1. Application of Adjoint-Derived Sensitivity Gradients to Tropical Cyclone Intensification
Brett T. Hoover and Michael C. Morgan
University of Wisconsin – Madison
20 August 2009

2. What is an Adjoint Model?
Starting with some differentiable function of model output, R:

3. 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

4. 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

5. 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.

6. 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?

7. Response Functions For TC Intensity
Define Response Function R for TC Intensity
Calculate Sensitivity of R With Respect to Initial Condition Vorticity
Observe Impact of Perturbations on Response Function at Final Time
Perturb Initial Condition Vorticity in Regions of High Sensitivity

8. 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.

9. Case Study – Low-Level Kinetic Energy
LEFT
RIGHT

10. Case Study – Low-Level Kinetic Energy
Perturbations introduced at model initialization produce a change in central minimum pressure of …

11. 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.

12. 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.

13. Case Study – Low Level Vorticity
Perturbations introduced at model initialization produce a change in central minimum pressure of …

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.