Numerical Modeling of Space Plasma Flows: ASTRONUM-2009
ASP Conference Series, Vol. 429, 2010
Nikolai V. Pogorelov, Edouard Audit, and Gary P. Zank, eds.
Grid Convergence of the HYB-Venus Hybrid Simulation
R. Jarvinen, E. Kallio, P. Janhunen and V. Pohjola
Finnish Meteorological Institute, Helsinki, Finland
I. Sillanpää
Southwest Research Institute, San Antonio, Texas
Abstract.
We study the spatial grid convergence of a Venus-solar wind interaction simulation. HYB-Venus, a 3-D hybrid simulation, models the interaction
in global scale including the ion kinetic effects. The selection of the spatial
resolution is an important factor and can affect the physics in the simulation.
Here we show that solutions of the nominal Venus run with cell sizes RV /10,
RV /15 and RV /20 (RV is the planet’s radius) share similar general characteristics. Especially, the planetary oxygen ion escape rate is not sensitive to the
cell size. More quantitatively, the higher resolution, which also means smaller
particle noise per physical volume unit, was found to introduce finer structures
which could be important and useful in detailed studies of the properties of the
Venusian plasma environment.
1.
Introduction
Venus is a non-magnetized terrestrial planet and its interaction with the solar
wind has been studied from the seventies (see, for example, Russell et al. 2006,
and references therein). The planet has an induced magnetosphere caused by
the highly conducting ionosphere which shields the atmosphere and the surface
from the magnetized super-magnetosonic solar wind flow. The structure of the
Venusian induced magnetosphere resembles somewhat the terrestrial counterpart: Venus has a low density magnetotail with magnetic lobes separated by a
central (or cross) tail current sheet (CTCS), a magnetosheath and a bow shock
(see Saunders & Russell 1986). Unlike in the Earth’s dipolar plasma interaction, the solar wind flows close to the Venus’ upper atmosphere. As a result,
the atmospheric ions are accelerated and escape from the planet, hydrogen and
oxygen being the most important species (see Barabash et al. 2007).
Global hybrid simulations are popular and efficient tools in the studies of
plasma interactions of unmagnetized or weakly magnetized planetary bodies
(see, for example, Ledvina et al. 2008, and references therein). The hybrid
approach treats ions as particles taking account the kinetic effects, and electrons
as a charge-neutralizing fluid. This is practical when the planet’s radius is close
to the ion gyroradii in the system.
Here we study how the nominal Venus solution from our HYB hybrid code
changes when we increase the spatial resolution from a typical run with ∆x =
RV /10 (605 km) to RV /15 (403 km) and to RV /20 (303 km), where RV is the
planet’s radius. The grid convergence is an important factor since the spatial
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gridding is an artificial feature compared to reality. The study is organized as
follows. First, we introduce the HYB code. Second, we present the results from
the simulation runs. Third, we discuss the results and summarize our findings.
2.
The HYB code
The HYB hybrid code has been used to study several objects (HYB-Mars (Kallio
& Janhunen 2002), HYB-Mercury (Kallio & Janhunen 2003), HYB-Titan (Sillanpää et al. 2007), HYB-Venus (Jarvinen et al. 2008), HYB-Moon (Kallio et
al. 2008)). The code solves the Lorentz force for the ions, the momentum equation for the electron fluid and the non-radiative Maxwell’s equations for the
electromagnetism in 3-D space. The electrons are assumed to be a massless,
charge-neutralizing and isothermal fluid. The leapfrog integration is used to
propagate ions and magnetic field in timesteps. The spatial gridding with cubic
cells is used for the field quantities. In this study we do not use the grid refinement feature of the code since the Venus simulations are usually done with a
constant sized grid. Table 1 lists the basic parameters of the simulation runs.
The resolution parameters of the runs are listed in Table 2. See Kallio & Janhunen (2003) for details of numerics. The HYB-Venus version is discussed in
Kallio et al. (2006). The simulation coordinate system is defined as follows: the
negative x-axis is along the incident solar wind flow, the IMF perpendicular
component to the flow is in the positive y-direction and the z-axis completes the
right-handed system. The Venus radius (RV ) is 6051.8 km.
Table 1.
Details of the HYB-Venus simulation.
Parameter
Value
Box size (x × y × z) [RV ]
∆x/∆t
Avg. macroparticles per cell
Solution snapshot time
Obstacle radius
Particle absorption radius
IMF [Bx , By , Bz ]
Solar wind H+ (V, n, T )
Electron temperature
Resistivity ηa
Total planetary H+ prod. rate
Total planetary O+ prod. rate
Solar EUV photo rates
2.1.
