Abstract

The role of asymmetric convection to the intensity change of a weak vortex is investigated with the aid of a “dry” thermally forced model. Numerical experiments are conducted, starting with a weak vortex forced by a localized thermal anomaly. The concept of wave activity, the Eliassen–Palm flux, and eddy kinetic energy are then applied to identify the nature of the dominant generated waves and to diagnose their kinematics, structure, and impact on the primary vortex. The physical reasons for which disagreements with previous studies exist are also investigated utilizing the governing equation for potential vorticity (PV) perturbations and a number of sensitivity experiments.

From the control experiment, it is found that the response of the vortex is dominated by the radiation of a damped sheared vortex Rossby wave (VRW) that acts to accelerate the symmetric flow through the transport of angular momentum. An increase of the kinetic energy of the symmetric flow by the VRW is shown also from the eddy kinetic energy budget. Additional tests performed on the structure and the magnitude of the initial thermal forcing confirm the robustness of the results and emphasize the significance of the wave–mean flow interaction to the intensification process.

From the sensitivity experiments, it is found that for a localized thermal anomaly, regardless of the baroclinicity of the vortex and the radial and vertical gradients of the thermal forcing, the resultant PV perturbation follows a damping behavior, thus suggesting that deceleration of the vortex should not be expected.

1. Introduction

It is generally accepted that the ubiquitous convective asymmetric features that accompany the symmetric hurricane circulation play a key role in modulating the intensity and the structure of the hurricane vortex. However, the exact manner in which the asymmetries influence the hurricane (e.g., intensifying or weakening the vortex) and their underlying dynamics is not fully understood. It is therefore central to obtain a better understanding of these processes.

Utilizing the generalized tendency of two-dimensional (2D) asymmetric vortex flows to return back to the symmetric state (axisymmetrization; Melander et al. 1987), Montgomery and Kallenbach (1997, hereafter MK97) proposed a new pathway for hurricane intensification through the axisymmetrization of convectively induced asymmetric vorticity. The axisymmetrization process is accompanied by the generation of outward-propagating vorticity filaments referred to as vortex Rossby waves (VRWs). These types of waves, with the background radial vorticity gradient serving as the restoring mechanism, are progressively sheared by the differential rotation and act to intensify the symmetric flow by transporting angular momentum radially inward. The radial propagation of VRWs revealed by the basic theory developed by MK97 (based on 2D nondivergent inviscid flows on an f plane) was found to be a robust feature in hurricane-like vortex flows in a shallow-water balance model. The results of MK97 and the significance of their proposed wave–mean flow interaction pathway to tropical cyclogenesis were further validated and extended by Montgomery and Enagonio (1998) using a three-dimensional (3D) nonlinear quasigeostrophic model, by Möller and Montgomery (1999) using a barotropic nonlinear asymmetric balance model for large Rossby numbers, in a baroclinic setting by Möller and Montgomery (2000) using a “dry” 3D asymmetric balance model, by Shapiro (2000) using a dry three-layer primitive equation model, and by Enagonio and Montgomery (2001) using a shallow-water primitive equation model. An additional “moist” experiment performed by Shapiro (2000) revealed significant differences between the dry and moist results. However, the pattern of acceleration and deceleration among the two cases was found to be qualitatively similar.

All the aforementioned studies considered the evolution of monochromatic and/or azimuthally localized initial asymmetric vorticity perturbations presumed to be the end result of adjustment to convective heating. Different to the above studies, Nolan and Grasso (2003, hereafter NG03) analyzed the evolution of instantaneous asymmetric temperature perturbations using a 3D nonhydrostatic linear model and found an overall negative effect, that is, a weakening of the symmetric vortex. Their results were validated by Nolan et al. (2007) using an improved version of the 3D linear model used in NG03 and more realistic heat forcing.

Given the diversity and inconsistency among previous studies on the overall impact of asymmetric convection to the intensity evolution of a symmetric vortex (intensification versus weakening), it is the main objective of this study to further illuminate on the current understanding. This will be done using a primitive equation nonlinear 3D dry model forced by asymmetric heat sources. Our emphasis will be on the wave radiation and the process of wave–mean flow interaction. A series of numerical simulations will be conducted, and then the results will be compared with previous findings. Special attention will be paid to clarify the underlying physical reasons for which the results obtained here may agree or disagree with earlier studies.

