Application of Adjoint-Derived
Sensitivity Gradients to Targeted
Observations for Tropical Cyclone
Steering: An Improved Methodology
Brett Hoover
Michael Morgan
University of Wisconsin - Madison
5 May 2009
An Adjoint-Based Targeting Strategy
assimilation system
y,x b
ob’s and bkgrnd
R R
,
y x b
forecast model
xa
xf
analysis
forecast
adjoint of
assimilation system
 R
x a
sensitivity to ob’s and
background

R
adjoint of forecast
model
sensitivity to analysis
R
x f
sensitivity
to forecast
Sensitivities of steering to the (potential) observations are informed by the
sensitivities to the analysis, the characteristics of the analysis errors, and the
nature of the assimilation system.


An Adjoint-Based Targeting Strategy
forecast model
The focus of this
talk: “How to define
best the response
function, R?”
xa
xf
analysis
forecast
 R
x a

adjoint of forecast
model
sensitivity to analysis

R

R
x f
sensitivity
to forecast
Steering Response Function
R1 = Average zonal wind in a
box centered on the TC:
R1 


300
850


A
300
850
u dx dy

A
dx dy
6
9
Steering Response Function
R1 = Average zonal wind in a
box centered on the TC:
R1 

300
850


A
300
850
u dx dy

dx dy
A
R1 represents the zonal

steering of the TC only when
the TC is centered in the
response function box, with the
symmetric circulation around
the TC being canceled out
6
9
Steering Response Function
R1 = Average zonal wind in a
box centered on the TC:
R1 

300
850


A
300
850
u dx dy

dx dy
A
A northward displacement of

the TC will result in a positive
contribution to zonal flow in the
box
6
9
What Happens to the “Environmental
Flow”?
R1

Use NOGAPS
adjoint model to
calculate
sensitivity of R1
with respect to
vorticity
'
r
V ' env
Scale sensitivities
at 500 hPa to
define
perturbations to
vorticity at
initialization
Calculate
perturbation to
“environmental
wind” between
perturbed and
control run

What Happens to the “Environmental
Flow”?
r
V ' env
1. Vorticity and Divergence …
What Happens to the “Environmental
Flow”?
r
V ' env
1. Vorticity and Divergence …
are removed from response
function box
What Happens to the “Environmental
Flow”?
r
V ' env
1. Vorticity and Divergence …
are removed from response
function box
2. Streamfunction and velocity
potential are calculated from
“environment”
What Happens to the “Environmental
Flow”?
r
V ' env
1. Vorticity and Divergence …
are removed from response
function box
2. Streamfunction and velocity
potential are calculated from
“environment”
3. Environmental flow is
calculated
What Happens to the “Environmental
Flow”?
“Environmental
flow” for control
run
“Environmental
flow” for run with
perturbed initial
conditions
Perturbation
“environmental
flow” (perturbed –
r
control)
V ' env
What Happens to the “Environmental
Flow”?
Perturbing the model
using sensitivities for R1
results in a perturbation
environmental flow with a
strong southerly
component
This southerly advection
pushes the TC slightly
north of the center of the
response function box,
allowing the TC’s own
circulation to contribute
positively to the response
function
6
9
We Need a New Response Function…
• Sensitivities of R1 are largely influenced by
small perturbations to the final-time
location of the TC in the response function
box
• This is evidenced by a strong southerly
component to the environmental flow
advecting the TC
• Why not just use the environmental flow
as our response function? RE1
R1 Perturbation
6
9
RE1 Perturbation
6
9
6
9
Perturbations to the “environmental flow” are significantly
different
R1 Perturbation
6
9
RE1 Perturbation
6
9
6
9
Perturbations to the “environmental flow” are significantly
different
R1 perturbations create a southerly advecting flow, while RE1
perturbations create a purely zonal advecting flow
Conclusions
• Adjoint models, coupled with measures of
statistical uncertainty in initial conditions, can be
used to objectively define targeting regions for
adaptive observations to improve specific
aspects of a model forecast (e.g. TC steering)
• Response functions currently employed to
define TC steering suffer from a complication:
perturbations to the final-time location of the TC
greatly influence these response functions
• New response functions defining the
“environmental flow” in the vicinity of the TC can
alleviate this problem