Introduction
The solar wind plasma density, velocity and temperature as well as the
orientation of its embedded interplanetary magnetic field (IMF) are
fundamental in determining the efficiency of solar wind–magnetosphere–ionosphere coupling processes. The plasma mass and momentum transport into
the magnetosphere are driven by local plasma processes, which have complex
dependencies on the large-scale configuration. Magnetic reconnection
close to the sub-solar point accounts for majority of
plasma transport during southward IMF, while viscous processes
such as the Kelvin–Helmholtz instability
or diffusion may play a strong role,
particularly when the IMF is directed northward .
Quantification of the relative importance of these processes under varying
solar wind conditions is a central element to developing capabilities to
predict space weather in the geospace.
Several overlapping and multi-spacecraft missions, such as Cluster
and MMS , are operational today,
offering statistically global coverage via their conjunction. However, it is
not feasible to use these for real-time space weather prediction. Instead,
global magnetohydrodynamic (MHD) models are often used to provide a
large-scale view of the dynamical evolution in real-time. The best known
models include the Lyon–Fedder–Mobarry (LFM) , BATS-R-US
, Open General Geospace Circulation Model (OpenGGCM)
, Ogino Model and the Grand Unified
Magnetosphere-Ionosphere Coupling Simulation (GUMICS) .
Global MHD simulations typically start with a dipole in an almost empty
space. In the initialisation process, the solar wind driver is turned on to
create the upstream bow shock and magnetopause structure as well as the
downstream magnetotail and plasma sheet configuration. Typically, constant or
slowly varying parameter values are used during the initialisation period
, and its duration is dictated by the
time needed (of the order of 1 h) to form the magnetosphere
and reset the magnetospheric memory (a few
hours) of past dynamical processes pp. 212–246.
While the different MHD simulations are based on the same plasma theory, the
exact form of the equations, the numerical solutions, and the initial and
boundary conditions as well as the ionospheric coupling solutions vary.
Several studies have focused on assessing the performance of the models
through comparisons of the simulation results with in situ or remote
observations of dynamic events or plasma processes . Recent comprehensive studies
statistically quantify the strengths and
weaknesses of each of the models, while they could not identify any of the
codes as clearly superior to the others.
As a part of validation process the simulation results are often compared
with point measurements made by satellites. As the magnetosphere includes
regions with large spatial and temporal gradients, even small errors in the
simulation configuration can cause large differences with respect to the
observations locally at a single point. In such a case, it is difficult to
interpret how well the simulation reproduces the large-scale dynamic
sequence. Moreover, local and large-scale effects of initialisation process
are not known. In order to quantify variations that may arise from the
initialisation procedure (history of the simulation), we investigate the
effect of artificial initialisation period containing stepwise, linearly or
sinusoidally changing IMF on a GUMICS-4 global MHD simulation run over a 9 h
period following the initialisation. The quantitative differences are
examined both locally considering difference in convergence of several
variables at grid points located in different parts of the magnetosphere and
globally studying differences in momentum transport over the magnetosphere.
This paper is constructed in the following way. Section describes
the features of GUMICS-4, and Sect. characterises the
chosen interval and discusses the ability of GUMICS-4 to reproduce the
large-scale magnetospheric boundary structure. Section
illustrates the initialisation methods. The model solutions and their
differences obtained from the multiple initialisation procedures are
presented in
Sect. –. The
paper ends with discussion and conclusions.
The GUMICS-4 global MHD model
The fourth edition of the Grand Unified Magnetosphere-Ionosphere Coupling
Simulation (GUMICS-4) is a global 3-D MHD simulation code for the
magnetosphere and the solar wind coupled to an electrostatic spherical
ionosphere region . The MHD solver utilises the finite-volume method. The simulation box has dimensions of +32 to
-224 RE in the xGSE direction, and -64 to
+64 RE in the yGSE and zGSE
directions. The inner boundary is spherical with a radius of
3.7 RE, with a gap separating the magnetospheric and ionospheric
simulation regions. The magnetosphere provides the ionosphere with the
field-aligned current pattern and the electron precipitation, while the
ionosphere gives the electric potential to the magnetosphere. This feedback
loop is updated every 4 s.
GUMICS-4 solves the ideal MHD equations with the separation of the magnetic
field to a curl-free (dipole) component and perturbed component created by
currents external to the Earth (B=B0+B1(t);
). Two important features of GUMICS-4 are its adaptive
grid and temporal subcycling. Both make the computations feasible on a single
PC. An adaptive grid enables the coupling of a high-resolution ionospheric
grid with coarse magnetospheric grid to execute computations in a reasonable
time. To limit the number of grid cells in the MHD region, the grid cell size
is varied depending on the local gradients (the plasma mass density, the
total energy density without the contribution of the dipole field, the
momentum density and the perturbed component of the magnetic field). Temporal
subcycling reduces number of MHD solutions an order of magnitude while
guaranteeing that the local Courant–Friedrichs–Lewy (CFL) constraint
pp. 121–151 is satisfied.
