Tropical Cyclones

1. Observations of tropical cyclones

1.1 Tropical-cyclone tracks

1.2 Structure

1.2.1 Formation and intensification of Hurricane Patricia

1.2.2 Flight level wind structure and temperature structure

1.2.3 Vertical cross sections in Hurricane Edouard (2015)

1.2.4 Low-level structure of Hurricanes Isabel (2003) and Earl (2010)

1.2.5 Thermodynamic structure of Hurricane Earl’s eye and eyewall

1.2.6 Intensity, strength and size

1.2.7 Asymmetries

1.2.8 Secondary eyewalls

1.3 Surface heat and moisture supply

1.4 Ocean interaction

1.5 Tropical cyclone genesis

1.5.1 Formation regions

1.5.2 Necessary conditions for formation

1.5.3 Highlights from recent field experiments

1.5.4 The formation of a tropical depression

1.5.5 The multi-scale nature of genesis in the real world

1.5.6 Practical outcomes

1.6 Synthesis

2. Fluid dynamics and moist thermodynamics

2.1 The equations of motion

2.2 Buoyancy and perturbation pressure

2.3 Thermodynamics

2.3.1 Equation of state

2.3.2 Thermodynamic energy equation

2.3.3 Potential temperature and specific entropy

2.3.4 Static energy

2.4 Prognostic and diagnostic equations

2.5 Moist processes

2.5.1 Equation of state for moist air

2.5.2 Saturation and latent heat release

2.5.3 Pseudo-adiabatic ascent

2.5.4 Equivalent potential temperature, moist entropy, moist static energy

2.6 Viscosity, Diffusion, Friction and Turbulence

2.7 Methods of solution

2.8 Kinetic energy and total energy

2.9 Vorticity and the vorticity equation

2.10 Vorticity-streamfunction method

2.11 Circulation

2.11.1 Kelvin’s theorem

2.11.2 Beyond barotropy

2.12 Potential Vorticity

2.13 Balance dynamics

2.14 PV global constraints

2.15 PV flux form and impermeability theorem

2.16 Vorticity flux equation

2.17 Coordinate systems

2.18 Exercises

2.19 Appendix: The membrane analogy

3. Tropical cyclone motion

3.1 The observations to be explained

3.2 The partitioning problem

3.3 Prototype problems

3.3.1 Symmetric vortex in a uniform flow

3.3.2 Vortex motion on a beta-plane

3.3.3 The effects of horizontal shear

3.3.4 More general environmental flows

3.4 Observations of the 𝛽-gyres

3.5 Exercises

3.6 Appendices

3.6.1 Appendix 1: Transformation of the momentum equation to an accelerating frame of reference

3.6.2 Appendix 2: Derivation of Eq. (3.14)

3.6.3 Appendix 3: Derivation of Eq. (3.21)

4. Vortex axisymmetrization, waves and wave-vortex interaction

4.1 Illustration of flow asymmetries

4.1.1 Examples of vortex axisymmetrization

4.1.2 The pseudo-mode

4.2 Vortex shear waves and vortex Rossby waves

4.2.1 Force balances in a circular vortex

4.2.2 Vortex waves and instabilities

4.2.3 Generalized Rayleigh and Fjortoft instability theorems

4.2.4 Solution to initial value problem

4.2.5 Case I: Bounded Rankine vortex: 𝑉 = Γ/𝑟, Ω = Γ/𝑟2, Γ = constant, 𝑎 ≤ 𝑟 ≤ 𝑏 . . .

4.2.6 More on vortex waves

4.2.7 Relevance to tropical cyclones

4.2.8 Case II: Unbounded Rankine vortex

4.2.9 Case III: Unbounded Rankine-like vortex with multiple discontinuities in 𝜁

4.3 Wave-vortex interaction

4.3.1 Effect of discrete VR wave only

4.3.2 Effect of exterior disturbance on outer flow

4.3.3 Effect of exterior disturbances on 𝑣𝑚𝑎𝑥 . .

4.3.4 Effect of near-core disturbances on 𝑣𝑚𝑎𝑥 . .

4.3.5 Model limitations applied to smooth vortices: quasi-modes

4.3.6 Resonant wave, vortex interaction

4.4 Synthesis

4.5 Enrichment topics

4.5.1 Vortex intensification by stochastic forcing with secondary circulation

4.5.2 Point vortex analogue of wave, vortex model

4.5.3 VR wave pathway to secondary eyewall formation?

4.6 Exercises

5. Axisymmetric Vortex Theory Fundamentals

5.1 Equations of motion in rotating cylindrical polar coordinates

5.2 The primary circulation

5.3 Interpretation of the thermal wind equation

5.4 Generalized buoyancy

5.4.1 Exercises

5.5 The tropical cyclone eye

