The Dynamic Response of the Hurricane Wind Field to Spiral Rainband Heating

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YUMIN MOON AND DAVID S. NOLAN Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida

(Manuscript received 6 April 2009, in ﬁnal form 28 January 2010)

ABSTRACT

The response of the hurricane wind field to spiral rainband heating is examined by using a three-dimensional, nonhydrostatic, linear model of the vortex–anelastic equations. Diabatic heat sources, which are designed in accordance with previous observations of spiral rainbands, are made to rotate with the flow around the hurricane-like wind field of a balanced, axisymmetric vortex. Common kinematic features are recovered, such as the overturning secondary circulation, descending midlevel radial inflow, and cyclonically accelerated tangential flow on the radially outward side of spiral rainbands. Comparison of the responses to the purely convective and stratiform rainbands indicates that the overturning secondary circulation is mostly due to the convective part of the rainband and is stronger in the upwind region, while midlevel radial inflow descending to the surface is due to the stratiform characteristics of the rainband and is stronger in the downwind region. The secondary horizontal wind maximum is exhibited in both convective and stratiform parts of the rainband, but it tends to be stronger in the downwind region. The results indicate that the primary effects of rainbands on the hurricane wind field are caused by the direct response to diabatic heating in convection embedded in them and that the structure of the diabatic heating is primarily responsible for their unique kinematic structures. Sensitivity tests confirm the robustness of the results. In addition, the response of the hurricane wind field to the rainband heating is, in the linear limit, the sum of the asymmetric potential vorticity and symmetric transverse circulations.

1. Introduction several hundred kilometers (e.g., Willoughby et al. 1984; Guinn and Schubert 1993). The observed radial distri- One of the most prominent features of tropical cy- bution of lightning within tropical cyclones agrees with clones is their banded structures of convection in circular this categorization (Molinari et al. 1999; Cecil et al. or spiral shapes called spiral rainbands. The importance 2002) in which inner rainbands are associated with the of spiral rainbands on tropical cyclones was recognized region of relative minimum in lightning frequency while early (e.g., Wexler 1947; Senn and Hiser 1959; Atlas et al. outer rainbands are associated with the region of larger 1963), and their cloud characteristics are often used to lightning frequency. Wang (2008) suggested the use of help diagnose the current intensity and forecast the future the rapid filamentation zone (Rozoff et al. 2006) as evolution of tropical cyclones (Dvorak 1975). a criterion to separate inner rainbands from outer rain- There are several ways to distinguish spiral rainbands. bands. The rapid filamentation zone is a strain-dominated They can be classified as inner or outer rainbands. Inner flow region just outside the radius of maximum wind rainbands are located near the vortex core, usually (RMW), where convection and its associated vorticity within 100 km from the center of the storm. They are are expected to be quickly filamented into thin strips, visible on radar images but are not always apparent on which are ultimately lost to diffusion. satellite images because of the dense overcast clouds Other classification methods exist as well. According to near the center. In comparison, outer rainbands often do Willoughby et al. (1984), spiral rainbands can also be not show much spiral curvature and can be quite long. classified as moving or stationary, based on their propa- They are typically visible on both radar and satellite gation relative to the mean tangential circulation. They images and located far from the center, on the order of are primary rainbands if they merge into the eyewall of the storm or secondary rainbands if they are disconnected Corresponding author address: Yumin Moon, 4600 Rick- from the eyewall. In addition, recent observational stud- enbacker Causeway, University of Miami, Miami, FL 33149. ies (e.g., Gall et al. 1998; Kusunoki and Mashiko 2006) E-mail: [email protected] have identified numerous smaller fine-scale bands with

DOI: 10.1175/2010JAS3171.1

Ó 2010 American Meteorological Society Unauthenticated | Downloaded 09/25/21 09:05 AM UTC 1780 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 67 cross-band scales of 2–10 km that are found near the the overall view that the diabatic heating structure in eyewall and have very little propagation relative to the spiral rainbands is primarily responsible for their kine- mean tangential circulation of the storm. matic structures. There exist contrasting views of the effect of spiral We first review previous observational studies of rainbands on the intensity of tropical cyclones. Obser- spiral rainbands in section 2. Section 3 introduces the vational studies have indicated that from a thermody- numerical model and the basic-state vortex. Section 4 namic viewpoint spiral rainbands have a negative impact describes how the structures of asymmetric heat sources on intensity because they intercept some of the moist used to represent spiral rainbands are represented in the low-level radial inflow into the eyewall and replace it numerical model. Section 5 describes the response of the with cooler, dry air originating from downdrafts (e.g., tropical cyclone wind field to the rainband heating and Barnes et al. 1983, hereafter BZJM83; Powell 1990, compares the results to observations. Section 6 presents hereafter P90; Cione et al. 2000). At the same time, sensitivity tests. Summary and discussions are provided however, spiral rainbands can also be considered to have in section 7. a positive impact on intensity if they are viewed as barriers to protect the storm core from dry air intrusion 2. Observed structures of spiral rainbands and environmental wind shear (Kimball 2006). From a. Previous observational studies a fluid dynamical point of view, spiral rainbands are viewed as potential vorticity (PV) bands, and they are A large portion of previous observational studies of believed to have a positive impact on intensity as they spiral rainbands examined the reflectivity of a single transport angular momentum inward through the ax- ground-based or airborne Doppler radar (e.g., Senn and isymmetrization process (e.g., Carr and Williams 1989; Hiser 1959; Atlas et al. 1963; BZJM83). Composites of Montgomery and Kallenbach 1997; Nolan and Farrell pseudodual Doppler radar data have also been used 1999). Full-physics numerical simulations of tropical before (e.g., P90; Barnes et al. 1991; Barnes and Powell cyclones have shown that PV anomalies are generated in 1995). More recent studies (e.g., May et al. 1994; Kim the middle troposphere within spiral rainbands (Chen et al. 2009) analyzed wind profiler observations to infer and Yau 2001) and that the PV generation is greater the structures of the spiral rainbands. in the stratiform portion of the rainbands (May and Previous studies have shown that spiral rainbands are Holland 1999; Franklin et al. 2006). Recently, Wang (2009) usually 10–40 km in width and often spiral radially in- examined the effects of outer spiral rainbands on trop- ward in a cyclonic direction. Their precipitation struc- ical cyclone structure and intensity by modifying the tures show that the upwind portion is mostly convective diabatic heating rate in outer spiral rainbands formed while the downwind side is mostly stratiform (Atlas within numerically simulated tropical cyclones. Wang et al. 1963; BZJM83), although there is a large var- (2009) found that the effect of the increased (decreased) iation in the degree of organization (e.g., Ryan et al. diabatic heating rate in rainbands is to weaken (strengthen) 1992; May 1996). A prominent rainband that extends the intensity of hurricanes but make them larger (smaller) outward toward one side of the vortex is often called in size. a principal rainband that sometimes joins the eye- To understand the effects of spiral rainbands on wall through a connecting rainband (Willoughby et al. tropical cyclones, it is necessary to fully incorporate 1984). their dynamic and thermodynamic aspects; however, BZJM83 hypothesized that the convective-scale cir- their dynamics and thermodynamics must be driven in culation within spiral rainbands is made up of two flows: large part by latent heat release in convection embedded an updraft accompanied by a low-level radial inflow and within them. Here, we propose that the effects of spiral an upper-level radial outflow, and a midlevel radial in- rainbands on the tropical cyclone wind field are caused flow that passes through the rainband and descends to by the direct response to diabatic heating in their con- the surface. This circulation was illustrated in Fig. 18 of vection. This approach is similar to that of Pandya and BZJM83, reproduced here as Fig. 1a. Also noted in the Durran (1996) and Pandya et al. (2000) in which the BZJM83 model is that the reflectivity tower is tilted mesoscale circulation around squall lines was approxi- radially outward with height. Another kinematic feature mated by the direct response to steady thermal forcing of the spiral rainband circulation is a midlevel jet of that resembles the latent heat release in a convective tangential wind located on the radially outward side of leading line. In this paper, it is also proposed that spiral the rainbands (P90; Samsury and Zipser 1995), as shown rainbands can be regarded as quasi-steady, asymmetric in the horizontal cross-sectional view of a spiral rain- heat sources that may rotate around the storm. It is the band circulation presented by P90 (Fig. 16a of P90, goal of this study to evaluate these approximations and reproduced here as Fig. 1b). This midlevel jet is also

