The Pennsylvania State University The Graduate School

THE ROLE OF ASYMMETRIC RAINBAND PROCESSES IN SECONDARY

EYEWALL FORMATION IN TROPICAL CYCLONES

A Dissertation in Meteorology and Atmospheric Science by Chau-Lam Yu

© 2020 Chau-Lam Yu

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2020

The dissertation of Chau-Lam Yu was reviewed and approved by the following:

Anthony C. Didlake, Jr. Assistant Professor of Meteorology Dissertation Adviser Chair of Committee

Paul Markowski Professor of Meteorology Associate Head, Graduate Program in Meteorology

Steven Greybush Associate Professor of Meteorology Associate Director, Center for Advanced Data Assimilation and Predictability Techniques Co-Hire, Institute for Computational and Data Sciences

Ashley N. Patterson Assistant Professor of Education

David Stensrud Professor of Meteorology Department Head and Professor of Meteorology

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ABSTRACT

Secondary eyewall formation (SEF) in tropical cyclones (TCs) can cause significant changes in both the intensity and storm structure. Through a systematic analysis of model simulations, this study examined the dynamical relationships between a sequence of important asymmetric rainband processes that contribute to the onset of SEF. In an idealized simulation where artificial rainband diabatic forcing is imposed, wind field broadening is observed, together with a newly found low-level spiral updraft that resembles the updraft structure observed in

Hurricanes Rita (2005) and Earl (2010). These features are also embodied clearly in a full-physics simulation of Hurricane Matthew (2016). Prior to the onset of the SEF, the simulated TC experiences a storm-scale wind field broadening accompanying with an intensifying spiral rainband. The associated wind acceleration is nearly 100 km in radial extent and covers the left- of-shear half of the storm with a slantwise descending structure. Quadrant tangential wind budget shows that this inward descending structure is due to the presence of a mid-level convergence zone associated with stratiform diabatic forcing, which draws angular momentum inward and accelerates the wind field. Collocated with stratiform cooling, the accelerated mid-level inflow turns into downdraft, forming a mesoscale descending inflow (MDI) that shapes the inward descending pattern of the acceleration field. At the upshear-left quadrant where this MDI reaches the surface, low-휃퐸 air is flushed into the boundary layer, forming a near surface cold pool. At the inner edge where this cold pool interacts with high 휃퐸 moist envelope of the TC inner core, a tight thermodynamic gradient is established, with intense convective updrafts being reinvigorated and maintained. An equivalent potential temperature budget shows that the inward intruding cold pool

iii destabilizes the atmospheric column by forming differential azimuthal warm advection, sustaining the intense convections at the vicinity of strong thermodynamic gradient.

To investigate whether the identified asymmetric rainband processes generally occur in other SEF cases, a nonlinear boundary layer model is used in conjunction with airborne observations of tangential wind in TCs before, after, and without undergoing SEF. The model simulation results show that among all quadrants, the downshear-left and left-of-shear quadrants exhibit the strongest secondary updraft signals that resembles an early signal of an incipient secondary eyewall. This finding aligns well with the asymmetric rainband processes identified in our full-physics simulation, and suggests that the dynamical processes in the left-of-shear quadrants, where the stratiform portion of the rainband typically lies, are of particular importance to the onset of SEF.

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TABLE OF CONTENTS

List of Figures ...... vii Acknowledgements ...... xv

Chapter 1. INTRODUCTION ...... 1 1.1 TC rainband organization and its impact on vortex wind field ...... 5 1.2 Hurricane boundary layer theories ...... 9

Chapter 2. IMPACT OF STRATIFORM RAINBAND HEATING ON THE TROPICAL CYCLONE WIND FIELD IN IDEALIZED SIMULATIONS ...... 15 2.1 Methodology ...... 17 2.2 Design of diabatic heat source ...... 19 2.3 MN10 stratiform profile ...... 23 2.4 Modified stratiform heating profile ...... 34 2.5 Buoyant updraft analysis ...... 38 2.6 Conclusions ...... 42

Chapter 3. OVERVIEW OF A FULL-PHYSICS SIMULATION OF HURRICANE MATTHEW ...... 45 3.1 WRF Model simulation of Hurricane Matthew (2016) ...... 45

Chapter 4. ASYMMETRIC WIND ACCELERATION PRIOR TO SECONDARY EYEWALL FORMATION IN A FULL-PHYSICS MODEL SIMULATION ...... 48 4.1 Overview of SEF in the Matthew simulation ...... 48 4.2 Storm-scale changes in tangential velocity and angular momentum ...... 52 4.3 Asymmetric Along-Band Structure ...... 59 4.4 Quadrant-Averaged Tangential Wind Budget Analysis ...... 62 4.5 Evolution of MDI ...... 72 4.6 Conclusions ...... 74

Chapter 5. THE ROLE OF BOUNDARY LAYER THERMODYNAMIC ASYMMETRY IN UPDRAFT MAINTENANCE ...... 77 5.1 The equivalent potential temperature 휃퐸 formulation and budget equation ...... 77 5.2 Boundary layer thermodynamic asymmetry ...... 79 5.3 Results ...... 83 5.4 Conclusions ...... 95

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Chapter 6. THE AXISYMMETRIC AND ASYMMETRIC SIGNALS OF SECONDARY EYEWALL FORMATION IN OBSERVATIONS-BASED MODELING OF THE TROPICAL CYCLONE BOUNDARY LAYER ...... 98 6.1 Data and Methodology ...... 99 6.2 Results ...... 107 6.3 Conclusions ...... 119

Chapter 7. CONCLUSIONS ...... 123 7.1 Summary and Discussion ...... 123 7.2 Ongoing and Future Works ...... 128

Appendix A: Reconstruction of The Rainband Diabatic Heating Structure from the Observed Secondary Circulation in Hurricane Rita ...... 131 Appendix B: Storm-Relative Tangential Wind Budgets ...... 134 Appendix C: Verification for the Forcing of 휃퐸 ...... 136

References ...... 140

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LIST OF FIGURES

Figure 1.1 The three important components involved in the SEF process, highlighted in different colors. The arrows indicate certain specific type of interaction that has been studied in the labelled chapter of this dissertation. A two-way arrow indicates the two-way interaction between the two components; a one-sided arrow indicates the one-way impact on one to the other component; a dashed arrow indicates only indirect implication is revealed in the chapter...... 4

Figure 1.2 a) Plan view schematic of the rainband complex and eyewall (reflectivity contours of 20 and 35 dBZ) under the influence of an environmental wind shear. The environmental wind shear vector points upward and defines the four storm quadrants. A mesoscale descending inflow (MDI) and an enhanced updraft (white dashes) both occur in the downshear-left (DL) quadrant within the stratiform sector of the rainband complex. From Didlake et al. (2018). b) Cross section of the azimuthally-averaged secondary circulation within the stratiform rainband sector of Hurricane Rita at 21 Sep 2005 1642 UTC. Updrafts (black) are contoured at every 0.3 푚푠−1 from 0.3 to 1.5 푚푠−1 , and downdrafts (gray dashed) are contoured at -0.1, -0.3, -0.6, and -0.9 푚푠−1. From Didlake and Houze (2013b). c) As in (b) but for cross section through the stratiform rainband of Hurricane Earl at 29 Aug 2010 2038 UTC. From Didlake et al. (2018)...... 6

Figure 1.3 Vertical profiles of (a) the three velocity components and (b–e) the radial and tangential velocity tendency equations at hour 18. The quantities presented correspond to 10 km radial averages near the secondary eyewall location (110–120 km radius). In the momentum budgets, the dotted lines correspond to the components of the material derivative, the solid lines correspond to the different forces in the budget. (From Fig. 4 of Abarca et al. 2015) ...... 10

Figure 1.4 Radial momentum budget for the nonlinear KW01 model for storm I in Kepert (2013). (a) is radial friction dissipation, (b) gradient wind residual, (c) vertical advection of radial momentum and (d) radial advection of radial momentum. (From Fig. 3a-d of Kepert 2013) 12

Figure 2.1 Vertical and radial distributions of (a) tangential wind and (b) potential temperature anomaly of the basic state vortex after the 24-hour spin-up period, contoured every 5 푚푠−1 and 1.69 퐾, respectively. Cross-section through the middle portion of MN10’s stratiform rainband diabatic heating is shown in (c), and plan view at 푧 = 4.6 푘푚 is shown in (d). (e) and (f) are same as (c) and (d), but for the modified stratiform profile. Contour spacings are 1 퐾ℎ푟−1 for (c) and (e), and 0.5 퐾ℎ푟−1 for (d) and (f)...... 22

Figure 2.2 a) Azimuthally averaged wind field response using MN10’s stratiform profile for the WRF simulation. Tangential velocity contours at intervals of 0.5 푚푠−1 are overlaid by the radial-vertical velocity vectors. Diabatic heating (cooling) of 0.15 퐾ℎ푟−1(−0.15 퐾ℎ푟−1)

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are highlighted in solid red (blue) contour. b) As in a) but for MN10’s simulation. From MN10...... 23

Figure 2.3 Plan view of full responses in vertical velocity (shading) and horizontal wind field (vectors) responses at (a) 6 푘푚, (b) 3.6 푘푚, and (c) 2 푘푚 altitudes from MN10’s stratiform heating using the WRF simulations. Solid contours of diabatic forcing with magnitude of 1 퐾ℎ푟−1 (heating in red and cooling in blue) at 5.6 푘푚, 3.6 푘푚 and 2.4 푘푚 are added respectively to indicate the rainband location. (d), (e), and (f) are the same as (a), (b) and (c), but for responses summing from azimuthal wavenumbers 0 to 4. (g), (h), and (i) are similar plots as (d), (e) and (f) but from MN10’s simulation at slightly different altitudes. Contours are at 0.03 푚푠−1 intervals...... 25

Figure 2.4 a) Plan view of vertical velocity at 2.2 km altitude from MN10’s stratiform heating using the WRF simulation. Red lines mark the cross sections shown in (b), (c), and (d). b) Tangential wind (shading) and second circulation (vectors) responses from the WRF simulation in the upwind cross section in (a). c) As in (b) but for the middle cross section. d) As in (b) but for the downwind cross section. Diabatic heating (cooling) of 1 퐾ℎ푟−1(−1 퐾ℎ푟−1) are highlighted in solid red (blue) contour...... 27

Figure 2.5 a) 880-hPa-level (near 푧 = 1.15 푘푚) plan view of the vertical velocity response from MN10’s stratiform heating in the WRF simulation in the lower right quadrant. b) Diagnosed vertical velocity from the buoyancy advection (BA) term in Eq. (5) corresponding to the vertical velocity field in (a). c) 880-hPa-level potential temperature anomalies and mean divergence between 880-hPa-level and the lowest pressure level using MN10’s stratiform heating profile. Divergence contours are at 3 × 10−5s−1 intervals. Diabatic cooling of −0.5 퐾ℎ푟−1 at 790 푚푏 are shown in solid blue contour...... 29

Figure 2.6 a) Cross-section showing the responses in the pressure field (shading) and vertical velocity (magenta contours; negative values are dashed) at downwind cross section from Fig. 5a. Vertical velocity is contoured at intervals of 0.082 푚푠−1. Diabatic heating (cooling) of 1 퐾ℎ푟−1(−1 퐾ℎ푟−1) are highlighted in red (blue) dashed contour. b) 3D Laplacian of the pressure response at the downwind cross section. c) the buoyancy pressure forcing 퐹퐵 푣휃 휕 휕푝′ at the downwind cross section. d) plan view of 0 (− ) (shading) and vertical 푔푟휌0 휕휆 휕푧 velocity response (magenta contours; negative values are dashed) at 푧 = 2 푘푚 of the rainband quadrant. Vertical velocity is contoured at 0.05 푚푠−1...... 32

Figure 2.7 Vertical velocity responses at 푧 = 2 푘푚 using MN10’s stratiform heating profile at different rainband rotation rates Ω: (a) no rainband rotation (Ω = 0); (b) Ω = 1.454 × 10−4 푟푎푑 푠−1 ; and (c) Ω = 2.856 × 10−4푟푎푑 푠−1 . Diabatic cooling of −1 퐾ℎ푟−1 at ′ 2.4 푘푚 are shown in solid blue contour. d) Anomalies of potential temperature response 휃푎 (shading) at 푧 = 1 푘푚. Positive vertical velocity response (푤′, magenta contours) and ′ −1 buoyancy advections (−풖풉 ∙ 훁풉휃 , black dashed contours) are contoured at 0.03 푚푠 and 1.2 × 10−5 퐾푠−1, respectively. e) same as (d), but for Ω = 1.454 × 10−4 푟푎푑 푠−1. f) same as (d), but for Ω = 2.856 × 10−4푟푎푑 푠−1...... 34

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Figure 2.8 Azimuthally averaged response of tangential wind (shading) and secondary circulation (vector) from the WRF simulation using the modified stratiform heating profile. Diabatic heating (cooling) of 0.15 퐾ℎ푟−1(−0.15 퐾ℎ푟−1) are highlighted in solid red (blue) contour...... 35

Figure 2.9 Plan view of vertical velocity (shading) and horizontal wind field (vectors) responses at (a) 8.4 푘푚 ; (b) 3.6 푘푚 and (c) 2 푘푚 altitudes for the WRF simulations using the modified heating profile. Solid contours of diabatic forcing with magnitude of 1 퐾ℎ푟−1 (heating in red and cooling in blue) at 8 푘푚 , 3.6 푘푚 and 2.4 푘푚 are added respectively to indicate the rainband location...... 36

Figure 2.10 Same as Fig. 2.4, but for the modified heating profile...... 38

Figure 2.11 Solid lines show the mean profiles of relative humidity for each moist sensitivity experiment, averaged over an annulus of 40-80 km radius, which covers the rainband region. Dashed lines show the specific humidity added to the Jordan mean hurricane sounding (훿푞) at each pressure level to initialize the experiments...... 40

Figure 2.12 Frequencies of buoyant convective updrafts occurring during simulation hour 2 to hour 6 using our modified heating profile with different relative humidity profiles: (a) 90%; (b) 93% and (c) 95%. Contours of diabatic heating (solid) and cooling (dashed) at z = 4 km are shown as reference, with spacing of 2 퐾ℎ푟−1...... 41

Figure 3.1 a) Observed and simulated tracks during the 72-hour simulation period from 0000 UTC 2 October to 0000 UTC 5 October 2016. The observed track comes from the National Hurricane Center best track data. b) Observed and simulated minimum sea-level pressure (MSLP, black) and maximum surface wind (Vmax, red). Observed intensities are dotted solid lines, simulated intensities are dashed lines. c)-d) Observed 89GHz brightness temperatures of the Advanced Microwave Scanning Radiometer 2 (AMSR2) before and after the simulated SEF...... 47

Figure 4.1 a) Hovmöller diagram of the simulated vertical velocity at z = 4.84 km. Vertical velocity of 0.25 푚푠−1 is contoured in black. The blue dotted lines indicate the times of the observed AMSR2 brightness temperature shown in Fig. 1c and d. Hours 17 and 19 are indicated by red dotted lines. b) As in (a), but for tangential velocity at z = 2.86 km. Black dotted lines indicate radius of maximum wind of the primary and secondary eyewalls...... 49

Figure 4.2 Plan views of (a-f) reflectivity averaged between z = 0.5 and 4km and (g-l) vertical velocity at z = 4km at simulation hours 15, 17, 19, 21, 22 and 23. Green arrows in (a-f) indicate the 850-200mb vertical wind shear vector. Black circles indicate 20, 40, and 60 km radii. All fields in subsequent figures are temporally averaged over the one-hour period of the corresponding simulation hour, unless otherwise specified...... 51

Figure 4.3 (a-f) R-Z cross-sections of azimuthally averaged vertical velocity (shading) and radial velocity (contoured in black at every 2푚푠−1 with 0 line thickened) at selected hours

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of 15, 17, 19, 21, 22 and 23; (g-l) as in (a-f) but for diabatic heating (shading) and tangential wind (contoured in black at every 5푚푠−1 with 30푚푠−1 line thickened)...... 52

Figure 4.4 (a-c) R-Z cross-sections of azimuthally averaged tangential wind acceleration at (a) hours 15, 17, and 19. Zero contour is plotted in black dotted line. Vertical black dashed lines indicate 20, 40, and 60 km radii. (d-f) Plan views of hourly storm-relative tangential wind changes averaged between z = 2 and 6 km at (d) hour 15, (e) hour 17 and (f) hour 19. Green arrows indicate the 850-200mb vertical wind shear vector, with black straight lines highlighting individual quadrants. Black circles indicate 20, 40 and 60 km radii...... 54

Figure 4.5 R-Z cross-sections of azimuthally averaged storm-relative angular momentum budget at hour 17: (a) actuall changes of storm-relative angular momentum; (b) integrated changes of angular momentum using 5-min model output; (c) contribution from friction; (d) advection by mean radial flow; (e) advection by mean vertical velocity; (f) sum of mean flow advection; (g) radial eddy flux; (h) vertical eddy flux; and (i) sum of eddy contributions. Vertical dashed lines indicate 20, 40 and 60 km radii...... 57

Figure 4.6 Same as Fig. 4.5, but for hour 19...... 58

Figure 4.7 Plan views of vertically averaged horizontal divergence between z = 2 and 5 km for (a) hour 17 and (b) hour 19. Green arrow indicates the 850-200mb shear vector. Red squares are the anchor points, with the third one aligning with the shear vector. Red spiral line is defined by the anchor points using cubic spline. Long solid black lines define the quadrants; short solid black lines define individual sectors labeled A-F. The three circles indicate 20, 40 and 60 km radii...... 59

Figure 4.8 Top row is the along-band cross-sections of hour 17 for (a) vertical motion (shading) and tangential wind acceleration (contoured at every 2ms-1hr-1, negative contours are dashed, smallest positive contour of 1ms-1hr-1 is thickened), and (b) the diabatic heating (shading) -1 and asymmetric tangential wind 푣푎 (contoured at every 0.6ms , negative contours are dashed, zero contour is thickened). (c) and (d) are the same as (a) and (b), but for hour 19. All fields are hourly and radially averaged over a 6 km radial range centered at the spiral shown in Fig. 4.7...... 60

Figure 4.9 a) Plan view of tangential wind acceleration vertically averaged between z = 2 and 6km at hour 17. Green arrow indicates the 850-200mb wind shear vector. Shear-relative quadrants are indicated by solid straight lines. Blue contour highlights the quadrant of interest. (b-i) The tangential wind budget averaged over the downshear-right (DR) quadrant: b) The actual hourly changes in tangential wind; c) the integrated hourly changes of tangential wind; d) radial advection; e) azimuthal advection; f) total horizontal advection; g) the contribution from pressure gradient force; h) the contribution from boundary layer friction; and i) the contribution from vertical advection. All vertical black dashed lines indicate 40 and 80 km radii...... 64

Figure 4.10 Same as Fig. 4.9, but for downshear-left (DL) quadrant at hour 17...... 65

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Figure 4.11 Quadrant and hourly averaged radial velocity and its flow components: a) Storm- relative radial velocity 푢푆푇 (shading) and vertical velocity 푤 (contoured at {0, ±1, ±2, ±4, … , ±64} × 0.08 ms−1 with 0ms−1 thickened) in the DR quadrant at hour 17; b) irrotational component 푢휒 (shading) and diabatic forcing (contoured at every {0, ±1, ±2, ±4, … , ±64} × 2.5 × 10−4 Ks−1 with 0 Ks−1 thickened.); c) nondivergent component 푢휓; d) environmental component 푢푒푛푣. (e-h) As in (a-d) but for DL quadrant at hour 17. (i-l) As in (a-d) but for UL quadrant at hour 19...... 66

Figure 4.12 Same as Fig. 4.9, but for upshear-left (UL) quadrant at hour 19...... 69

Figure 4.13 The decomposition of horizontal advection: (a) Plan view of vertically averaged (z = 2-6km) tangential wind acceleration in DL quadrant at hour 17. Green arrow indicates shear vector (850mb-200mb) and blue solid contour indicates the quadrant of interest. (b) Quadrant-averaged advection by the irrotational wind, (c) quadrant-averaged advection by nondivergent wind, and (d) quadrant-averaged total horizontal advection in DL at hour 17. (e-h) As in (a-d) but for DL quadrant at hour 17. (i-l) As in (a-d) but for UL quadrant at hour 19...... 72

Figure 4.14 Joint frequency distributions of 푢휒 and 푤 from hour 16 to 20 for (a) UL, (b) DL, (c) UR and (d) DR. Only 50th percentile contour lines are shown. e) The azimuthal and time −1 −1 evolution of 푢휒 (shading) and 푤 (contoured in red at every −0.3푚푠 from −0.3푚푠 to −1 −1 −1.5푚푠 ) for pixels that satisfy the criterion of 〈푤〉30푚𝑖푛 < −0.075푚푠 ...... 74

Figure 5.1 (a-b) Plan views of instantaneous fields for (a) 휃 at z = 300m and (b) diabatic heating at 4.3 km over the entire innermost domain (499×499 grid points) at hour 20. The two spirals highlight the region of the cold anomaly and region with mid-level diabatic cooling. (c-h) Plan views of hourly averaged 휃 (shading) and positive vertical velocity (contoured in black at every 0.1 푚푠−1) at z = 300m at selected hours: (c) hour 15; (d) hour 17; (e) hour 19; (f) hour 20; (g) hour 22 and (h) hour 23. The hourly averages are computed using 5-min WRF output. The three circles indicate radii of 20, 40 and 60 km. Shear vector in each hour is shown by green arrow...... 80

Figure 5.2 Plan views of hourly averaged 휃퐸 (shading) for (a) hour 19; (b) hour 20 and (c) hour 22. The 휃퐸 is also vertical averaged between z = 200 and 850m. Positive vertical velocity at z=1.5km is contoured in red at every 0.4 푚푠−1. Sectors enclosed by the two straight black lines are regions of focus in the subsequent analyses...... 82

Figure 5.3 a) Cross-section of 휃퐸 (shading), diabatic heating (contoured in magenta at every 10−3퐾푠−1 from 2 × 10−3 to 5 × 10−2 퐾푠−1) and secondary circulation (vectors) for hour 20. The averaging sector is indicated in Fig. 5.2. (b-e) Cross-section of the different terms 휕 휃 of the 휃 equation: (b) local tendency 푐 퐸; (c) horizontal advection −풗 ∙ 훁휃 ; (d) vertical 퐸 휕푡 푺푻 퐸 휕휃 advection −푤 퐸 ; (e) 휃 forcing (calculated as residual). Positive vertical velocity is 휕푧 퐸 contoured at 0.2 푚푠−1 from 0.4 to 10 푚푠−1...... 84

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Figure 5.4 Cross-sections of the different terms of the budget equation for 휕휃퐸/휕푧: (a-c) local tendency of 휕휃퐸/휕푧 ; (d-f) vertical gradient of horizontal contribution; (g-i) vertical gradient of vertical contribution; and (j-l) vertical gradient of 휃퐸 forcing. Vertical velocity is contoured in magenta at every 0.2 푚푠−1 from 0.4 to 10 푚푠−1. Updraft stronger greater than 0.6 푚푠−1 is thickened. The left column is for hour 19; middle column for hour 20; and right column for hour 22. Averaging region for each hour is shown in Fig. 5.2...... 87

Figure 5.5 (a-c) Cross-sections of the vertical gradient of (a) horizontal advection, (b) azimuthal advection and (c) radial advection for hour 20, averaged over the sector shown in Fig. 5.2b. Vertical velocity is contoured in magenta at every 0.2 푚푠−1 from 0.4 to 10 푚푠−1, with updraft stronger greater than 0.6 푚푠−1 thickened. (d-f) is the same as (a-c), but for plan views at z = 1.25 km. Positive vertical velocity is contoured in red at every 0.2 푚푠−1...... 89

Figure 5.6 a) Plan view of hourly averaged asymmetric 휃퐸 for hour 20, vertically averaged between 200m and 850m. Updraft at z=1.5km is contoured in red at every 0.4푚푠−1. Black solid lines indicate region of interest. (b) Azimuth-height cross-section of hourly averaged 휃퐸 for hour 20, radially averaged between 21 and 27 km radius. Updraft is contoured in −1 magenta at every 0.25푚푠 . Vectors indicate azimuthal advection of 휃퐸 , with warm advection pointing to the increasing azimuth direction...... 91

휕 Figure 5.7 Cross-section of sector and hourly averaged (a) ∇2푝∗; (b) buoyancy forcing (휌 퐵) 휕푧 표 ∗ 휌 휕푝표 and (c) dynamic forcing ∇ ∙ (−휌표풗 ∙ ∇풗 − (1 − ) 푟̂ − 휌표푓풌 × 풗 + 휌표푭). Vertical 휌표 휕푟 velocity is contoured in (a) and (b) in black (updraft in solid at every 0.2 푚푠−1 and downdraft in dashed lines at every 0.1 푚푠−1). Diabatic cooling is contoured in (c) in blue −4 −1 dashed lines at every 3 × 10 퐾푠 . Negative asymmetric 휃퐸 is contoured in (a-c) in dashed magenta lines at every 0.5 K...... 95

Figure 6.1 a) Azimuthal mean composite profiles for Non-, Pre- and Post-SEF groups. The sample count, and median values of 푟1 and 푟2 are indicated in the legend. The portion of profiles highlighted in red indicate extrapolated region. b) Schematics showing the orientations of the two set of shear-relative quadrants, Q1 and Q2. c) Quadrant averaged composite profiles for the Non-SEF group under Q1 quadrant definition (DL, UL, UR, and DR). d) is the same as c), but for the Q2 quadrant set of the Non-SEF group (DS, LS, US, and RS). e) and f) are the same as (c) and (d), but for the Pre-SEF group. g) and h) are the same as (c) and (d), but for the Pos-SEF group...... 102

Figure 6.2 a) The plan view of the filtered asymmetric tangential wind field 푣표 for the Pre-SEF group. The contour of 30 푚푠−1 is highlighted in red. The wind is contoured in black at every 0.5 푚푠−1 below 30 푚푠−1 and at every 1 푚푠−1 above 30 푚푠−1. b) The plan view of the observational variance obtained by the bootstrap procedure for the Pre-SEF group. The red circle indicates 150 km radius. The red arrow indicates the direction of the environmental wind shear vector. c) The tangential component of the nondivergent wind field 풗흍 after solving equation (6.2). Same contours are applied as in (a). d) The radial component of the asymmetric nondivergent wind field 풗흍...... 107

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Figure 6.3 Boundary layer responses for the axisymmetric Non-SEF (left column), Pre-SEF (middle column) and Post-SEF (right column) groups. Top row (a, d, g) displays vertical velocity (shaded) and radial velocity (contour), with radial velocity contoured every 0.8 푚푠−1 and the zero-line omitted. Middle row (b, e, h) displays tangential wind 푣 (shaded) and positive agradient wind (contour), contoured at every 0.6 푚푠−1 with zero-line omitted. Bottom row (c, f, i) displays radial velocity (shaded) and horizontal divergence (contour), contoured at (±2, ±4, ±8, ±16, ±32, ±64, ±128, ±256) × 10−5푠−1...... 109

Figure 6.4 a) The cross-section of azimuthal mean vertical velocity (shading), supergradient wind (magneta contours, at every 0.8 푚푠−1) and radial velocity (black contours, at every 0.6 푚푠−1 with 0 line omitted) of the asymmetric simulation using the asymmetric forcing of the Pre-SEF group. b) The plan view of the vertical velocity response at 푧 = 1.05 푘푚. (c-e) same as (b), but for vorticity forcing at the model top; divergence forcing at 푧 = 0.85 푘푚; and agradient wind at 푧 = 0.85 푘푚...... 112

Figure 6.5 a-h) The cross-sections of quadrant averaged mean vertical velocity (shading), supergradient wind (magneta contours, at every 1 푚푠−1 ) and radial velocity (black contours, at every 0.8 푚푠−1 with 0 line omitted) of the asymmetric simulation for the Q1 (left column) and Q2 (right column) quadrants. i-p) are the same as (a)-(h), but for the axisymmetric simulations that uses quadrant-averaged forcing...... 114

Figure 6.6 Box plot of the extreme values, 25, 50 and 75 quartiles of outer maximum updraft distribution for the Q1 and Q2 quadrants for the Pre-SEF group. The outer maximum updraft is defined as the maximum updraft at the radial range between 90 and 140 km radii...... 116

Figure 6.7 a) Cross-sections showing the composite of the vertical velocity (shading), radial inflow (contoured in black at every 1 푚푠−1 with 0 line omitted) and supergradient wind (contoured in magenta, at every 1 푚푠−1) for the strongest 8 members in Q1 quadrant set. b) is the same as a), but for the strong composite of the Q2 quadrant set. c) and d) are the same as (a) and (b), but the Q1 and Q2 quadrant sets of the weak composites...... 117

Figure 6.8 a) Cross-sections of the vertical velocity (shading), radial inflow (contoured in black at every 1 푚푠−1 with 0 line omitted) and supergradient wind (contoured in magenta, at every 1 푚푠−1) for the axisymmetric simulations that uses quadrant averaged forcings from the Q1 (left column) and Q2 (right column) of the Pos-SEF groups. b) is the same as a), but for the Non-SEF group...... 119

Figure 7.1 A schematic summarizing the impacts of asymmetric rainband processes to the wind and boundary layer thermodynamic fields prior to the onset of SEF. The storm is divided by the environmental wind shear into four quadrants, namely downshear right (DR), downshear right (DL), upshear left (UL) and upshear right (UR). The region between the eyewall and the updraft branch of streamlines designates the moat region, which is largely free of convection. The mesoscale descending inflow (MDI) feature is located at the left of shear quadrants where diabatic cooling due to melting and evaporation of hydrometeors

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occurs. The meanings of all other symbols are denoted by the legend at the top of the figure...... 125

Figure A.1 Reconstructed diabatic heating using the Sawyer Eliassen equation based on the observed secondary circulation within the stratiform rainband of Hurricane Rita (Fig. 1.2b). Zero contour is highlighted with thick solid black line...... 133

휃̇ 푚푝 퐿푣푞̇푚푝 Figure C.1 (a)-(e) Cross-section of forcing terms of 휃̂퐸: (a) microphysics: 휃̂퐸 ( + ); 휃 푐푝푇 휃̂퐸 퐿푣휃̂퐸 (b) boundary layer and radiation schemes ( 휃̇푝푏푙,푟푎푑 + 푞̇푝푏푙); (c) total forcing for 휃̂퐸; 휃 푐푝푇 휃̂퐸 퐿푣휃̂퐸 (d) 휃푚푝̇ ; (e) 푞̇푚푝. (f) Forcing for Bolton’s formulation (휃퐸), computed as residual ..138 휃 푐푝푇

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ACKNOWLEDGMENTS

This material is based upon work supported by the National Aeronautics and Space

Administration under Grants NNX16AI21G (New Investigator Program) and 17‐EARTH17F‐184 under the NASA Earth and Space Science Fellowship Program, and by the National Science

Foundation under Grants AGS-1810869 and AGS-1854607.