The input to the GUMICS-4 are solar wind plasma number density, temperature,
velocity and the IMF conditions at the upstream boundary X=32 RE. The smallest grid cell size in this study was
0.5 RE. While it would be desirable to use the maximum
resolution of 0.25 RE, due to the long simulation physical time
(10 days), this was not feasible, nor was it necessary for the purpose of
this study. Geocentric solar ecliptic (GSE) coordinates are used throughout
this paper.
Model inputs
We use solar wind parameters from the OMNI database at 1 min resolution as
inputs to GUMICS-4. OMNI data are a composite of solar wind measurements
typically measured at the Lagrangian point L1 and then subsequently
propagated to the bow shock nose using the
bow shock model. OMNI data were obtained through the NASA OMNIWeb service
(http://omniweb.gsfc.nasa.gov).
Simulation of a high-speed stream event
Event description
Continuous solar wind measurements are a necessary precondition for obtaining
a reliable simulation result of the magnetospheric dynamics. While such
periods are surprisingly uncommon, the period from 12 June, 04:00 UT, to
22 June, 04:00 UT, 2007 contains a mostly uninterrupted solar wind record
(from the Wind satellite during 12 June, 04:00 UT–13 June, 20:59 UT, 2007,
and from the ACE spacecraft thereafter) in the OMNI database with only short
data gaps. These gaps make up 14 % of the period. We have simulated the
full 10-day period that comprises a high-speed solar wind stream interval,
and we use the simulation results as a reference for further numerical
experiments.
Figure shows the IMF
BY and BZ components, solar wind speed, and density. A high-speed
stream was initiated during 14 June 2007, as indicated by the enhanced speed
(∼ 650 km s-1) around 00:00 UT on 15 June 2007. The leading
edge was associated with a BY reversal from slightly negative to positive
and rapidly fluctuating IMF BZ. The speed gradually decreased toward the
end of the interval from the peak value to about 400 km s-1. During
the high-speed stream, the density was low at a few particles per cubic
centimetre. The IMF BY was positive at a level of a few nT, and the
IMF BZ fluctuated around zero; 21 June features rapid density increase
with IMF BY and IMF BZ fluctuating between -10 and 10 nT.
We focus our attention particularly to the period when the solar wind speed
increases during 14 June, 12:00–24:00 UT, to cover both negative and positive
IMF BZ and a range of solar wind speed values.
Solar wind and IMF conditions during a 10-day period from 12 to
22 June 2007. Panels from top to bottom: (a) IMF BY and
(b) BZ in nT, solar wind speed (c) V in
km s-1, and density (d) n in cm-3. Vertical red lines
indicate 9 h data interval which was combined with initialisation periods.
Non-labelled panels at the top and bottom indicate data gaps longer than or
equal to 10 min.
In the magnetosphere, the high-speed stream did not cause major activity as
the IMF BZ did not include periods of intense southward fields after the
initial sheath region on 13 June (not shown). The Dst index remained above
-20 nT throughout the period. The sheath region leading to the
high-speed stream included a period of AE activity that exceeded
500 nT, but during the high-speed stream the AE activity was quite
low, with only moderate substorms until the end of the stream, which again
included a period of higher AE activity during 21 June 2007. Thus, from the
magnetospheric activity point of view, the period can be characterised as
driven by a fast solar wind stream, rather fluctuating IMF driver, which
resulted in only moderate magnetospheric and ionospheric responses.
The 1 min OMNI data were interpolated across short data gaps to obtain a
continuous simulation input. One hour of fixed solar wind values was
included at the start of the interval (12 June 2007 03:00–04:00 UT). In
order to preserve ∇⋅B=0, the IMF BX was set to
zero.
Bow shock and magnetopause locations
The location of the bow shock upstream of the magnetosphere is examined
utilising the Rankine–Hugoniot conditions . Assuming that
the plasma upstream and downstream of the bow shock is spatially uniform, we find that the
shock is time-stationary and planar, and its front is aligned with the
YZ plane; the shock can be located by examining the solar wind velocity
and magnetic field in the XY plane.
Rankine–Hugoniot relations make the assumption that the shock is a thin
layer, as it is assumed that dissipation takes place at collisional kinetic
scales below the fluid representation of the plasma. The plasma fluxes (mass,
momentum, energy) are conserved, resulting in jumps of the moments across the
shock transition layer. However, the layer can only be located up to the
smallest scale in the GUMICS-4 simulation grid, which in this study is
0.5 RE in the vicinity of the shock. Thus, the location of the
shock layer can be determined to an accuracy of ±0.5 RE. To
trace the changes, we cover sufficiently large distance away from the shock
region upstream and to beyond the shock front in the downstream direction
ending before reaching the magnetopause.
