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THEMATIC PROGRAMS |
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| April 23, 2024 |
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Takayuki Aoki, Tokyo Institute of Technology
and Hiroshi Yoshida, Tokyo Institute of Technology A high-accurate IDO (Interpolated Differential Operator) Scheme[1]
has been applied to shallow water equations in spherical geometry represented
by Williamson's test cases[2]. The IDO scheme required not only the
discretized physical value on the grid point, but also its spatial derivatives
as independent variables. The additional variables require solving more
equations derived from the given equations, so that it makes possible
to construct a high-accurate interpolation function around the local
area of the grid point. The interpolation function reduces to an approximation
solution of the given partial differential equation. For Williamson's
test case 2, our result keeps the initial balance much better than 4th-order
finite difference scheme. The result of the test case 5 has good agreement
with that of Pseudo-Spectral method. References [1] T. Aoki, "Interpolated Differential Operator (IDO) Scheme
for Solving
However, semi-Lagrangian methods in their simple form lack conservation properties. In many fields of application, e.g. climate modelling, or modelling of nonlinear phenomena with shocks, conservation of mass (and other physical constituents) is essential. We propose several modifications which add mass conservation to the
advection case of the semi-Lagrangian method. A comparison highlights
properties of the proposed schemes in different situations. Applications
in planar and spherical geometries are given. The effectiveness of mass
conservation is shown in test cases with diverging and converging wind
fields. Luca Bonaventura, Max Planck Institute
for Meteorology Hamburg The joint project of Max Planck Institut for Meteorology Hamburg and Deutscher Wetterdienst to develop a new dynamical core for climate and weather forecasting simulation is presented and outlined. The project has started in the spring of 2002 at MPI Hamburg and is expected to produce a new dynamical core by the end of 2005. A dynamical kernel based on the discretization of the fully compressible, non-hydrostatic equations is under development. The numerical model will use a icosahedral grid for the horizontal discretization and employ a height based vertical coordinate. Cell centered finite volume discretizations of the continuity equation on either the hexagonal cells or the dual triangular cells are going to be employed. As a first step towards the full model, conservative, semi-implicit numerical schemes to discretize the shallow water equations the sphere are being developed. Accuracy and benefits of various possible formulations will be analyzed by comparing results obtained on standard test cases. Preliminary results of numerical simulations will be presented. Elisabetta Cordero, Met Office The set of equations governing the atmospheric motions employed in the operational Met Office NWP and global climate model is the non-hydrostatic and deep-atmosphere form of the Navier-Stokes equation set. The equations are discretised on a uniform lat-long Arakawa C-grid in the horizontal and a Charney-Phillips grid in the vertical using a semi-implicit semi-Lagrangian finite difference time integration (``New Dynamics''). A key aspect in the development and testing of the New Dynamics was
the creation of 1D and 2D versions of the model. These versions were
used to investigate the numerical stability of the scheme, and to test
its behaviour in the presence of orography and for different top and
bottom boundary conditions. Some results from these idealised models
have been presented at previous ``PDEs on the sphere'' In order to bridge the gap between these experiments and various linear analyses, the linear normal modes of the exact 2D New Dynamics discretisation have been computed. The program has the flexibility to be used to analyse the 1D discretisation and different vertical staggering of the variables, and includes switches to change the top and bottom boundary conditions. At present it has been used to compute the normal modes of the 1D column model with both an isentropic and an isothermal basic state. In the isentropic basic state case, the numerical results are found to agree well with the analytic ones. The normal modes of the isothermal basic state 1D column model have been used to initialise the 1D New Dynamics: some of the numerical results will be presented. Future work will include: completing the analysis of the properties of the 1D and 2D versions of the existing discretisation, for different boundary conditions and vertical staggering of the variables; using the established normal mode model as a tool to analyse the properties of improved alternative discretisations, before implementing them in the full NWP model; and extending the program to include the trajectory calculation.
