The growth rate of the KH instability indicates how fast the instability evolves from small initial perturbations to vortices for a given plasma situation. The growth rate is obtained from the simulations in the following way. Since the perturbations grow exponentially in the first stage of the instability, we see a linear increase when we plot the time evolution of the maximum of the quantities in a logarithmic scale. The slope of the linear increase is the linear growth rate (see ). In this study, we used the time evolution of the vertical kinetic energy
to obtain the growth rates for the different cases discussed below.
Time evolution of the maximum of ln(Ey) of every time step for Π0 = B0 = 1.0 and for different ρ1.
For the magnetic field, we take three different profiles, one with B
= 0.0 and B
= 0.5 (ionopause cases) and one with B
0 (magnetopause case). However, for the cases with a non-vanishing magnetic field, the growth rates are only insignificantly larger than the ones for B
= 0.0 (see ), which is attributable to the assumed configuration. Since the magnetic field is perpendicular to the flow velocity and the wave vector, it does not greatly influence the evolution of the KH instability. Thus, we present the results for B
= 0.0, and assume that they are also valid for the magnetized ionopause and magnetopause cases. We investigate the effect of different density jumps, up to ρ
= 200 where numerically possible.
Fig. 7 Normalized growth rate as a function of normalized wave number for different densities for Π0 = B0 = 1.0. The asterisks and the diamonds give the values of the growth rate calculated for B1 = 1.1 (more ...)
We have the following situation on the boundary layer: On the one hand, the physical growth rate decreases for θ
→ 0° because the velocity shear decreases; on the other hand, the growth rate decreases for θ
→ 90° due to the stabilizing effect of compressibility. Therefore, the physical growth rate exhibits a maximum somewhere at an intermediate point on the boundary, as can be seen in
= 10 (upper plot) and ρ
= 100 (lower plot). This maximum lies around 40° SZA, where M
= 0.7. For our calculations, we thus took the case with Π
= 1.0, for which we get the maximum physical growth rates on the boundary.
Maximum growth rate γ as a function of the solar zenith angle θ for ρ1 = 10 (upper plot) and ρ1 = 100 (lower plot). The legend is valid for both plots.
Concerning the general evolution of the instability, we have three phases. First, there is a linear growth phase, which is clearly visible when we plot the maximum of the vertical kinetic energy
of every time step in a logarithmic scale (
). From the slope of the linear growth, a linear growth rate can be determined. After the linear growth phase, we have a saturation and the nonlinear evolution sets in. The end of the growth phase is identified with the first maximum of the evolution, which is an indicator of the linear growth time t
needed for a vortex to develop. Eventually, the vortex structure becomes irregular, and the boundary layer ends up disturbed.
An exemplary evolution of the KH instability is shown in
. These plots show the normalized mass density at different times during the simulation and they correspond to the case with ρ
= 10 and with the wave number for which we get the maximum growth rate, k
= 0.42 (L
= 15). The plasma flows from left to right. At time t
lin(10) = 93.33, a regular-structured vortex has evolved and the saturation sets in – ending in an irregular boundary layer.
Time series of the density.
shows the normalized growth rate as a function of the normalized wave number for four different values of ρ
1. The lines are polynomial fits of second order. We see a strong decrease of the growth rate with increasing ρ
1 and a shifting of the maximum to smaller wave numbers or larger wavelengths, respectively.
We found that the behaviour of the normalized maximum growth rate γ
m as a function of the normalized density of the lower layer, ρ
n, is well represented by a logarithmic fit of the form γ
1), with b
= 0.091 and c
= −0.014, which is depicted in the upper plot of
. The corresponding normalized linear growth time, using the same fit-function with b
= −2.09 and c
= 40.27, is shown in the lower plot. The dashed line represents the possible continuation for larger density jumps.
Fig. 8 Normalized maximum growth rate (upper plot) and corresponding normalized linear growth time (lower plot) as a function of the density ratio for Π0 = 1.0 and B0 = 1.0. The symbols denote the values obtained numerically, (more ...)
To interpret these results with respect to the situation at Venus, we have to look at observations. The upper ionosphere, the ionopause as well as the magnetopause are higly variable and time dependent. Of course, it would not be possible to fully account for this variability. However, we can take some average conditions for solar maximum and some average conditions for solar minimum.
Zhang et al. (2008a)
present ionopause and magnetopause altitudes derived from PVO and Venus Express observations. They give ionopause altitudes for solar minimum and maximum that are around 300 and 400 km at about 40° SZA, respectively. Miller et al. (1984)
report oxygen ion densities at solar maximum up to 20,000 cm−3
at around 400 km for a solar zenith angle of 65° ± 5° – so, there are even higher densities at lower SZA or lower altitudes. Thus, it seems we have to consider at least an ion density of some thousands of ions cm−3
when assuming the ionopause as the KH unstable boundary. With the density at the boundary at 40° SZA, taken from , ρ
≈ 8 × 10−20
, and an oxygen ion mass density of
, we get for ρ
= 6600. From we see that assuming such high density ratios across the boundary results in very small growth rates. This means that the instability or vortices might not be able to develop along the ionopause.
Zhang et al. (2008a)
also present magnetopause locations for solar minimum and maximum which are approximately 500 and 700 km at about 40° SZA, respectively. For solar maximum, the oxygen ion density at 700 km is about 8000 cm−3
(Miller et al., 1984
), which is still very high. Taking again the mass density at 40° SZA from , we get a density ratio ρ
≈ 2700. For such a density increase, the growth rate also becomes rather small, and it is questionable if we get any vortices from the KH instability at the magnetopause at such small SZA.
