Introduction
The interaction of the solar wind with a planet and its magnetic field, such
as the Earth's internal dipole field, causes the current system of the
magnetosphere. Thereby, currents in the outer region of the magnetosphere,
the magnetosheath, are related to the deflection of the solar wind plasma
around the planet. These currents, far from the planetary
surface,
can be observed by a spacecraft crossing the magnetosheath with its
boundaries being the bow shock and magnetopause. The influence of the solar wind on the
currents in the inner magnetosphere can also be directly observed by ground
magnetometer data. Modifications of the strength of these currents can be
expressed by the indices of geomagnetic activity depending on the solar wind
conditions.
This dependence was used by to reconstruct average
solar wind conditions and their gross temporal variations in the
pre-spacecraft era. Thereby, solar wind spacecraft observations were related
to the indices of geomagnetic activity using a multi-regression method. Then,
the solar wind conditions were estimated from ground magnetometer data back
to 1926. A different approach to relate solar wind conditions to the indices
of geomagnetic activity was introduced by . They use a
singular spectrum analysis to fill gaps of the solar wind data in the period
of 1972–2013. This analysis estimates hourly solar wind conditions during
periods where no in situ measured solar wind information is available. It
was shown that the reconstructed solar wind information can improve empirical
models such as the Tsyganenko model .
In contrast to the previous approaches, included
spacecraft observations to reconstruct solar wind conditions. During a
geomagnetic storm in 1989, the magnetopause was near the geostationary orbit
due to the extremely high solar wind dynamic pressure. The magnetopause
location was determined by the Geostationary Operational Environmental
Satellite (GOES). This magnetopause location was related to the solar wind
conditions with the magnetopause model by . This
reconstruction approach using the spacecraft data with a high time resolution
allows one to estimate the solar wind conditions more precisely on a much higher time resolution
compared to the previous approaches.
We extend this approach by using not only the magnetopause location observed
by a spacecraft, but by using spacecraft data obtained everywhere in the
magnetosheath. Analogously to the previous studies, the reconstruction can be
used to provide solar wind information if in situ solar wind observations are
not available. This determines the solar wind parameters, which are the
drivers of the magnetospheric current system and are usually required in
magnetospheric and ionospheric models
e.g.,
We reconstruct the solar wind conditions during a spacecraft magnetosheath
passage using the magnetosheath model introduced by . The solar
wind parameters are boundary conditions of the magnetosheath model and they
are varied until the model predictions match the magnetosheath data. The
corresponding adjoint magnetosheath model is derived to perform this
time-consuming procedure. Adjoint models are broadly used in fluid dynamic
optimization problems such as drag minimization
e.g.,, or in seismology
e.g.,. We will introduce and discuss the problems of how to deal
with a magnetosheath model when an adjoint approach is used. To verify
our method, we use data from the THEMIS (Time History of Events and Macroscale Interactions
during Substorms) mission to
reconstruct the solar wind conditions at Earth's orbit. The reconstructed
conditions are compared to OMNI solar wind monitor data (see http://omniweb.gsfc.nasa.gov/)
as well as THEMIS data from a second spacecraft
in the solar wind. This indicates the limitations of the magnetosheath model
used.
Solar wind reconstruction method
The forward magnetosheath model
For the solar wind reconstruction, we chose the zeroth order MHD (magnetohydrodynamics) model
presented in . This model uses series expansions of density,
velocity, gas pressure, and magnetic field up to the first order away from
the stagnation streamline to simplify the stationary MHD equations.
Consequently, the model is only valid close to the stagnation streamline. The
x axis of the coordinate system is chosen to be along the stagnation
streamline, the z axis is along the Earth's dipole axis, which is assumed to be
orthogonal to the x axis, and the y axis completes a right-hand orthogonal
system. Note that the y and z directions are called tangential
hereafter. The series expansion procedure provides the following set of
ordinary differential equations :
ρ0ux0′+ρ0uy10+uz01=0,Bz0ux0′+Bz0uy10=0,ρ0ux0ux0′+p0′+1μ0Bz0Bz0′-1μ0Bx01Bz0=0,p0ρ0-γ′=0.
