2 The incompressible Kelvin-Helmholtz instability

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1 Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect of any dynamical system. It is one thing to establish the dynamical equations for some system or other, it is another to establish that the system is stable. If it is unstable, then the system will evolve to some other state. For example, we showed, in dealing with shocks that there are two types of discontinuities - the shock discontinuity in which there is a mass flux across the discontinuity and the tangential contact discontinuity for which the mass flux is zero. The latter discontinuity is subject to a classical discontinuity - the Kelvin-Helmholtz discontinuity which is one of the subjects of this chapter. The incompressible Kelvin-Helmholtz instability The Kelvin-Helmholtz instability relates to the following situation. C54H Astrophysical Fluid Dynamics 1

2 ρ 1, V 1 z y x P 1 P ρ, V A large number of the properties of this instabilities can be understood in terms of the following incompressible analysis. Continuity equation for incompressible flow Incompressibility means that the density is constant along a streamline, so that dρ dt Perturbation and ρ + div( ρv) 0 ρ + V ρ + ρdivv 0 0 dρ + ρdivv 0 dt divv 0 To study stability, we perturb the above system as follows: C54 H Astrophysical Fluid Dynamics

3 where the 0 subscripts refer to the unperturbed situation. Thus In developing the equations which describe the development of the instability, we develop equations which refer to either z < 0 or z > 0 as far as possible, introducing either V 1 or V when necessary, when we come to consider the boundary conditions at the interface. Perturbation of the continuity equation This is simply Perturbation of the momentum equation The term V V 0 + V P P 0 + P V 0 ( V 1, 00, ) z > 0 V 0 ( V, 00, ) z < 0 divv 0 V V ( V 0 + V ) V V 0 V to first order, so that the perturbed momentum equation is: ρ V + V0 V ρ V + V 0, x V P Take the divergence of this equation ρ divv + V 0, x divv P C54H Astrophysical Fluid Dynamics 3

4 Since divv 0, Form of the perturbation Take all perturbed quantities to be of the form: where k x and k y are real components of the wave vector. (Note the use of a 0 superscript to indicate the amplitude, as distinct from the 0 subscript which characterizes the unperturbed initial state. Perturbation equation for the pressure Take P 0 f ( r, t) f 0 ( z) expik [ x x + k y y ωt] P P 0 expik [ x x + k y y ωt] P d P 0 ( k d x + k y )P 0 expik [ x x + k y y ωt] 0 z The combination k x + k y appears frequently in the following, so that we denote it by k. Hence d P 0 + k P 0 0 P 0 ( z) A dz 1 e kz + A e kz We take different parts of this solution on different sides of the interface. We require finite pressures at z ± and P 0 P 0 A 1 e kz z < 0 A e kz z > 0 4 C54 H Astrophysical Fluid Dynamics

5 We now impose the boundary condition that the perturbed pressures on both sides of the interface are equal so that A 1 and A A P 0 P 0 Ae kz z < 0 Ae kz z > 0 Perturbation equation for the velocity Taking, V z V z0 ( z) expik [ x x + k y y ωt] V z V z0 ( z) ikxexpik [ x x + k y y ωt] V z V z V z0 ( z) ( ik x ) expik [ x x + k y y ωt] V z0 ( z) ( iω) expik [ x x + k y y ωt] We substitute these expressions into the momentum equation to obtain: ρ 0 [ V z0 ( z) ( k x V 0 ω) ] expik [ x x + k y y ωt] ± expik [ x x + k y y ωt] Ae+ kz P z where the different signs refer to z > 0 and z < 0 respectively. This gives us the following solution for V z0 ( z) : V z0 ( z) Ae+ kz ± ρ 0 ( k x V 0 ω) C54H Astrophysical Fluid Dynamics 5

6 Displacement of the surface Perturbed interface ζ Unperturbed position of interface To proceed further, we need to consider the displacement of the fluid at the interface. Consider the displacement of a fluid element at any position in the fluid. A given fluid element satisfies: dr dt V so that putting r r 0 + r where r is the variation from the zeroth order flow as a result of the perturbation, gives d ( r0 + r ) V dt 0 + V d r V dt The differentiation on the left hand side is following the motion so that this perturbation equation is, in fact, 6 C54 H Astrophysical Fluid Dynamics

