A global shallow-water numerical model based ... - Wiley Online Library

Q. J . R. Meteorol. SOC. (1995), 121, pp. 1981-2005

A global shallow-water numerical model based on the semi-Lagrangian advection of potential vorticity* By J. R. BATES'+, Y. Li', A. BRANDT', S. F. McCORMICK3 and J. RUGE3 NASAICoddard Laboratory for Atmospheres, USA 2~eizmann Institute of Science, Israel University of Colorado, USA

'

(Received 10 January 1994; revised 20 March 1995)

SUMMARY A global shallow-water primitive equation model based on the semi-Lagrangian advection of potential vorticity

is presented. A modification of the basic advection scheme needed to stabilize Rossby waves is introduced. The divergence and continuity equations are the remaining governing equations. A two-time-level semi-Lagrangian semi-implicit numerical scheme that avoids forward extrapolation of non-linear terms is used. This leads to a set of non-linear implicit equations to be solved at each time-step. The wind field is expressed in terms of a streamfunction and velocity potential, and a spatial discretization based on second-order finite differences on an unstaggered grid is used. The implicit equations are solved simultaneously using a non-linear multigrid method. The model is integrated for periods of up to 50 days at various resolutions, using a variety of initial conditions including real data. Comparisons with an existing semi-Lagrangian finite-difference shallow-water model and an Eulerian spectral shallow-water model are carried out. The new model is found to integrate stably and efficiently, and to require no noise suppressors other than the inherent diffusivity associated with the interpolations. The model gives results that are, in general, very close to those of the comparison models, but a case of highly non-linear flow (where the true solution is unknown) is presented in which it gives results that stand notably apart.

KEYWORDS:Numerical methods Potential vorticity Semi-Lagrangian models

1. INTRODUCTION

Potential vorticity (PV) has been regarded as an important dynamical quantity by meteorologists since the discovery by Rossby (1936, 1940), and more generally by Ertel (1942), that it is a conserved quantity following a particle in frictionless adiabatic flow. In the early attempts to formulate atmospheric prediction models (Charney 1948), PV was at the centre of attention, and subsequently, throughout the era of quasi-geostrophic numerical weather prediction (NWP), the governing equations were of a form that could be recast as conservation laws for the quasi-geostrophic analogue of PV. At a later stage in the development of NWP, Charney (1962) formulated a balanced model, based on less restrictive assumptions than those of quasi-geostrophic theory, in which the evolution equation for PV was one of the governing equations. This approach never became operational, however, and for a period, as quasi-geostrophic models gave way to those based on the primitive equationss, PV occupied a less central place in the minds of modellers and of meteorologists generally. A n exception to the general trend was the work of Bleck (1973), in which a model based on the conservation of PV in isentropic surfaces, with the wind set equal to its geostrophic value, was formulated. A renewed interest in PV arose in the 1980s.In the modelling area, Arakawa and Lamb (1981) and Takano and Wurtele (1982) presented finite-difference numerical schemes for * This paper is dedicated to the memory of Andri Robert, who died on 19 November 1993. + Corresponding author: Geophysics Department, University of Copenhagen, Haraldsgade 6, DK-2200, Copenhagen, Denmark. There is an unfortunate lack of agreement in the literature, and in the glossaries, regarding the definition of the term 'primitive equations'. Some authors take it as meaning the equations expressed in a form where the velocity components are the primary prognostic variables. We use it in the sense, more generally accepted among meteorologists, of referring to thc equations of motion in which hydrostatic balance, hut no other form of balance, is assumed. In this definition, the prognostic variables can be chosen at will.

