Outline Horizontal Discretizations inAtmospheric Models

(1)

Schär, ETH Zürich

Horizontal Discretizations in Atmospheric Models

Christoph Schär ETH Zürich schaer@env.ethz.ch Supplement to Lecture Notes

“Numerical Modeling of Weather and Climate”

April 16, 2007

Schär, ETH Zürich

2 Outline

The pole problem

Regional models

Global spectral models

Other global grid approaches

(2)

Schär, ETH Zürich

The pole problem

The simplest model grid on the sphere is a regular latitude / longitude grid.

This grid suffers from the

“pole problem.”

The poles represent singularities in latitude / longitude grids.

As the grid spacing approaches zero near the poles, there are difficult stability issues (CFL criterion).

Schär, ETH Zürich

4 Outline

The pole problem

Regional models

Global spectral models

Other global grid approaches

(3)

Schär, ETH Zürich

The pole problem in limited-area models

Regular latitude / longitude grids suffer from the “pole problem.”

Latitude / longitude grids can be used at low and mid latitudes

Grids may be defined using map projections (here: polar stereographic projection)

Schär, ETH Zürich

6 Rotated latitude / longitude grids

Regular latitude / longitude grid Rotated latitude / longitude grid

This is a regular latitude/longitude grid based on rotated spherical coordinates.

It has a much more isotropic structure than the unrotated grid!

φ’=0 φ =50

°N

λ ’=0

λ =0

(4)

Schär, ETH Zürich

Outline

The pole problem Regional models Global spectral models

Other global grids

Schär, ETH Zürich

8 The “North pole problem” in a global ocean model

Madec and Imbard, 1996, Clim. Dyn., 12 (6), 381-388 A global ocean mesh to overcome the North Pole singularity (used in the MICOM ocean model and the Bergen climate model)

In ocean models, the Earth’s axes may be bended to place the North Pole over Asia.

This illustrates that the

pole problem is painful

and that there are no

simple solutions. A

more commonly used

approach is based on

the spectral method.

(5)

Schär, ETH Zürich

Spatial discretizations in one dimension

€

φ _i (t) = φ ( iΔx,t )

Local value of function (finite differences)

€

φ _i (t) = 1

Δx φ ( ) x,t

x _i −Δx 2 x _i +Δx 2

∫ ^dx

Mean over grid box (finite volume, finite differences)

€

φ ( ) x,t ⁼ ^ψ m ( ) t ^e ^{i k} ^m ^x

m

∑

Spectral representation (spectral method)

Schär, ETH Zürich

10 Linear advection with the spectral method

Governing equation

Discretized equation

€

φ ( ) x,t ⁼ ψ _m ( ) t ^e ^{i k} ^m ^x

m

∑

With Leapfrog time step

€

∂φ

∂t + u ∂φ

∂x = 0

Discretization

€

∂ψ _m

∂t + i u k _m ψ _m



 



  m=−M

M

∑ ^e ^{i k}

^m

^x ⁼ ⁰

€

∂ψ _m

∂t = − i u k _m ψ _m ( m =− M....M )

€

ψ _m ⁿ⁺¹ = ψ _m ⁿ⁻¹ − 2 i Δt u k _m ψ _m ⁿ

(6)

Schär, ETH Zürich

Comparison against centered finite differences

wavelength in Δx

10 5 3.3 2.5 2

∞

u _num / u

0 0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1 1.2 1.4

Leapfrog Spectral

Correct Solution

wavelength

^–1

in Δx

^–1

Schär, ETH Zürich

12 Nonlinear advection with the spectral method

Governing equation

Discretized equation

€

u x,t ( ) ⁼ ^ψ m ( ) t ^e ^{i k} ^m ^x

m

∑

Using interaction coefficients

Discretization

€

∂u

∂t + u ∂u

∂x = 0

€

∂ψ _m m=−M ∂t

M

∑ ^e ^{i k}

^m

^x ⁺ ^ψ j e ^{i k}

^j

^x j=−M

M

∑













 i k _l ψ _l e ^{i k}

^l

^x l=−M

M

∑













 = 0

€

∂ψ _m m=−M ∂t

M

∑ ^e ^{i k}

^m

^x ⁺ ^ψ j i k _l ψ _l e ^{i k}

^j+l

^x j,l=−M

M

∑ ⁼ ⁰

€

∂ψ _p

∂t = − I _j,l,p ψ _j

j ,l=−M

M

∑ ^ψ l ( p = −M....M)

€

I _j,l, _p = 1

L i k _l e ^{i k}

^j^+l^−p

0 L

∫ ^dx

with

The interaction matrix I j,l,p has size (2M+1) 3 and is prohibitively large!

