1. Introduction
Decades of observational studies have provided convincing evidence that sufficiently intense deep-layer vertical wind shear in the environment will hinder the development1 of a tropical cyclone (Gray 1968; McBride and Zehr 1981; DeMaria et al. 2001; Kaplan et al. 2010; Tang and Emanuel 2012). Complementary modeling studies have largely corroborated this empirical finding and have substantially advanced our knowledge of the underlying dynamics (e.g., Tory et al. 2007; Rappin and Nolan 2012; Tao and Zhang 2014). The precise quantitative impact of deep-layer shear has been shown to depend on details of the associated height-dependent wind profile, the surrounding distribution of moisture, and the sea surface temperature (ibid., Ge et al. 2013; Finocchio et al. 2016; Onderlinde and Nolan 2016). It is possible that circumstances exist under which weak-to-moderate shear can assist early development (Molinari et al. 2004; Davis and Bosart 2006; Musgrave et al. 2008; Nolan and McGauley 2012). However, it is more commonly inferred from modeling results that vortex misalignment (tilt) induced by shear plays an important role in frustrating the emergence of nearly saturated air and the generation of a robust symmetric component of convection over the central region of the lower-tropospheric circulation, which would otherwise expedite surface spinup.
A number of the aforementioned modeling studies have examined idealized scenarios in which an immature tropical cyclone on the f plane is exposed to a constant ambient shear flow of moderate amplitude. In this paradigm, a tilt develops pointing downshear and precesses toward an upshear orientation. Upon starting to tilt upshear, the vortex begins to realign. One might imagine that the decay of upshear tilt is primarily driven by differential advection, as in analogous simulations of certain dry adiabatic vortices that are otherwise unresilient (e.g., Fig. 8 of Reasor et al. 2004). Rios-Berrios et al. (2018) suggest that a more complex alignment mechanism involving both diabatic and adiabatic processes is more probable. In a general sense, their assessment seems consistent with some of the more realistically configured simulations of sheared tropical storm development, in which the convective generation and subsequent domination of a deep subvortex within the misaligned parent cyclone can be essential to the pertinent alignment process (Nguyen and Molinari 2015; cf. Chen et al. 2018). Regardless of the particular mechanism, the realignment of a tilted tropical cyclone during its development often appears to be a catalyst for the intensification of maximal winds.
The preceding discussion suggests that an integral part of understanding development is understanding alignment. As a first step toward improving current understanding of intrinsic vortex alignment mechanisms, it seems reasonable to remove environmental wind shear from the problem. Such removal eliminates the possibility that alignment is linked to precession of tilt into an upshear orientation. The remaining dynamics is nevertheless rich, and in need of further elucidation. Earlier studies of shear-free alignment have tended to focus on essentially adiabatic mechanisms found in simplified models where moisture effects are represented merely by the effects of reducing static stability (Polvani 1991; Viera 1995; Reasor and Montgomery 2001; Schecter and Montgomery 2003,2007). Expanding this work to the realm of cloud-resolving models is necessary to account for fundamentally distinct diabatic pathways of alignment that may occur with a more realistic treatment of moist convection. The potentially novel pathways revealed by an expanded study could very well have direct relevance to conceivable situations in nature where a substantial tilt of the vortex exists under fairly weak background shear [see, e.g., sections 4a–4b of Davis and Ahijevych (2012)]. The conditions under which such a system might rapidly transition to quasi-symmetric intensification or linger in a detrimental state of misalignment are largely unknown.
The general purpose of this paper is to begin a methodical study of the nature of shear-free development (alignment and intensification) in cloud-resolving numerical simulations of tropical cyclones with variable degrees of initial tilt. Intensification of both the maximum wind speed and kinetic energy of the primary surface circulation will be examined. An effort will be made to elucidate the variation of moist convection with tilt magnitude and the consequences on the angular momentum budget of the basic state of the vortex. Conclusions regarding the effects of tilt on the organization of convection and the temporal amplification of maximum wind speed will not stray far from those found in previous studies of developing tropical cyclones exposed to moderate deep-layer shear. On the other hand, our simulated vortices will follow somewhat distinct pathways toward the state of alignment that facilitates intensification. For example, the tilt of a strongly perturbed tropical cyclone will be shown to undergo an extended intermediate growth phase (following earlier decay) that cannot be attributed to forcing by domain-averaged shear. The importance of adiabatic and diabatic mechanisms in this and other stages of tilt evolution will be addressed.
The remainder of this paper is organized as follows. Section 2 describes our numerical experiments and our method for measuring the misalignment of a developing system. Section 3 examines the impact of misalignment on tropical cyclone development, and section 4 examines the tilt dynamics primarily in the context of an illustrative example. Section 5 recapitulates our findings. The appendixes provide supplemental details and explain a number of techniques used to analyze our simulations.
2. Basic methodology
The forthcoming discussions of our methodology and results require a preliminary remark on notation. In this paper, the properties of a tropical cyclone are usually described in a cylindrical coordinate system. The coordinates r, φ, and z respectively denote radius, azimuth, and height above sea level. The variables u, υ, and w respectively represent the radial, azimuthal, and vertical velocity fields. An arbitrary fluid variable is sometimes decomposed into an azimuthal mean dressed with an overbar, and a perturbation dressed with a prime. For example, the azimuthal velocity field is given by
a. Numerical simulations
The numerical simulations are conducted with Cloud Model 1 (CM1-r19.5; Bryan and Fritsch 2002) in an energy-conserving mode of operation. Unless stated otherwise, the model is configured with a variant of the two-moment Morrison microphysics parameterization (Morrison et al. 2005, 2009), having graupel as the large icy-hydrometeor category and a constant cloud-droplet concentration of 100 cm−3. The influence of subgrid turbulence above the surface is accounted for by an anisotropic Smagorinsky-type closure. The nominal mixing lengths are tailored for tropical cyclones on grids that are deemed insufficiently dense for a standard large-eddy-simulation scheme. The horizontal mixing length increases from 0.1 to 0.7 km as the underlying surface pressure decreases from 1015 to 900 hPa. The vertical mixing length increases asymptotically to 50 m with increasing z. Surface fluxes are parameterized with bulk-aerodynamic formulas. The momentum exchange coefficient Cd increases with the surface wind speed from 10−3 to 0.0024 above 25 m s−1 [compare with Fairall et al. (2003) and Donelan et al. (2004)]. The enthalpy exchange coefficient is given by Ce = 0.0012 roughly based on the findings of Drennan et al. (2007). Heating associated with frictional dissipation is activated. The computationally expensive parameterization of radiative transfer is turned off under the provisional assumption that the associated diabatic forcing is unessential to understanding the slowdown of intensification induced by misalignments of incipient tropical cyclones.
The model is set up on a doubly periodic oceanic f plane with a sea surface temperature of 28°C and a Coriolis parameter of f = 5 × 10−5 s−1 corresponding to a latitude of 20°N. The dynamical system is discretized on a stretched rectangular grid. The horizontal grid spans 2660 km in each orthogonal coordinate. The 800 × 800 km2 central region of the horizontal mesh has grid increments of 1.25 km. At the four corners of the horizontal domain, the grid increments are 13.75 km. The vertical grid has 73 points and extends upward to z = 29.2 km. The vertical grid spacing gradually increases from 50 to 400 to 750 m as z increases from 0 to 8 to 29 km. Rayleigh damping is imposed above z = 25 km.
