Analytical model for quasi-linear flow response to resonant magnetic perturbation in resistive-inertial and viscous-resistive regimes
aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug Analytical model for quasi-linear flow response to resonantmagnetic perturbation in resistive-inertial and viscous-resistiveregimes
Wenlong Huang
School of Computer Science and Technology,Anhui Engineering Laboratory for Industrial Internet Intelligent Applications and Security,Anhui University of Technology, Ma’anshan, Anhui 243002, China
Ping Zhu ∗ International Joint Research Laboratory ofMagnetic Confinement Fusion and Plasma Physics,State Key Laboratory of Advanced Electromagnetic Engineering and Technology,School of Electrical and Electronic Engineering,Huazhong University of Science and Technology, Wuhan, Hubei 430074, ChinaDepartment of Engineering Physics,University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
Hui Chen
Department of Physics, Nanchang University, Nanchang, Jiangxi 330031, China (Dated: August 12, 2020) bstract In this work, a quasi-linear model for plasma flow response to the resonant magnetic perturba-tion (RMP) in a tokamak has been rigorously developed in the resistive-inertial (RI) and viscous-resistive (VR) regimes purely from the two-field reduced MHD model. Models for plasma responseto RMP are commonly composed of equations for the resonant magnetic field response (i.e. themagnetic island) and the torque balance of plasma flow. However, in previous plasma responsemodels, the magnetic island and the torque balance equations are often derived separately fromreduced MHD and full MHD equations, respectively. By contrast, in this work we derive both themagnetic island response and the torque balance equations in a quasi-linear model for plasma flowresponse entirely from a set of two-field reduced MHD equations. Such a quasi-linear model canrecover previous plasma flow response models within certain limits and approximations. Further-more, the physical origins of quasi-linear forces and moments in the flow response equation are alsoaccurately calculated and clarified self-consistently. ∗ E-mail:[email protected] . INTRODUCTION Resonant Magnetic Perturbation (RMP) coils have been widely equipped in fusion de-vices due to their emerging and promising potential for controlling plasma properties andbehaviors [1–6]. For example, experiment and simulation results in J-TEXT [7] show thatRMP coils can be employed to control tearing modes and runaway electron activities [8, 9].In the last decades, edge localized modes (ELMs) suppression and mitigation by RMPs havebeen realized in various tokamaks [10–12].It is believed that the mechanism of ELM suppression or mitigation by RMP coils isclosely connected to the plasma response to external magnetic perturbations [13–17]. Pre-vious theory models on error field are often directly and heuristically applied to plasma re-sponse in both the viscous-resistive and the Rutherford regimes [14, 18], which have recentlybeen extended to include the two-fluid and neo-classical flow effects [15, 16]. Predictionsfrom those theory models are highly relevant to the interpretation of experimental results onthe RMP-induced ELM suppression (e.g. [14, 17, 18]). Nonetheless, most previous theorymodels are constructed mainly on heuristic bases instead of more rigorous or self-consistentderivations (e.g. [14, 19]). For example, in those models, the island evolution equation fornonlinear plasma response is derived from reduced MHD model, whereas the torque balanceequation is a direct outcome of the full MHD equations.In this work, we propose a more self-consistent approach to the derivation of the plasmaflow response model in both the resistive-inertial (RI) and viscous-resistive (VR) regimes incylindrical geometry within the framework of the two-field reduced MHD equations. Themodel is composed of the plasma response equation and the poloidal angular momentumequation in the spectral space of Bessel functions. By absorbing the rigid time-dependentflow into boundary perturbation and dropping the quasi-linear magnetic terms, we extendour previous linear plasma response solutions in slab configuration [19] to cylindrical geom-etry in presence of rigid time-dependent poloidal flow for both RI and VR regimes, whichcan reduce to the earlier solutions in the case of steady state flow, as well as the earliersteady state solutions of linear plasma responses in corresponding regimes for the same as-sumptions [20]. The extension of linear plasma response solutions to allow the presenceof time-dependent in addition to steady state flow, enables the construction of quasi-linearstresses that are more self-consistent with the dynamic nature of plasma flow in the plasma3omentum equation. After obtaining the linear plasma response solutions, we further ex-pand the poloidal angular momentum equation including the quasi-linear stresses in theBessel spectral space, where the quasi-linear forces retain the Maxwell torque without anyassumption on its radial profile [18]. Under certain approximations, the newly derived torquebalance equation can naturally reduce to its less complete versions in Refs. [16, 19]. Thenew derivation also allows us to clarify the physical meanings of the quasi-linear forces andmoments [22].It should be noted that the toroidal flow cannot be rigorously considered within the two-field reduced MHD framework adopted in this work. To study the dynamics of toroidalflow, at least the four-field reduced MHD mode or full MHD model should be used. Also,the quasi-linear magnetic effects are neglected here, which should be included in the highlynonlinear regime where the magnetic island width is much larger than the resistive tearinglayer width. Nonetheless, the self-consistent quasi-linear plasma flow response model withinthe framework of two-field reduced MHD equations should provide a solid foundation tothe building of the plasma response model including toroidal flow and quasi-linear magneticeffects next.The rest of the paper is organized as follows. In Sec. II, we introduce the reduced MHDmodel in the cylindrical geometry. In Sec. III, the linear plasma response solutions in the RIand VR regimes with time-dependent flow are obtained. Meanwhile, we derive the poloidalangular momentum equation in the Bessel spectral space and construct the relevant plasmaflow response model in Sec. IV. Finally, a summary and discussion is given in Sec. V.
