Page 1© Crown copyright 2005 Dynamics and Data Assimilation Richard Swinbank UTLS International School, Cargese

© Crown copyright 2005 Page 1

Dynamics and Data Assimilation

Richard Swinbank

UTLS International School, Cargese


Aim of lecture

The aim of the lecture is to give an overview of some concepts from atmospheric dynamics, and relate them to the way data assimilation systems are implemented in practice.

This lecture looks at some implications of dynamical theory, while my second lecture looks at practical interfacing of numerical models with data analysis.


Contents

Some dynamicsGeneral circulationMid-latitude weather systems

Implications for data assimilationBalanceControl variablesBackground error covariancesPractical issues

Use of assimilated dataStudies of dynamics and transport


Some Dynamical concepts

Zonal-mean circulation

The zonal-mean temperature structure highlights the conventional division of the atmosphere into troposphere, stratosphere and mesosphere.

The zonal mean westerly winds are essentially in balance with the temperatures.

July climatology from Met Office stratospheric analyses (plot from Andrew Bushell)


Wave-driven meridional circulation

The meridional circulation in the UTLS is characterised by ascent in the tropics and descent at mid- and high-latitudes.

This circulation is primarily driven by breaking planetary waves.

The Brewer-Dobson circulation extends to the mesosphere, driven by breaking gravity waves.

(Holton et al, 1995)


Extra-tropical cyclones

The typical lifecycle of an extra-tropical cyclone, from a frontal wave to an occluded vortex, is described by the Norwegian cyclone model.

(with thanks to Nigel Roberts)


Tropopause folding

Stratospheric air descends as a narrow tongue “into” the troposphere along a frontal zone.

This stratospheric air is dry (shows up as a “dry slot” on imagery) and has high PV.

The high PV can lead to explosive cyclogenesis.

A Joly (1986)

Filamentation

Coarse-grain PV maps from NWP data do not reflect the filamentary structure that occurs near the tropopause

The upper plot shows how a passive tracer (initialised as PV) is deformed after 4 days contour advection (on the 320-K isentrope)

A Meteosat image of upper-troposphere water vapour shows similar structures

(Appenzeller et al, 1996)


Polar vortex and surf zone

In the winter stratosphere the polar vortex is well-definedThe strong PV gradient

acts as a transport barrier (a bit like the tropopause).

The vortex is surrounded by a surf zone characterised by weak PV gradients.


Dynamics – some key points

The large-scale atmospheric circulation is characterized by geostrophic and hydrostatic balance.

The mean meridional circulation is a lot slower than the zonal circulation.

Stratosphere-troposphere exchange is dominated by ascent through the tropical tropopause and descent at middle to high latitudes.

Stratospheric air, dry with high PV, can be important in mid-latitude cyclogenesis.

Transport barriers – e.g. tropopause and polar vortex edge.

Dynamics – implications for data assimilation


The Data Assimilation Problem

The data assimilation problem can be summarized as the minimization of a cost function J

The analysis (xa), when J(x) is minimized gives the best fit to the information available: both background (Jb) and observations (Jo), subject to the specified observation (R=E+F) and background (B) error covariances.

1 11( ) ( ) ( ) ( ) ( )

2

Tb T b o o

b o

J H H

J J J

x x x B x x y x R y x


Data Assimilation and Balance

Atmospheric flow, at least on large scales, is characterized by hydrostatic and geostrophic balance. The “balanced flow” captures the most important dynamics.

In practical implementations of variational assimilation schemes, the background error covariance matrix B is structured to reflect this balance.

In some implementations, extra (dynamical) constraints are imposes by an additional Jc term in the cost function J.


Specifying background error covariance B

The obvious way of specifiying B would be in terms of the model variables

Instead, we define B using a set of different control variables, who errors can be assumed to be uncorrelated with one another

To achieve this, we choose a first control variable to represent the balanced flow, and others to represent unbalanced components of the flow

u v T q

u

v

T

q

ψ χ aprh

ψ

χap

rh


Balanced control variables

Three candidates for a balanced control variable have been suggested:

Rotational wind (i.e. streamfunction or vorticity) Mass field (e.g. pressure) Potential Vorticity (or PV-related streamfunction)

The first choice was originally used by Parrish and Derber (1992), and currently by many others, including Met Office

Geostrophic adjustment arguments suggest that pressure may be a better choice on large horizontal scales, shallow depths (where rotational wind adjusts to pressure). But PV may be best of all.

Work at DARC and Met Office is currently investigating possible use of a PV-type control variable (Ross Bannister, Mike Cullen)

Choice of other (unbalanced) control variables is determined by requirement for errors in each control variable to be uncorrelated.


How B is built up

We have seen how B is defined in terms of uncorrelated control variables

The errors in each control variables are in turn specified in terms of vertical and horizontal modes.

