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  • View in gallery

    (a) Horizontal and (b) vertical grid structure for the 3D splitting CSLR algorithm. Closed circles show the original Gaussian grid, and open squares and closed squares show the zonal and meridional interfaces, respectively. In the vertical direction, the interface is selected to be the half sigma level, where σ̇ is defined, and the cell-integrated interface is selected to be the full sigma level.

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    Idealized advection test 1: horizontal distribution (contour interval 2) of water vapor after circling the globe five times with the (a) CSLR and (b) SLF, and equatorial cross sections in (c) 6 h and (d) 6 months. The unit is 10−3 g g−1. Shaded areas denote the real solution in (a) and (b); the red line with closed circle shows the CSLR, and the blue line with open circle shows the SLF in (c) and (d). Partly enlarged curves are also plotted.

  • View in gallery

    Idealized advection test 2: horizontal water vapor distribution after one-round advection (solid: numerical approach; dashed: exact; unit: 10−3 g g−1): (a), (c), (e) the CSLR and (b), (d), (f) the SLF; α = 0.05 for (a) and (b), 0.5 for (c) and (d), and π/2 − 0.05 for (e) and (f). A scaled area of the center cosine bell is shown as the subdeck in the lower left of each panel.

  • View in gallery

    Error measurement (L2 and Linf) and the global integrated “water vapor” (sum) in test 2. (a), (c), (e) The CSLR and (b), (d), (f) the SLF; α = 0.05 for (a) and (b), 0.5 for (c) and (d), and π/2 − 0.05 for (e) and (f). Legend is plotted in (e).

  • View in gallery

    Seasonal (Mar–May) average of simulated precipitation rate with the (a) CSLR and (b) SLF schemes, in comparison with (c) the mean of CMAP analysis during 1979 and 2001. The panels are plotted in contours of 2, 4, 6, 8, 10, 12, and 16 mm day−1 and shaded colors. The differences in precipitation are plotted for (d) CSLR minus SLF, (e) CSLR minus CMAP, and (f) SLF minus CMAP.

  • View in gallery

    Dec–Feb average precipitation for the (a) CSLR and (b) SLF over a region of the southern Pacific Ocean, in comparison with (c) the same seasonal average of CMAP during Dec 1979 and Feb 2002. The rainfall differences for (d) CSLR minus SLF, (e) CSLR minus CMAP, and (f) SLF minus CMAP are shown in a contour interval of 1 mm day−1.

  • View in gallery

    Area-weighted zonal mean of (a) Jun–Aug and (b) Dec–Feb average precipitation (mm day−1). Simulations are shown as red (the CSLR) and blue (the SLF) in contrast to the CMAP analysis (black).

  • View in gallery

    Dec–Feb averages of area-weighted zonal average (a)–(c) relative humidity (%) and (d)–(f) temperature (K). (a) and (d) show the mean of ECMWF reanalysis during 1979 and 1993, (b) and (e) the simulation with the CSLR, and (c) and (f) the SLF. Differences for simulation minus the ECMWF reanalysis (shaded) are also plotted in the corresponding panels.

  • View in gallery

    Area-weighted zonal mean of Dec–Feb average cloud cover. Simulations with the CSLR (open circle) and SLF (closed circle) are shown in comparison with the ECMWF reanalysis (solid without mark), averaged from 1979 to 1993.

  • View in gallery

    Dec–Feb average, area-weighted zonal average (a), (c) longwave and (b), (d) shortwave radiative heating. The CSLR is shown in (a) and (b), and the SLF in (c) and (d). Difference for the CSLR minus the SLF is displayed in color shades.

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Conservative Semi-Lagrangian Transport on a Sphere and the Impact on Vapor Advection in an Atmospheric General Circulation Model

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  • 1 Earth Simulator Center, JAMSTEC, Yokohama, Japan
  • | 2 Department of Energy Sciences, Tokyo Institute of Technology, Yokohama, Japan
  • | 3 Earth Simulator Center, JAMSTEC, Yokohama, Japan
  • | 4 NEC Informatec Systems, Ltd., Kawasaki, Japan
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Abstract

A conservative semi-Lagrangian scheme with rational function for interpolation is implemented in spherical geometry and tested in an atmospheric general circulation model (AGCM). The new scheme, different from the conventional semi-Lagrangian method, is conservative and oscillation free. By introducing polar mixing and a time split computation of divergence, the scheme can compute advection transport correctly over the polar regions. Idealized advection tests with various velocity fields were carried out to demonstrate numerical accuracy and conservation in comparison with the spectral schemes. The impact of the advection computation on water vapor circulation in an AGCM is also investigated with numerical simulations on the Earth Simulator. Both pure advection tests and general circulation experiments show that the presented scheme is effective in improving the tracer transport property and the precipitation field in comparison with the leapfrog-spectral method.

Corresponding author address: Dr. Xindong Peng, Earth Simulator Center, 3173-25 Showamachi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan. Email: peng@es.jamstec.go.jp

Abstract

A conservative semi-Lagrangian scheme with rational function for interpolation is implemented in spherical geometry and tested in an atmospheric general circulation model (AGCM). The new scheme, different from the conventional semi-Lagrangian method, is conservative and oscillation free. By introducing polar mixing and a time split computation of divergence, the scheme can compute advection transport correctly over the polar regions. Idealized advection tests with various velocity fields were carried out to demonstrate numerical accuracy and conservation in comparison with the spectral schemes. The impact of the advection computation on water vapor circulation in an AGCM is also investigated with numerical simulations on the Earth Simulator. Both pure advection tests and general circulation experiments show that the presented scheme is effective in improving the tracer transport property and the precipitation field in comparison with the leapfrog-spectral method.