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Change Experiment field program (Houze et al. 2006). One of the goals of HH08 was to evaluate the main features of the BZJM83 and P90 conceptual spiral rainband models. HH08 examined the middle portion of a spiral rainband in Hurricanes Katrina and Rita (2005). The vertical cross sections (Figs. 6 and 7 of HH08) were taken through an intense updraft located on the radially inward edge of the rainband. They illustrate that the reflectivity of the convective cell slopes radially outward with height. In addition, there is an overturning secondary circulation that has radial inflow at lower levels and radial outflow at upper levels. The cross sections also reveal layers of convergence and divergence at the base and top of the reflectivity core, respectively, with the largest upward vertical motion found in between. A SHWM is located on the radially outward side of the spiral rainband. In addition, HH08 examined the vertical cross sections (Figs. 8 and 9 of HH08) through a downdraft found in the middle portion of the same rainband of Katrina and Rita. The cross sections show two downdraft circulations: a radial inflow (near height z 5 2 km) enters from the radially outward side of the rainband and descends to surface, while the other downdraft originates from upper levels (near z 5 8 km) but stays on the radially inward side of the reflectivity core. From examining the structures of the updrafts and downdrafts found in the middle portion of a spiral rainband (both Hurricanes Katrina and Rita), HH08 were able to find FIG. 1. Schematic diagrams of kinematic structures within spiral supporting evidence for the hypothesized features of the rainbands, originally presented as (a) Fig. 18 in BZJM83 and BZJM83 and P90 models: an overturning secondary (b) Fig. 16 in P90. Contours are for radar reflectivity (dBZ), and circulation associated with a convective updraft, a mid- the 25-dBZ contour marks the boundary of spiral rainbands. In (a), arrows represent convective-scale circulations, and potential level radial inflow that descends to surface, and a mid- temperature (K) is marked at the beginning and ending of each level SHWM. circulation. By using the retrieved three-dimensional velocity fields, HH08 also computed and examined the vertical profile of mass transport. Their analysis showed that the called a secondary horizontal wind maximum (SHWM). net vertical mass transport in the upwind and middle Samsury and Zipser (1995) found that the location of regions is positive everywhere throughout the tropo- a SHWM is highly correlated with the location of sphere, with a maximum value at middle levels, while spiral rainbands and speculated on the possibility that the downwind region has a nearly zero net vertical mass the SHWM is generated and maintained by spiral rain- transport. The magnitude of the upward vertical mass band processes, which were not clearly defined at that transport is greatest in the middle region of the spiral time. rainband. Figure 2 shows the conceptualized model of a spiral b. Recent dual-Doppler radar observations from the rainband circulation as presented in HH08. The hori- Hurricane Rainband and Intensity Change zontal view (Fig. 2a) shows that the rainband is con- Experiment program vective upwind and through the middle portion of the Hence and Houze (2008, hereafter HH08) analyzed rainband before becoming stratiform downwind. As the high-resolution dual-Doppler radar observations of spiral rainband becomes more stratiform, it also becomes wider. rainbands in Hurricanes Katrina and Rita (2005) col- A SHWM is located near the rainband and also on its lected during the Hurricane Rainband and Intensity radially outward side. A vertical cross section through

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nonhydrostatic, linear numerical model of vortex dynamics, now known as the Three-Dimensional Vortex Perturbation Analysis and Simulation (3DVPAS). It is based on the vortex–anelastic equations in cylindrical co- ordinates on the f plane (Hodyss and Nolan 2007), and its dynamics are linear and nonhydrostatic. It can simulate both asymmetric and symmetric motions on balanced axisymmetric vortices. Perturbations can have arbitrary radial and vertical structures but have har- monic variation in the azimuthal (tangential) direction. The symmetric response to purely asymmetric motions can also be computed by using the eddy flux divergences arising from asymmetric motions as forcing for symmetric motions. Free-slip, solid-wall boundary conditions are enforced on all boundaries, and Rayleigh damping regions (sponges) are placed at the upper and outer boundaries to suppress the reflection of gravity waves. The domain size of all simulations in this study is 300 km in the radial direction and 20 km in the vertical direction. The grids are stretched in the radial direction but equally spaced in the vertical direction. In the radial direction, the grid size is approximately 2 km in the inner-core region (0 km , r , 95 km) but larger (approximately 6 km) in the far outer-core region with a smooth transition in between. The grid size in the vertical direction is 1/3 km. More complete details of the 3DVPAS model can be found in Nolan and Montgomery FIG. 2. Schematic diagram of precipitation and kinematic structures within spiral rainbands, originally presented as Fig. 13 in (2002a), Nolan and Grasso (2003), and Nolan et al. (2007). HH08. (a) Horizontal cross-sectional view of the conceptual spiral rainband model. (b) Radius–height cross section along the thin b. Basic-state vortex solid line in (a). There are a number of idealized profiles that can provide a reasonably realistic representation of the tan- the middle part of the rainband (Fig. 2b) shows that the gential wind field of tropical cyclones (e.g., Willoughby convective cell leans radially outward with altitude and et al. 2006). As the control case in this study, we use the SHWM is located on its radially outward edge. the modified Rankine (MR) profile because it has a very There is an overturning secondary circulation coupled simple form yet provides a realistic representation of with an updraft located on the radially inward edge of tangential wind, as shown by Mallen et al. (2005). The the cell, with convergence and divergence at the base MR profile has a tangential velocity field defined by and top of the cell, respectively. Another radial inflow 8 r can be found slightly above the low-level radial inflow, <>y , r # RMW maxRMW and it descends to the surface while passing through the ~y(r) 5 a (1) :> RMW rainband. A downdraft that originates in the upper lev- ymax , r . RMW, els stays radially inward of the rainband. The boundary r of the rainband is marked by the reflectivity value of where y is the prescribed maximum tangential ve- 25 dBZ. max locity and a is the decay parameter. The radial structure of tangential wind field of tropical 3. The model and basic-state vortex cyclones is well documented because of the relative abundance of observations obtained from United States a. The numerical model: Three-Dimensional Vortex Air Force and National Oceanic and Atmospheric Ad- Perturbation Analysis and Simulation ministration aircrafts investigating a large number of To simulate the effects of spiral rainband heating on the hurricanes. The same cannot be said, however, about the tropical cyclone wind field, we use a three-dimensional, vertical structure of the tangential wind field. From

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC JUNE 2010 M O O N A N D N O L A N 1783 early studies (e.g., Frank 1977; Jorgensen 1984; Shea and Gray 1973), it has been known qualitatively for some time that the tangential wind decays away from the surface, and the RMW generally slopes radially outward with height. Some arbitrary vertical structures (e.g., Nolan and Montgomery 2002a; Nolan and Grasso 2003; Nolan et al. 2007) have been used before; however, the RMW of the wind ﬁelds used in these previous studies did not in fact slope outward and is thus unrealistic. A realistic radially outward slope of the RMW with height and a vertical decay rate of tangential velocity at the RMW can be quantitatively obtained by utilizing the steady-state axisymmetric hurricane model by Emanuel (1986, hereafter E86), which assumes slantwise moist neutrality. The E86 model has two components: an in- viscid, free atmosphere in hydrostatic and gradient wind balance and a boundary layer in which viscous and surface processes play an important role. Only the free atmosphere portion of the E86 model is used in our approach, which utilizes the fact that the RMW is very closely approximated by a surface of constant angular momentum (Stern and Nolan 2009). Details of this approach are outlined in the appendix.

