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Icarus. 2011 December; 216(2): 476–484.

PMCID: PMC3280700

Ute V. Möstl,^{a,}^{d,}^{} Nikolay V. Erkaev,^{b,}^{c} Michael Zellinger,^{a,}^{d} Helmut Lammer,^{a} Hannes Gröller,^{a} Helfried K. Biernat,^{a,}^{d} and Daniil Korovinskiy^{a}

Ute V. Möstl: ta.zarg-inu@ltseom.etu

Received 2011 April 28; Revised 2011 July 15; Accepted 2011 September 9.

Copyright © 2011 Elsevier Inc.

Open Access under CC BY-NC-ND 3.0 license

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to be such by Elsevier, is available for free, on ScienceDirect, at: http://dx.doi.org/10.1016/j.icarus.2011.09.012

This article has been cited by other articles in PMC.

► We study the Kelvin–Helmholtz instability at boundary layers around of Venus. ► The stability of the induced magnetopause and the ionopause is examined. ► The ionopause seems to be stable due to a large density jump across this boundary. ► The instability evolves into its nonlinear phase on the magnetopause at solar maximum. ► Loss rates are therefore lower than previously assumed.

The Kelvin–Helmholtz instability gained scientific attention after observations at Venus by the spacecraft Pioneer Venus Orbiter gave rise to speculations that the instability contributes to the loss of planetary ions through the formation of plasma clouds. Since then, a handful of studies were devoted to the Kelvin–Helmholtz instability at the ionopause and its implications for Venus. The aim of this study is to investigate the stability of the two instability-relevant boundary layers around Venus: the induced magnetopause and the ionopause. We solve the 2D magnetohydrodynamic equations with the total variation diminishing Lax–Friedrichs algorithm and perform simulation runs with different initial conditions representing the situation at the boundary layers around Venus. Our results show that the Kelvin–Helmholtz instability does not seem to be able to reach its nonlinear vortex phase at the ionopause due to the very effective stabilizing effect of a large density jump across this boundary layer. This seems also to be true for the induced magnetopause for low solar activity. During high solar activity, however, there could occur conditions at the induced magnetopause which are in favour of the nonlinear evolution of the instability. For this situation, we estimated roughly a growth rate for planetary oxygen ions of about 7.6 × 10^{25} s^{−1}, which should be regarded as an upper limit for loss due to the Kelvin–Helmholtz instability.

Since the observations of wave-like structures and plasma clouds in the vicinity of Venus by Pioneer Venus Orbiter (PVO), the Kelvin–Helmholtz (KH) instability has been discussed as a loss process for planetary particles (e.g. Amerstorfer et al., 2010; Brace et al., 1982; Elphic and Ershkovich, 1984; Lammer et al., 2006; Wolff et al., 1980). Generally, the KH instability develops at boundaries with velocity shears. Around planets, magnetopauses or ionopauses form such boundaries. Waves of initially small amplitudes grow and eventually reach the nonlinear stage, i.e. vortices are formed, on their way along the boundary from the subsolar point to the terminator. The vortex structures might be able to detach and form so-called plasma clouds, which contain ionospheric particles (Brace et al., 1982), and thus can contribute to the loss of ions. Recently, Pope et al. (2009) reported observations by the Venus Express (VEX) magnetometer VEXMAG indicating vortices of the magnetic field in the magnetosheath above the ionopause that might originate from nonlinear waves at the boundary.

