Geographical environment and problems of ethnocultural unity of the Ancient South-

Chapter 43. STATES OF SOUTHEAST ASIA IN ANCIENTITY

Southeast Asia is characterized by rugged terrain, alternating high mountains, usually overgrown with damp tropical forest, where small fast mountain rivers flow, with swampy valleys of large and medium-sized rivers. High temperatures and humidity, the richness of the flora led to an increased role of agriculture and gathering and a relatively small role of hunting and especially cattle breeding. One of the most ancient settlements of people who practiced it already in the 8th millennium BC was discovered here. e. agricultural production (cultivation of legumes and melons). The type of rice farming that then emerged in the Neolithic was more or less the same for ancient Southeast Asia, whose territory, which had similarities in the economy and partly in the cultural and anthropological appearance of its inhabitants, was somewhat larger in ancient times than it is now. It included the valleys of Xijiang and Yangtze with right tributaries; its periphery was the Ganges valley, where peoples related to the Mon-Khmers still live. The main ancient peoples of Southeast Asia are the Austroasiatics (Mons, Khmers and Thais) in its continental part and the Austronesians (Malays, Javanese, etc.) in the island part; together they are called the Austric peoples. The most developed were the Austroasiatic

region of the plains of Southern Indochina, where already in the 3rd millennium BC. e. the population independently switched to making tools from copper, and soon - from bronze. This ancient center of metallurgy had a profound influence on the western periphery and on the development of metallurgy in the Yellow River basin. But by the 2nd millennium BC. e. the economic development of Southeast Asia began to lag behind the development of neighboring regions. Hard mode big rivers Southeast Asia made it difficult to create large irrigation systems there as one of the most important conditions for the development of a specific rice culture. They learned to create such systems later. Long time Small rural communities engaged in rice growing remained the main unit of society.

Only in the late Bronze Age, during the famous Dong Son civilization of the 1st millennium BC. e.1,

in the valleys of large and medium-sized rivers of Ancient Southeast Asia, fairly large areas of compact agricultural population arose, which became the base early states. The development of plow farming and complex crafts entailed an increase in labor productivity and a more complex social structure of society. Fortified settlements appeared, and the first states began to take shape.

1 Named after the Vietnamese village of Dong Son, where the burial ground of this culture was first excavated. Its center is

Northern Vietnam.

The oldest written sources, written in peculiar hieroglyphs, typologically close to the early writings of Western Asia (although they arose thousands of years later), were discovered only recently, and their number is negligible. Valuable information is contained in ancient epigraphy in Sanskrit and in early medieval inscriptions in the languages ​​of the peoples of Southeast Asia. An important role in reconstructing the history of this region is played by early medieval chronicles (Vietian, Mon, etc.), as well as testimonies of ancient Chinese, ancient Indian and ancient authors.

The states that arose first of all among the ancient Austroasiatics and the ancient Viet, related to them in language, extended from Western Indochina through modern Northern Vietnam to the lower reaches of the Yangtze. Among them, four groups of states can be distinguished: the states of North-Eastern Indochina and North coast

South (modern South China) Sea; states of Southern Indochina; states of the ancients

Indonesians on the Malacca Peninsula and the Archipelago; states of the central part of the North

Indochina and surrounding northern regions, inhabited by Thai-speaking peoples.

East Asia - concept and types. Classification and features of the category "East Asia" 2017, 2018.

Forces acting in the atmosphere.

The forces acting in the atmosphere are divided into mass and surface:

Mass or volumetric forces.

Mass forces include those forces that act on each elementary volume of air, and are usually calculated per unit mass. These include:

Gravity represents the vector sum of two forces: the force gravity, directed towards the center of the Earth, and the centrifugal force arising due to the rotation of the Earth around its axis and directed along the radius of the circle of latitude passing through the point in question.

Coriolis force (deflecting force of the earth's rotation) is associated with the rotation of the Earth around its axis and acts on air particles moving relative to the Earth (air currents of the atmosphere). The Coriolis force arises as a result of the portable rotational motion of the Earth and the simultaneous movement of air particles relative to earth's surface.

Where? - angular velocity rotation of the Earth.

Using vector analysis formulas, we obtain the components of the Coriolis force along the coordinate axes.

Surface forces. Surface forces include those forces that act on the contacting surfaces of a layer of air.

Pressure force (baric gradient force) occurs due to uneven pressure distribution. The pressure gradient force vector is determined by the relation

and its components, related to a unit of mass, along the coordinate axes, have the following form:

Friction force occurs when air moves, when its different volumes have different speed movements. If we consider the movement of air as the movement of a viscous liquid, then when two adjacent layers of liquid move with different speeds, tangential forces of internal friction (tangential stress), or viscous forces, develop between them. Components of this force along the coordinate axes:

Kinematic coefficient of turbulent viscosity, and - dynamic coefficient of viscosity.

Equation of motion of the free atmosphere

As is known, the density of matter in physics is introduced by passing to the limit: , where in mechanics continuum should be understood as m as the mass of a substance contained in a volume W. Let's see what the law of conservation of mass will look like for an arbitrary moving volume of a continuous medium for which. From (1.12) it then follows:

or due to the arbitrariness of the volume W:

This equation is called the equation of continuity (continuity).

Geostrophic wind

The simplest type of air movement that can be imagined theoretically is rectilinear uniform movement without friction. Such movement with a deflecting force different from zero is called geostrophic wind.