By using the MR profile from Eq. (1) with ymax 5 45 m s21, RMW 5 31 km, and a 5 0.5 to prescribe the tangential velocity field at the lowest level (Fig. 3a), we obtain the vertical slope of the RMW and decay rate of tangential velocity along the RMW (Figs. 3b,c). The level at which tangential velocity becomes zero at the RMW is z 5 15.9 km. Then, the radial profile of tangential velocity at each vertical level is constructed by using Eq. (1). It turns out that the tangential wind field constructed this way decays too rapidly near the surface, such that the linear model supports unstable modes in that region. To remedy this problem, the lowest level of the tangential wind field is extended 0.5 km downward barotropically with the same radial structure, and the tangential wind field is smoothed in the vertical direction. Figure 4a shows the tangential wind field of this MR basic-state vortex. Although frictionally induced secondary circulations are important, they are neglected here, because above the boundary layer tropical cyclones can be approximated as vortices in hydrostatic and gradient wind balance (Willoughby 1990). After constructing the tangential wind field of the basic-state vortex, pressure and temperature fields that hold the vortex wind field in hydrostatic and gradient wind balance are computed, which is achieved with an iterative procedure described in Nolan et al. (2001) using the 21 FIG. 3. (a) MR radial profile of tangential velocity (m s ) with 21 Jordan (1958) mean hurricane season sounding as the ymax 5 45 m s and RMW 5 31 km. (b) Vertical profile of the far-field environment. Figure 4b shows the potential RMW of tangential velocity field constructed by using the free temperature anomalies required to keep the vortex in atmosphere component of the E86 model. (c) Vertical profile of balance. This MR vortex is stable to static and symmetric tangential velocity at the RMW from the same model.

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21 FIG. 4. Radius–height cross sections of (a) tangential velocity (m s ), (b) anomalous potential temperature (K), (c) square of strati- fication frequency (s22), and (d) PV fields [potential vorticity units (PVU)] of the basic-state MR vortex. Contour intervals are 5.0 m s21, 2.0 K, 5.0 3 1025 s22, and 2.5 PVU from the zero line, respectively. instabilities by ensuring ›u/›z . 0andPV. 0 over the technology have made it possible to observe and simu- whole domain (Figs. 4c,d). Stability analysis using the late tropical cyclones with convective features on the methods of Nolan and Montgomery (2002a) and Nolan same length scale as individual updrafts; however, it is and Grasso (2003) confirms that there are no exponen- beyond the ability of 3DVPAS to simulate the effect of tially growing modes in this MR vortex. individual updrafts and downdrafts in a spiral rainband. Rather, the goal of this study is to understand the response of the tropical cyclone wind field to the overall 4. Designing a spiral rainband heat source rainband heating in a simple framework that nonetheless can capture the time-dependent, nonhydrostatic a. The structure of an idealized spiral rainband dynamics of the response. heating The observational studies discussed in section 2 have Spiral rainbands are banded structures of convection shown that the upwind portion of spiral rainband heating and can be considered as a cluster of individual up- is convective and the downwind end is stratiform, with drafts and downdrafts where most of the strong verti- a transition between them. In addition, the rainband cal motions and associated latent heat release occur. heating spirals radially inward in a cyclonic direction Steady improvements in observational and computational and tilts radially outward with height. One simple way

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC JUNE 2010 M O O N A N D N O L A N 1785 to represent the overall diabatic heating structure of Nolan et al. (2007). First, we deﬁne the diabatic heating such rainbands is to overlap multiple heat sources sim- structures of purely convective and stratiform heat ilar to the individual heat sources previously used in sources as

8 >0, for z #"#z <> bc r r 2 z z Q_ (r, l, z, t) 5 Q_ exp bc sin bc p exp(inl invt), for z , z , z 1 s , (2) con > con max s s bc bc zc :> rc rc 0, for z $ zbc 1 szc 8 >0, for z "## zbs szs ! <> r r 2 z z Q_ (r, l, z, t) 5 Q_ exp bs sin bs p exp(inl invt), for z s , z , z 1 s (3) str > str max s s bs zs bs zs :> rs zs 0, for z $ zbs 1 szs

In Eqs. (2) and (3), rb(z) is the radial center position and and downwind (r 5 60 km) ends are purely convective sr and sz are the radial and vertical half-widths of the (Fig. 5a) and purely stratiform (Fig. 5b), respectively. heating, respectively. Subscripts c and s refer to purely The transition from the convective to stratiform diabatic convective and stratiform heat sources. For convective heating is deﬁned to be linear. As the rainband makes heat sources (Fig. 5a), zbc is chosen to be the lowest al- a transition from convective to stratiform, the diabatic titude where positive diabatic heating is present, and the heating structure becomes shallower and wider, and vertical depth of the heating is controlled by taking the an area of signiﬁcant cooling develops in the lower part half vertical wavelength szc only upward from zbc. For of the rainband. Figure 5c shows the vertical cross sec- stratiform heat sources (Fig. 5b), zbs is the level of zero tion in the middle part of the rainband, illustrating the heating, and its vertical depth is controlled by taking change in the vertical structure of diabatic heating during the half vertical wavelength szs both upward and down- this linear transition. At this location (r 5 70 km), the ward from zbs. Observations (e.g., BZJM83; P90) indicate heat source is a combination of 50% convective and 50% that rainband heating tilts radially outward with height, stratiform heat sources. The maximum heating rate of each which can be achieved by linearly varying rb with alti- individual convective heat source is chosen to be twice tude, such that rb(z) 5 rbsfc 1 z, where rbsfc is the radial the maximum heating rate of the stratiform heat source, _ 21 _ 21 center position at the surface. The complex exponential so Qcon max 5 3.0 K h and Qstr max 5 1.5 K h . term in Eqs. (2) and (3) rotates the heat source around the center of the vortex at an angular velocity v 5 y /r , rot c b. How 3DVPAS sees the spiral rainband diabatic where r is the radial location of the center point of the c heating rainband heating at the surface and yrot is the rotation speed of the heat source. To compute the impact of this idealized spiral rain- Now we create an idealized but realistic-looking spiral band in 3DVPAS, the idealized spiral rainband heating rainband by overlapping 40 heat sources. By ‘‘over- in Fig. 6a is separated into different azimuthal wave- lapping’’ heat sources, we mean that heat sources are number components by Fourier series decomposition. placed close to each other so that most points in the The magnitude of the purely asymmetric components rainband receive a heating contribution from multiple decreases as the azimuthal wavenumber increases (not heat sources. The sum of these overlapped heat sources shown). Figure 6b shows the sum of its Fourier de- is the rainband heating. Figure 6a shows the horizontal compositions from n 5 0 to 4, and it is a bit different cross section of this rainband at z 5 4.5 km. This rain- from the original rainband heating (Fig. 6a) because of band covers approximately one quarter of the circum- the truncation of the series at n 5 4. Each wavenumber ference in the lower right quadrant, but its upwind end is component of the heating is simulated separately in at r 5 80 km, while its downwind end is at r 5 60 km. 3DVPAS. To obtain the full response of the vortex to a The radial center location of each heat source is equal to rainband heating, it is necessary to consider the asym- rbc in Eq. (2) and rbs in Eq. (3), which varies linearly metric and symmetric responses to each asymmetric between the ends of the rainband (from r 5 80 km to component plus the symmetric response to the sym- 60 km). The heat sources at the upwind (r 5 80 km) metric part. Therefore, the full response of the vortex to

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the rainband heating can be approximated in the linear limit by the sum of the asymmetric and symmetric responses to its ﬁrst four asymmetric components plus the symmetric response to its symmetric part. Hereafter, the response refers to the sum of the asymmetric and symmetric responses to the purely asymmetric components (n 5 1–4) and the symmetric response to the symmetric part (n 5 0). In addition, all responses are rotated such that the rainband heating at the time the response is shown would always be in the same location as in Fig. 6a.

