Topographic Gravity Wave Parameterization Method Considering Variations in Mountain Sharpness

Abstract

This invention discloses a terrain gravity wave parameterization method considering variations in mountain sharpness, comprising the following steps: calculating high-resolution subgrid terrain ground truth values ​​within each grid of the numerical model; calculating subgrid terrain errors for terrain heights at each subgrid grid point represented by an elliptical mountain function; calculating the elliptical mountain sharpness within each model grid by minimizing the subgrid terrain error, and traversing all grids to obtain the global distribution of mountain sharpness; calculating the drag coefficient for each model grid after obtaining the mountain sharpness within each grid; and calculating the surface stress and vertical distribution of gravity wave stress in terrain gravity waves. This application develops a region-aware algorithm for subgrid mountain sharpness in terrain gravity wave parameterization, improving the terrain gravity wave parameterization scheme by incorporating regionally varying mountain sharpness factors, thus meeting the needs of high-precision numerical forecasting for complex terrain regions.

Description

Technical Field

[0001] This invention belongs to the field of meteorological technology, and in particular relates to a method for parameterizing topographic gravity waves that takes into account changes in mountain sharpness. Background Technology

[0002] Topography influences atmospheric circulation through a variety of complex processes at various spatial and temporal scales. For example, topographically excited gravity waves can significantly affect tropospheric circulation and stratospheric atmospheric states. However, stratospheric atmospheric states are crucial for medium-range weather forecasting, seasonal prediction, and climate projection. Under current limited computing power, numerical weather prediction models can generally only resolve topography at scales 3 to 8 times the grid spacing, failing to adequately represent the impact of small-scale topography on the atmosphere. Small-scale topographic effects include topographic gravity wave drag, airflow blocking drag, and turbulent topographic drag. The latter two processes mainly occur in the lower troposphere, while topographic gravity wave drag can not only slow down tropospheric winds but also propagate into the stratosphere, significantly affecting stratospheric circulation. In current numerical weather prediction models, parameterization schemes are commonly used to describe the topographic gravity wave drag process, and this has proven to be a key factor in improving forecast accuracy.

[0003] The topographic gravity wave parameterization scheme is based on the assumption of elliptical mountains within the subgrid and linear gravity wave theory. Due to the lack of constraints on the topographic parameterization process, the scheme contains numerous assumptions, empirical approximations, and uncertain parameters. Currently, when integrating elliptical mountains, mountain sharpness is treated as a constant, meaning that the mountain sharpness is the same across all grid frames. This uniform mountain sharpness causes it to lose its "regional awareness" capability and results in a constant topographic gravity wave drag coefficient. For example, Van Niekerk et al. (2018) showed that the topographic gravity wave drag coefficient was 0.5 in the UK MetUM model parameterization scheme, while it was 0.6 in the European Centre for Medium-Range Weather Forecasts (IFS) model parameterization scheme. The uncertainty of the topographic gravity wave drag coefficient and the loss of "regional awareness" in the mountain sharpness parameter introduce errors into the parameterization scheme and numerical models.

[0004] In the traditional topographic gravity wave parameterization scheme (Lott and Miller, 1997), the surface stress of topographic gravity waves is obtained by integrating the mountain function in the following equation.

[0005] (1);

[0006] in, The average density near the ground. The average wind speed near the ground. The average buoyancy frequency near the ground. It is an elliptical mountain function. The Fourier expression here Indicates the height of the subgrid mountain range. and These are the semi-minor axis and semi-major axis of the elliptical mountain range, respectively. Let be the sharpness of the elliptical mountain range. Integrating the mountain range function yields the expression for the surface stress caused by topographic gravity waves:

[0007] (2);

[0008] The effective mountain height is obtained by subtracting the airflow obstruction height from the subgrid mountain height. and It is an anisotropy function of elliptical mountain ranges. and These represent the components parallel and perpendicular to the wind direction, respectively. It is the angle between the azimuth of the mountain range and the wind direction. The drag coefficient is a unique function of the sharpness of the elliptical mountain range, as shown below:

[0009] (3);

[0010] in For gamma function. Due to the fact that in traditional topographic gravity wave parameterization schemes, the sharpness of elliptical mountains... The drag coefficient was set to a constant, thus causing the drag coefficient to be affected. It also becomes a constant, meaning it has the same drag coefficient under different terrain conditions, losing its ability to sense the surrounding terrain conditions, which obviously introduces errors.

