Modulation of tropical cyclone rapid intensification by mesoscale asymmetries

Computer model simulations are one of the most important tools in current use for understanding tropical cyclone (TC) formation and rapid intensification (RI). These include “idealized” simulations in which a TC‐like vortex is placed in a hypothetical environment with predefined sea surface temperature and vertical profiles of temperature, humidity, and wind that are either constant or slowly varying across a large domain. The vast majority of such simulations begin with a perfectly circular vortex as the precursor to a TC. However, most real TCs form or intensify while interacting with asymmetric wind fields either within or external to the vortex circulation. This study introduces a method to initialize idealized TC simulations with asymmetries, and investigates the extent to which such asymmetries might delay RI in favorable environments. It is shown that mesoscale asymmetries can delay RI and reduce the fastest rates of intensification, and that these effects are statistically significantly increased when relatively low values of vertical shear of the horizontal wind are present. In some cases the asymmetries tilt the vortex directly through advection. In other cases, the wind asymmetries increase the disorganization of the convection or increase the size of the inner‐core wind field, and thus make the weaker TC more susceptible to environmental wind shear. The results suggest that mesoscale asymmetries of the wind field could be useful predictors for delay of RI in otherwise favorable environments.


INTRODUCTION
Regarding tropical cyclones (TCs), rapid intensification (RI) refers to a rapid increase in the intensity of the storm, where the intensity is usually defined as the maximum 1-min sustained surface (10 − m) wind speed associated with the TC.While selected somewhat arbitrarily (Kaplan & DeMaria, 2003), a near-universally accepted definition for RI is when the intensity increases by 30 knots ( 15.4 m ⋅ s −1 ) in a 24-hour period.This change is equal to approximately the 95th percentile of 24 − hour intensity changes for Atlantic and East Pacific hurricanes.There remains some debate as to whether RI is the result of environmental factors, such as low vertical wind shear and high mid-level humidity (Kaplan et al., 2010;Rios-Berrios & Torn, 2017;Tang & Emanuel, 2012), or internal conditions, such as a symmetric and vertically aligned vortex with a small radius of maximum winds (Alvey III et al., 2020;Miyamoto & Nolan, 2018;Rogers et al., 2015;Stern et al., 2015), or whether it occurs simply due to the random confluence of these and many other environmental and internal factors (Kowch & Emanuel, 2015).Nonetheless, accurate prediction of RI remains of great scientific and practical importance in tropical meteorology (Cangialosi et al., 2020;DeMaria et al., 2021).
Mesoscale numerical modeling has been and remains today one of the most widely used tools for understanding TC intensification.Within mesoscale modeling, the idealized approach is the practice of simulating a hypothetical TC in a highly simplified or "idealized" environment, such as on the f -plane or -plane, with constant sea surface temperatures (SSTs) and vertical profiles of temperature, humidity, and wind that are either constant or slowly varying across the model domain.In the absence of synoptic weather features, land, or variations in SST, the modeling investigation can focus on the internal dynamics of the TC and its interaction with the nearby environment, which is also highly controlled.
Due to these advantages, numerous studies have used idealized modeling of TCs to better understand TC genesis (Montgomery et al., 2006;Nolan, 2007;Nolan & Rappin, 2008;Nolan & McGauley, 2012;Schecter, 2016), intensification (Alland et al., 2021a;Finocchio & Rios-Berrios, 2021;Gu et al., 2018;Onderlinde & Nolan, 2017;Ryglicki et al., 2018), and structure change (Dai et al., 2019(Dai et al., , 2021;;Rozoff et al., 2012;Terwey & Montgomery, 2008).A common feature of all the afore-mentioned studies is the placement of a symmetric vortex in a homogeneous environment with either no environmental flow, or with mean flow and wind shear that are constant across the model domain.In reality, pre-genesis disturbances, weak TCs, and even strengthening tropical storms are never close to being as symmetric as the initial conditions used in these studies.The prevalent use of symmetric vortices is for (at least) two reasons: first, a symmetric vortex is a conceptually simple initial condition with minimal parameters to be varied; and second, computing the initial temperature and pressure fields that put the associated wind fields in hydrostatic and gradient wind balance is much simpler for symmetric vortices than for asymmetric wind fields.
This paper introduces a balancing technique for asymmetric vortices (or any arbitrary, non-divergent wind field) in a rectangular model domain, on the f -plane or -plane.Similar methods were used in earlier studies, and also recently by Yu et al. (2023) to make changes to the symmetric structure of a developing TC, but to our knowledge have not yet been used to initialize asymmetric vortices for idealized TC simulations.We use our method to make a first exploration of the effects of mesoscale asymmetries on developing and possibly rapidly intensifying TCs.Here "mesoscale" refers to disturbances with scales of 100-1000 km that may be confined to the inner core, may be outside the inner core, or may be external to the initial vortex but that influence the inner core, possibly generating new mesoscale asymmetries.
To illustrate the possible effects of mesoscale asymmetries on RI, we present two examples.Figure 1 shows the post-season best-track analyses from the National Hurricane Center and real-time forecasts from the Decay Statistical Hurricane Intensity Prediction System (DSHIPS; DeMaria et al., 2005) for two Atlantic hurricanes: Dorian (2019) and Ian (2022).The forecasts for Dorian illustrate a range of poor to good outcomes.Some of the earliest forecasts (e.g., 1200 UTC 24 August), when

Dorian
had not yet been identified as a tropical depression (Avila et al., 2020), predict RI that did not verify.At 0600 UTC 27 August, DSHIPS predicts significant intensification that approximately matches a period that almost qualifies as RI.Later, Dorian did achieve and far exceeded RI, while the DSHIPS forecasts from 0000 UTC 31 August to 0000 UTC 1 September predicted little or no further intensification.
DSHIPS forecasts from the early stages of Hurricane Ian (2022) show positive bias.The disturbance that became Ian was first identified as an "invest" at 1200 UTC 21 September.
Although not yet a tropical depression, this designation triggers the production of intensity forecasts as shown in Figure 1.More so than for Dorian, some of these early forecasts predicted substantial intensification even before the storm was designated as a tropical depression at 0600 UTC September 23 (Bucci et al., 2023).This is not a failing of DSHIPS, as it is not designed to make predictions for disturbances that are not yet tropical depressions.However, we interpret the positive intensity change forecasts from DSHIPS as a sign that the surrounding environment was highly favorable for TC intensification, through some combination of high SSTs, low wind shear, or high mid-level humidity.Later, after Ian was named as a tropical storm on 0000 UTC 24 September, DSHIPS produced a series of forecasts with high positive biases on 1200 and 1800 UTC 24 September and 0000 UTC 25 September.These forecasts are an example of how unfavorable factors that are not known to a statistical forecasting system can lead to large "false alarm" errors.While such errors are not as concerning as the opposite type, they do contribute to average intensity errors and reduce overall confidence in such models and the official forecasts that rely on them.
A comparison of the mesoscale structures of Dorian and Ian suggests mesoscale asymmetries could partly explain their different evolutions and associated forecast errors.Figure 2 shows vertical vorticity on the 85 − hPa pressure level obtained from gridded 0.25 • × 0.25 • wind data from the European Centre for Medium-Range Weather Forecasts Reanalysis Version 5 (ERA5; Hersbach et al., 2020).While certainly not circular, the pre-RI vorticity distribution of Dorian on 26 August has a distinct central maximum and the surrounding contours have only weak low-wavenumber asymmetries (elliptical and triangular shapes).The vorticity pattern is very similar on the two subsequent days, with the area of strong vorticity increasing in size on the next day, and then the central peak vorticity increasing in magnitude on the day after that.At each time there is a clear central maximum and the asymmetries are relatively weak.
The pre-RI vorticity fields of Ian were generally much more asymmetric.Here we show the vorticity around the times of the afore-mentioned overly aggressive DSHIPS forecasts from 24 to 25 September.At 1200 UTC 24 September and 0000 UTC 25 September, there are multiple localized vorticity maxima and minima and surrounding asymmetric features.While the vorticity minima give something of the appearance of a developing eye, such a feature could not be resolved with these data and at 40 knots Ian was well below the typical intensity for eye formation.By 1200 UTC 25 September, the ERA5 reanalysis shows a nearly circular bullseye of vorticity and, not surprisingly, the first RI phase of Ian commenced 12 hours later.

