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  • View in gallery

    Vertical lines indicate noon LST. (top) Time evolution of maximum azimuthally averaged tangential wind speed for the simulation with constant solar radiation (CTRL) and diurnal cycle (20D) on an f plane at 20°N, and the simulation with diurnal cycle on an f plane at 25°N (25D). (middle) Radius (from minimum surface pressure) of maximum tangential winds plotted in (top). (bottom) Domainwide surface precipitation rate. Note that CTRL exhibits a nearly diurnal cycle from day 2 to 8 because of artificial gravity wave interference.

  • View in gallery

    Power spectral density (PSD) of upper-level (z = 14 km) azimuthally averaged temperature at a radius of 400 km from the TC center, calculated using only the data (top) between days 3 and 9 and (bottom) between days 20.5 and 26.5 for each simulation (solid black lines). The red-noise spectra (dashed black lines) and the 95% and 99% confidence levels (light gray and dark gray solid lines, respectively) are included for reference. Frequency (f) values (cycles per day) are labeled at the points where the PSD exceeds the 95% confidence level.

  • View in gallery

    (a) Schematic of IIGW excitation and propagation. Group velocity propagates away from the source, while phase speed propagates normal to , reflected along the vertical direction. (b) Schematic of wave-supporting regions for different ω in a TC. Simulations exhibit a “cap,” an upside-down cone in the upper troposphere, in the cutoff region as well as the lower-tropospheric cone. IIGWs can only propagate outside of the cutoff region corresponding to their frequency at an angle ±ϕ from the vertical.

  • View in gallery

    Each plot shows the azimuthally averaged surface where (thin black line) and (thick black line) for days 13, 15, and 17. At radii smaller than these surfaces, the corresponding wave cannot propagate. The filled black regions show transiently inertially unstable regions, —a region in which no IIGW can propagate. Gray dashed lines show azimuthally averaged surfaces of tangential wind speed. The 12 m s−1 surface is found to be a reasonable estimate of the surface for at least the lower 10 km of the atmosphere. The same is true for the 24 m s−1 surface and the surface.

  • View in gallery

    As in Fig. 4, but for days 19, 21, and 23.

  • View in gallery

    Hovmöller plots, where deep blue indicates −5 m s−1, and deep red indicates 5 m s−1 deviations from time-averaged and azimuthally averaged radial wind over the period 18.5–26 days for storm 25D, for three different radii from the storm center: (top) 200, (middle) 400, and (bottom) 600 km. The negative slope of the waves in the upper troposphere indicates a phase speed oriented downward, and the inverse is true for waves in the middle troposphere. Their juncture indicates the approximate vertical height of maximum solar forcing. Note the increasing coherence of lower- and midtropospheric waves with increasing distance from the storm center.

  • View in gallery

    Eddy temperature (K) fields from snapshots (0700 LST day 24) for three simulations. Time was chosen to show a particularly visible wave in 20D. (top) Eddy temperature from cross section of output. (bottom) Eddy temperature from azimuthally averaged output. The black line indicates the cutoff region; beyond it, propagation angles are superimposed on the temperature field. Angles are internally consistent with plot dimensions.

  • View in gallery

    “Diurnal eddies” of radiative heating rates (K day−1) for each simulation. The dark blue line shows the response at 100 km, and subsequent intervals of 100 km end at the yellow line for radius 700 km. The last 7 days of the simulation have been averaged hourly to create a representative mean 24-h cycle (x axis; LST). Fields shown are departures from the total mean value of each variable over all 7 days. See the appendix for details.

  • View in gallery

    As in Fig. 8, but for azimuthally averaged mean radial angular momentum flux eddies per unit mass. The calculation is provided in the appendix.

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Accessible Environments for Diurnal-Period Waves in Simulated Tropical Cyclones

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  • 1 Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, Israel
  • 2 Program in Atmospheres, Oceans and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts
  • 3 Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York, and Florida State University, Tallahassee, Florida
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Abstract

A recent observational analysis has reported significant repeating diurnal signals propagating outward at cloud-top height from tropical cyclone centers. Modeling studies suggest that the visible upper-level impacts reflect a diurnal cycle through the depth of the troposphere. In this study, the possibility of wavelike diurnal responses in tropical cyclones is characterized using 3D cloud-resolving numerical simulations with and without a diurnal cycle. Diurnal waves can only begin to propagate well beyond the storm core, and the outflow region is most receptive to near-core diurnal propagation because of its anticyclonic flow. The tropical cyclone structure appears particularly hostile to diurnal wave propagation during periods when the eyewall experiences a temporary breakdown similar to an eyewall replacement cycle.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Morgan E O’Neill, morgan.e.oneill@gmail.com

Abstract

A recent observational analysis has reported significant repeating diurnal signals propagating outward at cloud-top height from tropical cyclone centers. Modeling studies suggest that the visible upper-level impacts reflect a diurnal cycle through the depth of the troposphere. In this study, the possibility of wavelike diurnal responses in tropical cyclones is characterized using 3D cloud-resolving numerical simulations with and without a diurnal cycle. Diurnal waves can only begin to propagate well beyond the storm core, and the outflow region is most receptive to near-core diurnal propagation because of its anticyclonic flow. The tropical cyclone structure appears particularly hostile to diurnal wave propagation during periods when the eyewall experiences a temporary breakdown similar to an eyewall replacement cycle.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Morgan E O’Neill, morgan.e.oneill@gmail.com