Here, ρ0 represents the density, ux0 the velocity component in
x direction, Bz0 the magnetic field component along the Earth dipole
moment, and p0 the gas pressure. The index 0 indicates the zeroth order
approximation used. The derivatives marked by the prime are with respect to
the stagnation streamline direction x. The vacuum permeability is
μ0=4π×10-7N/A2. The tangential velocity
derivatives uy10 and uz01, as well as the magnetic field derivative
Bx01, are given by uy10=0.8Δu/xBS, uz01=Δu/xBS, and Bx01≈0 . Here, Δu denotes
the x component of the velocity difference across the bow shock, which is
calculated using Rankine–Hugoniot relations. The bow-shock distance to the
Earth's center is denoted by xBS. Figure sketches the workflow of this model.
Scheme of the forward magnetosheath model presented in
. The solar wind conditions are the input to the model. First,
the bow-shock distance is initially estimated. Then, the Rankine–Hugoniot
relations are solved and determine the post-shock values, which are used as
boundary conditions for the system of differential equations (DEQ). Solving
this system leads to a new estimate of bow-shock distance. After some
iterations, the bow-shock distance converges and the magnetosheath solution
is obtained.
The solar wind parameters are used as input parameters to obtain the
magnetosheath solution. Note that this evaluation procedure is called forward
model. In a later step, yet to be discussed, these boundary values are
modified to match any observed magnetosheath situation to our calculation
results using a so-called reverse model approach. The solar wind parameters
are the density ρSW, velocity's x component uSW, magnetic
field's z component BSW, and pressure pSW. Note that an
x component of the magnetic field does not affect the solution of the
zeroth order model used. This limitation of the magnetosheath model restricts
our solar wind reconstruction to the y and z components of the solar
wind magnetic field. The x component cannot be reconstructed using this
magnetosheath model. A y component of the magnetic field can be taken into
account in this model by replacing the z component Bz0 by the
magnitude of the tangential magnetic field, i.e., Bz0←sgn(Bz0)⋅(By02+Bz02)0.5. Here, sgn() denotes the
signum function. Then, the solar wind magnetic field BSW is replaced by
BSW←sgn(Bz,SW)⋅(By,SW2+Bz,SW2)0.5.
At the bow shock (x=xBS), the solar wind values are transformed into
post-shock magnetosheath conditions using Rankine–Hugoniot relations
e.g.,. The post-shock values ρ0(x=xBS),
ux0(x=xBS), Bz0(x=xBS), and p0(x=xBS) are the
boundary conditions for the ordinary differential Eqs. ()–() presented above. Furthermore,
the bow-shock distance needs to be known in order to calculate uy10 and
uz01. The bow-shock distance xBS is the sum of the magnetopause
distance to the Earth's center xMP and the magnetosheath thickness
ΔxMS. An initial estimator of the magnetopause distance denoted by
xMP0 is given by :
xMP0=f2M22μ0KρSWuSW216,
where the parameter of magnetopause geometry f=2.44, the flow deflection
parameter K=0.89, and a magnetic dipole moment M=8×1015Tm3
are valid for a terrestrial situation. The initial magnetosheath
thickness ΔxMS0 is estimated by an analytically derived formula
:
ΔxMS0=xMP045+mBSgu-1-1,
where the deceleration at the shock is gu:=uSW/ux0(x=xBS)
and mBS is a measure for the solar wind magnetization defined by
mBS:=1-11+γ2p0(x=xBS)pmag(x=xBS).
Here, pmag(x=xBS):=Bz02(x=xBS)/2μ0 is the post-shock
magnetic pressure and γ=5/3 is the ratio of specific heats. The sum of
Eqs. () and () estimates the initial bow-shock
distance:
xBS0=xMP0+ΔxMS0.
Next, the system of differential Eqs. ()–() is solved numerically.
Therefore, the differential equations are written as a matrix-vector
equation:
ux0ρ0000Bz0ux000ρ0ux0Bz0μ01-γp0ρ-γ-100ρ-γρ0′ux0′Bz0′p0′=-ρ0uy10+uz01-Bz0uy10Bx01Bz01μ0.
