a. The vorticity and divergence equations

The vorticity and divergence equation are derived from (2.2) and written in the form

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where

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The main parts of Ω_adv and D_adv are the horizontal and vertical advection, so that Ω_adv and D_adv are referred to as the vorticity and divergence advection terms, respectively. The terms K_ζ and K_δ are derived from K_u and K_v, and they represent the influence of unsolved horizontal scales on vorticity and divergence, respectively.

b. The vertical finite-difference form of the equations

Let U, V, Ω, and D be

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It is assumed that

m²m²₀m²(2.23)

where m²₀ is the averaged value of m² over the integration region and (m²)′ is its deviation. The vertical distribution of variables is shown in Fig. 1. Based on the vertical difference form derived in the appendix, the continuity equation (A.5) can be rewritten as

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where

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If the basic equations are written in column-vector form, (A.40) and (A.43) become

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and

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The thermodynamic equation (A.33) is rewritten as

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where T_adv becomes

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The hydrostatic equation (A.28) is

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Substituting it into (2.27), we have

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The equation for the mixing ratio (A.45) is

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The equations (2.24), (2.26), (2.28), (2.31), and (2.32) are the basic equations for the variables lnp∗, Ω, T, D, and q.

a. The separation of the scalar variables and wind

The limited-area model is defined in a closed region R shown by Fig. 2a. The region shown by Fig. 2b is an open region without the boundary, and it is also referred to as interior of R or within the region R. The closed line Σ shown by Fig. 2c is the boundary of region R. Based on the harmonic-sine series expansion (CK92a), a scalar variable A, such as temperature or geopotential height, defined on closed R can be divided into the harmonic and inner parts, A = A_i + A_h, where the harmonic part A_h satisfies the Laplace equation and Dirichlet boundary condition

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and the inner part A_i is determined by A_i = A − A_h. The inner part A_i vanishes at the boundary Σ automatically and can be expanded in a double sine series. The streamfunction and velocity potential on closed R are separated by

ψψ_hψ_iχχ_hχ_i(3.2)

where the inner parts ψ_i and χ_i satisfy the Poisson equations

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and they are easily solved by the double sine series. The internal wind on closed R is defined by

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and the external wind on closed R is derived by

U_EUU_IV_EVV_I(3.5)

The harmonic parts, ψ_h and χ_h, fulfill the Laplace equations

²ψ_h²χ_h(3.6)

in the interior of R, and at the boundary Σ they satisfy

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The separation of wind field into the internal and external winds based on (3.4) and (3.5) is different from that of scalar variables.

Because the inner parts are expressed by the double sine series, ∇²ψ_i and ∇²χ_i at the boundary Σ become

²ψ_iΣ²χ_iΣ(3.8)

If the vorticity and divergence need to be computed on closed R from the inner parts, they can be derived by

²ψ_iΣ²χ_iD_ΣD,(3.9)

where the symbol Ω|_Σ is defined on closed R and denoted by

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To avoid discontinuity of (3.9) at the boundary, the following method can be used. The vorticity and divergence are separated into their harmonic and inner parts expressed by

_ihDD_iD_h(3.11)

where Ω_h and D_h satisfy Laplace equations ∇²Ω_h = 0 and ∇²D_h = 0 with the Dirichlet boundary values Ω_h|_Σ = Ω|_Σ and D_h|_Σ = D|_Σ, respectively. Introducing two new variables, ψ_h,i and χ_h,i, they are a portion of the inner parts of streamfunction and velocity potential, respectively, and they satisfy the Poisson equations

²ψ_h,ih,²χ_h,iD_h,(3.12)

with zero Dirichlet boundary value. Based on (3.11), the inner parts Ω_i and D_i can be derived by Ω_i = ∇²ψ_i − ∇²ψ_h,i and D_i = ∇²χ_i − ∇²_h,i. Thus, the vorticity and divergence on closed R computed from the inner parts are expressed by

²ψ_iψ_h,ih²χ_iχ_h,iD_hD.(3.13)

b. The governing equations of the inner parts of the streamfunction and velocity potential

To simplify notations, let F denote a finite Fourier sine transform operator in the two-dimensional space, that is,

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The double Fourier sine series of f_I are

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where F⁻¹ is the inverse Fourier sine transform operator from wave to physical space.

