an immersed boundary method enabling large-eddy simulations of complex terrain in the wrf model

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551

This work was performed under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

An Immersed Boundary Method Enabling Large-Eddy

Simulations of Complex Terrain in the WRF Model

Performance Measures x.x, x.x, and x.x

Presented at:

University of Utah

April 4, 2012

Katherine A. Lundquist1,2, Fotini K. Chow2

1 Lawrence Livermore National Laboratory2 University of California, Berkeley

Numerical Weather Prediction in Complex Terrain

Vertical coordinate systems used in numerical weather prediction models.

Non-orthogonalorthogonal

sigma, or terrain-followingeta, or “step mountain”

Numerical Weather Prediction in Complex Terrain

Nested grids used in mesoscale models allow integration of physical processes across a range of scales

50 m grid1 km grid 300 m grid

Numerical Weather Prediction in Complex Terrain

Higher grid resolution leads to steeper slopes in complex terrain

Problems with Terrain-Following Coordinates in Complex Terrain

Grid skewness leads to significant errors in horizontal derivatives

• Advection• Diffusion• Pressure gradient

),(),(yxzzyxzzz

httop

httop

Mapping function for coordinate transformation (Gal-Chen and Sommerville ,1975)

New terms are introduced into the governing equations (metric terms)

zzppp z

When discretized, the metric terms have additional truncation errors

)(1 xOxF

xFF

i

ii

Problems with Terrain-Following Coordinates in Complex Terrain

In addition to truncation errors, numerical inconsistencies arise when:

zht

ht

Δzzx

zxx zzz

In the transformed coordinate η, the horizontal derivative is:

A first order finite difference approximation yields:

kixF

, xkiFkiF

),(),1(

zkiiFkiF

xz

),()1,1(

Problems with Terrain-Following Coordinates in Complex Terrain

Numerically inconsistent derivatives are more likely at large aspect ratios

Horizontal grid spacing of 1km with a vertical grid of 50 m leads to an aspect ratio of 20

Moral of the story: Don’t stretch grids too much

Example: 1 km horizontal spacing, 50 m vertical -- max allowed slope is 3 degrees

How can we quantify numerical errors due to terrain-following coordinates? Model fails at steep slopes, but when does solution quality

deteriorate?• What slope?• What grid aspect ratio?

Numerical Weather Prediction in Complex Terrain

Vertical coordinate systems used in numerical weather prediction models.

Non-orthogonalorthogonal

orthogonal

sigma, or terrain-followingeta, or “step mountain”

immersed boundary

Use immersed boundary method • Can eliminate the terrain-following coordinate

transformation• Quantify numerical errors through direct

comparison of the two solutions

Scalar Advection Test Case (Schär et al. 2002)

Domain Set-Up & Initialization Peak Height = 3 km Elevated Velocity Shear Layer at

5 km (terrain is very steep, but isolated)

Inviscid flow- no mixing Stable Stratification

Domain size 300 km x 25 km Grid spacing 1 km x 0.5 km 5th order horizontal and 3rd order vertical

advection scheme is used

Scalar Advection Test CaseGrid Configuration

WRF

IBM-WRF

Grid distortion

Scalar Advection Test CaseVelocity Comparisons

WRF

IBM-WRF

WRF

IBM-WRF

U (m/s) at t = 10000 s W (m/s) at t = 10000 s

Hei

ght

(km

)

x (km)

Hei

ght

(km

)

x (km)

Scalar Advection Test CaseScalar Concentration and Error

Effects of terrain slope in WRF

Change max mountain height

Error = max(|WRF-exact|)Er

ror

Terrain slope (degrees)

Atmosphere At Rest

3D hill, quiescent atmosphere, no forcing 10 degree slope (not very steep!) Stable atmosphere No flow should develop at all

U (m/s)

θ(K)

Atmosphere at RestSpurious Flow Develops

WRF coord. diffusion

WRF horiz. diffusion

IBM-WRF

U (m/s) U (m/s)

θ(K)θ(K)

Max velocity 1.7 m/s

Max velocity 0.28 m/s

Max velocity 3.8 e-5 m/s

Flow Over 3D Hill

Compare WRF and IBM-WRF• max slope of 10, 20, 30 deg, grid aspect ratio 1

Geostrophic pressure gradient forcing No-slip boundary condition Zero flux condition on temperature Run to steady state

Flow Over 3D HillVelocity Difference – WRF vs IBM-WRF

Absolute velocity difference – slice through peak of hill

x (km) x (km)x (km)

10 slope 20 slope 30 slope

Max u diff 1.0 m/s

Max u diff 1.8 m/s

Max u diff 3.1 m/s

ht (

km)

Increased turbulent eddy viscosity

Overwhelms numerical errors Absolute differences between WRF and IBM-WRF decrease

Low viscosity

Hei

ght

(km

)

x (km)

High viscosity

x (km)

Immersed Boundary Method

The effects of the body on the flow are represented by the addition of a forcing term in the momentum equation.

