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  • View in gallery

    Example of OLAM subdivided icosahedral mesh. Heavier lines denote original undivided icosahedron. In this example, N = 10.

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    Example of local mesh refinement applied to left portion of figure, and projection of a surface triangle cell to larger concentric spheres to generate multiple vertical model levels. In this example, N = 5.

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    (a) Horizontal computational stencil for a ρ value at location w0. The shaded area is the control volume. Numbered u locations indicate locations where transporting Ui fluxes for this control volume are defined. (b) Horizontal computational stencil for a Ui value at location u0. The arrow denotes the direction in which Ui is positive. The shaded area is the control volume. Other numbered u points denote the 12 neighboring locations where ui and/or Ui values are required on the right-hand side of Eq. (18) for prognosing Ui at u0, and numbered w points denote the six neighboring locations where ρ and p values are required.

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    Results for test case 1: (a) the initial scalar field, (b) the scalar field after 12 days for α = 0, (c) the error field for α = 0, and (d) the error field for α = π/2. Contour levels for (a)–(d) are −0.05, 0.05, 0.15, etc., and values less than −0.05 are shaded.

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    Error norms for test case 1 and dependence on N for α = 0. The bottom, middle, and top curves are the height 1 norm, the height 2 norm, and the height infinity norm, respectively.

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    Time-dependent normalized height field errors in shallow-water simulations with N = 32: (a) test case 2, α = 0, ε = 1.0; (b) test case 2, α = 0, ε = 0; (c) test case 2, α = π/2, ε = 1.0; and (d) test case 3, α = 0, ε = 1.0.

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    Normalized height field errors at 5 days for test case 3 with α = π/3 as a function of N. The bottom, middle, and top curves are the height 1 norm, the height 2 norm, and the height infinity norm, respectively.

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    Test case 5 height field after 15 days of integration. The contour interval (CI) is 50 m. (top) N = 32 and (bottom) N = 64.

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    Height field at 14 days in Rossby–Haurwitz wave simulation for (top) N = 32 and (bottom) N = 64. The CI is 200 m.

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The Ocean–Land–Atmosphere Model (OLAM). Part I: Shallow-Water Tests

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Abstract

The Ocean–Land–Atmosphere Model (OLAM) has been developed to extend the capabilities of the Regional Atmospheric Modeling System (RAMS) to a global model domain. OLAM adopts many features of its predecessor, including physical parameterizations, initialization methods, data assimilation, logic and coding structure, and I/O formats. However, its dynamic core is new and is based on a global geodesic grid with triangular mesh cells and a finite-volume discretization of the full compressible Navier–Stokes equations. Local mesh refinement can be activated in OLAM to cover selected geographic areas at very high resolution and hence to simulate atmospheric systems typically studied in regional models. This paper is one in a series whose purpose is to describe the formulation of OLAM and to validate the model’s performance through test results. This paper focuses on global-scale dynamics that can be represented by the shallow-water equations, and provides details of the spatial and temporal discretizations, which contain some unique algorithms that previously have not been applied to atmospheric models. Validation tests are performed using five standard shallow-water simulations that are commonly used benchmarks for global models. OLAM results for all tests are comparable to results from other global models.

Corresponding author address: Robert L. Walko, Department of Civil and Environmental Engineering, Duke University, P.O. Box 90287, Durham, NC 27708-0287. Email: robert.walko@duke.edu

Abstract

The Ocean–Land–Atmosphere Model (OLAM) has been developed to extend the capabilities of the Regional Atmospheric Modeling System (RAMS) to a global model domain. OLAM adopts many features of its predecessor, including physical parameterizations, initialization methods, data assimilation, logic and coding structure, and I/O formats. However, its dynamic core is new and is based on a global geodesic grid with triangular mesh cells and a finite-volume discretization of the full compressible Navier–Stokes equations. Local mesh refinement can be activated in OLAM to cover selected geographic areas at very high resolution and hence to simulate atmospheric systems typically studied in regional models. This paper is one in a series whose purpose is to describe the formulation of OLAM and to validate the model’s performance through test results. This paper focuses on global-scale dynamics that can be represented by the shallow-water equations, and provides details of the spatial and temporal discretizations, which contain some unique algorithms that previously have not been applied to atmospheric models. Validation tests are performed using five standard shallow-water simulations that are commonly used benchmarks for global models. OLAM results for all tests are comparable to results from other global models.

Corresponding author address: Robert L. Walko, Department of Civil and Environmental Engineering, Duke University, P.O. Box 90287, Durham, NC 27708-0287. Email: robert.walko@duke.edu

1. Introduction

The Ocean–Land–Atmosphere Model (OLAM) is a new numerical simulation model based on the Regional Atmospheric Modeling System (RAMS). RAMS is a limited-area model, designed to simulate mesoscale and cloud-scale processes for scientific study and numerical weather prediction (Pielke et al. 1992; Cotton et al. 2003). Since its inception at Colorado State University in the mid-1980s, RAMS has acquired a suite of state-of-the-art parameterizations that represent cloud and precipitation microphysical processes, land–vegetation–atmosphere interactions, radiative transfer, subgrid turbulent transfer, and subgrid cumulus convection. The RAMS grid is relocatable to any place on earth, and can be sized to cover an area from only a few meters across to as large as a hemisphere. Grid nesting is often used in RAMS to provide very high resolution in selected regions of a large computational domain. These features, plus a large selection of options for configuring and customizing any simulation, have made RAMS a highly versatile research tool.

