Secondary circulation of tropical cyclones in vertical wind shear: Lagrangian diagnostic and pathways of environmental interaction.

Transcription

1 Generated using version 3.2 of the official AMS L A TEX template 1 Secondary circulation of tropical cyclones in vertical wind shear: 2 Lagrangian diagnostic and pathways of environmental interaction 3 Michael Riemer, Institut für Physik der Atmosphäre, Johannes Gutenberg-Universität Mainz, Mainz, Germany 4 Frédéric Laliberté Department of Physics, University of Toronto, Toronto, ON, Canada Corresponding author address: Michael Riemer, Institut für Physik der Atmosphäre, Johannes Gutenberg-Universität Mainz, Mainz, Mainz, Germany. mriemer@uni-mainz.de 1

2 ABSTRACT This study introduces a Lagrangian diagnostic of the secondary circulation of tropical cyclones (TCs), here defined by those trajectories that contribute to latent heat release in the region of high inertial stability of the TC core. This definition accounts for prominent asymmetries and transient flow features. Trajectories are mapped from the three-dimensional, physical space to the (two-dimensional) entropy-temperature space. The mass flux vector in this space subsumes the thermodynamic characteristics of the secondary circulation. The Lagrangian diagnostic is then employed to further analyze the impact of vertical wind shear on TCs in previously-published, idealized numerical experiments. One focus of our analysis is the classification and quantitative depiction of different pathways of environmental interaction based on thermodynamic properties of trajectories at initial and end time. Confirming results from previous work, vertical shear significantly increases the intrusion of low-θ e air into the eyewall through the frictional inflow layer. In contrast to previous ideas, vertical shear decreases mid-level ventilation in our experiments. Consequently, the difference in eyewall θ e between the no-shear and shear experiments is largest at low levels. Vertical shear, however, significantly increases detrainment from the eyewall and modifies the thermodynamic signature of the outflow layer. Finally, vertical shear promotes the occurrence of a novel class of trajectories that has not been described 23 previously. These trajectories lose entropy at cold temperatures by detraining from the outflow layer and subsequently warm by K. Further work is needed to investigate in more detail the relative importance of the different pathways for TC intensity change and to extend this study to real-atmospheric TCs. 1

3 27 1. Introduction 28 a. Intensity change and shear-induced ventilation Predicting intensity changes of tropical cyclones (TCs) remains an important problem for operational forecast centers. While it is now apparent that intensity change is associated with appreciable inherent forecast uncertainty (e.g. Nguyen et al. 2008; Zhang and Tao 2013), insufficient understanding of the governing physical and dynamical processes still hamper our ability to identify limits of predictability and to produce the best forecasts that are physically possible. Vertical shear of the environmental horizontal winds (hereafter simply referred to as vertical shear ) is one main environmental contributor to intensity change (e.g. DeMaria and Kaplan 1999). The long-standing and currently prevailing view is that vertical shear affects TC intensity by promoting ventilation (Simpson and Riehl 1958; Frank and Ritchie 2001; Tang and Emanuel 2010; Riemer et al. 2010, 2013). The term ventilation usually describes the intrusion of environmental, low-entropy air into the TC s inner core but the literature lacks a clear definition. Fundamental to this prevailing view is the idea that TC intensity is governed to lowest order by the TC s heat engine (Riehl 1954, pg. 320ff). Assuming that the heat engine works akin to a Carnot cycle, Emanuel (1986, 1991) derived a theory for the upper bound of TC intensity. It is the TC s secondary circulation that transports the working fluid of the heat engine. Emanuel s theory makes the important assumption of a steady state and TC axisymmetry. Under these assumptions, the TC s secondary circulation can be depicted by the well-known in-up-out pattern of streamlines in the radius height (r z) plane (e.g. Fig in Riehl 1954). These streamlines denote air parcel trajectories and, therefore, a part of the assumed Carnot cycle. The simple picture of the secondary circulation breaks down in the presence of pronounced asymmetries and/ or time dependence of the flow. Vertical shear has long been recognized 2

4 to induce distinct asymmetries in the TC structure. These asymmetries affect in particular the secondary circulation, namely the eyewall convection (e.g. Reasor et al. 2000; Corbosiero and Molinari 2002; Rogers et al. 2003; Braun et al. 2006) and the low-level radial inflow (e.g. Riemer et al. 2013; Zhang et al. 2013). Often, vertical shear induces also temporal changes in TC structure and intensity that cast into doubt the validity of the stationarity assumption. It thus has to be expected that streamlines in the r z plane are a poor representation of air parcel trajectories in vertically-sheared TCs. A succinct analysis of the secondary circulation in asymmetric, time-evolving TCs is therefore necessary to examine ventilation associated with vertical shear in more detail. Even though the concept of ventilation is more than half a century old, there remain many important open questions about this concept. Distinct ventilation pathways are described in the literature but the relative importance of these pathways for TC intensity change 65 is unclear. It can be expected that the characteristics of different pathways depend on environmental conditions, e.g. the vertical profiles of wind and entropy, and on characteristics of the TC itself, e.g. intensity and the radial wind profile. So far, such relationships have not been addressed. The original idea of ventilation is that environmental air is brought into the TC circulation by the storm-relative radial flow at mid-tropospheric levels (4 5 km height, 70 Simpson and Riehl 1958). The importance of this mid-level pathway, however, may be considerably limited due to the significant deflection of intruding air masses by the swirling winds (Riemer and Montgomery 2011, pg. 9396, and references therein). Frank and Ritchie (2001) proposed that vertical shear erodes the TC s warm core in the upper troposphere. Such an upper-level pathway seems viable because the rotational constraint is much smaller than in the middle troposphere. However, Tang and Emanuel (2012) demonstrate that, from the thermodynamic perspective, ventilation at upper-levels is rather inefficient 1 thus putting the importance of such a pathway into question. Most recently, Riemer et al. (2010, 2013) 1 We note that Frank and Ritchie (2001) did not base their arguments on the framework of the TC heat engine. 3

