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- Author or Editor: Qiu-shi Chen x

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## Abstract

This paper examines further the problem of deducing the wind field from vorticity and divergence over a limited area with prescribed winds at the boundary. An earlier work showed that the wind field in a limited area can be partitioned into internal divergent, internal rotational, and harmonic wind components. Because the harmonic wind is both nondivergent and irrotational, it is demonstrated in this paper that the two harmonic wind components at the boundary must satisfy a consistency condition. Based on this properly, a direct method is developed to solve two Laplace equations with the prescribed two harmonic wind components at the boundary. If the prescribed harmonic wind components at the boundary satisfy the consistency condition, the solution of the two Laplace equations must be nondivergent and irrotational. The direct method is shown to be highly accurate and efficient. If the prescribed wind at the boundary does not satisfy the consistency condition, this implies a mismatch between the interior vorticity and divergence and the prescribed winds at the boundary. This inconsistency must be removed before the wind field can be reconstructed. A method to remove this inconsistency is discussed.

A harmonic-cosine series expansion is also developed for a function over a limited area. The application of the harmonic-cosine series expansion to the wind-field partitioning and reconstruction problem has two distinct advantages compared with the harmonic-sine series expansion. The first is that the internal and harmonic winds can be more accurately determined at the boundary. The second is that the partitioning of the wind field into streamfunction and velocity potential can be obtained more efficiently and accurately through an iterative method.

## Abstract

This paper examines further the problem of deducing the wind field from vorticity and divergence over a limited area with prescribed winds at the boundary. An earlier work showed that the wind field in a limited area can be partitioned into internal divergent, internal rotational, and harmonic wind components. Because the harmonic wind is both nondivergent and irrotational, it is demonstrated in this paper that the two harmonic wind components at the boundary must satisfy a consistency condition. Based on this properly, a direct method is developed to solve two Laplace equations with the prescribed two harmonic wind components at the boundary. If the prescribed harmonic wind components at the boundary satisfy the consistency condition, the solution of the two Laplace equations must be nondivergent and irrotational. The direct method is shown to be highly accurate and efficient. If the prescribed wind at the boundary does not satisfy the consistency condition, this implies a mismatch between the interior vorticity and divergence and the prescribed winds at the boundary. This inconsistency must be removed before the wind field can be reconstructed. A method to remove this inconsistency is discussed.

A harmonic-cosine series expansion is also developed for a function over a limited area. The application of the harmonic-cosine series expansion to the wind-field partitioning and reconstruction problem has two distinct advantages compared with the harmonic-sine series expansion. The first is that the internal and harmonic winds can be more accurately determined at the boundary. The second is that the partitioning of the wind field into streamfunction and velocity potential can be obtained more efficiently and accurately through an iterative method.

## Abstract

A harmonic-sine series expansion for a function in two-dimensional space is proposed to be a sum of two parts. The harmonic part is the solution of the Laplace equation with prescribed boundary values of this function. The inner part is the function from which the harmonic part has been subtracted; thus, it has zero boundary value and can be expanded by the double Fourier sine series. By using the harmonic-sine series expansion, it is shown that only simple operations are needed to solve the Laplace, Poisson, and Helmholtz equations with a given boundary condition.

The harmonic-sine series expansion is used to solve the wind partitioning and reconstruction problems in a limited area. The internal wind is computed from the inner parts of the streamfunction and the velocity potential. The harmonic wind is the difference between the observed wind and internal wind. In a limited region, the internal wind can be dealt with in the same way as the horizontal wind on the globe. The development of the vorticity and divergence in a limited area can be diagnosed from the inner parts of the streamfunction and velocity potential, and the corresponding internal rotational and divergent wind components. As long as the inner parts of the streamfunction and velocity potential are defined, the separation of the wind field into the internal rotational, the internal divergent, and the harmonic winds becomes completely definite. The harmonic wind is not only nondivergent but also irrotational in a limited region.

In both partitioning and reconstruction problems, the key is to solve the Laplace equations of the harmonic parts with the prescribed boundary value of the harmonic wind. The solution of the harmonic parts for the key problem is not unique, but the computed harmonic wind from the harmonic parts is. Based on this characteristic, an iterative method is developed. From a real-data example, it is demonstrated that the harmonic parts of the streamfunction and velocity potential and the computed harmonic wind can be accurately determined within 15 iterations. The iteration method by using harmonic-sine series expansion is very effective in solving the partitioning and reconstruction of problems in a limited region.

## Abstract

A harmonic-sine series expansion for a function in two-dimensional space is proposed to be a sum of two parts. The harmonic part is the solution of the Laplace equation with prescribed boundary values of this function. The inner part is the function from which the harmonic part has been subtracted; thus, it has zero boundary value and can be expanded by the double Fourier sine series. By using the harmonic-sine series expansion, it is shown that only simple operations are needed to solve the Laplace, Poisson, and Helmholtz equations with a given boundary condition.