In spite of RAMS’ assets, its limited-area domain has obvious disadvantages, particularly in limiting the range and scope of scientific problems that can be investigated. Thus, OLAM was developed as a global version of RAMS in order to eliminate these disadvantages and extend model applicability to global-scale circulations and to two-way dynamic interactions between global and local scales. Our approach in this effort has been to retain as many components of RAMS as feasible and to replace those that were not adaptable to the global domain. In particular, both models use identical or similar physical parameterizations, coding and logical structures, I/O file formats, and procedures for compilation, initialization, and execution.

A new dynamic core was developed for OLAM because some aspects of the RAMS dynamic core are unsuitable for global modeling. For example, RAMS uses a polar stereographic projection, which is not expandable to the globe, and it makes the Boussinesq approximation based on a horizontally homogeneous “reference state” that cannot accurately represent widely different environments (e.g., tropical and polar) within the same domain. However, in several aspects the OLAM dynamic core is similar to RAMS. OLAM has a local mesh refinement capability, which is essential for intended high-resolution simulations of mesoscale and microscale phenomena over limited geographic areas. OLAM retains the Arakawa-C grid stagger because of its superior coupling of mass and momentum fields. This is particularly advantageous for small-scale (3D) atmospheric motions, although the C grid is outperformed by other staggered schemes for some processes such as geostrophic adjustment (e.g., Arakawa and Lamb 1977) when the grid spacing is too coarse to resolve the Rossby radius. Since error in all schemes decreases with mesh size and reasonably high resolution is now easy to achieve even on global grids, OLAM adopts the stagger that performs best at the highest resolutions. Other RAMS features that were incorporated into OLAM are the momentum form of the Navier–Stokes equations, a finite-volume discretization of advective and turbulent transport, and explicit representation of acoustic waves using numerical time splitting.

OLAM is undergoing a wide variety of tests in order to verify that it performs as intended. The first of these tests have mostly focused on the dynamic core, which is the major new component of OLAM and incorporates some numerical techniques that to our knowledge have not been previously used in atmospheric models. This paper is one in a series whose purpose is to describe the formulation of OLAM and to validate its performance through test results. It focuses on global-scale dynamics that can be represented by the shallow-water equations. Section 2 describes the global grid structure used in OLAM, and section 3 describes the shallow-water equations and their spatial and temporal discretization. Section 4 presents results from five of the standard test simulations described in Williamson et al. (1992, hereafter W92). A companion paper (Walko and Avissar 2008, hereafter WA), focuses on the 3D nonhydrostatic compressible formulation of OLAM.

2. Computational mesh and coordinate system

OLAM’s global computational mesh consists of spherical triangles, a type of geodesic grid. Early applications of geodesic grids to atmospheric modeling are described in Sadourny et al. (1968) and Williamson (1968). Several studies have since refined the approach, including Heikes and Randall (1995a, b), Stuhne and Peltier (1996, 1999), Bonaventura (2000), Tomita et al. (2001, 2002), Ringler and Randall 2002, Majewski et al. (2002), Tomita and Satoh (2004), and Bonaventura and Ringler (2005). The geodesic grid offers important advantages over the commonly used latitude–longitude grid. It allows mesh size to be approximately uniform over the globe, and avoids singularities and grid cells of very high aspect ratio near the Poles.

There are many ways to construct a geodesic grid, a network of geodesics or arcs that follow great circles on the sphere. OLAM’s grid construction begins from an icosahedron inscribed in the spherical earth, as is the case for most other atmospheric models that use geodesic grids. An icosahedron is a regular polyhedron that consists of 20 equilateral triangle faces, 30 triangle edges, and 12 vertices, with 5 edges meeting at each vertex. The icosahedron is oriented such that one vertex is located at each geographic Pole, which places the remaining 10 vertices at latitudes of ±tan−1(1/2). Uniform subdivision of each icosahedral triangle into N × N smaller triangles, where N can be any integer greater than 1, is performed in order to construct a mesh of higher resolution to any degree desired. The subdivision adds 30(N2 − 1) new edges to the original 30 and 10(N2 − 1) new vertices to the original 12, with 6 edges meeting at each new vertex. All newly constructed vertices and all edges are then projected radially outward to the sphere to form geodesics. Figure 1 shows an example of the mesh at this step with N = 10. The projection causes most triangles to deviate from equilateral shape, which is impossible to avoid. Heikes and Randall (1995a, b) and Tomita et al. (2001, 2002) have shown that the accuracy of numerical computations on the mesh can be improved by adjusting the projected triangle shapes. Tomita et al. describe an adjustment method that is equivalent to relocating vertices on the sphere as if they are acted on by spring forces applied along each edge between the vertices that are located at the end points. It is assumed that the spring force is a linear function of edge length, and the equilibrium length is defined by
i1520-0493-136-11-4033-e1
where R is the earth radius and ε is an adjustable coefficient. When ε is set to 1, the equilibrium length is approximately equal to the global mean edge length. Vertices are allowed to move until forces sum to zero at each vertex. OLAM uses this spring adjustment method and solves the force balance equations numerically by an iterative method.