The remainder of this paper is organized as follows: section 2 describes briefly the numerical model used in this study and the initialization procedure, and section 3 revisits the generalized wave activity conservation laws and the Eliassen–Palm (EP) flux theorem in cylindrical and isentropic coordinates and the azimuthally averaged eddy kinetic energy budget equation in cylindrical coordinates. Diagnostic results are presented in section 4, and a summary and conclusions are discussed in section 5.

2. Numerical model and initialization

To investigate the role of asymmetric convection to the intensification process of an incipient vortex and the underlying dynamics of the axisymmetrization process, a dry but thermally forced version of the Weather Research and Forecasting Model (WRF) is used. WRF is a state-of-the-art 3D, nonlinear, fully compressible regional model of the atmosphere. By “dry,” we mean that all model physics are deactivated in these simulations, including cloud microphysics, cumulus parameterization, surface, planetary boundary layer, and atmospheric radiation processes. By “thermally forced,” we mean that a term that mimics some asymmetric features of a hurricane (e.g., the heating associated with sporadic bursts of convection) is added to the thermodynamic equation of the model.

a. Setup of the control experiment and initial conditions

Here the initialization procedure of the control experiment that deals with the changes in the intensity of a weak vortex that results from localized asymmetric convective bursts is described. It is pointed out that a series of sensitivity tests on the structure and the magnitude of both the initial basic-state vortex and the initial thermal forcing will be also performed. These experiments will be described and discussed in the following sections.

The model setup is a single domain with a horizontal grid mesh of 300 × 300 points and a grid spacing of 4 km. A total of 28 sigma levels are configured in the vertical. The simulation is carried out on the f plane with a constant Coriolis parameter f = 6.1635 × 10−5 s−1. The initial symmetric vortex is designed by first defining a relative vorticity (ζ0) radial profile that is given by a Gaussian vorticity distribution:

 
formula

Here ζm = 2.5 × 10−4 s−1 and Cα = 100 km. The maximum azimuthal velocity associated with this monopolar vorticity profile (vorticity decreasing monotonically with increasing radius) is close to 8 m s−1 and occurs at a radius of about 120 km. The azimuthal velocity (υ0) profile is then extended into the vertical by the use of an analytical function following Nolan and Montgomery (2002) as

 
formula

where Vs(r) is the surface velocity profile, Ht = 14 km is approximately the upper limit of the wind field, Cβ = 1.7, and the constant α1 = 2. Figure 1 shows the resultant initial symmetric vortex in terms of the azimuthal (or tangential) winds (Figs. 1a,b) and relative vorticity (Figs. 1c,d). The initial vortex is then brought into hydrostatic balance following the iterative procedure described in Nolan et al. (2001).

Fig. 1.

(left) Radius–height sections of the basic-state vortex used in the control experiment: (a) tangential winds (m s−1) and (c) relative vorticity (×10−4 s−1). (right) Surface profiles of the basic-state vortex used in experiment I: (b) tangential wind and (d) relative vorticity.

Fig. 1.

(left) Radius–height sections of the basic-state vortex used in the control experiment: (a) tangential winds (m s−1) and (c) relative vorticity (×10−4 s−1). (right) Surface profiles of the basic-state vortex used in experiment I: (b) tangential wind and (d) relative vorticity.

b. Thermal forcing

In the control experiment, the thermal forcing takes the shape of a double-cluster (or double blob) anomaly located outside the radius of maximum wind (RMW) of the initial symmetric vortex (Fig. 2). Such a configuration is intended to mimic a localized burst of convection near the vortex. Each blob is generated, following Nolan and Grasso (2003), by utilizing a function of the form

 
formula

where

 
formula

where (xcb, ycb) is the center of the vortex, is the heating rate amplitude, rcb is the radial location of each blob, δr = 20 km is the radial half-width, δz = 3 km is the vertical half-width, and zcb = 5 km represents the center of the heating in the vertical. The constants β1 and β2 are taken to be equal to 2. Figure 2b shows the vertical structure of the resultant double-cluster forcing. It is pointed out that the thermal forcing is active only during the first hour of the simulation (hereafter referred to as single pulse) and the model is then allowed to evolve freely. The vertical structure of the forcing bears some similarities with the simulated heating rate associated with vortical hot towers (Hendricks et al. 2004; Montgomery et al. 2006). The magnitude of 25 K h−1, however, is well below the diagnosed magnitudes obtained in Hendricks et al. (2004) , and those used in Nolan et al. (2007)  .