Axisymmetric initial conditions for preliminary priming of the model. (a) Azimuthal velocity distribution (color) and pressure (contours; hPa). (b) Relative humidity with respect to ice (T < 0°C) or liquid (T > 0°C), and selected positive (solid red) and negative (dashed blue) isolines of the potential temperature (θ) anomaly, here defined by
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
The misalignment experiments are initialized with conditions derived from the control simulation 99 h into its development, shortly before one might declare the genesis of a warm core tropical cyclone. Figure 2 depicts the azimuthally averaged fluid variables at this time, henceforth labeled t = 0. The absolute maximum of
Azimuthally averaged fields associated with the tropical disturbance after 99 h of priming and immediately before misalignment, when t is reset to zero. (a) Azimuthal velocity
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
The DSPD method first removes all asymmetry in φ from the vortex. Dissipative heating is eliminated, Cd is reduced to 10−4, and Ce is reduced to zero. The moisture parameterization is turned off and the water vapor is temporarily replaced with a passive tracer. The symmetric dry vortex is given one day to adjust toward a more balanced state, whereupon it is misaligned using variants of the ISPD method with adjustable values of τs, δτs and τd that typically correspond to slower forcing. At the end of the forcing period, the passive tracer is replaced with water vapor. The mixing ratio is reduced where necessary to 99.5% of its saturation value with respect to ice/liquid at altitudes above/below the local freezing level to prevent the minor shock of instantaneous cloud formation. All physics parameterizations of the control experiment are restored, and time t is reset to zero.
Table 1 lists all of the numerical experiments considered herein, except for the control run. The name of each experiment begins with the abbreviation of the misalignment method and ends with a code for two additional parameters. In this code, the symbol X represents 2Usτs, and Z represents zl; the rounded-down value (in kilometers) of each parameter is printed to the immediate right of the corresponding symbol. The value of 2Usτs ranges from 100 to 400 km, and can be viewed as a target magnitude for the tilt [Eq. (4)] that is initially forced on the vortex. The actual tilt magnitude moderately varies between IS/ISPD and DSPD simulations with the same parameter code. The actual tilt magnitude also depends to some degree on zl, which equals 5.25, 3.5, or (in one case) 1.75 km. It is found that the tilt dependence of tropical cyclone development in this study is basically consistent between groups of simulations with different values of zl. Appendix A illustrates the diversity of vortex states within our simulation set immediately after the misalignments are created.
List of experiments (excluding the control run) and the parameters used to generate the misalignments. See text for discussion.
b. Vortex center finding algorithm
Many of the measurements made for this study depend on a reasonable technique for establishing the center of the vortex in a given layer of the atmosphere. The technique employed herein first obtains a mass-weighted vertical average of the relative vertical vorticity distribution ζ in a layer defined by h− ≤ z ≤ h+. The vorticity distribution on a regular fine grid (with spacing lf = 1.25 km) is then mapped onto a regular coarse grid (with spacing lc = 23.75 km). The mapping involves redistributing the circulation of each fine grid cell to the larger cells associated with the four nearest coarse grid points by the method of area-weighting, and then dividing the accumulated circulation in each coarse grid cell by
Smoothing out small features in ζ prior to searching for xc is designed to reduce the probability of associating xc with the center of an intense but transient meso-γ-scale vortex. Once a relatively strong tropical storm develops, the probability of such dubious centering without smoothing of ζ is significantly reduced in the simulations considered for this study. At such a point of development, we therefore eliminate the use of a coarse grid and the smoothing step in the center finding algorithm.
The center of the surface vortex xcs is generally computed with h− = 0.025 km and h+ = 1.01 km. The center of the middle-tropospheric vortex xcm is generally computed with h− = 7.34 km and h+ = 8.13 km. The moving reference frame having coordinates centered at xcs will be called the surface-vortex-centered (SVC) reference frame. After misalignment, the velocity of the SVC reference frame relative to the ES reference frame is typically of order 1 m s−1.
Bear in mind that the description of tilt dynamics can vary to some extent with the definition of xc (e.g., Ryglicki and Hart 2015). The center finding technique described above is presently favored by the authors and (in essence) within the domain of conventional practice. For those interested, appendix B briefly addresses the consequences of using an alternative technique; the contents of this appendix are best read after section 4a.
3. Impact of misalignment on hurricane formation
a. Hindered intensification of the maximum surface wind speed
Let tITC denote the time during the evolution of an incipient tropical cyclone (ITC) when υm first reaches a modest pre–tropical storm value of 12.5 m s−1. Furthermore, let tCAT1 denote the time when υm first reaches a value of 32.5 m s−1, which approximately corresponds to the threshold wind speed of a category-1 (CAT1) hurricane. Henceforth, the time interval between the aforementioned events will be called the hurricane formation period (HFP). The duration of the HFP is given by τhf = tCAT1 − tITC. The time-averaged magnitude of the tilt vector over the HFP will be denoted by “tilthf.” It is worth remarking that the value of tilthf is closely linked to the maximum tilt magnitude applied within the first 6 h of simulation time (tilt0). A linear regression yields tilthf = −15.066 + 0.364tilt0 (km) and the Pearson correlation coefficient (PC) is 0.88.2
Figure 3 shows that τhf reliably grows with tilthf. The linear correlation is quantitatively robust in that PC = 0.92. The remainder of section 3 elaborates upon this central result. Sections 3b and 3c examine modifications of the tropical cyclones that coincide with increasing tilt magnitudes and slower development. Section 3d examines how the changes in vortex structure and moist convection associated with enhanced misalignment affect the angular momentum budget. Section 3e investigates how increasing the tilt magnitude affects the growth of alternative kinetic-energy-based measurements of vortex intensity.
Relationship between τhf and tilthf. The dotted linear regression line is given by τhf = 14.569 + 1.441tilthf in units corresponding to the plotted data.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
b. Reorganization of convection and enlargement of the surface vortex
The slowdown of hurricane formation in a misaligned system coincides with the reorganization of convection and enlargement of the surface vortex. Figure 4a demonstrates that greater values of tilthf correspond to greater values of the time-averaged precipitation radius rp. By definition, rp is the radius in the SVC coordinate system at which the azimuthally averaged 2-h precipitation (surface rainfall) distribution is maximized. Figure 4b shows that the time average of the radius rm at which υm occurs grows commensurately with rp.
(a) Precipitation radius rp averaged over the HFP vs tilthf. The dotted linear regression line is given by rp = 18.825 + 0.775tilthf (km) and the correlation coefficient is 0.96. (b) Relationship between rp and the radius of maximum surface-
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
In addition to moving outward, the precipitation grows increasingly asymmetric with enhanced tilt. Figure 5 depicts the tilthf dependence of the 2-h precipitation asymmetry during the HFP. The total (area integrated) 2-h precipitation in a 400-km circular disc centered at xcs is split into individual contributions from 4 quarter circles. The quarter circles are centered in azimuth at φ = 0°, 90°, 180°, and 270° (−90°), with φ = 0 corresponding to the downtilt direction (i.e., the direction of Δxc). Each absolute contribution to the 2-h precipitation is then divided by the total to form a fractional contribution. The plotted precipitation probability is the time average of the fractional contribution over the HFP. As tilthf increases from 10 to 60 km, the probability of precipitation in the downtilt quadrant dramatically grows from slightly above 25% to approximately 60%. The probability of precipitation in the uptilt quadrant (centered at 180°) decays to a value substantially less than 10%. The precipitation probabilities in the quadrants centered at −90° and 90° also diminish, but the former decays less than the latter.
