Inversion of global balance equation with spectral method using the spherical harmonics and double Fourier series

정형빈,권인혁,송영철 한국기상학회 2004 Asia-Pacific Journal of Atmospheric Sciences Vol.40 No.3

Global linear balance equation is solved with spectral method. Two kinds of expansion functions are considered, one the spherical harmonics and the other Double Fourier Series (DFS). For the Spherical Harmonics Method (SHM) a part of the recursion equations for the expansion coefficients are incomplete, and thus they are modified into three-term recursion relations just by combining two successive recursion equations with equal weight. For DFS method the balance equation is represented with tetra-diagonal matrix equations whose elements are the spectral coefficients. The matrix equations with the truncation-mismatch are modified into penta-diagonal matrices in an analogous manner to the SHM case. Test results showed that the accuracy is comparable for both methods. Normalized errors are in the order of machine rounding. DFS method is applied to the observed geopotential field of various pressure levels. It is shown that the balanced streamfunction and wind field can be obtained without the equatorial singularity. That the equatorial singularity of the global balance equation does not appear is further confirmed by the singular value decomposition of the spherical-harmonics coefficients matrix. In contrast to the previous study, all singular values were found to be nonzero. The sensitivity of singular values, particularly for the mode with the smallest value, to the resolution and zonal wavenumber is discussed. Although the balanced wind in the tropics is not very useful due to a large departure from the observed wind, it can be retrieved with accuracy only by a global treatment of the balance equation. 스펙트럴법을 이용한 전구 선형 균형 방정식(GLBE)의 해법을 연구하였다. 구면 조화 함수와 이중 후리에 급수를 각각 기저함수로 사용하는 두 종류의 스펙트럴법을 고려하였다. 구면조화함수를 사용한 스펙트럴법(SHM)에서는 전개계수의 일부분이 불완전하여 3-계수의 점화방정식(즉, 3-대각 행렬식)으로 변형하여 완전한 닫힌 방정식으로 처리하였다. 이중 후리에 급수의 스펙트럴법(DFS)를 사용하면 GLBE는 4-계수 행렬식으로 표현되는데, 일부분의 행렬식에서는 파수절단의 불일치가 발생한다. 이 경우에는 방정식을, 코리올리모수를 이용하여 5-계수 행렬식으로 변형하였다. 여러 함수를 사용하여 해법을 테스트한 결과, 두 방법 모두 매우 정확한 속도장을 역산할 수 있음을 확인하였다. 두 방법의 정확성은 거의 같았으며 정규화한 오차는 10-14-10-15 정도였다. DFS-스펙트럴법을 NCEP/NCAR의 관측자료에 적용하여 균형 속도장을 역산하였다. 본 연구에서 사용된 방법에서는 GLBE의 적도 특이성이 제거되어서 안정적으로 속도장을 계산 할 수 있었다. 적도-특이성이 제거됨을 확인하기 위하여 GLBE를 특이치 분해하여 특이치를 조사하였다. 모든 특이치가 양수로 판단되어서 본 연구에서 제시한 방법의 타당성을 확인 할 수 있었다. 보다 일반성을 확인하기 위하여 스펙트럴 전개의 분해능에 따른 특이치의 분포를 조사하였다. 선형 균형 속도장은 중-고위도에서는 관측치와 잘 일치하나 적도지역에서는 큰 차이를 보이는데, 적도를 중심으로 남북 5도 이내의 자료를 제외하면 매우 유용한 것으로 보인다.