Fig. 2.

(a) Horizontal cross section at about 4.3-km height of the double-cluster thermal anomaly (K h−1) used to force the simulation of the control experiment. (b) Vertical cross section of the double-cluster thermal anomaly (K h−1) taken along the gray line in (a).

Fig. 2.

(a) Horizontal cross section at about 4.3-km height of the double-cluster thermal anomaly (K h−1) used to force the simulation of the control experiment. (b) Vertical cross section of the double-cluster thermal anomaly (K h−1) taken along the gray line in (a).

3. Asymmetric dynamics in 3D vortex flows

To analyze the structure and the kinematics of the asymmetries that result from the asymmetric thermal forcing, the concept of wave activity conservation laws cast in cylindrical and isentropic coordinates are applied. The impact of these asymmetries on the primary vortex is determined utilizing the EP flux theorem. Finally, some eddy kinetic energy budget calculations are performed with more emphasis on the energy conversion terms from the mean vortex to the eddies and vice versa. In this section, the generalized wave activity conservation laws, wave–mean flow interactions, and the azimuthally averaged eddy kinetic energy equation in cylindrical coordinates are briefly reviewed.

a. Wave activity conservation laws and wave–mean flow interactions

The propagation of wave disturbances in shear flows and the resulting interaction with the background flow (wave–mean flow interactions) can be conveniently analyzed in terms of wave activities and their corresponding conservation laws. In the absence of source–sink terms, the conservation law is defined as a relation of the form

 
formula

in which A is the wave activity density that is a quadratic disturbance measure (in the small amplitude limit) and FA is the corresponding wave activity flux vector. By making the assumption that a hurricane vortex is in approximate hydrostatic balance, we can employ the 3D hydrostatic primitive equations in cylindrical and isentropic coordinates (see  appendix A). Assigning to each variable a mean state that is considered to be time and azimuthal invariant and an associated perturbation term, that is,

 
formula

two wave activities can be defined, namely, the pseudomomentum density and pseudoenergy density. Here s represents a generic flow field. We point out that in the present study, only the concept of pseudomomentum density (hereafter pseudomomentum) will be applied. The azimuthal invariant mean state allows for the derivation of the conservation law for pseudomomentum, that is,

 
formula

in which

 
formula

and

 
formula

Here is the pseudomomentum, q is the potential vorticity (PV), is the radial gradient of the mean state PV, F is the pseudomomentum flux vector, and S is the source–sink of pseudomomentum. The first term on the right-hand side (rhs) of Eq. (8) is related to the isentropic density perturbation σ′, and it is therefore referred to as the gravitational contribution to pseudomomentum g. The second term on the rhs of Eq. (8) contains γ0 and PV perturbation q′ is interpreted as the vortical or VRW contribution to pseudomomentum υ. For monotonic mean-state PV profiles (the case of the mean-state vortex in the control experiment), υ becomes sign definite. Such property permits the dynamical track of the propagation of υ across the flow domain. Taking the azimuthal average of Eq. (7), F reduces to the generalized EP flux vector FEP, given by

 
formula

The radial component of FEP is related to the radial transport of eddy cyclonic angular momentum and the vertical component is analogous to the radial eddy heat flux in pressure coordinates. When FEP points outward (upward) eddies transfer momentum (heat) radially inward and vice versa. The divergence of the EP flux ( · FEP) is connected in a straightforward manner to the time change of angular momentum through the angular momentum budget equation in cylindrical and isentropic coordinates

 
formula

Here D includes the diabatic heating terms. In the presence of friction, D includes also the friction term. The wave–mean flow interactions can be estimated directly from the divergence of the EP flux that is interpreted as an eddy forcing to the mean flow. When · FEP > 0 ( · FEP < 0), the eddies act to accelerate (decelerate) the mean flow.

b. Eddy kinetic energy prognostic equation

Following Wang (2002), the azimuthally averaged eddy kinetic energy (Kp) equation in cylindrical coordinates can be written as (see  appendix B)

 
formula

Equation (12) is slightly different than Eq. (7) of Wang (2002) because it is derived in cylindrical coordinates, while Wang (2002) adapted the cylindrical and pressure coordinates for the derivation.