In the end, B is transformed into a diagonal matrix

ψ χ aprh

ψ

χap

rh

ψ χ aprh

ψ

χap

rh


Transforms and preconditioning

The transforms just discussed can be used to define the background error covariance as B=UUT, where the U-transform is built up from the parameter, vertical and horizontal transforms U=UpUvUh

So, the covariance information is transferred into the transforms – and inverts B.

The variational problem can thus be restated in terms of a new, incremental, control variable v

B is effectively replaced by I, and the conditioning of the problem is very much improved

11( ) ( ) ( ) ( ,..) ( ,..)

2

Tb T b o oJ H H v v v v v y Uv R y Uv

Background Error Covariances in practice

(with thanks to Mike Keil and Bruce Ingleby)


Estimating background error covariances

When the first operational variational data assimilation system (SSI) was implemented, the B matrix was estimated from the differences between pairs of forecasts verifying at the same time. They originally used the differences between initialised T+0 and T+24, with rescaling of variances.

“This is a very crude first step in specifying the forecast-error covariance” – Parrish and Derber (1992).

The Met Office (and many other centres) still uses the same basic technique. We use differences between T+24 and T+48 forecasts for the global models.

The “NMC method”


Generating new background error covariances

When generating background error covariances for a new model configuration, initial data for the forecasts needs to be interpolated (“reconfigured”) from a previous model configuration.

A set of forecasts are run, and differences calculated between corresponding T+48 and T+24 fields.

Calculate error covariances from forecast differences in terms of analysis control variables, vertical and horizontal modes:

using 5 degree latitude bins; vertical regression applied to improve coherence of vertical

structures; rotate vertical modes to localise them in the stratosphere.

Procedure can be repeated as required.


Assimilation

Interpolate

T+0 T+0T+0

T+24 T+24T+24

T+48 T+48T+48

T+0

InterpolateInterpolateInterpolate

T+24

Calculation of Btime

NMC method – with reconfiguration


Assimilation using initial covariances

T+0 T+0T+0

T+24 T+24T+24

T+48 T+48T+48

T+0

T+24

timeRevised B

NMC method

Calculated background error covariances for first implementation of 3DVAR for the stratospheric (N48L50) model

The error covariances show that tropospheric errors are essentially balanced, but stratospheric errors are not(!)

(Before vertical regression, stratospheric errors were reasonably well balanced in the winter hemisphere.)

Error variances (illustrated for T) are larger in tropospheric mid-latitudes, and increase substantially in the stratosphere.


New Dynamics in stratospheric model

When New Dynamics was brought in, the NMC procedure was repeated, but… the assimilation trials kept failing.Large increments were observed at upper levels…due to very large background error variances at

upper levels.

The calculated error covariances needed adjustment at the upper levels


Assimilation using initial covariances to level 40

T+0 T+0T+0

T+24 T+24T+24

T+48 T+48T+48

T+0

T+24

timeRevised B

Forecast only above level 40

NMC method – modified at high levels


Assimilation using 2nd iteration covariances

T+0 T+0T+0

T+24 T+24T+24

T+48 T+48T+48

T+0

T+24

timeFinal B

NMC method – another iteration


Adjusted covariances

Trials were successful (they ran and gave improved results)

Currently used as the operational stratospheric covariances

But, there is a discontinuity around 10hPa

Is there a better way?


“Canadian Quick Covariances” (CQC) method

Error covariances are calculated from differences between adjacent six-hourly forecast fields in a single, extended, model integration.

Used in the Canadian Middle Atmosphere Model (CMAM) (Yves Rochon – see Polavarapu et al, 2005)

Covariances are calculated in exactly the same way as before, except using 6 hour differences instead of 24 hour differences

To minimise the tidal signal, means are accumulated at synoptic hours (0,6,12,18z) and subsequently removed from the appropriate forecasts.

Compared to the traditional NMC method, CQC is a lot quicker


CQC method

Continuous Forecast

Interpolate

Analysis

Subtract 0,6,12,18Z means

0 6 12 18 0 6 12 18

B calculated from 6-hour differences


Tidal signal is minimised


CQC v Current operational

Variances are comparable to current operational B; if anything, they seem more physically realistic

In trials, verification against analyses was improved, although verification against obs was poorer (but there are few obs used in the stratosphere for verification calculations).


A better approach…

ECMWF uses an ensemble of analyses to estimate the background error covariances (Fisher, 2003).

All the inputs to the analysis/forecast system are perturbed with random errors taken from the appropriate distribution.

The differences between pairs of forecasts will have the statistical characteristics of background error (but twice the variance).

Expensive, but practical where ensemble infrastructure is in place.

Analysis Forecastxb+εb



Analysis Forecastxb+ηb



Background differences

Stratospheric Dynamics and Transport


Transport studies

A key application of assimilated data, particularly in the UTLS region, is to diagnose transport and mixing of constituents (and aerosols).

