Chapter 5-Old. Kelvin-Helmholtz Instability

HTML
DOWNLOAD

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Chapter 5-Old. Kelvin-Helmholtz Instability"

Ἔραστος Μαρής
6 χρόνια πριν
Προβολές:

1 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Chapter 5-Ol. Kelvin-Helmholtz Instability Exercise 5.1 Rea section in the following textboo an erive symmetric Kelvin-Helmholtz instability occurre at a tangential iscontinuity (TD) ue to velocity shear on two sies of the TD. Pars, G. K., Physics of Space Plasmas: An Introuction, Aison-Wesley Publ. Co., Velocity shear at bounary of two meiums may be unstable to Kelvin-Helmholtz instability, which can result in large amplitue surface wave at the bounary. Kelvin-Helmholtz instability ue to win shear on water surface can lea to large amplitue surface wave. But the tension force of water surface can stabilize Kelvin-Helmholtz instability. Thus, large amplitue water wave can only be foun when the win spee is large enough to overcome the tension force. Kelvin-Helmholtz instability can also be foun in the atmosphere. An islan in ocean can isturb airflow above it. This isturbance can trigger Kelvin-Helmholtz instability an result in wavy clou pattern ownstream from the islan. Twisting of auroral arcs is another example of Kelvin-Helmholtz instability. In this lecture, we shall iscuss MHD Kelvin-Helmholtz instability occurre at a tangential iscontinuity, such as awn an us flans of magnetopause. The Kelvin-Helmholtz instability in this region can result in a mixing layer at low latitue bounary layer (LLBL). Basic equations of an ieal MHD plasma: Continuity equation ρ t + (ρv) = ( + V )ρ + ρ V = 0 (5.1) t Momentum equation ρ( + V )V = p + J B (5.) t Ol-5-1

2 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Energy equation ( t + V )(pρ γ ) = 0 (5.3) or ( t + V )p = γ p ρ ( t + V )ρ = γ p [ ρ V] = γ p V ρ (5.3') where Eq. (5.1) has been use to obtain Eq. (5.3'). Charge continuity equation J 0 MHD Ohm s Law E + V B 0 (5.4) Maxwell s equations E 0 B = 0 (5.5) B t = E (5.6) B J (5.7) Substituting Eq. (5.4) into Eq. (5.6) yiels B t or = E = (V B) = V B B V + B V + V B ( + V )B = B V + B V (5.6') t where Eq. (5.5) has been use to obtain Eq. (5.6'). Substituting (5.7) into (5.) yiels ρ( t + V )V = p + 1 ( B) B = p B B B + (5.') Define total pressure p tot = p + B (5.8) Eq. (5.') can be rewritten as Ol-5-

3 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall ρ( t + V )V = p tot + B B (5.") The time erivative of Eq. (5.8) in flui element moving frame is ( t + V )p tot = ( t + V )p + B [( + V )B] (5.9) t Substituting Eqs. (5.3') an (5.6') into Eq. (5.9) yiels ( t + V )p tot = (γ p + B ) V + B (B V) (5.10) We choose normal irection of the tangential iscontinuity (TD) to be the x irection. We consier a TD locate at x = 0. The bacgroun magnetic fiel an flow velocity are all in the tangent irection. To linearize above equations, we consier a small perturbation δ A superimpose on a bacgroun equilibrium state A 0 an assume the small perturbation δ A is in the following form δ A(x, y,t) = δ A exp[i( y t)] so that we have A(x, y,t) = A 0 + δ A exp[i( y t)] (5.11) The equilibrium state of the TD satisfies p 0tot = constant = p 0 + B 0 = p 0 + B + B 0z (5.1) an B 0x = 0 (5.13) V 0 = ˆ y V (5.14) Differentiating Eq. (5.1) once with respect to x yiels 0 = p 0 + B B + B 0z B 0z Applying Eq. (5.11) to Eq. (5.") yiels ( i)[ V ]δv x = δ p tot ( i)[ V ]δv y = i δ p tot + B (i ) δb y Ol-5-3 (5.15) + B (i ) δb x (5.16) + δb B x δv x V ( i)[ V ]δv z = 0 + B (i ) t δb z + δb x B 0z (5.17) (5.18)