Theoretical studies of the KH instability around unmagnetized planets, such as Venus and Mars, are rather rare. Wolff et al. (1980) and Elphic and Ershkovich (1984) investigated the stability of the ionopause of Venus analytically and came to the conclusion that the boundary can be unstable with regard to the KH instability. Thomas and Winske (1991) used a kinetic code to simulate the instability numerically at Venus. Their results show the detachment of plasma structures. Terada et al. (2002) studied the solar wind interaction with Venus using a global hybrid simulation code. Waves at the ionospheric boundary are also clearly visible in their results. Penz et al. (2004) studied the KH instability analytically for Mars, assuming a tangential discontinuity as a boundary, and compared the ion loss to other loss processes. With a simple estimation they found that the loss rate due to the KH instability is comparable to other non-thermal loss processes. Gunell et al. (2008) analyzed ion and electron ASPERA-3 data from Mars Express and found oscillations in the electron and ion densities as well as in the ion velocities in the magnetosheath at large solar zenith angle (SZA). Comparison with a one-dimensional linear model gave some agreement in the occurring frequencies but was not able to explain the observations satisfactorily. By solving the linear magnetohydrodynamic (MHD) equations numerically, Amerstorfer et al. (2007) investigated the influence of an increasing density, when approaching the planet, on the linear growth rate of the instability. Amerstorfer et al. (2010) studied the nonlinear evolution of the KH instability and vortices. The results of both studies show that the growth rate and instability evolution is rather sensitive to the density jump across the boundary layer. Borisov and Fränz (2011) investigated the excitation of fast magnetosonic perturbations in the vicinity of a magnetopause due to the KH instability. They mention that the observation of fast magnetosonic waves in a high *β* supersonic plasma within a magnetosheath could be an indication of the occurrence of the KH instability at the magnetopause.

Zhang et al. (2008b) studied the induced magnetosphere and its upper boundary, the induced magnetopause, at Venus. They mention that magnetosphere-like and magnetopause-like features seem to occur at every planet, no matter if the planet possesses an intrinsic magnetic field or not. Their Fig. 2 shows a nice sketch of the boundary situation at Venus, where we have the bow shock, followed by the induced magnetopause, and finally, the ionopause, which marks the lower boundary of the induced magnetosphere. The induced magnetopause is a well-defined boundary, characterized by the drop in magnetosheath wave activity. It is interesting that there seems to be no signature in the magnetic field strength across this boundary layer, but rather a gradual increase (Zhang et al., 2008a). A similar boundary can also be found at Mars, where it is called the magnetic pile-up boundary or the magnetospheric boundary (e.g. Dubinin et al., 2006, 2008). The lower boundary of the induced magnetosphere is the ionopause. There are different definitions of the ionopause. It was defined, for example, as the altitude where the plasma density drops below 100 cm^{−3} or where the thermal pressure of the ionosphere first is equal to the magnetic pressure piling up in the magnetic barrier above the ionopause (see Knudsen (1992), and references therein). The average magnetic state of the ionosphere depends upon the solar activity. Whereas the ionosphere is magnetized most of the time during solar minimum, it is most of the time unmagnetized during solar maximum (Zhang et al., 2008a). At solar minimum, both the Venus ionopause and magnetopause altitudes are lower than at solar maximum. Concerning the thickness of the magnetic barrier, Zhang et al. (2008a) found that it stays the same for both solar minimum and maximum activity. The real obstacle to the solar wind seems to be the magnetopause, which is at higher altitudes than the ionopause. The same situation is found at Mars, where the magnetic barrier deflects the magnetosheath solar wind (Dubinin et al., 2006).

In principal, all the aforementioned theoretical investigations give some reasonable arguments why we can assume that the KH instability can occur at some boundary around Venus. The question is, what boundary is unstable – the induced magnetopause or the ionopause. The plasma situation is different at each of these boundaries. On the one hand, from the induced magnetosphere to the ionosphere, the mass density can perform an increase up to even 10,000 times. On the other hand, if the induced magnetopause is thought to be the unstable boundary, then the density increase might even be lower than 100-fold.

This work aims to shed some light on the matter of what boundary is more likely to be unstable to the KH instability and on what boundary vortices, which are thought to be necessary for plasma clouds, can evolve. For this purpose, we study the nonlinear evolution of the 2D KH instability by solving the MHD equations numerically.

