Micro-scale Modelling: examination of two different approaches. - - PowerPoint PPT Presentation

• Micro-scale Modelling: examination of two

different approaches.

Sarah-Jane Lock, Alison Coals, Alan Gadian and Stephen Mobbs NCAS, University of Leeds The presentation will examine:- 1. some of the latest approaches using the terrain following Gal Chen (and immersed boundary) approaches using preconditioning techniques .. (first 5 minutes of the talk) 2. Some of the latest work from Sarah-Jane Lock and Alison Coals, who prepared these later slides, with the cut cell devleopments .. (last 7 minutes of the talk)

• For NWP and atmospheric modelling, grid point (finite difference) and spectral

methods using some sort of Gal-Chen (height) or Sigma (pressure) co-ordinate systems have been dominant over the past 50 years. (Phillips, 1956, 2 level GCM model) There are now challenges to this status-quo, partly due to the advent of the rise

• f new engineering cfd approaches, and partly because essentially the

atmospheric models have used a “vector” type approaches, which are not compatible with the “massively parallel” type architectures now being constructed. Different grid ideologies, such as hexagonal grids, unstructured grids are also knocking at the door. (ICON) Engineering (CFD) codes are being used to look at atmospheric flows (e.g. at Southampton) These are ideal for neutrally (i.e. unstratified) flows or heated (i.e . unstably stratified) flows. The argument has been, to date that perhaps they are not ideally suited to examine dispersive waves (i.e., stably stratified non-hydrostatic flow situations), which are found in the atmosphere, whether NWP or e.g. cloud process studies. The jury is still out (in my opinion!) The work at Leeds has examined the used of cut cell technology to represent the lower boundary. This has started with the work of David Woodhead (& Stephen Mobbs), and more recently with Sarah-Jane Lock and Alison Coals ( and I ) This talk will look at results from the Smolarkiewicz model and the cut cell work.

• There are questions,

whether with good newer iterative methods, for the Helmholtz equation, and even with terrain following co-ordinate system, results can be obtained which are good.

• !

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Plan of Gillygate, York - Dashed outline defines the model domain. Two lamp-posts G3 and G4 are marked on the diagram

Set-up for Gillygate experiment: (Dixon et al, 2005, Atmos Env ) These results use the Smolarkiewicz model, 256 processors on HPCx

periodic domain,

with uniform z0 = 0.1m

grid-boxes: 231 x 261 x 60,

for dx = dy = dz = 1m

time: 1200 time steps,

for dt = 0.025s

model spin up ~ 30s - results computed

for 30s run time – it is anticipated that the spin up and run times need to be longer for statistically significant outputs

Rayleigh damping sponge above 50m Neutral, (constant potential

temperature), u0= 5ms-1 from right to

• left. The periodic boundaries develop a

logarithmic type surface layer

• #$%

Upper left frame is Vertical velocity. All other frames are concentra- tions. The two lamp-posts are marked.

• &

Exploring a cut-cell approach for model simulations of flow over hills in a microscale model

• S. Lock, A. Coals, A. Gadian

University of Leeds

With particular thanks to H.-W. Bitzer, U. Schaettler, J. Steppeler DWD, Germany

• '
• Background to cut-cell approach
• Method(s) in the microscale model
• Results from microscale model for idealised flows
• Future work - microscale model and beyond!

• (

Adcroft, Marshall, Hill (1997):

• explored shaved cells in representing

irregular topography in an ocean model

• 2 approaches: “partial steps” and “piecewise

linear” (illustrated)

• proposed a finite-volume approach to solve

flow in shaved cells

• solved 2D Boussinesq equations in flux-form

=> Gauss’s divergence theorem applied to fluxes in a (grid) volume

• concluded piecewise linear gives a superior

representation, especially at course resolutions

Figure taken from Adcroft et al. (1999

• )
• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred spatial

• grid:

Arakawa-C (horizontal) Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

• *
• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred spatial

• grid:

Arakawa-C (horizontal) Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

Prognostic variables: u v wind velocity w π ’ Exner pressure pert. θ ’ potential temp. pert. q (moist version) Diagnostic variables: ρ density Π Exner pressure field Θ potential temp. p pressure T in situ temperature

• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred spatial

• grid:

Arakawa-C (horizontal) Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

Based on Klemp & Wilhelmson (1978): Equations take the form where sφ = fast modes fφ = slow modes n n+1 Update slow modes on long time-step (∆t)… …and fast modes

• n intermediate

short steps (∆τ)

• !"