[0011] In summary, to meet the demand for high-precision numerical weather prediction in complex terrain areas, it is necessary to develop a region-aware algorithm for subgrid mountain sharpness in terrain gravity wave parameterization. Based on this, the terrain gravity wave parameterization scheme should be improved, and a new scheme should be developed to incorporate the region-varying mountain sharpness factor. Summary of the Invention

[0012] To address the issue that current topographic gravity wave parameterization schemes typically use constant mountain sharpness parameters, leading to a loss of regional awareness in the topographic gravity wave drag coefficient, this application combines high-resolution topographic elevation data and model topographic data to study a high-precision calculation method for subgrid mountain sharpness. A novel topographic gravity wave parameterization method considering variations in mountain sharpness is proposed to meet the requirements of high-precision numerical simulation and improve the numerical model's ability to simulate tropospheric and stratospheric circulation.

[0013] To achieve the above objectives, this application proposes a topographic gravity wave parameterization method that considers variations in mountain sharpness, comprising the following steps:

[0014] Using global 1km high-resolution terrain elevation data and model grid terrain data, high-resolution subgrid terrain true values ​​are calculated within each grid of the numerical model.

[0015] Within each grid, for The number of subgrid points is used to calculate the subgrid terrain error based on the elliptical mountain range function representation;

[0016] After minimizing the subgrid terrain error, the sharpness of the elliptical mountain range within each pattern grid is calculated. And by traversing all grids, the global distribution of mountain sharpness is obtained;

[0017] Based on the mountain sharpness within each pattern grid of the bureau The gravity wave drag coefficient of each model grid was calculated. The gravity wave drag coefficient It varies with the pattern grid area;

[0018] Calculate the surface stress of topographic gravity waves and the vertical distribution of gravity wave stress.

[0019] Preferably, the high-resolution subgrid topographic ground truth value is calculated within each grid of the numerical model using global 1km high-resolution topographic elevation data and model grid topographic data, as mathematically represented below:

[0020] ;

[0021] It is global 1km high-resolution terrain and elevation data. It is pattern grid terrain data. These are grid coordinates. and It is located in The first grid within the pattern frame The index of each subgrid point.

[0022] Preferably, within each grid, for The number of subgrid points is used to calculate the subgrid terrain error based on the elliptical mountain range function, which is mathematically represented as follows:

[0023] ;

[0024] This is a nonlinear optimization problem. Indicates the height of the subgrid mountain range. and These are the semi-minor axis and semi-major axis of the elliptical mountain range, respectively. For the sharpness of the mountain range, yes Total number of points in the subgrid within the pattern grid frame. and It is located in The first grid within the pattern frame The index of each subgrid point.

[0025] Preferably, the logarithm of the subgrid terrain error formula for the elliptical mountain function is taken, resulting in:

[0026] ;

[0027] , , , Then, the nonlinear optimization problem is transformed into a least squares problem, resulting in:

[0028] ;

[0029] The azimuth angle of an elliptical mountain range, that is, the angle between the principal axis and the meridional coordinate axis, is... Therefore, when calculating mountain sharpness, the rotating coordinate system maintains consistency between the pattern coordinates and the elliptical mountain function coordinates, i.e. , , It is the angle between the azimuth of the mountain range and the wind direction.

[0030] Preferably, the sharpness of the elliptical mountain range within each pattern grid is... The calculation expression is:

[0031] ;

[0032] This is the number of grid points. It is the grid point number.

[0033] Preferably, the drag coefficient of each pattern grid is calculated using the following formula. :

[0034] ;

[0035] in This is the gamma function.

[0036] Preferably, after obtaining the gravity wave drag coefficient within each pattern grid frame, the following formula is used to obtain the terrain gravity wave surface stress:

[0037] ;

[0038] The average density near the ground. The average wind speed near the ground. The average buoyancy frequency near the ground. The effective mountain height is obtained by subtracting the airflow obstruction height from the subgrid mountain height. and It is an anisotropy function of elliptical mountain ranges. and These represent the components parallel and perpendicular to the wind direction, respectively. It is the angle between the azimuth of the mountain range and the wind direction. This is the drag coefficient.

