A baroclinic instability test

case for dynamical cores

of GCMs

Christiane Jablonowski (University of Michigan / GFDL)

David L. Williamson (NCAR)

AMWG Meeting, 3/20/06

Overview

• Basic Idea & Design Goals

• Derivation of the test case & discussion of the initital conditions

• 4 dynamical cores: NCAR CAM3 & DWD’s GME

• Results of the test case

– Steady-state conditions

– Evolution of the baroclinic wave

– Uncertainty of the ensemble of reference solutions

• Conclusions

The Idea & Design Goals

• Goal: development of a dynamical core test case (without physics, dry & prescribed orography) that – is easy to apply

– is idealized but as realistic as possible

– gives quick results

– starts from an analytic initial state, suitable for all grids

– triggers the evolution of a baroclinic wave

• Designed for primitive equation models with pressure-based vertical coordinates (hybrid, sigma or pure pressure coordinate)

Derivation of the Initial Conditions

• Initial data required: u, v, T, ps, s

• Find a steady-state, balanced solution of the PE eqns: prescribe u, v and the surface pressure ps

• Plug prescribed variables into PE equations and derive the– Geopotential field : based on the momentum equation for v

(integrate), calculate surface geopotential s

– Temperature field: based on the hydrostatic equation

dv

dtu2 tana

1

a

RdT ln p

fu

RdT

p

p

'

RdT

'

Initial Conditions• v = 0 m/s

• ps = 1000 hPa

• u & s

Initial Temperature Field

Perturbation

• Overlaid perturbation (at each model level) triggers the evolution of a baroclinic wave over 10 days

• Suggested: pertubation of the zonal wind field ‘u’ orthe vorticity and divergence (for models in - form)

Characteristics of the Initial Conditions

• Instability mechanisms:– Baroclinic instability - vertical wind shear

– Barotropic instability - horizontal wind shear

• But:– Statically stable

– Inertially stable

– Symmetrically stable

Characteristics

• Static stability

Characteristics

• Symmetric stability & Inertial stability

Test Strategy

• Initialize the dynamical core with the analytic initial conditions (balanced & steady state)

• Let the model run over 30 days (if possible without explicit diffusion)

• Does the model maintain the steady state?

• Perturb the initial conditions with a small, but well-resolved Gaussian hill perturbation

• 10-day simulation: Evolution of a baroclinic wave

Step 1:

Step 2:

Model Intercomparison

• Eulerian dynamical core (EUL), spectral

• Semi-Lagrangian (SLD), spectral

• Finite Volume (FV) dynamical core (NCAR/NASA/GFDL)

• Icosahedral model GME (2nd order finite difference approach)

NCAR CAM3:

German Weather Service (DWD):

Resolutions

• Default: 26 vertical levels (hybrid) with model top ≈ 3hPa.In addition: 18 and 49 vertical levels were tested.

• EUL & GME: varying horizontal diffusion coefficients K4, no explicit diffusion in SLD and FV simulations.

EUL / SLD (truncation)

FV (lat x lon, degrees)

GME(min. grid distance)

T21 4 x 5 ≈ 440 km (ni = 16)

T42 2 x 2.5 ≈ 220 km (ni = 32)

T85 1 x 1.25 ≈ 110 km (ni = 64)

T170 0.5 x 0.625 ≈ 55 km (ni =128)

T340 0.25 x 0.3125 ≈ 26 km (ni = 256)

Steady-State Test Case

• Maintenance of the zonal-mean initial state (u wind)

Wave number 5 effect

Decentering

parameter effect,with =0 EUL is matched

Steady-State Test Case: GME• GME shows a truncation error with wave number 5

• Artifact of computational grid and low-order num. method

Steady-State Test Case as a Debugging Tool

• Discovery of a flaw in the SLD dynamical core (old CAM2 version):

• systematic decrease in the zonal wind speed over time

• Here: plotted for 30 days with an older version of the test case (umax = 45m/s)

Baroclinic Waves: 30-day Animation

Movie: Courtesy ofFrancis X. Giraldo, NRL, Monterey

Surface pressure

Surface temperature

North-polar stereographicprojection

Evolution of the Baroclinic Wave

• Perturbation gets organized over the first 4 days and starts growing rapidly from day 6 onwards

FV 0.5 x 0.625 L26 dycore

T (850 hPa)ps

Evolution of the Baroclinic Wave• Explosive cyclogenesis after day 7• Baroclinic wave breaks after day 9

FV 0.5 x 0.625 L26 dycore

T (850 hPa)ps

Convergence with Resolution• Surface pressure starts converging at 1 x 1.25 degrees

FV L26 dycore, Day 9

Model Intercomparison at Day 9

• Second highest resolutions, L26

• ps fields visually very similar

• Spectral noise in EUL and SLD

Model Intercomparison at Day 9

• ps fields visually almost identical

• Differences only at small scales

850 hPa Vorticity at Day 7

• Differences in the vorticity fields grow faster than ps diff.

850 hPa Vorticity at Day 9• Small-scale differences easily influenced by diffusion• Spectral noise in EUL and SLD (L26)

Impact of explicit diffusion• EUL T85L26 with K4 increased by a factor of 10 (1 x 1016 m4/s)• No spectral noise, but severe damping of the circulation

Model Intercomparisons: Uncertainty

• Estimate of the uncertainty in the reference solutions across all four models using l2(ps)

Single-Model Convergence• Single-model uncertainty stays well below the uncertainty

across models

• Models converge within the uncertainty for the resolutions T85 (EUL & SLD), 1x1.25 (FV), GME (55km / ni=128)

Uncertainty in the Relative Vorticity• Estimate of the uncertainty in the reference solutions

using l2[ (850 hPa)]

• Errors grow faster, but conclusions are the same

Vertical Resolutions• Model runs with 18 and 49 levels at mid-range horizontal

resolutions are compared to the default 26-level runs

• Uncertainty stays well below the uncertainty across the models

Phase Errors• Phase errors diminish with increasing resolutions

• Phase error at lower resolutions is substantial for GME, attributes to the relatively ‘late’ convergence of GME (55 km)

Energy Fixer: SLD Dynamical Core

• Baroclinic wave test revealed problem in the energy fixer of the SLD dynamical core (old CAM2 version)

SLD problem with the energy fixer

corrected

Conclusions

• Goal: Development of an easy to use baroclinic wave test case that serves as a debugging tool and and fosters model intercomparisons

• Test of the models as they are used operationally (no extra diffusion, no special tuning of parameters)

• Established an ensemble of reference solutions and their uncertainty

• Models converge within the uncertainty at the resolutions EUL & SLD T85, FV 1 x 1.25, GME (55km/ni=128)

• Convergence characteristics the same for T or variable• Accessibility: We make the ensemble of solutions (ps)

available to all interested modeling groups and offer to compute l2(ps) norms

Publications

• Jablonowski, C. and D. L. Williamson, 2006a: A Baroclinic

Wave Test Case for Dynamical Cores of General Circulation

Models: Model Intercomparisons, NCAR Technical Note TN-

469+STR, 89 pp. (available online at

http://www.library.ucar.edu/uhtbin/cgisirsi/TRm6NSmtE3/0/261320020/503/7631

(shortly: also available on NCAR’s CAM3 web page)

• Jablonowski, C. and D. L. Williamson, 2006b: A Baroclinic

Instability Test Case for Atmospheric Model Dynamical Cores,

Quart. J. Roy. Meteor. Soc., in review