This equation can be represented by
A(q)q′=b(q),
with q:=ρ0,ux0,Bz0,p0T, the
coefficient matrix A, and the inhomogeneity vector b. Note
that b depends not only on q but also on uy10 and
uz01, initially determined by xBS0. For numerical calculation, the
magnetosheath solution q is discretized along the x direction:
q(x)→q(xn)=qn,
where the position index n is space discretization. Consequently, Eq. () transforms into
Aqnqn+1-qnΔx=bqn,
where Δx is the step size along the x direction. Equation ()
is solved with respect to qn+1 using a Gaussian
elimination algorithm. Starting from the bow shock (x=xBS), the solution
is evaluated earthward until the magnetopause is reached, i.e., until the
flow velocity vanishes (ux0=0). The distance in space between bow shock
and magnetopause yields a new estimator of the magnetosheath thickness
ΔxMS1.
The planetary magnetic field is approximated by a dipole with its moment
along the z direction. The dipole is included via inner boundary
conditions; i.e., the calculated total pressure ptot (sum of gas
pressure, magnetic pressure, and dynamic pressure) at the magnetopause is
related to the planetary field as follows :
ptot=f⋅M22μ0xMP3.
Note that due to the iteration procedure used, the actual stagnation pressure
at the magnetopause will differ from the stagnation pressure KρSWuSW2 in Eq. () when initiating the iteration. Now, Eq. ()
is solved with respect to xMP. This gives a new estimator
xMP1 for the magnetopause distance. Together with the calculated
magnetosheath thickness ΔxMS1, a new estimator for the bow-shock
distance is obtained: xBS1=xMP1+ΔxMS1. As depicted in
Fig. , the newly estimated bow-shock distance is used to
run the scheme again until the bow-shock distance converges. Then, the
magnetosheath solution qn is obtained.
The magnetosheath solution is determined along the stagnation streamline.
According to , extending the solution to locations off the
stagnation streamline requires one to shift the corresponding x coordinate with
respect to the paraboloid coordinates used. To transfer the spacecraft's
location (xSC, ySC, zSC) into the parabolic-shifted coordinate
system of the model indicated by xSC,shift, the following equation is
used:
xSC,shift=xSC+cBS,y+ΔcyxSCxMPySC2+cBS,z+ΔczxSCxMPzSC2,
where Δcy:=cMP,y-cBS,y and Δcz:=cMP,z-cBS,z. The curvature parameters are determined to be cBS,y=0.4/xBS, cBS,z=0.5/xBS, cMP,y=0.4/xMP, and cMP,z=0.5/xMP. However, due to the assumptions made to derive the
magnetosheath model, the solution is only valid in the vicinity of the
stagnation streamline.
Data assimilation
The spacecraft at a position (xSC, ySC, zSC) in the
magnetosheath measures the density ρSC, velocity uSC, magnetic
field BSC, and pressure pSC. Values for these quantities at a
certain position form a single data point. Each data point is related to
certain solar wind parameters individually by the model introduced above and
the succession of steady-state states approximates any dynamical behavior of
the magnetosheath. To find the appropriate solar wind parameters, the
magnetosheath solution needs to match the data point at its position.
Starting from an arbitrary first guess of the solar wind parameters, the
parameters are modified until the solution matches the data, which is called
data assimilation. It seems natural to start from typical solar wind
conditions at Earth's orbit, such as ρSW,t=8.0⋅mP cm-3, uSW,t=400 kms-1, BSW,t=5nT, and
pSW,t=1×10-11Pa. Note that the proton mass is mP=1.67262178×10-27kg. To simplify the notation, we set
mP=1 in the following, so that the mass density is presented as a particle
density.
During the process of varying the solar wind parameters, it is important to
quantify the agreement between data and model. Therefore, a cost function is
introduced. Consider, for example, the density ρSC measured at
(xSC, ySC, zSC) and assume the model proposes a density ρ.
The difference of these two values is a measure of the density agreement of
model and data. The square of this difference is taken to obtain a function
with a minimum: ρ-ρSC2. In order to sum up such
terms for different quantities, these terms are normalized to avoid effects
from different units. The measured values themselves can be taken as
normalization. The cost function K sums up all contributions from the
different quantities:
K=ρ-ρSCρSC2+u-uSCuSC2+B-BSCBSC2+p-pSCpSC2.
The cost function K is minimized with respect to the solar wind parameters
ρSW, uSW, BSW, and pSW. Thereby, the step size is
proportional to the value of the cost function. If the model used is able to
represent the physics of the situation considered and data errors are
negligible, the cost function vanishes in its (global) minimum. Taking data
errors and limitations of the model into account, the global minimum of the
cost function gives the best estimator for the solar wind parameters.
