# Presentation: 014919

## Model Uncertainty in Ensemble Design:

- I: Time-Step Truncation Error
- II: Stochastic Parameterizations
- João Teixeira(1,2), Carolyn A. Reynolds(1) , Kevin Judd(1,3) and Craig Bishop(1)
-  Naval Research Laboratory, Monterey, CA, USA
-  NATO Undersea Research Centre, La Spezia, Italy
-  University of Western Australia, Perth, Australia

## I:Truncation error: introduction – 3 questions

- 3 models of different levels of complexity to answer the questions:
- 1) What are the consequences in terms of numerical convergence of using different time-steps in chaotic models such as Lorenz63? (not discussed here).
- 2) What are the differences between truncation error (time-step) and initial-condition error in models such as the QG model?
- 3) What is the impact of the truncation error in terms of the predictability of an operational system such as NOGAPS?

- Extreme sensitivity of models to initial conditions
- major source of uncertainty in weather and climate prediction

- What is the sensitivity of chaotic systems to truncation error?

## QG model- Truncation error growth

- Quasi-geostrophic (QG) potential vorticity (PV) model (Marshall and Molteni, 1993)
- T21 horizontal resolution – 3 vertical levels (800, 500 and 200 hPa).
- Forcing - Source terms of PV, varying in space but not in time, NH winter climatology

- Log10 of L2 norm error in non-dimensional stream function state vector: different time-steps versus control time-step of 1.25 min.

- Initial condition error

- Truncation (Δt)  error

- Different growth properties between truncation and initial-condition error

- Regular decoupling – Log relation

## Simple model of truncation error growth in chaotic systems

- Simple model of error growth is constructed, by considering relative error growth rates in the stable and unstable directions:

- σ - rate of decay of stable modes
- λ - sensitivity to initial conditions
- εΔt - typical one-step truncation error

- Assuming                    we obtain:

- Truncation error in non-linear systems is a simple combination of:
- Stable linear error
- Unstable exponential error

## NOGAPS Ensemble – the impact of time-step

- Ensemble spread of 12-h accumulated precipitation (mm) - May 2005                            32 member ensemble, T119L30 resolution
- Solid lines - ensemble with varying initial conditions (IC) but same time-step of 900 s. Dashed lines - ensemble with varying IC and two different time-steps: 900 s and 450 s.

- Tropics (TROP)                     NH extratopics (NHX)    SH extratropics (SHX)

- Spread increases with different time-steps
- Different time-steps may be a simple way of introducing model error in ensemble design

## Truncation error: Summary

- Solutions with different time-steps decouple from each other (time of decoupling depends on log of time-step)
- Numerical convergence in chaotic models is not forever
- 3 stages of truncation error growth in chaotic systems (QG):    1) rapid growth, 2) saturation, 3) exponential growth
- Truncation_error=stable error + unstable error (simple model)
- NOGAPS: truncation error is substantial part of total error
- Using different time-steps is a simple and potentially useful way of introducing model (truncation) error into ensembles

**Notes:**
For NOGAPS different time step ensemble experiments.  There is considerably less sensitivity at the resolution of T119 then at the resolution of T79.  Not sure if we should mention this.

## II: Stochastic nature of physical parameterizations in ensemble prediction

- _In ensemble prediction systems:_
-  Physical parameterizations should be viewed/used as stochastic  (rather than deterministic as usually done)
-  This can be done within the context of current parameterizations (without imposing artificial stochastic terms)
- _Parameterizations:_
-  Typically used to predict the evolution of grid-mean quantities (due to unresolved sub-grid scale processes)
-  Can also provide estimates of higher moments                        (which can be used to constrain the random determination of a variable)

- _For parameterizations: Ensemble prediction is fundamentally different from deterministic prediction_

## Stochastic nature of physical parameterizations in ensemble prediction

- _Parameterizations in ensemble prediction systems:_
- No a priori reason for parameterizations to produce evolution of grid-mean
- Each ensemble member should provide a probable value of a variable
- Random values should be constrained by parameterization PDFs            (e.g. variance given by parameterization)
- Introducing additional stochastic terms may not be necessary

*Parameterizations in ensemble prediction should be used in a stochastic manner, based on PDFs produced by the parameterizations themselves*

## Parameterization problem in weather models 

_*The problem of parameterization*_
- How to better estimate:
- 1) the pdf of each variable        		2) the subgrid flux
-  => estimating

- Temperature (K)

- probability

- pdf of temperature in the grid-square

*longitude*

*latitude*

- Climate and weather prediction models: Δx=Δy~100 km

- Δx

- Δy

## Variance/standard-deviation equations 

- How to determine the standard deviation
- 1) Prognostic equation:
- 2) Diagnostic equation:

- Eddy-diffusivity (PBL)

- Mass-flux (convection)

## Stochastic parameterizations: methodology

- _Methodology for stochastic parameterizations based on the variance_
- A variable after being updated by a parameterization (e.g. moist convection) can be written as:
-         - mean value of the variable after convection
-         - stochastic value after convection
-         - normally distributed stochastic variable with
-                 mean
-                 standard deviation
-         - standard deviation due to moist convective processes
- After discretizing the first term on the rhs, the following equation is obtained
-        -  mean value before the moist convection parameterization

## Stochastic convection: simple approach

- - constant of proportionality

- - normally distributed stochastic variable with mean

-  and standard deviation

- leads to

- Simple vertical correlation:  single random number per column
- No horizontal or temporal correlations:
-  Simpler
-  Perturbations assumed much smaller than grid-size/time-step
-  Parameterization variance already possesses a certain degree of correlation
-  Physically unclear how to construct correlations

- _Assuming standard deviation proportional to convection tendency leads to:_

## Stochastic convection in NOGAPS

- Proposed methodology compared to Buizza et al. (1999):
- Based on using formalism of convection parameterization                           (i.e. variance given by convection parameterization)
- Stochastic approach only for moist convection parameterization
- Assumes that PDF is well approximated by a Normal distribution

- NOGAPS: Navy Operational Global Atmospheric Prediction System - global spectral model in horizontal and energy conserving finite difference in vertical
- NAVDAS: NRL Atmospheric Variational Data Assimilation System

- NOGAPS ensemble system:
- 32-members, 10 day forecasts, T119L30 resolution
- Twp types of experiments:
-  Stochastic convection only
-  Initial condition perturbations only                                                                   (based on randomly sampling NAVDAS analysis error variance estimate)

## Ensemble spread due to stochastic convection only: 500 hPa Z

-  Perturbations grow in time
-  At 24 h: mostly in Tropics/Sub-tropics  – At 144 h: mostly in Mid-latitudes
-  Similar for U at 250 and 850 hPa, T at 850 hPa

**Notes:**
Square root of ensemble variance for 1 case, 32-member ensemble, for 2005050100 (results are typical of other days). For different forecasts times from 24 hours to 144 hours.

## Stochastic convection: ensemble spread

- Two different ensemble experiments are analyzed:
-  Only initial condition perturbations
-  Only stochastic convection perturbations
-  32-Member ensembles, T119l30, May ‘05

- Ensemble spread in Tropics:
- Mean-sea-level pressure (hPa)
- Temperature at 850 hPa (K)
- Wind at 250 hPa (m/s)

- _Initial condition (IC) ensembles: initial decay or slow growth._
- _Stochastic convection (SC) ensembles: spread increases substantially from  start - comparable to initial condition spread for 250 U and SLP_

- TROPICS

## Stochastic convection: ensemble spread

- Two different ensemble experiments are analyzed:
-  Only initial condition perturbations
-  Only stochastic convection perturbations
-  32-Member ensembles, T119l30, May ‘05

- Ensemble spread in 30-90N:
- Mean-sea-level pressure (hPa)
- Temperature at 850 hPa (K)
- Wind at 250 hPa (m/s)

- _Initial condition (IC) ensembles: Some saturation after 10 days_
- _Stochastic convection (SC): Still significant impact in extra-tropics                       Less saturation at 10 day forecast range_

- EXTRA-TROPICS

## Stochastic parameterization in ensemble systems

- _Stochastic nature of parameterizations in ensemble systems: _
- Parameterizations in ensemble prediction should be stochastic
- Stochasticity constrained by parameterization PDF
- _A simple stochastic convection approach:_
- Standard deviation proportional to convection tendency
- Perturbations grow in time + ‘migrate’ (tropics to extra-tropics)
- Tropics: stochastic convection spread ~ init. cond. spread
- NEAR FUTURE...
- Stochastic convection  +  Initial-condition perturbations
- Feasibility of this type of approach at NCEP

## Extra Slides

- Preliminary Results:
-  Stochastic convection + initial condition perturbations.
- May 2005, T119L30, 32-member ensembles.

## STOCHASTIC PARAMETERIZATION OF DEEP CONVECTION

**Stochastic physics enhances ensemble spread, particularly in the tropics.**

## Control (initial perturbations only)		Initial + Stochastic Perturbations

- Control (initial perturbations only)		Initial + Stochastic Perturbations

- On average, stochastic perturbations increase spread in track by 15-20%. Appear to have less of an impact during recurvature.