In our current study, we use initial conditions that are constant along the boundary layer, which is of course some simplification. When following the magnetopause boundary to higher SZA, the boundary altitude increases up to 1700 km for solar maximum (Zhang et al., 2008a
). This means that the density jump will strongly decrease as we follow the boundary to the terminator. Especially at the altitudes corresponding to solar maximum, there should be at least an order of magnitude lower ion densities, which would result in a lower density ratio across the boundary (maybe around ρ
≈ 200–300), such that vortices might be able to be produced by the KH instability.
Taking the case Π
= 1.0 and ρ
= 200, we see that the normalized linear growth time is
. When we assume that the perturbations travel with velocity 0.5v
, they need the following distance D
along the boundary to develop into vortices,
where we have to keep in mind that here t
is the physical, unnormalized linear growth time, which is
. Inserting this expression for t
we get for the distance D
With the half-thickness of the magnetopause of about a
= 100 km (Zhang et al., 2008b
), we thus obtain D
= 10,500 km. This is the distance that the initial perturbation travels while evolving into a vortex. We want to compare this distance to the length of the boundary layer from the subsolar point to the terminator, L
. This length is given by L
), where R
is the radius of the planet and h
is some average boundary layer altitude. With R
≈ 6000 km and h
≈ 1000 km, we get L
≈ 11,000 km. Thus, when the perturbation starts at the subsolar point, it would reach the nonlinear vortex phase around the terminator, when it always grows with the maximum growth rate on the boundary. However, the maximum growth rate is only available around θ
≈ 40°. It seems that for this set-up, the vortex would need more time, respectively distance, to develop.
However, we assume a very small amplitude of the initial perturbation with dv
= 0.01, which is likely to be larger from the beginning and which is definitely larger around θ
≈ 40°, where the perturbations grow with the maximum growth rate. We made thus simulation runs with larger initial perturbations, taking dv
= 0.03, dv
= 0.05 and dv
= 0.1 for the case Π
= 1.0, ρ
= 200 and L
= 28. The resulting linear growth times for these runs are
, respectively. With these growth times we get D
= 7750 km, D
= 5800 km and D
= 4650 km, respectively, which are approximately half of the boundary layer length L
. Therefore, we conclude that for these cases a vortex is able to develop and we can make a rough estimation of a maximum loss rate of ions.
Concerning estimations of the loss rates of ionospheric ions due to the Kelvin–Helmholtz instability, we follow the approach adopted in an earlier paper (Amerstorfer et al., 2010
), where we estimated the amount of particles in each cloud which has spatial scales comparable to the size of the produced vortices. The loss rate due to one cloud is then
is the number of ions in the cloud produced within time t
and is given by
(see Amerstorfer et al., 2010
), with n
as the ion number density in the lower plasma layer and L
as the spatial scale of the perturbation in all directions. From our simulations we get L
. Using the same arguments as in Amerstorfer et al. (2010)
concerning the estimation of the number of clouds present around the terminator, we arrive at 12 clouds. This number is the same as that given by Brace et al. (1982)
, where they estimate from observations that there should be approximately 12 clouds around the terminator at any given time. Thus, the total loss rate Γ
due to all clouds amounts to
, we obtain
And with a
= 100 km, v
= 210 km/s and n
≈ 600 cm−3
(assuming that all ions in the lower layer are oxygen ions and having a jump of 200 in the mass density), we finally get an estimation of the total oxygen ions loss rate of Γ
≈ 7.6 × 1025
. This number has to be regarded as a maximum loss rate due to the KH instability. First, the stabilizing effect of a magnetic field parallel to the flow is neglected in our simulation. Secondly, we took the maximum possible velocity shear which is for sure smaller in real situations. And finally, it is not sure if the detachment of plasma clouds due to the KH instability is really a continuous process.
Some of the detached plasma structures reported by Brace et al. (1982)
were seen in a very large distance to the ionopause, for example at altitudes of about 1500 or 2000 km above the ionopause altitude. This strengthens the idea that the boundary layer from which the plasma clouds originate might not be the ionopause but some layer at higher altitudes, e.g. the induced magnetopause. However, there are also observations with plasma clouds not having such high altitudes. They might be produced by KH vortices at the ionopause when conditions are temporally in favour of the instability at this boundary. Anyway, it might be possible that the origin of these detached plasma structures is not connected to the KH instability and its vortices.
The PVO observations were conducted during high solar activity. Venus Express has been orbiting the planet in a low solar activity phase. Until today, no observations of plasma clouds, similar to what was found in PVO measurements, are reported. Only one publication gives rise to speculations of vortices in the magnetic field, based on observations of the VEX magnetometer (Pope et al., 2009
). The results presented in this paper would fit quite well into this situation, since we found that the KH instability and vortices are most likely to occur during a solar maximum activity phase. Recently, Borisov and Fränz (2011)
mentioned another way of detecting the KH instability. They show that fast magnetosonic waves in the magnetosheath can be caused by the KH instability excited at the magnetopause. Thus, the observation of such fast magnetosonic waves might be an indication of the occurrence of the instability.
We hope that the future VEX mission, extended until the end of 2012, will provide observations and the possibility to understand the solar activity dependence of the appearance of the KH instability.
This study uses initial conditions that do not vary along the boundary layer. Further investigations will need to include more realistic initial plasma conditions.