1. Introduction

The diurnal cycle of incoming solar radiation is one of the most fundamental sources of variability in Earth’s weather and climate. Many efforts have been directed toward improving the understanding of diurnal variations in convection in response to the diurnal cycle. Observational studies have shown that the amplitude of diurnal oscillations in deep convection and heavy rainfall in the tropics is larger over land than over the open ocean (Gray and Jacobson 1977). Tropical rainfall rates typically exhibit an afternoon maximum over land, while they peak in the morning over ocean (Nesbitt and Zipser 2003). Tropical cyclones (TCs) also exhibit diurnal behavior because of their characteristic, high-altitude outflow canopy and significant internal circulation. In this work, we examine cloud-resolving TC simulations to elucidate the impact of the diurnal cycle on quasi-steady TCs. A better understanding of how TCs react to diurnal insolation may lead to improved intensity forecasts, which are of significant societal interest upon landfall.

Observational studies of the TC diurnal cycle have largely focused on cloud-top changes (e.g., Browner et al. 1977; Muramatsu 1983; Lajoie and Butterworth 1984; Steranka et al. 1984; Kossin 2002; Dunion et al. 2014, hereafter DTV14; Wu and Ruan 2016) or precipitation (e.g., Bowman and Fowler 2015; Wu et al. 2015; Leppert and Cecil 2016), because of the comparative difficulty of detecting midtroposphere changes with current observational tools. Kossin (2002) examined variations in azimuthal-mean cloud-top temperature at various radii from TC centers. The author found diurnal oscillations in consistent with variations in the areal extent of the cirrus canopy identified by previous studies. However, he did not find a significant diurnal signal near the TC inner convective region where the coldest cloud tops exist (radius < 200 km). This result challenged previous hypotheses that related the diurnal cycle of the cirrus canopy to oscillations in deep convection near the TC center (e.g., Browner et al. 1977). Instead, a semidiurnal cycle was identified near the central convection and hypothesized to be associated with the solar semidiurnal atmospheric tide.

Recent analysis of satellite imagery by DTV14 revealed diurnal pulselike features emanating from the center of mature TCs. Consistent with Kossin’s observation that propagating signals do not occur close to the storm core, DTV14 noted a diurnal signal that began at a radius of 200 km, propagating out to at least 700 km. These large, outward-propagating rings appear as periodic oscillations of cooling and warming in the infrared brightness temperature field. The diurnal pulsing is also discernible in microwave imagery, which is able to penetrate cirrus clouds, suggesting that this phenomenon involves a deep tropospheric layer.

These observations prompt a number of questions: where is the primary diurnal forcing occurring? If the eye and surrounding eyewall are most strongly forced by a radiative cycle, why do these diurnal rings only appear in satellite observations well beyond the storm core? Is the diurnal signal muted throughout the depth of the storm interior, or is it only the outflow that resists diurnal response until large radii? DTV14’s observations strongly suggest a propagating wave feature, which must be supported by an appropriate restoring force of some kind. Large-scale gravity waves have been observed emanating from TC cores with periods of 5–13 h (Chane Ming et al. 2014), though diurnal-period waves were not reported in that study.

There have been only a few modeling studies that investigated the diurnal cycle of TCs. The impact of the diurnal cycle of solar radiation on TC genesis, intensification, and structure was investigated in several case studies (Melhauser and Zhang 2014; Tang and Zhang 2016) and idealized simulations (Hobgood 1986; Ge et al. 2014; Nicholls 2015; Ge et al. 2015; Zhou et al. 2016). These studies suggested that TC genesis is accelerated by the diurnal cycle, particularly the nighttime phase. Navarro and Hakim (2016) found a spatially coherent diurnal signal in temperature, wind, and latent heating tendency throughout the depth of the troposphere in idealized 2D axisymmetric TC simulations. These studies, however, did not directly address the mechanisms behind the propagation of diurnal signals. Because of the difficulty of comprehensive observations in the hostile environment of a TC, idealized 3D cloud-resolving simulations provide a valuable laboratory for exploring diurnal heating and wave responses through the full depth of the troposphere.

We search for wave-supporting regions in idealized TC simulations in a radiative–convective equilibrium (RCE) environment and compare our findings with those from the observational and modeling studies described above. A distinction is drawn between in situ evanescent diurnal responses and gravity waves that are able to propagate long distances from the site of forcing, as in Willoughby (2009). We show that the DTV14 signal is consistent with an internal inertial gravity wave (IIGW), which is prohibited from the core region as a result of either inertially unstable flow or inertially stable flow of sufficiently high frequency.

The rest of the paper is organized as follows. Section 2 describes the model setup and simulations. Section 3 describes the evolution of the simulations and the power spectrum analysis. Section 4 briefly reviews IIGWs and discusses wave environments and responses. Section 5 provides a discussion, caveats, and potential future work.

