The paper analyzes the statistical and temporal seasonal and decadal variability of the atmospheric pressure field in the Arctic region of Russia. Schemes for the frequency analysis of probability transitions for characteristics of stochastic-diffusion processes were used as the main research method. On the basis of the given series of 60 years long from 1948 to 2008, such parameters of diffusion processes as the mean (drift process) and variance (diffusion process) were calculated and their maps and time curves were constructed. The seasonal and long-term variability of calculated fields was studied as well as their dependencies on a discretization of the frequency intervals. These characteristics were analyzed and their geophysical interpretation was carried out. In particular, the known cycles of solar activity in 11 and 22 years were revealed. Numerical calculations were performed on the Lomonosov-2 supercomputer of the Lomonosov Moscow State University.
Time series analysis (ATS) is one of the most well-developed and widely used areas in
mathematical statistics. ATS methods are successfully applied in geophysics,
economics, engineering and other types of human activity related to the study of data sets. For
example, one of the first applications of ATS methods was the analysis of harvest data
in England in the 18th century [
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). However, this one requires the use of a large amount of computing power, computer time and memory, solving problems of visualizing the results, and many related problems. In recent years, thanks to significant progress in the field of computing systems and numerical modeling, accumulation and processing of big data, experiments on ATS have become available to many research groups and individual users belonging to a certain scientific community. This, in turn, contributes to the further development of numerical modeling, analysis of modeled data and their obtained results with the further comparison.
Research on ATS is also widely used directly in probability theory and mathematical
statistics. One of the methods of analysis is the representation of the series in the form
of a Markov chain and / or a Markov process. Since the literature on Markov processes
is very extensive, we will mention only a few of the most famous works in this area,
which, however, set out all the necessary theoretical provisions and practical methods
for calculating the characteristics necessary for further research. For example, [
In this work, the behavior of the atmospheric pressure field is modeled on the basis
of the Markov diffusion process. Such processes describe well the behavior of the
characteristics of fields that change under the influence of two forces - a short-period one,
called process diffusion, and a long-period one, called drift. These models generalize
the decomposition of a time series into a trend, periodic and random component,
presented earlier in the literature, mentioned in [
The characteristics of those processes are adequately described by such models if two basic conditions are met. First, the increment (that is, the difference between two sequential points in time) should be much less than the total length of the row, and secondly, the field of these characteristics should be sufficiently uniform, that is, the behavior at neighboring points in space does not differ much from each other, especially if this behavior is viewed over long intervals. For the atmospheric pressure field in a relatively small region, which we are considering, these conditions are satisfied. The length of the row is 60 years, while the time step, that is, the increment, is one day. And the size of the cyclonic atmospheric formation, which basically forms the pressure field, is comparable to the dimensions of the entire area under consideration, that is, inside the area for one formation it does not change much. It is important to investigate to what extent the result depends on the division of the actually observed pressure interval (that is, the difference between the maximum and minimum pressure in the entire area) into separate sub-intervals, which are used to calculate the frequency (statistical) characteristics when analyzing the variability of this field.
Methods of diffusion stochastic processes were previously used for various
problems, including for data assimilation problems, in the methods proposed by the authors
[
The work did the following: - the characteristics of the models are built, their features are described, in particular the features of the seasonal and long-term course, the analysis of the features is carried out;
- time graphs and spatial maps of these characteristics were built, and their analysis was carried out.
- the analysis of resistance to the division of the entire pressure interval (maximum minus minimum of the field over the entire region) into frequency sub-intervals was carried out. 2
The variability of a random process, (in our case this is a pressure field), is represented
in the form
dX a(t, X )dt b(t, X )dW ,
(1)
(2)
where X is a pressure value at moment t and at point with given coordinates, this is
not explicitly shown, t is a time, dW is a standard notation of Gaussian ‘white noise’,
that is the generalized random process with zero average and variance equaled 1. Its
covariance function is equaled to delta-function, that is the following
EdW(t)dW ( ) (t ) . Hereafter, (t ) 1, if t or zero otherwise. Besides
that, a(t, x) , b(t, x) are some functions which are calculated according to the work [
tt tt X (t t) X (t) a(u, X )du b(u, X )[W (u du) W (u)].
t t
In (2) the expression represents the Gaussian random variable W (u du) W (u) with
zero average and variance du . The stochastic integral theory and all definitions needed
to understand formulae (1) and (2) can be found in [
According to [
To statistically determine the conditional probability, one need to have a sample of
observations (values) x and y at a fixed point in space. However, since the area under
consideration is homogeneous, as noted above, points with these values can be marked
throughout this area. Namely, the technique for determining these probabilities is as
follows: at step t, all points in the region are marked where X (t) x (xmin x xmax ) .
For simplicity, the values xmin , xmax can be considered the same for all t . Let there be
n(x) such points. Further, at step t dt at those and only at those points where
X (t) x all points are selected where X (t dt) y . Let there be m( y) such points.
Then p( y | x)dt m( y) / n(x) . Obviously it really is a probability. Further, the
calculation of the coefficients is carried out according to formulas (3) and (4). Such a method
for determining the coefficients was previously published for a slightly different
problem in [
The paper considers the field of atmospheric pressure in the area bounded by
coordinates 62°N-80°N and 30°E-90°E, that is, the region of Russia, from the Baltic coast
near St. Petersburg to Severnaya Zemlya and the Yenisei in Siberia. On the one hand,
this region is wide enough to neglect the local features of atmospheric processes; on the
other hand, it is sufficiently homogeneous, since the sizes of large atmospheric
formations are comparable to the dimensions of the entire region. By time, pressure data
were recorded from January 1, 1948 to December 31, 2008, daily in a one-degree grid.