5. The response of the hurricane wind field to rainband heating a. A purely convective spiral rainband heating A typical spiral rainband heating contains both convective and stratiform components; however, to better understand the combined response to these two different heating profiles, it is helpful to first consider the response to a rainband heating that is entirely convective or stratiform. Figures 6c and 6d show the horizontal cross sections at z 5 4.50 km of the idealized purely convective rainband heating and its sum of Fourier decompositions from n 5 0 to 4. Just like the mixed rainband described in the previous section, a purely convective rainband spirals inward (from r 5 80 to 60 km), but the structure of its diabatic heating remains the same along the band (see Fig. 5a). The first four purely asymmetric components plus the symmetric part are rotated around the MR vortex for 24 h. Since rainbands are observed to move more slowly than the mean circulation speed (Anthes 1982), the rotation speed is set to be 70% of the maximum tangential wind at the middle part of the 21 rainband at r 5 70 km, so yrot 5 21.1 m s . Figure 7 shows time evolution of kinetic energy (KE) of the sum of the asymmetric responses to the asymmetric components (n 5 1–4, hereafter KEaa), the symmetric response to the symmetric part (n 5 0, hereafter KEss), and the sum of the symmetric responses to the asymmetric components (n 5 1–4, hereafter KEsa). It shows that KEaa increases rapidly in the first 8 h and then gradually levels off although it never quite reaches a steady state, which is due to the excitation of a weakly damped n 5 1 inner-core mode, as KE for n $ 2 reaches FIG. 5. (a),(b) Vertical cross sections of normalized purely con- a nearly steady state by approximately t 5 15 h (not 5 vective and stratiform diabatic heat sources centered at r 80 and shown). After t 5 16 h, the change in KE only results 60 km, respectively. (c) Vertical cross section of normalized mixed (50% convective, 50% stratiform) diabatic heat source centered at from the accumulation of KE in the n 5 1 inner-core r 5 70 km. For (a), rbsfc 5 80 km [so rbc(z) 5 80 km 1 z in Eq. (2)], mode. Unlike KEaa, however, after a period of adjust- zbc 5 1 km, src 5 2 km, and szc 5 7 km are used; for (b), rbsfc 5 ments in the first several hours KEss increases at a lin- 60 km [so rbs(z) 5 60 km 1 z in Eq. (3)], zbs 5 4 km, srs 5 6 km, ear rate after t 5 5 h. The period of initial adjustment s 5 5 and zs 2 km are used. For (c), rbsfc 70 km. in KEsa is longer but it increases linearly after t 5 15 h.

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21 FIG. 6. (a) The horizontal cross section at z 5 4.50 km of the mixed spiral rainband heating (K h ). (b) The sum of its Fourier decompositions from n 5 0 to 4. (c),(d) As in (a),(b), but for the purely convective spiral rainband heating (K h21). (e),(f) As in (a),(b), but for the purely stratiform spiral rainband heating (K h21). Forty heat bubbles are used to construct each rainband heating, which spirals cyclonically inward from r 5 80 to 60 km. The zero line is not plotted. Maximum and minimum values can be found above each panel. See section 4a for explanations.

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heating. Comparing the location of vertical motion at each cross section with the location of the rainband heating (Figs. 6c,d) indicates that vertical motion is well coupled to the heating, and this is found to be true for all subsequent cases. Hereafter, the location of vertical motion of the response is used interchangeably with the location of the rainband heating. Examining the horizontal velocity components of the response at the lower levels (Fig. 8c) shows an accelerated tangential circulation on the radially outward side of the rainband heating. However, this feature becomes weak at the middle levels (Fig. 8b), and at the upper levels (Fig. 8c) there is instead a decelerated tangential circulation on the radially outward side of the rainband heating. Figure 9 shows vertical cross sections through the downwind, middle, and upwind regions (see white lines FIG. 7. Time evolution of normalized KE of the asymmetric responses to the asymmetric components (KEaa; solid), the sym- in Fig. 6a) of the response to the convective rainband metric response to the symmetric part (KEss; dashed), and the heating. Arrows depict radial and vertical velocity com- symmetric responses to the asymmetric components (KEsa; ponents of the response, while contours show the tan- dashed–dotted) of the convective rainband. KE values are nor- gential velocity component of the response. All cross malized by their respective maximum value during the 24-h period, which are 7.90 3 1012 J for KEaa, 4.44 3 1014 J for KEss, and sections show that tangential circulation around the lo- 8.30 3 1012 JforKEsa. cation of the heating and on its radially outward side is accelerated at the lower levels (from the surface to z 5 4 km) but decelerated at the upper levels between z 5 5 KEsa decreases1 for the first 6 h as the asymmetric com- and 10 km. In addition, there is radial inflow toward the ponents of the rainband heating interact with the basic- rainband from its radially outward side at the lower state vortex and extract KE from it (see section 4b of levels but radial outflow away from the rainband at the Nolan et al. 2007). At t 5 16 h and thereafter, the re- upper levels. Upward vertical motion is found between sponse to the rainband heating is expected to change radial inflow and accelerated tangential velocity at the only quantitatively. Unless noted otherwise, the re- lower levels and between radial outflow and decelerated sponse hereafter is evaluated at t 5 16 h for all cases tangential velocity at the upper levels. The strongest considered. rising motion is found between z 5 4 and 6 km. The axis Figure 8 shows horizontal cross sections of the re- of the largest upward velocity moves radially inward in sponse to the convective rainband heating at z 5 1.83 the downstream direction, in accordance with the in- (lower level), 4.50 (middle level), and 7.17 km (upper wardly spiraling structure of the convective rainband level); z 5 1.83 and 7.17 km are located slightly above heating. the bottom of the heating and slightly below the top of Although full understanding of the response to the the heating respectively, while z 5 4.50 km is at the rainband heating requires examining both asymmetric middle level of the heating. Arrows and contours rep- and symmetric responses to the asymmetric and sym- resent horizontal and vertical velocity components of metric components of the heating, the symmetric re- the response. All cross sections show that an inner-core sponses alone capture a large part of the full response. mode is excited by the rainband heating, and this mode Figure 10a shows the vertical cross section of the sum of has a strong n 5 1 structure. Despite the presence of an the symmetric responses to both asymmetric and sym- inner-core mode, the response to the rainband heating metric parts (n 5 0–4) of the convective rainband, while can be identified in all cross sections. They show that Fig. 10b shows the vertical cross section of the symmetric there is upward vertical velocity at the location of the response to the symmetric part (n 5 0) alone. Both cross sections clearly show that around the rainband location and on its radially outward side there is low-level radial inflow and upper-level radial outflow, which are con- 1 KEsa can in fact be negative, as it is defined as the difference in nected by upward vertical motion through the rainband total KE between the initial symmetric vortex and the symmetric vortex at later times. If interactions with the evolving asymmetries heating. On its radially inward side, there is weak low- change the symmetric vortex to a flow with less KE, then KEsa can level radial outflow that enters the updraft and some of be negative. the updraft gets diverted into weak radial inflow at the

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upper levels. Clearly, there is convergence and divergence at the base and top of the updraft, respectively. These vertical cross sections also indicate that tangential circulation at the lower levels is accelerated at the location of the heating, as well as on its radially outward side, but decelerated at the upper levels. This symmetric response is qualitatively similar to transverse circulations presented in Eliassen (1951), Shapiro and Willoughby (1982), and Hack and Schubert (1986). All vertical cross sections from Fig. 9 exhibit these similar features of the symmetric responses shown in Fig. 10. There are, however, some notable differences among different parts of the rainband. The low-level radial inflow toward the rainband is deeper and stronger in the downwind region in comparison to the upwind region. This radial inflow after passing the rainband continues to travel toward the vortex center in the downwind region, while there is actually a radial outflow toward the rainband in the upwind region. At the upper levels, the radial outflow from the rainband is weaker and shallower in the upwind region than in the downwind region. In addition, on the radially inward side of the rainband, there is radial inflow in the upwind region but radial outflow in the downwind region. These differences may be understood conceptually by considering the PV of the asymmetric response. In a linearized system such as 3DVPAS, the perturbation PV is 1 1 ›u h ›u ›u q 5 (v $u 1 v $u ) 5 j 1 n 1 z n r n n r n ›r r ›l n ›z ›u h ›u ›u 1 j n 1 n 1 (f 1 z) n , (4) ›r r ›l ›z