We concentrate on locating the shock distance at the Sun–Earth line from the
Rankine–Hugoniot relations. By letting both the upstream and downstream
plasma flows to be perpendicular to the magnetic field the MHD equations in the
rest frame of the shock yield a perpendicular shock:
r=BdBu=ρdρu=VuVd,
where B is the magnetic field, ρ is the plasma density, and V is the
plasma speed. Subscripts u and d refer to regions upstream and downstream of
the shock, respectively, and the ratio r is given by
r=(γ+1)Mu22+(γ-1)Mu2,Mu=Vuγ(pu/ρu)1/2,
where p is the plasma pressure, γ=5/3 is the ratio of the specific
heats, and Mu the upstream sonic Mach number.
Equation () provides three parameters (magnetic field,
density and flow velocity) for identifying the bow shock stand-off distance.
To compute the shock position from the simulation sufficiently far from the
Earth averaged over several grid points, we first evaluate the sonic Mach
number (Eq. ) by using values of V, P and ρ
averaged over a 4 RE distance upstream of the shock between X=24 and 28 RE, which gives us the ratio r relating the upstream
and downstream values in Eq. (). For example, plasma flow
speed of 493 km s-1 results in r≈3.92, thus suggesting
Bd=3.92Bu, ρd=3.92ρu and
Vd=Vu/3.92. The values of B, ρ and V are
evaluated along the Sun–Earth line towards the Earth until they reach the
predicted downstream value; the bow shock position is defined as the first
grid point that satisfies Eq. () for each parameter
individually providing three estimates of the shock position. Note that this
method identifies the inner edge of the shock, as we trace the first point
inside of the shock layer. Due to 0.5 RE maximum resolution the
accuracy of each shock position estimate is less than or equal to
0.5 RE. This procedure is repeated at 1 min temporal
resolution throughout the simulation run, but for slight smoothing we use
10 min sliding averages both for the simulation results and the OMNI data.
The standard deviation of each 10 min ensemble is used as an uncertainty
estimate.
To provide a reference to the three shock position estimates and to be
consistent with the used OMNI solar wind data, the estimates are compared
with empirical values from the OMNI database derived from the
bow shock model. A comparison of the shock location
between OMNI and GUMICS-4 is shown in panel (a) of
Fig. together with the relative difference between
the shock position obtained using the three variables B, ρ and V
(panel b). The standard deviations over 10 min intervals used as a proxy for
uncertainty are shown by vertical bars. The relative difference in the bow
shock standoff distance between GUMICS-4 and OMNI is mostly positive with
88 % of the values in the range of 0–20 % for B and over 99 %
for both ρ and V, thus suggesting that using ρ and V results as
lower difference compared to OMNI. Panel (b) in Fig. shows that the blue plot (V) is consistently closer to zero line than
green (ρ) and especially magenta (B), which is confirmed by calculated
average relative differences over the 10-day period that are 10.8, 10.7 and
6.7 % for B, ρ and V, respectively. Regardless of the chosen
variable, the applied method consistently predicts the bow shock position
closer to the Earth by up to several RE than that obtained from
OMNI. The general trend is expected given the 0.5 RE spatial
resolution, which leads to a wider making MHD shock in the simulation than in
reality. Furthermore, global MHD models generally underestimate the ring
current in the inner magnetosphere, which positions the magnetopause and
hence the shock closer to the Earth , especially during
times of strong geomagnetic activity. However, GUMICS-4 predicts the temporal
evolution of the bow shock location quite accurately over timescales of tens
of minutes. This holds even when abrupt changes in solar wind conditions take
place (see number density values on 13, 14 and 21 June 2007 in
Fig. ) as such effects are transient in nature.
Bow shock and magnetopause positions: (a) bow shock standoff
distance in RE from the OMNI data (black) and GUMICS-4 (coloured
lines), (b) relative difference between the shock position obtained
using GUMICS-4 and OMNI, (c) magnetopause standoff distance in
RE from the Shue model (black) and GUMICS-4 (magenta), and
(d) relative difference between the GUMICS-4 magnetopause position
and the Shue model. The relative differences are computed from 100⋅(reference-GUMICS)/reference. Vertical red lines
indicate the 9 h data interval which was combined with initialisation periods.
The magnetopause subsolar point position given by GUMICS-4 is compared to the
empirical model, which along the Sun–Earth line is reduced
to the form
r0=(a+bBZ)pdyn-1/6.6,
where r0 is the magnetopause standoff distance at the subsolar point,
pdyn is the solar wind dynamic pressure in nPa, a=11.4,
b=0.013 for BZ≥0 and b=0.14 for BZ<0. The position of
the subsolar magnetopause in GUMICS-4 is computed using the current density
JY component. Starting from the bow shock position obtained from
Eq. () and ending at X=5 RE, we identify
the maximum value of JY as the magnetopause position. Ten-minute
averages similar to the bow shock location determination are used both in
evaluating the Shue model and the GUMICS-4 magnetopause position.