With geostrophic wind, except driving force gradient G = - 1/?*dp/dn, the deflecting force of the Earth’s rotation A = 2?*sin?*V also acts on the air. Since the movement is assumed to be uniform, both forces are balanced, that is, equal in magnitude and directed mutually opposite. The deflecting force of the Earth's rotation in the northern hemisphere is directed at right angles to the speed of movement to the right. It follows that a gradient force equal in magnitude must be directed at right angles to the velocity to the left. And since the isobar lies at right angles to the gradient, this means that the geostrophic wind blows along the isobars, leaving low pressure on the left (Fig. 4.21).

Fig.4.21. Geostrophic wind. G-- pressure gradient force, A -- deflection force of the Earth's rotation, V -- wind speed.

IN southern hemisphere, where the deflection force of the Earth's rotation is directed to the left, the geostrophic wind should blow, leaving low pressure to the right. The speed of the geostrophic wind can be easily found by writing the equilibrium condition of the acting forces, i.e., equating their sum to zero. We get

from where, having solved the equation, we find for the speed of the geostrophic wind

This means that the speed of the geostrophic wind is directly proportional to the magnitude of the pressure gradient itself. The greater the gradient, i.e., the denser the isobars, the stronger wind.

Let us substitute into formula (2) numerical values ​​for air density under standard conditions of pressure and temperature at sea level and for the angular velocity of the Earth’s rotation; Let's express the wind speed in meters per second, and the pressure gradient in millibars per 100 km. Then we obtain formula (2) in a working form convenient for determining the speed of the geostrophic wind (at sea level) from the magnitude of the gradient.

The atmosphere envelops the entire globe, putting pressure on every square meter surfaces. Consequently, on the surface of the Earth and at any altitude, a certain amount of pressure is created at each point, i.e., a pressure field, or baric field. This field can be described as a system of surfaces of equal pressure, so-called isobaric surfaces, for example: 1000 hPa, 850 hPa, 500 hPa, 200 hPa, etc. At sea level, intersections with isobaric surfaces form lines of equal pressure - isobars.

The distribution of pressure on the globe is very heterogeneous, it varies from point to point and changes over time. The heterogeneity of pressure distribution is explained by the uneven distribution of air masses within each column of the atmosphere, which in turn depends on the temperature distribution. If in one geographical area pressure is high, and in another - low, then the air will move from the area more high pressure to the area more low pressure. Moreover, the greater the pressure difference, the greater acceleration acquires air. The pressure difference per unit distance normal to the isobar is called the horizontal pressure gradient. Otherwise, this is the force that moves the air. In addition to the pressure gradient force, inertial forces (Coriolis and centrifugal forces), as well as frictional forces, act. All air currents are considered relative to the Earth, which rotates around its axis. You can understand how the Coriolis force (CF) works if you remember that linear speed rotation of each stationary body on Earth is equal to the product of the angular velocity of rotation of the Earth si by the distance to the axis of rotation r, i.e. u = wr. Let us consider the action of the Coriolis force using the example of the movement of a body of unit mass along a meridian. Let us assume that 1 kg of air in the Northern Hemisphere is located at latitude f and begins to move along the meridian to the north with wind speed V. Due to inertia, this kilogram of air will maintain the linear rotation speed u that it had at latitude f. As a result of moving north, it will be at increasingly higher latitudes, where the distance to the Earth's rotation axis is smaller and the linear speed of the Earth's rotation is less. Thus, this body will be ahead stationary bodies, located on the same meridian, but at higher latitudes, that is, an observer on Earth will be able to note that this body, under the influence of some force, will deviate to the right. This force is the action of the Coriolis force. Similar reasoning shows that in the Southern Hemisphere such a kilogram of air will deviate to the left of the direction of movement. The magnitude of the horizontal component of the Coriolis force acting on 1 kg is equal to SC = 2wVsin. In the Northern Hemisphere, it is directed at a right angle to the right of the wind speed V. From the formula it follows that if the body is at rest, then there is no Coriolis force. It only works when the air is moving.

On our planet, the forces of the horizontal pressure gradient and the Coriolis force are of the same order, so they often almost balance each other. Then the air acceleration is small and the movement is close to rectilinear and uniform. In this case, the air does not move along the pressure gradient, but along or close to the isobar, leaving low pressure on the left in the Northern Hemisphere.
Air currents in the atmosphere have a vortex nature: usually the trajectories of air particles are curved, and the particles move either counterclockwise or clockwise. With this movement, every kilogram of air is acted upon by a centrifugal force V2/R, where V is the wind speed, and R is the radius of curvature of the trajectory. In the atmosphere the force is always less than the pressure gradient force. The force of friction arises between the surface of the Earth and the air moving above it. Unevenness of the earth's surface traps lower volumes of air. The transfer of air volumes with low horizontal speed upward from lower levels delays movement upper layers air. Thus, friction against the earth's surface is transmitted upward, gradually weakening. The force of friction slows down the wind speed. It is noticeable in the 1 - 1.5 km layer, which is called the planetary layer boundary layer. The wind here, due to friction, is deflected from the isobars towards low pressure. Above 1.5 km, the influence of friction is significant, so the higher layers are called a free atmosphere.

Features of the manifestation of gravity in the atmosphere.

Gravity. One of the mass forces is the force of gravity acting on any particle of air, both stationary and moving relative to the Earth.

The force of gravity g is the vector sum of two forces: the force of gravity g, directed towards the center of the Earth, and the centrifugal force c, arising from the rotation of the Earth around its axis and directed along the radius of the latitudinal circle passing through the point in question (Fig). In the figure it is impossible to maintain the correct ratio of the magnitudes of these two forces, since the centrifugal force is too small compared to the gravity force. Indeed, the value centrifugal acceleration is determined by: where - v 2 per transfer speed, a rᵩ - the distance of the particle from the earth's axis.