Thus, the solutions of (3.15) are

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and

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where

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and L_mn is referred to as a horizontal scale of the wave and expressed by

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 Utilizing (3.18) and (3.19), (2.26) and (2.27) can be solved with zero Dirichlet boundary values, and their solutions are

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Because the terms K_ζ, K_δ, K_T, and K_q represent the influence of unsolved horizontal scales, their treatment involves a horizontal diffusion solved by the spectral method based on the splitting procedure. The method for solving these terms is described in section 4e, and thus, these terms are temporarily omitted from (3.22) and (3.23).

c. The equation of the inner part of the generalized geopotential height on the constant σ surface

Now we introduce a variable

ϕΦRT₀p(3.24)

where ϕ′ is referred to as the generalized geopotential height in σ coordinates. Because the geopotential height on the constant σ surface depends on the temperature through the hydrostatic equation (2.30), the generalized geopotential height is expressed by

ϕϕRBTRT₀p(3.25)

Taking the partial derivative with respect to t, (3.25) becomes

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Substituting (2.24) and (2.28) into (3.26), the equation of the generalized geopotential becomes

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where matrix A is

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Here,

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denotes the variation of the generalized geopotential height caused by the advection and diabatic heating.

Based on (3.14), (3.27) can be rewritten into two equations as

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and

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Based on (3.10), (3.11), and (3.12), (3.30) is rewritten in the form

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where

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is the harmonic part of the divergence in the closed region with zero boundary value. Equations (3.22), (3.23), and (3.32) are the basic equations of the inner parts of the three variables: the streamfunction, velocity potential, and generalized geopotential.

d. The equation of the generalized ageostrophic geopotential

If the inner parts of the streamfunction and geopotential in p coordinates are denoted by ψ_p,i and ϕ_p,i, the ageostrophic deviation in the atmospheric motion can be expressed by ϕ_p,ia = ϕ_p,i − f₀ψ_p,i, where ϕ_p,ia is referred to as an inner part of the ageostrophic geopotential on isobaric surface. In σ coordinates, the generalized ageostrophic geopotential at σ surface is approximately defined by

ϕ′_iaϕ′_if₀ψ_iϕRT₀p_if₀ψ_i(3.34)

From (3.22) and (3.32), the equation of the generalized ageostrophic geopotential is

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where

Φ′_had,iaΦ′ _had,if₀ψ_adv,i(3.36)

is referred to as the variation of the generalized ageostrophic geopotential caused by advection and diabatic heating.

The velocity potential equation (3.23) can be rewritten as

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The basic variables for describing the atmospheric motion now become the inner parts of the streamfunction, velocity potential, and generalized ageostrophic geopotential.

a. The inner part equations of the vertical modes

For this purpose, the eigenvectors of the matrix A are derived and expressed by E_j, j = 1, . . ., N. In (4.1), the matrix E represents the eigenvector matrix with each column representing an eigenvector E_j. The matrix E⁻¹ is the inverse of the matrix E. The matrix G is a diagonal matrix with the diagonal elements given by the N eigenvalues, (gh₁, gh₂, . . ., gh_N), of the matrix A. The values h₁, h₂, . . ., and h_N are the equivalent depth of the vertical modes.

Equations (3.22), (3.37), and (3.35) are multiplied from left by the matrix E⁻¹; then

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where

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The equations (4.4)–(4.6) can be rewritten as

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for each of the vertical modes separately, where C_k is the gravity wave phase speed for the kth vertical mode, and L_0k is the radius of deformation of the kth mode, and they are expressed by

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respectively. The equations for the vertical mode k have the same form as the shallow water equations with equivalent depth, h_k.

b. The solution of the gravity–inertia wave equation in the semi-implicit time-integration scheme

Now introducing a leapfrog scheme,

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to analogize the time derivative of a variable X, and using an implicit treatment of the linear gravity wave terms in (4.8)–(4.10), we obtain the following set of prognostic equations:

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where

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From (4.17), (4.13) becomes

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Thus, (4.14)–(4.16) can be rewritten as

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Eliminating ϕ^′_ia*k^t from (4.20) and (4.21), we obtain

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where the forcing term, Ft*k, is expressed by

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Equation (4.22) is the semi-implicit time integration scheme of the gravity–inertia wave equation, and it describes the averaged velocity potential between two time steps in vertical mode space.