[0039] Preferably, during the upward propagation of topographic gravity wave stress, the following steps are performed: First, the gravity wave amplitude is calculated layer by layer:

[0040] ;

[0041] Indicates the first Gravity wave amplitude on each mode layer Indicates the first Gravity wave stress on each mode layer, , , They are respectively the first Density, buoyancy frequency, and wind speed at each mode layer;

[0042] Calculate the local Richardson number:

[0043] ;

[0044] in The average Richardson number of the airflow. Where Fr represents the gravity wave frequency and U is the average near-surface wind speed. It is the amplitude of local gravity waves. The average buoyancy frequency near the ground;

[0045] Determine whether gravity waves have broken up in this layer; compare the local Richardson number with a critical value. If it is greater than the critical value, the topographic gravity wave stress in this layer continues to propagate upwards; if it is less than the critical value, calculate the gravity wave amplitude at which gravity waves have broken up and the gravity wave stress in this layer:

[0046] ;

[0047] in, The critical wave amplitude is obtained by taking the local Richardson number as equal to the critical value; the vertical distribution of gravitational wave stress is determined by iteratively propagating upwards.

[0048] This application develops a subgrid-based regional perception algorithm for mountain sharpness in topographic gravity wave parameterization, improves the topographic gravity wave parameterization scheme by incorporating mountain sharpness factors that vary with the region, and develops a new topographic gravity wave parameterization scheme that meets the needs of high-precision numerical forecasting for complex terrain regions. Attached Figure Description

[0049] Figure 1 The gravity wave parameterization method flow provided in the embodiments of this application;

[0050] Figure 2 Probability density of the global distribution of the subgrid terrain gravity wave drag coefficient;

[0051] Figure 3 Global zonal wind distribution. Among them, 3(a) is ERA5, 3(b) is the control experiment, 3(c) is the control experiment minus ERA5, and 3(d) is the new experiment minus the control experiment. Detailed Implementation

[0052] The present invention will be further described below with reference to the accompanying drawings, but this is not intended to limit the present invention in any way. Any modifications or substitutions made based on the teachings of the present invention shall fall within the protection scope of the present invention.

[0053] As mentioned earlier, using the same drag coefficient under different terrain conditions loses its ability to perceive regional terrain information, which obviously introduces errors. Therefore, this application studies a high-precision calculation method for subgrid mountain sharpness by combining high-resolution terrain elevation data and model terrain data, and proposes a new terrain gravity wave parameterization method that considers the variation of mountain sharpness.

[0054] First, it is necessary to utilize global 1km high-resolution topographic elevation data. and pattern grid terrain data In each grid of the numerical model The internal calculation of high-resolution subgrid terrain ground truth is shown below:

[0055] (4);

[0056] Then, within each grid, for The calculation of the number of subgrid points is based on the subgrid terrain error represented by the elliptical mountain function:

[0057] (5);

[0058] This is a nonlinear optimization problem. However, taking the logarithm of the above equation, we get:

[0059] ;

[0060] set up , , , After that, except Since is the unknown quantity and everything else is known, this becomes a least squares problem. We can obtain:

[0061] (6);

[0062] The azimuth angle of an elliptical mountain range, that is, the angle between the principal axis and the meridional coordinate axis, is... Therefore, when calculating mountain sharpness, it is necessary to rotate the coordinate system to maintain consistency between the pattern coordinates and the elliptical mountain function coordinates, i.e. , .

[0063] Finally, the expression for calculating the sharpness of the elliptical mountain range within each pattern grid is obtained as follows:

[0064] (7);

[0065] This is the number of grid points. It is the grid point number.

[0066] By traversing all grids using this formula, the global distribution of mountain sharpness can be obtained. The mountain sharpness in this subgrid is mainly distributed in complex terrain regions, such as the Tibetan Plateau and the Rocky Mountains. The mountain sharpness within each grid is then obtained. Then, the drag coefficient of each pattern grid can be obtained through equation (3). Thus making It is no longer a constant for different regions, but becomes a more accurate terrain gravity wave drag coefficient that varies with the model grid region.

[0067] The probability density distribution of the global distribution of the gravity wave drag coefficient of subgrid terrain is as follows: Figure 2 As shown, the sharpness of the subgrid mountain range is mainly below 1, with a peak value of around 0.6.