Isotropic High-Order Filter for a Local Domain on the Sphere

정형빈,남예진,이충희,김재진 한국기상학회 2021 Asia-Pacific Journal of Atmospheric Sciences Vol.57 No.1

High-order filtering with isotropy of a local-domain data defined at an arbitrary location on the sphere, with an application to scale-decomposition of mean sea level pressure field, was investigated. The equation of high-order filter was based on the implicit hyper-diffusion of 8th-order, which was discretized by the Fourier-finite element method (FFEM). In order to apply the FFEM high-order filter to a local-domain data which is not represented with two-dimensional array, a rotation of coordinates was performed. The geometry of the rotated local domain was made to be either a polar cap (polar cap domain) or a symmetric window with respect to the Equator (Equatorial symmetric domain) to reduce the anisotropy of the local domains caused by the metric effect of spherical coordinates. The rotated local domains are extended by padding artificial data to match the boundary condition of vanishing gradient. The performance of the filters for the symmetric- or asymmetric- domain, was compared for both initial fields defined with an analytic function and the observed meteorological data. It turned out that the filters on the local domains with symmetry, i.e., the polar cap domain and the Equatorial symmetric domain, produced improved results over the local domains with asymmetry: Asymmetric error pattern between the northern and southern boundary, which is typical of the window domain defined on the middle latitude (or between the pole and the Equator), is reduced appreciably and the improvement is shown to become more significant for the window-domain closer to the pole. Through tests with wind fields, the performance of the filters on the polar cap domain and the Equatorial symmetric domain was found comparable to each other. The advantage of polar cap filter, being not available for the Equatorial symmetric domain filter, was illustrated by conducting a scale decomposition and composite analysis of the mean sea level pressure that requires the data manipulations such as a zonal averaging as well as azimuthal-phase shifting.

이중퓨리에 급수를 이용한 천해파방정식의 해법

정형빈 한국기상학회 1999 한국기상학회 학술대회 논문집 Vol.1999 No.-

Composite Analysis of the Beta-gyre and Rossby Wave Induced by Tropical Cyclones

정형빈,남예진,이충희 한국기상학회 2023 Asia-Pacific Journal of Atmospheric Sciences Vol.59 No.2

Beta gyre and Rossby wave train induced by tropical cyclones were identified from the ERA5 global-reanalysis data ofthe recent 30 years using a composite method. To pick up the disturbances relevant to the beta gyre and Rossby wave trainsurrounding tropical cyclones, the disturbances were decomposed into three distinct horizontal scales including small,intermediate, and planetary-scale. Composite map of the disturbances containing small- and intermediate-scale showed awell-organized Rossby wave train. The orientation of wave train was found to depend on the translation direction of tropicalcyclones, and also appeared to split into two orientations except for those translating in the west-northwestward direction. The wave energy of the wave train was shown to propagate along the wave train axis, which was inferred from the amplitudechange with time within the wave train. The wave train shows a weak upward-westward tilt and increasing amplitude withheight, implying the wave energy propagating upward. A dipole circulation cell, bearing a close resemblance to the betagyre depicted in the theories and numerical models, was found from the Rossby wave train. The strength and orientation ofthe beta gyres were revealed to vary with the translation direction of the tropical cyclones, with the weakest and strongestamplitudes being found for the westward- and northward-translation cases, respectively. It was shown that the orientationof the beta gyre obtained by a lag-composite method rotates clockwise with time regardless of the translation direction oftropical cyclones.

Construction of the Spherical High-Order Filter for Applications to Global Meteorological Data

정형빈,Han-Byeol Jeong 한국지구과학회 2015 韓國地球科學會誌 Vol.36 No.5

The high-order Laplacian-type filter, which is capable of providing isotropic and sharp cut-off filtering on the spherical domain, is essential in processing geophysical data. In this study, a spherical high-order filter was designed by combining the Fourier method with finite difference-method in the longitude and latitude, respectively. The regular grid system was employed in the filter, which has uniform angular spacing including the poles. The singularity at poles was eliminated by incorporating variable transforms and continuity-matching boundary conditions across poles. The high-order filter was assessed using the Rossby-Haurwitz wave, the observed geopotential, and observed wind field. The performance of the filter was found comparable to the filter based on the Galerkin procedure. The filter, employing the finite difference method, can be designed to give any target order of accuracy, which is an important advantage being unavailable in other methods. The computational complexity is represented with diagonal matrices solver with being the target order of accuracy. Along with the availability of arbitrary target-order, it is also advantageous that the filter can adopt the reduced grid to increase computational efficiency.

