Numerical Frameworks

1. Numerical Frameworks 11-13 September 2012

2. Thematic Outline • Considerations related to grid-point methods for numerical weather prediction • The underlying formulation of spectral methods for numerical weather prediction

3. Grid Point Methods • Grid-point methods rely on the atmosphere being represented by some form of a computational grid. • Such a grid typically takes one of two forms… • Structured: defined by an array of cells arranged in a regular two- or three-dimensional pattern • Unstructured: defined by collections of polygonal elements in an irregular pattern

4. Structured Grid Example (Image obtained from Penn St. University E-Education: “A World of Weather”)

5. Unstructured Grid Example

6. A Note on Unstructured Grids • Unstructured grids are quite flexible in nature, particularly for complex domains. • They are useful for models using adaptive meshes. • In an adaptive mesh, a higher density of grid points is clustered in a region of particular interest • Contrast to a fixed mesh, where grid points are evenly spaced across the entire model domain • But, they also require information to be carried in the model about how their elements are connected.

7. How do we relate our choice of computational grid to the geography of the Earth? Via the selection of an appropriate map projection!

8. Note that we aren’t (yet) saying anything about how we solve the primitive equations on a chosen computational grid. That comes later when we discuss spatial derivative methods.

9. Map Projection Fundamentals • The Earth is approximately spherical, with known area, shape, and angle properties. • We desire to convert the sphere to a “flat” grid. • But, in so doing, we cannot preserve all of the properties of the sphere! • Instead, we must choose what is most important to us to preserve and minimize the impacts of the rest.

10. Map Projection Fundamentals • For meteorological applications, we want to preserve angles first and foremost. • Why? Our kinematic fields (wind, vorticity, etc.) are all vector fields – in other words, they are angle-dependent. • Aside: here, “preserve” means that angles on the map projection equal those on the sphere/Earth.

11. Map Projection Fundamentals • Conformal map projections preserve angles, making them a popular choice for modern NWP. • Areas are not preserved, however, while shapes are only approximately conserved when the area covered by the map is relatively small. • Examples of conformal projections: Mercator Lambert Conic Polar Stereographic

12. Mercator Projection • Wrap the globe in a cylinder tangent to the Earth at the standard parallel (typically the equator). • Project the Earth’s surface onto the cylinder. • Distance and shape distortion is greatest at high latitudes. It is zero at the standard parallel.

13. Mercator Projection Example Note the increasingly “stretched” look at higher latitudes as shape and distance are distorted.

14. Lambert Conic Projection • “Top” the Earth with a cone that bisects the Earth at two latitudes (the standard parallels). • Project the Earth’s surface onto the surface of the cone. • Distance and shape distortion is greatest far from the standard parallels.

15. Lambert Conic Projection Example Note the curved shape to the map with minimal distortion. The standard parallels used to create this map are 43.75°N and 60°N.

16. Polar Stereographic Projection • Place a horizontal plane tangent to the Earth at either pole. • Example at left is for a more generic stereographic projection secant to the Earth at sub-polar latitudes. • Project the Earth’s surface onto the surface of the horizontal plane. • Distortion is largest at lower latitudes well away from the poles.

17. Northern Polar Stereographic Example Note the similar shape to that given by the Lambert conic projection.

18. Map Projection Fundamentals • Each map projection is associated with distance distortion. • For a structured computational grid, we specify the distance between grid points on the chosen grid. • But, the actual distance on the Earth between these grid points varies with latitude as a result of projecting the data onto the grid.

19. Projection Distances • ∆xg, the distance between grid points, is fixed. • ∆xe, the distance between the corresponding points on the Earth, varies with latitude.

20. Map Projection Fundamentals • To address this, we utilize a measure of the distortion known as the map-scale factor: • This ratio varies between each of the different types of map projections as a function of latitude. • Why? Because of how the Earth is projected onto each planar surface with the various projections.

21. Map-Scale Factors by Projection m = 1 (unity) indicates no distortion from projection.

22. Map-Scale Factor Considerations • Since the map projection results in varying ∆xe over the computational grid, we must modify the primitive equations to account for this. • a/k/a “coupling” the equations to the map projection • Example: equation (3.3)… time deriv. and total advection density/pres. gradient term Coriolis terms (e and f) curvature diffusion

23. Map-Scale Factor Considerations • We desire m ≈ 1 to minimize distance and shape distortion due to projection. • We also desire the spatial derivative of m to be ≈ 0 to minimize numerical impacts due to grid “stretching.” • Recommendations for practical NWP experiments: • Tropics (30°S-30°N): Mercator • Mid-latitudes (30°-60° N/S): Lambert conic • Polar latitudes (60°-90° N/S): Polar stereographic

24. Map-Scale Factor Considerations • Because of the map projection, we must also ensure that observations are appropriately mapped to the model / computational grid. • This is particularly important for directional / vector fields, namely our u and v wind components.

25. Map-Scale Factor Considerations • Finally, recall the CFL stability criterion for advection terms in an Eulerian framework: • ∆x is this case is equal to ∆xe and is latitude-variant. • For a given ∆xg, ∆xe will be largest away from the standard parallels of the chosen map projection. • Thus enabling the CFL criterion to be more easily violated!

26. Latitude-Longitude Grids • Partition the Earth into grid cells by increments of latitude and longitude. • Because the distance between longitudes decreases with increasing latitude, these grids have very densely packed grid points near the poles. • The poles themselves are singularities that must also be dealt with in some fashion.