(a) Variation of the precipitation probability distribution with tilthf. (b) The four quarter-circles [colored to match the symbols in (a)] over which the precipitation probability is distributed; the angles are defined such that the time-dependent tilt vector Δxc always points toward 0°.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
For illustrative purposes, Figs. 6 and 7 depict the asymmetric structure of the vortex in simulation DSPD-X400Z5 at a time during the HFP when |Δxc| = 240 km. DSPD-X400Z5 is among a handful of simulations most worthy (in our view) of detailed examination, because the misalignment coincides with severely hindered development of the tropical cyclone. The structure of the vortex in DSPD-X400Z5 is similar to that found in earlier studies of real-world and simulated tropical cyclones that are tilted by moderate environmental wind shear prior to achieving hurricane status (e.g., Rappin and Nolan 2012; Nguyen et al. 2017). Figure 6 shows the misalignment of quasi-circular lower-tropospheric (lw) and middle-tropospheric (md) streamlines in the slowly moving SVC reference frame, superimposed on a complementary plot of the magnitude of the local shear velocity, defined by umd − ulw.3 The lower- and middle-tropospheric flows specifically correspond to elevations of 1.2 and 7.7 km above sea level. The green ray represents the φ = 0 axis and therefore points exactly downtilt.
Depiction of the horizontal flow in the SVC reference frame 48 h after the initial misalignment in simulation DSPD-X400Z5. The black and white streamlines respectively correspond to z = 1.2 and 7.7 km. Colors show the magnitude of the local shear velocity measured between the two levels. The green ray points in the direction of the tilt vector.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Selected fields associated with convection 48 h after the initial misalignment in simulation DSPD-X400Z5. All fields are rotated such that the tilt vector points directly to the right. (a) Logarithm of the 2-h accumulated precipitation P normalized to P0 = 1 cm. The accumulation is measured over the interval 47 ≤ t ≤ 49 h. (b) Vertical average of equivalent potential temperature in the 1-km surface boundary layer, denoted θeb. Surface streamlines are shown in red. (c) Vertical average of the relative humidity between z = 2.3 and 7.7 km. (d) Vertical velocity field (red and blue) and streamlines of the horizontal flow (black) in the lower-middle troposphere. The vertical velocity field w is averaged between z = 2.3 and 7.7 km, and is smoothed in the horizontal plane using a Gaussian kernel with a standard deviation of 6.2 km in both dimensions. The colormap is logarithmic. The streamlines correspond to z = 5.6 km and thus belong to a transition layer where the center of rotation is in between xcs and xcm. (e),(f) Height perturbation about z = 5.52 km (color) and horizontal streamlines on the θυ = 325.4-K isosurface in (e) the CM1 simulation and (f) a system with equivalent ζ in a state of nonlinear balance. The streamlines in (e) and (f) are shaded such that black indicates |u| < 7.5 m s−1, gray indicates 7.5 ≤ |u| ≤ 15 m s−1 and white indicates |u| > 15 m s−1. All streamlines in (b) and (d)–(f) correspond to u in the slowly moving SVC reference frame. Grid circles are spaced 50 m apart in (a)–(d); the dotted circles in (e) and (f) show where r = 200 m.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figure 7a shows the 2-h precipitation field rotated such that downtilt is now directly to the right. Consistent with Fig. 5, much of the precipitation is seen in the downtilt quadrant, whereas the uptilt quadrant is relatively quiet. Figure 7b shows the surface streamlines superimposed on the boundary layer equivalent potential temperature θeb, defined here as the vertical average over the lowest 1 km of the troposphere. The distribution of θeb has relatively low values at and downwind of where low-entropy downdrafts are expected in association with strong convective activity. Note that the deficit of θeb in the vicinity of the active precipitation region coincides with a pronounced cold pool having surface values of ordinary potential temperature θ down to 3.3 K below the domain average (not shown). Figure 7c shows the horizontal distribution of relative humidity averaged between z = 2.3 and 7.7 km, taking values with respect to ice/liquid at altitudes above/below the freezing level. Humidification that could facilitate vigorous deep convection has failed to develop uptilt. Figure 7d depicts a lower-middle-tropospheric flow pattern suggesting that an influx of relatively dry air from the outer part of the vortex and a weak meso-α-scale downdraft (in conjunction with subsidence warming) contribute to maintaining low relative humidity uptilt.
Although more than one factor may contribute to the predominant downdraft between φ = 0 and 180°, its magnitude notably agrees with an estimate that assumes middle-tropospheric downgliding of unsaturated air along a nearly frozen virtual potential temperature isosurface. Figure 7e depicts one such isosurface in the CM1 simulation, which compares favorably to that of a system with equivalent ζ that has adjusted to a state of nonlinear balance (Fig. 7f, see appendix C). The downgliding velocity is estimated as wdg ~ δzU/L = −0.01 m s−1, in which δz ~ −5 × 102 m is the vertical displacement along a streamline path that traverses the interval 0° ≤ φ ≤ 180°, U ~ 10 m s−1 is the characteristic horizontal wind speed, and L ~ 5 × 105 m is the horizontal pathlength.
c. Variation of moist-thermodynamic parameters
The convective and structural modifications associated with misalignment coincide with changes to several bulk moist-thermodynamic parameters in the surface-centered core of the developing system. Figure 8a verifies that strong negative correlations (PC = −0.92 ± 0.01) exist between tilthf and spatiotemporal averages of the relative humidity distribution measured as in Fig. 7c. The time averaging covers the entire HFP. The spatial averaging is over a cylindrical volume defined in the SVC coordinate system by 2.3 ≤ z ≤ 7.7 km and 0 ≤ r ≤ R, in which R is either 100 or 250 km. Qualitatively similar anticorrelations have been verified for R = 25 and 400 km (not shown). The association of enhanced tilt with drier air above the surface vortex is notable in view of prior studies suggesting that such low humidity alone can hinder the onset of rapid intensification (e.g., section 3 of Schecter 2016).
(a) Spatiotemporal average of lower-middle-tropospheric relative humidity (RH) vs tilthf. The spatiotemporal average is within a radius R of 100 km (main plot) or 250 km (inset) from xcs during the HFP. The dotted linear regression line is given by RH = 95.899 − 0.177tilthf (main plot) or RH = 89.896 − 0.141tilthf (inset). (b) Similar spatiotemporal average of the surface enthalpy flux Fk vs tilthf for (top) R = 100 km and (bottom) R = 250 km. The dotted linear regression line in the top plot is given by Fk = 2.476 − 0.011tilthf. (c) Similar spatiotemporal average of CAPE vs tilthf for (top) R = 100 km and (bottom) R = 250 km. The dotted linear regression line in the top plot is given by CAPE = 1.146 + 0.008tilthf. Note that the units of the variables in each regression equation equal those in the corresponding plot. See Fig. 3 for the symbol legend.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
One might reasonably ask whether slower development in a strongly misaligned system also coincides with a reduction in the rate at which the vortex extracts the sum of latent and sensible heat from the sea surface. Figure 8b addresses the preceding question by showing the relationship between the tilt magnitude and the parameterized surface enthalpy flux. The enthalpy flux Fk is defined as in Eq. (3) of Zhang et al. (2008). As for relative humidity, the plotted value of Fk is a spatiotemporal mean taken within a variable radius R of the surface vortex center during the HFP. A strong anticorrelation (PC = −0.93) exists between the mean value of Fk and tilthf when R = 100 km; a qualitatively similar result is found for R = 25 km (not shown). The anticorrelation becomes far less convincing for broader averaging discs, as shown for the case in which R = 250 km, where PC = −0.44.
