1. Rapid Intensification of Tropical Cyclones by Organized Deep Convection Chanh Q. Kieu, and Da-Lin Zhang Department of Atmospheric and Oceanic Science University of Maryland, College Park

2. Motivations Growth of the 850-hPa tangential wind of Hurricane Diana (1984) during a 24-h period. From Willoughby (1990). Solid - t = 0 h Dashed - t = 24 h Time series of the minimum surface pressure (hPa) and maximum surface wind (m s -1 ) associated with Hurricane Wilma (2005). Solid - Best track Dashed - Simulation Why does V grows much more rapidly in the Inner core than the outer regions? p MIN V MAX Why do hurricanes often undergo a rapid deepening stage?

3. Growth of the area-averaged peak vertical motion (dotted), maximum tangential wind (dashed), and surface central-pressure drop (solid) associated with the simulated Wilma (2005). Why do W MAX, V MAX and P MIN grow at different rates? Is there any relationship between them? W MAX V MAX P MIN

4. Objectives Develop an analytical theory of TC intensification due to organized deep convection; and Determine the structures of TC vortices for hurricane models, and the surface pressure and wind relationship, given the vertical motion field.

5. Consider the complete primitive Eqs in the pseudo-height cylindrical coordinates Basic equations All variables have their traditional meteorological meaning.

6. TCs are axi-symmetric; A positive feedback between latent heating J and vertical motion w, i.e., N 2 w = J/T, and First order friction F (u,v ) =  (u,v). Simplified Equations: Assumptions: (6) (7) (8)

7. Solution for secondary circulation Given the vertical motion w (or latent heating) profile: where   (  - N 2 ) 1/2. Substitute w into the continuity, we obtain solutions for the radial wind where H(z)=W 0 sin( z), S = N 2 /g, and Q(z)=W 0 [S sin( z)- cos( z)]/2. Region I Region II (9) (10)

8. Solutions for tangential flow (region I) Substituting w 1 and u 1 into the Eq. (7) gives The only separable solution that this Eq. can accept is of the form v 1 = F 1 (z,t) r. Upon substituting this form into the above Eq, we get where  =  W 0 G 1 /  G 0. - Integration constant (11)

9. Solutions for tangential wind (region II) Following a step similar to that in region I, we substitute w 2 and u 2 into Eq. (7) and get Unlike region I, the disappearance of the vertical advection makes the tangential wind Eq. in region II much simpler. The only separable solution that this Eq. can accept is of the form v 2 = F 2 (z,t)/r. Upon substituting this form into the above Eq. and performing an integration, we get where (12)

10. Solutions for geopotential height  Solution for  in region I can be obtained by substituting u 1, w 1, and v 1 into Eq. (6), followed by an integration: where  a (z) is the geopotential perturbation at r = a and will be given by the solution for region II. Solution for region II is: where  0 is a constant that is chosen so that the perturbation geopotential  2 = 0 at r = R m. (13) (14)

11. Tangential winds in the inner-core region could grow double- exponentially while they could grow exponentially in the outer region; the latter is due to the absence of upward transport of absolute angular momentum Verification - growth rates

12. Given the vertical motion as a function of time, the exact solution reproduces the growth of the surface central pressure and the maximum wind during the 18-h deepening period of Hurricane Wilma (2005), starting from 1200 Z 18 Oct 2005. Growth rate for Wilma (2005) P MIN V MAX

13. External parameters

14. Vertical structure of the tangential flows depend on that of vertical motion Vertical structures Comparison of the vertical profile of the area-averaged tangential wind between the observed (dashed, from McBride 1981) and the exact solution (solid).

15. Application: a 3D TC vortex Given the vertical profile of W, a 3D vortex can be constructed by setting t = 0 in all the exact solutions. Radius-height cross section of the tangential wind (contoured) and pressure perturbations (shaded), superimposed by in-plane flow vectors, as constructed from the exact solutions with V max = 30 m s -1 at a = 100 km and z = 1 km, and W 0 =0.12 m s -1.

16. Application: Pressure-wind relationship A common approach to obtaining the pressure-wind relationship is to use the statistical method by postulating some relationship based on physical arguments. The regressive method will then give us a set of recursive coefficients. With the exact solutions, all we need to do is to combine solution for V and  taken at r = a and r = 0, z = 0. After some manipulations, we get where

17. Summary and conclusions  Given the vertical motion field, we have derived he evolution and structures of TCs.  Given the vertical motion field, we have derived a set of exact solutions that can capture the evolution and structures of TCs.  TCs grow more rapidly in the core than the outer region due to the nonlinear advection of absolute angular momentum;  TCs grow more rapidly in the core than the outer region due to the nonlinear advection of absolute angular momentum;  Given the exponential growth of the secondary circulation, the primary circulation would tend to grow at a double-exponential rate;  Given the exponential growth of the secondary circulation, the primary circulation would tend to grow at a double-exponential rate;  The vertical distribution of rotational flow is closely related to that of vertical motion; and  The vertical distribution of rotational flow is closely related to that of vertical motion; and   The analytical solutions can be used to construct 3D vortices and derive the pressure-wind relationship.