NWP | Spectral Method

Spectral methods

Spectral methods in numerical weather prediction (NWP) are advanced computational techniques that use global basis functions to solve the atmospheric governing equations with high accuracy. These methods are particularly effective for simulating large-scale weather patterns due to their exponential convergence properties when solutions are smooth. Here's a detailed breakdown:

Core Concept of Spectral Methods

Spectral methods approximate meteorological variables (e.g., wind, temperature) by expanding them into series of orthogonal basis functions, such as spherical harmonics for global atmospheric models. For example, a variable \[ u(x,t) \] might be written as:
\[ u(x,t) = \sum_{n=0}^N a_n(t) \phi_n(x) \]
where \(\phi_n(x)\) are pre-defined basis functions (e.g., Fourier modes or Chebyshev polynomials), and \(a_n(t)\) are time-dependent coefficients. Derivatives are computed analytically using the properties of these functions, avoiding the numerical errors common in finite-difference approximations.

ECMWF: Spherical harmonics

Spherical harmonics are a set of orthogonal basis functions defined on the surface of a sphere. They are used to represent atmospheric variables such as wind, temperature, and pressure in terms of a series expansion.

How to find the resolution of a spherical harmonic field in GRIB

$ grib_get -p J t1000.grib
1279

In this case, the value of J=1279 indicates the triangular truncation used is T1279.

It is not possible to relate the resolution of the spherical harmonic data directly to the resolution of the corresponding grid as this depends on the specific relationship between the spectral representation and the Gaussian grid used by the IFS. Three different relationships have been used by ECMWF over the years. For a triangular truncation of T, the corresponding Gaussian grid resolutions are:

Relationship	Gaussian grid resoluion
linear grid: the shortest wavelength is represented by 2 grid point	N = (T + 1) / 2
quadratic grid: the shortest wavelength is represented by 3 grid points	N = 3 (T+1) / 4
cubic grid: the shortest wavelength is represented by 4 grid points	N = T + 1

where N corresponds to the number of latitude lines between the poles and the equator. However, to determine which relationship was used when the data were produced requires interrogation of a corresponding grid point field.

Gridpoint Equivalency of Spectral Resolution

• Gridpoint Equivalency of Spectral Resolution

It takes about five to seven grid points to get reasonable approximations of most weather features. Still more points per wave feature are often necessary to get a good forecast. Because spectral and gridpoint models preserve information in different ways, no precise equivalent grid spacing can be given for a spectral model resolution.

The approximate grid spacing with the same accuracy as a spectral model can then be represented as \(\Delta X \sim 360° / 3N\); where \(N\) is the number of waves.

\[ \Delta X \sim 360° / 3N \]

For a T80 model, this results in maximum grid spacing for equivalent accuracy of about 1.5° ~ 160 km (assuming 1° ~ 111 km).

We can approximate the grid spacing to obtain equivalent accuracy to a spectral model with a fixed number of waves using a simple approach. First, we assume that three grid points are sufficient to capture the information contained in each of a series of continuous waves.

The dynamics of spectral models retain far better wave representation than gridpoint models with this grid spacing. However, the spectral model physics is calculated on a grid, with about three times as many grid lengths as number of waves used to represent the data. Since it takes five to seven grid points to represent 'wavy' data well and even more for features that include discontinuities, the resolution of the physics is poorer than the above formulation indicates and degrades the quality of the spectral model forecast.

TxxLyy

For example, T42L9 refers to a specific configuration of a global spectral model used in numerical weather prediction (NWP). Here's a breakdown of what each part means:

• T42: This indicates the triangular truncation of the spectral model. In this case, the maximum wavenumber (or degree) is 42. This means that the model includes spherical harmonics up to degree 42, which corresponds to a horizontal resolution of approximately 2.8 degrees (about 300 km) at the equator.

• L9: This denotes the vertical resolution of the model. In this case, the model has 9 vertical layers. The vertical layers are typically defined by a set of levels in the atmosphere, often using a hybrid coordinate system that combines pressure and height coordinates.

The T42L9 model is a relatively low-resolution model compared to modern standards, which often use higher resolutions like T1279L91 (used by ECMWF's IFS) or T574L64 (used by some versions of the GFS). However, the T42L9 model can still be useful for research purposes or as a dynamic kernel in simpler forecasting systems, such as the self-memorial T42 model (SMT42) mentioned in some studies.