First of all, I want to express my wholehearted gratitude to my PhD advisor Dr. Anthony

Didlake, who gave me this enjoyable and fruitful graduate experience. His insightful guidance and mentorship in the last three years has led me through many research obstacles and challenges, and taught me the value of patience and perseverance in producing quality work. He is an open-minded mentor, who would grant his student freedom and time, and encourage them to explore and come up with new ideas. His kindness and trust are truly what make this journey enjoyable.

I also want to dedicates this research to my father and my co-advisor Dr. Fuqing Zhang, without whom this journey of PhD study would not be possible. I could not stress enough how big an influence my Dad's scientific philosophy has on my appreciation of science and my determination to pursue an academic career. His intense love, support and blessing, even during his passing, is truly something that I can never repay. While facing my Dad’s passing and many other obstacles of life, Fuqing helped me with the connection with Anthony, who together gave me a second chance to pursue my dreams. Fuqing has always been so kind, energetic and open- minded. And I treasure his mentorship and value every single moment of our conversation.

I thank Dr. Jeff Kepert for his advice and discussion on my work, and for his help in

xv performing multiple runs of asymmetric boundary layer model simulations in Chapter 6. I also thank Robert Nystrom for his help in performing the early runs of the WRF model simulations.

I thank all my committee members for their helpful comments and suggestions on my work.

I also want to thank Dr. Kerry Emanuel for his generosity in sharing with me his former student

Dr. Diamilet Perez-Betancourt’s thesis as learning material. And I thank Dr. Sukyoung Lee for her help and encouragement.

I also thank all my groupmates Chelsey Laurencin, Nicholas Barron and Jonathan Unger and everyone in the ADAPT group for their friendship and support.

I thank Texas Advanced Computing Center for providing computing resources. I also want to thank the Department of Meteorology and Atmospheric Science at Penn State for all the support and research facilities.

Last but not least, I want to thank my mother, brother and my girlfriend. Their unconditional love, care and companionship in all these years are what support me to sail through all the difficult moments during this journey. I cannot thank them enough.

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CHAPTER 1

INTRODUCTION

Tropical cyclones (TCs) are extreme weather events that can cause severe economic damage and numerous casualties. Over the past few decades, forecasts of hurricane tracks have gained substantial improvement, owing to the much-improved model resolution, data assimilation and physical parameterizations. The intensity forecasts of hurricane, however, have made relatively slow progress. A major reason is our incomplete understanding on the nonlinear vortex internal dynamics and how TC vortex interacts with the ever-changing environment.

One of the dynamical processes of hurricane that can cause significant structural and intensity fluctuations is secondary eyewall formation (SEF). SEF generally begins with convection organized into spiral bands that populate outside of the TC eyewall (Houze 2010). Under certain circumstances, these rainbands can coalesce into a distinct outer eyewall that has a closed convective ring with clear reflectivity maximum, and is often associated with a secondary wind maximum (Willoughby et al. 1982). While a hiatus of intensification usually occurs during SEF, the resulted storm after the eyewall replacement cycle (ERC) generally has a broader wind field and a larger eye, and thus may lead to increased wind damage and higher risk of storm surge when making landfall.

The dynamics after a secondary eyewall is clearly established is well known. Once the

1 incipient outer eyewall has a significant projection onto the azimuthal mean to give a clear wave number-zero signature, balanced dynamics proposed by Eliassen (1951), as applied to tropical cyclone by Willoughby (1979), Shapiro and Willoughby (1982) and Schubert and Hack (1983), is capable of explaining the inward contraction of the outer eyewall and the subsequent evolution of the ERC process. The initiation mechanism for the SEF process, however, is not fully understood.

In the last few decades, many theories and hypotheses have been proposed to explain the initiation mechanism of SEF, including barotropic vorticity interactions (Kuo et al. 2004, 2008), Vortex

Rossby waves dynamics (Montgomery and Kallenbach 1997; Menelaou et al. 2012), rapid filamentation zone (Rozoff et al. 2006), beta-skirt axisymmetrization (Terwey and Montgomery

2008), etc., most of which consider the dynamics only within the free troposphere.

Besides processes in the free troposphere, some studies (Huang et al. 2012; Abarca and

Montgomery 2013; Kepert 2013) have pointed out the importance of boundary layer processes in the feedback that leads to the initiation of SEF. Abarca and Montgomery (2013) proposed a boundary layer control mechanism in which a sequence of unbalanced dynamical events within the TC boundary layer can lead to the spin-up of a secondary eyewall. On the other hands, Kepert

(2013) applied a nonlinear, steady-state hurricane boundary layer model (Kepert 2001; Kepert and

Wang 2001) to investigate how boundary layer updraft can be driven by forcing imposed at the boundary layer top, and proposed a feedback process between boundary layer updraft and diabatic heating release in the free troposphere. While both theories include supergradient wind (i.e., the net positive outward force after the balance between centrifugal, Coriolis and pressure gradient forces), a key discrepancy is whether the supergradient flow within the boundary layer is the driving force that leads to the localized convergence and updrafts (Huang et al. 2012; Abarca and

Montgomery 2013), or simply the consequence of the response of an otherwise balanced vortex to

2 surface friction (Kepert 2013).

In addition to the aforementioned axisymmetric theories, observations and modelling studies have identified a number of asymmetric processes essential to the SEF. These asymmetric processes occur in the form of asymmetric rainband convection that persistently populates the outer core region prior to SEF. Observations frequently found an outward expansion of the tangential wind field at the outer core prior to SEF (Sitkowski et al. 2011, 2012; Didlake and Houze

2013a, b; Sun et al. 2013; Tang et al. 2018). This feature has later been confirmed in a number of modeling studies (Fang and Zhang 2012; Zhang et al. 2017; Rozoff et al. 2012), which further propose a pathway emphasizing the roles of the rainband convection and the expanded vortex circulation on initiating the formation of secondary eyewall. The pathway proposed in these studies aligns with recent observational studies (Didlake and Houze 2013b; Didlake et al. 2018), which revealed a consistent pattern of Mesoscale Descending Inflow (MDI) in the stratiform sectors of the spiral rainband in hurricane Rita (2005) and Earl (2010) prior to their corresponding SEF events.

These studies demonstrate that this MDI feature is capable of drawing the high angular momentum air from the environment inward and contribute to the expansion of the storm-scale wind field. In addition to the dynamical impacts on the wind field expansion, a number of studies also showed that the latent cooling at the downwind stratiform portion the TC rainband is critical for enhancing outer core convection, and ultimately increasing the likelihood of SEF (Li et al. 2014, 2015; Zhu et al. 2015; Tyner et al. 2018; Chen 2018; Chen et al. 2018).

Figure 1.1 summarizes the different components discussed so far. The general consensus among the research community is that preceding SEF there is an expansion of tangential wind field, which is followed by enhanced boundary layer convergence. The enhanced convergence triggers new convections, which reinforce the tangential wind of an incipient secondary eyewall (Rozoff

3 et al. 2006, 2012; Fang and Zhang 2012). However, exactly how the interaction and feedback processes occur are still unclear. Building upon this whole body of research, the goals of the dissertation are to provide further understanding on the linkages between the different components of SEF, including both the dynamic and thermodynamic aspects of the asymmetric rainband processes that lead to the initiation of SEF. Specifically, this research aims to address the following questions:

1. How does the rainband diabatic forcing result in the observed MDI? And what is the role

of the MDI in causing storm-scale wind field expansion prior to SEF?

2. How does the boundary layer respond to the changes in the free-tropospheric wind field to

create localized convergence and updrafts, which can feedback to cause the amplification

of the secondary eyewall?

3. Besides the dynamical processes happened in the TC boundary layer, what is the role of

storm-scale thermodynamic asymmetry in sustaining the intense updrafts during the SEF

process?

Fig. 1.1. The three important components involved in the SEF process, highlighted in different colors. The arrows indicate certain specific type of interaction that has been studied in the labelled chapter of this dissertation. A two-way arrow indicates the two-way interaction between the two components; a one-sided arrow indicates the one-way impact on one to the other component; a dashed arrow indicates only indirect implication is revealed in the chapter.

This dissertation is organized as follows. The rest of this chapter provides brief reviews on the three components discussed in Fig. 1.1. As the first part of the study, chapter 2 examines the

4 one-way response of vortex wind field given the rainband diabatic heating using idealized WRF-

ARW experiments. Specific attentions are placed on the similarities between the simulated responses in this dissertation and the observational evidence presented in Didlake and Houze

(2013b, DH13), Didlake et. al. (2018) and Wunsch and Didlake (2018).

Chapter 3 introduces the full-physics model simulation of hurricane Matthew (2016). In chapter 4 and 5, the dynamic and thermodynamic aspects of the two-way interactions between the three aforementioned components in the Matthew simulation are examined. In chapter 6, observations-based modeling using a nonlinear boundary layer (Kepert and Wang 2001, Kepert

2018) in conjunction with observational composite (Wunsch and Didlake, 2018) is employed to examine whether or not those features discussed in previous chapters are consistent with observations. Chapter 7 concludes the main findings of this dissertation. For clarity purposes, the interactions that have been studied in each chapter is labelled in Fig. 1.1.

1.1 TC rainband organization and its impact on vortex wind field

Spiral rainbands of tropical cyclones are banded precipitation features that populate the region outside of the TC eyewall, forming a spiral or circular arc of clouds and precipitation observed in radar and satellite images (Willoughby et al. 1984; Houze 2010; Hence and Houze

2012). Under the influence of environmental wind shear, rainbands often form an organized complex that remains quasi-stationary with respect to the TC center, and thus is termed a

“stationary band complex” (SBC; Willoughby et al. 1984). The SBC tends to align with the wind shear vector such that the more active convective upwind end lies in the right-of-shear half, while a broad swath of predominantly stratiform precipitation lies in the left-of-the-shear half, resulting in a quasi-stationary asymmetry for times on the order of days (Corbosiero and Molinari 2002,

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2003; Chen et al. 2006; Hence and Houze 2012; Didlake et al. 2018).

Figure 1.2a depicts the precipitation structure of an SBC when the larger storm is embedded in moderate or strong deep-layer environmental wind shear. In the upwind portion of the SBC, new convection is triggered, forming discrete convective cells or a connected band (Barnes et al.

1983; Powell 1990; May 1996). These active convective cells are noted to be associated with clear overturning circulations and convective-scale tangential wind jets, where the depth of the jet and reflectivity tower largely depends on distance from the storm center (Didlake and Houze 2013a).

Traveling along the rainband, convection matures and collapses, while slowly falling ice particles originating from the active convection upwind are advected even further downwind. In these downwind portions of the rainband complex, ice particles fall out to form a broad, homogenous precipitation band that predominantly displays stratiform characteristics (May and Holland 1999;

Hence and Houze 2008, 2012; Didlake and Houze 2013b; Didlake and Kumjian 2017).

Fig. 1.2. a) Plan view schematic of the rainband complex and eyewall (reflectivity contours of 20 and 35 dBZ) under the influence of an environmental wind shear. The environmental wind shear vector points upward and defines the four storm quadrants. A mesoscale descending inflow (MDI) and an enhanced updraft (white dashes) both occur in the downshear-left (DL) quadrant within the stratiform sector of the rainband complex. From Didlake et al. (2018). b) Cross section of the azimuthally- averaged secondary circulation within the stratiform rainband sector of Hurricane Rita at 21 Sep 2005 1642 UTC. Updrafts (black) are contoured at every 0.3 푚푠−1 from 0.3 to 1.5 푚푠−1, and downdrafts (gray dashed) are contoured at -0.1, -0.3, -0.6, and -0.9 푚푠−1. From Didlake and Houze (2013b). c) As in (b) but for cross section through the stratiform rainband of Hurricane Earl at 29 Aug 2010 2038 UTC. From Didlake et al. (2018).

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Riemer (2016) theorized that this asymmetry of having new convection being triggered in the right-of-shear quadrants was due to the overlapping of anomalous low-level moisture and anomalous low-level positive vorticity due to distortion of the moisture envelope and vortex vertical structure by the environmental shear. As these convective cells travel from the upwind to the downwind portions of the rainband, they display the life cycle of convection, ending with decaying stage stratiform precipitation in the left-of-shear quadrants (Didlake and Houze 2013b).

The horizontally invariant nature and large spatial coverage of the downwind stratiform rainband marks its difference from the stratiform portions near upwind rainbands that have embedded convective cells (Barnes et al. 1983).

Early studies also noted the importance of the stratiform sector of spiral rainbands on the intensity and structural evolution of TCs. By using Doppler wind profiler and sounding observations, May et al. (1994) found that the rainband of Tropical Storm Flo had prominent features characteristic of stratiform precipitation, including a mesoscale updraft and downdraft above and below the melting level, maximum horizontal convergence at the melting level, and a midlevel jet. Hence and Houze (2008) also found similar mesoscale vertical transports in

Hurricane Katrina’s left-of-shear portions of the rainband complex. Franklin et al. (2006) later confirmed this association between the mid-level jet and the stratiform portion of the rainband.

They further showed that the evaporative cooling associated with the stratiform precipitation is crucial in setting up the required buoyancy gradient for the potential vorticity (PV) generation and the existence of the mid-level tangential jet.

This is consistent with the conclusions of Didlake and Houze (2013b), who suggested that the diabatic cooling associated with sublimation, melting and evaporation of Hurricane Rita’s stratiform rainband were responsible for driving the observed secondary circulation within the

7 rainband. As shown in Fig. 1.2b, within the stratiform rainband of Hurricane Rita, a robust layer of mesoscale inflow descended towards the boundary layer as it advanced radially inward [termed the mesoscale descending inflow (MDI)]. This negatively buoyant MDI brought in high angular momentum air from the storm environment and led to the emergence of a midlevel tangential jet.

This process was also demonstrated by Moon and Nolan (2010) using idealized simulations. They found that the mesoscale overturning circulations and midlevel jet developed in response to stratiform heating project strongly onto the azimuthal mean of the storm. Their study showed the potential impact of a stratiform heating profile on the vortex-scale structure and that it differs from convective-type heating patterns. In a full-physics simulation, Moon and Nolan (2015) highlighted the structure of a downwind stratiform region of the rainband and also found descending inflow but at a lower altitude than that seen in observations.

Examining airborne Doppler radar observations of Hurricane Earl (2010), Didlake et al.

(2018) again found a clear signature of an MDI and enhanced tangential jet in the downwind stratiform rainband (Fig. 1.2c). As shown by the black contour in Fig. 1.2c, just inward from the radii where the MDI reached the boundary layer, an intense low-level updraft occurred. This updraft was likely the result of low-level convergence induced by the advancing negatively buoyant MDI. Since these three features (i.e., MDI, enhanced tangential jet and adjacent updraft) persistently occurred, accelerated the low-level tangential winds, and appeared just prior to the occurrence of secondary eyewall formation, Didlake et al. (2018) hypothesized that they sufficiently projected onto the azimuthal mean and led to the eventual development of Earl’s secondary eyewall. The dynamics connecting these rainband features have yet to be thoroughly investigated using model simulations.

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1.2 Hurricane boundary layer theories

Once the outer vorticity anomalies give a significant projection onto the azimuthal mean, the couplings between boundary layer frictional convergence, boundary layer updraft and diabatic heating release in free troposphere become important to the subsequent development of SEF, as highlighted by a number of studies (Huang et al. 2012; Huang et al. 2018; Abarca and Montgomery

2013; Kepert 2013; Wang et al. 2016; Zhang et al. 2017). This section gives a brief overview on the two leading, but contrasting SEF theories that emphasize on the role of boundary layer processes, and discuss certain aspects of these theories as related to the studies of this dissertation.

1.2.1 Unbalanced dynamics of SEF

By analyzing axisymmetric dynamics of WRF simulation of Hurricane Sinlaku (2008),

Huang et. al. (2012) attributes the intensification and the formation of the secondary eyewall to a sequence of events occurred within the boundary layer, which constitutes a progressive boundary layer control pathway. The sequence of event first begins with the broadening of the tangential wind field (a precursor that has been widely accepted among the community). This broadened tangential wind field induces an increased boundary layer inflow at the boundary layer underneath

(due to increased frictional dissipation). Supergradient flow associated with this enhanced boundary layer inflow then results in deceleration of the inflow, causing an eruption of air out of the boundary layer. To understand the dynamics at the region of supergradient wind, Huang et. al.

(2012) and Abarca et al. (2015) analyzed the evolution of agradient force (퐴퐺퐹), as defined below

푣2 1 휕푝 퐴퐺퐹 = + 푓푣 − . (1.1) 푟 휌 휕푟 Their analysis, as shown in Fig.1.3 (Fig. 4 of Abarca et al. 2015), concludes that the presence of

9 agradient force is essential in causing the deceleration of the inflow near the secondary wind maximum. They argue that at the region of supergradient wind, all the radial forces in the radial momentum equation is directed outward: Outward pointing AGF Outward pointing 퐷푢 휕푢 휕푢 휕푢 푣2 1 휕푝 = + 푢 + 푤 = ( + 푓푣 − ) − 퐹푟𝑖푐푡𝑖표푛 , (1.2) 퐷푡 휕푡 휕푟 휕푧 푟 휌 휕푟

Thus, at this region, the radial inflow is unbalanced and experiences rapidly deceleration, which enhances localized convergence and leads to the eventual eruption of air out of the boundary layer.

Fig. 1.3: Vertical profiles of (a) the three velocity components and (b–e) the radial and tangential velocity tendency equations at hour 18. The quantities presented correspond to 10 km radial averages near the secondary eyewall location (110–120 km radius). In the momentum budgets, the dotted lines correspond to the components of the material derivative, the solid lines correspond to the different forces in the budget. (From Fig. 4 of Abarca et al. 2015)

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1.2.2 Nonlinear Ekman balance: a steady state limit

In contrast to interpretation provided by Huang et. al. (2012) and Abarca and Montgomery

(2013, 2015), Kepert (2001) and Kepert and Wang (2001) alternatively considered the supergradient flow developed near both the primary and secondary eyewalls as the near-steady state response of the boundary layer to the free-tropospheric forcing. Building upon previous studies (Eliassen and Lystad 1977; Rosenthal 1962; Ooyama 1986; Schubert and Hack 1983),

Kepert and Wang (2001) developed a boundary layer model (hereafter the KW01 model) that can simulate the steady-state hurricane boundary layer flow given a constant forcing (constant in time) aloft by incorporating a more realistic boundary condition at the surface (a slip condition with bulk formulation with drag coefficient).

Kepert (2013) applied the KW01 model to simulate the boundary layer flow of a hurricane undergoing SEF. His result shows that the model is able to reproduce realistic flow characteristics of SEF, such as a secondary maximum of radial inflow, updraft and tangential jet. As shown in

Fig. 1.4 (his Fig. 3a-d), we see that above the inflow layer, the two nonlinear advection terms, in particular the vertical advection (Fig. 1.4c), play an important role in balancing the positive

휕푢 tendency from the AGF term (Fig. 1.4b). Here, since 푢 is in steady state, i.e. = 0. The two non- 휕푡

푑푢 휕푢 zero advection terms together implies that is non-zero (i.e. + (풖 ∙ 훁)푢 = (풖 ∙ 훁)푢 ≠ 0). 푑푡 휕푡

Thus, air parcels at the location of the supergradient wind do experience outward acceleration, which is in agreement with Huang et. al. (2012) and Abarca et al. (2015). However, the radial flow is actually in steady state, so the flow is clearly not ‘unbalanced’. In fact, Kepert (2001) demonstrated that under such steady-state limit, this outward accelerating radial flow and the updrafts are maintained by self-advection, whose location is determined by the structure of the forcing profile imposed at the boundary layer top.

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Fig. 1.4.: Radial momentum budget for the nonlinear KW01 model for storm I in Kepert (2013). (a) is radial friction dissipation, (b) gradient wind residual, (c) vertical advection of radial momentum and (d) radial advection of radial momentum. (From Fig. 3a-d of Kepert 2013)

1.2.3 Discrepancies between the two boundary layer theories

As discussed in previous sections 1.2.1 and 1.2.2, it is clear that the two SEF theories have many features in common, including the existence of supergradient wind, and outward acceleration of the returning inflow. The major disagreements between the two theories are that the former

12 treated a flow with non-zero outward acceleration (i.e., positive Lagrangian time derivative 퐷푢/퐷푡) as ‘unbalanced’; while the latter showed that such outward acceleration can also exist in a steady- state flow (not necessarily being unbalanced). Furthermore, the former regards the supergradient wind as the cause of the updrafts near the vicinity of the secondary eyewall; while the latter interprets the updrafts and supergradient wind as the near steady-state response of the boundary layer flow to the vorticity forcings imposed by the free-troposphere aloft.

Based on the above discussion, we see that the disagreements between the two theories arises mainly because of the nuance in their interpretations and are largely “cause and effect” in nature. Perhaps a more meaningful question would be whether or not a TC boundary layer is close to a steady state during the SEF period. Upon close examination of Fig. 1.3 (Fig. 4 of Abarca et al.

2015), we note that the local tendency of radial wind (휕푢/휕푡, blue dashed curve in Fig. 1.3b) in the full-physics WRF experiment of Abarca et al. (2015) is indeed close to zero throughout the

SEF region, indicating a near steady state has been established in the radial direction. The tangential wind, on the other hand, does have positive tendency (휕푣/휕푡, blue dashed curve in Fig.

1.4d), but is small compared to the other balancing forces in the tangential momentum equation.

Thus, we see that while the steady state assumption made in the KW01 model renders it only suited for diagnostic purposes, a number of modelling studies (Kepert and Nolan 2014; Zhang et al. 2017) showed that the KW01 simulated steady-state boundary layer response realistically compares to the boundary layer structure simulated in full-physics model, such as the WRF-ARW model, and thus can be a useful diagnostic tool to understand the one-way response of the boundary layer to the tropospheric forcing in a steady state limit.

Kepert (2018) later modified the KW01 model to incorporate an asymmetric vorticity forcing that represents the forcing of a spiral rainband, and examined the asymmetric, steady-state

13 boundary layer response to the rainband forcing. His finding showed that the boundary layer response to an asymmetric rainband vorticity forcing is in fact structurally similar to that of an annular vorticity forcing that represents a secondary eyewall, which suggests a close dynamical linkage between rainbands and secondary eyewalls. This finding aligns with previous observations that secondary eyewall often develops from the asymmetric convections of an intensifying spiral rainband (Didlake and Houze 2013b; Didlake et al. 2018). In chapter 6, we will further explore the possibility of using this modified KW01 model to examine the asymmetric boundary layer structure that develops prior to SEF.

1.2.4 The potential importance of boundary layer thermodynamics

While the importance of boundary layer dynamics to the SEF process has been widely accepted, the current paradigms of the boundary layer theories that attempt to explain the updraft mechanism near the SEF radii in a large part only take into account the dynamical component of the boundary layer evolution; while the role of boundary layer thermodynamics in the SEF process has not been fully considered. Given the fact that tropical cyclone is natural heat engine that relies on the enthalpy exchange between the underlying warm ocean surface and the atmosphere as its supply of energy, the question about the roles of boundary layer thermodynamic in the SEF process will be explored in chapter 5.

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CHAPTER 2

IMPACT OF STRATIFORM RAINBAND HEATING ON THE TROPICAL CYCLONE

WIND FIELD IN IDEALIZED SIMULATIONS

As discussed in chapter 1, owing to its large spatial coverage and close proximity to the eyewall, the stratiform rainband likely plays an important role in the dynamical and structural evolution of TCs, especially for secondary eyewall formation (e.g., Tyner et al. 2018; Didlake et al. 2018). To better understand the dynamical role of rainbands on the wind field evolution, Moon and Nolan (2010; MN10 hereafter) used idealized simulations to study the storm-scale wind field response of a hurricane-like vortex to idealized convective and stratiform heating patterns rotating around the hurricane inner core. Their stratiform heating experiments used a typical, idealized heating structure for convection-generated stratiform precipitation, having latent heating above and latent cooling below the melting level to match the water phase changes at these levels. The vortex response to this heating profile included the generation of midlevel radial inflow and a midlevel tangential wind jet. This vertical profile of heating, as with convection outside of TC environments, generated potential vorticity (PV) anomalies in the midlevels (Raymond and Jiang

1990; May et al. 1994; May and Holland 1999). These features are consistent with stratiform rainband characteristics found in some full-physics modeling studies (e.g., Franklin et al. 2006;

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Moon and Nolan 2015).

In an observational study using airborne Doppler radar, Didlake and Houze (2013b; DH13 hereafter) noted many of these same structures in the rainband complex stratiform sector in

Hurricane Rita. These included the midlevel tangential wind jet and a robust inflow layer, in their case was clearly descending throughout the width of the rainband. The reflectivity signature suggested ongoing microphysical processes that supported a structure of diabatic heating and cooling, aligning well with the MN10 experiments.

While MN10’s finding is in good agreement with the observational evidence presented in

DH13, certain aspects of the stratiform rainband still deserves further exploration. For instance, the rainband heating used in MN10 was rotating about the vortex center, mimicking the diabatic forcing imposed by a non-stationary spiral rainband. In addition, the stratiform heating profile used in MN10 is common among typical observed stratiform precipitation. However, whether this stratiform structure remains realistic in a gradient wind environment, such as at the stratiform rainband region of a hurricane, is open to question. MN10 also considered a stationary rainband with a mixed heating profile, but no significant results were discussed. Therefore, fundamental differences associated with quasi-stationary rainbands of stratiform heating still need further investigation.

This chapter aims to explore these unaddressed questions by using Weather Research and

Forecasting (WRF) model idealized simulations to study the response of a dry, idealized hurricane- like vortex to the presence of stationary stratiform rainband heating.

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2.1 Methodology

2.1.1 Numerical Model

The Advanced Version of the Weather Research and Forecast (WRW-ARW) model version 3.9.1.1 is used to simulate the response of an idealized, dry hurricane-like vortex to the presence of a prescribed diabatic heating and cooling effects of a hurricane stratiform rainband.

The model is configured to run at an 푓-plane of 5 × 10−5푠−1 with triply nested domains with horizontal resolutions of 1, 3 and 9 km, and with 38 vertical levels up to 25 km altitude. The innermost, middle and outermost domains have 253 x 253, 139 x 139 and 301 x 301 grid points.

Periodic boundary conditions are used at the lateral boundaries, and the outermost domain is designed to be large enough to prevent any boundary effect (if any) from reaching the middle and innermost domains within the analysis time window. The effect of rainband diabatic heating is represented by a prescribed diabatic heating term 푄 [퐾푠−1] added to the thermodynamic equation of the WRF model (Skamarock et al. 2008) as

휕휇휃 + ∇ ∙ (휇풖휃) = 휇퐹 + 휇푄 , (2.1) 휕푡 휃 where 휃 is potential temperature; 휇 represents the mass per unit area; 풖 is the three-dimensional wind vector; 퐹휃 is the model forcing from microphysics. To isolate the storm scale response to the added diabatic forcing, all parameterization schemes, including microphysics, cumulus and boundary schemes, are turned off. Without boundary layer friction, the lower boundary is a free- slip boundary condition. Therefore, the microphysics forcing term 휇퐹휃 in Eq. (2.1) is zero.

Sensible heating and moisture fluxes at the lower boundary are also turned off.

17

2.1.2 Basic state vortex

In order to compare our results with MN10’s findings, the basic state vortex used in the simulations is the same modified Rankine (MR) vortex wind profile as in MN10. The MR vortex is defined as

푟 푣 (푧) ( ) , 푟 ≤ 푅푀푊(푧) 푚푎푥 푅푀푊(푧) 푣(푟, 푧) = 푎 (2.2) 푅푀푊(푧) 푣푚푎푥(푧) ( ) , 푟 > 푅푀푊(푧) { 푟 where r is radius, z is altitude, 푣푚푎푥(푧) and 푅푀푊(푧) are the magnitude and radius of the maximum tangential wind; and the parameter 푎 (taken as 0.5) controls the decay of the wind magnitude outside of 푅푀푊. Both 푣푚푎푥(푧) and 푅푀푊(푧) are derived by following the procedure described in the Appendix of MN10. The thermodynamic fields that hold the vortex wind profile are then constructed using the iterative procedure of the built-in WRF initialization module for idealized hurricane simulations together with the Jordan (1958) mean hurricane sounding as the environmental temperature profile. With this vortex, we observed some weak gravity waves and vortex-Rossby waves radiating outward from the inner core of the vortex due to weak hydrostatic adjustment at the beginning. In order to prevent the interaction between this wave activity and the prescribed diabatic forcing, a 24-hour spin-up period is performed to allow the waves to decay or propagate out of the region where diabatic forcing is added. During the spin-up period, the intensity of the vortex slightly weakens (surface maximum wind decreases from 43 푚푠−1 to 40 푚푠−1) due to the presence of a sponge layer at the top of the domain (between 푧 = 20 − 25 푘푚). The storm structure remained unchanged. Sensitivity tests confirm that these small intensity changes do not impact the response induced by the rainband. Figures 2.1a-b show the profiles of the tangential wind and the potential temperature anomaly after the spin-up period. After this spin-up period, two

24-hour-long experiments are conducted: one with the prescribed diabatic heating and the other

18 without (control experiment). Following Kwon and Frank (2005), the response induced by the diabatic forcing is computed by subtracting the model fields of the two experiments. To compare with the results of MN10, all of the analyses of the response are evaluated at hour 18 after the spin period, unless otherwise specified.