Since the earth rotates around its axis with a constant angular velocity where T*- day, then at a distance rᵩ from the axis the transfer speed is equal to w rᵩ,. The size rᵩ, is equal rᵩ= r cosᵩ ( r- the distance of the particle from the center of the earth). Taking all this into account, we can write the formula for centrifugal acceleration as follows:

Where w 2 = 7.297 10 -5 1/s is the angular velocity of the Earth’s rotation; r- the distance of the particle from the center of the earth, ᵩ - geographical latitude.

Centrifugal force With very small compared to the force of gravity g, and as it approaches the pole it decreases to zero, and the force of gravity g increases with increasing latitude.

The action of gravity determines the shape of the surface of the world's oceans and, to a large extent, also the shape of the land surface. Obviously, in the absence sea ​​currents the surface of the sea must be perpendicular to the direction of gravity everywhere (otherwise the tangential component of gravity will begin to move water particles). Such surfaces are called level surfaces and approximately represent ellipsoids of revolution, the minor axis of which coincides with the axis of rotation of the earth.

Elastic stress tensor. Connection with viscosity.

Stress tensor is a second-rank tensor consisting of nine quantities representing mechanical stresses in arbitrary point loaded body. These nine quantities are written in the form of a table in which along the main diagonal there are normal stresses in three mutually perpendicular axes, and in other positions - tangential stresses acting on three mutually perpendicular planes.

Complete mechanical stress tensor of the elementary volume of a body. The letter σ denotes normal mechanical stresses, and tangent stresses - the letter τ.

Components of the stress tensor in Cartesian system coordinates (i.e.) are entered as follows. Consider an infinitesimal volume of a body (continuous medium) in the form rectangular parallelepiped, the faces of which are orthogonal to the coordinate axes and have areas. Surface forces act on each face of the parallelepiped. If we denote the projections of these forces on the axis as , then the components of the stress tensor are the ratio of the projections of the force to the area of ​​the face on which this force acts:

There is no summation by index here. Components , , , also denoted as , , are normal stresses; they represent the ratio of the projection of the force onto the normal to the face in question:

Components , , , also denoted as , , are tangential stresses; they represent the ratio of the projection of the force on the tangent directions to the face in question:

In case linear theory elasticity, the stress tensor is symmetrical (the so-called law of pairing of tangential stresses).

Level of continuity.

The continuity equation, often called the continuity equation, is a special form common law conservation of mass, established by Lomonosov, specialized for the case of a continuous medium.

Let us consider the elementary mass of liquid δm, filling the volume δτ . If we trace the movement of liquid particles that make up a given liquid volume, then the constancy of mass can be expressed by the relation
(1)

Because
, then from (1) it follows that
(2)

Substituting this expression into (2) and reducing by δτ, we obtain the continuity equation

It is also possible to translate the continuity equation into another form, more convenient for further conclusions. To do this, we will reveal the expression for the individual derivative of density and for the divergence of speed

This form of the continuity equation is most often used in meteorological research.

For an incompressible fluid, and equation (4) takes the form

The validity of this relationship can also be verified directly by recalling the physical meaning of velocity divergence.

Let us give another expression for the continuity equation in a spherical coordinate system (to derive it it is enough to express it in these coordinates)

13 Turbulence in the atmosphere. Changes in equations

Due to the uneven distribution of pressure in the atmosphere, its air masses move horizontally, causing wind.

Wind speed and direction are constantly changing. Average wind speeds are 5-10 m/s, but can reach 50 m/s or more. In the upper layers of the atmosphere in jet streams wind speed can exceed 100 m/s.

The movement of air in the atmosphere is turbulent. The essence of the phenomenon of turbulence is that vortex flows are formed in a mass of air in motion. These vortices cause chaotic fluctuations in the characteristics of moving air masses, i.e. their speed, direction, temperature, pressure and density. One of the sources of turbulence is the difference in wind speeds in adjacent layers. Turbulence is especially great in the lower layers of the troposphere: in the ground layer at a height of 50-100 m and in the friction layer extending to a height of 1000-1500 m. Turbulence caused by differences in velocities in adjacent layers is called dynamic.

In addition to horizontal movements of air masses, there are also vertical movements in the atmosphere. The speeds of vertical movements are significantly lower than horizontal ones. Under normal conditions, vertical movements are measured in centimeters per second. The development of these movements is associated with the presence of Archimedean or hydrostatic force. Air, warmer near the earth's surface and, therefore, less dense than the environment, moves upward, and cooler air sinks in its place.

Vertical movement of air is called convection. With weak development, convection has a chaotic turbulent character. With developed convection, powerful ascending and descending air currents arise over individual areas of the heated earth's surface, reaching the stratosphere. Downdrafts are usually less intense, but cover much larger areas.

Turbulent mixing is tens and thousands of times greater than molecular mixing or molecular diffusion.

Turbulent diffusion leads to the propagation of heat and moisture in the atmosphere in the vertical direction. The consequence of turbulence is the transfer of momentum from top to bottom, which leads to some equalization of the distribution of wind speed over height. The amount of motion is given by

Where m is the mass of air, v is the speed of movement of this mass.

Since in higher layers of the atmosphere the wind speed is higher than near the earth's surface, when mixing, air masses with higher speeds move to lower levels, resulting in turbulent friction.

In addition to the main components, the composition of air includes variable parts: water vapor, carbon dioxide, ozone, as well as various impurities, i.e. the smallest solid and liquid particles called aerosols. The quantity of any substance is characterized by its specific content s, i.e. mass fraction of the substance.

In the process of turbulent exchange of air, any substance spreads in the direction in which it decreases. The change in a substance per unit distance is called its gradient. In the atmosphere, a decrease in substance is usually observed in the direction from bottom to top.