where j, h, and z are, respectively, the radial, azimuthal, and vertical components of the three-dimensional vorticity vector v; r is density; u is potential temperature; the overbar and subscript n (azimuthal wavenumber) denote the axisymmetric (basic state) and asymmetric perturbation quantities, respectively; and f is a Coriolis parameter of 5.0 3 1025 s21. In a baroclinic vortex without secondary circulations, such as the MR vortex here, a thermal perturbation such as the rainband heating produces a PV perturbation that has contributions from FIG. 8. Horizontal cross sections at z 5 (a) 7.17, (b) 4.50, and (c) 1.83 km of the response to the convective rainband heating (Fig. the fourth and sixth terms in Eq. (4). For the convective 6c). Contours represent vertical velocity (m s21) of the response rainband, the fourth term creates positive PV on its ra- with a contour interval of 0.01 m s21 from the zero line, which is dially inner side and negative PV on its radially outer not plotted. In this and all subsequent grayscale ﬁgures, dark colors side, while the sixth term produces positive PV at the are more positive than light colors, and positive values are in- lower levels and negative PV at its upper levels. The sum dicated by solid contours, while negative values are indicated by dashed contours. Arrows show horizontal velocity (m s21) of the of these four PV components results in the PV forcing response. Maximum and minimum values of the contoured ﬁeld as that has positive PV at the low levels and negative PV at well as the magnitude of the largest arrow (jmaxj) can be found the upper levels (Fig. 11a). above each panel.

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FIG. 10. Vertical cross sections of (a) the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0– 4) and (b) the symmetric response to the symmetric component (n 5 0) of the convective rainband heating (Fig. 6c). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.10 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21)of the response. Maximum and minimum values of the contoured ﬁeld as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.

In response to this PV forcing, positive PV and cyclonic circulations are expected to develop at the lower levels, but negative PV and anticyclonic circulations should develop at the upper levels. Figure 12 shows the sum of the asymmetric responses to the asymmetric components of the convective rainband, and it clearly FIG. 9. Vertical cross sections through the (a) downwind, (b) illustrates these features; the full response also hints at middle, and (c) upwind regions of the response to the convective them (Figs. 8a,c). Figure 11b also shows the vertical rainband heating (Fig. 6c). Contours represent tangential velocity 2 2 cross section of PV of the sum of the asymmetric re- (m s 1) of the response with a contour interval of 0.20 m s 1 from the zero line, which is not plotted. Arrows show radial and vertical sponses to the asymmetric components (n 5 1–4) to the velocity components (m s21) of the response. Maximum and mini- convective rainband. Careful examination of Figs. 10 mum values of the contoured ﬁeld as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.

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21 FIG. 11. Vertical cross sections through the middle region of (a) PV forcing (PVU h ) associated with the convective rainband heating (Fig. 6c) and (b) PV (PVU) associated with the response to the convective rainband heating. (c),(d) As in (a),(b), but for the stratiform rainband (Fig. 6e) case. Note that different contour interval values are used, and the zero line is not plotted. Maximum, minimum, and contour interval can be found above each panel.

and 12 indicates that the asymmetric PV and symmetric upper levels between the upwind and downwind regions transverse responses cooperate in some places but act in can be attributed to the different superimpositions of the opposite directions in other places. For example, at the asymmetric response with the symmetric response. lower levels in the upwind region, the asymmetric and The collocation of radial inflow with increased tan- symmetric responses are in the same direction on the gential velocity at the lower levels and radial outflow radially inward side of the rainband, producing radial with decreased tangential velocity at the upper levels outflow into the rainband, but in the opposite direction suggests that the change in tangential wind field is simply on its radially outward side, resulting in weak radial the result of an angular momentum-conserving radial inflow into the rainband. At the lower levels in the flow. The rainband heating is located outside the RMW downwind region, the asymmetric and symmetric re- where the tangential wind decays as r2a (a 5 0.5 in the sponses are in the same direction on the outward side, MR vortex); thus, angular momentum associated with but they are in the opposite direction on the inward side tropical cyclone tangential wind field increases radially and result in radial inflow toward the center because the outward. Therefore, radial flow from a region of higher asymmetric response is stronger. Similarly, the differ- angular momentum to a region of lower angular mo- ences in the strength and direction of the radial flow at the mentum (e.g., low-level radial inflow) will increase

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height, more outward than the updraft. The change in tangential wind field due to angular momentum- conserving radial flow, however, is larger in comparison to that due to angular momentum-conserving vertical flow, because the angular momentum transport by radial flow is larger than that by vertical flow by a factor of 3 or higher. Comparing Figs. 6d and 6b suggests that the error from truncating the Fourier series at n 5 4 is larger in the convective rainband forcing (Fig. 6c) than in the mixed rainband forcing (Fig. 6a). The convective rainband case was repeated with the truncation at n 5 8, but the response of the hurricane wind field to the n 5 0–8 convective rainband heating remained essentially the same (not shown). b. A purely stratiform spiral rainband heating Figures 6e and 6f show the horizontal cross sections at z 5 4.50 km of the idealized purely stratiform rainband heating and its sum of Fourier decompositions from n 5 0 to 4. The purely stratiform rainband has heating above and cooling below the level of zero heating at z 5 4.0 km (see Fig. 5b). Figure 13 shows horizontal cross sections of the response at z 5 2.17 (lower level), 3.83 (middle level), and 5.83 km (upper level). Note that z 5 2.17 and 5.83 km are located slightly above the bottom of the cooling component and slightly below the top of the heating component of the stratiform rainband heating, respectively, while z 5 3.83 km is slightly below the level of zero heating. All cross sections show that there are inner-core modes excited by the heating. While the modes from the convective rainband case have a strong wavenumber-1 signature at all levels (Fig. 8), different FIG. 12. Horizontal cross sections at z 5 (a) 7.17 and (b) 1.83 km wavenumber structures tend to dominate at different of the sum of the asymmetric responses to the asymmetric com- levels in the stratiform case. Despite the excitation of ponents of the convective rainband (Fig. 6c). Contours show PV (PVU) and arrows show horizontal velocity (m s21). Note that the inner-core modes, it is easy to identify the response different contour interval values are used, and the zero line is not to the stratiform rainband heating. Since this stratiform plotted. Maximum and minimum values of the contoured field as rainband heating has heating above and cooling below well as the magnitude of the largest arrow (jmaxj) can be found z 5 4.0 km (see Fig. 5b), there are rising and sinking above each panel. motions above and below this level, indicating vertical divergence at this level. Also present near this level tangential wind, while radial flow from a region of lower (Fig. 13b) is accelerated tangential circulation on the angular momentum to a region of higher angular mo- radially outward side of the rainband heating. Near the mentum (e.g., upper-level radial outflow) will decrease top of the heating component (Fig. 13a), there is anti- tangential velocity. In addition, the updraft component cyclonic radial outflow emanating away from the rain- of the overturning secondary circulation flows from band where rising motion is present. At the base of the a region of higher angular momentum to a region of cooling component (Fig. 13c) there is weak anticyclonic lower angular momentum, leading to positive tangential radial outflow away from the rainband where sinking velocity change at the upper levels but negative tan- motion exists. It is interesting to note that the low-level gential velocity change at the lower levels (not shown). horizontal circulation is substantially weaker than that This process occurs because the tangential velocity field at middle and upper levels. of the MR vortex decreases with height, so the angular Figure 14 shows vertical cross sections through the momentum surfaces are sloped radially outward with downwind, middle, and upwind regions (see white lines