Taken together, the first and second terms on the rhs of Eq. (12) represent the flux divergence of by the mean flow (FDMF). The third and fourth terms represent the flux divergence of by the eddies (FDE). The fifth and sixth terms represent the barotropic energy conversion from the mean vortex that is associated with the azimuthal-mean flow (BTA), while the seventh and eighth terms are the barotropic energy conversion associated with the mean radial inflow convergence (BTR). Together, BTA and BTR account for the energy conversion by all barotropic processes (BT). The ninth and tenth terms represent the baroclinic energy conversion that is associated with the mean radial flow (BCR) and the mean azimuthal flow (BCA), respectively. Similarly, BCR + BCA define the energy conversion by all baroclinic processes (BC). Finally, the last two terms on the rhs of Eq. (12) are associated with the transfer of eddy potential energy into eddy kinetic energy (PTK). It is pointed out that more emphasis will be given on the energy conversion terms. These terms appear in the same form in both the azimuthal-mean eddy kinetic energy equation and the mean kinetic energy equation, but with opposite signs (see  appendix B). As such, by calculating these terms, one can indicate the flow of kinetic energy between the mean flow and the eddies. In addition, a direct comparison between the barotropic and the baroclinic energy conversion terms further allows us to determine the nature of the dominant energy exchange process.

4. Diagnostic results

a. Wave activities

The symmetric vortex in the control experiment is perturbed by a single pulse (lasts 1 h) of a double-blob thermal anomaly. As part of the adjustment process to this source of imbalance, radiation of gravity waves and the generation of vorticity perturbations are expected. Figure 3 shows a radius–time Hovmöller diagram drawn from the center of the vortex toward the southwest of the two components of pseudomomentum, g (Fig. 3a; gravity waves) and υ (Fig. 3b; VRWs). As a reminder, pseudomomentum maps can be used to dynamically track the propagation of wave activities across the domain. In addition, the comparison between the two components of pseudomomentum can indicate the type of waves that are more dynamically active (VRWs versus gravity waves). Fast outward gravity waves can be clearly seen during the first few hours of the simulation that propagate throughout the whole extent of the plotted domain. After about 10 h, however, no clear signal of gravity waves can be identified. In contrast, the generated rotational-dominated asymmetry is radially confined while subject to an axisymmetrization process that is sheared up by the differential rotation. The outward-propagating filaments become finer in spatial scale and their speed slows down consistent with the VRW theory. It is pointed out that the magnitudes of υ do not vary linearly. The green contours are one order of magnitude (on the order 109 kg K−1 s−1) larger than the red contours (on the order 108 kg K−1 s−1) and two orders of magnitude larger than the black contours (on the order 107 kg K−1 s−1). Such magnitudes are more than three orders of magnitude larger than that of g (on the order 105 kg K−1 s−1), indicating that the response of the symmetric vortex to the double-blob thermal anomaly is dominated by the excitation of VRWs. The fact that the magnitude of υ decreases rapidly in a few hours strongly suggests a critical layer damping of these sheared VRWs.

Fig. 3.

Radius–time Hovmöller diagram drawn from the center of the vortex toward the southwest of (a) the gravitational contribution to pseudomomentum at θ = 308 K (contours of ±0.4–8.0 × 105 kg K−1 s−1 with ±0.4 × 105 kg K−1 s−1 interval) and (b) the vortical contribution to pseudomomentum θ = 300 K (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval).

Fig. 3.