5 used idealized numerical experiments to demonstrate that shear-induced downdrafts may significantly reduce entropy in the TC s inflow layer. While the air in the inflow layer spirals towards the eyewall updrafts, its entropy will be replenished by surface enthalpy fluxes. It has however not been fully determined whether replenishment is small enough to make this low-level pathway a leading-order process that governs intensity evolution. 83 b. A Lagrangian perspective of the secondary circulation To address these open issues, we present here a Lagrangian description of the secondary circulation (Sec. 2). From an axisymmetric perspective, the secondary circulation can generally be considered to comprise all of the flow in the r z plane. As motivated above, the focus here is on ventilation, i.e. the systematic dilution of the inner-core updrafts by environmental, low-entropy air. For our analysis, we define inner-core updrafts based on the notion that the contribution of latent heat release to the maintenance or intensification of the primary circulation increases significantly with increasing inertial stability (I 2, e.g. Schubert and Hack 1982; Nolan et al. 2007, I 2 (ζ + f 0 ) (f 0 + 2v/r), where ζ is relative vorticity, f 0 the (constant) Coriolis parameter, v the tangential wind component, and r radius). In the following, the only part of the secondary circulation that will be considered is the part where trajectories contribute to latent heat release in the high inertial-stability region of the inner core. As a proxy for latent heat release we use (near-)saturated ascent and the threshold for I 2 is set high enough to exclude contributions from outer rain bands. The time period over which air parcels are tracked is set to 12 h, a typical time scale over which intensity changes occur in vertically-sheared TCs. In the cases considered below, the precise choice of this time period is not crucial. To diagnose the large number (approximately half a million) of considered trajectories a thermodynamic transform (Kjellsson et al. 2014; Laliberté et al. 2015, here described in 102 Sec. 3) is performed. This transformation maps the trajectories from three-dimensional 103 physical space to two-dimensional entropy temperature space. By depicting the mass flux 4

6 vector in this thermodynamic space, the secondary circulation can be examined in entropy temperature diagrams familiar from the study of thermodynamic processes. In Section 4, this framework is applied to study the impact of vertical shear on the secondary circulation in the previously-published, idealized numerical experiments of Riemer et al. (2010). Using simple thermodynamic criteria, first strides towards objectively identifying distinct ventilation pathways are made in Sec. 5. A summary of our results concludes this study (Sec. 6) Numerical experiment and trajectory calculation This study further analyzes the idealized numerical experiments of Riemer et al. (2010), hereafter referred to as RMN10, from the Lagrangian perspective. The next subsection first provides a brief summary of these experiments and of the employed numerical model. For more information concerning these matters, the reader is referred to RMN10. Subsection 2b then details the trajectory calculation, on which the results of the current study are based. 117 a. Idealized numerical experiment of TCs in vertical shear The idealized numerical experiments of RMN10 revisited the now-standard experimental setup of Bender (1997) and Frank and Ritchie (2001), in which a vertical-shear flow is superimposed on a model TC after the vortex spin-up phase (here: after 48 h). The experiments are on an f-plane over a constant sea surface temperature of 28.5 C. When shear is imposed, the TCs in RMN10 are comparable to category 3 hurricanes, with an intensity of 68 m s 1 (Fig. 1). The intensity metric used here (and in RMN10) is the maximum of the azimuthally-averaged tangential wind at 1 km height. In addition to the high intensity, the TCs exhibit a rather broad radial profile (see Fig. 4 in RMN10). It can thus be expected that the TCs in this experimental setup possess relatively high inertial stability. 127 The current study examines the 15mps and 20mps shear case of RMN10. 5 In these