Terry Davies, Met Office, Bracknell, UK There are a wide range of idealised tests available in 1 or 2 dimensions for testing atmospheric code but only limited 3 dimensional tests. The dynamical core tests of Held and Suarez need to be run for several hundred days to approach quasi-equilibrium. Full 3 dimensional balanced flows can be easily set up on an f-plane in limited area models where the lateral boundaries can be used to maintain a constant (in time) large scale flow. Here I will describe a similar approach for global tests where the flow is close to geostrophic and hydrostatic balance when orography is absent. Orography can be introduced inducing perturbations to the flow which increase with either the size of the orography or the speed of the flow. These tests are 3 dimensional analogues to some of the shallow-water test set and are useful in testing dynamics algorithms and the response to orographic forcing. The tests need to be run for no more than 10 days to provide meaningful results making them ideal for testing at high resolution. The use of the Schmidt transformation in a semi-implicit, semi-Lagrangian formulation for shallow water equations in spherical geometry gives the ability to focus resolution over one particular area of interest. The spectral transform computation is modified, to account for the grid transformation, by solving a linear system in spectral space. This paper will present a formlation for advection of vorticity and divergence that fits well with smooth grid transformations. Numerical results using this formulation for the shallow water test cases will be presented and generalizations to other types of grid transformations will be discussed. A. Fournier, University of Maryland,
G. Beylkin, University of Colorado, V. Cheruvu, University
of Colorado, S. Sumetkijakan, University of Maryland We present a high-order, adaptive numerical method for numerically solving advection-diffusion PDEs, and demonstrate the advantage of this method for fluid dynamics problems, especially that generate quasi-singular or coherent structures. There are four aspects of the method that we will elucidate. First,
the spatial discretization employs piecewise-polynomial spaces similar
to Second, the adaptivity criterion for a set of elements is the norm
of the projection onto the corresponding multiwavelets. Using this simple Third, we take advantage of the spatial discretization, to improve the time-stepping scheme. We use the ``exact linear part'' scheme (Beylkin et al. 1998), which amounts to using scaling and squaring to compute the exponential propagator due to the diffusive linear operator. This adaptive operator exponentiation is made feasible by our spatial discretization. It significantly improves convergence for high Reynolds numbers and large time steps. Finally, we use block-sparse data structures to achieve efficiency. We will demonstrate these features using the standard shallow-water
test suite of Williamson et al. (1992). Francis X. Giraldo, Naval Research Laboratory A new dynamical core for NWP based on the spectral element semi-Lagrangian
method developed in [1] and [2] is presented. [1] F.X. Giraldo, The Lagrange-Galerkin spectral element method on
[2] F.X. Giraldo and J.B. Perot, A spectral element semi-Lagrangian
(SESL) Francis X. Giraldo, Naval Research Laboratory
and Thomas E. Rosmond A new dynamical core for NWP based on the spectral element method is
presented. In this work, the 3D primitive hydrostatic atmospheric equations
are written, discretized, and solve in 3D Cartesian space. The advantages
of this approach are: the pole singularity which plagues all gridpoint
methods disappears, the horizontal operators can be approximated by
local high-order elements, and any grid can be used including lat-lon,
icosahedral, hexahedral, and adaptive unstructured grids. The locality
property of spectral elements means that the method will scale efficiently
on distributed-memory computers. In order to validate our 3D atmospheric
dynamical core, [1] F.X. Giraldo, A spectral element shallow water model on spherical
geodesic grids, International Journal for Numerical Methods in Fluids,
Vol. 35, 869-901 (2001). Max Gunzburger, Iowa State University,
Lili Ju, Iowa State University and Qiang Du, Penn State
University Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations
for which the generators are also the centers of mass (with respect
to a given density function) of the the corresponding Voronoi regions.