Corresponding author address: Dr. Xindong Peng, Earth Simulator Center, 3173-25 Showamachi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan. Email: peng@es.jamstec.go.jp

1. Introduction

The advection process plays an essential role in fluid dynamics. The properties of an advection scheme are the key for proper representation of fluid dynamical phenomena in numerical models. As a matter of fact, the so-called high-resolution schemes that were originally devised for aerodynamics have received increasing attention from researchers in meteorological and climatic modeling. Common features of the high-resolution schemes are that 1) they conserve the advected quantity, and 2) they are of second-order or higher accuracy in the smooth region and effectively suppress the numerical oscillations that are associated with discontinuities or large gradients. Many implementations of the high-resolution schemes can be found in aerodynamic codes and various numerical models for regional or global simulations. Some examples are the total variation diminishing (TVD) scheme (Harten 1983; Moorthi et al. 1995), the monotone upwind schemes for scalar conservation laws (MUSCL) (van Leer 1977; Allen et al. 1991; Russell and Lerner 1981), and the piecewise parabolic method (PPM) scheme (Colella and Woodward 1984; Rood 1987; Carpenter et al. 1990; Lin and Rood 1996). Hourdin and Armengaud (1999) compared some of these schemes in a general circulation model in terms of numerical accuracy and computational cost.

Since the publication of the pioneering work of Wiin-Nielsen (1959), the semi-Lagrangian scheme has received much attention for its excelling in computational stability. It has also been applied to atmospheric models, for instance, Ritchie (1985) and Rood (1987). Staniforth and Côté (1991) gives a detailed review of this method and the application to meteorological modeling. This class of advection method can be adapted to large Courant numbers with the concept of “point remapping.” It has been observed that the semi-Lagrangian schemes are very accurate in numerical dispersion. A highly accurate spatial approximation can be easily constructed if a high-order interpolation function is used. Williamson and Rasch (1989) provided a series of two-dimensional, shape-preserving, semi-Lagrangian schemes with cubic and rational Hermite interpolations. Modification of the derivative estimates ensured monotonic interpolant in their work. Generally, the conventional semi-Lagrangian schemes do not conserve the advected quantity.

In fact, the formulation of flux form used in the high-resolution conservation schemes is equivalent to a “volume remapping” if we consider that the remapped quantity is a volume-averaged value rather than the point value in the conventional semi-Lagrangian schemes. This observation leads to a sort of conservative semi-Lagrangian scheme (Allen et al. 1991; Lin and Rood 1996), which is actually based on a finite-volume formulation, but stable for a large Courant–Friedrichs–Lewy (CFL) number.

Based on the underlying idea of the cubic-interpolated pseudoparticle (CIP; Yabe and Aoki 1991) method, Yabe et al. (2001) suggested another type of conservative advection scheme by predicting the cell-integrated value through a volume remapping while the interface value is predicted by a point remapping. The resulting methods, which are called CIP–conservative semi-Lagrangian (CIP–CSL) schemes, require fewer computational stencils or grid points compared with conventional schemes of the same accuracy. The methods also retain numerical properties of conventional semi-Lagrangian schemes and of high-resolution conservative schemes, that is, accurate numerical dispersion and precise conservation, respectively. As an improvement to Yabe et al. (2001), Xiao et al. (2002) constructed a CIP–CSL-type scheme by using rational function (CIP–CSLR, referred to as CSLR hereafter), which has been proved to be convexity preserving and oscillation free. Numerical experiments were also reported in Xiao et al. (2002) to show better numerical dispersion compared to the PPM scheme. CSLR is a finite-volume advection scheme, but uses the surface (or interface) value as another model variable in addition to the cell-integrated average used in other conventional finite-volume methods. In the sense of using “multimoments,” the CSLR is similar to Russell and Lerner (1981) and Prather (1986) schemes, but using different state variables and a simpler algorithm with a semi-Lagrangian nature. Also, the additional variables are predicted according to the corresponding control equations in the CSLR algorithm.

Because of its importance in meteorological modeling, numerical advection remains an active field, and various advection schemes have been developed and tested in atmospheric models. Previous research (Peng et al. 2003) shows that the sophisticated advection scheme can significantly improve mesoscale numerical results. In the global high-resolution simulation, the nature of the advection scheme should become increasingly important to the numerical products, especially when a large gradient or discontinuity exists, because the fine-mesh model resolves a larger gradient than a coarse-grid model. Concerning the applications in global models, Ritchie (1987), Williamson and Rasch (1989), and Hundsdorfer and Spee (1995) have presented the implementation of semi-Lagrangian schemes in spherical geometry.

Our main goal of this research is to improve the tracer transport in the AGCM for the Earth Simulator (AFES) model by implementing the CSLR scheme. Before the practical implementation, the scheme should be extended to multidimensional problems in the Gaussian grid on spherical geometry. AFES is a global hydrostatic model in which a spectral representation is used for the spatial discretization, and a leapfrog-type integration is used for the time stepping (referred to as SLF hereafter). Besides the fact that the computational cost of the Legendre transformation is high, there is a well-known problem associated with the spectral representation called the Gibbs effect (Navarra et al. 1994). The spectral discretization with limited number of waves tends to create oscillations near large gradients or discontinuities. For positively defined physical variables such as water vapor, rainwater, aerosol density, or other microphysics quantities, negative values cause nonphysical solutions or lead to computational instability, in some cases. By implementing the CSLR scheme as the advection solver for vapor, we expect to improve the numerical conservation and shape preservation and computational efficiency of transport computation. To apply the CSLR in a global model on a spherical coordinate, we introduce polar mixing and a divergence correction to dimensional splitting in this paper to make the CSLR scheme well suited for applications over a sphere.

In section 2, the AFES model and the CSLR scheme are briefly described. How to adapt the CSLR scheme to spherical geometry and incorporate it into the AFES model is then discussed. Verifications with numerical experiments are presented in section 3 via idealized advection tests in the AFES; a further analysis of the impact of the conservative semi-Lagrangian scheme on global vapor circulation is given in section 4. Finally, the paper ends with a summary.

2. Implementation of the CSLR to the AGCM model AFES

a. A brief description of the AFES

The AFES is a hydrostatic primitive equation model, constructed on a sigma vertical coordinate of Lorenz grid and spherical geometry. It is developed on the Earth Simulator based on the AGCM model version 5.4.02 of the Center for Climate System Research, University of Tokyo–National Institute for Environment Studies (CCSR/NIES AGCM) of Japan (Numaguti et al. 1997). The model equations are generally solved with spectral discretization with triangular truncation for dynamical parts and on a Gaussian grid for physical processes. Therefore, forward and inverse spectral transformation is needed for each time step integration. Nonlinear advection is first computed on a Gaussian grid and then transformed to spectral space. In the original model, the leapfrog scheme is used for explicit temporal integration, except for gravitational wave-related terms, which are integrated implicitly. The Asselin (1972) time filter is adopted to deal with the computational mode. Zero sigma velocity is set to both the lower and upper boundaries.