7 experiments the magnitude of the shear flow, as described by the difference in wind speed at the surface and at 12 km height, is 15 and 20 m s 1, respectively. The shear profile is unidirectional with a cosine structure in the vertical. In both cases, the TCs are resilient and, after a period of weakening, re-intensify in the later stage of the experiment (Fig. 1). This study focuses on the initial weakening phase. The time when vertical shear is imposed in the shear experiments defines the reference time 0 h. In the benchmark experiment, the no-shear case, the TC continues the intensification of the spin-up phase and reaches an intensity of 88 m s 1 at 16 h. The TC in the 15mps case also continues to intensify at first and then starts to weaken significantly at 8 h, reaching a minimum intensity of 62 m s 1 at 14 h. The TC in the 20mps case starts weakening at 2 h reaching a minimum intensity of 51 m s 1 at 18 h. RMN10 s idealized experiments were performed with the Regional Atmospheric Modeling System (RAMS Pielke and Coauthors 1992; Cotton and Coauthors 2003). RAMS is a stateof-the-art, three-dimensional, non-hydrostatic numerical modeling system. The model has been run in a two-way interactive multiple nested grid configuration with a horizontal grid spacing of 5 km in the inner-most nest and 38 vertical levels were employed with the lowest half-level at 49 m. At the lower boundary, the vertical velocity is required to vanish. The vertical grid spacing is 100 m between the lowest two levels and increases with height by a stretching factor of Above 17 km height, a Rayleigh friction layer was included to minimize the reflection of gravity waves from the top of the model, which resides at 24.6 km. Deliberately, RMN10 employed a simple set of parameterizations. Notably, radiation was neglected in these experiments for simplicity. As an important consequence for the current study, there is no entropy sink in the outflow layer. Cloud microphysics are represented by a warm-rain scheme (Kessler 1969). Due to the lack of ice microphysics, the increase of entropy above the freezing level associated with the latent heat release of sublimation (e.g. Fierro et al. 2009) is neglected. Vertical and horizontal subgrid-scale mixing is represented by a standard first-order turbulence scheme (Smagorinsky 1963), with modifications by Lilly 6

8 (1962) and Hill (1974) that enhance diffusion in unstable conditions and reduce diffusion in stable conditions. This scheme can be expected to be very active in the frictional boundary layer and in the eyewall region. Importantly, turbulent mixing constitutes a source/ sink term of entropy along trajectories of the resolved flow. Enthalpy and momentum fluxes from 159 the underlying ocean surface are parameterized by the bulk aerodynamic formulas. The exchange coefficient of enthalpy is set equal to the drag coefficient, which is calculated using Deacon s formula. Employing a more realistic representation of the exchange coefficients and ice microphysics does not have a qualitative impact on the results of the idealized experiments (Riemer et al. 2013). 164 b. Calculating trajectories of the secondary circulation The trajectories diagnosed in this study are calculated as offline trajectories from stored model output. Data is available every 360 s. A second-order Runge-Kutta scheme with a time step of 5 s and linear interpolation in space and time has been employed. The 5 s time step virtually yields the same results as a time step of 15 s. It can thus be assumed that, with the available data, the solutions for the trajectory calculations have converged with the employed time step. The trajectory results are stored at the same dates as the available model data, i.e. every 360 s. To represent the secondary circulation, as introduced in Sec. 1b, the seeding locations of the trajectories need to fulfill three criteria: robust ascent, near saturation, and high inertial stability. The criteria are defined by thresholds that need to be exceeded in vertical motion, relative humidity, and inertial stability, namely 1 m s 1, 0.99, and 10 6 s 2, respectively. The sensitivity of our results to the threshold of relative humidity between values of 0.9 and 1 is negligible. Our results are robust also to reasonable choices of the other two thresholds, examined between m s 1 and s 2, respectively. Thresholds towards the lower end of these ranges, however, appear to partly include trajectories from rain bands also so that the results in these cases exhibit somewhat more noise (not shown). 7

9 The horizontal distribution of seeding locations is illustrated for the no-shear and the 20mps case at 6 h in Fig. 2a) and b), respectively. In the no-shear case, the distribution exhibits a large degree of axisymmetry and clearly indicates the eyewall and two inner rainbands. In the 20mps case, representative for the shear cases, the distribution exhibits a prominent wavenumber-1 pattern with a distinct maximum in the downshear-left quadrant. This bias in seeding locations reflects the well-known, shear-induced asymmetry of the innercore convection. A small portion of the air parcels are seeded within the updrafts of the stationary band complex in the downshear semicircle at a radius beyond 50 km (see RMN10 and Riemer and Montgomery (2011) for a more detailed discussion of the shear-induced convective activity outside of the eyewall in this experimental setup). The current study investigates the fate of air parcels rising within the given asymmetric inner-core convection and does not attempt to further examine the causes of these asymmetries. 193 Trajectories are initialized on a horizontal grid of km. The lowest vertical 194 level considered is at 835 m, which is near the top of the frictional inflow layer. In the vertical, trajectories are initialized every 12.5 hpa up to a height of 12 km, which is within the outflow layer in this experimental setup. The vertical distribution of the seeding locations is illustrated in r z sections in Fig. 2c) and d). Above 3 km height, both distributions tilt radially outward with height, consistent with the general outward tilt of the eyewall. In the 20mps case, a small portion of the air parcels are seeded at relatively large radii at low levels and at small radii at upper levels. The number of seeding locations is fairly constant with height. Air parcels with the dimensions given above possess a mass of kg. Seeding in pressure coordinates renders the trajectories divergence-free in physical space due to continuity. With the given spatial resolution and the above thresholds, O(10 5 ) trajectories are computed. From their seeding locations, trajectories are integrated forwards and backwards in time for 6 h. Hereafter, the end of the backward integration will be referred to as initial time. At the end of the forward and backward integrations, the bulk of the trajectories are located 8

