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 this analysis is the classification and quantitative depiction of different pathways of environmental interaction based on thermodynamic properties of trajectories at initial and end times. Confirming results from previous work, vertical shear significantly increases the intrusion of low–equivalent potential temperature () air into the eyewall through the frictional inflow layer. In contrast to previous ideas, vertical shear decreases midlevel ventilation in these experiments. Consequently, the difference in eyewall 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 previously. These trajectories lose entropy at cold temperatures by detraining from the outflow layer and subsequently warm by 10–15 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.
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 hampers our ability to identify limits of predictability and to produce the best forecasts that are physically possible.
Vertical shear of the environmental horizontal winds (referred to herein 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, hereafter RMN10, 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, p. 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 assumptions 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., Figure 11.33 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 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 also induces 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 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 midtropospheric levels (4–5-km height) (Simpson and Riehl 1958). The importance of this midlevel pathway, however, may be considerably limited because of the significant deflection of intruding air masses by the swirling winds (Riemer and Montgomery 2011, p. 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, RMN10 and Riemer et al. (2013) 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 toward 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.
b. A Lagrangian perspective of the secondary circulation
To address these open issues, we present here a Lagrangian description of the secondary circulation (section 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: that is, 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 [e.g., Schubert and Hack (1982); Nolan et al. (2007); , where ζ is relative vorticity, is the (constant) Coriolis parameter, υ is the tangential wind component, and r is the 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 is set high enough to exclude contributions from outer rainbands. 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, described in section 3) is performed. This transformation maps the trajectories from three-dimensional physical space to two-dimensional entropy–temperature space. By depicting the mass flux 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 RMN10. Using simple thermodynamic criteria, first strides toward objectively identifying distinct ventilation pathways are made in section 5. A summary of our results concludes this study (section 6).
2. Numerical experiment and trajectory calculation
This study further analyzes the idealized numerical experiments of 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. Section 2b then details the trajectory calculation on which the results of the current study are based.
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.
The current study examines the 15 and 20 m s−1 shear cases of RMN10. In these 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, reintensify in the latter 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 15 m s−1 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 20 m s−1 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 et al. 1992; Cotton et al. 2003). RAMS is a state-of-the-art three-dimensional nonhydrostatic 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 innermost 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 1.09. 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). Because of 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 (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 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 [e.g., Eq. (15) in RMN10]. 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).
b. Calculating trajectories of the secondary circulation
The trajectories diagnosed in this study are calculated as offline trajectories from stored model output. Data are 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 section 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 for reasonable choices of the other two thresholds, examined between 0.25 and 1 m s−1 and between and , respectively. Thresholds toward the lower ends of these ranges, however, appear to partly include trajectories from rainbands as well, so the results in these cases exhibit somewhat more noise (not shown).
The horizontal distribution of seeding locations is illustrated for the no-shear and the 20 m s−1 case at 6 h in Figs. 2a and 2b, 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 20 m s−1 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 inner-core 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.
Trajectories are initialized on a horizontal grid of 1.25 km × 1.25 km. The lowest vertical 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 Figs. 2c and 2d. Above 3-km height, both distributions tilt radially outward with height, consistent with the general outward tilt of the eyewall. In the 20 m s−1 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 . Seeding in pressure coordinates renders the trajectories divergence free in physical space because of continuity. With the given spatial resolution and the above thresholds, trajectories are computed.
From their seeding locations, trajectories are integrated forward and backward in time for 6 h. Here, 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 outside a radius of 200 km (not shown). Trajectories are seeded at 6 and 8 h in all cases and also at 10 h in the shear cases. Each seeding time actually comprises five calculations starting at five 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 represent 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 are identified. Furthermore, we consider a very large number of trajectories such that, 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), as well as 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.
3. Mapping trajectories into entropy–temperature space
One of the main goals of this study is to propose a succinct depiction of the TC’s secondary circulation in the presence of asymmetries and time-dependent flow features. Analogous to the prevailing lowest-order intensity theory (Emanuel 1986, 1991) and its extension to include the impact of ventilation (Tang and Emanuel 2010), we propose depiction in entropy–temperature (–T) space. The (moist) entropy s is here expressed in terms of equivalent potential temperature : , where is the specific heat capacity of dry air at constant pressure. For tropical tropospheric values, say from 330 to 370 K, s is approximately a linear function of . For simplicity, we therefore use [calculated as in Bolton (1980)] as the entropy variable.