Because the lateral boundary value of the inner part of the velocity potential is homogeneous, the solution of the Helmholtz equation (4.22) can be derived from the double sine series. Based on (3.16) and (3.17), the solution is written in the form

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where

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After the vertical mode of the velocity potential is obtained from (4.24), the streamfunction can be computed from (4.19) by

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After ψ_i*^t is derived, we have

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 From (4.17) and (4.18), the inner parts of the variables in the next time step are

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c. The semi-implicit scheme for surface pressure, temperature, and mixing ratio

After introducing a leapfrog finite difference for the time derivatives and an implicit treatment of the linear gravity wave terms, (2.24), (2.28), and (2.32) can be written as

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Based on (3.14), (4.29)–(4.31) can be separated into two parts. The equations for the inner parts are

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and the equations for the harmonic parts are

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where D_i^t in (4.32) and (4.33) can be calculated from (3.13) by

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and D^t_h is obtained by the prediction of the global model. The values for the step t + Δt can be deduced by

X^t+ΔtX^tX^t−Δt(4.39)

d. The computation of the initial time step

For the initial time steps, we perform three initial steps: a forward step of Δt/4 and a centered one of Δt/2, then a centered one of Δt. After these three initial time steps, the basic time scheme with 2Δt is started. This method is an effective way of reducing the initial shock, especially when using the large time steps permitted by use of the semi-implicit method.

The first forward time step is

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Thus,

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The second and third steps are

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Then the basic time step is started at t = Δt.

e. Horizontal diffusion

“Horizontal” smoothing of streamfunction, velocity potential, and specific humidity is represented by a simple linear fourth-order diffusion applied along the σ coordinate surface:

K_XiK⁴X_i(4.44)

where X_i = ψ_i, χ_i, or q_i. It is applied in spectral space to t + Δt values such that if (x_i) ^m_n is a spectral coefficient of X_i computed for time step t + Δt prior to diffusion, then the diffused value x^m_in is given by an implicit scheme as

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or

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where L_mn is determined by (3.21). A modified diffusion is used for temperature to avoid an unrealistic warming of mountain tops. A computationally convenient form that approximates diffusion on pressure surfaces is used. Equation (4.46) becomes

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where

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and (p∗∂p/∂p∗ ∂T/∂p)_ref denotes reference values, and it varies only with the model level based on the standard ICAO atmosphere.

a. The wind reconstruction and its lateral boundary condition

The basic property of the wind separation into the internal and external winds discussed by CK92a and Chen et al. (1996) will be utilized to solve the lateral boundary problem of a limited-area model. As pointed out in section 1, if a limited-area model is nested in a global model, only the internal wind in the limited area can be predicted from the vorticity and divergence equations of the limited-area model, while the external wind in the limited area must be derived from prediction of the global model.

Based on the equations in section 4, the inner parts, ψ^t+Δt_i and χ^t+Δt_i, are predicted by the limited-area model as shown in the right column of Fig. 3. The internal wind is computed by

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The external wind, U_E and V_E, on closed R can be obtained by the natural method (Chen et al. 1996) from the predicted wind (or vorticity and divergence) of the global model as shown in the middle column of Fig. 3. In this natural method, the internal wind on closed R is also derived based on the global model prediction, then the external wind is computed by (3.5). The external wind derived by the natural method satisfies the consistency condition (Chen et al. 1996) automatically.