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the primitive equations that, while taking the velocity components as prognostic variables, gave an improved treatment of PV advection for non-divergent flow and at the same time gave global conservation of the related quantity, potential enstrophy. A resurgence of interest in PV in the meteorological field at large was initiated by the landmark paper of Hoskins et al. (1983, which has re-established the central importance of PV as a dynamical variable. This paper has given rise to a large number of observational and modelling studies that focus on PV diagnostics (see Browning (1993) and Norton (1994) and references therein). It has also been followed by a number of modelling studies that use the PV equation as one of the governing equations in conjunction with various forms of balance condition (Lynch 1989; Egger 1990; Lynch and McDonald 1990; Raymond 1992; McIntyre and Norton personal communication). Davis et al. (1993) integrated the primitive equations with the velocity components as prognostic variables while simultaneously integrating the PV equation in a passive manner for diagnostic purposes. The increasing awareness of the importance of PV leads us to consider its use as a prognostic variable in a full primitive-equation model. In our model formulation, we use the divergence equation as a companion to the PV equation and thus refer to our governing set as the PV-D equations. It is reasonable to expect, particularly in situations of highly non-linear flow, that an integration scheme based directly on the PV equation will give a more faithful simulation of PV evolution than one based on the velocity components or on vorticity and divergence as the primary prognostic variables. Temperton and Staniforth (1987) have formulated a limited-area shallow-water primitive-equation model that, though motivated differently, is similar to our model in anumber of respects. It uses the PV equation as one of the governing equations to avoid damping of the model’s slow modes. Insofar as the linear aspects of wave propagation are concerned, it has been shown that spatial discretizations based on the vorticity-divergence form of the equations of motion on an unstaggered grid are generally better than those based on the velocity component form on staggered or unstaggered grids, both for geostrophic adjustment (Williams 1981; Randall 1994) and for Rossby-wave propagation (Neta and Williams 1989). For the particular situation studied by these authors-shallow-water equations linearized about a state of rest with continuous time variation-the PV-D equations are equivalent to the equations in vorticity-divergence form. Since we also choose an unstaggered grid (see subsection 2(d) for further discussion), our equations partake of the advantages pointed out above. There are a number of additional factors that make it opportune to consider the development of a PV-based model. The semi-Lagrangian method of integration, which has been gaining increasing use amongst modellers since the work of Robert (1981,1982) and Bates and McDonald (1982)* offers high accuracy for advection and is ideally suited to the numerical treatment of quantities that obey a conservation law following a particle; in addition, it aliows one to express the primitive divergence-equation in a much simplified form. Multigrid methods, which provide an eficient means of solving the non-linear implicit equations that arise in a PV-D model, are reaching a high state of development (Brandt 1977,1988; McCormick 1992). The potential of isentropic coordinates, which are apposite for an extension of a PV-D model to three dimensions, has been demonstrated for the primitive equations by a number of modelling groups (Hsu and Arakawa 1990; Zhu et al. 1992; Johnson et al. 1993; Bleck and Benjamin 1993). A feasibility study to investigate the applicability of multigrid methods to solving the equations that arise in a PV-D shallow-water model on a #?-planehas already been carried out (Bates et al. 1993b). The results encourage us to proceed to the case of the PV-D shallow-water equations on the sphere, which is the subject of the present paper. * For reviews see Staniforth and Cote (1991) and Bates et al. (1993a).

MODEL BASED ON PV ADVECTION

2. THEPV-D

1983

EQUATIONS

( a ) Basic formulation Our governing equations are the potential-vorticity, divergence and continuity equations for a global shallow-water primitive-equation model with a free surface and a horizontal lower boundary, expressed in spherical coordinates (A, 4). The equations are discretized in a two-time-level semi-Lagrangian semi-implicit (SLSI) manner. We formulate them in as simple a form as is possible while avoiding Rossby-wave instability and forward extrapolation of the non-linear terms. The most convenient and accurate form of the PV equation is obtained by directly discretizing the analytical PV equation. To follow the same route for the divergence equation would lead to a complicated set of implicit non-linear terms; we therefore follow an alternative route for this equation, as explained below. The winds used in the trajectory calculations are, with an exception to be described below, still obtained by forward extrapolation (Vn+1/2= 1.5V" - OSV"-'). In the forerunners to the present paper (McDonald and Bates 1989; Bates et al. 1990; Bates et al. 1993a; hereafter referred to as MB, BSHB and BMH respectively) forward extrapolation of the non-linear terms was a source of gravity-wave noise, which had to be controlled by divergence damping or by uncentring of the numerical schemes (see Gravel et al. (1993) for an analysis). Uncentring has been shown by Semazzi and Dekker (1994) to be a source of inaccuracy in the Rossby-wave solutions, imposing an undesirable restriction on the choice of time-step. In the present model, neither divergence damping nor uncentring is found to be necessary. The shallow-water PV equation (Rossby 1936) is

1. For all other values of a we have a physical and a computational mode: 1

A = -[(I - iawRAt) f {(I

2

+ iauWRAt)’ - 4 i ~ A t } ’ / ~ ] .