A quantitative characteristic of turbulent exchange is the flow of the substance, i.e. the amount of substance transferred through a unit area per unit time.

In accordance with the theory, the substance transferred in the process of turbulent exchange must satisfy the following conditions.

1. The amount of substance in an individual air particle during its movement, until it is mixed with the surrounding air, must remain unchanged.

2. When mixing two masses of air, it must remain total quantity substances.

3. The substance must be a passive impurity, i.e. do not influence turbulent movement.

Under these conditions, the flow of the substance is proportional to the gradient mass fraction substances. In the case of vertical transport of a substance, its flow can be expressed by the formula Sв = -А* dS/dZ, where Sв is the vertical transport of the substance, -dS/dZ is the vertical gradient of the substance, A is the turbulent exchange coefficient, depending on atmospheric conditions and the nature of the underlying surfaces.

Turbulent heat transfer in the atmosphere is more complex character. Due to the compressibility of air and the adiabatic changes in its temperature that continuously occur within its thickness, the direction of heat transfer cannot be judged by the direction of the temperature gradient. In a dry adiabatic process, the remaining characteristic of the thermal state of the air mass is its potential temperature.

14. Scalar, vector, tensor quantities

Scalar quantity(from Latin scalaris - stepped) in physics - a quantity, each value of which can be expressed by one real number. That is, a scalar quantity is determined only by its value, in contrast to a vector, which has a direction in addition to its value. Scalar quantities include length, area, time, temperature, etc.

Vector quantity- a quantity is called a vector (vector) if it is determined by two elements of different nature: algebraic element- a number indicating the length of the vector and being a scalar, and geometric element, indicating the direction of the vector.

Vector quantities are indicated by the corresponding letters with an arrow at the top or in bold. Examples of vector physical quantities:

strength; speed; pulse.

Vectors are represented by directed segments. The beginning of the vector is the point where the directed segment begins (point A in Fig. 1), the end of the vector is the point at which the arrow ends (point B in Fig. 1).

Tensor quantities- objects linear algebra, linearly transforming elements of one linear space into elements of another. Special cases of tensors are scalars, vectors, bilinear forms etc. The term "tensor" is also often shortened for the term "tensor field", which is the study of tensor calculus. Many tensor quantities whose tensor rank is 2 are determined by an equation of the form, where and are two vector physical quantities, related by transformation. Examples: inertia tensor; effective mass tensor; dielectric constant tensor.

15. Similarity theory. Scale.

The doctrine of physical research. phenomena based on the concept of physical. likeness. Two physical phenomena are similar if, from the numerical values ​​of the characteristics of one phenomenon, it is possible to obtain the numerical values ​​of the characteristics of another phenomenon by simple recalculation, which is similar to the transition from one system of units of measurement to another. For any set of similar phenomena, all corresponding dimensionless characteristics (dimensionless combinations of dimensional quantities) have the same numerical value. The converse conclusion is also true, i.e. if all the corresponding dimensionless characteristics for two phenomena are the same, then these phenomena are physically similar.

Dimensional analysis and dimensional analysis are closely related to each other and form the basis of experiments with models. In such experiments, the study of a certain phenomenon in nature is replaced by the study of a similar phenomenon on a model of a smaller or larger scale (usually under special laboratory conditions).

After establishing a system of parameters that define a selected class of phenomena, conditions for the similarity of two phenomena are established. Namely, let the phenomenon be determined by independent parameters, some of which may be dimensionless. Let, further, the dimensions of the defining variables and physical. constants are expressed through the dimensions k of these parameters with independent dimensions (). Then from n quantities it is possible to compose only n-k independent dimensionless combinations. All the desired dimensionless characteristics of the phenomenon can be considered as functions of these n-k independent dimensionless combinations made up of defining parameters. Among all the dimensionless quantities, composed of the defining characteristics of a phenomenon, it is always possible to indicate a certain base, i.e., a system of dimensionless quantities, which determine all the others.

The class of phenomena defined by the corresponding statement of the problem contains phenomena that are generally dissimilar to each other. The identification of a subclass of similar phenomena from it is carried out using the following condition.

For the similarity of two phenomena, it is necessary and sufficient that the numerical values ​​of dimensionless combinations, composed of a complete list of defining parameters that form the base, in these two phenomena are the same. Conditions for the constancy of the base of abstract parameters composed of given values, defining the phenomenon, called. similarity criteria. In hydrodynamics, the most important similarity criteria are the Reynolds number, which characterizes the relationship between inertial forces and viscous forces, the Mach number, which takes into account the compressibility of the gas, and the Froude number, which characterizes the relationship between inertial forces and gravity forces. The main criteria for the similarity of heat transfer processes between a liquid (gas) and a streamlined body are: Prandtl number characterizing the thermodynamic. state of the environment; Nusselt number characterizing the intensity of convective heat exchange between the surface of the body and the flow of liquid (gas); Peclet number characterizing the relationship between convective and molecular processes of heat transfer in liquids; Stanton number characterizing the intensity of energy dissipation in a flow of liquid or gas. For the distribution of heat in a solid body, the similarity criteria are the Fourier number, which characterizes the rate of change in thermal conditions in environment and the rate of restructuring of the temperature field inside the body, and the Biot number, which determines the nature of the correspondence between the temperature conditions of the environment and the temperature distribution inside the body. In processes that change over time, the main criteria of similarity, characterizing the similarity of processes over time, are the criteria of homochrony. In problems of aerohydromechanics this criterion is called. Stroukhalya in number. The mechanical similarity criterion motion is a Newton number. When studying elastic deformations, the similarity criterion is Poisson's ratio. If the similarity conditions are met, then for the actual To calculate all characteristics in situ using data on dimensional characteristics on a model, it is necessary to know the transition scales for all relevant quantities. If a phenomenon is determined by parameters, of which k have independent dimensions, then for quantities with independent dimensions the transition scales can be arbitrary and they must be set taking into account the conditions of the problem, and in experiments, taking into account the experimental conditions. Transitional scales for all other dimensional quantities are obtained from formulas expressing the dimensions of each dimensional quantity through the dimensions of k quantities with independent dimensions, for which the scales are suggested by the conditions of the experiment and the formulation of the problem.