0

2

U

FUpUUtU

The immersed boundary method is a technique for representing boundaries on a non-conforming grid

Immersed Boundary MethodFormulation of the Forcing Term

Source: Peskin (1977)RHStUVFn

IBM was first used by Peskin (1972) and (1977) to simulate blood flow through the mitral valve of the heart.

Direct (or Discrete) Forcing proposed by Mohd-Yusof (1997) and used to model laminar flow over a ribbed channel

where VΩ is the desired Dirichlet boundary condition

FUpUUtU

2

tUV

tUU nnn

1

Immersed Boundary MethodBoundary reconstruction

Boundary is coincident with computational nodes

Boundary effects must be interpolated to

computational nodes

Stair step or nearest neighbor grid

Immersed Boundary Method grid

Immersed Boundary MethodBoundary Reconstruction

Using a ghost cell method, the forcing term is applied within the solid domain.

UΩ2

U2U1

Ughost cell

cellghostdcellghostccellghostbacellghost

d

c

b

a

xzwzwxwwUUUUU

wwww

xzzxxzzxxzzxxzzx

____

2

1

2

1

222

111

222

111

1111

UΩ1

Scalar Fluxes at the Immersed Boundary

Bt FFVt

2

Use the immersed boundary method to impose boundary conditions on temperature, moisture, passive scalars, etc.

Or a flux boundary condition can be imposed with IBM.

fn ˆ

A Dirichlet boundary condition

Immersed Boundary MethodBoundary Reconstruction

Neumann boundary conditions are set by modifying the interpolation matrix to include the boundary condition

δΦ/δnΩ1

Φ2Φ1

Φghost cell

cellghostdcellghostccellghostbacellghost

n

n

d

c

b

a

xzwzwxww

wwww

xzzxxzzxzxzx

____

2

1

2

1

222

111

2222

1111

11

sincoscossin0sincoscossin0

fn ˆ

δΦ/δnΩ2

Idealized Valley Simulations

Thermal Slope flow induced by diurnal heating

Uncoupled simulations with specified surface heating

Coupled simulations using atmospheric parameterizations

WARM

Idealized Valley Simulations

Set-up and Initialization ΔX = ΔY = 200 m, ΔZ ~100 m (U,V,W) = (0,0,0) Stable Potential Temp. 40% Relative Humidity Sandy Loam, Savannah Soil Moisture, 20% saturation

rate Soil Temperature, equal to

atmospheric temperature

Uncoupled Ideal Valley

Integrate from 6:00 to 18:00 UTC

Specified heat flux Zero moisture flux at

surface No atmospheric physics No surface properties Constant t

Uncoupled Ideal Valley Evolution of Potential Temp.

Potential Temp at Valley Center

Coupled Ideal ValleyComparison of Velocity Profiles

Comparison of instantaneous velocity profiles for IBM-WRF (red) and WRF (black)

Coupled Ideal Valley

Fully Coupled Model• RRTM Longwave Radiation• MM5 Shortwave Radiation• MM5 Surface Layer Model• NOAH Land Surface Model

Each atmospheric physics module has been modified to account for the immersed boundary

Coupled Ideal Valley Initialization

Date: March 21, 2007 Location: 36°N, 0°E Sandy Loam, Savannah Soil Moisture, 20% saturation rate Soil Temperature, equal to atmospheric temperature

Coupling IBM to the Atmospheric Physics Models

Surface physics models interact with the lowest coordinate level when terrain-following grids are used

The atmospheric physics models used here are column models

For radiation models the vertical integration limits are modified to exclude any portion of the atmosphere below the terrain

For surface physics a modified reference height is calculated and used with similarity theory

Coupled Ideal Valley Radiation Models

Domain averaged incoming radiation (longwave and

shortwave) differ by less than 0.43% during the simulation.

Spatial variation in radiation at 12:00. Error from IBM coupling is negligible in

comparison to changes with terrain height.

Coupled Ideal Valley Surface Physics

Domain averaged heat flux differs by less than 5.4%, and moisture fluxes by less than 0.74%

Coupled Ideal ValleyLand-Surface Properties

IBM provides boundary condition to both WRF and NOAH simultaneously.

Complex Terrain Owens Valley, CA

Valley terrain can be extremely steep with slopes

of up to 60 degrees.