To compute the advection terms at the time step t + Δt, it is necessary to know U^t+Δt, V^t+Δt, Ω^t+Δt, and D^t+Δt on closed R. The total wind on closed R at each time step is the sum of the internal and external winds expressed by

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Based on (3.9), the vorticity and divergence, Ω^t+Δt and D^t+Δt, on closed R are calculated from ψ^t+Δt_i and χ^t+Δt_i by

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where Ω|^t+Δt_Σ and D|^t+Δt_Σ can be obtained from the global model prediction. For avoiding discontinuity of (5.3) at the boundary, based on (3.13), (5.3) can be replaced by

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By this boundary method, the predicted total wind at the boundary of the limited region is not equal to the predicted wind of the global model. Because the vorticity and divergence within the region have been changed by the limited-area model prediction, the wind at the boundary should also change for consistency.

b. The lateral boundary condition for the variables ln p∗, T, and q

The harmonic parts of the surface pressure, temperature, and mixing ratio, lnp^t+Δt∗,_h, T^{t+Δ t}_h, and q^t+Δt_h, are different from the external wind because they can be predicted not only from the global model but also from the limited-area model by (4.35)–(4.37). The harmonic parts computed from the global model are denoted by (lnp∗,_h)^t+Δt_gm, (T_h)^t+Δt_gm, and (q_h)^t+Δt_gm, while those calculated from the limited-area model based on (4.35)–(4.37) are expressed by (lnp∗,_h)^t+Δt_lm, (T_h)^t+Δt_lm, and (q_h)^t+Δt_lm. In general, the harmonic parts derived from the limited-area model have higher resolution (or shorter horizontal scales) than those derived from the global model, especially when higher resolution topography is used in the limited region.

The harmonic part of a scalar variable used as the boundary value of a limited-area model is referred to as a boundary harmonic part. There are two methods to determine the boundary harmonic parts. One method is that the boundary harmonic parts are the same as the harmonic parts derived from the global model, and thus,

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This method is referred to as a simple method.

The other method is called a weighted mean method. In this method, the boundary harmonic parts are weighted mean values of two harmonic parts derived from both the global and limited-area models, and they are expressed by

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where α + β = 1. The simple method is a special case of the weighted mean method when α = 0 and β = 1 in (5.6).

The predicted variables, T, lnp∗, and q at t + Δt in the limited region, are the sum of the inner and boundary harmonic parts expressed by

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The method shown by (5.2), (5.3), and (5.7) is referred to as the external wind lateral boundary method, and its goal is to guarantee two properties of the predicted variables at the boundary for the limited-area model: one is consistency or continuity at the boundary, and the other is accuracy.

The inner parts are predicted by the limited-area model in the form of the double sine series. The boundary effects from the global model are represented by the harmonic parts added to the inner parts. The harmonic parts are all harmonic functions, and so are their linear combinations shown by (5.6). The components of the external wind are also harmonic functions. The harmonic functions are very smooth in the closed region up to the boundary, and thus, the sum shown by (5.2) and (5.7) cannot cause any discontinuity near the boundary. The accuracy of the prediction of the limited-area model depends upon the accuracy of the provided boundary values. The weighted mean method given by (5.6) is designed with a view to obtain more accurate boundary values for the limited-area model than by the simple method.

The main purpose of this paper is to show the property of the consistency or continuity for the external wind lateral boundary method, and thus, only the tests of the simple method are discussed in the next section.

c. The basic idea of a two-way coupling between limited-area and global models

If the feedback from the limited-area model to the global model is needed, the basic idea of a two-way coupling method is also shown in Fig. 3. In the two-way coupling method, the global surface is divided into two parts as shown in section 1: one is the limited area R and the other is the region Q outside the area R. The boundary line between the regions R and Q is denoted by Σ. During two-way coupling, the wind predicted by the limited-area model will be used to update the wind predicted by the global model in the limited area R including the boundary Σ. From the global perspective, the external wind of the region R is related to the vorticity and divergence in the region Q, and inversely, the external wind of the region Q is also related to the vorticity and divergence in the region R. After the vorticity, divergence and wind are updated in the region R, the external wind of the region Q should also be updated in order to obtain the continuous wind at the global surface. The external wind of the region Q can be derived from the updated wind at Σ. A similar method can also be used to treat the variables lnp∗, T, and q.