2.2. Design of diabatic heat source

2.2.1 Stratiform heating profile in Moon and Nolan (2010) We reexamine the storm-scale wind field response using the stratiform rainband heating profile in MN10. The radial and vertical structure of MN10’s stratiform rainband is computed as

푄푀푁10(푟, 휆, 푧)

0, for 푧 ≤ 푧푏푠 − 휎푧푠 2 8 (2.3) 푟 − 푟푏푠 휆 + 휋/4 푧 − 푧푏푠 = 푄max exp (− ( ) − ( ) ) sin ( 휋) , for 푧푏푠 − 휎푧푠 < 푧 < 푧푏푠 + 휎푧푠 휎푟푠 휋/4 휎푧푠

{ 0, for 푧 ≥ 푧푏푠 + 휎푧푠 where 휆 is azimuthal angle; 푟푏푠(휆, 푧) is the radial center location of the stratiform heating; 푧푏푠 is the vertical center location of the stratiform heating, which is also the level where transition from cooling to heating occurs (and therefore is of zero heating at 푧푏푠 ); 휎푟푠 and 휎푧푠 are the half- wavelengths that control the radial and vertical extents of the heating profile, respectively. The

−1 heating magnitude is controlled by 푄max, which is taken to be 4.24 퐾 ℎ푟 . Note that 푟푏푠(휆, 푧) is a function of height and azimuth and is designed to mimic the spiral structure and outward tilt of a tropical cyclone rainband. Following MN10’s definition, 푟푏푠(휆, 푧) is given by 푟푏푠(휆, 푧) =

λ 푟 (λ) + 푧 , where 푟 (휆) = 60 − 10 [푘푚] . This definition of 푟 (휆) gives a spiral 푏푠푓푐 푏푠푓푐 π/4 푏푠푓푐 structure of the rainband that linearly varies from 60 km at the downwind end (휆 = 0) to 80 km at the upwind end (휆 = −휋/2). The parameters 푧푏푠, 휎푟푠 and 휎푧푠, are taken to be 4 푘푚, 6 푘푚 and

2 푘푚, respectively. The radial-height structure of MN10’s stratiform profile at the middle of the

19 rainband (i.e., 휆 = −π/4) is shown in Fig. 2.1c.

Note that the azimuthal coverage of the rainband heating in Eq. (2.3) is controlled by a

휆+휋/4 8 stationary envelope, exp (− ( ) ), which confines the rainband in the lower-right quadrant 휋/4 of the storm, as shown in Fig. 2.1d. This is different from MN10, whose rainband heating rotates at a constant rate of 70% of the tangential wind at the rainband region. Observational studies

(Willoughby et al. 1984; Hence and Houze 2012) show that the stratiform sector of a rainband complex does not generally rotate at this rate, but rather remains quasi-stationary and aligns with the environmental wind shear. While MN10 intensively studied the storm-scale wind field response with convective and stratiform rainbands that rotate with the tangential wind, rainbands that exhibit a stationary nature have not been fully explored. Therefore, in this study our stratiform rainband heating is designed to be stationary in the lower right quadrant of the storm. Due to the presence of the stationary rainband forcing, we do observe some weak wobbling motion of the TC vortex, with a scale smaller than 1km. Such small-scale wobbling of the storm center does not impact the evolution and analysis of the response. Therefore, in all the WRF simulations the rainband forcing is fixed in Cartesian space relative to the domain center, unless otherwise specified.

2.2.2 A modified stratiform diabatic heating profile based on DH13 The MN10 idealized structure captures the basic heating profile of convection-driven stratiform precipitation, but its exact structure in a tropical cyclone may not be realistically represented by this idealized structure. We therefore design a modified structure that better matches the observed structures of a tropical cyclone rainband complex.

Our modified stratiform heating profile is based on the secondary circulation in the

20 stratiform rainband of Hurricane Rita documented by DH13. DH13 found a seemingly closed secondary circulation that consists of a clear MDI and a rising outflow above and inward of it. We use the Sawyer-Eliassen equation (Eliassen 1951) to reconstruct the stratiform heating and cooling structure based on the azimuthally averaged secondary circulation shown in Fig.1.2b. The details of the reconstruction process are presented in the Appendix A.

Based on the reconstructed diabatic heating and cooling derived from observational data, we designed the modified stratiform rainband heating and cooling profile shown in Figs. 2.1e-f.

The radial coverage of the modified profile is confined to a radial range similar to MN10’s stratiform profile, while the maximum heating rate and azimuthal coverage were kept the same.

The modified profile captures the diagonal structure of the diabatic heating and cooling as shown in Fig A.1, with the low-level cooling located slightly radially outward from the upper-level heating. The altitude of the maximum heating is at 6 km, which is slightly higher than that of the stratiform profile in MN10, while the maximum cooling is at similar altitude. Given these differences, the modified profile structure at 4.6 km altitude (Fig. 2.1f) appears in a dipole pattern rather than a singular band of heating.

21

Fig. 2.1. Vertical and radial distributions of (a) tangential wind and (b) potential temperature anomaly of the basic state vortex after the 24-hour spin-up period, contoured every 5 푚푠−1 and 1.69 퐾 , respectively. Cross-section through the middle portion of MN10’s stratiform rainband diabatic heating is shown in (c), and plan view at 푧 = 4.6 푘푚 is shown in (d). (e) and (f) are same as (c) and (d), but for the modified stratiform profile. Contour spacings are 1 퐾ℎ푟−1 for (c) and (e), and 0.5 퐾ℎ푟−1 for (d) and (f).

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2.3. MN10 stratiform profile

2.3.1 Axisymmetric response

Fig. 2.2. a) Azimuthally averaged wind field response using MN10’s stratiform profile for the WRF simulation. Tangential velocity contours at intervals of 0.5 푚푠−1 are overlaid by the radial-vertical velocity vectors. Diabatic heating (cooling) of 0.15 퐾ℎ푟−1(−0.15 퐾ℎ푟−1) are highlighted in solid red (blue) contour. b) As in a) but for MN10’s simulation. From MN10.

MN10 found that once the vortex reaches a steady state, the wavenumber-0 response of the primary circulation is a large component of the full tangential wind response. Figure 2.2a shows the axisymmetric responses of the tangential wind field and the secondary circulation on the 푟 − 푧 plane at simulation hour 18. Consistent with MN10 (as shown in Fig. 2.2b), the WRF simulation shows a clear axisymmetric tangential wind enhancement in the mid-levels, with a maximum at

푧 = 4 km where the lower-level cooling transitions to upper-level heating. The axisymmetric rainband heating occurs approximately at 70 km radius. On the outward side of the axisymmetric heating is a clear mid-level inflow which is collocated with the tangential wind enhancement.

Radially inward, there is a slight decrease in tangential wind in the mid-levels, which collocates with weak outflow between 푟 = 45 and 65 km. This couplet of mid-level inflow and outflow and the associated horizontal convergence are driven by the rising and sinking motions induced by the diabatic heating and cooling. Conservation of angular momentum then results in the corresponding

23 changes in the tangential wind. Both above and below the mid-level responses, vertical motion subsides and the flow diverges in weaker inflow-outflow couplets. This couplet, along with the corresponding changes in tangential wind, has an opposite configuration than that in the mid- levels. While small differences do occur between the two axisymmetric responses in Fig. 2.2, the overall patterns are in good agreement.

2.3.2 Plan view and cross-section analysis

Figure 2.3 shows the individual plan views at different altitudes for the WRF and MN10 simulations. For the WRF simulation, we present both the full wind response (Figs. 2.3a-c) as well as the sum of responses from azimuthal wavenumbers 0-4 (Figs. 2.3d-f) in order to easily compare to the wavenumbers 0-4 response from MN10 (Figs. 2.3g-i). At the middle level (z=3.6 km; Fig.

2.3b, e and h), the rainband regions (lower right quadrant) for both simulations are dominated by sinking motion, while at the upper level (z=6 km; Fig. 2.3a, d and g) rising motion is dominant.

Some differences do occur at these two levels. For instance, a moderate downdraft not seen in

MN10 occurs radially inward of the rainband heating at the upper level, while at the middle level, a widespread but weak updraft is present in MN10’s simulation. Otherwise, the responses at the middle and upper levels are generally in good agreement between the two experiments.

More noticeable differences can be found at the lower level of 푧 = 2 km. As shown in Fig.

2.3f, the WRF simulation produces a band of downward motion along the spiral rainband arc, which is caused by the diabatic cooling below 푧 = 4 km. Along the inner edge of this downdraft is a prominent band of low-level updraft. Both the updraft and downdraft appear to be stronger in the WRF full response (Fig. 2.3c). In MN10’s simulation (Fig. 2.3i), this low-level band of upward motion is mostly absent, while the outer downdraft near the region of rainband cooling is also

24 considerably weaker. As we will show in the next section, the main reason for these discrepancies in low-level vertical motion is the difference in rotation rates of the rainband heating.

Fig. 2.3. Plan view of full responses in vertical velocity (shading) and horizontal wind field (vectors) responses at (a) 6 푘푚, (b) 3.6 푘푚, and (c) 2 푘푚 altitudes from MN10’s stratiform heating using the WRF simulations. Solid contours of diabatic forcing with magnitude of 1 퐾ℎ푟−1(heating in red and cooling in blue) at 5.6 푘푚, 3.6 푘푚 and 2.4 푘푚 are added respectively to indicate the rainband location. (d), (e), and (f) are the same as (a), (b) and (c), but for responses summing from azimuthal wavenumbers 0 to 4. (g), (h), and (i) are similar plots as (d), (e) and (f) but from MN10’s simulation at slightly different −1 altitudes. Contours are at 0.03 푚푠 intervals.

Focusing on the WRF full wind field response, we look at cross-sections slicing through the upwind, middle and downwind portions of the rainband heating region, as shown in Fig. 2.4.

For the tangential wind response, a uniform enhancement of tangential wind occurs across all three

25 cross-sections at about the same radial locations with similar magnitudes. The responses in the secondary circulation, on the other hand, have some variations across the rainband. The upwind portion shows the weakest response as this is the first region where the primary circulation enters the rainband and becomes modified by the heating structure. In the middle portion, the sinking motion associated with the mid-level inflow becomes significantly stronger, extending all the way to the lowest model level near 푟 = 60 km. Consistent with Fig. 2.3c, radially inward between 푟 =

40 and 60 km lies the low-level updraft, which extends vertically upward to 푧 = 4 km and then becomes part of the mid-level outflow at the inward side of the rainband heating. Going further downwind (Fig. 2.4d), the mid-level descending inflow gets even stronger and shifts inward, following the spiral structure of stratiform rainband.

Fig. 2.3a shows that extending downwind of the rainband heating region, vertical motion responses occur along an inward spiral (highlighted by the gray shading). These wave responses also exist in the filtered response (Fig. 2.3d-f), which appear to be stronger than the 3DVPAS simulation from MN10 (Fig. 2.3g-i). In the inner core region, the WRF simulations also show inner-core wave modes, which appear to be in similar magnitude, but with different orientations from MN10’s simulations. These differences are due mostly to the difference in rainband rotation rates between the two studies and the nonlinearity of the WRF model. However, since there is no moisture and microphysics in the current experiment setting, it is difficult to quantify the actual impacts of these wave modes and their interaction with the rainband on the storm evolution. Thus, for the remainder of this study, we focus our analyses on the vortex responses within and surrounding the rainband region. Also, given the similarity between the WRF full (all wavenumbers) and filtered (sum of azimuthal wavenumbers 0-4) wind field responses, in the rest of the analysis we focus only on the full wind field responses, unless otherwise specified.

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Fig. 2.4. a) Plan view of vertical velocity at 2.2 km altitude from MN10’s stratiform heating using the WRF simulation. Red lines mark the cross sections shown in (b), (c), and (d). b) Tangential wind (shading) and second circulation (vectors) responses from the WRF simulation in the upwind cross section in (a). c) As in (b) but for the middle cross section. d) As in (b) but for the downwind cross section. Diabatic heating (cooling) of 1 퐾ℎ푟−1(−1 퐾ℎ푟−1) are highlighted in solid red (blue) contour.

2.3.3 Omega equation analysis

Using MN10’s stratiform profile, the midlevel descending inflow in our WRF simulation is consistent with both MN10’s result and stratiform rainband observations (Didlake and Houze

2013b; Didlake et al. 2018). Additionally, a low-level updraft is found on the radially inward side of the rainband. Didlake et al. (2018) found a similar low-level updraft in Hurricane Earl and

27 hypothesized that this updraft occurred in response to the negatively buoyant MDI reaching the high-휃푒 boundary layer. This persistent low-level updraft likely played an important role forming the eventual secondary eyewall. In this section, we aim to further explore the physical mechanism that generates this low-level updraft.

We employ the generalized omega equation derived by Krishnamurti (1968) based on a balanced assumption (equation (16) of Krishnamurti 1968; equation (23) of Zhang et al. 2000).

Similar to the quasi-geostrophic (QG) omega equation, the generalized omega equation is a diagnostic equation for vertical motion. The generalized omega equation assumes hydrostatic balance and neglects the time-derivative of divergence and advection by the divergent component of the wind field, while it includes the curvature effect of the flow, and therefore is applicable to flow regimes with relatively large Rossby number (푅표~1). Neglecting the effects of differential deformation and differential divergence (small compared to other terms) and assuming a constant

푓-plane, the generalized omega equation takes the following form:

휕2휔 휕 휕 휕 휕휓 훻2(휎2휔) + 푓2 − 푓 (휔 훻2휓) − 푓 (휵 휔 ∙ 휵 ) ℎ 휕푝2 휕푝 휕푝 ℎ 휕푝 풉 풉 휕푝 (2.4) 푅 2 휕 푅푇 2 = 훻ℎ 퐻 + 푓 (풖풉 ∙ 휵풉휁) + 훻ℎ (풖풉 ∙ 휵풉휃) 푐푝푝 휕푝 푝휃

2 2 2 where 휵풉 = 휕푥풊 + 휕푦풋 and 훻ℎ = 휕푥 + 휕푦 are the horizontal gradient and Laplacian operators, respectively; 푝, 푇, and 휃 are pressure, temperature, and potential temperature, respectively; 휔 =

푑푝 is the pressure velocity; 푓 is the Coriolis parameter; 푅 and 푐 are the ideal gas constant and 푑푡 푝 specific heat capacity at constant pressure; 휓 is the streamfunction of the nondivergent horizontal

푅푇 휕휃 wind; 휎2 = − is the static stability parameter; 퐻 is the diabatic heating per unit mass of air; 푝휃 휕푝

풖풉 is the horizontal wind vector and 휁 is relative vertical vorticity. Similar to the QG omega

28 equation, the major forcings on the right-hand-side of Eq. (2.4) are from diabatic heating (first term), differential vorticity advection (DVA; the second term) and buoyancy advection (BA; the third term). Equation (2.4) is solved numerically with initial guess 휔0 ≡ 0 using the Damped

Jacobi Method. The diagnosed 휔 is then converted into vertical velocity 푤. Figures 2.5a-b show a comparison of the vertical velocity from the WRF simulation and the diagnosed vertical velocity from the BA term at 880 ℎ푃푎 (about 푧 = 1.15 푘푚). It is clear that at this low level where the contribution of diabatic forcing is small, the BA term is the dominant, while DVA is at least two orders smaller (not shown).

Fig. 2.5. a) 880-hPa-level (near 푧 = 1.15 푘푚) plan view of the vertical velocity response from MN10’s stratiform heating in the WRF simulation in the lower right quadrant. b) Diagnosed vertical velocity from the buoyancy advection (BA) term in Eq. (5) corresponding to the vertical velocity field in (a). c) 880-hPa-level potential temperature anomalies and mean divergence between 880-hPa-level and the lowest pressure level using MN10’s stratiform heating profile. Divergence contours are at 3 × 10−5s−1 intervals. Diabatic cooling of −0.5 퐾ℎ푟−1 at 790 푚푏 are shown in solid blue contour.

Buoyancy advection can be understood by examining the potential temperature anomaly

′ (휃푎), which is the deviation of potential temperature response (휃′) from its azimuthal average

′ response (as a function of radius and height). The 880-hPa 휃푎 (near 푧 = 1.15 푘푚, which is below the level of rainband cooling) is shown in Fig. 2.5c. Since tangential wind is the dominant component of the horizontal wind, the azimuthal advection of 휃′ (which is equal to the azimuthal

′ advection of 휃푎) is the dominant component of the buoyancy advection. Thus, the distribution of

29

′ 휃푎 clearly displays the azimuthal gradient of 휃′ that causes buoyancy advection. Downwind and

′ inward of the rainband cooling (as indicated by the blue contour in Fig. 2.5c), 휃푎 shows a stationary negative anomaly, which is caused by the cold advection of the inward advancing

′ descending inflow. In contrast, the 휃푎 is positive upwind and radially inward of the rainband heating, indicating that air coming from the upwind side is positively buoyant in a relative sense.

As this upwind warm air advances into the rainband region, convergence occurs between these two air-masses, as shown by the 880-1000 hPa layer mean horizontal divergence in Fig. 2.5c. This results in a localized updraft on the inner edge of the rainband diabatic cooling. This mechanism shows that the 휃′ gradient and the resulting advection of 휃′ is essential to generate the enhanced low-level updraft.

2.3.4 Pressure field analysis

We next build on the dynamical analysis of the low-level updraft by examining the pressure responses associated with the stratiform heating. Figure 2.6a shows the cross-section of pressure response along the downwind portion of the rainband. Below 푧 = 2 푘푚, the rainband cooling induces a high-pressure anomaly, while a predominant low-pressure anomaly is present between

푧 = 2 to 6 푘푚 where the transition between cooling and heating occurs. This is an expected pressure pattern of hydrostatic adjustment given diabatic cooling in the mid-to-lower troposphere and has also been demonstrated in other rainband modeling studies (Wang 2009). Figure 2.6a also shows the low-level updraft of interest between 40-55 km radius. To see exactly how this pressure field response helps to maintain the low-level updraft, we first decompose the pressure response,

푝′, following the Eq. 7.4 from Houze (2014):

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휕 푣2 ∇2푝′ = (휌 퐵) − ∇ ∙ (휌 풖 ∙ ∇풖 + 휌 0 푟̂) = 퐹 + 퐹 (2.5) 휕푧 0 0 0 푟 퐵 퐷

휕 푣2 where 퐹 = (휌 퐵) is the buoyancy pressure forcing; 퐹 = −∇ ∙ (휌 풖 ∙ ∇풖 + 휌 0 푟̂) is the 퐵 휕푧 0 퐷 0 0 푟 dynamic pressure forcing that accounts for the control experiment being in gradient wind balance;

′ 휌 ′ 퐵 = − 푔 is buoyancy with 휌 being the density response and 휌0 the density field of the control 휌0 experiment; 풖 is the three-dimensional wind vector of the heating experiment; 푣표 is the tangential wind of the control experiment; 푟 and 푟̂ are radius and the unit vector in radial direction. Figures

2 ′ 2.6b-c show the cross-sections of ∇ 푝 and 퐹퐵 along the downwind portion of the rainband. The pressure field response is largely due to buoyancy pressure forcing, while the dynamic pressure forcing is two orders smaller in magnitude (not shown). This buoyancy-driven nature suggests that the pressure response is the result of negatively buoyant air (induced by the rainband cooling) accumulating near the surface. A scale analysis (not shown) shows that the dominant component of ∇2푝′ is the vertical component. Thus, Equation (2.5) can be approximated as 휕2푝′/휕푧2 ≈

휕(휌0퐵)/휕푧. This indicates that the pressure and density responses are close to hydrostatic balance,

1 휕푝′ 휌′ 휃′ which can be approximated as ≈ − 푔 ≈ 푔, where 휃0 is potential temperature of the 휌0 휕푧 휌0 휃0 control experiment (Markowski and Richardson 2010). Recall that vertical motion and buoyancy advection fields share very similar structure (Fig. 2.5b); this is because the dominant terms in Eq.

푅푇 (4) (i.e., 훻2(휎2휔) on left-hand-side and 훻2(풖 ∙ 휵 휃) on right-hand-side) both contain the 훻2 ℎ 푝휃 ℎ 풉 ℎ operator. Hence, we have the following approximated relationship between 푤 and the azimuthal gradient of vertical pressure gradient force:

′ ′ 푣 ∂휃 푣휃0 휕 휕푝 푤 ∝ − ≈ − ∙ ( ) . (2.6) 푟 휕휆 푔푟휌0 휕휆 휕푧 where 휆 is the azimuthal angle. Figure 2.6d shows that the distribution of vertical velocity and

31

푣휃 휕 휕푝′ − 0 ∙ ( ) correspond well. This indicates that as the air on the upwind side travels 푔푟휌0 휕휆 휕푧

푣 휕 휕푝′ downwind, it experiences an enhancing vertical pressure gradient force (i.e., (− ) > 0) 푟 휕휆 휕푧 associated with the cold and dense air accumulated at the downwind portion of the rainband, which provides the lifting force for the observed low-level updraft.

Fig. 2.6. a) Cross-section showing the responses in the pressure field (shading) and vertical velocity (magenta contours; negative values are dashed) at downwind cross section from Fig. 5a. Vertical velocity is contoured at intervals of 0.082 푚푠−1. Diabatic heating (cooling) of 1 퐾ℎ푟−1(−1 퐾ℎ푟−1) are highlighted in red (blue) dashed contour. b) 3D Laplacian of the pressure response at the downwind cross section. c) the buoyancy pressure forcing 퐹퐵 at the downwind cross section. d) plan view of 푣휃 휕 휕푝′ 0 (− ) (shading) and vertical velocity response (magenta contours; negative values are dashed) 푔푟휌0 휕휆 휕푧 at 푧 = 2 푘푚 of the rainband quadrant. Vertical velocity is contoured at 0.05 푚푠−1.

2.3.5 Rotating rainband experiments

The physical mechanism discussed in sections 2.3.3 and 2.3.4 suggests that the stationary

32 nature of the rainband diabatic forcing is essential for generating the enhanced low-level updraft.

To test this hypothesis, two more experiments are performed with the rainband heating structure rotating at an angular velocity of 1.454 × 10−4 푟푎푑 푠−1 and 2.856 × 10−4 푟푎푑 푠−1, the latter of which is similar to the rotation rate used in MN10’s simulation. Figure 2.7 shows a comparison of

′ the vertical velocity and 휃푎 at 푧 = 2 푘푚 between the three experiments. All three experiments have positive vertical motion radially inward from the rainband region, with decreasing magnitude

′ at faster rainband rotation rates. Also, the faster the rainband rotates, the more the negative 휃푎 shifts towards the upwind side of the rainband, and the more widespread they become.

Consequently, the azimuthal temperature gradient is much reduced and the resulting region of positive buoyancy advection (black dashed contours) becomes narrower and weaker, and so does the low-level updraft (solid magenta contours). These results demonstrate that the stationary nature of the rainband in the WRF simulations is the major cause of the differences in the low-level updraft response between the WRF simulation and MN10’s findings.

We also note that with a rotation rate similar to MN10, the WRF model still produced noticeable low-level upward motion radially inward of the rainband, (Fig. 2.7c), which is absent in the 3DVPAS simulation from MN10 (Fig. 2.3i). Additionally, the downdraft induced by the rainband diabatic cooling is also stronger compared to that in Fig. 2.3i. These weak vertical motions also exist in the WRF filtered response (wavenumbers 0-4, not shown), indicating that the differences are not caused by wave filtering. These differences may partly be due to the nonlinearity of the WRF model, but the exact reason remains unclear. Also, both the orientation and structure of the inner core wave modes vary across the three experiments, which is likely caused by the nonlinear interaction between these wave modes and the induced circulation by the rainband heating.

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Fig. 2.7. Vertical velocity responses at 푧 = 2 푘푚 using MN10’s stratiform heating profile at different rainband rotation rates Ω: (a) no rainband rotation (Ω = 0); (b) Ω = 1.454 × 10−4 푟푎푑 푠−1; and (c) Ω = 2.856 × 10−4푟푎푑 푠−1. Diabatic cooling of −1 퐾ℎ푟−1 at 2.4 푘푚 are shown in solid blue contour. ′ d) Anomalies of potential temperature response 휃푎 (shading) at 푧 = 1 푘푚. Positive vertical velocity ′ response (푤′, magenta contours) and buoyancy advections (−풖풉 ∙ 훁풉휃 , black dashed contours) are contoured at 0.03 푚푠−1 and 1.2 × 10−5 퐾푠−1 , respectively. e) same as (d), but for Ω = 1.454 × 10−4 푟푎푑 푠−1. f) same as (d), but for Ω = 2.856 × 10−4푟푎푑 푠−1.

2.4. Modified stratiform heating profile

2.4.1. Axisymmetric response

While the design of MN10’s stratiform profile is typical among most stratiform clouds due to dying convection (Houze 1997), whether or not this heating/cooling structure would remain the same within a strong gradient-wind environment, such as a TC rainband region, is open to question.

Based on the secondary circulation observed in Hurricane Rita (2015), DH13 hypothesized a diagonal pattern of latent heating and cooling (see their Fig. 17b), which is confirmed in our reconstruction of heating structure based on the Sawyer-Eliassen equation (see Appendix A). This

34 diagonal pattern of heating and cooling enhances the radial gradient of diabatic forcing, which may further enhance the secondary circulation (as indicated by Sawyer-Eliassen equation (Eliassen

1951; Schubert and Hack 1983)), as well as the mid-level descending inflow as seen in the previous simulations. Therefore, in this section we further explore how a TC-like vortex would respond to this modified stratiform heating structure.

Fig. 2.8. Azimuthally averaged response of tangential wind (shading) and secondary circulation (vector) from the WRF simulation using the modified stratiform heating profile. Diabatic heating (cooling) of 0.15 퐾ℎ푟−1(−0.15 퐾ℎ푟−1) are highlighted in solid red (blue) contour.

Figure 2.8 shows the azimuthal average response in the tangential and secondary circulations with the modified stratiform heating/cooling profile. Overall, the azimuthal average response shares some similarities with the MN10 profile (Fig. 2.2a), but important differences do occur. At the mid-levels near 푧 = 4 푘푚, we again see strong inflow and a weaker outflow on the two sides of the rainband heating, which converge near 푟 = 70 푘푚. Compared to the MN10 profile, these responses are deeper, with a larger portion of the mid-level inflow being directed downward towards the low levels and forming a clear descending inflow. This feature is in good

35 agreement with the MDI observed in Hurricanes Rita (2005; Didlake and Houze 2013b) and Earl

(2010; Didlake et al. 2018). Following angular momentum conservation, a layer of tangential wind enhancement, which is noticeably deeper than that of MN10’s stratiform rainband heating, is collocated with the simulated MDI. Owing to the noticeably stronger descent, the tangential wind jet also shows a descending pattern with its jet core located in the lower half of inflow layer.

2.4.2. Plan view and cross-section analysis

Fig. 2.9. Plan view of vertical velocity (shading) and horizontal wind field (vectors) responses at (a) 8.4 푘푚; (b) 3.6 푘푚 and (c) 2 푘푚 altitudes for the WRF simulations using the modified heating profile. Solid contours of diabatic forcing with magnitude of 1 퐾ℎ푟−1(heating in red and cooling in blue) at 8 푘푚, 3.6 푘푚 and 2.4 푘푚 are added respectively to indicate the rainband location.

More differences with MN10 are revealed when examining plan views of different levels and cross-sections. Figure 2.9 shows that the overall vertical velocity pattern near the rainband heating/cooling appears similar to that seen in the case of MN10 (Fig. 2.3). One exception is that the modified heating pattern produces stronger vertical velocities. In Figure 2.10, the mid-level inflow across all three cross sections shows a very clear descending pattern near 푟 = 80 푘푚, as with the axisymmetric response. This pattern allows the accompanying high angular momentum air from larger radii to reach the surface more readily, resulting in enhanced tangential winds that skew towards the lowest levels. In the middle and downwind cross sections, a branch of mid-level

36 rising outflow branch can also be seen around 푧 = 4 to 6 km, as shown in Fig. 2.10d. Conserving its angular momentum, this rising outflow branch is associated with the upper level anticyclonic circulation, which is also significantly stronger than that in MN10’s stratiform profile, as shown in Fig. 2.4.

Inward of the descending inflow lies a low-level updraft similar to that from MN10’s stratiform heating, but with a noticeably larger magnitude, especially in the middle portion (Fig.

2.10c). Also different from MN10’s heating profile, this updraft band strongly curves inward, eventually separating from the mid-level rising outflow, as can be seen in Fig. 2.9c and Fig.

2.10c and d. This likely indicates that the two updrafts at low levels and at mid-to-upper levels have different origins. Further investigation of the diagnostic omega equation terms (Eq. 2.4) from the low-level updraft (not shown) showed that the BA term alone reproduced most of the low-level updraft, as in Fig. 2.5b. The associated 휃′ anomalies showed that negatively buoyant air from the modified stratiform cooling extended to smaller radii, causing a convergence zone between the warm and cold air masses and hence an updraft to spiral inward more strongly.

One should note that the current modified profile has slightly larger radial and vertical extent compared to the MN10’s profile, and hence a larger total integrated absolute diabatic forcing. We have performed a parallel simulation of the modified heating structure but with a total integrated heating equal to that of the MN10 heating profile. Our results show that while the magnitude of the circulation response varies with the integrated heating magnitude, the structure of the circulation response remains the same.

37

Fig. 2.10. Same as Fig. 2.4, but for the modified heating profile.

