Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

Appl. Math. Mech. -Engl. Ed. 6, 79 7 00 DOI 0.007/s048-00-06-z c Shanghai University and Springer-Verlag Berlin Heidelberg 00 Applied Mathematics and Mechanics English Edition Boundary-layer eigen solutions for multi-field coupled equations in the contact interface Lei HOU,, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, Lin QIU,. Department of Mathematics, Shanghai University, Shanghai 00444, P. R. China;. Division of Computational Science, E-Institute of Shanghai Universities at SJTU, Shanghai 0040, P. R. China;. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 0040, P. R. China Communicated by Xing-ming GUO Abstract The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how constraints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact interface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This paper describes the field affected by the gravity constraints. Applying the multi-scale analysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analysis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic perturbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface. Key words coupling dynamic equations, boundary problem, eigenvalue, asymptotic perturbation analysis, turning point Chinese Library Classification 65N0 000 Mathematics Subject Classification O4. Received Jan., 00 / Revised May, 00 Project supported by the National Natural Science Foundation of China No. 0875 and the Pujiang Talent Program of China No. 06PJ446 Corresponding author Lei HOU, Professor, E-mail: houlei@shu.edu.cn

70 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU Introduction Back to the early 8th century, Faraday and other scholars realized that the solid or fluid moving in a magnetic field will experience an EMF electromotive force effect. If the moving solid or fluid is conductive, then the body is guided to form an electric circuit, or the body together with the outside world forms an electric circuit. In this loop, there will exist current flow. Thus, there exists the interaction between the current and magnetic fields, that is, due to the moving conductor, the magnetic field induces current, called induced current; on the other hand, the induced current also produces a magnetic field, which is induced by the external magnetic field and affect the original field. When the movement body is a fluid conductor, the problem is more complicated. Therefore, coupling dynamics is to study interactions between velocity and constraint fields in the contact interface. The regular and singular perturbation in micro-phenomena of laboratories and macrophenomena of nature requires us to use the multi-scale analysis. In the paper, a complex threedimensional coupling dynamic system is transformed into a boundary layer problem in the onedimensional complex space by the multi-scale analysis. Finally, we carry out the asymptotic perturbation analysis to give the solution to the problem of the boundary-layer perturbation characteristic function in a multi-field coupling system. Problem formulation We begin with the simple slab model of an infinite plane current layer specified by the equilibrium shear vector fields and B 0 = e x B 0x y+e z B 0z y V 0 = e x V 0x y+e z V 0z y subject to a gravitational acceleration e y g acting in the positive y-direction. The equilibrium flow V 0 is parallel to B 0. The equilibrium density ρ is assumed to vary only in the y-direction. We consider the resistive constraint-field equations for an incompressible equilibrium velocity with a uniform resistivity and including only the perpendicular component of the collision part of the viscous tensor. The dynamic system of the equilibrium vector field, which is constrained by the gravitational field and the coupling field, consists of the following four equations [ 4] : V ρ t + V V = c j B+ρg + μ V, B t ρ + ρv =0, t = V B+ η 4π B, B = V =0. 4 Here, is the momentum conservation equation, is the diffusion equation, is the continuity equation, and 4 reflects the incompressibility constraint field in fact, the model allows a slight deformation. In the above formulations, ρ and V are the fluid density and velocity, j is the current density, c is the speed of light mixed Guassian units, and η and μ are the resistivity and perpendicular coefficients of viscosity, which are assumed uniform. The flow speed, the equilibrium constraint vector field, and the fluid density consist of the equilibrium quantities and perturbed quantities, i.e., V = V 0 + V, B = B 0 + B, ρ = ρ 0 + ρ,