II. TWO-FIELD REDUCED MHD MODEL
In the cylindrical coordinate system ( r , θ , z ), we consider the plasma response to exter-nal magnetic field perturbation in a low- β , large aspect ratio, periodic “straight” tokamakequilibrium. Introducing the flux function ψ and stream function φ , the magnetic field andvelocity can be written as ~B = B z ~e z + ~e z × ∇ ψ , where B z is the constant toroidal magneticfield, and ~v = ~e z × ∇ φ . In the low β plasma, the perturbed toroidal components of magneticfield and velocity can be neglected. Then, the incompressible two-field reduced MHD model4overning ψ and F are given, respectively, by[23] ∂ψ∂t + ( ~e z × ∇ φ ) · ∇ ψ − B z ∂ z φ = ηj z , (1) ρ ( ∂∂t + ~v · ∇ ) F = ~B · ∇ j z + ν ⊥ ∇ F , (2)where ρ , η , and ν ⊥ are plasma density, resistivity, and viscosity, respectively. In addition,the vorticity F = ~e z · ∇ × ~v = ∇ ⊥ φ = (cid:20) r ∂∂r ( r∂ r ) + ∂ θ r (cid:21) φ, and the toroidal component of current density j z = ~e z · ~j = 1 µ ∇ ⊥ ψ = 1 µ (cid:20) r ∂∂r ( r∂ r ) + ∂ θ r (cid:21) ψ. In the cylindrical geometry, any quantity can be written as f = f eq + δf , where f eq = f eq ( r )and δf = δf ( r, θ, φ, t ) are the equilibrium and perturbation parts of f , respectively. Weexpand the perturbed quantity as δf = P ∞ l = −∞ δf l e il ( mθ − n zR ) , where m ( n ) is the poloidal(toroidal) mode number, and a ( R ) is the minor (major) radius of plasma. Adopting thesingle helicity approximation, neglecting the quasi-linear magnetic terms, and keeping onlythe quasi-linear flow effects, Eqs. (1) and (2) can be reduced to the following ∂ t δψ + δ v · ∇ ψ eq − B z ∂ z δφ + v · ∇ δψ = ηδj z , (3) ρ [ ∂ t δF + v · ∇ δF + δ v · ∇ F ] = B eq · ∇ δj z + δ B · ∇ j z eq + ν ⊥ ∇ δF , (4) ρ∂ t ∆Ω θ = Mr + Rr + ν ⊥ r ∂∂r (cid:18) r ∂∂r ∆Ω θ (cid:19) , (5)where B θ = dψ eq /dr , B eq = B z ~e z + ~e z ×∇ ψ eq = B z ~e z + B θ ~e θ , F = F eq + δF , v = v eq + δ v =( v eq + δv θ ) ~e θ = r Ω θ ~e θ , and δv θ = r ∆Ω θ (see also Appendix A for detail). The Maxwell andReynolds stresses, M and R , satisfy M = − mr Im { δψ ∗ δj z − δψ δj ∗ z } ,R = ρ mr Im { δφ ∗ δF − δφ δF ∗ } . III. PLASMA RESPONSE SOLUTIONS WITH TIME DEPENDENT FLOW INRI AND VR REGIMES
In this section, we extend the previous plasma response solutions in slab configuration [19]to cylindrical geometry in presence of time-dependent poloidal flow for both RI and VR5egimes. This part of the work is also an extension to the previous work on linear plasmaresponse solution in Ref. [20], where the steady state flow instead of time-dependent flowis considered. The latter extension is more consistent with the quasi-linear plasma flowresponse model developed later in Sec. IV, where the plasma flow evolves in response toRMP and is indeed time-dependent.