To diagnose the pathways of air parcels through –T space, one preferably considers an isentropic mass streamfunction [as, for example, in Pauluis and Mrowiec (2013)]. Such an approach, however, is inadequate for our purposes because of the large divergence of the flow in –T space2 (illustrated below). To properly handle divergent flows in –T space, we instead perform the thermodynamic transform introduced by Kjellsson et al. (2014) and discussed in more detail by Laliberté et al. (2015):
where D/Dt denotes the material derivative, and the Dirac delta distribution δ is approximated by a top-hat function:
The vector is a vector in –T space that depicts the material rate of change of 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 at which mass flows through specific parts of the –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 is set to 0.5 K, both for T and . The material rate of change of () 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.
For a reversible heat engine (e.g., working akin to a Carnot cycle) can be related to the rate at which work is performed.3 In this study, however, we do not intend to use 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 –T space with physical space in our experimental setup, a vertical cross section of and T is shown in Fig. 3. This figure represents the and T distributions just before vertical shear is imposed and may serve as a reference for the interpretation of figures presented below. Figure 3 exhibits well-known features of the distribution in the vicinity of a TC. High values () are found in the frictional inflow layer4 (below 1-km height), in the eyewall (at a radius of 30–50 km) and in the upper part of the outflow layer (around 14–15-km height in Fig. 3). Note the sharp gradients at the edge of the eyewall and at the top of the inflow layer that help distinguish these regions from the environmental free troposphere. The frictional inflow layer is mostly confined to and the outflow layer mostly to . As examples, the 255- and 290-K isotherms are located in the environment at approximately 8- and 2-km heights. The lowest values () 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 values of 340–350 K, indicative of the TC’s moist envelope (Willoughby et al. 1984; Riemer and Montgomery 2011). The highest values () are found in the eye at low levels (within a radius of 25 km) and in the model stratosphere (above 16-km height). The tropopause temperature in our experimental setup is approximately 200 K.
4. Characterization of the secondary circulation by the thermodynamic mass flux vector
In this section, 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 20 m s−1 case with trajectories seeded also at 6 h. Unless otherwise noted, all features below are representative for the no-shear and shear cases, respectively.
a. Lagrangian, thermodynamic perspective of the no-shear experiment
In the no-shear case, one prominent feature of is the distinct dipole pattern in its divergence (Fig. 4a). In –T space, has units of kilograms per second and thus can be interpreted as the rate at which air masses with specific values are depleted () or accumulated () over the time scale of the trajectory calculation. The regions of prominent divergence and convergence exhibit a clear localization in –T space with (, T) values representative of the frictional inflow layer and the outflow layer, respectively. Not surprisingly, 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. 4a indicates an approximate linear relationship between and T with a slope of approximately , consistent with the idea that air parcels rising with higher values in the eyewall reach colder outflow temperatures than their counterparts with lower values. Of interest from the heat-engine perspective, the distinct dipole in is evidence that, over the considered 12-h period, the TC clearly constitutes an open system.
The vector itself indicates that the air from the region of large divergence (i.e., from within the frictional inflow layer) increases its value by several kelvins before the air starts to rise in the eyewall, as identified by the onset of significant cooling at approximately 290 K in Fig. 4a. The initial value averaged over all air parcels is 357.5 K, while the average value of the parcels rising through the 290-K isotherm (at approximately 2-km height) is 365 K (Fig. 5a). This increase in occurs in the small T range between 300 and 295 K. Rising in the eyewall, indicated by the pronounced cooling of air parcels, depicts a decrease of by several kelvins between K and T = 260–250 K. At colder T, the ascent is approximately pseudoadiabatic. A further small decrease in is noted for . Finally, a further increase in is noted at the lowest T values () for some air parcels. Arguably, this signal is an indication of air parcels penetrating into the model stratosphere and acquiring high stratospheric values by irreversible mixing.
Next, the 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 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 values of rising (i.e., cooling) air parcels are depicted in Fig. 5a. Between and (at approximately 5.5-km height), the rising air loses a large amount () of the acquired since the initial time: the average value at is 360 K. This loss in is consistent with the results of Fierro et al. (2009), who performed an online 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 to entrainment from the environment. Our results, presented below in section 5, are consistent with their interpretation. Between and , a further decrease by occurs. At colder T, values are approximately constant (i.e., the ascent can be considered to be pseudoadiabatic)5 up to . Qualitatively, the no-shear case at 8 h is very similar (Fig. 5a).