5.6 Spin up of the primary circulation

5.7 Stability

5.7.1 Barotropic vortices

5.7.2 Exercises

5.7.3 Exercises

5.7.4 Baroclinic vortices

5.7.5 Exercises

5.8 Scale analysis

5.8.1 Continuity equation

5.8.2 Momentum equations

5.8.3 Thermodynamic equation

5.8.4 Exercise

5.9 The secondary circulation

5.9.1 Exercises

5.10 Solutions of the Eliassen equation

5.10.1 Boundary effects in the membrane analogy

5.10.2 Scale effects in the membrane analogy

5.10.3 Other anisotropic effects in the membrane analogy

5.10.4 Point source solutions in an unbounded domain

5.10.5 Point source solutions in a partially bounded domain

5.11 Representation of the diabatic heating rate, 𝑄

5.11.1 Exercise

5.12 Buoyancy relative to a balanced vortex

5.13 Buoyancy in axisymmetric balanced vortices

5.14 Enrichment topics

5.14.1 Toroidal vorticity equation

5.14.2 Eliassen equation and toroidal vorticity equation

5.14.3 Geopotential tendency equation

5.14.4 Deductions from the spin-up function

5.14.5 The linear approximation, the Eliassen equation and extension to include unbalanced forcing

6. Frictional effects

6.1 Vortex spin down

6.2 Scale analysis of the equations with friction

6.2.1 𝑤-momentum equation

6.2.2 𝑢- and 𝑣-momentum equations .

6.2.3 Boundary layer depth scale

6.2.4 Boundary layer equations

6.3 The Ekman layer

6.4 The linear approximation

6.4.1 Physical interpretation

6.4.2 Mathematical solution

6.4.3 Vertical structure of the solution

6.4.4 Observed wind structure

6.4.5 Radial-vertical structure

6.4.6 Interpretation, torque balance

6.4.7 Factors determining the inflow and vertical motion

6.4.8 Dependence on vortex size

6.4.9 Supergradient winds in the linear solution

6.4.10 Exercises

6.4.11 Limitations of the linear theory

6.5 A nonlinear slab boundary layer model

6.5.1 The boundary layer equations

6.5.2 Representation of surface and top fluxes

6.5.3 The final equations

6.5.4 Starting conditions at large radius

6.5.5 Exercise

6.5.6 Slab boundary layer solutions

6.5.7 Physical interpretation

6.6 The boundary layer spin up mechanism

6.7 Limitations of the two boundary layer models

6.7.1 Advantages of the slab model

6.7.2 Limitations of boundary-layer theory in general

6.7.3 Balanced boundary layer approximation

6.8 Importance of the tropical cyclone boundary layer

6.9 Appendices

6.9.1 Appendix 1: Radial variation of 𝜈, 𝐼2, 𝑎1 and 𝑎2 in the linear boundary layer solution . .

6.9.2 Appendix 2: What determines the vertical velocity in the linear boundary layer?

6.9.3 Appendix 3: The upper boundary condition

7. Estimating boundary layer parameters

7.1 Boundary layer structure, supergradient winds

7.2 Subgrid-scale parameterizations

7.2.1 Vertical diffusivity in the boundary layer

7.3 Horizontal diffusivity in the boundary layer

7.4 Air-sea interaction, drag coefficient, enthalpy coefficient

8. A prognostic balance theory for vortex evolution

8.1 Solutions for the evolution of a balanced vortex

8.1.1 Diabatic heating, no friction

8.1.2 Friction, no heating

8.1.3 Diabatic heating and friction

8.2 Interpretation: The classical spin up mechanism

8.2.1 Exercises

8.3 Rotational stiffness, latitude dependence and vortex size evolution

8.3.1 A laboratory experiment

8.3.2 Balance considerations

8.3.3 Idealized balance simulations

8.3.4 Effects of friction on vortex size growth

8.3.5 Vortex intensity and size metrics

8.3.6 Dependence of frictionally-driven inflow on latitude

8.3.7 Summary

8.4 Interplay between diabatic heating and friction

8.4.1 Flow structure at the initial time

8.4.2 Flow structure at later times

8.4.3 Summary: the issue of convective ventilation

8.4.4 Pathological nature of the balanced boundary layer

8.4.5 Utility and limitations of the prognostic balance model

8.5 Appendix

9. Moist convection

9.1 Convective instability

9.2 Aerological diagrams

9.2.1 𝐶𝐴𝑃𝐸 and 𝐶𝐼𝑁 . . .

