TROPICAL CYCLONE RESEARCH REPORT TCRR (2016) No 1: 1–24 Meteorological Institute Ludwig Maximilians University of Munich Recent Developments in the Fluid Dynamics of Tropical Cyclones Michael T. Montgomerya 1and Roger K. Smithb a Dept. of Meteorology, Naval Postgraduate School, Monterey, CA. b Meteorological Institute, Ludwig Maximilians University of Munich, Munich, Germany Abstract: This paper reviews progress in understanding the fluid dynamics and moist thermodynamics of tropical cyclone vortices. The focus is on the dynamics and moist thermodynamics of vortex intensification and structure and the role of coherent eddy structures in the evolution of these vortices. We have sought to give an appraisal of previous ideas on many facets of the subject and have articulated also some open questions. The advances reviewed herein provide new insight and tools for interpreting complex vortex-convective phenomenology in simulated and observed tropical cyclones. KEY WORDS Tropical cyclone intensification, rotating deep convection, boundary layer control, size-growth, potential intensity, quasi steady-state hurricanes Date: April 12, 2016 1 Introduction sea, make tropical cyclones a particularly interesting and challenging scientific problem to understand. Practical con- Tropical cyclones are fascinating large-scale, organized, siderations, such as saving human life and property in the convective vortices that continue to hold many scientific path of these storms are another important driving factor secrets regarding their birth, intensification, mature evolu- in the quest for knowledge about them. Atlantic Hurricane tion and decay. These moist convective vortices comprise Sandy (2012) is a reminder that even tropical storms (max- arguably all facets of classical fluid dynamics ranging from imum near-surface wind speed < 32 m s−1) can wreak the microscale flow in and around small droplets, the coa- havoc on populated coastal communities, maritime assets lescence of smaller droplets into larger ones, precipitation and even inland populations (e.g. Lussier et al. 2015). As and evaporation processes, to the larger scales of buoyant coastal communities continue to grow in tropical cyclone thermals in a rotating environment, their aggregate effects affected regions, there is an increasing demand for more on the vortex circulation, and to the even larger scale of accurate tropical cyclone forecasts. vortex waves and eddies, such as inertia-buoyancy waves, There are two main aspects of the forecasting prob- vortex Rossby waves, eyewall mesovortices and their inter- lem. The first is to forecast the storm track, and the second action with the vortex circulation. The large Reynolds num- is to forecast its intensity, characterized typically by the bers of these flows implies that turbulence of the Kol- maximum near-surface wind speed. Track forecasts have mogorov kind will be an element at the small scales, but improved significantly in the past 25 years, but progress the presence of strong, spatially variable, vertical rotation in intensity forecasting has shown comparatively little in these systems suggests that quasi-two dimensional fluid improvement (DeMaria et al. 2005; Rogers and Coauthors dynamics and its associated turbulence phenomenology 2006). Because the track depends mainly on the large-scale should be an important element also with modifications due flow in which the vortex is embedded, the improvement in to the presence of deep moist convection, which is intrin- track forecasting may be attributed largely to the improve- sically three-dimensional. The more intense manifestations ment in the representation of the large-scale flow around of these vortices (maximum near-surface wind speed > 32 −1 the vortex by global forecast models. In contrast, the inten- m s ) are called hurricanes in the Atlantic and Eastern sity appears to depend on processes of wide ranging scales Pacific basins and typhoons in the Western North Pacific spanning many orders of magnitude as noted above. region. Because of the challenges of forecasting tropical Although flow speeds are well below the sound speed cyclone intensity change, the problem of understanding (typically < 100 m s−1), non-conservative effects, prin- how intensity change occurs has been at the forefront of cipally associated with friction at the ocean surface and tropical cyclone research in recent years, especially in the wind-forced transfer of moisture and heat from the warm context of the rapid intensification or decay of storms. These challenges are motivated by the recent instigation 1Correspondence to: Michael T. Montgomery, Naval Postgraduate School, 159 Dyer Rd., Root Hall, Monterey, CA 93943. E-mail: of the Hurricane Forecast Improvement Project (HFIP) [email protected] by the National Oceanic and Atmospheric Administration Copyright c 2016 Meteorological Institute 2 M. T. MONTGOMERY AND R. K. SMITH (NOAA) and other U. S. Government agencies to coordi- cylindrical polar coordinates, (r, λ, z). The rotation of the nate hurricane research necessary to accelerate improve- Earth is incorporated