DATA ASSIMILATION IN SHELF CIRCULATION MODELS
PIs: Alexander L. Kurapov, J. S. Allen, G. D. Egbert,
R. N. Miller
2.
Results using linear
models and optimal, variational data assimilation
3.
Results using a full primitive equation model and sub-optimal,
sequential optimal interpolation (OI)
4.
Ongoing research: using variational
Generalized Inverse Method with nonlinear models
5.
References
6.
Links
For
a circulation model to be relevant to the
-
Application of
optimal data assimilation schemes to simplified linear models, and
-
Application of
simplified, sub-optimal data assimilation schemes to a full primitive equation
model.
Presently, we are working
towards merging these two data assimilation approaches, assessing utility of
the variational, representer-based
Generalized Inverse Method (GIM) (Chua and Bennett 2001, Bennett 2002) for data
assimilation into nonlinear models of coastal ocean dynamics.
Applications have included:
-
Wind-driven shelf
circulation
-
Internal tide on
the shelf
-
Currents over beach
topography
2. Results using linear models and optimal, variational
data assimilation:¿
Studies
with linearized models and the GIM have included the
analysis of the spatial and temporal structure of the representers
(proxies for the model or forecast error covariances)
using analytical dynamical models (Scott et al. 2000, Kurapov
et al. 1999, 2002, 2005a). The existence of the traveling components in the representers explains how the correction made to the
forecast solution propagates alongshore, providing solution improvement at the
distances exceeding the forecast error decorrelation
length scales.
Numerical
approaches with linearized models on realistic
topography proved that these models can be usefully applied to real data. A
linear, baroclinic, spectral in time tidal inverse
model has been developed for assimilation of surface currents from HF radars (Kurapov et al. 2003). A series of inverse solutions
provides a uniquely detailed picture of the spatial and temporal variability of
the M2 internal tide on the mid-Oregon shelf. In that study specifics of data
assimilation correcting open boundary conditions has also been discussed.
Studies using suboptimal
sequential optimal interpolation (OI) and the nonlinear primitive equation
Princeton Ocean Model (POM) (Blumberg and Mellor, 1987) focused on aspects of
modeling the subinertial wind-driven circulation.
Formulation of suitable open boundary conditions for the nonlinear hydrostatic
primitive equations that would yield a well-posed mathematical problem remains
an outstanding issue (Oliger and Sundström
1978, Bennett 2002). In our POM implementation for circulation on the Oregon
shelf, we have avoided this problem by imposing alongshore periodic boundary
conditions. For shelf flows that are strongly
wind-driven with the mesoscale behavior dominated by
flow-topography interactions, the model in this geometry represented important
aspects of the observed flows for limited time periods (45 to 90 days). The
OI data assimilation scheme sequentially updates state variables based on
data-model differences and stationary estimates of forecast and data error
statistics. Correction to the state variables is introduced gradually over time
to overcome the re-initialization problem. The POM-OI system was developed and
implemented initially for assimilation of HF radar surface current data (Oke et al. 2002), demonstrating the value of surface
information for constraining the model solution at depth.
The study with POM has been
extended to assimilation of velocity profiles from an array of acoustic Doppler
profiler (ADP) moorings. We find that data assimilation improves prediction of
currents at an alongshore distance of 90 km from the site where data were
assimilated, both to the north and to the south (Kurapov
et al. 2005a). Improvement to the south in part results from advection of
corrections with the southward upwelling jet. Improvement to the north may be
an effect of northward propagation of information by coastally trapped waves.
Moored velocity data assimilation also improves prediction of other oceanic
fields of interest, such as SSH, temperature, salinity, turbulent dissipation
rate in the bottom boundary layer, and bottom stress, as confirmed by
comparisons with data that are not assimilated (Kurapov
et al. 2005b). Understanding that moored velocity data assimilation improves
modeled variability near the bottom facilitated the study of temporal and
spatial variability in the bottom mixed layer (BML) on the mid-Oregon shelf
using the data assimilation solution (Kurapov et al.