Figure shows the comparison of the magnetopause
subsolar point location (panel c) and the relative difference between the
position obtained using GUMICS-4 JY and the Shue model (panel d). The
general trend is similar to the bow shock position: GUMICS-4 magnetopause
position is closer to the Earth than the predicted by the Shue model, in good
agreement with previous studies , and results mostly from
the ring current underestimation. However, the relative difference is smaller
and fluctuates between -10 and 20 % with 98.5 % of the data in
the range of 0–10 % and 77.4 % of the data in the range of
0–5 % during the 10-day period. Average relative difference over the
10-day period is 3.4 %. The relative difference is more enhanced when the
number density and the solar wind speed (Fig. )
increases and thus dynamic pressure increases and IMF BZ fluctuates with
an amplitude up to 10 nT (14 and 21 June 2007).
GUMICS-4 simulation initialisation
Initialisation sequences
In order to study the effects of simulation initialisation, we ran a set of
five simulations during the 10-day period, starting at 12:00 UT on
14 June 2007. Input solar wind data in each simulation consists of 9 h
identical subset of the 10-day run input data section (15:00–24:00 UT) preceded by varying
3 h synthetic initialisation period (12:00–15:00 UT). The 3 h length was
chosen to distinctly exceed 1 h time needed for the magnetosphere to form
. To quantify the effect of the different initialisation
procedures in GUMICS-4, we compare the model solutions obtained from each
initialisation procedure by examining time series of several variables in
different points across the magnetosphere (Sect. )
and surface plots of the magnitude of the relative difference in ρVX
variable between the 10-day reference run and the runs using different
initialisations (Sect. ).
Five different initialisations for the simulation. Panels from top
to bottom: IMF (a) BY and (b) BZ in nT, and solar wind
velocity (c) VX in km s-1 and (d) density in
cm-3 at x=32 RE. The initialisations are shown with
different colours (st1 in blue, st2 in blue with symbols, lin1 in magenta,
lin2 in magenta with symbols and sin in green, 10-day (10d) run in black). The
10-day run data are from OMNI. The yellow background highlights the 1 h interval during
which the magnetosphere should be formed .
Magnetospheric configuration and the Earth's dipole field in
14 June 2007 at 16:30 (a) and 19:40 (b). Both the Earth's
magnetic field and the IMF field as well as ρVx flow lines are
showing. White dots show the time series points in the XZ plane. Labels (a–i)
refer to Figs. , , and
. Note that the figures are cut planes of 3-D plots.
We performed five simulations with constant solar wind parameters (n, T,
VX, VY, VZ) but variable IMF during the initialisation
periods. The fixed values for density and velocity are chosen to be typical
values in the solar wind, while the value for temperature is an average over
the 10-day period. Two simulations employ a stepwise-varying BZ with
constant BY, two linearly varying BZ with constant BY, and one
simulation uses a sinusoidally varying BZ together with a sinusoidally
varying but anti-phase (90∘ phase difference) BY. The chosen
initialisation methods both utilise simple functions to determine the time
evolution of IMF BZ and are used in previous studies (see e.g.
). As the IMF components do not change
spatially at the solar wind inflow boundary (XZ plane, X=32 RE), the magnetic field BX component is fixed (zero in
the present study) to preserve the ∇⋅B=0 condition at
the boundary. The IMF BY component is fixed to 2 nT for the four
first simulations. The IMF BZ component is varied between -5 and
5 nT, values typically observed in the solar wind
. The first stepwise-varying initialisation starts from
the BZ maximum, while the second one uses the minimum as a starting
point. The same applies to the two linearly varying initialisations.
Sinusoidally varying BZ is initially at 5 nT, combined with
sinusoidally varying BY starting from 0 nT.
Figure illustrates the initialisation conditions covering
the 3 h initialisation period and the first hour with the observed
solar wind data. From this point, we refer to the different initialisations
as st1 (stepwise varying starting from +5 nT), st2 (stepwise
varying starting from -5 nT), lin1 (linearly varying starting from
+5 nT), lin2 (linearly varying starting from -5 nT), and
sin (sinusoidally varying starting from +5 nT).
Tables and summarise the
numerical values of the solar wind and IMF parameters.
Fixed solar wind parameters with the associated numerical values.
Note that BY was not fixed in sinusoidally varying BZ method.
Fixed parameters
Numerical value
n
2 cm-3
T
9475 K
Vx
-450 kms-1
Vy, Vz, Bx
0
By
2 nT
The first 3 h of the GUMICS-4 simulation were run using the initialisations
described above. Following the 3 h period, the simulation was run
using the observed solar wind parameters similarly to the 10-day run. This
produces five 9 h simulation periods that we compare with the 10-day
simulation that starts from 04:00 UT on 12 June 2007. The period was chosen
to comprise abrupt changes in the solar wind parameters that result in
dynamic changes in the magnetosphere (Fig. ); not only
do the polarities of BY and BZ change multiple times but also the
plasma speed undergoes a rapid change from 300 to 600 km s-1.
Magnetotail plasma parameters in the lobe (a, c) and plasma
sheet (b, d) at X=-8 RE Y=0 RE
Z=Zsheet (a, b) and X=-15 RE (c, d).
Zsheet refers to varying Z coordinate in the plasma sheet. In
each panel, subpanels from top to bottom: plasma flow speed in km s-1,
absolute difference in plasma velocity VX in km s-1, density in
cm-3 and BY in nT, all in GSE coordinates. Absolute difference is
calculated as 10d - x, where x = st1, st2, lin1, lin2, sin.