2.5. Buoyant updraft analysis

Thus far, we have analyzed the generation mechanism of the low-level updraft response to the idealized stratiform heating structures. One important aspect of interest is understanding how this low-level updraft impacts the evolution of the larger TC. The motivation for this interest is that the low-level updraft may be capable of triggering convectively buoyant updrafts that can further influence the intensity and structure of the TC, such as the updrafts found in Hurricane Earl

(Fig. 1.2c). We found that the low-level updraft mostly does not overlap with the region of imposed

38 diabatic heating and cooling. Given our current model setup with no moisture or microphysics, a rising air parcel here is forced upward and eventually becomes negatively buoyant in a stably stratified atmosphere. Thus, the potential evolution of this updraft in a real atmosphere has not been fully explored. To answer this question, we conducted a new set of experiments that determines the convective buoyancy of air parcels being forced upward. To do this, we reinitialize the WRF model with added moisture of various degrees of relative humidity (90%, 93% and 95%) and with a microphysics parameterization (WSM 6-class scheme) turned on (in addition to the prescribed modified heating profile, as in Fig. 2.1e and f). As in the previous simulations, there is no boundary layer or cumulus parameterizations. The profiles of mean relative humidity averaged over an annulus covering the rainband region (40-80 km radius) are shown in Fig. 2.11. These modified profiles were based on the Jordan mean hurricane sounding. The amount of specific humidity added to the Jordan sounding at each level to achieve the desired relative humidity below

600 mb is shown by the dashed lines in Fig. 2.11. Due to the added moisture, we did not perform a 24-hour spin-up in the following experiments; we verified that this change does not modify the overall dynamic response. The model is integrated for a short period of 6 hours to collect statistics of the convectively buoyant updrafts (as defined below).

We define a parcel at a given grid point (x, y, z) as convectively buoyant if it meets the following criteria. First, the grid point must have a positive vertical velocity. Second, the

1 associated vertical kinetic energy 푤2 must be greater than the convective inhibition (퐶퐼푁) that it 2 will experience. 퐶퐼푁 is calculated as

푝퐿퐹퐶 ′ ′ 퐶퐼푁 = − ∫ 푅푑푟푦 (푇푒푣푛(푝 ) − 푇푝푎푟푐푒푙(푝 )) 푑푙푛푝′ (2.8) 푝0 where 푝 is pressure, 푝0 is the initial level of the parcel under consideration, 푝퐿퐹퐶 is the level of

39 free convection, 푅푑푟푦 is the gas constant for dry air. 푇푝푎푟푐푒푙(푝) is the temperature the air parcel of interest would have if it is lifted upward from 푝0 to 푝, and 푇푒푛푣(푝) is the temperature profile of the atmospheric column at the same horizontal location as the air parcel. 퐶퐼푁 and vertical kinetic energy are calculated for all grid points below 600 ℎ푃푎 height. If 푝퐿퐹퐶 for a given air parcel is higher than 400 ℎ푃푎, the parcel is automatically flagged as non-convective, since above this altitude the 퐶퐼푁 is likely to be large enough to suppress the air parcel from being convectively buoyant. At any given time instant, if any grid point within a column is found to be convectively buoyant, we will count the entire column as a convectively buoyant column.

Fig. 2.11. Solid lines show the mean profiles of relative humidity for each moist sensitivity experiment, averaged over an annulus of 40-80 km radius, which covers the rainband region. Dashed lines show the specific humidity added to the Jordan mean hurricane sounding (훿푞) at each pressure level to initialize the experiments.

Figure 2.12 shows the frequency of convectively buoyant columns between simulation hours

2 and 6 for each humidity profile. All three experiments have a high frequency of buoyant columns

40 at the locations of the low-level updraft. Expectedly, the higher relative humidity environments

(Fig. 2.12c) were more likely to have convectively buoyant columns. The locations with larger frequencies lie entirely inward of the radii where upper-level heating (indicated by the black solid line) is located, showing that most of the induced buoyant updrafts are not directly associated with the imposed upper-level diabatic heating. Close inspection of individual updrafts (not shown) indicate that the low-level updraft identified in previous simulations becomes slightly stronger in magnitude due to diabatic heating release from the microphysics parameterization and that the location of the maximum updraft shifts slightly downwind and coincide with the regions of maximum frequency of buoyant columns, as in Fig. 2.12.

Fig. 2.12. Frequencies of buoyant convective updrafts occurring during simulation hour 2 to hour 6 using our modified heating profile with different relative humidity profiles: (a) 90%; (b) 93% and (c) 95%. Contours of diabatic heating (solid) and cooling (dashed) at z = 4 km are shown as reference, with spacing of 2 퐾ℎ푟−1.

These results show that the low-level updrafts discussed in previous sections are capable of triggering convectively buoyant updrafts or new convective cells that may further impact the intensity and structure evolution of the TC. These convectively buoyant updrafts tend to be generated at the inward side of the stratiform rainband and may provide a plausible pathway for how a quasi-stationary stratiform rainband can produce sustained buoyant updrafts along its inner edge. In a related study, Chen (2018) also found that by continuously augmenting the low-level

41 diabatic cooling at an annulus region outside of the primary eyewall, it is possible to synthetically produce a ring of persistent updrafts that eventually form an apparent secondary eyewall. Their findings align with previous observations (Didlake and Houze 2013a; Didlake et al. 2018) which indicate that diabatic cooling associated with the stratiform rainband is likely to play an important role in secondary eyewall formation. This same mechanism may also play a role in forming a wavenumber-1 asymmetry in a mature secondary eyewall through interaction with a stratiform rainband located just radially outward (Didlake and Kumjian 2017). Based on our analysis, we believe that our findings may provide support to these studies from a dynamical stand point.

2.6. Conclusions

In this study, we use idealized simulations of the Advanced Research version of the

Weather Research and Forecasting model to examine the one-way dynamic response of a dry hurricane-like vortex to stratiform rainband diabatic forcing. Two stratiform rainband heating profiles have been investigated. One is the same stratiform heating profile used in Moon and Nolan

(2010; MN10), which represents typical diabatic forcing in stratiform precipitation having heating above and cooling below the melting level. The second is a modified stratiform heating profile derived using the Sawyer Eliassen equation and the observational data collected from Hurricane

Rita (2005) as presented in Didlake and Houze (2013b). In contrast to MN10’s stratiform profile, the modified heating profile consists of a heating and cooling dipole in a diagonal pattern, creating an enhanced negative radial gradient of diabatic heating. Both stratiform heating profiles are stationary with respect to the vortex to mimic the diabatic forcing imposed by the stratiform portion of a quasi-stationary rainband complex.

Using MN10’s stratiform profile, we reproduce secondary circulation and enhanced

42 tangential wind responses that are largely consistent with the findings of MN10. When using the modified stratiform heating profile, both the mid-level inflow and enhancement of tangential flow are deeper in comparison. Owing to the larger radial gradient heating associated with the modified heating profile, most of the mid-level inflow descends towards the surface. This circulation pattern is similar in structure to the MDI observed in Hurricanes Earl (2010) and Rita (2005).

For both heating profiles, a low-level updraft response occurred radially inward of the

MDI. A pressure field analysis shows that the low-level updrafts lie in a region of enhanced upward-pointing pressure gradient force induced by buoyancy forcing. Using the generalized omega equation for balanced flow, we show that this low-level updraft is driven by the buoyancy advection term. This term indicates that the low-level updraft is caused by the azimuthal gradient of the near-surface temperature induced by the stratiform cooling and its advection by the vortex tangential flow. In experiments with the rainband rotating with the tangential flow rather than remaining stationary, the azimuthal temperature gradient was reduced and thus the low-level updraft magnitude was reduced. In experiments with moist thermodynamics and microphysics, the forced low-level updraft from the stationary rainband was sufficient for triggering persistent buoyant updrafts along the inner side of the rainband diabatic heating profile.

These experiments provide pertinent context for the features observed by Didlake et al.

(2018) in the stratiform sector of Hurricane Earl (2010). Didlake et al. (2018) found a persistent deep updraft occurring at the terminus of the MDI. They proposed that the descent of the MDI disturbed the boundary layer and forced the adjacent deep updraft through buoyancy dynamics.

Given the similarities with the simulated low-level updrafts, our experiments support their hypothesis, indicating that the MDI becomes a boundary layer cold pool that, through buoyancy advection by the tangential flow, triggers the adjacent buoyant updraft observed in the spiral

43 rainband complex of Hurricane Earl.

The idealized experiments in this study provide insight to a potential mechanism for secondary eyewall formation in TCs. The persistent updraft in Earl accelerated the low-level tangential flow and was a precursor for the formation of an axisymmetric secondary tangential wind maximum and secondary eyewall. The stationary rainband complex from our experiments yield a persistent, spatially continuous, and deep updraft that spans the azimuthal extent of the stratiform rainband sector. Such an updraft would create a localized region of heating and enhanced low-level vorticity that would substantially project onto the azimuthal mean, as evidenced by the updraft and low-level tangential wind in the axisymmetric responses in Figs. 2.2a and 2.8. These responses could initiate secondary eyewall formation via several hypothesized axisymmetric dynamical mechanisms, such as upscale cascading of vorticity anomalies along a background radial vorticity gradient (Terwey and Montgomery 2008) or coupling with the boundary layer and inducing a positive feedback of enhancing convection (Wu et al. 2012; Huang et al. 2012, 2018; Abarca and Montgomery 2013, 2014; Kepert 2013, 2018; Zhang et al. 2017).

The current study presents a one-way response of a hurricane-like vortex to stratiform rainband heating without the presence of boundary layer and microphysics parameterizations. In addition, the diabatic forcing is prescribed and remains constant throughout the simulation, so the circulation response does not alter the structure of the diabatic forcing. To fully investigate the coupled interactions between the boundary layer and the free troposphere, in the chapters 4 and 5 we will examine at an SEF event in a full-physics WRF simulation of hurricane Matthew (2016).

Specific attentions will be placed on understanding the interplay between the rainband diabatic forcing and the storm-scale wind field, and how stratiform rainband convection is modulated and maintained by the boundary layer thermodynamics to undergo the subsequent axisymmetrization.

44

CHAPTER 3

OVERVIEW OF A FULL-PHYSICS SIMULATION OF HURRICANE MATTHEW (2016)

To understand the roles of rainband processes in SEF of real hurricanes, in the next two chapters we examine an SEF event in a high-resolution full-physics simulation of Hurricane

Matthew (2016) during its time of peak intensity (maximum winds of 67 ms-1 and minimum pressure of 940 hPa) located in the Caribbean Sea. In this chapter, we introduce the model and simulation configurations, and validation for the simulated storm.

3.1 WRF Model simulation of Hurricane Matthew (2016)

The simulation of Matthew (2016) uses WRFv3.5.1 (Skamarock et al. 2008) and consists of four two-way nested domains (D1-D4; the three inner domains D2-D4 are vortex following) with horizontal grid spacings of 27, 9, 3, and 1 km, covering 10200km × 6600km,

2700km×2700km, 900km×900km and 500km×500km, respectively. The model has 43 vertical levels. This domains configuration is identical to that in Zhang and Weng (2015) with the exception of the innermost domain (499x499 grid points), which has been added for the forecast to better resolve the TC inner core. The suite of model physics includes the WSM6 microphysics scheme (Hong et al. 2004), the YSU boundary layer scheme (Noh et al. 2003), the Monin–

Obukhov scheme for the surface layer, and the Grell-Devenyi Ensemble cumulus scheme (Grell

45 and Freitas 2014; only in the outermost D1 domain), which are also identical to that in Zhang and

Weng (2015).

This SEF event of the simulated Matthew occurs near 0000 UTC 3 October. The model simulation is initialized from an EnKF analysis mean at 0000 UTC 2 October 2016, which is 24 hours before the SEF event. The 60-member EnKF cycling data assimilation that produced the

EnKF analysis mean begun at 1800 UTC 25 September 2016, with ensemble members initialized with the NOAA Global Forecast System (GFS) operational analysis and added ensemble perturbations generated using the background error covariance discussed in Barker et al. (2004).

The data assimilation system assimilated the synthetic hurricane position and intensity observations, all available conventional observations, and airborne Doppler radar super observations (Gamache et al. 1995; Weng and Zhang 2012) and dropsonde observations at a 3- hour cycling frequency. This assimilation cycling procedure is identically to previous studies using the Pennsylvania State University WRF EnKF System (e.g., Munsell and Zhang 2014, Weng and

Zhang 2016, Nystrom et al. 2018). To better resolve the observed small eye of Matthew (radius of maximum wind of about 10 km; Stewart 2017), the innermost domain of 1 km horizontal grid spacing is added for the purposes of this study, as in Nystrom and Zhang (2019). The boundary conditions for this simulation are from the GFS forecast initialized from 0000 UTC 2 October.

As shown in Figs. 3.1a-b, the simulated storm matched the real storm well, having a similar track during our period of focus. The simulated storm has a similar minimum sea-level pressure but slightly weaker maximum winds than the real storm. Figures 3.1c and d show the observed microwave brightness temperatures of Matthew before and after the times of our simulated ERC.

While conclusive observations of a secondary eyewall and ERC are lacking, the satellite images show an appreciable increase in the eye diameter from about 9 km near 1900 UTC 2 October, to

46

17 km near 0700 UTC 3 October. As will be demonstrated in Chapter 4, a comparable eye size increase is well-captured in the simulated storm near the same timing period (Fig. 3.1c-d), which undergoes an ERC near 0100 UTC on 3 Oct. Despite any differences with observations, detailed analyses of the simulated storm will be provided in the chapters 4 and 5 to obtain useful insights about its SEF mechanisms.

Fig. 3.1. a) Observed and simulated tracks during the 72-hour simulation period from 0000 UTC 2 October to 0000 UTC 5 October 2016. The observed track comes from the National Hurricane Center best track data. b) Observed and simulated minimum sea-level pressure (MSLP, black) and maximum surface wind (Vmax, red). Observed intensities are dotted solid lines, simulated intensities are dashed lines. c)-d) Observed 89GHz brightness temperatures of the Advanced Microwave Scanning Radiometer 2 (AMSR2) before and after the simulated SEF.

47

CHAPTER 4

ASYMMETRIC WIND ACCELERATION PRIOR TO SECONDARY EYEWALL FORMATION IN A FULL-PHYSICS MODEL SIMULATION

In this chapter, we investigate the asymmetric rainband processes leading into the formation of a secondary eyewall of the Matthew simulation that we discussed in the last chapter.

Prior to the SEF period, the simulated storm develops a shear-aligned stationary rainband complex that contributes dynamically to the subsequent formation of a secondary eyewall. This study will primarily focus on the asymmetric convective and stratiform processes that contribute to the local and axisymmetric tangential wind acceleration of the evolving storm.

4.1 Overview of SEF in the Matthew simulation

Figure 4.1 shows the azimuthally-averaged vertical motion at z=4.84 km and tangential wind at z=2.86 km for the first 36-hour simulation period. After 7 hours of spin-up period, the storm stabilizes at an intensity of ~110 kts, with a radius of maximum wind of ~10 km. The storm maintains its intensity until hour 16, after which a clear axisymmetric projection of the updraft near 60 km radius emerges (Fig. 4.1a). This signal marks the beginning of SEF, and it steadily

48 contracts inward. After hour 20, the inner eyewall weakens substantially and is subsequently replaced by the newer eyewall at hour 25, which continues at a larger radius (~20 km).

Fig. 4.1. a) Hovmöller diagram of the simulated vertical velocity at z = 4.84 km. Vertical velocity of 0.25 푚푠−1 is contoured in black. The blue dotted lines indicate the times of the observed AMSR2 brightness temperature shown in Fig. 1c and d. Hours 17 and 19 are indicated by red dotted lines. b) As in (a), but for tangential velocity at z = 2.86 km. Black dotted lines indicate radius of maximum wind of the primary and secondary eyewalls.

Figures 4.2 and 4.3 show plan views and axisymmetric cross sections at selected hours during this SEF period. Soon after the deep-layer environmental wind shear stabilized at a moderate value of about 6푚푠−1 near hour 15, a stationary band complex (SBC) develops in the downshear quadrants, which exhibits strengthening maxima of reflectivity and vertical velocity

(Figs. 4.2b-c, h-i). During hours 15 to 19, the azimuthal mean projections of the vertical velocity and diabatic heating of the SBC not only display an increase in magnitude and areal coverage, but also a clear inward contraction from about 60 km to 30 km radius, as shown in Fig. 4.3a-c and g-

49 i. Along with this enhancing diabatic heating, both the tangential wind and the mid-level inflow strengthen progressively, as in seen Figs. 4.3c and i. Consistent with previous studies (Zhang et al.

2017; Wang et al. 2019), the diabatic heating and updrafts of the SBC largely maximize at the mid- to-upper level above 4 km during hour 15 to 19. These maxima then descend towards the low- levels and become connected to the boundary layer after hour 21 (Figs. 4.3d-f, j-l). As shown in

(Fig. 4.1a, this inward contraction gives rise to the typical evolution of SEF, demonstrating that the secondary eyewall indeed stems from this intensifying SBC.

Near hour 21, the rainband starts to axisymmetrize when the updrafts continuously propagate and wrap around the storm, (Figs. 4.2d-e j-k). During this axisymmetrization period, the azimuthal mean updraft and diabatic heating interact and seemingly connect with the boundary layer quickly strengthen, eventually forming the secondary eyewall near 20 km radius at hour 23

(Figs. 4.3d-f, j-l).

50

Fig. 4.2. Plan views of (a-f) reflectivity averaged between z = 0.5 and 4km and (g-l) vertical velocity at z = 4km at simulation hours 15, 17, 19, 21, 22 and 23. Green arrows in (a-f) indicate the 850-200mb vertical wind shear vector. Black circles indicate 20, 40, and 60 km radii. All fields in subsequent figures are temporally averaged over the one-hour period of the corresponding simulation hour, unless otherwise specified.

51

Fig. 4.3. (a-f) R-Z cross-sections of azimuthally averaged vertical velocity (shading) and radial velocity (contoured in black at every 2푚푠−1 with 0 line thickened) at selected hours of 15, 17, 19, 21, 22 and 23; (g-l) as in (a-f) but for diabatic heating (shading) and tangential wind (contoured in black at every 5푚푠−1 with 30푚푠−1 line thickened).

4.2. Storm-scale changes in tangential velocity and angular momentum

4.2.1. Tangential velocity analysis

Prior to the simulated SEF, an organized tangential wind broadening is found to be associated with the intensifying asymmetric rainband complex during hours 15-20. Fig. 4.4 shows

52 the azimuthally-averaged cross section and vertically-averaged plan view (between 푧 = 1-5 km) of the storm-following tangential velocity changes at the selected SEF hours. Early at hour 15 when the rainband starts to develop, the azimuthally-averaged tangential velocity acceleration is weak and has no clear organization (Figs. 4.4a,d). The strongest acceleration is located in the downshear-right quadrant near 50-60 km radii, where the rainband updrafts are more convectively active.

After just two hours, at hour 17, the azimuthally-averaged acceleration becomes significantly more intense and broader in scale (Fig. 4.4b). The region of positive acceleration shows an inward descending pattern from mid-to-upper levels (between z=5-10 km) and maximizes between 푟 = 40 and 60 km. Looking at the plan view (Fig. 4.4e), once the rainband intensifies, the acceleration in both the DL and DR quadrants become more organized and project strongly onto the azimuthal mean. Specifically, the acceleration in the DR quadrant is strong but mostly concentrated at 50-60 km radii, while the acceleration in the DL quadrant is similarly strong but noticeably more widespread. Quadrant averages of the acceleration field (not shown) indicate that the acceleration in the DL quadrant extends beyond 120 km radius. This analysis demonstrates that while the acceleration maximum near 50 km radius in Fig. 4.4b receives contribution from both DR and DL quadrants, the broad-scale descending pattern in the mean acceleration largely comes from the DL quadrant.

At hour 19, positive acceleration extends from the DL quadrant towards the upshear quadrants, while acceleration in the DR quadrant weakens. The resulting pattern is a more axisymmetric acceleration band that maximizes near the 30 to 40 km radii and projects strongly onto the azimuthal mean, as shown in Figs. 4.4c, f. This axisymmetric acceleration indicates that the storm wind field is becoming more axisymmetric during hour 19. From the reflectivity (Fig.

53

4.2c) and updrafts (Fig. 4.2i) of hour 19, we see that the upshear quadrants are mostly free of strong convection but is covered with moderate reflectivity about 30-40 dBz. As will be shown in section

4.3, this acceleration pattern also has a descending trend towards its downwind end. The mechanisms of the wind acceleration during hour 17 and 19 will be explored in more details in later sections.

Fig. 4.4. (a-c) R-Z cross-sections of azimuthally averaged tangential wind acceleration at (a) hours 15, 17, and 19. Zero contour is plotted in black dotted line. Vertical black dashed lines indicate 20, 40, and 60 km radii. (d-f) Plan views of hourly storm-relative tangential wind changes averaged between z = 2 and 6 km at (d) hour 15, (e) hour 17 and (f) hour 19. Green arrows indicate the 850-200mb vertical wind shear vector, with black straight lines highlighting individual quadrants. Black circles indicate 20, 40 and 60 km radii.

4.2.2. Angular Momentum Budget Analysis

Given the distinct patterns of positive tangential velocity acceleration at hours 17 and 19, we now focus our attention on these particular hours leading into SEF. As shown in Fig. 4.4e, hour

17 is the earliest hour when systematic storm-scale tangential velocity acceleration first occurs

54 accompanying the intensifying rainband, while at hour 19 the tangential wind field acceleration starts to extend towards the upshear quadrants and becomes more axisymmetric. To quantify the dynamical processes that contribute to the acceleration, the axisymmetric storm-relative angular momentum budget is performed for these two selected hours. The budget is calculated using:

휕 푀̅ 휕푀̅̅̅̅̅ 휕푀̅̅̅̅̅ ̅̅̅̅̅̅휕̅̅푀̅̅′̅̅ ̅̅̅̅̅휕̅푀̅̅̅′̅̅ ̅1̅̅휕푝̅̅̅ 푐 푆푇 = −푢̅̅̅̅̅ 푆푇 − 푤̅ 푆푇 − 푢′ 푆푇 − 푤′ 푆푇 − + 푟퐹̅̅̅̅ (4.1) 휕푡 푆푇 휕푟 휕푧 푆푇 휕푟 휕푧 휌 휕휆 휆

휕 , where 푐 denotes local time derivative that follows the storm center; ̅∙ denotes azimuthal average 휕푡 about storm center; the prime symbol denotes deviation from the azimuthal mean; (푟, 휆, 푧) represents storm following cylindrical coordinate; 푢푆푇 and 푣푆푇 are the storm relative radial and

1 tangential winds; 푀 = 푟푣 + 푓푟2 is the absolute angular momentum of the storm relative 푆푇 푆푇 2 wind; 푤 is vertical motion; 푝 is pressure; 휌 is density; 퐹휆 is friction along the tangential direction.

The terms at the right-hand-side of (4.1) are the mean radial and vertical advections, radial and vertical eddy momentum fluxes, and contributions from the pressure gradient force and the boundary layer friction, respectively.

Fig. 4.5 shows the actual and integrated hourly changes of absolute angular momentum during hour 17, and the corresponding terms at the right-hand side of equation (4.1). The baroclinic pressure gradient term is negligible and is not shown. The diagnosed 푀푆푇 changes (Fig. 4.5b) are forward integrated using 5-minute model output. As shown in Fig. 4.5a, during hour 17 the absolute angular momentum exhibits a broad-scale increase between 40 km to 120 km radii. This tendency displays a descending trend towards inner radii and has a local maximum at the mid- to low-levels near 40 to 60 km radii, which is consistent with the tangential acceleration shown in

Fig. 4.4b. Looking at the integrated M changes shown in Fig. 4.5b, it is clear that the all major tangential acceleration patterns are nicely represented.

55

During hour 17, the mean-flow contribution to 푀푆푇 is largely positive throughout the mid- troposphere (Fig. 4.5f). In contrast, the total eddy contribution (Fig. 4.5i) is largely negative at the same region, resulting in slight cancellation with the mean flow contribution. However, positive eddy contribution is found outside of 70 km radius above 7km altitude, as well as near 40 km radius within the boundary layer. Further breaking down the mean flow advection term, both mean radial (Fig. 4.5d) and vertical advection (Fig. 4.5e) are important to the total mean flow contribution at hour 17. Specifically, the mean radial advection has a broad positive contribution that covers from 50 to 100 km radius between 3 to 5 km altitude. On the other hand, the vertical advection term has positive contribution that concentrates near 50 km radius. Recall from Figure

4.4a that the DR quadrant has a concentrated positive acceleration centered between 50 to 60 km radii, whereas the acceleration in the DL quadrant is similarly strong but covers a much wider radial range. Figs. 4.5d and e suggest that the different acceleration patterns in these two quadrants may be due to two distinct modes of mean flow accelerations. This possibility will be further examined in more detail in section 4.4.

56

Fig. 4.5. R-Z cross-sections of azimuthally averaged storm-relative angular momentum budget at hour 17: (a) actuall changes of storm-relative angular momentum; (b) integrated changes of angular momentum using 5-min model output; (c) contribution from friction; (d) advection by mean radial flow; (e) advection by mean vertical velocity; (f) sum of mean flow advection; (g) radial eddy flux; (h) vertical eddy flux; and (i) sum of eddy contributions. Vertical dashed lines indicate 20, 40 and 60 km radii.

Moving on to hour 19 when the wind field starts to undergo axisymmetrization, Fig. 4.6a shows that the increase in the angular momentum is noticeably more extensive than in hour 17.

The integrated 푀푆푇 changes shown in Fig. 4.6b are again in good agreement with the actual changes. Looking into individual terms, the total mean flow contribution (Fig. 4.6f) is almost uniformly and strongly positive outside r=30 km radius below 7 km altitude, and is the dominant component of the positive total tendency in the mid-troposphere. Between 50 to 80 km radii, this mean flow contribution has a local maximum descending from 6 km altitude towards low levels at inner radii. This pattern largely is reflected in the mean radial advection (Fig. 4.6d), which shares the same descending structure. The mean vertical advection term (Fig. 4.6e) is also positive, but

57 has an overall weaker magnitude near this region. At hour 19, we note that the regions with positive mean radial advection cover a larger radial range compared to that in hour 17, which is due to an increase of the midlevel mean radial inflow (Fig. 4.3c).

Fig. 4.6. Same as Fig. 4.5, but for hour 19.

During hour 19, eddies also have positive contributions outside 70 km radii above 5 km altitude and between 30 to 40 km radii at mid-levels. Specifically, the positive eddy contribution near 30-40 km radii comes largely from the vertical eddy flux term. This region sits at the inner edge of positive mean flow contribution, and is seemingly connected to the boundary layer. While this positive eddy contribution is somewhat weak, it actually is an important contribution to the positive tendency in the total integrated tendency within 40 km radius. Consistent with Wang et al. (2019), this suggests that the boundary layer eddy processes and the associated updrafts are

58 important to the axisymmetrization during hour 19.

4.3. Asymmetric Along-Band Structure

To further understand the three-dimensional structure of the rainband complex during the

SEF period, we examine the variations in kinematic variables along the spiral rainband in a fashion similar to that in Wang et al. (2019). Using cubic splines with natural boundary conditions

(Vetterling et al. 1992), we define spirals using 7 subjectively-determined anchor points that pass through the rainband-induced convergence maxima shown by the vertically-averaged (2-5 km) horizontal divergence fields for hours 17 and 19 (Fig. 4.7). The azimuthal direction of the 3rd anchor point always aligns with the wind shear vector, while the azimuthal coordinates of the remaining six anchor points are defined accordingly with a 30-degree spacing. The resulting spirals span an azimuthal range of 180 degrees divided into six sectors, labeled A-F.

Fig. 4.7. Plan views of vertically averaged horizontal divergence between z = 2 and 5 km for (a) hour 17 and (b) hour 19. Green arrow indicates the 850-200mb shear vector. Red squares are the anchor points, with the third one aligning with the shear vector. Red spiral line is defined by the anchor points using cubic spline. Long solid black lines define the quadrants; short solid black lines define individual sectors labeled A-F. The three circles indicate 20, 40 and 60 km radii.

59

Fig. 4.8. Top row is the along-band cross-sections of hour 17 for (a) vertical motion (shading) and tangential wind acceleration (contoured at every 2ms-1hr-1, negative contours are dashed, smallest positive contour of 1ms-1hr-1 is thickened), and (b) the diabatic heating (shading) and asymmetric -1 tangential wind 푣푎 (contoured at every 0.6ms , negative contours are dashed, zero contour is thickened). (c) and (d) are the same as (a) and (b), but for hour 19. All fields are hourly and radially averaged over a 6 km radial range centered at the spiral shown in Fig. 4.7.

Figures 4.8a-b show the along-band cross-section of the rainband at hour 17. During hour 17, the rainband is steadily intensifying, with growing convective cells in the downshear quadrants. This is reflected in the vertical velocity field (Fig. 4.8a), which has more intense vertical motions in sectors A and B of the DR quadrant. Traveling into the DL quadrant, the updraft magnitudes decrease significantly, and a weak downdraft develops below 2.5 km altitude in sectors D-F. The diabatic heating (Fig. 4.8b) displays the same overall pattern. The magnitudes and pattern of the vertical velocity and heating indicate that the rainband in the DL quadrant is more stratiform in nature, consistent with previous observational (Hence and Houze 2012; Didlake and Houze 2013) and modeling studies (Wang et al. 2019).

The tangential wind acceleration also displays significant structural changes along the rainband. In the DR quadrant (sectors A-B), owing to the more intense vertical motions which

60 advect strong tangential winds to the upper levels, the tangential wind acceleration above z=10 km is strongly positive, as highlighted by the contours in Fig. 4.8a. Traveling downwind into sectors

C-F, the positive acceleration region descends towards 3km altitude. Because of this descending acceleration field, the associated asymmetric tangential wind (푣푎) also shares a similar structure.