Note that slower development with enhanced tilt is not associated with a reduction of convective available potential energy (CAPE) in the central region of the surface vortex. Figure 8c shows the spatiotemporal mean value of the 500-m mixed layer CAPE, calculated under the assumption of undiluted pseudoadiabatic ascent with liquid-only condensate. The averaging is identical to that of Fk. For R = 100 km, the mean CAPE reliably grows (PC = 0.95) with increasing values of tilthf; the same is true when R is reduced to 25 km (not shown). When R is extended to 250 km, no meaningful correlation between the mean CAPE and tilthf can be established (PC = −0.26).
d. Modification of the angular momentum budget
It is now appropriate to delve deeper into how the geometrical restructuring of the vortex and the coupled reorganization of convection affect the mechanism of surface spinup in the vicinity of maximal winds. A comprehensive investigation would extend beyond the scope of this article, but a limited analysis seems fitting. The following compares the
Time series of υm in simulation DSPD-X400Z5 (dashed) and the control experiment (solid) during the 6-h periods of early development depicted in Figs. 10–12. Time t′ is measured from the start of each 6-h period.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figure 10 depicts the time-averaged state of each system during the aforementioned 6-h analysis periods. In the azimuthal mean, the misaligned vortex with highly asymmetric convection (DSPD-X400Z5) has a shallower cyclonic circulation, a larger value of rm, and two distinct updrafts sprouting from the lower troposphere. The virtually symmetric convection in the control simulation is characterized by a single strong updraft peaked near the relatively small radius of maximum surface wind speed.
(a),(b) Structure of the tropical cyclone in simulation DSPD-X400Z5 during a selected 6-h period of early development (37 ≤ t ≤ 43 h). (a) Streamlines of the 6-h time-averaged horizontal flow in the SVC reference frame at z = 1.2 km (black) and z = 7.7 km (white) superimposed over the logarithm of the 6-h accumulated precipitation P normalized to P0 = 1 cm. (b) The 6-h time averages of
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
(a),(b) Comparison of simulation DSPD-X400Z5 to SE dynamics in the SVC reference frame. (a) The 6-h time averages of
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figures 12a–d show the individual contributions to
(a)–(d) SE-based decomposition of
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figures 12e–h show the contributions to
Perhaps the main result from the foregoing analysis is that the restructuring of the vortex and the reorganization of convection in the misaligned system rendered the mp-circulation less effective in accelerating the maximum of
Note that reasonable accuracy of the SE-based analysis relied partly on the small fractional difference (0.3 or less) between the gradient wind and
e. Surface kinetic energy growth
One might wonder whether increasing the tilt magnitude has the same qualitative effect on all measures of vortex intensity. Herein, we address the preceding question by comparing time series of several intensity parameters. Figure 13 (left) shows the temporal growth of υm for all simulations over the time scale for the virtually aligned control vortex to mature into a well-developed hurricane. Each thick curve covers the spread in a group of vortices that are initially perturbed with similar target misalignments (2Usτs) and equivalent values of zl. Consistent with Fig. 3, the time series exhibit considerable variation; enhanced tilt markedly slows the intensification of υm.
(a) Time series of the (left) maximum and (center, right) mean-squared values of the azimuthally averaged tangential velocity field (in the SVC coordinate system) at the sea surface for several groups of simulations in which the misalignment is created by splitting the vortex at zl = 5.25 km. The radial averaging intervals for
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
4. Tilt dynamics
Because tropical cyclones with minimal misalignment are generally more efficient in accelerating the cyclonic surface winds, understanding how tilt decays is an important part of understanding intensification. Figure 14 shows that the evolution of tilt is generally nonmonotonic in systems with initially forced tilt magnitudes exceeding approximately 150 km.5 To facilitate discussion, we divide the evolution into three consecutive stages. Stage 1 involves a rapid reduction of the misalignment. Stage 2 entails partial regrowth of the tilt magnitude. Stage 3 is eventually characterized by gradual decay of the tilt magnitude, but may begin with a repetition of the preceding cycle (see, e.g., the dotted curve in Fig. 14b). The remainder of this section will examine the three stages of evolution in detail for simulation DSPD-X400Z5, which is distinguished by having the largest initial tilt. The pathways of tilt decay and transient amplification that operate in DSPD-X400Z5 are considered illustrative of many (but not all) of the possibilities. Some of the known similarities and differences with other simulations will be noted as the narrative proceeds.
(a) Time series of the tilt magnitude for several groups of simulations in which the misalignment is created by splitting the vortex at zl = 5.25 km. As in Fig. 13, the asterisk in the legend stands for all prefixes (IS, ISPD, and DSPD) among those existing for the given suffix. The dotted black curve belongs to the member of the gray group having the greatest tilt. (b) As in (a), but for zl = 3.5 km. Note that each group in (b) has only two members.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Before discussing the intricacies of tilt evolution in DSPD-X400Z5, it is worthwhile to briefly consider the potential relevance of ambient wind shear that may arise over time despite our elimination of the extraneous forcing that created the initial misalignment. Figure 15 shows time series of the magnitude and angle of the ambient shear vector, defined by ⟨um − us⟩xy, in which um (us) is the vertically averaged horizontal velocity in the middle-tropospheric (near-surface) layer corresponding to where xcm (xcs) is measured. It is found that the shear magnitude remains weak (0–0.35 m s−1) and undulates over the course of the simulation (Fig. 15a). Ambient wind shear possessing one-half the maximum intensity seen here—acting in a direction parallel (antiparallel) to the tilt vector—would amplify (diminish) the tilt of the tropical cyclone at a rate of 15 km day−1. Such a rate is too small to account for the tilt tendencies found at any stage of evolution. Moreover, during the final and slowest stage of alignment, the shear vector rotates anticyclonically on a time scale that is short (an inertial period) compared to the precession period of the tilt vector (Fig. 15b). It follows that extrinsic forcing by ambient shear is not only weak but inefficient.
(a) Magnitude of the ambient shear vector in simulation DSPD-X400Z5. (b) Angles of the ambient shear vector and tilt vector relative to the eastward direction.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
a. Stage 1
The first stage of tilt evolution is characterized by rapid decay of the measured misalignment. Such decay commonly coincides with the migration of xcs toward an area of vigorous deep cumulus convection in the general direction of xcm. Figure 16 provides a minimal depiction of the process during the first 8 h of simulation DSPD-X400Z5. At any arbitrary instant during this 8-h period, the surface center of rotation sits roughly in the middle of a 100-km scale patch of cyclonic vorticity. Figure 16a shows the configuration at t = 6 h. Figure 16a also illustrates how the aforementioned vorticity patch is exposed to irrotational winds that converge toward an (initially) outward moving band of convection. The irrotational velocity field uχ has a magnitude of approximately 2 m s−1 in the vicinity of xcs. One might therefore hypothesize that advection of the central vorticity patch by westerly irrotational winds has a nonnegligible role in the 75-km eastward drift of xcs over an 8-h period. On the other hand, advection (by any part of u) is not the entire story. Downtilt convection also reshapes and rescales the nondivergent (rotational) velocity field of the surface vortex that determines the location of xcs (Fig. 16b). The nondivergent winds become enhanced in the east relative to the west. Furthermore, the radius of maximum surface wind speed rm increases from 52.5 to 70 km (131.25 km) over the first 8 h (16 h) of development. Thus, during its eastward migration, xcs transitions from representing the center of a modest meso-β-scale vortex core to representing the center of a core that is 2–3 times larger.