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 7 where V, B, and ρ are perturbed quantities. From the Maxwell equations [],wehave j = c 4π B. Thus, c c j B = c 4π B B = 4π B B B = B B. 4π Then, the momentum equation can be written as V ρ t + V V = 4π B B + ρg + μ V. 5 Followed by an application of the operator e y to the curled momentum equation 5, the linearization [] yields two coupled equations describing the y-components of the first-order velocity V y and the equilibrium constraint field B y. Take all perturbed quantities to vary like the single Fourier harmonic function exp ik x x + k z z+ωt, where k = k x, 0,k z } is the horizontal wave vector, and ω is the perturbed quantity growth rate. Then, from 5, we find e y ρ 0 t V + V 0 V + V V 0 = e y 4π B 0 B +B B 0 + ρg + μ V. 6 We analyze each term in 6 as follows: e y ρ 0 t V = ω[k ρ 0 V y ρ 0 V y ], e y ρ 0 V 0 V =i[k ρ 0 k V 0 V y ρ 0 k V 0 V y ρ 0k V 0 V y ], e y ρ 0 V V 0 = i[k V 0 ρ 0 V y +k V 0 ρ 0 V y ], where the prime denotes differentiation with respect to y. The terms on the right-hand side of 6 are simplified as e y B 0 B =i[k k B 0 B y k B 0 B y k B 0B y ], e y B B 0 = i[k B 0 B y +k B 0 B y ], e y ρ g = k ρ g, e y μ V = μ V y = μ y k V y. In the gravitational term, the density perturbation ρ can be estimated using the linear form of, namely, ρ [ω +ik V 0 ] + ρ 0 V y =0. Thus, k ρ g = k ρ 0 g V y ω +ik V 0.

7 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU Following the methods of magnetohydrodynamic MHD experts, Furth, Killeen, and Rosenbluth see [4], we introduce the standard multi-scale variables: Here, ψ = B y B, F = k B 0 kb, S = τ R, ρ = ρ 0 τ H ρ, G = g ρ 0 τ ρ H, N =4π μ 0 η, W = ikτ R V y, α = ka, P = ωτ R, k = kx + k z, y = aμ, R = τ R k V a y. τ R = 4πa η and τ H = a 4πρ B are the resistive and constraint coupling dynamic time scales of the layer in the contact interface, k = k x, 0, k z is the wave vector, S is the Lundquist Reynolds number, ρ and B are measures of the density and the constraint field strength, and a is the characteristic dimension of the current layer. F denotes the scaling of the constraint shear, R is the rescaled fluid shear on the resistive diffusion time scale, and N is the rescaled viscosity. Thetimescalesτ R and τ H can vary considerably according to the problem under consideration. For example, in the interior of stars, τ R 0 9 a a is the unit of year; in Sun spot regions, τ R 50 a, while in laboratory thermonuclear fusion plasmas, τ R 0 7 s; for typical neutral hydrogen HI clouds of 0 4 solar masses and magnetic field of 0 0 T, τ H 0 7 a, whereas in laboratory thermonuclear fusion plasmas, τ H 0 s see [4 6]. The ratio of the resistive diffusion to hydrodynamic time scales is very large. The values of S for laboratory thermonuclear fusion plasmas are typically in the range of 0 0 7. In astrophysical applications, where the characteristic dimension a is extremely large, S is found similarly to be a large number. The significance of a high Lundquist Reynolds number is that resistive diffusion effects are small, and the flux-freezing may be considered to be a good approximation. In terms of these variables, for the left-hand side of 6, we have e y ρ 0 t V + V 0 V + V V 0 = ω[k ρ 0 V y ρ 0 V y ]+i[k ρ 0 k V 0 V y ρ 0 k V 0 V y +k V 0 ρ 0 V y ] =[ω +ik V 0 ][k ρ 0 V y ρ 0 V y ]+iρ 0 k V 0 V y = α ikτr [ωτ R +iτ R k V 0 ] a ρ 0V y a a ρ 0V y ikτ R + ikτ RV y kτr ρ 0 k V 0 τ R = ika τr P +ir [ α ρ 0 W +ρ 0 W W ]+i ikτr ρ 0R a = ika τr [P +ir ρ 0 W α ρ 0 W +iρ 0 R W ], where the prime denotes differentiation with respect to μ. For the right-hand side of 6, we have [ ] e y 4π B 0 B +B B 0 + ρg + μ V = i 4π [k k B 0 B y k B 0 B y +k B 0 B y ]+k ρ g μ y k V y = ik B 0 k B y B y 4π + k B 0 B y + k ρ g μ k B 0 a 4 μ α W ikτ R i [ = ka τr ψ ψ α + F α S F + α S GW F P +ir + N μ α ] W.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 7 Since the left-hand and right-hand sides of 6 are equal, we have ika τr [P +ir ρ 0 W α ρ 0 W +iρ 0 R W ] [ = ika τr ψ ψ α + F α S F + α S GW F P +ir + N μ α ] W. Hence, the momentum equation finally becomes P +ir ρ 0 W α ρ 0 W +iρ 0 R W = ψ ψ α + F α S F + α S GW F P +ir + N d dμ α W. 7 Likewise, linearizing the diffusion equation and taking the y-component yield ωb y =ik B 0 V y ik V 0 B y + η 4π B y. 8 Therefore, the dimensionless form of the diffusion equation is P +ir ψ + FW = ψ α ψ. 9 The system of equations describing the resistive tearing and gravitational modes in the presence of an equilibrium shear flow is given by 7 and 9. Examples for multi-field coupling boundary layer solution to the perturbation problem According to the preceding analysis, when S is large, the impact on the impedance of the proliferation is small, and the steady-state constraint field outside the boundary layer is considered to be a good approximation. We rewrite 7 by dividing both sides with S when G is sufficiently small. From S 0, we can obtain the external solution of the differential equation ψ ψ α + F =0. F We can obtain the asymptotic solution by use of the conventional analysis methods. Obviously, g = α is a solution of the equation. Its substitution yields } e αμ F + α, μ > 0 ψ = e αμ =e α μ F + α, F + α, μ < 0 αe αμ ψ F + α+e αμ F, μ > 0, = αe αμ F + α+e αμ +F, μ < 0. Then, we can get the jump condition matching the external solution and the internal solution: ψ 0+ = α +, ψ 0 =α. Therefore, Δ ext = ψ 0+ ψ 0 ψ0 = α + α = α α.