For example, in the problem of a steady flow of an incompressible viscous fluid around a body, all dimensionless quantities characterizing the movement as a whole are determined by three parameters: angles a, b (the direction of the translational velocity of the body relative to its surface) and the Reynolds number R. Physical conditions. similarities - similarity criteria - are represented by the relations:

Homochrony number. Application example

Similarity criteria are dimensionless numbers made up of dimensional physical. quantities that determine the considered physical. phenomenon. Any physical the quantity is the product numerical value per unit of measurement and, thus, always depends on the choice of system of units of measurement. The values ​​of the similarity criterion do not depend on the units of measurement. Equality of all criteria of the same type for two physical phenomena (processes) or systems - necessary and sufficient condition physical similarities of these systems.

The homochrony number characterizes the nonstationary nature of the motion process and is used when studying heat transfer in nonstationary (for example, pulsating) flows. The Euler number determines the similarity of pressure fields. In some systems, this number is a single-valued function of the Reynolds number.

VT/L=But, where V is the characteristic speed, T is the characteristic time of process change, L is the characteristic linear size.

The Strouhal number is a special type of homochrony criterion used in hydroaeromechanics.

The homochrony number Ho and the Fourier number Fo are the determining criteria for non-stationary processes. The number, or Fourier criterion, is one of the criteria for the similarity of non-stationary thermal processes. Characterizes the relationship between the rate of change in thermal conditions in the environment and the rate of restructuring of the temperature field inside the system (body) under consideration, which depends on the size of the body and the coefficient of its thermal diffusivity:

where a = l/rc - thermal diffusivity coefficient, (l - thermal conductivity coefficient, r - density, c - specific heat), l is the characteristic linear size of the body, t0 is the characteristic time of change in external conditions.

Since the criteria establishing the relationship between development rates various effects, are called homochronicity criteria, the Fourier number is a criterion for the homochronicity of thermal processes, i.e. relates the times of various effects.

Froude number. Application example

The Froude number (), or the Froude criterion, is one of the criteria for the similarity of the movement of liquids and gases, and is a dimensionless quantity. It is used in cases where the influence of external forces is significant. Introduced by William Froude in 1870.

Froude number in hydrodynamics

The Froude number characterizes the relationship between the force of inertia and external force, in the field of which movement occurs, acting on the elementary volume of liquid or gas:

where v is the characteristic velocity scale, g is the acceleration characterizing the action of the external force, L is the characteristic size of the region in which the flow is considered.

For example, if we consider the flow of liquid in a pipe in a gravity field, then the value g is understood as the acceleration of free fall, the value v is the flow velocity, and L can be taken as the length of the pipe or its diameter.

In shipbuilding, another version of the Froude number is used - the root of the above-mentioned hydrodynamic Froude number.

The Froude number makes it possible to compare wave conditions for ships of different sizes. For large displacement vessels, the Froude number is usually 0.2-0.3, and for small planing vessels it is usually greater than 1, but is usually selected from the range of 2-3.

The Froude Number is also used when modeling water flows in open channels and testing models of hydraulic structures.

Froude number in heat transfer

In heat transfer, the Froude criterion also characterizes the relationship between the force of inertia and the force of gravity, but is expressed differently:

g - free fall acceleration,

l- defining (characteristic) size,

w is the flow rate of liquid or gas.

How larger number Fr, the less the influence of gravity on the properties of motion.

L(10 2 – 2 10 6 m) and speed V g v= 1.5*10 -5 m2/s, for similarity criteria we obtain the following values ​​of the upper and lower limits possible values Froude numbers:

Upper limit 50 2 /10*10 22 =2.5

Lower limit 10/2*7*10 -5 *2*10 6 =4*10 -2

Number of deviations from geostrophicity. Application example

V-characteristic speed, L-characteristic size, ω-angular velocity

The higher the number De, the less the rotational deflection force influences the movement

At large values numbers De on the properties of motion great influence have si-

inertia lines determined by the convective term in the equations of motion.

Taking into account length change intervals L(10 2 – 2 10 6 m) and speed V(10 - 50 m/s) and approximately taking g≈ 10 m/s 2, ω =7*10 -5 1/s and v= 1.5*10 -5 m2/s, for similarity criteria we obtain the following values ​​of the upper and lower limits of possible values:

Upper limit 50/2*7*10 -5 *10 2 =4*10 3

Lower limit 10*10 2 /1.5*10 -5 =7*10 7

Euler's number. Application example

The Euler number (Eu) is a dimensionless coefficient that occurs in the Navier-Stokes equations, describing the relationship between pressure forces on a unit volume of liquid (or gas) and inertial forces.

where ρ is density, Δр is pressure difference spent on overcoming hydraulic resistance, v is speed.

Reynolds number. Application example.

Reynolds number– one of the similarity criteria (dimensionless quantities characterizing the relationships various forces, acting in liquid (gas).