[0068] After obtaining the gravity wave drag coefficient within each pattern grid frame, it is substituted into formula (2) to obtain the surface stress of the terrain gravity wave. Then, during the upward propagation of the terrain gravity wave stress, the following steps are performed: First, the gravity wave amplitude is calculated layer by layer:

[0069] (8);

[0070] Then calculate the local Richardson number:

[0071] (9);

[0072] in The average Richardson number of the airflow. Where Fr represents the gravity wave frequency and U is the average near-surface wind speed. The average buoyancy frequency near the ground. This represents the local gravity wave amplitude. Then, it's determined whether the gravity wave breaks up at that layer. The local Richardson number is compared to a critical value. If it's greater than the critical value, the topographic gravity wave stress at that layer continues to propagate upwards; if it's less than the critical value, the gravity wave amplitude indicating breakup and the gravity wave stress at that layer are calculated.

[0073] (10);

[0074] in The critical wave amplitude is obtained by estimating the local Richardson number to a critical value. This is then iteratively propagated upwards to determine the vertical distribution of the gravitational wave stress.

[0075] The invention will be further explained below using a simulated global zonal wind from January 1st to 10th, 2023 as an example, in conjunction with the accompanying drawings. Figure 3 As shown, ERA5 data indicates the presence of tropospheric and stratospheric jet streams in the mid-latitude regions of the Northern Hemisphere. Figure 3 a) The controlled experiment successfully simulated the structural characteristics of the jet stream, but the wind speed intensity was greater than that of ERA5. Figure 3 b). The differences between the control experiment and ERA5 also show that the control experiment using the original mountain sharpness and drag coefficient has a strong positive bias for both stratospheric and tropospheric jet streams. Figure 3 c). However, by considering the regional variations in mountain sharpness and using the newly calculated drag coefficient, the new experiment mitigated the positive bias in tropospheric and stratospheric wind speeds in the controlled experiment ( Figure 3 d).

[0076] The beneficial effects of this application are as follows:

[0077] This application develops a subgrid-based regional perception algorithm for mountain sharpness in topographic gravity wave parameterization, improves the topographic gravity wave parameterization scheme by incorporating mountain sharpness factors that vary with the region, and develops a new topographic gravity wave parameterization scheme that meets the needs of high-precision numerical forecasting for complex terrain regions.

[0078] As used herein, the term "preferred" is meant as an example, illustration, or illustration. Any aspect or design described herein as "preferred" need not be construed as being more advantageous than other aspects or designs. Rather, the use of the term "preferred" is intended to present the concept in a specific manner. As used in this application, the term "or" is intended to mean an inclusive "or" rather than an exclusionary "or." That is, unless otherwise specified or clear from the context, "X uses A or B" naturally includes either of the permutations. That is, if X uses A; X uses B; or X uses both A and B, then "X uses A or B" is satisfied in any of the foregoing examples.

[0079] Furthermore, although this disclosure has been shown and described with respect to one or more implementations, equivalent variations and modifications will occur to those skilled in the art based on a reading and understanding of this specification and the accompanying drawings. This disclosure includes all such modifications and variations and is limited only by the scope of the appended claims. In particular, with respect to the various functions performed by the aforementioned components (e.g., elements, etc.), the terminology used to describe such components is intended to correspond to any component (unless otherwise indicated) that performs the specified function of said component, even if structurally not equivalent to the disclosed structure performing the functions in the exemplary implementations of this disclosure shown herein. Moreover, although a particular feature of this disclosure has been disclosed with respect to only one of several implementations, such feature may be combined with one or more features of other implementations that may be desirable and advantageous for a given or particular application. Furthermore, with regard to the use of the terms “comprising,” “having,” “containing,” or variations thereof in the Detailed Description or claims, such terms are intended to be included in a manner similar to the term “including.”

[0080] The functional units in this invention embodiment can be integrated into a processing module, or each unit can exist physically separately, or multiple units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. The storage medium mentioned above can be a read-only memory, a disk, or an optical disk, etc. The aforementioned devices or systems can execute the storage methods in the corresponding method embodiments.

[0081] In summary, the above embodiments are one implementation of the present invention, but the implementation of the present invention is not limited to the embodiments described above. Any changes, modifications, substitutions, combinations, or simplifications made that deviate from the spirit and principle of the present invention should be considered equivalent substitutions and are included within the protection scope of the present invention.