Modeling framework
All simulations in this study are produced with version 4.3.0.3 of the Weather Research and Forecasting Model (WRF; Skamarock et al., 2019).As noted above, we seek to simulate TC development and intensification in carefully specified environments, where here the "environment" refers to the vertical profiles of temperature, moisture, wind speed and direction that are homogeneous across a large computational domain.For this purpose we used the method of "point downscaling" (PDS) described in Nolan (2011).In this framework, the model domain is a large rectangular f -plane with a constant SST at the lower boundary.The lateral boundaries use periodic boundary conditions in both the zonal and meridional directions.
As in many previous studies (Finocchio et al., 2016;Gu et al., 2018;Nolan, 2007;Rios-Berrios, 2020), the initial condition of the model is first defined with vertical profiles of temperature, moisture, and zonal wind that were constant across the model domain.In PDS, the vertically varying zonal flow is not balanced by meridional temperature and pressure gradients as required by the thermal wind equation (e.g., Holton & Hakim, 2013, pp. 82-83).Rather, its Coriolis acceleration is balanced by an artificial pressure gradient force equal to the initially required value.This modification to the true equations of motion is equivalent to applying the Coriolis force only to the perturbations from the initial wind profile and has been used with other atmospheric models (Alland et al., 2021a,b;Ryglicki et al., 2018;Skamarock et al., 1994).The wind field of an axisymmetric TC-like vortex is then added to the flow, along with the axisymmetric temperature and pressure perturbations that hold this vortex in gradient and thermal wind balance.
This study requires extending our initialization capability beyond axisymmetric vortices.The procedure is similar in principle to the axisymmetric balancing procedure.Working in height coordinates, we first hold the density constant and compute the pressure field p(x, y, z) at each height level that balances the horizontal wind field.This updated pressure field deviates from the hydrostatic balance of the initial sounding, so the density (and temperature) at each model point is corrected to meet hydrostatic balance exactly according to the new pressure field.Due to the small changes in density, the pressure field no longer balances the wind field, so it is recomputed using the new density field.These two steps are repeated until some convergence is reached.
For an axisymmetric vortex, the pressure step only requires a straightforward numerical integration of the gradient wind balance equation, inward from large radius.For arbitrary two-dimensional flows, the procedure is more demanding.The full details are provided in the Appendix, but a conceptual outline is given here.The equations of motion are used to derive an equation for the time rate of change of horizontal divergence.Setting this rate of change to zero and assuming that horizontal variations of density can be neglected leads to a Poisson equation for the pressure on each height level: ( where u and v are the zonal and meridional winds,  v is the virtual density, and the source term S is a functional of those fields.This equation is solved by successive underrelaxation.The horizontal and vertical balances are coupled by regularly implementing the hydrostatic balance correction to  v and updating the source terms in equation (1) after a series of horizontal iterations.The horizontal iterations and hydrostatic updates are repeated until satisfactory convergence is reached.Here, "satisfactory" means that if the model is initialized with no moist physics or surface friction, the initial, possibly very asymmetric vortex, embedded in shear, will emit only weak gravity waves with vertical velocities on the order of 0.01 m ⋅ s −1 .A demonstration is shown in the Appendix.

Model grids, resolution, and parameterizations
All simulations use the vortex-following nested grid capability of WRF.The outer domain is 240 × 180 points with 18 − km grid spacing, and two nested grids are used with 180 × 180 grid points and 6 − km and 2 − km grid spacing.While 2 − km inner-nest grid spacing is larger than what has been used in many recent studies of RI, this is required to enable the use of ensembles as described below.Sixty vertical levels are used between the surface and 20 km, using the same vertical distribution of levels as in Nolan et al. (2013).Turbulence and vertical diffusion are parameterized with the Yonsei University boundary layer scheme (YSU;Hong et al., 2006) and using the "TC" option with the surface roughness formula capped at wind speeds over 25 m ⋅ s −1 as in Dudhia et al. (2008).Microphysical processes use the Morrison scheme (Morrison et al., 2005).The Tiedtke cumulus parameterization (Tiedtke, 1989;Wang et al., 2007) is active on the 18 − km grid.Short-wave and long-wave radiation schemes are not activated.While recent studies have shown that long-wave radiation has significant positive feedbacks on developing TCs (Ruppert Jr & O'Neill, 2019;Wu et al., 2021), long-wave radiative cooling would also immediately begin to change the domain-wide temperature profile away from its initial states, increasing instability.As they are, the sounding and wind profiles are already very favorable for intensification and almost all cases proceed quickly to RI.The sea surface temperature is fixed at 27.5 • C and the Coriolis parameter is a constant 5.0 × 10 −5 s −1 .