Differences in lobe and plasma sheet dynamics
In order to quantify the simulation differences, we have selected points in
the plasma sheet and in the lobe at four locations along the magnetotail and
one point at the dayside magnetosheath. By using these points we cover three
important regions in the magnetosphere. Choosing identical X coordinate
values for both the plasma sheet and the lobe points allows us to compare the
conditions in the two regions approximately equal distance away from the
Earth in the X direction. On the other hand, selecting a point in the
magnetosheath several Earth radii away from the Sun–Earth line can actually
lead to studying a point that is constantly moving between the solar wind and
the magnetosphere and thus contribute to revealing how GUMICS-4 reproduces
the bow shock and the magnetopause positions using a different approach than
what was described in Sect. . At these locations,
we produce absolute difference time series of several parameters. For the
plasma sheet and the lobe, these parameters are VX, n and BY,
while in the magnetosheath we examine time series V, B, plasma beta
β, and EY. The selected points are shown in
Fig. .
Figure shows the absolute difference time series in
the tail at X = -8 and -15 RE (panels a2, a3, a4, b2,
b3, b4, c2, c3, c4, d2, d3, d4). Panels (a1, b1, c1, d1) illustrate plasma
flow speed at the selected points. Time series plots at X = -10 and
-20 RE are found in Fig. in
Appendix A. The general trend is that, by 16:00 UT, the convergence has taken
place almost everywhere for the three variables, thus being consistent with
earlier studies see e.g. which state that the
formation of the magnetosphere takes approximately 1 h.
Some interesting features remain, though: while generally VX shows similar
time evolution for each simulation, there are significant differences
(several hundreds of km s-1) in the lobe close to the Earth at
-8 RE lasting several hours (Fig. ).
The largest differences are produced by the sin initialisation. The
differences decay gradually and eventually diminish by 17:00 UT (23:00 UT)
at X=-8 RE. Further away at X = -15 RE
the differences in VX get as high as 50–100 km s-1 and vanish by
23:00 UT. Remarkably low number density values and the fact that similar
effects are not observed in the plasma sheet, where the number density is
higher suggest that differences on the large-scale mass transport in the
magnetotail are small.
To further test the origin of the differences, we performed yet another
simulation with sinusoidally varying BZ but constant BY. This led
to substantially smaller differences in the lobe plasma transport values (not
shown). Thus, the differences arise from the configurational changes
associated with IMF BY due to the significant role of BY in tail
rotation. In GUMICS-4, the magnetotail BY is usually approximately
50 % of the solar wind value , which roughly
corresponds to observations . This result highlights the
importance of using a BY that is close to the observed value in the
initialisation process, as the effects are clearly visible in the simulation
results for several hours after the initialisation period.
Initialisation methods over a 3 h period on 14 June,
12:00–15:00 UT, 2007
Initialisation
BZ
BY
No initialisation (10d)
measured
measured
Stepwise (st1)
+5, -5, +5 nT
constant 2 nT
Stepwise (st2)
-5, +5, -5 nT
constant 2 nT
Linear (lin1)
+5→-5→+5→-5 nT
constant 2 nT
Linear (lin2)
-5→+5→-5→+5 nT
constant 2 nT
Rotation (sin)
+5sin(2πt(hours))
5cos(2πt(hours))
Panels from top to bottom: (a) bow shock and magnetopause
nose positions in RE, (b) plasma speed V in
km s-1, (c) magnetic field magnitude B in nT,
(d) plasma β=2μ0p/B2 and (e) electric field
EY in mV m-1. Parameter time series in panels (b–e) are
produced at X=9 RE Y=6 RE Z=0 RE (in
GSE coordinates). Yellow background highlights 1 h interval during which
the magnetosphere should be formed .
Figure shows the bow shock and the
magnetopause position (panel a), the V (panel b), B (panel c), plasma
beta (β=2μ0p/B2, panel d), and EY (panel e) at X=9 RE, Y=6 RE, Z=0 RE. Note
that since the magnetopause and the bow shock are in motion as depicted in
panel (a), this point is not necessarily always located within the
magnetosheath but fluctuates relative to the magnetopause position crossing
to the upstream solar wind when the magnetopause moves inward. This can be
seen in the time series as small values of the magnetic field magnitude in
panel (c) that often coincide with the bow shock and the magnetopause moving
closer to the Earth in panel (a). The time series measured at X,Y,Z=[9,6,0]
show only minor differences between the runs following the first hour of
similar solar wind input, indicating no locally introduced dynamic evolution
that would carry the memory of the previous driving conditions.
(a) Scaled (by solar wind value) ρVx in the 10-day run at
18:30 on 14 June and (b–f) the magnitude of the relative difference
(defined as |(ρVxi-ρVx10d)/ρVx10d|⋅100) in ρVx in the noon–midnight
meridian plane. Note that the scaling of (a) is not shown.