The paper is organized as follows. In Section 2 we briefly describe the MHD equations and numerical issues. The initial plasma configurations and input parameters are listed in Section 3. Section 4 deals with the results and the discussion, before we draw our conclusions in Section 5.

To study the development and evolution of the nonlinear Kelvin–Helmholtz instability, we solve numerically the equations of magnetohydrodynamics. A detailed description of the numerical procedure and the algorithm can be found in Amerstorfer et al. (2010). Here, we briefly describe the most important points.

We use the conservative system of the MHD equations

$$\frac{\partial}{\partial t}\rho +\nabla \xb7(\rho \mathbf{v})=0\text{,}$$

(1)

$$\frac{\partial}{\partial t}(\rho \mathbf{v})+\nabla \xb7\left(\rho \mathbf{vv}+\Pi \mathbf{I}-\frac{\mathbf{BB}}{{\mu}_{0}}\right)=0\text{,}$$

(2)

$$\frac{\partial}{\partial t}e+\nabla \xb7\left((e+\Pi )\mathbf{v}-\frac{(\mathbf{B}\xb7\mathbf{v})\mathbf{B}}{{\mu}_{0}}\right)=0\text{,}$$

(3)

$$\frac{\partial}{\partial t}\mathbf{B}+\nabla \xb7(\mathbf{vB}-\mathbf{Bv})=0\text{,}$$

(4)

where *ρ* is the mass density, **v** the plasma velocity, **B** the magnetic field, *e* the total energy density, *Π* the total pressure, *μ*
_{0} the permeability in vacuum and **I** is the unit matrix. The total energy density *e* is given by

$$e=\frac{p}{\kappa -1}+\frac{\rho {\mathbf{v}}^{2}}{2}+\frac{{\mathbf{B}}^{2}}{2{\mu}_{0}}\text{,}$$

(5)

which is the sum of thermal, kinetic and magnetic energies, with *κ* as the ratio of specific heats (in our case, *κ*
= 5/3; the numerical results discussed later do not change significantly for other values, e.g. for *κ*
= 2). The total pressure *Π* is the sum of thermal and magnetic pressures,

$$\Pi =p+\frac{{\mathbf{B}}^{2}}{2{\mu}_{0}}\text{.}$$

(6)

Additionally, we have the divergence-free condition of the magnetic field,

$$\nabla \xb7\mathbf{B}=0\text{.}$$

(7)

For the numerical treatment, we use normalized quantities, as there are

$$\tilde{\rho}=\frac{\rho}{{\rho}_{n}}\text{,}\phantom{\rule{1em}{0ex}}\tilde{\mathbf{v}}=\frac{\mathbf{v}}{{v}_{n}}\text{,}\phantom{\rule{1em}{0ex}}\stackrel{\sim}{\Pi}=\frac{\Pi}{{\rho}_{n}{v}_{n}^{2}}\text{,}\phantom{\rule{1em}{0ex}}\stackrel{\sim}{\mathbf{B}}=\frac{\mathbf{B}}{\sqrt{{\mu}_{0}{\rho}_{n}{v}_{n}^{2}}}\text{,}$$

where subscript *n* denotes the dimensional quantities used for normalization, representing the values near the planetary boundary layer. The spatial scales are normalized with *a*, which is the half width of the layer across which the plasma changes its properties, and the time is normalized with *a*/*v*
_{n}. We will skip the tilde for better readability in the following, and if not otherwise stated, we mean the normalized plasma quantities.

To solve the MHD equations, we use the total variation diminishing Lax–Friedrichs (TVDLF) scheme as described and tested in Tóth and Odstrčil (1996) and as already used for previous studies (Amerstorfer et al., 2010). This TVD scheme has the advantage of solving Riemann problems without the need to know the local characteristic waves. Thus, the implementation is not as complex as for exact or approximate Riemann solvers, and the code can be easily adapted to other systems of equations without having to calculate new specific eigenvectors and eigenvalues.