The model physics includes a two-stream K-distribution scheme for radiative transfer (Nakajima and Tanaka 1986), simplified Arakawa–Schubert cumulus parameterization (Arakawa and Schubert 1974; Moorthi and Suarez 1992; Numaguti et al. 1997), large-scale condensation with prognostic cloud water (Le Treut and Li 1991), orographic gravity wave drag (McFarlane 1987), the Mellor–Yamada (1972) level 2 turbulence scheme with simple cloud effect, and a bulk scheme for surface fluxes (Louis 1979; Uno et al. 1995). Horizontal diffusion for variable q is represented through hyperviscosity type as −(−1)nK2nq, where n = 2 ∼ 8 (2 is selected in this paper), and K = 4 is the horizontal diffusion coefficient.

In the original spectral model, an artificial moisture compensation is conducted where negative vapor appears (AFES Team 2002, p. 34). The supplementing water vapor is extorted from its neighbors. If the amount of water vapor on the neighboring grid points is not enough to compensate for the negative value, a global adjustment is then conducted. This treatment will inevitably introduce false sinks or sources to the model atmosphere and affects the quality of model products.

b. Flux-form water vapor equation for the CSLR method

As presented in Xiao et al. (2002), the CSLR method is a conservative solver for the flux-form advection equation. Two kinds of quantities (or moments) of the advected variable are used in the interpolation reconstruction. We call them the cell-integrated average and the interface value. They are separately treated as the prognostic variables according to the dynamical equations. The resulting scheme requires fewer computational stencils and is simpler and more accurate in numerical dispersion than conventional conservative schemes. The rational functions used in the CSLR scheme effectively suppress the numerical oscillation even without the explicit limiters that prove to be necessary in high-resolution schemes.

Next, we discuss the implementation of the CSLR scheme on the dynamical equations for the tracers in a spherical geometry. The full 3D flux-form equation for tracers, including water vapor and liquid water in a spherical coordinate, is given by
i1520-0493-133-3-504-e1
where μ = sin ϕ; U = u cos ϕ, and V = υ cos ϕ; u and υ represent zonal and meridional wind components; and λ and ϕ represent the longitude and latitude, respectively. Here, D(q) represents the horizontal diffusion term, and P(q) the physical effects, such as radiation, cumulus convection, planetary boundary layer, and surface processes. The variable q may be water vapor, or liquid water or any other tracer, and δ denotes horizontal divergence, which is defined as
i1520-0493-133-3-504-e2
Equations (1) and (2) can also be rewritten as
i1520-0493-133-3-504-e3
i1520-0493-133-3-504-e4
where ũ = U/[a(1 − μ2)] and υ̃ = V/a. The 3D advection terms are solved by the CSLR method based on the finite-volume formulation; the divergence term (δ + ∂σ̇/∂σ) is computed by a finite-difference method.

c. Grid structure for the CSLR in the model

A dimensional-splitting procedure based on the 1D CSLR is adopted for the 3D advection. In the CSLR scheme, both the cell-integrated average and the cell-interface value are required for the interpolation reconstruction (Xiao et al. 2002) based on the Arakawa C grid (Arakawa and Lamb 1977) as in Fig. 1a. Variables in the AFES, however, are distributed on a Gaussian grid, that is, the Arakawa A grid (Arakawa and Lamb 1977) in the horizontal direction and the Lorenz (1960) grid in the vertical direction. Redistribution of the horizontal wind is required for using the CSLR. In this paper, average wind components on the interfaces are used to compute the tracer advection. It should be noted that the integrated variable is conservative while the interface one is not. The cell-integrated quantity is what we need as the dependent variable in the AFES model for the dynamic core and physics computation, and the interface variable serves only for the flux computation. The cell-integrated quantity is defined on the original Gaussian grid points. The interface point is located at the center of two adjacent Gaussian–Lorenz grids in each direction of λ, μ, and σ (Fig. 1). There are J + 1 and K + 1 interfaces in μ and σ directions, respectively, where the Gaussian grid numbers are defined as J and K. The top and bottom interfaces in a vertical direction are put to the highest (σ = 0) and lowest (σ = 1) half-sigma levels (Fig. 1b), where sigma velocity σ̇ = 0. To deal with transport across the poles in a global conservative advection, two grid points for integrated variables are added to both poles, respectively. The two polar points are treated just as on the Gaussian grid, but serve only for advection computation. The values at the polar points are predicted with the governing equation, not interpolated from the values at the surrounding grid points. Additional interfaces in the meridional direction are simply put at the midpoint of the first/last Gaussian grid point and Antarctic/Arctic pole, which are shown in Fig. 1 as the interface points between j = 1 and 90°S, j = J and 90°N.

d. Application of splitting transport with the CSLR

The dimensional-splitting method has been widely adopted in multidimensional transport computation (Strang 1968; Purnell 1976; Bott 1992; Hundsdorfer and Spee 1995) because of its simplicity. Conceptually, any 1D advection algorithm can be straightforwardly extended to 3D by using dimensional splitting, even if additional numerical correction is sometimes necessary (Strang 1968; Hundsdorfer and Spee 1995; Clappier 1998). Since the CSLR is different from the conventional conservative schemes and the interface value needs to be updated at each step, an efficient dimensional-splitting approach for the CSLR scheme can be constructed [refer to Xiao et al. (2002) for details].

For implementation of the CSLR to the tracer transports in the AFES model, the temporal integration of the tracers is modified into an Eulerian forward scheme from the original leapfrog one, while the other parts of the model remain as they are in the original AFES model.