Such tessellations have many remarkable properties and are especially
well suited for efficiently generating very high-quality grids. We discuss
CVTs on the sphere and their properties, and also present some very
efficient, eminently parallelizable, probabilistic algorithms for their
construction. We demonstrate the superior uniformity of "uniform"
CVT grids and also how locally refined grids can be systematically constructed
within the CVT framework. We then study, both theoretically and computationally,
the use of uniform and locally refined CVT grids for the solution of
model PDE problems on the sphere by finite volume and finite element
discretization methods. Among the result we present are optimal error
estimates for the approximate solutions and computational results which
demonstrate that optimally accurate results can be obtained through
the use of both uniform and nonuniform CVT grids. For example, for finite
volume discretizations, CVT grids yield second-order accuracy (with
respect to L2 norms). Thomas
Heinze, Technische Universitaet at Muenchen, Germany In 1988 L.L.Takacz introduced the flow over an isolated mountain as a test problem for the shallow water equations (SWE). After Williamson et.al. added it as fifth test case to the NCAR test suite for the SWE many researchers investigated this problem. One of the big challenges for the next generation of global circulation models (GCM) will be adaptive meshes. The most suitable test case for this kind of grid generation out of the test suite is test case 5. Because of this it will become more and more important to understand the inherent problems of test case 5. We investigate test case 5 with three different models (GME, SEAM, FEMmE) and compare the results. Depending on the model we observe different problems. After the discussion of them we will propose an additional test and a modified mountain to get a deeper understanding of the model numerics. Christiane Jablonowski, University
of Michigan Idealized test cases for primitive equation based dynamical cores have become a standard assessment tool for the strengths and weaknesses of current dynamics packages in GCMs. But although tests like the Held-Suarez test or the breaking polar vortex pattern show important model characteristics, additional test cases will be needed for future model intercomparisons. The talk will propose a new standard test sequence for dynamical cores with pressure-based vertical coordinates. The test suite comprises two parts. First, the model will be initialized with steady state, balanced initial conditions that are an analytical solution to the primitive equations. This initial flow consists of a zonally symmetric basic state with two jets in midlatitudes and a realistic temperature profile. The careful design guarantees static, inertial and symmetric stability properties, but is unstable with respect to baroclinic or barotropic instability mechanisms. Model integrations over thirty days reveal how well the model can keep its initial state. Second, a well-resolved small amplitude Gaussian hill perturbation is superimposed on the initial state. This triggers baroclinic wave activities that lead to explosive cyclogenesis. The test sequence has been applied to NCAR's newest dynamical cores (CAM2 model framework), to NASA's finite volume model (developed at GSFC) and the icosahedral model GME (developed at DWD). In addition, model results based on cubed meshes and reduced grids are available. The intercomparison reveals interesting characteristics that have already led to model improvements. The diagnostics include error norm statistics, grid point data comparisons and wave number analyses. In the future the test will serve as a standard test case for adaptive grid simulations that are capable of tracking localized phenomena at high resolutions. A brief outline of this project will be given.
A new and very accurate cell-integrated semi Lagrangian (CISL) two time level scheme has been formulated and tested for the shallow water equations in a plane channel model with realistic variation of the Coriolis-parameter and including topography. The implementation includes semi-implicit time stepping of the gravity wave terms. Furthermore, two-dimensional advection of passive tracers has been tested in idealized flows as well as in a fully general flow. Regarding basic features, the new formulation is based on step-functions, which implicitly are advected with the flow across cell boundaries. The scheme is exact in case of passive advection by a flow that is constant in time and space. This means that the accuracy is of indefinite order. In addition to this attractive feature the scheme shows very small numerical dispersion in fully non-linear and divergent flows and in such flows the scheme is locally (and globally) mass conserving, positive definite and monotonic. The high order of accuracy is achieved by introducing two additional variables. For a given prognostic variable the memory carrying variables are: the grid-cell integrals, the values at the grid cell corner points and the functional average on the intersections between grid cell corner points in either of the two spatial directions. The penalty of running the new scheme is an increase in memory consumption by a factor of three. In the present preliminary formulation of the scheme only the mass field is treated with the CISL scheme, while a traditional semi-Lagrangian scheme is used for the momentum equations. It is, however, possible to use the CISL scheme also for the cell-integrated form of the momentum equations. In this case also momentum is conserved. Results of test integrations will be presented and compared to integrations based on finite difference and spectral formulations of traditional semi-implicit Eulerian and semi Lagrangian shallow water models. The very convincing tests of passive advection by idealized flows and by a fully divergent flow simulated by the shallow water model are also shown. The presentation is completed by a discussion of the generalization of the scheme to fully three-dimensional flows on the sphere. Briefly, the semi-implicit implementation follows Machenhauer and Olk (1997) and the application on the sphere will follow the formulation presented in Nair and Machenhauer (2002) with some modifications. Following Machenhauer and Olk (1998) the generalization to three dimensions is based on two-dimensional CISL advection on model levels in combination with a diagnosing of the vertical advection under the assumption of hydrostatic balance. This formulation ensures a more consistent treatment of the vertical advection problem than is the case in the traditional hydrostatic semi-Lagrangian models based on three dimensional trajectories. REFERENCES Machenhauer, B. and M. Olk, 1998: Design of a semi-implicit cell-integrated
semi-Lagrangian model. MPI Workshop on Conservative Nair, R. and B. Machenhauer, 2002: The Mass-Conservative Cell-Integrated Semi-Lagrangian Advection Scheme on the Sphere, Mon. Wea. Rev., Vol 130, No.3, 649-667.