Selected images of the boundary layer flow of simulation DSPD-X400Z5 in the fixed domain-centered coordinate system during the initial migration of xcs toward vigorous convection downtilt. (a) Relative vertical vorticity ζ (color) and streamlines of the irrotational velocity field uχ in the boundary layer at t = 6 h. Streamline thickness is proportional to the local irrotational wind speed; gray (black) indicates wind speeds greater than (less than) 2.0 m s−1. Thin black arrowless contours of w at z = 8.9 km are superimposed on the plot to show where deep convection occurs at the time of the snapshot; the outermost-to-innermost contour levels are 0.2, 1, 3, and 5 m s−1. The dashed green ray points in the direction of the tilt vector. The “rms” statistic above the top-right corner of the plot corresponds to the root-mean-square value of |uχ| in the depicted subregion of the simulation domain. The integers printed inside the plot show the location of xcs at t = 0, 4, 6, and 8 h, with the present location in black and all others in gray. (b) Streamlines and magnitude (color) of the nondivergent velocity field uψ in the boundary layer at (left) t = 4 h and (right) t = 8 h; here the streamlines are given uniform thickness and shading. The plus sign shows the instantaneous location of xcs. All fields but w in (a) and (b) are vertical averages between z = 0.025 and 1.01 km. The ζ and w fields in (a) are smoothed in x and y using a Gaussian kernel with a standard deviation of 6.25 km. See appendix C for the precise definitions of uχ and uψ.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Whereas the early convection-seeking drift of xcs substantially reduces tilt in a number of other simulations, northeastward drift of xcm largely counters such an effect in DSPD-X400Z5 (not shown). Instead, rapid reduction of tilt occurs through a sudden jump of the middle-tropospheric center of rotation to the area of deep convection (Fig. 17a). The jump apparently results from the emergence of intense middle-tropospheric vorticity anomalies within the updraft region of the mesoscale convective system (MCS; see Fig. 17c). Appendix B (Fig. B1b) illustrates the relatively strong rotational winds associated with the emergent disturbance. The sources of the vorticity anomalies are analyzed in appendix E. Notable vorticity anomalies are also found in the northern sector of the boundary layer of the MCS (Fig. 17b), but they are insufficiently strong in the aggregate to abruptly relocate xcs.6
Selected images associated with the abrupt reduction of tilt during stage 1 of simulation DSPD-X400Z5. (a) Surface and middle-tropospheric centers of rotation [xcs (red +) and xcm (blue ×)] plotted every 30 min over the interval 8.0 ≤ t ≤ 9.5 h. The center markers are superimposed on a plot of the logarithm of the accumulated precipitation P (normalized to P0 = 1 cm) during the aforementioned 1.5-h time interval. Yellow symbols highlight xcm immediately before (circle) and after (diamond) it jumps to an area of enhanced precipitation. (b),(c) Vertical averages of the relative vertical vorticity distribution ζ immediately after the rapid reduction of tilt (t = 9.5 h) in (b) the 1-km surface boundary layer and (c) the middle-tropospheric layer between z = 7.34 and 8.13 km. A contour plot of the vertical velocity field w at z = 8.9 km (averaged over 8 ≤ t ≤ 9.5 h) is superimposed on each vorticity distribution. Both ζ and w are smoothed in the horizontal plane using a Gaussian kernel with a standard deviation of 6.25 km in both dimensions. The + and × in (b) and (c) respectively mark xcs and xcm at the time of the vorticity snapshot. The origin of the coordinate system in each plot corresponds to the fixed center of the simulation domain.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
b. Stage 2
The subsequent regrowth of tilt in experiment DSPD-X400Z5 occurs gradually over a 2-day period of modest intensification of υm. The first issue we address is whether moisture has an essential role in the process. Figure 18a compares the growth of |Δxc| to that found in two dry adiabatic restarts at t = 28 h. Two methods are used for restarting the model to demonstrate that details have little consequence on the result. The first restart eliminates cloud microphysics, dissipative heating and the surface enthalpy flux without any additional modifications. The second restart also reduces Cd to 2.5 × 10−5 and refines the fluid variables. The velocity field is refined by zeroing w, the irrotational component of u, and the z-dependent horizontal mean of u. The nondivergent component of u is obtained by inverting ζ, adjusted to have zero vertical gradient in a 303-m layer adjacent to the sea surface. The final refinement involves enforcing conditions of nonlinear balance on θ and the pressure field (see appendix C). Both restarts demonstrate that the system would have a propensity to increase tilt under dry adiabatic dynamics somewhat faster and more effectively than the actual process occurs amid moist convection.
Stage 2 of tilt evolution in experiment DSPD-X400Z5. (a) (bottom) Time series of the tilt magnitudes in the moist CM1 simulation, an unrefined dry restart and a dry balanced restart. (top) Attendant growth of υm in the moist simulation. (b) Surface and middle-tropospheric centers of rotation at t = 28, 34, and 60 h. Squares show xcs (red) and xcm (blue) from the moist simulation. Triangles show xcs (light pink) and xcm (light blue) from the unrefined dry restart (left pointing) and the dry balanced restart (right pointing) initialized at t = 28 h. Circles show xcs (dark pink) and xcm (cyan) predicted by 2D vortex-in-cell simulations of the boundary layer and middle-tropospheric layer starting at t = 28 h. The position markers are superimposed on a plot of the logarithm of the accumulated precipitation P (normalized to P0 = 1 cm) between t = 28 and 34 h. (c) Relative vertical vorticity ζ (color) and streamlines of uχ in the middle-tropospheric layer containing xcm (×) in the moist simulation at (left) t = 31 h and (right) t = 60 h. Vorticity is smoothed in the horizontal plane using a Gaussian kernel with a standard deviation of 6.25 km in both dimensions. Streamline thickness is proportional to |uχ|; gray (black) indicates irrotational wind speeds greater than (less than) 1.5 m s−1. (d) Corresponding snapshots of the streamlines and magnitude (color) of uψ; here the streamlines are given uniform thickness and shading. All plotted fields in (c) and (d) are vertical averages over the interval 7.34 ≤ z ≤ 8.13 km. The origin of the coordinate system in (b)–(d) is located at the fixed center of the simulation domain.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figure 18b compares the moist and dry trajectories of xcs and xcm. The middle-tropospheric vortex centers of the dry CM1 simulations move northward with their moist counterpart, but drift farther to the west. Early on, the surface center of the moist system is strongly inhibited from following any dry inclination to move southwest. Such inhibition is consistent with the common attraction of surface centers toward areas of vigorous deep convection, here situated to the northeast of xcs. Figure 18b also shows the trajectories predicted by ideal 2D fluid dynamics. The 2D results come from two separate vortex-in-cell simulations (Leonard 1980) initialized with ζ distributions obtained from the pertinent surface and middle-tropospheric layers of the atmosphere at t = 28 h. Each vortex-in-cell simulation has roughly 108 vorticity elements, a rectangular mesh with 0.65-km grid spacing, and doubly periodic boundary conditions equivalent to those of the CM1 simulations. The middle-tropospheric trajectory predicted by 2D dynamics remains relatively close to that of the moist system. Such closeness may be somewhat coincidental, but reproduction of the basic northward drift suggests some relevance of the 2D model. By contrast, the 2D surface trajectory strays considerably from its moist counterpart, and ends up far west of all 3D systems.