Construction of the global mesh in the manner described above enables logically structured 2D horizontal indexing of grid cells (Sadourny et al. 1968). However, logical structuring severely limits the possible topologies that the mesh can have. Local mesh refinement, which is a required capability of OLAM, is generally not feasible unless the mesh is unstructured. Therefore, OLAM uses an unstructured approach and represents each grid cell with a single horizontal index. Required information on local grid-cell topology is stored and accessed by means of linked lists.

If local horizontal mesh refinement is required, it is performed at this step of mesh construction. Refinement follows a three-neighbor rule that each triangle must share the finite length of its edges with exactly three other triangles. The range of possible topologies that obey this rule is enormous. We show a simple example of local mesh refinement in Fig. 2 where resolution is exactly doubled in a selected geographic area by uniformly subdividing each of the previously existing triangles into 2 × 2 smaller triangles. Auxiliary edges are inserted at the boundary between the original and refined regions for adherence to the three-neighbor rule. Each auxiliary line in this example connects a vertex that joins seven edges with a vertex that joins five edges. More generally, a transition from coarse to fine resolution is achieved by use of vertices with more than six edges on the coarser side and vertices with fewer than six edges on the finer side of the transition. It is not necessary for these vertices to be concentrated along a narrow band as shown in Fig. 2. A more gradual mesh refinement can be achieved by distributing these vertices more sparsely over a wider perimeter. A more sudden mesh refinement can be achieved by using vertices with more than seven and/or fewer than five edges, as seen for example in Wenneker et al. (2002, hereafter WSW02) and Bacon et al. (2000). However, such vertices force triangles that include them to have higher aspect ratios that may decrease numerical accuracy. Therefore, it seems preferable for atmospheric modeling to limit vertices to more than seven and no fewer than five edges. Secondary spring adjustment is applied following the mesh refinement step. Local mesh refinement will be discussed in greater detail in a forthcoming paper, but is mentioned here because of its relevance to the unstructured grid approach and the numerical scheme described below. Simulation results presented in this paper use only the quasi-uniform global grid.

The final step of mesh construction is definition of its vertical levels. To do this, the lattice of surface triangular cells is projected radially outward from the center of the earth to a series of concentric spheres of increasing radius. The vertices on consecutive spheres are connected with radial line segments (Fig. 2). This creates prism-shaped grid cells having two horizontal faces (perpendicular to gravity) and three vertical faces. The horizontal cross section of each grid cell and column expands gradually with height. The vertical grid spacing between spherical shells may itself vary and usually is made to expand with increasing height. The largest sphere is usually set more than 30 km above the first, where atmospheric pressure is less than 1 mb.

Recently, Bonaventura and Ringler (2005) presented a geodesic shallow-water model that uses a C-staggered arrangement of variables with scalars defined at triangle centers and velocity defined on triangle edges. Their study elucidates fundamental mathematical properties of discrete numerical operators applied on this type of grid. Their numerical operators are developed for discrete mass values defined at nodal points of the Voronoi dual grid (see Bonaventura and Ringler 2005), not at barycenters of the triangles. The barycentric formulation is less restrictive because it allows a larger range of triangle shapes as explained in WSW02 and Wenneker et al. (2003, hereafter WSW03). This is important when considering local mesh refinement, which forces some triangles to depart significantly from an equilateral shape.

OLAM uses a C-staggered grid discretization that is based in part on a scheme introduced by WSW02 and WSW03 for an unstructured mesh of triangular cells. Scalar properties are defined and evaluated at triangle barycenters, and the velocity component normal to each triangle edge is defined and evaluated at the center of each edge. The numerical formulation allows for nonperpendicularity between the line connecting the barycenters of two adjacent triangles and the common edge between the triangles. Control volume surfaces for horizontal momentum are the same as for scalars. This is accomplished by defining the control volume for momentum at any triangle edge to be the union of the two adjacent triangular mass control volumes (Fig. 3). This means that no spatial averaging is required to obtain mass flux across momentum control volume surfaces. WSW02 applied their scheme with local mesh refinement to 2D adiabatic compressible flow around an aircraft wing and, therefore, they did not consider the full Navier–Stokes equations in a 3D rotating coordinate system. Hence, application in OLAM required the addition of turbulence, Coriolis force, and gravity terms in the momentum equations, diabatic heat and moisture sources, and spatial discretization within a 3D spherical geometry.