Radius–time Hovmöller diagram drawn from the center of the vortex toward the southwest of (a) the gravitational contribution to pseudomomentum at θ = 308 K (contours of ±0.4–8.0 × 105 kg K−1 s−1 with ±0.4 × 105 kg K−1 s−1 interval) and (b) the vortical contribution to pseudomomentum θ = 300 K (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval).

The azimuthal propagation of the radially confined damped sheared VRW is depicted also in Fig. 4b, an azimuth–time Hovmöller diagram drawn at r = 140 km. Different from the evolution of the VRW, the fast gravity waves radiate away after the first few hours. Finally, the axisymmetrization process can be clearly seen in Fig. 5, in which snapshots of absolute vorticity perturbation ξ′ (shaded) superimposed with υ (contoured) are shown. As expected, the response of the vortex to the initial perturbation is largely projected onto an azimuthal wavenumber m = 2 sheared VRW that is well correlated with ξ′.

Fig. 4.

Azimuth–time Hovmöller diagram at 140-km radius of (a) the gravitational contribution to pseudomomentum at θ = 308 K (contours of ±0.1 − 3.1 × 106 kg K−1 s−1 with ±0.3 × 106 kg K−1 s−1 interval) and (b) the vortical contribution to pseudomomentum at θ = 300 K (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval).

Fig. 4.

Azimuth–time Hovmöller diagram at 140-km radius of (a) the gravitational contribution to pseudomomentum at θ = 308 K (contours of ±0.1 − 3.1 × 106 kg K−1 s−1 with ±0.3 × 106 kg K−1 s−1 interval) and (b) the vortical contribution to pseudomomentum at θ = 300 K (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval).

Fig. 5.

Horizontal cross section at θ = 300 K of the absolute vorticity perturbation (×10−5 s−1) (azimuthal wavenumber 1–4, shaded) superimposed with υ (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval). Simulation times are (a) t = 6, (b) t = 12, (c) t = 18, and (d) t = 24 h.

Fig. 5.

Horizontal cross section at θ = 300 K of the absolute vorticity perturbation (×10−5 s−1) (azimuthal wavenumber 1–4, shaded) superimposed with υ (black contours of 5–45 × 106 kg K−1 s−1 with 5 × 106 kg K−1 s−1 interval; gray contours of 50–100 × 106 kg K−1 s−1 with 10 × 106 kg K−1 s−1 interval; red contours of 12–20 × 107 kg K−1 s−1 with 2 × 107 kg K−1 s−1 interval; purple contours of 40–90 × 107 kg K−1 s−1 with 5 × 107 kg K−1 s−1 interval; green contours of 10–20 × 108 kg K−1 s−1 with 2 × 108 kg K−1 s−1 interval). Simulation times are (a) t = 6, (b) t = 12, (c) t = 18, and (d) t = 24 h.

b. EP flux and its divergence

The axisymmetrization process of the dominant sheared VRW points to a possible wave–mean flow interaction pathway and an acceleration of the latter by the wave activity. Such hypothesis can be further evaluated by utilizing the EP flux and its divergence, interpreted as the flux of wave activity and as an eddy forcing to the mean flow, respectively. Figure 6 shows a radius–height cross section of the horizontal component of the EP flux vector (Fig. 6a) that is associated with the radial transport of eddy angular momentum and the EP flux divergence (Fig. 6b) at time t = 7.5 h, along with their profiles at θ = 300 K. It is pointed out that the present study focuses on the underlying dynamics of the dominant convectively induced waves. As such, the time interval for the analysis was chosen to be after the initial forcing is turned off and at a time interval in which the waves are dynamically active. Through the EP flux divergence, we can assess how the sheared VRW may affect the mean flow. The vertical component of the EP flux vector is negligible at this time, and it is therefore not shown. The EP flux shows a strong maximum in the lower levels, indicating that eddy angular momentum is transported radially inward. Note that since the initial symmetric vortex is a “surface” vortex (the strongest winds occur at the surface) the radial wind shear of the tangential winds is highest at the surface. As such, the largest impact of the sheared VRW is expected to occur at the lower levels. The divergence of the EP flux shows a maximum close to the initial RMW of the symmetric vortex, thus verifying the acceleration of the mean flow induced by the axisymmetrization of a sheared VRW and the inward transport of angular momentum. It is pointed out that although the degree of acceleration produced by the VRW may be considered to be small, one has to note that we are dealing with a very weak initial vortex and that this acceleration is the result of a single pulse of VRW that is generated by a moderate heat source anomaly. Such eddy acceleration was found to be of the same order of magnitude with the acceleration induced by the dominant mean terms in Eq. (11), that is, the mean radial transport of mean absolute vorticity (second and third terms; not shown). Sensitivity tests on the structure and intensity of the initial heat source to be presented next will show that the acceleration induced again by a single pulse of VRW can be more than doubled.