Claims

1. A topographic gravity wave parameterization method considering variations in mountain sharpness, characterized in that, Includes the following steps: Using global 1km high-resolution terrain elevation data and model grid terrain data, the high-resolution subgrid terrain true value is calculated in each grid of the numerical model. Within each grid, for The number of subgrid points is used to calculate the subgrid terrain error based on the elliptical mountain range function representation; After minimizing the subgrid terrain error, the sharpness of the elliptical mountain range within each pattern grid is calculated. And by traversing all grids, the global distribution of mountain sharpness is obtained; Based on the mountain sharpness within each pattern grid The gravity wave drag coefficient of each model grid was calculated. The gravity wave drag coefficient The sharpness of the elliptical mountain range varies with the pattern grid area; the sharpness of the elliptical mountain range within each pattern grid varies. The calculation expression is: It is the pattern grid number. These are grid coordinates. and They are respectively The semi-minor and semi-major axes of the elliptical mountain range within the pattern network, and It is located in The index of the i-th subgrid point within the pattern network. The angle between the principal axis and the meridional coordinate axis. It is the true value of high-resolution subgrid terrain. Indicates the height of the subgrid mountain range; Calculate the surface stress of topographic gravity waves and the vertical distribution of gravity wave stress.

2. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 1, characterized in that, The method utilizes global 1km high-resolution terrain elevation data and model grid terrain data to calculate high-resolution sub-grid terrain ground values ​​within each grid of the numerical model. The mathematical representation is as follows: It is global 1km high-resolution terrain and elevation data. It is pattern grid terrain data. These are grid coordinates. and It is located in The index of the i-th subgrid point within the pattern network.

3. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 2, characterized in that, Within each grid, for The number of subgrid points is used to calculate the subgrid terrain error based on the elliptical mountain range function, which is mathematically represented as follows: E The above equation represents a nonlinear optimization problem. Indicates the height of the subgrid mountain range. and They are respectively The semi-minor and semi-major axes of the elliptical mountain range within the pattern network, For the sharpness of the mountain range, yes Total number of points in the subgrid within the pattern grid frame. and It is located in The first grid within the pattern frame The index of each subgrid point.

4. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 3, characterized in that, Taking the logarithm of the subgrid terrain error formula for the elliptical mountain function, we get: Least squares coefficients Least squares coefficients , Then, the nonlinear optimization problem is transformed into a least squares problem, resulting in: The azimuth angle of an elliptical mountain range, that is, the angle between the principal axis and the meridional coordinate axis, is... Therefore, when calculating mountain sharpness, the rotating coordinate system maintains consistency between the model coordinates and the elliptical mountain function coordinates, i.e., the coordinates under the model coordinates. and They become , .

5. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 4, characterized in that, The drag coefficient for each pattern grid is calculated using the following formula. : in This is the gamma function.

6. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 5, characterized in that, After obtaining the gravity wave drag coefficient within each pattern grid frame, substituting it into the following formula, the terrain gravity wave surface stress is obtained: in, The average density near the ground. The average wind speed near the ground. The average buoyancy frequency near the ground. The effective mountain height is obtained by subtracting the airflow obstruction height from the subgrid mountain height. and It is an anisotropy function of elliptical mountain ranges. and These represent the components parallel and perpendicular to the wind direction, respectively. It is the angle between the azimuth of the mountain range and the wind direction. This is the drag coefficient.

7. The topographic gravity wave parameterization method considering changes in mountain sharpness according to claim 6, characterized in that, During the upward propagation of topographic gravity wave stress, the following steps are performed: First, the gravity wave amplitude is calculated layer by layer. Indicates the first Gravity wave amplitude on each mode layer Indicates the first Gravity wave stress on each mode layer, , , They are respectively the first Density, buoyancy frequency, and wind speed at each mode layer; Calculate the local Richardson number: in The average Richardson number of the airflow. ,in, Indicates the frequency of gravitational waves. It is the average wind speed near the ground. It is the amplitude of local gravity waves. It is the average buoyancy frequency near the ground; Determining whether a gravitational wave is in the 1st... The first pattern layer breaks down; the local Richardson number is compared with a critical value; if it is greater than the critical value, then the first... The topographic gravity wave stress of the first model layer continues to propagate upwards; if it is less than the critical value, then the gravity wave amplitude at which the gravity wave breaks up and the first model layer are calculated. Gravity wave stress in each mode layer: in, The critical wave amplitude is obtained by taking the local Richardson number as equal to the critical value; the vertical distribution of gravitational wave stress is determined by iteratively propagating upwards.

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