2. Model simulations

Three tropical cyclone simulations were performed using the System for Atmospheric Modeling (SAM) cloud-resolving model, version 6.8.2 (Khairoutdinov and Randall 2003). SAM employs the anelastic equations of motion on a three-dimensional domain that, in this case, spans 2048 × 2048 km2 horizontally. The horizontal grid size is 2 km, and the lateral boundary conditions are doubly periodic. There are 64 levels in the vertical with a rigid lid at 28 km and stretched grid spacing, ranging from 75 m near the surface to 500 m above 3.5 km. A sponge layer in the upper 30% of the vertical domain is employed to minimize the reflection and buildup of gravity waves. The bottom boundary is assumed to be an ocean with a fixed sea surface temperature (SST) of 301 K, representative of the tropical oceans under current climate conditions. While SST is kept constant, surface turbulent (sensible and latent heat) fluxes are allowed to vary depending on the atmospheric conditions above. The fluxes and exchange coefficients are calculated iteratively according to Monin–Obukhov similarity theory.

The control run (CTRL) is on an f plane at approximately 20°N (f = 5.00 × 10−5 s−1) and under a constant solar insolation of ~413 W m−2, achieved by fixing a solar constant of 650.83 W m−2 and a zenith angle of 50.5° [following Bretherton et al. (2005)]. Two additional f-plane experiments (20D and 25D) were run with a diurnal cycle of insolation equal to that at 19.45°N and perpetual Julian day 80.5, yielding an average solar insolation equal to that of the control run. The value of the Coriolis parameter for these two experiments corresponds to that at approximately 20° (f = 5.00 × 10−5 s−1) and 25°N (f = 6.16 × 10−5 s−1), respectively. The shortwave and longwave radiative fluxes are computed interactively using the Rapid Radiative Transfer Model (RRTM; Mlawer et al. 1997; Clough et al. 2005; Iacono et al. 2008).

The domain size, which was chosen to be large enough to contain the tropical cyclone while remaining computationally feasible, has implications for the propagation of gravity waves. The first deformation radius at 20°N is approximately 3000 km, larger than our domain, which implies that barotropic gravity waves may propagate without significant modification by the Coriolis force. In a doubly periodic domain, gravity waves can additionally return to their source and cause constructive or destructive interference. This was the motivation behind the 25D simulation, in which there is a larger local Coriolis parameter and therefore a smaller deformation radius. The small change in latitude is expected to provide a control or comparison in the event that the CTRL and 20D simulations experience latitude-dependent resonances with the domain size.

The simulations are initialized with an environmental temperature and moisture profile from the horizontal average of the last 25 days of a small-domain simulation (96 × 96 km2) run to radiative–convective equilibrium. To initialize the TC, a warm and moist bubble is inserted into an initially calm, horizontally homogeneous environment, following the approach of Fovell et al. (2010). The center of the bubble is in the middle of the domain, at a height of 4 km. It has a horizontal radius of 250 km and a vertical radius of 2 km. The temperature and moisture perturbations are +10 K and +7.8351 g kg−1, values such that the air in the bubble is saturated. During a 1-day spinup period, the model rapidly reacts to the perturbation to create a vortex, which then intensifies into a TC. Each simulation is integrated for 26 model days, with hourly output.

While initialized with a stationary centered bubble, the TCs are free to wander away from the domain center. At each output time step, the center of the TC is defined as the location of the minimum surface pressure (Yang et al. 2007). The results are insensitive to whether only the surface pressure defines the center or whether the center is defined at each vertical level by the local pressure minimum. For analysis purposes, the 3D data are horizontally shifted to recenter the storm in the domain and then interpolated onto a cylindrical grid.

3. Storm evolution

Because of the unbalanced thermal bubble initialization, all three storms immediately exhibit a strong gravity wave that propagates through the doubly periodic domain and back toward the storm. In the CTRL simulation, the gravity wave signal weakens over time. The domainwide thermodynamic impact of the diurnal cycle can be seen in the bottom panel of Fig. 1, which shows the mean surface precipitation rate over time. The control storm (CTRL) exhibits a similarly large, periodic domainwide variation in precipitation as the two experiments (20D and 25D). Such self-interference due to the doubly periodic domain was also observed by Nolan (2007), who was able to remove gravity wave self-interference with a much larger domain. The two diurnal storms in this study quickly settle into the familiar oceanic cycle of peak precipitation in the early mornings (Nesbitt and Zipser 2003), and the control experiment’s early periodic precipitation becomes noisy and unpredictable.

Fig. 1.
Fig. 1.