For a variable velocity field, the solution from a simple splitting cannot be monotonic even if the velocity field is multidimensionally divergence free. This problem becomes even more serious in spherical geometry with polar singularity. Extra modifications (Hundsdorfer and Spee 1995; Lin and Rood 1996; Clappier 1998) are required to correct these splitting errors. The advection equation for the specific humidity q is extracted from (3) as
i1520-0493-133-3-504-e5
We can straightforwardly devise a corrected splitting by rewriting the above equation into the following multistep form:
i1520-0493-133-3-504-e6
i1520-0493-133-3-504-e7
i1520-0493-133-3-504-e8

The left-hand sides of (6)(8) are separately solved by the 1D CSLR scheme. The right-hand sides of the corresponding equations are the divergence terms in each direction, which also provide the corrections to the splitting errors. In the CSLR schemes, the interface value of the advected quantity is also treated as a dependent variable. The cell-integrated value q is defined at the Gaussian grid of the AFES model and is used as the prognostic quantity in the model. An interface grid, where the interface value f is defined, locates at the midpoint of two adjacent points of the original Gaussian grid.

Denoting the cell-integrated value as q and the interface value as f, the computational procedure based on the above multistep splitting can be summarized as follows.

  • Start with qtijk, fti+1/2jk, ftij+1/2k and ftijk+1/2 at time t,
  • λ direction (variables denoted with superscript * represent those after the advection in λ direction):
    1. Compute q*ijk, f*i+1/2jk from qtijk, fti+1/2jk with the 1D CSLR scheme for Eq. (6). According to (6), the interface value f*i+1/2jk is updated only for the semi-Lagrangian computation.
    2. Make divergence correction
      i1520-0493-133-3-504-eq1
    3. Update the interface values in the other two directions as
      i1520-0493-133-3-504-eq2
      i1520-0493-133-3-504-eq3
  • Based on the values of q*ijk, f*i+1/2jk, f*ij+1/2k and f*ijk+1/2, solve the variables denoted with superscript ** via the similar steps but in the μ direction.
  • With the provisional values after the computations in the μ direction q**ijk, f**i+1/2jk, f**ij+1/2k and f**ijk+1/2, carry out the 1D computation in the σ direction and end the all advection computations with the updated variables superscripted with ⋄.

The computational procedure presented above precisely conserves the cell-integrated value q if the wind field is three-dimensionally divergence free.

On a spherical coordinate, the zonal lateral boundary condition is naturally periodical. As shown in Fig. 1, if there are I (i = 1, 2, . . . , I) points for the integrated value in zonal direction, I + 1 interfaces (i = ½, 1½, · · · , I − ½, I + ½, where i = ½, and i = I + ½ indicate the same one) are adopted in the 1D CSLR computation.

Unlike the zonal grid, the grid in the meridional direction is bounded by the two poles. As illustrated in Fig. 1a, additional grid points for the cell-integrated values are defined at the poles, and the tendency at each polar grid point is computed via the flux through the two neighboring interface grid points that are symmetrically distributed with respect to the pole,
i1520-0493-133-3-504-e9
i1520-0493-133-3-504-e10
where qsp and qnp denote the cell-integrated quantities on Antarctic and Arctic poles; f υ̃ represents the meridional flux of f through the interface for a unit Δμ during Δt on the sphere with
i1520-0493-133-3-504-eq4
The divergence on the poles (δsp and δnp) is
i1520-0493-133-3-504-eq5
where Δμsp or Δμnp shows polar cell length. The initial values of the specific humidity at the poles, as well as at the interface grid points j = 1/2 and j = J + 1/2, are given with the zonal average on j = 1 or j = J. The inner interfaces are initially assigned the meridional average of the Gaussian neighbors. In practice, fia1/2k and fiaJ+1/2k are selected to be the boundaries of the ith column interface variable. Therefore, J + 2 cells and J + 3 interfaces are employed in μ-direction advection.

Concerning the vertical boundary in a sigma coordinate, the velocity boundary condition along the σ axis is imposed as σ̇ = 0 on the bottom and top levels of the model. This means than no vapor flux passes through the model top and bottom; thus the total mass of the transported quantity should be conserved by large-scale advection.

e. Updating the interface value due to the nonadvection phase

The interface value f serves as an auxiliary variable for computing the cell-integrated value q in the advection, but is treated as a quantity that needs to be advanced at each time step in the CSLR scheme. In the present study, the computation for a tracer is divided into the advection phase and the nonadvection phase (operator splitting). The 3D advection is solved by the procedure discussed in the above subsection. Suppose the provisional values of the cell-integrated quantity and the interface quantities are denoted as qijk and fi+1/2jk, fij+1/2k, fijk+1/2, respectively. The cell-integrated quantity is further updated with the physical parts, called the nonadvection phase, by the following simple Euler forward integration,
i1520-0493-133-3-504-e11
where the subscriptnadυ represents the nonadvective phase, including the horizontal diffusion and physical processes.
When q is updated, the new interface values can be computed with finite differences
i1520-0493-133-3-504-e12
i1520-0493-133-3-504-e13
i1520-0493-133-3-504-e14

The above treatment makes the implementation of a CSLR-type scheme very efficient, even if the interface values are computed as the extra variables.

f. Polar mixing

In spherical geometry, as shown in (1), the scaled horizontal wind vector becomes u/(a cos ϕ) and υ cos ϕ/a. The pole is a singular point because cos ϕ = 0. Regardless of the real (u, υ) velocities in the Cartesian system, the zonal and meridional velocities are scaled to infinity and zero at the pole. As a result, fluxes at the poles cannot be correctly computed. Considering the grid points concentrated within a narrow region at the poles, as in Hundsdorfer and Spee (1995), polar mixing is introduced. In this paper, the cell-integrated quantity q on the poles (j = 0 and j = J + 1) and grid next to the poles (j = 1 and j = J), and the interface quantity f for j = 1/2 and j = J + ½, are substituted with the corresponding averages,
i1520-0493-133-3-504-e15
i1520-0493-133-3-504-e16
Using these polar mixing and splitting correction techniques, the advection over the poles is greatly improved.