Oswald Knoth, Institute for Tropospheric
Research and Detlef Hinneburg, Institute for Tropospheric Research
The implementation of a global nonhydrostatic anelastic model on the sphere in a lat-lon-z grid is presented. The representation of the orography is realized by cut cells, which describe approximately the intersection of the orography boundary with the grid cells. Therefore the free fluid part in each grid cell is characterized by the free cell volume and the free face area of the six cell faces. Since the lat-lon grid is locally orthogonal the discretization in space can be expressed with the above defined cell characteristics. The spatial discretized system is solved adaptive in time by a combination
of a Rosenbrock-method for the advcetion-diffusion part and a Chorin-type
projection method for the pressure. This implicit time integration procedure
avoids time step restrictions which are caused by small volume cells
at the poles and at the cutting boundary and by fast but unimportant
physical processes. The resulting linear systems are solved by preconditioned
CG-like methods. For the advection-diffusion part the BICGStab method
is used. The preconditioner is built up by an approximate matrix factorization
of the transport terms (advection, diffusion) and the source terms (Coriolis,
curvature, buoyancy). The code is parallelized by a decomposition of the computational domain in rectangular blocks in horizontal as in vertical direction. These blocks are distributed than on the available processors. The decomposition allows also to apply different grid resolution in the individual blocks. One possible application is the use of a coarser resolution in the polar regions. The parallelization requires changes in the solvers for the linear systems. Especially for the pressure computation we use an extended pressure equation with additional flux variables (pressure gradients) at the boundaries between neighbouring blocks. We will present computational results for simplified test scenarios. L.M. Polvani, Princeton University and
R.K. Scott, Columbia University
At present, the only widely used test case for dry dynamical cores is the one proposed by Held & Suarez (1994), in which simply defined parametrizations of two key physical processes (thermal relaxation and surface drag) are added to the dynamical core. The model behavior is then tested by performing a 1,000-day integration and comparing the time-averaged fields to those in the Held & Suarez test case. One major drawback of this test case, however, is that a large amount of averaging needs to be performed before the results of a given model can be compared to those of the test case. It is conceivable that subtle coding errors or noisy features may not reveal themselves owing to the averaging. A second drawback is that numerical convergence was not demonstrated by the authors. A final, though minor drawback, is that new parameterizations need to be added to the dynamical core itself. We here propose a new test case, meant to address the shortcomings of the Held & Suarez test case, and serve as a complementary tool for testing dynamical cores. Our new test case is an initial-value problem, with the zonal winds specified to be those of a baroclinically unstable, midlatitude zonal jet, analytically specified to be very close to the LC1 paradigm described by Thorncroft et al (1993). This zonal jet is slightly perturbed, and its evolution is integrated for 10 days. We present snapshots of the fields at various time intervals (e.g. the vorticity near the surface) as the baroclinic instability develops. We also show the time evolution of several diagnostic quantities over the 10 days of integration, and provide tables of these to serve as numerical benchmarks against which new dynamical cores can be precisely and quantitatively compared. Unlike the Held & Suarez test case, our test case involves the
addition of no additional parametrizations to the dynamical core, and
requires a much shorter integration time. Also, the instantaneous fields
computed with a new dynamical core can be compared directly to those
of the benchmark, without the need for temporal or spatial averaging.