10 outside a radius of 200 km (not shown). Trajectories are seeded at 6 h and 8 h in all cases, and also at 10 h in the shear cases. Each seeding time actually comprises 5 calculations starting at 5 subsequent 360 s steps. This procedure is designed to increase the representativeness of our results for the respective seeding time. The results presented below thus represents information from approximately half a million trajectories. As with all offline trajectories calculated from stored model data, the question of how accurately these trajectories represent model trajectories arises. Our temporal resolution of 360 s is marginally sufficient to represent transient convective activity, including downdrafts. Our focus, the environmental interaction of the TCs, somewhat alleviates the need for very high temporal resolution. Details of the inner-core convection will not be considered. Our Lagrangian framework reproduces several features that are expected from previous studies and distinct differences between the shear cases and the no-shear benchmark experiment 220 are identified. Furthermore, we consider a very large number of trajectories such that, 221 at least in a statistical sense, our results can be expected to be robust. Degrading the temporal resolution to 720 s yielded results that exhibit the same salient features as our results depicted in Fig. 4 below, and also quantitative similarities. In contrast, degrading the data to a temporal resolution of 1 h yielded results that did not show any physically meaningful differences between shear and no-shear cases. We are therefore confident that, for the purpose of this study, our results have adequately converged for the available 360 s data Mapping trajectories into entropy temperature space 229 One of the main goals of this study is to propose a succinct depiction of the TC s sec- 230 ondary circulation in the presence of asymmetries and time-dependent flow features. In analogy to the prevailing lowest-order intensity theory (Emanuel 1986, 1991) and its exten- sion to include the impact of ventilation (Tang and Emanuel 2010), we propose depiction in 9

11 entropy temperature (θ e T ) space. (Moist) entropy, s, is here expressed in terms of equivalent potential temperature, θ e : s = c p ln θ e, where c p is the specific heat capacity of dry air at constant pressure. For tropical tropospheric values, say from K, s is approximately a linear function of θ e. For simplicity, we therefore use θ e (calculated as in Bolton (1980)) as the entropy variable. To diagnose the pathways of air parcels through θ e T space, one preferably considers an isentropic mass-streamfunction, as e.g. in Pauluis and Mrowiec (2013). Such an approach, however, is inadequate for our purposes because of the large divergence of the flow in θ e T 241 space 2 (illustrated below). To properly handle divergent flows in θ e T space we instead perform the thermodynamic transform introduced by Kjellsson et al. (2014) and discussed in more detail by Laliberté et al. (2015): F θet (θ b e, T b ) = 1 τ N τ n=1 t=1 D D t (θ e, T )(t, n) δ ( ) ( θ e (t, n) θe b δ T (t, n) T b ) ) (1) where D/Dt denotes the material derivative, and the Dirac delta distribution δ is approxi- mated by a top-hat function: δ(x) = 1 for 0 x < x (2) δ(x) = 0 elsewhere. (3) 247 The vector F θet is a vector in θ e T space that depicts the material rate of change of θ e and T, integrated over all N air parcels and averaged over all time steps τ of the trajectory calculation. This vector thus measures the rate with which mass flows through specific parts of the θ e T space. Here, we will refer to this vector as the thermodynamic mass flux vector. The right-hand side of Eq. 1 describes the mapping of the material tendencies into discrete bins of the thermodynamic space. The superscript b denotes the respective bin value and the bin size x is set to 0.5 K, both for T and θ e. The material rate of change of (T, θ e ) of an air parcel along its trajectory is computed by centered finite differences between output times. Summation over the air parcels here implies integrating over mass. 2 Large vorticity of the flow in θ e T space prohibits the use of a flow potential also. 10

12 256 For a reversible heat engine, e.g. working akin to a Carnot cycle, F θet can be related 257 to the rate with which work is performed 3. In this study, however, we do not intend to 258 use F θet in a quantitative sense to calculate work performed by a TC. Rather, we employ this framework to analyze and compare the thermodynamic properties of the secondary circulations of TCs in a quiescent environment and under the impact of vertical shear. To help relate the θ e T space with physical space in our experimental setup, a vertical cross-section of θ e and T is shown in Fig. 3. This figure represents the θ e and T distributions just before vertical shear is imposed and may serve as reference for the interpretation of figures presented below. Figure 3 exhibits well-known features of the θ e distribution in the vicinity of a TC. High θ e values (355 < θ e < 370 K) are found in the frictional inflow layer 4 (below 1 km height), in the eyewall (at a radius of km), and in the upper part of the outflow layer (around km height in Fig. 3). Note the sharp θ e gradients at the edge of the eyewall and at the top of the inflow layer that help distinguish these regions from the environmental free tropopsphere. The frictional inflow layer is mostly confined to T > 295 K and the outflow layer mostly to T < 215 K. As examples, the 255 and 290 K isotherms are located in the environment at approximately 8 and 2 km height. The lowest θ e values (θ e < 340 K) are found from just above the frictional inflow layer up to 8 9 km outside a radius of approximately 150 km. Within this radius, the eyewall is surrounded by somewhat enhanced θ e values of K, indicative of the TC s moist envelope (Willoughby et al. 1984; Riemer and Montgomery 2011). The highest θ e values (θ e > 370 K) are found in the eye at low levels (within a radius of 25 km) and in the model stratosphere (above 16 km 3 For a reversible process, the specific work w = T ds. The curve C can be parameterized by a function of C time, say γ(t). Representing the (homogeneous) working fluid by a single air parcel with values (s(t), T (t)), the curve C can be defined as the path of that air parcel in s T space and thus simply γ(t) = (s(t), T (t)). Performing the parameterization and applying the fundamental theorem of calculus, it can be shown that the instantanous work rate equals the product of T and the magnitude of the instantaneous mass flux: d w/d t = T (t) d (s, T )/d t 4 Under the assumption that the height scale for turbulent mixing of momentum is the same as for enthalpy, the frictional inflow layer can be associated with the high θ e in the boundary-layer. 11