The TVDLF scheme uses slope limiters to obtain the left and right states of the Riemann problem. To avoid problems due to negative pressures, we do not limit the conserved variables but the primitive ones, i.e. **v**, *ρ*, *p* and **B**. To avoid problems due to negative pressures for the cases of large density variations across the boundary or small initial pressures, we combined two limiters: The magnetic field and the density are limited with the Woodward limiter (Tóth and Odstrčil, 1996), whereas the pressure and velocities are limited with the minmod limiter (see also Balsara (2004) for this concept of using two different limiters). In those cases where we still got problems with negative pressures, we used the minmod limiter only, which is more diffusive than the Woodward limiter, and thus, the growth rates of the instability are a little bit, but not significantly, lower.

As suggested by Tóth and Odstrčil (1996), a Strang-type operator splitting with alternate dimension sweeps is implemented to extend the numerical scheme to two dimensions. The time step Δ*t* obeys the Courant–Friedrichs–Levy condition,

$$\mathrm{\Delta}t=C\mathrm{min}\left[\left(\frac{\mathrm{\Delta}x}{{c}_{x}^{\mathit{max}}}\right)\text{,}\left(\frac{\mathrm{\Delta}y}{{c}_{y}^{\mathit{max}}}\right)\right]$$

(8)

where *C*
= 0.8, Δ*x* and Δ*y* are the space steps in *x*- and *y*-direction, respectively, *c*
^{max} is the maximum propagation speed of information on the mesh in the corresponding direction and is given as the sum of the macroscopic flow velocity and the fast magneto-sonic wave,

$${c}_{d}^{\mathit{max}}=|{v}_{d}|+\frac{1}{\sqrt{2}}\sqrt{\frac{\kappa p+{\mathbf{B}}^{2}}{\rho}+\sqrt{{\left(\frac{\kappa p+{\mathbf{B}}^{2}}{\rho}\right)}^{2}-4\frac{\kappa {\mathit{pB}}_{d}^{2}}{{\rho}^{2}}}}\text{,}$$

(9)

where index *d* stands for either the *x*- or the *y*-component of *v* and *B*, depending on the current dimension sweep. We use periodic boundary conditions in *x*-direction and fixed boundary conditions in *y*-direction. The computational domain extends from 0 to *L*
_{x} in *x*-direction and from −20*a* to 20*a* in *y*-direction.

Around Venus, the boundaries are layers rather than discontinuities. These layers have some finite thickness and represent the transition from one plasma to the other. Within such a layer, the plasma quantities change. For the background parameters we choose tanh-profiles, which represent a smooth approximation to the change of the plasma quantities from one side of the boundary layer to the other. The following initial configuration is assumed,

$$\begin{array}{cc}\hfill & {v}_{x}(y)=0.5{v}_{0}[1+\mathrm{tanh}(y)]+0.5{v}_{1}[1-\mathrm{tanh}(y)]\text{,}\hfill \\ \hfill & \rho (y)=0.5{\rho}_{0}[1+\mathrm{tanh}(y)]+0.5{\rho}_{1}[1-\mathrm{tanh}(y)]\text{,}\hfill \\ \hfill & {B}_{z}(y)=0.5{B}_{0}[1+\mathrm{tanh}(y)]+0.5{B}_{1}[1-\mathrm{tanh}(y)]\text{,}\hfill \end{array}$$

(10)

where *v*
_{0}, *ρ*
_{0} and *B*
_{0} denote the (normalized) velocity, mass density and magnetic field in the upper layer, respectively, and *v*
_{1}, *ρ*
_{1} and *B*
_{1} the (normalized) quantities in the lower layer. We always assume *v*
_{1}
= 0.0, *v*
_{0}
= 1.0 and *ρ*
_{0}
= 1.0. For *ρ*
_{1}, we take values up to 200, which represents the situation of an increasing mass density when approaching the planet from the magnetosheath to the ionosphere. If we assume the ionopause to be the unstable boundary, we assume on the one hand *B*
_{1}
= 0.0, representing the unmagnetized ionosphere at solar maximum, and on the other hand *B*
_{1}
= 0.5, representing the magnetized ionosphere at solar minimum. If we assume the magnetopause to be the unstable boundary, *B*
_{z} actually slightly increases by a factor of 1.1, due to the pile up of magnetic field in the induced magnetosphere (Zhang et al., 2008a).