(25)

In Fig. 1, 1 A1 corresponding to the physical mode in (25) is plotted as a function of Iw~Atl for various values of a. For the usual forward extrapolation of the trajectory wind, i.e. a = 1.5, we see that the Rossby wave is unstable for all IwRAtl (just as for ct = 1.0). The instability increases with lwRAt1, i.e. maximum instability occurs for the longest waves near the equator. (As an example, for zonal wave-number one at the equator with e = 0 and At = 1h, we have IwRAtl = 0.52 and 1A1 = 1.033). The existence of this instability was verified by replacing (10) by (2) in the full PV-D model; numerical integrations with a stable Rossby-Haurwitz-wave initial state (zonal wave-number one) then showed large energy growth and distortion of the wave pattern in integration periods of a few days.


1987

Figure 1 shows that for a = 2.0 the Rossby wave is damped in part of the wave spectrum. This suggests that perhaps one could obtain a stable solution (at the expense of some additional truncation error) by setting (r = 1.5 E R , where E R is a small parameter. We performed numerical experiments to test this possibility for various values of E R . The results showed that this option always gave energy growth for At in excess of 15 min, and led to wave distortions for even smaller values of At. In view of these results, the unmodified PV-equation (2) was abandoned. The Rossby-wave solutions given by the modified PV-equation (10) are examined next. Under the present assumptions, (10) linearizes to

+

Assuming a solution of the form (22) we have A=

1 - itWR 1 itWR'

+

i.e. 1 A 1 = 1, indicating that the Rossby waves are now neutrally stable for all wavelengths. The result of this simple analysis is borne out in practice: numerical integrations with the PV-D equations in the form (lo), (11) and (12) lead to very satisfactory results, as will be described in section 3. We point out that the stabilization of the Rossby waves in the modified PV-equation has been accomplished by what is equivalent to an SLSI treatment of the linear part of the Coriolis term. Insofar as the Rossby waves are concerned, it is the modified rather than the original formulation of the PV equation that is analogous to the SLSI formulation of the shallow-water equations in (u, v ) form used in BSHB (see appendix). ( d ) Spatial discretization and method of solution Having opted for finite-difference discretization, the first question to be decided upon is the choice of grid. In MB, BSHB and BMH, where the equations of motion were integrated with the velocity components as the primary prognostic variables, an Arakawa C-grid was chosen. In that context, the C-grid had considerable advantages, particularly in avoiding the two-grid-interval noise characteristic of an unstaggered grid and in being well adapted to the polar numerics. A major disadvantage of the C-grid in the semi-lagrangian context, however, is the expense of calculating a triple family of trajectories (separate trajectories at the u, v and points) and of performing the associated triple family of interpolations at the upstream points. For this reason, among others, Purser and Leslie (1988) have advocated the use of an unstaggered grid in semi-Lagrangian models. In the PV-D formulation, an unstaggered grid, with grid points at the poles, becomes the natural choice. Because of the way the variables (9, x , @) interact with each other, this choice involves no red-black separation of solutions with its attendant grid-scale noise, and, because (9,x , a) are true scalars, the polar singularity is less of an issue. At the same time, one takes advantage of having a single family of trajectories and of the beneficial wave propagation characteristics referred to in the introduction. In spatially discretizing (lo), (11) and (12), all differential operators are approximated by their most straightforward centred-difference analogues on the unstaggered grid. At the poles, the divergence, curl and Laplacian operators are approximated using their integral definitions. The trajectory calculations are performed as in MB (apart from the fact that we now have a single family), with the rotated spherical-coordinate system used poleward of 60"