IBM allows explicit resolution of this terrain.

Verification with Field Campaign Dataset

Joint Urban 2003 Oklahoma City Field Campaign

Source: Allwine and Flaherty (2006)

Joint Urban 2003 Oklahoma City Terrain

Problems with Boundary Reconstruction

Matrix is singular

Flux is prescribed in the incorrect direction

Cannot find eight appropriate neighbors

cellghostdcellghostccellghostbacellghost

d

c

b

a

xzwzwxwwUUUUU

wwww

xzzxxzzxxzzxxzzx

____

2

1

2

1

222

111

222

111

1111

UΩ2

U2U1

Ughost cell

UΩ1

Immersed Boundary MethodBoundary Reconstruction

p

n

nn RR

RRw

max

max

nnn

nn

o Fww

F 1

Inverse distance weighing is an interpolation method developed for scattered data (Franke 1982)

Immersed Boundary MethodBoundary Reconstruction

Inverse Distance Weighting is used for the interpolation which determines the forcing applied at the ghost node to

enforce the Dirichlet boundary condition

p

n

nn RR

RRw

max

max

nnn

nn

image Uww

U 1

imageghost UUU 2

Immersed Boundary MethodBoundary Reconstruction

Inverse Distance Weighting preserves local maximum and minimum values. For Neumann boundary conditions, the probe length must be extended, so that the ‘image’ point is

surrounded by neighbors.

p

n

nn RR

RRw

max

max

nnn

nn

image ww

1

nd probeimageghost ˆ

Verification with Flow Over 3D Hill

Geostrophic pressure gradient forcing No-slip boundary condition Zero flux condition on temperature Run to steady state

VerificationGeostrophic Flow over a 3D Hill

Differences are larger for inverse distance weighing than for trilinear interpolation, however both methods produce accurate results. Inverse distance weighting has the added advantages of being easier to implement and using a flexible interpolation stencil.

VerificationGeostrophic Flow over a 3D Hill

Changing the vertical grid in WRF produces much larger differences than those seen between WRF and IBM-WRF

Joint Urban 2003 Oklahoma City Terrain

Joint Urban 2003 Oklahoma CityNested Domain

Mesoscale models are usually run in a nested mode. Here the mesoscale domain is nested down to an urban domain,

Joint Urban 2003 Oklahoma CityOne-way Nested Domain

One-way nesting is used to run the Oklahoma City domain within a channel flow simulation

Parent Domain Nested Domain

Joint Urban 2003 Oklahoma CitySet-Up and Parent Domain Flow

• IOP 3• Outer Domain: ΔX, ΔY = 6 m and ΔZ is stretched form 1 to 3 m• Inner Domain: ΔX, ΔY = 2 m and uses the same ΔZ• Δt = 1/20 s on the outer domain, and 1/60 s on the inner domain• Domains are run in concurrent mode• No atmospheric physics• Static Smagorinsky closure

Joint Urban 2003 Oklahoma City Instantaneous Velocity Field

IBM-WRF (LES)

FEM3MP (RANS)

Joint Urban 2003 Oklahoma City Verification with Observations

IBM-WRF

FEM3MP

Joint Urban 2003 Oklahoma City Verification with Observations

FACx = Predictions within a factor of XFB = Fractional biasMG = Geometric mean biasNMSE = Normalized mean square errorSAA = Scaled average angle

Joint Urban 2003 Oklahoma City Verification with Observations

IBM-WRF

FEM3MP

Joint Urban 2003 Oklahoma City Verification with Observations

FACx = Predictions within a factor of XFB = Fractional biasMG = Geometric mean biasNMSE = Normalized mean square error

Summary

An immersed boundary method was developed in WRF that eliminates numerical errors caused by terrain-following coordinates

Errors arising from terrain-following coordinates were quantified

Two different interpolation cores were developed for use in the IBM, each with different strengths

The IBM has been verified for use in both 2D and 3D terrain through canonical cases

JU2003 was used for verification for real urban terrain Atmospheric physics parameterizations have been

coupled to the IBM to provide surface fluxes of heat and moisture

Immersed Boundary MethodNearest Neighbor Algorithm

A) Numerical instabilities in previous IBMs are avoided by choosing the nearest neighbors of an image point (instead of the ghost point)

B & C) By choosing boundary points (and not ghost points) as nearest neighbors the solution for the ghost point is independent. Iterative procedures used in previous IBMs are not necessary.

D) With isobaric coordinates this point often moves between the fluid and solid domain. Flexibility is added to the algorithm, so that a fluid point in close proximity to the boundary can be a ghost point.