A method for joint statistical processing of the forecast fields of geopotential height and temperature over the Northern Hemisphere with a lead time up to 72 hours is described. These forecast fields were obtained from several operational hydrodynamic models. The approach reduces essentially the forecast errors for both meteorological fields for all lead times at standard pressure levels in the troposphere and lower stratosphere as compared to the best hydrodynamic schemes. Along with statistical methods, the variational adjustment of the vertical profiles of geopotential height and temperature is applied and reduces the forecast error when using the approximation of the profiles by cubic splines. The temperature forecast error increases approximately linearly with lead time, and the geopotential height error grows quadratically.

Abstract The turbulent exchange in boundary layer models is usually characterized by a scalar eddy viscosity coefficient assumed to be a positive function of the vertical variable. We introduce a more general form for the turbulence exchange description, which includes two functions that describe the turbulence without any assumption about their positivity. We construct a model of the Akerblom–Ekman type, but with a complex coefficient of turbulent exchange. The basic quality criterion for these models and algorithms is the maximal agreement with meteorological observations. We optimize the agreement between the global meteorological archive of high-resolution wind observations that are provided by World Meteorological Organization (WMO) in Binary Universal Form for the Representation (BUFR). The main result of our work is that agreement between model solutions and observations will be much better if the turbulent exchange coefficient is optimized in the space of all complex-valued functions, and not limited to the cone of real positive functions.

The changes in temperature regime on the territory of Russia and adjacent areas are evaluated using the 20-year archive data of the Hydrometeorological Research Center of the Russian Federation. The dynamics was evaluated for the average annual temperature, the amplitude and phase of its seasonal fluctuations, the dates of the last and first freeze. The results significantly depend on geography and do not give grounds for the conclusion that the nature of the changes is anthropogenic.

For an elliptic equation of the second order with variable discontinuous coefficients and the right side, a scheme of the fourth order of accuracy is constructed. On the jump line, the docking conditions (Kirchhoff) are assumed to be satisfied. The use of Richardson’s extrapolation, as the numerical experiments show, increases the order of accuracy to about the sixth order. It is shown that relaxation methods, including multigrid methods, are applicable to solve such systems of linear algebraic equations (SLAEs) corresponding to a compact finite-difference approximation of the problem. In comparison with the classical approximation, the accuracy increases by a factor of about 100 with the same complexity. Various variants of the equation and boundary conditions are considered, as well as the problem of determining the eigenvalues and functions for a piecewise constant coefficient of the equation.

Получен критерий для проверки монотонности неявных схем, аппроксимирующих линейные эволюционные уравнения в частных производных. В некоторых случаях его можно применять и к нелинейным уравнениям. Рассмотрено применение этого критерия к простейшим схемам. Критерий может использоваться при построении схем, обладающих улучшенными свойствами точности и устойчивости при обязательном выполнении свойства монотонности.

The Hydrometeorological Center of Russia receives agrometeorological information from about 950 stations one time per ten days and the remote sensing Advanced Scatterometer (ASCAT) data from three Meteorological Operational (MetOp) satellites. We suggest a combined objective analysis (OA) of the available water content based on the available water content measurements at agrometeorological stations and on remote sensing data. The new version of OA is constructed using two neural networks and the backpropagation of error to learn it simultaneously. The first neural network is used to convert the ASCAT data into the available water content values, and the second network is used to estimate the inhomogeneities of soil moisture fields. We use the optimal interpolation (OI) method for assimilation of the ground-based data. In the new version, we evaluate the correlation functions (CFs) of inhomogeneous non-Gaussian fields, not from sample statistics but from machine learning methods. The method takes into account the combining of various datasets: ASCAT data, Food and Agriculture Organization (FAO) soil types, European Space Agency (ESA) GlobCover, and National Center for Atmospheric Research (NCAR) climate data.

