Prediction of Ocean Circulation in the
Gulf of Mexico and Caribbean Sea
An application of the ROMS/TOMS
Data Assimilation Models
Hernan G. Arango (IMCS, Rutgers University)
Emanuele Di Lorenzo (Georgia Institute of Technology)
Arthur J. Miller, Bruce D. Cornuelle
(Scripps Institute of Oceanography, UCSD)
Andrew M. Moore (PAOS, Colorado University)
Ocean Observations
Gulf of Mexico and Caribbean Seas
plus satellite data (SSH, SST) and radar
Ocean Modeling Framework
ROMS/TOMS
PROPAGATOR
NL_OCEAN
M
A
S
T
E
R
AIR_OCEAN

TLM eigenmodes

Forcing singular
vectors

Stochastic optimals

Pseudospectra
TL_OCEAN
RP_OCEAN
OCEAN

Optimal pertubations

ADM eigenmodes
INITIALIZE
AD_OCEAN
RUN
ADSEN_OCEAN
FINALIZE
S4DVAR_OCEAN
KERNEL
NLM, TLM, RPM, ADM

physics

biogeochemical

sediment

sea ice
IS4DVAR_OCEAN
ESMF
W4DVAR_OCEAN
ENSEMBLE_OCEAN
Re
io
M o d e li n
Ocean
g S
na l
y
st
em
g
SANITY CHECKS
R es
e a rc h C o mm u ni
ty
TLCHECK_OCEAN
PICARD_OCEAN
GRAD_OCEAN
PERT_OCEAN
Ocean Modeling of North Atlantic
Gulf of Mexico
Ocean Model Grid
Ocean Model Surface Currents and Sea Level
Ocean Modeling Applications in
Gulf of Mexico and Caribbean Seas
• Develop a real-time data assimilation and
prediction system for the Gulf of Mexico and
Caribbean Seas based on a continuous upper
ocean monitoring system
• Demonstrate the utility of variational data
assimilation in a real-time, sea-going environment
• Demonstrate the value of collecting routine ocean
observations from specially equipped ocean
vessels (Explorer of the Seas)
• Develop much needed experience in both the
assimilation of disparate ocean data and ocean
prediction in regional ocean models.
• Add platform oceanic measurements (a
possibility)
Ensemble
Prediction
For an appropriate forecast skill measure, s
Predictable
Unpredictable
High
Spread
Low
Spread
s
s
t
time
t
time
Ocean Adjoint Modeling Applications
Kelvin Wave
Pattern
SSH
Time=t0
Maximum transport
SSH
Time=tN
Example from the Caribbean model run, of sensitivity of the transport
through the Yucatan Strait given a particular realization of the circulation.
In this case the maximum transport at time tN, indicated by the strong
gradients in sea surface height (SSH), is sensitive to a pattern of Kelvin
waves at previous time t0. These types of sensitivity, computed using the
non-linear and Adjoint models of ROMS, will be applied for the Florida
Strait to explore how different topographic shapes affect the transport
during different circulation regimes.
4D Variational Data Assimilation
Platforms (4DVAR)
• Strong Constraint (S4DVAR) drivers:
 Conventional S4DVAR: outer loop, NLM, ADM
 Incremental S4DVAR: inner and outer loops, NLM,
TLM, ADM (Courtier et al., 1994)
 Efficient Incremental S4DVAR (Weaver et al., 2003)
• Weak Constraint (W4DVAR) - IOM
 Indirect Representer Method: inner and outer loops,
NLM, TLM, RPM, ADM (Egbert et al., 1994; Bennett
et al, 1997)
Normalized misfit
Strong Constraint 4DVAR from IOM
misfit variance reduced 62%
8
1st guess
IOM solution
6
4
2
(Di Lorenzo et al., 2005)
0
-2
-4
T
-6
-8
0
2000
S
U
4000
V
6000
8000
Free Surface
Surface NS Velocity
SST
0.5
36
TRUE
0.1
36
0
0
34
-0.1
32
-128
-0.2
-126
-124
-122
-120
-118
34
32
-128
34
-0.5
-126
-124
-122
-120
-118
32
-128
-126
-124
-122
-120
-118
-126
-124
-122
-120
-118
-126
-124
-122
-120
-118
0.5
1 st
GUESS
36
0.1
36
0
34
-0.1
32
-128
36
-0.2
-126
-124
-122
-120
-118
0
34
32
-128
34
-0.5
-126
-124
-122
-120
-118
32
-128
0.5
IOM
solution
36
0.1
36
0
34
-0.1
32
-128
36
-0.2
-126
-124
-122
-120
-118
0
34
34
-0.5
32
-128
-126
-124
-122
-120
-118
32
-128
Strong and Weak Constraint 4DVAR
(Southern California Bight)