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+

latitude. The trajectory calculations use (n 1/2) fonvard-extrapolated winds, with one iteration to achieve space centring. Because forward extrapolation of the wind components is not possible at the initial time, the uncentred level n values are used at that time. Bilinear interpolation is used in the trajectory calculations and bicubic Lagrange-interpolation is used to obtain values at the departure points of the trajectories. When the forcing functions rl, r2,r3 have been calculated, the PV-D equations (lo), (11) and (12) are solved iteratively as a simultaneous set using a non-linear multigrid method. The multigrid principle is based on the use of a relaxation cycle applied to the basic grid (whose grid points are here the trajectory arrival points) and a succession of coarser grids. The highest wave-number components of the solution are approximated on the fine grid and the lower wave-number components on the coarse grids. Thereby, one achieves a much greater efficiency of solution than is possible by successive over relaxation on the fine grid alone. Optimizing the solver places some restrictions on the choice of grid: the number of points on a parallel of latitude is two raised to an integral power, while on a half-meridian (including the poles) the number of points is one plus two raised to another integral power. Due to the anisotropy of the grid intervals on the sphere, a computationally demanding line-relaxation is needed poleward of f85"latitude. Elsewhere, point relaxation is used. It was found that a full multigrid cycle followed by one V-cycle was sufficient to reduce the residuals to at least an order of magnitude smaller than the truncation errors. Full details of the multigrid solver have been given by Ruge, J., Li, Y., McCormick, S. F., Brandt, A. and Bates, J. R. (personal communication). After the PV-D equations have been inverted to give (Q, x , using the above method, the wind at non-polar gridpoints is retrieved from (9) using simple centred differences. The wind at the poles is then calculated using the first Fourier component of the un+I at the gridpoints closest to the poles, in the manner of appendix B of MB.

3. NUMERICAL RESULTS In order to examine the performance of the PV-D model, we integrated it with a number of different initial conditions as described below. Comparison integrations were carried out using two other models. The first (hereafter referred to as the (u,u ) model) was a shallow-water version of the BMH model. In this, an uncentring parameter E was used to control noise. The continuity equation was discretized as

Here, N is the nonlinear term (- @'D),and fi"+' denotes the fonvard-extrapolated value (2N" The above discretization is similar to that used in the momentum and thermodynamic equations of BMH except that here the uncentring parameter E occurs only in the linear term. Omitting the uncentring in the non-linear term was found to give a reduction of noise and to allow the use of a smaller value of E in the linear term. The second model used for comparison (hereafter referred to as the spectral model) was an Eulerian spectral shallow-water model provided by the United States' National Center for Atmospheric Research (NCAR) (Hack and Jacob 1992). This model contained a V4 diffusion to control noise. Unless otherwise stated, we use a 512 x 257 uniform latitude-longitude grid ( A h = A@= 0.703') for the finite-difference models, and a corresponding Gaussian grid (spectral resolution T170) for the spectral model. At T170 resolution, a coefficient of 2x m4 s-' in the V4 diffusion has been found to give realistic energy-spectra in the


1989

spectral model (Jakob et al. 1993). We refer to this value of the coefficient as the recommended value. The initial states were in all cases initialized using a digital filter (Lynch and Huang 1992). The respective models were integrated forwards and backwards for six hours from the uninitialized state, and the filter, with a cut-off period of six hours, was applied to the fields thus generated. (a) Hough mode

The first initial state was the gravest symmetrical rotational Hough-mode of zonal wave-number one with a mean height of 10 km. From linear theory (Kasahara 1976) it is known that this mode has a period of five days. For a sufficiently small perturbationamplitude the analytical solution provides an exact reference solution for the numerical models. It is particularly useful for testing phase-propagation accuracy on the sphere. We chose a perturbation amplitude of 200 m for our numerical experiments. (Decreasing this by a factor of 10 did not affect the nature of the results.) The initial geopotential field is shown in Fig. 2.

Longitude Figure 2. Initial global field of geopotential height for the five-day Hough mode, in metres. Contour interval 30 m.

