Incorporation of dynamic balance in data assimilation and application to coastal ocean Zhijin Li and Kayo Ide SAMSI, Oct. 5, 2005,

Incorporation of dynamic balance

in data assimilation and application to coastal ocean

Zhijin Li and Kayo Ide

SAMSI, Oct. 5, 2005,

First Numerical Weather Predictionby Richardson: 1922

Richardson’s raw data Filtered data (Lynch,2000)

Richardson’s forecast might well have been realistic with the filtered data

Balanced flow: the 3D velocity field functionally related to mass field

Chinese Yin-YangMass field: YangVelocity field: Yin

A flow is said to be balanced if the three-dimensionalvelocity field v(x,t) is functionally related to the massfiled p(x,t) or the spatial distribution of mass. Such a functionalrelation is called “balance relation”. (McIntyre 2003)

There exist various balance equations: P(p,v)=0

Initialization, nonlinear normal modes, and the slow manifold

),(

),(

ZWNZidt

dZ

ZWNWidt

dW

ZZ

WW

),()(

),(

1bZYb

bYZ

ZYNiZ

ZYNYidt

dY

W: slow Rossby modes

Z: gravity modes

),(),()( bZZbZ ZYNZYNZZidt

dZ

),( bZYLocus of : slow manifold

Fast gravity modes: noise

Quasi-geostrophic flow (Leith 1980)

Slow manifold and data assimilationSlow manifold and data assimilation

S

Y

Z

D

A

L

N O

Schematic slow manifold diagram for a comprehensive model (after Leith 1980; Daley 1980)

Z: fast manifoldY: Rossby manifoldS: slow manifoldD: data manifold => estimate manifold

A: optimal estimateL: linear normal mode initializationN: non-linear normal mode initializationO: optimal estimate with dynamic balance

Data assimilation is suggested to seek O, rather than A.

Why the optimal estimate not on the slow manifold

The true atmospheric and oceanic state weak components of fast gravity modes. Observation observational errors and poor distribution in space and time. A primitive equation model permission of fast gravity modes. Unbalanced error covariance inaccuracy and invalid linearization

A strategy: control variable transform

p

vx

p

vvx u

uup

),(),(

),(),(

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

ppCvpC

pvCvvCB

yHxRyHxxxBxxxJ oTofTf

),( vpC

),(),(

),(),(

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

ppCvpC

pvCvvCB

yHxRyHxxxBxxxJ

u

uuuu

oTofuuu

Tfuuu

p

vx v: velocity field; p: mass field

0),( pvpP

Problem: The balance relation works through It may be violatedbecause of inaccurate B, and it is always violated since P is nonlinear.

The balance relation. uv is the unbalanced velocity.

An example for the control

Variable transform: NCEP 3DVAR

Tsfc

f

Wp

QT

Z

DD

)ln(

“The balanced components of the mass and momentum fields have been combined into a single variable. This allows the balance between the mass and momentum fields to be implicitly included.” (Parrish and Derber 1992)

Z is the linear balance operator. The DA control variables: ς, D and f

A refinement:Incremental control variables

p

vx

p

vx

pp

vvvx u

fup

f

)()()( 11 oTopv

T yHxRyHxxBxxJv

v: velocity field; p: mass field

Cost function

Control variables

0),( pvpP Incremental balance relation

Weak geostrophic and hydrostatic balance in a 3DVAR system for coastal ocean: ROMS-DAS

aaTSuv xxx

TSGuv xx

aTS xxx

TSS xx

Geostrophic balance

Vertical integral of the hydrostatic equation

TSfTS

aaTSfuv

aTSf

TS

uv

xx

xxx

xxx

x

x

x

S

T

v

u

x

ax Unbalanced streamfunction and velocity potential

Experiment with a single observation:v-component

ROMS-DAS: Configuration forReal time experiments

12-hour forecast

Time

Aug.100Z

Aug.118Z

Aug.112Z

Aug.106Z

Initialcondition

6-hour forecast

Aug.200Z

Xa = xf + xf

Xa

xf

3-day forecast

y: observationx: model

6-hour assimilation cycle

y

y

Integrated Ocean Observing and Prediction Systems

15 Aug

16 Aug

17 Aug

0

AUV Remus ROMS Reanalysis

18 Aug

Distance (km) 23Subsurface salinity minimalDistance (km) 23Subsurface salinity minimal

Conclusions

With the concept of the slow manifold, it is demonstrated that an additional dynamic constraint is needed to keep the analysis in dynamic balance.

A strategy is suggested that the unbalanced components should be used as control variables to incorporate dynamic balance implicitly in DA.

ROMS-DAS, a 3DVAR system for coastal oceans, has been developed using the strategy. Among the DA control variables are then non-dynamic SSH, adjusted ageostrophic streamfunction and velocity potential.