7 r + V r r + V 0, x r V The component of this set of equation which is of the most use to us, is the z component. Denoting the z component of r by ζ, ζ ζ + V z V 0, x Ae+ kz ± ρ 0 ( k x V 0 ω) As with all other functions, we put, ζ ζ 0 expik [ x x + k y y ωt] ζ ζ iωζ 0 expik [ x x + k y y ωt] ik x ζ 0 expik [ x x + k y y ωt] Therefore, the equation for ζ becomes: iζ 0 ( k x V 0 ω) expik [ x x + k y y ωt] ζ 0 Ae+ kz ± ρ 0 ( k x V 0 ω) expik [ x x + k y y ωt] Ae+ kz ± ρ 0 ( k x V 0 ω) Now, at z 0, the displacements calculated from either side of the interface should be identical. Therefore, ρ 1 ( k x V 1 ω) ρ ( k x V ω) ρ 1 ( k x V 1 ω) + ρ ( k x V ω) 0 Expanding out the quadratic terms: C54H Astrophysical Fluid Dynamics 7

8 ρ 1 [ ω ωk x V 1 + k x V 1 ] + ρ [ ω ωk x V + k x V ] 0 ( ρ 1 + ρ )ω ωk x [ ρ 1 V 1 + ρ V ] + k x [ ρ 1 V 1 + ρ V ] 0 ( ρ 1 + ρ ) ω k ω [ ρ1 V k x x 1 + ρ V ] + [ ρ 1 V 1 + ρ V ] 0 and the solution of this quadratic is ω ---- k x ( ρ 1 V 1 + ρ V ) ± iv ( 1 V )( ρ 1 ρ ) 1 / ρ 1 + ρ This is our main result, the dispersion relationship between frequency and wave number. The important feature of this solution is that it has both real and imaginary parts: ω R k x ρ 1 V 1 + ρ V ρ 1 + ρ ( V 1 V ) ( ρ 1 ρ ) 1 / ρ 1 + ρ ω I k x The complex part corresponds to both exponentially decaying and growing solutions, since An arbitrary set of initial conditions will give both decaying and growing solutions, so that the above solution enables us to identify a growth rate, where, the density ratio, expi[ ω R ± iω I t] expiω R exp+ ω I t ( ω g ( V 1 V ) ρ 1 ρ ) 1 / η k 1 / ρ 1 + ρ x ( + η V V )k 1 x η ρ ρ and 8 C54 H Astrophysical Fluid Dynamics

9 k x k cosφ is the component of the wave number in the direction of flow. y k ( k x, k y ) Plan view of interface φ x Features Growth depends upon there being a velocity difference. Growth rate proportional to k x (component of wave number in the direction of flow) so that the smallest waves (largest k x ) grow the fastest. The growth rate reduces to zero for waves perpendicular to the direction of motion. The growth rate is a maximum for η 1. All perturbations diminish exponentially away from the interface. ( Perturbation exp± kz) This is a characteristic feature of surface waves. These features are also characteristic of the KH instability for compressible flows (as we show in the next section). C54H Astrophysical Fluid Dynamics 9

10 3 Compressible Kelvin-Helmholtz instability 3.1 General comments When we deal with compressible flow, the main complicating factor is that we have to deal with are sound waves when we perturb the flow. In one sense we can think of the KH instability as sound waves in an inhomogeneous medium. (Sound waves are emitted as the surface is disturbed.) The consideration of the compressible KH instability is similar to that of the incompressible KH instability with some complications related to compressibility. The steps in the development are: Determine the dispersion relations for waves in each medium. Apply boundary conditions at z ± and at the interface. This leads to polynomial equations and conditions on the roots of these leads to useful information. 3. Dispersion relations in each medium We start with the usual equations: ρ + ( ρv i ) 0 i ρ V i V i + V j j P + 0 i and perturb them according them according to the usual recipe: ρ ρ 0 + ρ V i V 0, i + V i 10 C54 H Astrophysical Fluid Dynamics

11 ρ 1, V 1 z y x P 1 P ρ, V These perturbations imply that ρv i ρ 0 V 0, i + ρ 0 V i + ρ V 0, i (used in the continuity equation) and j V i x j V i V 0 V i ( V 0, j + V j ) x j (used in the momentum equation). Make these substitutions ρ 1 + ρ 0 V j + V 0, j ρ 0 j j ρ 0 V i + V 0 V i + P 0 i In the following, we adopt the pressure as the primary variable since it is continuous across the interface rather than the density which may be discontinuous. In doing so we use, C54H Astrophysical Fluid Dynamics 11

