Application of Adjoint-Derived Sensitivity Gradients to Tropical Cyclone Intensification

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Application of Adjoint-Derived Sensitivity
Gradients to Tropical Cyclone Intensification

• Brett T. Hoover and Michael C. Morgan
• University of Wisconsin Madison
• 20 August 2009

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What is an Adjoint Model?

• Starting with some differentiable function of
  model output, R

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What is an Adjoint Model?

• Starting with some differentiable function of
  model output, R

We can define the gradient of that response
function with respect to the model state at
verification time, Xout
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What is an Adjoint Model?

• Starting with some differentiable function of
  model output, R

ADJ
The adjoint model then integrates these gradients
backward through time to calculate the gradient
of that same response function with respect to
model input, Xin
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What is an Adjoint Model?

• Starting with some differentiable function of
  model output, R

For a response function R that describes the
intensity of a tropical cyclone, this quantity
describes the sensitivity of TC intensity to
various parts of the model initial conditions.
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Response Functions For TC Intensity

• An appropriate response function for TC intensity
  can take a number of forms
• Low-level kinetic energy? (Doyle et al. 2008)
• Low-level vorticity? (Vukicevic and Raeder 1995)
• Vertical motion?
• How can we be sure that the response function,
  and the sensitivity of that response function, is
  appropriate?

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Response Functions For TC Intensity
Define Response Function R for TC Intensity
Calculate Sensitivity of R With Respect to
Initial Condition Vorticity
Perturb Initial Condition Vorticity in Regions of
High Sensitivity
Observe Impact of Perturbations on Response
Function at Final Time
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Case Study
36-Hour simulation of Hurricane Noel (2007)
during extra-tropical transition
NOGAPS model is run at T159L30 resolution to
define basic state. Sensitivities are calculated
for a number of response functions using the
NOGAPS adjoint model.
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Case Study Low-Level Kinetic Energy
LEFT
RIGHT
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Case Study Low-Level Kinetic Energy
Perturbations introduced at model initialization
produce a change in central minimum pressure of
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Case Study Low-Level Kinetic Energy
Perturbations introduced at model initialization
produce a change in central minimum pressure of
4 hPa. The cyclone weakens and is pushed
slightly to the west.
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Case Study Low-Level Kinetic Energy
LEFT
RIGHT
Unperturbed
Perturbed
Perturbations resulted in an increase in KE
through a strengthening of the pressure gradient
on the east side of the cyclone.
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Case Study Low Level Vorticity
Perturbations introduced at model initialization
produce a change in central minimum pressure of
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Case Study Low Level Vorticity
Perturbations introduced at model initialization
produce a change in central minimum pressure of
-8 hPa. The cyclone intensifies and is pushed
slightly to the east.
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Case Study Low Level Vorticity
LEFT
RIGHT
Unperturbed
Perturbed
Perturbations resulted in an increase in
vorticity through a reduction of negative
vorticity in the box. Intensification is
incidental.
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Case Study Column Stretching

• The stretching term of the vorticity equation
• Can be used as a response function.
  Perturbations in regions of strong sensitivity to
  stretching should intensify the cyclone.

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Case Study Column Stretching
Perturbations yield a change of -4 hPa, which
appears to be a non-incidental change. However,
the methodology is flawed By defining the
response function this way, we constrain the
adjoint to focus on this one process of
intensification. The purpose is to define an
appropriate response function for TC intensity,
and allow the adjoint model to determine what
processes are important for intensification.
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Conclusions

• While the adjoint model is a powerful tool for
  dynamical research, caution must be taken when
  defining a response function.
• The lack of dynamical interpretation of
  adjoint-derived sensitivity gradients makes the
  diagnosis of these kinds of methodological
  problems difficult.
• There is, as yet, no response function for TC
  intensity that satisfies our conditions for
  appropriateness. Research is ongoing with
  response functions designed to overcome observed
  difficulties.