# Presentation: 091390

## Can we expect to forecast Weeks 3 and 4?

- Matt Newman
- CIRES/Climate Diagnostics Center, U of Colorado and NOAA/ESRL/PSD
- Collaborators: Prashant Sardeshmukh, Cécile Penland, Philip Sura, Chris Winkler, Jeff Whitaker

**NOAA 32nd Annual Climate Diagnostics and Prediction Workshop, Tallahassee**

## Outline of talk

- Study of the mechanics of the atmosphere
- Search for elusive empiric periodicities

## Daily vs. extended range forecast skill

- Most skill is lost for “short-waves” for a 5-day forecast.
- “long-waves” skillful well into week 2.
- Obvious point: Weeks 3 and 4 forecasts will mostly be of large-scale dynamics

- Forecast skill

- Figure courtesy Jeff Whitaker

## Teleconnections: high-frequency vs. low-frequency

-  The maps show the
-  correlation of the “band-pass
-  filtered” 500 mb anomaly
-  height time series at all
-  points on the hemisphere with
-  the time series at a
-  north Pacific “base point”.
-  Positive correlations
-  are indicated by red and
-  negative correlations by
-  blue colors.

**Differences between**
**synoptic and climate variability appear on timescales as short as a week**

- Figure courtesy Gil Compo

**Synoptic**

**Weekly**

**Monthly**

**Seasonal**

## Separation of scale: baroclinic and barotropicVariance-conserving spectra of 500 mb height (left) and omega (right), averaged in high and mid latitudes

- Figure courtesy Gil Compo

- 500 mb 

- 500 mb 

## Motivation

- Consider the dynamical system describing the variable x,
- 	dx/dt = N(x) + F     (N is a nonlinear operator and F is external forcing)
- This can always be rewritten as
- 	dx/dt = slow nonlinearity + fast nonlinearity
- If:
- we are only interested in the slowly evolving portion of x
- and there is a big difference between “fast” and “slow”
- this may be usefully approximated as
- 	dx/dt = Lx + white noise

## Barotropic eigenmodeAssumedx/dt = LBx + Fswhere LB represents linear barotropic dynamics.Then weekly variations result from the least damped (closest to neutral) eigenmodes of LB.

- Simmons, Wallace, and Branstator 1983

## Problem: Barotropic dynamics alone can’t explain time evolutionSolution (?): A different linear operator:dx/dt = Lx + F[but a linear baroclinic L is only slightly better]and/or a second model for the forcing: dF/dt = MF + s

- c.i.=0.15; blue < 0

- Winkler, Newman, and Sardeshmukh 2001

## Change L: include linear parameterization of transient eddy feedbackSynoptic eddy feedback favors the development of some anomalies (left)but not others (right) inperpetual January GCM run

- Change **L**: include linear parameterization of transient eddy feedbackSynoptic eddy feedback favors the development of some anomalies (left)but not others (right) inperpetual January GCM run

- Branstator 1995

**Notes:**
From Branstator (1995). Results from 200 month perpetual January run.
Observationally, it’s hard to separate cause/effect with feedback.

## Candidates for “boundary forcing” to extratropical troposphere

- Surface anomalies
- SST
- Soil moisture anomalies (warm season)
- Sea ice/snow cover anomalies (seasonal)
- Tropical heating
- Driven by SST (e.g., ENSO)
- Atmospheric phenomena (e.g., MJO)
- Stratospheric anomalies
***	***
***These act all at the same time and not independently.***

## Two ways to determine L

- “Forward method” -- derive **L** from a physical model (including linear parameterization of nonlinear terms)
- 	This may be difficult.
- “Inverse method” -- derive **L** from observed statistics of both extratropical anomaly and “forcings”
- 	This may be easier, but is not pain-free.

## Linear inverse model (LIM)

- d**x**/dt = **Lx** + **F**s
- Then 0-lag and zero-lag covariance are related as
**C**(0) = exp(**L**0) **C**(0)
- So we can solve for **L**.
- Test of linearity: for much longer lags , is
**C**() = exp(**L**) **C**(0)    ?

