Non-hydrostatic atmospheric grazing-incidence integral numerical calculation method

Abstract

The application discloses a non-hydrostatic atmospheric grazing incidence integral numerical calculation method, which constructs a grazing incidence integral function under a non-hydrostatic equilibrium condition, analyzes singular point conditions of the grazing incidence integral function, converts the grazing incidence integral function into an integrable function form through a truncation interval method, expands a smooth part of an integrated function in the integrable function form by using Chebyshev polynomials, and calculates coefficients of the polynomials by using a fast Fourier transform, so as to obtain a columnar content of air particles along an optical path. The application makes up for the deficiency of traditional Chapman function calculation based on an atmospheric density index attenuation assumption, can effectively solve the calculation problem of the columnar content of gas particles along the optical path under the condition of not satisfying the hydrostatic equilibrium, and has extremely important significance for accurate quantitative analysis of diffusion, chemical and heating processes in the upper atmosphere and revealing the physical mechanism behind the same.

Description

TECHNICAL FIELD

[0001] The present application relates to a non-hydrostatic atmosphere grazing incidence integral numerical calculation method, and belongs to the technical field of middle and high atmosphere dynamics. BACKGROUND

[0002] In recent years, the research on near space has been a hot topic in the world, which is within the scope of the middle and high atmosphere research. By accurately grasping the changes of various components in the atmosphere, it is helpful for us to understand various physical and chemical mechanisms of the middle and high atmosphere, especially the accurate quantitative analysis of the diffusion, chemistry and heating processes in the high atmosphere, which reveals the physical mechanism behind it, and has extremely important scientific, military and economic significance.

[0003] The traditional atmosphere grazing incidence integral is based on Chapman function, which is used to calculate the column content of gas particles along the light path, and it is realized based on certain "simplification" and "idealization". Specifically, the atmosphere is assumed to be spherically symmetric, the atmosphere satisfies the hydrostatic equilibrium condition, the temperature and gravity acceleration do not change with the altitude. This assumption makes the ideal gas density change exponentially with the height.

[0004] The scale height of the actual earth atmosphere changes significantly with the change of the altitude, and in many cases, it does not satisfy the hydrostatic equilibrium condition. Therefore, it is very necessary to develop a method for calculating the column content of gas particles along the light path under non-hydrostatic equilibrium condition, which can obtain more accurate calculation results. SUMMARY

[0005] The technical problem to be solved by the present application is to provide a non-hydrostatic atmosphere grazing incidence integral numerical calculation method, which can accurately calculate the column content of gas particles along the light path under non-hydrostatic equilibrium condition, and provide technical support for accurate calculation of atmospheric composition.

[0006] The present application adopts the following technical solution to solve the above technical problem:

[0007] The non-hydrostatic atmosphere grazing incidence integral numerical calculation method, the method specifically comprises:

[0008] The grazing incidence integral function under the non-hydrostatic equilibrium condition is constructed, the singular point condition of the grazing incidence integral function is analyzed, the grazing incidence integral function is converted into an integrable function form by the method of truncating interval, and the integrable function form is solved by using the modified matrix method, so as to obtain the column content of air particles along the light path;

[0009] The grazing incidence integral function is:

[0010]

[0011] Where n(z) is the vertical profile of the atmospheric density, z is the height of the point on the light path, z0 is the height of the pre-set starting point of the light path, z∈[z0,∞), and χ is the zenith angle;

[0012] PR(z; z0,χ) is the PR kernel function, which is expressed as:

[0013]

[0014] Where R is the radius of the Earth.

[0015] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0016] The present invention proposes a method for calculating the columnar content of gas particles along a light path under non-hydrostatic equilibrium conditions, and provides a solution including a truncation interval for the problem of singular points. Specifically, the smooth part of the integrand in GII is expanded using Chebyshev polynomials, the coefficients of the polynomials are calculated using FFT, and finally the column content is calculated. This method makes up for the shortcomings of the traditional Chapman function calculation based on the atmospheric density exponential decay assumption, and can effectively solve the problem of calculating the columnar content of gas particles along a light path under non-hydrostatic equilibrium conditions. It is of great significance for the precise quantitative analysis of diffusion, chemical and heating processes in the upper atmosphere, and for revealing the physical mechanisms behind them. DETAILED DESCRIPTION

[0017] The embodiments of the present invention are described in detail below. The embodiments are exemplary and are only used to explain the present invention, and are not to be construed as limiting the present invention.