Starting from the typical solar wind conditions presented above, a
gradient-based minimization algorithm is used to minimize the cost function.
We use a steepest descent method without line search as minimization
algorithm. If the gradient vanishes, a minimum of the cost function is
determined. Using an adjoint approach the gradient required for this
minimization scheme is determined as illustrated in the next section.
The reverse magnetosheath model – adjoint approach
An adjoint approach offers an efficient way to calculate the gradient of a
cost function e.g.,. The numerical implementation of the
magnetosheath model provides a source code, which calculates the cost function
from the initial solar wind parameters. Here, the adjoint approach is
performed with an automatic differentiation tool. Such a tool is able to
transfer the source code into a new modified source code, which automatically
calculates the derivative of the cost function . Therefore,
the automatic differentiation tool dissembles the code into elementary
operations, i.e., simple arithmetic operations. To illustrate this, consider
a simple example of a cost function similar to defined by
K(ρSW,uSW)=2ρSW+sin(uSW),
which depends on the two parameters ρSW and uSW. The example
discussed here illustrates how the tool is processing the code. An
appropriate source code implementation to calculate this function with
elementary operations only is
l1=ρSWl2=uSWl3=2l1l4=sin(l2)l5=l3+l4.
Using the chain rule, the derivative of the cost function (Eq. ) can be
expressed as follows:
∂K∂(ρSW,uSW)=∂l5∂(l3,l4)⋅∂(l3,l4)∂(l1,l2)⋅∂(l1,l2)∂(ρSW,uSW).
Now, the automatic differentiation tool inserts calculation of the matrices
∂(l1,l2)/∂(ρSW,uSW), ∂(l3,l4)/∂(l1,l2), and ∂l5/∂(l3,l4) needed to calculate the derivative. This gives
l1=ρSWl2=uSW∂K∂(ρSW,uSW)←∂(l1,l2)∂(ρSW,uSW)l3=2ρSWl4=sin(uSW)∂K∂(ρSW,uSW)←∂(l3,l4)∂(l1,l2)⋅∂K∂(ρSW,uSW)l5=l3+l4∂K∂(ρSW,uSW)←∂l5∂(l3,l4)⋅∂K∂(ρSW,uSW).
Here, the arrow to the left (e.g., a←a⋅b) means that the
expression on the right side (a⋅b) is calculated first, and then, the
variable on the left side (a) is replaced by the result of the right side.
The entities of the matrices are elementary derivatives such as
sin(l2)′, which are substituted from a library used by the automatic
differentiation tool: sin(l2)′→cos(l2). This procedure
has the advantage whereby the calculated derivatives are not subjected to errors
of finite difference approximations because analytical expressions are
inserted.
The new code (Eq. ) calculates the derivative parallel to the
execution of the forward model. This procedure is called forward
differentiation. However, this is not the most efficient way to calculate
the derivative. It is more efficient to use the automatic differentiation in
reverse mode, which is nothing else but the adjoint approach. A short
introduction of the mathematical motivation of adjoint methods is given in
Appendix . Executing the code above performs the calculation
of the derivative as shown in Eq. () from the right to the
left. During this procedure, 12 elementary multiplications are performed.
However, the calculation from the left to the right in Eq. ()
needs only 8 elementary multiplications. Consequently, the reverse mode is
more time efficient. The transpose of Eq. () reverses the
order similar to the calculation described above. Note that adjoint and
transpose are equivalent in real space and this is why the reverse approach
is called the adjoint approach.
