A Simplified Dynamical System for Understanding the Intensity-Dependence of Intensification Rate of a Tropical Cyclone Yuqing Wang International Pacific Research Center and Department of Atmospheric Sciences, University of Hawaii at Manoa, Honolulu, Hawaii, USA Jing Xu State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, CMA, Beijing, China International Workshop on High Impact Weather Research 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 (V max ) with red and black curves indicating the smoothed 50 th and 95 th percentiles of IR for the given storm intensity for Atlantic TCs during Xu and Wang (2014)

EYE Outflow Inflow Eyewall 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 V max of the storm, the intensification rate (IR) should increase with the increase in TC intensity. B: heating source V: momentum forcing Heating efficiency A, B, C, D, E indicate increasing in the inner core inertial stability

Thermodynamic Control of TC Intensity and its change A Carnot Heat Engine View SST T out Carnot heat engine Emanuel 1988

Energy Budget in the Carnot Heat Engine  The thermodynamic efficiency of the Carnot heat engine C k The surface exchange coefficient |V| The near surface wind speed k* o Enthalpy of the ocean surface k a Enthalpy of the atmosphere near the surface  Air density near the surface C D The surface drag coefficient Rate of Intensity Change = Rate of Energy Input – Dissipation Rate

|V||V| Energy change rate |V mpi | MPI Energy input Dissipation rate 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 V mpi, 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||V| Energy change rate |V mpi | MPI Energy input - Dissipation rate Intensification |V mpir | Wang and Xu (2015) EGRmpi If we consider that 5% of TCs could reach their MPI of kt. The lifetime maximum IR could be kt, very close to the peak for the 95th percentile IR in observations.

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 term Second term: limits the maximum wind to an upper bound (V mpi ) 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 V mpi. Letting, we can find that IR reaches a maximum value when the storm intensity is around V mpir given by

We use the lifetime maximum intensity of each storm as an estimate for the steady state intensity V s (a)Scatter diagram of the lifetime maximum 24-h IR (IR max ) against the averaged storm intensity during the 24-h IR max period normalized by the lifetime maximum intensity of the storm (namely V/V s ). (b)The frequency distribution of the lifetime IR max as a function of the corresponding normalized storm intensity.

Conclusions Observations 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.

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