matrix differential equations
Theorem A.1.
Let be a first-order matrix ordinary differential equation of the form dd t x ( t ) = A ( t ) x ( t ) (A 1) with x a n × vector and A a n × n matrix. If A ( t ) commutes with its integral (cid:82) t A ( s )d s then the general solution of the differential equation is x ( t ) = e (cid:82) t A ( s )d s c , (A 2)2 V. Maquet et alwhere c is an n × constant vector.Proof. Using the definition of the matrix exponential, the solution (A 2) can be written x ( t ) = e (cid:82) t A ( s )d s c = ∞ (cid:88) n =0 (cid:16)(cid:82) t A ( s )d s (cid:17) n n ! c . (A 3)Its derivative reads dd t x ( t ) = dd t + ∞ (cid:88) n =0 (cid:16)(cid:82) t A ( s ) ds (cid:17) n n ! c , (A 4) = A ( t ) + ∞ (cid:88) n =0 (cid:16)(cid:82) t A ( s ) ds (cid:17) n − ( n − c (A 5) = A ( t ) e (cid:82) t A ( s )d s , (A 6)where the second equality follows from the fact that, if (cid:82) t A ( s )d s commutes with its t derivative A ( t ) , one can write dd t (cid:18)(cid:90) t A ( s ) ds (cid:19) n = n A ( t ) (cid:18)(cid:90) t A ( s ) ds (cid:19) n − . (A 7)We finally have ˙ x ( t ) = A ( t ) x ( t ) . (A 8) Appendix B. Proof of Equation (2.25)
We will here provide a proof of the identity
Im[ α ( x )] (cid:12)(cid:12)(cid:12)(cid:12) x =+ κx = − κ (cid:39) − πk | (cid:15) (cid:48) | . (B 1)To prove this assertion, one has to notice that, in fact, the LH resonance is not exactlylocated at x = 0 . Because of the collisions arising in the plasma, the frequency ω is notreal but possesses a small, positive, imaginary part: ω = Re ω + i Im ω, | Im ω | (cid:28) | Re ω | and Im ω > . (B 2)Consequently, (cid:15) can be expanded as (cid:15) ( ω ) = (cid:15) (Re ω + i Im ω ) = (cid:15) (Re ω ) + i Im ω ∂(cid:15) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω =Re ω + O (cid:0) (Im ω ) (cid:1) (B 3)and also exhibits a small imaginary part, Im (cid:15) (cid:39) Im ω ∂(cid:15) ∂ω . Recalling ourselves that (cid:15) ( ω ) = 1 − (cid:80) α ω pα ω − ω cα , one can show that Im (cid:15) is indeed positive: Im (cid:15) (cid:39) Im ω ∂(cid:15) ∂ω (cid:39) ω Im ω ∂(cid:15) ∂ω (cid:39) ω Im ω (cid:88) α ω pα (cid:104) (Re ω ) − ω cα (cid:105) > . (B 4)In the following, we will simply write ω instead of Re ω . The main consequence of thediscussion above is that (cid:15) ( x ) doesn’t vanish anymore at x = 0 , but at x = ¯ x ≡ − i Im (cid:15) (cid:15) (cid:48) . nalytic power loss at the lower hybrid resonance in ANTITER IV x = 0 , but at x − ¯ x = 0 .This fact is easily implemented in (2.24) by replacing Im α ( x ) (cid:12)(cid:12) x =+ κx = − κ by Im α ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x − ¯ x =+ κx − ¯ x = − κ = 1 k Im (cid:20) log( κ + ¯ x ) − log( − κ − ¯ x ) (cid:15) (cid:48) (cid:21) (B 5) (cid:39) k arg ( κ + ¯ x ) − arg ( − κ − ¯ x ) (cid:15) (cid:48) . (B 6)Noticing that, for | κ | (cid:29) (cid:12)(cid:12)(cid:12) Im (cid:15) (cid:15) (cid:48) (cid:12)(cid:12)(cid:12) , one has arg ( κ + ¯ x ) (cid:39) , (B 7) arg ( − κ − ¯ x ) (cid:39) (cid:26) π, (cid:15) (cid:48) > − π, (cid:15) (cid:48) < . (B 8)Equation (B 6) becomes Im α ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x − ¯ x =+ κx − ¯ x = − κ (cid:39) − πk | (cid:15) (cid:48) | , (B 9)which is the desired result. REFERENCESBerro, E. A. & Morales, G. J.
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