(7)

Schär, ETH Zürich

Pseudospectral method with

Spectral space

€

u x,t ( ) ⁼ ^ψ m n e ^{i k} ^m ^x

m

∑ ^€

∂u

∂t + u ∂u

∂x = 0

Computation of derivatives

€

∂ ∂x : ψ _m ⁿ → ik _m ψ _m ⁿ

Time step

€

ψ _m ⁿ⁺¹ = ψ _m ⁿ⁻¹ + 2Δt ψ ˜ _m ⁿ

Physical space

€

u ⁿ _j = u x ( _j , t ⁿ )

Computation of tendencies

€

u ˜ ⁿ _j := ∂u

∂t



  

 

j n

= − u ⁿ _j ∂u

∂x



  

 

j

€ n

i k _m ψ _m ⁿ

FT

⁻¹

   → ∂u

∂x



  

 

j n

€

ψ ˜ _m ⁿ ←  _FT  ˜ u ⁿ _j

€

ψ _m ⁿ

FT

⁻¹

   → u ⁿ _j Transformations

Schär, ETH Zürich

14 Spectral methods in different geometries

€

∂ ² f

∂x ² = –µf

€

µ = n 2π L _x



  

 

2 €

f ( x) = e ^ik ⁿ ^x with k _n = n2 π L _x 1D:

0 ≤ x ≤ L

_x

Geometry Eigenvalue problem Eigenvalue Eigenfunction (base functions)

€

∂ ²

∂x ² + ∂ ²

∂y ²





 





  f = – µ f

€

µ = n 2 π L _x



  

 

2 + m 2 π L _y





 





 

2 €

f ( x, y) = e ^{i k} ( ⁿ ^x+l ^m ^y )

= e ^ik ⁿ ^x e ^il ^m ^y 2D:

0 ≤ x ≤ L

_x

0 ≤ y ≤ L

_y

€

∇ ² f = –µf

€

µ = n n ( + 1 )

a ²

€

f ( λ , φ ) = Y _n ^m ( λ , φ )

= e ^imλ P _n ^m [sin(φ)]

Sphere:

Radius a

(8)

Schär, ETH Zürich

Spherical Harmonics

Family Y ³ (order m=3)

Family Y ¹⁰ (order m=10)

€

Y ₃ ³

€

Y ₄ ³

€

Y ₅ ³

€

Y ₆ ³

€

Y ₁₀ ¹⁰

€

Y ₁₅ ¹⁰

€

Y ₂₀ ¹⁰

Schär, ETH Zürich

16 φ λ,ϕ,t ( ) = ψ

nm

( ) t

n=m N(m) m=−M

∑

M

∑ ^Y

ⁿ^m

⁽ ^λ ^,ϕ ⁾

Spectral method on the sphere

Two-dimensional fields φ on the sphere are represented as an expansion using spherical harmonics : Y

_n^m

( λ , ϕ )

€

Y _n ^m ( λ , ϕ ) ⁼ ^e ^{i m} ^λ ^P n m [ sin( ϕ ) ]

€

P _n ^m

Order m and determines variations in λ -direction, degree n variations in ϕ -direction.

= associated Legendre polynomials

n

m n

m triangular truncation

n ≤ M

rhomboidal truncation n ≤ M + |m|

Truncation of series (example M=4)

(9)

Schär, ETH Zürich

Outline

The pole problem Regional models Global spectral models

Other global grids

Schär, ETH Zürich

18 Spherical cube grid

Adcroft et al. 2004, MWR, 132 (12): 2845-2863; William Sawyer, PhD ETH

The spherical cube grid is

formed by projecting a cube

onto the sphere. There are 8

non-orthogonal grid points

corresponding to the corners of

the original cube.

(10)

Schär, ETH Zürich

Geodesic grids

Geodesic grids are formed from an icosahedron by iteratively bisecting the edges and projecting the new points onto the sphere.

Delaunay (triangular) grid Voronoi (hexagonal–pentagonal) grid Resulting

grids

William Sawyer, PhD ETH Bonaventura and Ringler 2005, MWR 133; Ringler et al. 2000, MWR, 128

Schär, ETH Zürich

20 Global Gridpoint Models: Icosahedral Global Mesh

(German Weather Service, DWD)

GME / ICON (DWD)

This grid is constructed from a projection of an icosahedron onto the sphere, and

subsequent refinement of the 20 triangles.

Current operational grid spacing

as used for NWP applications

by the German Weather

Service (DWD): about 60 km

(11)

Schär, ETH Zürich

Global Models with variable Resolution

(CAS 2003, Annecy; Schmidt 1977, 1982, Beiträge Phys. Atmosph.)