The adjoint approach of the more complex cost function with the magnetosheath
model presented in the previous section is derived by the OpenAD/F (open-source tool for the automatic generation of adjoint code from Fortran 95 source code) tool
. The code contains a Fortran subroutine evaluating the
magnetosheath model as shown in Fig. with the solar
wind parameters and the cost function as input and output variables. To
prepare the subroutine for the OpenAD/F tool, solar wind parameters, such
as rhoSW, are declared as independent variables by a
!$openad INDEPENDENT(rhoSW) statement. The cost function
(cost) is declared as dependent by a !$openad DEPENDENT(cost) statement. The tool transfers an ordinary variable y into
a structure, which contains the value of the variable y%v and the value of
the derivative of the cost function with respect to y denoted by y%d. In
order to
enable the code to operate in reverse mode, the derivative of the cost
function is initialized by 1 because the derivative of the cost function with
respect to the cost function gives 1. Then, the OpenAD/F tool can be applied
in reverse mode to the source code. First, the tool executes a lexical,
syntactic, and semantic analysis, which results in an intermediate
representation, called whirl. Then, the OpenAnalysis module of the
OpenAD/F tool produces call and flow graphs. This information is combined
with the whirl code into a new representation of the code called
xaif. This code representation is transferred into a code that
calculates the derivative parallel to the execution via the xaifBooster
module. Afterwards, the tool transfers this representation back to ordinary
Fortran code. A call of the new subroutine is able to calculate the gradient.
For the four independent parameters that we consider, i.e., ρSW,
uSW, BSW, and pSW, the adjoint approach can be 4 times
faster. To calculate in reverse mode requires one to execute the code before the
derivatives are calculated because the derivative terms require the results
calculated later in the code. Therefore, all variables used during the
execution of the code to calculate the cost function need to be stored. The
implementation of checkpointing, i.e., splitting the adjoint calculation into
separate parts to reduce memory requirements, was not necessary for our
adjoint code.
A technical problem and its solution
The cost function has a lot of very small local minima on top of the
monotonous cost function with its global minimum. These minima result from
the iterative modification of the bow shock as described in Fig. .
The bow-shock distance is used to calculate uy10
and uz01. Small modifications change the value of the cost function
differently compared to the global shape of the cost function. If a finite
difference approach is applied, the local minima can be ignored using a step
size for the finite difference, which is larger than the extent of such a
local minimum. However, it is difficult to determine the correct finite
difference size because the size varies depending on the solar wind
conditions.
The situation is different for an automatic differentiation approach because
it is based on analytical expressions. Unfortunately, without code
modifications, the gradient is calculated with respect to the local minima.
Consider, e.g., the one-dimensional cost function h(x):=hq(x)+ho(x)
with the quadratic function hq(x):=x2 and the oscillation function
ho(x):=(cos(10x)-1)2. The global minimum is determined by the quadratic
function, which decreases monotonically on both sides towards the minimum.
However, the oscillation function superposes local minima to the function.
The derivative ∂xh(x)=2x-20sin(10x)(cos(10x)-1) oscillates
around the global minimum due to the latter contribution ∂xho(x).
Following the gradient usually gives a local minimum. This problem can be
solved by replacing ho(x) by a sum of step functions h̃o(x)=∑iaiθ(x-xi) with the coefficients ai and xi, which are
determined to fit ho(x). Here, θ(x) denotes a step function. The
analytical derivative of h̃o vanishes if x≠xi. Consequently,
∂xh(x)=∂xhq(x)+∂xh̃o(x)=∂xhq(x)=2x and the global minimum is found by following the gradient. In
our magnetosheath model, the modifications of uy10 and uz01 produce
the local minima and we excluded them by introducing step functions. This was
done by rounding the tangential velocities. Note that the rounding error is
chosen to be very small on the order of the numerical error, essentially not
effecting the result of the cost function. Consequently, the calculation of
uy10 and uz01 does not contribute to the calculated gradient and a
gradient with respect to the global shape is obtained.
Figure shows the resulting global shape of the
cost function depending on the solar wind density ρSW and solar wind
velocity uSW.
Color-coded value of the cost function with respect to solar wind
velocity and density. The solar wind magnetic field and gas pressure is set
to zero. The global minimum can be obtained using a gradient-based
minimization.
The solar wind magnetic field and the solar wind pressure are zero. The
spacecraft values are set to ρSC=20.4 cm-3, uSC=48.0 kms-1,
BSC=0.1nT, and pSC=0.59nPa at
the spacecraft's position xSC=12.8RE, ySC=0.9RE, and
zSC=3.1RE, with the Earth's radius RE=6371km.
The red areas on the lower left and upper right corner are cost function
values where the spacecraft's position is outside the calculated magnetosheath.
Consequently, the cost function is high due to the bad matching of data and
model. Note that the magnetospheric values, i.e., the values outside the
magnetosheath on the earthward side, are set to the magnetopause values except
from the magnetic field, which increases earthward. The middle area of the
plot offers a smooth global minimum as sketched in Fig. , which can be determined by a gradient minimization.
