5. Determination of the Meridional and Zonal Wind
From the Longitudinal Variations in W
[10] The velocity of the plasma vertical drift caused by the neutral
wind is described by the known relation:
 | (2) |
where
U is the zonal (positive eastward) and
V is the meridional
(positive northward) components of the wind, and
I and
D are the
inclination and declination of the magnetic field, correspondingly.
Then one can try to solve the inverse problem: using the known
variations in the plasma vertical drift
W with longitude,
l, to
determine both components of the neutral wind with the accuracy
up to the first Fourier harmonics. To do this, we expand the right-hand
and left-hand parts of equation (2) into a finite Fourier series:
 | (3) |
[11] It is known that the geomagnetic field parameters are well
enough determined by two harmonics of the Fourier expansion:
 | (4) |
[12] Then according to (2), the variations in the plasma vertical drift
W should be described by three harmonics of the Fourier series. If
one substitutes relations (3) and (4) into equation (2) and equalizes
the corresponding terms, we obtain the equation system:
 | (5) |
where
v =(V0, Vc, Vs, U0, Uc, Us)T,
w = (W0, W1c, W1s,
W2c,
W2s,
W3c, W3s)T and
A is the matrix of the
7 
6 dimension, its elements depend only on the magnetic field
parameters.
[13] System (5) consists of 7 linear algebraic equations in 6
unknowns. In a classical sense it can have an infinite number of
solutions, one solution (when one of the equations is a linear
combination of 6 others) or no solutions at all. Note that vector w
is determined experimentally and contains some errors. As a result,
the classical solution of the system (even if it does exist) may
describe the physical situation inadequately. However, we can try
to find a normal solution
[Tikhonov and Arsenin, 1986].
The vector
with a minimal norm among those vectors for which the difference
between the right-hand and left-hand parts of the system is
minimal (such vectors are called pseudosolutions) is called a
normal solution. It is known
[Tikhonov and Arsenin, 1986]
that the
normal solution for system (5) exists and is unique. However, a
determination of the normal solution is an incorrect problem: small
changes (errors) in the input data (i.e., in the w vector) may cause
rather large changes in the solution. To find a normal solution
stable to small perturbations of the right-hand side of system (5),
the Tikhonov regularization method
[Tikhonov and Arsenin, 1986]
was applied.
|
|
Figure 2
|
[14] The calculations performed have shown that the regularization
method provides a stable solution for the meridional component of
the wind
V in the entire latitudinal belt considered, whereas for
adequate determination of the zonal component
U the accuracy in
determination of the longitudinal variations in
hmF2 is not
sufficient. Therefore the neutral wind model HWM
[Hedin et al., 1991]
is used in subsequent calculations. Currently, it is the only
global empirical model of the neutral wind so it is often used for
calculations and comparisons with measurements. As a rule, a
good agreement with both calculations and other measurements is
noted. That raises some doubts, taking into account large
inaccuracy of measurements of the wind velocity (by all methods)
and insufficiently large data set used for the model elaboration
(especially in the Southern Hemisphere)
[Hedin et al., 1991].
Therefore the calculations were performed in order to determine
how accurate the winds components obtained from the HWM
model describe the longitudinal and latitudinal variations in
hmF2.
For this purpose the direct problem was solved: first longitudinal
variations in the vertical drift
W were calculated on the basis of the
model values of the wind components, and then, using the servo
model the longitudinal variations in
hmF2 were calculated. The
|
|
Figure 3
|
results of the calculations are shown in Figures 2 and 3. Comparing
Figures 2a and 3a to Figures 1a and 1c, correspondingly, one can
see that the longitudinal variations in
hmF2 obtained on the basis of
HWM model differ strongly in shape from the experimental ones
(especially at high latitudes of the Southern Hemisphere) as one
could have expected. Thus the HWM model, on the whole,
inadequately reproduces the longitudinal variations in the neutral
wind. On the other hand, the considered average values of
hmF2 are similar, and the local maximum in the meridional wind at
longitudes about
180-240o in the Southern Hemisphere is again
observed (see Figure 3c).
[15] We tried to correct the HWM model using reliable
measurements of
hmF2 and applying the thermospheric and
ionospheric models well recommended. We used the fact that the
main contribution into the
hmF2 variations is provided by the
meridional component, whereas even strong changes in the zonal
component of the wind weakly influence these variations
[Karpachev and Gasilov, 2000].
The longitudinal variations in the
zonal component of the wind calculated using the HWM model,
were presented by one first harmonic, because taking into account
the higher harmonics would exceed the measurement accuracy.
Using the smoothed zonal wind and applying the regularization
method, we determined the variations in the meridional component
of the wind describing most accurate the experimental values of
hmF2.
|
|
Figure 4
|
|
|
Figure 5
|
[16] Figures 4 and 5 show the longitudinal variations in
hmF2 according to the data of the Intercosmos 19 satellite (Figures 4a
and 5a), calculated longitudinal variations in
W (Figures 4b and 5b),
calculated meridional component of the wind (Figures 4c and 5c),
and smoothed zonal component of the wind for invariant
latitudes 40o, 50o,
60o
and 65o in the Northern and Southern
hemispheres, respectively (Figures 4d and 5d). The obtained
system of the winds reproduces the longitudinal variations in
hmF2 in the entire region of the considered latitudes fairly well. For
comparison, Figure 4b shows also the value of
W obtained for the
considered conditions at the Millstone Hill radar (54o
L,
289oE)
[Buonsanto and Witasse, 1999].
This value is approximately in the
middle between the values calculated for 50o
L and 60o
L,
that is a
partial proof of the correctness of the calculations performed. Thus
the global topside sounding data can be used for a correction of the
neutral wind model.
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