[0018] The column content of air particles arriving at a certain point in the atmosphere along the light path can be expressed as the path integral as follows:

[0019]

[0020] Where n(s) is the air density along the light path, the parameter is s, and s0 is a point in the atmosphere. Pressly first converted it into:

[0021]

[0022] Where z0 is the altitude of a point in the atmosphere (set as needed), z is the altitude of a point on the light path, R is the radius of the Earth, χ is the zenith angle, and n(z) is the vertical profile of atmospheric density, provided by the NRLMSIS 2.0 thermospheric atmospheric model. This formula is only valid when χ < π / 2.

[0023] For the case of χ≥π / 2, Rees proposed:

[0024] N(z0,χ)=2N(z0,π / 2)-N(z0,π-x)

[0025] The present application proposes a column content integral formula GII (Grazing Incidence Integral) of air particles along an optical path to a certain point in the atmosphere under non-hydrostatic atmosphere conditions:

[0026]

[0027] Only the integral formula needs to be solved, and the column content of air particles along the optical path to a certain point in the atmosphere can be obtained.

[0028] The detailed expression of the PR kernel is:

[0029]

[0030] The integral result can be obtained by solving the above integral formula using the modified matrix method.

[0031] (1) When , the PR kernel function can be expressed as:

[0032]

[0033] wherein z∈[z0,∞), it is obvious that when z=z0, the PR kernel function denominator is 0, and there is a singular point; when z→∞, there is also a singular point, and the following variable substitution is used to more intuitively display the singular point.

[0034] Let x=[(z0+R) / (z+R)] 2 , then on the definition domain of z, x∈(0,1], and the GII integral formula after substitution can be obtained:

[0035]

[0036] When x=1, i.e. z=z0, there is obviously a singular point, and it is integrable; when x→0, i.e. z→∞, there is an integrable singular point. The method of truncating the interval is used to make the integral formula integrable, i.e. the integral interval is truncated from [z0,∞) to [z0,z T ], and the problem is solved by losing part of the precision. The upper limit z T of the integral can be set according to actual needs and precision requirements, for example, it can be set to 1000 km.

[0037] Let x T =[(R+z0) / (R+z T )] 2 Again, variable substitution is performed, and let x=(s T+ t)(1 - x T ) / 2, introduce a new variable t, t ∈ [-1, 1], s T = (1 + x T ) / (1 - x T )>1, the GII integral becomes:

[0038]

[0039] where,

[0040] (2) When GII integral formula can be written as:

[0041]

[0042] Variable substitution, let x = sin 2 χ[(z0+R) / (z+R)] 2 , we get:

[0043]

[0044] where, x0 = sin 2 χ, x T = sin 2 χ[(z0+R) / (z T +R)] 2 . Obviously, x = 1 has a singularity, at this time, let x(z c ) = 1, then z c = (z0+R) sinχ-R, so z c < z0, not in the range of the domain [z0, z T ]. There is no singularity in theory, but in fact when the zenith angle ε≤0 appears a singularity. The lower limit of the integral interval from z0 to z c , can eliminate this singularity.

[0045]

[0046] There are singularities at x = 1 in both integrals, and the subtraction of the two integrals can eliminate this singularity; but when x0→1, the upper and lower limits of the integral are both 1, then the integral is 0. Only the first integral degenerates into the case (1) discussed.

[0047] Similarly, in t ∈ [-1, 1], we have:

[0048]

[0049] where Here * ∈{T,0},s * =(1+x * ) / (1-x * ).

[0050] (3) When PR kernel function becomes a regular function, the integral formula becomes a definite integral. By variable substitution, z(t) = (z T -z0)(τ+t) / 2, where τ = (z T +z0) / (z T -z0), t ∈ [-1, 1], then for GII integral formula is:

[0051]

[0052] where,

[0053] The above integral formula is solved by using the modified matrix method:

[0054] 1) For the case with singular points, i.e. cases (1) and (2)

[0055] This method uses Chebyshev polynomial expansion for the smooth part of the integrand function in GII, and its expression is as follows:

[0056]

[0057] In the formula, * ∈{T,0},ak is the polynomial coefficient, which can be written as:

[0058]

[0059] a k can be calculated by fast Fourier transform (FFT).

