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Advantages of data assimilation in coastal ocean circulation models:

Oregon perspective

Alexander L. Kurapov, J. S. Allen, G. D. Egbert, R. N. MillerAlexander L. Kurapov, J. S. Allen, G. D. Egbert, R. N. Miller COAS/Oregon State University

In cooperation with P. M. Kosro, M. D. Levine, T. Boyd, J. A. Barth, J. N. Moum, P. T. Strub, S. Erofeeva

29 January 2004, AGU/Ocean Sciences

http://www.coas.oregonstate.edu/po/research/kurapov/main.html

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wind stress (upwelling favorable) is dominant forcing

strong effects of flow-topography interactions

energetic internal tide

Summer circulation on the Oregon shelf:

HF radarsHF radarsMoorings Moorings

(ADP, T, S)(ADP, T, S)

currents: 3D+time density:

3D+time

Summer 2001: DA system is implemented with data from COAST observational program

Data assimilation:

improves prediction of the ocean state,

provides solution error estimates,

is used as a tool for data synthesis,

helps to design an observational system (e.g., suggests optimal observational locations)

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Dual approach:

Variational

(generalized inverse)

DA method Simpler, sequential

(optimal interpolation)

Linearized Dynamics Fully non-linear

Internal tides Application Wind-driven circulation

Objectives:

• to develop practical, but still nearly optimal methods for the assimilation of data into coastal circulation models

• to apply these methods to measurements from the Oregon shelf

• to utilize DA to increase scientific understanding of shelf circulation

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Model of of M2 internal tide [Kurapov et al., JPO 33, 2003]

- linearized, primitive eqns, 3D, periodic in time [~exp(it)]

- terrain following coordinates

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oMgH H K

i g dH H

u

u f u

e.g., momentum equations:

HF

(P. M. Kosro)

HF

ADPADP

Model domain: 40 60 km, x=1 km, 21 -layers

- Zone of coverage of 2 HF radars (May-July 1998)

- Efficient model solver (direct factorization of the model operator)

- Address open boundary issues

Most internal tide comes from outside the computational domain

DA: corrects open boundary baroclinic flux

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Generalized Inverse Method (GIM):

Solution minimizes a cost function:

Cost Function = || Model error ||2 + || BCond error ||2 + || Obs error ||2 min

- Explicit statistical assumptions about errors in the inputsExplicit statistical assumptions about errors in the inputs

- Statistics in the output (prior model and inverse solutions) are computed Statistics in the output (prior model and inverse solutions) are computed [[Bennett, 1992, 2002]]

State vector: v = {velocity, sea surface elevation, density}

Model+BCond: S v = f + em

Data: L v = d + ederrors in model forcing and data

( ) ( ) ( ) ( )CF 1 1m dSv f C Sv f Lv d C Sv d

Cov( , ) Cov( , ) m m d dm dC e e C e e

specified prior to assimilation

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Use of Representers:

1

1

Inverse solution = Prior Solution +

Representer: , where is the row of the data functional

K

k kk

k k k

b

1m

r

r S C S l l L

Model+BCond: S v = f + em

Data: L v = d + ed

Adjoint solverFwd solver

1 1( ) ( )o

1m db LS C S L C d Lv

Reduce burden of representer computation with:

- reduced basis representer approach

- indirect representer approach [Egbert et al., JGR, 1994]

HF radars: K=900 locations where radial velocity components are available

Standard feature in Inverse Ocean Modeling system [IOM, Chua and Bennett, Ocean Modeling, 2001]

vo

Strongly constrained dynamics:

0 0

0

INTERIORm

OB

CC

C

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Solution sensitivity to the choice of model error covariance COB (in an experiment with synthetic data)

-”true” solution: forced at open boundary (OB) with a significantly baroclinic flux