Differences in the noon–midnight meridian plane
To gain a large-scale view of the simulation differences, we focus on mass
transport X component ρVX in the XZ plane.
Figure shows time instance 18:30 UT on 14 June 2007, which
represents the time interval when the highest differences occurred over the
course of the 12 h simulation period.
The top left panel (a) in Fig. shows the ρVX
scaled by the upstream value at X=14.5 RE for the 10-day run
and reveals the large-scale characteristics of the solar wind, magnetosheath,
and magnetosphere in the form of colour-coded relative plasma transport
(scaling not shown). The black sphere in the middle of each panel
(3.7 RE radius) masks the region not covered by the MHD
simulation. The white sphere inside the black sphere depicts the Earth.
The following panels (b–f) show the magnitude of the relative difference in
ρVX between the 10-day run and st1, st2, lin1, lin2 and sin,
respectively, with the colour scale shown indicating the differences as
percentages. The pattern of the large relative differences approaching and in
excess of 100 % in the inner part of the magnetosphere (shown with red
colours) is qualitatively similar in each of the panels (b–f) in
Fig. (18:30 UT). These differences are mainly caused by
differences in VX. The localised nature of the differences indicates
that these arise at the boundaries of plasma regions, where a slightly different
orientation of the overall magnetospheric configuration can lead to large
differences in the local values of the plasma parameters. While such effects
may not affect the large-scale evolution of the magnetotail dynamics (as
demonstrated by the convergence of the time series above), they can cause
large differences at local point measurements. Such differences are important
when simulation results are compared with spacecraft observations – just a
slight change in the satellite orbit might lead to significantly different
time series along the spacecraft trajectory.
Appendix Figs. – show time instances
16:03, 17:12 and 19:30 UT on 14 June 2007 in a format similar to
Fig. . The overall pattern of the large relative
differences in the inner part of the magnetosphere is the same in all four
figures. However, panel (e) in Fig. shows a bright region
in the nightside tail lobe extending from -15 to -25 RE in
the X direction, indicating an absolute difference in the range of 30 to
40 %. It appears to be an extension to one of the bright regions in the
inner tail, and is mainly caused by the decrease in the tailward flow speed
VX in the lin2 run with respect to the 10-day run. Such differences imply
differences in the overall magnetotail configuration and dynamic evolution,
for example, generation of flow bursts in the magnetotail. As the differences
move around the noon midnight meridian plane and do not grow as a function of
time, the results indicate that the simulations converge towards similar
long-term dynamical evolution.
Quantification of the differences
Above, we have shown that there are large, localised differences and smaller,
wider-spread differences in the plasma transport properties. In order to
quantify the overall convergence of the different initialisations with the
10-day reference run, we compute kernel density functions of the differences
integrated over all grid cells and over 4 h covering the time period
16:00–20:00 UT. This time period includes all four instances studied in
Figs. and –. The grid
used is evenly spaced, interpolated from the original non-uniform GUMICS-4
grid, including 13 015 grid points at 0.5 RE resolution,
covering the spatial volume of -25<X<29 RE, -30<Z<30 RE.
Kernel density functions of the magnitude of the relative difference
in ρVx for st1, st2, lin1, lin2 and sin runs integrated over 4 h from 16:00 to 20:00, 14 June. The kernel density values are normalised
by the total grid volume.
Figure shows the kernel density functions for the
differences for each of the initialisations. It is clear that for most grid
points the differences are very close to zero, below 5 %. Such points show as dark-blue regions in Figs. and
–. Furthermore, it is also evident that
only a small minority of grid points show relative differences over 5 %
which cover the large localised errors (extreme values shown in red) and the
region in between (error between 5 and 30 %), which includes the smaller
errors over larger regions (shown in light blue) in Figs.
and –.
Probability for having the magnitude of the relative difference in
ρVx in % in the range of 0–5, 5–30 and 30 or more during
14 June 16:00–20:00 UT and 16:00–24:00 UT, 2007.
16:00–20:00
16:00–24:00
Simulation
0–5 (%)
5–30 (%)
30– (%)
0–5 (%)
5–30 (%)
30– (%)
st1
93.6
5.68
0.72
92.5
6.73
0.73
st2
94.5
4.67
0.85
93.6
5.58
0.77
lin1
92.8
6.35
0.84
93.3
5.93
0.74
lin2
94.9
4.28
0.82
93.1
6.07
0.81
sin
94.4
4.70
0.92
92.1
7.06
0.87
The very thin tails of the distribution functions with over 5 % errors
demonstrates that the large and medium-sized errors are statistically
insignificant and do not lead to large-scale differences in the temporal
evolution of the simulations.