The initial total pressure *Π*
_{0} is constant. The initial thermal plasma pressure is calculated from

$$p(y)={\Pi}_{0}-\frac{{B}_{z}^{2}(y)}{2}\text{.}$$

Seed perturbations for the KH instability are given through the *y*-component of the velocity,

$${v}_{y}(x\text{,}y)=\delta {v}_{y}\mathrm{sin}\left(\frac{2\pi}{{L}_{x}}x\right){e}^{-{y}^{2}}\text{,}$$

(11)

where *δv*
_{y}
= 0.01 is the amplitude of the initial perturbation, and *L*
_{x} is the length of the computational box in *x*-direction. We assume that the perturbations propagate only in *x*-direction. Thus, the wave number *k*
_{x} is perpendicular to the magnetic field *B*
_{z}.

Regarding the estimation of the total pressure and the magnetic field, we have to take into account that both quantities vary along the surface of the obstacle. The variation of the total pressure can be described by the so-called Newtonian formula (e.g. Petrinec and Russell, 1997; Phillips et al., 1988)

$$\Pi =({\Pi}_{\mathit{sp}}-{\Pi}_{t})\mathrm{cos}{(\theta )}^{2}+{\Pi}_{t}\text{,}$$

(12)

where ${\Pi}_{\mathit{sp}}=K{\rho}_{\infty}{v}_{\infty}^{2}$ is the pressure at the stagnation point, *θ* is the angle between the normal to the obstacle surface and the undisturbed solar wind velocity vector, and *Π*
_{t} is the sum of the thermal and magnetic pressures near the flow terminator (please note that we take the physical unnormalized values for the following derivation). The subscript ∞ denotes the upstream solar wind values, and *K* is an empirical constant, approximately 0.88 for Venus conditions (Zhang et al., 1991). The pressure term *Π*
_{t} is only a fraction of *Π*
_{sp}, *Π*
_{t}
=
*p*
_{t}
*Π*
_{sp}. For *Π*
_{sp} we take the average value of 5.5 nPa (Zhang et al., 1991). Fig. 1
shows the variation of the total pressure as a function of the solar zenith angle (SZA) *θ* along the obstacle surface and for different values of *p*
_{t}.

Using formula (12), we describe the behaviour of the normalization quantities along a streamline in the magnetosheath at the obstacle surface. In order to get the velocity along the obstacle surface, we use the Bernoulli equation for a 2D MHD plasma with the velocity being perpendicular to the magnetic field, which is

$$\frac{{v}^{2}}{2}+\frac{\kappa}{\kappa -1}\frac{p}{\rho}+\frac{{B}^{2}}{{\mu}_{0}\rho}=\frac{\kappa}{\kappa -1}\frac{{p}_{\mathit{sp}}}{{\rho}_{\mathit{sp}}}+\frac{{B}_{\mathit{sp}}^{2}}{{\mu}_{0}{\rho}_{\mathit{sp}}}\text{,}$$

(13)

where the subscript *sp* refers to the values at the subsolar point, *κ* is the adiabatic coefficient, and where we set *v*
_{sp}
= 0.0. To determine the profiles of the mass density, the plasma pressure and the magnetic field, we start with the adiabatic relation between mass density and plasma pressure,

$$\rho ={\rho}_{\mathit{sp}}{\left(\frac{p}{{p}_{\mathit{sp}}}\right)}^{1/\kappa}\text{,}$$

(14)

and with the proportionality relation of the magnetic field and the mass density (valid for 2D),

$$B=\rho \frac{{B}_{\mathit{sp}}}{{\rho}_{\mathit{sp}}}\text{.}$$

(15)