The harmonic-sine series expansion is used to solve the wind partitioning and reconstruction problems in a limited area. The internal wind is computed from the inner parts of the streamfunction and the velocity potential. The harmonic wind is the difference between the observed wind and internal wind. In a limited region, the internal wind can be dealt with in the same way as the horizontal wind on the globe. The development of the vorticity and divergence in a limited area can be diagnosed from the inner parts of the streamfunction and velocity potential, and the corresponding internal rotational and divergent wind components. As long as the inner parts of the streamfunction and velocity potential are defined, the separation of the wind field into the internal rotational, the internal divergent, and the harmonic winds becomes completely definite. The harmonic wind is not only nondivergent but also irrotational in a limited region.

In both partitioning and reconstruction problems, the key is to solve the Laplace equations of the harmonic parts with the prescribed boundary value of the harmonic wind. The solution of the harmonic parts for the key problem is not unique, but the computed harmonic wind from the harmonic parts is. Based on this characteristic, an iterative method is developed. From a real-data example, it is demonstrated that the harmonic parts of the streamfunction and velocity potential and the computed harmonic wind can be accurately determined within 15 iterations. The iteration method by using harmonic-sine series expansion is very effective in solving the partitioning and reconstruction of problems in a limited region.

## Abstract

In *σ* coordinates, a variable *ϕ*_{e}(*x, y, σ, t*) whose horizontal gradient −**∇***ϕ*_{e} is equal to the irrotational part of the horizontal pressure gradient force is referred to as an equivalent isobaric geopotential height. Its inner part can be derived from the solution of a Poisson equation with zero Dirichlet boundary value. Because −**∇***ϕ*(*x, y, p, t*) is also the irrotational part of the horizontal pressure gradient force in *p* coordinates, the equivalent geopotential *ϕ*_{e} in *σ* coordinates can be used in the same way as the geopotential *ϕ*(*x, y, p, t*) used in *p* coordinates. In the sea level pressure (SLP) analysis over Greenland, small but strong high pressure systems often occur due to extrapolation. These artificial systems can be removed if the equivalent geopotential *ϕ*_{e} is used in synoptic analysis on a constant *σ* surface, for example, at *σ* = 0.995 level. The geostrophic relation between the equivalent geopotential and streamfunction at *σ* = 0.995 is approximately satisfied.

Because weather systems over the Tibetan Plateau are very difficult to track using routine SLP, 850-hPa, and 700-hPa analyses, equivalent isobaric geopotential analysis in *σ* coordinates is especially useful over this area. An example of equivalent isobaric geopotential analysis at *σ* = 0.995 shows that a secondary high separated from a major anticyclone over the Tibetan Plateau when cold air affected the northeastern part of the plateau, but this secondary high is hardly resolved by the SLP analysis. The early stage of a low (or vortex), called a southwest vortex due to its origin in southwest China, over the eastern flank of the Tibetan Plateau is more clearly identified by equivalent isobaric geopotential analysis at *σ* = 0.825 and 0.735 than by routine isobaric analysis at the 850- and 700-hPa levels. Anomalous high and low systems in the SLP analysis over the Tibetan Plateau due to extrapolation are all removed by equivalent isobaric geopotential analysis at *σ* = 0.995.

Use of equivalent geopotential *ϕ*_{e} in the vorticity and divergence equations is presented, and the equivalent geopotential equation is derived. These equations can be used in numerical models, initializations, and other dynamical studies. As an example, it is shown how these equations are used to derive a velocity potential form of the generalized *ω* equation in *σ* coordinates. As a check, retrieval of precipitation over Greenland using this *ω* equation shows that the computed precipitation distributions for 1987 and 1988 are in good agreement with the observed annual accumulation.

## Abstract

In *σ* coordinates, a variable *ϕ*_{e}(*x, y, σ, t*) whose horizontal gradient −**∇***ϕ*_{e} is equal to the irrotational part of the horizontal pressure gradient force is referred to as an equivalent isobaric geopotential height. Its inner part can be derived from the solution of a Poisson equation with zero Dirichlet boundary value. Because −**∇***ϕ*(*x, y, p, t*) is also the irrotational part of the horizontal pressure gradient force in *p* coordinates, the equivalent geopotential *ϕ*_{e} in *σ* coordinates can be used in the same way as the geopotential *ϕ*(*x, y, p, t*) used in *p* coordinates. In the sea level pressure (SLP) analysis over Greenland, small but strong high pressure systems often occur due to extrapolation. These artificial systems can be removed if the equivalent geopotential *ϕ*_{e} is used in synoptic analysis on a constant *σ* surface, for example, at *σ* = 0.995 level. The geostrophic relation between the equivalent geopotential and streamfunction at *σ* = 0.995 is approximately satisfied.

