9B.4 UNDERSTANDING THE GENESIS OF HURRICANE VINCE THROUGH THE SURFACE PRESSURE TENDENCY EQUATION Kwan-yin Kong City College of New York 1 1. INTRODUCTION 20°W Hurricane Vince was one of the many extraordinary hurricanes that formed in the record-breaking 2005 Atlantic hurricane season. Unlike Katrina, Rita, and Wilma, Vince was remarkable not because of intensity, nor the destruction it inflicted, but because of its defiance to our current understandings of hurricane formation. Vince formed in early October of 2005 in the far North Atlantic Ocean and acquired characteristics of a hurricane southeast of the Azores, an area previously unknown to hurricane formation. Figure 1 shows a visible image taken at 14:10 UTC on 9 October 2005 when Vince was near its peak intensity. There is little doubt that a hurricane with an eye surrounded by convection is located near 34°N, 19°W. A buoy located under the northern eyewall of the hurricane indicated a sea-surface temperature (SST) of 22.9°C, far below what is considered to be the 30°N minimum value of 26°C for hurricane formation (see insert of Fig. 3f). In March of 2004, a first-documented hurricane in the South Atlantic Ocean also formed over SST below this Figure 1 Color visible image taken at 14:10 UTC 9 October 2005 by Aqua. 26°C threshold off the coast of Brazil. In addition, cyclones in the Mediterranean and polar lows in sub-arctic seas had been synoptic flow serves to “steer” the forward motion of observed to acquire hurricane characteristics. These hurricanes. cyclones, like typical tropical cyclones, are apparently warm Note that the first statement does not include the core in structure, and appeared to be energized by well-accepted criterion of 26°C SST as a pre-condition for convection. The formation of these cyclones over hurricane formation. This has become a criterion in doubt in anomalously low SST motivates us to re-examine our basic light of Vince forming over water at 23°C, and the Brazilian understandings of hurricane genesis and intensity change. hurricane that formed over 24°C water. Our basic understandings on hurricanes stated 2. WHAT WE KNOW ABOUT HURRICANE FORMATION above are admittedly rather sketchy. What we are uncertain 2.1 Our basic understandings about is regarding the details of the above statements, To a large degree, our basic understandings on specifically, their importance and relation with one another, hurricane formation are still primarily based on deduction from and whether their importance differs in different situations. long-term observations. The following statements summarize This is largely due to the fact that there has not been a what we confidently know about hurricanes and their intensity proven theoretical framework of hurricane formation and changes. intensification with which we can confidently rely on as a method of diagnosing hurricane intensity change. This is • Hurricanes are warm-core cyclones that form and develop over warm ocean waters. No hurricanes have ever formed demonstrated by the fact that meteorological computer and intensified over land away from large bodies of water. models have been unable to handle hurricanes despite their relative success in simulating and forecasting higher latitude • Hurricanes and their link with moist convection are weather systems. However, recent advances in numerical inseparable. They appear to intensify in response to methods and computing power appear to be overcoming this development of moist convection near or around their difficulty. For the first time, they appear capable of center of circulation. simulating hurricane structures in realistic details. • Synoptic environmental factors are recognized to greatly Nevertheless, even if computer models can simulate influence hurricane formation, intensity, and motion. For hurricanes perfectly, they function like a black box from which example, vertical wind shear is recognized as a strong no information about the fundamental processes governing modulator of hurricane intensity, whereas the upper-air hurricane intensity is offered. There remains the necessity to formulate a theoretical framework for the purpose of Corresponding author address: Kwan-yin Kong, City College of understanding and diagnosing hurricane formation and New York, Dept. of Earth & Atmospheric Sciences, New York, NY intensification. 10031; e-mail: [email protected]. 