Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 233

strong-constraint of all methods is used in both data poor and data rich regimes
to examine the efficacy. To accomplish these experiments, a system that is weakly
nonlinear during the assimilation window is required. The Lorenz ( 1963 ) (hereafter
referred to as Lorenz63) system provides a nonlinear, dynamic system with particu-
lar sensitivity to the initial state, x

. The Lorenz63 system is a well used test case
in dynamical systems due to its highly nonlinear but basic structure. Prediction is a
difficult problem with its strong sensitivity to changes in the initial conditions.
One of the first adjoint-based assimilation experiments using Lorenz63 was
performed by Gauthier ( 1992 ). This work was followed by other experiments using
both adjoint and Kalman filter techniques by Evensen and Fario ( 1997 )and Miller
et al. ( 1994 ). More recently, Ngodock et al. ( 2007 ) examined how well the weakly-
constrained Representer method can be used to approximate strongly nonlinear
flows in an observation-rich environment.
The Lorenz63 system is a simplified model of convective atmospheric dynamics.
It is unforced ( f

.t 0 /

to
describe the convective motion intensity, temperature difference between vertical
currents, and vertical temperature deviation from linearity, respectively. The model
parameters

.t i / D 0 ) and uses a three-dimensional state vector .x;y; z /

describe the Prandtl number, ratio of the Rayleigh number to
criticality, and convective period. Parameter values of

.;r;b/

are chosen to
provide a strongly nonlinear flow as in Gauthier ( 1992 ). The forward integration of
the model is performed with the standard fourth-order Runge-Kutta method. The
tangent-linear operator M is implemented with the tangent-linearization of both
the nonlinear model and Runge-Kutta integrator. Likewise, the adjoint model is
represented by M T .
Gauthier ( 1992 ) examined two regimes of the Lorenz63 system: “regular,” in
which the system remains within a single attractor during the time window and
is weakly nonlinear, and the “transition” case that changes attractor for the other
during the time window and is strongly nonlinear. For each case, Gauthier ( 1992 )
integrated from

.10;28;8=3/

; however, these periods are too long for linear methods.
As shown by Gauthier ( 1992 )and Evensen and Fario ( 1997 ), the local minima (due
to nonlinearity) in the transition period limit the effectiveness of the assimilation.
Miller et al. ( 1994 ) showed that these issues were due to the length of time window
used.
Determining the length of the window to keep the system weakly nonlinear
requires a metric to evaluate any differences between the linear and nonlinear
solutions. A simple cost-function to compare a perturbed trajectory against the
unperturbed is given by

t D Œ0;8

x x t / T Q 1 . ,where x is a perturbed model

trajectory sampled at every timestep, x t is the unperturbed model trajectory, and Q
is the prescribed measure of estimate error chosen to be diagonal with values of 2.
Perturbations were randomly chosen with a variance of Q and integrated through
the tangent-linear model (linearized about x t ). These perturbations were also added
to x t and integrated through the nonlinear model ( 10.1 ). Any differences between
the nonlinear and linear cost functions, A , are due to unresolved nonlinearity.
Figure 10.1 shows the dramatic difference between two different time windows.
In Fig. 10.1 a, the system is weakly nonlinear over 1.9 time units; however, over the

A D . x x t /