Large-scale environments, control-case vortex, and ensembles
All simulations in this study use the same atmospheric sounding.It is based on the "moist tropical" sounding from Dunion (2011), except that the atmosphere is slightly drier in the middle levels.Specifically, the water vapor q v is reduced by a maximum amount of 10% centered at 600 hPa, with this 10% reduction factor shaped like a Gaussian with a vertical decay parameter  z = 150 hPa.The same profile of q v is shown in figure 1.6 of Nolan and McGauley (2012).The purpose of this mid-level drying is to control the timing and rate of RI in the unperturbed simulations.It also reduces the development of unrelated cumulus convection across the model domain.
Two sets of simulations are used.The first set have no background flow and no wind shear, that is, the classic "hurricane in a box" as used in some earlier studies (Kurihara & Tuleya, 1974;Montgomery et al., 2006;Nolan, 2007).While zero mean flow or shear virtually never occurs in the atmosphere (Nebylitsa et al., 2023;Nolan & McGauley, 2012), the purpose of the no-flow simulation is to isolate the effects of mesoscale asymmetries on TC intensification from interactions with mean flow or shear-induced asymmetries.The second set of simulations have weak easterly flow at the surface with u = −2 m ⋅ s −1 at p = 1000 hPa, varying like a cosine function in log-pressure to u = +3 m ⋅ s −1 at p = 200 hPa, in other words, approximately 5 m ⋅ s −1 of westerly wind shear above 2 m ⋅ s −1 of easterly flow at the surface.The wind profile formula is shown as equations ( 1) and (2) in Finocchio et al. (2016).
The initial vortex is meant to represent a weak tropical storm and is a variation on initial vortex structures used in many previous studies (Finocchio & Rios-Berrios, 2021;Stern & Nolan, 2011).The initial radial profile of the tangential wind uses the classic modified Rankine vortex formula: with V max = 16.35 m ⋅ s −1 , r max = 72 km, and the decay parameter a = 0.3.In a doubly-periodic modeling framework, it is necessary for the tangential wind to go exactly to zero at finite radius inside the model domain.This is achieved with a sixth-order exponential decay function: using r cut = 700 km.Furthermore, as equation ( 2) produces a wind profile with a sharp cusp at r max , the wind profile is then smoothed 30 times with a 1-2-1 filter on a radial grid with Δr = 4 km.This ameliorates the cusp but increases the actual radius of maximum winds (RMW) to 90 km and decreases V max to 15 m ⋅ s −1 .The tangential wind field has its maximum value at z = 1.5 km and decreases both upward and downward with height using the vertical decay formula and parameters of Stern and Nolan (2011;p. 2083).The initial vertical wind profile, vortex structure, and the domain-scale flow (including the 2 m ⋅ s −1 of easterly flow at the surface) are shown in Figure 3.
All results in this study are derived from mini-ensembles of three simulations for each set of initial conditions.To create these ensembles, random perturbations of temperature are added to the initial temperature field on the WRF 18 − km grid.The noise has a maximum standard deviation of 0.5 K at distance r = 100 km from the vortex center.The noise variance decays as a Gaussian away from r = 100 km with decay scale  r = 50 km, and perturbations are only added to model levels below 3 km.The perturbations cause different evolutions of the initial moist convective updrafts, leading to different timings and rates of RI, but they are not intended to represent the errors of any particular instrument or analysis.

Asymmetries
Two types of asymmetries are added to the initial conditions: secondary vortices and low-wavenumber asymmetries.
The secondary vortices are intended to represent small circulations that are sometimes generated by bursts of convection outside of the primary convection near the vortex center; secondary vortices also sometimes spin outward from the center after being produced by inner-core convection.They are constructed as modified Rankine vortices that are smaller (reduced r max ), weaker (reduced V max ), and narrower (increased a) than the primary vortex.These are modified by a single parameter s with values ranging from 0.25 to 1.0, that is, and a ′ = a∕s.The same large-radius cutoff, smoothing of the tangential wind profile, and extensions into the vertical are applied to the secondary vortices before they are added to the three-dimensional wind field.The secondary vortices are initially centered at distances 450,360, and 270 km west of the primary vortex, corresponding to five, four, and three times the actual RMW (90 km).The six initial conditions with secondary vortices are shown in Figure 4.
The low-wavenumber asymmetries are intended to represent changes to the flow that can be caused by asymmetric convection or caused by the large-scale flow.For example, wavenumber 1 asymmetries are associated with a displacement of the inner-core vortex relative to the surrounding, larger-scale circulation.This can be caused by asymmetric convection or even vortex reformation.Wavenumber 2 asymmetries can be caused by the horizontal deformation associated with the horizontal shear of the environmental flow, and also by convection, and are associated with an elliptical shape of the wind field.Here the low-wavenumber asymmetries to the initial wind field are derived from asymmetric stream functions with the form: where  is the angle around the azimuth, C is a constant, n is the azimuthal wavenumber,  is a phase angle that controls the orientation, r d controls the radial size of the asymmetry, and k is a shape parameter for the exponential cutoff.The exponential function works similarly as in equation ( 3), providing a finite size to the asymmetric wind field.Asymmetries with values of n of 1, 2, and 3 are used.Asymmetry values n of 3 are included as a logical extension beyond n = 1 and 2, and to be representative of all higher wavenumbers.For n = 2 and 3, the value of r d is chosen so that the maximum perturbation radial wind occurs at one of three distances r p = 1× RMW, 2× RMW, or 6× RMW, where RMW= 90 km.These values generate "inner-core," "outer-core," and "far-field" asymmetries.For the inner-core and outer-core asymmetries, the cutoff parameter k = 2.For r p = 6×RMW, a sharper cutoff with.k = 4 is needed to limit the size of the far-field asymmetries.The value of C is chosen so that the maximum radial wind is equal to half the tangential wind at the specified value of r p .In all cases  = 0, except in section 4c.The asymmetric winds are derived from u = −∕y and v = ∕x.
At n = 1 asymmetries are structured differently in that the maximum perturbation radial wind occurs when r = 0, where, unlike for all higher wavenumbers, there is flow across the center axis.For an example and comparison to n = 2, see figure 5 of Nolan and Farrell (1999) (note that paper used k for wavenumber).For n = 1, the asymmetry size parameter r d is set to the same three values as r p above, and the same values of k are used.C is chosen so that the maximum radial velocity (at r = 0) is equal to half the tangential wind at r d .
Thus, for each of the asymmetric wavenumbers n = 1, 2, and 3, asymmetries of three different sizes are generated.All nine combinations are shown in Figure 5.As for the secondary vortices, the amplitudes of the asymmetries vary with height in the same manner as the symmetric vortex.For this reason, the inner-core and outer-core asymmetries at n = 1 represent displacement of the vortex, but not tilt.The far-field n = 1 asymmetry will generate an early tilt of the vortex because the asymmetric large-scale flow varies with height.This will be discussed further below.