Differential relations include both differential operators and solvers for boundary value problems. The formulas of compact finite difference approximations for first- and second-order differential relations of the form $${{P}_{1}}[u] = {{P}_{2}}[f]$$ are obtained. An approximation on three-point stencils is used. Its implementation, as in the case of classical difference schemes, requires the inversion of a tridiagonal matrix. However, compact schemes provide significantly higher accuracy and the fourth order of approximation instead of the second.

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Pade approximation is used. Numerical experiments confirm the high "transparency" of these boundary conditions and determine the conditional stability regions for finite-difference scheme. (C) 2020 Elsevier Inc. All rights reserved.

Compact finite-difference schemes are well known and provide high accuracy order for differential equation with constant coefficients. Algorithms for constructing compact schemes of the 4-th order for boundary value problems with variable (smooth or jump) coefficient are developed. For the diffusion equations with a smooth variable coefficient and the Levin – Leontovich equation, compact finite-difference schemes are also constructed and their 4-th order is experimentally confirmed. The method of constructing compact schemes of the 4-th order can be generalized to partial differential equations and systems with weak nonlinearity, for example, for the Fisher – Kolmogorov – Petrovsky – Piskunov equation, for the nonlinear Schrödinger equation or for the Fitzhugh – Nagumo system. For such nonlinear problems, a combination of simple explicit schemes and relaxation is used. Richardson’s extrapolation increases the order of the circuits to the 6-th. To approximate multidimensional problems with discontinuous coefficients, for example, the two-dimensional stationary diffusion equation in inhomogeneous media, it is necessary to estimate the possible asymptotics of solutions in the vicinity of the boundary line’s breaks. To do this, we use generalized eigen-functions in the angle, which can be used as a set of test functions and build compact difference schemes approximating the problem on triangular grids with high order of accuracy. The asymptotics along the radius of these generalized eigen-functions (in polar coordinates in the vicinity of the vertex of the angle) have irrational indices which can be found from a special dispersion equation and which determine the indices of the corresponding Bessel functions along the radius. For a number of difference schemes approximating the most important evolutionary equations of mathematical physics, it is possible to construct special boundary conditions imitating the Cauchy problem (ICP) on the whole space. These conditions depend not only on the original equation, but also on the type of the difference scheme, and even on the coefficients of the corresponding differential equation. The ICP conditions are determined with accuracy to a gauge. But the choice of this gauge turns out to be essential with numerical implementation. The role of rational approximations of the Pade – Hermite type of the symbol of the corresponding pseudo-differential operator is important. Examples of movie solutions of problems with ICP conditions for various finite-difference schemes approximating the basic mathematical physics equations, see https://cs.hse.ru/mmsg/transbounds. The study was realized within the framework of the Academic Fund Program at the National Research University – Higher School of Economics (HSE) in 2016–2017 (grant No. 16-05-0069) and by the Russian Academic Excellence Project «5–100».

Изучены пространственные корреляции станционных данных измерений запасов продуктивной влаги в пахотном и 10-сантиметровом слоях почвы и спутниковых данных об относительной влажности верхнего слоя почвы (ИСЗ MetOp-А и В, скаттерометр ASCAT) по европейской территории России. Для этого в тестовом режиме использовалась версия компьютерной технологии, при помощи которой осуществлялся контроль станционных данных, а затем выполнялся объективный анализ.

We describe our scheme of the operative wind (and possible gusts) forecasts with the lead time up to 3 days and evaluate its success. It uses joint statistical processing of the results of several best operative forecasting hydrodynamic weather forecasting schemes. This approach allowed us to reduce the error with respect to original schemes. The operative forecast and its evaluation for the period of 2014–2016 is implemented for ~ 2800 cities of Russia, Belarus, and Central Asia. The update results are represented at the official site of the Hydrometeorological Center of Russia every day at 8:30 (a.m. and p.m.), Moscow time.

We develop a new compact scheme for the second-order PDE (parabolic and Schrodinger type) with a variable time-independent coefficient. It has a higher order and smaller error than classic implicit scheme. The Dirichlet and Neumann boundary problems are considered. The relative finite-difference operator is almost self-adjoint. (C) 2018 Elsevier Inc. All rights reserved.