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC JUNE 2010 M O O N A N D N O L A N 1793 in Fig. 6a) of the response to the stratiform rainband heating. All cross sections show that there is radial inflow toward the rainband at the middle levels (between z 5 3 and 5 km). After passing through the rainband, this inflow splits into two branches: one descends to the surface and continues to travel toward the vortex center, and the other ascends to the upper levels (near z 5 6km). This midlevel radial inflow pattern is collocated with the positive tangential velocity component of the response, indicating that angular momentum conservation leads to the accelerated tangential circulation. Also present at these levels is radial outflow that enters the rainband from its radially inward side and decreased tangential velocity response. In addition, on the radially outward side of the rainband, there is negative tangential velocity response above and below the midlevel radial inflow layer, respectively. Figure 15 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the stratiform rainband heating and the symmetric response to its symmetric part (n 5 0) alone. Both cross sections clearly show that at the location of the rainband heating there are rising and sinking motions above and below z 5 4 km, respectively. In addition, on the radially outward side there is radial inflow toward the rainband at this altitude. It appears that the symmetric responses are very similar to the full response through the middle portion (Fig. 14b). The collocation of positive and negative tangential velocity responses with radial inflow and outflow indicates that angular momentum conservation leads to these tangential velocity responses. The angular momentum transport by radial flow is larger than that by vertical flow by a factor of 4 or higher. Examining Fig. 14 in detail reveals that the response to the stratiform rainband heating does not show as much variation along the band as the response to the convective rainband (Fig. 9), which indicates that the symmetric responses to the stratiform rainband capture an even larger portion of the full response in comparison to the convective case. There are, however, still some notable differences, especially in the radial flow pattern. For example, there is radial inflow toward the rainband from its radially outward side at the middle levels in the downwind region but weak radial outflow in the upwind region. In addition, the radial flow on the radially outward side at the upper levels is oppositely directed between the downwind and upwind regions. FIG. 13. As in Fig. 8, but for the response to the stratiform These differences may be attributed to the asymmet- rainband heating (Fig. 6e) at z 5 (a) 5.83, (b) 3.83, and (c) 2.17 km, with a contour interval of 0.03 m s21. ric PV responses. Figure 11c shows the vertical cross section of PV forcing associated with the stratiform rainband. There is negative PV at the lower and upper levels but positive PV at the middle levels. In response,

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FIG. 15. As in Fig. 10, but for the stratiform rainband heating (Fig. 6e), with a contour interval of 0.50 m s21.

positive PV and cyclonic circulations develop at the middle levels and negative PV and anticyclonic circulations prevail at the lower and upper levels, as illustrated in the horizontal cross sections of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband (Fig. 16). Figure 11d also shows the vertical cross section of PV of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband. Comparing Figs. 15 and 16 suggests that the asymmetric PV and symmetric transverse responses are in the same direction in some regions but in the opposite direction in other places. For FIG. 14. As in Fig. 9, but for the response to the stratiform rainband example, on the radially outward side of the rainband, 21 heating (Fig. 6e), with a contour interval of 0.50 m s . the asymmetric and symmetric responses are in the same direction in the downwind region but in the opposite direction in the upwind region. Because the radial velocity of the asymmetric response is stronger than that of

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the symmetric response, there is weak midlevel radial outﬂow in the upwind region. A more pronounced midlevel tangential velocity response in the downwind region is due to the fact that the cyclonic asymmetric response strongly projects into the azimuthal direction in the downwind region but more into the radial direction in the upwind region.

c. A mixed spiral rainband heating With the previous results in mind, we now present the response to a mixed spiral rainband heating. The detailed configurations of the mixed rainband were described in section 4a. The mixed rainband heating spirals radially inward in a cyclonic direction from r 5 80 to 60 km, and the upwind and downwind regions of the heating are convective and stratiform, respectively, with a linear transition between them. Figure 17 shows horizontal cross sections of the response at z 5 2.50 (lower level), 4.17 (middle level), and 6.83 km (upper level). Despite the usual excitation of an inner-core mode, it is not difficult to isolate the response to the rainband heating. At the lower levels (Fig. 17c), the rainband heating is positive upwind and negative downwind, and accordingly there are rising and sinking motions in those respective regions. On the radially outward side of the rainband, there are some signs of accelerated tangential circulation. This positive tangential velocity response becomes more evident at the middle levels (Fig. 17b), especially on the downwind region. At these altitudes, the heating is positive everywhere along the rainband, and accordingly there is rising motion everywhere. At the upper levels (Fig. 17a), there is instead a negative tangential velocity response. Also present at these altitudes is radial flow that diverges away from the rainband in an anticyclonic direction. It seems evident from these cross sections that the asymmetric PV response is cyclonic at the lower and middle levels but anticyclonic at the upper levels. Figure 18 shows vertical cross sections through the upwind, middle, and downwind regions (see white lines in Fig. 6a) of the mixed rainband. All cross sections show that on the radially outward side of the rainband there is accelerated tangential velocity at the lower levels, especially between z 5 3 and 4 km. Above the positive tangential velocity response, there is a decelerated tangential circulation, centered near z 5 7 km. In the up- FIG. 16. Horizontal cross sections at z 5 (a) 5.83, (b) 3.83, and (c) wind region, there is weak radial inflow toward the 2.17 km of the sum of the asymmetric responses to the asymmetric rainband from its outward side at the lower levels and components of the stratiform rainband (Fig. 6e). Contours show PV (PVU) with a contour interval of 0.60 PVU from the zero line, radial outflow from the radially inward side. They con- which is not plotted. Arrows show horizontal velocity (m s21). verge at the base of the rainband and then rise through it Maximum and minimum values of the contoured field as well as the at approximately r 5 80 km, and most of it becomes magnitude of the largest arrow (jmaxj) can be found above each panel.

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC 1796 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 67 radial inflow at the upper levels. In the downwind region, there is deep (from the surface to z 5 4 km) radial inflow toward the rainband from its outward side, and most of it continues to flow toward the center after passing the rainband. Between z 5 3 and 4 km, some of the radial inflow that enters the rainband descends to lower levels between r 5 80 and 60 km. Radial inflow between z 5 4 and 5 km meets radial outflow that enters the rainband from its inward side and rises to upper levels, where there is mostly radial outflow. The tangential velocity response in the downwind region is greater than in the upwind region. The middle region appears to be a mix of the upwind and downwind regions. Figure 19 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the heating and the symmetric response to the symmetric part (n 5 0) alone. Both cross sections show that there is deep radial inflow toward the rainband from its radially outward side, and this inflow is stronger at z 5 4 km than at the surface. Most of this inflow rises through the rainband and becomes radial outflow away from the rainband between z 5 5 and 9 km; however, some of this inflow descends from z 5 3 to 2 km and continues to travel radially inward. Near z 5 6 km, some of the updraft diverts toward the vortex center. On the radially inward side, there is radial outflow at z 5 4 km toward the rainband. Clearly, there is convergence at the location of the rainband heating at this altitude, and divergence above it. This symmetric response to the mixed rainband also appears to be a mix of the convective and stratiform symmetric responses presented earlier, and it explains a large portion of the full response. As in the previous cases, the asymmetric and symmetric responses act in the same direction in some locations, such as on the radially outward side at the lower and middle levels in the downwind region, but in the opposite direction in other locations, such as on the radially outward side at the upper levels in the upwind region. d. Comparison to observations Previous observational studies of spiral rainbands indicated that there are an overturning secondary circulation, an accelerated tangential circulation on the radially outward side of the rainband, and a midlevel radial inflow that descends to the surface, intertwining FIG. 17. As in Fig. 8, but for the response to the mixed rainband with the lower branch of the overturning circulation. heating (Fig. 6a) at z 5 (a) 6.83, (b) 4.17, and (c) 2.50 km, with 21 Figure 2 illustrates a conceptualized model of a spiral a contour interval of 0.03 m s . rainband circulation as presented in HH08. The idealized spiral rainband heating structures considered so far are designed to represent only the gross effects of latent heat release within the spiral rainband. Nonetheless, the

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FIG. 19. As in Fig. 10, but for the mixed rainband heating (Fig. 6a), with a contour interval of 0.20 m s21.

response of the MR vortex wind ﬁeld to the mixed spiral rainband heating captures all of the aforementioned features with the contributions from the responses to both purely convective and stratiform rainband heating. It appears that the midlevel radial inﬂow and its descending motion to the surface closely resemble the stratiform response and are stronger in the downwind region. The overturning secondary circulation feature belongs to the convective response and is stronger in the upwind region. An accelerated tangential circulation on the radially outward side of the rainband shares the characteristics of both the convective and stratiform FIG. 18. As in Fig. 9, but for the response to the mixed rainband responses, but it tends to be stronger in the downwind 21 heating (Fig. 6a), with a contour interval of 0.40 m s . region where stratiform response prevails. In addition, this accelerated tangential circulation feature extends further radially away from the rainband than what observations indicate (see Figs. 1b and 2).