The respective distributions at 290, 270, and 250 K and at initial and end times are shown in Fig. 6. The sharp peak at in the distribution of the initial values signifies that most of the air comprising the secondary circulation stems from the small range of values characteristic of the frictional inflow layer. The distributions at 290, 270, and 250 K exhibit a sharp drop toward high and are clearly skewed toward lower values. Interestingly, the sharp peak at initial time broadens considerably while air is rising in the eyewall (i.e., at and, in particular, at T = 270 and 250 K). In contrast to the eyewall distributions (at T = 290, 270, and 250 K), the distribution at end time exhibits a fairly sharp drop toward low values and is skewed toward high .
Indicated also by in Fig. 4a is the intrusion of some low- air () into the TC’s secondary circulation from low levels above the frictional inflow layer (). The long tail in the distribution of the initial 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.
b. Modifications by vertical wind shear
Not surprisingly, the general characteristic that the TC depletes high- air at warm T and accumulates air masses at low T at similar values holds for the sheared TCs also (Fig. 4b). There are, however, several important differences that will be discussed in the following.
1) 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 of the air rising in the eyewall is significantly lower than in the no-shear case. The difference between the no-shear and the 20 m s−1 case is a maximum at (, Fig. 5a) and decreases considerably up to (). Figure 5a summarizes the average values for all cases and seeding times considered and confirms that the reduction of eyewall , with a maximum reduction at 290 K in the shear cases, is a very robust feature.
The eyewall distribution (Fig. 6b) drops off toward high 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 in the shear cases do not exhibit such a pronounced skewness toward lower values. As a consequence, eyewall 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 15 m s−1 case at 6 h at 215 K (Fig. 5b). This is a remarkable result, because the initial distributions of the values drawn into the secondary circulation in the shear cases exhibit a bimodality and, thus, a significantly larger standard deviation than without shear (Figs. 6 and 5). This bimodality in the initial distribution will be the focus of further examination below.
For the reduction of the tail toward low in the shear case, we offer the following explanation, based on two results that will be presented below in section 5. For one, the shear cases exhibit reduced mid- to upper-tropospheric ventilation that brings low- air into the eyewall updrafts and arguably enhances the tail toward low in the no-shear case. Furthermore, the shear cases exhibit enhanced detrainment from the eyewall that tends to detrain low- air from the eyewall updrafts and thus tends to reduce the tail toward low .
2) Modification of the low-level inflow
For TCs in the same thermodynamic environment, as is the case here, it is expected that eyewall values are lower in a weaker TC, as compared to a more intense TC. The diagnosed reduction of eyewall could therefore be merely a consequence of the intensity decrease of the vertically sheared TCs by 10–20 m s−1, rather than the cause for the weakening. Figure 4b, however, provides evidence that indeed vertical shear leads 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. First, the intrusion of low- air at (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 to the secondary circulation of air masses that originate from above the frictional inflow layer. Second, the increase of at high values of for is much reduced in the shear cases. In the no-shear case in Fig. 4a, increases up to 370 K, whereas in the shear case exemplified in Fig. 4b, values hardly exceed 363 K. This difference is not only because of a decrease of the saturation of the sea surface as a result of its pressure dependency, as is the upper bound that eyewall may acquire by surface fluxes. Rather, at least half of this difference can be attributed to an increase of the difference between and the actual of trajectories before rising in the eyewall. For inflowing air parcels at a 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 a 50-km radius, this difference is 4.7 K (or 25%) higher. We will argue below [section 5b(2)] that it is the intrusion of low- 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.
3) Modifications of the outflow layer
Further differences between the shear and no-shear cases in 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 –T space (i.e., a linear relationship between and T, as noted above, would not be as well defined). The range of low T at which high- air accumulates is larger (195–215 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 values () at very low temperatures (). These features are consistent with the finding of Riemer et al. (2013, their section 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 values of the inner-core updrafts at [section 4b(1)]. Apparently, in a vertically sheared TC, the height of the convection (and therefore the outflow temperature) is not just determined by the 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: that is, 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).