LHS = fast modes: acoustic & gravity RHS = slower modes:

• inc. advection, Coriolis, …

(as in KW78) Fast/slow mode split - as for COSMO & WRF models, & Cullen (1990)

• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred spatial

• grid:

Arakawa-C (horizontal) Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

Time-differencing: For the long step, 2nd-order leapfrog

• inc. Robert-Asselin filter

For the short step, 1st-order forward- backward t t-∆t t+∆t

• !
• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred space

• grid: Arakawa-C (horizontal)

Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

Arakawa-C grid (horizontal): Charney-Phillips grid (vertical): x y x z

• equation set:

3D, Cartesian, nonhydrostatic, fully compressible

• prognostic/diagnostic variables
• time-stepping:

time-splitting method

• numerical schemes:

fully explicit (horizontal & vertical) 2nd-order leapfrog 2nd-order centred spatial

• grid: Arakawa-C (horizontal)

Charney-Phillips (vertical)

• lower boundary:

COSMO-DE cut-cell approach

Cut-cell approach:

• Vertical levels remain horizontal
• Orography cuts through grid-cells -

terrain-intersecting

• Orographic surface represented by

continuous bilinear function

From Steppeler et al. (2006)

• Finite-volume

method used to compute divergence term in cut-cells

• Approx. zero-

normal-flow condition

• &

Orography set-up:

• Cartesian grid, i.e. horizontal vertical levels
• define orography height at column-centres
• interpolate orography heights at column-corners
• define unique continuous piecewise bilinear

bilinear surface from four column-corner heights

• 3 types of grid-cell:

i) “pure Earth” cells ii) “pure air” cells iii) “cut-cells” treated with finite-volume approximation

• compute “weights” associated with relative air surfaces/volumes in grid-cells
• relatively simple since bilinear function means intersections are straight lines
• “weight” = ratio of area/volume in air to area/volume of full grid-cell

Microscale model cut-cell approach:

From Steppeler et al. (2006)

• '

2D/3D bell-shaped hill: hydrostatic flow (a=10km) 2D bell-shaped hill: benchmark case, a=1km 2D bell-shaped hills: increasingly narrow/steep hills

• (

2D/3D bell-shaped hill: hydrostatic flow (a=10km)

• Gallus & Klemp (2000) - benchmark test case
• Stratified flow, N = 0.01s-1, U = 10ms-1
• Hill: Gaussian, H = 400m, a = 10km, ∆x = 2km
• 2D results compared with analytical solution, step-approach

models (Gallus & Klemp, 2000) & COSMO models

• 3D results compared with COSMO (terrain-following & cut-cell)

2D bell-shaped hill: benchmark case, a=1km 2D bell-shaped hills: increasingly narrow/steep hills

• )

#$%+*

1) Microscale model, ∆z = 200m 2) Analytical solution (smooth hill) 4) Step approach, ∆z = 200m 3) Step approach, ∆z = 10m

Figures 2,3,4 from Gallus & Klemp (2000)

• *

#$%+*

Figures 2,3 from Steppeler et al. (2002)

1) Microscale model, ∆z = 200m 2) COSMO model, cut-cell method 3) COSMO model, terrain-following

• &$%+*

Figures 2,4 from Bitzer & Steppeler (2004)

1) Microscale model, xz slice (centre) 3) Microscale model, xy slice (z=1200m) 2) COSMO-DE, xz slice (centre) 4) COSMO-DE, xy slice (z=1164m

• 2D/3D bell-shaped hill: hydrostatic flow (a=10km)

2D bell-shaped hill: benchmark case, a=1km

• Gallus & Klemp (2000) - benchmark test case
• Stratified flow, N = 0.01s-1, U = 10ms-1
• Hill: Gaussian, H = 400m, a = 1km, ∆x = 200m
• Results: comparison with analytical solution and step-approach

models (Gallus & Klemp, 2000) 2D bell-shaped hills: increasingly narrow/steep hills

• +

2) Analytical solution (smooth hill)

Figures 2,3,4 from Gallus & Klemp (2000)

1) Microscale model, ∆z = 200m 3) Step approach, ∆z = 10m 4) Step approach, ∆z = 200m

• !

2D/3D bell-shaped hill: hydrostatic flow (a=10km) 2D bell-shaped hill: benchmark case, a=1km 2D bell-shaped hills: increasingly narrow/steep hills Continuing with similar set-up for steeper, narrower hills:

• Stratified flow, N = 0.01s-1, U = 10ms-1
• Hill: Gaussian, H = 400m, a = <1km, ∆x = 0.2a

• !

Exploring steeper gradients:

• What gradients is microscale model capable of?
• Rosatti, Cesari & Bonaventura (2005) & Yamazaki & Satomura (2008) - 2D

idealised flows over semi-circular obstacles (i.e. flow up ~vertical walls) Microscale moisture (A. Coals):

• schemes complete for latent heat exchange for water vapour <-> liquid

transitions (based on LEM schemes)

• results for moist (no orography) idealised tests agree well with benchmarks
• looking for good idealised moist orographic flows for validation

Longer term (Sarah-Jane Lock):

• incorporate a no-slip surface boundary condition
• explore turbulence schemes

Cut-cells in WRF (NCAR DeveL/ Testbed Center) (Sarah-Jane Lock):

• January 2009 - 5 weeks at NCAR to implement cut-cells
• test against traditional WRF, & with real data (e.g. COPS)

• &

'(%)*%#++#"

Dry simulation: results

VHREM max = 2.1K Wmax = 14.5 m s-1 Bryan &Fritsch (Fig.1, MNW 2002) max = 2.0K Wmax = 14.8 m s-1

• '

THANK YOU!