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC 1798 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 67 e. The effects of the inner-core modes All cross sections through the responses indicate that the response to the rainband heating occurs not only at the location of the heating but also around the RMW. In the convective rainband case, horizontal cross sections (Fig. 8) indicate that there are strong azimuthal wavenumber-1 signatures in the inner-core response. Vertical cross sections (Fig. 9) show that the tangential wind ﬁeld at the lower levels (from the surface to z 5 3 km) is decelerated inside the RMW but accelerated just outside the RMW. In addition, there is increased tangential velocity response near the RMW at the upper levels (between z 5 6 and 10 km). In the linear limit, this process would suggest that the effect of the inner-core modes excited in response to the purely convective rainband is to vertically stretch the tangential circulation on the rainband side and pull the whole tangential circulation toward the rainband. The inner-core modes in the response to the purely stratiform rainband are not as severe as in the convective case. The inner-core modes in the mixed rainband case show similar low-level structures as in the convective case but are disorganized in the upper levels. The overall effects of the asymmetric inner-core modes in all cases, however, are small because the differences in the wind speed change between the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the rainband and the sum of the symmetric response to the symmetric component (n 5 0) of the rainband are less than 5% (see Figs. 10, 15, and 19).

6. Sensitivity tests FIG. 20. Vertical cross sections through the (a) downwind and (b) upwind regions of the response to the mixed rainband heating We must consider whether the responses described in whose convective heat bubbles are 10 km deep in the vertical di- the previous section are highly sensitive to the particular rection. Contours represent tangential velocity (m s21) of the re- parameters that define the structure and movement of sponse with a contour interval of 0.30 m s21 from the zero line, the rainband heating. As described in section 4a, there which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum are a large number of parameters involved in controlling values of the contoured field as well as the magnitude of the largest the evolution of imposed rainband heating. Tests are arrow (jmaxj) can be found above each panel. performed for only parameters that appear to have a large variation based on observations. heating, the upper bound of z 5 8kmisverycloseto a. Depth of convection in the upwind region the 75th percentile of the cumulative density function Cecil et al. (2002) created the composites of radar of radar reflectivity. reflectivity over tropical cyclones as observed by the To evaluate the effects of the different depths of up- Tropical Rainfall Measuring Mission satellite (see their wind convection, the mixed rainband case is repeated

Fig. 3). Their statistics indicate that the convective with sz 5 9 and 11 km so the upper bound of upwind region of spiral rainbands sometimes has convection convection is z 5 10 and 12 km, respectively. Figure 20 penetrating deeper than z 5 8 km, which is the upper shows vertical cross sections of the response through the boundary of the convective heat forcings used in the upwind and downwind region (see white lines in Fig. 6a) previous section (see section 4a and Fig. 5a). By using for the sz 5 9 km case (thus 10 km deep). It is evident 20 dBZ to deﬁne the boundary for a spiral rainband that the effect of deeper upwind convection is to elevate

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FIG. 21. Vertical cross sections through the (top) downwind and (bottom) upwind regions of the response to the mixed rainbands that are (left) 50% convective and (right) 50% stratiform. See section 6b for details. Contours represent tangential velocity (m s21) of the response with a contour interval of 0.50 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel. the radial outflow of the overturning secondary circu- schemes. One scheme (hereafter 50% convective) keeps lation feature; however, the midlevel radial inflow and the rainband purely convective over the first half of the its descending motion, as well as the accelerated tan- rainband from its upwind end and then makes a linear gential circulation on the radially outward side of the transition to purely stratiform regime over the second rainband, remain unaffected by the presence of deeper half, while the other scheme (hereafter 50% stratiform) upwind convection. This point further supports the makes a linear transition from purely convective to conclusion from the previous section that the over- purely stratiform over the first half and then remains turning circulation is due to the response to purely purely stratiform over the second half. Three common convective parts of the rainband. rainband circulation features are present in the response to both cases (Fig. 21). Notable differences from the b. Organization of convection within rainbands mixed rainband case are that the overturning circulation All rainband cases examined so far have the linear is strengthened in the 50% convective case while the transition occurring over the whole rainband, which may 50% stratiform case shows enhanced descending mid- not be representative for all rainbands. The mixed rain- level radial inflow. Both cases show increased tangential band case is repeated but with two different transition circulation on the radially outward side of the rainband,

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC 1800 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 67 but the altitude of the largest positive tangential velocity response is higher for the 50% stratiform case than the 50% convective case. While a large number of observed rainbands are convective upwind and stratiform downwind, it is possible to have highly disorganized rainbands, which can be represented by constructing a rainband only with heat sources that have equal convective and stratiform contributions at all locations along the band (see Fig. 5c). Other parameters for this disorganized rainband are the same as in the mixed rainband. The response to the disorganized rainband (Fig. 22) recovers the overturning circulation and accelerated tangential circulation on its radially outward side. The descending midlevel radial inflow, however, is not present in the downwind region response, likely due to the reduced area of cooling in the downwind region of this disorganized rainband in comparison to the mixed rainband (cf. Figs. 5c and 5b). We also considered another way to change the transition between the upwind and downwind ends, which is _ _ to modify the values of Qcon max in Eq. (2) and Qstr max in Eq. (3). Their ratio represents the different contributions of purely convective and purely stratiform heat _ _ sources. When Qcon max: Qstr max is increased, the overturning secondary circulation is strengthened as in the 50% convective case. Decreasing the ratio leads to enhanced descending midlevel radial inflow as in the 50% stratiform case. It also appears that the effect of stronger convective heat sources in the rainband in the upwind region is to vertically stretch the positive tangential velocity response near the rainband because of the stronger updraft. In the downwind region, stronger stratiform heat sources lead to enhanced tangential circulation at FIG. 22. Vertical cross sections through the (a) downwind and (b) the middle levels on the radially outward side of the upwind regions of the response to the disorganized rainband (see 21 rainband. section 6b). Contours represent tangential velocity (m s ) of the response with a contour interval of 0.50 m s21 from the zero line, c. Transient rainband which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum So far we have examined the response of the hurricane values of the contoured field as well as the magnitude of the largest wind field to rainband heating that is persistent over arrow (jmaxj) can be found above each panel. time. However, observations (e.g., Anthes 1982) indicate that the duration of individual convective cells embedded within rainband heating is typically on the shapes the rainband heating to approximately last for order of several hours. The rainbands themselves are 2tforc h. The structure of this exponential function is transient, but the precise mechanisms controlling their very similar to that used in Nolan et al. (2007; see their initiation, duration, and decline remain unknown. To Fig. 5b). Another simple function to represent the tran- represent the transient effects of the rainband heating, sient effects is to use a piecewise linear function such that we now multiply Eqs. (2) and (3) by the rainband heating rate reaches its peak at t 5 tpeak and "# then decays until it dissipates at t 5 tdiss. 4 t t The mixed rainband case from the previous section is F(t) 5 exp forc . (5) ptforc repeated with the heating now following the exponential and linear functions described previously. For the ex-

In Eq. (5), tforc is the time at which the rainband heating ponential function, tforc 5 6 h is used, and for the linear is at its maximum. Setting the coefﬁcient p 5 0.55163 function, tpeak 5 4 h and tdiss 5 3tpeak 5 12 h are used.