Here, the asymmetric tangential wind is computed by first removing the environmental component

(풗푒푛푣) from the storm-relative wind (풗푆푇), and then removing the azimuthal mean component. The environmental wind is computed by solving the following equation (Davis et al. 2008) over the innermost storm-following domain:

∇2휓 = 휁; ∇2휒 = 훿;

휓|휕 = 0; 휒|휕 = 0 (4.2)

풗휓 = 푘̂ × ∇휁; 풗휒 = ∇휒; 풗푒푛푣 = 풗푺푻 − 풗휓 − 풗휒

2 where ∇ is the horizontal Laplacian operator; ∙ |휕 denotes evaluation at the boundary of the 1-km

̂ innermost domain; 휁 = 푘 ∙ ∇ × 풗푺푻 and 훿 = ∇ ∙ 풗푺푻 are the relative vorticity and divergence of the storm relative horizontal wind field 풗푺푻; 휓 and 휒 are streamfunction and velocity potential; 풗휓 =

(푢휓, 푣휓) and 풗휒 = (푢휒, 푣휒) are the nondivergent and irrotational winds. As shown in Fig. 4.8b, the asymmetric storm tangential wind (푣푎) follows the same descending trend as the wind acceleration in the DL quadrant. This, therefore, demonstrates that the storm tangential wind field is top-heavy in the DR, but bottom-heavy in the DL. The descending wind pattern is also found in the rainband complex simulated in a quiescent environment (Wang et al. 2019), indicating that this is common feature with a TC rainband complex.

At hour 19, the rainband has intensified to near its peak intensity. As shown in Figure 4.8c, intense vertical updrafts extend more into the DL and span more uniformly the DR quadrant. The associated diabatic heating structure (Fig. 4.8d) is also significantly stronger than hour 17. In the

61 downwind sectors of the rainband, weakening in both upward motion and diabatic heating is still clear but shifted more towards the downwind end (sector D-F).

Accompanying the structural changes in updrafts and diabatic heating, both the tangential wind acceleration and asymmetric tangential wind 푣푎 in hour 19 show a clear downwind shift. The tangential wind acceleration becomes largely concentrated in the downwind sectors D-F (Fig.

4.8c)), while the asymmetric tangential wind 푣푎 displays a strong maximum that extends into sector E, peaking near 8 km altitude (Fig. 4.8d). Near sector E, the asymmetric 푣푎 quickly descend towards the surface and the storm wind field becomes strongly bottom-heavy at sector F. At hour

19, the wind acceleration is mostly offset from the asymmetric wind, indicating that the asymmetric wind mostly reaches its peak intensity and axisymmetrization process of the wind field is underway. The dynamics of the wind acceleration along the rainband will be explored next.

4.4. Quadrant-Averaged Tangential Wind Budget Analysis

As demonstrated in the along-band analysis in section 4.3, during the intensification period of the quasi-stationary rainband, near and outside of the rainband is widespread tangential wind acceleration that extends from the upper levels of the DR quadrant toward the low levels of the

DL (Fig. 4.8a). This acceleration is accompanied by descending asymmetric tangential wind within the rainband. To further understand the dynamics of this storm-scale wind acceleration, quadrant- averaged tangential wind budgets are performed to quantify the contributions of individual processes. The quadrant-averaged tangential velocity equation (B.8 of appendix) is:

휕 푣̿ 푢̿̿̿̿̿̿휕̿̿푀̿̿̿̿ 푣̿̿̿̿̿̿휕̿푣̿̿̿̿ ̿̿̿휕̿̿푣̿̿̿̿ ̿1̿̿̿휕푝̿̿̿ 휕 푣̿ 푐 푆푇 = − 푆푇 푆푇 − 푆푇 푆푇 − 푤 푆푇 − + 퐹̿ − 푓̿̿̿푢̿̿ − 푐 푐 (4.3) 휕푡 푟 휕푟 푟 휕휆 휕푧 푟휌 휕휆 휆 푐 휕푡

, where ̿∙ denotes quadrant averaging; 풗푐 = (푢푐, 푣푐) is the vector of storm translation. All other variables bear the same physical meaning as in equation (4.1). Note that our goal here is to

62 investigate the quadrant-averaged acceleration from a dynamical perspective instead of looking at wave-mean-flow interaction within the quadrant, therefore in equation (4.3) we do not separate the nonlinear terms into mean and eddy contributions. We focus on selected quadrants at specific hours to highlight details of the ongoing dynamical processes.

4.4.1. Downshear-right quadrant at hour 17

We first examine the acceleration in the DR at hour 17. Figure 4.9 shows the corresponding quadrant-averaged tangential wind budget. The last two terms in equation (4.3) are both small and are therefore not shown. Figures 4.9b-c show a comparison between the actual and integrated tangential wind changes during this hour. Similar to results presented in section 4.2.2, the integrated tendency captures most of the acceleration pattern outside r=20 km.

Figure 4.9b shows that the DR quadrant in the low-to-midlevels experiences significant positive acceleration between 40 km to 60 km radii. The pattern of the strongest acceleration is rather upright and radially confined. The major contribution of the tendency pattern comes from vertical advection (Fig. 4.9i) associated with the strong diabatic heating and convective updrafts located in this region (Fig. 4.2h). In addition to vertical advection, the horizontal advection is also positive below 3km altitude (Fig. 4.9f). This positive tendency arises mostly from radial advection, which is due to low-level inflow at these radii (Fig. 4.9d, f). But this low-level horizontal advection pattern is slightly offset by the pressure gradient acceleration (Fig. 4.9g). Overall, vertical advection most dominantly shapes the total tangential acceleration in DR quadrant low-to- midlevels. This pattern significantly projects onto the axisymmetric acceleration, as suggested by the mean vertical advection of angular momentum (Fig. 4.5e).

63

Fig. 4.9. a) Plan view of tangential wind acceleration vertically averaged between z = 2 and 6km at hour 17. Green arrow indicates the 850-200mb wind shear vector. Shear-relative quadrants are indicated by solid straight lines. Blue contour highlights the quadrant of interest. (b-i) The tangential wind budget averaged over the downshear-right (DR) quadrant: b) The actual hourly changes in tangential wind; c) the integrated hourly changes of tangential wind; d) radial advection; e) azimuthal advection; f) total horizontal advection; g) the contribution from pressure gradient force; h) the contribution from boundary layer friction; and i) the contribution from vertical advection. All vertical black dashed lines indicate 40 and 80 km radii.

4.4.2. Downshear-left quadrant at hour 17

We next examine the acceleration in the DL quadrant and compare with the concurrent DR fields at hour 17. As shown in Fig. 4.10a, during this 1-hour period, the DL quadrant experiences

64 an organized acceleration that covers from 40 km to near 120 km radii.

Fig. 4.10. Same as Fig. 4.9, but for downshear-left (DL) quadrant at hour 17.

Fig. 4.10b and 4.10c show that the widespread acceleration in DL has a clear slantwise pattern that descends from upper levels at outer radii towards low levels at inner radii, capturing the general pattern in Fig. 4.5a. Among the individual terms, the total horizontal advection (Fig.

4.10f) is the dominant contributor of this descending pattern, which is an outward sloping band of acceleration covering 40 to 80 km radii. Both radial and azimuthal advection contribute positively to this acceleration band. Specifically, near 60 to 80 km radii, azimuthal advection is positive

65 between 4-11 km altitudes. This positive azimuthal advection is in good agreement with the top- heavy wind structure (positive 푣푎 above 5 km altitude, as in Fig. 4.8b) in the DR quadrant, which implies that high angular momentum air is transported from the upwind (DR) quadrant into the

DL quadrant. Near the bottom and underneath the positive azimuthal advection, radial advection is positive, which indicates that angular momentum is drawn inward by the radial inflow below 4 km altitude near 40 to 70 km radii, as shown in Fig. 4.10d.

Fig. 4.11. Quadrant and hourly averaged radial velocity and its flow components: a) Storm-relative radial velocity 푢푆푇 (shading) and vertical velocity 푤 (contoured at {0, ±1, ±2, ±4, … , ±64} × −1 −1 0.08 ms with 0ms thickened) in the DR quadrant at hour 17; b) irrotational component 푢휒 (shading) and diabatic forcing (contoured at every {0, ±1, ±2, ±4, … , ±64} × 2.5 × 10−4 Ks−1 with −1 0 Ks thickened.); c) nondivergent component 푢휓; d) environmental component 푢푒푛푣. (e-h) As in (a- d) but for DL quadrant at hour 17. (i-l) As in (a-d) but for UL quadrant at hour 19.

To further understand the origin of this mid-level radial inflow, we decompose the radial velocity into irrotational (푢휒), nondivergent (푢휓) and environmental (푢푒푛푣) components using

66 equation (2), as shown in Figs. 4.11e-h. As expected, the environmental 푢푒푛푣 (Fig. 4.11h) is largely uniform in the horizontal direction, with structure consistent with the direction of the environmental wind shear. As shown in Fig. 4.11e and f, the mid-level inflow flow (near z=4km) between 40 and 70 km radii receives a large contribution from the irrotational radial inflow (Fig.

4.11f). This mid-level irrotational inflow begins radially outward of the stratiform diabatic forcing at mid-levels (contours in Fig. 4.11f), with its inner edge collocated with cooling and downdraft

(dashed contour in Fig. 4.11e), and thus resembles the mesoscale descending inflow (MDI, Didlake and Houze 2013). In contrast, the low-level irrotational inflow in DR (Fig. 4.11b) resides outward of convective heating patterns. Thus, we see clearly that the irrotational inflows in these two quadrants are driven by convergence associated with distinct diabatic heating/cooling structures.

Looking back at Figs. 4.10d and 4.10f, the stratiform-induced MDI also appears associated with the inward radial advection and the outward sloping band of positive horizontal acceleration in the same region.

The vertical advection in the DL quadrant (Fig. 4.10i) also contributes positively to the tangential wind tendency due to the corresponding deep updrafts seen in Fig. 4.11e. These updrafts all populate the inner edges of the rainband and are located immediately inward of the aforementioned descending inflow near 40 to 60 km radii (Fig. 4.10f). However, at this early phase of SEF, a large portion of these updrafts undergo outward acceleration once they leave the boundary layer, thus resulting in a near cancellation with the total horizontal advection due to angular momentum conservation. The net effect of this cancellation, together with the contribution from pressure gradient force term, is a relatively weak positive contribution at the inner edge of the aforementioned acceleration band, (Fig. 4.10c, f). Overall, the findings in sections 4.4.1 and

4.4.2 suggest that the distinct acceleration patterns in DR and DL in hour 17 are indeed due to the

67 two different acceleration processes.

4.4.3. Upshear-left quadrant at hour 19

As discussed in section 4, hour 19 is the earliest hour when the main region of tangential wind acceleration extends from the downshear quadrants towards the upshear quadrants (Fig. 4.4c).

It, therefore, marks the beginning of the axisymmetrization of the wind acceleration of the secondary eyewall. Figure 4.12 shows the tangential wind budget during this hour in the UL quadrant. As can be seen in Figs. 4.12b-c, near 30 km radius, there is a band of wind acceleration that extends vertically from the boundary layer to 10 km altitude. At outer radii, a broad but weaker positive acceleration covers outside of 60 km radius.

Vertical advection is an important contributor to the total acceleration pattern near 30 km.

Here, organized updrafts bring high tangential momentum air from the boundary layer to the free troposphere, resulting in a clear peak of tangential acceleration outside of the primary eyewall.

Another important contribution of the acceleration near 30 km radius is the pressure gradient force term, which accelerates the tangential wind below 5 km. This positive pressure gradient force is due to the buildup of asymmetric high pressure in the DL quadrant (not shown) owing to hydrostatic adjustment to stratiform cooling.

Horizontal advection (Fig. 4.12f) also has important contributions to the total tendency in this quadrant. Above 3 km and near 30 km radius, the horizontal advection term shows a positive contribution, which is also a major contribution to the acceleration of the incipient secondary eyewall. Outside of 40 km radius, positive horizontal acceleration extends beyond 100 km radius.

This storm-scale acceleration has a descending pattern towards inner radii and seemingly connects to the boundary layer at the outer edge of the updraft region of the developing secondary eyewall.

While this acceleration is partially offset by the vertical advection term, its general pattern remains

68 in the total tangential wind tendency.

Fig. 4.12. Same as Fig. 4.9, but for upshear-left (UL) quadrant at hour 19.

4.4.4. Decomposing the horizontal advection of tangential wind

In the previous quadrant analyses, horizontal advection of tangential wind has a major contribution in the total tangential wind acceleration in each quadrant. However, when looking into the radial and azimuthal advections in the UL quadrant (Fig. 4.12d and 4.12e), we found that these two terms have strong cancellation in various regions. Such cancellation is due to the wind field asymmetry, which results in projection of M-conserving flow onto both the radial and

69 tangential directions. As confirmed in Figs. 4.11i-l, the radial flow in the UL quadrant has an upper-level inflow, lower-level outflow structure that largely comes from the nondivergent (푢휓) and environmental (푢푒푛푣) components. These two components obscure the mid-level irrotational inflow (푢휒) that is collocated with both the stratiform cooling (contours in Fig. 4.11j) and the positive horizontal advection (Fig. 4.12f). Therefore, examining the irrotational and non-divergent winds, rather than radial and tangential components, may further help inform our current analysis on the horizontal advection of the tangential wind and link these patterns to convection processes.

Furthermore, it was also shown in Figs. 4.5d and 4.6d that the mean radial advection (i.e.,

휕̅푀̅̅̅̅̅ −푢̅̅̅̅̅ 푆푇) has an important contribution to the azimuthal mean tangential wind acceleration. 푆푇 휕푟

Using Green’s theorem, we know that 푢̅푆푇 = 푢̅휒 since both 푢휓 and 푢푒푛푣 are divergence-free:

1 1 1 (4.4) 푢̅ = ∮ 풗 ∙ 푟̂ 푟푑휆 = ∬ ∇ ∙ 풗 푑퐴 = ∬ ∇ ∙ 풗 푑퐴 = 푢̅ 푆푇 2휋푟 푺푻 2휋푟 푺푻 2휋푟 휒 휒

휕̅푀̅̅̅̅̅ This means that the mean radial advection −푢̅̅̅̅̅ 푆푇 only receives contribution from the 푆푇 휕푟 irrotational radial velocity 푢̅휒 . Therefore, it is useful to further quantify the contribution of irrotational wind to the horizontal advection in different quadrants.

We decompose the horizontal advection into the advection by irrotational and nondivergent flows:

풗푆푇 Horizontal Advection = − ∙ 훻푀 푟 푆푇 (4.5) 풗휒 풗휓′ = − ∙ 훻푀 − ∙ 훻푀 푟 푆푇 푟 푆푇

Here, we have grouped the nondivergent 풗휓 and environmental 풗푒푛푣 components into 풗휓′, which is still divergence-free. Figure 4.13 shows the decomposition of the horizontal advections of the three selected quadrants/times. Figs. 4.13c, g, k show that the advection by the nondivergent wind

70

풗휓′ is significant in the quadrant averages. This tendency can only contribute in an axisymmetric sense to the radial eddy fluxes (Fig. 4.5g and 4.6g), which appear to play a role in the total angular momentum tendency at the top of the boundary layer. Consistent with Figs. 4.11b, f, which demonstrated the distinct driving forces of the irrotational flows in DR and DL, here in Fig. 4.13b and f we can further see how the difference in their driving forces result in the distinct patterns in their contribution to the axisymmetric wind acceleration. The irrotational flow in the DR is driven by convergence (not shown) underneath convective heating near 40 to 60 km radii, thus its contribution resides mostly in the lower troposphere and near surface (Fig. 4.13b). On the other hand, the irrotational flow in the DL (Fig. 4.11f) is driven by the mid-level convergence induced by the stratiform diabatic forcing. As this irrotational inflow passes through stratiform cooling near

4km altitude, it becomes negatively buoyant (Yu and Didlake 2019) and forms an MDI, which draws angular momentum inward (Fig. 4.13f) and results in the descending trend in the acceleration pattern (Fig. 4.13h). At the beginning of axisymmetrization at hour 19, the stratiform cooling in UL (Fig. 4.11j) is more pronounced, and so is the MDI. The MDI seemingly connects to the boundary layer at its inner edge and has important contribution to the horizontal advection

(Fig. 4.13l). This analysis demonstrates that the irrotational flow, specifically the MDI induced by the stratiform diabatic forcings, indeed plays an important role in shaping and initiating the descending storm-scale wind acceleration in the left-of-shear half of the storm and contributing to the axisymmetric wind acceleration.

71

Fig. 4.13. The decomposition of horizontal advection: (a) Plan view of vertically averaged (z = 2-6km) tangential wind acceleration in DL quadrant at hour 17. Green arrow indicates shear vector (850mb- 200mb) and blue solid contour indicates the quadrant of interest. (b) Quadrant-averaged advection by the irrotational wind, (c) quadrant-averaged advection by nondivergent wind, and (d) quadrant-averaged total horizontal advection in DL at hour 17. (e-h) As in (a-d) but for DL quadrant at hour 17. (i-l) As in (a-d) but for UL quadrant at hour 19.

4.5. Evolution of MDI

Given the importance of the MDI shown at hours 17 and 19, we next examine the prominence and evolution of this pertinent feature. Based on Fig. 4.11, we note that this MDI feature is best identified with the negative irrotational radial velocity coinciding with sinking motion. Therefore, we examine the joint frequency distributions of 푢휒 and 푤 in each quadrant from hours 16 to 20 (Figs. 4.14a-d). A 30-degree azimuthal averaging is first applied to both 푢휒

72 and 푤 fields to filter out small-scale waves and to extract the mesoscale signals. Then, all 푢휒 and

푤 values between 40-80 km radius and 2-6 km altitude in each quadrant at each hour are collected.

Among the quadrants, the distributions in the DR are most widespread and largely have positive 푤 and negative 푢휒 due to the vigorous convective rainband activity. Downwind in the DL quadrant, occurrences of negative w and negative 푢휒 have a more significant contribution to the distributions at all times, suggesting the existence of an MDI. The UL quadrant also has more distribution centered in the diagram space of negative 푤 and 푢휒.

To further demonstrate the spatial and temporal evolution of MDI, we focus on the lower- half plane of the 푢휒-푤 phase diagram and show the azimuth-time evolution of the 푢휒 (shading) and negative 푤 (contour) in Figure 4.14e. A downdraft threshold of −0.075푚푠−1 on the 30-

−1 minute averaged 푤 field (i.e., 〈푤〉30푚𝑖푛 < −0.075푚푠 ) is used to determine the existence of

MDI. A volume-weighted average is then performed on the 푢휒 and 푤 fields where the downdraft criterion is satisfied. It is clear that the majority of downdrafts (red contours) correspond well with negative 푢휒, indicating existence of an MDI. A clear MDI feature first emerges at the DL quadrant at hour 17, which also coincides (both temporally and spatially) with the emergence of broad wind acceleration shown in Fig. 4.4e. This MDI feature then propagates downwind and remains persistent in the UL quadrant during hour 18 and 19, and later extends to the UR at hour 20. This progression is also consistent with the lags in the downwind shift of the 50 percentile contours towards the negative 푤 and 푢휒 space between UL and DL (Figs. 4.14a-b).

73

Fig. 4.14. Joint frequency distributions of 푢휒 and 푤 from hour 16 to 20 for (a) UL, (b) DL, (c) UR and (d) DR. Only 50th percentile contour lines are shown. e) The azimuthal and time evolution of 푢휒 (shading) and 푤 (contoured in red at every −0.3푚푠−1 from −0.3푚푠−1 to −1.5푚푠−1) for pixels that −1 satisfy the criterion of 〈푤〉30푚𝑖푛 < −0.075푚푠 .

4.6. Conclusions

In this study, we investigated the detailed dynamics of the evolving tangential wind field prior to and during a secondary eyewall formation (SEF) event in a convection-permitting model simulation of Hurricane Matthew (2016). We focus primarily on the role that asymmetric rainband processes have on the local and axisymmetric tangential wind acceleration. Embedded in moderate environmental wind shear, the simulated storm develops a rainband complex that remains largely stationary in the downshear quadrants prior to SEF. As this rainband complex intensifies, the TC wind field experiences a broadening of the axisymmetric tangential wind field associated with the rainband complex. Soon after, a low-level axisymmetric wind maximum develops within the

74 incipient secondary eyewall. We highlight specific times of the simulation that correspond to the intensification period of this rainband complex (hour 17) and also the initial axisymmetrization stage that leads to SEF (hour 19).

At hour 17, the angular momentum axisymmetric tendency field has an inward-descending positive tendency pattern that extends from 120 km to 50 km radii and maximizes at low levels.

Mean vertical advection contributes to a more concentrated positive tendency region near 50 km radius, while mean radial advection is more widespread and captures the overall descending pattern, indicating the possibility of having distinct modes of tangential acceleration.

A close examination of the tangential acceleration field reveals distinct characteristics in different quadrants of the storm. In the downshear-right (DR) quadrant where the rainband complex is more convective, the acceleration is more upright and concentrates within a band of 20 km wide centered at 50 km radius. A quadrant-averaged tangential wind budget over the DR confirms that vertical advection is the dominant contributor here, resulting in a local acceleration pattern that resembles the mean vertical advection in the aforementioned angular momentum budget.

In the downshear-left (DL) quadrant, the acceleration is similarly intense but more widespread, extending from 40-120 km radius. Here, horizontal advection plays a more important role in shaping the total acceleration structure than vertical advection. This horizontal advection is a slantwise band of positive acceleration, with azimuthal import of high angular momentum air at the upper levels and inward advection of angular momentum by a persistent mid-level inflow underneath. This mid-level inflow is collocated with diabatic cooling and downward motion, resembling the mesoscale descending inflow (MDI) that has been examined in observational studies (Didlake and Houze 2013; Didlake et al. 2018). This emergence of the MDI in the DL

75 quadrant is also the primary reason of the inward descending pattern in the axisymmetric acceleration outside of 40 km radius. By decomposing the horizontal flow into irrotational and nondivergent components, we show that this MDI is best identified with the irrotational velocity as irrotational inflow coinciding with sinking motion. This feature, closely associated with diabatic cooling in stratiform precipitation, suddenly emerges at hour 17 in the DL quadrant, consistent with the timing and location of the first appearance of the widespread tangential acceleration there.

By tracking the evolution of downdrafts, we show that throughout the SEF period, the MDI strengthens and travels downwind over time. As the MDI traverses into the upshear-left (UL) quadrant at hour 19, it becomes more widespread, and so as the associated acceleration at large radii. We note that the UL is also the quadrant where the MDI seemingly interacts with the boundary layer. At the inner edge of the descending inflow, new convective updrafts are triggered, which are found to be important to the axisymmetric tangential acceleration at low levels and the following axisymmetrization of the secondary eyewall. This stratiform-to-convective transition is also similar to the observed pattern in Didlake et al. (2018), and that the acceleration mechanism and the MDI structure agree well the findings of idealized experiment presented in chapter 2.

While the results presented in this chapter provide important insight about how the asymmetric rainband processes shape the three-dimensional structure in the storm wind and acceleration field, several important aspects of the SEF processes have not been fully examined.

These include the process that modulate the stratiform-to-convective transition at the downwind end of the spiral rainband, as well as the coupling between the tropospheric and the boundary layer processes. To answer these questions, in the next chapter, we will closely examine the thermodynamic structure at the TC boundary layer, with a specific focus on how convections can be maintained before they axisymmetrize into an outer eyewall.

76

CHAPTER 5

THE ROLE OF BOUNDARY LAYER THERMODYNAMIC ASYMMETRY IN UPDRAFT

MAINTENANCE

In chapter 4, we have examined the detailed dynamics of the asymmetric rainband processes that lead to the storm-scale wind field acceleration occurred prior to the SEF of the simulated storm. As we mentioned in Fig. 4.6, just before the axisymmetrization of the secondary eyewall occurs, vertical eddy fluxes associated with asymmetric updrafts contribute significantly to the spin-up of the low-level tangential wind near the secondary eyewall radius. Given the importance of these eddy processes, in this chapter we focus our attention on these asymmetric updrafts and examine the mechanism that leads to their occurrence and maintenance.

5.1. The equivalent potential temperature 휽푬 formulation and budget equation

In this chapter, the budget for equivalent potential temperature 휃퐸 is examined to understand the processes involved in the sustained release of diabatic heating in the organized updrafts that lead to the subsequent axisymmetrization of the secondary eyewall. Similar to chapter

4, the 휃퐸 budget integration is performed in a storm-relative framework, but uses WRF output with

77 higher temporal resolution at every 1 minute. The equivalent potential temperature 휃퐸 is an integrated variable that composes of both entropy and latent energy, and is conserved in saturated pseudo-adiabatic processes that involves only the vapor and liquid phases of water (Bolton 1980;

Holton 1972). The derivation of the conservation relation in such processes can be found in Bolton

(1980) and Holton (1972). Based on the suggestion of Bolton (1980), the following approximated form of 휃퐸 (his equation 38) is used, which has a maximum error of about 0.05 K against the iterative solution under various atmospheric conditions:

3.376 −3 휃퐸 = 휃 exp [( − 0.00254) × 푞(1 + 0.81 × 10 푞)] (5.1) 푇퐿 where 휃 is dry potential temperature, 푞 is vapor mixing ratio, 푇퐿 is the temperature of the air parcel at the lifted condensation level. The value of the above 휃퐸 formulation, its local tendency and advections are tracked at every time step and output from the WRF model at every minute

퐷휃 휕 휃 휕휃 퐸 = 푐 퐸 + 풗 ∙ 훁휃 + 푤 퐸 = 퐹 (5.2) 퐷푡 휕푡 푺푻 퐸 휕푧 휃퐸

퐷 휕 휕 where is Lagrangian time derivative; 푐 = + 풗 ∙ 훁 is the Eulerian time derivative in storm- 퐷푡 휕푡 휕푡 풄 relative coordinate; 풗푺푻 = 풗 − 풗풄 is the storm-relative horizontal wind, with 풗 and 풗풄 being the

full horizontal wind and storm translation vector, respectively; 퐹휃퐸 is the forcing of 휃퐸, which is calculated as residual. Here, we would like to emphasize that the WRF model is not designed to precisely conserve the above formulation of 휃퐸, nor the real atmosphere (since phase change in atmosphere is not actually pseudo-adiabatic). Nevertheless, the evolution of 휃퐸 contains important information about how energy is converted between the forms of latent energy and entropy, and therefore its budget can provide important insight about the concurrent dynamics of the moist processes. While the conservation property of the 휃퐸 formulation in equation 5.1 is good when the required assumptions are satisfied, the expression of the external forcing for this formulation is

78 less straightforward to derive. On the other hand, Rotunno and Emanuel (1987) also illustrated the derivation of the conventional, but less accurate, form of equivalent potential temperature (here denoted as 휃̂퐸), which takes the following form

휃̂퐸 = 휃 exp[퐿푣푞/푐푝푇] (5.3)

where 퐿푣 is the latent heat of vaporization; 푐푝 is specific heat of dry air. This formulation has a simple forcing of the following form

퐷휃̂퐸 휃̂퐸 휃̂퐸퐿푣 = (휃̇푚푝 + 휃̇푝푏푙 + 휃̇푟푎푑) + (푞̇푚푝 + 푞̇푝푏푙) (5.4) 퐷푡 휃 푐푝푇푘 where 휃푚푝̇ , 휃̇푝푏푙 and 휃̇푟푎푑 are diabatic heating from microphysics, sensible heat flux from boundary layer and radiative forcing, respectively; 푞̇푚푝 and 푞̇푝푏푙 are sources and sinks of water vapor due to microphysics scheme and water vapor fluxes from boundary layer parameterization.

The derivation of the forcing terms of the above 휃̂퐸 equation, together with a verification of the residual forcing for 휃퐸 (Bolton’s formulation), are provided in Appendix C.

5.2. Boundary layer thermodynamic asymmetry

Fig. 5.1a-b show the potential temperature 휃 at the TC boundary layer at 푧 = 300m and the diabatic forcing at 4.3km at the inner most domain (spatial coverage of 499km x 499km) at hour 20 of the model simulation. From Fig. 5.1a we see a wide-spread low 휃 anomaly covering the north half of the storm in a spiral pattern. Meanwhile, from the diabatic heating near the melting level at 4.3 km altitude (Fig. 5.1b), we see the overall structure of the spiral rainband. The spiral rainband is populated with intense heating at its inner edge. Outside of the heating, we see a wide- spread mid-level cooling that covers about 120 km in radial extent. This mid-tropospheric diabatic

79 cooling of the rainband takes the same spiral structure as the cold anomaly of the low-level 휃, as shown by the region enclosed by the two spirals in Fig. 5.1a and b, demonstrating that cold anomaly in low-level 휃 is a storm-scale feature associated with the rainband, but not an environmental feature.

Fig. 5.1 (a-b) Plan views of instantaneous fields for (a) 휃 at z = 300m and (b) diabatic heating at 4.3 km over the entire innermost domain (499×499 grid points) at hour 20. The two spirals highlight the region of the cold anomaly and region with mid-level diabatic cooling. (c-h) Plan views of hourly averaged 휃 (shading) and positive vertical velocity (contoured in black at every 0.1 푚푠−1) at z = 300m at selected hours: (c) hour 15; (d) hour 17; (e) hour 19; (f) hour 20; (g) hour 22 and (h) hour 23. The hourly averages are computed using 5-min WRF output. The three circles indicate radii of 20, 40 and 60 km. Shear vector in each hour is shown by green arrow. 80

Figure 5.1c-f show the hourly averaged 휃 field at selected hours over a smaller radial range.

We first note that the cold anomaly covers the north half of the storm and remains largely stationary throughout the SEF period. Overlaying on top of the low-level 휃 field is the hourly averaged vertical velocity (black contours) at the same level of z=300 m. Averaging over one-hour period using 5-min model outputs, these time averaged updrafts take the form of organized bands that collocate with regions of strong thermodynamic gradient. Unlike the transient updrafts at each instantaneous frame (not shown), these localized updraft bands are organized features that indicate regions supportive of persistent convective activities and continuous diabatic heating release. One of these zones locates at the downshear-right quadrant of the storm and remains quasi-stationary throughout hour 15 to hour 20 with a spiral pattern that separating the warm air at inner radii from the colder air at its immediate downwind outer edge. Another branch of organized updrafts develops later at the north part of the storm. This north branch first only appears scattered updrafts at hour 15 and 17. At hour 19, the cold air masses from the North of the storm has strengthened and pushed further inward, resulting in a noticeably stronger thermodynamic gradient at the left- of-shear quadrant, as well as a localized band of updraft at the Northern-side of the storm (Fig.

