Divergence can be used to characterize the divergence degree of vector fields at various points in space. Physically, the meaning of divergence is the field's activeness.When div F>0, it means that there is a positive source (divergent source) of flux at this point;When div F<0, there is a negative source (hole or sink) of absorption flux at this point;When div F=0, it means that the vector field line of the point is not emitted or converged.[6]
Divergence is the amount that describes the degree to which air converges or diffuses from the surroundings.Horizontal divergence is the rate of change of horizontal area of gas in unit time.If the area increases, the divergence is positive, which is horizontal divergence;If the area is reduced, the divergence is negative, which is horizontal convergence.The divergence of three-dimensional space represents the change rate of unit volume of any gas block in unit time.The volume expansion of the gas block is called divergence, and the volume contraction of the gas block is called convergence[1]。
In atmospheric science, divergence refers tophysical quantity。The unit is/second.It represents the expansion rate of volume in unit time.In incompressible fluid, the divergence is 0, so there is divergence or convergence in the horizontal direction, and compensatory contraction and extension will occur in the vertical direction, resulting in vertical movement.Therefore, you can use theHorizontal divergenceCalculate the vertical velocity in the atmosphere[2]。
For a vector field
As for divergence, there are two different definitions.
The first definition is independent of the coordinate system:[3]
The second definition is carried out in the rectangular coordinate system:[4]
Set vector field
Is represented by
Of which
They are the unit vector in the X, Y and Z directions, and the component of the field
With the first order continuous partial derivative, then the vector field
The divergence of is:
It can be proved that the two definitions are equivalent when the limit exists.Therefore, it is also often used directly
representative
The divergence of.
From the definition of divergence,
Represents the vector emitted from the unit volume at a point
The density of the flux source is described.For example, if the thermal radiation intensity vector of each point in space is regarded as a vector field, then the thermal radiation intensity vector around a thermal radiation source (such as the sun) points outward, indicating thatsunlightIt is the source of new thermal radiation, and its divergence is greater than zero.
It can also be seen from the definition that divergence isVector fieldAn intensity property of, just like density, concentration and temperature, its corresponding extensive property is the flux on the surface of a closed area.
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(a, b are constants) (linear operation)
(
Is a scalar field)
(passive curl field)[3]
Divergence expressions in different coordinate systems
Expression of divergence of vector V in spherical coordinates:
Expression of divergence of vector V in cylindrical coordinates:
Divergence Theorem
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sinceVector fieldThe divergence at a certain place is the volume density of the flux near the vector field at that place. If the divergence within a certain volume is integrated, the total flux within the volume should be obtained.This inference can be proved to be correct, called the Gauss divergence theorem, orGauss formula。It is expressed in mathematical language as:[3]
Gauss formula indicates that if the volume
Vector field in
With divergence, then divergence
The volume fraction of is equal to the vector field in
Surface of
Area score of.
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In electromagnetics and electrodynamics
electrostatic field
Its divergence is not zero and its curl is zero. It is an active irrotational field.
Static magnetic field
The divergence of is zerocurlIt's not zero, it's rotatingPassive field。
Meteorology
Divergence can represent the rate of change per unit volume when a fluid moves.In short, the concentrated area of fluid in motion is convergence, and the divergent area in motion is divergence.When the divergence value is negative, it is convergence, which is conducive to the development and enhancement of convective weather systems such as cyclones. When the divergence value is positive, it means divergence, which is conducive to the development of weather systems such as anticyclones.
Often, wind speed is the most widely used in meteorology
Horizontal divergence of.
The expression of horizontal divergence is:
Where u is the wind speed in the x-axis direction and v is the wind speed in the y-axis direction.
In general, the x-axis represents the tangent direction of the latitude circle (positive from west to east), and the y-axis represents the tangent direction of the longitude circle (positive from south to north).
fluid mechanics
A vector field with zero divergence is called a passive field or a tubular field.In fluid mechanics, the fluid with zero density divergence is called incompressible fluid, that is, the total amount of fluid flowing into a tiny volume element in each tiny time interval is equal to the total amount of fluid flowing out of this volume element in this time interval.[5]
For compressible fluid, the following equation holds:
That is, the rate of change of density is equal to the divergence of momentum.