CONTROL SIMULATIONS
We first consider RI from symmetric vortices, with and without mean flow and wind shear.Time series of minimum surface pressure and maximum 10 − m wind speed are shown in Figure 6, for each of the three mini-ensemble members, and for each ensemble mean.In these figures the wind speed values have been smoothed two times with a 1-1-1 filter; the pressure is not smoothed.After about 30 hours, all six simulations proceed to periods of RI that last about 36 hours.The first 30 hours are quite different between the two cases.For the NoFlow set, the peak winds and surface pressures change relatively little for the first 24 hours, followed by a brief increase in surface wind speeds, which is then followed by a long period of steady and strong intensification.The ShearFlow simulations evolve quite differently, showing sudden increases of intensity in both pressure and wind speed after 18-20 hours, followed by short periods of reduction in the surface winds while the pressure continues to fall.The two sets meet almost perfectly at 43 hours, after which all members intensify rapidly, but the rate is slightly slower for the ShearFlow ensemble.
The earlier intensification of the weak TCs when wind shear is present is due to the dynamical interaction between the initial vortex and the wind shear.Although the vortex is initially in almost perfect gradient and hydrostatic balance, differential advection of its vertical vorticity generates upward motion on its downshear side and downward motion upshear, as predicted by the quasi-geostrophic omega equation (e.g., Holton & Hakim, 2013, pp. 198-199) and more advanced theoretical frameworks for tilting vortices (Dörffel et al., 2021; Jiang & Raymond, 1995;Raymond, 1992).To assess the vertical motions caused only by tilting of the vortex by shear, we performed a simulation of the control case on the 18 − km grid only, and with the microphysics and boundary layer parameterizations deactivated.This simulation revealed that after 12 hours the shear-induced upward motion reaches its maximum value of 0.03 m ⋅ s −1 near a z = 5 km (not shown).With physics activated, this generates an organized cluster of deep moist convection that rotates around to the downshear left quadrant while generating increased surface winds and greater surface pressure falls.
The timings and amplitudes of the RI events in these simulations are measured from the time series of peak surface wind speed S10 max .For each ensemble member Abbreviations: MaxRI, the largest intensity change in 24 h; TFirstRI, the beginning of rapid intensification; TMaxRI, the beginning of the maximum 24 h intensification period.
we compute the beginning hour of the first period of RI ( 15.4 m ⋅ s −1 in 24 hours), designated TFirstRI.In addition to the 15.4 m ⋅ s −1 increase in 24 hours, there must also be an increase of at least 2.6 m ⋅ s −1 (5 knots) in the first six hours of the 24-hour period.Without this condition, very rapid intensifications could appear to start before intensity actually began to increase.For example, if the intensity were constant for 12 hours, and then increased by 16 m ⋅ s −1 in the next 12 hours, then it would appear that RI had started 12 hours earlier than any intensification had begun.The maximum intensification (change in maximum surface wind) in any 24 − hour period is designated MaxRI, and the time when MaxRI begins is designated TMaxRI.Values for all cases are shown in Table 1 (NoFlow) and Table 2 ( ShearFlow).Each entry shows the mean value across the three members and the estimated standard deviation.Here we first only consider the values for the NoFlow and ShearFlow control cases, which show that the values of TFirstRI, TMaxRI, and MaxRI for the two environments with symmetric vortices are very similar.Even though their statistics are similar, it is worthwhile to compare the flow structures for both cases in the hours leading up to TFirstRI.Figure 7 shows simulated reflectivity and wind vectors for one ensemble member from each of these cases at 24 hours and 36 hours.At 24 hours, NoFlow has relatively low reflectivities (and thus rain rates, not shown) that are very symmetrically distributed around a rain-free center.ShearFlow has a much stronger cluster of reflectivity with higher than 50 dBZ in its northeast quadrant.At 36 hours, NoFlow has a few updrafts with dBZ above 40, evenly spaced around what may be a developing eye.ShearFlow is still very asymmetric, although there is some reflectivity scattered around all sides of the center, and the strongest convection has moved closer to the center.The ShearFlow case is much more similar to reality in that most TCs at the start of an RI period still have asymmetric convection and partially formed eyewalls (Reasor et al., 2009;Stevenson et al., 2014).

Evolution of asymmetric initial vortices with no mean flow
The ensemble-mean intensity evolutions for the control (symmetric) case and every case with asymmetric initial conditions are shown in Figure 8.At first glance, it is apparent that there are no large nor consistent differences between the asymmetric cases and their symmetric counterparts.The blue curves are the control cases, and all other cases generally lie on either side of and close to the blue curves.A few of them do show some Abbreviations: MaxRI, the largest intensity change in 24 h; TFirstRI, the beginning of rapid intensification; TMaxRI, the beginning of the maximum 24 h intensification period.
non-trivial delay in intensification as compared to the control.These include the far-field asymmetry at n = 1, the outer-core asymmetry at n = 2, and the stronger secondary vortices with s = 0.75 and s = 1.0.In all four cases, the asymmetries in the initial wind field lead to either a trochoidal-like motion of the vortex, or increased asymmetrical rainbands, or both.Reflectivity and wind fields for these four cases are shown in Figure 9.The results for n = 1 have the appearance of a sheared storm, and this is due to the motion of the low-level vortex compared to the nearly stationary environment.For the other three cases, the main difference is due to the larger size of the wind field, due to either the increased rain bands or the merger of the two vortices, leading to greater inertial stability and reduced intensification rates (Schubert & Hack, 1982).
For some of the cases, the initial asymmetries lead to slightly earlier values for TFirstRI and TMaxRI, and slightly greater values of MaxRI.We note that adding the asymmetries actually increases the overall kinetic energy of the wind field, and perhaps more importantly, it leads to greater maximum surface wind speeds during the first 24 hours as shown by the red and magenta curves in Figure 8a-c.The localized areas of increased surface wind cause greater moisture fluxes and enhanced frictional convergence, accelerating the development of deep convection (not shown).
The timings and intensities of RI for the NoFlow cases are summarized in Table 1.Entries that are statistically significantly different (p = 0.05) from the control case (the first row), as determined from a two-sided Welch's t-test, are italicized.As will be discussed below, the bold entries show significant differences between equivalent NoFlow and ShearFlow cases; some entries are both italicized and bold.Of the 45 numbers in Table 1 that summarize the asymmetric NoFlow results, only eight are italicized, indicating that the large majority of them are not significantly different from the control results.

Results with asymmetries and wind shear
Figure 10 shows the intensity time series for the ShearFlow simulations.The results show that the extent to which mesoscale asymmetries lead to delay of RI is strongly dependent on whether there is also wind shear in the TC environment.With wind shear, every non-blue curve in Figure 10 (which is the mean of three simulations) falls near or to the right of the blue curves, indicating delayed RI, weakened RI, or both, as compared to the control case for ShearFlow.It is also apparent from Figure 10c that the asymmetries for n = 3 lead to only slight reductions in intensification.Physical evolutions are presented in Figure 11 for two ShearFlow cases with significant delays: the far-field asymmetry at n = 2 and the secondary vortex for x = −4, and s = 0.75.Reflectivity and wind vectors with z = 1.5 km are shown at t = 36, 60, and 84 hr for n = 2 on the left and for s = 0.75 on the right.For n = 2 at 36 hours, the convection is not surprisingly asymmetric, but it is also apparent that the surrounding wind field possesses an axis of horizontal wind shear from the northwest to southeast, or alternatively, an axis of shearing deformation in the north-south direction.This deformation field is the influence of the far-field asymmetry in Figure 5f that is decaying slowly as it is advected around the vortex while also being radially sheared.The inner wind field has a mesoscale asymmetry with n = 2 with broader and weaker winds spreading out to the southwest of the vortex center.Similar features of the outer and inner wind fields are still present at 60 hours (Figure 11c).By 84 hours, the wind field is becoming more symmetric but the inner-core convection is still confined to the western quadrant.Although it does not have the large-scale deformation, the evolution of the case at s = 0.75 is otherwise quite similar.At 36 hours the inner-core convection is highly asymmetric and the wind field is intensified on the east side and spread out and weaker on the west side.This pattern continues at 60 hours, but by 84 hours the wind and reflectivity fields are more symmetric and RI is about to begin.
Timings and intensities of RI for the ShearFlow simulations are summarized in Table 2.As above, the values which are statistically different from the control case are italicized.In this case, 20 of the 45 values are significantly different, showing that when shear is present, the Furthermore, based on the earlier findings regarding the similarities of the intensity evolutions for the NoFlow and ShearFlow control cases, it appears that the introduction of asymmetries leads to notably different outcomes for simulations with or without shear.These differences are highlighted by the bold numbers in Tables 1 and 2, which indicate when the presence of wind shear leads to significantly different results with the same initial conditions (note that each bolded value in the NoFlow section is matched with a bolded value in the ShearFlow section).Without balanced asymmetries in the initial condition, the NoFlow and ShearFlow ensemble-mean intensities are not significantly different, and in fact, they are nearly identical.With asymmetries, out of the 45 comparable values, 23 significantly indicate delayed or weakened RI.We note that one of the bolded values, MaxRI for the far-field asymmetry at n = 1, is significantly different but in the wrong sense (faster intensification instead of slower), and is not included in this count of 23 significantly later or weaker RIs.