5.1e). At hour 20, this branch of updrafts becomes more organized and intense, and appears to be separated from the spiral band at the downshear-right quadrant. About two hours after the emergence of this branch of updrafts, the convections axisymmetrized into a secondary eyewall.

Throughout all the hours shown in Fig. 5.1, the low-level organized updraft remains collocated with regions of strong thermodynamic gradient, indicating certain dynamical relationship between them. This implied relationship also can be shown with equivalent potential temperature (휃퐸) field, as highlighted in Fig. 5.2 for selected hours prior to SEF. In addition to the close association between 휃퐸 gradient and updrafts, the low-level 휃퐸 (vertical averaged between

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200 to 850m) has a strong asymmetric structure of a “comma” shape. This asymmetric structure is frequently seen in sheared tropical cyclone, and has been referred as the “moisture envelope” in

Riemer and Montgomery (2011), who also demonstrated that this asymmetric 휃퐸 structure is the largely the consequence of the wavenumber-one forcing imposed by a steady, environmental wind shear. Riemer (2016) later also explored the dynamical relation between the moist envelope and the quasi-stationary rainband, and proposed an updraft mechanism based on frictional convergence driven by vorticity forcing associated with a tilted vortex in a sheared environment. However, different from his updraft hypothesis, which focuses on the dynamical driving force of updraft, our analysis in this chapter aims to explore how the observed thermodynamic asymmetry contribute to an environment supportive of continuous convective activity. Specifically, we will focus on the intense updraft region at the north/northwest side of the storm (enclosed by black straight lines in

Fig. 5.2). This intense updraft region first emerges at hour 19 (Fig. 5.2a), continues to intensify in subsequent hour 20 (Fig. 5.2b) and subsequently leads to the axisymmetrization of the secondary eyewall. These updrafts, in fact, are also updrafts that responsible for the strengthening vertical eddy momentum flux discussed in previous chapter 4, as shown in Fig. 4.6.

(a) : Hr 19 (b) : Hr 20 (c) : Hr 22

Fig. 5.2 Plan views of hourly averaged 휃퐸 (shading) for (a) hour 19; (b) hour 20 and (c) hour 22. The 휃퐸 is also vertical averaged between z = 200 and 850m. Positive vertical velocity at z=1.5km is contoured in red at every 0.4 푚푠−1. Sectors enclosed by the two straight black lines are regions of focus in the subsequent analyses.

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5.3. Results

휕휃 5.3.1 The budgets for 휃 and 퐸 at the updraft regions 퐸 휕푧

In order to understand the role of thermodynamic gradient in sustaining these persistent updrafts, we first look at the 휃퐸 structure and how it evolves within the intense updraft region.

Figure 5.3a shows the radial-height cross-section of the 휃퐸 and secondary circulation, quadrant averaged over the 90-degree sector at hour 20, as highlighted by the black lines in Fig. 5.2b.

Consistent with previous studies (Houze 2010), the first thing we note is that below 4 km 휃퐸 is largely decreasing with height, indicating that the TC outer core environment is moist absolutely unstable (i.e., unstable to vertical perturbation for saturated air parcel). As expected, at the region of strong diabatic heating release (magenta contours), upward motion is strong and the secondary circulation has clear cross 휃퐸 contour component. As will be demonstrated in Appendix C, since

휃퐸 is largely conserved at the region of diabatic heating release, air flow with a strong cross-휃퐸 component from the warm to low 휃퐸 region indicates that the air parcels would become more positively buoyant as they ascend. Therefore, we see that the intense diabatic heating release and updrafts are due to the release of moist absolute instability.

Figure 5.3b-e show the terms of 휃퐸 budget within the same quadrant. Integrated over one hour using 1-min outputs from the WRF model, we see that the local 휃퐸 tendency (in storm following coordinate) is uniformly small compared to the other terms, indicating the 휃퐸 structure is only slowly varying within an hour with no substantial change in vertical structure. Below 0.5 km within the boundary layer, the dominant process is the enthalpy exchange between warm ocean surface and the low-휃퐸 air aloft, which is represented by a strong balance between positive tendency associated with surface enthalpy fluxes and negative tendency of the horizontal advection

83 of low-휃퐸 air from the environment. Within the updraft region, vertical advection of 휃퐸 by the updraft brings high 휃퐸 air from low to upper levels. This tendency is largely offset by horizontal advection, resulting in only small changes in 휃퐸 within the quadrant over the one-hour period. This relatively small in the 휃퐸 structure also implies that the moist unstable environment is largely maintained throughout this hour, even under such strong instability release within the quadrant.

−3 −1 Fig. 5.3 a) Cross-section of 휃퐸 (shading), diabatic heating (contoured in magenta at every 10 퐾푠 from 2 × 10−3 to 5 × 10−2 퐾푠−1) and secondary circulation (vectors) for hour 20. The averaging sector is indicated in Fig. 5.2. (b-e) Cross-section of the different terms of the 휃퐸 equation: (b) local 휕 휃 휕휃 tendency 푐 퐸 ; (c) horizontal advection −풗 ∙ 훁휃 ; (d) vertical advection −푤 퐸 ; (e) 휃 forcing 휕푡 푺푻 퐸 휕푧 퐸 (calculated as residual). Positive vertical velocity is contoured at 0.2 푚푠−1 from 0.4 to 10 푚푠−1.

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To further understand how is the instability maintained within the intense updraft region, we take vertical derivative of equation 5.2, which becomes a budget equation of the moist

휕휃 instability parameter 퐸: 휕푧

휕 휕휃 휕 휕 휕휃 휕퐹휃 푐 ( 퐸) = − (풗 ∙ 훁휃 ) − (푤 퐸) + 퐸 (5.3) 휕푡 휕푧 휕푧 푺푻 퐸 휕푧 휕푧 휕푧

휕휃 Figure 5.4 shows the budget terms of the 퐸 at hour 19, 20 and 22 over the same sectors 휕푧

휕휃 highlighted in Fig. 5.2. Consistent with the 휃 budget, the local time derivative of 퐸 is small 퐸 휕푧 within the updraft region, demonstrating that the instability within the atmospheric column remains slowly varying at these updraft regions. Below 1km altitude, the balance is between horizontal advection, boundary layer fluxes and vertical advection. Near the surface, the terms representing horizontal advection and boundary layer fluxes are strongly opposing each other. Boundary layer enthalpy fluxes from the warm moist ocean surface continuously destabilizing the surface layer aloft, while this enthalpy is immediately carried away by the horizontal advection, thus stabilizing the boundary layer air at the vicinity.

Focusing on the intense updraft above z = 1km, we see that the dominant balance is between vertical gradients of the vertical and horizontal advections, consistent with the previous

휃퐸 budget. The vertical advection of 휃퐸 has a stabilization effect by continuously bringing the high

휃퐸 air from low level upward, which tends to reduce the instability at the vicinity by making local

휕휃 퐸 less negative. Such effect is particularly strong at the strong updraft region between 1 to 2 km 휕푧 altitude (thick magenta contours). Simultaneously, this stabilization effect of updraft is balanced

휕 by the differential horizontal advection (e.g., (ℎ표푟𝑖. 퐴푑푣) ; Fig. 5.4g-i), which has a 휕푧 destabilization effect at these strong updraft regions. As will be demonstrated in next section, this

85 differential horizontal advection results in differential warming that decreases with height, thus tends to strengthen the negative slope of 휃퐸 and destabilizes the atmosphere. As shown in Fig.

5.4g-i, above 푧 = 1km this destabilization effect is the only term that can balance the stabilization effect inherent to the vertical advection of the intense updrafts, and is therefore essential to sustain the continuous release of diabatic heating by maintaining the instability at the vicinity. This is confirmed from Fig. 5.4g-i that the destabilization effect of the horizontal advection indeed persists at the intense updraft regions throughout the hours prior to the axisymmetrization of the secondary eyewall.

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Fig. 5.4 Cross-sections of the different terms of the budget equation for 휕휃퐸/휕푧: (a-c) local tendency of 휕휃퐸/휕푧 ; (d-f) vertical gradient of horizontal contribution; (g-i) vertical gradient of vertical contribution; and (j-l) vertical gradient of 휃퐸 forcing. Vertical velocity is contoured in magenta at every 0.2 푚푠−1 from 0.4 to 10 푚푠−1. Updraft stronger greater than 0.6 푚푠−1 is thickened. The left column is for hour 19; middle column for hour 20; and right column for hour 22. Averaging region for each hour is shown in Fig. 5.2.

5.3.2 Structure leading to the destabilization effect of horizontal advection

Given the importance destabilization effect, we focus next on hour 20 and investigate the component of the horizontal advection responsible for destabilization effect. Figure 5.5 shows the

87 decomposition of the vertical gradient of horizontal advection into its azimuthal and radial advection parts. Comparing Fig. 5.5b and c, the first thing we note is that the two components of advection have strong opposite effects. Focusing at the updraft core between 1 to 2 km altitude, the major destabilization is largely the result of the azimuthal advection, while the radial advection at this region has only small contribution, or even opposite stabilization effect (below 1 km). Recall from Fig. 5.4m-o that the boundary layer fluxes from the warm ocean surface strongly destabilizes the surface layer, the stabilization effect from the radial advection near the surface is largely to oppose the effect of surface fluxes, as discussed in previous section 5.3.1. Focusing on the azimuthal advection term near the core of the updrafts, we see that the destabilization effect actually extends towards the surface and remains strong, but part of this effect is offset by the radial advection near the surface. The net effect of the cancellation between the two parts is a destabilization effect that passes through the core of the updraft near 1 km altitude and upward.

Looking at the hourly averaged plan views near z = 1.25km (Fig. 5.5d-f), we see that the region of destabilization indeed collocates nicely with region of persistent updrafts. From the azimuthal advection (Fig. 5.5e), we see a distinct spiral band of strong destabilization, passing through the strong updraft region at the north of the storm, as well as the spiral rainband at the downshear right quadrant. This destabilization effect is partially offset by the stabilization effect of the radial advection.

Here, it should be emphasized that the strong updraft region shown in the hourly averaged vertical velocity field is not the result of a single updraft or convective event, but is the composited result of many convective updrafts that continuously being fired off and intensified at that region throughout this one-hour period (confirmed by examining instantaneous field at 1-min frequency, not shown). Once these intense updrafts leave the strong destabilization region at the left-of-shear

88 quadrant and travel downwind into the upshear quadrants, the updraft magnitude becomes weaker, leaving only a scarce distribution there. This is simply because without the support of the destabilization effect, the stabilization effect of the vertical advection quickly results in local reduction instability at the vicinity of the updraft and thus ceases the diabatic heating release. As a consequence, the intense updrafts always appear to be collocated with region of strong destabilization.

Fig. 5.5 (a-c) Cross-sections of the vertical gradient of (a) horizontal advection, (b) azimuthal advection and (c) radial advection for hour 20, averaged over the sector shown in Fig. 5.2b. Vertical velocity is contoured in magenta at every 0.2 푚푠−1 from 0.4 to 10 푚푠−1, with updraft stronger greater than 0.6 푚푠−1 thickened. (d-f) is the same as (a-c), but for plan views at z = 1.25 km. Positive vertical velocity is contoured in red at every 0.2 푚푠−1.

To visualize the effect of destabilization more clearly, we next look at the azimuth-height cross-section that passes through the strong updraft region at hour 20 (as shown by the black solid lines in Fig. 5.6a). The azimuth-height cross-section is radially averaged between 21 to 27 km

89 radius over the intense outer updraft region shown in Fig. 5.4b. Figure 5.6 shows the 휃퐸 structure within the azimuthal height section. Temperature warmer than 357 K are also contoured in black.

We see that below z=1.25 km a strong azimuthal gradient in 휃퐸 is located near 100-degree azimuth, where warm 휃퐸 air is located at the upwind side of the cross-section. Collocated at this zone of strong temperature gradient, updraft also reaches its maximum. Overlaying on top of the fields are

푣 휕휃 vectors representing azimuthal advection of 휃 (i.e., − 퐸 ; not tangential wind), with warm 퐸 푟 휕휆 advection pointing in the cyclonic direction (increasing azimuths). As expected, azimuthal advection is strong at the region of strong azimuthal 휃퐸 gradient. But more importantly, this warm advection of 휃퐸 is decreasing with height, with gradient being strongly negative near 1 to 1.5 km altitude. Recall from previous 휃퐸 budget that the vertical advection of 휃퐸 at the strong updraft region brings high-휃퐸 air from low-level upward, thus always tends to neutralize the unstable atmosphere. In order to sustain these organized updrafts and their continuous release of latent heating, destabilization effect must exist to oppose the stabilization effect of vertical advection. At the left-of-shear quadrant of hour 20, this is accomplished by having a differential 휃퐸 advection that decreases with height to keep supplying the atmospheric column with warmer 휃퐸 at lower level than aloft, thus maintaining the instability of the column.

Figure 5.6a also shows the plan view of the low-level asymmetric 휃퐸, vertically averaged between 200m to 850m. Slightly downwind and outward of the intense updraft region, we see that an anomalous low-휃퐸 air that is associated with a surface cold pool. The core of this cold pool can also be seen in Fig 5.6b near 140-degree azimuth. By comparing Fig. 5.6a with the full 휃퐸 field shown in Fig. 5.2b, we also see that the warm 휃퐸 air at the upwind side of the azimuthal-height cross-section (Fig. 5.6b) actually belongs to the TC inner core, which is largely the symmetric part of the 휃퐸 and thus cannot be seen in the asymmetric plan view of 휃퐸 in Fig. 5.6a. Therefore, this

90 demonstrates that the azimuthal gradient of the asymmetric 휃퐸 in this intense updraft region is mainly set up by the presence of the cold pool. As a result, the associated region of destabilization always located at the inner edge of asymmetric 휃퐸. Looking back at Fig. 5.1c-e, we see that as this cold pool strengthens and advances towards the TC inner core between hour 15 to hour 19, the thermodynamic gradient strengthens and the associated updrafts intensify and contract inward (as in hour 20 shown in Fig. 5.1f).

Fig. 5.6 a) Plan view of hourly averaged asymmetric 휃퐸 for hour 20, vertically averaged between 200m and 850m. Updraft at z=1.5km is contoured in red at every 0.4푚푠−1. Black solid lines indicate region of interest. (b) Azimuth-height cross-section of hourly averaged 휃퐸 for hour 20, radially averaged between 21 and 27 km radius. Updraft is contoured in magenta at every 0.25푚푠−1. Vectors indicate azimuthal advection of 휃 , with warm advection pointing to the increasing azimuth direction. 퐸

Furthermore, from the asymmetric 휃퐸 structure (Fig. 5.6a), we also see that the asymmetric

휃퐸 at the upwind ends of the spiral rainband (downshear-right quadrant) has different origin from its downwind counterpart. At the downshear-right quadrant where the upwind side of the TC

91 rainband lies, the asymmetric 휃퐸 responsible for setting up the azimuthal 휃퐸 gradient is a band of strong, warm 휃퐸 anomaly. As seen in Fig. 5.2b, this warm 휃퐸 anomaly is part of the high-휃퐸 air of the TC inner core, which has an asymmetric structure of a “comma” shape. As mentioned earlier, this asymmetric 휃퐸 structure is forced by the environmental wind shear (Riemer and Montgomery

2011). Riemer (2016) also discuss the possibility that the updrafts at the upwind end of a quasi- stationary rainband could be due to frictional convergence forced by vorticity forcing associated with a tilted storm vortex in a sheared environment. However, his analysis and proposed mechanism did not fully explain why the most intense updrafts always align with region of strong

휃퐸 gradient. Our analysis discussed in Fig. 5.5e further demonstrated that the 휃퐸 asymmetry also result in a destabilization zone that supports continuous convective activity and diabatic heating release. In addition, the region of the destabilization at the downshear-right quadrant is located at the radially outward and downwind edge of this asymmetric warm 휃퐸, in distinct contrast to its downwind counterpart (at the left-of-shear quadrant).

5.3.3 Pressure decomposition

Previous sections demonstrated how thermodynamic asymmetry associated with surface cold pool can result in regions of strong azimuthal gradient that is favorable for the maintenance of persistent updraft. In this section, we further examine the relationship between the perturbation pressure and density response to understand the origin of this surface cold pool. Here, we define perturbations of pressure and density as the deviations of these fields from their azimuthal and hourly time mean. Smith et al. (2005) discussed in details about the ambiguity in defining buoyancy in a gradient wind environment. Specifically, in a gradient wind environment, the reference state for defining the buoyancy of an air parcel needs to take into account the “system

92 buoyancy” of the basic state vortex. This requires the basic state pressure and density to be functions of both height and radius, or even azimuth if the storm is strongly asymmetric. In our case, the azimuthally varying portion of the thermodynamics is the target of investigation, therefore we do not treat the asymmetric component as part of the basic state, and therefore the basic state only captures the radial and height varying parts.

Following the derivation of Houze (2014), we decompose the pressure perturbation into buoyancy and dynamically driven components using the following equation

휕 휌∗ 휕푝 ∇2푝∗ = (휌 퐵) + ∇ ∙ (−휌 풗 ∙ ∇풗 − (1 − ) 표 푟̂ − 휌 푓풌 × 풗 + 휌 푭) 휕푧 표 표 휌 휕푟 표 표 표 (5.4)

= 퐹푏푢표푦 + 퐹푑푦푛

, where 훻 is the three-dimensional gradient operators; 휌표(푟, 푧) and 푝표(푟, 푧) are the axisymmetric

∗ ∗ basic state density and pressure; 휌 = 휌 − 휌표 and 푝 = 푝 − 푝표 are the perturbation density and

휌∗ pressure; 퐵 = −푔 is buoyancy; 풗 is the three-dimensional wind field; 푭 is friction; and 푓 is the 휌표

Coriolis parameter. The first term on the right-hand-side of equation (5.4) is buoyancy pressure forcing, while the second term being the dynamical pressure forcing. Figure 5.7 shows the hourly averaged decomposition result for hour 20, azimuthally averaged over the same quadrant shown in Fig. 5.6. Comparing Fig. 5.7 a with b and c, it is clear that a large part of the perturbation pressure (deviation from both azimuthal and time mean) are responding to buoyancy forcing within the quadrant. As shown by the magenta contours in Fig. 5.7a and b, the asymmetric 휃퐸 within the quadrant is anomalously cold (and dense) compared to the azimuthal mean. The radially outward side of the cold pool are connected with downdrafts (black dashed contours in Fig. 5.7a and b) that originates from the mid-troposphere near 4 km altitude. These downdrafts are cooling driven, as shown by the blue dashed contours in Fig. 5.7c. From the configuration of the cold pool,

93 downdrafts and diabatic cooling, it clearly shows that this surface cold pool is driven by the stratiform cooling at the downwind portion of the spiral rainband, which flushes cold and dense air into the TC boundary layer (consistent with Fig 5.1a and b). At the inner edge of the cold pool, intense updraft (black solid contours in 5.7a and b) is found, which is maintained by the mechanism discussed in previous section.

We also note that the dynamical pressure forcing is closely related to supergradient force.

푣2 To see this, consider for a simple case of a steady-state axisymmetric vortex, then −풗 ∙ ∇풗 = 푟̂. 푟

The dynamical pressure forcing 퐹푑푦푛 can be approximated as

푣2 1 휕푝 퐹 ≈ ∇ ∙ (휌 ( + 푓푣 − + 퐹 ) 푟̂) (5.5) 푑푦푛 푟 표 푟 휌 휕푟 푟 where ∇푟 is the radial component of the divergence operator; 퐹푟 is the radial component of friction.

Equation (5.5) is essentially the radial divergence of the deviation from the classical Ekman balance. As shown in Fig. 5.7c, the strongest dynamical pressure forcing is located at the leading edge of the negative 휃퐸 anomaly, i.e. the cold pool. The overall structure of cold pool and dynamical pressure forcing is similar to that at the leading edge of a density current, but with the additional complication of being in a gradient wind environment. This analysis also demonstrates that the existence of supergradient wind response is indeed related to the convergence between two distinct air masses, which in turns helps to modulate updrafts to exit the boundary layer (Huang et al. 2012; Abarca and Montgomery 2013).

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휕 Fig. 5.7 Cross-section of sector and hourly averaged (a) ∇2푝∗; (b) buoyancy forcing (휌 퐵) and (c) 휕푧 표 ∗ 휌 휕푝표 dynamic forcing ∇ ∙ (−휌표풗 ∙ ∇풗 − (1 − ) 푟̂ − 휌표푓풌 × 풗 + 휌표푭). Vertical velocity is contoured 휌표 휕푟 in (a) and (b) in black (updraft in solid at every 0.2 푚푠−1 and downdraft in dashed lines at every 0.1 푚푠−1). Diabatic cooling is contoured in (c) in blue dashed lines at every 3 × 10−4퐾푠−1. Negative asymmetric 휃퐸 is contoured in (a-c) in dashed magenta lines at every 0.5 K.

5.4. Conclusions

In this chapter, we continue our analysis of the Matthew simulation and focus on understanding the mechanism of how the intense low-level updrafts that occur at the left-of-shear half of the storm prior to SEF are sustained, which subsequently axisymmetrized into a secondary eyewall.

Previous SEF theories that emphasizing boundary layer processes (Huang et al. 2012;

Abarca and Montgomery 2013; Kepert 2013) largely focus on the dynamical impact of the boundary layer on the feedback process leading to SEF. However, when looking at the thermodynamics at the TC boundary layer prior to SEF, we notice a strong spatial correlation between the intense updraft signals and the regions with strong low-level temperature gradient, indicating certain important dynamical relationship between them. Budgets analysis on the equivalent potential temperature 휃퐸 and the moist stability parameter 휕휃퐸/휕푧 at the updraft regions show that throughout an hourly-long period the moist instability at the strong updraft region largely is slowly varying, and is under the strong balance between horizontal advection,

95 vertical advection and boundary layer forcing. Among the balance of these terms, it is shown that the vertical advection at strong updraft region constantly advects high-휃퐸 air from low level towards upper troposphere, which tends to reduce to moist instability within the column and thus exerting a local stabilization effect. For strong updrafts above 1km altitude, the only term that can balance the stabilization effect is the horizontal advection, which, in contrast, continuously destabilizes the atmosphere by providing a differential warm-휃퐸 advection. Near the top of the boundary layer, this differential warm advection decays with height, and maintains an unstable 휃퐸 structure that can support the continuous release of diabatic heating at the updraft region.

Further decomposition of the differential horizontal advection into its azimuthal and radial components showed that, in the left-of-shear quadrants where the band of intense updrafts lies, the dominant destabilization effect comes from the azimuthal advection. At this region, the azimuthal gradient of 휃퐸 necessary to result in the warm advection is setup by an inward intruding cold pool.

This low-휃퐸 air of the cold pool comes directly from the stratiform-cooling at its immediate outward side near 4 km altitude. The cooling-induced MDI continuously flushes cold and dense air-masses into the TC boundary layer. At the leading edge of the cold pool where the low-휃퐸 air meets the TC inner core warm air, strong 휃퐸 gradient is established, with supergradient wind and updrafts collocated at the same place. The analysis provided herein demonstrated how cold pool effect due to stratiform cooling can induce strong updraft at its inner edge and contribute to SEF process, consistent with previous modelling studies (Chen et al. 2018) and the idealized experiment presented in chapter 2 (Yu and Didlake 2019).

The thermodynamic picture depicted above contributes to the current understanding on the

SEF phenomenon in the following ways. First, the aforementioned updraft mechanism is capable of explaining not only the spiral nature of the rainband, but also can distinguish the differences in

96 the destabilization origins at the upwind and downwind ends of the rainband. Second, previous modelling study (Wang et al. 2019) showed an enhanced eddy kinetic energy at the boundary layer just prior to the axisymmetrization process. However, no specific explanation has been provided in their study about the source of the eddy forcing. In this study, we note that the destabilization

̅푣̅휕̅̅휃̅̅ 푣̅̅′̅휕̅̅휃̅̅′ mechanism discussed in this chapter is purely an asymmetric forcing (∵ − 퐸 = − 퐸) that 푟 휕휆 푟 휕휆 only exists at a specific sector of the storm when suitable thermodynamic conditions are met. Thus, the resulted intense low-level updrafts occurred during hour 19-22 are also highly asymmetric and located primarily at the left-of-shear to upshear-left quadrants. Potential vorticity (PV) budget (not shown) confirms that the asymmetric diabatic heating releases at these regions are indeed the main asymmetric PV source that leads to the increasing wave activity above the boundary layer.

97

CHAPTER 6

INVESTIGATING AXISYMMETRIC AND ASYMMETRIC SIGNALS OF SECONDARY

EYEWALL FORMATION USING OBSERVATIONS-BASED MODELING OF THE

TROPICAL CYCLONE BOUNDARY LAYER

In chapter 4 and 5 we respectively examined the dynamical and thermodynamic aspects of the asymmetric rainband processes that lead to the SEF event of a hurricane Matthew simulation.

We revealed a number of asymmetric processes in the downwind end of the spiral rainband that play vital roles in the early preconditioning of the wind and thermodynamic fields, which subsequently lead to the intense updraft at the left-of-shear half of the storm. In this chapter, we seek further evidence in observations to see whether such asymmetric organizations in the wind field and updrafts generally exist in TCs that undergo SEF.

Wunsch and Didlake (2018, hereafter WD18) performed a composite analysis on aircraft

700-hPa-level observations for 17 years of Atlantic basin TCs. Besides finding an axisymmetric broadening of the outer tangential wind field in the composite of storms prior to SEF, they also showed that these storms experience the largest change of the tangential wind field in the left-of- shear quadrants. This finding aligns with previous studies on the importance of the left-of-shear rainband structures in SEF (Didlake and Houze 2013; Didlake et al. 2018). Given these results,

98 they hypothesized that the evolution of these wind-shear-induced asymmetries were indicative of recurring asymmetric rainbands that initiate SEF.

Building upon the results from WD18 and other previous studies, in this chapter we use a nonlinear hurricane boundary layer model (Kepert 2018, here after the K18 model) to examine the steady-state boundary layer response to the tropospheric forcing represented by the observational composites of WD18. Both axisymmetric mean and asymmetric forcings are used to investigate how boundary layer responds to the free tropospheric forcing aloft during SEF events. As emphasized in Kepert (2018), since the nonlinear K18 model simulates the steady-state boundary layer response under the constant pressure forcing at the boundary layer top, the simulated response is only a one-way response of an apparent two-way interaction. However, it is because no feedback to the imposed tropospheric forcing is included in the model, the one-way boundary layer response can therefore be isolated from an apparent two-way interaction, and thus can provide insight about what asymmetric rainband features can reinforce the coupled interactions between boundary layer and tropospheric processes, and are conducive to the subsequent development of the secondary eyewall. By applying the K18 model with an asymmetric pressure forcing prior to SEF, our analysis aims to provide insights about where the earliest SEF-like structures actually develop.

6.1 Data and Methodology

6.1.1. The K18 Model

To examine the boundary layer response to the observed free tropospheric wind distribution before and after SEF, we use the nonlinear boundary layer model (K18 model)

99 developed by Kepert and Wang (2001) and later modified by Kepert (2018). The K18 model simulates the steady-state boundary layer response by integrating a set of prognostic three- dimensional nonlinear primitive equations to nearly steady state. The model is dry and is only forced by the pressure forcing from the free-tropospheric wind at the top of the integration domain.

No feedback to the forcing field is included in the model, so the simulated response is a one-way response of the boundary layer to the free tropospheric forcing. Different from the axisymmetric

KW01 model, the pressure forcing used in the K18 model top can be either axisymmetric or asymmetric. When axisymmetric forcing is used, the simulated result is equivalent to the KW01 model. When asymmetric forcing is used, it allows the investigation of the boundary layer response to wind asymmetries induced by environmental influences, such as environmental wind shear. The procedures of deriving this asymmetric nondivergent wind field will be discussed in section 6.1.3.

More details of the model design can be found in Kepert (2018).

The momentum equations in the K18 model include the horizontal and vertical momentum diffusion terms, which can be parameterized by various choice of schemes. Following the recommendation of Kepert (2012), the neutral Louis boundary layer scheme is used (Louis 1979) to parameterize the momentum diffusions. The model has 20 vertical levels, covering from 10 m near the surface to 2.25 km. The maximum vertical spacing is 200 m and the horizontal grid spacing is 3 km. All experiments are integrated for 48 hours to a nearly steady state.

6.1.2. Tangential Wind Composite from WD18

WD18 used the Extended Flight-Level Dataset for Tropical Cyclones (FLIGHT+; Vigh et al. 2016) to perform a composite analysis for Atlantic hurricanes from year 1999 to 2015. They first identified storms that underwent SEF and marked the SEF time at the first instance when the

100 axisymmetric tangential wind radial profile exhibited a secondary wind maximum. The 700-hPa level flight legs are then selected, and divided into groups without, prior to, and after SEF (the

“Non-SEF,” “Pre-SEF” and “Pos-SEF” groups). In order to accommodate the different storm sizes and flight leg coverages for the composite analysis, the flight legs of each group were normalized.

For the “Non-SEF” group, the wind profiles are normalized by the radius of maximum wind (푟1,

RMW). For the Pre-SEF and Pos-SEF groups, the flight legs were normalized by two length scales.

For radii less than the RMW, the wind profile was normalized by the RMW (푟1), while outside the

RMW, the wind profile was normalized by the width of the moat (푚), which is the distance between the primary and secondary eyewalls. The radius of the secondary maximum wind (푟2) is equal to 푟1 + 푚. More detailed description about the data set and the normalization can be found in WD18.