The Reynolds number is used in dynamics before sound flows (flows with velocities lower than the speed of sound) and is determined by the formula where U is the flow velocity, L is the characteristic linear size of the flow. (this can be either a vertical dimension H or a horizontal dimension L, depending on the specifics of the flow under consideration and the need to separate linear dimensions vertically and horizontally), v m is the kinematic viscosity of the liquid (traditionally (when considering laminar flows) under this the value is understood as molecular viscosity, but in meteorology, where turbulent flows are studied, it most often means its “turbulent” analogue ) . The Reynolds number (named after the English physicist Osborne Reynolds) characterizes the relationship between inertial forces and friction forces in a fluid flow. Other formulations of the Reynolds number are quite often used, e.g. where is the difference in flow velocity at the boundaries of the region under consideration, - gradient

velocity in the liquid layer under consideration. The most commonly used Reynolds number is

when studying the patterns of movement of liquids and gases in channels in the absence of rotation.

The higher the Re, the less the viscous force affects the properties of motion.

The value of the Reynolds number, at cat. laminar flow gives way to turbulent flow is called the critical Reynolds number. If , then the flow occurs in laminar mode, and if , then turbulence may occur. Physically this means that friction forces with increasing inertial forces, it is not able to maintain the dynamic equilibrium characteristic of laminar flow, and it gives way new form dynamic equilibrium, at cat. the structure of currents becomes time dependent.

The Re number is used in hydraulics (eg calculating the hydraulic radius of pipes and channels).

21. Defining and internally determined criteria. Examples.

Similarity criteria are divided into 2 groups:

a) Similarity criteria containing defining parameters, i.e. externally determined characteristic quantities and physical constants. The physical constants of a liquid are the characteristic density and kinematic viscosity coefficient. The angular velocity of the Earth's rotation and the acceleration of gravity are also among the determining parameters.

The presence of these criteria imposes additional conditions to externally determined values. Indeed, motion will be similar only when externally determined quantities simultaneously satisfy criteria formed from the equations of motion and from boundary conditions. In other words, each such criterion limits the possibility of carrying out similar movements and is, thus, decisive.

b) Similarity criteria containing at least one of the internally determined quantities, phenomena. non-defining. If all conditions are met. similarities resulting from defining criteria and boundary conditions, then these criteria are necessarily satisfied for the entire class of similar movements.

Thus, when the corresponding dimensionless number is determined for a particular case, the non-determining criterion is a relationship connecting the characteristic values.

When calculating the number of defining criteria, one must be observed important rule– the criteria must be brought to such a form that each internally determined value occurs in only one of them. Obviously, this can always be achieved by multiplying or dividing the criteria containing. the same internally determined quantity. If this rule is not observed, then no conclusions about what criteria are to be found. Of course, it cannot be determined.

To clarify the definition. and indeterminate similarity criteria, we will analyze some issues related to modeling the flow of a steady air flow around a mountain range. Let's direct the axis X in the direction of undisturbed flow, axis z vertically, and let it be away from the array u=u(z), v=0, w=0. The height of the obstacle at a point with coordinates x, y is described by the equation z=h(x,y) at x>0.

Then the “sticking” condition will be written as:

From these boundary conditions it follows that when modeling the motion it is necessary. reproduce the profile of a rock mass in certain specific proportions

and the flow approaching the obstacle, i.e. L and V values ​​in datum. case of phenomenon externally determined.

It follows that out of 5 dimensionless similarity criteria, three will be decisive - the coincidence of the Froude and Reynolds numbers and the wind deviation from the geostrophic one.

If, for example, you set a certain characteristic size of the model, then the ratio L 1 / L 2 will be a known value, then the defining criteria will be the same three numbers - Fr, Re, De. The coincidence of numbers But will be carried out automatically, since in the case of steady motion the period is equal to infinity, then no new conclusions can be drawn from here.

The coincidence of the numbers Eu in in this case leads to a very important result. If we determine from experience the pressure difference between two points of the model, then the pressure difference at the corresponding points during natural flow can be found from the relation

Consequently, the non-defining criterion gives the rules for converting the results of the experiment to reality. It should be noted that such a ratio of criteria, when Fg, Re, De are the defining criteria, and Ho and Eu are the non-defining criteria, occurs in very many problems. hydromechanics. However, in a number of meteorological problems, the value of L turns out to be not externally, but internally determined by the size. This leads to a radical change in the defining criteria of similarity.

Dot product of vectors. An example in the Department of Meteorology.

Dot product- an operation on two vectors, the result of which is a number (scalar) that does not depend on the coordinate system and characterizes the lengths of the factor vectors and the angle between them. This operation corresponds to multiplication length given vector A on projection another vector b on given vector A.

Cross product of vectors. Example in dynamics. Meteorology

If for determining a physical quantity. In addition to the numerical value, it is necessary to indicate the direction in space; such quantities are called vectors.

The vector product AxB of two vectors is the vector C = A*B (Fig.), directed perpendicular to the plane of the factor vectors in the direction from which the rotation from the first factor to the second by a smaller angle counterclockwise and equal in size to the area of ​​the parallelogram built on these vectors, i.e. |C|=|A*B|=ABsin(A,B)

The cross product of vectors is defined the following conditions:

1). Vector modulus |C| is equal to ABsin(A,B), where (A,B) is the angle between vectors A and B;

2). Vector |C| perpendicular to each of vectors A and B;

3). Vector direction |С| corresponds to the "rule" right hand" This means that if vectors A, B and |C| brought to general beginning, then the vector |C| should be directed as directed middle finger the right hand, the thumb of which is directed along the first factor (that is, along vector A), and the index finger - along the second (that is, along vector B).

The vector product depends on the order of the factors, namely: .

The necessary and sufficient condition for the parallelism of vectors has the form: A*B=0.