Changes to the asymmetry angles
All simulations presented so far have their asymmetries oriented in the same direction.For the NoShear set, there is no external direction vector and the results are expected to be identical (other than numerical differences) for different asymmetry angles.For ShearFlow, there could be different outcomes based on the location of the secondary vortex or phase angle of the asymmetry.
To investigate this possibility, additional simulations were performed for two of the cases with largest effects: the far-field asymmetry with n = 2 and the secondary vortex at s = 0.75, for three additional configurations.For n = 2 the phase angle  was changed to ∕2, , and −∕2.The initial locations of the secondary vortices were changed to x = +4 × RMW, y = +4 × RMW, and y =−4 × RMW.Only one simulation was performed for each case.
The intensity evolutions are shown in Figure 12, with original ensembles in red and the changed angle simulations in black.For reference, the three ensemble members of the symmetric case and the respective asymmetry are also shown as blue curves in each plot.Comparing the red and black curves, the results are quite different for some of the asymmetry orientations, most notably for the case where  =  for n = 2, and the y = −4 case for s = 0.75.In both, their intensity evolutions revert back to the same pattern as the symmetric case.
Most of the other curves are similar to the original set.None of these alternate asymmetries lead to intensification that is faster than the symmetric ensemble.Nonetheless, from these simulations, we understand that all the afore-mentioned results must come with the caveat that some orientations of the asymmetries can lead to significantly different timing of RI.