Figures 3 and 4 show the five-day forecasts given by the PV-D and ( u , v) models respectively, with time-steps of 30 minutes, one hour and two hours. The uncentring parameter E for the ( u , v) model was set to 0.02. It is seen that the wave is well simulated by the PV-D model, with its phase speed showing little sensitivity to time-step. (What difference there is indicates a slightly more accurate representation of the phase speed for the larger values of At). The forecasts given by the ( u , v) model show a result very similar to that of the PV-D model for At = 30 min, but exhibit a greater sensitivity to time-step, with the phase speed becoming less accurate as At increases (with At = 1hour there is a 20", i.e. 5.6%, phase error after five days). It was found that decreasing the spatial resolution by factors of two and four led to little change in the results presented in Figs. 3 and 4. The spectral model, with a resolution of T42 and At = 20 min, was found to give a virtually exact representation of the phase speed. This is to be expected, since the Hough mode is composed of spherical harmonics whose linear phase-speeds are free of spatial

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Longitude Figure 3. Five-day global forecasts from the initial state of Fig. 2 using the PV-D model. Mean height H 10 km. Contour interval 30 m. Intervals of latitude and longitude 30". Spatial resolution 512 x 257. Time resolution: (a) 30 minutes; (b) one hour; (c) two hours.


Longitude Figure 4. As Fig. 3 but using the (u,u ) model. Uncentring parameter E = 0.02.

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truncation-error in the spectral case, and since T42 provides a virtually complete representation of this mode. With the time-step and amplitude used, errors due to time truncation and non-linearity are insignificant. (b) Geostrophically-balancedstate with strong cross-polar flow One of the most troublesome problems traditionally encountered with Eulerian finitedifference models has been computational noise induced by the polar numerics in the presence of strong cross-polar flow. This problem has been overcome in semi-Lagrangian finite-difference models by discretizing the momentum equation in vector form, though it has previously been necessary (as in our ( u , u ) model) to retain some form of noise suppressor to control gravity-wave noise arising from orographic effects or forward extrapolation of the non-linear terms. To assess the performance of the PV-D model in the presence of a strong cross-polar flow, we used the initial state of MB, where the initial geopotential field is given by

@(A, 4 , o ) = 5 + 2 ~ 2 a usin3 ~ 4 cos 4 sin A

(29) and the initial u and u are derived from (29) using the geostrophic relationship. Here, 52 and a denote, respectively, the earth's rotation rate and radius. The values 5 = 5.789 x lo4 m2 sp2and uo = 50 m s-' were chosen (uo is the initial cross-polar wind strength). The initial field of geopotential height corresponding to these parameters is shown in Fig. 5.

Figure 5. Initial hemispheric field of geopotential height (m) given by Eq. (29) with 5 = 5.789 x lo4 m2 s-l and uo = 50 m s-'. Contour interval 150 m; NP = north pole; GM = Greenwich meridian; ID = 180" longitude.

The ten-day forecast given by the PV-D model with At = 30 minis shown in Fig. 6(a). It is seen that no computational noise occurs near the pole or elsewhere, despite the absence of polar filtering, divergence damping, uncentring or any explicit diffusion. The ( u , u ) model with the same resolution requires an uncentring parameter of E = 0.15 to give a noise-free integration; the ten-day forecast in this case is shown in Fig. 6(b). The spectral model, with a spatial resolution of T170, a time-step of eight minutes (required for stability), and V4 diffusion with the recommended value of the coefficient, is found to give a smooth integration, as shown in Fig. 6(c). Visual comparison shows that the three forecasts are very similar.


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Figure 6 . Ten-day forecasts from the initial state of Fig. 5. Contour interval 150 m. (a) PV-D model (resolution 512 x 257; At = 30 minutes). (b) ( u , u ) model (resolution 512 x 257; At = 30 minutes; uncentrin parameter E = 0.15). (c) Spectral model (resolution T170; At = eight minutes; diffusion 2 x 1013m4 s - ~ V ~ ) .

(c) Rossby-Haurwitz wave The Rossby-Haurwitz (R-H) wave is an exact non-linear solution to the non-divergent shallow-water equations, but it has been traditional since the work of Phillips (1959) to use it as an approximate solution for the free-surface case. Williamson et al. (1992) have recommended that it be used as a standard test-case for shallow-water models on the sphere. The R-H wave of zonal wave-number four is stable and we here compare the simulations of this wave by the three models. The initial geopotential height field is shown in Fig. 7. The spatial resolution we use (512 x 257 for the PV-D and ( u , u ) models and T170 for the spectral model) is considerably higher than that generally regarded as sufficient for spatial convergence of R-H wave four (Puri and Bourke 1974; Naughton et al. 1993). The five-day forecasts given by the PV-D, (u, v ) and spectral models are shown in