Vertical lines indicate noon LST. (top) Time evolution of maximum azimuthally averaged tangential wind speed for the simulation with constant solar radiation (CTRL) and diurnal cycle (20D) on an f plane at 20°N, and the simulation with diurnal cycle on an f plane at 25°N (25D). (middle) Radius (from minimum surface pressure) of maximum tangential winds plotted in (top). (bottom) Domainwide surface precipitation rate. Note that CTRL exhibits a nearly diurnal cycle from day 2 to 8 because of artificial gravity wave interference.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

a. Spectral analysis

A spectral (Fourier) analysis is applied to the data to identify the dominant periodicities more robustly. Figure 2 shows the power spectrum of the azimuthally averaged temperature at a radius of 400 km and height of 14 km (near the outflow layer) for each of the simulations. The top panel corresponds to the early stages of the simulations (time series between days 3 and 9), while the bottom panel corresponds to the last 6 days of integration (days 20.5–26.5). Following DTV14, we test for statistical significance against a null hypothesis defined as the theoretical first-order autoregressive [AR(1)] red-noise spectrum (Gilman et al. 1963). Here the lag-1 autocorrelation coefficient indicates the hour-to-hour “memory” of the system. The 95% and 99% confidence levels are estimated using the chi-squared statistical test (Wilks 2006).

Fig. 2.
Fig. 2.

Power spectral density (PSD) of upper-level (z = 14 km) azimuthally averaged temperature at a radius of 400 km from the TC center, calculated using only the data (top) between days 3 and 9 and (bottom) between days 20.5 and 26.5 for each simulation (solid black lines). The red-noise spectra (dashed black lines) and the 95% and 99% confidence levels (light gray and dark gray solid lines, respectively) are included for reference. Frequency (f) values (cycles per day) are labeled at the points where the PSD exceeds the 95% confidence level.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

The initial strong gravity wave appears latitude dependent, as expected (top row of Fig. 2: 20.5 h in CTRL, 20.5 h in 20D, and 18 h in the more poleward 25D). Because it appears equally in the constant-insolation control simulation, we can rule out a diurnal origin. Rather, the model’s response to the unbalanced thermal bubble is a strong gravity wave that self-interferes in the doubly periodic domain. Neither of these initial periods corresponds to the inertial period at each latitude. The doubly periodic domain allows these waves to return to the storm center, and the subsequent “sloshing” of the fluid at the domain scale can be seen in movies of convective features at early times, including in the control simulation. A long integration time is required to ensure that the early gravity wave signal has dissipated. DTV14 speculate that their observed diurnal pulse may occur only in mature TCs, and the type of simulation in this paper is not suited to exploring the existence of a diurnal signal in intensifying TCs. We note another good reason to integrate for a long time: the initial simulated TC intensity (maximum tangential wind speed) is much higher than that of the equilibrium TC, as observed by Hakim (2011) and Chavas and Emanuel (2014).

Near the end of the simulation (bottom row of Fig. 2), the two diurnal experiments exhibit exactly diurnal signals, and 20D additionally exhibits a semidiurnal (12 h) signal. CTRL no longer experiences a statistically significant low-frequency signal, though it does now exhibit an 8–10-h period (25D also has an ≈8-h periodicity). This is consistent with the understanding that a diurnal cycle of insolation is not required to excite periodic behavior, and gravity waves can be a dynamical response to convective bursts.

Phenomenologically, there is a big distinction between the catchall phrase “diurnal cycle/signal,” which can include any local evanescent responses, and the subset of such a cycle that is the wave response. A wave requires not only a periodic forcing, but also a restoring force in the medium that will allow the wave to propagate away from the forcing site. The spectral analysis used here, and in DTV14 and Navarro and Hakim (2016) with respect to the diurnal cycle, cannot distinguish between evanescent and wave responses to diurnal solar forcing. In the following sections, as we examine wave responses with diurnal and semidiurnal periods, we look for conditions that are required to support such waves (Willoughby 2009) and compare simulated propagation angles with those predicted by the appropriate dispersion relation.

b. Eyewall replacement

All three storms experience at least one event that is similar to, though not exactly, an eyewall replacement cycle (ERC; top and middle panels of Fig. 1). A better description of the event is a temporary breakdown of the eyewall, resulting in a rapid widening of the radius of maximum winds (RMW) while the maximum mean tangential wind decreases. For convenience, we will refer to this as an “ERC-like event,” though it only occasionally exhibits a simultaneous secondary wind maximum. Interestingly, all ERC-like events are preceded, by 2–3 days, by the appearance of a (very weak) secondary tangential wind maximum in the vertical, at 3 km or higher and at large radii (300–700 km, shown in section 4). A related feature, at a higher altitude and a smaller radius from the center, can be seen in the simulations of Wang (2009, their Fig. 3) and Zhou and Wang (2009, their Fig. 4). Those simulations are quadruply nested, suggesting that this phenomenon is not due to the artifice of a doubly periodic domain.

TC 25D undergoes an ERC-like event first, at 12 days, followed by 20D at 13.5 days and CTRL at 15.5 days. TC 25D also experiences the most disruptive and long-lasting ERC-like event (approximately 4 days), in which multiple rings of local-max tangential flow appear to contract consecutively. It appears as though 25D is starting to undergo a second ERC-like period at the end of the simulation, and further integration of the two other storms might yield slightly delayed ERC-like events as well. Whether the ERC-like onset is robustly sensitive to diurnal heating and latitude will require a much larger suite of simulations over a broad range of latitudes. The following analysis examines the regions in a TC environment where diurnal and semidiurnal waves can propagate. These regions are very sensitive to such ERC-like events.