If the coordinate axes system is right and vectors A and B are specified in this system by their coordinates:

, ,

That vector product vector A to vector B is determined by the formula

Or C=A*B=(A 1 i 1 +A 2 i 2 +A 3 i 3)*(B 1 i 1 +B 2 i 2 +B 3 i 3)=i 1 (A 2 B 3 -A 3 B 2)+i 2 (A 3 B 1 -A 1 B 3)+i 3 (A 1 B 2 -A 2 B 1)

Example in dynamic meteorology:

Such vectors, the direction of which is established by agreement and which change their direction when replaced right system coordinates to the left are called axial, for example moment of force and angular velocity. Vectors whose direction is determined physical meaning and which do not change their direction when the coordinate system changes are called polar, for example force and speed.

24. The concept of a tensor. Example in dynamics. Meteorology

Tensor(from lat. tensus, "tense") - an object of linear algebra that linearly transforms elements of one linear space into elements of another. Special cases of tensors are scalars, vectors, bilinear forms, etc.

Scalar or tensor of rank zero - physical quantity, completely determinable in any coordinate system one number (or function) that does not change when the spatial coordinate system changes. A scalar has one component.

Thus, if φ is the value of a scalar in one coordinate system, and φ" in another, then φ"=φ.

If the nature of air currents depended only on the thermal heterogeneity of the earth's surface and air masses, then the wind would be determined by the horizontal pressure gradient and the movement of air would occur along this gradient from high pressure to low. In this case, the wind speed would be inversely proportional to the distance between lines of equal pressure, i.e., isobars. The smaller the distance between the isobars, the greater the pressure gradient and, accordingly, the wind speed.

Pressure gradient force. In theoretical meteorology, forces are usually related to a unit of mass. Therefore, in order to express the force of the pressure gradient acting on a unit of mass, the value of the pressure gradient should be divided by the air density. Then the numeric value pressure gradient forces(G) will be determined by the expression:

where ρ is the air density, dρ/ dn– pressure gradient.

Under the influence of the force of the pressure gradient (baric gradient), wind arises. This means that if an excess of air mass is formed in a certain area (high pressure), then it must flow out into an area with a lack of air (low pressure). This outflow is stronger, the greater the pressure difference.

So the main driving force the occurrence of air movement is the baric gradient. If only the force of the pressure gradient acted on air particles, then their movement would always occur in the direction of this gradient, like the flow of water from a higher level to a lower one. In reality this does not happen.

In large-scale processes, the thermal root cause of air currents is combined with the action of a number of other factors that significantly complicate atmospheric circulation. Therefore, both monsoon and interlatitudinal circulation, due to the actions of a number of forces and the vortex nature of atmospheric circulation, are incomparably more complex.

The deflecting force of the Earth's rotation. Changes in the direction and speed of air currents are primarily caused by the deflecting force of the Earth's rotation, or, as it is commonly called, the Coriolis force. The emergence of this force is associated with the rotation of the Earth around its axis. Under the influence of the Coriolis force, the wind does not blow along the pressure gradient, i.e., from high pressure to low, but deviating from it to the right in the northern hemisphere, and to the left in the southern hemisphere.

In the diagram (Fig. 29, A) It is clearly shown how the deflecting force of the Earth's rotation affects the change in the direction of air movement, which began along a pressure gradient with a gradually increasing speed. The influence of other forces is not taken into account here.

Let us assume that, under the influence of the pressure gradient force, an air particle (indicated by a circle) begins to shift in the direction of the gradient (G). In the first moment, as soon as speed appears V 1 there will be an acceleration of the deflecting force of the Earth's rotation A 1 directed perpendicular and to the right in relation to the speed V 1 . Under the influence of this acceleration, the particle will not move along the gradient, but will deviate to the right; in the next instant the speed of the air particle will become equal to V 2 . But at the same time, the Coriolis force will change to A 2. Under the influence of this rotational acceleration, the speed of the air particle will further change, becoming equal to V 3 . The Coriolis force will not be slow to change, etc. As a result, the pressure force and the deflecting force of the Earth's rotation are balanced and the movement of the air particle occurs along the isobars. The effect of the Coriolis force increases with increasing particle speed and latitude. It is defined by the expression:

where ω is angular velocity, φ is geographic latitude, V- movement speed.

The acceleration of the deflecting force of the Earth's rotation is measured in quantities from zero at the equator to 2ωV at the pole.

Geostrophic wind. The simplest type of motion is linear and uniform motion without friction. In meteorology it is called geostrophic wind. However, such a movement can only be allowed theoretically. With geostrophic wind, it is assumed that, in addition to the gradient force (G), only the deflecting force of the Earth’s rotation acts on the air (A). When the movement is uniform, then both of these forces, acting in opposite directions, are balanced and the geostrophic wind is directed along the isobars (Fig. 29, b). In this case, low pressure is on the left in the northern hemisphere, and on the right in the southern hemisphere.

When the forces of the pressure gradient and the deflecting force of the Earth's rotation are in balance, their sum will be equal to zero. This is expressed by the following relationship:

whence we obtain that the speed of the geostrophic wind

It follows that the speed of the geostrophic wind is directly proportional to the magnitude of the horizontal pressure gradient. Therefore, the denser the isobars on pressure maps, the stronger the wind. Although in actual atmospheric conditions a purely geostrophic wind is almost never observed, observations show that at an altitude of about 1 km and above, air movement occurs approximately along the isobars, with slight deviations caused by other reasons. Therefore, in practical work Instead of the actual wind, geostrophic wind is also used. In addition to the force of the pressure gradient and the Coriolis force, the air movement is affected by the friction force and centrifugal force.