12 i ρ c s i P Hence, the continuity equation becomes c 0 P + V 0 P + ρ0 V x j 0 j Summary of perturbation equations Combine ρ 0 and c 0 in continuity equation and put together with momentum equation: P + V 0 P ρ0 c + V 0 x j 0 j ρ 0 V i + V 0 V i + P 0 i Similar to the compressible case, we take perturbations of V i proportional to expik [ x x + k y y+ k z z ωt] P and where ( k ẋ, k y ) is real and k z and ω may be complex. Compare this with the z-dependence for the incompressible Kelvin-Helmholtz instability e kz. Take P Aexpi[ k x x + k y y+ k z z ωt] V i A i expik [ x x + k y y+ k z z ωt] With these dependencies: 1 C54 H Astrophysical Fluid Dynamics

13 P + V 0 P i ( ω kx V 0 )Aexpi[ k x x + k y y+ k z z ωt] V i + V 0 V i i( ω k x V 0 )A i expik [ x x + k y y+ k z z ωt] V ik A exp ik [ x + k y + k z ωt ] j j j x y z j Putting all of this together, using ρc 0 γ P 0, i( ω k x V 0 )A + γ P 0 ( ik j A j ) 0 iρ 0 ( ω k x V 0 )A i + ik i A 0 Notation for wave vector At this point we introduce the following notation for the wave vector k ( k x, k y, k z ) ( k, k z ) k x k cosφ k y k sinφ k k + k z k i k i where k is the component of the wave vector parallel to the interface. With this notation the perturbation equations become: i( ω k V 0 cosφ)a + γ P 0 ( ik j A j ) 0 i( ω k V 0 cosφ)ρ 0 A i + ik i A 0 Take the scalar product of the second of the above equations with. This gives k i C54H Astrophysical Fluid Dynamics 13

14 i( ω k V 0 cosφ)ρ 0 k i A i + ik A 0 ik i A i ik A ρ 0 ( ω k V 0 cosφ) Substitute this result back into the first of the perturbation equations. i( ω k V 0 cosφ)a + γ P 0 ( ik j A j ) i( ω k V 0 cosφ)a 0 + γ P 0 ik A ρ 0 ( ω k V 0 cosφ) Solving, and dividing out the common factor of A γ P ( ω k V 0 cosφ) k c 0 k c 0 ( k + k z ) ρ 0 Note that we have essentially recovered the dispersion equation for sound waves! For the two different sides of the interface, ( ω k V 1 cosφ) k ( ω k V cosφ) c k Perturbation of the surface As before consideration of the perturbation of the surface is important. The z-component of the displacement is the same as for the incompressible case and is given by ζ ζ + V 0 V z A z expik [ x x + k y y+ k z z ωt] 14 C54 H Astrophysical Fluid Dynamics

15 Putting ζ B z expik [ x x + k y y + k z z ωt] gives i[ ω k V 0 cosφ]b z A z B z ia z [ ω k V 0 cosφ] Now we have from the perturbation equations: i( ω k V 0 cosφ)ρ 0 A i + ik i A 0 so that A z k z A ρ 0 ( ω k V 0 cosφ) B z and therefore,, the coefficient of the z component of displacement is given in terms of A, the coefficient of the pressure field, by B z ik z A ρ 0 ( ω k V 0 cosφ) This is a generic equation which applies to either region. However, both the displacement, and the pressure are continuous at the interface. Therefore B z A 1 is continuous and k 1, z ρ 1 ( ω k V 1 cosφ) k, z ρ ( ω k V cosφ) Now go back to the dispersion relations which we derived for the two regions, viz, C54H Astrophysical Fluid Dynamics 15

16 ( ω k V 1 cosφ) k ( k + k 1, z ) ( ω k V cosφ) c k c ( k + k, z ) We make k z the subject of these equations, k 1, z k, z ( ω k V 1 cosφ) k ( ω k V cosφ) k c In the equation derived from the boundary condition, we put giving us γ ρ 1 P 0 γ ρ P c k 1, z γ 1 ( ω k V 1 cosφ) k, z γ ( ω k V cosφ) c Squaring, and substituting for k 1, z γ 1 ( ω k V 1 cosφ) 4 4 k z k, z γ ( ω k V cosφ) 4 c 4 16 C54 H Astrophysical Fluid Dynamics