5. Distinct pathways of environmental interaction
Comparison of and 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 . Ventilation pathways will be defined based on trajectories that intrude into the secondary circulation from the environment. The environment is here defined by values that are distinctly lower than those in the inflow 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 –T space and is detailed below. We remind the reader that all trajectories were 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.
a. Classification of pathways
1) Ventilation pathways
The distribution of the initial positions in –T space is depicted in Figs. 7a and 7b. By visual inspection, four distinct clusters of initial positions may be identified. Most trajectories, in particular in the no-shear case, start with and (i.e., from within the frictional inflow layer outside a radius of 50 km). Many fewer trajectories start with and . Apparently, these values represent trajectories that are drawn into the eyewall convection from the eye at low levels. The remaining trajectories start with . From these trajectories, the majority exhibit and thus represent air masses at low levels but above the frictional inflow layer. The remaining air parcels are dominated by , representing mid- to upper-tropospheric air masses.
Based on these clusters, we define “low-level ventilation” as the subset of those trajectories that have initial values of and . We define “free-tropospheric ventilation” as the subset of those trajectories that have initial values of and . We define “frictional inflow” as the remaining trajectories with initial . 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.
2) 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 () values that can be attributed clearly to the outflow layer of the TCs ( and T < 210–215 K; Figs. 7c,d). As expected from Fig. 4, the localization of the end positions in –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 . 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 are included (cf. the sum of the values depicted in Figs. 10c–e).
In the no-shear case, the distribution of end positions with clearly indicates three distinct regimes (Fig. 7e). The regime of highest values () 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 and are, therefore, classified as air parcels detrained from the eyewall into the environment. Two distinct regimes can be separated along the 255–260-K isotherm, indicating detrainment of air into the environment at midtropospheric and upper-tropospheric levels.
In the shear cases, we cannot clearly distinguish between eye entrainment and environmental 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 radial7 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 a 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.
We performed a preliminary examination of for trajectories that end in the upper troposphere () that indicated two distinct pathways (not shown): (i) detrainment directly from the eyewall and (ii) detrainment from the outflow layer, in the sense that air parcels reach low T first and then descend (i.e., warm) to reach . 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 can be identified in the shear cases. In the no-shear case, the threshold is .
In summary, we define “upper-level detrainment” as the subset of those trajectories that end with and , “outflow detrainment” as those with and , and “midlevel detrainment” as those with .
b. Frictional inflow and ventilation pathways
1) Shear-induced low-level ventilation
Low-level ventilation is depicted in Figs. 9a and 9b. In the shear case, as well as in the no-shear case, along this subset of trajectories increases significantly (by 10–30 K) before rising in the eyewall updrafts. Interestingly, this increase occurs at temperatures representative of the upper part of the frictional inflow layer () but distinctly cooler than the surface temperature in our experimental setup of . A qualitative difference between low-level ventilation in the shear and in the no-shear case is indicated by . The distinct maximum of at very low () 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 same distinct difference in initial is discernable also in Fig. 6, which depicts the initial 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- 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 15 and 20 m s−1 case, respectively.
2) Limited entropy increase of frictional inflow
By depicting 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 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, even decreases (at θe = 355–360 K and T = 295–300 K). This distinct difference is corroborated by the 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 on average. By contrast, values at this temperature in the shear case are somewhat lower than that indicated by the initial peak.
The suppressed increase of in the frictional inflow is arguably due to turbulent mixing with the low- 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 distinctly away from the ocean surface and the associated heat fluxes. Consequently, low-level ventilation trajectories increase their at the expense of the inflow-layer trajectories, and surface enthalpy fluxes are apparently not sufficiently strong to compensate for this mixing effect.
Figure 11 further illustrates the exchange between the low-level ventilation and the frictional inflow trajectories for the representative shear case. At initial time, the 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 (Fig. 11b). The shapes of the respective distributions at this temperature are rather similar, and the difference in the mean values has already decreased to 4–5 K. At , the differences between the distributions have further decreased and resemble those at the end positions, with a difference in mean 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 occurred 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 because of 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.
3) Free-tropospheric ventilation
Free-tropospheric ventilation is depicted in Figs. 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 cool significantly. Based on , most of the intrusion into the eyewall occurs in the range of , representing a height range of 3–8 km in our experimental setup. We can thus consider the free-tropospheric ventilation analyzed here to be mostly equivalent to the commonly assumed pathway of midlevel ventilation. Figure 9 does not indicate that the outflow characteristics of this pathway are distinct from that of the frictional inflow and low-level ventilation pathways.