## Extra Slides

## Non-linear models – Decoupling Log-relation

-  Decoupling time as a function of time-step for 3 models:L63, QG, NOGAPS

- Critical time of decoupling TC: when error for a certain Δt is above a fixed threshold

- Lorenz63

- slope (QG):-16.7

- slope (NOGAPS):-2.9

## NOGAPS – Truncation error growth

- NOGAPS - Navy Operational Global Atmospheric Prediction System
- Global spectral model – resolution:  T79L30
- Parameterizations: boundary layer turbulence, moist convection, clouds, subgrid orography, land-surface, solar and longwave radiation.

- Log10 of L2 norm error in global dry total energy: different time-steps versus control time-step of 60 s. Also shown: operational (T239, 300 s) error (forecast-analysis)

- Different from QG model: initial fast growth towards saturation

- Parameterizations constrain accuracy to first-order

- Truncation error can be a substantial part (40-60%) of total forecast error

## NOGAPS Ensemble – the impact of time-step

- Ensemble spread of 500 hPa geopotential height (m) - T79L30 NOGAPS 20-members.
- Solid lines - ensemble with varying initial conditions (IC) but same time-step of 900 s. Dashed lines - ensemble with varying IC and two different time-steps: 900 s and 450 s.

- Globe                      NH extratopics (NHX)    SH extratropics (SHX)

- Spread increases with different time-steps
- Different time-steps may be a simple way of introducing model error in ensemble design

## Parameterization problem in weather and climate prediction models

- Atmospheric/oceanic model equation for a generic variable can be written as:

- Using Reynolds decomposition and averaging
-  to get an equation for the mean:

- vertical sub-grid flux needs to be _parameterized_

- Where it is assumed that S is linear and the horizontal divergence terms of the sub-grid fluxes can be neglected

## Eddy-diffusivity (ED) approach  

- In ED closure the sub-grid flux is parameterized as
- where k is the diffusivity coefficient. The mixing length approach (e.g. Taylor, Prandtl) is
- where  ***w******t****** ***is a turbulent velocity and ***l ***is a mixing length.
- ED is successful in representing:
- Surface layer (MO theory), momentum mixing
- Neutral/stable boundary layers => Logarithmic-law:

- Surface layer (constant  flux):

- and

- leads to

## Mass-flux (MF) approach 

*Mass-flux* (MF) represents  large-scale eddies

*Eddy-diffusivity* (ED) represents small-scale eddies

- Mass-flux (MF) approach

- MF closure is based on parcel ideas (e.g. Stommel 1947) and tries to represent strong upward/downward convection:

*a* - updraft/downdraft area *w**u/d* - upward/downward vertical velocity in *a            **φ**u/d* - variable value in *a*.

- MF is typically used for parameterization of deep convection

## Variance/standard-deviation equations 

- How to determine the standard deviation
- 1) Prognostic equation:
- 2) Diagnostic equation:

- Eddy-diffusivity

- Mass-flux

## Ensemble spread due to stochastic physics only: 250-hPa U

## Ensemble spread due to stochastic physics only: 850 hPa T

## Ensemble spread due to stochastic physics only: 850 hPa U

## Stochastic convection: Tropics versus Extra-tropics 

- Tropics

- NH Extra-tropics

- Total Energy difference: one ensemble member - control

- NOGAPS stochastic convection after 5 days:
-  Starts to saturate in Tropics
-  Starts to exhibit synoptic (sub-synoptic) peak in NH Extra-tropics

- 1	Total Wave Number           80

- 1	Total Wave Number           80

**Notes:**
Key gives corresponding forecast hour for each color. Lowest curve is for 6 hours, then 12, 24, 48,72,96, and 120 h for curves of increasing magnitude. Note the in contrast to the tropics, the NH extratropics has (1) significant synoptic/subsynoptic scale peak and 2) still has substantial perturbation growth between 96 and 120h.

## STOCHASTIC PARAMETERIZATION OF DEEP CONVECTION

**Ensemble mean error is comparable or slightly larger when stochastic physics included.  Significantly larger for stochastic physics only. (Results for April 2005)**

## STOCHASTIC PARAMETERIZATION OF DEEP CONVECTION

**Number of outliers is decreased by inclusion of stochastic physics, particularly in the tropics (although the number of outliers is still too large).**

## STOCHASTIC PARAMETERIZATION OF DEEP CONVECTION

**In the tropics, addition of stochastic physics improves the spread-skill relationship (minimal impact in the mid-latitudes, not shown).**