A Simplified Dynamical System for Understanding the Intensity-Dependence of Intensification Rate of a Tropical Cyclone

Yuqing WangInternational Pacific Research Center and Department of

Atmospheric Sciences, University of Hawaii at Manoa, Honolulu, Hawaii, USA

Jing XuState Key Laboratory of Severe Weather, Chinese Academy of

Meteorological Sciences, CMA, Beijing, China

International Workshop on High Impact Weather Research20-23 January 2015, Ningbo, China

Outline

• Motivation

• A Carnot heat engine view

• An alternative view based on a simplified dynamical system for intensity forecast (LGEM)

• Conclusions

Different intensification rate (IR)

RI of a TC is often defined as an increase in the peak 10-m wind speed of 30 knots (or roughly 15m/s) in 24 h

Frank Marks (Director of Hurricane Research Division)

“I have often wondered how quickly a TC can intensify and have questioned my smarter brethren in the TC community to provide some theoretical basis for a maximum intensification rate for a TC, such has been proposed and debated for something like MPI. It seems that it should be straightforward to use the Navier-Stokes equations and determine the peak intensity change possible. ….

The question is what controls that rate and what parameters determine it. That would provide potential bounds to the problem that would inform modelers as well as forecasters.”

Scatter diagram of the subsequent 24-h IR against the storm intensity (Vmax) with red and black curves indicating the smoothed 50th and 95th percentiles of IR for the given storm intensity for Atlantic TCs during 1988-2012. Xu and Wang (2014)

EYE

Outflow Outflow

Inflow

EyewallEyewall

Tropopause

Schematic diagram showing the dynamical processes in a strong TC

Wang （ 2014）

Schubert & Willoughby(1982): If the TC structure is given, the inner-core inertial stability is proportional to Vmax of the storm, the intensification rate (IR) should increase with the increase in TC intensity.

B: heating sourceV: momentum forcing

Heating efficiency

A, B, C, D, E indicate increasingin the inner core inertial stability

Thermodynamic Control of TC Intensity and its change

A Carnot Heat Engine View

SST

Tout

Carnot heat engine

Emanuel 1988

Energy Budget in the Carnot Heat Engine

e The thermodynamic efficiency of the Carnot heat engineCk The surface exchange coefficient

|V| The near surface wind speedk*o Enthalpy of the ocean surfacek a Enthalpy of the atmosphere near the surface Air density near the surface

out

out

T

TSST

CD The surface drag coefficient

Rate of Intensity Change = Rate of Energy Input – Dissipation Rate

|V|

Energy change rate

|Vmpi|

MPI

Energy input

Dissipation rate

)( **pi ao

D

km kk

C

CV

Intensification

Wang (2013)

At MPI,

Emanuel (1997),

During the intensification stage, the energy growth rate (EGR) of the dynamical system can be written as

Using the express of Vmpi, the above equation can be rewritten as

The storm IR depends on the storm intensity (the maximum near-surface wind speed). IR reaches a maximum when

This will lead to maximum IR to occur at an intermediate intensity

The corresponding maximum EGR will be

|V|

Energy change rate

|Vmpi|

MPI

Energy input

- Dissipation rate

)( **pi ao

D

km kk

C

CV

Intensification

|Vmpir| Wang and Xu (2015)

EGRmpi

If we consider that 5% of TCs could reach their MPI of 120-140 kt. The lifetime maximum IR could be 69-81 kt, very close to the peak for the 95th percentile IR in observations.

< 30[30, 40)> 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Vmax MPI

>= 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Vmax MPI

>= 0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Vmax MPI

>= 0.7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Vmax MPI

>= 0.8

Steady state solution

An alternative Dynamical System based on a logistic growth equation model (LGEM) of DeMaria (2009)

In DeMaria’s system, IR is mathematically given as

First term: a linear growth termSecond term: limits the maximum wind to an upper bound (Vmpi)κ is the time-dependent growth rate, and β (1/24h) and n (2.5)are positive constants that determine how rapidly and how close the solution for V can come to Vmpi.

Letting , we can find that IR reaches a maximum value when the storm intensity is around Vmpir given by

We use the lifetime maximum intensity of each storm as an estimate for the steady state intensity Vs

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10

20

30

40

50

60

70

80

90

V Vs

IRm

ax (

kt/2

4h

)

(a)95th50th

0.3 0.4 0.5 0.6 0.7 0.8 0.9

V Vs

5%

10%

15%

20%

25%

30%

35%(b)

(a)Scatter diagram of the lifetime maximum 24-h IR (IRmax) against the averaged storm intensity during the 24-h IRmax period normalized by the lifetime maximum intensity of the storm (namely V/Vs).

(b)The frequency distribution of the lifetime IRmax as a function of the corresponding normalized storm intensity.

ConclusionsObservations show a strong dependence of IR on TC intensity and

the existence of a preferred intermediate intensity for RI to occur. This was previously explained as a result of a balance between heat efficiency and the MPI.

Based the Carnot heat engine, we have developed a simplified dynamical system model to explain the observed intensity-dependence of IR. In this view, the energy input and energy dissipation rates increase with the storm intensity at quite different rates, namely linear versus a cubic power of wind speed.

In addition, an alternative simplified dynamical system for TC intensity change previously developed by DeMaria (2009) was also used to further demonstrate the nature of the dynamical system that we newly developed.

Thank you for your attention!

Questions and comments!