ANALYSIS
The results so far suggest that, in the context of idealized simulations of RI, the presence of mesoscale asymmetries To understand the physical mechanisms for the delays or weakening of RI, we first consider the time evolutions of the sizes of the vortices, as defined by the RMW, and the distribution and intensity of the moist convection as a function of radius.Many theoretical (Hack & Schubert, 1986;Nolan et al., 2007) and observational (Guimond et al., 2010;Houze Jr. et al., 2009;Rogers et al., 2015) studies have shown that the arrival or formation of deep convection and concomitant latent heat release inside the RMW is critical for significant intensification of TCs.
To explore the intensity and radial distribution of the moist convective heating we compute radial profiles of the azimuthal-mean surface rain rate, and the tangential wind at 2 km, at all times for each ensemble member of each simulation.The centers of the TCs are computed from an algorithm to find the centroid of the negative surface pressure anomaly, similar to that proposed by Nguyen et al. (2014).A 100 km × 100 km box is first located at the center of the 6-km grid, and we find the centroid of the negative surface pressure anomaly in this box.A new box is centered on the first computed centroid, and the process is repeated four times to provide the TC center.For both wind and rain rate, the data from the three ensemble members are averaged together to make composite radius-time Hovmöller diagrams for each case.around 5 mm ⋅ hr −1 .This is consistent with the relatively low values of reflectivity inside the RMW noted at 36 hours for the control case as in Figure 7b.While snapshots of rain rate do show convective cores with rain rates locally exceeding 40 mm ⋅ hr −1 around this time (not shown) in all three ensemble members, the areal coverage of these rain rates at any one time is very small and thus the azimuthally averaged values are drastically smaller.
The next plot, Figure 13b, shows the rain rate with the far-field asymmetry at n =1.As can be seen in Figure 8a and Table 1, TFirstRI is delayed by eight hours in this case.However, Figure 13b is very similar to Figure 13a, and in fact, the similar contraction of the RMW and the earlier appearance of slightly stronger rain rates would suggest that RI should have begun sooner for that case.We will return to this point.
Two other cases for NoFlow with significantly delayed RI, outer core asymmetry at n = 2, and a secondary vortex at s = 0.75, are shown in Figure 13c,d.In these cases, the delay of RI appears to be caused by the larger RMWs which in turn are caused by more substantial convection in the rainbands between r = 90 and 180 km.TFirstRI occurs at 40 and 39 hours in these two cases, respectively, which is shortly after the RMW has decreased to 80 km and rain rates inside of that have exceeded 5 mm ⋅ hr −1 .
Rain rates and RMWs for some of the ShearFlow cases are shown in Figure 14.The control case (Figure 14a) is quite different from NoFlow (Figure 13a).While both show steadily decreasing RMWs, ShearFlow shows much greater rainfall rates, well inside the RMW, even before RI begins.The first burst of convection from 21 to 27 hours corresponds with the initial burst of organized convection that is forced by the tilting of the vortex by wind shear.RI does not begin until a second burst of inner-core convection starts at 34 hours.Interestingly, the rain rates in this second burst are far greater than those associated with TFirstRI for the NoFlow case (Figure 13a).The color values saturate at 10 mm ⋅ hr −1 , but the azimuthal-mean rain rate in this second burst of rainfall reaches 12 mm ⋅ hr −1 , over twice the rate seen in the burst that initiated RI for NoFlow.However, as shown in previous studies (e.g., Alland et al., 2021a;Rappin et al., 2010), sheared convection in the early stages of a TC is often followed by a significant reduction in low-level equivalent potential temperature  e due to downdrafts and near-surface evaporational cooling, and this is also the case in these simulations (not shown).These reductions in  e and the delay of further convection due to the time needed for boundary layer recovery may partly explain why the the subsequent rate of intensification for ShearFlow is less than for NoFlow.
Figure 14 also shows results for ShearFlow with the n = 1 far-field asymmetry, the n = 2 far-field asymmetry, and the secondary vortex at s = 0.75.In the first of these, the RMW contracts steadily and there are strong bursts of convection inside the RMW from five to 32 hours and then beginning again at 42 hours.However, TFirstRI does not begin until 51 hours.For the outer-core asymmetry for n = 2, there are several strong periods of convection but the RMW does not contract until 80 hours, and its TFirstRI is at 86 hours.For s = 0.75, there are also several bursts of inner-core convection and the size of the RMW oscillates around 90 km until 68 hours; its TFirstRI is at 75 hours.Contraction of the RMW to sizes typically less than 100 km is a well-established condition for significant intensification (Miyamoto & Nolan, 2018;Stern et al., 2015;Wu & Ruan, 2021).In some of the simulations above, asymmetries in the wind field lead to asymmetric convection and larger RMWs, which in turn delay RI.However, the results are not consistent: for Figure 13b  This leads us to consider if there is another factor that delays RI in some cases.An obvious candidate is tilt of the vortex.For all simulations, we compute the tilt (horizontal displacement) between the surface vortex, as found by the pressure anomaly centroid, and the center of the vortex at 8 km.To compute the center of the upper-level circulation, we first compute the square of the vorticity, but with the sign preserved, that is,  sq =  |  |, and then iteratively find the centroid of this field in a 100 km × 100 km box following the same procedure as above for the surface center.
The ensemble-mean tilts for selected cases of interest are shown in Figure 15, with NoFlow in the top panel and ShearFlow in the bottom.Even for the control (symmetric) vortex in NoFlow (blue curve in Figure 15a), mean tilt does not remain exactly zero, but increases to almost 20 km during the first 24 hours, due to small displacements caused by randomly located bursts of convection (not shown).As the TCs intensify, the tilt decreases to less than 10 km.The three other cases in this plot are those with NoFlow that showed non-trivial delays of RI.Of these three, two show development of larger tilts during the first 24 hours.In these cases, for the far-field asymmetry for n = 1 and the secondary vortex at s = 0.75, the initial asymmetries in their wind fields lead to motion of the low-level vortex.As stated in Section 2d, the asymmetries all decay in the vertical with the same decay function as the symmetric vortex, so that the upper-level flow is relatively undisturbed by the asymmetries.For s = 0.75, the time scale of the motion of the secondary vortex is fast enough that the upper and lower-level centers re-align and then separate again before the secondary vortex is completely symmetrized.
When shear is present, and for the symmetric vortex, the evolution of tilt follows a pattern that has been shown with many previous studies using idealized simulations (Alland et al., 2021a,b;Finocchio et al., 2016;Rios-Berrios, 2020;Yu et al., 2023).The blue curve in Figure 15b shows that during the first 12 hours the surface to 8 − km tilt increases to 90 km, the size of the RMW, and then decreases to 50 km at 24 hours.As noted above, due to the adjustment of the vortex to shear and the development of moist convection, the first 24 hours are really a "spin-up" period and the TC at 24 hours is more like the true initial condition.After another six hours, the tilt begins to decrease again and by 48 hours the tilt is less than 20 km and TC is already engaging in RI.This emphasizes that, at least for initially symmetric vortices, 5 m ⋅ s −1 of wind shear is not prohibitive to RI.The evolution of one ensemble member is shown in the left column of Figure 16, in terms of the surface pressure field and wind vectors and vorticity at 7.9 km.The white star marks the surface center (based on pressure) and the magenta star marks the upper-level center (based on vorticity).In the figures, both the pressure and vorticity fields have been smoothed 10 times with 1-1-1 filters in both directions.This is part of the reason why the white stars do not fall at the center of the innermost contour, because the pressure center is pulled toward low pressure anomalies caused by convection under the mid-level center, but these are not visible after the smoothing, which is necessary for clarity of the images.
After the initial tilting and convective adjustment, the control case "overcomes" the wind shear and proceeds quickly to RI.However, as shown in Figure 15b, some of the asymmetric cases require another 12-72 hours of development.The example of one ensemble member of the far-field asymmetry case at n = 2is shown in the right column of Figure 16.At 24 hours, the tilts and structures of the control case and the asymmetric case are nearly identical.However, at 48 hours, the asymmetric case has more tilt and the upper-level vortex is still further downshear than for the control case at 36 hours.At 72 hours, the asymmetric case is still strongly tilted downshear left.A band of positive vorticity stretching to the north is the leftover effects from a stratiform rainband stretching out northeast from the center in the previous 24 hours (not shown).This interaction between a stratiform rainband and the mid-level center, leading to a delay in alignment and intensification, is similar to the "precession hiatus" process discussed by Yu et al. (2023).In this case, the increased convective asymmetries and inner-core wind shear caused by the initial asymmetry have a similar effect as the moderate value of large-scale wind shear shown to lead to the precession hiatus in Yu et al.
To see further examples of these differences in evolution, the plots in Figure 17 show the temporal evolutions of the surface to 8 − km tilt vectors on an (x, y) plot for three of the afore-mentioned ensembles: ShearFlow control, far-field asymmetry at n = 2, and x = −4, s = 0.75.For ShearFlow control, all three vortices are quickly tilted to about 90 km ENE of the surface center.After 20-30 hours, all three begin to process upshear to a point about 50 km NNE of the surface center, where they experience a brief precession hiatus.However, by 40 hours (purple colors) the vortices begin to rapidly align.For the far-field set at n = 2, the early evolution is similar, with the three members arriving at a similar precession hiatus between 30 and 40 hours.However, all three vortices then tilt further to the west and north and meander for 24 − 48 hours before coming back to the surface center.These three members are the same as the three red curves in Figure 12a.For the set of s = 0.75, the initial tilting is followed by rapid re-alignment (presumably as the secondary vortex comes around to the other side of the primary vortex), followed by wild looping motions for all three members before alignment (compare to Figure 12b).Thus we find that these cases with significant delays of RI are associated with re-amplification of the vortex tilts after the initial tilt and partial re-alignment, presumably by some combination of asymmetric convection or inner-core wind shear.Vortex tilt and environmental wind shear are closely related.Shear causes tilt, but it is also true that a tilted vortex can appear to have wind shear that is different and even substantially greater than for the larger environment (Dai et al., 2021;Yu et al., 2023).Ensemble-mean 850-200 hPa wind shear was computed for all the cases in Figure 15, both inside a circle of radius 500 km (hereafter, inner shear) and in a 200-800 km annulus (hereafter, outer shear).Following operational practices, each of these are centered at the low-level circulation center and do not account for tilt of the vortex.Beginning with the NoFlow control cases, we can see that both inner and outer shear are mostly very small throughout the simulations.The far-field case at n = 1 and the s = 0.75 case actually begin with non-zero shear because of the asymmetric low-level wind fields at t = 0 (see Figures 4e and 5c).As noted above, the asymmetric flow causes the lower-and upper-level vortices to move around each other, resulting in briefly large tilts despite shear values remaining less than 2 m ⋅ s −1 .The outer shears all remain less than 2 m ⋅ s −1 , while the inner shear reaches a surprising value of 5 m ⋅ s −1 at 72 hours for the case of n = 1.Although RI is already occurring at this time, this large value of inner shear is caused by an asymmetric outflow jet associated with a cluster of deep convection north of the TC at this time.This asymmetry is surprisingly robust across the three ensemble members (not shown), such that the shear values for each ensemble member are not far from the mean value shown in the plot.
For the ShearFlow cases, both shears begin at exactly 5 m ⋅ s −1 as expected.For the control case, the outer shears and inner shears increase to 6 and 7 m ⋅ s −1 respectively.For each of the asymmetric ShearFlow cases, both the inner and outer shears increase to notably larger values for some period.The largest is for the s = 0.75 case for which the mean inner core shear increases to over 10 m ⋅ s −1 from 60 to 72 hours.The large inner shear at this time is due to outflow from a large rainband extending northwest and then north of each of the three ensemble members at this time, similar to the rainband outflows discussed in Dai et al. (2019Dai et al. ( , 2021)).Both this case and the far-field case at n = 1 for NoFlow show that, at least in the world of idealized TC simulations, it is possible to have large values of 0 to 500 km wind shear even when the vortex has very small tilt.This is due to the presence and dynamical evolution of mesoscale asymmetries.