13 277 height). The tropopause temperature in our experimental setup is approximately 200 K Characterization of the secondary circulation by the thermodynamic mass flux vector In this section, F θet is used to diagnose the impact of vertical shear on the TC s secondary circulation in our set of idealized numerical experiments. As a benchmark, we examine first the no-shear case with trajectories seeded at 6 h. The shear cases are exemplified by the 20mps case with trajectories seeded also at 6 h. Unless otherwise noted, all features below are representative for the no-shear and shear cases, respectively. 285 a. Lagrangian, thermodynamic perspective of the no-shear experiment 286 In the no-shear case, one prominent feature of F θet is the distinct dipole pattern in its divergence (Fig. 4a). In θ e T space, F θet has units of kg s 1 and thus can be interpreted as the rate with which air masses with specific (θ e, T ) values are depleted ( F θet > 0) or accumulated ( F θet < 0) over the timescale of the trajectory calculation. The regions of prominent divergence and convergence exhibit a clear localization in θ e T space with (θ e,t ) values representative of the frictional inflow layer and the outflow layer, respectively. Not 292 surprisingly, F θet signifies the transport of air from the inflow to the outflow layer of the TC. Consequently, the localized region of large convergence can be interpreted as the thermodynamic signature of the outflow in our numerical experiment. In this region of large convergence, Fig. 4 indicates an approximate linear relationship between θ e and T with a slope of approx. 0.5 K/K, consistent with the idea that air parcels rising with higher θ e values in the eyewall reach colder outflow temperatures than their counterparts with lower 298 θ e values. Of interest from the heat-engine perspective, the distinct dipole in F θet is 299 evidence that, over the considered 12 h period, the TC clearly constitutes an open system. 12

14 The vector F θet itself indicates that the air from the region of large divergence, i.e. from within the frictional inflow layer, increases its θ e value by several Kelvin before the air starts to rise in the eyewall, as identified by the onset of significant cooling at approx. 290 K in Fig. 4a. The initial θ e value averaged over all air parcels is K while the average value of the parcels rising through the 290 K isotherm (at approx. 2 km height) is 365 K (Fig. 5a). This increase in θ e occurs in the small T range between 300 and 295 K. Rising in the eyewall, indicated by the pronounced cooling of air parcels, F θet depicts a decrease of θ e by several K between T = 290 K and T = K. At colder T, the ascent is approximately pseudoadiabatic. A further small decrease in θ e is noted for 215 K > T > 205 K. Finally, a further increase in θ e is noted at the lowest T values (T < 205 K) for some air parcels. Arguably, this signal is an indication of air parcels penetrating into the model stratosphere and acquiring high stratospheric θ e values by irreversible mixing. Next, the θ e distribution of the eyewall ascent is examined in some more detail, which will be important later when compared with the shear cases. For the ascending air, the θ e values are weighted by the cooling rate of the trajectories, which can be interpreted as a vertical mass flux in the current context. These (weighted) mean θ e values of rising, i.e. cooling, air parcels are depicted in Fig. 5a. Between T = 290 K and T = 270 K (at approx. 5.5 km height), the rising air loses a large amount ( θ e 5 K) of the θ e acquired since the initial time: the average θ e value at T = 270 K is 360 K. This loss in θ e is consistent with results of Fierro et al. (2009) who performed an on-line trajectory analysis of a higher-resolution simulation (with a horizontal grid spacing of 750 m) of a tropical squall line. These authors attributed the decrease in θ e to entrainment from the environment. Our results, presented below in Sec. 5, are consistent with their interpretation. Between T = 270 K and T = 250 K, a further decrease by θ e 2 K occurs. At colder T, θ e values are approximately constant, i.e. the ascent can be considered to be pseudo-adiabatic 5, up to T = 215 K. Qualitatively, 5 Fierro et al. (2009) found a subsequent increase in θ e above the freezing level due to the latent heat release from ice processes. As ice microphysics are not included in our idealized numerical experiment, pseudo-adiabatic ascent above the freezing level in our experimental setting is again consistent with their 13

