We explore two new modifications of the magnetotail magnetic flux
(

Magnetotail magnetic flux is one of the key parameters in
magnetospheric physics

Based on a large data set of simultaneous solar wind and tail lobe
measurements, PR96 constructed an empirical model, giving the
magnetotail radius

The scheme, illustrating the procedure of

Assuming

According to PR96:

In the present study this empirical dependence is reexamined, and now it looks as follows:

Formula (2b) will be explained in more detail in Sect. 4 and in the Appendix. As is shown there, formula (2b) suits the tests made in this study better than (2a), so hereafter we everywhere use it in our calculations.

Finally, the magnetic flux value is calculated as follows:

The algorithm is described in more detail in

As the algorithm is based on pressure balance, it was applied in
the region where the “tail approximation” is valid, i.e.,
approximately at

All comparisons are based on GMHD simulations made in the
Community Coordinated Modeling Center (CCMC;

Results of the simulation BATSRUS_Gordeev_110309_1
in the meridional
magnetosphere cross section

Results of the simulation BATSRUS_Gordeev_110309_1.
Top panel: IMF

To broaden the working area, we make further modifications to the
method. Notice that the open magnetic flux is formed by the
“external” part of magnetic field

Analogously to

Note that the procedure of

However, the usage of the

The magnetopause proxy, corresponding to the

So we have three magnetotail magnetic flux proxies,

The quantities that may be calculated in the MHD simulations are

In this section we present the results of the

Figure 3 presents results based on the simulation
BATSRUS_Gordeev_110309_1, which was already presented in
Fig. 2. The only variable input parameter
IMF

Figure 3 shows that the

Another CCMC simulation studied is BATSRUS_Sergeev_060508_1.
The input parameters (including the dipole tilt) were taken from a
real event, 00:00–16:00 UT 5 March 2008 (Fig. 4). The event includes
different geomagnetic situations with variable IMF

The input parameters of the simulation
BATSRUS_Sergeev_060508_1. From top to bottom: ion density
(

Results of the simulation BATSRUS_Sergeev_060508_1.

Real event simulation BATSRUS_Sergeev_060508_1.
Distributions of the average over the simulation flux
value, correlation and regression coefficients and the free term,
describing the relationship of

Figure 5a presents the “merging electric field”

We made a regression analysis of

Note that the major discrepancies between algorithm results and

According to Fig. 5 the variations and absolute values of

However, until now we considered only separate observation points. Below, we discuss the global distribution of different magnetic flux estimates.

Figure 6 presents the global distribution of results of regression
analysis of all three (

The comparison is carried out using the equation

Figure 6 presents the distribution of the following statistical
parameters: average over the simulation

The most prominent feature of Fig. 6 is the scope of the

The upper row shows that the average

In summary, the

The magnetopause radius in the terminator plane (

The formula is valid for all tilt angles in the SW

Summary for methods application.

To check whether the new formula improves the result, we compared
calculations based on the PR96 terminator radius and on the new

The values from Eq. (8) turned out to be positive for all algorithm modifications, indicating slightly higher accuracy of the flux estimates. So the new formula for the tail radius at terminator is preferable.

The value of the regression coefficient

To emphasize the ability of our modified

The results for Events 1 and 2 are presented in panels a and b
correspondingly. The sampling points are taken at

High uniformity of the regression coefficient

It is important to note that the algorithm presumes zero dipole
tilt. In Event 2 the dipole tilt varied between

We present an algorithm for magnetotail magnetic flux

Also a new empirical dependence for terminator radius

To find the analytical representation of

Top panel – scatterplot of terminator radius dependence
on dipole tilt angle. Bottom panel – demonstration of results of
the data correction procedure (Eq. A2) that eliminates the

The figure demonstrates significant

Since we obtained a clear and strong

The relationship between solar wind dynamic pressure and terminator radius with subtracted dipole tilt dependence. The red curve is the regression line corresponding to Eq. (A4). The black line is the moving average (with the window size 1 nPa, step size 0.2 nPa).

Now we can determine the

Dividing the

To reveal the IMF

Relationship between IMF

Substituting the

Remember that in order to “move” the original data to the
terminator plane we used the analytical magnetopause shape from
PR96. To verify how much this procedure could affect the results,
we tried another model developed by

The developed empirical model of the terminator radius has a
series of interesting features. It demonstrates a strong enough
perturbation of magnetopause position in the

Terminator radius dependence on solar wind dynamic
pressure for different empirical models (see the legend). For each
model, two profiles corresponding to IMF

However, the most interesting observation is the IMF

We thank V. A. Sergeev for useful discussions and Marianna Holeva for her help in the manuscript preparation. We are grateful to the topical editor C. Owen for their objective assessment of this paper and to the referees for their constructive criticism.

Global MHD simulation results were provided by the Community
Coordinated Modeling Center at the Goddard Space Flight Center through
their public Runs on Request system (

The work was carried out as part of EU FP7 ECLAT project. Data on
magnetotail magnetic flux (