9.2.2 Height-temperature-difference diagram

9.2.3 More on aerological diagrams

9.2.4 The use of 𝜃𝑒 for assessing convective instability .

9.3 Types of penetrative convection

9.3.1 Shallow convection

9.3.2 Intermediate convection

9.3.3 Deep convection

9.3.4 Convective downdraughts

9.4 Understanding the effects of deep convection on the tropical circulation

9.5 Buoyancy in a finite horizontal domain

9.6 Quantification of effective buoyancy

9.7 Implications for 𝐶𝐴𝑃𝐸 . .

9.8 More on 𝐶𝐴𝑃𝐸 . .

9.9 Cloud structure in tropical cyclones

9.9.1 Ventilation by deep convection in tropical cyclones

9.10 Exercises

9.11 Appendices

9.11.1 Appendix 1: Effective buoyancy per unit volume

9.11.2 Appendix 2: Numerical solution of Eq. (9.15)

9.11.3 Appendix 3: Forcing of 𝑝′ by 𝐹𝑑 in Eq. (9.15) on the upper domain axis .

10. Tropical cyclone formation and intensification

10.1 The prototype problem for genesis and intensification

10.2 A simplified numerical model experiment

10.3 The numerical simulation

10.3.1 A summary of vortex evolution

10.3.2 Evolution of vorticity

10.4 Moist instability and 𝜃𝑒 .

10.5 Azimuthal mean view of vortex evolution

10.6 Modified view of spin up

10.7 A system-averaged perspective

10.8 Predictability issues

10.9 Inclusion of ice processes

10.10 Vortex evolution with and without ice

10.11 Moist instability and 𝜃𝑒 .

10.12 An azimuthal mean view of vortex evolution

10.13 Mid-level vortex development with ice microphysics

10.13.1 Increasing influence of the boundary layer

10.13.2 Synthesis

10.14 Boundary layer control

10.14.1 Boundary layer coupling in brief

10.14.2 A demonstration of boundary layer coupling

10.15 Towards a conceptual model for tropical cyclogenesis

11. The rotating-convection paradigm

11.1 Flux form of the vorticity equation

11.2 Axisymmetric flow

11.3 Non-axisymmetric flow

11.4 Azimuthally-averaged tangential and radial wind tendency

11.4.1 Characterizing eddy processes

11.4.2 Attributes of the mean-eddy flow partitioning

11.4.3 Eddy effects of an isolated deep convective cloud

11.5 Applications to a numerical model simulation

11.5.1 Tangential velocity tendency analysis

11.5.2 Spin up at later times

11.5.3 Radial velocity tendency analysis

11.5.4 Summary of radial velocity analysis at 30 h

11.6 Other features of the numerical simulation

11.6.1 Upper level inflow jets

11.6.2 Centrifugal recoil effect

11.7 Summary of the rotating-convection paradigm

12. Emanuel’s intensification theories

12.1 The intensification theories

12.1.1 The Emanuel 1989 theory

12.1.2 The Emanuel 1995 theory

12.1.3 The later theories

12.1.4 Specifics of the E97 theory

12.2 The air-sea interaction intensification theory, WISHE

12.3 The E12 theory

12.3.1 Specifics of the E12 theory

12.4 A boundary layer explanation for spin up

12.5 Congruence of 𝑀 and 𝜃∗𝑒 surfaces during spin up? .

12.6 Appraisal of the Emanuel intensification theories

12.7 Can the E12 model quantify the likelihood of more rapidly intensifying hurricanes in a warmer world?

12.8 Appendix A: Derivation of 𝜕𝑣𝑚/𝜕𝑡 in the E12 theory, Eq. (12.4) . .

13. Emanuel’s maximum intensity theory

13.1 The E86 steady-state model

13.1.1 Dissipative heating

13.1.2 High resolution tests of the E86 PI theory

13.2 Unbalanced effects

13.3 A revised theory

13.4 Three dimensional effects

13.5 Summary of Emanuel’s steady-state PI theories

13.6 Appendix A: Derivation of extended PI formulation, Eq. (13.7)

13.7 Boundary layer closure

13.7.1 Gradient wind balance limit

13.7.2 General case

13.8 Appendix B: Construction of E86 steady-state hurricane solution