15 the no-shear case at 8 h is very similar (Fig. 5a). The respective θ e distributions at [290, 270, 250] K and at initial and end time are shown in Fig. 6. The sharp peak at θ e 360 K in the distribution of the initial values signifies that most of the air compromising the secondary circulation stems from the small range of θ e values characteristic of the frictional inflow layer. The distributions at [290, 270, 250] K exhibit a sharp drop towards high θ e and are clearly skewed towards lower values. Interestingly, the sharp peak at initial time broadens considerably while air is rising in the eyewall, i.e. at T = 290 K and in particular at T = 270 and 250 K. In contrast to the eyewall distributions (at T = [290, 270, 250] K), the distribution at end time exhibits a fairly sharp drop towards low values and is skewed towards high θ e. Indicated also by F θet in Fig. 4a is the intrusion of some low-θ e air (330 < θ e < 355 K) into the TC s secondary circulation from low levels above the frictional inflow layer (290 < T < 295 K). The long tail in the distribution of the initial θ e values in Fig. 6a corroborates this notion. This feature will be discussed in more detail below, in comparison with the respective feature in the shear cases. 340 b. Modifications by vertical wind shear Not surprisingly, the general characteristic that the TC depletes high-θ e air at warm T and accumulates air masses at low T at similar θ e values holds for the sheared TCs also (Fig. 4b). There are, however, several important differences that will be discussed in the following ) Characteristics of inner-core convection Most importantly, Fig. 4b demonstrates that vertical shear indeed acts as a constraint on the thermodynamics of the TC: the mean θ e of the air rising in the eyewall is significantly results. 14

16 lower than in the no-shear case. The θ e difference, θ e, between the no-shear and the 20mps case is a maximum at T = 290 K ( θ e = 6.5 K, Fig. 5a) and decreases considerably up to 350 T = 250 K ( θ e = 3.5 K). Figure 5a summarizes the average θ e values for all cases and seeding times considered and confirms that the reduction of eyewall θ e with a maximum reduction at 290 K in the shear cases is a very robust feature. The eyewall-θ e distribution (Fig. 6b) drops off towards high θ e at lower values than in the no-shear case, as could be expected based on the lower mean value in the shear cases. Interestingly, however, the distributions for T < 290 K in the shear cases do not exhibit such a pronounced skewness towards lower values. As a consequence, eyewall θ e exhibits a sharper distribution than in the no-shear case. Consideration of the standard deviation of the distributions verifies that this is a consistent feature in the shear cases, with the only exception being the 15mps case at 6 h at 215 K (Fig. 5b). This is a remarkable result because the initial distributions of the θ e values drawn into the secondary circulation in the shear cases exhibit a bimodality and thus a significantly larger standard deviation than without shear (Fig. 6 and Fig. 5). This bimodality in the initial distribution will be the focus of further examination below. 364 For the reduction of the tail towards low θ e in the shear case we offer the following explanation, based on two results that will be presented below in Sec. 5. For one, the shear cases exhibit reduced mid- to upper tropospheric ventilation that brings in low-θ e air into the eyewall updrafts and arguably enhances the tail towards low θ e in the no-shear case. Furthermore, the shear cases exhibit enhanced detrainment from the eyewall that tends to detrain low-θ e air from the eyewall updrafts and thus tends to reduce the tail towards low θ e ) Modification of the low-level inflow For TCs in the same thermodynamic environment, as is the case here, it is expected that eyewall θ e values are lower in a weaker TC as compared to a more intense TC. The diagnosed 15

17 reduction of eyewall θ e could therefore be merely a consequence of the intensity decrease of the vertically-sheared TCs by m s 1, rather than the cause for the weakening. Figure 4b, however, provides evidence that vertical shear leads indeed to structural changes of the secondary circulation, in particular at low levels. Comparing Figs. 4a and 4b, two modifications at low levels, i.e. high T, are readily evident. Firstly, the intrusion of low-θ e air at 295 > T > 285 K, i.e. from just above the frictional inflow layer, is considerably enhanced. Such an inflow pathway is indicated in the no-shear case also. It is obvious, however, that vertical wind shear promotes this contribution of air masses to the secondary circulation that originate from above the frictional inflow layer. Secondly, the increase of θ e at high values of θ e for T > 290 K is much reduced in the shear cases. In the no-shear case in Fig. 4a, θ e increases up to 370 K whereas in the shear case exemplified in Fig. 4b, θ e values hardly exceed 363 K. This difference is not only due to a decrease of the saturation θ e of the sea surface (θes) due to its pressure dependency, as θes is the upper bound that eyewall θ e may acquire by surface fluxes. Rather, at least half of this difference can be attributed to an increase of the difference between θes and the actual θ e of trajectories before rising in the eyewall. For inflowing air parcels at 35 km radius, this difference is 3.5 K (or 15 %) higher in the shear case than in the no-shear case (not shown). At 50 km radius, this difference is 4.7 K (or 25 %) higher. We will argue below (Sec. 5b2) that it is the intrusion of low-θ e air down into the frictional inflow layer that prevents the sheared TC from gaining the same relative saturation level as in the no-shear case ) Modifications of the outflow layer Further differences between the shear and no-shear cases in F θet and its divergence can be found in the region of strong convergence at low T, i.e. in the thermodynamic signature of the outflow layer. As compared to the no-shear case, this convergence region does not exhibit the same distinct localization in θ e T space, i.e. a linear relationship between θ e and T as noted above would not be as well defined. The range of low T at which high-θ e air 16