The rounding procedure introduced above is used to neglect the small local
minima due to iterative bow-shock modifications for the calculation of the
gradient. To ensure that the minimum found by the gradient-based minimization
is the local minimum, which corresponds to the global minimum, we calculate
the cost function value on a parameter grid a couple of steps in each
direction around the minimum. We choose the grid step sizes Δρ=0.035 cm-3, Δu=2 kms-1, ΔB=0.035nT,
and Δp=1.5×10-3nPa. Note that the step sizes
determine the accuracy of the reconstructed parameters.
Application
In the magnetosheath model, the time resolution is limited
because of the steady-state approximation in the model. A disturbance due to
a change of solar wind conditions needs to pass the magnetosheath to obtain a
stationary state. A typical subsolar magnetosheath thickness is about 2.5RE
and the average flow velocity in the magnetosheath is about 50 kms-1.
Then, the transit time for the disturbance is approximately
5min. Consequently, we use spacecraft data with a time resolution
of 5min for our reconstruction.
The temperature is often very difficult to measure precisely. Therefore, we
assume a cold plasma approximation for the solar wind. A variation of the
solar wind temperature does not affect the reconstruction results much for
the results presented here. Note that the temperature T0 is related to the
pressure used in the magnetosheath model via ρ0kBT0=p0,
with the Boltzmann constant kB=1.3806488×10-23J/K.
We have chosen THEMIS magnetosheath transitions where a second THEMIS
spacecraft observes the solar wind conditions to validate our method.
Reconstruction close to the stagnation streamline
The reconstruction method is applied to magnetosheath transitions along the
stagnation streamline. First, the magnetosheath transition of the THEMIS
spacecraft THC on 1 August 2009 is discussed. Starting at the magnetopause
close to the stagnation point, the spacecraft traverses the magnetosheath and
crosses the subsolar point at the bow shock about 2.5h later.
The THC magnetosheath data and the best model prediction, which could be
found in the magnetosheath by varying the solar wind boundary parameters, are
shown in Fig. . Note that the data are presented in the
same coordinates as the model with the z direction along the dipole axis
(see Sect. 2.1).
The magnetosheath density (top panel), the ion velocity (middle
panel), and the magnetic field (bottom panel), which are observed by THEMIS on
1 August 2009, are presented in red. The best match for the data to the
magnetosheath model for any initial choice of the solar wind boundary
parameters was found for the magnetosheath model predictions, which are shown
in blue.
It is seen that the observation and model prediction agree very well. Only
at the magnetopause and at the bow shock do the model and the observations
significantly differ. Our magnetosheath model represents the bow shock by a
step function profile using Rankine–Hugoniot relations to calculate the jump
conditions. This approximation is not correct close to the shock where instead a
smooth transition is observed. Our algorithm fits the values within
this transition region with values of either pre- or post-shock values, and
thus the reconstruction fails there. The magnetopause is approximated as a
rigid boundary by a vanishing flow velocity in the model. This neglects,
e.g., single particle processes where high energy particles can penetrate the
magnetopause boundary and diffusion processes. Our magnetosheath model is
extended into the magnetosphere using the magnetopause values of velocity,
density, and magnetic field. Therefore, our approach is able to reproduce the
magnetospheric values observed approximately.
The results of the solar wind reconstruction are displayed together with OMNI
solar wind data and the solar wind observations by the THEMIS spacecraft THB
in Fig. .
The reconstructed solar wind boundary conditions of the model
density (top panel), the ion velocity (middle panel), and the magnetic field
(bottom panel) for the magnetosheath data presented in Fig.
are shown in blue. Additionally, the THB (red) and OMNI
(green) solar wind data during this time interval is shown.
The calculated solar wind conditions show overall a good agreement with the
solar wind data. The first 10min interval corresponds to a
reconstruction from magnetospheric observations. Due to the magnetospheric
extension of our magnetosheath model, the magnetosheath model is able to
estimate the solar wind parameters roughly during this time interval. For
example, a different choice of the solar wind magnetic field parameter would
lead to a different magnetopause magnetic field, and consequently the
magnetospheric field of the model does not fit the magnetospheric
observations. A higher solar wind dynamic pressure is related to a different
magnetopause location, and again, the model predictions would differ from the
observations. However, this reconstruction method is valid only in the
vicinity of the magnetopause. Our approach cannot be used to reconstruct
accurate solar wind parameters far in the magnetosphere.