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Figure 23 shows horizontal cross sections at z 5 4.17 km heat sources were designed in accordance with previous of the response to the transient mixed rainband heating observations of spiral rainbands, and they were embedded using Eq. (5) at different times. Comparison with the in the hurricane wind field and rotated around the vortex. response to the persistent mixed rainband heating (Fig. The response to a typical mixed rainband captures many 17) indicates that cyclonic tangential acceleration on the common features of spiral rainband kinematic structures, radially outward side of the rainband can be seen locally such as an overturning secondary circulation, an enhanced around the rainband heating as early as t 5 4 h (Fig. tangential circulation on the radially outward side of the 23a). The overturning secondary circulation and de- rainband (or secondary horizontal wind maximum), and scending midlevel radial inflow are also present in the a midlevel radial inflow that descends to the surface. response by this time (not shown). As the heating rate Comparison of the responses to the purely convective becomes larger, the response becomes more easily dis- and stratiform rainbands indicates that the overturning cernable (Fig. 23b, t 5 7 h), and this tangential accel- secondary circulation mostly results from the convective eration starts to propagate in the downstream direction part of the rainband and is stronger in the upwind re- relative to the rainband heating. As the heating rate gion, while midlevel radial inflow descending to surface ramps down (Fig. 23c, t 5 10 h), the response starts to results from the stratiform characteristics of the rainband weaken, especially in vertical velocity response; how- and is stronger in the downwind region. The secondary ever, the cyclonic tangential acceleration feature re- horizontal wind maximum is exhibited in both convective mains coherent long after the heating is turned off (Figs. and stratiform parts of the rainband, but it tends to be 23e,f). The evolution of the transient mixed rainband stronger in the downwind region. Sensitivity tests con- that follows the piecewise linear forcing is qualitatively firm that the response is robust and that the latent similar (not shown). heating structure of convection embedded in rainbands is primarily responsible for their kinematic structures. In d. Rotation speed and radial location of rainband addition, the response of the hurricane wind field to the heating rainband heating is, in the linear limit, the sum of the Previous observational studies suggested that spiral asymmetric PV and symmetric transverse circulations. rainbands move around the center more slowly than the Frictionally induced secondary circulations are an im- mean circulation. There are observations that some portant part of the tropical cyclone wind field. The basic- spiral rainbands (e.g., principal bands) stay at the same state MR vortex used in this study, however, lacks them, region for a long period of time. The mixed rainband which is justified by the work by Willoughby (1990) that case was repeated but with yrot prescribed to be 60% and showed in the free atmosphere above the boundary 80% of the maximum tangential wind at the middle part layer tropical cyclones are approximately cyclonic vor- of the rainband (r 5 70 km). The corresponding values tices in hydrostatic and gradient wind balance. The 21 of yrot are 18.1 and 24.1 m s , respectively. A stationary secondary circulations are closely related with the dis- 21 mixed rainband case (yrot 5 0.0 m s ) was also consid- tribution of convection around tropical cyclones, which ered. All responses (not shown) recovered the aforemen- are highly asymmetric and transient; however, the re- tioned main features of rainband circulation, indicating sponses to the rainband in all cases considered develop that the sensitivity of the response to the rotation speed their own secondary circulations, so the qualitative as- is not significantly large. pects of the response appear not to be significantly af- The sensitivity of the response to the radial location fected by the lack of the secondary circulation. In of the rainband was examined by considering the mixed addition, the location of the rainband secondary circu- rainband case with the convective upwind end at r 5 lation above the boundary layer is different from the 120 km and the stratiform downwind end at r 5 100 km. location of the mean secondary circulation. It is possible Sixty heat sources were used to construct the rainband, that incorporating the secondary circulation into the which still makes a linear transition in the downstream basic-state vortex would lead to a more vigorous response direction. The response was qualitatively similar to the to rainband heating because the mean anticyclonic cir- mixed rainband except that the features are shifted 20 km culation at the upper levels reduces inertial stability. The radially outward (not shown). excitation of inner-core modes could be less severe because of the upward and outward advection of the PV anomalies (Nolan and Montgomery 2002b). 7. Summary and discussions This study examines the response of the hurricane The response of the hurricane wind field to rainband wind field to imposed rainband heating whose structures heating has been examined by using a three-dimensional, are already prescribed. A fundamental question that still nonhydrostatic, linear model of vortex dynamics. Diabatic remains unanswered concerns the factors or processes

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FIG. 23. Horizontal cross sections at z 5 4.17 km of the response to the transient mixed rainband heating at t 5 (a) 4, (b) 7, (c) 10, (d) 12, (e) 14, and (f) 16 h. Contours show vertical velocity (m s21) of the response, with a contour interval shown above each panel from the zero line, which is not plotted. Note that different contour interval values are used for different plots. Positive and negative values are marked by solid and dashed contour lines, respectively. Arrows show horizontal velocity (m s21) of the response. Maximum and minimum values of the contoured ﬁeld as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.

Unauthenticated | Downloaded 09/25/21 09:05 AM UTC JUNE 2010 M O O N A N D N O L A N 1803 that control the evolution of these diabatic heating tangential wind field is closely approximated by an an- structures of convection in tropical cyclones. The answer gular momentum surface (Stern and Nolan 2009), the appears to be closely related to the evolution of the rates at which the RMW slopes radially outward with boundary layer structures of hurricanes, which are not altitude and the tangential velocity decays along the represented in 3DVPAS. Comparisons to spiral rainbands RMW may be obtained from these expressions. that are seen to develop ubiquitously during full-physics First, the tangential wind field at the lowest level of simulations of tropical cyclones should be made as a the domain, which should be considered to be the top of next step. Attempts to isolate the impact of spiral rain- the boundary layer, is prescribed by a profile such as the bands on the surrounding wind field in high-resolution MR vortex in Eq. (1), and the pressure and temperature simulations should also be made. fields are initialized with constant values p0 and TB, re- The response of the hurricane wind field to transient spectively. Then, we compute the pressure field that mixed rainbands is intriguing. It suggests that the accel- would hold the prescribed wind field in gradient wind erated cyclonic tangential circulation can wrap around balance the entire vortex if the heating lasts long enough and that the change in the tangential circulation would re- 1 ›p y2 5 f y 1 , (A1) main coherent for a long time after the rainband heating r ›r r dissipates. This process may help us better understand secondary eyewall formation, a phenomenon not well where r is density, p is pressure, r is radius, f is the understood at the moment. Kuo et al. (2004) and Wang Coriolis parameter, and y is tangential velocity. Equa- (2009) also suggested that the presence of strong rain- tion (A1) is combined with the equation of state— bands may favor the formation of secondary eyewalls. p Our efforts to understand the role of spiral rainbands in r 5 , (A2) R T the formation of secondary eyewalls will continue with d numerical models of increasing complexity. where Rd is the dry gas constant and T is temperature— Acknowledgments. This research was supported by to eliminate r and then is integrated inward from the the National Science Foundation Grants ATM-0432717 outer boundary using a Crank–Nicholson scheme while and ATM-0756308. We also would like to sincerely holding T fixed. The density field is computed by en- thank Drs. Brian E. Mapes, Robert F. Rogers, and Hugh forcing Eq. (A2) with fixed T. E. Willoughby for their helpful comments and feedback Next, we compute the environmental thermodynamic on this study. Detailed and constructive reviews by properties consistent with the prescribed tangential wind Drs. Yuqing Wang, Shigeo Yoden, and two anonymous field by following an iterative procedure outlined in reviewers are greatly appreciated. Thanks are due to E86 (597–598). Then, Eq. (51) of E86 is evaluated at the Mr. Daniel P. Stern for sharing his version of the code RMW to obtain the radius of an outward sloping angular for the free atmospheric portion of the E86 model and momentum surface that is congruent to the RMW. The also to Mr. David Painemal for help with figures. altitude of this angular momentum surface can be found from Eq. (56) of E86. Vertical decay of tangential velocity along the RMW can be found by inverting the APPENDIX definition of absolute angular momentum M [ ry 1 fr2/2

at each altitude. We use values of p0 5 1015.1 mb, TB 5 Obtaining the Slope of the RMW and Decay Rate of 26.38C, and T0 5270.08C. Tangential Velocity at the RMW from E86

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