Because weather systems over the Tibetan Plateau are very difficult to track using routine SLP, 850-hPa, and 700-hPa analyses, equivalent isobaric geopotential analysis in *σ* coordinates is especially useful over this area. An example of equivalent isobaric geopotential analysis at *σ* = 0.995 shows that a secondary high separated from a major anticyclone over the Tibetan Plateau when cold air affected the northeastern part of the plateau, but this secondary high is hardly resolved by the SLP analysis. The early stage of a low (or vortex), called a southwest vortex due to its origin in southwest China, over the eastern flank of the Tibetan Plateau is more clearly identified by equivalent isobaric geopotential analysis at *σ* = 0.825 and 0.735 than by routine isobaric analysis at the 850- and 700-hPa levels. Anomalous high and low systems in the SLP analysis over the Tibetan Plateau due to extrapolation are all removed by equivalent isobaric geopotential analysis at *σ* = 0.995.

Use of equivalent geopotential *ϕ*_{e} in the vorticity and divergence equations is presented, and the equivalent geopotential equation is derived. These equations can be used in numerical models, initializations, and other dynamical studies. As an example, it is shown how these equations are used to derive a velocity potential form of the generalized *ω* equation in *σ* coordinates. As a check, retrieval of precipitation over Greenland using this *ω* equation shows that the computed precipitation distributions for 1987 and 1988 are in good agreement with the observed annual accumulation.

## Abstract

Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (*x*, *y*) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width *D*, and analyze only the Fourier sums in the smaller domain (*x*, *y*) ∈ [*D*, 1 − *D*] ⊗ [*D*, 1 − *D*]. It is shown that the error in a one-dimensional Fourier series with *N* terms or in a two-dimensional series with *N*
^{2} terms, is *smaller* by a factor of *N* on a *boundary-buffer-discarded* domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.

## Abstract

Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (*x*, *y*) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width *D*, and analyze only the Fourier sums in the smaller domain (*x*, *y*) ∈ [*D*, 1 − *D*] ⊗ [*D*, 1 − *D*]. It is shown that the error in a one-dimensional Fourier series with *N* terms or in a two-dimensional series with *N*
^{2} terms, is *smaller* by a factor of *N* on a *boundary-buffer-discarded* domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.

## Abstract

In comparison to the Tatsumi’s spectral method, the harmonic-Fourier spectral method has two major advantages. 1) The semi-implicit scheme is quite efficient because the solutions of the Poisson and Helmholtz equations are readily derived. 2) The lateral boundary value problem of a limited-area model is easily solved. These advantages are the same as those of the spherical harmonics used in global models if the singularity at the pole points for a globe is considered to be the counterpart of the lateral boundary condition for a limited region.

If a limited-area model is nested in a global model, the prediction of the limited-area model at each time step is the sum of the inner part and the harmonic part predictions. The inner part prediction is solved by the double sine series from the inner part equations for the limited-area model. The harmonic part prediction is derived from the prediction of the global model. An external wind lateral boundary method is proposed based on the basic property of the wind separation in a limited region. The boundary values of a limited-area model in this method are not given at the closed boundary line, but always given by harmonic functions defined throughout the limited domain. The harmonic functions added to the inner parts at each time step represent the effects of the lateral boundary values on the prediction of the limited-area model, and they do not cause any discontinuity near the boundary.

Tests show that predicted motion systems move smoothly in and out through the boundary, where the predicted variables are very smooth without any other boundary treatment. In addition, the boundary method can also be used in the most complicated mountainous region where the boundary intersects high mountains. The tests also show that the adiabatic dynamical part of the limited-area model very accurately predicts the rapid development of a cyclone caused by dry baroclinic instability along the east coast of North America and a lee cyclogenesis case in East Asia. The predicted changes of intensity and location of both cyclones are close to those given by the observations.

## Abstract

In comparison to the Tatsumi’s spectral method, the harmonic-Fourier spectral method has two major advantages. 1) The semi-implicit scheme is quite efficient because the solutions of the Poisson and Helmholtz equations are readily derived. 2) The lateral boundary value problem of a limited-area model is easily solved. These advantages are the same as those of the spherical harmonics used in global models if the singularity at the pole points for a globe is considered to be the counterpart of the lateral boundary condition for a limited region.

If a limited-area model is nested in a global model, the prediction of the limited-area model at each time step is the sum of the inner part and the harmonic part predictions. The inner part prediction is solved by the double sine series from the inner part equations for the limited-area model. The harmonic part prediction is derived from the prediction of the global model. An external wind lateral boundary method is proposed based on the basic property of the wind separation in a limited region. The boundary values of a limited-area model in this method are not given at the closed boundary line, but always given by harmonic functions defined throughout the limited domain. The harmonic functions added to the inner parts at each time step represent the effects of the lateral boundary values on the prediction of the limited-area model, and they do not cause any discontinuity near the boundary.

Tests show that predicted motion systems move smoothly in and out through the boundary, where the predicted variables are very smooth without any other boundary treatment. In addition, the boundary method can also be used in the most complicated mountainous region where the boundary intersects high mountains. The tests also show that the adiabatic dynamical part of the limited-area model very accurately predicts the rapid development of a cyclone caused by dry baroclinic instability along the east coast of North America and a lee cyclogenesis case in East Asia. The predicted changes of intensity and location of both cyclones are close to those given by the observations.