A compact difference scheme on a three-point stencil for an unknown function is proposed. The scheme approximates a second-order linear differential equation with a variable smooth coefficient. Our numerical experiment confirms the fourth order of accuracy of the solution of the difference scheme and of the approximation of the eigenvalues of the boundary problem. The difference operator is almost self-adjoint and its spectrum is real. Richardson extrapolation helps to increase the order of accuracy.

The statistical scheme is proposed for the forecast of surface air temperature and humidity using operative weather forecasts with 3–5-day lead time from the best forecasting hydrodynamic models as well as the archives of forecasts of these models and observational data from 2800 weather stations of Russia, Eastern Europe, and Central Asia. The output of the scheme includes the forecasts of air temperature for the standard observation moments with the period of 6 hours and extreme temperatures with the lead times of 12–120 hours. The accuracy of temperature and humidity forecasts for the period from July 2014 till June 2017 is much higher than that for the forecasts of original hydrodynamic models. The skill scores for extreme temperature forecasts based on the proposed method are compared with the similar results of the Weather Element Computation (WEC) forecasting scheme and forecasts by weathermen.

Internal gravity wave (IGW) data obtained during the passage of atmospheric fronts over the Moscow region in June–July 2015 is analyzed. IGWs were recorded using a group of four microbarographs (developed at the Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) located at distances of 7 to 54 km between them. Regularities of variations in IGW parameters (spatial coherence, characteristic scales, propagation direction, horizontal propagation velocity, and amplitudes) before, during, and after the passage of an atmospheric front over the observation network, when the observation network finds itself inside the cyclone and outside the front, are studied. The results may be useful in studying the relationships between IGW effects in different physical fields at different atmospheric heights. It is shown that, within periods exceeding 30 min, IGWs are coherent between observation points horizontally spaced at distances of about 60 km (coherence coefficient is 0.6–0.9). It is also shown that there is coherence between wave fluctuations in atmospheric pressure and fluctuations in horizontal wind velocity within the height range 60–200 m. A joint analysis of both atmospheric pressure and horizontal wind fluctuations has revealed the presence of characteristic dominant periods, within which cross coherences between fluctuations in atmospheric pressure and wind velocity have local maxima. These periods are within approximate ranges of 20–29, 37–47, 62–72, and 100–110 min. The corresponding (to these dominant periods) phase propagation velocities of IGWs lie within an interval of 15–25 m/s, and the horizontal wavelengths vary from 52 to 99 km within periods of 35 to 110 min, respectively.

The following data was used: the archives of measurements of available water capacity carried out at Roshydromet network of stations and satellite measurements of relative humidity of the upper soil layer from ASCAT data (from the MetOp satellites). The statistical structure of the field of available water capacity in the upper 10- and 20-cm soil layers is assessed. The correlations between the Earth remote sensing data and data from agrometeorological stations are revealed. The procedure of automatic data checking from ground-based observations is developed. The algorithm is suggested for statistically optimal conversion of the Earth remote sensing data to the estimate of moisture content in the upper soil layer.

The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schrödinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we need to solve a non-linear system. We use for this step a simple explicit difference scheme and then Newton – Raphson iterations, which are implemented by the double-sweep method. Numerical experiments confirm the 4-th order of an algorithm. The Richardson extrapolation improves it up to the 6-th order.

The statistical scheme using the results of the best foreign global schemes and the COSMO-RU7 regional scheme is proposed for forecasting surface temperature for five days and the amount of precipitation for three days. Presented are the estimates of prognostic values of the surface air temperature and amount of precipitation for the period of July 2010–June 2013. The joint statistical taking account of different types of systematic errors in the complex forecasting scheme enables excelling all initial schemes in quality. A scheme of the complex forecast is operationally used and its results are daily updated on the website of the Hydrometcenter of Russia at 09:15. The forecasts of extreme temperature, dew-point temperature, and wind speed near the surface are also presented on the website.