3. Idealized numerical tests in a spherical geometry

In the Cartesian system, the CSLR has been validated to be efficient for multidimensional transport (Xiao et al. 2002). Some advantages are shown in comparison with the PPM scheme (Colella and Woodward 1984; Lin and Rood 1996). To verify the CSLR scheme in a spherical system, idealized tests are carried out with the AFES code. Divergence-free wind is selected to drive the numerical transport. To better understand the performance of the advection scheme, all the physical processes and additional numerical treatment—that is, modification for global-mass conservation and correction on negative water vapor in the original model (Numaguti et al. 1997; AFES Team 2002)— are turned off. As a result, D(q) = P(q) = 0 in (3) is specified. The advection computing in the model can be switched between the CSLR and the SLF scheme easily. T106L20 with 320 × 160 transform grids is utilized in this section.

a. Experiment design

The first numerical test is a passive transport of a square wave by a steady and uniform wind, in order to show the capability of the CSLR and the SLF in capturing a large gradient or discontinuity. The velocity for advection transport is specified as u = 15 cos ϕ m s−1, where ϕ is the latitude, and υ = σ̇ = 0. Courant number 0.432 is adopted here. The initial water vapor is defined as
i1520-0493-133-3-504-e17
where r = (ii0)2 + (jj0)2 and i0 = 160.5, j0 = 80.5 denotes the point (180°, 0°); i and j are the grid indexes in longitude and latitude directions.
The second numerical test is an advection of a “cosine” bell to illustrate the schemes in dealing with passive advection with asymmetric divergence-free wind in the spherical coordinate. The wind field is designed just the same as test case one in Williamson et al. (1992). In this case,
i1520-0493-133-3-504-eq6
u0 = 2πa/(12 days) (approximately 40 m s−1) and a is the radius of the earth. The time step is selected as 1200 s in this case. The initial cosine bell is given by
i1520-0493-133-3-504-eq7
and
i1520-0493-133-3-504-eq8
where (λc,ϕc) = (3π/2,0) and q0 = 0.01. Advection was computed with rotational velocity fields of different orientations, defined respectively by α = 0.05, 0.5, and π/2−0.05. We note that the Courant number is 0.4 over the equator. The maximum Courant number, however, is much larger than unity in polar regions. It is 1.94 for α = π/2 − 0.05 in case of the CSLR.

The three different cases of α help us to validate the performance of the schemes in computing zonal transport, zonal propagation with large meridional variation, and the pole-across meridional advection. In a spherical geometry system, the “pole issue” becomes serious and is the key for the model to deal with high-latitude weather phenomena. Opposite the steep distribution, a smooth cosine bell-like initial condition is selected because it favors the spectral method. The asymmetric horizontal wind, which possesses strong divergence in each individual direction but is free of divergence in the two-dimensional field, is crucial for the dimensional-splitting algorithm; therefore, it is useful to verify the CSLR method.

In (18), the analytic divergence of the vector field is zero, and there should be no shape change during advection in these cases. In the numerical model, however, the divergence is not completely zero, and errors are observed in the high latitudes. The deformation is then shown as the key to evaluating the new scheme. As suggested in Williamson et al. (1992), some global measures of the numerical error are defined. The global integral is defined as
i1520-0493-133-3-504-e18
and two types of error measurement,
i1520-0493-133-3-504-e19
i1520-0493-133-3-504-e20
will be used to evaluate the numerical errors in this paper.

b. Results

In test 1, pure advection is carried out by using the AFES model of T106L20 for a 6-month integration with the initial condition (17). Polar mixing is not activated in this test because there is no meridional wind anywhere. Figure 2 shows the numerical results with the CSLR and SLF, respectively. During the transport, water vapor is found to be expanding rapidly across the whole globe with oscillation by the SLF, even though only zonal wind is being specified. This is because the spectral method always expresses a local quantity with global function. The CSLR, however, gives an accurate tracer distribution with the steep jumps—namely, the square wave—being well resolved. The status after transportation around the globe five times is plotted in Figs. 2a and 2b. In 6-h and 6-month advection tests (Figs. 2c,d), the CSLR is superior to the SLF with regard to shape-preserving and numerical oscillation. High-frequency oscillations in the SLF decrease with time as the numerical tracer distribution becomes diffusive. At hour 6, the maximum overshoot and undershoot in the SLF are displayed to be ±7.69 × 10−4, 7.69% of the exact peak value 0.01. In Fig. 2d, the overshoot of the SLF gets to 17.36% of the real peaks.

In the numerical model, negative water vapor becomes a problem in both computational and physical fields. The sophisticated advection scheme largely relieves the issue and provides a more accurate result for the advection of tracers where large gradients or discontinuities exist. The one-dimensional CSLR is proved to be a positive-definite scheme (Xiao et al. 2002). In practical computation, negligible negative value in an order of 10−20, the order of roundoff error of the computer, is observed for double-precision programming. We also compared the global integration of the negative water vapor between the schemes by the end of this test, which is −6.564 51 × 10−19 with the CSLR and −2.706 24 × 109 with the SLF.

For test 2, integration is executed for one revolution for each of the three α cases with the CSLR and SLF. Polar mixing is utilized with the CSLR scheme. Numerical results are listed in Fig. 3 in comparison with the exact solution. The cosine bell feature is the most appropriate problem for the SLF scheme, because it can be exactly represented with the triangular function. In the first case (Figs. 3a,b), α = 0.05 defines a zonal current with small meridional vibration. Both schemes show realistic results, though the SLF seems to be diffusive in the center area. Also, oscillation and dissipation appear to be stronger in Fig. 3b.

In the second case, α = 0.5 describes a large-amplitude wavelike propagation, with the same order of u and υ components. Figures 3c and 3d show the results after one revolution over the sphere. The CSLR provides a natural numerical solution. It shows the successful implementation of the time-splitting CSLR method in computing multidimensional advection in the spherical coordinate.

Figures 3e and 3f illustrate situations of case 3 (α = π/2 − 0.05). In this case, the meridional current goes across the poles directly. Without the polar mixing and convergence correction, that is, the divergence term in (5) being computed in each splitting step as in (6)(8), the “solid body” is broken at the poles (figure omitted). Even with the numerical treatment, the CSLR appears to be inferior (Fig. 3e) in this case, in comparison with the SLF (Fig. 3f).