In this two-way coupling, changes are not only for the wind and the scalar variables lnp∗, T, and q in the region R, but also for the external wind and the harmonic parts of these scalar variables in the region Q. The predicted results will be very smooth over the global surface for this two-way coupling method. Only some basic ideas of this two-way coupling method are mentioned, and how the external wind and harmonic parts of the region Q are computed based on the boundary values at Σ will not be discussed in this paper.

d. Some discussion on computation and boundary value problems

The semi-implicit time integration scheme is applied to the inner part equations of the model, and these inner part equations are solved by the double sine series, such as (4.24). During the computation of the spatial derivatives, if the lateral boundary values are homogeneous, for example, the computation of (5.1) and ∇²ψ_i in (5.3), the double sine series are directly used; if the spatial derivatives of the harmonic function at the boundary need to be computed, such as (5.4), the solution of the harmonic function (CK92a) is used. As pointed out in section 1, the Tatsumi’s (1986) modified Fourier series are useful in computation of the spatial derivatives, this method is also used in computing the spatial derivatives of the horizontal advection terms in this paper. The nonlinear advection terms are computed by the transform method (Orszag 1970; Eliassen et al. 1970), in which products are performed by transforming to the values at grid points, performing the product, and transforming back to wave space.

In the spectral method, the wavenumber truncation for avoiding the aliasing error was discussed by Canuto et al. (1987, p. 84), which is sometimes referred to as the 3/2 rule. Chen (1993) found that using the 3/2 rule in the spectral model with the transform method is not necessary and the nonlinear computational instability induced by the aliasing terms is not present in the long-term integration. In our computations, it is also found that wavenumber truncation for avoiding the aliasing error is not necessary and the predicted results without the wavenumber truncation are better. The method without wavenumber truncation is used in this paper.

If a limited-area model is nested in a global model, there are two kinds of interaction at the boundary. One is the boundary consistency problem, which concerns whether the variables at the boundary predicted by two models are compatible (or continuous) with each other at the same time step t = t₀ + Δt. The boundary treatments studied by Benwell et al. (1971), Okamura (1975), and Davies (1976) are all aimed at solving this consistency problem, but the external wind lateral boundary method is very effective for this purpose.

The other kind of interaction at the boundary between two models is caused by horizontal transports of physical quantities, such as sensible heat and momentum fluxes. This problem may be referred to as a boundary forcing problem. This kind of interaction is a long-term effect, and it can be studied only by time integration of two models after the boundary consistency problem has been solved.

a. Rapid cyclonic development along the east coast of North America

The first example is a rapid development of a cyclone near the east coast of North America. The topography of the model domain is shown in Fig. 4. In the vertical, 16 σ-levels at σ = 0.015, 0.045, 0.075, 0.105, 0.140, 0.200, 0.285, 0.390, 0.510, 0.630, 0.735, 0.825, 0.900, 0.950, 0.980, and 0.995 are used. The grid spacing is 180 km, and the total number of grid points is 61 × 51. A time step Δt of 30 min is used for the semi-implicit scheme.

The observed sea level pressure (SLP) and the geopotential height, temperature, and wind fields at the 500-hPa level at the initial time, 1200 UTC 4 January 1985, are shown in Figs. 5a and 5b; they are based on the analyses from the European Center for Medium-Range Weather Forecasts (ECMWF) at 2.5° × 2.5° resolution [TOGA Archive II from the National Center for Atmospheric Research (NCAR)]. The SLP for 0000 and 1200 UTC 5 and 0000 UTC 6 January 1985 and the geopotential height, temperature, and wind fields at the 500-hPa level for 1200 UTC 5 January 1985 are presented in Figs. 5c–f, respectively. The observed results show that a cyclone rapidly develops in the 24-h period from 0000 UTC 5 (Fig. 5c) to 0000 UTC 6 (Fig. 5e) January 1985 along the east coast of North America. In this 24-h period, the central pressure of the cyclone fell by about 27 hPa, which is greater than 1 bergeron [1 bergeron = 24 hPa day⁻¹ (sinϕ/sin60°), where ϕ, the latitude of the cyclone center, is about 50°]. This example was studied by Kuo and Low-Nam (1990), who found no significant difference in predicted intensity and location of this cyclone by using the NCAR–Penn State MM4 model with and without latent heat release. Thus, dry baroclinic instability is an extremely important factor for this explosive cyclogenesis case. This example is chosen with a view to assess the performance not only for the adiabatic dynamical part of the H-F spectral limited-area model but also for the external wind lateral boundary method.