2005c). Model results suggest that the response of the BML thickness to
upwelling and downwelling favorable winds differs
qualitatively between the area of “simple” bathymetric slope (45N)
and a wider shelf area east of Stonewall Bank (44.5N). On the wider shelf,
where the alongshore current is separated from the coast, increased BML in
response to upwelling favorable winds is explained to be associated with the
enhanced bottom Ekman transport convergence and local
convective mixing at the top of the bottom boundary layer. Data assimilation is
shown to control both the intensity and timing of the events of large BML
thickness.
4. Ongoing
research: using variational GIM with nonlinear models
We
are currently assessing the utility of GIM for coastal applications, in which
nonlinearity is essential. Using GIM, optimal solutions are obtained
iteratively through a sequence of solutions to the linearized
Euler-Lagrange equations (i.e., the conditions for the minimum of the quadratic
penalty functional, a sum of data and model error terms), using tangent linear
(TL) and adjoint (AD) counterparts of the nonlinear
model. The indirect representer algorithm (Egbert et al. 1994, Chua and Bennett 2001) has made GIM
practical for the large three-dimensional and time-dependent circulation
problems (e.g., Bennett et al. 1998).
GIM has many advantages over
other data assimilation methods. Compared to sequential OI, it better
represents model error statistics and propagation of assimilated information
under changing oceanic conditions (such as jet meandering, eddy formation,
transition from upwelling to downwelling, etc.) since
the state-dependent model (forecast) error covariance is implied. Unlike any
sequential method (OI, or variants of the ensemble or reduced space Kalman Filter), variational
methods are capable of projecting observations back in time, providing
correction to model input errors in the recent past. Compared to variational minimization in the state space (the “adjoint” method or “4DVAR”), GIM is
potentially more efficient since it searches for the solution in the generally
much smaller data space, spanned by representers
(Bennett, 2002). GIM is especially flexible and explicit in the specification
of the error covariances in the inputs, allowing
control over dynamical consistency in corrections to the multivariate fields at
initial time (Kurapov and Di
Lorenzo, 2005) or at the open boundary (Kurapov et
al. 2003). GIM also includes methodology for prior and posterior model error
analysis, enabling the observational array assessment (even before the actual
observational platforms and instruments are deployed).
We
are currently testing utility of TL and AD codes for the Regional Oceanic
Modeling System (ROMS), using a beta-version of the TL and AD ROMS codes
provided to us by A. Moore (U. Colorado), H. Arango (Rudgers U.) and colleagues. Successful tests using GIM and
the two-dimensional (barotropic) mode of the TL and
AD ROMS have been performed, assimilating synthetic data in a problem of
nonlinear jet instability (Kurapov and Di Lorenzo, 2005). That study has shown that the choice of
the initial condition error covariance affects convergence of the GIM
algorithm. The optimal covariance provides dynamically consistent corrections
to initial pressure and velocity fields.
We have
also developed our own TL and AD codes for the nonlinear shallow water
equations, using some algorithmic features and recipes from ROMS. These codes will be used as a
“workhorse” to test new ideas and address a number of fundamental issues
in data assimilation, such as:
-
Correction of open boundary
conditions and space- and time-varying forcing;
-
Definition of inputs, outputs, and
data functionals (matching data and model output)
providing economical and efficient minimization
-
Effective and dynamically
consistent covariance smoothers
-
Approaches for overcoming
non-smoothness of open boundary conditions due to logical switching between
inflow and outflow conditions
-
GIM implementation for assimilation
over long time intervals in flows with instabilities
-
Testing utility of the Inverse
Ocean Modeling (IOM) system, which is an interface for GIM implementation,
currently under development by A. Bennett (OSU) and colleagues.
The
new shallow-water TL and AD models are being applied to the problem of flows
over beach topography forced by gradients in the radiation stress tensor
resulting from breaking waves (Slinn et al., 2000).
These experiments will help us resolve some fundamental issues in coastal ocean
data assimilation with regard to open boundary conditions and physical
instabilities in the TL model. While the growth of instabilities is constrained
by the nonlinear advection in the fully nonlinear model, it is not similarly
constrained in the companion TL model, potentially posing threat to the
stability of GIM. Our study, using synthetic data, will progress from a case
with large dissipation, in which the ocean flow is stationary in response to
the stationary forcing, to cases of relatively smaller dissipation, in which
the model exhibits instabilities, including a regular equilibrated wave pattern
or irregular, “turbulent” behavior. For these classes of flows, we
will determine the utility of GIM and requirements for adequate data
coverage.