-synthetic data (velocity harmonic constants) are sampled from true solution

-prior model: forced with depth-averaged OB current

-DA: corrects OB baroclinic fluxes

Depth-ave RMS error with respect to true solution

Prior DA, COB (Type I) DA, COB (Type II)

these two solutions allow for OB b/clinic correction of the same magnitude (but

different correlation structure)

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DA COB (Type I) is obtained by nesting approach:

In a large domain, compute representers for small domain boundary data

then sample these representers along the OB of small domain

COB (covariance for the errors on the OB of the small domain, with a dynamically consistent spatial structure)

COB controls radiation at an open boundary

representer column of prior solution error covariance matrix

COB (Type II): our best guess w/out nesting

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A series of M2 tidal solutions, May-July 1998

Internal tide intermittence: analysis in 2-week overlapping time windowsDA: in each time window

Validation ADP

DA solution

No DA

deviations from depth-ave. (CW)

depth-ave (rotating CCW)

Assimilation of HF surface currents improves prediction at depth

Tidal ellipses of horizontal currents at ADP location, vs depth: (a) observed, (b) prior model, (c) DA.

ADPADP

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M2 tidal ellipses on the surface: internal tide velocities can be twice as large as barotropic tidal velocities

CCW rotation

CW rotation

Depth-aveDeviations from depth-ave (time window centered on day 139)

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Energy balance is closed : Data assimilation corrects only boundary inputs

40 W m-1

Most baroclinic signal comes into the computational domain from outside

Some persistent features are found: e.g., baroclinic phase and energy propagation is from NW.

Terms in the baroclinic energy equation (time and space averaged)

Baroclinic energy flux (depth-integrated and time-ave.)

day, 1998

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Baroclinic KE

averaged over a series of days 139-167: a) surface, b) bottom, c) cross-section north of Stonewall Bank, d) cross-section through Stonewall Bank.

Zones of higher KE variability are aligned along the coast, consistent with energetic of a internal Poincare wave

interaction with bathymetry

Dominance of 1st baroclinic mode

beams over Stonewall B

A series of tidal solutions (constrained by HF radar data) provides a uniquely detailed description of spatial and temporal variability of M2 internal tide

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Model of wind-driven circulation:

AV

HR

R S

ST

, o C[c

ou

rte

sy P

.T.

Str

ub

]

-Princeton Ocean Model: 220350 km, periodic OB conditions (south-north), x~2 km, 31 -layers

-Forcing: alongshore wind stress, heat flux

-Data assimilation: Optimal Interpolation

-Initial implementation (summer 1998): assimilation of HF radar data improves modeled circulation at depth [Oke et al., JGR-Oceans, 2002]

-Data from COAST program (summer 2001): assimilate moored ADP currents

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Optimal Interpolation (3DVAR):

,a ft t t POMν ν

( )a f ft t t t ν ν G obs Hν

1Gain matr x: i

f f

dG P H HP H C

matrix matching observations to state vector

||Error||

Time

model w/out DA

DAforecast

analysis

Forecast error covariance (stationary in OI): Pf = Pm F (lagged Pm, Cd) where Pm is the covariance of errors in the model solution not constrained by the data (in contrast, Pf is conditioned upon previously assimilated data) [Kurapov et al., Mon. Wea Rev., 2002]

Pf has a shorter horizontal scale in the alongshore direction than Pm (effect of propagation)

Pm: could be obtained as representer calculation, if an adjoint model were available

Presently, Pm is computed from an ensemble of model solutions

Incremental approach: correction is applied gradually over the analysis time window (1/4 of inertial period)

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Spatial structure of Pf:

NMS,

12m

SSB,

16m

[cm2 s-2]

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Time- and depth-ave terms in the momentum eqn. (along-jet direction)

no DA

DA (ADPs in south)

Dominant dynamical balance is preserved

Smooth, large scale correction (in this case, DA tends to reduce upwelling intensity)

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Assimilation of moored ADP velocities (May-Aug 2001):