A striking result of our study is that midlevel ventilation is less prominent in our shear cases than in the no-shear case. Only 3%–7% of the air parcels follow this pathway in the shear cases, as compared to 8%–10% in the no-shear case (Fig. 10b), with the exception of the 15 m s−1 case at 6 h. This reduced ventilation is consistent with a smaller decrease of eyewall between 290 and 270 K in the shear cases [cf. section 4b(1)]. This result is again consistent with results derived from the more accurate trajectory calculations by Fierro et al. (2009).
We offer the following hypothesis for the observed differences between the shear and the no-shear cases. The no-shear TC is analyzed during a 36-h period of intensification (see Fig. 3 in RMN10). For such a consistently intensifying TC, convergence above the frictional inflow layer may be expected (e.g., Ooyama 1969). It stands to reason that such a storm-scale convergence promotes intrusion of air masses from the free troposphere into the eyewall. By contrast, the TCs in the shear cases do not intensify but rather decrease their intensity. It is thus more likely to find systematic divergence above the inflow layer (Eliassen and Lystad 1977; Montgomery et al. 2001; Riemer et al. 2013): that is, a configuration that can be expected to hinder free-tropospheric ventilation. In the 15 m s−1 case, the TC is still intensifying for a short period of time after shear is imposed. The larger amount of free-tropospheric ventilation in this case at 6 h is therefore consistent with our hypothesis. Furthermore, we will show in the next subsection that the TCs in the shear cases (again, except for the 15 m s−1 case at 6 h) exhibit larger detrainment of air masses at midtropospheric levels, consistent also with our hypothesis.
c. Detrainment pathways
Detrainment pathways are depicted in Fig. 12. A general feature in this figure is that the different detrainment pathways in the shear and no-shear case, respectively, do not exhibit qualitative differences for . This feature suggests that a specific detrainment pathway is not directly linked to a specific inflow pathway. Consistently, the outflow characteristics of the frictional inflow and of the individual ventilation pathways are similar (see above).
1) Midlevel detrainment
Midlevel detrainment is exemplified in Figs. 12a and 12b. At T ≈ 290 K, the decrease of along this pathway becomes more prominent than the average decrease of in the eyewall (cf. Fig. 4). This notion indicates that midlevel detrainment is a process that begins at low levels when the air starts rising out of the boundary layer. The prominent decrease of continues until trajectories reach their end positions (here, mostly at and T = 270–280 K). Arguably, midlevel detrainment is a part of the commonly assumed midlevel ventilation process. In addition, midlevel detrainment can be influenced also by low-level ventilation: reducing the buoyancy of boundary layer air parcels arguably increases their likelihood to detrain at midlevels.
Midlevel detrainment accounts for only a small percentage of all trajectories (Fig. 10c). For the no-shear cases and the intensifying 15 m s−1 case at 6 h, it is less than 1% of trajectories. For the remaining shear cases, the relative increase of midlevel detrainment is large and now corresponds to 2%–5.5% of trajectories. It remains an important task for future work to evaluate the importance of midlevel detrainment for the intensity evolution, given that this pathway accounts for only a very small percentage of trajectories.
2) Upper-level detrainment
The trajectories that constitute upper-level detrainment (Figs. 12c,d) exhibit a fairly continuous decrease of while rising in the eyewall. In the shear case, trajectories detrain from the eyewall over a broad range of . The decrease of just above the boundary layer, say from 290 to 270 K, is similar to that found for all trajectories (cf. Figure 4). This similarity indicates that upper-level detrainment is governed by processes in the mid- to upper troposphere, in contrast to midlevel detrainment. The upper-level detrainment diagnosed here is reminiscent of the warm-core erosion process described by Frank and Ritchie (2001). In the no-shear case, only ~1% of the trajectories constitute this pathway (Fig. 10d). The relative increase in upper-level detrainment with shear is again large, increasing to 3.5%–9%. These numbers indicate that upper-level detrainment might be more important for intensity modification than midlevel detrainment. The results of Tang and Emanuel (2012), however, indicate that the impact of ventilation on intensity change decreases with height. The relative importance of upper-level detrainment thus remains unclear, and, as for the midlevel detrainment pathway, further research is needed to clarify its role in the intensity evolution of vertically sheared TCs.