In the present study, oscillation is observed over the polar region. In one revolution, the peak of the bell is 0.010 415 8, with an overshoot of 4.158% in this particular test (α = π/2 − 0.05). It is worth noting that oscillation is mainly the physical product at the pole where divergence exists in the discrete Gaussian grid. Even though the analytical divergence is null, the numerical value shows an order of 10−4, which is the same as the overshoot. The dimensional splitting produces also numerical deformation due to the polar singularity. The undershoot −2.713 13 × 10−16 is observed in the third case. It is much smaller than that (i.e., −1.475 72 × 10−4) produced by the SLF method due to the intrinsic Gibbs errors of the spectral expression. The global sum of the negative water vapor is calculated as well. A fraction of −3.553 736 × 10−13 is shown using the split two-dimensional CSLR, and −2.718 005 × 10−4 using the SLF, which demonstrates that negative water vapor is not as serious in the CSLR as in the SLF.

Although the extra numerical errors in the polar regions remain a problem requiring further investigation, the overall performance of the CSLR scheme is quite promising, especially because of its mass conservation and computational efficiency. The CSLR scheme can be expected to give good numerical results in spherical geometry for practical simulations.

As presented in Eqs. (18)–(20), the errors based on different norms are plotted in Fig. 4. Figures 4a, 4c and 4e show the measures for the CSLR, and Figs. 4b, 4d and 4f show the measures for the SLF. For cases 1 and 2 in this test, the CSLR illustrates much smaller L2 and Linf errors than the SLF. The L2 error of the SLF is approximately 3–4 times that of the CSLR; Linf error is 2 times larger. For the third case (α = π/2 − 0.05), errors in the CSLR clearly grow; L2 and Linf in the CSLR are about 2 times larger than those in the SLF in a half migration (hour 142). In this case, the errors of the CSLR increase quickly in the first half-pass and decrease in the second half. This is mainly caused by the systematic phase shift and strong divergence error in the polar regions as a result of time splitting. Longer integration is also performed for α = π/2 − 0.05; L2 and Linf in both schemes are at the same level after fourth revolutions (figure omitted).

The precise conservation of the transported quantity is also shown in Figs. 4a, 4c and 4e. For the SLF, however, it is not assured, as is shown in Figs. 4b, 4d and 4f, especially for case 3 in test 2. We may owe these conservation errors to two causes. The first is due to the singular point of the pole, where the water vapor flux cannot be calculated in an Arakawa A grid. The global water vapor integration therefore shows a large error in the polar region. The second cause is the spectral truncation error of the nonlinear advection term, because the numerical advection with a finite-difference method is transformed to spectral space every time step. In Figs. 4b and 4d, there is no visible variation in the global integral water vapor, but the variation is easily confirmed with the printout data. In Fig. 4f, a relatively large increase of global integrated water vapor is observed at both poles.

4. The impact on global vapor circulation

After verification with the idealized advection tests, general circulation simulations are also carried out with AFES T106L20. Though used only for water, the CSLR is expected to improve the distribution of global water vapor and thus have an impact on the model precipitation and other variables via complex interactions. In the following experiments, the same Δt is used for the semi-Lagrangian scheme as for the original spectral method. Because the CSLR is applied with an Euler forward temporal integration scheme, the Asselin (1972) time filter is turned off for water vapor and rainwater equations. Other equations remain the same as they were in the original model.

a. Model initialization and data

The model is first initialized with the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data (2.5° × 2.5° resolution) of an arbitrary date and integrated for 5 yr with spectral discretization and full physics so as to get a well-spun-up model field. The output is then taken as the initial condition of the numerical experiments to test the SLF and CSLR. The monthly mean SST data arranged in the Hadley Centre for Climate Prediction and Research (1° × 1° resolution), after linear interpolation, are used as the bottom forcing over the ocean. Ozone distribution is prescribed according to Liang and Wang (1996).

b. Experiments

The “relaxed” Arakawa–Schubert scheme (Moorthi and Suarez 1992; Numaguti et al. 1997), with a prescribed adjustment time scale of 7200 s, is selected in the model configuration to deal with the cumulus convection. Large-scale condensation and dry-convective adjustments are also active. With the advection schemes of the SLF and CSLR, 20-yr integration of the T106L20 is performed on the Earth Simulator. Ten nodes with 80 processors are employed. Though the CSLR can be used for large time steps, the same time step of 360 s is selected for both SLF and CSLR for fair comparisons. During the integration, however, the meridional Courant number is restricted to be less than unity for the SLF, but not for the CSLR. As mentioned in section 3, the Courant number is larger in the case of the CSLR because the size of polar cells is smaller. In all cases of the leapfrog-type temporal integration, the Asselin (1972) time filter, that is, ft = (1 − 2ν)ft + ν(ft−Δt + ftt), is used with a fraction of ν = 0.05.

c. Results

Besides the difference of water vapor distribution, the variation of model precipitation is a reasonable identifier for the impact of advection scheme on vapor transport. In this paper, the enhanced Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP) data (Xie and Arkin 1996) with NCEP–NCAR reanalysis (2.5° × 2.5° resolution) during January 1979 and February 2002 is taken as a reference. The seasonal average precipitation from the model output will be listed in comparison with the CMAP analysis to show the effect of advection computation. Though some other physical processes such as cumulus convection and longwave radiation greatly affect the result, the difference in precipitation shows the integral effect that is based on the different vapor distribution between the advection schemes. The cumulus convection and radiation themselves are dependent on the vapor supply in the model.

Two 20-yr experiments are carried out on the Earth Simulator with both the spectral method SLF and the semi-Lagrangian method CSLR. A 6-h average of prognostic and diagnostic variables—such as relative humidity, temperature, longwave and shortwave radiative heating, total cloud cover, and model precipitation—are output in an interval of 6 h. Based on these output data, the seasonal average of precipitation and zonal average of temperature, relative humidity, radiative heating, and cloud cover are analyzed.

Figure 5 shows the March–May average precipitation of the 20-yr model output, in comparison with the CMAP analysis. The CMAP analysis is averaged from 1979 to 2001. A well-distributed rainband is shown in the equatorial region with the CSLR in Fig. 5a, where it is known as the intertropical convergence zone (ITCZ). A successive rainband is simulated over the tropical Pacific (Fig. 5a). In the SLF case, however, a broken belt is illustrated clearly over 150°W in Fig. 5b. The strong precipitation in the western tropical Pacific is displayed with a 10 mm day−1 center by the CSLR, which agrees well with the analysis. In the Southern Hemisphere, the southern Pacific convergence zone (SPCZ) is improved by the semi-Lagrangian scheme. A proper peak value and the location are displayed at the Solomon Islands (10°S, 160°E) in Fig. 5a, while the original-scheme SLF shows the maximum at 10°S, 175°W, 25° departure from the analysis. Both amount and position are significantly improved in the simulation with the new scheme.