Advantages in NWP

• Exponential Convergence: For smooth solutions, spectral methods achieve errors decreasing faster than any polynomial of the grid size, making them highly efficient for large-scale simulations.
  
• Global Representation: Unlike finite-element or finite-volume methods, which approximate solutions locally, spectral methods inherently model atmospheric waves (e.g., Rossby waves) across the entire domain, crucial for global weather prediction.
  
• Computational Efficiency: Operations like differentiation are performed in spectral space via transforms (e.g., Fast Fourier Transforms), reducing computational cost compared to traditional grid-based methods.

Implementation in NWP Models

Most global NWP models (e.g., ECMWF’s IFS) use spectral methods for horizontal discretization, combined with finite-difference or finite-volume techniques in the vertical direction. The process involves:

1. Transform: Convert grid-point data to spectral coefficients using spherical harmonic transforms.
   
2. Spectral Computation: Solve dynamical equations (e.g., momentum, continuity) in spectral space.
   
3. Inverse Transform: Convert results back to physical space for parameterizations (e.g., cloud physics).

For example, the PEAK model (Primitive-Equation Atmospheric Research Model Kernel) demonstrates how spectral discretization handles the shallow-water equations and primitive equations, balancing accuracy and computational cost.

Limitations

• Localized Features: Sharp gradients (e.g., thunderstorms) are poorly resolved due to the global nature of basis functions, often requiring hybrid approaches.
  
• Spectral Truncation: Retaining only a finite number of basis functions introduces aliasing errors in nonlinear terms, mitigated via techniques like dealiasing.

Comparison to Finite Methods

Aspect	Spectral Methods	Finite-Difference/Volume
Accuracy	Exponential for smooth solutions	Polynomial (e.g., \(O(h^2)\))
Domain Handling	Global	Local
Computational Cost	Lower for high accuracy	Higher for equivalent resolution
Shock Capture	Poor (no built-in dissipation)	Better with flux limiters

Applications

Spectral methods dominate global NWP due to their efficiency in simulating planetary-scale dynamics. Regional models, which require localized precision, often prefer finite-volume methods. In summary, spectral methods excel in global NWP by leveraging smooth basis functions for accurate, efficient simulations of large-scale atmospheric behavior, while hybrid models address their limitations in resolving localized phenomena.

References

1. Ehrendorfer, Martin. Spectral numerical weather prediction models. Society for Industrial and Applied Mathematics, 2011.
2. A new grid for the IFS | 2016
   1. The new ECMWF interpolation package MIR | 2017
3. How to access the data values of a spherical harmonic field in GRIB | 2023
4. Wedi, Nils P., Mats Hamrud, and George Mozdzynski. "A fast spherical harmonics transform for global NWP and climate models." Monthly Weather Review 141.10 (2013): 3450-3461. | ECMEF
5. The spectral technique | Reading | 1983 | M. Jarraud and A.J. Simmons
6. Chu, Peter C., Xiong-Shan Chen, and Chenwu Fan. "Comparison between wavenumber truncation and horizontal diffusion methods in spectral models." Monthly weather review 129.1 (2001): 152-158.
7. Spectral Technique in Atmospheric Modelling | 07 October, 2021
8. 中央氣象局第二代全球波譜模式之介紹 | 1993
9. 截谱模式 | 《中国⼤百科全书》第三版⽹络版 | 2022
10. 国家气象中心业务数值预报发展的回顾与展望 | 2010
    1. 从T42L9到T213L31，中期预报模式从建立到逐步发展完善，模式在分辨率的提高，物理过程的改进、完善，数值计算技术及计算机应用技术等方面都取得了明显的进步，客观分析、初值化、资料同化方法及资料的有效利用技术等也有所提高，模式的有效预报时效明显延长，降水落区预报的可用性明显改善。
    2. 目前，国家气象中心的业务数值预报已建立起比较完整的数值天气预报业务体系，中期预报模式已由T213L31 (水平分辨率约为60 km)升级为TL639L60 (水平分辨率约为28 km)；区域预报模式已由我国自行研制的 GRAPES-MESO 中尺度预报模式(水平分辨率为0.15°×0.15°)取代了原区域预报模式HLAFS；
       1. \((360°/213/3)*111km \sim 62.5km\)
11. 中央气象台全球中期数值预报业务系统的发展 | 2021 | 沈学顺