(−3...3) × (−4...4) × (−4...4)
15130 km/s
25
300 s
RV + 300 km
RV + 200 km
[-8.09, 5.88, 0] nT
430 km s−1 , 14 cm−3 , 105 K
104 K
constant for r > RV , zero r < RV
6.42 × 1024 s−1
1.409 × 1025 s−1
solar minimum
The Hall term
In the hybrid model the electric field comes usually from the equation (ignoring
the electron pressure and resistive terms)
~ = −U
~ e × B,
~
E
(1)
HYB-Venus Hybrid Simulation
195
where the electron velocity is defined as
~e =
U
J~
−
qe n e
P
~
i qi n i Vi
qe n e
.
(2)
From these and the Amperè’s law we get
P
~i × B
~
~ ×B
~
qi n i V
(∇
×
B)
~ = −
+ i
.
E
µ0 qe ne
qe n e
{z
} |
{z
}
|
~ Hall
E
(3)
~ ion
E
The relative magnitude of the Hall term is
EHall
Eion
∼
B2
qe n e
vA λ i
×
=
,
Lµ0 qe ne qi ni Vi B
Vi L
(4)
where vA is the Alfvén velocity, Vi the ion velocity, λi the ion inertial length and
L the length scale of the magnetic field. Minimum L is limited by the spatial
grid resolution of the simulation. When the cell size is about the ion inertial
length, the system can produce Hall terms comparable to the ion (convection)
term. Typically, L ≈ ∆x at sharp boundaries whereas elsewhere in the system
L is not limited by ∆x. This raises a question does the global solution depend
on the fine scales of the Hall term?
Table 2.
3.
Resolution parameters of the HYB-Venus simulation runs.
Parameter
Run 1
Low
Run 2
Nominal
Run 3
Higher
Run 4
Highest
Grid cell size [RV ] [km]
Grid cells (nx × ny × nz )
Timestep [ms]
1/5 (1210)
30 × 40 × 40
80
1/10 (605)
60 × 80 × 80
40
1/15 (403)
90 × 120 × 120
27
1/20 (303)
120 × 160 × 160
20
Results
Figures 1, 2 and 3 give a comparison of the qualitative properties of the simulation runs with different spatial resolutions. We show only the y = 0 planes but
the drawn results are valid for the global 3-D solutions in general. The solar
wind H+ density is shown in Figure 1, the planetary O+ in Figure 2 and the
magnetic field in Figure 3.
The basic structure of the solar wind H+ density is similar in the runs. The
densities are highest at the bow shock nose (max. ∼ 50 cm−3 ) and lowest in
the magnetotail (≤∼ 0.5 cm−3 ), in the magnetosheath the density is about 5
cm−3 . The proton dropout boundary (PDB) between the magnetosheath and
the wake of the planet (see the light blue and the dark blue colors in Figure 1)
is better developed in the highest resolution run when compared to the nominal
run. The location of the PDB does not vary much between the runs.
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Jarvinen et al.
Figure 1.
Number density [m−3 ] of the solar wind H+ at the y = 0 plane
in three runs with different resolutions. The color scale is linear. The gray
spherical surface represents Venus and the arrows show the direction of the
solar wind velocity and electric field. The cell sizes are (from the right):
RV /10, RV /15 and RV /20.
Figure 2.
Number density [m−3 ] of the planetary O+ at the y = 0 plane
in three runs with different resolutions. The color scale is logarithmic. The
figure is in the format as Figure 1.
The oxygen densities show very consistent features in the runs: the asym~SW × B
~ IMF k +z)
metry in the direction of the convection electric field (= −V
+
and the three tail rays (”fingers” or extensions of the ionospheric O densities,
see the red color in Figure 2). The high resolution runs show more planetary
O+ in the upstream of the bow shock than the nominal and low resolution runs
which is a result from the better particle statistics. Low altitudes (< 2000 km)
are more structured with higher resolution because the ionospheric boundary is
formed in the first grid cell above the obstacle boundary. Further, the energization of the pickup O+ particles in the upper (+z) half of the simulation box,
~ ×B
~ drift caused by the solar wind, is alike in all the
when they undergo the E
runs (figure not shown).
HYB-Venus Hybrid Simulation
197
Figure 3.
Magnetic field strength [T] at the y = 0 plane in three runs with
different resolutions. The color scale is linear. The figure is in the format as
Figure 1.
