International Conference “Recent Problems in Computational Mathematics and Mathematical Modeling” November, 29 – 30, 2010, Moscow, Russia Past, Present and Future of Atmospheric Circulation Studies at INM RAS V.P. Dymnikov V.P. Dymnikov Institute for Numerical Mathematics, RAS

Introduction - I In the atmospheric sciences there are two main problems – weather prediction and climate change. These problems are connected each other. Let’s imagine that we have ideal mathematical model of climate system. Let A- global attractor of this model, m* – probabilistic ergodic invariant mesure on A. Then : Weather prediction : convergence of m(t) = B(m(t0)) to m*

Introduction - II Climate change : delta m* Mean rate of convergence m(t) to m* on A is climatic characteristic and defines the weather predictability. The predictability time is defined, in particular by energy and enstrophy transfer along spectra. Therefore I’l consider our achievements in both problems. Because the roots of INM belong to Computer Centre of SB of AS I’l begin the consideration of past from the period of our work in CC.

Large-scale hydro-thermodynamics of the atmosphere Subgrid-scale processes: parameterization

Parameterization of subgrid-scale processes Turbulence in the atmospheric boundary layer, upper ocean layer and bottom boundary layer Convection and orographic waves Diabatic heat sources (radiative and phase changes, cloudiness, precipitation, etc.) Carbon dioxide cycle and photochemical transformations Heat, moisture and solute transport in the vegetation and snow cover Production and transport of the soil methane Etc.

Weather prediction ( )-I Model: Primitive hydrostatic hydrodynamic equations, Transport of humidity fields, Boundary layer (calculation of air surface temperature and wind), Precipitation

Weather prediction ( )-II What was original (complitly new)? 1. Splitting-up method of solution (transport, adaptation, physics) 2. Reduction on three-dimensional adaptation system to the few two-dimensional problems 3. New equation for the transport humidity, condensation and precipitation 4. SOS-scheme (control energy scheme)

Weather prediction ( )-III Project PASP (Hydrometeorological Institute, Novosibirsk, ): Data –analysis – model –postprocessing (Computer Vesna, 200 Kflops, Levin) Present time: Semi-Lagrangian operative model of medium- range weather forecast (INM – HMC – Meteo-France )

Coupled model of general circulation of the atmosphere and ocean ( ) - I What was original (new )? 1. Symmetrization of primitive equations 2. Exact conservation of energy by finite-difference approximation (adiabatic approximation ) 3. Original parameterizations of all sub-grid physical processes 4. Splitting-up method 5. Factorization of implicit scheme 6. Reduced grid for iteration processes

Conservation laws 1. Mass conservation 2. Angular momentum balance 3. Total energy balance If then there is total energy conservation law

Для адиабатической атмосферы интегральный закон сохранения полной энергии не является квадратичным, так что выполнение его в конечномерном аналоге не обеспечивает вычислительной устойчивости. Если сделать замену переменных то закон сохранения будет иметь вид так что систему уравнений термогидродинамики атмосферы можно представить в виде:

Схема Кранка-Николсон обеспечивает точный закон сохранения энергии для конечно-мерной аппроксимации, если оператор задачи аппроксимируется кососимметрической матрицей (что делается элементарно): Если то метод расщепления также обеспечивает такое сохранение полной энергии (квадратичной формы), что обеспечивает и вычислительную устойчивость. Модель циркуляции атмосферы и океана на основе этой идеи была построена и описана в монографии (Г.И. Марчук, В.П. Дымников, В.Б. Залесный, В.Н. Лыкосов, В.Я. Галин. Математическое моделирование общей циркуляции атмосферы и океана. – Л.: Гидрометеоиздат, 1984 г., 320 с.)

Coupled model of general circulation of the atmosphere and ocean ( ) - II First version -10 degrees in longitude, 6 degrees in latitude, 3 vertical levels (10x6x3) Shortcomings: 1. Upper boundary condition (much better for so crude approximation to have the boundary condition on tropopouse) 2. Wrong transfer energy along spectra for “real” orography 3. No local balances G.I. Marchuk, V.P. Dymnikov, V.B. Zalesny, V.N. Lykossov, V.Ya. Galin “Mathematical modeling of general circulation of the atmosphere and ocean“, 1984, Leningrad, Gidrometeoizdat

Coupled model of general circulation of the atmosphere and ocean ( ) - III Next version: 1. conservation of potential enstrophy in two- dimensional approximation 2. Semi-explicit scheme in time Shortcoming: broad stencil (smoothing of high-frequency mid-scale variability) Resolution: 5x4x21 (good results for 1x1x21) AMIP 1, AMIP2

Среднегодовая ошибка ТПО в модели

Давление на уровне моря в декабре- феврале по данным модели (вверху) и наблюдений (внизу)

Температура в декабре- феврале по данным модели (вверху) и наблюдений (внизу)

U-компонента скорости ветра в декабре- феврале по данным модели (вверху) и наблюдений (внизу)

Среднеквадратичное отклонение ТПО в районе Эль-Ниньо по данным модели (вверху) и наблюдений (внизу)

Mathematical problems of climate theory - I 1.Global solvability 2.Existence of global attractor. Estimates of dimension 3.Dynamics on attractor. Local and global Lyapunov exponents 4. Stability of attractor (set and ergodic measure) 5.Response operator to small external forcing (FDT-theorem, periodic orbits)

Mathematical problems of climate theory - II 6. Possibility of calculations of response operator for real climate system 7. Approximation of attractor 8. Calculation of stationary points and periodic orbits on attractor 9. Approximation of atmospheric regimes by stationary points and periodic orbits 10. Stabilization of trajectories

Оператор отклика для первого момента (нелинейная теория) Нелинейная модель: ( - белый шум по времени) «Возмущенная» модель:

Стационарный отклик Уравнение Фоккера-Планка для плотности инвариантной меры имеет единственное стационарное решение.

В первом приближении по В случае нормального распределения

Восстановление отклика CCM0 на синусоидальную аномалию нагревания

If we know М we can find the forcing producing given response in the system statistics

Model response (right) onto the heating (top left) was multiplied by the inverse response operator. Reconstructed forcing is shown on the (bottom left) Apply the forcing to the model, get Reconstruct the forcing as How it works for CCM0?

Projects Modeling of climate and climate change Mathematical modeling of regional climate processes Construction of computational core for Earth System Model of new generation (petascale and exascale computations) Mathematical problems of climate theory

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