4. Gravity waves in a TC environment

Internal gravity waves propagate energy both horizontally and vertically away from a wave source. In a homogeneous fluid, the group velocity points radially away from the source location, to satisfy the radiation condition. The phase speed is perpendicular to the group velocity, such that their vertical components are antiparallel (Fig. 3a). In a barotropic, rotating, stratified fluid, gravity waves are IIGWs. The IIGW frequency ω is bounded in the vertical by the Brunt–Väisälä frequency N and in the horizontal by the inertial frequency I. For a quiescent fluid (planetary rotation only), is the local Coriolis frequency ( is twice the angular frequency of planetary rotation):
e1
Fig. 3.
Fig. 3.

(a) Schematic of IIGW excitation and propagation. Group velocity propagates away from the source, while phase speed propagates normal to , reflected along the vertical direction. (b) Schematic of wave-supporting regions for different ω in a TC. Simulations exhibit a “cap,” an upside-down cone in the upper troposphere, in the cutoff region as well as the lower-tropospheric cone. IIGWs can only propagate outside of the cutoff region corresponding to their frequency at an angle ±ϕ from the vertical.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

The local Coriolis frequency at θ = 30°N is 1 day−1, equivalent to the diurnal frequency (where the subscript denotes a period of 24 h), so 30° is the upper latitude bound for the propagation of diurnal IIGWs in a quiescent fluid. The total inertial frequency includes rotational flow and is higher than in the strongly sheared, cyclonic environment of a TC. The local Rossby number is O(1), and the hurricane is in approximate gradient wind balance. Total angular momentum is given by , and the total inertial stability is given by for radius from the storm center r and azimuthally averaged tangential wind . The near-core inertial frequency is dominated by the rotating wind:
e2
The first factor on the left-hand side of Eq. (2) is the absolute vorticity of the mean flow ζ; the second factor is the mean-flow inertial parameter ξ (Willoughby 1977). Because of the very strong flow curvature near the TC core, it is impossible for the near-core flow to support a diurnal IIGW. The diurnal frequency is lower than the near-core inertial frequency. Equation (2) provides a dynamic lower limit on waves supported by the TC environment. Semidiurnal, 12-h period waves have a larger , so the core cutoff region is much smaller. Beyond the radius of critical frequency for a diurnal or semidiurnal wave, the region outside can be approximated as .

Equation (2) is still only an approximate calculation, assuming a barotropic vortex. TCs are baroclinic, and the complete passband filter for IIGW in a TC was worked out by Willoughby [2009, his Eqs. (10) and (11)], for a study examining wave periods up to 16 h. Because the diurnal and semidiurnal waves considered here only occur well beyond the most baroclinic region of the inner TC core, the difference that the baroclinic terms make is negligible for both the cutoff regions and wave propagation angles examined here (not shown). This permits the simplifying barotropic assumption.

Figure 3 shows a schematic in the rz plane of a localized periodic forcing (imagine there is a small cylinder vibrating up and down), and the resulting IIGW field (often referred to as “St. Andrew’s cross”). In the case of no flow (Fig. 3a), waves propagate away both above and below the height of the forcing, at a specific angle ϕ from the vertical for each wave frequency. In a quiescent, constant-N2 fluid, there are no regions in which such waves are prohibited. This is not the case for the complicated rotational flow of a hurricane (Fig. 3b), where diurnally varying radiative heating aloft has the potential to act as an analog to the bobbing-cylinder forcing. Because of the wind field, waves cannot immediately propagate away from the forcing. They are blocked by the flow-induced cutoff regions (one could also consider them high-pass filters). The present frequencies of interest, and , have cutoff regions that are roughly cone shaped when the TC is in steady gradient wind balance. The cutoff region for includes an upward-tilted cone as well, stretching above the upper-eyewall diurnal forcing.

a. Cutoff regions and waves

The simulated, instantaneous cutoff regions and their corresponding tangential flow fields are depicted for a series of days in Figs. 4 and 5 for each experiment. An additional barrier to wave propagation is depicted by a black filled-in region in some of the plots. As seen before in both observations and simulation studies (e.g., Molinari and Vollaro 2014), our simulations all transiently experience inertial instability (I2 < 0). Instead of acting as a frequency-dependent filter, these black areas prohibit all inertial waves, including IIGWs.

Fig. 4.
Fig. 4.

Each plot shows the azimuthally averaged surface where (thin black line) and (thick black line) for days 13, 15, and 17. At radii smaller than these surfaces, the corresponding wave cannot propagate. The filled black regions show transiently inertially unstable regions, —a region in which no IIGW can propagate. Gray dashed lines show azimuthally averaged surfaces of tangential wind speed. The 12 m s−1 surface is found to be a reasonable estimate of the surface for at least the lower 10 km of the atmosphere. The same is true for the 24 m s−1 surface and the surface.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for days 19, 21, and 23.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

In all cases, the plots corroborate DTV14’s observation that waves are not observed propagating outward until a radius of 200 km or so. These simulations suggest that the diurnal waves appear as close to the center of the TC as they are permitted, and at smaller radii they simply do not exist.