A. Plasma response equations in inner and outer regions
Neglecting the flow shear terms, the linearized governing equations for δψ and δφ , i.e.Eqs. (3) and (4), can be reduced to (cid:18) ∂∂t + im Ω s (cid:19) δψ + δ v · ∇ ψ eq − B z ∂ z δφ = ηδj z , (6) ρ (cid:18) ∂∂t + im Ω s (cid:19) δF = B eq · ∇ δj z + δ B · ∇ j z eq + ν ⊥ ∇ δF , (7)where Ω s = Ω θ ( r s , t ), r s represents the m/n rational surface. Note that Ω s in Eqs. (6) and(7) can be time-dependent.In the outer region, Eq. (7) becomes B z R (cid:18) q − q s (cid:19) δj z = 1 r dj z eq dr δψ , (8)where q = rB z / ( R B θ ) is the safety factor and q s = m/n . Following Refs. 24 and 25, wedefine δψ ( r, t ) = δψ s ( r ) ψ s ( t ) + δψ c ( r ) ψ c ( t ), where ψ c = δψ ( a, t ) and ψ s ( t ) = δψ ( r s , t )represent the external RMP field and the corresponding plasma response in magnetic fieldon the resonant flux surface, respectively. Besides, δψ s and δψ c satisfy that δψ s ( r s , t ) = 1 , δψ s ( a, t ) = 0 , δψ c ( r s , t ) = 0 , δψ s ( a, t ) = 1 . (9)Then, the index ∆ ′ = [ d ln δψ /dx ] r s can be rewritten as ψ s ∆ ′ = ∆ ′ ψ s +∆ ′ c ψ c = [ dδψ s /dx ] r s ψ s +[ dδψ c /dx ] r s ψ c , where [ f ] r s ≡ f ( r s +) − f ( r s − ) is the jump across the resonant flux surfaceat r = r s .To proceed, we define ˆ ψ ≡ δψ e iϕ temp ( t ) and ˆ φ ≡ δφ e iϕ temp ( t ) , where ϕ temp ≡ R t m Ω s ( t ′ ) dt ′ [19].In the inner region, assuming ∂ x ≫ ( m/r, n/R ), one can simplify Eqs. (6) and (7) as ∂ ˆ ψ ∂t + i B z R C x ˆ φ = η ˆ j z , (10) ρ ∂ ˆ F ∂t = − i B z R C x ˆ j z − i mr dj z eq dr ˆ ψ + ν ⊥ ∂ ˆ F ∂x , (11)6here x = r − r s , C = mq ′ s /q s . We further Laplace transform Eqs. (10) and (11) [26] andneglect the second term on the right hand side of Eq. (11) [27]. Then, one arrives at ∂ ∂χ Ψ = δ ΩΨ s ( ΨΨ s + χξ ) , (12) δ V R δ ∂ ∂χ ξ − δ RI δ ∂ ∂χ ξ + χ ξ + χ ΨΨ s = 0 , (13)where χ = x/δ layer = x/ ( δr s ), δ layer ≡ δr s is the resistive tearing layer width, and˜ ψ = L [ ˆ ψ ] = Z ∞ ˆ ψ e − st dt, ˜ φ = L [ ˆ φ ] = Z ∞ ˆ φ e − st dt,τ R = µ r s η , τ H = R B z √ µ ρr s C , τ V = r s ρν ⊥ ,δ RI = sτ H τ R , δ V R = τ H τ R τ V , ν = isC ǫ s δ , Ω = δτ R s, ǫ s = r s R , Ψ = C B z ˜ ψ, U = − ˜ φ/ν, Ψ s = Ψ( r s ) , Ψ c = Ψ( a ) , ξ = U/ Ψ s . Equations similar to Eqs. (12) and (13) are first proposed in Ref. [21], where the formulasof solutions are given for the steady states. Here, we extend previous results in Refs. [19, 21]to the solutions of linear plasma response in cylindrical geometry with time-dependent rigidflow in both RI and VR regimes.