CONCLUSIONS
This paper introduced a method to initialize a mesoscale model with idealized initial conditions containing TC-like vortices with non-trivial asymmetries.This method was then used to perform numerous simulations of TCs with varying degrees of intensification.The purpose was to determine whether mesoscale asymmetries lead to delays in the beginning of rapid intensification (TFirstRI), the period of fastest intensification (TMaxRI), and reductions in the peak intensification rate (MaxRI).We used two classes of asymmetries: low-wavenumber asymmetries of the inner-core, outer-core, and far-field wind fields, and secondary vortices of varying strengths and distances from the primary vortex.All simulations were performed both in environments with zero background flow (NoFlow) or with weak easterly mean flow at the surface and 5 m ⋅ s −1 of westerly wind shear from 850 to 200 hPa (ShearFlow).Furthermore, each individual case of a particular asymmetry and mean flow was performed three times with different random temperature perturbations to generate three-member ensembles for every case.
For the NoFlow cases, mesoscale asymmetries led to statistically significant delays in TFirstRI for only five of the 15 cases (note italicized numbers in Table 1).MaxRI was significantly changed in only three cases.The conclusion for the NoFlow cases is that mesoscale asymmetries do not usually cause meaningful changes to RI.For the ShearFlow cases (Table 2), results were quite different.TFirstRI was significantly delayed for seven of the 15 cases, TMaxRI was delayed for seven cases, and MaxRI was reduced for six of the cases.
Alternatively, we can compare how two ensembles with identically asymmetric vortices evolve without or with wind shear.Unlike for the symmetric "control" cases, many asymmetric simulations had significant differences in RI when wind shear was present (note bolded numbers in Table 1).In some cases the differences were quite large, with TFirstRI changing from 34 hours to times such as 70, 75, and 86 hours.From either perspective, the primary conclusion is that in the presence of wind shear, mesoscale asymmetries in the wind field or convection of developing TCs are likely to delay the beginning and intensity of RI.However, additional simulations showed that the extent of this effect on RI can vary significantly in some cases based on the orientation of the asymmetries relative to the large-scale wind shear vector.An even larger set of ensembles, or a statistical analysis of asymmetries and RI outcomes, will be required to identify the properties of asymmetries that tend to delay or weaken RI.
We should emphasize here that 5 m ⋅ s −1 (about 10 knots) of 850 to 200 hPa wind shear is generally considered light to moderate wind shear that is not prohibitive for TC intensification.Kaplan and DeMaria (2003) found the mean value of shear for Atlantic RI was 4.9 m ⋅ s −1 .Hendricks et al. (2010) found much larger values of 8.2 and 10.4 m ⋅ s −1 of wind shear for Atlantic and West Pacific RI cases, respectively.Most recently, Nebylitsa et al. (2023) found the mean wind shear for RI cases in the Atlantic was 6.0 m ⋅ s −1 .Thus, modulation of RI by mesoscale asymmetries is likely to be relevant for most TCs in favorable environments.A corollary of this finding is that idealized simulations with no mean flow or wind shear will likely give misleading results regarding RI and asymmetries, and possibly for other physical processes.
Two physical mechanisms for the causes of delays in RI by mesoscale asymmetries were explored.The first was the extent to which asymmetries led to changes in the distribution of convection and precipitation relative to the center and relatedly the size of the radius of maximum winds (RMW) and the outer wind field.In some cases, with or without shear, asymmetries led to an increase in the RMW through an increase in precipitation in rain bands in the outer core.However, especially when shear was present, azimuthal-mean precipitation rates inside the RMW were often quite large, comparable to those occurring even after RI had begun.The second physical mechanism was the tilt of the vortex.We found that ensembles with longer delays of RI had larger and more persistent vortex tilts that were prohibitive to RI.This was even true for two ensembles with NoFlow, where the asymmetries themselves caused displacements between the low-level and upper-level circulations.For ShearFlow, large tilts delayed RI in more of the ensembles and for longer times.
Not surprisingly then, we find that asymmetries of the TC wind field and wind shear in the environment will combine to have a more prohibitive effect on RI than either will alone.For some of the ensembles presented here, the effect is clearly "non-linear" with much larger (and significantly different) delays of TFirstRI and TMaxRI.This result is similar to other findings using idealized simulations of TCs in light to moderate wind shear.Rios-Berrios (2020) found that, with 5 m ⋅ s −1 of wind shear, the timing of intensification was highly sensitive to the strength of cold pools caused by convective downdrafts.Alland et al. (2021a) found that the combination of wind shear and reduced environmental humidity leads to increasing delays of intensification.Similarly, the "ventilation index" of Tang and Emanuel (2012), which is identical to the "incubation parameter" of Rappin et al. (2010), describes the combined negative effects of shear and dry air as multiplicative.
Current RI prediction schemes, such as SHIPS-RII, consider the asymmetries of TCs as a negative factor for imminent RI, but they do so through measured asymmetries of the convection as seen in infrared satellite images (Kaplan et al., 2015).This is not necessarily the same as the asymmetry of the wind field.Fortunately, due to the increasing resolution of global models, it is possible to resolve mesoscale asymmetries of the wind field on the scales used here both in reanalyses and operational analyses, as can be seen in the examples shown in Figure 2. The results here suggest that asymmetries of the surrounding wind field on scales of 100-1000 km from the TC center may be deleterious to RI and should be considered in future development of statistical models for RI.

APPENDIX
This appendix describes the numerical scheme used to compute the pressure and temperature fields that achieve gradient and hydrostatic wind balance for the initial wind fields.The method is an extension to three dimensions of the balancing scheme used for axisymmetric vortices described in Nolan et al. (2001).In contrast to previously reported methods for balanced initialization on sigma () coordinates (e.g., Kurihara et al., 1993;Sunqvist, 1975;Wang, 1995), the method here works for three-dimensional data in height (z) coordinates.At the model initialization stage, the wind, pressure, temperature, and water vapor fields are interpolated to the WRF eta (η) coordinates (Klemp et al., 2007;Laprise, 1992).Our scheme is similar to the balancing method recently presented by Yu et al. (2023).which is designed to add a balanced perturbation to the existing flow, with minimal change to the unbalanced part.In our case, it is designed to provide a fully nonlinearly balanced state to an initial condition that is divergence-free.
The first step is to define vertical profiles for temperature T(z) and water vapor q(z) from some reference sounding such as the Dunion (2011) moist tropical sounding used in this paper.The vertical levels do not need to be evenly spaced but this makes computation of vertical derivatives simpler.The initial hydrostatic pressure p(z) is computed by vertical integration of the hydrostatic equation, and for increased accuracy the semi-implicit form is preferred: where g is gravitational acceleration,  is the vertical distance between height levels k and k + 1, R is the dry gas constant, T v is the virtual temperature.
The three dimensional horizontal wind fields u(x, y, z) and v(x, y, z) are provided on an unstaggered three-dimensional grid.They are presumed to be smooth and non-divergent and there is no vertical velocity.If the point-downscaling method (Nolan, 2011) is to be used, then the "background" flow (mean zonal wind and shear) should not be included at this stage.However, if they are included at this stage, the method will work and will also provide the background pressure and temperature gradients needed for simulations with balanced shear.
The method needs a first guess solution which is provided by geostrophic balance: The critical assumption for the balance computation is that the time rate of change of divergence should be very small (or zero).Assuming that horizontal derivatives of density are very small, we drop the terms on the LHS of (6.3) which leaves an elliptic equation to be solved:  [u(x, y, z), v(x, y, z), (x, y, z)] where S is the source term.
Starting with the geostrophic pressure as the first guess, we solve for p with the following relaxation scheme.On each vertical level, a corrected value for p at each grid point is computed by reversing the discrete Laplacian: where  = 0.5 was used here.This scheme is repeated N times on each vertical level.When updating p at the boundaries, we use zonally periodic boundary conditions on the east and west boundaries.Updated p values on the north and south boundaries are computed by second-order extrapolation of the interior grid points.This boundary condition makes the scheme equally capable of solving for the pressure when not using the point-downscaling method and the large-scale shear is balanced by meridional temperature gradients.After every horizontal level has been iterated N times, the virtual density  v is corrected by requiring hydrostatic balance to be exactly true again: After this step, the source terms in (6.4) are recomputed, and then horizontal iterations on each level are resumed.This combined process is repeated K times.Through trial and error, we found that N = 40 and K = 80 produced good results.For the last step, T v is computed from  v , and then T is computed from T v , where we assume q(z) has not changed.
As an example, we consider the case of the moderately strong secondary vortex at x = − 360 km with s = 0.75, with no mean flow or shear.The simulation was performed on the 18 km grid only, with no microphysics or boundary layer parameterizations.motion in the centers of both vortices.Imbalance also leads to radiation of gravity waves as shown by the vertical oscillations at the fixed point, which are also improved by the balancing.The upward and downward motions of about 0.02 m ⋅ s −1 at six hours correspond to physical motions associated with vortex tilting.In this case, each vortex is tilting due to the vertical shear induced by the other vortex.
U R E 1 Best-track intensity analyses (black) from the National Hurricane Center and intensity forecasts (colors becoming redder with time) from the Decay SHIPS model (DSHP) for (a) Hurricane Dorian (2019); and (b) Hurricane Ian (2022).The absence of black dots at the initialization times indicates that the system was still in the "invest" category at that time.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 3 Initial conditions for symmetric vortices: (a) radial profile of maximum tangential wind; (b) vertical cross-section of meridional wind; (c) horizontal cross-section of wind vectors at the lowest model level and surface pressure, when mean flow and wind shear are included.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 6 Ensemble members and ensemble-mean (a) minimum surface pressure and (b) maximum surface wind speed for the control (symmetric) vortices with no environmental flow (NoFlow) and with mean wind and wind shear (ShearFlow).[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 7 Simulated radar reflectivity and wind vectors at z = 1.4 km for the control vortex simulations; (a) NoFlow at 24 hours; (b) NoFlow at 36 hours; (c) ShearFlow at 24 hours; (d) ShearFlow at 36 hours.A vector that reaches the base of its neighbor represents 20 m ⋅ s −1 .[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 8 Ensemble-mean intensities (maximum surface winds) for the asymmetric simulations with no mean flow or shear: (a) for n = 1 asymmetries; (b) n = 2; (c) n = 3; (d) for secondary vortices.Blue curves show the result for symmetric initial conditions.[Colour figure can be viewed at wileyonlinelibrary.com] asymmetries are more likely to lead to delayed or weakened RI.