OLAM uses a rotating Cartesian system with origin at the earth’s center, the z axis aligned with the geographic North Pole, and x and y axes intersecting the equator at 0° and 90°E, respectively. The three-dimensional geometry of the computational mesh, particularly relating to terms in the momentum equation and involving relative angles between proximate grid cell surfaces, is worked out in this Cartesian system. The procedure involves computation and storage of the unit vector normal to each surface, and solution of linear systems that contain the unit vector coefficients. Bacon et al. (2000) and others have pointed out that the lack of local alignment of triangle edges on a geodesic grid with any set of global coordinates removes any advantage of horizontal coordinate transformations or curvilinear coordinate systems, such as spherical, that require metric terms and may possess singularities.

3. Shallow-water formulation

a. Continuous equations

The simulations performed for this study are based on the shallow-water equations (see W92), expressed in the 3D earth Cartesian coordinate system by
i1520-0493-136-11-4033-e2
i1520-0493-136-11-4033-e3
where t represents time, ρ is the depth of the fluid, p is the height of the free surface of the fluid above sea level, Ω is the earth’s angular velocity vector, u is the three-dimensional velocity vector, Uρu is velocity times fluid depth, and subscript i represents a vector component in the xi direction. It is to be understood that all dependent variables are invariant in the earth radial direction and radial motion is zero, so all gradients and motion are locally horizontal. Spherical metric terms are not represented by means of spherical coordinates or a map projection, but directly in Cartesian space, as described below in the discretized form of the equations.

b. Finite-volume discretization

OLAM uses the finite-volume method of discretization (Hirsch 1990) in which the continuous dynamic equations are integrated over individual grid cells, each of which represents a control volume Ψ bounded by a surface of area σ. Advective transport terms, which appear as vector divergences in the governing conservation equations, are transformed via the Gaussian divergence theorem into integrals of surface fluxes over each control volume surface. Transport between grid cells is proportional to the surface area of their common face, which is explicitly represented in the equations. In the full dynamic core of OLAM including multiple vertical levels and vertical motion, scalar control volumes are prisms with triangular horizontal upper and lower surfaces and three rectangular vertical sides. Full details are provided in WA.

For the shallow-water system considered here, all fluxes are horizontal and they occur only through the vertical sides of each prism grid cell. Furthermore, the depth of the fluid is incorporated into U in Eqs. (2) and (3), so the vertical dimension of the control volumes and surfaces is arbitrary and divides out of the equations. Thus, the finite-volume discretization can be described from a horizontal cross section, as is shown in Fig. 3. In the C-staggered grid, horizontal velocity and height-weighted velocity are defined at numbered u points, while fluid depth and surface height are defined at numbered w points. Control volumes for ρ and Ui are shaded, and the full computational stencil for each is shown. The discretized shallow-water equations are written as
i1520-0493-136-11-4033-e4
i1520-0493-136-11-4033-e5
where j is an index for summation over each planar surface of a control volume, σj is the area of each planar surface, Ψ is the volume, U is defined as normal to a control volume face, uij has a direction parallel to Ui but is interpolated to face j, and overbars represent control volume mean values. Referring to Fig. 3a, j loops over locations u1, u2, and u3 for Eq. (5). Figure 3b, which applies to Eq. (4), illustrates the momentum discretization scheme developed by WSW02. Index j loops over locations u1, u2, u3, and u4. Interpolated velocity uij at these locations is obtained by projecting values at neighboring points onto xi. For example, velocities at u0, u5, and u6 are used to obtain uij at location u1, and velocities at u0, u7, and u8 are used to obtain uij at location u2. The height gradient term at u0 is obtained from p values at all six numbered w locations, since the line segment connecting points w1 and w2 is not in general parallel to Ui at u0.

OLAM required the following modifications and additions to the WSW02 scheme for use in global atmospheric simulation:

  1. In contrast to the first-order upstream advection algorithm tested in WSW02, atmospheric simulation requires a less dissipative scheme. Thus, OLAM uses the second-order method of Crowley (1968), shown here in connection with Eq. (4),
    i1520-0493-136-11-4033-e6
    where C is the Courant number, which can be positive or negative depending on Uj, and provides the upwind bias of the Crowley scheme. Velocity values uIN and uOUT represent values inside and outside the u0 control volume that are parallel to Ui at location u0. The inside value is simply ui at u0, but the outside value must be constructed from values at outside points, as described next.
  2. Triangles shown in the stencils of Fig. 3 are spherical triangles in OLAM, not planar triangles as used in WSW02. This means, for example, that horizontal velocities at locations u5 and u6 are not in the same plane as horizontal velocity at u0, so construction of uOUT to obtain uij at location u1 requires the following 3D projection:
    i1520-0493-136-11-4033-e7
    where vertical velocity w3 happens to be zero in the shallow-water system.Projection coefficients a, b, and c are determined by solving the matrix equation
    i1520-0493-136-11-4033-e8
    where N() is a column vector that contains the (x, y, z) components of the unit vector aligned in the positive velocity direction for the given stagger point in parentheses. The unit vector components are defined relative to the earth Cartesian system. In spherical geometry, N(w3) has a nonzero projection onto N(u0). These 3D metric coefficients, and the spherical geometric relationship between triangle sides and angles, together constrain the flow to a spherical domain without the explicit use of spherical coordinates or a map projection.
  3. WSW02 developed their scheme for small-scale flow in a nonrotating coordinate system. Thus, it was necessary to add the Coriolis term for OLAM. This is done using a linear equation analogous to Eq. (7), where Coriolis force at u0 uses horizontal velocities at u1, u2, u3, and u4, and vertical velocities at w1 and w2, the latter two being zero in the shallow-water system.Discretization of the shallow-water system is a special case of the more general discretization of the full 3D equations, described in WA.

