## A Proposal for the Intercomparison of GCM Dynamical Cores with Stochastic Perturbations

with Stephan Juricke, Peter Düben, Tim Palmer

## Motivation

• Large-scale numerical models use an approximation: Deterministic primitive equations are discretized and solved on grids.

• We don’t often account for this uncertainty in our models.

• Represent the inherent uncertainty in our solutions as stochastic terms.

• Stochastic terms also represent physics (e.g. Brownian motion)

## Exact equations for a GCM

The exact equations are known except for some caveats.

## Approximations

 Spherical geoid Quasi-hydrostatic Anelastic Shallow atmosphere

## Dynamical Cores

Dynamical core intercomparison to evaluate their performance should also consider their inherent uncertainty.

## Sources of uncertainty

• Parameterized diffusion
• Numerical discretization
• $$\frac{\partial u_j}{\partial t} + \frac{\phi_{j+1} - \phi_{j-1}}{2\Delta x} = 0$$

• Choice of grids

• Physics tendencies of dynamics, orography, sub-grid processes.

## Our proposal

• In our inter-comparison test cases, include uncertainty representation in the dynamic tendencies.

• This uncertainty can be represented as an additive stochastic term in the equations

• Dynamical cores which differ in performance despite this representation of uncertainty can be evaluated to perform differently.

## Held-Suarez Testcase

• 1500-day integrations : assessment of climate statistics
• Simplified physics - temperature relaxation function and Rayleigh friction for the wind in lower levels.

## Simplified physics

$$Q = -\kappa(T-T_{eq})$$

• Warmer than Teq => cooling
• Cooler than Teq => warming

Model initialised at T = 255 K uniformly everywhere.

U, V = 0 everywhere. Resolution: T159.

## Zonal winds and temperature

Mean winds and temperature over last 1000 days of 1500 days simulation

## Generic form of Explicit Diffusion Mechanism

$$F_\psi = (-1)^{q+1} K_{2q}\nabla^{2q}\psi$$

where $q>1$ is a positive integer, 2q - the order of the diffusion, $K_{2q}$ - diffusion coefficient and $\nabla$ is the gradient operator.

We now compare the use of biharmonic (q=2) and hyper (q=4) diffusion in a Held-Suarez testcase.

## Zonal wind comparison

Uncertainty in the diffusion operator translates into uncertainty in model solution

## Meridional Eddy Heat Transport: $\overline{V'T'}$

Uncertainty in the diffusion operator translates into uncertainty in model solution.

## Kinetic Energy Spectra

700 hPa Kinetic energy spectra (bi-harmonic vs hyper diffusion)

The kinetic energy spectra are comparable except at large wavenumbers, the hyper diffusion has lesser energy.

## Stochastic parameterization to represent model error

The net parameterized physics tendency: $$𝑿= 𝑋_U, 𝑋_𝑉, 𝑋_𝑇, 𝑋_𝑄$$ are perturbed with multiplicative noise $$𝑿′ = (1 + \mu 𝑟)𝑿$$

SPPT is currently implemented at ECMWF, NCEP, NCAR, UKMO, JMA and others.

## SPPT Zonal Wind Tendencies

Stochastic tendencies of zonal and meridional wind saved from an aquaplanet simulation of ECMWF IFS

## Additive stochastic forcing for dynamical core

$$\frac{\partial V}{\partial t} = - U . \nabla(V) + Mixing + \sigma \mathcal{N}(0,1)$$


!===================================================================================
!     Stochastic Perturbations to U, V
!===================================================================================

IF (LHELDSUAREZ_STOCH) THEN
!    WRITE(NULOUT,*) 'Starting Loop for stochastic forcing IBL:',IBL
!    CALL FLUSH(NULOUT)
DO JK=1,KLEV
DO JL=KIDIA,KFDIA
PTENU(JL,JK) = PTENU(JL,JK) + TRLXQ(JL,JK,1,IBL)
PTENV(JL,JK) = PTENV(JL,JK) + TRLXQL(JL,JK,1,IBL)
ENDDO
ENDDO
!    WRITE(NULOUT,*) 'Stoch update sucessfully finished'
!    CALL FLUSH(NULOUT)
ENDIF


## Zonal wind comparison

Stochastic (SPPT) perturbations to dynamical tendencies do not change the mean climate significantly.

## Eddy Heat Transport: $\overline{V'T'}$

Stochastic (SPPT) perturbations to dynamical tendencies do not change the eddy variability significantly either.

## Kinetic Energy Spectra

Kinetic energy spectra (700 hPa)

Stochastic perturbation run (right) comparable to the deterministic run (left)

## Stochastic perturbations to diffusion coefficient

$$F_\psi = (-1)^{q+1} K_{2q}\nabla^{2q}\psi$$

where $q>1$ is a positive integer, 2q denotes the order of the diffusion, $K_{2q}$ stands for the diffusion coefficient and $\nabla$ is the gradient operator.

Stochastic diffusion coefficient is given by

$$F'_\psi = (1 + r \mu) F'_\psi$$ where r and $\mu$ are the random perturbations and standard deviation terms.

## Zonal wind comparison

Stochastic perturbations to diffusion coefficient do not change the mean climate significantly either. Same order of magnitude as using a higher order diffusion

## Eddy Heat Transport: $\overline{V'T'}$

Stochastic perturbations to diffusion coefficient do not change the eddy variability significantly either. Same order of magnitude as using a higher order diffusion

## Kinetic Energy Spectra

Kinetic energy spectra (700 hPa)

Stochastic perturbations to $F_\psi$ (right) is comparable to the deterministic run (left)

## Summary

• Stochastic representation of uncertainty in dynamical tendencies from an operational weather forecasting model was used in an idealized test case.

• Changes in model mean and eddy variability due to stochastic forcing is of similar magnitude as using a different diffusion scheme

• We propose that uncertainty in dynamical tendencies be represented as "an additive stochastic term" in all dynamical cores in an inter-comparison study

• This will help evaluate the performance of dynamical cores in these tests in the presence of inherent uncertainty in the system

• Dynamical cores which differ in evaluation metrics despite this representation of uncertainty can be identified as performing differently

## More accurate weather prediction with less precision

• The stochastic chip / reduced precision emulator is used on 50% of numerical workload: All floating point operations in grid point space.
• All floating point operations in the Legendre transforms between wavenumbers 31 and 85
• Imprecise T85 cost approx that of T73
Thank you

Many thanks to: Sylvie Malardel, Filip Vana, Linus Magnusson (ECMWF)

## Dynamical Core of IFS

• Hydrostatic primitive equations
• Reduced Gaussian grid (Horizontal) & Hybrid in vertical
• Spectral with a semi-Lagrangian, semi-implicit time discretisation
• Uniform spherical geometry and finite-element vertical discretization
• Horizontal Diffusion : dampen energy and enstrophy accumulation, absorb vertical waves at top of model, subgrid-scale mixing

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