B. Plasma response solution in the RI regime
In the RI regime, δ = δ RI ≫ δ V R , i.e. s ≫ τ R / ( τ H τ V ), which is equivalent to t ≪ ( τ H τ V ) /τ R , the viscosity term in Eq. (13) can be neglected [21]. In addition, the constant- ψ assumption in the inner region is valid when δ Ω ≪
1, i.e. s ≪ / ( τ R τ H ), so that Ψ / Ψ s ≈ ∂ χ Ψ should be kept in Eqs. (12) and (13). Thus in the constant- ψ RI regime, Eq. (13)reduces to ∂ ∂χ ξ RI − χ ξ RI = χ, (14)where ξ = ξ RI . Eq. (12) becomes ∂ ∂χ Ψ = δ RI ΩΨ s (1 + χξ RI ) , (15)in the inner region. After the asymptotic matching, one arrives at α Ω r s Ψ s = ∆ ′ Ψ s + ∆ ′ c Ψ c , (16)7here α = R ∞−∞ (1 + χξ RI ) dχ ≈ .
12 [19, 28].Using the inverse Laplace transform, Eq. (16) can be transformed to ψ s ( t ) = − ∆ ′ c ∆ ′ e − iϕ temp ( t ) Z t G ( t − t ′ ) ψ c ( t ′ ) e iϕ temp ( t ′ ) dt ′ , (17)where G ( t ) = 1 τ RI ( −
45 [ P A e P A τ + P B e P B τ ] − λ √ π Z ∞ e − uτ u (1 − √ λ u + λ u ) du ) ,τ = t/τ RI , τ RI = τ R τ H , λ = − α / ( r s ∆ ′ ), and P A,B = λ − exp( ± πi/ λ and ∆ ′ c / ∆ ′ , the expression of ψ s in Eq. (17) is nearlythe same as in the slab geometry [19]. C. Plasma response solution in the VR regime
In the VR regime [21], δ = δ V R ≫ δ RI , i.e. s ≪ τ R / ( τ H τ V ), which is equivalent to t ≫ ( τ H τ V ) /τ R , the second term on the left hand side of Eq. (13) can be ignored. Additionally,the constant- ψ assumption is equivalent to δ Ω ≪
1, i.e. s ≪ τ V / ( τ R τ H ). Then Eq. (13) canbe simplified as ∂ ∂χ ξ V R + χ ξ V R = − χ, (18)where ξ = ξ V R . Eq. (12) becomes ∂ ∂χ Ψ = δ V R ΩΨ s (1 + χξ V R ) , (19)in the inner region. Asymptotic matching leads to the following relation α Ω r s Ψ s = ∆ ′ Ψ s + ∆ ′ c Ψ c , (20)where α = R ∞−∞ (1 + χξ V R ) dχ ≈ .
103 [29, 30]. Similar to the RI regime, Eq. (20) can beinverse Laplace transformed to the following α δ V R τ R r s [ ddt + im Ω s ] ψ s = ∆ ′ ψ s + ∆ ′ c ψ c , (21)which yields the following ψ s = − ∆ ′ c ∆ ′ e − iϕ temp ( t ) Z t G ( t − t ′ ) ψ c ( t ′ ) e iϕ temp ( t ′ ) dt ′ , (22)8here G ( t ) = P C τ V R e − P C τ , τ = t/τ V R , τ V R = δ V R τ R , λ = − α / ( r s ∆ ′ ), and P C = λ − .Note that Eq. (21) has also been heuristically derived by Fitzpatrick [14] and furtherinvestigated by Beidler et al. [31, 32]. As previously claimed in the appendix of Ref. [14],such an equation is derived from the following relations in Ref. [33] ω = m Ω θ ( r s ) − n Ω φ ( r s ) , (23) P = τ R τ V , Q = τ H τ R ω, (24) δ FV R = τ H τ R τ V r s , ˆ∆ = ∆ δ FV R r s , (25)ˆ∆ = − . e − iπ/ P Q, (26)where the index ∆ = ∆( ω ) is also conventionally named as ∆ ′ in our work. The rest ofdefinitions are conventional and can be found following Eq. (17) of Ref. [33]. It should benoted that the relations in above Eqs. (23)-(26) are meant for steady state. In particular,Eq. (26) is the steady state solution of Eq. (20). Thus, Eq. (8) in Ref. [14] does not directlyderive from Eqs. (23)-(26). In this work, we transfer the effect of time-dependent flow intothe phases of ˆ ψ and ˆ φ , thus the resulting linearized reduced MHD equations (10) and (11)can be solved using Laplace transform as before [19, 26]. Our linear response solutions withtime-dependent flow can be straightforwardly extended to various parameter regimes. IV. QUASI-LINEAR PLASMA FLOW RESPONSE TO EXTERNAL MAGNETICPERTURBATION
In previous section, we have developed a systematic derivation of the plasma responsesolutions with time-dependent flow in the RI an VR regimes. To close the plasma flowresponse model, the quasi-linear equations for the flow evolution are further derived in thissection.
A. The quasi-linear angular momentum equation in the Bessel function spectralspace
We expand the poloidal angular velocity as ∆Ω θ = P ∞ k =0 a k J ( µ k ˆ r ) / ˆ r , where ˆ r = r/a , J is the first order Bessel function, and µ k are the k -th zero points of J . Then, Eq. (5) can9e transformed to ρ∂ t a k = C k M k + C k R k − ν ⊥ a µ k a k , (27) M k = 2 mJ ( µ k ˆ r s ) Im Z J ( µ k ˆ r ) { δψ δj ∗ z } d ˆ r, (28) R k = 2 mρJ ( µ k ˆ r s ) Im Z J ( µ k ˆ r ) { δφ ∗ δF } d ˆ r, (29)where C k = J ( µ k ˆ r s ) a N k and N k = J ( µ k ).Since δj z , δφ , and δF are localized around the rational surface, the Maxwell andReynolds stresses should be nonzero only in the inner region. Using Taylor expansion at therational surface, M k and R k can be approximated as M k = 2 mJ ( µ k ˆ r s ) a Im Z r s + r s − J ( µ k ˆ r ) δψ δj ∗ z dr = 2 mJ ( µ k ˆ r s ) a Im Z r s + r s − [ J ( µ k ˆ r s ) + µ k J ′ ( µ k ˆ r s ) x ] δψ δj ∗ z dr = F m + D k N m , (30)and R k = 2 mρJ ( µ k ˆ r s ) a Im Z r s + r s − J ( µ k ˆ r ) δφ ∗ δF dr = 2 mρJ ( µ k ˆ r s ) a Im Z r s + r s − [ J ( µ k ˆ r s ) + µ k J ′ ( µ k ˆ r s ) x ] δφ ∗ δF dr = F r + D k N r , (31)where D k = µ k J ′ ( µ k ˆ r s ) J ( µ k ˆ r s ) , and F m = 2 ma Im Z r s + r s − δψ δj ∗ z dr, (32) F r = 2 mρa Im Z r s + r s − δφ ∗ δF dr, (33) N m = 2 ma Im Z r s + r s − xδψ δj ∗ z dr, (34) N r = 2 mρa Im Z r s + r s − xδφ ∗ δF dr. (35)Similar to Ref. [22], we define the quasi-linear forces and moments as the zeroth and firstmoments of the relevant stresses, where F m and F r are the Maxwell and Reynolds forces,and N m and N r are Maxwell and Reynolds moments, respectively. They are the lowest and10he next order terms in Taylor expansion series of the relevant stresses in the Bessel spectralspace.Combining Eqs. (27), (30), and (31), one obtains the following equation ρ∂ t a k = C k ( F m + F r + D k N m + D k N r ) − ν ⊥ a µ k a k . (36)The above equation reduces to the previous poloidal torque balance equation in Besselspectral space if one neglects F r , N r , and N m [16, 25]. Different from the conventional modelin Ref. [18], the torque balance equation in real space is not needed, nor is any assumptionon the radial profile of Maxwell torque. In contrast, we construct the quasi-linear plasmaflow model from the plasma response solution and the poloidal angular momentum equationin the Bessel spectral space, which can also be extended to include the toroidal flow. Onthe other hand, Cole et al. [22] argue that the forces tend to cause the tearing mode lockingwhereas the moments determine the evolution of the flow shear. From Eq. (36), one findsthat the plasma flow as well as its shear can be modified by both the forces and the moments.We further neglect the current gradient and flow shear terms, and adopt the constant- ψ assumption. Since δj z is an even function of x , and δφ and δF are odd functions of x [22, 30], one can simplify Eqs. (32)-(35) as F m = 2 mµ a | ψ s | Im ∆ ′ = 2 mµ a | ψ s | ∆ ′ c Im (cid:26) ψ c ψ s (cid:27) , (37) F r = 2 mρa Im Z r s + r s − δφ ∗ δF dr, (38) N m = 0 , (39) N r = 0 . (40)Note that the quasi-linear moments in this work are exactly zero. When effects such as flowshear in the inner region are considered, the plasma flow evolution could be significantlyinfluenced by the Maxwell and Reynolds moments [22]. B. Quasi-linear forces in the RI and VR regimes
In the steady state RI regime with constant- ψ assumption, ψ s satisfies the followingequation α im Ω s δ RI τ R r s ψ s = ∆ ′ ψ s + ∆ ′ c ψ c . (41)11ombing Eq. (41) and the relationship between δφ and ψ s [19], the steady state Maxwelland Reynolds forces can be written as F m = 2 mµ a | ψ s | ∆ ′ c Im (cid:26) ψ c ψ s (cid:27) = 2 mµ a | ψ s | α r s | m Ω s τ RI | sgn( m Ω s ) sin 58 π, (42) F r = 2 mρa Im Z r s + r s − δφ ∗ δF dr = − mµ a | ψ s | α r s | m Ω s τ RI | sgn( m Ω s ) sin 18 π, (43)where α = R ∞−∞ [1 + χξ RI ] dχ ≈ .
12 and α = R ∞−∞ ξ RI ∂ χ ξ RI dχ ≈ .
54. Obviously, theReynolds force F r is opposite sign to the Maxwell force F m and F r < F m . Note that theratio of F r /F m is a constant independent of equilibrium, which is similar with the case inslab geometry [19].On the other hand, within constant- ψ assumption, ψ s in the steady state VR regimesatisfies α im Ω s δ V R τ R r s ψ s = ∆ ′ ψ s + ∆ ′ c ψ c . (44)Similar to the case in the steady state RI regime, the Maxwell and Reynolds forces can bewritten as F m = 2 mµ a | ψ s | ∆ ′ c Im (cid:26) ψ c ψ s (cid:27) = 2 mµ a | ψ s | α r s | m Ω s τ V R | sgn( m Ω s ) sin 12 π, (45) F r = 2 mρa Im Z r s + r s − δφ ∗ δF dr = 0 . (46)Note that the Reynolds force in the steady state VR regime is exactly zero. In fact, theabove expressions for the Reynolds force in the steady state RI and VR regimes differ fromprevious results [22]. This is because our Reynolds force is defined from δφ and δF inthe inner region, whereas the F r in the previous work is calculated using the approximatedexpressions of perturbed stream function and vorticity in the outer region. On the otherhand, the above expressions of the Maxwell force in both the steady state RI and VR regimesrecover the previous results except the geometry factors [22]. C. Analytical plasma flow model in presence of RMP
When the island width W is still much smaller than the resistive tearing layer width δ layer , quasi-linear magnetic terms may be neglected. Based on the Laplace transform andBessel expansion, we propose a rigorous derivation for the plasma flow model in response12o RMP in cylindrical geometry. In this model, the island evolution is determined by thelinear plasma response solutions in the constant- ψ RI and VR regimes, i.e. Eqs. (17) and(22), which are used to obtain the equation for plasma flow in the Bessel spectral space.We further demonstrate that quasi-linear moments are exactly zero and Reynolds force canalways be neglected. We summarize the model for plasma flow belowΩ s = ∞ X k =0 a k J ( µ k ˆ r s )ˆ r s + Ω eq ( r s ) , (47) ρ∂ t a k = C k F m − ν ⊥ a µ k a k , (48) F m = 2 mµ a | ψ s | ∆ ′ c Im (cid:26) ψ c ψ s (cid:27) , (49)where C k = J ( µ k ˆ r s ) a N k , and N k = J ( µ k ).Different from previous work, the above equations (47)-(49) are naturally derived fromthe two-field reduced MHD model. Note that the plasma response solutions in the RI andVR regimes, i.e. Eqs. (17) and (22), are appropriate only in the small island regime. When W ≫ δ layer , quasi-linear magnetic terms should be included. V. SUMMARY AND DISCUSSION