The data were obtained at the Hydrometeorological Center of Russia (HMC) and were
used earlier in some works, for example, in [
Fig. 1(a-d) presents the pressure fields in the considered area on January 1, with the time-interval 20 years (1948–2007).
Figures 1 demonstrate that the pressure fields have the long-term variability but homogeneous enough with respect to the space, since the areas of approximately equaled pressure take most of space domain. Some exemption can be seen in 1987 but even in this case the pressure gradient is not large, its value is approximately 10 gpa per 1000 km. Therefore, this field may be considered as homogeneous with the large confidence level and the aforementioned methods can be applied.
According to the given observational data, using formulas (3) and (4), the coefficients were constructed for each X as the pressure values in the considered area for a specific day. In this case, the division of the entire range of pressure values was carried out into 20 and 60 intervals. We present only the 60 sub-interval discretization.
The next figures illustrate the behavior of average value of coefficients a(t, x) and b(t, x) , their seasonal and decadal variability.
Fig. 2(a-d) shows the behavior of the average coefficient a(t, x) for 2007 when divided into 20 intervals. From formula (1) it is seen that the average for the process dX will be a(t, x) 0 . However, the sample mean a(t, x) may not coincide with the theoretical mean, and this difference needs to be analyzed. 2007 is taken as indicative, in other years the picture is similar. In these figures, it is noteworthy that the spread of the coefficient a(t, x) around the mean value equal to zero is very small in summer, in July, and rather large in the transitional months – April and October, especially in October. Moreover, the deviation from zero is generally positive. This means that atmospheric formations (cyclones and anticyclones) in the area under consideration mainly change in the direction of increasing pressure, that is, the incoming cyclone (and most of them) does not deepen, that is, the pressure does not decrease. 200,00 180,00 160,00 140,00 120,00 100,00 80,00 60,00 40,00 20,00 0,00 b(15 Jan) Figs. 3 and 4 demonstrate the tendencies in ice fraction for 2 different time-periods. Fig. 3 shows the model prediction on 2027 started from 2000 and Fig. 4 shows the difference between 1965–2016 for low resolution model. It is clearly visible the tendency to the ice thickness everywhere in Arctic except some zones in the East of Russia and to the eastward from Novaya Zemlya. The low resolution model shows the global fall of ice thickness during 40-year period everywhere except Severnaya Zemlya archipelago. Some questions arise the slight increase of the ice thickness in Baltic Sea but this can be explained because the increase is not significant, it is about 30 cm and this is really observed in January.
1948 1958 1968 1978 1988 1998 2008
In Figs. 4 and 5 one can see that the coefficient b2 (t, x) quite well corresponds to the coefficient a(t, x) , with some differences. So, from Fig. 5 that the seasonal variation for the coefficient b2 (t, x) is less pronounced, for example, it is almost invisible in April or July, and the interannual variation reflects the 11-year cycle worse (although it also exists) and the quasi-biannual cycle is better than the coefficient a(t, x) . Neither coefficient a(t, x) nor b2 (t, x) contain any linear trends. There is also a strong surge in 1948, explained above.
Fig. 6 shows the spatial location of the coefficient a(t, x) , a total of 4 values for January 15, 1948, 1968, 1988 and 2008, interval 20 years.
It is seen that the jumps in the values of this coefficient are sufficiently localized and do not exceed 5 gpa/day with different signs. The spatial arrangement of this coefficient over the area is uniform, no noticeable localization zones are observed, and it should also be noted that the isolines of the values of the coefficient a(t, x) are quite local in comparison with the dimensions of the area itself (shown in different colors). This indicates the local cause of pressure changes on a large scale.
The coefficient b2 (t, x) (Fig. 7) is more chaotic, its distribution over space is not as common as for the coefficient a(t, x) . In addition, it can be concluded that the spatial location of the coefficient b2 (t, x) is more localized and concentrated in continuous zones (except for the values for 2008). In terms of amplitude, the range of this coefficient is significantly larger than for the coefficient a(t, x) , but locally it occupies a smaller size of the entire area. b) d)
It can also be noted that with an increase in the discretization into the number of intervals, graphs 3 and 5 become smoother, Figs. 4 and 6 are more pronounced, but they qualitatively coincide. Therefore, the results of the work are qualitatively independent of the number of divisions. However, for the correctness and reliability of the calculations, it is required that, when the whole interval is broken down, each sub-interval must contain at least one observation, so that the conditional probabilities can be correctly calculated using formulas (3) and (4). 4
Several characteristics were obtained in this study that reflect both long-term and shortterm behavior of the pressure increment in the Northern region of Russia over 60 years. Knowledge of such characteristics is very useful for medium and long-term forecasts of weather and climate change, as well as for modeling the dynamics of currents in the North Seas of Russia, especially when calculating pilotage along the Northern Sea Route. In addition, the knowledge and forecast of the characteristics obtained in the work will make it possible to calculate and determine the confidence limits of possible pressure values, and hence a number of derivatives of this value, for example, geostrophic wind, which will allow applying this knowledge in determining extreme values, such as strong winds, extreme waves and a number of other characteristics.
This work was carried out with partial support from the Russian Foundation for Basic Research, project 18-29-10085 mk and within the framework of the topics 01492019-0004, and ‘Mathematical methods for data analysis and forecasting’.