27. Latitude-Longitude Grids Density near the poles is computationally inefficient: too many points are clustered over a very small area.

28. Latitude-Longitude Grids To address this, we can filter the data to remove fast modes near the poles or utilize non-constant longitude intervals (above).

29. Spherical Geodesic Grids • Start with an icosahedron (20 faces, 30 edges, 12 vertices) and subdivide it using multiple polygons. • Triangles, pentagons, and hexagons are the most common polygonal choices. • Unstructured grid allows for local mesh refinement by adding polygons in an area (or areas) of interest. • Best-known example: the NOAA/ESRL FIM model.

30. Other Considerations: Nesting • Oftentimes, we have a reason for focusing in on a specific area within a model simulation… • To focus upon a specific region of interest • To focus upon a specific feature of interest • To better represent processes in regions of complex terrain • The easiest way to do so: use a small enough horizontal grid spacing over the entire domain to sufficiently resolve the feature / region of interest. • This is very computationally inefficient, however!

31. Other Considerations: Nesting • The most common way to do so: embed higher-resolution domains inside of a larger, coarser domain • Outer domain provides initial and lateral boundary conditions for the inner domain • Both two-way and one-way feedback between domains is possible within current models • The change in horizontal grid spacing between domains should be relatively small to minimize wave reflection issues at the domain edges. • Typical nest ratios: 3:1, 5:1, 7:1

32. Nested Domain Example 3:1 ratio

33. Other Considerations: Nesting • Another means of doing so: stretched grids. • Feature gradual, rather than abrupt, changes of horizontal grid spacing. • Grid stretching can remain static or evolve during a model simulation. • “Grid” need not be a grid; recall: spherical geodesic

34. Stretched Grid Example

35. Other Considerations: Nesting • Most modern models use vertical grid stretching to concentrate vertical levels over specified layers. • This often impacts the boundary layer, tropopause, and other areas of greatest interest and horizontal / vertical gradients in meteorological fields. • How this is accomplished varies slightly based upon the choice of vertical coordinate, however.

36. Vertical Grid Spacing • Since the atmosphere is three-dimensional, we need to also discuss vertical grid spacing selection. • Why is vertical grid spacing important? • Many atmospheric features are either sloped (fronts, etc.) or have substantial vertical structure. • If you decrease horizontal grid spacing to better resolve these features, you must also decrease the vertical grid spacing to better resolve the features’ vertical structure(s). • Not doing so may result in spurious and/or noisy solutions.

37. Vertical Grid Spacing Example 75 layers 25 layers Note that inappropriate vertical grid spacing is not the only possible source of forecast degradation when only the horizontal grid spacing is decreased!

38. Vertical Grid Spacing • The ideal relationship between horizontal and vertical grid spacing remains unclear, however… • Simulations often encompass features with varying slopes • Horizontal numerical diffusion limits the “effective” horizontal resolution of simulated features • Vertical grid structure is often far more unstructured than the horizontal grid structure (choice of vertical coordinate) • While nested grids can have finer horizontal grid spacing, the same is not necessarily true in the vertical • Resource issue: double levels, double computations

39. Vertical Grid Spacing • General recommendation: increase the number of vertical levels as you decrease horizontal grid spacing • Practical manifestation: • For coarse horizontal grid spacing (∆x ≥ 30 km), many simulations use ~30 vertical levels in the troposphere. • For storm-scale horizontal grid spacing (∆x ≤ 4-10 km), many simulations use 50-70+ vertical levels. • As in the horizontal, just because we can resolve features does not mean that we have the data to accurately represent/ initialize or verify them!

40. Spectral Methods

41. Background • Primarily used for global NWP, though some regional spectral models do exist • Pioneered in the 1970s in response to limitations in then-state of the art grid-point methods • Singularity and computational inefficiency near the poles • Numerical issues (aliasing, representation of derivatives) • Most current global models (namely the GFS and ECMWF) are spectral models

42. Overview • To first order, spectral methods involve the transform of continuous, dependent atmospheric fields (u, v, T, etc.) from physical into spectral space. • Methods: Fourier series and Legendre functions • This enables us to solve spatial derivatives (in x and y) analytically, wherein the structure of the field is stated in a mathematical form.

43. Example • To illustrate the spectral transform methods, let us first consider a generic one-dimensional field… • Equation (3.5), also below, is a Fourier series representation for a generic field A(x). • The structure of A in the x direction is given as the linear superposition of an infinite number of waves.

44. Example • am and bm are Fourier coefficients (amplitudes) • m is the zonal wavenumber (and is an integer) • k = 2π/L, where L is the domain length

45. Example • Euler’s identity: • If we let θ = mkx, then… where...

46. Example • If we add these expressions, we get… • If we subtract these expressions, we get… • Now, substitute these expressions into (3.5)…

47. Example • Rearrange this result to group like exponentials… • Let us consider the special case of m=0. With m=0, the exponentials are all 1. With simplification, • We’ll let a0 = C0.

48. Example • Now, consider the positive and negative exponential. Combine the fractional coefficients on each, i.e., • We define the coefficients on each as Cm and C-m, respectively (note slight difference from text).

49. Example • This allows us to write: • Multiply the index in the second summation by -1… • Put C0 back into sum form and combine to obtain…

50. Example • Thus, A(x) is represented analytically by the summation of an infinite number of waves. • For meteorological applications, we allow the Fourier coefficients (am, bm, Cm) to vary in time. • We also need to limit the number of waves for computational reasons; this is known as truncation.