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I G U R E 10 Ensemble-mean intensities (maximum surface winds) for the asymmetric simulations with mean flow and shear: (a) for n = 1 asymmetries; (b) n = 2; (c) n = 3; (d) for secondary vortices.Blue curves show the result for symmetric initial conditions.[Colour figure can be viewed at wileyonlinelibrary.com]in the initial vortex can lead to delays in the start of RI (TFirstRI), the beginning of the strongest RI (TMaxRI), and the peak RI rate (MaxRI).Furthermore, in the presence of weak 850-200 hPa wind shear (5 m ⋅ s −1 or about 10 knots), these changes are often significantly increased, with more delay of RI and more weakening of RI, occurring in more cases.
Results for the NoFlow set are shown in Figure 13.The colors show rain rate in mm ⋅ hr −1 and the black line shows the RMW = 2 km.Considering first the control F I G U R E 11 Simulated reflectivities for single ensemble members of ShearFlow simulations.Left column: (a-c), for n = 2, r a = 6 at 36, 60, and 84 hours; right column: (d-f), for x = −4, s = 0.75 at 36, 60, and 84 hours.[Colour figure can be viewed at wileyonlinelibrary.com] (symmetric) case, the results are as expected: the RMW begins to decrease after 24 hours and higher values of rain rate begin to appear 20 − 30 km inside the RMW around 35 hours.This is consistent with a TFirstRI of 33 hours.The reader may be surprised at the rather small values of the rain rate when RI begins, being only F I G U R E 12 Intensity evolutions for individual ensemble members for selected cases from the ShearFlow set, and with changes in the position of the initial asymmetries: (a) for the n = 2 far-field asymmetry; (b) for the s = 0.75 secondary vortex.Note that the blue curves are for the control cases for each set.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 14 Ensemble-mean, azimuthal-mean rain rate for selected ShearFlow cases: (a) control case; (b) n = 1, r a = 6; (c) n = 2, r a = 6; (d) x = −4, s = 0.75.Note the time axes are different for (c, d).TFirstRI does not occur until 86 hours for case (c).[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 15 Ensemble-mean vortex tilts and environmental wind shears for selected cases for (a) NoFlow tilts; (b) NoFlow shears; (c) ShearFlow tilts; (d) ShearFlow shears.Note different ranges for the NoFlow and ShearFlow shear values.The ensemble-mean start times of the first RI for each case are marked by the triangles.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 16 Structural evolution of single ensemble members from the ShearFlow sets.Colors show smoothed vertical vorticity and horizontal wind vectors on model level 37 (z = 7.9 km) and black contours show smoothed surface pressure every 1 hPa.Left column: control case, (a-c) at 24, 36, and 60 hours; right column: n = 2, r a = 6, (d-f) at 24, 48, and 72 hours.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 17 Evolution of the surface to 8 − km tilt vectors for three cases: (a) ShearFlow control; (b) n = 2, r a = 6; (c) x = −4, s = 0.75.The color bar indicates the time for each data point increasing from blue (0 hours) to red (96 hours).The circles, pluses, and stars indicate the tilts for seed1, seed2, and seed3, respectively.[Colour figure can be viewed at wileyonlinelibrary.com] p x =  v fv and p y = − v fuwhere f is the Coriolis parameter and the virtual density  v = p∕RT v .On each vertical level, p(x, y, z) is computed by integrating these two equations from an arbitrary reference point, for example, the lower-left corner of the domain, horizontally to every other point.Using the simplified two-dimensional horizontal equations of motion, including the Coriolis force, we multiply through each equation by  v , and then take ∕x of the u equation and add it to ∕y of the v equation.Dropping the superscripts on  v for clarity leads to: Figure A.1.shows the vertical velocities at z = 2.9 km for the first six hours.The curves show the minimum w anywhere in the domain, the maximum w in the domain, and also a time series of w at a grid point 30 points (540 km) west of the primary center for simulations initialized with the geostrophic pressure, or balanced with N = 10 and K = 20, N = 20 and K = 40, and N = 40 and K = 80.For these three sets of iterations, the global maximum value of the error in (6.4) is reduced to 6.9%, 1.6%, and 0.42% of its initial value.Initializing with geostrophic balance greatly underestimates the necessary minimum pressure, leading to strong downward F I G U R E A.1 Vertical velocities in the first 6 h of single domain simulations with no physics and for varying degrees of balancing, using the secondary vortex at x = −360 km with s =0.75.Thick solid curves show minimum w on model level 22 (z = 2.9 km) anywhere in the domain, thick dashed curves show the maximum w, and the thin lines show w time series at a fixed point 540 km west of the domain center.[Colour figure can be viewed at wileyonlinelibrary.com] Ensemble-mean RI start times and maximum intensity changes for NoFlow cases.
TA B L E 1 Ensemble-mean RI start times and maximum intensity changes for ShearFlow cases.