1. Introduction
Over the past few decades there have been notable improvements in forecasting the tracks of tropical cyclones (TCs) but not in their intensity (e.g., Rogers et al. 2006, 2013; DeMaria and Mainelli 2005). A recent review paper from Montgomery and Smith (2014) highlighted many of the existing theories on intensity change including conditional instability of the second kind (CISK) (Charney and Eliassen 1964), wind-induced surface heat exchange (WISHE) (Emanuel 1986; Rotunno and Emanuel 1987), the theory of cooperative intensification (Ooyama 1982), and the role of rotating deep convection (e.g., Montgomery et al. 2006; Van Sang et al. 2008; Smith et al. 2009; Montgomery and Smith 2014). Of the four paradigms, the first three are axisymmetric mechanisms and the fourth allows for asymmetric components. However, none of these theories have been widely accepted and the understanding of intensity changes remains an important problem in tropical cyclone research.
Recently, another hypothesis, the so-called efficiency approach, has received some attention. The basic idea is that a TC derives its major source of energy from the release of latent heat in the eyewall region owing to the condensation of water vapor. Part of this latent heat is converted into potential energy to keep the vortex in hydrostatic and thermal wind balance. The other part is converted into kinetic energy that is manifested in the acceleration of the primary circulation (e.g., Nolan et al. 2007). Using balanced models, Vigh and Schubert (2009) and Pendergrass and Willoughby (2009) showed that the response of the vortex to diabatic heating depends on the radial location of the heating relative to the radius of maximum wind (RMW) and on the structure of the vortex itself. These characteristics determine the efficiency that the vortex converts heat energy into kinetic energy (Schubert and Hack 1982; Hack and Schubert 1986)—the higher the inertial stability, the stronger the resistance.
Although this is a feasible idea, recently it has been used without justification of its inherent assumptions. Specifically, the inference of a one-to-one relationship between inertial stability and efficiency implies a strict balanced regime. When this aspect is overlooked, the wrong conclusion may result in relating the location of the heating with respect to inertial stability to intensity change.
For example, many studies often use the efficiency framework to interpret observations (e.g., Rogers 2010; Rogers et al. 2015; Xu and Wang 2015) and model simulations (e.g., Rozoff et al. 2012; Kanada and Wada 2015). In particular, Xu and Wang (2015) used various datasets over the North Atlantic to study 24-h TC intensification rates. They found a positive correlation between the subsequent 24-h rate of intensification and the initial 10-m tangential wind speed for vortices of sustained winds less than 70–80 knots (1 kt = 0.51 m s−1; see their Fig. 1a). Their interpretation was “The initial increase in IR [intensification rate] with the storm intensity is due to the increasing efficiency of the intensification to eyewall heating because of the increasing inner-core inertial stability” (p. 699). However, caution must be exercised in applying the efficiency hypothesis as the inherent assumption of balance was not demonstrated in their study. As will be shown here, other processes can be invoked to explain the same results.
Additionally, Musgrave et al. (2012) investigated how the width of a “vorticity skirt”1 influences the lower-level intensification rate for a specified heat source. Using a gradient wind balanced model, they showed that as the eyewall-like heating was brought radially inward into the vorticity skirt region, the magnitude of the tangential wind increased. They explained that the heating located inside the highest inertially stable region leads to a larger intensification using the efficiency approach of Hack and Schubert (1986). However, it should be noted that the total amount of heat energy injected into their system was constrained to remain constant in all of their experiments. Thus the maximum heating rate increased as the heat source shifts radially inward. In this regard, it is unclear whether the acceleration of the tangential winds is due to the nature of the specific vorticity skirts or due to the increase in the amplitude of the maximum heating.
An alternate hypothesis, which will be called the dynamic approach, of how inertial stability influences intensification was discussed in a recent review paper from Smith and Montgomery (2016). They show that an imposed annular heat source will generate a meridional overturning circulation on either side of its core (refer to their Fig. 1a). If the source is located outside the radius of maximum wind (i.e., outside the higher-vorticity region), then the low-level flow associated with the inner branch of this circulation will transfer angular momentum surfaces radially outward. On the other hand, if the source is located inside the region of enhanced vorticity, the low-level flow associated with the outer branch of the meridional circulation will move these surfaces radially inward. This will eventually spin up the vortex and contribute to the contraction of the radius of maximum wind. More generally, a direct relationship between enhanced vorticity and inertial stability is inherent in the definition [refer to Eq. (1)]. A region of enhanced I implies a region of larger ζ. Therefore, for a heat source inside of a larger inertially stable region, there will be an enhanced inward radial transport of vorticity leading to an intensification.