Rewriting Eq. (14) for the pressure and taking the magnetic field from Eq. (15), we get for the total pressure

$$\Pi ={\rho}^{\kappa}\frac{{p}_{\mathit{sp}}}{{\rho}_{\mathit{sp}}^{\kappa}}+{\rho}^{2}\frac{{B}_{\mathit{sp}}^{2}}{2{\mu}_{0}{\rho}_{\mathit{sp}}^{2}}\text{.}$$

(16)

Making the simplifying assumption that *κ*
≈ 2 (which is only roughly true for an adiabatic change of state), we get for the mass density the following approximate expression

$$\rho =\sqrt{\Pi {\left(\frac{{p}_{\mathit{sp}}}{{\rho}_{\mathit{sp}}^{2}}+\frac{{B}_{\mathit{sp}}^{2}}{2{\mu}_{0}{\rho}_{\mathit{sp}}^{2}}\right)}^{-1}}\text{.}$$

(17)

Now, we can also determine the plasma pressure and the magnetic field along the obstacle surface. From Eq. (13) we get the expression for the velocity,

$$v=\sqrt{2\frac{\kappa}{\kappa -1}\left(\frac{{p}_{\mathit{sp}}}{{\rho}_{\mathit{sp}}}-\frac{p}{\rho}\right)+\frac{2}{{\mu}_{0}}\left(\frac{{B}_{\mathit{sp}}^{2}}{{\rho}_{\mathit{sp}}}-\frac{{B}^{2}}{\rho}\right)}\text{.}$$

(18)

We take *B*
_{sp}
= 80 nT (Zhang et al., 2009), *ρ*
_{sp}
= 1 × 10^{−19}
kg m^{−3} (Biernat et al., 2007) (which is 3*ρ*
_{∞}, with *ρ*
_{∞}
= 3.3 × 10^{−20}
kg m^{−3}, determined from *n*
_{∞}
≈ 20 cm^{−3}, see Luhmann et al. (1997)), and ${p}_{\mathit{sp}}={\Pi}_{\mathit{sp}}-{B}_{\mathit{sp}}^{2}/(2{\mu}_{0}$), with *Π*
_{sp}
= 5.5 nPa as an average pressure value (Zhang et al., 1991). The variations of the plasma parameters along the obstacle surface, given by Eqs. (14), (15), (17) and (18), are plotted in Fig. 2
as a function of the solar zenith angle and for different terminator total pressures. We should note here that the results discussed in Section 4 are not very sensitive to the magnetic field value due to the considered configuration.

Mass density, velocity, magnetic field and plasma pressure along the obstacle surface as a function of the solar zenith angle and for different terminator total pressures.

From theory and observations we know that the mass density and velocity approximately reach their undisturbed solar wind values around the terminator. From Fig. 2 we see that this is the case only for *p*
_{t}
= 0.1. Also, the magnetic field values agree with observations for this case (Zhang et al., 1991, 2008a).

With the magnetosheath values for *p*
_{t}
= 0.1, the normalized total pressure and the normalized magnetic field exhibit the behaviour as shown in Fig. 3
. The normalized total pressure and the magnetic field at the subsolar point are infinite due to the fact that the velocity at the boundary, which is the normalization velocity, is zero there. Thus, we start the plot at a distance of 500 km along the surface from the stagnation point, which corresponds to a solar zenith angle of about 5°.

Normalized total pressure and normalized magnetic field as a function of solar zenith angle along the obstacle surface.

In the unnormalized case, the flow velocity increases and the mass density decreases along the streamline from the subsolar point to the terminator. In the normalized case, the velocity and the mass density in the magnetosheath are always 1.0 – the velocity and density variations are inherent in the variations of the normalized total pressure and of the normalized magnetic field.