Finally, by demonstrating numerical convergence, our new test case de
facto provides a new, non-trivial, exact -- albeit numerically derived
-- solution to the time-dependent primitive equations in spherical coordinates.
Janusz Pudykiewicz, Meteorological
Service of Canada The set of equations governing reactive flow in spherical geometry
describes a complex system in which the fluid dynamics is coupled with
local processes such as chemical reactions or phase changes. In mathematical
terms the reactive flow equations consist of Navier-Stokes equations
coupled with continuity equations for a number of interacting scalar
fields characterizing the chemical composition of fluid. In order to
solve this very complex set of equations in an efficient manner we first
discretise the spatial derivatives on an icosahedral mesh defined in
Cartesian coordinates. The system of ordinary differential equations
obtained following this procedure is then solved using several different
numerical methods including the third and fourth order Runge-Kutta schemes.
A discussion of constraints which are required in order to maintain
the monotonicity of the the scheme will be also presented. In particular,
it will be shown that the solver is both mass conserving and nonoscillatory
which make it ideally suited for the solution of the complex reactive
flow problems in spherical geometry. This property will be illustrated
by the examples of application of the scheme for the simulation of tropospheric
chemistry. In conclusion, the advantages of using geodesic grids for
the simulation of reactive flows will be summarized.
Abdessamad Qaddouri, Meteorological Service
of Canada and Jean Côté, Meteorological Service
of Canada Numerical weather prediction using the Canadian GEM model involves the solution of separable elliptic boundary value (EBV) problems in spherical geometry. This type of equation arises either as Helmholtz or as horizontal diffusion problems. The direct solution of the EBV problem involves a transform, in the variable-mesh case a full-matrix multiplication, where the cost per grid point rises linearly with the number of grid points along the transform direction. In order to improve the performance of the EBV problem, the matrix product in the direct solution is accelerated by using either the Strassen method or by exploiting the symmetry of the mesh. An iterative solution of the EBV has also been implemented and compared to the direct solution. The purpose of this presentation is to report on the details of these
implementations and to show the improved performance of the EBV problem
solution either by accelerating the full-matrix in the direct solver
or by using a preconditioned Conjugate Gradient iterative solver. The
direct and iterative solvers have been tested on the NEC SX-4 and SX-5. Todd D. Ringler, Colorado State University
and David A. Randall, Colorado State University This talk will review a new dynamical formulation of the governing equations that has recently been implemented into the CSU AGCM. The new dynamical formulation is characterized by its conservation of mass, tracers, and total energy. Furthermore, the formulation exhibits a consistency between the momentum form of the governing equations and the vorticity-divergence form of the equations. We will compare simulations that use this new dynamics package to simulations that use an older formulation. We will compare and contrast the results in terms of energy and enstrophy spectra. Furthermore, using a full physics AGCM, we will show that the new formulation leads to an changes not only in the dynamical fields, but also in the parameterized fields such a precipitation and cloud cover. We have extended the new dynamical scheme to include the discretization of second order tensors by extending the definition of the discrete gradient operator to operate on vectors, as well as scalars. In addition, we have developed a conservative form of the divergenceof second order tensors. We will present results on the impact of this discretization in simulations of the 2-D turbulence. Masaki Satoh, Frontier Research System for