From the above discussion, it is clear that the question of why TCs become more intense when the heating is located inside the RMW remains unanswered. Is it because higher inertial stability reduces adiabatic cooling to increase the heating efficiency? Or, is it because higher inertial stability results in larger transfer of vorticity? The idea of increased transfer of relative vorticity was discussed in Montgomery and Smith (2014), which provided a simpler explanation for the increase in spinup rate when the diabatic heating is located inside the RMW. However, as was discussed above, it is clear that many papers do not use this mechanism to explain the acceleration of the tangential winds of a TC. Additionally, there has never been a direct comparison of the spinup due to the heating efficiency versus simply the inward transport of relative vorticity. It is the purpose of this study to shed light on which of the two above mechanisms may be dominant.
The rest of the paper is organized as follows. Section 2 describes the specifications of the runs performed using a 3D model. To compare the efficiency and dynamic frameworks in a manner similar to the original papers, the Sawyer–Eliassen equation (SEQ) model was run during a specified time of the simulation. The diagnosed vertical and horizontal winds from the 2D model were then used to calculate the tangential wind acceleration according to the two approaches. These calculations are compared with one another and then with the full 3D simulation in sections 3 and 4. Future work is discussed in section 5 and the final conclusions are presented in section 6.
2. Experimental design and model details
a. Three-dimensional model setup and specifications
The numerical simulations are conducted with the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2005). WRF is a conventional fully compressible nonhydrostatic primitive-equation atmospheric simulation system suitable for use in a broad range of applications. For this particular study, the model is cast in an idealized configuration in that all model physics such as cloud microphysics, cumulus parameterization, atmospheric radiation, surface processes, and planetary boundary layer processes are switched off.
The fields evolve on an f-plane computational domain. The specified Coriolis parameter f is 5 × 10−5 s−1, which is representative of 20°N. The horizontal domain spans 600 km in each orthogonal direction and the corresponding grid spacing is 2 km. The vertical grid has 28 sigma levels and extends upward to z = 25 km, with z denoting the height above sea level. A traditional Rayleigh damping layer is imposed above z = 20 km to prevent unphysical wave reflection from the upper boundary. The lateral boundary conditions in both directions are periodic.
Figure 1b depicts the time evolution of the maximum heating rate with 
(a) Normalized azimuthally averaged tangential winds (m s−1) at time t = 0. (b) Evolution of the heating maximum (K h−1) vs time (min) of the simulation. The maximum heating increases linearly for the first hour and remains at 10 K h−1 thereafter. (c) Horizontal cross section of the heating function (K h−1) at z = 5 km after it has reached its maximum value. (d) A vertical cross section through the center of the domain depicting the vertical structure of the heating function (K h−1).
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
In total, nine simulations are performed. They are all initialized with a basic-state vortex with the same spatial structure [
To avoid contamination from any transient gravity wave activity, our diagnostic results will be presented 3 h after the initial heat injection. To compare the two hypotheses discussed in the previous section, selected output fields were input into a balanced 2D model. The initialization and specifications of the balanced model will be explained in the following section.
b. Two-dimensional model setup
In this study, the terms required to calculate the coefficients on the left-hand side of Eq. (4) were obtained from the WRF output at a time interval of 5 min. Specifically, the 
c. Two ways of calculating the tangential wind acceleration from a balanced model
1) Thermodynamic efficiency hypothesis
2) Surface dynamic hypothesis
To shed light on which of these two hypotheses better account for the change in intensity of a tropical cyclone, we compare and contrast the results from the two hypotheses particularly with regard to the magnitudes and spatial structures of the terms. Any discrepancies will be discussed and then compared to the full 3D simulation. The following section concentrates on the results from the balanced SEQ approach unless otherwise specified.
3. Comparing the balanced acceleration
To begin, we present the response of the secondary circulations to a constant heat source in experiments with increasing inertial stability. Figure 2 shows only four of the possible nine vortices. The first and last columns represent the weakest and strongest vortices, respectively, whereas the third column was chosen as the control case (υmax = 40 m s−1) and the second column (υmax = 30 m s−1) was chosen to show a more gradual increase in inertial stabilities. From the definition of inertial stability [refer to Eq. (1)] it can be seen that as the vortices intensify,3 the heating (contoured) is located in regions of increasing inertial stability.
Sensitivity to different inertial stabilities for (left)–(right) υmax = 15, 30, 40 (the control run), and 55 m s−1. (top)–(bottom) The initial inertial stability (s−2), with the location of the heating contoured, the balanced response of the radial winds (m s−1), and the balanced response of the vertical winds (m s−1) diagnosed from the SEQ. The numbers in the panels correspond to the (middle) minimum induced radial winds (m s−1) and (bottom) maximum induced vertical winds (m s−1).
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
A weakening of the secondary circulation occurs in response to an increase in the inertial stability. This feature can be found in the induced radial winds (middle row) where the inflow at the lowest level of the weakest vortex is approximately −1 m s−1 and that of the strongest vortex is −0.5 m s−1. This finding is consistent with previous studies that showed that an increase in inertial stability results in a reduction in the radial motion and, through mass continuity, to a weakening of the vertical wind (e.g., Shapiro and Willoughby 1982; Holland and Merrill 1984). As the results for the nine vortices are qualitatively similar, we focus on a discussion of the control case below.
Figure 3 depicts the accelerations calculated from the surface and efficiency hypotheses in the control case. Both frameworks have similar spatial patterns with a positive (red) and negative (blue) dipole in the acceleration at both the lower levels and around 8 km. In these idealized experiments, a heating bubble will create an overturning circulation both in the inner-core and outer-core regions of the vortex (refer to the induced secondary circulation in the second and third rows of Fig. 2). In the surface dynamical framework, this induction of inner-core winds pointing radially outward, more specifically in the region from approximately r = 10 to 20 km and z = 0 to 4 km, will result in an outward advection of relative vorticity, as can be seen from the negative tangential wind tendency around 20 km. On the other hand, the outer-core winds point radially inward, causing a positive tangential acceleration around 30 km. In the efficiency framework, the acceleration at the surface inside the region of increasing inertial stability is dominated by the warming in the midatmosphere. Specifically, the change in sign at the location between approximately 15 and 20 km at the surface coincides with the change in sign of ∂θ/∂r in the midatmosphere (not shown).
The tangential wind accelerations (m s−2) calculated from the control run averaged from t = 3 h to 3 h 20 min for (a) the surface dynamic hypothesis from the SEQ and (b) the efficiency hypothesis from the SEQ. The red colors represent an acceleration of the tangential winds and the blue colors denote a deceleration. An outline of the heating function is shown by the dotted black lines.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
The dipoles in acceleration in the upper levels can be understood by the fact that the case is a barotropic vortex with a nontilted heat source. Thus the outflow at the top of the heat source in combination with the lack of baroclinicity creates a strong double gyre on either side of the heat source. When baroclinicity is present, it has been shown that the inner gyre is significantly reduced (Pendergrass and Willoughby 2009).
Although these two hypotheses produce some general similarities, there are notable differences such as the location and strength of the maximum accelerations. For example, at around approximately 23 km the two frameworks in fact disagree as to whether the accelerations are positive or negative. If these two processes are thought to be independent, then one question that may arise is would it be possible to have a heat source structure or location where the two approaches are positively correlated? If so, could this create a stronger storm for a given initial inertial stability and maximum value of heating? Simple sensitivity experiments (not shown here) were conducted by moving only the radial location of the heat source. The results were similar to what is seen in Fig. 3. That is the efficiency framework is offset by the surface dynamical framework by approximately 5 km. Thus a simple radial shift in the heat source is not capable of aligning the accelerations such that they are in phase and can amplify each other. Possible tangential acceleration responses to changes in the structure of the heat source will be further discussed in section 5.
a. Comparison of the SEQ calculations to the WRF run
Since the accelerations calculated from the two hypotheses disagree in some locations, it is desirable to compare with the WRF simulation.
Figure 4 depicts the tangential acceleration from the WRF Model (Fig. 4a), the acceleration from the SEQ for the surface dynamic hypothesis (Fig. 4c), and the efficiency hypothesis calculation from the SEQ (Fig. 4d) for the control vortex. Qualitatively, the WRF Model indicates dipoles in acceleration at the lower and upper levels, similar to the SEQ calculations. However, the low-level maximum acceleration in WRF occurs between the radii of about 25–35 km and is in better agreement with the surface dynamic hypothesis than the efficiency hypothesis that places the maximum between 20 and 27 km. In terms of the magnitude of the acceleration, the maximum from both the efficiency and surface dynamic hypotheses are on the same order of magnitude as the WRF values. On the other hand, the vertical extent of the positive acceleration in the low levels shows better agreement between WRF and the efficiency hypothesis.
The tangential accelerations (m s−2) calculated from the control run averaged from t = 3 h to 3 h 20 min from (a) the WRF Model, (b) the complete dynamic approach from the SEQ, (c) the surface dynamic approach from the SEQ, and (d) the efficiency approach from the SEQ. The red colors correspond to an acceleration of the tangential winds and the blue colors to a deceleration.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
The calculation for the surface dynamic hypothesis only takes into account the horizontal advection of absolute vorticity, while the vertical advection of tangential momentum [the second term on the rhs of Eq. (13)] is not taken into consideration. We calculated this term and added it to the acceleration obtained from the surface dynamic hypothesis to yield the acceleration in the so-called complete dynamic framework (Fig. 4b). Since the vertical advection of tangential momentum has little influence at the lower levels, the low-level acceleration is very similar to the surface dynamic hypothesis (Fig. 4c). However, the tangential acceleration in the complete dynamic framework extends farther into the midatmosphere on account of the addition of the vertical advection of momentum. Thus the complete dynamic framework reproduces better the acceleration from the WRF output both in terms of the magnitude and the spatial extent than the efficiency framework.
All of the above results were calculated under the assumption of strictly balanced winds. The limitations of this assumption will be discussed in the following section.
b. Balanced and unbalanced contributions
Although the complete dynamic framework appears to better capture the dynamics of the WRF simulation some differences still remain. Specifically, the magnitude of the surface tangential wind acceleration is stronger in the complete dynamic framework than the WRF output (Figs. 4a,b). One major difference between the calculations using the SEQ and the WRF output is the assumption of balance. The 2D SEQ model is under the constraint that the diagnosed vertical and radial winds from the streamfunction must be in strict thermal and gradient wind balance. On the other hand, the 3D WRF Model has no such constraint. To study this further the direct output from the WRF Model [i.e., the lhs of Eq. (13)] is compared first to the sum of the terms from the rhs of Eq. (13) for the SEQ (balanced) and then to the sum of the terms on the rhs of Eq. (13) for the WRF (no constraint of balance) model. It should be reiterated that the background fields (such as ζ and 
Figures 5a and 5b show the differences between the WRF acceleration and the rhs of Eq. (13) calculated with the SEQ and with the WRF output, respectively. Because of the time truncation error there is some residual between the lhs and rhs of Eq. (13) (Fig. 5b). It can be noted that the rhs calculated from the balanced response significantly underestimates the WRF acceleration especially in the region between 5 and 8 km in height at a radius of approximately 30–35 km (Fig. 5a). The difference between the WRF acceleration and the terms calculated from the SEQ secondary circulation is on the same order of magnitude as the acceleration itself. The implication is that in our simulation results the response due to the heat source is not well captured by the balanced linear dynamics. This is in contrast to some previous studies (e.g., Stern et al. 2015; Heng and Wang 2016) but is consistent with others (e.g., Bui et al. 2009; Abarca and Montgomery 2015). For this reason, the next section will focus on the changes in the tangential wind acceleration with inertial stability using the fully nonlinear WRF run.
The difference, from the control run, between the WRF acceleration and the sum of the rhs terms using the radial and vertical winds diagnosed from (a) the SEQ model and (b) the WRF Model. The red colors correspond to the lhs of Eq. (13) being stronger and the blue colors to the addition of the terms on the rhs dominating.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
4. Investigation of the effect of inertial stability using the WRF simulations
Before calculating the contributions to the wind budget to diagnose the relationship between the spinup of the lower-level tangential winds and the inertial stability, we wish to justify our use of the azimuthal-mean operator defined by Eq. (13) in section 2. It is not obvious that discussing the dynamics in an azimuthal-mean sense is justifiable while using a model with both symmetric and asymmetric components. Figure 6 depicts the hourly potential vorticity (PV) of the control case at a height of 4 km after the heat source has reached its maximum amplitude. It can be seen that the stable response of PV to the given heat source is highly symmetric. Similar results are found at other levels, thus justifying the use of the azimuthal mean in displaying our results.
Horizontal cross section of potential vorticity (PV units; 1 PVU = 10−6 K kg−1 m2 s−1) from the WRF output for the control run at t = (a) 2, (b) 3, (c) 4, and (d) 5 h at a height of 4 km.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
a. Contributions from the tangential wind budget
We calculated the tangential wind budget for four vortices: υmax = 15, 30, 40, and 55 m s−1. To compare the contributions from the radial advection of absolute vorticity and vertical advection of tangential momentum, we display in the first two columns of Fig. 7 the actual tangential wind acceleration from the WRF Model and the sum of the terms on the rhs of Eq. (13). It can be seen that the tangential wind budget from Eq. (13) accurately captures both the strength and spatial structure of the WRF output. Some differences exist as a result of the time truncation error of using 5-min output data.
Radius vs height plots that represent (left)–(right) the direct output of the tangential wind acceleration from the WRF data, the addition of the terms on the rhs of Eq. (13), the contribution from the radial advection of absolute vorticity, and the vertical advection of tangential momentum. (top)–(bottom) The same four vortices that were used in Fig. 2: υmax = 15, 30, 40 (the control run), and 55 m s−1. The red colors correspond to an acceleration of the vortices while the blue colors represent a deceleration of the tangential winds.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
The breakdown of the contributions from the radial advection of absolute vorticity and the vertical advection of tangential momentum is shown in the last two columns of Fig. 7. Below approximately 5 km the main contribution to the acceleration comes from the radial advection of absolute vorticity in all four cases. In addition, the contribution in the midatmosphere from the vertical advection of tangential acceleration is as strong as that of the lower levels by the third hour. However, at the location where the vertical advection of tangential momentum is strongest, the radial advection of absolute vorticity is negative and counteracts part of the tangential wind acceleration resulting in weaker total accelerations in the midatmosphere as compared to the surface.
b. Normalized lower-level response to increased inertial stability
Figure 8 shows the relationship between normalized inertial stability and the normalized lower-level tangential winds at the third hour. The normalization is done with respect to the control case at the location where the tangential wind acceleration is maximized. The normalization of the inertial stability is calculated at the start of the simulation and the tangential acceleration is calculated 3 h into the simulations. It can be seen that when the vortex weakens and the inertial stability decreases, there is a reduction in the acceleration of the tangential winds. More specifically, the magnitude of the acceleration of the weakest storm almost halved compared to the control. Additionally, the strongest initial vortex was almost 1.2 times that of the control case. To gain better insight into which term is responsible for the larger acceleration of the tangential winds, a further breakdown of Eq. (13) is performed.
Normalized inertial stability vs tangential acceleration from the WRF simulations for the nine vortices. The point 
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
Figure 9a demonstrates the influence of an increase in inertial stability on the radial flow at the lowest model level for all nine vortices. It is evident that as the initial vortex is strengthened there is a continual decrease in the magnitude of the radial winds. As expected, because of the nature of the stationary heat source, the changes in inertial stability do not alter the location of the maximum radial inflow for the different vortices which is always around r = 36 km. The other panels of Fig. 9 are all calculated in a similar manner to Fig. 8 but illustrate the normalized inertial stability versus the normalized radial winds (Fig. 9b), the normalized relative vorticity (Fig. 9c), and the normalized relative vorticity advection (Fig. 9d). This figure demonstrates that although the increase in inertial stability reduces the radial winds it also increases the relative vorticity. When comparing Figs. 9b and 9c with the normalized relative vorticity advection, it can be seen that this decrease in radial winds is more than compensated by the increase in relative vorticity, which in consequence leads to higher values of the advection of absolute vorticity. Through the aid of this figure, as well as the spatial structures shown in Fig. 7, it can be concluded that, for our simulations, when the heating is located inside regions of enhanced inertial stability there is an increase in radial absolute vorticity advection, which acts to accelerate the low-level tangential winds.
(a) Surface radial winds for all the vortices at t = 3 h. (b)–(d) As in Fig. 8, but for (b) radial winds vs I2, (c) relative vorticity vs I2, and (d) advection of relative vorticity vs I2. The normalization is carried out as in Fig. 8.
Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0298.1
5. Discussion
It was shown (refer to Fig. 4) that the tangential wind accelerations from both the surface dynamic and efficiency frameworks were on the same order of magnitude but radially slightly offset. The surface dynamic hypothesis was a simple calculation from the first term on the rhs of Eq. (13) and has its maximum dominated by the location of the strongest inward radial winds. The relationship between the location of the strongest inward radial winds and the maximum tangential wind acceleration can be seen by comparing the middle rows of Figs. 2 and 4a. On the other hand, the efficiency framework is dominated by the local changes in θ. The shift in the location of the maximum acceleration was explained by Schubert and Hack (1982) as being due to the gradient of the inertial stability in the region of the heating. Specifically, the fluid particles that are located in regions of higher inertial stability would experience a larger acceleration, which would result in the appearance of an inward shift of the RMW such that it coincides with the inner edge of the heating (Schubert and Hack 1982). This contraction mechanism was also discussed in some of the earlier papers such as Shapiro and Willoughby (1982) and Willoughby et al. (1982). This tendency for the acceleration to align itself with the inner edge of the heating can be seen in Fig. 3b, which shows the efficiency hypothesis with the original heating contoured.
The thermal efficiency approach, which was first brought forward by Schubert and Hack (1982), compared the amount of injected heat Q to the local warming of the column 
The assumption that the local potential temperature tendency is only a function of the source/sink terms and the vertical advection [refer to Eq. (8)] is unrealistic for a simulation with enhanced baroclinicity or azimuthal asymmetries. In the case of the WRF simulations presented here, the last term on the rhs of Eq. (7) was calculated and was found to be about one-fifth of the total potential temperature tendency. Although the contribution of this term in our specific case may not be that significant, it may become nonnegligible for a different vortex. In addition, a simulation that is not completely symmetric would have contributions from the eddy terms. Thus for vortices with enhanced complexity, the efficiency approach introduced by Schubert and Hack (1982) may not be valid since there is no longer a one-to-one relationship between heating and adiabatic cooling to the local potential temperature change. In addition, the balanced response of the SEQ to the heating was found to be inaccurate when it is compared to the induced secondary circulation from the WRF simulation. Specifically, the diagnosed vertical winds from the WRF Model were approximately 50% stronger than those calculated from the balanced model (not shown). This implies that the balanced approach may not be suitable for capturing the dynamics that occur in a more complex simulation.
One main purpose of this study was to understand the response of a heat source to varying background inertial stabilities. However, it should be noted that only one possible configuration was studied. Changes in the basic-state vortex, such as the addition of baroclinicity, as well as changes to some of the parameters in the heating function [Eq. (3)] could change these results. It may be that the efficiency and dynamical hypotheses can in fact positively influence each other such that the total acceleration is stronger.
A recent review paper by Smith and Montgomery (2016) considered the assumption of a fixed heating rate. It was argued that, through the connection of vertical velocity and the latent heating of condensation, a reduction in the secondary circulation from an enhancement of inertial stability would also reduce the diabatic heating rate. Some words of caution were “For these reasons alone, we would argue that the gain in efficiency resulting from holding the magnitude and spatial structure of the heating rate fixed should not be applied to interpret the behaviour of real or model storms” (p. 2084). This is one inherent limitation of the present study and caution must be exercised when applying a similar methodology to both longer-duration storms and more complicated physics simulations.
It was shown in Figs. 7 and 9 that in these idealized tropical cyclone simulations the leading term for the intensification at the lower levels of a storm is the radial advection of absolute vorticity. However, it can also be seen that above the lower levels the vertical advection of tangential momentum becomes important. This term is connected to the lower-level advection of relative vorticity in that as the tangential winds spin up and the shear increases, added momentum is transported vertically allowing for a spinup of the vortex at higher altitudes. Because of the simplicity of some of the models used in previous studies, (e.g., Schubert and Hack 1982; Musgrave et al. 2012), this term was not accounted for. However, even as early as 3 h it is evident that it can be as important as the horizontal advection of relative vorticity in spinning up the vortex.
It is clear from Eq. (1) that there are three parameters that affect the inertial stability: the Coriolis parameter f, the radius r, and the strength of the winds υ. In the previous sensitivity tests the tangential winds was the only variable that was altered in order to adjust the basic-state inertial stability. A variety of tangential velocities ranging from 15 to 55 m s−1 every 5 m s−1 was tested as was explained in section 2. Other sensitivity tests were performed by changing the Coriolis parameter. It was found (not shown) that this parameter only became important in altering the induced circulations for unrealistic values of f not associated with a tropical cyclone. The sensitivity of the RMW has not been thoroughly studied. Vigh and Schubert (2009) performed some sensitivity experiments on the RMW using two different values such that for one case the heating was inside the RMW and for the other it was outside. However, to study a range of different inertial instabilities by keeping the tangential wind speed and Coriolis parameter constant while only changing the RMW can lead to a different set of challenges. Increasing the RMW while keeping the heating function in a fixed location does not lend itself to a realistic simulation as, in most storms, the locations of the heating and RMW are relatively close to one another. However, moving the heat source radially inward (outward) would increase (decrease) the total amount of heating injected into the system. To compensate for this, a reduction in the magnitude of the heating would be required [as was done in Musgrave et al. (2012)], although a relationship between the inertial stability and maximum induced heating would not be as clear.
It should be stated that the purpose of this study was not to confirm or reject any of the previous tropical cyclone hypotheses as discussed in the recent review paper from Montgomery and Smith (2017). Rather, it was to clarify whether the efficiency hypothesis first brought forward from Schubert and Hack (1982) can be used in the general context of tropical cyclone intensification for more complicated simulations. Further studies need to be conducted to address the problem of genesis and intensification.
6. Conclusions
It was found that for both the balanced Sawyer–Eliassen equation and the 3D WRF Model for a constant heat source, an increase in the background inertial stability resulted in a reduction of the secondary circulation. A comparison of the surface dynamic versus the efficiency hypotheses using the 2D balanced equation model demonstrated that the dynamic hypothesis lends agreement with the WRF simulation at the lower levels, as was described in Smith and Montgomery (2016). Furthermore, the assumption of thermal and gradient wind balance, in addition to neglecting some of the terms in the thermodynamic tendency equation, may not be valid for realistic tropical cyclones. For these reasons, it is believed that the simpler surface dynamic hypothesis may offer a more suitable explanation as to why the lower-level tangential wind acceleration increases when a heat source is placed in a region of enhanced inertial stability.
Using the 3D WRF Model, a further breakdown of the azimuthal wind budget was performed. It was shown that although the radial winds decreased with increasing inertial stability, the larger values of relative vorticity more than offset this change resulting in a net increase in the advection of relative vorticity. The radial advection of relative vorticity was then shown to be the dominating term in the spinup of the lower-level winds. In addition, it was demonstrated that after 3 h the vertical advection of tangential momentum was the dominating spinup mechanism in the higher levels.
To extend the findings of this paper to more realistic tropical cyclones, more complexity needs to be added to the simulations. Some key missing features include the effects of the planetary boundary layer as well as baroclinicity. Future work will concentrate on some of these areas.
Acknowledgments
The research reported here is supported by the NSERC/Hydro-Quebec Industrial Research Chair Program Grant IRCPJ/381215-14. The authors would like to also thank Zhenduo Zhu with the assistance of the SEQ model in addition to three anonymous reviewers for their thorough and helpful comments.
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In their case a vorticity skirt refers to the smooth transition between the inner core (constant relative vorticity) and far field (zero relative vorticity). Refer to their Fig. 12c for a schematic.
It should be stated that because of the lack of a planetary boundary layer this calculation is not actually occurring at the surface. This calculation mimics the conventional spinup mechanism described in Smith and Montgomery (2016), which describes the acceleration through the lower-tropospheric advection of AAM.
For the rest of this paper, this refers to the increase in the initial υmax.