c. Time integration method

The method used in OLAM to integrate the shallow-water equations is a very simplified form of the scheme for integrating the full 3D compressible model, which involves time splitting and implicit vertical differencing. The details and motivations of that scheme are fully described in WA. Here, the single vertical layer and the absence of explicit vertical motion in the shallow-water system reduce the algorithm to an explicit solution method, which we describe below. It is important to integrate the shallow-water equations in this manner so that the simulations described in this paper serve as a valid test of the methods applied in the unsimplified OLAM dynamic core.

Given horizontal momentum (advecting mass flux) values at current time step t and previous time step t − Δt, a future mass flux value is estimated by simple extrapolation in time according to
i1520-0493-136-11-4033-e9
Coefficient A (and also B, used below) is a constant approximately equal to ½, but may be increased slightly to provide damping of the solution (in practice, A and B are usually set to 0.55). Terms that contain A and B in their superscripts thus apply approximately midway through the time step. Next, mass (or fluid depth) is updated from the current to the future time step using
i1520-0493-136-11-4033-e10
The future height of the free surface is then calculated from
i1520-0493-136-11-4033-e11
where h is topography height, and free surface height at an intermediate future time is determined from
i1520-0493-136-11-4033-e12
Finally, future momentum is updated from
i1520-0493-136-11-4033-e13
where GU represents the Coriolis force.

Note that advecting mass flux Ut+AΔt is identical between Eqs. (10) and (13). Consistency of Ut+AΔt for all advective transport is a chosen requirement for the OLAM dynamic core, as described in WA. Advected velocity components ut that appear on the right hand side of Eq. (13) at the current time step t are evaluated using the second-order Crowley (1968) method. Advective contributions from both horizontal mass flux components are applied simultaneously, without including cross-derivative terms, even though the Crowley method is known to be potentially unstable in this form at higher Courant number. However, Courant numbers are low in this system because gravity wave modes propagate much faster than maximum fluid velocities. Application of the Crowley method for the full compressible system is discussed more fully in WA.

4. Test simulations

W92 describe a series of shallow-water simulations that are intended as fundamental tests of global model dynamic cores. These tests are designed to examine basic processes such as advection, gravity wave propagation, and geostrophic adjustment, without added complications of moist processes, radiative transfer, surface fluxes, turbulence, or even explicit vertical motion. They have become a standard benchmark for model evaluation. We have performed W92 test cases 1, 2, 3, 5, and 6, and the results are presented in this section. All tests are an application of OLAM configured with only a single vertical level. Cases 1–3 pertain to flows that have known analytic solutions in continuous space, which enables direct evaluation of model error. OLAM results for all cases are compared against other models, particularly for cases 5 and 6, which do not have analytic solutions for a divergent shallow-water model. Results are also evaluated for convergence as resolution is increased.

To compare OLAM test results with other models, most of the test simulations are run with N = 32. This corresponds to the same triangular mesh as the TWIG10242 grid of Heikes and Randall (1995a, hereafter HR), and the glevel = 5 grid of Tomita et al. (2001, hereafter TO). However, the C-staggered OLAM mesh provides more degrees of freedom than the unstaggered mesh of those studies. This is because triangle centers, where scalars are defined in OLAM, are twice as numerous as triangle vertices, where all variables are defined in the unstaggered mesh. Also, triangle edges, where horizontal velocity is defined in OLAM, are three times as numerous as vertices, although each triangle edge holds only a single (normal) velocity component while the unstaggered schemes hold both horizontal components at vertices. Thus, the C-staggered scheme holds 50% more horizontal velocity values than the corresponding grid of the unstaggered scheme. It is therefore difficult to directly compare accuracy and resolution between different staggering configurations, although equating scalar points would be approximately equivalent to N = 23 in OLAM, and equating velocity points would be approximately N = 26 in OLAM.

a. Test case 1: Advection of a cosine-shaped bell

Test case 1 is a simulation of scalar field advection by a specified global wind field that is held constant in time and is in solid-body rotation relative to the earth. The scalar value is initially zero everywhere except within a circular region of radius 1/3 that of the earth, and follows a cosine bell profile inside this region with a maximum value of 1 at the center. The bell is centered across the wind speed maximum, which causes it to be advected along a great circle path. To test model sensitivity to direction, the wind field axis of rotation is set to coincide with the earth’s axis (α = 0), the equator (α = π/2), and angles in between. After the bell travels one complete rotation, which takes 12 days at the specified wind speed, the advected field is compared to the initial field. In a perfect model, they should be identical. The experimental parameters are described in greater detail in Eqs. (75)–(80) of W92.