(Publications
from this project are marked
***)
Bennett, A. F., B. S. Chua, D. E. Harrison, and
M. J. McPhaden, 1998: Generalized inversion of the
Tropical Atmosphere-Ocean (TAO) data and a coupled model of the tropical
Pacific, J. Climate, 11, 1786-1792.
Blumberg, A. F., and G.
L. Mellor, 1987: A description of a three-dimensional coastal ocean circulation
model, in Three-dimensional Coastal Ocean Models, Coastal Estuarine Sci. Ser., Vol.
4, edited by N. Heaps, 1-16, A.G.U.,
Chua, B. S., and A. F.
Bennett, 2001: An inverse ocean modeling system, Ocean Modelling,
3, 137-165.
Egbert, G. D., A. F. Bennett,
and M. G. G. Foreman, 1994: TOPEX/POSEIDON tides estimated using a global
inverse model, J. Geophys.
Res., 99 (C12), 24 821–24 852.
Kurapov, A. L., J. S. Allen, R. N. Miller, and G. D. Egbert, 1999: Generalized inverse for baroclinic coastal flows. Proc. 3rd Conference on Coastal Atmospheric and Oceanic Prediction and Processes, 3-5 November 1999, New Orleans, LA, 101-106. ***
Kurapov, A.L., G.D. Egbert, R.N. Miller, & J.S. Allen, 2002: Data assimilation in a baroclinic coastal ocean model: ensemble statistics and comparison of methods, Mon. Wea. Rev., 130, 1009-1025. ***
Kurapov, A.L., G.D. Egbert, J.S. Allen, R.N. Miller, S.Y. Erofeeva & P.M. Kosro, 2003: M2 internal tide off Oregon: inferences from data assimilation, J. Phys. Oceanogr., 33, No. 8, 1733-1757. ***
Kurapov, A. L., J. S. Allen, G. D. Egbert, R. N. Miller, P. M. Kosro, M. Levine, and T. Boyd, 2005a: Distant effect of assimilation of moored currents into a model of coastal wind-driven circulation off Oregon, J. Geophys. Res., 110, C02022, doi:10.1029/2003JC002195. ***
Kurapov, A. L., J. S. Allen, G. D. Egbert, R. N. Miller, P. M. Kosro, M. Levine, T. Boyd, J. A. Barth, and J. Moum, 2005b: Assimilation of moored velocity data in a model of coastal wind-driven circulation off Oregon: multivariate capabilities, J. Geoph. Res. – Oceans, in press (COAST special issue) ***
Kurapov, A. L., J. S. Allen, G. D. Egbert, and R. N. Miller, 2005c: Modeling bottom mixed layer variability on the mid-Oregon shelf during summer upwelling, J. Physic. Oceanogr., in press ***
Kurapov, A. L., and E. Di Lorenzo, 2005: A data assimilation test of the tangent linear and adjoint ROMS: shallow water channel flow, unpublished manuscript ***
Moore, A. M., H. G. Arango, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. 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.
Oke, P.R., J.S. Allen, R.N. Miller, G.D. Egbert, & P.M. Kosro, 2002: Assimilation of surface velocity data into a primitive equation coastal ocean model, J. Geophys. Res., 10.1029/2000JC00511. ***
Oliger, J.,
and A. Sundström, 1978: Theoretical and practical
aspects of some initial boundary value problems in fluid dynamics,
Scott, R. K., J. S. Allen, G. D. Egbert, and R. N. Miller, 2000: Assimilation of surface current measurements in a coastal ocean model, J. Phys. Oceanogr., 30, 2359-2378. ***
Slinn, D. N., J. S. Allen, and R. A. Holman (2000), Alongshore currents over variable beach topography, J. Geophys. Res., 105(C7), 16,971–16,998.
OSU
Tidal Data Inversion (G. D. Egbert & S. Y. Erofeeva)
A. F. Bennett's Inverse Ocean Modeling project
(IOM)
COAS physical
oceanography page
Ocean Modeling and Data Assimilation at COAS
Coastal Ocean
Advances in Shelf Transport (COAST) project
Funding for
this project is provided by the Office
of Naval Research through Grant #N00014-98-1-0043.
Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of
the Office of Naval Research.
This page is
last updated July 22, 2005