From 03:20 to 04:00 UT the density of the solar wind
data from OMNI, THB, and the reconstructed solar wind parameters of THC
differ slightly. This might be due to spacecraft potentials
e.g., or due to a varying spatial density distribution
of the solar wind. Note that the THB spacecraft is located about 12RE from THC
in y direction. The density variation between
05:15 and 5:40 UT observed by the solar wind monitor
data is well reconstructed. The average value of the solar wind velocity is
well reconstructed; however, the velocity reconstruction shows more
variability than the solar wind data from THB and OMNI. A slight increase of
the solar wind magnetic field towards the bow shock is seen in all solar wind
data in the lower panel.
We ensure that our algorithm produces the same results for different initial
solar wind parameters. As an example, the minimization of the cost function
of the data point at 05:30UT in Fig. for
three different initial values of the density is presented in Fig. .
The value of the cost function at each iteration step is shown for
three different initial densities. Independent of the starting value, the
cost function converges to the identical value.
A different initial choice leads to the same reconstructed solar wind
parameters, only the number of iteration steps differ. Note that we have
chosen a cold plasma approximation to fix the solar wind pressure.
To examine the reconstruction method in more detail, we present a second
magnetosheath transition close to the stagnation streamline, which contains
more solar wind variations.
The magnetosheath density (top panel), the ion velocity (middle
panel), and the magnetic field (bottom panel), which are observed by THEMIS on
28 September 2008, are presented in red. The best match for the data to the
magnetosheath model for any initial choice of the solar wind boundary
parameters was found for the magnetosheath model predictions, which are shown
in blue.
On 28 September 2008, the THC spacecraft crosses the bow shock and traverses
the magnetosheath. The corresponding THC observations and the model fit are
shown in Fig. with the bow shock on the left-hand side
and the magnetopause on the right-hand side. Both, model and observations,
fit very well, except for the bow-shock region similar to the previous
example. The reconstruction of the solar wind parameters are compared to THB
and OMNI solar wind data in Fig. .
The reconstructed solar wind boundary conditions of the model
density (top panel), the ion velocity (middle panel), and the magnetic field
(bottom panel) for the magnetosheath data presented in Fig.
are shown in blue. Additionally, the THB (red) and OMNI
(green) solar wind data during this time interval are shown.
One expects that the variations of the THB solar wind data are in general
more precise than OMNI solar wind data because THB is located close to the
subsolar bow shock and observes the local solar wind conditions. Note that
THB enters the magnetosheath at 20:15 UT; therefore, afterwards no THB
solar wind observations are available. The solar wind density varies between
4 and 7 cm-3 and the reconstructed solar wind data
reproduce the solar wind variations seen on timescales of about
15min. For example, the density dip around 16:25 UT, the
increase and decrease around 17:55UT, and the increase at
18:15UT agree very well. The reconstructed density shows a small
offset of about 0.2 cm-3 compared to the solar wind monitor data.
Such an offset can be produced by a small spacecraft potential
. Comparing THC and THB solar wind observations, typical
offsets of about 10 % can be observed. Thus, we assume that 10 % is a
typical data error for an offset. The solar wind reconstruction fails at the
bow shock and at the magnetopause as discussed before. The velocity
reconstruction reproduces the solar wind variations, e.g., observed around
16:50UT. However, similar to the previous example, there are
additional small variations on timescales less than 10min in the
reconstruction, which are less pronounced or not present at all in the solar
wind data, e.g., at 18:10UT. Both solar wind data, as well as the
reconstruction, provide a slightly different offset. The magnetic field data
are directed approximately along the z direction during the transition and
varies between 3 and 6nT. Similar to the density
variations, the magnetic field variations on scales larger than
15min are well reconstructed from the THC magnetosheath data,
e.g., the magnetic field decreases at 15:40, around
18:00, and at 19:00UT.
Finally, a third THC magnetosheath crossing is used to investigate our
reconstruction method. The THC data from 16 September 2009 were used to
reconstruct the solar wind data during the magnetosheath transition. The
magnetosheath data and model predictions agree very well, similar to the
previous examples discussed. THB and OMNI solar wind data are compared to the
THC reconstruction in Fig. .