Geopotential Field in Nonlinear Balance with the Sectoral Mode of Rossby-Haurwitz Wave on the Inclined Rotation Axis

정형빈,박자린 한국지구과학회 2007 韓國地球科學會誌 Vol.28 No.7

Analytical geopotential field in balance with the sectoral mode (the first symmetric mode with respect to theequator) of the Rosby-Haurwitz wave on the inclined rotation axis was derived in presence of superrotation backgroundflow. The balanced field was obtained by inverting the divergence equation with the time derivative being zero. Theinversion consists of two steps, i.e., the evaluation of nonlinear forcing terms and the finding of analytical solutions basedon the Poisson’s equation. In the second step, the forcing terms in the form of Legendre function were readily inverteddue to the fact that Legendre function is the eigenfunction of the spherical Laplacian operator, while other terms weresolved either by introducing a trial function or by integrating the Legendre equation. The balanced field was found to beexpressed with six zonal wavenumber components, and shown to be of asymmetric structure about the equator. Inassociation with asymmetricity, the advantageous point of the balanced field as a validation method for the numericalmodel was addressed. In special cases where the strength of the background flow is a half of or exactly the same as therotation rate of the Earth, it was revealed that one of the zonal wavenumber components vanishes. The analyticalbalanced field was compared with the geopotential field which was obtained using a spherical harmonics spectral model.It was found that the normalized diference lied in the order of machine rounding, indicating the reliability of theanalytical results. The stability of the sectoral mode of Rossby-Haurwitz wave and the associated balanced field wasdiscussed, comparing with the first antisymmetric mode........... . : .... .... .. ..... , ... .... .... .... .. ... ... ........... .... (... .. .. .... . .. .. ). ... ... ...... ..... ..... . .

전구균일격자체계에 대한 라플라시안 연산자의 고유치와 고유벡터

정형빈,강현규,이재덕 한국기상학회 2014 한국기상학회 학술대회 논문집 Vol.2014 No.5

전지구 수치예보 모델의 역학코어

정형빈 한국기상학회 2017 한국기상학회 학술대회 논문집 Vol.2017 No.10

이중후리에급수 스펙트럴법의 최적 파수절단에 관한 연구

정형빈,권인혁,박성욱 한국기상학회 2007 한국기상학회 학술대회 논문집 Vol.2007 No.-

Tropical Cyclone Track and Intensity Prediction with a Structure Adjustable Balanced Vortex

정형빈,권인혁,강현규,박자린,한현준,김재진 한국기상학회 2011 Asia-Pacific Journal of Atmospheric Sciences Vol.47 No.3

A new Tropical Cyclone (TC) initialization method with the structure adjustable bogus vortex was applied to the forecasts of track, central pressure, and wind intensity for the 417 TCs observed in the Western North Pacific during the 3-year period of 2005-2007. In the simulations the Final Analyses (FNL) with 1°× 1° resolution of National Center for Environmental Prediction (NCEP) were incorporated as initial conditions. The present method was shown to produce improved forecasts over those without the TC initialization and those made by Regional Specialized Meteorological Center Tokyo. The average track (central pressure, wind intensity) errors were as small as 78.0 km (11.4 hPa, 4.9 m s^(−1)) and 139.9 km (12.4 hPa,5.5 m s^(−1)) for 24-h and 48-h forecasts, respectively. It was found that the forecast errors are almost independent on the size and intensity of the observed TCs because the size and intensity of the bogus vortex can be adjusted to fit the best track data. The results of this study indicate that a bogus method is useful in predicting simultaneously the track, central pressure, and intensity with accuracy using a dynamical forecast model.