17 ( ω k V 1 cosφ) k γ 1 ( ω k V 1 cosφ) 4 4 ( ω k V cosφ) k c γ ( ω k V cosφ) 4 c 4 Instabilities This can actually be simplified! We divide the numerators by and the denominators by k 4 ω and put V ph ---, the phase velocity of the wave. This gives, k k ( V ph V 1 cosφ) γ 1 ( V ph V 1 cosφ) ( V ph V cosφ) c --- γ ( V ph V cosφ) c 4 Furthermore, we can make a Galilean transformation in the x direction in which the velocity,, of the lower stream is zero, i.e. and this transforms V 1 x x V 1 t k x x ωt to [ k x ( x + V 1 t) ωt] [ k x x ( ω k x V 1 )t] Therefore, in the new frame, and where ω ω k x V 1 ω ω + k x V 1 ω k x V cosφ ω k x ( V V 1 ) cosφ ω k x V cosφ C54H Astrophysical Fluid Dynamics 17

18 is the difference in velocity between the two streams. Hence, V V V 1 ω --- V k 1 cosφ ω --- V k cosφ ω k ω V cosφ k i.e. where V ph ω k V ph V 1 cosφ V ph V ph V cosφ V ph V cosφ is the phase velocity in the primed frame, the one in which the velocity of the lower stream is zero. The equation for the KH instability becomes: V ph γ 1 V ph ( V ph V cosφ) c γ ( V ph V cosφ) c 4 Finally, we express all velocities in terms of ratios with respect to the sound speed in medium 1, i.e. x V ph m V cosφ and this gives 18 C54 H Astrophysical Fluid Dynamics

19 x x 4 γ c ---- γ c 1 ( x m) c c ( x m) 4 This is the basic dispersion relation for the compressible KH instability. It is a sixth order polynomial equation when multiplied out. Condition on the solution Since we want the solutions to vanish at z ± following important conditions on the solution: then we have the Re( k 1, z ) < 0 and Re( k, z ) > Physical interpretation of x and m V cosφ φ V x m V ph Phase velocity of perturbation in frame of lower stream relative to speed of sound V cosφ Relative Mach number of streams in direction of perturbation C54H Astrophysical Fluid Dynamics 19

20 3.4 Special cases Take γ 1 γ a c ---- then the dispersion equation becomes x a ( x m) a ( x m) 4 Factoring the second term on the right hand side: x a a x x 4 ( x m) ( x m) 4 1 a a x ( x m) ( x m) 4 x x a ( x m) 1 a a ( x m) ( x m) x x and factorising the equation: x a ( x m) x a ( x m) so that either x a ( x m) (quadratic) or x a (quartic) ( x m) 0 C54 H Astrophysical Fluid Dynamics

21 Roots of quadratic m x , 1 a Since a x ( x m) x mx + m ( a 1)x + mx m 0 m a m V cosφ a c ---- then x ω k V cosφ , V cosφ c + c ω --- k V c cosφ, V + c cosφ Note that both of these roots are real and therefore neither correspond to an instability. Roots of quartic Special case: x a ( x m) a c x + ( x m) ( x m) + x x ( x m) C54H Astrophysical Fluid Dynamics 1

22 In order to solve this equation, put y m y --- m y x m --- m y m y --- y m y m making the equation into the following quadratic in : y m y 4 m y m y 4 m y m 4 m m y ± ( 1 + m 4 ) 1 / y Now the roots for y are always positive when 1 m > 1 + m m m > 1 + m 16 m > 8 If y > 0 then y is real and so is x, so that there is no instability. Hence, the condition for there to be no instability is: m V cosφ > 8 or, equivalently, the perturbation is unstable, if C54 H Astrophysical Fluid Dynamics

23 M rel cosφ < 8 where M rel V is the relative Mach number of the two streams. Note that for any Mach number there is a critical angle for the wave vector for which instability occurs given by: cosφ crit M rel Perturbations with φ > φ crit are unstable. Unstable φ crit V V Stable cosφ C54H Astrophysical Fluid Dynamics 3