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Longitude Figure 7. Initial global field of geopotential height (m) for the Rossby-Haurwitz wave. Contour interval 120 m. s-'; ho = 8 x lo3 m, using the notation of Williamson et af. (1992)). (Parameters: R = 4; o = K = 7.848 x

Fig. 8. It can be seen that in this instance there is a considerable difference between the PV-D model solution and the other two. The PV-D model closely maintains the identity of the R-H wave, while the (u,u ) and spectral models show a vacillating behaviour. The amplitude of the wave used above is such that the flow is highly non-linear. To examine whether the solutions become closer when the non-linearities are smaller, we reduced the initial amplitude of the wave by a factor of four. The five-day forecasts given by the three models then, all parameters being otherwise the same, are shown in Fig. 9. The three solutions have become virtually identical, all three maintaining the wave's initial shape. Experiments using different spatial and temporal resolutions show that, in the fullamplitude case, the qualitative difference described above between the PV-D solution and those of the other two models is a feature that persists. The true evolution from the R-H wave initial state is unknown. Traditionally, highresolution-model integrations have been regarded as providing a good estimate of the truth, and convergence has been measured by carrying out 'identical twin' experiments. In these, the resolution of the model in question is gradually increased towards that of the high-resolution control run, and convergence is assumed to occur if the difference from the control integration tends to zero as the resolution tends towards that of the control. The general validity of this measure of convergence has been called into question by the numerical experiments of Semazzi and Dekker (1994). These authors have shown, using R-H wave four as the prescribed analytical solution to a set of modified shallowwater equations, that estimates of truncation error provided by identical-twin experiments can be misleading: for a given spatial resolution, the most accurate solution may occur for a time-step longer than that which identical-twin experiments would indicate to be optimal. The difficulties associated with identical-twin studies of convergence are also clear from the thorough study of Naughton et al. (1993), who have shown that questions of convergence, in addition to being influenced by the amplitudes of the motions concerned, are also dependent upon initialization, the removal of small scales, and the formulation of diffusion. We have carried out experiments to test the effect of varying the coefficient of the V4 diffusion on the spectral-model solution for the full-amplitude R-H wave described above. Counter to intuition, it is found that the strength of the vacillation is substantially


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Figure 8. Five-day forecasts from the Rossby-Haunvitz wave initial state shown in Fig. 7. Contour in1.erval 120 m. (a) PV-D model (resolution 512 x 257; At = 30 minutes). (b) (u. u ) model (resolution 512 x 257; A t = ?10 minutes; uncentring parameter E = 0.2). (c) Srectral model (resolution T170; A? = 7.5 mirlutes; diffusion 2 x lo1 m4 s-' V4).

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Figure 9. As Fig. 8, but with the initial amplitude of the Rossby-Haurwitz wave reduced by a factor of four. Contour interval 30 m.


1997

increased by increasing the diffusion coefficient by factors of ten and a hundred above its recommended value. Since the largest spherical-harmonic components of the solution do not feel the diffusion directly (to any extent), the changes in the solution must result from the non-linear-cascade interaction of these components with the short scales where the diffusion is felt. It is found that the period and phase of the vacillation are also dependent on the diffusion coefficient. In view of these experiments, it is clear that measures of difference such as r.m.s. height differences between the PV-D and spectral integrations will substantially depend on the diffusion used in the spectral model (whose true physical form is unknown, but whose presence is necessary to control noise, particularly in the vorticity and divergence fields). For this reason, we choose not to present such measures of difference here. The fact that the spectral solution for the R-H wave becomes qualitatively closer to the PV-D solution as the diffusion coefficient is reduced can perhaps be interpreted as evidence that the PV-D solution is not unrealistic. ( d ) Real data The fourth initial state used to test the PV-D model was a real data analysis. Part of our motivation in using this was to see how well the PV-D model represents small scale features. The analysis of 0000 UTC 1 December 1992 by the European Centre for Medium-Range Weather Forecasts was chosen. The initial state is shown in Fig. 10.

Figure 10. ECMWF analysis of global field of 500 hPa geopotential height for 0000 UTC 1 December 1992. Contour interval 60 m.