Figure 4 shows the initial, final, and error fields from the OLAM simulation for N = 32 and α = 0, and the error field for α = π/2. A time step of 300 s was used in these simulations. The advected field is somewhat distorted from the initial circular shape, and the peak values lag slightly behind the analytically correct location. Maximum errors exceeding 15% with α = 0 and 25% with α = π/2 are mostly associated with this lag. Numerical dispersion leads to negative scalar values on the trailing edge of the bell and a weakening of the maximum value, although the maximum exceeds 98% of the original value. Comparing OLAM results with Fig. 4 of HR shows that dispersion is somewhat weaker in the OLAM results, with better preservation of the maximum and minimum values and less retardation of the bell center. Interestingly, the distortion pattern of the two models is different, with OLAM contours being more nearly circular and HR contours having a much smaller radius of curvature on the trailing side than on the leading side of the bell. This difference is presumably a direct consequence of the different base grids, one using triangles with staggering and the other using hexagons without staggering, and not of effective resolution.

Normalized OLAM global errors at 12 days are plotted as a function of N in Fig. 5 for α = 0, using a constant time step of 300 s for all N. The 1-, 2-, and infinity-norm errors are defined in Eqs. (82)–(84) of W92. It is apparent that errors at N = 32 are less than half those at N = 16, indicating that convergence over this range is faster than linear. Our errors at N = 16 are nearly identical to the final-time errors shown in Fig. 7c of HR for a mesh corresponding to N = 32. Even accounting for the additional degrees of freedom provided by the C-staggered grid, OLAM gives superior performance for this case. In the interval from N = 64 to N = 128, errors in the OLAM solution decrease by nearly a factor of 4, which is a quadratic convergence rate.

b. Test case 2: Global steady-state nonlinear geostrophic flow

Test case 2 contains an initial global wind field in solid-body rotation identical to the previous experiment, and an initial height field in geostrophic balance with it. Unlike test case 1, test case 2 (and also case 3, 5, and 6) exercises all terms in the shallow-water equations. The initial conditions are defined in Eqs. (90)–(96) of W92. Model integration is carried out for 5 days with a time step of 400 s. The ideal solution should remain in steady state, and any departure from this state during model integration represents a degree of error in the model’s representation of the flow.

Normalized global height errors from OLAM test case 2 simulations are plotted as a function of time in Figs. 6a–c. The smallest errors occur for α = 0, and somewhat larger errors occur for α = π/2. These results can be compared with Fig. 8 of HR, Fig. 2 of Stuhne and Peltier (1999, hereafter SP), and Figs. 9 and 10 of TO. OLAM’s results are similar to the results of these other models, although somewhat smaller than the vsw5 result of SP. A weak dependence on the equilibrium spring length parameter ε can be seen by comparing Figs. 6a,b. Tomita et al. (2002) and TO noted a similar dependence in their results.

The HR model uses a modified mesh structure, which they term the twisted icosahedral grid. The modification is accomplished by longitudinally shifting the mesh in one hemisphere relative to the other by π/5, which longitudinally aligns the nonpolar five-edged vertices in the two hemispheres and produces a mesh that is symmetric across the equator. To test how this modification would impact the test case 2 results in OLAM, we performed simulations with the twisted grid and α = 0 for both ε = 0 and ε = 1.0. Error growth in these simulations (not shown) was slightly smaller than for the untwisted grid. Examination of meridional winds revealed that cross-equatorial flow occurred on the untwisted grid but not on the twisted grid. Evidently, the symmetry of the twisted grid helped to maintain the flow field symmetry specified for test case 2.

c. Test case 3: Steady-state nonlinear zonal geostrophic flow with compact support