[0060] Then GII can be written as:

[0061]

[0062] where,

[0063] 2) For the case without singular points, i.e. case (3)

[0064] Let Similarly, a k can be calculated by fast Fourier transform toolbox.

[0065] Finally, summing up, we get:

[0066]

[0067] wherein:

[0068] The above examples only illustrate the technical ideas of the present application, and cannot limit the protection scope of the present application. Any modification made according to the technical ideas of the present application on the basis of the technical solutions falls within the protection scope of the present application.

Claims

1. A non-hydrostatic atmospheric grazing-incidence integral numerical calculation method, characterized by, The method is specifically: The grazing incidence integral function under the non-static balance condition is constructed, the singular point condition of the grazing incidence integral function is analyzed, the grazing incidence integral function is converted into an integrable function form through the truncation interval method, the integrable function form is solved by using the modified quadrature method, and thus the columnar content of the air particles along the light path is obtained; The grazing incidence integral function is: In the formula, n(z) is the vertical profile of atmospheric density, z is the height of a point on the light path, z0 is the height of a starting point of the light path, z is [z0, ∞), and χ is the zenith angle; PR(z; z0, χ) is a PR kernel function, and the expression is: In the formula, R is the radius of the earth.

2. The non-hydrostatic atmospheric grazing-incidence integral numerical calculation method according to claim 1, characterized in that, The singular point condition of the grazing incidence integral function is analyzed, and the grazing incidence integral function is converted into an integrable function form through the truncation interval method, and the specific process is as follows: (1) when the PR kernel function is represented as: When z = z0, the PR kernel function has a singular point; when z → ∞, the PR kernel function also has a singular point. The method of truncating interval is used to make the grazing incidence integral function integrable, that is, the integral interval is truncated from [z0, ∞) to [z0, z T ] and the upper limit of integration z T is set according to actual needs and accuracy requirements; The grazing incidence integral function is converted into: wherein t, x, x T and s T are variables, t∈[-1,1], x=(s T +t)(1-x T ) / 2, x T =[(R+z0) / (R+z T )] 2 , s T =(1+x T ) / (1-x T ), s T >1; (2) when the grazing incidence integral function is represented as: The grazing incidence integral function is converted into: wherein t∈[-1,1], s * = (1 + x * ) / (1 - x * ), *∈{T,0} (3) When the PR kernel function is regular, by variable substitution, z = (z T -z0)(τ + t) / 2, where τ = (z T +z0) / (z T -z0), t ∈ [-1, 1], the grazing incidence integral function is converted to: In the formulae, 3. The non-hydrostatic atmospheric grazing-incidence integral numerical calculation method according to claim 2, characterized in that, The modified quadrature method is specifically: the smooth part of the integral function in the integrable function form is expanded by using the Chebyshev polynomial, and the coefficients of the polynomial are calculated by using the fast Fourier transform, so that the columnar content of the air particles along the light path is obtained.

4. The non-hydrostatic atmospheric grazing-incidence integral numerical calculation method according to claim 2, characterized in that, The integral function in the integrable function form is solved by using the modified quadrature method, and for the singular point condition, that is, for (1) and (2), the smooth part of the integral function is expanded by using the Chebyshev polynomial, and the expression is as follows: where *∈ {T,0}, a k are polynomial coefficients, T k (t) is a polynomial; a k is represented as: The fast Fourier transform is used to calculate a k ; The grazing incidence integral function is expressed as: In the formulae, 5. The non-hydrostatic atmospheric grazing-incidence integral numerical method according to claim 2, characterized in that, The integral function in the integrable function form is solved by using the modified quadrature method, and for the singular point condition, that is, for (1) and (2), the smooth part of the integral function is expanded by using the Chebyshev polynomial, and the expression is as follows: By analogy, The a is calculated using a fast Fourier transform k ; The grazing incidence integral function is expressed as: In the formulae,

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