6.3. On the Physics of the Kelvin–Helmholtz Instability
A relatively simple case for which we can find analytical results will help to understand the physics of the instability. We consider two hydrodynamic fluids with the same unperturbed characteristics (same density , same pressure as required from the equilibrium at the contact discontinuity, and same sound velocity ) and work in the frame where the two fluids move with opposite velocities (with positive and the upper sign corresponding to ).
The dispersion relation becomes , where . Unstable modes (with ) exist for , with purely imaginary . One way to find this result is to substitute when the dispersion relation reduces to with solutions . Actually, the angle has an important meaning connected to the argument of the resulting complex wavenumbers in the direction, which are .
Although the results apply for any angle between
and
, to simplify the expressions, from now on we restrict ourselves to the case
, i.e., we consider a disturbance with
(with positive
). The characteristics of the unstable mode, the Lagrangian displacement, and the perturbations of the pressure, density, and velocity (written in a way to show the phase difference between each quantity with
) are
Suppose we initially disturb the interface between the two fluids by . The Lagrangian displacement along at any later time and for any other fluid element has the form ; the goal is to find and . Using the Lagrangian displacement and the Lagrangian perturbations of the velocity, pressure, density, which are , , and , respectively, the equations that lead to that goal are the momentum and the continuity (coming from ). Since all perturbations are proportional to , a time derivative has the meaning , and the divergence , with the total wavenumber in the two fluids (their components are the same, but the components differ).
The momentum equation along connects the displacement with the pressure perturbation . Requiring and to be continuous at the interface between the two fluids, we arrive at a first relation between the unknowns .
The pressure gradient along the interface is connected to the corresponding Lagrangian displacement through the momentum equation along , which gives . (The latter means that , a characteristic of longitudinal waves.)
The continuity also connects the two components of the displacement. It gives . Substituting and , we arrive at the second relation between the unknowns (one for each fluid) .
The perturbation essentially consists of two sound waves in the two fluids, with wavenumbers . In the fluid rest frames, the two waves move with different velocities, satisfying , a relation equivalent to , with . The two waves meet at the contact discontinuity and as a consequence they have the same and such that their phases are continuous at the interface. As already presented, for the case of fluids with the same density, in a frame in which they move with opposite velocities , and is parallel to the relative velocity, the resulting unstable mode has and , with , .
The value of the complex wavenumber (in each part) is directly connected to compressibility through S. In the incompressible limit it is purely imaginary, since in that limit , , and . (Note that in the incompressible limit the continuity gives . Since the two vectors are complex, this does not mean . It rather means , using from the momentum equation. In addition, the relation does not mean that the vector is zero.) If we consider cases with decreasing S, i.e., decreasing keeping the same, in which the compressibility becomes more and more important, the first part of the expression of affects the value of in two ways. Firstly, the imaginary part decreases, meaning that the perturbation survives at longer distances from the interface. Secondly, the increases, contributing more to the phase of the perturbation, which is . Both effects are expressed through the angle , which decreases with decreasing S (increasing M). The growth rate is also connected to S through . It decreases as the result of compressibility, from in the limit to zero when , corresponding to the minimum S for which the perturbation is unstable.
The vorticity is a key quantity in the Kelvin–Helmholtz instability. Its undisturbed value is the reason behind the development of the instability. Even with the perturbation included it is nonzero only in the interface between the two fluids. Nevertheless, the motion of the fluids along the interface redistributes the vorticity compared to the unperturbed state. The related velocity inside the upper/lower fluid, just above/below the interface , is . The mean value shows that fluid accumulates near the positions of lower pressure, where the vorticity increases (in the direction) by . This accumulation of vorticity further increases the displacement , leading to instability. It becomes stronger for larger , i.e., smaller M, since the mean value of increases with .
Three example solutions are shown in
Figure 2. The upper panel corresponds to an incompressible case with
, for which
and
. (For
the approximate expression for
is
.) The exponential decrease in the perturbation as we move away from the interface is evident. The perturbed interface is shown, along with the circulation around points of minimum pressure (the mean velocity of the fluids on the interface points toward these minimum pressure points).
The two other panels correspond to cases in which compressibility is important. The solution in the middle panel has , , and , and in the lower panel , , and . The now has a nonnegligible part and as a result the iso-phase planes are constant, tilted as shown in the panels.
For cases approaching the maximum value of M for which the perturbation is unstable (the ), the values of , , and approach zero, and the perturbation practically consists of two standing sound waves (one in each fluid) with real wavenumbers. (For the approximate expression for is .)
6.4. Influence of Magnetic Field
For simplicity we consider again two homogeneous fluids with the same unperturbed characteristics, and work in the frame where the fluids move with opposite velocities . Now, there is also a constant magnetic field in the unperturbed state, the same in both fluids.