This test is similar to test case 2 except that instead of a global wind field in solid-body rotation, test case 3 contains zonal winds that are confined to a narrower band of latitude and are not symmetric across the equator. As in test case 2, initial wind and height fields are specified from analytically balanced fields, and the simulation tests the model’s ability to maintain a steady-state balanced solution. Model integration is carried out for 5 days. Equations (101)–(115) in W92 provide the details of the initial conditions. We ran OLAM simulations using different values of ε, both with and without twisting of the icosahedral grid. The results of the simulations were not sensitive to these changes. The twisted grid showed no apparent advantage for the asymmetric flow field. Thus, we show time-dependent height errors for only one case (Fig. 6d), that uses N = 32, α = π/3, ε = 1.0, and no twisting. Error convergence rates are determined by comparing errors at 5 days with N = 16, 32, 64, and 128 (using time steps of 400, 400, 200, and 100 s, respectively) which are plotted in Fig. 7. OLAM error values are close to those of TO Fig. 12 for N = 16 and 32, but are somewhat larger at higher resolution. OLAM error values are smaller than SP’s Fig. 3 for N = 16 and 32, but fairly similar at N = 128. Error convergence rates in OLAM test case 3 simulations exceed 1.5 in all cases, but are generally lower than the quadratic rate achieved in TO and SP. At 10 days of integration, OLAM errors (not shown) are all approximately 50% larger than the values shown in Fig. 7 for 5 days. OLAM’s 10-day error values are smaller than the 10-day values in Bonaventura and Ringler’s (2005, hereafter BR), Table 3, while convergence rates in the two models are very similar.

d. Test case 5: Zonal flow over an isolated mountain

This test simulation begins with geostrophically balanced zonal flow in solid-body rotation around the earth as in test case 2, except that the atmospheric layer is deeper (5960 m at the equator) and zonal flow is slower (20 m s−1 at the equator). A cone-shaped mountain of height 2000 m and base radius 10° of latitude–longitude [see Eq. (134) of W92] is located at 30°N, 90°W. Waves develop and propagate as the flow impinges on the topographic barrier. Integration is carried out for 15 days, and height fields are compared between different models and different resolutions because no analytic solutions exist.

Figure 8 shows OLAM results after 15 days for N = 32 and N = 64, using time steps of 400 and 200 s, respectively. Comparison with a reference solution produced by the National Center for Atmospheric Research (NCAR) spectral model at T106 resolution (shown in Fig. 4b of BR) shows close agreement for both OLAM values of N, although as expected, N = 64 shows better agreement. Similar results are obtained from other geodesic models [Fig. 14 of HR, Fig. 4 (top) of SP, Fig. 19a of TO, and Fig. 4a of BR].

Test case 5 is also used to evaluate how well models conserve quantities that should remain globally invariant for this experiment (i.e., total energy, potential enstrophy, and mass). The time evolution of these quantities was examined for the N = 32 and N = 64 OLAM simulations. We used the same definition for the total energy integral as HR, SP, and TO where the relatively large potential energy of the initial state is subtracted. This accentuates any errors that occur in the much smaller fields of kinetic energy and potential energy of the height deviation from the initial state. The energy error in OLAM reached a magnitude of 0.003 at 15 days for both N = 32 and N = 64, which is about an order of magnitude smaller than Fig. 11c of HR and Fig. 6 (top) of SP, although it is comparable to Fig. 17 glevel 5 and glevel 6 of TO. The OLAM error appears larger than Fig. 7 of BR, although they do not state whether they subtract the initial state potential energy. Relative error in potential enstrophy in the OLAM simulations has a magnitude a little less than 0.0003 for N = 32 and a little less than 0.0002 for N = 64. These errors are a few times smaller than Fig. 6 of SP and Fig. 7 of BR, close to but larger than Fig. 11d of HR, and a few times larger than Fig. 17b glevel 5 and glevel 6 of TO. Schemes that are specifically designed to conserve quadratic quantities perform better in this regard, as shown, for example, in Lipscomb and Ringler (2005), although they are generally more expensive and/or subject to oscillatory behavior.

As pointed out by TO, the finite-volume formulation of mass conservation ensures that mass will be conserved to machine accuracy. Pressure and density are represented in OLAM with double precision so that pressure can vary in sufficiently small increments relative to its full atmospheric value. However, all other quantities, including mass fluxes between grid cells, use native single precision, which was 4 bytes for the simulations presented in this paper. Total mass in this OLAM simulation fluctuated between positive and negative values with a relative error magnitude usually less than 10−10, which is consistent with the single precision mass fluxes.

e. Test case 6: Rossby–Haurwitz wave

Initial conditions for test case 6 represent a Rossby–Haurwitz wave of wavenumber 4 and are given by Eqs. (143)–(149) of W92. In a nondivergent barotropic model, this wave maintains constant shape and amplitude, and propagates eastward at a speed given by Eq. (142) of W92. In BR this wave was shown to be an unstable solution to the shallow-water equations. Nevertheless, it has been simulated successfully in several shallow-water models with the original wave pattern mostly intact out to 14 days. For example, Fig. 20 of TO shows solutions at day 14 for three different resolutions of their geodesic model. OLAM results for day 14 are shown in Fig. 9 for resolutions N = 32 and N = 64 (using time steps of 400 and 200 s, respectively), which correspond to the top two panels of Fig. 20 in TO. The solutions are very similar between the models for each resolution, although the wave amplitude is a little better preserved in the TO results. We have not computed height error norms as in HR and TO, which require reference solutions generated by a spectral model.