2.2 Recent researches on tropical cyclone transitions surface pressure tendency. This is in contrast to the The intriguing subject of tropical cyclone transitions (potential) vorticity approach, which has become the has attracted attention in recent years. Tropical cyclones, dominant idea since the 1950s. being inherently warm-core and vertically stacked, often A classic method of deriving an equation of surface transform into vertically tilted cold-core extratropical cyclones pressure tendency is by differentiating with respect to time the after encountering higher-latitude baroclinity. The reverse integrated form of the hydrostatic equation, followed by process—extratropical cyclones transforming into tropical substitution of the continuity equation to yield the result cyclones—is seen less frequently. But in the years 2000 to ∂ ∞ ps r 2003, nearly half of the Atlantic tropical cyclones were = −g (∇ ⋅ ρv) dz . (0) ∂t baroclinically initiated, according to NOAA’s Tropical ∫ 0 Prediction Center. Davis and Bosart (2004) classified the This result states that the surface pressure tendency is due tropical transitioning cyclones into two main groups—strong simply to the integrated mass divergence of the entire depth extratropical pre-cursors (SEC) and weak extratropical pre- of the overlying atmospheric column. Although the physical cursors (WEC). The occlusion of an old frontal cyclone is meaning of this result is easy to understand, it does not offer recognized as the main characteristics of SEC cases. This further insights into identifying the thermodynamic processes cold-cored, vertically stacked occluded cyclone must persist that undoubtedly affect the surface pressure. Adiabatic over warm SST and has surface winds strong enough to cooling and latent heat release are examples of important induce surface heat exchange (WISHE; Emanuel 1987). The thermodynamic processes during cyclogenesis, especially increase of low-level moisture must then trigger the growth of tropical cyclogenesis. Using another approach, it will be convection close to the cyclone center. This must then trigger shown in the next section that it is possible to derive a surface a mutual intensification process in which growth of convection pressure tendency equation that reveals the thermodynamic leads to intensification of the cyclone. Diabatic heating due to factors affecting the surface pressure. the growth of convection is thought to reduce the vertical 3.1 Derivations shear wherefore the mutual intensification process may be We begin by considering the hydrostatic equation: nurtured. ∂p As for WEC cases, Davis and Bosart defined them = −ρg . (1) ∂ as either (1) occluding cyclones with weak surface winds that z > fail to trigger WISHE, or (2) the presence of mid-tropospheric First, we integrate it with respect to z from z1 to z2 , ( z2 z1 ). convectively driven mesoscale vortex or vortices. These z2 − = − ρ initial cyclonic disturbances eventually develop into well- p2 p1 g dz (2) defined tropical cyclones. ∫ z1 where p and p are pressures at z and z respectively. Regardless of classification, the spin-up of a pre- 1 2 1 2 We then differentiate it with respect to t with x, y, z and z existing vortex in apparent relation to development of 1 2 convection is recognized as the most significant characteristic held constant. of tropical transitioning cyclones. Hendricks, et al. (2004) ∂p ∂p z2 ∂ρ 2 − 1 = −g dz (3) called the convectively driven vortices “vortical hot towers”. ∂t ∂t ∂t ∫ z1 Davis and Bosart (2003) attributed the vortex spin-up to x, y ,z x, y, z x, y,z ∂ ∂ redistribution of potential vorticity, which, Hendricks, et al. Here p2 t means the local pressure tendency at ∂ ∂ understood as the result of vertical diabatic heating gradient. (x, y, z2 ) and likewise, p1 t the local pressure tendency Despite the differences in terminology and ways of relating at (x, y, z1 ) . The integral on the right-hand-side (RHS) convection to cyclogenesis, the crucial point that remains represents the integrated value of the local density tendency largely unexplained is exactly how the initial vortex acquired from z1 to z2 at point (x, y) . vorticity and how convection may trigger further vortex spin- − If we divide through this equation by z2 z1 and up. The answer to this key point not only pertains to tropical − → take the limit as (z2 z1 ) 0 , we find that transitioning cyclones but also tropical cyclogenesis in ∂ ∂p ∂ρ general. = −g (4) ∂z ∂t ∂t x,y,z x,y,z 3. UNDERSTANDING CYCLOGENESIS BY MEANS OF which is actually the continuity equation in disguise! THE SURFACE PRESSURE TENDENCY The traditional derivation approach from this point To considering the problem of tropical cyclogenesis, onward is to substitute the continuity equation in place of the as well as tropical and extratropical cyclone transformations, local density tendency, integrate, and yield a surface pressure let us take a step back and consider the problem more tendency equation in terms of mass divergence, i.e. equation broadly as just cyclogenesis. In order to approach the (0). But instead, we will proceed by expressing local density problem theoretically, we must define cyclogenesis by means tendency in terms of total density tendency and advections. of a physical quantity. The physical quantity we will use is the Before doing so, we need to express the partial derivative of density on the RHS in Cartesian-pressure coordinates (x, y, p) as ∂ ∂ ∂ρ ∂ ∂ρ p = − + p g . (5) ∂z ∂t ∂t ∂t ∂p x, y,z x,y, p x,y,z x,y,t Next, local density tendency is expressed in terms of total and partial derivatives in (x, y, p) coordinates as ∂ ∂p ∂ρ ∂p ∂ρ dρ + g = gvr ⋅ ∇ ρ + ω − .