74 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU To calculate the internal boundary layer solution, [ ] introduced a linear change in MHD equations. By the Fourier harmonic analysis, the scale changes, and the introduction of new parameters, a complex three-dimensional coupling dynamic system is transformed into a boundary layer problem in the one-dimensional complex space of the forth-order differential equations P +irθ d H dθ θ H + G H ir F P +irθ N d4 H dθ 4 = Pθ F ΔF, 0 Δ =Δ P + θhdθ, where P is a complex eigenvalue, H is a complex eigenfunction, R is a shear flow characteristic parameter, G is a gravitational parameter, and N is the viscosity parameter. Equation is the matching condition of the moving boundary on the contact interface. The asymptotic perturbation method of the eigenfunction for the boundary layer internal solution is as follows: On the boundary layer equation 0, for the Fourier transform of Hθ, we define hk = Hθe ikθ dθ. Then, the equation in the original physical field has been transformed into that in the Fourier space. The asymptotic differential operator eigenfunction is used to obtain the solution. Let ε = R. For the viscous tearing mode, we can obtain the differential operator[9 0] Lh = h + ε k h k P + k 4 Nh and the third-order differential operator viscous tearing G-mode operator d Mh = ε dk P Lh G F h =πip ε δ k π ip RF PF + ΔF δ G k+π ΔF i εf δk. Now, we discuss the eigenfunction asymptotic perturbation approach: εh +k h εp k h =0, ε 0, k 0 <k 0: h Pk h =0 h k e ± P k, k <k 0 < 0: εh +k h =0 h e ε k, k. For k 0 <k 0, we can ignore the flow field terms. However, for large k, the flow terms play a leading role. Therefore, when k, we can ignore the P inertial term. The turning point of the solution is the transition from the index oscillation solution to the algebraic solution when the ε flow term approximately balances the P inertial term, that is, ε k h Pk h. In the perturbation region, the solution behaves like h kp h so that ε k 0 P h Pk 0 h. Thus, we obtain that the turning point from the index oscillation solution to the algebraic solution is k 0 ε P.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 75 Hence, when 0 χ π for Mh =0, <k<0, and 0 <k<+, we can classify the eigenfunctions of the contact interface dynamic problems into four regimes according to the values of the real eigenvalue ReP andn: i When N =0andReP > 0, e ε k, k +, hk k e εp k, k. ii When N =0, ReP < 0, and ε 0, e ε k, k +, hk k, k. iii When N =0andReP < 0, e ε k, k +, hk k e ± Pk, k 0 <k 0, k, <k< k 0. iv When N 0, hk exp 6 exp R 6 R + ε +4N k }, ε +4N k }, k +, k. In case iv, since we introduce viscous effects, the structure of hk undergoes another qualitative change. The solution hk decays more rapidly when N 0andk +; when k, hk is independent of positive k or negativek, and contains an exponentially decaying behavior rather than the oscillatory algebraic decay in the non-viscous case. To account for the behavior of the growth rate P when the flow ε becomes small, we carry out an asymptotic study, which enables us to estimate P in the small ε limit analysis. The viscous G-mode equation takes the form R d dk P Lh G F h =0, <k<0, 0 <k<, where R = ε, and Lh = d h dk + R d dk k h k P + k 4 Nh is a viscous tearing mode operator. The associated boundary and jump conditions at the origin k =0are h± =0, h0+ =, h 0± = ie ±iχ, 4 h 0+ h G 0 = πi F 5 together with the eigenvalue relation πp = h0 h 0 e iχ h0+ h 0+ eiχ, π <χ<π. 6

