Non-linear evolutions of the Kelvin-Helmholtz instability in stratified shear layers
Yosuke Matsumoto and Masahiro Hoshino
Department of Earth and Planetary Science, University of Tokyo.
Abstract
A two-dimensional MHD simulation of the Kelvin-Helmholtz (K-H) instability in a stratified shear layer shows a strong development of turbulence through non-linear instabilities. The difference in density between the two media plays a crucial role on the fast turbulent mixing and transport. The onset of the turbulence is triggered not only by secondary K-H instability but also by Rayleigh-Taylor (R-T) instability at the density interface inside the normal K-H vortex, whose onset mechanisms are attributed to the centrifugal force of the rotating motion. The secondary R-T instability alters the macroscopic structure by transporting dense fluids to tenuous region, while the secondary K-H instability is just a seed for the turbulence. The onset mechanism and the formation of the broad mixing layer give a new understanding of the mixing process in a variety of geo- and astrophysical phenomena.
Introduction
The interface between two media flowing relative to each other is unstable and the resulting growing wave is well known as the Kelvin-Helmholtz (K-H) instability (e.g. Chandrasekhar, 1961). The non-linear evolution of the K-H instability, which is characterized by momentum exchange and mixing of the two media, has been studied for a long time and applied to several fields with greatly varying scales of phenomena: Wind-induced water wave is a general common phenomenon that can be seen anywhere; The sheared E¡ßB drift instability of plasmas inhibits the turbulent transport in tokamaks of laboratory plasmas; The K-H instability is a candidate along with alternative explanations of planetary phenomena such as the formation of the low latitude boundary layer (LLBL) at Earth's magnetosphere and the magnetic flux ropes observed in Venus ionosphere. Recent observational and computational studies reveal the deformation of the astrophysical jets by the K-H instability.
Where there is a velocity shear, there is usually a density interface. A representative example, which is our most interesting issue, is the interaction between the shocked solar wind plasma and the tenuous magnetospheric plasma at the earth. In particular, the transport mechanism of the solar wind plasma into the earth's magnetosphere via the velocity shear interface is still an unsolved issue and numerous studies with direct numerical simulations have attempted to explain it. Magneto-hydro dynamic (MHD) simulations of the K-H instability have been carried out with a transverse magnetic field configuration and they have showed that there is momentum transport across the shear layer by means of the Reynolds stress (e.g. Miura, 1984). Mass transport mechanisms have also been investigated by a number of kinetic particle simulations (Thomas and Winske, 1993; Fujimoto and Terasawa, 1994). The finite gyro-radius effect of ions is expected to enhance the mixing rate with time. In both MHD and kinetic simulations, however, no one has succeeded in obtaining the transport over the K-H vortex mixing layer nor in explaining the broad mixing layer lying in the low latitude region of the earth's magnetosphere. In this context, turbulence is a candidate mechanism for explaining the formation of such broad mixing layer. It is well known that turbulence enhances the mixing and is generated in a three-dimensional hydrodynamical K-H instability (Fritts et al., 1996). In the present paper we show, for the first time, the onset of turbulence from a normal K-H vortex in a two-dimensional plane. We also propose the mechanism for obtaining the enhanced mixing layer, which can be applied to a variety of situations and in particular to the interaction between the solar wind plasma and the earth's magnetosphere.
Simulation settings
We carried out a two-dimensional direct numerical simulation solving conventional MHD equations: the equation of continuity, the momentum equation, the equation of state and Faraday's law with the frozen-in condition. The initial equilibrium is maintained by constant total (thermal and magnetic) pressure with a sheared velocity profile which varies as Vx = 0.5V0tanh(y/¦Ë), where ¦Ë is a shear scale length and V0 is a jump in velocity across the shear layer. In the following simulation runs we adopt Alfvén Mach number MA = V0/VA = -1.0. We initially set the homogeneous transverse magnetic field Bz so that the initial thermal pressure is constant in the system and plasma beta is set equal to 0.3. The number density profile is provided with N = N0/2{(1+¦Á)+(1-¦Á)tanh(y/¦Ë)}, where ¦Á is the asymptotic number density ratio. We varied ¦Á between 0.2 and 1.0 in order to explore how the inhomogeneity in density affects the non-linear evolution of the K-H instability.
Results
In the linearly growing stage, each simulation run shows the development of the eigen-mode with the growth rate as expected from the linear analysis. In the non-linear stage, however, the eigen-mode develops differently in each run for different ¦Á. In the simulation run with ¦Á=0.2, the secondary instabilities start growing at a time t=84.38¦Ë/V0 (Fig. 1). Newly-induced waves grow at the density interface at the outer edge and inside the normal K-H vortex. As a consequence of such developments of secondary instabilities, the normal vortex structure collapses and the system proceeds to the turbulent flow stage. In the final stage of the simulation run at t=156.25¦Ë/V0 fine structures appear with turbulent flows and the mixing layer approaches the boundary at yb = -10¦Ë. In this simulation run, two characteristic properties are present: One is the collapse of the normal vortex structure and the other is that the position of the mixing layer moves toward the -y direction from the center of the simulation domain. The former leads the system to the turbulent structure and enhances the mixing with time. The latter introduces the diffusion of fluids from the dense to the tenuous region. (animation: From the left to the right, the results show the non-linear evolutions with density ratios 0.2, 0.4, 0.8, respectively.)