Inspecting the dispersion relation (
22), we see that the magnetic field enters in two ways. Firstly, through its pressure
, or equivalently the square of the Alfvén velocity
. Secondly, through its tension, manifest in the terms
, i.e., its component parallel to
, or equivalently the component of the Alfvén velocity
.
The magnetic pressure
enters in the expression of
S and increases its value. Thus, it moves the dynamics toward the incompressible limit and, according to the discussion of the previous section, destabilizes (the second terms inside the square roots equal
, so an increase in
leads to the square roots being closer to unity). Actually, if we include magnetic field normal to
only, the dispersion relation is exactly equivalent to its hydrodynamic analogue, with the only difference that
M now represents the fast magnetosonic Mach number
.
Figure 3 shows the resulting growth rates. Notably it includes as subcases the purely hydrodynamic case (
) considered in the previous section, and the cold case (
), with only the
component of the magnetic field.
The magnetic tension enters in the dispersion relation through (essentially the component of the magnetic field along ) in two places: inside A and S. It always has a stabilizing effect since the tension is a restoring force; we have seen that in the previous examples of Alfvén waves and the Rayleigh–Taylor instability, even in the incompressible limit. It also affects incompressibility through its appearance inside S. Since it decreases S, it moves the dynamics away from the incompressible limit, something that also, in general, stabilizes.
Another related connection is that the magnetic tension affects the perturbation of the vorticity inside each fluid, which, using the expressions of the velocity perturbations given in
Appendix A, can be expressed as
.
In the following, we present the methodology to find numerical results.
The dispersion relation (
22) in the case under consideration can be written as
and can be solved numerically. However, for the case of fluids with the same characteristics considered here it is possible to proceed analytically. As shown in
Appendix C, the dispersion relation can be transformed to the following quartic polynomial equation for the square of the growth rate,
using the parametrization
Note that the angle
is connected to the angle between the magnetic field and the wavenumber since
.
The left panel of
Figure 4 shows the growth rate for cases where the magnetic field has only component along
. In general, the field decreases the growth rate, and if it is sufficiently strong it completely suppresses the instability. Similar behavior is shown in the right panel of
Figure 4 for the cold case. For all strengths of the magnetic field, if its orientation is sufficiently close to the wavevector, the magnetic tension completely suppresses the instability.
6.5. Range of Instability
It is interesting to explore the regions of
M for which the Kelvin–Helmholtz instability is present, as shown in
Figure 4. The question is: Under what conditions does the dispersion relation (
27) have purely imaginary roots? For simplicity, we consider a disturbance with
parallel to the velocities. The results can be easily generalized.
One might naively think that the extreme values of these instability regions can be found by setting
in the dispersion relation (
27) and requiring both numerators to vanish (since the denominators are opposite real numbers). However, this implies
, resulting in no
x-dependence of the disturbance. This case corresponds to magnetosonic waves in the frame of each fluid with total wavevectors
and
, with
(see
Section 3.1). The absence of
x-dependence makes these cases unrelated to the unstable modes.
However, there are two other possibilities. The vanishing of both denominators in the dispersion relation (
27) for
needs to be considered as a possible limiting case. This corresponds to Alfvén waves with
, with
(see
Section 3.2). Note that
does not appear in the dispersion relation. Nevertheless, its value is nonzero, given by
. The
x-dependence of the disturbance allows for a possible connection with unstable modes.
A third possibility is that the limiting values may correspond to bifurcations of the dispersion relation. It is actually evident from inspecting
Figure 3 that the slope
becomes infinity when
. In general, a dispersion relation depends on various parameters, and the slope is defined as the derivative with respect to one of these parameters, while keeping all others constant. For a dispersion relation of the form
constant, its differential
shows that bifurcation corresponds to
. Thus, extreme values of the instability regions may be connected to the condition
.
In our case, there are three parameters, and we can choose
,
,
. The algebra gives that
leads to
or the following cubic for
:
or
.
For the left panel of
Figure 4 we have
,
, and the extreme values of
M are given by
. The roots of this cubic are
,
,
, and the corresponding wavenumbers
,
,
, respectively. Thus, the value
corresponds to a magnetosonic wave without
x-dependence (unrelated to instability), while the other two solutions
,
are the extreme values of
M related to the unstable modes.
The result is that for the Kelvin–Helmholtz instability occurs only if and for velocities in the interval . As we approach the limits, the wavenumbers approach real values and the instability is transformed to magnetosonic waves with constant amplitudes and wavenumbers .
For the right panel of
Figure 4 we have
and
and the maximum value of
M corresponds to bifurcation for which
. The nontrivial root is
, and the corresponding wavenumbers
.
The lower limit of the instability region corresponds to Alfvén waves , so .
The result is that for the cold case the Kelvin–Helmholtz instability occurs only if . As we approach the limits, the wavenumbers approach real values and the instability is transformed to waves (Alfvén waves in the lower limit and magnetosonic in the upper) with constant amplitudes and wavenumbers .