90 k

m

Central part of model domain with mooring locations, Bathymetry each 100 m (black) and 10 m (half-tone, from 0 to 200 m)

Moorings: Lines N and S – COAST (Kosro, Levine, Boyd), NH10 – GLOBEC (Kosro)

Study is focused on:

- Distant effect of data assimilation- Multivariate capabilities (effect on SSH, isopycnals, temperature, salinity transport, turbulent dissipation rate

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Case 1: assimilate currents at Northern Line improve currents at NH10, SSB

Correction can be advected by a predominantly southward current

90 k

m

ADP sites, May-Aug 2001Assimilated ADP sitesSites where DA is better than model only solution (smaller model-data rms error, larger correlation)

NH10

SSB

rmse: 7.8 5.8 cm s1, corr: 0.18 0.71

rmse: 9.6 7.1 cm s1, corr: 0.36 0.70

Alongshore depth-ave current: obs, no DA, DA

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Case 2: assimilate ADP currents at Southern Line improve currents up North

Correction can be propagated northward with coastal trapped waves

NMS

NH10

rmse: 11.3 7.9 cm s1, corr: 0.46 0.79

rmse: 7.8 6.9 cm s1, corr: 0.18 0.63

Alongshore depth-ave current: obs, no DA, DA

ADP sites, May-Aug 2001Assimilated ADP sitesSites where DA is better than model only solution (smaller model-data rms error, larger correlation)

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Posterior error statistics analysisE.g., compare expected and actual analysis rms error as a consistency test for Pf

Expected performance

diag (Pm) and (Pa) are compared, where Pa = Pf – G H Pf is the analysis error covariance

Actual performance

Assimilated siteDA is better than model only solutionDA is worse than model only solution

Discrepancy between expected and actual outcome when assimilating inner-shelf data :

artificially large decorrelation scale in Pf

inclusion of a more realistic spatially varying wind stress is a necessity

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Multivariate capabilities

no DA DA (South)SeaSoar measurements (Barth et al.)

e.g., effect on SSH (validation - tide gauge data):

effect on isopycnal slope:Model-data Corr.: 0.51 0.78, rmse: 5.4 3.8 cm

SSH: obs, model only, DA (Lines N+S)

(white contours are measured 24, 25, and 26 kg m-3)

+ improvement in temperature correlations, surface salinity transport

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Turbulent Dissipation rate ()

Microstructure data [J. Moum, A. Perlin] No DA DA (North)

12 transects on Line N

32 3

2

POM:

(m s ),16.6

where is TKE 2

is turb. length scale

q

L

q

L

yearday, 2001

Time series of averaged near bottom (in box area)

DA correction in near-bottom velocity field yields improvement in

Analysis of BBL dynamics is extended for the whole study period – presentation OS52I-08

23http://www.coas.oregonstate.edu/po/research/kurapov/main.html

SUMMARY:

Progress has been made on both aspects of the dual approach to coastal ocean DA

Linearized dynamics, variational DA (internal tides)

-has provided unique information on spatial and temporal variability of internal tide from HF radar measurements of surface currents

-has given us experience in open boundary DA

Nonlinear dynamics, sequential OI DA (wind-driven circulation)

- has shown the value of assimilation of currents from HF radar and from moored ADPs (distant effect, multivariate capabilities, BBL analysis)

-has provided information on optimal ADP mooring locations and on effective alongshore scales of ADP current measurements

In both cases, formulation of error hypotheses is the science and art of DA

DA is utilized to increase scientific understanding of shelf circulation

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PLANNED RESEARCH:

Merger of approaches: use tangent linear and adjoint codes for a fully non-linear ocean circulation model (ROMS)

Use data assimilation to help provide open boundary conditions for high-resolution limited-area coastal models

Tidal research: study effect of wind-forced subinertial flows on internal tide propagation

Study of wind-forced upwelling circulation: analyze cross-shelf transport, bottom boundary layer processes, dynamical balances