The reconstructed solar wind boundary conditions of the model
density (top panel), the ion velocity (middle panel), and the magnetic field
(bottom panel) for the magnetosheath transition on 16 September 2008 are
shown in blue. Additionally, the THB (red) and OMNI (green) solar wind data
during this time interval are shown.
The solar wind densities show a small offset. It is seen that a density and
magnetic field variation at 19:35 UT is observed by THB and also
seen in the reconstruction, but not in the OMNI data. This indicates that
this variation is a local solar wind phenomenon. The reconstruction can
reproduce such local solar wind variations better than the OMNI solar wind
predictions for the subsolar point. Closer to the bow shock and magnetopause,
the reconstruction fails as discussed previously.
Reconstruction alongside the stagnation streamline
The magnetosheath model used is limited to the stagnation streamline region.
To evaluate the applicability of the reconstruction using observations at
some distance off the stagnation streamline, we consider a magnetosheath
traversal on 24 August 2008 where the THC spacecraft is positioned 5–6 RE
off the stagnation streamline. The corresponding density, velocity,
and magnetic field observations are shown in Fig.
together with the best magnetosheath model predictions for any choice of the
solar wind parameters.
The magnetosheath density (top panel), the ion velocity (middle
panel), and the magnetic field (bottom panel), which are observed by THEMIS on
24 August 2008, are presented in red. The best match for the data to the
magnetosheath model for any choice of the solar wind boundary parameters was
found for the magnetosheath model predictions, which are shown in blue.
The density differs closer to the magnetopause location, seen on the
right-hand side of the plot. The reconstruction method takes not only the
density, velocity, and magnetic field magnetosheath data into account, but
also the spacecraft's position where the data point was measured. Thus, more
measured parameters are used (xSC, ySC, zSC, ρSC,
uSC, BSC, and pSC) than free solar wind boundary parameters
(ρSW, uSW, BSW, and pSW), which need to be determined.
The misfit of model prediction and data can be due to limitations of the
magnetosheath model itself or due to data errors.
The reconstructed solar wind parameters and the OMNI, as well as the THB solar
wind data, are presented in Fig. .
The reconstructed solar wind boundary conditions of the model
density (top panel), the ion velocity (middle panel), and the magnetic field
(bottom panel) for the magnetosheath data presented in Fig.
are shown in blue. Additionally, the THB (red) and OMNI
(green) solar wind data during this time interval are shown.
The solar wind density of THB and OMNI show an offset of about 15 %.
Because the electron and ion densities are slightly different, a spacecraft
potential might be present, which can explain such an offset. The
reconstructed solar wind density agrees with the THB density. However, a
significantly different density behavior can be observed at
00:50UT, where the density increases in the reconstruction but not
in the solar wind monitor data. The corresponding density peak can be
observed in the magnetosheath data in Fig. . This might
be either a very local solar wind density enhancement or a time-dependent
process, which is not covered by our model. Apart from that, in the first
1.5h the differences in the reconstruction of the solar wind
data are similar to the previous results for the stagnation streamline
events. Closer to the magnetopause, the calculated solar wind parameters
predict a continuous velocity increase during the transition. However, the
OMNI and THB solar wind data offer a nearly constant solar wind velocity.
More magnetosheath transitions in the neighborhood of the stagnation
streamline were investigated. Nearly all solar wind reconstructions from
magnetosheath data alongside the stagnation streamline offer a systematic
velocity increase. The increase is more significant far away from the
stagnation streamline. The solar wind velocity distribution is not correlated
with a magnetosheath transition of a spacecraft. Consequently, one would not
expect an increase of the velocity in all transitions alongside the stagnation
streamline. Thus, the systematic velocity increase is due to model
limitations. Note that this limitation can be examined without solar wind
monitor data because of the systematic increase in the reconstructed solar wind
velocity. This is not surprising because the series expansions used to derive
the magnetosheath model restricts the model to the stagnation streamline
. The model predicts an approximately continuous linear
decrease of the velocity in the magnetosheath on the stagnation streamline.
However, alongside this streamline, the velocity also decreases, but it still
jumps at the magnetopause from a certain value to zero, which is outside the
scope of the model used here .