Fig. 6.

(a) Horizontal component of the EP flux vector (contours of ±0.5–5 × 106 kg m K−1 s−2 with 0.5 × 106 kg m K−1 s−2 interval), (b) EP flux divergence (contours of ±0.1–1 m s−1 day−1 with 0.1 m s−1 day−1 interval), and (c) profile of the EP flux divergence (gray) and the EP flux (black) at θ = 300 K. Simulation time is 7.5 h.

Fig. 6.

(a) Horizontal component of the EP flux vector (contours of ±0.5–5 × 106 kg m K−1 s−2 with 0.5 × 106 kg m K−1 s−2 interval), (b) EP flux divergence (contours of ±0.1–1 m s−1 day−1 with 0.1 m s−1 day−1 interval), and (c) profile of the EP flux divergence (gray) and the EP flux (black) at θ = 300 K. Simulation time is 7.5 h.

c. Sensitivity tests on the structure and magnitude of the thermal forcing

The control experiment revealed an intensification mechanism for the weak vortex, that is, through the transport of eddy angular momentum to the mean flow by a convectively induced sheared VRW (wave–mean flow interaction). Although a double-cluster asymmetry has been observed in both real hurricanes and simulated ones, the number of individual convective towers as well as the magnitude of the latent heat release associated with each tower may vary significantly. Sensitivity tests are designed here to further assess how changes made in the initial structure and the magnitude of the thermal asymmetry may affect the wave–mean flow interaction.

Figure 7a shows the structure of the new thermal asymmetry that is a quadruple cluster. Three sensitivity tests are performed by changing the magnitude of the forcing, as is shown in Fig. 7b in terms of the vertical profile of the heating rate taken from the center of a tower. Similar to the double-cluster case, the forcing is active only during the first hour of the simulation. As expected, the response of the symmetric vortex is again dominated by the radiation of VRWs that are subject to axisymmetrization (not shown). Figure 8 shows the wave–mean flow interaction in terms of the EP flux divergence at θ = 300 K and t = 7.5 h for the three cases. Of interest to note is that the acceleration of the mean flow induced by the single pulse of a VRW that is convectively induced by a moderate heat source (in terms of magnitude) can be as large as 2.4 m s−1 day−1. As such, the results here leave little room for speculation about the importance of the wave–mean flow interaction mechanism in the intensification of a weak vortex.

Fig. 7.

(a) Horizontal cross section at about 4.3-km height of the quadruple-cluster thermal anomaly (K h−1) used to force the sensitivity simulations in experiment I. (b) Vertical profile of the heating rate taken from the center of a tower for the three sensitivity tests.

Fig. 7.

(a) Horizontal cross section at about 4.3-km height of the quadruple-cluster thermal anomaly (K h−1) used to force the sensitivity simulations in experiment I. (b) Vertical profile of the heating rate taken from the center of a tower for the three sensitivity tests.

Fig. 8.

EP flux divergence (m s−1 day−1) profile at θ = 300 K and t = 7.5 h for the three sensitivity tests in experiment I.

Fig. 8.

EP flux divergence (m s−1 day−1) profile at θ = 300 K and t = 7.5 h for the three sensitivity tests in experiment I.

d. Eddy kinetic energy budget

The sources–sinks of the eddy kinetic energy (VRW kinetic energy here), as well as the exchange of kinetic energy between the eddies and the mean flow, can be examined by performing some eddy kinetic energy budget calculations. As expected, the dominant source of eddy kinetic energy in the present dry thermally forced simulations is realized through the conversion from eddy potential energy (associated with the asymmetric diabatic heating) (PTK; not shown).