4 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Applying Eq. (5.11) to Eq. (5.3') yiels ( i)[ V ]δ p = γ p 0 [ δv x Applying Eq. (5.11) to Eq. (5.6') yiels + i δv y ] δv x p 0 (5.19) ( i)[ V ]δb x = 0 + B (i )δv x (5.0) ( i)[ V ]δb y = B [ δv x ( i)[ V ]δb z = B 0z [ δv x Applying Eq. (5.11) to Eq. (5.10) yiels + i δv y ] + B (i )δv y +δb x V δv x B + i δv y ] + B (i )δv z δv x B 0z ( i)[ V ]δ p tot = [γ p 0 + B 0 ][ δv x + i δv y ] + B B (i )δv y + B 0z B (i )δv z + B δb x V (5.1) (5.) (5.3) Governing equations to be solve inclue Eq. (5.16): {δv x,δb x, δ p tot Eq. (5.17): {δv x,δb x,δb y Eq. (5.18): {δv z,δb x,δb z Eq. (5.0): {δv x,δb x Eq. (5.1): {δv x,δb x,δb y, δv x Eq. (5.): {δv x,δv z,δb z, δv x Eq. (5.3): {δv y,δv z,δb x, δv x Substituting Eq. (5.0) into Eqs. (5.16)~(5.18), (5.1) an (5.3) to eliminate δb x, it yiels Ol-5-4

5 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Eq. (5.0) + Eq. (5.16) to eliminate δb x Eq. (5.4): {δv x, δ p tot Eq. (5.0) + Eq. (5.17) to eliminate δb x Eq. (5.5): {δv x,δb y Eq. (5.0) + Eq. (5.18) to eliminate δb x Eq. (5.6): {δv z,δb z Eq. (5.0) + Eq. (5.1) to eliminate δb x Eq. (5.7): {δv x,δb y, δv x Eq. (5.0) + Eq. (5.3) to eliminate δb x Eq. (5.8): {δv x,δv z, δv x We have eliminate δb x. Now we wish to eliminate δb y an δb z. Eq. (5.5) + Eq. (5.7) to eliminate δb y Eq. (5.9): {δv x, δv x Eq. (5.6) + Eq. (5.) to eliminate δb z Eq. (5.30): {δv x,δv z, δv x Inee, we foun that Eq. (5.30) is an expression without δv x. That is we have Eq. (5.30): {δv y,δv z, δv x In summary, after eliminating δb x, δb y, an δb z, we are left with the following four governing equations Eq. (5.4): {δv x, δ p tot Eq. (5.8): {δv y,δv z, δv x Eq. (5.9): {δv x, δv x Eq. (5.30): {δv x,δv z, δv x The following proceure is for constructing governing equation of δ p tot Eq. (5.8) + Eq. (5.30) to eliminate δv z Eq. (5.31): {δv x, δv x Ol-5-5

6 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Eq. (5.9) + Eq. (5.31) to eliminate δv y Eq. (5.3): {δv x, δv x Eq. (5.4) Eq. (5.33): { δv x, δ p tot, δ p tot Eq. (5.3) + Eq. (5.33) to eliminate δv x Eq. (5.34): {δv x, δ p tot, δ p tot Eq. (5.4) + Eq. (5.34) to eliminate δv x Eq. (5.35): {δ p tot, δ p tot The Eq. (5.35) is the governing equation of δ p tot., δ p tot The following proceure is for constructing governing equation of δv x Eq. (5.3) Eq. (5.36): {δv x, δv x, δv x, δ p tot Eq. (5.4) + Eq. (5.36) to eliminate δ p tot The Eq. (5.37) is the governing equation of δv x. Eq. (5.37): {δv x, δv x, δv x Similary, we can obtain the governing equations of other functions δv y, δb x, δb y, an δb z. Here we briefly list Eqs. (5.4)~(5.37) δv x = t where [ i δ p tot ] (5.4) {[ ] B (5.4a) [ ] δv y = [ ]δ p tot [ ] B δb y i{ B 1 B + [ ] ρ 0 V δv x (5.5) Ol-5-6

7 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall [ ] δv z = [ ] B δb z i[ B 1 B 0z ]δv x (5.6) [ ] B δb y = i B 1 δv x i[ B 1 V + B 1 B ]δv x (5.7) [ ]δ p tot + i[ + B + B 0z ] 1 δv x +i B 1 V δv x [ + B 0z ]δv y = B B 0z δv z (5.8) [ ] δv y 1 V = [ ]δ p tot +i B 1 δv x iδv x t (5.9) δv z = i B B 0z 1 δv x B B 0z δv y (5.30) [ ]δ p tot + i{ + B + +i B 1 V δv x { + B 0z [ ] [ ] B B 0z [ ] 1 δv x [ ] B δv y = 0 (5.31) Ol-5-7