In summary, we have developed an analytical model for the quasi-linear plasma flowresponse to RMP in the RI and VR regimes within the framework of two-field reduced MHDequations. Neglecting the quasi-linear magnetic effects, previous linear solutions on plasmaresponse in magnetic field [19, 20] are extended to cylindrical geometry in presence of time-dependent rigid poloidal flow for both RI and VR regimes. The extension to the linear plasmaresponse solutions in presence of steady state flow [20] is to allow time-dependent flow, whichis more consistent with the quasi-linear plasma flow response model developed in this work,where the plasma flow evolves in response to RMP and is indeed time-dependent. Thecorresponding plasma flow response equation including the quasi-linear forces and momentsis derived in the Bessel spectral space, without invoking any assumption on the Maxwelltorque or its radial profile. Different from previous works, our analytical model is builtpurely from the two-field reduced MHD equations, which allow us to accurately calculateand clarify the physical origins of the quasi-linear forces and moments self-consistently.Due to the limitation of the two-field reduced MHD model, many physics elements for the13MP induced plasma response have not been included. For example, two-fluid, neo-classical,finite-orbit-width, and finite-Larmor-radius effects are known to have strong influence overtearing modes as well as plasma response to RMPs near resonant surfaces[16, 33–35]. Fur-thermore, our derivation is appropriate only in the small island regime ( W ≪ δ layer ). Weplan to address these important issues in future studies. ACKNOWLEDGMENTS
This work was supported by the Fundamental Research Funds for the Central Universitiesat Huazhong University of Science and Technology Grant No. 2019kfyXJJS193, the NationalNatural Science Foundation of China Grant No. 11775221, 11763006, and 51821005, theYoung Elite Scientists Sponsorship Program by CAST Grant No. 2017QNRC001, and U.S.Department of Energy Grant Nos. DE-FG02-86ER53218 and DE-SC0018001.
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed inthis study. 14 ppendix A: Derivation of equation for ∆Ω θ We write any quantity as f = f eq + δf , where f eq = f eq ( r ) and δf = δf ( r, θ, φ, t )are the equilibrium and perturbation parts of f . We expand the perturbed quantity as δf = P ∞ l = −∞ δf l e il ( mθ − nφ ) . Using the single helicity assumption, the quasi-linear equationfor the 0 / δF , i.e. δF , can be obtained from Eq. (2) ρ ∂∂t δF + ρ ( δ~v ∗ · ∇ δF + δ~v · ∇ δF ∗ ) = δ ~B · ∇ δj ∗ z + δ ~B ∗ · ∇ δj z + ν ⊥ ∇ ⊥ δF , (A1)where δv θ = ∂∂r δφ = r ∆Ω θ , (A2) δF = ∇ ⊥ δφ = 1 r ∂∂r ( r ∂∂r δφ ) = 1 r ∂∂r ( r ∆Ω θ ) , (A3) ∇ ⊥ δF = 1 r ∂∂r (cid:26) r ∂∂r δF (cid:27) = 1 r ∂∂r (cid:26) r ∂∂r [ 1 r ∂∂r ( r ∆Ω θ )] (cid:27) = 1 r ∂∂r (cid:26) r ∂∂r ( r ∂∂r ∆Ω θ ) (cid:27) , (A4) δ ~B · ∇ δj ∗ z + δ ~B ∗ · ∇ δj z = imr ∂∂r { δψ ∗ δj z − δψ δj ∗ z } = − mr ∂∂r Im { δψ ∗ δj z − δψ δj ∗ z } , (A5) δ ~v · ∇ δF ∗ + δ ~v ∗ · ∇ δF = imr ∂∂r { δφ ∗ δF − δφ δF ∗ } = − mr ∂∂r Im { δφ ∗ δF − δφ δF ∗ } . (A6)Here, higher harmonics are neglected in the quasi-linear approximation. SubstitutingEqs. (A3)-(A6) into Eq. (A1) and performing integral R r rdr on both sides of Eq. (A1),we obtain ρ∂ t ∆Ω θ = Mr + Rr + ν ⊥ r ∂∂r ( r ∂∂r ∆Ω θ ) , (A7)where M = − mr Im { δψ ∗ δj z − δψ δj ∗ z } ,R = ρ mr Im { δφ ∗ δF − δφ δF ∗ } .
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