In general, dry baroclinic instability is an important necessary condition for explosive oceanic cyclogenesis. If the dry baroclinic instability is strong enough, the baroclinic development alone can cause explosive cyclogenesis. In this case, the strong baroclinic instability is not only a necessary condition, but also a sufficient one. If the dry baroclinic development is not strong enough, other diabatic heating processes (latent heat release, sensible heat, and moisture fluxes at the lower boundary, and convective condensation feedback) are also necessary for the explosive development. In this latter case, the baroclinic instability together with the diabatic heating processes becomes the sufficient condition. The example chosen here belongs to the former case.

In the tests, the limited-area model is not nested in a global model. The boundary values are provided by the analyzed data instead of those from the prediction of a global model. The analyzed data are available only every 12 h, but the boundary values are required at each time step. The boundary values at each time step are linearly interpolated from analyzed data at 12-h intervals.

The external wind lateral boundary method discussed in section 5 is used in the tests, and there is no other boundary treatment in the model. As an example, the components of the external wind, U_E and V_E, at σ = 0.510 level for the initial time 1200 UTC January 4 1985 are shown in Figs. 6a and 6b, respectively. The boundary harmonic parts of the surface pressure, lnp∗_h, and the temperature, T_h, at σ = 0.825 level are determined by the simple method (5.5) except that the harmonic parts are derived from the analyzed data, and their values for the initial time are shown in Figs. 6c and 6d, respectively. Figures 6 show that they are all harmonic functions and attain their largest and smallest values at the boundary.

The 12-, 24-, and 36-h predictions of SLP are shown in Figs. 7a–c, respectively. It is seen from Fig. 7c that the predicted location of the developing cyclone is the same as the observed one in Fig. 5e. The predicted positions of all systems in Fig. 7c are very good. This is due to the fact that there is no systematic phase error in the spectral method. The rapid development of the cyclone is predicted very well by the adiabatic dynamical part of the model. The central pressure change predicted in 24 h from 0000 UTC 5 to 0000 UTC 6 January 1985 is about 25 hPa. The central pressure of the cyclone in Fig. 7c differs by only a small amount from that of the observations. The 12-, 24-, and 36-h predictions of the geopotential, temperature, and wind fields at the 500-hPa level are better than those of the SLP, but only the 24-h prediction is shown in Fig. 7d for comparison with the observations in Fig. 5f.

It is seen from Figs. 5a, 5c, and 5d that a cyclone moves into the region through the western boundary. In Fig. 5e, this cyclone is already within the region, and is followed by another cyclone coming through the same boundary. The 12-, 24-, and 36-h predicted results presented in Figs. 7a–c show that the cyclone gradually and smoothly moves into the region across the western boundary and is smoothly followed by the second cyclone (Fig. 7c). Thus, the boundary method is capable of transmitting motion systems smoothly into and out of the limited region. All the fields near the boundary shown in Figs. 7a–d are very smooth, and there is no discontinuity at the boundary.

Comparing Figs. 5e and 7c, some errors occur near Greenland. These errors are caused by two factors. One is due to the values being interpolated at 12-h intervals at the boundary where high mountains are cut off. The value of lnp∗_h is very large at the boundary where high mountains are cut off, as shown in Fig. 6c over Greenland. If the boundary values are linearly interpolated from analyzed data at 12-h intervals, and the evolution of the true boundary values is produced by the nonlinear equations, some errors must be generated at the boundary. Thus, the errors of lnp∗_h must be relatively large at this boundary. Tests show that the boundary errors expressed by harmonic functions do not produce any discontinuity at the boundary even though the errors are very large. This is an important advantage of the external wind lateral boundary method. However, the accuracy of the prediction is affected by the errors at the boundary. If the 12-h interpolation is removed by using the limited-area model nested in a global model or in a coarser limited-area model over a larger domain, the errors in prediction can be greatly reduced, which will be shown in the next example.