3) Outflow detrainment
The defining characteristic of outflow detrainment is that trajectories lose entropy at cold temperatures () and subsequently warm to 220–230 K (Figs. 12e,f). This pathway exhibits the largest variability in the shear cases, with 2%–16% of trajectories contributing. In the no-shear case, it is ~3%. Even with the high variability, outflow detrainment appears to increase with increasing magnitude of the vertical shear (Fig. 10e).
We are unaware of a previous description of such a trajectory behavior in the TC context. We propose two potential mechanisms by which air might descend from the outflow layer (in the absence of radiation): 1) the balanced response of the TC vortex to the vertically sheared environmental flow, leading to a shear-forced secondary circulation that may feature subsidence on the upshear side (Zhang and Kieu 2006) and 2) a symmetric instability of the outflow layer (Molinari and Vollaro 2014) that may induce local secondary circulations in the regions where the instability is released. The high variability of outflow detrainment indicated in Fig. 10e may speak for the local release of instability as the main process. Future research needs to clarify the mechanism that governs the outflow detrainment pathway.
6. Summary and conclusions
This study introduces a Lagrangian diagnostic of the secondary circulation of TCs that can account for transient flow features and prominent asymmetries. We define the secondary circulation as those trajectories that contribute to latent heat release in the region of high inertial stability of the TC core. The explicit focus of this diagnostic is on thermodynamic properties. Trajectories are mapped from the three-dimensional physical space to a two-dimensional thermodynamic space. In accordance with the prevailing, lowest-order intensity theory for TCs, the entropy–temperature (–T) space is considered. The mass flux vector in thermodynamic space is introduced to describe the (, T) characteristics of the secondary circulation. This vector signifies the rate at which air masses flow through specific parts of the –T space, averaged over the time period of the trajectory calculation.
The Lagrangian diagnostic is employed to further analyze the impact of vertical wind shear on TCs in the idealized numerical experiments of RMN10. For the current study, it is important to note that RMN10 have deliberately chosen a very simple set of parameterizations that neglects ice microphysics and radiative processes. In the benchmark no-shear experiment, and its divergence depict depletion of air masses with high and high T—values representative of the frictional inflow layer—and accumulation of air masses with similarly high at low T, representative of the outflow layer. Not surprisingly, thus denotes that the secondary circulation transports air from the frictional inflow to the outflow layer. From a heat-engine perspective, and over the 12-h time frame of the trajectory calculation, the TC considered here clearly constitutes an open system.
The (, T) values in the region of large convergence of can be interpreted as the thermodynamic signature of the outflow. In the no-shear case, this region exhibits a clear localization in thermodynamic space with an approximate linear relationship between and T, consistent with the idea that higher values in the eyewall lead to colder outflow temperatures. In the inflow layer, a large increase of values is diagnosed before air parcels rise and cool in the eyewall. Consistent with the results of Fierro et al. (2009), values decrease notably during eyewall ascent at . After some further decrease of to values of , eyewall ascent at colder T is approximately pseudoadiabatic.
In the shear cases, the average eyewall is several degrees (~4–7 K) smaller than in the no-shear case. This smaller average confirms that vertical shear here acts as a constraint on the thermodynamics of the TC. The differences in eyewall between shear and no-shear cases are largest at low levels when the air starts rising in the eyewall. Our quantitative analysis thus reveals that the most prominent impact of vertical shear in this set of experiments is the intrusion of low- air from low levels in the free troposphere down into the frictional inflow layer and then into the eyewall updrafts, as suggested in RMN10 and Riemer et al. (2013). What has not been shown in these previous studies and is shown here is that the replenishment by surface enthalpy fluxes is not sufficient to counteract this intrusion: the difference between the saturation of the sea surface and the actual of the frictional inflow increases considerably in the shear cases as compared to the no-shear case. Clearly, the reduced values of eyewall are the original cause, and not merely a consequence, of the observed intensity decrease of 10–20 m s−1.
Based on the trajectories’ values at initial and end time, respectively, different pathways for intrusion into and detrainment from the secondary circulation are classified. To the authors’ knowledge, this is the first objective classification of distinct pathways of TC–environment interaction. The trajectory analysis by Cram et al. (2007) focused on eye–eyewall interaction and considered a single (midlevel) ventilation pathway only. The different pathways identified in the current study are summarized schematically in Fig. 13.