In order to convert the normalized azimuthal mean composite wind profiles into input for the K18 boundary layer model, the following preprocessing steps are performed. First, each normalized flight legs are rescaled back to physical space using the median value of 푟1 and 푟2 of each group (only 푟1 is used for Non-SEF group). Then, for the inner core region (푟 < 푟1 = RMW), the legs were linearly interpolated to zero due to incomplete coverage of the flight legs. Individual legs that terminated at 푟 < 150km were removed from the composites to ensure that unphysical signals due to data unavailability were not displayed in the composite. The wind composites at large radii (푟 > 150km) become noisy as the sample size drops at these large radii. Therefore, the profiles at 푟 > 150km are replaced by 퐶푟−0.5, where 퐶 is a constant to ensure continuity at the extrapolation radii. Figure 6.1a shows the resulted azimuthal mean composite after the above rescaling procedure is performed.

101

(a) (b)

Q1 Q2

LS UL DL US DS UR DR RS

(c) (d)

(e) (f)

(g) (h)

Fig. 6.1. a) Azimuthal mean composite profiles for Non-, Pre- and Post-SEF groups. The sample count, and median values of 푟1 and 푟2 are indicated in the legend. The portion of profiles highlighted in red indicate extrapolated region. b) Schematics showing the orientations of the two set of shear-relative quadrants, Q1 and Q2. c) Quadrant averaged composite profiles for the Non-SEF group under Q1 quadrant definition (DL, UL, UR, and DR). d) is the same as c), but for the Q2 quadrant set of the Non- SEF group (DS, LS, US, and RS). e) and f) are the same as (c) and (d), but for the Pre-SEF group. g) and h) are the same as (c) and (d), but for the Pos-SEF group.

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6.1.3. The axisymmetric and asymmetric forcing for K18 model

The pressure forcing at the K18 model top is represented by a single input of a two- dimensional nondivergent wind field, which can be either axisymmetric or asymmetric. For the three sets of azimuthal mean composites discussed in section 6.1.2, their corresponding two- dimensional realization is an axisymmetric tangential wind field, which is nondivergent by definition and therefore can be directly used as the forcing for the K18 model.

To utilize the capability of the K18 model to incorporate an asymmetric forcing, an asymmetric nondivergent wind field is required. This asymmetric nondivergent wind field can be derived from the flight leg composites by the following procedure. First, to ensure the derived wind asymmetries in the composites are induced by environmental wind shear, flight legs that have

850-200 hPa wind shear under 3푚푠−1 are removed. Here, the 850-200 hPa wind shear for each flight legs are extracted from the Statistical Hurricane Intensity Prediction Scheme (SHIPS) database (DeMaria et al. 2005). Next, a running quadrant average is applied to the set of rescaled flight legs at every 45 azimuthal degrees. The purposes of this 90-degree averaging are two-folded.

First, it reduces the small-scale fluctuations in the wind profiles by incorporating information from multiple nearby flight legs. Second, this procedure assigns a relatively uniform weight to each flight legs despite their irregular azimuthal spacings. The running quadrant averaging results in 8 overlapping quadrant averages that are named based on the shear relative position. The first quadrant set (referred as Q1) is oriented such that the shear vector lies at the dividing line between two quadrants, namely Downshear-Left (DL), Upshear-Left (UL), Upshear-Right (UR), and

Downshear-Right (DR). The second quadrant set (referred as Q2) is shifted 45 degrees from the

Q1 set, namely Downshear (DS), Left-of-Shear (LS), Upshear (US), and Right-of-shear (RS). The orientation of these two sets of quadrants are shown in Fig. 6.1b. The rescaled profiles of these

103 two quadrant sets (Q1: DL, UL, DR, UR; and Q2: DS, LS, US, RS) for each of the three groups are shown in Fig. 6.1c-h. These 8 quadrant averages constitute an asymmetric tangential wind field. Then, a low-pass filter is applied to the asymmetric wind field to extract signals from wavenumber 0 to 3 (hereafter referred as the filtered tangential wind field). Further testing using more frequent averaging window (i.e., at every 5 azimuthal degrees) show that the wavenumber 0 to 3 structures of the tangential wind and vorticity are insensitive to the azimuthal spacing at which the averaging is applied.

To derive the nondivergent wind component from the filtered tangential wind field, the method of optimal interpolation is used, which solves for the streamfunction 휓 of the nondivergent wind 풗흍 that minimizes a cost function 퐽(휓) of the following form:

1 1 퐽(휓) = (퐷 휓 − 푣 )푇푂−1푅(퐷 휓 − 푣 ) + (−푅−1퐷 휓)푇퐵−1푅(−푅−1퐷 휓) (6.1) 2 푟 표 푟 표 2 휆 휆

, where 푣표 is the filtered tangential wind; 퐷푟 and 퐷휆 are matrix representations of radial and azimuthal differential operators; 푅 is radius in a diagonal matrix form; 푂 is the variance of observational uncertainty; 퐵 is the variance of uncertainty of the initial guess. The first term measures the distance square of the analysis to the observed values, taken into account the uncertainty of the observation. The second term is essentially a regularizer that constrains the magnitude of the asymmetries introduced into the analysis. For simplicity, in this study both 푂 and

2 2 2 퐵 are taken to be diagonal (i.e., 푂 = 푑𝑖푎푔(휎표 ) ; 퐵 = 푑𝑖푎푔(휎퐵 )), with 휎표 having radial and

2 azimuthal dependence and 휎퐵 being a constant. It can be shown that when assuming a diagonal 푂 and 퐵, the 휓 that minimizes equation (6.1) satisfies the following generalized Poisson equation

1 휕 1 휕휓 1 휕 1 휕휓 1 휕 푣표 (푟 ( 2) ) + 2 ( 2 ) = (푟 ( 2)) (6.2) 푟 휕푟 휎표 휕푟 푟 휕휆 휎퐵 휕휆 푟 휕푟 휎표

104

휕휓 1 휕휓 The relationship between the 휓 and the nondivergent wind field 풗 is 풗 = ( , − ) . 흍 흍 휕푟 푟 휕휆

Equation (6.2) is derived by first computing ∇퐽 and then equate ∇퐽 to 0. This yields equation (6.2), the solution of which can minimize (6.1). Equation (6.2) is similar to the nondivergent component of the typical Helmholtz decomposition but incorporates the information about the variance of

2 2 observational uncertainty 휎표 . The information of the variance 휎표 (푟, 휆) is obtained by the following bootstrapping procedure. At every azimuthal direction where the quadrant averaging is performed, the set of flight legs within that quadrant are bootstrapped by 1000 times, and then the variance of the bootstrap mean is computed. After performing this bootstrap procedure for all quadrants, the resulted variance field is passed to the same low-pass filter to retain only signals from wavenumber 0 to 3. Given the filtered tangential wind field 푣표 and observational variance

2 휎표 , equation (6.2) is solved in a circular domain of 300 km radius with Neumann boundary condition:

휕휓 | = 푣표(푟 = 300km) (6.3) 휕푟 푟=300km

The above procedure can be applied to all three groups of composites. However, since the primary interest of the asymmetric analysis to examine the asymmetries that occur prior to the SEF events, therefore only the results of the Pre-SEF group are shown. Figure 6.2a and 6.2b show the plan

2 views of the filtered asymmetric tangential wind (푣표) and observational variance (휎표 ) for the Pre-

2 2 SEF group. Multiple values of 휎퐵 have been tested, and it is found that the magnitude of 휎퐵 mainly affects the magnitude of 푢휓, but have little impacts on 휓, 푣휓 and the simulated updraft response

2 2 −2 of the K18 model. For the results shown in Fig. 6.1, the value of 휎퐵 is taken to be 1.93푚 푠 ,

2 which equals to be the mean value of 휎표 at 푟 = 150 푘푚 (highlighted by red circle in Fig. 6.2b).

From Fig. 6.2a, we see that the tangential wind field in the downshear half of the storm is

105 noticeably broader than the upshear half. This wind field broadening is in good agreement with

WD18 and previous observational studies about the orientation of spiral rainband at the downshear quadrants in a sheared environment. A signature of “flattening” is seen near 100 km radius around the storm. This flattening signal is strongest at the LS quadrant, where the wind field almost appears as having a secondary wind maximum. This structure of the tangential wind will later be shown to be corresponding to distinct band of local vorticity maximum that cover the entire DL and LS quadrants. Figure 6.2c and 6.2d show the tangential and radial components of the nondivergent wind 풗흍 for the Pre-SEF group after solving equation (6.2). Overall, the tangential component of the nondivergent wind is more axisymmetric compared to the filtered asymmetric tangential wind (Fig. 6.2a), but has a similar “flattening” signature at the LS quadrant. Note also that the radial component (Fig. 6.2d) is about an order smaller than the tangential component, and mostly concentrates at the primary eyewall.

106

-1 2 -2 (a) Full V: Pre-SEF [ms ] (b) V variance [m s ] 10

8

6

Y [km] Y 4

2

0

(c) Nondivergent V: Pre-SEF [ms-1] (d) Nondivergent U: Pre-SEF [ms-1] Y [km] Y

X [km] X [km] Fig. 6.2. a) The plan view of the filtered asymmetric tangential wind field 푣표 for the Pre-SEF group. The contour of 30 푚푠−1 is highlighted in red. The wind is contoured in black at every 0.5 푚푠−1 below 30 푚푠−1 and at every 1 푚푠−1 above 30 푚푠−1. b) The plan view of the observational variance obtained by the bootstrap procedure for the Pre-SEF group. The red circle indicates 150 km radius. The red arrow indicates the direction of the environmental wind shear vector. c) The tangential component of the nondivergent wind field 풗흍 after solving equation (6.2). Same contours are applied as in (a). d) The radial component of the asymmetric nondivergent wind field 풗 . 흍

6.2 Results

6.2.1 Axisymmetric forcing

WD18 showed a clear axisymmetric broadening of tangential wind field in both the Pre- and Post-SEF groups, but not in the Non-SEF group. In this section, we first examine the distinctions in the K18 simulated axisymmetric boundary layer response using the axisymmetric

107 mean forcing of the three groups.

6.2.1.1 Non-SEF group

Figures 6.3a-c show the boundary layer response of the Non-SEF group simulated by the

K18 model. Near the radius of 20 km lies the updraft maximum of the primary eyewall, which is consistent with the RMW of the Non-SEF group shown in Fig. 6.1a. Fig. 6.3b shows the boundary layer tangential wind. The positive agradient wind, calculated as the difference between the actual tangential wind and the gradient wind, is shown by the magenta contours. This field reveals the tangential wind jet of the primary eyewall, which is supergradient in nature (KW01; Kepert 2013).

The associated inflow maximum lies beneath and just slightly outward (near 푟 = 30 푘푚) from the updraft maximum, as shown in Fig. 6.3c. As expected, no secondary eyewall signature is found outside of the primary eyewall. The updraft, supergradient wind, and horizontal convergence maxima of the primary eyewall are the dominant features from each field.

6.2.1.2 Pre-SEF group

Figures 6.3d-f show the boundary layer response of the Pre-SEF group. Similar to the Non-

SEF group (Fig. 6.3a), in Fig. 6.3d we see the updraft maximum of the primary eyewall near the radius of 27 km, which is consistent with the RMW of the forcing profile (푟1 in Fig. 6.1a). Outside the primary eyewall, a secondary updraft maximum (near the 100 km radius) is also clear, consistent with the 푟2 value shown in Fig. 6.1a. Associated with this secondary updraft maximum is a local outflow maximum at the same level. This overturning circulation is a preceding signature of the incipient secondary eyewall.

108

(a) (g) (d) [ms-1]

(b) (e) (h)

[ms-1] height (km)

(c) (f) (i) [ms-1]

radius (km) Fig 6.3. Boundary layer responses for the axisymmetric Non-SEF (left column), Pre-SEF (middle column) and Post-SEF (right column) groups. Top row (a, d, g) displays vertical velocity (shaded) and radial velocity (contour), with radial velocity contoured every 0.8 푚푠−1 and the zero-line omitted. Middle row (b, e, h) displays tangential wind 푣 (shaded) and positive agradient wind (contour), contoured at every 0.6 푚푠−1 with zero-line omitted. Bottom row (c, f, i) displays radial velocity (shaded) and horizontal divergence (contour), contoured at (±2, ±4, ±8, ±16, ±32, ±64, ±128, ±256) × 10−5푠−1.

Figure 6.3e shows the tangential wind structure within the boundary layer. Compared to the Non-SEF group (Fig. 6.3b), this boundary tangential wind field is more broadened, and contains a secondary supergradient tangential jet (magenta contour in Fig. 6.3e). Figure 6.3f shows the radial flow and divergence within the boundary layer. Within the inflow layer, enhanced horizontal convergence occurs at 100 km due to a slight local weakening of the boundary layer inflow near there. This airflow then turns into the updraft seen in Fig. 6.3d.

Overall, the existence of secondary maxima of boundary layer updraft, tangential wind jet, and convergence signal the existence of an incipient secondary eyewall in the Pre-SEF group,

109 which is not seen in the Non-SEF group.

6.2.1.3 Post-SEF group

Figures 6.3g-i show the boundary layer response of the Post-SEF group. Compared to the

Pre-SEF group, the overall signatures of the now-developed secondary eyewall are significantly stronger. As shown by the vertical velocity in Fig. 6.3g, the secondary eyewall is now located between 푟 = 80 and 90 km, signaling a clear inward contraction compared to the Pre-SEF group.

This contraction is expected once the secondary eyewall is well established, which acts as an axisymmetric heat source within a vortex (Shapiro and Willoughby 1982). The secondary updraft maximum also becomes noticeably stronger and deeper. In the radial wind (Fig. 6.3i), a secondary inflow maximum emerges with a magnitude similar to that of the primary eyewall. This associated convergence maximum is located just radially inward, which is stronger and occupies a broader radial range compared to the Pre-SEF group.

Figure 6.3h shows the tangential wind field and the associated tangential jet within the boundary layer. Compared to the Pre-SEF group, the tangential winds of the secondary eyewall are strengthened significantly, and almost appears as an isolated tangential wind maximum. The calculated tangential jet (magenta contours) of the secondary eyewall is also more substantial.

6.2.2 Asymmetric simulation of the Pre-SEF group

Even though the axisymmetric mean profile of the Pre-SEF does not have a clear secondary wind maximum (Fig. 6.1a), the forcing profile was sufficient to force a clear but weak separated secondary maximum of inflow, updraft and supergradient tangential wind jet within the boundary layer (Fig. 6.2d-f). We now seek evidence to determine if there is a specific azimuthal region that

110 contribute more to this incipient secondary eyewall signal by using K18 model in conjunction with the asymmetric nondivergent wind field of the PreSEF group, as shown in Fig. 6.2c-d. The results can provide insight to which quadrant possesses the earliest asymmetric features that can induce boundary layer responses that are favorable to the subsequent SEF process.

6.2.2.1 Azimuthal average and plan views

Figure 6.4 summarizes the results of the asymmetric response simulated by the K18 model using the two-dimensional asymmetric forcing (Fig. 6.2c-d). Fig. 6.4a shows the azimuthal average of the response. Comparing Fig. 6.4a with 6.3d, it is clear that the azimuthal mean response to the asymmetric forcing is in nice agreement with the response simulated using the axisymmetric forcing (Fig. 6.3d-f). Despite the fact that the asymmetric simulation is driven by vorticity forcing that has significant asymmetry, the radial location, structure and magnitude of the azimuthal mean secondary updraft and radial inflow maxima in the asymmetric simulation are consistent with the axisymmetric simulation that uses the axisymmetric forcing.

Fig. 6.4b shows the plan view of updraft response near 1km altitude. From the plan view of the updraft structure, it is clear that most of the updrafts at the secondary eyewall location near r = 100 km come from the left-of-shear half of the storm, which is covered with an outer distinct band of updraft. This distinct band of outer updraft begins at the down-shear direction, strengths downwind and maximizes at the LS quadrant of the storm. This updraft band also coincide both radially and azimuthally with a distinct vorticity band in the forcing field, which is shown in Fig.

6.4c. This collocation of the updraft response and vorticity forcing is consistent with the theoretical analysis presented in Kepert (2001), which showed that in the steady-state limit the updraft response is directly linked to the vorticity forcing aloft.

111

Fig. 6.4d and e also showed the plan view of convergence and agradient wind at z = 850m, which is underneath the updraft shown in Fig. 6.4b. It is clear that the convergence field follows the pattern of the updraft response, which is expected from the continuity equation. However, it is also interesting to see that compares to the azimuthal locations of the outer updraft and convergence, the strongest supergradient wind shown in Fig. 6.4d displays a noticeable downwind shift towards the updraft quadrants.

Fig. 6.4. a) The cross-section of azimuthal mean vertical velocity (shading), supergradient wind (magneta contours, at every 0.8 푚푠−1) and radial velocity (black contours, at every 0.6 푚푠−1 with 0 line omitted) of the asymmetric simulation using the asymmetric forcing of the Pre-SEF group. b) The plan view of the vertical velocity response at 푧 = 1.05 푘푚. (c-e) same as (b), but for vorticity forcing at the model top; divergence forcing at 푧 = 0.85 푘푚; and agradient wind at 푧 = 0.85 푘푚.

6.2.2.2 Quadrant averaged analysis

Figures 6.5a-h shows the quadrant-averaged cross-sections of vertical motion and radial

112 velocity across the Q1 and Q2 quadrant sets. Consistent with the plan view of the updraft response

(Fig. 6.4b), the outer updraft maximum first emerges at the DL quadrant (Fig. 6.5b). In the same quadrant, a clear secondary inflow maximum underneath the outer updraft can be seen. Travelling downwind, the updraft and inflow maxima reach their strongest intensity at the LS quadrant (Fig.

6.5c). Further downwind from the LS quadrant, these secondary eyewall signals start to weaken, as seen in the UL and US quadrants (Fig. 6.5d and e). It is also important to note that the secondary updraft and inflow maximum at the DL, LS and UL are noticeably stronger than both the azimuthal mean response (Fig. 6.4a) and the response in the axisymmetric simulation (Fig. 6.3d). This demonstrates that the incipient secondary eyewall signal found in Fig. 6.4a and 6.3d indeed comes largely from the left-of-shear quadrants.

Meanwhile, it is also noted that both the supergradient wind and radial outflow maxima exhibit noticeable downwind shift compared to the azimuthal locations of the outer updraft and boundary layer inflow. As shown in Fig. 6.5c and d, the supergradient wind is strong at the LS quadrant (Fig. 6.5c) and become strongest at the UL quadrant (Fig. 6.5d), while strong updrafts begins more upwind at DL, and maximizes at the LS quadrants. Furthermore, the downwind shift in the outflow is even more apparent. For instance, the strong outflow actually begins at UL, and then becomes strong at the US and UR quadrants.

Besides looking at the quadrant averaged responses of the asymmetric simulation, it is also important to examine how much of the response is locally driven by the forcing aloft. To examine this, we first compute the quadrant averages of the nondivergent tangential wind (as in Fig. 6.2c), which are then use as axisymmetric forcings to drive the K18 model. Figure 6.5i-p show the axisymmetric boundary layer responses using the quadrant-averaged forcing for each quadrant. By comparing Fig. 6.5i-p with Fig. 6.5a-h, it is clear that the axisymmetric responses using quadrant-

113 averaged forcing are able to capture the radial location and magnitude of the secondary updraft and inflow maxima. This is particularly clear for the DL, LS and UL quadrants (Fig. 6.5m, j, and n). Difference in the updraft response can be found at the US and UR quadrants, where some weak, intermittent updrafts at the moat region near 60 to 80 km radii in the axisymmetric forcing simulation (Fig. 6.5k) are less pronounced in the asymmetric simulations (Fig. 6.5e). The generally good agreement in the updraft responses between Fig. 6.5a-h and 6.5i-p indicates that the quadrant averaged updraft and inflow at the secondary eyewall location, to a large degree, are driven by the local vorticity band aloft.

(a) (b) (i) (m)

(c) (d) (j) (n)

(e) (f) (k) (o)

(g) (h) (l) (p)

Fig. 6.5. a-h) The cross-sections of quadrant averaged mean vertical velocity (shading), supergradient wind (magneta contours, at every 1 푚푠−1) and radial velocity (black contours, at every 0.8 푚푠−1 with 0 line omitted) of the asymmetric simulation for the Q1 (left column) and Q2 (right column) quadrants. i-p) are the same as (a)-(h), but for the axisymmetric simulations that uses quadrant-averaged forcing.

By comparing the supergradient wind and radial outflow between Fig. 6.5a-h and 6.5i-p, we see that the downwind shift signatures are stronger in the supergradient wind and radial outflow

114 than in the updraft and radial inflow. In the axisymmetric simulation (Fig. 6.5i-p), strong supergradient wind and radial outflow maxima are only seen in the DL and LS quadrant (Fig. 6.5m and j), while in the asymmetric simulation the supergradient wind and radial outflow clearly shift downwind and extend into the upshear quadrants (Fig. 6.5d-f). This demonstrates that the downwind shifts in the supergradient wind and outflow are mainly the consequence of the asymmetries in the flow field, and that the supergradient wind and outflow are dynamically connected (as consistent with Abarca and Montgomery 2013).

6.2.2.3 Sensitivity of updraft strength to the variability of the vorticity forcing

The results from previous section demonstrated that the early secondary eyewall signature present in the azimuthal mean response of the Pre-SEF group largely comes from an outer band of updraft maximum located at the left-of-shear half of the storm. This distinct band of updraft is in close correspondence to the outer vorticity maximum near the same region. In order to further examine the robustness of the updrafts at this region, this section examines the sensitivity of the updraft signature to the variabilities in forcing structures across different quadrants of the storm.

In Fig. 6.5, it is demonstrated that the quadrant-averaged vertical velocity is largely responding to the local forcing aloft, and that the updraft response forced by an axisymmetric forcing with structure same as the quadrant-averaged forcing can be a good approximation. In addition, Fig. 6.2b also demonstrated that using the bootstrapping method it is possible to assess the variability in the observed tangential wind structure. Therefore, using the bootstrap sample, it is possible to further examine the relative robustness of outer updrafts at different quadrants of the storm. To do that, for each of the overlapping quadrants, 30 representative set of bootstrap samples are selected. These selected bootstrap samples are chosen based on more restrictive criteria. In

115 each bootstrap sample, any member can only repeat by at most one time, and exactly a quarter of the members are repeated. The requirement on the maximum number of reoccurrences of the members is to avoid over-emphasizing the forcing profiles of any particular member. And the requirement of having a quarter of member being repeated is to ensure that the set of selected bootstrap samples at each quadrant are subject to the same degree of variability. Next, the average of each bootstrap sample is computed, which is then used as an axisymmetric forcing for the K18 model. In total, 240 times of K18 runs are performed.

Fig. 6.6. Box plot of the extreme values, 25, 50 and 75 quartiles of outer maximum updraft distribution for the Q1 and Q2 quadrants for the Pre-SEF group. The outer maximum updraft is defined as the maximum updraft at the radial range between 90 and 140 km radii.

Figure 6.6 summarizes the result of the sensitivity experiments by showing the quartiles and the extreme values of the maximum outer updraft in all quadrants. Here the maximum outer updraft is defined as the maximum value of the vertical velocity between 90 to 140 km radial range. From Fig. 6.6, it is quite clear that the updraft distribution of the DL and LS quadrants are noticeably higher than that of the other quadrants. Specifically, the medium values of DL

(0.1337 푚푠−1) and LS (0.1393 푚푠−1) are both as high as, if not higher than, the strongest member

116 of the other quadrants (0.1357푚푠−1 at RS). The 25 percentiles of DL and LS quadrant, on the other hand, are both higher than the 75 percentiles of most of the other quadrants, except the RS.

This result demonstrates that the outer updraft signal at the DL and LS quadrants are noticeably stronger than the other quadrants, and that these outer updraft signals are robust features that emerge earliest prior to the SEF.

(a) (b) (c) (d)

Fig. 6.7. a) Cross-sections showing the composite of the vertical velocity (shading), radial inflow (contoured in black at every 1 푚푠−1 with 0 line omitted) and supergradient wind (contoured in magenta, at every 1 푚푠−1) for the strongest 8 members in Q1 quadrant set. b) is the same as a), but for the strong composite of the Q2 quadrant set. c) and d) are the same as (a) and (b), but the Q1 and Q2 quadrant sets of the weak composites.

To further demonstrate the variability in the updraft structure among the bootstrapping samples, in Fig. 6.7 we show the composites of the strongest and weakest 8 members of the 30 bootstrap samples in each of the quadrants (herein referred as the strong and weak composites, respectively). Consistent with the previous results, in the strong composites (Fig. 6.7a and b) both the DL and LS quadrants have a clean distinct outer updraft maximum and a clear moat region.

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The outer inflow and supergradient maxima are also distinct. Transit into the quadrants downwind, the outer updraft maxima becomes noticeably weaker (UL) with a less clear moat that has weak intermittent updrafts. The radial inflow and supergradient wind also do not have a clear isolated maximum at these quadrants.

Similarly, for the weak composites (Fig. 6.7c and d), the secondary updraft maxima in the

DL and LS quadrants are still clear and are noticeably stronger than that of the other quadrants.

The moat remains well-defined with no clear intermittent updrafts. The result herein demonstrates that the isolated outer maxima of updraft and radial inflow in the DL and LS quadrants are robust features that exist cross both the strongest and weakest subsets of the bootstrap samples. In contrast, the signals in the other quadrants is less consistent in comparison. For instance, the isolated outer updraft maximum in the strong composite of UR is found absent in the weak composite, indicating that the signal seen in strong composite could be due to a small subset of flight legs not included in the weak composite.

6.2.3 Quadrant analyses for Non-SEF and Pos-SEF groups

As demonstrated in Fig. 6.3, the axisymmetric responses to the azimuthal mean profiles of

Non-SEF and Pos-SEF groups display a distinct contrast at the outer core. Specifically, the Pos-

SEF group show clear outer maxima of updraft, supergradient tangential jet, and convergence; while the Non-SEF group has none of these features. In this section, we perform quadrant by quadrant analyses to these two groups to examine whether this distinction exist across different quadrants of the storm.

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(a) (b)

Fig. 6.8. a) Cross-sections of the vertical velocity (shading), radial inflow (contoured in black at every 1 푚푠−1 with 0 line omitted) and supergradient wind (contoured in magenta, at every 1 푚푠−1) for the axisymmetric simulations that uses quadrant averaged forcings from the Q1 (left column) and Q2 (right column) of the Pos-SEF groups. b) is the same as a), but for the Non-SEF group.

Figure 6.8 shows the comparison of the axisymmetric responses using the quadrant averaged wind profiles as forcing for the Non-SEF and Pos-SEF groups. Consistent with the forcing profiles shown in Fig. 6.1g and h, the axisymmetric response of the Pos-SEF group (Fig.

6.8a) using the quadrant-averaged forcings display clear secondary maxima of updraft, inflow and supergradient outflow across all eight over-lapping quadrants of the storm. Comparing the strength of these signals, we see that the strongest secondary eyewall signature lies at the left-of-shear half of the storm (LS and UL quadrants) and extends towards the US quadrant, consistent with the results of the Pre-SEF group shown in previous sections.

Looking into the Non-SEF response (Fig. 6.8b), we note that while most of the quadrants only have signal at the primary eyewall, noticeable outer updrafts are found in the US and UR

119 quadrants. As shown in Fig. 6.1c and 6.1d, the quadrant averaged profiles for these two quadrants do exhibit discernible wind field broadening, where the associated vorticity anomalies induce local updraft response at these quadrants. However, when taking the averaged response of the quadrants, the mean response (not shown) are nearly identical to the axisymmetric response using the azimuthal mean profile, as in Fig. 6.3a-c. This demonstrate the outer updraft response shown in the US and UR quadrants in the Non-SEF group are localized and are not strong enough to yield a significant projection onto the azimuthal mean, unlike the Non-SEF group.

6.3 Conclusions

In this chapter, we examined the tropical cyclone boundary layer response to imposed free- tropospheric forcings that are derived based on observations to investigate the axisymmetric and asymmetric boundary layer response during secondary eyewall formation (SEF). Observed tangential wind composite profiles of storms without SEF, prior to SEF, and after SEF (Non-, Pre- and Post-SEF; Wunsch and Didlake 2018) are used to the force the Kepert (2018) boundary layer model (the K18 model), which is integrated to a steady state.

The axisymmetric Post-SEF profile exhibited the expected secondary wind maximum of the secondary eyewall, and the boundary layer response to this profile further demonstrated the canonical structure of an existing secondary eyewall, with strong maxima of updraft, boundary layer inflow, and convergence. Although the axisymmetric Pre-SEF composite did not have a noticeable secondary wind maximum, the profile exhibited a notably broadened wind field compared to the Non-SEF composite, and was sufficient to induce a clear boundary layer response associated with the incipient secondary eyewall. As expected, the Non-SEF profile did not induce any notable response outside of the primary eyewall.

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To examine the asymmetries in the Pre-SEF composite that are responsible for the inducing the secondary eyewall signal in the axisymmetric simulation, an asymmetric nondivergent wind field is derived from the Pre-SEF composite using optimal interpolation. The nondivergent wind field is then used to drive the K18 model. This asymmetric nondivergent wind field displays a clear broadening of the wind field in the downshear quadrants. At the downwind portion of the broadening, the wind field exhibits a noticeable flattening signature, which is associated with a distinct vorticity band that covers the left-of-shear half of the storm. Using the asymmetric nondivergent forcing, the K18 simulated boundary layer response has a clear band of outer updraft that primarily occupies at the left of shear quadrants. Azimuthal and quadrant averages of the response confirm that updraft and boundary layer inflow signals at the left-of-shear quadrants are clearest among all quadrants, and that the preliminary secondary eyewall signature present in the axisymmetric simulation comes largely from the left-of-shear quadrants.