76 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU The values of h and h at both sides of the origin are required to obtain the asymptotic estimate of P. We first perform an asymptotic analysis to determine the eigen solution h, and then predict the eigenvalue P for large values of the flow parameter R to support the numerical analysis. With the asymptotic approximation given by Lh = 0, we can determine the growth rate for the tearing mode in. Bondeson and Persson [7] found that for the tearing mode, when R becomes large, the growth rate P has the form P πε eiχ + cos χγ, ε 0 7 πε in the non-viscous limit N = 0. It follows from this formula that the growth rate ReP increases as ε decreases. Some authors pointed out that large flow ε 0 could exist with ReP > 0for χ <πso that the tearing mode corresponding to Δ < 0 i.e., π < χ <π can be driven unstablely. 4 Asymptotic estimate of P for the viscous tearing mode in the large flow limit An asymptotic approach to the problem of estimating the eigenvalue P for the viscous tearing equation can be expressed in terms of a singular perturbation expansion h = h 0 + h + h + h +, ε 0, 8 where h n,n=,,,, are nth-order perturbation solutions with h n 0forε 0. The asymptotic recurrence relations of the nth-order solutions h n and n th-order solutions h n are εh 0 +k h 0 =0, 9 εh n +k h n = εk P + k 4 Nh n, n =,,,. For sufficiently large values of the flow parameter R, the solution h n and its derivatives h m n m =0,, are assumed to decay as we know from the qualitative discussion of the eigenfunction hk. The leading-order solution h 0 k andthenth-order solutions h n k satisfy the boundary conditions h0 ± =h 0± =0, h n ± =h n ± =0. 0 Thus,wemaysolve9fork>0andk<0, respectively. 4. Asymptotic estimate of the leading-order solution h 0 k A straightforward general form of the leading-order solution from the homogeneous equation is obtained as h 0 k =c 0 e ξ + c e ξ e ξ dk, ξ = ε k, where c 0 and c are arbitrary constants. The first-order equation is from 9 with n =. The integrations of the first-order equation over [k, + fork>0andover,k]fork<0 yield h + ε k h = P k h 0 dk + N k 4 h 0 dk, k > 0, h + ε k h = P 0 k h 0 dk + N k 4 h 0 dk, k < 0.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 77 By use of the L Hospital rule, we can obtain h 0 k =c 0 e ξ, k > 0. h 0 k =c e ξ e ξ dk = ok, k < 0, k. It follows from and that the values of the solution h 0 k at the origin k = ±0 are h 0 0+ = c 0, h 0 0+ = 0, 4 h 0 0 = c ε Γ, h 0 0 =c. 5 4. Asymptotic estimate of the high-order solution h n k By use of the leading-order solution h 0 k, we now estimate the derivatives of the first-order solution h n k at the both sides of the origin. From 9, we obtain where h 0+ = P k h 0 dk + N h 0 =Pc k 4 h 0 dk = Pc 0 k e ξ dk + Nc 0 k 4 e ξ dk = εp c 0 Nc0 ε 5 Γ = Pc ε 4 k e ξ, 6 e ξ dk dk + Nc 0 e ξ Γ dξ,ξ + Nc ε k 4 e ξ e ξ dk dk ξ e ξ Γ,ξ dξ = OPε 4,Nε, 7 Γa, x = x u a e u du, a > 0, x > 0 is the incomplete Gamma function. To estimate the first-order solution h k at the both sides of the origin, we multiply by e ξ and integrate it over [0,k]fork>0andover,k]fork<0to find the convergent solution. Therefore, we have h e ξ = P e ξ k h 0 dkdk + N e ξ k 4 h 0 dkdk, k > 0, 0 0 8 h e ξ = P e ξ k h 0 dkdk + N e ξ k 4 h 0 dkdk, k < 0, where h satisfies the assumed condition h ± =0. Then, at the positive side of the origin, in terms of the normalized solution h0+ = c 0, h is given by h 0+ = 0. 9