8 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall 1 V δ p tot [ t ] + i δv x +iδv x t = 0 (5.3) where = + B [ ] + B 0z B [ ] 1 (5.3a) i δv = 1 x V [ ][ 1 δ p tot ] + [ 1 δ p tot ] (5.33) δ p tot [ t ] + 1 δ p tot + 1 δ p tot 1 V [ ] 1 V +iδv x t = 0 (5.34) 1 δ p tot + 1 δ p tot 1 [ 1 ] + δ p F tot 0 = 0 (5.35) i δ p tot = 1 [ V ] 1 δv x { [ ] + δv x 1 V (5.36) δv x + 1 [ V ] 1 δv x { [ ] + δv x 1 V = 0 (5.37) Ol-5-8

9 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall In Summary: The governing equation of δ p tot is 1 δ p tot + 1 δ p tot 1 The governing equation of δv x is δv x + 1 where [ 1 ] + δ p F tot 0 = 0 (5.35) [ ] 1 δv x { [ ] + δv x 1 V = 0 (5.37) {[ ] B (5.4a) = + B [ ] + B 0z B [ ] 1 (5.3a) Remar 1: in Eq. (5.3a) can be written as [ ] 4 = t 1 R 0 where R 0 = { γp 0 + B + B 0z It can be shown that [ ] 4 R 0 [ ] B = [ ] 4 { γp 0 + B + B 0z [ ] B = {[ ] V F {[ ] V SL Ol-5-9 where V F0 y an V SL0 y are, respectively, the phase velocity of fast-moe an

10 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall slow-moe waves which propagate parallel or anti-parallel to the surface wave irection ( = ˆ y ). Thus > 0 inicates or [ ] > V F [ ] < V SL Remar : in Eq. (5.4a) can be written as = {[ ] V A where V A0 y is the phase velocity of shear Alfven wave which propagates parallel or anti-parallel to the surface wave irection ( = ˆ y ). Thus > 0 inicates [ ] > V A Ol-5-10

11 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Remar 3: For neutral flui ynamics B = B 0z = 0, it yiels = [ ], [ ] = t 1, an δ p tot = δ p. Thus, Eqs. (5.35) an (5.37) are reuce to 1 δ p + 1 δ p [ 1 ] { 1 [ ] [ ] + δ p { t 1 = 0 (5.35n) [ ]δv x + 1 [ V ] 1 δv x {ρ 0 [ ] + δv x 1 V 1 = 0 (5.37n) Eqs. (5.35n) an (5.37n) are similar to Eqs. (7) an (8), respectively, in the classical paper by Blumen (1970), in which ensity istribution are assume to be uniform in space. Exercise 5. Rea the following classical papers, which stuy K-H instability in neutral flui: Blumen, W., Shear layer instability of an invisci compressible flui, J. Flui Mech., 40, 769, 1970 Blumen, W., P. G. Drazin, an D. F. Billings, Shear layer instability of an invisci Ol-5-11

12 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall compressible flui. Part, J. Flui Mech., 71, 305, Drazin, P. G., an A. Davey, Shear layer instability of an invisci compressible flui. Part 3, J. Flui Mech., 8, 55, Exercise 5.3 Rea the following classical paper by Miura an Pritchett (198): Miura, A. an P. L. Pritchett, Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophys. Res., 87, 7431, 198. Ol-5-1

13 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Remar 4: For surface wave propagating perpenicular to the ambient magnetic fiel, i.e., B = 0. It yiels = [ ], an [ ] = t + B 1. 0z Thus, Eqs. (5.35) an (5.37) are reuce to 1 δ p tot + 1 δ p tot [ 1 ] { 1 [ ] [ ] + δ p tot { t + B 1 = 0 0z (5.35 ) [ ]δv x + 1 [ ] 1 δv x {ρ 0 [ ] + B 1 0z + δv x 1 V = 0 (5.37 ) Eqs. (5.35 ) an (5.37 ) are similar to Eqs. (5.35n) an (5.37n). Thus, solutions obtaine by Drazin an Davey (1997) are applicable to Eqs. (5.35 ) an (5.37 ). Ol-5-13

14 Nonlinear Space Plasma Physics (I) [SS-841] Chapter 5-Ol by Ling-Hsiao Lyu 008 Fall Remar 5: For uniform bacgroun, = {[ ] B = [ ] + B + B 0z B [ ] 1 [ 1 ] = 0, V = 0 Eqs. (5.35) an (5.37) are reuce to 1 δ p tot + δ p tot = 0 (5.35u) 1 1 δv {δv x + x = 0 (5.37u) V Thus, for δ p tot 0, it yiels + xi xr = 0 That is fast-moe or slow-moe. For the shear Alfven moe, we have δ p tot = 0 For δv x 0, it yiels = 0 [ ] = B (Shear-Alfven moe) or + xi xr where R 0 = ( = 0 {( ) V F + B + B 0z )( ) B {( ) V SL = xr xi R 0 Ol-5-14

Σχετικά έγγραφα

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset