## 1. Introduction

Although documentation of the tropical cyclone (TC) diurnal cycle has received much attention in the literature, both the dynamical mechanism and the impact to storm structure and intensity remain unknown. Oscillations observed in the cirrus canopy of real storms have been linked to changes in storm intensity (Browner et al. 1977); however, the reason for this relationship is poorly understood. Previous numerical modeling studies disagree on the relative importance of the diurnal cycle in the lifetime of a TC. This lack of consensus has been attributed to model constraints, such as initial and boundary conditions, as well as to radiative parameterization schemes. Here, we diagnose the diurnal signal in a statistically steady-state TC and investigate its role on storm structure and intensity.

Early work on observing the diurnal cycle of convection in TCs focuses mainly on documenting storm high cloudiness. Browner et al. (1977) show that, in the Atlantic basin, the maximum area coverage of high clouds in storms occurs at 1700 local time (LT), with a minimum at 0300 LT. In the Pacific, Muramatsu (1983) finds a maximum in the early morning (near 0600–0730 LT) and a minimum in the evening (around 1800–2100 LT). For TCs near Australia, Lajoie and Butterworth (1984) demonstrate a maximum at 0300 LT and a minimum at 1800 LT. Steranka et al. (1984) clarify the inconsistency between ocean basins by showing that these calculations are sensitive to the chosen brightness temperature, as well as to the radial distance from storm center. By organizing storms into sets of concentric rings, Steranka et al. (1984) demonstrate that the diurnal cloud maximum is a propagating feature that emanates from the storm center. Estimated propagation speeds are 10–15 m s^{−1} for tropical storms and 2 m s^{−1} for hurricanes.

Recent observational studies have also focused on the upper-level clouds and further document a diurnal, as well as a semidiurnal, signal in the TC cirrus canopy. Using Hovmöller diagrams of infrared brightness temperature, Kossin (2002) demonstrates a pronounced diurnal oscillation in the areal extent of the TC cirrus canopy. Power spectrum analysis reveals that the diurnal signal is often absent near the convection in the inner core of the storm, but a significant semidiurnal signal is occasionally observed in this region. Kossin (2002) shows that the semidiurnal signal is in phase with the semidiurnal atmospheric tide *S*_{2} and hypothesizes that this relationship results in an oscillation of the lapse rates that modulates the convection. Similar to Steranka et al. (1984), Dunion et al. (2014) also document an outward-propagating TC diurnal signal. Using 6-h differences in brightness temperatures, Dunion et al. (2014) show that, for North Atlantic major hurricanes from 2001 to 2010, diurnal “pulses” originate near the storm center around sunset, strengthen overnight, and propagate outward in the early morning, with propagation speeds near 5–10 m s^{−1}. Significant warming of the cloud tops is observed after the passing of the diurnal pulse, which is coincident with an observed radial expansion of the storm’s overall structure.

The TC diurnal cycle has also been explored using numerical experiments, with varying results. Sundqvist (1970) and Hobgood (1986) demonstrate a more rapid growth rate in the developing stages of the simulation of a single storm with a diurnal cycle, with no effect thereafter. Hack (1980) and Tuleya and Kurihara (1981) find no influence on the initial growth rate, but rather an earlier onset of storm intensification. For axisymmetric TCs, Craig (1996) demonstrates that a modest increase in overall storm intensity is observed with realistic radiation but shows no impact on the storm growth rate or the timing of intensification. Melhauser and Zhang (2014) show, using modified radiation experiments in the Advanced Research version of the Weather Research and Forecasting (ARW) Model that TC development is highly sensitive to the diurnal cycle, exhibiting suppressed formation in daytime-only and no-radiation experiments and accelerated intensification in nighttime-only experiments as compared with a control. Tang and Zhang (2016) demonstrate a similar result for Hurricane Edouard (2014), showing that the impact of the diurnal cycle is mainly concentrated on the rate of intensification for the development stage of the TC and on the structure and strength in the rapid intensification and mature stage. In the mature stage, Tang and Zhang (2016) note that the difference in radiation between the daytime and nighttime phases has a minor impact on storm intensity.

Presently, there is no consensus in the literature on the impact of the diurnal cycle of radiation on TC structure and intensity. This lack of consensus motivates the need for a study that quantifies the role of the diurnal signal in the storm. The current hypothesized mechanisms can be summarized as follows: 1) radiative destabilization at cloud top overnight, which steepens the lapse rate and causes an increase in nocturnal convection (Hobgood 1986); 2) differential cooling between the cloudy and clear-air environment of the TC, which enhances low-level convergence (Gray and Jacobson 1977; Melhauser and Zhang 2014; Tang and Zhang 2016); and 3) radial contraction of the upper branch of the TC secondary circulation by outgoing longwave radiation, shrinking the extent of the cirrus canopy overnight (Kossin 2002). While this disagreement motivates the need for a dynamical explanation of this phenomenon, the direct evaluation of these hypotheses is not the objective of the current study.

The goal of this work is to quantitatively describe and document the TC diurnal cycle in a numerical simulation. A steady-state TC is produced with no environmental influences to isolate the internal storm response to the diurnal cycle of radiation. Previous studies have used this framework to explore the impact of radiation on TC variability (Hakim 2011, 2013; Brown and Hakim 2013); however, they did not include a diurnal cycle. Based on the lack of conceptual understanding of this phenomenon, we first evaluate the TC diurnal cycle using an axisymmetric model. The remainder of this paper is organized as follows: model configuration, methods for analysis, and the time-mean storm are described in section 2; results using time series analysis to identify a diurnal cycle are described in section 3a; results using composite averaging to describe the structural evolution of the TC diurnal cycle are described in section 3b; and section 4 gives a concluding summary.

## 2. Method

### a. Model configuration and analysis

The model used here is the axisymmetric, nonhydrostatic cloud model of Bryan and Fritsch (2002) and Bryan and Rotunno (2009b) (CM1). This model includes the National Aeronautics and Space Administration (NASA) Goddard radiation scheme (Chou and Suarez 1994), simulating the effects of both longwave and shortwave radiation, and uses the NASA Goddard version of the Lin et al. (1983) ice microphysics scheme. The domain is 1500 km in the horizontal direction and 25 km in the vertical direction. A fixed horizontal resolution of 4 km is used; resolution in the vertical is 250 m from 0 to 10 km, with a gradual increase to 1 km at the top. A damping layer is applied to momentum and potential temperature within 5 km of the model top and to momentum within 100 km of the outermost radius. The horizontal and vertical length scales for the turbulence parameterization are 500 and 200 m, respectively, and are consistent with both previous axisymmetric modeling studies in CM1 (Bryan and Rotunno 2009a; Rotunno and Bryan 2012; Hakim 2011), as well as recent work measuring the horizontal mixing length in the low-level region of intense hurricanes (Zhang and Montgomery 2012). The exchange coefficients for the surface fluxes of energy and momentum are given a ratio of unity, and initial temperature and moisture profiles are taken from Rotunno and Emanuel (1987).

Following the approach established by Hakim (2011, 2013) and Brown and Hakim (2013), we integrate the simulation to 340 days, of which approximately 40 days are required for the storm to reach radiative–convective equilibrium with the environment. This simulation excludes all external influences, such as varying sea surface temperature (SST) and wind shear, which allows for an identification of storm structure and intensity with the diurnal cycle of radiation. The sea surface temperature is fixed at 26.3°C, and the Coriolis parameter is constant at a value corresponding to 20° latitude. The solution is obtained for a fixed calendar day of 10 September 2015. Since angular momentum is lost in the radial inflow layer as a result of frictional dissipation, a source of angular momentum is required in the closed system to maintain a statistically steady state.^{1} This is accomplished at the lateral boundary, where anticyclonic (i.e., negative *υ* component) parcels in the outflow reach the damping layer near the lateral edge of the domain, which increases their angular momentum. The damping coefficient is fixed in time and therefore does not vary periodically with respect to the diurnal cycle.

Using the last 300 days of the simulation, power spectrum analysis is performed to identify the diurnal harmonics, and time series reconstruction is used to quantify the diurnal cycle. Composite analysis is then used to evaluate the horizontal and vertical structure of the TC diurnal cycle. To compute the composites, the time mean over the 300-day sample is first removed; then the anomalies from this mean are composited at each hour of the day (e.g., all the azimuthal wind anomalies at 0100 LT, 0200 LT, etc.).

### b. Equilibrium storm

Figure 1a shows the time series of the maximum surface wind for the full 340-day simulation. After a transient spinup period lasting approximately 40 days, the storm reaches radiative–convective equilibrium about a statistically steady state. The steady-state portion of the TC simulation will be referred to as the “equilibrium storm.” The mean intensity at the surface for the full 340-day simulation is 36 m s^{−1}, with a standard deviation of 3.9 m s^{−1}. Peak values of maximum surface wind speeds exceed 50 m s^{−1}. The location of the radius of maximum wind also exhibits large variability (Fig. 1b); this radius quickly transitions from the time-mean value of 53 km to values near 100–120 km approximately 40 times, indicating possible eyewall replacement cycles. Properties of the transient, superintense storm and eyewall replacement cycles are discussed in detail in Hakim (2011, 2013); here, we will only discuss properties related to the diurnal cycle.

The time-mean values of the equilibrium storm for the azimuthal wind, temperature, radial wind, and latent heating tendency fields are plotted in Figs. 2a–d. These fields serve as the baseline for comparison with the composite anomalies presented in this study. A maximum in azimuthal wind is observed at a height of 1.1 km and a radius of 50 km (Fig. 2a), demonstrating the mean location of the radius of maximum wind for the equilibrium storm. The average magnitude at this level is 38 m s^{−1}. Winds decay rapidly with both radius and height and the azimuthal wind becomes negative at radii greater than 300 km. Absolute temperatures in the troposphere decrease nearly linearly with height (Fig. 2b); a mean lapse rate of −6.1 K km^{−1} is observed from the surface up to 12.5 km. The mean height of the tropopause is located at 14.5 km and is indicated by the abrupt transition in the gradient of absolute temperature. In the lower stratosphere, absolute temperature is more homogeneous with a lapse rate of 0.3 K km^{−1}. Negative values of radial wind are observed near the surface, indicating radial inflow (Fig. 2c). Magnitudes approach −10 m s^{−1} near the storm core and extend from the radius of maximum wind to approximately 400 km. Positive values, indicating radial outflow, are observed at a height of 12.5 km with an average magnitude of 4 m s^{−1}. In the outer environment, descending midlevel radial inflow is observed near a height of 5 km and from radii of 225–400 km. The average magnitude of this inflow is −2 m s^{−1}. A maximum in the latent heating tendency is observed in the troposphere (Fig. 2d); tendencies are highest at the radius of maximum wind with magnitudes near 11 K h^{−1}. These values decay rapidly with increasing radius and height, reaching zero at 75 and 8 km, respectively. Negative latent heating tendency is observed at a radius of 100 km and 4-km height and is as a result of the melting of ice particles near the freezing level (not shown). Negative latent heating tendency in the boundary layer demonstrates evaporation of liquid water into the subsaturated air.

## 3. Results

Results from time series analysis of the experiment are presented first, followed by results based on composite analysis of the diurnal cycle.

### a. Time series analysis

Figure 3a shows the time series in the azimuthal wind field for the equilibrium storm at a radius of 50 km and a height of 1.1 km, which is near the mean location of the radius of maximum wind. The time mean has been removed, and a high-pass filter has been applied to remove low frequencies corresponding to periods greater than 5 days, as this variability is not a subject of this study. The high-pass-filtered time series has a standard deviation of 2.3 m s^{−1} at this location, with peak amplitudes of the wind at individual times exceeding 7 m s^{−1}. Magnifying a 30-day portion of the high-pass-filtered time series shows that multiple high frequencies are present (Fig. 3b); however, examining the time series on days 19, 21, and 22, for example, suggests an underlying diurnal signal.

Figure 4 shows the power spectrum of the high-pass-filtered time series in the azimuthal wind at a radius of 102 km and height of 1.1 km, which is approximately double the distance of the radius of maximum wind. The peak of the first diurnal harmonic, corresponding to a period of 24 h, is the dominant peak of the spectrum. Statistical significance of the peaks is determined by comparing the power spectrum to a null hypothesis defined by a first-order autoregressive (AR-1) process, based on the 1-h autocorrelation. Frequencies where the power exceeds the 95% confidence bound on the AR-1 process, defined by a chi-squared test (Wilks 2005), are potentially significant. The first diurnal harmonic clearly meets this requirement. Peaks corresponding to periods of 48, 20, 12, and 8.5 h also meet this requirement; however, these peaks are within the expected uncertainty range for the 95%-confidence-bound threshold.^{2} All remaining frequencies are considered to be consistent with the red-noise null hypothesis.

Calculating the power spectrum and applying the null hypothesis to each grid point on the domain demonstrates the spatial distribution of the first diurnal harmonic in the azimuthal wind field (Fig. 5a). Two main regions are highlighted: the first region is in the upper troposphere at a radius of 58 km and a height of 13.5 km, at radii between 100 and 300 km; the second region is near the surface at a radius of 75 km and up to a height of 5 km, from 150 km in radius extending to 400 km. Weak significance is also indicated at radii between 100 and 200 km and at 11.5-km height and from radii between 150–400 km and at 19.5-km height. Everywhere else on the domain the diurnal signal is not statistically significant. Significant power in the absolute temperature (Fig. 5b) and the radial wind (Fig. 5c) fields also highlights the same two main regions, with maxima in the upper troposphere. A third region of importance in the midtroposphere is indicated by the latent heating tendency (Fig. 5d); this region extends from a height of 1.1 km to a height of 8 km and is narrow, spanning the eyewall region from approximately 40 to 90 km. Power in these highlighted regions suggests two things: 1) a coherent diurnal signal is detected at the top of the storm and 2) the diurnal cycle exhibits an interior storm response. The low-level signal that extends from the inner core of the storm (*r* < 150 km) to the outer environment (*r* > 300 km) in the azimuthal wind, temperature, and radial wind fields suggests the outer TC environment is also responding to the diurnal cycle of radiation.

Table 1 summarizes the fraction of the variance explained by the diurnal cycle for the azimuthal wind, temperature, radial wind, and latent heating tendency fields. A substantial amount of variance is explained in the TC outflow layer, where the diurnal cycle accounts for 62% of the overall variance in the temperature and 37% of the variance in the radial wind field. For the azimuthal wind, the diurnal cycle accounts for 36% of the variance in this region. In the boundary layer, the diurnal signal accounts for 28% of the variance in the temperature and 25% in the radial wind fields. The midlevel response is limited to the latent heating tendency, which explains 3% of the variance in this field. These results suggest that the diurnal cycle contributes most significantly to variability of the TC outflow layer, with impacts on the lower levels and the region outside the eyewall.

The variance explained by the diurnal cycle for the temperature, wind, and latent heating tendency fields for specific vertical levels in the storm. Data have been high-pass filtered to remove low-frequency variability. Percentages represent the maximum amount of variance explained in this region.

### b. Composite analysis

Anomalies from the time-mean equilibrium storm are composited at each hour to determine the evolving structure of the fields related to the diurnal cycle. The radial structure of the composite azimuthal wind field reveals a symmetric, horizontally broad signal near the surface (Fig. 6a). This field is a vertical average over the boundary layer, from the surface to 2-km height, and corresponds to the highlighted region at low levels previously discussed in (Fig. 5a). Starting at 0300 LT, a positive azimuthal wind anomaly is observed in the region near the eyewall. This positive anomaly has a maximum magnitude of 0.8 m s^{−1} at 1100 LT and is apparent mainly over 50–100-km radius. A weak negative anomaly is observed inside of the radius of maximum wind near 40 km, with average magnitude of 0.3 m s^{−1}. Beyond 100 km, the positive signal weakens, with an average magnitude of 0.3 m s^{−1}. This positive azimuthal wind anomaly persists mainly until 1500 LT, except for radii inward of 50 km, where a positive anomaly remains until 2100 LT. A negative azimuthal wind anomaly forms at radii of 50–350 km at 1500 LT, with a maximum value of 0.9 m s^{−1} at a radius of 54 km observed at 2100 LT. These signals are periodic and suggest a diurnal response in intensity is occurring at the radius of maximum wind. The broad, weak signal at large radii indicates a diurnal, nearly stationary, “pulsing” of the intensity of the storm in the far field, which implies an effective change in storm size.

The radial structure of the latent heating tendency suggests a diurnal cycle of anomalous convection (Fig. 6b). This field is a vertical average from the surface to 10 km. A positive anomaly is observed beginning at 2300 LT from 50 to 125 km in radius, which persists until the early morning. Maximum (minimum) values in latent heating tendency anomalies are 0.4 (−0.4) K h^{−1}. Positive latent heating tendency anomalies weaken substantially after 0900 LT, which coincides with the onset of solar heating. Negative anomalies then form and persist throughout the day. In real storms, diurnal pulses in the high clouds form in the inner core near sunset, propagate away from the storm center overnight, and reach radii near 300–400 km in the outer environment the following afternoon (Steranka et al. 1984; Dunion et al. 2014). A significant warming near cloud top is observed behind this propagating feature. These results are consistent with the timing and location of the diurnal pulses from Steranka et al. (1984) and Dunion et al. (2014), suggesting that convection is favored overnight, with local maxima occurring near the eyewall region and inhibited throughout the day. These results also suggest that convection in the lower troposphere is responding to the diurnal cycle.

The mean vertical structure of the TC diurnal cycle is depicted using the composite temperature field (Fig. 7a). A wavelike pattern is observed in both the lower troposphere and the stratosphere and is connected to anomalies that arise in the TC outflow layer. At a height of 12 km, positive temperature anomalies appear at 1100 LT and last until 2000 LT, which is consistent with warming by absorption of solar radiation. These anomalies slowly propagate downward through the troposphere and reach a maximum at the top of the boundary layer near 2 km approximately 6 h later. Beginning at 2200 LT, negative temperature anomalies are observed at a height of 12.5 km, consistent with cooling due to longwave emission from cloud top. Again, downward propagation of the signal is observed. Similar propagating disturbances are also observed in the stratosphere; beginning at 0500 LT, negative temperature anomalies propagate from a height of 18 km down to the TC outflow layer at 12.5-km height throughout the day. As will be discussed below, such phase propagation is consistent with upward energy propagation for inertia–gravity waves.

*x*–

*z*) gravity waves in a rotating, stably stratified atmosphere at rest is given bywhere

*ν*is the frequency,

*f*is the Coriolis parameter,

*N*is the Brunt–Väisälä frequency, and

*k*and

*m*are the zonal and vertical wavenumbers, respectively. Note that the meridional wavenumber

*l*is set to zero. For a TC, rapid rotation in the core of the storm requires consideration of the centrifugal force; however, far from the storm in the surrounding environment, the ratio of the centrifugal force to the Coriolis force is small, and the centrifugal term can be neglected. In this region, we can locally apply this dispersion relationship in a Cartesian framework to qualitatively estimate inertia–gravity wave features. Rearranging Eq. (1) for

*k*and recalling thatwhere

*τ*is the wave period. Estimating a propagation speed of 0.06 m s

^{−1}from the downward-propagating negative signal in the stratosphere of Fig. 7 and using the phase speed relationship

^{−4}s

^{−2}for

*N*

^{2}in the stratosphere and

*τ*= 24 h yields a horizontal wavelength of 2124 km and a phase speed

^{−1}. For the troposphere, taking a value of 1 × 10

^{−4}s

^{−2}for

*N*

^{2}yields a horizontal wavelength of 1062 km and a phase speed of 12.3 m s

^{−1}. These results are similar to estimates of diurnal propagating features in the high clouds of real storms, which have propagation speeds of 5–15 m s

^{−1}(Steranka et al. 1984; Dunion et al. 2014), and suggest the response in the lower stratosphere is twice as fast as the response in the lower troposphere.

Examining the radial wind field also shows downward-propagating features (Fig. 7b); these disturbances are observed from the lower stratosphere near 20-km height down to 10 km. Anomalous radial inflow is observed at the base of the TC outflow layer near 10 km from 1100 LT until 2200 LT, with magnitude of 0.8 m s^{−1}. Anomalous radial outflow is observed at the top of the outflow layer near 13.5 km, with magnitude near 1 m s^{−1}. Overnight, the signal changes, with anomalous radial inflow observed above the TC outflow layer and anomalous radial outflow observed below 12.5 km, with magnitudes of −1 and 0.8 m s^{−1}, respectively. Durran et al. (2009) have shown using numerical experiments that heating of cirrus clouds in the tropical tropopause layer can produce vertically propagating disturbances; these perturbations extend well beyond the region of heating as a result of gravity wave dynamics. Durran et al. (2009) also demonstrate that one characteristic of cirrus-generated gravity waves is that they drive radial inflow at the base of the heat source and radial outflow at the top, similar to the features described here. At the surface, anomalous radial outflow is observed from 1400 to 2000 LT, with magnitudes near 0.6 m s^{−1}. Radial inflow of similar magnitude is then observed overnight and persists through the morning. These features are in phase with the radial wind anomalies above the outflow layer and propagate upward from the surface to the midtroposphere.

The horizontal and vertical structure of the TC diurnal cycle suggests two distinct responses to radiation: 1) a radiatively driven response in the outflow layer and 2) a convective response (driven by latent heating) in the lower troposphere. We propose that each level of heating drives a circulation, and we proceed to discuss these qualitatively. Figure 8 shows the composite mean radial–vertical wind vectors as well as the net radiative tendency for both 1500 and 0300 LT. In the afternoon, radiative heating is observed at a height of 12 km (Fig. 8a), which is associated with both anomalous upward motion in the region of heating as well as anomalous radial outflow near this level. Peak amplitudes of anomalous vertical motion approach 5 cm s^{−1} near the center of the heating, with anomalous radial outflow on the order of 1 m s^{−1} at the leading edge of the net radiative response. Below this heating, anomalous radial inflow on the order of 0.8 m s^{−1} is observed. This effect is local and only affects the winds in close proximity to the region. In the early morning, the opposite circulation is observed (Fig. 8b), which we interpret as radiative cooling driving weak anomalous inflow into the TC outflow layer near a height of 12.5 km and subsidence through this region. Peak amplitudes of anomalous downward motion of 1 cm s^{−1} near the center of cooling are observed, which is slightly weaker than the daytime response. A layer of anomalous radial outflow on the order of 1 m s^{−1} is also present beneath this layer at 10-km height, which also suggests a local circulation at this level.

Figure 9 shows the same wind vectors along with the anomalous latent heating tendency. During the day, subsidence is observed from 50 to 400 km, consistent with heating at the top of the layer from absorption of solar radiation. Minimum values of latent heating tendency anomalies are −0.6 K h^{−1} near the storm core and coincide with downward motion on the order of 5 cm s^{−1}. Anomalous radial outflow is present in the boundary layer from a radius of 50 km to a radius of 400 km with magnitudes near 1 m s^{−1}. At night, convection drives an overturning response throughout the troposphere (Fig. 9a). Maximum values of latent heating tendency anomalies approach 1.2 K h^{−1} at a radius of 50 km. Anomalous upward motion is observed through the region of heating for radii of 75–150 km and from the top of the boundary layer to a height of 10 km. This anomalous upward motion has a magnitude that exceeds 5 cm s^{−1} and is associated with a deep layer of anomalous radial inflow on the order of 0.8 m s^{−1} that extends to outer radii near 400 km. Anomalous radial outflow of magnitude 0.6 m s^{−1} is observed for radii of 100–400 km and from the top of the boundary layer near 4-km height. These results suggest a wind response in the low levels, linked to anomalous latent heating, that is a maximum near the core of the storm.

## 4. Summary and conclusions

The goal of this work is to quantify a diurnal signal for TCs and investigate its relationship to storm structure and intensity. A numerical experiment is conducted in an axisymmetric framework with minimal environmental influence to explicitly identify the internal response of the storm to the diurnal cycle of radiation. Time series analysis reveals a coherent TC diurnal signal that dominates the variance in the outflow layer and accounts for 62% of the variance in temperature in this region. At low levels, the diurnal cycle accounts for up to 25% of the variance in the radial wind field and 21% of the variance in the azimuthal wind field. A weak diurnal signal is also present in the midtroposphere, accounting for 3% of the variance in the latent heating field. These features are all statistically significant at the 95% confidence level.