78 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU At the negative side of the origin, h is given by h 0 =Pc + Nc = 4 Pc ε 5 e ξ + Nc ε 7 e ξ k e ξ ξ e ξ k 4 e ξ ξ e ξ e ξ dk dkdk e ξ dk dkdk e ξ Γ,ξ dξdξ ξ e ξ Γ,ξ dξdξ = OPε 5,Nε 7, ε 0. 0 In the same way, we can get the asymptotic values of the second-order and third-order solutions: h 0+ P k Pc 0 ε ke ξ dk + N k 4 P c 0 ε ke ξ dk = P c 0 ε 7 Γ +NPc 0 ε, h 0 h 00 = c, ε 0, k < 0. Thus, h 0 may be neglected when ε becomes small. The second-order solution h is estimated as which results in h =e ξ P 0 e ξ k h dkdk + N 0 0 e ξ k 4 h dkdk =e ξ OP ε,npε, ε 0, K > 0, h 0+ = P k h dk + N 0 k 4 h dk = OP ε 4,N Pε 8. 4 5 Estimate for the growth rate P when the flow ε becomes small Finally, taking the normalized leading-order solution with c 0 =,wehavethesolutionsat k = ±0: h0+ = h 0 0+ =, and h0 =h 0 0 +h 0 = ε c0 Γ h 0+ = h 0+ + h 0+h 0+ = εp c 0 ε 5 Nc0 Γ + Oε 4 P,ε 8 N P, + OPε 5,Nε 7, + 7 ε P c 0 Γ +ε NPc 0 h 0 =h 00 =c. 5

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 79 Substituting h0+ h 0+ terms, we obtain h0 and h 0 πp = h0 h 0 e iχ h0+ h 0+ eiχ into the eigenvalue relation 6 and balancing the dominant R e iχ P + ε NΓ ε 4 P Γ +ε NP + ε Γ By use of the binomial theorem, we have πp Pε eiχ ε NΓ P + 4 8 ε P Γ Rewriting 7, we obtain + ε 4 P Γ + ε Γ + 4ε 4 N Γ 4ε NPΓ Γ P P e iχ. 6 e iχ + Oε 5 P. 7 πp ε eiχ + ε P Γ cosχ Re iχ 4ε NΓ +. 8 P 6 Estimate for the growth rate P of the G-mode In the section, we perform an asymptotic estimate for the G-mode growth rate P when ε becomes small. From, we have h + G h ε k Pεh P k h = εh F P k. 9 Make the perturbation expansion h = h 0 + h + h + h +, ε 0, 40 where the leading-order solution and the nth-order solutions satisfy the conditions h0 ± =h 0± =h 0± =0, h n ± =h n ± =h n ± =0. 4 We solve 9 perturbed for k>0andk<0 by the same method as in Section 4, respectively. The transformation of 9 into the asymptotic recurrence relation εh 0 + k h 0 =0, εh n + k h n = Pε h n εp k h n + G F P k ε h n, n =,,, is obtained by putting lower-order differential terms into the inhomogeneous term. The method has been tested in Section 4, which yields a good estimate for the viscous tearing mode growth rate. When estimating P, we only use the values of h and h at both sides of the origin k = ±0. However, the nth-order expansion solutions h n, h n,andh n must satisfy the boundary condition so that these solutions decay when k. 4

70 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU Same as the previous asymptotic analysis, the G-mode solution is obtained by the perturbation method. Collecting the results for h0+, h0, h 0+, h 0, h 0+, and h 0, we finally obtain the expansion of h and its derivatives at k =0± for ε 0intheform h0+ =, h0 = ε c0 Γ ε c Γ + oε 5 P, h 0+ = εp + ε 7 P Γ +ε 8 P Γ + oε 5 G, h 0 =c 0 + c 0 ε 4 P Γ +oε 5 P, h 0+ = ε 4 G F Γ ε 8 P Γ + oε G, h 0 =c. The growth rate P can be estimated by balancing the leading-order terms in 4. Thus, we substitute P into the eigenvalue relation 4 πp = h0 h 0 e iχ h0+ h 0+ eiχ, π <χ<π. The ratios h0 h 0 and h0+ h 0+ are obtained as h0 h 0 = ε c 0 Γ ε c Γ c 0 + c 0 ε 4 P Γ ε Γ c Γ ε 4 ε P Γ c 0 44 and h0+ h 0+ = εp ε 4 P Γ ε, 45 5 Γ where the higher-order terms in the binomial expansion are negligible when ε is sufficiently small. We substitute 44 and 45 the higher-order terms in 45 are neglected into the eigenvalue relation to yield πp εp eiχ + ε Γ cos χ ε Γ + ε 5 P Γ e iχ c ε Γ e iχ. 46 P 4 c 0 To estimate ce iχ c 0 in 46, π G F = h 0+ h 0+ eiχ h 0 h 0 e iχ, 47