Besides the frictional inflow, a low-level and a free-tropospheric ventilation pathway can be identified. In the no-shear experiment, frictional inflow accounts for ~80% of the trajectories. In the shear cases, this number drops to 50%–65%, decreasing with increasing shear magnitude. The low-level pathway is characterized by the intrusion of environmental air from the low troposphere into the frictional inflow layer. By mixing with high- air from the frictional inflow layer, values increase by 10–30 K before rising in the eyewall. This pathway has been emphasized by RMN10 and Riemer et al. (2013). The percentage of trajectories composing the low-level pathway increases significantly with increasing shear magnitude, from ~10% in the no-shear case to 25%–45% in the shear cases. Furthermore, low-level ventilation in the shear cases exhibits a more systematic intrusion of air masses from outside of the TC’s moist envelope (i.e., with very low values), supporting the results of Riemer and Montgomery (2011) based on a quasi-stationary framework.
Direct intrusion of low- air into the eyewall from the free troposphere occurs over a large range of T. A maximum of this intrusion, however, can be identified between 280–260 K. This pathway can thus be considered as the commonly assumed midlevel ventilation pathway. In the no-shear case, 8%–10% of the trajectories constitute this pathway. Importantly, in contrast to previous ideas, this number decreases in the shear cases, down to 3%–7% in the strong-shear case. Detrainment at midlevels, however, does increase with increasing shear magnitude, from below 1% in the no-shear case to ~4% in the strong-shear case.
It is a general feature in our experimental setup that many fewer trajectories end up in the textbook outflow layer in the shear cases than in the no-shear case. Less than 5% of the trajectories constitute detrainment pathways in the no-shear case, whereas this number increases to 13%–24% in the shear cases. Besides midlevel detrainment, detrainment at upper levels also increases substantially (from 1% to 3%–9%). This pathway may be likened to upper-level processes discussed by Frank and Ritchie (2001). Finally, trajectories have been identified that detrain from the outflow layer back into the upper troposphere. These trajectories lose entropy at cold T and subsequently warm by 10–15 K. We are unaware of a previous description of such a trajectory behavior in the TC context. This pathway exhibits a large variability but appears to become more prominent with increasing shear magnitude.
The Lagrangian diagnostic presented here provides a tool to investigate different pathways of environmental interaction. The specific results of this study, however, cannot be directly generalized. The TCs analyzed here exhibit very large inertial stability, and it stands to reason that the characteristics of different pathways, as well as their relative importance for intensity change, may vary considerably with different TC characteristics and also with different environmental conditions. Such dependencies are largely unexplored, and we hypothesize that the vertical-shear profile, the ratio of shear magnitude and TC intensity, the environmental distribution, and the radial structure of the TC vortex are the most important controlling factors. Testing this hypothesis in specifically designed numerical experiments or in carefully selected real atmospheric TCs constitutes a worthwhile task for future research.
The numerical experiments investigated in this study were performed while the first author (MR) held a National Research Council Research Associateship Award at the Naval Postgraduate School in Monterey, California, United States. MR is grateful to M. T. Montgomery for his inspiration and guidance during this time and to M. E. Nicholls for aiding the design of and performing the numerical experiments. MR thanks Elisa Spreitzer for assisting with the calculation of the trajectories. We furthermore appreciate the comments of two anonymous reviewers and of Brian Tang, which helped to improve our manuscript during revision.
We note that Frank and Ritchie (2001) did not base their arguments on the framework of the TC heat engine.
Large vorticity of the flow in –T space prohibits the use of a flow potential as well.
For a reversible process, the specific work . The curve can be parameterized by a function of time, say . Representing the (homogeneous) working fluid by a single air parcel with values , the curve can be defined as the path of that air parcel in s–T space and, thus, simply . Performing the parameterization and applying the fundamental theorem of calculus, it can be shown that the instantaneous work rate equals the product of T and the magnitude of the instantaneous mass flux: .
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 in the boundary layer.
Fierro et al. (2009) found a subsequent increase in above the freezing level as a result of the latent heat release from ice processes. As ice microphysics are not included in our idealized numerical experiment, pseudoadiabatic ascent above the freezing level in our experimental setting is again consistent with their results.
A detailed Lagrangian analysis of entrainment of air parcels into the eye has been performed by Stern and Zhang (2013).
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.
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 (dashed line in Fig. 8a).
As a further aside, we note that the min(T) distribution for is unimodal with, not surprisingly, distinctly higher T, indicating that these trajectories stay relatively warm before detrainment.