T
Assimilated data:
TS 0-500m
Free surface
Currents 0-150m
S
Normalized
Misfit
V
U
Datum
True Synthetic Data
SST
0-500 m
data
1st Guess
Annual
Climatology
Strong Constraint
SST
Weak Constraint
SST
SST
CalCOFI
Sampling
grid
Adjoint Sensitivity
• Given the model state vector:
• Consider a Yucatan Strait transport index, J ,
defined in terms of space and/or time integrals
of  :
• Small changes d in  will lead to changes dJ
in J where:
 J
 J 
 J 
 J 
 J 
dJ    du    dv  
 dT    dS  
 u 
 v 
 T 
 S 
 
• We will define sensitivity as

 d +

…
 J   J   J 
 ,  , 
,
 u   v   T 
etc.
Publications
Arango, H.G., Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003: The ROMS Tangent Linear and
Adjoint Models: A comprehensive ocean prediction and analysis system, Rutgers Tech. Report.
http://marine.rutgers.edu/po/Papers/roms_adjoint.pdf
Di Lorenzo, E., A.M. Moore, H.G. Arango, B. Chua, B.D. Cornuelle, A.J. Miller and A. Bennett, 2005: The Inverse Regional
Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies, Ocean
Modelling, In preparation.
Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2004: A comprehensive ocean prediction
and analysis system based on the tangent linear and adjoint of a regional ocean model, Ocean Modelling, 7, 227-258.
http://marine.rutgers.edu/po/Papers/Moore_2004_om.pdf
Moore, A.M., E. Di Lorenzo, H.G. Arango, C.V. Lewis, T.M. Powell, A.J. Miller and B.D. Cornuelle, 2005: An Adjoint Sensitivity
Analysis of the Southern California Current Circulation and Ecosystem, J. Phys. Oceanogr., In preparation.
Wilkin, J.L., H.G. Arango, D.B. Haidvogel, C.S. Lichtenwalner, S.M.Durski, and K.S. Hedstrom, 2005: A Regional Modeling
System for the Long-term Ecosystem Observatory, J. Geophys. Res., 110, C06S91, doi:10.1029/2003JCC002218.
http://marine.rutgers.edu/po/Papers/Wilkin_2005_jgr.pdf
Warner, J.C., C.R. Sherwood, H.G. Arango, and R.P. Signell, 2005: Performance of Four Turbulence Closure Methods
Implemented Using a Generic Length Scale Method, Ocean Modelling, 8, 81-113.
http://marine.rutgers.edu/po/Papers/Warner_2004_om.pdf
Background Material
Overview
• Let’s represent NLM ROMS as:
• The TLM ROMS is derived by considering a small
perturbation s to S. A first-order Taylor expansion yields:
A is real, non-symmetric
Propagator Matrix
• The ADM ROMS is derived by taking the inner-product with
an arbitrary vector
, where the inner-product defines an
appropriate norm (L2-norm):
Tangent Linear and Adjoint Based GST
Drivers
• Eigenmodes of R (0, t ) and
• Singular vectors:
T
R (t,0)
T
R (t,0) XR(0, t )
T





