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

First Numerical Weather Prediction by 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-Yang Mass field: Yang Velocity field: Yin A flow is said to be balanced if the three-dimensional velocity field v(x,t) is functionally related to the mass filed p(x,t) or the spatial distribution of mass. Such a functional relation is called “balance relation”. (McIntyre 2003) There exist various balance equations: P(p,v)=0

Initialization, nonlinear normal modes, and the slow manifold W: slow Rossby modes Z: gravity modes Locus of : slow manifold Fast gravity modes: noise Quasi-geostrophic flow (Leith 1980)

Slow 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 manifold Y: Rossby manifold S: slow manifold D: data manifold => estimate manifold A: optimal estimate L: linear normal mode initialization N: non-linear normal mode initialization O: 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 v: velocity field; p: mass field Problem: The balance relation works through It may be violated because of inaccurate B, and it is always violated since P is nonlinear. The balance relation.is the unbalanced velocity.

An example for the control Variable transform: NCEP 3DVAR “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

A refinement: Incremental control variables v: velocity field; p: mass field Cost function Control variables Incremental balance relation

Weak geostrophic and hydrostatic balance in a 3DVAR system for coastal ocean: ROMS-DAS Geostrophic balance Vertical integral of the hydrostatic equation Unbalanced streamfunction and velocity potential

Experiment with a single observation: v-component

ROMS-DAS: Configuration for Real time experiments 12-hour forecast Time Aug.1 00Z Aug.1 18Z Aug.1 12Z Aug.1 06Z Initial condition 6-hour forecast Aug.2 00Z X a = x f +  x f XaXa xfxf 3-day forecast y: observation x: 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

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.