The bow shock gets sharper and moves closer to the planet when the resolution is increased which is seen in the magnetic field and H+ properties (see
the lower right corner of the panels in Figure 3 and 1). The change in the
shock-terminator distance is of the order of 0.1RV . Similarly, the CTCS shows
a sharper gradient and a small displacement in the high resolution runs. The
particle noise makes the shock more blurry when the resolution decreases, which
is seen especially in the +z hemisphere in the H+ density. Lastly, the magnetic
barrier shows more extended structure towards the +z hemisphere with higher
resolution (see the red color in Figure 3).
3.1.
Escape rate
Figure 4 shows the time evolution of the planetary O+ escape rate from the
simulation box. In addition to the three runs in the previous figures also a low
resolution run with ∆x = RV /5 (1210 km) is plotted.
The escape enters a quasi-stationary stage of development at about t = 200 s
(after the initial transient). The increased resolution causes the runs to stabilize
somewhat slower which is seen as a shift of the initial peak in the escape rate.
After that the rates change in a longer time scale and the absolute values do not
vary much. Our test runs suggest that the value at about t = 300...400 s is close
to the stationary solution. The biggest difference between the runs is caused by
the particle noise, which creates the strongest fluctuations in the low resolution
runs. However, the average value of about 3 × 1024 s−1 is the same for all the
runs. The average energy of the escaping planetary O+ ions is about 12 keV in
all the runs.
3.2.
The Hall term
Figure 5 illustrates how the magnitude of the Hall term of the electric field (see
Equation 3) depends on the resolution. The maximum strength and the fine
scales of Hall term at the bow shock and CTCS increase in concert with the
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Jarvinen et al.
Figure 4.
Total escape rate of the Venusian planetary O+ in runs with four
different resolutions.
cell size. Notably, the wake region displays more structured features with the
highest resolution.
4.
Discussion
Qualitatively, the structure of the Venusian induced magnetosphere in the HYBVenus simulation does not seem to depend drastically on the spatial resolution
in the analyzed range from ∆x = RV /10 to RV /20. The smaller grid size allows
sharper gradients which introduce finer structures in the system, especially at the
boundaries, but this does not change drastically the big picture. Examples of the
boundaries discussed in this study are the PDB, the bow shock and the CTCS.
All of these are located realistically and are in agreement with the observations
(comparisons with the data not shown). However, the higher resolution creates
sharper density and field gradients at the PDB and CTCS and, for example,
higher density jump at the bow shock. In a more quantitative inspection it
is found that the bow shock is pushed somewhat closer to the planet and the
escape of the planetary ions stabilize slower with increasing resolution.
The grid size has an indirect effect on the particle noise since in the simulation the average amount of macroparticles per cell is constant. There are more
macroparticles per physical volume unit in the high resolution runs. This means
that the velocity space in a physical volume unit is sampled more accurately,
which can be seen, for example, as more abundant planetary particles in the
upstream of the bow shock.
The total escape rate of the planetary ions is very immune to the exact
resolution in the analyzed runs. Only the particle noise is seen as larger statistical fluctuations in the nominal and lower resolution escape rates. Perhaps since
we do not model the self-consistent ionospheric physics but use a fixed emission
HYB-Venus Hybrid Simulation
199
Figure 5.
Magnitude of the electric field Hall term [Vm−1 ] (see Equation 3)
in three runs with different resolutions. The color scale is linear. The figure
is in the format as Figure 1. For a comparison, the magnitude of the solar
wind convection electric field is 2.5 × 10−3 Vm−1 .
profile, the better resolution at low altitudes does not change the escape rates.
With this method it is important to use a realistic emission rate (see Jarvinen
et al. 2009).
The Hall term of the electric field shows a dependence on the resolution
in the studied runs. The qualitative picture is similar in the low and high
resolution runs but the finer scales change. Since the term is only a small
fraction of the total electric field (cf. Figure 5) one should study more carefully
its importance, for example, to the planetary ion escape. Even if the fraction is
small the morphology of the Hall term in global scale may allow a considerable
energization of the ions or occurrence of plasma instabilities.
5.
Summary
We presented a study of a grid convergence of the HYB-Venus Venus-solar wind
interaction hybrid simulation using grid cell sizes from RV /10 to RV /20. While
the higher resolution was found to introduce finer structures, which could be
important and useful in detailed studies of the properties of the Venusian plasma
environment, the high resolution solutions were found to share the same general
characteristics than the nominal run.
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