We can convert the normalized total pressure into the magnetosonic Mach number *M*
_{f} for the magnetosheath flow with the following relation:

$${M}_{f}=\sqrt{\frac{1}{2{\Pi}_{0}}}\text{.}$$

(19)

Miura and Pritchett (1982) have shown that the larger the magnetosonic Mach number, the smaller the growth rate of the KH instability, which actually means the same as the fact that the smaller the total pressure, the smaller the growth rate. The magnetosonic Mach number characterizes the importance of compressibility for the plasma flow. The incompressible case is represented with the limit *M*
_{f}
→ 0.

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 Fig. 5). In this study, we used the time evolution of the vertical kinetic energy ${E}_{y}=0.5\rho {v}_{y}^{2}$ to obtain the growth rates for the different cases discussed below.

Time evolution of the maximum of ln(*E*_{y}) of every time step for *Π*_{0} = *B*_{0} = 1.0 and for different *ρ*_{1}.

For the magnetic field, we take three different profiles, one with *B*
_{1}
= 0.0 and *B*
_{1}
= 0.5 (ionopause cases) and one with *B*
_{1}
= 1.1*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*
_{1}
= 0.0 (see Fig. 7), 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*
_{1}
= 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*
_{1}
= 200 where numerically possible.

Normalized growth rate as a function of normalized wave number for different densities for *Π*_{0} = *B*_{0} = 1.0. The asterisks and the diamonds give the values of the growth rate calculated for *B*_{1} = 1.1 **...**

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 Fig. 4
for *ρ*
_{1}
= 10 (upper plot) and *ρ*
_{1}
= 100 (lower plot). This maximum lies around 40° SZA, where *M*
_{f}
= 0.7. For our calculations, we thus took the case with *Π*
_{0}
=
*B*
_{0}
= 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 *ρ*_{1} = 10 (upper plot) and *ρ*_{1} = 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 ${E}_{y}=0.5\rho {v}_{y}^{2}$ of every time step in a logarithmic scale (Fig. 5
). 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*
_{lin} 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 Fig. 6
. These plots show the normalized mass density at different times during the simulation and they correspond to the case with *ρ*
_{1}
= 10 and with the wave number for which we get the maximum growth rate, *k*
_{x}
= 0.42 (*L*
_{x}
= 15). The plasma flows from left to right. At time *t*
=
*t*
_{lin}(10) = 93.33, a regular-structured vortex has evolved and the saturation sets in – ending in an irregular boundary layer.

Fig. 7
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*
_{1}
=
*ρ*
_{i}/*ρ*
_{n}, is well represented by a logarithmic fit of the form *γ*
_{m}
=
*b*
+
*c*
ln(*ρ*
_{1}), with *b*
= 0.091 and *c*
= −0.014, which is depicted in the upper plot of Fig. 8
. 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.

Normalized maximum growth rate (upper plot) and corresponding normalized linear growth time (lower plot) as a function of the density ratio for *Π*_{0} = 1.0 and *B*_{0} = 1.0. The symbols denote the values obtained numerically, **...**

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 Fig. 2, *ρ*
_{n}
≈ 8 × 10^{−20}
kg m^{−3}, and an oxygen ion mass density of ${\rho}_{i}=20\text{,}000\times {10}^{6}\xb716\xb7{m}_{{H}^{+}}=5.3\times {10}^{-16}\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-3}$, we get for *ρ*
_{1}
=
*ρ*
_{i}/*ρ*
_{n}
= 6600. From Fig. 8 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 Fig. 2, we get a density ratio *ρ*
_{1}
≈ 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*
_{1}
≈ 200–300), such that vortices might be able to be produced by the KH instability.