Figures 18c and 18d show snapshots of the moist middle-tropospheric vortex near the start and end of the northward drift of xcm that is largely responsible for the regrowth of tilt. It is seen that the drift coincides with considerable reshaping of an asymmetric vertical vorticity distribution with multiscale structure and a prominent band extending outward from the core. The process occurs amid continual 3D-adiabatic and diabatic perturbations of ζ. The associated irrotational winds represented by uχ are nontrivial (Fig. 18c), but as for any predominantly vortical flow, the nondivergent component of the velocity field uψ is characteristically stronger (Fig. 18d). The root-mean-square (rms) value of |uψ| is 3.4 times the rms value of |uχ| over the depicted area in both snapshots.7
The relative strength of uψ combined with the qualitatively successful prediction of the vortex-in-cell simulation in the middle troposphere motivate further consideration of how nondivergent 2D dynamics may contribute to the drift of xcm. Figure 19 splits the middle-tropospheric nondivergent velocity field into two parts during the northward drift period. The first part
(a)–(c) Decomposition of the nondivergent winds in the middle troposphere near the beginning of the tilt restoration period (t = 31 h) in simulation DSPD-X400Z5. (a) Relative vertical vorticity smoothed using a Gaussian kernel with a standard deviation of 30 km in both x and y. The closed black curve is a contour where the smoothed vorticity distribution equals 5 × 10−6 s−1. The predominantly red area enclosed by this contour contains the (unsmoothed) vorticity that generates
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Although regrowth of the tilt magnitude in a cloud-resolving simulation of tropical cyclone development without assistance from environmental wind shear may seem surprising, the spontaneous misalignment of a vortex is not an entirely novel phenomenon. Previous discussions of spontaneous misalignment have often dealt with tilts arising from essentially adiabatic, three-dimensional circular shear flow instabilities (Gent and McWilliams 1986; Schecter et al. 2002; Reasor et al. 2004). Jones (2000a) examined what might be viewed as the nonlinear stage of an instability contributing to the growth of tilt in a dry vortex simultaneously interacting with environmental winds. The associated dynamics happened to resemble that described above, in which an outer anticyclonic vorticity anomaly acts to drive the cyclonic core of the vortex in one layer of the atmosphere away from the core in another layer. While the similarity is intriguing, we caution against extending the analogy too far. Events leading up to the foregoing scenario in DSPD-X400Z5 are distinct in part by having a substantial diabatic element (see section 4a and appendix E). There is also evidence, provided below, suggesting that convection and its associated irrotational winds significantly modulate the drift of xcm during the tilt amplification period.
The aforementioned evidence is found by viewing the drift of the middle-tropospheric cyclone from a perspective that is more directly connected to the measurement of xcm. Recall that xc is essentially the point in a specified layer where the centering of a polar coordinate system yields the largest peak value of
Changes of middle-tropospheric
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
c. Stage 3
The final stage of tilt evolution brings the system to a state of virtual alignment. Dry adiabatic restarts cannot reproduce the sustained alignment trends found in the moist simulation (Fig. 21a). It stands to reason that diabatic cloud processes are essential. Figure 21b shows the near-surface and middle-tropospheric vortex trajectories over the period between hours 165 and 171 of development, which essentially covers the final surge to an aligned state. The trajectories are superimposed over a depiction of the attendant moist convection. Whereas xcm shifts little within the broader updraft region of a convective complex,11 xcs darts westward toward its counterpart.
Overview of stage 3 of tilt evolution in simulation DSPD-X400Z5. (a) (bottom) Time series of the tilt magnitude in the moist CM1 simulation (solid), two unrefined dry adiabatic restarts (dotted), and one dry adiabatic restart initialized in nonlinear balance (dashed). (top) Attendant growth of υm in the moist simulation. (b) Surface and middle-tropospheric centers of rotation [xcs (red +) and xcm (blue ×)] plotted hourly over the shaded time interval (165 ≤ t ≤ 171 h) in (a). Yellow circles (squares) indicate the start time (end time). The background shading shows the logarithm of the accumulated precipitation P (normalized to P0 = 1 cm) over the aforementioned 6-h time interval. The contours show the corresponding 6-h time-averaged vertical velocity field at z = 8.9 km. The origin of the coordinate system corresponds to the fixed center of the simulation domain.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figure 22 illustrates the nature of the fluid dynamics during the westward motion of xcs. The top row shows that the near-surface cyclone consists of multiple meso-γ-scale vortices immersed in diffuse, predominantly cyclonic background vorticity. The meso-γ-scale vortices are typically products of convection in either the core of the parent cyclone or peripheral rainbands. Some may travel far away from their points of origin over the course of a lifetime; the three prominent eastern vortices at t = 165 h notably emerged from the main area of convection west of xcs. The irrotational velocity field consists of broad inflow from the outer part of the cyclone and zones of confluence near active convection. Relatively strong western confluence of uχ may contribute significantly to the westward shift of xcs. On the other hand, the rms wind speed of the nondivergent velocity field is 5.6 times that of uχ in each depicted snapshot of the evolving system. Generic mixing processes typical of nondivergent 2D flows seem to have a nontrivial role in reshaping the vorticity distribution. The bottom row of the figure shows the nondivergent velocity field and helps clarify why the measurement of xcs moves westward. The intensity distribution becomes highly skewed to the west before the strongest winds become more evenly distributed about a 100-km scale circle whose center is substantially displaced (to the west) from where the surface vortex center resided 6 h earlier.
Selected snapshots of the boundary layer flow of simulation DSPD-X400Z5 in the fixed domain-centered coordinate system during the final surge toward alignment (the shaded time interval in Fig. 21a). (a) Relative vertical vorticity ζ (color) and streamlines of the irrotational velocity field uχ. Streamline thickness is proportional to the local irrotational wind speed; gray (black) indicates wind speeds greater than (less than) 2.5 m s−1. (b) Streamlines and magnitude (color) of the nondivergent velocity field uψ; here the streamlines are given uniform thickness and shading. All fields in (a) and (b) are vertical averages over the interval 0.025 ≤ z ≤ 1.01 km but are unsmoothed in x and y. The instantaneous location of xcs is marked by a small +. The thick semitransparent gray circle in (b) is centered on the large +, which marks the location of xcs at t = 171 h and serves as the fixed coordinate center for the analysis in Figs. 23a and 23b.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
To elaborate, the westward drift of xcs coincides with the amplification of maximal
(a) Change of boundary layer
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
As a final remark, the third stage of tilt evolution coincides with fairly slow growth of υm (Fig. 21a). The onset of rapid intensification does not occur until nearly one full day after the final surge of alignment that brings |Δxc| down to approximately 20 km.
d. Comment on alignment paradigms based on vortex Rossby wave dynamics
There exists a sizable body of literature on the potential importance of vortex Rossby (VR) wave dynamics in contributing to the alignment process (Reasor and Montgomery 2001,2015; Reasor et al. 2004; Schecter et al. 2002; Schecter and Montgomery 2003,2007; Schecter 2015). In quasi-balanced linear perturbation theory, a relatively weak tilt decomposes into a set of discrete and sheared VR waves.12 If stability conditions are satisfied, free alignment may occur by the outward propagation and spiral windup of sheared VR waves, or by the negative feedback that a discrete VR wave will receive upon exciting a potential vorticity (PV) perturbation in a critical layer.