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface 7 where, from 4, h 0+ h 0+ = 4 ε G εp F Γ ε 8 P Γ +ε 4 P Γ +ε 5 Γ + oε 8 P ε GΓ ε 4 P Γ F P F GΓ, 48 h 0 h 0 = c ε 4 P Γ + oε 8 P c 0 c ε 4 P Γ. 49 c 0 Thus, from 47 49, we have c [ e iχ π G c 0 F + ε GΓ ε 4 P Γ P F GF Γ e iχ] + ε 4 P Γ. 50 Combining 46 and 50 yields πp εp eiχ + ε Γ ε Γ cos χ + P ε 5 P Γ 4 e iχ + ε Γ π G F +π ε 4 GΓ F + ε GΓ P F eiχ ε 5 P Γ = R P eiχ + ε Γ cos χ +π ε GΓ F e iχ +π ε GP Γ Γ F + Oε 7 6. 5 Neglecting the higher-order ε terms in 5, we obtain P πε eiχ + ε P Γ cos χ ε ε eiχ ε πgpγ + F ε eiχ = πε eiχ + πε eiχ ε eiχ ε Γ cosχ + πε eiχ ε eiχ ε πgγ F. Thus, the final form of the asymptotic estimate of the eigenvalue of the N-G-mode for large flow is given by P πε eiχ + ε Γ cosχ + ε GΓ π F ε NΓ, ε 0. 5 Thus, we obtain the relationships of the eigenvalue spectrum P with the parameters R, G, N, and χ. The main parameter χ denotes the convective coupling angle in the contact interface. From the eigenvalue spectrum, we can obtain an optimization characteristic function-base for

7 Lei HOU, Han-ling LI, Jia-jian ZHANG, De-zhi LIN, and Lin QIU the multi-field coupling problem. With the asymptotic forms, we can analyze the contact interface problems in the mechanical impact model with multi-scale parameters. The asymptotic estimates of the instability growth rate P nonetheless showed the multi-field physical coupling effects of parameters such as perturbation ε = R, contact angle χ, and slipn onthe contact interface dynamics. References [] Hou, L. Resistive Instabilities in the Magnetohydrodynamics, Ph. D. dissertation, University of Abertay Dundee, Dundee, UK 994 [] Hou, L., Paris, R. B., and Wood, A. D. Resistive interchange mode in the presence of equilibrium flow. Physics of Plasmas, American Institute of Physics, 47 48 996 [] Hou, L., Han, Y. H., and Li, J. L. The exploration of boundary layer solution in magneticcurrent coupling problem in Chinese. Journal of East China Normal University Natural Science 45, 0 6 008 [4] Furth, H. P., Killen, J., and Rosenbluth, M. N. Finite-resistivity instabilities of a sheet pinch. Physics of Fluids 64, 459 484 96 [5] Paris, R. B. and Sy, W. N.-C. Influence of equilibrium shear flow along the magnetic flows on the resistive tearing instability. Physics of Fluids 60, 966 975 98 [6] Persson, M. and Bondeson, A. Oscillating magnetic islands in a rotating plasma. Physics of Fluids B 0, 5 990 [7] Bondeson, A. and Persson, M. Resistive tearing modes in the presence of equilibrium flows. Physics of Fluids 99, 997 007 986 [8] Paris, R. B. and Wood, A. D. Exponentially-improved asymptotics for the gamma function. J. Comput. Appl. Math. 4-, 5 4 99 [9] Su, Y. C. and Wu, Q. G. Singular Perturbation Problem Cited Theory of Numerical Methods in Chinese, Chongqing Press, Chongqing 99