18 accumulates is larger ( K) than in the no-shear case. More inner-core updrafts in the shear cases than in the no-shear cases apparently penetrate the model tropopause and acquire very high, stratospheric θ e values (θ e > 365 K) at very low temperatures (T < 195 K). These features are consistent with the finding of Riemer et al. (2013, their Sec. 3.7) that vertical shear tends to increase the variability of updraft amplitude. It is interesting to note that this increased variability occurs with a decreased variability of the θ e values of the inner-core updrafts at 290 > T 215 (Sec. 4b1). Apparently, in a vertically-sheared TC the height of the convection (and therefore the outflow temperature) is not just determined by the θ e value of air parcels. Arguably, the increased variability is due to additional dynamical forcing that ensues from the interaction of the TC with vertical shear, e.g. vertical motion that is associated with the adjustment of the TC vortex to a balanced state (Jones 1995; Reasor et al. 2000; Zhang and Kieu 2006; Davis et al. 2008) Distinct pathways of environmental interaction Comparison of F θet and F θet for the no-shear and shear cases in Fig. 4 indicates a prominent ventilation of the sheared TCs at low levels. This section will investigate this and other ventilation pathways more systematically and will visualize these pathways by their F θet. Ventilation pathways will be defined based on trajectories that intrude into the secondary circulation from the environment. The environment is here defined by θ e values that are distinctly lower than those in the in- and outflow layers and in the eye and eyewall. We will consider detrainment pathways also, defined based on those trajectories that exit the textbook secondary circulation and end in the environment. The classification of the different pathways will be based on the initial and end positions of the trajectories in θ e T space and is detailed below. We recall to the reader that all trajectories where seeded in regions of high inertial stability and saturated ascent and thus are, by design and definition, part of the inner-core convection at the time of their seeding. 17

19 425 a. Classification of pathways 426 1) Ventilation pathways The distribution of the initial positions in θ e T space is depicted in Fig. 7a,b. By visual inspection, four distinct clusters of initial positions may be identified. Most trajectories, in particular in the no-shear case, start with 365 K > θ e > 355 K and T > 295 K, i.e. from within the frictional inflow layer outside a radius of 50 km. Much fewer trajectories start with θ e > 365 K and T > 280 K. Apparently, these values represent trajectories that are drawn into the eyewall convection from the eye at low levels. The remaining trajectories start with θ e < 355 K. From these trajectories, the majority exhibits 295 > T > 280 K and thus represent air masses at low levels but above the frictional inflow layer. The remaining air parcels are dominated by 260 > T > 240 K, representing mid- to upper-tropospheric air masses. Based on these clusters, we define low-level ventilation as the subset of those trajec- 438 tories that have initial values of θ e < 355 K and 297 K T 280 K. We define free tropospheric ventilation as the subset of those trajectories that have initial values of θ e < 355 K and T < 280 K. We define frictional inflow as the remaining trajectories with initial θ e 355 K. For simplicity, we include in the frictional inflow the few trajectories that start in the eye. We do not consider these trajectories separately because our focus here is on environmental interaction and the very small contribution of eye parcels does not affect the interpretation of the diagnostics for the frictional inflow shown below ) Detrainment pathways The classification of detrainment pathways is more involved than that of the ventilation pathways. First, we reiterate that the vast majority of the end positions exhibit (θ e, T ) 448 values that can be attributed clearly to the outflow layer of the TCs (θ e > 350 K and 449 T < K, Figs. 7c,d). As expected from Fig. 4, the localization of the end positions in 18

20 θ e T space in the shear cases is not as pronounced as in the no-shear case. In the following we focus on trajectories that end with warmer T than typically found in the outflow layer. For a clear distinction we focus on end values of T > 215 K. For these values, less than 5% of the trajectories in the no-shear case are included but 12 25% of the trajectories in the shear cases (cf. the sum of the values depicted in Figs. 10c e). In the no-shear case, the distribution of end positions with T 215 K clearly indicate three distinct regimes (Fig. 7e). The regime of highest θ e values (θ e > 360 K) signifies air parcels that entrain into the eye. These air parcels shall not be considered further in this study 6. The vast majority of the remaining parcels have θ e < 350 K and are, therefore, 459 classified as air parcels detrained from the eyewall into the environment. Two distinct regimes can be separated along the K isotherm indicating detrainment of air into the environment at mid-tropospheric and at upper-tropospheric levels. In the shear cases, we cannot clearly distinguish between eye entrainment and environ- 463 mental detrainment, in particular at low T (Fig. 7f). The separation between mid- and upper-level detrainment, however, is apparent also. To distinguish environmental detrainment from eye entrainment, we refer to the radial 7 distribution of the end positions. On average (Fig. 8a), as well as in all individual shear cases (not shown), a local minimum in the frequency distribution is found around 50 km radius. For the detrainment pathways defined below, we thus consider trajectories with end positions outside of this radius only 8. It should be kept in mind, however, that a definite separation between eye entrainment and environmental detrainment, in particular at upper levels, is not accomplished by this method. 6 A detailed Lagrangian analysis of entrainment of air parcels into the eye has been performed by Stern and Zhang (2013). 7 The TC center is identified by a vorticity centroid as described in RMN10. The center is defined as a function of height. Center positions are usually found up to 12 km height. Above, the center is defined by the center position at the greatest height possible. 8 As an aside, we note that the radial end positions of the detrainment pathways are confined more closely to the center than those of the textbook outflow with T 215 K (dashed line in Fig. 8a). 19