by the addition of Coriolis and cen- ments in hurricane track and intensity forecasts (Gall et al. trifugal forces in the usual manner (Gill 1982; Holton 2004) 2013). and, because of the relatively limited horizontal scale of the There have been significant advances in understanding tropical cyclone circulation, the rotation rate is assumed to tropical cyclone behaviour since the earlier reviews of the be independent of latitude (i.e., a so-called f-plane, where topic by Emanuel (1991) and Chan (2005) and the field f is the Coriolis parameter given by f = 2Ω sin φ, Ω is the has broadened significantly. As a result, a comprehensive Earth’s rotation rate, and φ is latitude). The governing equa- review of all fluid dynamical aspects is not possible in tions are: the space available to us. For this reason we have chosen to focus on the dynamics and thermodynamics of the ∂u ∂u v ∂u ∂u v2 1 ∂p + u + + w fv = + Fr, vortex when viewed as a coherent structure with embedded ∂t ∂r r ∂λ ∂z − r − −ρ ∂r substructures. To begin, for those readers working in other (1) fields, we review briefly in section 2 the equations of ∂v ∂v v ∂v ∂v uv 1 ∂p + u + + w + + fu = + Fλ, motion and some other basic concepts involving zero-order ∂t ∂r r ∂λ ∂z r −ρr ∂λ force balances, moist thermodynamics and deep convective (2) ∂w ∂w v ∂w ∂w 1 ∂p clouds in a rotating environment. This material provides a + u + + w = g + Fz, (3) reference for much of the later discussion. Some readers ∂t ∂r r ∂λ ∂z −ρ ∂z − may wish to skip this section. ∂ρ 1 ∂ρru 1 ∂ρv ∂ρw In section 3 we survey progress that has been made + + + =0, (4) ∂t r ∂r r ∂λ ∂z in understanding tropical cyclone intensification and struc- ture from the perspective of the prototype intensification ∂θ ∂θ v ∂θ ∂θ + u + + w = θ˙ + F , (5) problem, which considers for simplicity the spin up of an ∂t ∂r r ∂λ ∂z θ initially balanced, axisymmetric, cloud-free, conditionally- 1 −1 unstable, baroclinic vortex of near tropical storm strength ρ = p∗π κ /(Rθ) (6) in a quiescent tropical environment on an f-plane. Here, where u,v,w are the velocity components in the three coor- paradigms for vortex intensification including emerging dinate directions, θ is the potential temperature, θ˙ is the ideas pointing to the importance of boundary layer control diabatic heating rate (1/cpπ)Dh/Dt, h is the heating rate − − κ in vortex evolution are discussed. In sections 4 we examine per unit mass expressed as J kg 1 s 1, π = (p/p∗) is more deeply the role of cloud-generated vorticity in sup- the Exner function, p the pressure, g the effective gravi- porting vortex spin up. Progress in understanding mature tational force per unit mass, R the specific gas constant for vortex intensity is reviewed in section 5 and the steady-state dry air, cp the specific heat at constant pressure, κ = R/cp problem is reviewed in section 6. The conclusions are given and p∗ = 1000 mb is a reference pressure. The tempera- in section 7. ture is given by T = πθ. The terms (Fr, Fλ, Fz) represent Because of space constraints, we are unable review unresolved processes associated with turbulent momentum aspects of vortex motion, vortex Rossby waves and their diffusion. In the case of numerical models, these terms contribution to vortex resilience, nor the early stages of specifically represent a divergence of sub-grid scale eddy storm formation. For the same reason we cannot address momentum flux associated with unresolved processes such topics such as: the interaction of storms with ambient ver- as convection for a model that cannot resolve clouds and/or tical shear; helicity; secondary eyewall formation; ocean frictional stress at the lower surface and related mixing pro- feedback effects; the interaction with neighbouring weather cesses in the frictional boundary layer. Similarly, Fθ repre- systems including fronts and upper troughs; the extra- sents the effects of turbulent heat transport (again possi- tropical transition when storms move into the middle lati- bly including those associated with convection for a coarse tudes; cloud microphysics; boundary layer rolls; wind wave resolution model). The foregoing equations comprise the coupling; and details of the surface layer (or emulsion three components of the momentum equation, the conti- layer). nuity equation, the thermodynamic equation and the ideal gas equation of state, respectively. In the foregoing equa- 2 Preliminaries tions, the traditional approximation is made of neglecting the horizontal component of the earth’s rotation rate and To provide a common framework for this review, we other metric terms associated with the approximate spheric- present the equations of motion pertinent to understanding ity of the Earth that are small on account of the limited tropical cyclone behaviour. horizontal scale of a typical hurricane vortex compared to the mean radius of the Earth. In the above equation set, it 2.1 The equations in cylindrical polar coordinates is assumed that the origin of coordinates is located at some Because an intensifying tropical cyclone exhibits some suitably defined vortex centre.