Comparing the kernel density functions in Fig. shows that
the performance using the different initialisations shows some differences,
but overall all initialisations work reasonably well. However, two simulations
starting with positive IMF BZ and utilising constant IMF BY (st1, lin1)
seem to produce fewer errors in the below 5 % range (lower peak values)
and thus produce larger errors in the 5–30 and > 30 % ranges,
indicating some differences in the magnetotail dynamics. The third simulation
starting with positive IMF BZ (sin) differs from the two in terms of peak
value in Fig. . However, as was shown in
Sect. , it produced largest localised differences
partly because of the usage of fluctuating IMF BY. This would indicate
that it is useful to start the initialisation with strongly reconnecting
magnetosphere driven by negative IMF BZ and constant (close to observed
value) or near-zero IMF BY.
We classify the simulations by computing probabilities for having 0–5, 5–30
and > 30 % differences during two time intervals, 16:00–20:00 and
16:00–24:00 UT, on 14 June 2007. Table shows the
probabilities for having the relative difference in ρVX in the
aforementioned percentage ranges during the indicated time intervals. The
first set of probabilities (16:00–20:00 UT) contains the same data as
Fig. . As the integration times are much longer than the
solar wind transit time through the simulation box, the differences in the
numbers between the two integration times give an indication of the
statistical errors (a few percent in each direction).
About 5–7 % of the time the relative errors fall between 5 and 30 %,
indicating that there are instances when the large-scale patterns are
slightly different, but these do not last for extended periods of time. The
small localised errors are of the order of less than 1 %, and do not
generate larger errors as the simulation continues. Based on these
statistics, it is not easy to pick one initialisation method over another.
However, we would slightly prefer st2 and lin2 initialisations as they have
not only the lowest probability during 16:00–20:00 UT but also lowest and third
lowest probability during 16:00–24:00 UT of medium-size errors, which tend
to indicate larger-volume differences in the configuration.
Discussion
In this paper we study the performance of the GUMICS-4 global MHD simulation.
First, we compare the bow shock standoff distance and magnetopause subsolar
point position with empirical models using the OMNI database containing the
empirical shock model by and the
empirical model for the magnetopause location. The performance was evaluated
using a simulation covering a 10-day period during 12–22 June 2007, when
nearly continuous solar wind observations were available from the OMNI
database.
We conclude that depending on the method used (B,ρ,V) the GUMICS-4 shock
position stays within 20 % of the empirical value 88 or 99 % of the
inspected time interval, while the magnetopause position is closer to the
empirical estimate, within 10 % for 98.5 % of the 10-day period. The relative differences averaged
over the 10-day period are 10.8–6.7 % (depending on used method) for
the bow shock position and 3.4 % for the magnetopause position. Using our
methodology and the GUMICS-4 simulation position, we find the shock and the
magnetopause consistently closer the Earth than the empirical references,
consistent with earlier findings . As relative
differences of the order of 10 % suggest absolute differences of the
order of 1 RE, the differences almost fall within the errors
introduced by the 0.5 RE spatial resolution in the simulation
run. The differences between empirical and simulation estimates did not
increase even when abrupt changes in the solar wind parameters take place,
indicating that GUMICS-4 is correctly capturing the large-scale dynamics.
Next, we took a 9 h subset of the 10-day data interval, combined it with
five different 3 h synthetic initialisation phases, simulated the
resulting five 3 + 9 h intervals, and compared the results to the
original 10-day run in order to identify the effect of using different
initialisation methods. Magnetospheric response to synthetic data in global
MHD simulation has been studied previously using GUMICS-4 by
. Our results demonstrate that, regardless of the
initialisation parameters, the GUMICS-4 simulations converge toward a common
evolution within 1 h.
However, comparing difference images produced on the noon–midnight meridian
plane (XZ plane) using ρVx variable suggests that some differences
in the magnetospheric state remain for extended periods following the
initialisation phase. These differences depend on the initialisation
parameters used and can affect the results several hours afterwards, as is shown
in Table . The table shows that 5.1–7.2 % of the
grid points show differences above 5 % even 9 h after the initialisation
stage has ended. On the other hand, it can also be seen that the relative
differences in ρVx greater than 30 % are statistically
insignificant on timescales of several hours. This implies that the small
localised deep red regions visible in Figs. and
– have only minor effects on differences
between the simulations.
While the results do not clearly point to any of the initialisations as being
superior to the others, there are couple of indications that may be useful
for future studies. First, the dayside magnetosheath does not show any
differences between the different initialisations, indicating that the
magnetosheath has no internal processes that would carry the memory of the
past driving conditions. Second, introducing a sinusoidally varying IMF BY
during the initialisation caused strong twisting of the tail, whose effects
remained in the magnetotail for several hours. Thus, we would recommend using
constant and/or near-zero IMF BY preferably close to the observed value in the simulation
initialisation. Third, the initialisations starting with strongly
reconnecting magnetosphere driven by negative IMF BZ seem to produce more
reliable results in the magnetotail. As shown for the BY case, the
magnetotail has a long memory that can retain effects of past configurations
for quite some time. Strong reconnection has the effect of removing these by
enhancing the convection cycle. Thus, we would recommend starting the
initialisation with negative IMF BZ.