The variation in cutoff regions between days is significant, and it shows some correlation to the ERC-like periods. The tangential wind field echoes the diversity of structures. The location of azimuthally averaged 24 m s−1 tangential wind roughly approximates the 12-h cutoff, and the 12 m s−1 wind roughly approximates the 24-h cutoff. Across the simulations, the frequent nonmonotonicity in the vertical of both the far wind field and the ω12 and ω24 cutoff regions is striking and accompanies the weakening of the peak winds at the RMW. From day 13 to day 17, all three storms are experiencing an ERC-like event, and so their cutoff regions are correspondingly not conical. At day 22, 25D starts to experience an increasing and oscillating RMW again, indicating another ERC-like event onset. Thus, at day 21, both the far tangential wind field and the cutoff region again lose their vertical monotonicity, because of the large-radius secondary tangential wind maximum in the vertical. The other storms are not yet experiencing another eyewall disruption, and so correspondingly they maintain conical cutoff regions. Because of 25D’s longer, more disruptive ERC-like periods, the corresponding diurnal cutoff regions are more common and, because of 25D’s larger size (see the RMW in Fig. 1), frequently much larger.

For axisymmetric wavenumber the associated hydrostatic (mk) dispersion relation for IIGWs is
e3
In this relation, we have assumed that is a good approximation for , which is true beyond the cutoff regions, and that waves are approximately two-dimensional, neglecting variation in the azimuthal direction. Equation (3) can be rearranged to provide the propagation angle ϕ:
e4
For the low frequency of diurnal waves, this angle is virtually horizontal; slightly greater than 89° from the vertical, for waves above the wave source, or slightly less than 91° from the vertical, for waves below the wave source. The height of the hurricane’s upper anticyclone is coincident with the height of the diurnal forcing at 13–14 km (Fig. 8 shows the average diurnal heating rate). The anticyclone significantly reduces the inertial frequency of the flow. The nominal diurnal forcing region is therefore adjacent to a region that is particularly favorable for diurnal IIGW propagation. Thus, diurnal waves can be excited with little attenuation and propagate away to large distances in the free troposphere. The cutoff regions are rather inconsequential because they slope downward toward the surface at a lower angle: ϕ = 92°.1

That narrative changes when the cutoff region extends into the troposphere at a larger angle than that of diurnal propagation, as visible in the shelf-like and anvil-like structures of Figs. 4 and 5. In some cases, a diurnal wave can be excited near the primary diurnally forced region in the upper atmosphere and then absorbed quickly by an outcropping cutoff region. Further spelling the demise of diurnal waves in such an environment, these outcrops tend to be concurrent with regions of inertial instability, which would immediately scramble all inertial wave signals. The net result is an environment that is extremely hostile to low-frequency IIGWs.

At heights of 16–18 km, the evanescent region for diurnal frequencies outcrops as far as 400 km radially. Diurnal waves that are forced below from the core heating and cooling are filtered by this blocking region and cannot propagate farther upward into the lower stratosphere. The dominant waves in the upper troposphere have periods of 6–15 h. This is true for all three simulations, indicating that there is ample forcing of near-semidiurnal waves regardless of the diurnal cycle. Figure 6 is a Hovmöller diagram that shows the deviation in radial wind from the time and azimuthal average for simulation 25D for three different radii from the storm center. The removal of the mean secondary circulation in Fig. 6 shows a clear propagating diurnal wave in the middle troposphere and a more semidiurnal wave in the upper troposphere. The slopes of the features indicate IIGW that are propagating away from a forcing region located at a height of approximately 14 km, persisting as far as 600 km from the storm center. The notable role of the diurnal cycle in the propagation of 6–15-h waves [like those observed by Chane Ming et al. (2014)] appears to be bringing such near-semidiurnal waves into a 12-h resonance.

Fig. 6.
Fig. 6.

Hovmöller plots, where deep blue indicates −5 m s−1, and deep red indicates 5 m s−1 deviations from time-averaged and azimuthally averaged radial wind over the period 18.5–26 days for storm 25D, for three different radii from the storm center: (top) 200, (middle) 400, and (bottom) 600 km. The negative slope of the waves in the upper troposphere indicates a phase speed oriented downward, and the inverse is true for waves in the middle troposphere. Their juncture indicates the approximate vertical height of maximum solar forcing. Note the increasing coherence of lower- and midtropospheric waves with increasing distance from the storm center.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

Semidiurnal cutoff regions, being much smaller, provide a more welcoming environment for wave propagation. Because semidiurnal waves can be excited by the second harmonic of the diurnal forcing, they may be ubiquitous in TC environments. Figure 7 shows a snapshot of the eddy temperature field for each simulation. The eddy field is the departure from the average temperature in the rz plane, for an average taken from 12 h before the snapshot to 12 h after. This moving average allows instantaneous waves to be visible at any time during the simulations, even though in every simulation the domain-mean temperature rises steadily. The top row of Fig. 7 shows the eddy temperature field taken from a cross section of the simulation output. The bottom row, in contrast, shows the eddy temperature field from azimuthally averaged fields, and the resulting waves are muted. In some cases, strong waves that appear in the cross-sectional plots fail to appear in the azimuthally averaged plots at all.