To understand the development of the turbulent flows, we take an energy spectrum in the x direction with integrating in the y direction. As results, the energy spectra (Figs. 2A, 2B, 2C) with the final density profiles of each run (Figs. 2D, 2E, 2F) show that their development depend on ¦Á. When ¦Á=0.2, while the power of the normal mode remains constant, the energy cascades rapidly to the shorter wave modes as time goes on. When ¦Á=0.6, the energy also cascades to the shorter wave modes, although their peak values are rather small. When ¦Á=1.0, the energy does not cascade to the shorter wave modes and the normal mode dominates the system. We fitted the spectra at t=156.25¦Ë/V0 (thick dark red line) with power functions and evaluated the powers for ¦Á=0.2,0.6,1.0 as -1.32, -1.38, -2.49, respectively.
Examining in detail, we find that the onsets of the turbulence are triggered by two kinds of secondary-induced instabilities. The one is the K-H instability and the other is, more importantly, the Rayleigh-Taylor instability (Sharp, 1984). The secondary K-H instabilities are excited by the strong shear flows inside the normal vortex. In the linear stage of the normal K-H instability, the unstable eigen-mode of the perturbed total pressure accelerates the fluid elements, which makes the light fluid turn faster and the heavy fluid turn slower. The other secondary instability is also attributed to the vortex motion. The centrifugal force of the rotating motion can act as a radial effective gravity force "geff" on the fluid medium. Under the presence of the gravity force, the well-known Rayleigh-Taylor instability grows. Consequently, the density interface inside the vortex is unstable to the R-T instability due to the radial centrifugal force. Both secondary instabilities show that the growth rates depend on the inhomogeneity in density. Furthermore, the growth rates are positive functions of the wave number, which means that the free energy of the inhomogeneity in density is transported to shorter wave modes and as a result, turbulence takes place.
Turbulence enhances the mixing of the two media. The width of the mixing layer increases with time through two phases in the run for ¦Á=0.2, 0.3 (Fig. 3A). The first phase, which can be seen in all simulation runs until t=60¦Ë/V0, corresponds to a linear growth of the normal K-H instability. (Slight differences in time evolution are due to the slight differences in growth rates.) The second phase is clearly seen in the run for ¦Á=0.2 and ¦Á=0.3 after the first phase and the beginning of the second phase at t=80¦Ë/V0 in ¦Á=0.2 corresponds to the onset of the secondary instabilities. The width of the mixing layer increases from the first saturation level while in other simulation runs it oscillates within a certain level. A second point to be noted is the time development of the position of the mixing layer (Fig. 3B). In all simulation runs except for ¦Á=1.0 the position of the mixing layer shifts toward the -y direction until t=60¦Ë/V0 and the displacement is larger for smaller ¦Á. This corresponds to the linear growth of the normal K-H instability whose rotating motion weakly induces the R-T instability at the outer edge of the vortex. The stability condition of the R-T instability introduces the asymmetry in the y direction, that is, it is unstable to the R-T instability in the negative y region of the outer edge of the vortex whereas the opposite side is stable. Hence the mixing layer seems to shift toward the -y direction. After that the displacement again starts increasing in the run for ¦Á=0.2 and weakly in ¦Á=0.3 at t=100¦Ë/V0 while stays almost constant in other runs. This additional displacement starts after the onset of turbulence. Examining in detail, we find that the displacement is accompanied by the R-T instability at the outer edge of the normal K-H vortex. At first, the secondary K-H instabilities grow at the two interfaces since there are velocity differences across the interfaces. Accordingly, the flow is bent and the resulting wavy structure induces the R-T instability. Mass exchange is a basic element of the R-T instability and this characteristic introduces the transport of the dense fluid to the tenuous region at the edge of the normal vortex. (There is no mass transport at the opposite side of the normal vortex since it is R-T stable.)
Summary
Two-dimensional MHD simulations of the Kelvin-Helmholtz instability have shown that the inhomogeneity in fluid induces turbulence and enhances the mixing of the two media. The triggers of the onset are the secondary K-H and the Rayleigh-Taylor instabilities which grow inside the normal K-H vortex. Furthermore, the R-T instability also plays an important role in the mass transport to tenuous regions while the secondary K-H instability is just a seed for the turbulence. Although these two secondary instabilities themselves are classical, the excitation of secondary instabilities, in particular, the R-T instability by the rotating motion of a normal K-H instability is a new concept. Once the onset is triggered, turbulence is generated since the subsequent higher order instabilities are induced by the pre-excited K-H and R-T instabilities. This chain-reaction continues until the dissipation overcomes the instabilities. The mechanism is therefore a strong non-linear coupling. In space and astrophysical phenomena there is also a magnetic field interface where there is a velocity shear interface. To elucidate the effect of inhomogeneity in the magnetic field (thermal pressure) on the non-linear development we have also carried out the simulation of an inhomogeneous transverse magnetic field case maintained by inhomogeneous temperature fluids with a homogeneous density profile. As a result, no development of turbulence is confirmed and we conclude that inhomogeneities in the magnetic field are not important for the onset of turbulence in a ideal MHD regime, even though kinetic approaches are required for a more appropriate expression. The results presented here and what we suggest are significant in a variety of fields, since the density interface with a velocity shear are observed in plenty of natural phenomena. In particular, the present mechanism, which is fundamentally hydrodynamic, well explains the formation of the broad mixing layer lying in the low latitude region of the earth's magnetosphere, providing a new understanding of the Sun-Earth connection model from the view point of the transport of solar wind plasma into the earth's magnetosphere.
References