To further examine the robustness of the asymmetric outer updraft signals in the asymmetric simulation of the Pre-SEF group, a bootstrapping procedure is performed, in which 30 representative bootstrapping sample sets are selected for each quadrant. The mean each of the bootstrap sample set is then used as an axisymmetric forcing for the K18 model. The results show that the outer updraft at the downshear-left and left-of-shear quadrants are significantly stronger than that of the other quadrants, confirming that the updraft signature seen in the asymmetric simulation is a robust signature. This is an intriguing but profound result given that the left-of- shear half is often populated by a broad swath of stratiform precipitation as part of a larger rainband complex (Hence and Houze 2012a), with an accompanying kinematic structure of a mesoscale descending inflow (MDI) that both broadens the tangential wind field and helps build a low-level tangential wind maximum, as consistent with results shown in previous chapters 4 and 5 and

121 previous studies (Didlake and Houze 2013; Didlake et al. 2018). This analysis also indicates that if any feedback process initiating the SEF occurs [as those proposed by Kepert (2013), Kepert and

Nolan (2014), Zhang et. al. (2017)], these are likely to take place earliest in the DL and LS quadrants.

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CHAPTER 7

CONCLUSIONS

7.1. Summary and Discussion

Secondary eyewall formation (SEF) is an internal dynamical process of tropical cyclones

(TCs) that can cause abrupt intensity and structural changes, resulting in temporary hiatus or even reversal in the intensification rate while replacing the storm with a larger eye and broader coverage of gale-force wind. Frequently observed in mature TCs, the dynamics and exact mechanism that initiates SEF has not been fully understood. Studies in recent decades have identified a number of important processes contributing to the occurrence of SEF, including an earlier axisymmetric broadening of the tangential wind field, close association with the asymmetric convections of an intensifying spiral rainband, and the coupling between rainband dynamics and boundary layer processes. However, the dynamical relationship and causal linkages between these processes has not been fully studied. Therefore, this dissertation aims to investigate the interactions between these different components of SEF from both dynamic and thermodynamic perspectives, with a specific focus on the roles of the asymmetric rainband processes in the initiation of SEF.

In chapter 2, we explore the question about how the rainband diabatic forcing can lead to

123 the observed MDI and wind field expansion prior to SEF. This is done in the context of an idealized experiment to examine the one-way response of a dry, hurricane-like vortex to a prescribed stratiform heating profiles that mimic the forcing of a quasi-stationary spiral rainband. The first profile was taken from the study by Moon and Nolan (2010, MN10), which is typical of stratiform precipitation that has heating above and cooling below the melting level. The simulated vortex response displays a mesoscale descending inflow (MDI) and a midlevel tangential jet, consistent with the MN10 study and other previous observational studies. An additional response, which has not been found in previous modelling studies, was an inward spiraling low-level updraft at the radially inward side of the rainband heating. This newly found low-level spiral updraft shares a similar structure with the observed low-level updraft in hurricanes Rita (2005) and Earl (2010).

The second profile was a modified stratiform heating structure derived from observations and consisted of a diagonal dipole of heating and cooling. The same low-level updraft feature was found, but with stronger magnitude and larger vertical extent.

In order to further understand the dynamics that lead to the low-level updraft, we use the generalized Omega equation to diagnose the forcing that drives the vertical velocity response. The results demonstrated that this updraft was driven by buoyancy advection due to the stratiform- induced cold pool at the low level. Sensitivity experiments that have rainband forcings rotating at different rates show that the stationary nature of the rainband diabatic forcing is an important factor in modulating the temperature and pressure anomalies to drive this updraft. Further sensitivity experiment with added moisture and full microphysics shows that this low-level updraft response was robust and could trigger sustained deep convection that could further impact the storm evolution, and thus may play potential roles in secondary eyewall formation.

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Fig. 7.1. A schematic summarizing the impacts of asymmetric rainband processes to the wind and boundary layer thermodynamic fields prior to the onset of SEF. The storm is divided by the environmental wind shear into four quadrants, namely downshear right (DR), downshear right (DL), upshear left (UL) and upshear right (UR). The region between the eyewall and the updraft branch of streamlines designates the moat region, which is largely free of convection. The mesoscale descending inflow (MDI) feature is located at the left of shear quadrants where diabatic cooling due to melting and evaporation of hydrometeors occurs. The meanings of all other symbols are denoted by the legend at the top of the figure.

With the insights learnt from the idealized study, in chapter 3-5 we focus on a secondary eyewall formation in a full-physics model simulation to examine the wind field and updraft response in a more realistic environment. Specifically, we focus on a SEF event that happens in a simulation of Hurricane Matthew (2016) using the WRF-ARW model. To aid the illustration,

Figure 7.1 summarizes the major findings regarding the dynamic and thermodynamic impacts of an asymmetric stationary band complex (SBC) to the SEF process. Prior to SEF, the SBC in the downshear quadrants intensifies and the simulated Matthew experiences an axisymmetric

125 broadening of the tangential wind field, which is associated with an inward-descending acceleration that covers nearly 100 km radius. In the upwind quadrant of the rainband complex, convections are more vigorous and the local acceleration is more upright and concentrated. In the downwind portion of the rainband where precipitations are more stratiform-like, the acceleration is similarly intense but more widespread, with horizontal advection playing a more important role in shaping the slantwise, descending structure of the acceleration.

Quadrant-averaged tangential wind budget at the downwind portion of the rainband showed that this broad slantwise pattern of positive acceleration is due to a mesoscale descending inflow (MDI) that is driven by midlevel cooling within the stratiform regions and draws absolute angular momentum inward. Decomposing the wind field nondivergent and irrotational components, we found that the MDI indeed extended downwind all the way into the upshear-left quadrant. In the upshear-left quadrant, the MDI connects and interacts with the boundary layer, where new convective updrafts are triggered along its inner edge. These new upshear-left updrafts are found to be important to the subsequent spin-up and axisymmetrization of the low-level tangential wind maximum near the secondary eyewall radii.

Given the importance of asymmetric updrafts that emerges after the MDI starts interacting with the boundary layer, in chapter 5 we pay specific attention on the mechanism that leads to their preferred quadrants of emergence and how they are maintained. Noticing the strong spatial correlation between the intense updraft signals and the regions with strong low-level temperature gradient, we perform budgets on the equivalent potential temperature 휃퐸 and the moist stability parameter 휕휃퐸/휕푧 at the updraft regions to understand the thermodynamic structure necessary for sustaining the continuous diabatic heating release there. The budget for the moist stability parameter 휕휃퐸/휕푧 indicates that the moist instability at the strong updraft region has largely been

126 maintained (only slowly varying) throughout the one-hour period. Looking at the balancing terms, we found that the vertical advection constantly advects high-휃퐸 from low level towards upper troposphere, attempting to reduce to local moist instability. The only term that can balance the stabilization effect is the horizontal advection, which constantly destabilizes the atmosphere by having a differential warm-휃퐸 advection that decays with height, tending to maintain the unstable

휃퐸 structure within the column. Thus, we conclude that the destabilization effect from the horizontal advection is essential in maintaining the continuous diabatic heating release in the intense updraft region. Further decomposing the differential horizontal advection into its azimuthal and radial components, we found that in the left-of-shear quadrants, the dominant destabilization effect comes from the azimuthal advection. This azimuthal advection relies on the azimuthal gradient of 휃퐸 that is setup by the cold pool, which is directly underneath the MDI and the associated stratiform cooling at the downwind end of the rainband.

The in-depth studies presented in chapter 4 and 5 about the wind acceleration and the updraft mechanism in the Matthew simulation reveals an asymmetric organization that is driven by external forcing of the environmental wind shear. In order to further examine if such asymmetric organization can be found in real observations, in chapter 6 we examined both the axisymmetric and asymmetric signals of SEF by using a nonlinear boundary layer model (Kepert

2018, the K18 model) with composite observational data (Wunsch and Didlake 2018, WD18). The model simulates the steady-state boundary layer response to the free-tropospheric pressure forcing derived from observed tangential wind fields during SEF. The composite tangential wind profile prior to SEF (the Pre-SEF group) displayed a broadened wind field, while the axisymmetric K18 model response showed a secondary updraft maximum associated with the incipient secondary eyewall. To analyze the asymmetric SEF signal, the K18 model is run in asymmetric mode using

127 a two-dimensional nondivergent wind as forcing. The results showed that, relative to the environmental shear vector, the response in the downshear-left (DL) and left-of-shear (LS) quadrants showed the clearest secondary peaks of updraft, tangential jet, and inflow, displaying early indicators of the incipient secondary eyewall. Sensitivity experiments using bootstrapping technique further confirms that these asymmetric outer updraft signal seen in the DL to LS quadrants are robust features. These findings have two important implication. First, since the K18 model simulates only the one-way interaction of boundary layer updraft to the forcing at the boundary layer top, the outer updraft response seen in the left-of-shear quadrants demonstrates that how the boundary layer processes can feedback to the forcing in the free troposphere by triggering more updrafts at the secondary eyewall radii. Second, these results demonstrated that under the asymmetric forcing imposed by the environmental wind shear, the earliest signal of SEF indeed emerges first near the left-of-shear half of the storm, which is consistent with the asymmetric organizations in the wind field and updraft that we demonstrated in the Matthew simulation.

7.2. Future Work and Broader Impacts

A number of the research findings discussed above are in fact closely linked to previous investigations regarding the SEF problem. The insights provided by some of the findings may also highlight certain unexplored areas that deserve further attention from the research community, and thus provide a broader contribution towards a more comprehensive understanding of the SEF process in general.

7.2.1 SEF in sheared versus in quiescent environment

The present study identified and quantified the role of asymmetric rainband processes in

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SEF for TCs in a vertically sheared environment, and we have demonstrated in chapter 6 that similar asymmetric organization can generally be observed in the storm-scale wind field of real

SEF cases. However, a number of idealized modelling studies (Qiu and Tan 2013; Abarca and

Montgomery 2014; Wang et al. 2019; Chen et al. 2018; Chen 2018; Wang and Tan 2020; etc.) also demonstrated that SEF can occur in quiescent environment without the present of environmental mean wind and shear. In fact, many of these studies (Qiu and Tan 2013; Wang et al. 2019; Chen et al. 2018; Chen 2018; Wang and Tan 2020) also attribute the onset of SEF to the dynamics at the downwind portion of the TC rainband, even though the rainbands in those experiments are externally forced. Given the general importance of rainband processes in SEF, it would be valuable to further analyze if there are noticeable differences in the rainband dynamics, such as the rainband rotation rate, the magnitude and placement of the MDI and boundary layer cold pool, in SEF events that occur in a quiescent environment. The answer to these questions could provide important insight about the environmental influence that is conducive to the onset of the SEF process.

7.2.2 The spatial and temporal scales of the thermodynamic asymmetry

The investigation on the boundary layer thermodynamic asymmetry in chapter 5 show that low-level thermodynamic gradient plays an important role in sustaining the low-level updrafts, which provides strong potential vorticity (PV) forcing that eventually leads to the subsequent SEF.

However, most of the current SEF theories that rely on boundary layer processes mainly focus on the dynamical effect of boundary layer processes, with little attention placed on the importance of boundary layer thermodynamics. The findings presented in chapter 5 thus motivates an important question about the minimum thresholds of the strength, length and time scales of the thermodynamic asymmetries in order for axisymmetrization of the secondary eyewall to take place.

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The answers to this unexplored question may potentially yield direct indications about the onset timing and radial location of the secondary eyewall formation.

7.2.3 Wave-mean-flow interaction theory

One of the ongoing works following the research presented in chapter 5 is a PV budget, which shows that the diabatic heating associated with those intense updrafts generates sufficient low-level PV that is immediately brought upward to the mid-to-upper troposphere in the form of eddy PV fluxes. The convergence of these fluxes is the major process that allows these eddies to be converted into azimuthal mean, i.e., the axisymmetrization of the secondary eyewall. Given these preliminary findings, it seems that wave-mean-flow interaction theory in TCs (Menelaou et al. 2013), as related to the Eliassen-Palm Flux divergence, could be a useful tool to more quantitatively assess the conversion process from wave activity into the mean flow. However, the current theoretical framework proposed by Menelaou et al. (2013) is under 휎-coordinate (using potential temperature as vertical coordinate), which may not be optimal for analyzing strongly convective storms, such as tropical cyclones. Extension of that theoretical framework into more appropriate vertical coordinate would be necessary to make the above proposed assessment feasible.

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APPENDIX A

RECONSTRUCTION OF THE RAINBAND DIABATIC HEATING STRUCTURE FROM SECONDARY CIRCULATION IN HURRICANE RITA

In the chapter, we present the procedure of reconstructing the modified stratiform rainband heating from the secondary circulation observed in Hurricane Rita (2005) using the Sawyer

Eliassen equation (Eliassen 1951). The circulation data used is the azimuthally-averaged radial and vertical velocity (푢, 푤) fields (Fig. 1.2b) from Leg 1 presented in DH13. A detailed description of the dataset can be found in section 2 of DH13. We used the Sawyer Eliassen equation in 푧-coordinates derived by Pendergrass and Willoughby (2009):

휕푏 휕2휓 휕푏 휕2휓 휕2휓 − 2 + 퐼2 휕푧 휕푟2 휕푟 휕푧휕푟 휕푧2

퐼2 1 휕푏 3휉푆 휕훾 휕휓 − ( − − − 푁2 ) A.1 퐻휌 푅휌 휕푟 푟 휕푟 휕푧

휕푄 휕푄 휕(휉푀) = 푟휌 ( + 훾 − ) 휕푟 휕푧 휕푧 where 휓 is the mass streamfunction of the secondary circulation; 푏 is buoyancy; 퐼2 is the inertial stability; 푅휌 and 퐻휌 are radial and vertical scale lengths of density 휌; 훾 is the ratio of radial pressure gradient force to gravitational acceleration; 푆 is the vertical shear of the background wind

2푣 푣 ; 휉 = 푓 + 0 is twice the absolute angular velocity; 푀 and 푄 are the momentum and heat 0 푟

131 sources, respectively.

The density field from the WRF model (as in chapter 2) after the one-day spin-up simulation is used to compute the mass flux (휌푢, 휌푤). To avoid the impact of frictional dissipation from boundary layer, all data below 2 푘푚 altitude are removed. To obtain the mass streamfunction

휓, the mass flux is decomposed into nondivergent (휌푢휓, 휌푤휓) and irrotational (휌푢휒, 휌푤휒) parts following Bijlsma et al. (1986):

휕 1 휕휓 휕2휓 푟 ( ) + = −푟휁 휕푟 푟 휕푟 휕푧2

1 휕 휕휒 휕2휒 A.2 (푟 ) + = 훿 푟 휕푟 휕푟 휕푧2

(푢휓, 푤휓)|휕 + (푢휒, 푤휒)|휕 = (푢, 푤)|휕 ; 휒|휕 = 0

휕휓 휕휓 where 휓 is the mass streamfunction satisfying = −푟휌푢 , = 푟휌푤 ; 휒 is velocity potential 휕푧 휓 휕푟 휓

휕휒 휕휒 휕휌푢 휕휌푤 1 휕 휕휌푤 of (휌푢 , 휌푤 ) satisfying = 휌푢 , = 휌푤 ; 휁 = − and 훿 = (푟휌푢) + are the 휒 휒 휕푟 휒 휕푧 휒 휕푧 휕푟 푟 휕푟 휕푧 vorticity and divergence of (휌푢, 휌푤) on the r-z plane; ∙ |휕 denotes evaluation at the boundary of the computation domain.

Results indicate that the irrotational part is small compared to the nondivergent part and the structure of 휓 is insensitive to small variations of 휌. 휓 is then passed to the (A.1) to compute the total forcing term. To obtain the heating structure, the second and third terms in the right-hand- side of (A.1) are neglected, because the frictional effect is generally small in the computational domain and 훾 is small except near the radius of maximum wind (confirmed by a scale analysis not

휕푄 shown). The forcing 푟휌 is then integrated radially inward from the outer boundary at 푟 = 휕푟

190 푘푚 to obtain the heating structure 푄. Such a radial integration requires an initial vertical profile of heating 푄0(푧) at 푟 = 190 푘푚. Since in general the diabatic forcing is concentrated more

132 at smaller radii in a tropical cyclone, this vertical heating profile 푄0(푧) at the outer boundary (푟 =

190 푘푚) is assumed to be zero (i.e., 푄0(푧) = 0).

Fig. A1 shows the reconstructed diabatic forcing 푄. Several features are prominent. It has a predominant heating region near 6 푘푚 altitude at the inner side, a low-level cooling region near and below 4 푘푚 altitude at the outer side, and the two of which form a clear diagonal pattern. We developed our modified stratiform heating profile (Fig. 2.1e) to capture the general pattern and prominent features of the reconstructed heating. Smoothed patterns were preferred to allow for a meaningful comparison with the MN10 heating profile. The exact structure for the modified heating is available in the supplementary materials.

Fig. A1. Reconstructed diabatic heating using the Sawyer Eliassen equation based on the observed secondary circulation within the stratiform rainband of Hurricane Rita (Fig. 1.2b). Zero contour is highlighted with thick solid black line.

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APPENDIX B

STORM-RELATIVE TANGENTIAL WIND BUDGETS

Storm-relative tangential wind budgets are often calculated in many studies by adapting the cylindrical coordinate tangential momentum equation to storm-relative tangential winds. But such an approach does not fully account for the effect of storm motion. Here, we re-derive the storm-relative tangential wind momentum equation, with particular interest in using for storm sector budget calculations and for storms with appreciable translation speed. We begin with horizontal momentum equation

퐷풗 1 = − 훁푝 − 푓푘̂ × 풗 + 푭 (B.1) 퐷푡 휌

퐷 휕 휕 where = + 풗 ∙ 훁 + 푤 is Lagrangian derivative; vector 풗 = (푢, 푣) is the horizontal wind, 퐷푡 휕푡 휕푧 with 푢 and 푣 being the radial and tangential components; 푭 is the external forcing. Next,

휕 휕 defining the storm-following local time derivative as 푐 = + 풗 ∙ 훁, where 풗 = (푢 , 푣 ) is the 휕푡 휕푡 풄 풄 푐 푐 storm translation vector, equation (B.1) becomes

휕푐풗 휕풗 1 = −(풗 − 풗 ) ∙ ∇풗 − 푤 − 훁푝 − 푓푘̂ × 풗 + 푭 (B.2) 휕푡 풄 휕푧 휌

We then take the dot product of (B.2) with the unit vector along the azimuthal unit vector 휆̂:

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휕푐푣 휕푣 1 휕푝 = −((풗 − 풗 ) ∙ ∇풗) ∙ 휆̂ − 푤 − − 푓푢 + 퐹 (B.3) 휕푡 풄 휕푧 푟휌 휕휆 휆

̂ Using product rule, −((풗 − 풗풄) ∙ ∇풗) ∙ 휆 can be rewritten into two terms

̂ ̂ −((풗 − 풗풄) ∙ ∇풗) ∙ 휆 = −(풗 − 풗풄) ∙ 훁푣 + (풗 − 풗풄) ∙ 훁휆 ∙ 풗 (B.4)

The second term arises due to curvature of the polar coordinate

̂ ̂ 휕휆 (푣 − 푣푐) 휕휆 (푣 − 푣푐) (풗 − 풗 ) ∙ 훁휆̂ = ((푢 − 푢 ) + ) ∙ 풗 = − 푢 (B.5) 풄 푐 휕푟 푟 휕휆 푟

휕휆̂ 휕휆̂ where we use the fact that = 0 and = −푟̂. Hence, 휕푟 휕휆

휕 푣 (푣 − 푣 ) 휕푣 1 휕푝 (B.6) 푐 = −(풗 − 풗 ) ∙ 훁푣 − 푐 푢 − 푤 − − 푓푢 + 퐹 휕푡 풄 푟 휕푧 푟휌 휕휆 휆

Defining storm-relative wind as 풗푆푇 = 풗 − 풗풄 = (푢푆푇, 푣푆푇) and storm-relative absolute angular

1 momentum as 푀 = 푟푣 + 푓푟2, (B.6) may be rewritten as 푆푇 푆푇 2

휕 푣 푢 휕푀 푣 휕푣 휕푣 푣 푣 휕(푣 ) 1 휕푝 푐 푆푇 = − 푆푇 푆푇 − 푆푇 푆푇 − 푤 푆푇 + [− 푆푇 푢 − 푆푇 푐 ] − + 퐹 휕푡 푟 휕푟 푟 휕휆 휕푧 푟 푐 푟 휕휆 푟휌 휕휆 휆 (B.7) 휕 푣 − 푓푢 − 푐 푐 푐 휕푡

휕(푣 ) 휕(풗 ∙휆̂) 휕(휆̂) Note that because 푐 = 풄 = 풗 ∙ = −풗 ∙ 푟̂ = −푢 , the two terms inside the square 휕휆 휕휆 풄 휕휆 풄 푐 bracket cancel. Equation (4.3) in section 4.4 is derived by taking sector averages to equation

(B.7)

휕 푣̿ 푢̿̿̿̿̿̿휕̿̿푀̿̿̿̿ 푣̿̿̿̿̿̿휕̿푣̿̿̿̿ ̿̿̿휕̿̿푣̿̿̿̿ ̿1̿̿̿휕푝̿̿̿ 휕 푣̿ 푐 푆푇 = − 푆푇 푆푇 − 푆푇 푆푇 − 푤 푆푇 − + 퐹̿ − 푓̿̿푢̿̿̿ − 푐 푐 (B.8) 휕푡 푟 휕푟 푟 휕휆 휕푧 푟휌 휕휆 휆 푐 휕푡

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APPENDIX C

VERIFICATION FOR THE FORCING OF 휃퐸

The Bolton (1980) formulation of 휃퐸 (equation 5.1) is a relatively accurate, regression- based expression of equivalent potential temperature, assuming the phase change occurs pseudo- adiabatically between liquid and vapor. However, because the Bolton’s formulation involves regression of the exact equivalent potential temperature (solved by numerical iteration) against mixing ratio, the forcing for his formulation may not have a simple expression that conveys clean physical meaning. On the other hand, Rotunno and Emanuel (1987) also illustrated a simple derivation of the conventional formulation of equivalent potential temperature (denoted as 휃̂퐸), which has a relatively simple forcing expression. This forcing formula, therefore, maybe used to understand the structure of the residual forcing in the 휃퐸 budget that uses Bolton’s formulation.

Following the derivation in Rotunno and Emanuel (1987), the conservation law for 휃̂퐸 =

휃 exp(퐿푣푞/푐푝푇) can be derived as follows. Beginning from the first law of thermodynamics and the conservation law of water vapor

푑휃 = 휃̇ + 휃̇ (C.1) 푑푡 푚푝 푝푏푙,푟푎푑

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푑푞 = 푞̇ + 푞̇ (C.2) 푑푡 푚푝 푝푏푙

where 휃 is the dry potential temperature; 푞 is mixing ratio for water vapor; 휃푚푝̇ and 푞̇푚푝 are diabatic forcing for 휃 and 푞 from microphysics; 휃̇푝푏푙,푟푎푑 is the sum of forcing for 휃 from boundary layer scheme and radiation; 푞̇푝푏푙 is the forcing for 푞 from boundary layer scheme.

퐿푣 Multiplying (C.2) with (퐿푣 is latent heat of vaporization, 푐푝 is specific heating for dry 푐푝푇 air at constant pressure; 푇 is temperature), and (C.1) with 1/휃, and then adding the result together yields

1 푑휃 퐿 푑푞 1 퐿푣 + = (휃푚푝̇ + 휃̇푝푏푙,푟푎푑) + (푞̇푚푝 + 푞̇푝푏푙) (C.3) 휃 푑푡 푐푝푇 푑푡 휃 푐푝푇

Ignoring the variation of temperature 푇, combining the two terms on the right-hand side yields

퐿푞 ̂ 푑 ln (휃 exp ( )) 푑 ln 휃퐸 푐푝푇 1 퐿푣 (C.4) = = (휃푚푝̇ + 휃̇푝푏푙,푟푎푑) + (푞̇푚푝 + 푞̇푝푏푙) 푑푡 푑푡 휃 푐푝푇

Multiplying both sides with 휃̂퐸, we have

̇ 푑휃̂퐸 휃̂퐸 퐿푣휃̂퐸 휃푚푝 퐿푣푞̇푚푝 = ( 휃̇푝푏푙,푟푎푑 + 푞̇푝푏푙) + 휃̂퐸 ( + ) (C.5) 푑푡 휃 푐푝푇 휃 푐푝푇

If irreversible process is not involved, then 휃̇푝푏푙,푟푎푑 = 푞̇푝푏푙 = 0; and if microphysical processes

휃̇ 퐿 푞̇ happen pseudo-adiabatically between liquid and vapor, then 푚푝 + 푣 푚푝 = 0 . Therefore, 휃 푐푝푇

̇ 휃̂퐸 퐿푣휃̂퐸 휃푚푝 퐿푣푞̇푚푝 separately inspecting the two forcing terms (i.e., ( 휃̇푝푏푙,푟푎푑 + 푞̇푝푏푙) and 휃̂퐸 ( + )) 휃 푐푝푇 휃 푐푝푇 can tell us respectively about 1) the source of external forcing from boundary layer and radiation

137 schemes; 2) how well is 휃̂퐸 being conserved in the microphysical process.

휃̇ 푚푝 퐿푣푞̇푚푝 Fig. C1. (a)-(e) Cross-section of forcing terms of 휃̂퐸: (a) microphysics: 휃̂퐸 ( + ); (b) 휃 푐푝푇 휃̂퐸 퐿푣휃̂퐸 boundary layer and radiation schemes ( 휃̇푝푏푙,푟푎푑 + 푞̇푝푏푙); (c) total forcing for 휃̂퐸; (d) 휃 푐푝푇 휃̂퐸 퐿푣휃̂퐸 휃푚푝̇ ; (e) 푞̇푚푝. (f) Forcing for Bolton’s formulation (휃퐸), computed as residual. 휃 푐푝푇

Figure C1a-d shows each forcing terms on the left-hand side of equation, as well as the two

휃̂퐸 휃̂퐸퐿푣 separate microphysics terms, 휃푚푝̇ and 푞̇푚푝 . All terms are hourly averaged using 1-min 휃 푐푝푇

WRF output at hour 20, and azimuthally averaged over the same sector shown in Fig. 5.2b. Shown

휃̇ 푚푝 퐿푣푞̇푚푝 in Fig. C1a is the total forcing term from microphysics 휃̂퐸 ( + ). It is clear that the most 휃 푐푝푇 of the non-zero forcing occur above 4 km altitude, where ice-phase microphysical processes are involved. Looking into the two contributing terms individually (as in Fig. C1d and e), we see that the conversion between latent energy and entropy forcing indeed strongly cancelling each other,

138 indicating that a large part of the microphysical process indeed conserves 휃̂퐸, especially below 4 km altitude where only liquid and vapor phases are not involved. Forcing from boundary layer scheme and radiation scheme is concentrated at the boundary layer (Fig. C1b), which is expected

(forcing from radiation is small). The sum of forcings from microphysics, boundary layer and radiation schemes is shown in Fig. C1c.

Figure C1f shows the hourly averaged residual forcing of 휃퐸 using Bolton’s formulation at

푑휃 hour 20, sector averaged the same quadrant. The residual forcing 휃̇ is computed as 휃̇ = 퐸 = 퐸 퐸 푑푡

휕휃 퐸 + 풗 ⋅ 훁휃 . Value of 휃 is computed within the WRF model and tracked every time step to 휕푡 퐸 퐸

휕휃 compute its local tendency 퐸. Comparing Fig. C1f and C1c, we see that the analytic expression 휕푡 of forcing for 휃̂퐸 (right-hand-side of eq. C.5) is able to explain the major features in 휃̇퐸 .

Differences do exist near 10 to 30 km radii where updrafts are strong. Part of these difference is attributed to the fact that the derivation of 휃̂퐸 neglected the variation of temperature 푇 following the air parcel, while the assumption is not explicitly made in the Bolton’s formulation of 휃퐸.

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Vita

CHAU LAM YU

Pennsylvania State University Phone: +1 8019063823 Department of Meteorology and Atmospheric Science Email: [email protected] 402 Walker Building University Park, PA 16802

EDUCATION

Penn State University 2017 – present Ph.D. in Atmospheric Sciences University of Utah 2015 – 2017 Master in Atmospheric Sciences University of Reading 2012 – 2013 Master in Mathematics: Numerical modeling of the Atmosphere and Oceans University of Hong Kong 2009 – 2012 Bachelor of Science, double majors in Mathematics and Physics

PUBLICATIONS

Yu, C.-L., A. C., Didlake Jr., and F. Zhang (2020), Secondary eyewall formation in hurricane Matthew (2016): The impact of stratiform cooling on boundary layer thermodynamics. In preparation. Yu, C.-L., A. C., Didlake Jr., F. Zhang, and J. D., Kepert (2020), Investigating axisymmetric and asymmetric signals of secondary eyewall formation using observations-based modeling of the tropical cyclone boundary layer. In preparation. Yu, C.-L., A. C., Didlake Jr., F. Zhang and R. G. Nystrom (2020), Asymmetric rainband processes leading to secondary eyewall formation in a model simulation of Hurricane Matthew (2016). Submitted. Yu, C.-L., and A. C., Didlake Jr. (2019), Impact of stratiform rainband heating on the tropical cyclone wind field in idealized simulations. J. Atmos. Sci., 76, 2443–2462. Pu, Z., C.-L. Yu, V., Tallapragada, J., Jin, and W., McCarty (2019), The Impact of Assimilation of GPM Microwave Imager Clear-Sky Radiance on Numerical Simulations of Hurricanes Joaquin (2015) and Matthew (2016) with the HWRF Model, Mon. Weather Rev., 147, 175-198. Yu, C.-L. and M. A. C., Teixeira (2015), Impact of non-hydrostatic effects and trapped lee waves on mountain wave drag in directionally sheared flow. Quart. J. Roy. Meteorol. Soc., 141 (690). pp. 1572-1585. ISSN 1477-870X doi: 10.1002/qj.2459 Teixeira, M. A. C. and C.-L. Yu (2014), The gravity wave momentum flux in hydrostatic flow with directional shear over elliptical mountains. European Journal of Mechanics & Fluids B: Fluids, 47. pp. 16-31. ISSN 0997-7546 doi: 10.1016/j.euromechflu.2014.02.004