Differences in precipitation between the two schemes in Fig. 5d also illustrate the numerical effect of advection computation on ITCZ and SPCZ modeling, where water vapor is concentrated. We note that advection is not the only important process in the tropical region, where cumulus convection and solar and thermal radiation all play important roles. But the water vapor distribution significantly affects the activity of cumulus via conditional instability of the second kind (CISK), hence the radiation through clouds. Just as in Fig. 5, the sophisticated advection scheme results in obvious amelioration of tropical precipitation. The difference between the simulated precipitation and the CMAP analysis is also shown in Figs. 5e and 5f. The rainfall is underestimated in the ITCZ by the CSLR. The rainband is also a little northward in east Asia and the eastern Indian Ocean in comparison with the CMAP. The SLF shows even larger bias in these regions. Overestimation of rainfall in the SPCZ and underestimation in the ITCZ are observed in Fig. 5f.

These results indicate an improvement of vapor transportation within the strong convergence zone, as in Peng et al. (2003), where a sharp variation of horizontal winds exist. Taking the CMAP analysis as the truth, we calculated the global L2 error of the simulations with (19), which is shown to be of 0.40 in the case of CSLR, and 0.43 in the case of SLF.

Let us focus on the SPCZ, where major changes of the precipitation are observed. Figure 6 shows the average precipitation and corresponding errors over the southern Pacific Ocean during December and February. Within the similarly simulated rainband, rainfall peaks are shown to be quite different between the two schemes. Much more precipitation is illustrated in 10°–15°S, 160°–170°W by the SLF. In a comparison with the CMAP analysis, the L2 error of the simulation over this region is 0.20 for the CSLR and 0.32 for the SLF, respectively. The difference between the simulation and the CMAP (Figs. 6e,f) manifests the same conclusion.

Considering the ITCZ rainband variation between the hemispheres during the summer and winter, we choose to display the area-weighted zonal mean of the seasonal average rainfall for June–August and December–February (Fig. 7) in a comparison with the analysis data. Improvements in the Tropics and high-latitude region are observed even in this widely averaged illustration. The simulated precipitation is still somewhat insufficient in the Tropics and overestimated in some mid- and high-latitude regions. The L2 error of the simulations of seasonal average zonal mean rainfall is shown in Table 1. It is 0.25 for the CSLR and 0.30 for the SLF in the June–August case, 0.23 for the CSLR, and 0.25 for the SLF in the December–February case, which shows that the simulation with the new scheme is more consistent with the CMAP analysis.

In comparison with the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA), the simulated seasonal averages of area-weighted zonal means of relative humidity and temperature for December–February are listed in Fig. 8. In Figs. 8a, 8b and 8c, a large gradient of relative humidity is observed above 300 hPa in both simulations. The meridional distribution, however, is quite different in the tropopause. A sharp variation is displayed along the tropopause in the CSLR, and the different height of the tropopause layer is clearly displayed in Fig. 8b between the tropical and extratropical regions, which is similar to the ERA (Fig. 8a) with a moister equatorial area and a drier polar one. In Fig. 8c, the relative humidity is meridionally uniform in the upper layer above 200 hPa for the SLF and therefore shows a great positive difference with the ERA in the top layer between 300 and 100 hPa in high-latitude regions. Also, a negative difference is shown on 150 hPa of the equator. The relative humidity bias in Fig. 8c is about twice that in Fig. 8b in the top troposphere. We also noticed that both simulations show drier bias in lower and moister bias in upper levels.

The simulated temperature field (Figs. 8e or 8f) is quite similar to the reanalysis in Fig. 8d. Differences are also shown in the tropopause and Antarctic area between the schemes. Warmer bias in the troposphere of the Antarctic region in Fig. 8f is eliminated by the semi-Lagrangian transport. Negative temperature bias is shown on the tropopause in both cases, and it is much larger in Fig. 8f than in Fig. 8e on 200 hPa. It is worth noting that the temperature bias might be indirectly affected by the vapor transport computation. A direct semi-Lagrangian transport of temperature in Williamson et al. (1998) shows a colder bias in the tropopause in comparison with the Eulerian transport. Both the relative humidity and temperature show a positive impact of the CSLR on the simulation.

The cloud cover, which is closely related to the moisture distribution and convective activity, is also examined. Figure 9 shows the zonal mean of December–February average cloud cover in both simulations and the ECMWF reanalysis. No obvious improvement in the cloudiness is observed in the midlatitudes and Tropics by this diagnosis. In the high-latitude regions (60°S to the Antarctic pole and 60°N to the Arctic pole), almost full cloud cover (97%) is predicted by the SLF. On the other hand, proper spatial variation and amount are simulated with the CSLR. The difference in cloud cover in turn affects solar and thermal radiation and thus temperature. However, both schemes show more cloud cover than the ERA. As a result, radiative heating makes a warmer temperature bias in the global troposphere by both schemes, which can be found in Figs. 8e and 8f.

Longwave and shortwave radiative heating of the model output is also displayed (Fig. 10). Just as with the temperature, the December–February average radiative heating shows the main differences appear in the Tropics and the high-latitude region near the Antarctic pole. In the Tropics, the CSLR shows stronger longwave radiative cooling in the top troposphere and weaker cooling in the lower levels (under σ = 0.4) than in the SLF. In the high-latitude region near the Antarctic pole, obvious weaker longwave radiative cooling in the CSLR is shown between σ = 0.3 and 0.6. It manifests to be a result of more cloud generation by the SLF in the polar region. In the equatorial troposphere, from σ = 0.4 to 0.9, a weaker radiative cooling region of the CSLR is also displayed, which may be a result of more lower stratiform clouds by the SLF. On the other hand, the CSLR shows more clouds over the upper tropical troposphere. The shortwave radiative heating, as illustrated in Figs. 10b and 10d, is found at the same location. Stronger heating over the upper tropical troposphere and weaker heating in the Antarctic troposphere (under σ = 0.3) are simulated by the CSLR. This reveals that the transport of water vapor can affect the thermodynamical process through convection and radiation of clouds. The CSLR scheme improves not only the tropical precipitation but the simulations concerning thermodynamical processes in the high-latitude region.