13.8.1 Conceptual overview

13.8.2 Deductions from thermal wind balance and moist neutrality

13.8.3 Boundary layer constraints

13.8.4 Solution for ln 𝜋 in Region III

13.8.5 Solution for ln 𝜋 in Regions I + II

13.8.6 Solution for 𝑣2𝑚 and ln 𝜋𝑠 at 𝑟 = 𝑟𝑔𝑚. . . . .

13.8.7 The complete solution

13.8.8 The tropical cyclone as a Carnot-like heat engine

13.9 Exercises

14. Global budgets and steady state considerations

14.1 The numerical simulation

14.2 Budget calculations

14.2.1 Water budget

14.2.2 Kinetic energy budget (Gill form)

14.2.3 Kinetic energy budget (Anthes form)

14.2.4 Kinetic energy budget calculations

14.2.5 Total energy budget

14.3 Role of surface enthalpy fluxes

14.3.1 Contributions to 𝜃𝑒 changes .

14.3.2 Some observations

14.4 Absolute angular momentum budget

14.5 Exercises

14.6 Global steady-state requirements

15. Tropical cyclone life cycle

15.1 Newtonian cooling

15.2 Life cycle metrics

15.3 Vortex asymmetries

15.3.1 Genesis and RI phases

15.3.2 First mature phase

15.3.3 Decay phase

15.4 Azimuthally-averaged view of vortex evolution

15.4.1 Mature phase

15.4.2 Temporary decay and reintensification phase

15.4.3 A new pathway to inner-core rainband formation

15.4.4 Decay phase

15.5 Interpretations of the life cycle

15.5.1 Important kinematical features

15.5.2 Boundary layer dynamics

15.5.3 Boundary layer coupling

15.5.4 Ventilation of the boundary layer inflow

15.6 Life cycle summary

16. Applications of the rotating-convection paradigm

16.1 Minimal conceptual models for vortex intensification

16.1.1 A general prognostic balance model

16.1.2 Zero-order model

16.1.3 A minimal representation of friction

16.1.4 First-order model

16.1.5 Exercises

16.1.6 Beyond the minimal representation of friction

16.1.7 Cumulus parameterization in minimal models

16.1.8 Role of the WISHE feedback?

16.1.9 Important caveats

16.1.10 Synthesis

16.2 Comparison between three-dimensional and axisymmetric tropical cyclone dynamics

16.2.1 Synthesis

16.3 The effects of latitude on tropical cyclone intensification

16.3.1 The Smith et al. simulations

16.3.2 Vortex evolution at different latitudes

16.3.3 Slab boundary layer solutions

16.3.4 Thermodynamic support for deep convection

16.3.5 Diabatically-forced overturning circulation

16.3.6 Quantifying the effects of rotational stiffness

16.3.7 Flow asymmetries

16.3.8 Summary of latitudinal dependence

16.4 The effects of sea surface temperature on intensification

16.4.1 Interpretation of the SST dependence

16.4.2 Summary of SST effects

16.5 The effects of initial vortex size on genesis and intensification

16.5.1 Numerical experiments on vortex size

16.5.2 Synthesis

16.6 Tropical cyclogenesis at and near the Equator

16.6.1 An idealized numerical study

16.6.2 Synthesis

16.7 Observational tests of the new spin up model

16.8 Tropical lows over land

16.8.1 A tropical low case study

16.8.2 Synthesis

16.9 Polar lows, medicanes and tropical cyclones

16.10 The rotating-convection paradigm in the research of others

16.10.1 An idealized numerical study

16.10.2 Formation of a thermodynamic shield in a category 5 hurricane, but not in a category 3 hurricane.

16.10.3 Invocation of WISHE-like positive feedback mechanism to explain the rapid intensification of Hurricane Michael (2018)

16.10.4 Synthesis

16.11 Vertical shear regimes

16.11.1 Synthesis

17. Epilogue

17.1 Examples of recent events

17.1.1 Formation and intensification of Hurricane Fiona (2022)

17.1.2 Formation and intensification of Hurricane Ian (2022)

17.2 Applications and future directions