Global Change/ Saitama Institute of Technology, Japan For the equation set of the non-hydrostatic icosahedral global climate model being developed at Frontier Research System for Global Change, we have devised a conservative scheme of the compressible non-hydrostatic equations. The scheme is based on the flux form equations of total density, three components of momentum, total energy, and densities of water substances. Time-splitting is used with the leap-frog or 2nd order Runge-Kutta schemes for the large time step integration, and sound and gravity waves are treated implicitly in the vertical direction and explicitly in the horizontal directions for the small time step integration. Energy correction is introduced to ensure the conservation of total energy at every small time integration. The hydrological process including the warm rain cloud processes is also incorporated with the flux form equations. The scheme is tested with the non-hydrostatic core of the new global model. Besides the conservations, two improvements have been devised in the model. First, we use more accurate formulas of the thermodynamics of the moist atmosphere by taking account of the effects of specific heats of water substances and the dependence of latent heat on temperature. These effects are generally neglected in most numerical models. Second, we introduced a conservative semi-Lagrangian method for the transportations due to rain. We have performed experiments of a squall-line and direct calculation
of a radiative-convective equilibrium to confirm the conservations of
mass and energy. We found that, if the accurate moist thermodynamics
are used, the total rain is reduced more than 10\% in the squall-line
experiment in comparison to the case when the usual simplified thermodynamics
are used. We also found that the change in energy due to transportation
of rain are very large and cannot be negligible in the flux-form formulation,
while that in momentum could be M. Shoucri, Institut de Recherche Hydro-Québec
(IREQ), It has been pointed out some time ago by Yakimiw and Robert (Mon. Wea. Rev., 1986) that the method of fractional steps for the numerical solution of the shallow water equations has the advantage of reducing the multidimensional matrix inversion problem into an equivalent one-dimensional problem, so the technique becomes very simple and very attractive to apply provided it is accurate and stable enough. We apply this fractional steps technique by splitting the shallow water equations, and successively integrating in every direction along the characteristics using the Riemann invariants (which are constant quantities along the characteristics), associated with cubic spline interpolation (Shoucri, J. Comp. Phys. ,1986). The method is tested on simple advection models as well as the full shallow-water equations, and shown to be accurate and stable. The linear analysis of the equations show the method is unconditionally stable, reproducing exactly the frequency of the slow mode, while the frequency of the fast modes is exact to second order. Extension to the three-dimensional weather forecast equations will be discussed. Reiji Suda, Nagoya University We are developing a library routine set called FLTSS (Fast Legendre Transform with Stable Sampling), with which the spherical harmonics transform can be done in time $O(T^2 \log T)$. This is the first report of developments and applications of the FLTSS. Our algorithm is based on approximate fast polynomial interpolation using the FMM (Fast Multipole Method). FLTSS provides routines for fast evaluations and expansions of the associated Legendre functions and those derivatives. The evaluation points can be the Gauss points. We will discuss two topics. The first topic is the performance. The speed-up rate in the floating operation counts reaches 9.1 for $T = 4095$ and $\epsilon = 10^{-6}$. However, our first implementation does not provide such high speed in CPU time, perhaps because of the fine granularity of computations. We are now developing several versions of the code with various schemes of tuning, and the results will be shown in the presentation. The second topic is the error. Because of the use of the FMM, our algorithm
has trade-off between the computational costs and the approximation
error. We are evaluating the effects of the errors of the transform
on the solutions using shallow water equations and non-divergent flow
equations. Our code controls the error with weights about $n$ (i.e.
requires lower precision for higher wave numbers), and the effects of
the weights on the results will be also discussed. Hirofumi
Tomita, Masaki Satoh, and Koji Goto, Frontier
Research System for Global Change When the horizontal resolution becomes high in atmospheric general
circulation model, the spectral method may have some difficulties In order to radically overcome the pole problem, it is needed to use another type of grid which is distributed as homogeneously as possible on the sphere. The Next Generation Model Research Group in Frontier Research System for Global Change ( FRSGC ) has started to develop a new model based on the icosahedral geodesic grid, which is one of the quasi-homogeneous grids. Using the shallow water equations, we have overcome some difficulties in the icosahedral grid configuration[1]. When a simulation with high resolution in the horizontal direction
is performed, the equation in the vertical dynamics should be also reconsidered
from the usually used hydrostatic equation to the non-hydrostatic one.