Taking the case *Π*
_{0}
=
*B*
_{0}
= 1.0 and *ρ*
_{1}
= 200, we see that the normalized linear growth time is ${\tilde{t}}_{\mathit{lin}}=210$. When we assume that the perturbations travel with velocity 0.5*v*
_{n}, they need the following distance *D* along the boundary to develop into vortices,

$$D=\frac{{v}_{n}}{2}{t}_{\mathit{lin}}\text{,}$$

(20)

where we have to keep in mind that here *t*
_{lin} is the physical, unnormalized linear growth time, which is ${t}_{\mathit{lin}}=a/{v}_{n}{\tilde{t}}_{\mathit{lin}}$. Inserting this expression for *t*
_{lin} we get for the distance *D*

$$D=\frac{a}{2}{\tilde{t}}_{\mathit{lin}}\text{.}$$

(21)

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*
_{B}. This length is given by *L*
_{B}
=
*π*/2(*R*
_{pl}
+
*h*
_{B}), where *R*
_{pl} is the radius of the planet and *h*
_{B} is some average boundary layer altitude. With *R*
_{pl}
≈ 6000 km and *h*
_{B}
≈ 1000 km, we get *L*
_{B}
≈ 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*
_{y}
= 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*
_{y}
= 0.03, *dv*
_{y}
= 0.05 and *dv*
_{y}
= 0.1 for the case *Π*
_{0}
=
*B*
_{0}
= 1.0, *ρ*
_{1}
= 200 and *L*
_{x}
= 28. The resulting linear growth times for these runs are ${\tilde{t}}_{\mathit{lin}}=155\text{,}\phantom{\rule{0.35em}{0ex}}{\tilde{t}}_{\mathit{lin}}=116$ and ${\tilde{t}}_{\mathit{lin}}=93$, 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*
_{B}. 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

$${\Gamma}_{\mathit{cloud}}=\frac{{N}_{\mathit{cloud}}}{{t}_{\mathit{lin}}}\text{,}$$

(22)

where *N*
_{cloud} is the number of ions in the cloud produced within time *t*
_{lin} and is given by ${N}_{\mathit{cloud}}\approx 0.5{n}_{i}{L}_{V}^{3}$ (see Amerstorfer et al., 2010), with *n*
_{i} as the ion number density in the lower plasma layer and *L*
_{V} as the spatial scale of the perturbation in all directions. From our simulations we get *L*
_{V}
≈ 10*a*. 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

$$\Gamma =12{\Gamma}_{\mathit{cloud}}\text{.}$$

(23)

Inserting *t*
_{lin}
≈ 100*a*/*v*
_{n}, we obtain

$$\Gamma =60{a}^{2}{n}_{i}{v}_{n}\text{.}$$

(24)

And with *a*
= 100 km, *v*
_{n}
= 210 km/s and *n*
_{i}
≈ 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 × 10^{25}
s^{−1}. 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.

The KH instability has been discussed as a loss process for planetary ions by various authors (e.g. Amerstorfer et al., 2010; Brace et al., 1982; Lammer et al., 2006). The idea is that waves on the dayside of a planetary boundary evolve into vortices around the terminator, where the vortices detach and form plasma structures, called plasma clouds. These plasma clouds contain ionospheric particles, which are therefore lost to the solar wind.

We find that the KH instability is not able to reach the nonlinear phase on the ionopause as well as on the magnetopause during solar minimum, due to low boundary layer altitudes and corresponding high density jumps. For solar maximum conditions, the ionopause seems still to be stable. The induced magnetopause, however, might become unstable with regard to the KH instability, especially for higher SZA. We estimated some maximum loss rate of oxygen ions to be about 7.6 × 10^{25}
s^{−1}, which is lower than that estimated from observations (Brace et al., 1982) and in a previous study (Amerstorfer et al., 2010). This loss rate has to be taken with care and has to be regarded as an upper limit of loss by the KH instability since the conditions used for our simulation and for the estimation are for sure not fulfilled all the time.

This work is supported by the Austrian Science Fund Project P21051-N16 and also by RFBR Grant No. 09-05-91000-ANF_a. H.L. and H.G. are supported by the Helmholtz Association through the research alliance “Planetary Evolution and Life” and by the Austrian Science Fund Project I199-N16. M.Z. and D.K. are supported by the Austrian Science Fund Project I193-N16.

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