Stochastic subgrid scale modeling

Transcription

1 Stochastic subgrid scale modeling Daan Crommelin Centrum Wiskunde & Informatica, Amsterdam joint work with Eric Vanden-Eijnden, New York University NDNS+, 15 april 21 p. 1/24

2 The parameterization problem: how to incorporate the effect of unresolved (subgrid scale) processes on resolved model variables NDNS+, 15 april 21 p. 2/24

3 A well-known problem in atmosphere-ocean science Example: Convection processes and cloud formation: scales < 1 km Weather forecasting models: grid scale 1 4 km Climate models: grid scale 1 km Other parameterizations: radiation transfer, boundary layer turbulence, surface fluxes, subgrid orography, gravity wave drag,... NDNS+, 15 april 21 p. 3/24

4 Spatial scales in atmospheric flow Courtesy of Pier Siebesma, KNMI NDNS+, 15 april 21 p. 4/24

5 Model resolution Courtesy of Frank Selten, KNMI NDNS+, 15 april 21 p. 5/24

6 Stochastic parameterization For given macrostate, range of subgrid scale responses instead of just one Subgrid scales not always diffusive or damping ("backscatter") Underdispersive ensembles NDNS+, 15 april 21 p. 6/24

7 Incomplete model d X = Ẋ = F(X,B) dt X(t): state vector B(t): fluxes / subgrid scale stresses /... For example: Ẋ k = f(x k,x k±1 ) + B k k: grid point index Parameterization / closure: need model for B NDNS+, 15 april 21 p. 7/24

8 Construct deterministic function B = B(X) (systematic derivation / physical arguments / ad hoc / data driven). Computational approach: B = B(Y ), Ẏ = ε 1 g(x,y ). To reduce computational cost, use short simulations of Y with X fixed to produce B. Relies on ε 1. Model B as stochastic process conditional on X. Data-driven: infer process from (X,B) data. NDNS+, 15 april 21 p. 8/24

9 Data-driven stochastic parameterization: Assumptions: (i) B is a Markov process (ii) the process is conditional on X Use conditional transition probability P(B(t + t) B(t),X(t)), inferred from (X,B) data, to evolve B in time Variations on a theme: e.g. P(B(t + t) B(t),X(t),Ẋ(t)), etc. (C. and Vanden-Eijnden 28) NDNS+, 15 april 21 p. 9/24

10 For variables on a lattice, X = (X 1,X 2,...,X K ) and B = (B 1,...,B K ): Use P(B k (t + t) B k (t),x k (t)) instead of P(B(t + t) B(t), X(t)) for tractability i.e., replace K-dim process B by collection of 1-dim. processes B k NDNS+, 15 april 21 p. 1/24

11 Algorithm: Given X(t),B(t) (i) integrate Ẋ = F(X,B) from t to t + t with B fixed at B(t) (ii) obtain B(t + t) by sampling from distribution P(B k (t + t) B k (t),x k (t)) for each k separately Practical implementation: small Markov chains for B k Hybrid stochastic-deterministic model, ODEs for X coupled to Markov chain(s) for B. NDNS+, 15 april 21 p. 11/24

12 Test case: the Lorenz 96 model Nonlinear, forced-dissipative model on a 1-dim. periodic lattice. Often used for parameterization and predictability studies in atmospheric science. Ẋ k = f(x k,x k±1,x k 2 ) + B k B = B(Y ) Ẏ = ε 1 G(Y,X) with X R K, Y R K J NDNS+, 15 april 21 p. 12/24

13 X: "large scale" variables Y : "small scale" variables model parameters: ε =.5 (i.e., no real time scale separation), K = 18, J = 2 Aim: construct closed model for X by parameterizing B NDNS+, 15 april 21 p. 13/24

14 B k versus X k (statistically identical for all k) B k X k NDNS+, 15 april 21 p. 14/24

15 Deterministic: fit curve through (X k,b k ) data B k X k NDNS+, 15 april 21 p. 15/24

16 Stochastic: B k jumps between s, using P(B k (t + t) B k (t),x k (t),ẋ k (t)) B k X k NDNS+, 15 april 21 p. 16/24

17 Comparison: CMC: Conditional Markov chain scheme DTM: deterministic, B k (t) = g(x k (t)), curve fit for g(x k ) AR1: B k (t) = g(x k (t)) + ξ(t) with ξ(t) an AR(1) process fitted to timeseries of B k (t) g(x k (t)) NDNS+, 15 april 21 p. 17/24

18 Probability density functions.1 CMC L DTM L Probability density AR1 L X k NDNS+, 15 april 21 p. 18/24

19 Autocorrelation functions 1.5 CMC L DTM L ACF 1.5 AR1 L time NDNS+, 15 april 21 p. 19/24

20 Wave amplitudes and variances (X FFT U) Mean wave amplitude < u m > L96 CMC DTM AR1 Variance < u m <u m > > L96 CMC DTM AR Wavenumber m Wavenumber m NDNS+, 15 april 21 p. 2/24

21 Ensemble integrations (5 members): RMSE and anomaly correlation RMSE CMC 2 DTM AR Lead time Anomaly Correlation CMC DTM AR Lead time NDNS+, 15 april 21 p. 21/24

22 Ensemble integrations (2 members): Rank histogram.25 CMC.25 DTM.25 AR NDNS+, 15 april 21 p. 22/24

23 Outlook: Parameterization of atmospheric convection Joint project with KNMI and TU Delft (Selten, Siebesma, Jonker) Why convection / clouds? "Cloud feedbacks remain the largest source of uncertainty in climate sensitivity estimates" (IPCC 4th Assessment Report, 27) NDNS+, 15 april 21 p. 23/24

24 From L96 to realistic convection L96: at each gridpoint k, scalar X k and B k Convection: at each k, 5 functions X k (vertical profiles for velocities, temp. and humidity) and B k (turbulent fluxes) Data: generated with Large Eddy Simulation (LES) NDNS+, 15 april 21 p. 24/24