5. Summary

The CIP-based conservative semi-Lagrangian scheme CSLR (Xiao et al. 2002) is implemented in a spherical coordinate and used in the AFES model for tracer transports. A dimensional-splitting algorithm with divergence correction is utilized in 3D advection computation. A flux-form formulation is used in the meridional direction across the two poles to ensure the exact conservation of the advected quantity. With the polar-mixing technique, the presented scheme can get around the numerical difficulty over the polar regions and produce adequate numerical results for practical simulations.

Numerical properties such as conservation, oscillation suppression, accuracy in dispersion, and shape preserving of the original CSLR scheme are reconfirmed in the spherical geometry with various numerical experiments. The idealized experiments reported in this paper show that the CLSR proves to be a superior alternative to the original SLF in the AFES model.

In the realistic simulations of T106L20, the CSLR and the SLF are separately used as advection schemes for vapor transport. The seasonal precipitations simulated with both advection schemes are compared with the CMAP analysis. Improvements of the rainband are observed over the intertropics and the SPCZ with the CSLR scheme, though no obvious amelioration is found in midlatitude regions. The area-weighted zonal average precipitation shows a better agreement between the CSLR simulation and the CMAP through the four seasons. The particularly significant correlations are found between the CSLR simulation and the ECMWF reanalysis on the zonal average relative humidity and temperature. Much amelioration is found in the upper troposphere and high-latitude region. The spurious radiative heating and warmer bias near the Antarctic pole during December and February in the output of the SLF are not found in the result of the CSLR simulation.

The real-case simulations presented in this paper show again that computation of the vapor transport essentially affects the numerical results, whose impact on the AGCM output is worth more attention and should be investigated further.

Acknowledgments

We thank Profs. T. Yabe and T. Sato for their valuable comments and encouragement of this research. We also thank the referees for their constructive comments. We acknowledge Messrs. S. Shingu and M. Yamada for their help on the AFES run and Ms. Mary Golden for grammatical editing of the manuscript. The AFES model is developed on the Earth Simulator based on the CCSR/NIES AGCM. The GrADS software is used for plotting.

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Fig. 1.
Fig. 1.

(a) Horizontal and (b) vertical grid structure for the 3D splitting CSLR algorithm. Closed circles show the original Gaussian grid, and open squares and closed squares show the zonal and meridional interfaces, respectively. In the vertical direction, the interface is selected to be the half sigma level, where σ̇ is defined, and the cell-integrated interface is selected to be the full sigma level.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 2.
Fig. 2.

Idealized advection test 1: horizontal distribution (contour interval 2) of water vapor after circling the globe five times with the (a) CSLR and (b) SLF, and equatorial cross sections in (c) 6 h and (d) 6 months. The unit is 10−3 g g−1. Shaded areas denote the real solution in (a) and (b); the red line with closed circle shows the CSLR, and the blue line with open circle shows the SLF in (c) and (d). Partly enlarged curves are also plotted.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 3.
Fig. 3.

Idealized advection test 2: horizontal water vapor distribution after one-round advection (solid: numerical approach; dashed: exact; unit: 10−3 g g−1): (a), (c), (e) the CSLR and (b), (d), (f) the SLF; α = 0.05 for (a) and (b), 0.5 for (c) and (d), and π/2 − 0.05 for (e) and (f). A scaled area of the center cosine bell is shown as the subdeck in the lower left of each panel.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 4.
Fig. 4.

Error measurement (L2 and Linf) and the global integrated “water vapor” (sum) in test 2. (a), (c), (e) The CSLR and (b), (d), (f) the SLF; α = 0.05 for (a) and (b), 0.5 for (c) and (d), and π/2 − 0.05 for (e) and (f). Legend is plotted in (e).

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 5.
Fig. 5.

Seasonal (Mar–May) average of simulated precipitation rate with the (a) CSLR and (b) SLF schemes, in comparison with (c) the mean of CMAP analysis during 1979 and 2001. The panels are plotted in contours of 2, 4, 6, 8, 10, 12, and 16 mm day−1 and shaded colors. The differences in precipitation are plotted for (d) CSLR minus SLF, (e) CSLR minus CMAP, and (f) SLF minus CMAP.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 6.
Fig. 6.

Dec–Feb average precipitation for the (a) CSLR and (b) SLF over a region of the southern Pacific Ocean, in comparison with (c) the same seasonal average of CMAP during Dec 1979 and Feb 2002. The rainfall differences for (d) CSLR minus SLF, (e) CSLR minus CMAP, and (f) SLF minus CMAP are shown in a contour interval of 1 mm day−1.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 7.
Fig. 7.

Area-weighted zonal mean of (a) Jun–Aug and (b) Dec–Feb average precipitation (mm day−1). Simulations are shown as red (the CSLR) and blue (the SLF) in contrast to the CMAP analysis (black).

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 8.
Fig. 8.

Dec–Feb averages of area-weighted zonal average (a)–(c) relative humidity (%) and (d)–(f) temperature (K). (a) and (d) show the mean of ECMWF reanalysis during 1979 and 1993, (b) and (e) the simulation with the CSLR, and (c) and (f) the SLF. Differences for simulation minus the ECMWF reanalysis (shaded) are also plotted in the corresponding panels.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 9.
Fig. 9.

Area-weighted zonal mean of Dec–Feb average cloud cover. Simulations with the CSLR (open circle) and SLF (closed circle) are shown in comparison with the ECMWF reanalysis (solid without mark), averaged from 1979 to 1993.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Fig. 10.
Fig. 10.

Dec–Feb average, area-weighted zonal average (a), (c) longwave and (b), (d) shortwave radiative heating. The CSLR is shown in (a) and (b), and the SLF in (c) and (d). Difference for the CSLR minus the SLF is displayed in color shades.

Citation: Monthly Weather Review 133, 3; 10.1175/MWR-2869.1

Table 1.

The L2 error of the simulated seasonal average, area-weighted zonal mean of precipitation for the CSLR and SLF schemes.

Table 1.
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