The five-day forecasts given by the three models, PV-D, ( u , u ) and spectral, are shown in Fig. 11. it can be seen that the forecast fields are very alike. From Table 1it can also be seen that the sensitivity to spatial and temporal resolution of the five-day forecasts given by the PV-D model is small. (According to a common measure, r.m.s. differences of 3 m day-' between forecasts are regarded as insignificant.) This test can be regarded as a proof-of-concept for the PV-D model as relates to its ability to forecast from real data.

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Figure 11. Five-day forecasts from the initial state of Fig. 10. Contour interval 60 m. (a) PV-D model (resolution 512 x 257; At = 30 minutes). @) ( u , v) model (resolution 512 x 257; At = 30 minutes; uncentring arameter 8 = 0.02). (c) Spectral model (resolution T170; At = 7.5 minutes; diffusion 3 x 1014 m4 s-'V'$.

1999

MODEL BASED ON PV ADVECTION TABLE 1. R.M.S.

DIFFERENCES IN GEOPOTENTIAL HEIGHT BETWEEN THE FIVE-DAY FORECASTS (REAL DATA CASE) GIVEN BY THE PV-D MODEL AT VARIOUS SPATIAL AND TEMPORAL RESOLUTIONS

r.m.s. difference Time-step Fixed at 30 min Fixedat30min 1 h-15 min 30 min-15 min

Spatial resolution

0-4

128x65 -512x257 256x129-512x257 Fixed at 512x257 Fixed at 512x257

27.09 13.64 28.33 14.32

(e) Some additional numerical results Because of the way in which the @-termis expressed, our modified PV equation (10) does not give a zero value for the PV residual R , defined as

It is of interest to see how this quantity compares in the PV-D and ( u , v ) models. Figure 12 shows the residual at day seven for the respective models, in integrations starting from an initial state defined by (29), with uo = 20 m s-'. A resolution of 128 x 65 with At = 1 h is used for both models. It can be seen that over most of the domain the residual is about an order of magnitude smaller in the PV-D case than in the ( u , v) case. In the immediate neighbourhood of the pole, the residual in the PV-D case remains small, while in the ( u , v ) case it assumes comparatively large positive and negative values. These results increase our confidence that the PV-D model better simulates the evolution of PV. The global variations in mass, total energy (which we here define as the kinetic energy plus the perturbation potential energy) and potential enstrophy from 0 to 50 days for integrations of the PV-D and ( u , u ) models at a resolution of 512 x 257 with At = 30 min are shown in Fig. 13. It can be seen that while the conservation properties of both models are quite good, the variations in all three quantities are smaller in the PV-D case.

TABLE 2. CPU TIMES FOR TEN-DAY Resolution 64 x 128 x 256 x 512 x

33 65 129 257

INTEGRATIONS

PV-D model (sec)

( u , u ) model

(set)

PV-D model ( u , u ) model

5.09 13.46 42.37 157.17

2.73 9.91 38.00 151.03

1.86 1.36 1.12 1.04

Times are for ten-day integrations of the PV-D and ( u, u ) models on a single processor of the CRAY C98 computer. At = 1 hour.

Finally, we present in Table 2 the computer central processor unit (CPU) times required to integrate the PV-D and ( u , v) models at various resolutions. It can be seen that the PV-D model takes longer to integrate at coarse resolution, but about the same time

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Figure 12. PV residual R at day seven for the northern hemisphere.(a) PV-D model (resolution 128 x 65; At = one hour). Contour interval 3 x m-%. (b) ( u , u ) model (resolution 128 x 65; A? = one hour; uncentring parameter E = 0.05). Contour interval 30 x m,-’s.

as the ( u , v ) model at high resolution. The reason for the greater relative reduction in expense of the PV-D model with increasing resolution is that the computationally demanding line-relaxation used near the poles in the multigrid solver then vectorizes more efficiently.

4. CONCLUSIONS A global shallow-water numerical model, termed a PV-D model, has been developed. It uses the evolution equation of the basic dynamical quantity potential vorticity as one of its governing equations. The semi-lagrangian scheme with bicubic interpolation used to integrate this equation gives high accuracy for PV advection. A slight modification needed to keep the Rossby-wave solutions numerically stable has been shown not to interfere sensibly with PV conservation along the trajectories.

MODEL BASED ON PV ADVECTION 10

'

8-

"

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