A reasonable estimate of the time scale for outward propagation and spiral windup of a sheared (tropical cyclone scale) VR wave is τυ ≡ 2πrυ/υυ, in which rυ and υυ respectively denote the characteristic radius and azimuthal velocity of the vortex. The time scale for damping of a discrete VR wave is sensitive to the average value of a quantity proportional to the radial gradient of basic-state PV in the critical layer, centered on the surface where
As time advances, alternative (nonlinear and diabatic) mechanisms can diminish the tilt, cloud coverage may become more diffuse, and the tropical cyclone may intensify to some degree. Cloud-related reduction of N and growth of υυ both decrease
5. Conclusions
This paper has examined the development of tropical cyclones with variable degrees of misalignment in a cloud-resolving model set up over a moderately warm ocean on an f plane. The study was distinguished from its predecessors in virtually eliminating ambient vertical wind shear. Our numerical simulations showed that increasing the initial tilt of an incipient tropical cyclone from a negligible value to several hundred kilometers can extend the time scale of hurricane formation from 1 to 10 days. The dramatic slowdown of development in a strongly perturbed system was linked to an extended duration of misalignment resulting from incomplete early decay and subsequent transient growth of the tilt magnitude. Prolonged misalignment coincided with a prolonged period of asymmetric convection peaked far from the rotational center of the surface vortex xcs. Conventional wisdom holds that such a state of affairs generally frustrates the intensification of the maximum tangential wind speed.
To elaborate, the incipient tropical cyclones were found to immediately begin a major realignment phase lasting up to 1 day. The mechanism was examined in detail for a selected simulation starting with an exceptionally large (367 km) tilt. As in other simulations, xcs migrated toward an area of vigorous deep cumulus activity in the general direction of the middle-tropospheric rotational center xcm. Moreover, xcm abruptly jumped to a pronounced vorticity anomaly that formed within the same mesoscale convective system.
Setting the initial tilt magnitude to appreciably exceed the 100-km length scale of the vortex core generally prevented a direct route to alignment, and enabled substantial regrowth of the tilt magnitude over a 1–2 day time scale following its initial decay. For the aforementioned case study, a similar propensity for the restoration of misalignment was found in two dry adiabatic restarts with slightly different initializations and computational configurations. Furthermore, a nondivergent 2D fluid model correctly predicted the direction and magnitude of the drift of xcm away from xcs that largely accounted for the regrowth of tilt. Analysis (of the moist 3D system) suggested that an anticyclonic component of the nondivergent velocity field uψ associated with an external negative vorticity patch nudged the cyclonic vorticity core containing xcm in a manner that could assist the observed drift. That being said, diabatic processes persisted and affected the evolution of the tilt vector. Deeper analysis revealed that convection and irrotational winds modulated the specific changes of uψ that our center-finding algorithm used to determine the motion of xcm. Moreover, the moist system restored the misalignment more slowly and less completely than the dry adiabatic systems.
The final stage of the alignment process lasted several days in tropical cyclones initialized with relatively large tilts. In the simulation selected for detailed examination, diabatic processes were clearly essential to the ultimate reduction of tilt; the removal of moisture at different times arrested or reversed the decay trend. The final stage of alignment seemed to involve intricate mechanics, as illustrated during a late 6-h westward surge of the surface center toward an area of deep convection containing its middle-tropospheric counterpart. The surge coincided with the mixing of subvortices and filaments within the broader surface cyclone, amid convective forcing weighted to the west. The motion of xcs was linked to a modest amplification of maximal (lower-tropospheric)
The extent to which the foregoing dynamics of a misaligned tropical cyclone depends on details of the experimental setup remains to be seen. Sensitivity to variation of the sea surface temperature and the inclusion of a radiation parameterization merits further investigation. Of equal interest would be a study on sensitivity to the basic structure of the vortex prior to the initial misalignment, which is a well-known issue in the related dry adiabatic problem (Jones 2000a,b; Schecter and Montgomery 2003; Reasor et al. 2004).
Overall, the results of our numerical experiments appear to have corroborated inferences drawn from earlier studies (with sustained shear) on how misalignment reorganizes convection and inhibits tropical cyclone development. A robust positive linear correlation was found between the time-averaged tilt of a developing system (tilthf) and the time required for the surface-maximum tangential velocity υm to reach the threshold wind speed of a hurricane. Relatively large values of tilthf and slower development were associated with enhanced asymmetry (downtilt bias) of the precipitation probability distribution and larger values of the characteristic precipitation radius rp in the surface-vortex-centered coordinate system. The growth of rp with increasing tilthf coincided with a commensurate growth of the radius rm at which υm is located.
The consequences of tilt-related modifications to tropical cyclone structure on the mean-vortex spinup mechanism were briefly examined in the context of a Sawyer–Eliassen (SE) model. The SE-based analysis provided a reasonably accurate decomposition of the early azimuthal velocity tendency in the vicinity of υm, in selected systems with low and high degrees of misalignment. Although no single factor completely explained slower intensification in the strongly misaligned system, greater tilt was linked to weaker positive azimuthal velocity forcing near υm by the component of the mean secondary circulation attributed to heating by microphysical cloud processes.
One notable aspect of the present simulation set was minimal system-to-system variation in the growth of kinetic energy contained outside the core of the primary surface circulation over the time scale required for an aligned vortex to mature. In other words, the detrimental effect of tilt on surface kinetic energy growth was confined to the inner region of the surface vortex.
We conclude with a final remark on the relevance of this paper to forecasting. Although a strong correlation was found between tilthf and τhf, our results suggest that an instantaneous measurement of the tilt magnitude may not be a good predictor of the time scale for future intensification of υm, even in a quiescent environment. Such is evident from the nonmonotonic evolution of the tilt magnitude. Distinct time scales would be found for equivalent magnitudes measured during different growth or decay phases of the misalignment.
Acknowledgments
The authors thank Dr. George Bryan of the National Center for Atmospheric Research (NCAR) for providing the cloud-resolving model (CM1) that was used for the numerical experiments. The authors also thank several anonymous reviewers for their generally constructive comments on the submitted manuscript. This paper was primarily supported by the National Science Foundation under Grant AGS-1743854. Additional support came from the Natural Sciences and Engineering Research Council of Canada, and from Hydro-Quebec through the IRC program. The computational resources required to conduct the simulations were provided by NCAR’s Computational and Information Systems Laboratory (doi:10.5065/D6RX99HX), and by Compute Canada.
