1. Water Problems Institute of Russian Academy of Science
 Nonlinear Processes in Geophysics

2. IN NONLINEAR HYDROLOGY»
Naidenov Vyacheslav and Shveikina Valentina
«STOCHASTIC METHODS IN
NONLINEAR HYDROLOGY»

3. The development of the thermodynamic theory of irreversible processes and the theory of nonlinear dynamic systems and classical mechanics led to a qualitatively new understanding of complex phenomena of nature.
A common mathematical property of dynamical systems is the nonlinearity of balance equations of pulse, heat, and matter, due to which positive feedback can be accomplished between the determining factors of the system, which leads to bifurcation, instability, and phase transitions.
The most urgent problems of modern hydrology and climatology are examined from the position of nonlinear mechanics and heat physics.

4. Instability and bifurcation in fluctuations of enclosed water bodies level. The Caspian Sea phenomenon.

5. Year-to-year oscillations in the level of enclosed water bodies without consideration of underground water flow are determined by a water-balance equation, averaged with respect to area
is the level (depth) of the water body ;
is the area of the evaporation surface. ,
are the river flow and layer of visible evaporation (the amount of evaporation with subtraction of precipitation);

6. The stochastic differential equation
Let
and
be fluctuations.
be regular values and
For simplicity we assume processes
statistically independent.
The stochastic differential equation
(1)
are the average river flow and layer of visible evaporation;
are the intensities of these random processes;
are the Wiener measures of inflow and the layer of visible evaporation;

7. is the two-dimensional probability density of the level.
The following Fokker-Plank-Kolmogorov equation corresponding to (1) during the interpretation of the stochastic integral in the Stratonovich sense.
(2)
where
is the two-dimensional probability density of the level.
The initial condition for problem (2) has the form
where
is the Dirac function

8. is significantly nonlinear
For steady-state problems (
i.e., the probability flux from the
state space is equal to zero) the following equation is obtained for extrema of the probability density of the water level:
(3)
Analysis of (3) shows that
and
dependences also exist,
which yield several extrema of the probability density. The dependence of the
water body level
on river flow
is significantly nonlinear
and ambiguous: several levels can correspond to the same value of river flow.

9. The hysteresis dependence of H on P

11. Mathematical model of fluctuation of river discharge
(Gamma distribution of river runoff)
Consider the following model of river runoff variations, comprising the equation of water budget for a basin and the exponential dependence of runoff Q on water reserves W.

12. Changing variables by the formulas
We obtain the equation
- is standard Wiener process

13. The Fokker-Plank-Kolmogorov equation corresponding to the above random differential equation has the form
The stationary solution of the latter equation is gamma distribution

14. This nonlinear model can be used to solve new problems of stochastic hydrology, which cannot be studied by methods of the correlation theory of random processes.

15. Stochastic Dynamics of the Hydrosphere
We would like to demonstrate how chaotic fluctuations can appear in global processes in the hydrosphere and climate without external casual influences..

16. Balance of the thermal energy
The basic laws of conservation
Balance of the thermal energy
The water balance of dry land
The change in the kinetic energy of water upon its flow from the continents
(1)
Balance of the carbon dioxide ; ;;

17. - the planetary albedo;
Accordingly
– the solar constant; ;
- the amount of outgoing thermal radiation;
- the planetary albedo;
- the heat capacity of the system;
- precipitation;
- evaporation from the surface of continents;
- the effective acceleration due to gravity;
- the characteristic time of subsurface water discharge delay in the active water exchange zone; -
- the characteristic time of carbon dioxide relaxation;
- a value characterizing the rate of carbon dioxide output from the earth’s surface into the atmosphere;
- stationary concentration carbon dioxide and temperature.

18. The reflect energy is a highly variable factor:
Mass of ice and snow on dry land;
Clouds in the sky;
World ocean area;
The land’s humidity;
Kind of vegetation.
Water has maximum heat capacity and the greatest solar energy absorption capability.
Heat capacity:
deserts 0.8 – 1
ocean J/g
Albedo ( ratio of size of the reflected energy to falling)
deserts 0.28
ocean

19. Land’s albedo depends on:
The best albedo reflectors are glaciers The worst albedo reflectors are oceans
Land’s albedo depends on:
the type of soil;
color and structure of soil;
soil humidity;
vegetation.
Albedo of:
Peat weakly podzolic supersandy – 0.18 – 0.24
Humid soil – 0.18
Wet land – 0.16
Land soaked through – 0.11

21. Instability of global temperature of atmosphere

22. T

23. W

24. changes in the Earth’s climate.
The analysis of the planetary thermal regime showed that synchronous and in-phase increase (decrease) of moisture reserve of all continents
leads
to a decrease (increase) in the planetary albedo
and a sharp spontaneous increase (decrease) in the global temperature
as well as
changes in the Earth’s climate.
Thus, global warming and cooling, as well as sharp changes in the concentration of carbon dioxide in the atmosphere, are explained by natural processes.

25. The reason of global climatic changes is a nonlinear dependence of
heat-physics properties of dry land on water capacity
Hydrosphere, atmosphere, lithosphere, cryosphere
are connected by uniform global process,
which cannot be divided on
climatic and hydrological processes