• Forcing Singular vectors:   R(t, )dt  X   R(t , )dt 
0

0

 
|t t '|/ tc T
'
e
R
(

,
t
) XR(t , )dt ' dt
• Stochastic optimals:  
0 0
• Pseudospectra:
 I  A
H
 I  A
1
Two Interpretations
• Dynamics/sensitivity/stability of flow to
naturally occurring perturbations
• Dynamics/sensitivity/stability due to
error or uncertainties in the forecast
system
• Practical applications:
 Ensemble prediction
 Adaptive observations
 Array design ...
GSA on the Southern California Bight (SCB)
SST and Surface
currents
Free-Surface
Eigenmodes
• TLM eigenvectors (A): normal modes
• ADM eigenvectors (AT): optimal excitations
Real Part
Imag Part
SCB coastally trapped waves
Optimal Perturbations
• A measurement of the fastest growing of all
possible perturbations over a given time interval
diffluence
confluence
SCB maximum growth of perturbation energy over 5 days
Stochastic Optimals
Provide information about the influence of stochastic
variations (biases) in ocean forcing
SCB patterns of stochastic forcing that maximizes the
perturbation energy variance for 5 days
Singular Vectors
Open Boundary Sensitivity: errors growth quickly and appear
to propagate through the model domain as coastally trapped
waves.
Ensemble
Prediction
• Optimal perturbations / singular vectors and
stochastic optimal can also be used to generate
ensemble forecasts.
• Perturbing the system along the most unstable
directions of the state space yields information
about the first and second moments of the
probability density function (PDF):
 ensemble mean
 ensemble spread
• Adjoint based perturbations excite the full
spectrum
Data Assimilation Overview
• Cost Function:
where
model,
background,
observations,
inverse background error covariance,
inverse observations error covariance
• Model solution depends on initial conditions (
boundary conditions, and model parameters
• Minimize J to produce a best fit between model
),
and observations by adjusting initial conditions,
and/or boundary conditions, and/or model
parameters.
Minimization
• Perfect model constrained minimization (Lagrange
function):
We require the minimum of
,
,
at which:
,
yielding
• AT is the transpose of A, often called the adjoint
operator. It can be shown that:
The adjoint equation solution
provides gradient information
4D Variational Data Assimilation
Platforms (4DVAR)
• Strong Constraint (S4DVAR) drivers:
 Conventional S4DVAR: outer loop, NLM, ADM
 Incremental S4DVAR: inner and outer loops, NLM,
TLM, ADM (Courtier et al., 1994)
 Efficient Incremental S4DVAR (Weaver et al., 2003)
• Weak Constraint (W4DVAR) - IOM
 Indirect Representer Method: inner and outer loops,
NLM, TLM, RPM, ADM (Egbert et al., 1994; Bennett
et al, 1997)
RP:
Forward and Adjoint MPI Communications
AsendE
TILE L
ArecvE
ArecvW
AsendW
TILE R
receive
Jend
Jend
Forward
i-2 i-1
i
i+1
i-2 i-1
i
i+1
Jstr
Jstr
send
Istr
V iL =
V iL+1 =
TILE L
Iend
R
Vi
V iR+1
AsendE
Istr
V iR-2 =
V iR-1 =
ArecvE
ArecvW
AsendW
Iend
L
V i-2
V iL-1
TILE R
ad_send
Jend
Jend
Adjoint
i-2 i-1
i
i+1
i-2 i-1
i
i+1
Jstr
Jstr
ad_receive
Istr
ad_V iL-2 =
Iend
L
ad_V i-2 + ad_V iR-2
Istr
R
; V i-2 = 0
ad_V iL-1 = ad_V iL-1 + ad_V iR-1 ; V iR-1 = 0
ad_V iR
=
ad_V iR
+
Iend
L
i
L
; Vi
=0
ad_V iR+1 = ad_V iR+1 + ad_V iL+1 ; V iL+1 = 0