21 We performed a preliminary examination of F θet for trajectories that end in the upper troposphere (255 K > T 215 K) that indicated two distinct pathways (not shown): a) detrainment directly from the eyewall and b) detrainment from the outflow layer, in the sense that air parcels reach low T first and then descend, i.e. warm, to reach T 215 K. The minimum T along a trajectory (min(t )) is a simple metric to distinguish these two pathways (exemplified in Fig. 8b). A bimodal distribution of min(t ) was found in all cases 9. The characteristics of the two modes exhibit large variability in the individual cases, but the bimodality itself is a very robust feature. To distinguish the modes, a robust threshold T min = 210 K can be identified in the shear cases. In the no-shear case, the threshold is T min = 220 K. In summary, we define upper-level detrainment as the subset of those trajectories that end with 215 K T < 255 K and min(t ) T min, outflow detrainment as those with 215 K T < 255 K and min(t ) < T min, and mid-level detrainment as those with T 255 K. 485 b. Frictional inflow and ventilation pathways 486 1) Shear-induced low-level ventilation Low-level ventilation is depicted in Fig. 9a,b. In the shear case, as well as in the no-shear case, θ e along this subset of trajectories increases significantly (by K) before rising in the eyewall updrafts. Interestingly, this increase occurs at temperatures representative of the upper part of the frictional inflow layer (T 295 K), but distinctly cooler than the surface temperature in our experimental setup of T = K. A qualitative difference between lowlevel ventilation in the shear and in the no-shear case is indicated by F θet. The distinct maximum of F θet at very low θ e (< 335 K) in the shear case indicates the main origin of the low-level ventilation trajectories. The no-shear case does not exhibit such a feature. The 9 As a further aside, we note that the min(t)-distribution for T 255 K is unimodal with, not surprisingly, distinctly higher T, indicating that these trajectories stay relatively warm before detrainment. 20

22 same distinct difference in initial θ e is discernable also in Fig. 6, which depicts the initial θ e distribution of all trajectories. Our trajectory analysis thus corroborates results based on a quasi-stationary framework (Riemer and Montgomery 2011) that vertical shear organizes the systematic intrusion of environmental, very-low-θ e air from outside of the moist envelope into the TC s frictional inflow layer. A further striking feature of low-level ventilation is its distinct increase with vertical shear (Fig. 10a). In the no-shear case, only approximately 10 % of the trajectories contribute to low-level ventilation. This percentage increases approximately to 30 % and 40 % in the 15mps and 20mps case, respectively ) Limited entropy increase of frictional inflow 505 By depicting F θet of the frictional inflow (Fig. 9c,d) we can reveal a clear qualitative difference between the shear and the no-shear cases. In the no-shear case we see that θ e values increase distinctly from their initial values before trajectories start rising in the eyewall but we do not notice such an increase in the shear case. Partly, θ e even decreases (at θ e = K and T = K). This distinct difference is corroborated by the θ e distribution of all trajectories shown in Fig. 6. The sharp peak around 362 K in the initial distributions can be identified with the frictional inflow at initial time. In the no-shear case, air masses rise at 290 K with higher θ e on average. By contrast, θ e values at this temperature in the shear case are somewhat lower than that indicated by the initial peak. The suppressed increase of θ e in the frictional inflow is arguably due to turbulent mixing with the low-θ e air of the prominent low-level ventilation in the shear case. Vertical turbulent mixing, in particular, can be expected to play the most significant role. A strong indication of such a mixing process is that the low-level ventilation trajectories increase their θ e distinctly away from the ocean surface and the associated heat fluxes. Consequently, low-level ventilation trajectories increase their θ e at the expense of the inflow layer trajectories and surface enthalpy fluxes are apparently not sufficiently strong to compensate for this mixing 21

23 effect. Figure 11 further illustrates the θ e exchange between the low-level ventilation and the frictional inflow trajectories for the representative shear case. At initial time, the θ e distributions of both pathways are clearly distinct. The end distributions, on the other hand, exhibit very similar shapes and the difference in the mean values has reduced by an order of magnitude from 20 K at initial time to 2 K. This large decrease in the difference indicates an almost complete mixing of the air masses along this part of the TC s secondary circulation. Figure 9 confirms that the outflow characteristics of both pathways are similar. Most of the mixing occurs below T = 290 K (Fig. 11b). The shapes of the respective distributions at this temperature are rather similar and the difference in the mean θ e values has already decreased to 4 5 K. At T = 270 K, the differences between the distributions have further decreased and resemble those at the end positions with a difference in mean θ e of 1 2 K. Interestingly, the mean value of the frictional inflow exhibits a larger decrease between 290 and 270 K than that of the low-level ventilation trajectories. This larger decrease indicates that a further exchange of entropy by turbulent mixing between the two air masses has occured between 290 and 270 K. The features discussed above are robust in all shear cases (not shown). We would like to stress, however, that the above analysis of turbulent mixing is impaired by uncertainties in the accuracy of our trajectory calculation due to the available temporal and spatial data resolution. We do not expect our trajectories to follow individual air parcels exactly. Therefore, the mixing diagnosed above has to be partly attributed to such deviations from true (model) trajectories. The use of online trajectories in future analyses will allow for a more accurate examination of the role of turbulent mixing ) Free-tropospheric ventilation Free-tropospheric ventilation is depicted in Fig. 9e,f. In these figures it is apparent that air parcels tend to warm before they intrude into the eyewall and hence before they start to 22