Fig. 7.
Fig. 7.

Eddy temperature (K) fields from snapshots (0700 LST day 24) for three simulations. Time was chosen to show a particularly visible wave in 20D. (top) Eddy temperature from cross section of output. (bottom) Eddy temperature from azimuthally averaged output. The black line indicates the cutoff region; beyond it, propagation angles are superimposed on the temperature field. Angles are internally consistent with plot dimensions.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

Significant, domainwide semidiurnal waves are seen emanating from the upper core of the TCs (Fig. 7). The cutoff contour is provided, and the wedges depict the angle ±ϕ at which waves are expected to propagate outside the cutoff region. The angles are a function of the local static stability and inertial stability, as determined by the dispersion relation above, and prove to be a good fit. As suggested in Fig. 2, CTRL has near-semidiurnal waves that propagate at upper levels despite the lack of a diurnal cycle of insolation. Because these waves have a higher frequency than waves, they propagate at a slightly larger angle than the angles provided.

This indicates a difficult, although not surprising, aspect of the search for diurnal and semidiurnal waves in TCs. Like vortex Rossby waves (VRWs; Montgomery and Kallenbach 1997), these IIGWs may not propagate away from the TC core as a ring, but rather as a spiral. It depends on whether the storm is symmetric and responding symmetrically to radiative forcing or whether localized convective towers asymmetrically excite the waves. Waves can be spontaneously generated from convection, dynamic instabilities, and VRWs themselves. Fortunately, spiral waves can be seen to propagate in azimuthally averaged fields anyway, even though their angle of propagation may be distorted by such an average.

The diurnal wave can be most clearly seen upon averaging spatial rz fields over multiple days, which removes the impact of nondiurnal and nonsemidiurnal waves. Figure 8 shows the radiative heating rate for all three simulations, calculated as “diurnal eddies,” or departures from a composite “mean day” (calculation described in the appendix). In the upper troposphere near the core of the TC in 20D and 25D, there is anomalous heating centered about local noon and anomalous cooling in the nighttime. At low levels, there is only a diurnal variation in radiative heating in the outer regions of the storm, where the lack of a dense cloud canopy allows for atmospheric absorption of shortwave radiation around local noon. As expected, there is no such temporal evolution of radiative heating in CTRL, because it lacks a diurnal cycle of insolation. Figure 9 shows attendant waves in angular momentum flux. At the highest levels, the 20D waves are semidiurnal, perhaps because diurnal waves were filtered by the “cap” exhibited by the region. The 25D simulation fails to exhibit a significant semidiurnal wave at upper levels, consistent with the power spectrum at late times (Fig. 2). The shorter-period signal, if indeed a wave, would not be seen in these wave plots because 8.5 is not a factor of 24. Over 1 day, perhaps such waves would be visible, but the daily phase shift would smear the wave out in our mean day.

Fig. 8.
Fig. 8.

“Diurnal eddies” of radiative heating rates (K day−1) for each simulation. The dark blue line shows the response at 100 km, and subsequent intervals of 100 km end at the yellow line for radius 700 km. The last 7 days of the simulation have been averaged hourly to create a representative mean 24-h cycle (x axis; LST). Fields shown are departures from the total mean value of each variable over all 7 days. See the appendix for details.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

Fig. 9.
Fig. 9.

As in Fig. 8, but for azimuthally averaged mean radial angular momentum flux eddies per unit mass. The calculation is provided in the appendix.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0294.1

The angular momentum flux propagation at 12 km is the closest model analog that we have found to DTV14’s upper-tropospheric wave in brightness temperature change. The timing is roughly the same, though the phase should not be considered corroborating evidence until the exact forcing mechanism is better understood. At lower levels, the wave stalls, and at 3 km and below the diurnal wave actually propagates (and decays to zero) toward the storm.

b. Limiting latitude for propagation

DTV14 reported that two TCs examined in their study were poleward of 30°N: TC Erin was examined at latitude θ = 33.4°N and TC Maria was examined at θ = 31.6°N. Both TCs exhibited the usual diurnal cycle passing through 300 km. This is possible in TCs well poleward of the quiescent-flow 30°N limit for diurnal IIGWs. Using the tangential winds of the 25D TC, but supplying higher latitudes of 35° and 40°N to the dispersion relation, plots similar to Fig. 4 exhibit a shrinking but sizable region at the outflow level where diurnal waves continue to be supported (not shown).

Inertial-period IIGWs do not appear to dominate any region of the atmosphere in any of the three simulations. This is consistent with the modeling results of Shibuya et al. (2014), who find that diurnal forcing in the boundary layer only resonates with the inertial period at 30° and 90°N. Power spectra favor 24- and 12-h periods after the initial geostrophic adjustment during cyclogenesis, and there is no systematic presence of 35-h waves (for 20D) or 28-h waves (for 25D).