Two approaches:Kinematic – use analysed (3-D) windDiabatic – use heating rates to get an alternative

estimate of vertical velocity


2-D transport study - Permeability of Polar Vortex edge

Does Polar Vortex act more like a “containment vessel” or a “flowing processor”?

Addressed by Chen (1994) 40-day contour advection experiments for the Antarctic

vortex Contours initialised from PV for 21 July 1993 Mixing quantified by how quickly the contours lengthen


Line of separation at edge of vortex

Average e-folding time as a function of potential vorticity

“Line of separation” is the contour that lengthens most slowly

425K

500K

PV

Time – log(length) graph of the lines of separation on different isentropic levels

350K

375K

400K

500K

425K

450K


Mixing at the edge of the vortex

Grey-scale images of the “line of separation on 31 Aug 93, after 40 days contour advection

350K 375K 400K

425K 450K 500K


NH Ozone depletion

Although chemical depletion had been well established in the SH, Met Office analyses helped to confirm it in the NH (Manney et al, 1994, 1995)

Observed vortex-mean ozone was compared with vortex-mean ozone derived from passive advection of particles

Particle trajectories were calculated from Met Office winds (in horizontal) and MIDRAD (in vertical)

Ozone loss observedby MLS

MLS observations of ozone at 465 K show a reduction in the ozone mixing ratio through the spring (1993)

The polar vortex is indicated by PV contours, taken from Met Office stratospheric analyses

However, this is not conclusive evidence for chemical ozone loss

(after Manney et al, 1994)


The smoking gun…

Presence of ClO (left panels) is consistent with chemical destruction of ozone (right panels)


Calculated Ozone loss

Vortex average O

zone mixing ratio (ppm

v)

1992

1993

1994

1995

1996

Ozone treated as a tracer – green

MLS measurements - blue circles

Calculated loss - cyan triangles


Stratospheric transport and age of air

Schoeberl et al (2003) used 3-D trajectories from two different data assimilation systems (DAS) and a free-running GCM to compute age spectra in the lower stratosphere.

GCM results showed isolated tropics (as observed), but DAS results showed too much mixing between tropics and mid-latitudes, and also (for kinematic trajectories) excessive vertical dispersion.


3-D trajectory calculations after 50 days

From Schoeberl et al, 2003


Diagnosing transport

Assimilated data form a good basis for mid- and high-latitude studies of transport and mixing. They form an invaluable dynamical framework for the interpretation of constituent measurements.

However, assimilated data does not properly capture slow processes, such as the Brewer-Dobson circulation. Data must be used with caution (e.g. diabatic trajectories).

Low latitude dynamics are not so well captured, mainly reflecting the lack of wind observations, and reduced mass-wind coupling.


Summary

We have given an overview of some key dynamical concepts relevant to the UTLS and data assimilation.

Dynamical balance influences the way in which background errors are treated in data assimilation.

Some practical difficulties with the current treatment of background errors have been highlighted.

Finally, we have illustrated the use and limitations of assimilated data for studies of stratospheric dynamics and transport.


Questions?


Further reading

ReferencesAppenzeller, C., H.C. Davies and W.A. Norton, 1996: Fragmentation of stratospheric

intrusions, J. Geophys. Res., 101(D1), 1435-1456.Chen, P., 1994: The permeability of the Antarctic vortex edge, J. Geophys. Res., 99,

20563-20571.Fisher, M., 2003: Background error covariance modelling. In Recent developments in data

assimilation for atmosphere and ocean, ECMWF Seminar proceedings, 45-63.Holton, J.R., P.H. Haynes, M.E. McIntyre, A.R. Douglass, R.B. Rood, L. Pfister, 1995:

Stratosphere-troposphere exchange, Rev Geophys., 33 (4), 403-439.Manney, G.L., et al., 1994: Chemical depletion of ozone in the Arctic lower stratosphere

during winter 1992-93, Nature, 370, 429-424.Manney, G. L., R. W. Zurek, L. Froidevaux, J. W. Waters, A. O'Neill, and R. Swinbank,

1995: Lagrangian transport calculations using UARS data. Part II: Ozone. J. Atmos. Sci., 52, 3069-3081.

Polavarapu, S., S. Ren, Y. Rochon, D. Sankey, N. Ek, J. Koshyk and D. Tarasick, 2005: Data Assimilation with the Canadian Middle Atmosphere Model, Atmosphere-Ocean, 43(1), 77-100.

Schoeberl M.R., A.R. Douglass, Z.X. Zhu, et al., 2003: A comparison of the lower stratospheric age spectra derived from a general circulation model and two data assimilation systems J. Geophys. Res., 108 (D3): Art. No. 4113

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Page 1© Crown copyright 2005 Dynamics and Data Assimilation Richard Swinbank UTLS International School, Cargese