**x(t) = 86-component vector whose components are the time-varying**
**coefficients of the leading slp, *****T* (250 and 750 hPa)****H, and *****S* (30 hPa)
**PCs of 7-day running means.**
**L is thus a **
**86x86 matrix**
**Trained on 5-day lag**

## Dynamics are effectively linearObserved 21-day lag covariance(left) reproduced by the LIM (right)

- Newman and Sardeshmukh 2007

- 250 hPa 

- slp

- Heating

## Heating is responsible for most of the persistent variability captured by this LIMTop:LIM 21-day lag covarianceMiddle:LIM 21-day lag covariance, effects of H removed from LBottom: LIM 21-day lag covariance, effects of sremoved from L

- Heating is responsible for most of the persistent variability captured by this LIMTop:LIM **21-day lag covariance**Middle:LIM **21-day lag covariance**, effects of H removed from **L**Bottom: LIM **21-day lag covariance**, effects of sremoved from **L**

- Newman and Sardeshmukh 2007

- 250 hPa 

- slp

- LIM

- LIM
- w/o
- Heating

- LIM
- w/o
- Strat.

## Remaining persistence due to “internal” extratropical dynamicsLeading eigenmodes of “troposphere-only” portion of L correspond to remaining persistence

- Newman and Sardeshmukh 2007

- Remaining persistence due to “internal” extratropical dynamicsLeading eigenmodes of “troposphere-only” portion of **L** correspond to remaining persistence

## Week 3 250 hPa skill, LIM and Reforecast (1979-2000)

- SUMMER

- WINTER

- LIM
- (cross-validated)

- Reforecast

- Newman, Sardeshmukh, Winkler, and Whitaker 2003

## Even if the climate system is exactly effectively linear, why does the LIM outperform the GCM at Week 3?[Or, are there sources of skill that may be exploited in future GCMs?]

- Tropical heating forecast skill in LIM
- No climate drift in LIM

## Week 2 TROPICAL skill, LIM and Reforecast

- LIM

- LIM

- Reforecast

- Reforecast

- Newman, Sardeshmukh, Winkler, and Whitaker 2003

## Week 2 skill from tropical “C-LIM” 

- Diabatic heating forecast skill severely degraded without air-sea coupling, but this is mostly due to lack of ENSO SST forcing.

- Different state vector: 3 levels each of tropical heating, streamfunction, velocity potential, plus SST (38 PCs)

## Power spectra of leading SST, heating PCs

**Test of linearity: LIM trained on 1-week lag reproduces 40-50 day and 4-7 yr spectral peaks.**

- Without coupling:
- SST variability is
- weakened and
- peak period is shorter.
- Heating variability on subseasonal timescale is minimally altered.

## Week 2 June climate drift

## Seasonal cycle of LIM, Reforecast skill

- Skill of LIMs (red line) constructed
- for two-month “seasons” compared to reforecast skill (blue line).
- Skill measure is pattern correlation of 250 hPa streamfunction in Northern Hemisphere between 120E-60W.

## Predicting skill within the LIM

## Predicting Week 3 skill

- Comparing predicted LIM forecast skill with actual LIM and reforecast skill (pattern correlation of NH 250 hPa )

- Poorer LIM summer skill: reality or due to missing land obs?

## LIM is skillful even where the PDF is non-Gaussian 

- Color Shading:
- Wintertime Week 3 LIM forecast skill, 250 hPa 
- Hatching:
- where the streamfunction PDF is non-Gaussian (by the K-S criterion, 95% significant)

- Sura, Newman, Sardeshmukh, and Penland 2004

## Observed departures from Gaussianity can be mimicked by linear multiplicative noise (can be part of L)

- Departure from Gaussianity of joint pdf derived from least damped barotropic mode (period=33 days, eft=14 days) with some *stochastic damping* and steady forcing

- Departure from Gaussianity of
- Joint p.d.f of first two EOFs of
- 750 mb streamfunction, DJF
- 1950-2002

- Sura, Newman, Sardeshmukh, and Penland 2004

## Can we expect to forecast Weeks 3 and 4?

- Yes, but:
- Empirical models (LIM + others in Tropics) appear to still have better forecast skill than GCM
**	*****But there are may be areas of nonoverlapping skill (particularly for skewed distributions), and/or we may use more skillful tropical LIM forecasts to nudge GCM.***
- Extratropical forecast skill is modest on average
**	*****But there are cases when the skill is relatively high. These cases can to some extent be identified a priori and provide forecasts of opportunity.***
- As GCM skill surpasses LIM, predictability estimates from LIM may remain useful
**	*****But in the mean time, ***_***Week 3 and Week 4 forecasts may be made now.***_

## Solid: LIM (Actual)  Circles: LIM (predicted)  Dotted: Reforecast
ENSO: Red arrows warm events; Blue arrows cold events

- ENSO: Red arrows warm events; Blue arrows cold events

- Week 3 skill not **all** ENSO

- Predictability Variations:  Winter vs. Summer