5. Discussion

DTV14 proposed that the outflow-level propagating signal they observe may indicate a wave that is propagating through the full thickness of the atmosphere. The simulations here provide supporting evidence that they are correct, but the picture is complicated by flow-induced cutoff regions that prohibit diurnal wave propagation. These regions are latitude dependent, such that higher-latitude TCs are less likely to exhibit diurnal waves throughout the free troposphere. Disruptive ERC-like events can also significantly restrict the ability of diurnal waves to leave the core forcing region.

The relationship between ERC-like behavior and the far tangential wind field is an interesting by-product of this study. A statistical comparison of secondary wind maxima in the vertical at 300–600 km from the storm, and imminent ERCs, would be a valuable future study, perhaps using long time integration simulations with multiple confirmed ERCs like that of Hakim (2013). Because the simulations described in this paper do not replicate the classical dynamics of real ERCs sufficiently, another model or set of experiments is necessary to explore this question further.

These simulations are restricted in applicability because of the domain size. That was not anticipated during experimental design, but the apparent ability of diurnal and semidiurnal waves to propagate all the way to the domain edge indicates that there is opportunity for significant positive or negative interference from the doubly periodic setup. Indeed, that appears to be occurring in the early days of TC spinup, when all three simulations exhibit remarkable full-depth gravity waves well before the diurnal cycle has a dynamical impact on 20D and 25D.

Ideally these simulations would be run for much longer, on much larger domains. Chavas and Emanuel (2014) tested the sensitivity of axisymmetric TCs to domain size and found that the radial wind profile at height z = 1.5 km did not converge until the domain was at least 3000 km in radius. For the full-physics, 3D simulations presented here, such a large domain was not feasible, but their findings suggest that the outer region of the storm is likely self-interacting even at late times. This may explain why the larger 25D storm does not exhibit a 12-h signal at late times (Fig. 2).

The control simulation in particular exhibited more near-semidiurnal waves than we expected to see, throughout the integration. These need not be excited by diurnal solar forcing (Chane Ming et al. 2014), and an identification of the specific convective events and flow asymmetries that trigger the waves is beyond the scope of this paper. However, strictly diurnal, 24-h waves do not appear in the control simulation, as expected.

While we found evidence of diurnal wave propagation in the upper troposphere in fields like eddy temperature, we were not able to identify the direct analog of DTV14’s cloud-top temperature. The relevant 2D output fields from SAM—cloud-top temperature and cloud-top height—both failed to show a statistically significant diurnal wave. This might be because of the relatively coarse 500-m vertical resolution at these heights. It is of interest to perform similar simulations with much higher upper-tropospheric vertical resolution. The 2D output fields are also hourly averages, instead of snapshots like the 3D fields. Another possibility is the role of tropopause GW reflection in the real world. SAM’s stratospheric sponge layer absorbs GWs rapidly, so the interaction of waves with their reflection is not permitted.

Both this work and the axisymmetric simulations of Navarro and Hakim (2016) find significant semidiurnal signals in the vicinity of the TC. Semidiurnal tides have been hypothesized by Kossin (2002) as the cause of a semidiurnal signal at upper levels of TCs. However, here and in Navarro and Hakim (2016) the damped stratospheric layer precludes thermal tides, and the semidiurnal signal remains. We suggest that the much smaller corresponding cutoff region actually favors semidiurnal waves away from the TC core, and the diurnal cycle drives near-semidiurnal waves into a 12-h resonance.

Acknowledgments

The authors thank Kerry Emanuel, Timothy Cronin, David Nolan, Jason Dunion, Leif Thomas, and Olivier Pauluis for insightful discussions and suggestions. The authors are also thankful to Marat Khairoutdinov for providing SAM, the cloud-resolving model. The first author was supported by the Weizmann Institute of Science Koshland Prize. She acknowledges the Israeli Science Foundation through Grant 1819/16 to Yohai Kaspi. The second author was supported by the National Science Foundation (NSF) Graduate Research Fellowship Program (2388357) and AGS-1520683. The third author acknowledges financial support from an NSF Atmospheric and Geospace Sciences Postdoctoral Research Fellowship (Award 1422351). We acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc; Computational Information Systems Laboratory 2012), provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the NSF.

APPENDIX

Calculation of Diurnal Waves

A synthetic “mean day” is calculated from an average of the last 6 days of each simulation. The azimuthally averaged output is binned such that all of the 1200 LST snapshots are averaged together, all of the 1300 LST snapshots are averaged together, and so forth. From this synthetic day, the 6-day time average is then removed, leaving an average departure from a diurnal mean at each hour. Figure 9 depicts the diurnal departures from the azimuthal-mean radial angular momentum flux per unit mass, where the flux is
ea1
Here, is the azimuthally averaged radial wind (positive indicates wind moving away from the storm center); is the azimuthally averaged tangential wind (positive indicates cyclonic flow); r is the radial distance from the storm center; and the square brackets indicate a time average over 6 days. Finally, each radial value in Figs. 8 and 9 has been averaged over ±10 km radially to smooth the waves.

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1

These regions do matter for waves that are forced at lower altitudes in the core and will act to dampen and absorb the waves.

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