In almost of the existing non-hydrostatic models, In this Workshop, we introduce the formulation of used scheme in our
non-hydrostatic global model and show the first result. [1] H.Tomita, M.Tsugawa, M.Satoh, and K.Goto (2001), [2] M.Satoh (2002), Motohiko Tsugawa, Yukio Tanaka and Seong Young
Yoon, Frontier Research System for Global Change We are planning to apply eddy-resolving global ocean model to climate simulations on the Earth Simulator, a massively parallel vector computer. The model will be with about 0.1 deg resolution and longitude-latitude grid model will be adopted. However, in longitude-latitude models, meridians converge around the poles and it causes inefficient computation. The computational efficiency is critical in ocean climate simulations because it takes more than thousand years to spin up the global ocean. In order to construct numerically efficient model without deterioration of physical performance, we are exploring both numerical schemes and computational techniques. We are now trying to introduce a quasi-uniform grid. Currently, we
are using quasi-homogeneous cubic grids of Purser and Rancic(1998).
This kind of cubic grid has non-orthogonal coordinate and has 8 singular
points on the sphere. However, by using B-grid and by employing an adequate
definition of metric tensors, we found that the dynamical core of the
cubic grid is simple and treats those drawbacks appropriately. In the
presentation we will talk about the development of global ocean model
in FRSGC. We will discuss finite difference schemes on the cubic grid,
treatment of the singular points and will show preliminary results of
the cubic grid global model. Agathe Untch and Mariano Hortal (ECMWF) Recently ECMWF has implemented in its operational semi-Lagrangian model a finite element discretization for the vertical based on cubic B-splines. This provided the infrastructure also for the use of cubic splines for the vertical interpolations in the semi-Lagrangian advection scheme. Semi-Lagrangian schemes using cubic splines are known to be less diffusive and less dispersive then schemes using cubic Lagarange interpolation but can suffer from noise due to large over- and undershooting near sharp gradients. We will present results from our investigations into the benefit or otherwise of using cubic spline interpolation for the vertical advection of the dynamical variables (winds, temperature and humidity) and for tracers in the ECMWF model and discuss ways of alleviating the noise problem associated with spline interpolation. Nigel Wood, Met Office, J. Coté,
Meteorological Service of Canada, Andrew Staniforth Co-authors Andrew Staniforth and Jean Cote. NWP and climate models are viewed as consisting of two distinct Recent developments mean that the dynamical components of the models
are generally very accurate and numerically stable while considerable
effort continues to be expended on improving the accuracy, and also
the scope, of the various physical parametrisations. However, the recent
work of Caya, Laprise and Zwack (1998) [CLZ98] well illustrates that
the way the parametrisations are coupled to the dynamics can significantly
limit the The complexity of the physics schemes and the non-linear nature of
the coupled system makes this a non-trivial task. However, CLZ98 have
shown that useful progress can be made by judiciously reducing the problem
to its essence. They presented a highly simplified model, or paradigm,
of a physics-dynamics coupling and used it to diagnose the source of
a problem in their global model. We have extended their paradigm to
allow for the effects of uniform advection and any number of temporally
and spatially varying forcings. These extensions permit the study of
rather more complex physics-dynamics coupling strategies (as well as
numerical issues such as spurious semi-Lagrangian orographic resonance)
but in an analytically tractable framework. This extended framework
will be presented as will results of its application to four coupling
strategies in the context of a given spatio-temporal forcing coupled
with a diffusive process. K.S. Yeh, NASA Goddard Space Flight Center,
M. Fox-Rabinovitz and S.J. Lin The NASA/NCAR finite-volume General Circulation Model has been generalized
to variable resolution with multiple areas of interest for both regional
and global applications. As its original uniform-resolution version,
the variable-resolution version also conserves the air mass, absolute
vorticity and thermodynamic energy both locally and globally. One unique
advantage of implementing variable resolution with finite-volume dynamics
is the prevention of computational noise by the intrinsic diffusion
associated withthe monotonicity constraints. While the numerical solution
also benefits from the scale selectivity of the intrinsic diffusion
in high-resolution areas, the diffusion effect in low-resolution areas
is, however, overly excessive when the grid is aggressively stretched.
A so-called "semi-stretched grid" is thus introduced to mitigate
the deterioration of the numerical solution in low-resolution areas,
and to reduce the amplitude of dispersion due to the irregularity of
resolution. The semi-stretched grid is also designed with multiple areas
of interest which can be used to focus the resolution on multiple sources
of forcing. Various experiments have been conducted with the variable-resolution
model, and preliminary results are quite encouraging.
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