Friction force. The friction force is always directed in the direction opposite to the movement and is proportional to the speed. By reducing the speed of air flows, it deflects them to the left of the isobars, and the movement occurs not along the isobars, but at a certain angle to them, from high pressure to low. Through turbulent mixing air, the influence of friction is transmitted to the overlying layers, up to approximately 1 km above the surface of the earth.

The effect of friction on the direction and speed of air movement is shown in the diagram (Fig. 30, a). The diagram shows the pressure field and air movement under the influence of the pressure gradient force, the deflecting force of the Earth's rotation and friction. Under the influence of the Coriolis force, air moves not along the pressure gradient G, but at right angles to it, i.e., along isobars. The actual wind is shown by arrow B, friction force T deviated slightly to the side from the wind direction. The Coriolis force is shown at right angles to the actual wind by an arrow TO. As we can see, the angle between the actual wind IN and friction force T is greater than 90°, and the angle between the actual wind B and the force of the pressure gradient G less than 90°. Since the gradient force is perpendicular to the isobars, the actual wind turns out to be deflected to the left of the isobars. The magnitude of the angle between the isobar and the direction of the actual wind depends on the degree of roughness of the earth's surface. The deviation occurs to the left of the isobars, usually at an angle of 20-30°. Over land, friction is greater than over the sea; near the surface of the earth, the influence of friction is greatest, and with height it decreases. At a height of about 1 km the frictional force almost stops.

Centrifugal force. If the isobars are curvilinear, i.e., have, for example, the shape of an ellipse or circle, then the movement

air has an effect centrifugal force. This is the force of inertia, which is directed from the center to the periphery along the radius of curvature of the air movement path. Under the influence of centrifugal force (in the absence of friction), movement occurs along isobars. In the presence of friction, the wind blows at an angle to the isobars in the direction of low pressure. The magnitude of the centrifugal force is determined from the equality

Where V - air speed (wind speed), r - radius of curvature of its trajectory.

If we assume that the movement of air occurs in a circle, then its speed at any point of the trajectory will be directed tangentially to the circle (Fig. 30, b and c). As follows from this diagram, the Coriolis force (A) directed (in the northern hemisphere) at right angles radially to the right of the wind speed( V). Centrifugal force (C) is directed from the center of the cyclone and anticyclone to their periphery, and the gradient force (G) balances geometric sum the first two forces and lies on the radius of the circle. All three forces in this case are related by the equation

Where r - radius of curvature of isobars.

From this equation it follows that the wind is directed perpendicular to the pressure gradient. This special case winds with circular isobars in the cyclone system. This wind is calledgradient.

In the northern hemisphere in the cyclone system (Fig. 31, b) the pressure gradient force is directed towards its center, and the centrifugal and Coriolis forces that balance it are in the opposite side. In the case of an anticyclone (Fig. 30, c), the Coriolis force is directed towards its center, and the centrifugal force and the pressure gradient force are in the opposite direction and balance the first.

The gradient wind equation in the case of an anticyclone has the following form:

In the southern hemisphere, where the deflection force of the Earth's rotation is directed to the left of the air speed, the gradient wind is deflected from the pressure gradient to the left. Therefore, in the southern hemisphere, the wind in a cyclone is directed clockwise, and in an anticyclone it is directed counterclockwise.

Outside the influence of friction force, i.e. above 1 km, The wind is approaching a gradient wind in direction and speed. The difference between actual and gradient wind is usually small. However, these small deviations of the actual wind from the gradient wind play an important role in changing atmospheric pressure.

Air pressure is determined by its mass in the atmospheric column by cross-section, equal to one area. When air moves unevenly due to changes in air thermal properties And active forces there is a decrease or increase in the mass of air in the column, and, accordingly, a decrease or increase in atmospheric pressure.

The main factor in changing the pressure field (pressure field) is the deviation of the actual wind from the gradient one (at altitudes). When the direction and speed of the actual wind correspond to the gradient one, an increase or decrease in air mass and a change in pressure occur, and atmospheric vortices - cyclones and anticyclones - can arise and develop (see below).

Wind deviations are significant in areas of convergence of air flows in the troposphere and with large curvature of moving air flows.

Pressure field. Pressure field structure, or pressure field The atmosphere is quite varied. In extratropical latitudes, near the surface of the earth and at altitudes, you can always find large or relatively small cyclones and anticyclones, troughs, ridges, and saddles.

Cyclones are the largest atmospheric vortices, with low pressure in the center. The movement of air in their system in the northern hemisphere occurs counterclockwise. Anticyclones are vortices with high pressure in the center. The movement of air in their system in the northern hemisphere occurs clockwise.

In the southern hemisphere, in both systems the air circulation is reversed, i.e. the winds in a cyclone blow clockwise, and in an anticyclone they blow counterclockwise. A ridge is an area of ​​high pressure extended from the central part of an anticyclone with an anticyclonic circulation system. A trough is an area of ​​low pressure extended from the central part of the cyclone with a cyclonic circulation system. A saddle is a form of pressure relief between two cyclones and two anticyclones located crosswise.

Figure 31 shows the pressure field at the surface of the earth with a wind system. In addition to two cyclones and two anticyclones, there are troughs, ridges and a saddle. The wind direction is shown by arrows, the speed is shown by the tail. The greater the distance between the isobars, the lower the wind speed and the smaller the plumage. This image of isobars and wind is accepted on weather maps (see below).

The structure of the pressure field on the globe is diverse and complex. Therefore, the regime of air currents is different in winter and summer, at the surface of the earth and at altitudes, over continents and over the oceans, not to mention its great variability in the middle and high latitudes from day to day. Typically, average monthly pressure and wind maps show only the prevailing air mass transport during the month and hide many of the interesting features of atmospheric processes that are revealed in daily weather maps.