Composite analysis shows a complex response in the storm structure linked to anomalous latent heating in the boundary layer and to absorption of radiation in the TC outflow layer. In the boundary layer, a periodic overturning circulation drives anomalous upward motion in the region near the radius of maximum wind; this feature is associated with a deep layer of anomalous radial inflow. This anomalous inflow drives larger values of angular momentum from the storm environment into the inner core of the storm, which, by conservation principles, must result in an increase in tangential winds. This is consistent with the positive azimuthal wind response observed in the early morning that lags the positive response in the latent heating field by approximately 5 h. This suggests that the TC diurnal cycle favors storm intensification in the early hours of the morning and storm weakening in the late afternoon and evening.

In the outflow layer, observed disturbances are downward- and outward-propagating features; in the stratosphere, these disturbances have estimated vertical wavelengths of 5.6 km, and horizontal phase speeds are 24.6 m s^{−1}. In the troposphere, phase speeds are 9.8 m s^{−1}. These phase speed are similar to speeds documented by both Steranka et al. (1984) and Dunion et al. (2014), who estimate values near 5–10 m s^{−1} for real storms. Comparing these results to perturbations introduced in a rotating, stably stratified atmosphere suggests correspondence to an inertia–gravity wave response. One characteristic of inertia–gravity waves is that the phase speed and group velocity are orthogonal; in this case, downward-propagating phase implies upward-propagating group velocity, and thus wave energy propagates out and away from the storm. It is unclear at this time what impact, if any, these waves have on the overall storm structure. The presence of gravity waves in TCs is an active area of research and is the subject of future study.

A hypothesis is drawn from these results that the TC diurnal cycle is a combined response from two periodic heat sources: one in the TC outflow due explicitly to radiation, and one in the boundary layer due to latent heating, which derives indirectly from radiative cooling in the upper troposphere. These heat sources each correspond to a local circulation that drives an anomalous tangential wind response and affects storm intensity. The present findings are qualitative and will be tested quantitatively in a follow-up study exploring the effects of periodic heating in an idealized vortex.

One additional result from this work is the presence of a semidiurnal signal in the power spectrum. A semidiurnal signal has been observed near the convective region in the inner core of real storms and has been attributed to a modulation of lapse rates by the atmospheric semidiurnal tide *S*_{2} (Kossin 2002). This model contains a damping layer in the stratosphere, which inhibits the effect of an atmospheric tide. We are thus not capable of evaluating the influence of *S*_{2} in this model. We suggest a further evaluation of the semidiurnal signal in a numerical simulation, as well as a testing of the hypothesis, as possible avenues of future work.

The present study is limited to the configuration of the model, which only includes the NASA Goddard longwave and shortwave radiation schemes. Future work should evaluate the robustness of these results to changes in radiative parameterization, as well as the sensitivity to the chosen microphysics scheme. Features such as the amount of pristine ice in the upper troposphere may quantitatively affect the results, as this would alter the radiative tendencies, which may affect the strength of the hypothesized circulations. In addition, three-dimensional analysis is required to evaluate the role of asymmetries, as cloud fields may be overestimated in the axisymmetric framework. However, given the qualitative agreement of the results presented here and observations, we believe such changes may not significantly affect the results.

The authors thank Dr. Dale Durran and Dr. Angel F. Adames-Correleza for conversations related to this research. Comments from three anonymous reviewers also brought significant insight and improvements to the manuscript. This research was supported by the Hurricane Forecast Improvement Project (HFIP) through Award NA14NWS4680031 made to the University of Washington. This work was part of the first author’s doctoral thesis at the University of Washington.

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^{1}

In real storms, this angular momentum source is supplied by lateral mixing with the storm environment.

^{2}

The present study only considers the diurnal signal (i.e., the first diurnal harmonic). Further investigation of the potential significance of these peaks is beyond the scope of this work.