APPENDIX A
Vortex States Generated by Artificial Misalignment Forcing
Figure A1 depicts a number of selected vortices immediately after their misalignments are generated by methods explained in section 2a. Figures A1a and A1b show a vortex that is misaligned using the DSPD method, with zl = 5.25 km and a target tilt magnitude of 400 km. The top panel shows relative vertical vorticity and relative humidity in a vertical plane parallel to the tilt vector [Eq. (4)] and passing through the surface vortex center. Both fields are 500-km centered averages over the coordinate y′ that measures horizontal distance perpendicular to the visualization plane. The relative humidity assumes ice (liquid) condensate above (below) the freezing level. The bottom panel shows horizontal slices of lower- and middle-tropospheric vorticity. Note that the misalignment process 1) reduces middle-tropospheric relative humidity over the central and uptilt regions of the surface vortex, and 2) reweights the negative component of the vorticity distribution toward the east in the outer part of the middle-tropospheric circulation. Figures A1c and A1d are similar to Figs. A1a and A1b, but for a DSPD experiment with a target tilt magnitude of 100 km. Here, the aforementioned consequences of the misalignment process are considerably less pronounced.
Vortex states immediately after misalignments are generated [t = 0 h (6 h) for DSPD (ISPD) experiments]. (a) Relative vertical vorticity (color) and relative humidity (contours; %) in a vertical plane parallel to the tilt vector and passing through the surface vortex center in experiment DSPD-X400Z5. (b) Relative vertical vorticity at z = 1.0 km (ζ1; color) and z = 7.7 km (ζ7.7; contours). The contour labels are in units of 10−4 s−1. The locations of the surface and middle-tropospheric vortex centers (xcs and xcm defined in section 2b) are marked by the symbols + and ×, respectively. The green x′ (y′) axis runs parallel (perpendicular) to the tilt vector, which points from + to ×. The fields in (a) are averaged over the 500-km distance between the depicted end-bars of the y′ axis; in addition, they are smoothed using a Gaussian kernel with standard deviations of 6.25 km in x′ and 0.15 km in z. The fields in (b) are smoothed using a Gaussian kernel with a standard deviation of 30 km in both x and y. (c),(d) As in (a) and (b), but for experiment DSPD-X100Z5. (e),(f) As in (a) and (b), but for experiment ISPD-X400Z5. (g),(h) As in (a) and (b), but for experiment ISPD-X300Z3. The color bar to the right of (g) applies to all panels in the top row. The contour levels in (d) are as in (b), whereas the contour levels in (h) are as in (f). The origin of the coordinate system in each plot corresponds to the center of the surface vortex.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figures A1e and A1f show the vortex from ISPD-X400Z5, in which the tilt is generated while convection is active over the first 6 h of the simulation. The misalignment is moderately smaller than its counterpart in the DSPD experiment with equivalent settings for the target tilt magnitude and zl (Figs. A1a,b). The convection has also generated a dipole-asymmetry within the predominantly cyclonic core of the middle-tropospheric vortex. The intensity of the negative vorticity hole within the core is found to decay with decreasing values of the target tilt magnitude (not shown). Of further note, as in the aforementioned DSPD experiment, the misaligned vortex has diminished negative vorticity in the outer western region of the middle troposphere. Figures A1g and A1h are similar to Figs. A1e and A1f, but for ISPD-X300Z3.
APPENDIX B
Sensitivity to the Center-Finding Algorithm
The “vortex center” at a specified altitude is not uniquely defined in tropical cyclone meteorology. One might wonder how changing the definition would affect the description of the dynamics. This appendix aims to remove some of the mystery for the present study without an exhaustive sensitivity test.
Figure B1a compares the tilt magnitude in DSPD-X400Z5 computed with our standard method to that obtained by replacing xcs/cm with
(a) Time series of the tilt magnitude in experiment DSPD-X400Z5 computed with standard vortex centers (see section 2b) and alternative vortex centers defined by Eqs. (B1) and (B2) with ϵζ = 0.1 and ϵζ = 0.02. (b)–(d) Snapshots of DSPD-X400Z5 during the shaded time interval in (a), in the domain-centered ES reference frame. (b) Computed vortex centers (see legend) superimposed on the streamlines and magnitude of the nondivergent velocity field uψ at t = 9.5 h, vertically averaged over the middle-tropospheric layer defined by 7.34 ≤ z ≤ 8.13 km. The inset magnifies the flow within the dashed white square. (c),(d) As in (b), but for t = 14 and 18.5 h, respectively.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
Figures B1b–d show snapshots of the streamlines and magnitude of the middle-tropospheric nondivergent velocity field after an early event when the single vortex model of the middle-tropospheric flow becomes questionable. The standard location of xcm (found as in section 2b) and the locations of
APPENDIX C
Helmholtz Velocity Decomposition and Nonlinear Balance
APPENDIX D
Sawyer–Eliassen-Based Analysis of Vortex Spinup
The main text considers the integral of each term in Eq. (D8) over a relatively short time period (δt = 6 h) using data stored every 10 min. The time integrals of
APPENDIX E
Middle-tropospheric
Figure E1 depicts the ζ budget in the middle troposphere prior to the sudden jump of xcm toward a region of vigorous cumulus activity during stage 1 of experiment DSPD-X400Z5 (see Fig. 17). Figure E1a (E1b) corresponds to the time interval between hours 5 and 6.5 (8 and 9.5) of the simulation. The far-left panel shows the sum of all terms contributing to the right-hand side of Eq. (E1) averaged over the 1.5-h analysis period. The panels to the right show similar time averages of specific contributions. Contours of ζ at the end of the analysis period (henceforth ζe) are superimposed on each plot. The analysis is conducted in the domain-centered ES reference frame.
(a) Analysis of ∂tζ in the middle troposphere during the period 5 ≤ t ≤ 6.5 h in experiment DSPD-X400Z5. Colors show time averages of (left to right) the sum of all terms on the right-hand side of Eq. (E1), the sum of the vertical advection and tilting terms, the sum of the horizontal advection and stretching terms, and the stretching term alone. The contours show a snapshot of ζ at the end of the averaging period. The positive (solid) and negative (dashed) contour levels are ±[0.05, 0.25, 0.5, 0.75]ζmax, in which ζmax = 1.21 × 10−3 s−1 is the maximum magnitude of ζ. All depicted fields are taken at z = 7.7 km and smoothed with a Gaussian kernel whose standard deviation parameter is 6.25 km in both horizontal dimensions. (b) As in (a), but during the period 8 ≤ t ≤ 9.5 h and with ζmax = 1.55 × 10−3 s−1; the location of xcm at t = 9.5 h is marked with an ×.
Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0074.1
One can readily see that the sum of all tendency terms is well-correlated with ζe in the displayed vicinity of vigorous convection. Moreover, the “forcing” of ζ by the combination of vertical advection and tilting tends to have the same sign as ζe; rarely does the opposite sign occur where ζe is substantial. The combination of horizontal advection and stretching partly reinforces and partly opposes vertical advection and tilting. The sign of the stretching term alone does not consistently match that of its combination with horizontal advection or that of all terms summed together. Note that all displayed tendencies have substantial positive and negative parts (see Haynes and McIntyre 1987). Considering the partially negative forcing of ζ associated with convection, along with the initial eastern crescent of outer negative vorticity (Fig. A1b), it is unsurprising to find a prominent anticyclonic vorticity patch east of the cyclonic core of the middle-tropospheric circulation at the start of stage 2 of the simulation (see section 4b).
APPENDIX F
Helmholtz-Based Decomposition of
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