Action de Recherche Petite Echelle Grande Echelle

ARPEGE (Meteo France)

Representation on the globe with the spectral approach, but using

- grid stretching (Schmidt transform) - pole rotation

to enhance resolution over the region of interest

Current resolution in NWP applications:

spectral resolution: T

_L

358 stretching factor: C2.4 Resulting grid:

Δ = 23 km (France) Δ = 133 km (antipodes)

SCHMIDT F, BEITR PHYS ATMOS 55 : 335 1982 SCHMIDT F, BEITR PHYS ATMOS 50 : 211 1977

Schär, ETH Zürich

22 Unstructured grids

Unstructured grids allow for the enhancement of coast-lines, topography, etc. They require the explicit definition of each computational cell.

http://vortex.atgteam.com/

(12)

Schär, ETH Zürich

Adaptive grids

Refined topography Refined coasts

Refined meteorology Final grid _D.P.

Bacon et al., 2000, MWR 128, 2044-2076: Operational Multiscale Environment Model with Grid Adaptivity (OMEGA). http://vortex. atgteam .com/

Outline Horizontal Discretizations inAtmospheric Models

Horizontal Discretizations in Atmospheric Models

Christoph Schär ETH Zürich schaer@env.ethz.ch Supplement to Lecture Notes

“Numerical Modeling of Weather and Climate”

April 16, 2007

2

Outline

The pole problem

Regional models

Global spectral models

Other global grid approaches

The pole problem

The simplest model grid on the sphere is a regular latitude / longitude grid.

This grid suffers from the

“pole problem.”

The poles represent singularities in latitude / longitude grids.

As the grid spacing approaches zero near the poles, there are difficult stability issues (CFL criterion).

4

Outline

The pole problem

Regional models

Global spectral models

Other global grid approaches

The pole problem in limited-area models

Regular latitude / longitude grids suffer from the “pole problem.”

Latitude / longitude grids can be used at low and mid latitudes

Grids may be defined using map projections (here: polar stereographic projection)

6

Rotated latitude / longitude grids

Regular latitude / longitude grid Rotated latitude / longitude grid

This is a regular latitude/longitude grid based on rotated spherical coordinates.

It has a much more isotropic structure than the unrotated grid!

φ’=0 φ =50

°N

λ ’=0

λ =0

Outline

The pole problem Regional models Global spectral models

Other global grids

8

The “North pole problem” in a global ocean model

Madec and Imbard, 1996, Clim. Dyn., 12 (6), 381-388 A global ocean mesh to overcome the North Pole singularity (used in the MICOM ocean model and the Bergen climate model)

In ocean models, the Earth’s axes may be bended to place the North Pole over Asia.

This illustrates that the

pole problem is painful

and that there are no

simple solutions. A

more commonly used

approach is based on

the spectral method.

Spatial discretizations in one dimension

€

φ i (t) = φ ( iΔx,t )

Local value of function (finite differences)

€

φ i (t) = 1

Δx φ ( ) x,t

x i −Δx 2 x i +Δx 2

∫ dx

Mean over grid box (finite volume, finite differences)

€

φ ( ) x,t = ψ m ( ) t e i k m x

m

∑

Spectral representation (spectral method)

10

Linear advection with the spectral method

Governing equation

Discretized equation

€

φ ( ) x,t = ψ m ( ) t e i k m x

m

∑

With Leapfrog time step

€

∂φ

∂t + u ∂φ

∂x = 0

Discretization

€

φ _i (t) = φ ( iΔx,t )

φ _i (t) = 1

x _i −Δx 2 x _i +Δx 2

∫ ^dx

φ ( ) x,t ⁼ ^ψ m ( ) t ^e ^{i k} ^m ^x

φ ( ) x,t ⁼ ψ _m ( ) t ^e ^{i k} ^m ^x

∂ψ _m

∂t + i u k _m ψ _m

∑ ^e ^{i k}

^x ⁼ ⁰

∂ψ _m

∂t = − i u k _m ψ _m ( m =− M....M )

ψ _m ⁿ⁺¹ = ψ _m ⁿ⁻¹ − 2 i Δt u k _m ψ _m ⁿ

u _num / u

u x,t ( ) ⁼ ^ψ m ( ) t ^e ^{i k} ^m ^x

∂ψ _m m=−M ∂t

∑ ^e ^{i k}

^x ⁺ ^ψ j e ^{i k}

^x j=−M

 i k _l ψ _l e ^{i k}

^x l=−M

∂ψ _m m=−M ∂t

∑ ^e ^{i k}