The results suggest that choosing the switching moment (14 June 15:00 UT)
from artificial initialisation to non-artificial input data differently would
have no impact on our recommendation by creating qualitative differences
between the runs. Such change would naturally alter the behaviour of IMF
BZ at 15:00 UT for each simulation. However, the current study is already
covering the switching of IMF BZ from +5 to -7 nT (st1, lin2)
and from -5 to -7 nT (st2, lin1, sin) as depicted in
Fig. . If the change in the switching time would cause
qualitative differences the simulations would be divided in the
aforementioned two groups by the behaviour of IMF BZ at 15:00 UT. Such a
division is not apparent in Table , however.
We identified two types of errors that arise in the simulations. Large but
localised errors are observed in regions of strong gradients, at the plasma
sheet boundary, when the parameters change significantly over a short
distance. Such errors were shown to be statistically insignificant, but can
cause major discrepancies when the simulation results are compared with
in situ measurements along the spacecraft orbit. Thus, caution is necessary
when interpreting such comparisons.
Medium-size relative errors in the range 5–30 % occur over larger
portions of space and imply dynamic events that do not occur simultaneously
in the two simulations. Such errors vary in location and do not have a
continuous temporal evolution, which again points out that, despite such
differences, the simulations on the large scale converge toward a common
solution.
In order to examine the cause of the differences, we plot in
Fig. time series of the relative errors in the three
error ranges for the time interval 16:00–20:00 UT on 14 June 2007. The top
panel shows the IMF clock angle for reference. The second panel shows the
energy flux incident at the magnetopause evaluated from
K=u+p-B22μ0V+1μ0E×B,
where u is the total energy density, p pressure, B magnetic field,
V flow velocity and E×B the Poynting flux. The
three bottom panels show the errors in the ranges 0–5, 5–30, and
> 30 % for each of the initialisation runs. The grey bars note periods
when the accuracy for at least one of the initialisation runs falls below the
typical 93–95 % (panel c), indicating periods when errors are observed
between the 10-day run and the different initialisation runs.
IMF clock angle (a), energy flux (b), normalised
(by the total volume) grid volume with difference in ρVx between 0
and 5 % (c), between 5 and 30 % (d) and over
30 % (e) during 14 June, 16:00–20:00 UT. The grey background
highlights the occurrence of peaks in (c–e).
Dynamics in the magnetosphere is controlled by the energy entering through
the magnetopause in the form of Poynting flux . The total
energy flux of the solar wind expressed by Eq. () is a
strong function of BZ and thus moderate difference in BZ value
results in moderate difference in transferred energy amount from the solar
wind to the magnetosphere. The first two difference peaks are associated with
a peak in the energy flux, featuring an about 70–80 % increase in the
incident energy flux. Interestingly enough, the associated simulations lin2
and
st1 are the only ones with northward IMF BZ at the end of the
initialisation period. It is possible that the two first peaks are associated
with the rapidly changing energy entry to the magnetosphere with a profile
different from the 10-day simulation.
The last two maxima are preceded by a rapid change in sign of IMF BY from
positive to negative and back to positive. The actual peaks coincide with
rapid rotations between northward and southward orientations. However, the
start of the increase in the errors is not marked by distinct changes in the
IMF or the energy flux variations.
The present study has utilised the relative differences in ρVX to
quantify the global differences in the simulation runs at the maximum grid
resolution of 0.5 RE. This configuration has been used
extensively in previous studies and therefore any
differences we observe are likely induced by the initialisation as opposed to
the chosen model setup. For completeness, we should mention that an
alternative approach would be to examine global variables such as the
cross-polar cap potential in the ionosphere. However, this would require
increasing the adaptation level to 5 (0.25 max grid resolution) rather than
4,
which we have used here. Even so, even with a finer grid resolution, GUMICS-4
produces relatively low polar-cap values due to all the
currents closing through the polar cap; this is a result from too weak region
2 currents. Therefore, the validity of this approach with regards to the
initialisation phase is unclear, and more investigation is required. For
that reason, we have focused exclusively on the magnetospheric region which
has been validated extensively, and reported in the existing literature.
Having said that, the investigation of the polar-cap potential will be the
focus of future studies, but this is beyond the scope of the present paper.
It should be noted that a global MHD simulation does not necessarily converge
towards the same solution even if resolution is improved .
In fact, if resolution is improved, numerical diffusion decreases, which means
that the role of MHD physics as an error source grows. Our results imply that
the differences in the simulation runs arise from the different initial
conditions rather than numerical accuracy, as the solutions converge towards
the same large-scale state in the long run. Thus, the results might be
generalisable to all the MHD codes applying similar physics (ideal MHD) to
that of GUMICS-4.
Analysis of the temporal behaviour of differences appearing in both large and
small spatial scales indicates that differences in the detailed locations of
current and flow systems in the (inner part) of the magnetotail maintain a
memory of about 3 h. Thus, while for most purposes the different
initialisations lead to rapid convergence toward nearly the same solution,
there are local- and large-scale differences that can be seen hours after the
initialisation period. Future work should extend the investigation of the
large-scale dynamics causing the observed differences in the ρVx
variable, including computation of the energy budget of the magnetosphere.