5. Summary and conclusions

This paper has described the formulation of the Ocean–Land–Atmosphere Model (OLAM), which is based on the Regional Atmospheric Modeling System (RAMS) but covers the global domain. Many components of RAMS were incorporated into OLAM with little modification, which saved considerable development effort. However, OLAM’s dynamic core is new and consists of 1) a global triangular-cell grid mesh with local refinement capability; 2) the full compressible nonhydrostatic Navier–Stokes equations; 3) a finite-volume formulation of conservation laws for mass, momentum, and potential temperature; and 4) numerical operators that include time splitting for acoustic terms. The global domain greatly expands the range of atmospheric systems and scale interactions that can be represented in the model, which was the primary motivation for developing OLAM.

The new dynamic core includes some elements that have not previously been applied to atmospheric modeling. These include 1) a spatial discretization scheme introduced by WSW02 and WSW03 and previously applied to flow around an aircraft wing; 2) modifications and additions to the WSW02 discretization to accommodate 3D spherical earth geometry, the Coriolis force, and a higher (second) order advection scheme; and 3) a time discretization scheme that combines time splitting with consistent use of advecting mass flux between all transported quantities. Therefore, an important purpose of this study was to subject OLAM to a series of numerical tests to determine whether the new dynamic core could successfully reproduce expected features of global dynamics.

The present study focused on the shallow-water system of equations in order to evaluate how well OLAM’s new dynamic core simulates fundamental processes such as advection, gravity wave propagation, and geostrophic adjustment. The relatively simple shallow-water system tests only a subset of the full OLAM dynamic core, but the numerical methods described and used in this paper are themselves a proper subset of those used for flows with vertical motion and multiple vertical levels. Therefore, the numerical simulations in this study provide a valid test of the full system. OLAM was tested with five of the seven well-known global shallow-water simulations described by W92. OLAM performed each test successfully, giving results that compare well with other models and with analytic solutions where applicable. Tests at different resolution showed error convergence rates to generally lie between 1.5 and 2.0. The simulation tests give confidence that OLAM’s dynamic core is correctly formulated to represent basic features of the global circulation. A companion paper (Walko and Avissar 2008) describes and tests additional aspects of the OLAM dynamic core, focusing on vertical nonhydrostatic motion and the acoustic time-splitting scheme. We are in the process of examining OLAM’s performance with moist and radiative processes, surface water and energy exchange, and local mesh refinement. Results will be reported in forthcoming papers.

Acknowledgments

The development of OLAM was supported by the Edmund T. Pratt Jr. School of Engineering, Duke University, by NASA Grant NAG5-13781, and by the Gordon and Betty Moore Foundation. This paper benefited significantly from the anonymous reviewers’ comments.

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Fig. 1.
Fig. 1.

Example of OLAM subdivided icosahedral mesh. Heavier lines denote original undivided icosahedron. In this example, N = 10.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 2.
Fig. 2.

Example of local mesh refinement applied to left portion of figure, and projection of a surface triangle cell to larger concentric spheres to generate multiple vertical model levels. In this example, N = 5.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 3.
Fig. 3.

(a) Horizontal computational stencil for a ρ value at location w0. The shaded area is the control volume. Numbered u locations indicate locations where transporting Ui fluxes for this control volume are defined. (b) Horizontal computational stencil for a Ui value at location u0. The arrow denotes the direction in which Ui is positive. The shaded area is the control volume. Other numbered u points denote the 12 neighboring locations where ui and/or Ui values are required on the right-hand side of Eq. (18) for prognosing Ui at u0, and numbered w points denote the six neighboring locations where ρ and p values are required.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 4.
Fig. 4.

Results for test case 1: (a) the initial scalar field, (b) the scalar field after 12 days for α = 0, (c) the error field for α = 0, and (d) the error field for α = π/2. Contour levels for (a)–(d) are −0.05, 0.05, 0.15, etc., and values less than −0.05 are shaded.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 5.
Fig. 5.

Error norms for test case 1 and dependence on N for α = 0. The bottom, middle, and top curves are the height 1 norm, the height 2 norm, and the height infinity norm, respectively.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 6.
Fig. 6.

Time-dependent normalized height field errors in shallow-water simulations with N = 32: (a) test case 2, α = 0, ε = 1.0; (b) test case 2, α = 0, ε = 0; (c) test case 2, α = π/2, ε = 1.0; and (d) test case 3, α = 0, ε = 1.0.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 7.
Fig. 7.

Normalized height field errors at 5 days for test case 3 with α = π/3 as a function of N. The bottom, middle, and top curves are the height 1 norm, the height 2 norm, and the height infinity norm, respectively.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 8.
Fig. 8.

Test case 5 height field after 15 days of integration. The contour interval (CI) is 50 m. (top) N = 32 and (bottom) N = 64.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

Fig. 9.
Fig. 9.

Height field at 14 days in Rossby–Haurwitz wave simulation for (top) N = 32 and (bottom) N = 64. The CI is 200 m.

Citation: Monthly Weather Review 136, 11; 10.1175/2008MWR2522.1

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