The second factor may be due to the strong cooling at the ice surface in winter over the Greenland Ice Sheet because of radiation and land–ice–atmosphere interactions in the PBL, which is not taken into account in the prediction of the adiabatic dynamical part of the limited-area model.

b. Lee cyclogenesis over East Asia and other phenomena related to the Tibetan Plateau

Another example involves lee cyclogenesis over East Asia. Orographically forced cyclones, known as Alpine lee cyclones in Europe and Colorado and Alberta cyclones in North America, formed in the lee of the Rocky Mountains, have been studied by many authors (e.g., Palmen and Newton 1969; Buzzi and Speranza 1983). Although many major mountain barriers exist in East Asia, very few studies on lee cyclogenesis in this area have been done. Figure 8 shows the topography of the region defined for the limited-area model. Figures 4 and 8 are all computed from the U.S. Navy high resolution (10′ × 10′) dataset provided by NCAR. In the region shown by Fig. 8, the largest mountain is the huge Tibetan Plateau, whose maximum height is greater than 5000 m, and to the southwest of Lake Baikel, there are the Altai–Sayan Mountains, whose height is about2200 m. The height and horizontal scale of the Altai–Sayan Mountains are comparable to those of the Alps. The topography of this region is the most complex in the world. The chosen example is to check whether or not the adiabatic dynamical part of the H-F spectral model can be used in this most orographically complicated region and to assess whether the external wind boundary values can be applied to the boundary where high mountains are cut off, as shown on the western boundary in Fig. 8.

c. Lee cyclogenesis

The observed SLP and the geopotential, temperature, and wind fields at the 500-hPa level for the initial time 1200 UTC 14 April 1988, based on the data from ECMWF, are shown in Figs. 9a,b, respectively. At 1200 UTC 14 April (Fig. 9a), a mature cyclone moves eastward to the north of Lake Baikal, which may be referred to as the parent cyclone according to Palmen and Newton (1969). A front trailing along the cyclone is oriented from northeast to southwest with a cold high behind. At the 500-hPa level (Fig. 9b), a major trough is located over the Urals with a diffluence of the height contours over the Altai–Sayan region. The SLP for 0000 and 1200 UTC 15 and 16 April 1988 and the geopotential, temperature, and wind fields at the 500-hPa level for 1200 UTC 15 and 16 April 1988 are presented in Figs. 9c–h, respectively. In Fig. 9c, a pressure ridge is formed on the windward side of Altai–Sayan when the front moving eastward is retarded by mountains. At the same time, the sea level pressure on the lee side of the mountains starts to fall and a low pressure trough appears over the Mongolian Plateau ahead of the front. Twelve hours later, as shown in Fig. 9d, a lee cyclone was formed with the central pressure about 996 hPa. At 0000 and 1200 UTC 6 April, as shown in Figs. 9e and 9f, this cyclone has grown to its mature state. However, the central pressure does not decrease, but increases and then stays unchanged at 999 hPa. The lee cyclone separates from the parent cyclone as the latter moves away from the limited region.

This example was simulated by Chen and Lazic (1990) with two experiments using a finite-difference limited-area model, of which the model description can be found in Mesinger et al. (1988). One experiment is performed with a “step mountain” (eta) coordinate (Mesinger and Janjic 1986) and the other with standard sigma coordinates. The horizontal resolution used for these experiments was 0.5° × 0.5° with 16 levels in the vertical. It was found that the ETA experiment produced the cyclogenesis in a way similar to that in the analyses both at the surface and in the midtroposphere.

In our numerical simulation of this example, 16 σ levels in the vertical are the same as given in the last subsection. The grid length is 120 km, and the total number of grid points is 61 × 51. The time step Δt is 20 min for the semi-implicit scheme.

As pointed out in the above example, the accuracy of the prediction is affected by the errors at the boundary due to the linear interpolation from 12-h intervals, especially by the errors in lnp∗_h at the boundary where high mountains are cut off, as shown on the western side of Fig. 8. To avoid this error at the boundary, a coarser limited-area model over a larger domain is used to provide the boundary values to the fine-mesh limited-area model. The coarse-mesh model is the same spectral limited-area model with the same total grid points 61×51, but it is over a larger domain than Fig. 8 with the grid length of 240 km. The coarse-mesh model runs first for 48-h prediction with the lateral boundary condition derived from the linear interpolation of the analyzed values of 12-h intervals. Then the boundary values of the fine-mesh model are interpolated from the predicted values of the coarse-mesh model every 3 h. This boundary method is used in the simulation for the region shown in Fig. 8.

The predicted SLP for 12, 24, 36, and 48 h and predicted geopotential, temperature, and wind fields at 500-hPa level for 24 and 48 h from the initial time are shown in Figs. 10a–f, respectively. In Fig. 10a, a predicted pressure ridge is also formed on the windward side of Altai–Sayan and a low pressure trough appears over the Mongolian Plateau ahead of the front, which is the same as the observed situation in Fig. 9c. In the prediction for 12 h later shown in Fig. 10b, a lee cyclone is formed and its central pressure is 994 hPa, about 2 hPa lower than the observed. The predicted cyclone at 0000 and 1200 UTC 16 April shown in Figs. 10c,d develops to its mature stage. However, the predicted central pressure of the lee cyclone does not decrease but increases to 998 and 999 hPa, respectively. Its simulated maturation is the same as that analyzed in Figs. 9d and 9e. From the above, it is seen that the position, intensity, and maturation process of the predicted lee cyclone are very similar to those of the analyzed. Compared to the simulated results of Chen and Lazic (1990), a more accurate position, intensity, and maturation process of the surface lee cyclone are predicted by the H-F spectral model with less horizontal resolution and without the step-mountain coordinate.

d. Other phenomena

From 1200 UTC 15 (Fig. 9d) to 1200 UTC 16 (Fig. 9f) April 1988, the anticyclone behind the lee cyclone moves toward the southeast, and it causes cold air to affect the northeastern part of the Tibetan Plateau. It can also be seen from the wind and temperature fields at 500-hPa level that cold-air advection appears in advance of the upper trough (Fig. 9g) and ultimately develops into a small midtropospheric cutoff low (Fig. 9h).

In the predictions for 1200 UTC 15 (Fig. 10b) and 16 (Fig. 10d), the anticyclone to the west of the lee cyclone also moves toward southeast. The positions of the high pressure in Figs. 9f and 10d are similar, but the predicted central pressure is higher than the observed. The predicted wind and temperature fields at the 500-hPa level also show that cold-air advection occurs in advance of the upper trough (Fig. 10e) and ultimately develops into a small midtropospheric cutoff low (Fig. 10f). Chen and Lazic (1990) found that the experiment with the standard sigma coordinate failed to simulate the upper cutoff low. It is seen from Fig. 10f that the upper cutoff low can also be simulated the H-F spectral model with the standard sigma coordinate. The cold air affecting the northeastern part of the Tibetan Plateau is predicted but not as strongly as that observed.

e. Phenomena over the Tibetan Plateau

Weather analysis over the Tibetan Plateau is a difficult and unsolved problem, and so is numerical weather prediction near this region. The synoptic analysis at sea level and at the 850- and 700-hPa levels over the Tibetan Plateau are all extrapolated from the upper troposphere. Weather systems in these analyses, including the analyzed data from ECMWF, are very difficult to identify. Both analysis methods and numerical weather prediction over this region should be improved.

In the SLP analysis from ECMWF shown in Figs. 9a, 9c, 9d, and 9e, there is a high pressure area over the western region of the Tibetan Plateau. The first step for modeling over the Tibetan Plateau is to test the model predictions against the analyses from ECMWF including the SLP. It is found that the spectral limited-area model with the external wind boundary values can smoothly run over this most orographically complicated region. The predicted results shown in Figs. 10a, 10b, and 10c also have the high pressure area over the western region of the Plateau.

Routine weather analysis method in the lower troposphere cannot identify weather systems moving across the Tibetan Plateau, but such weather systems do exist and often produce precipitation over Sichuan Basin and other downstream regions after they move out of the plateau. Comparing Figs. 9g and 10e, a trough in the lower troposphere moving to the east of the Tibetan Plateau is predicted by the H-F spectral model. This trough in the 24-h prediction is correct, but it is too strong in the 48-h prediction (Figs. 9h and 10f). Recently, an isobaric geopotential height in σ coordinates (Chen and Bromwich 1996) has been proposed. It can improve the performance of the H-F spectral limited-area model over the Tibetan Plateau and other mountainous regions, and this will be discussed elsewhere.