We are mostly interested in the kinetic energy conversion terms (BT and BC) through which the flow of kinetic energy between the eddies and the mean flow can be assessed. Figure 9 shows the energy conversion terms associated with all barotropic (BT; Fig. 9a) and all baroclinic (BC; Fig. 9b) processes, taken at t = 7.5 h. Before proceeding to the results, we emphasize that in the dry vortices under investigation, a symmetric secondary circulation is absent (except during the first hour in which the heat source is active). As such, the terms FDMF, BTR, and BCR that involve either the mean radial wind component or the mean vertical wind component are negligible relative to the rest of the budget terms and therefore are not discussed. The positive values of BT seen at higher levels (Fig. 9a) indicate that the barotropic conversion of mean kinetic energy associated with the mean azimuthal flow (recall that BTR is negligible) is a source of eddy kinetic energy in the upper levels. This finding is consistent with that of Anthes (1972) and Wang (2002). In contrast, at lower levels, an upscale energy cascade mechanism can be realized, that is, the transfer of kinetic energy from the eddies to the mean flow as a result of the radial shear of the mean azimuthal flow. This is a manifestation of the axisymmetrization mechanism discussed earlier and is another way of verifying our previous findings. The largest eddy damping occurs at the lower levels, consistent again with the fact that the strongest horizontal shear of the mean azimuthal flow occurs at the surface. On the other hand, the baroclinic processes are not as important as the barotropic processes in converting kinetic energy. Only at upper levels do the eddies receive some kinetic energy from the mean vortex through the baroclinic conversion related to the vertical wind shear of the azimuthal flow (Fig. 9b; recall that BCR is negligible).

Fig. 9.

Radius–height cross section of the kinetic energy conversion by (a) all barotropic processes (BT) and (b) all baroclinic processes (BC) at t = 7.5 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1]. (c) As in (a), but with a focus on the lower 2-km height [positive contours (thick gray), negative contours (thick black); contour interval 0.2 m−2 s−2 day−1].

Fig. 9.

Radius–height cross section of the kinetic energy conversion by (a) all barotropic processes (BT) and (b) all baroclinic processes (BC) at t = 7.5 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1]. (c) As in (a), but with a focus on the lower 2-km height [positive contours (thick gray), negative contours (thick black); contour interval 0.2 m−2 s−2 day−1].

Apart from the energy conversion terms, the flux divergence of eddy kinetic energy by the eddies themselves contribute significantly to the budgets, in that the eddies transport eddy kinetic energy radially inward from outside the RMW at lower levels and outward at higher levels (Fig. 10). Finally, it is pointed out that the axisymmetrization process described here remains valid for several hours in the simulation. Figure 11 shows BT at t = 12 h. The pattern of BT obtained here bears a great similarity with that seen in Fig. 9a.

Fig. 10.

(a) Radius–height cross section of the flux divergence of eddy kinetic energy by the eddies (FDE) at t = 7.5 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1]. (b) As in (a), but with a focus on the lower 2-km height [positive contours (thick gray), negative contours (thick black); contour interval 0.2 m−2 s−2 day−1].

Fig. 10.

(a) Radius–height cross section of the flux divergence of eddy kinetic energy by the eddies (FDE) at t = 7.5 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1]. (b) As in (a), but with a focus on the lower 2-km height [positive contours (thick gray), negative contours (thick black); contour interval 0.2 m−2 s−2 day−1].

Fig. 11.

Radius–height cross section of the kinetic energy conversion by all barotropic processes (BT) at t = 12 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1].

Fig. 11.

Radius–height cross section of the kinetic energy conversion by all barotropic processes (BT) at t = 12 h [positive contours (purple), negative contours (green); contour interval 0.05 m−2 s−2 day−1].

e. Comparison with the results of NG03 and Nolan et al. (2007)

Our findings emphasize the importance of the VRWs radiated by localized convective anomalies to the intensification of an incipient vortex. However, they stand in direct contrast with the findings of NG03 and Nolan et al. (2007), who pointed that thermal anomalies do not lead to intensification and have rather an overall negative effect. It is therefore important to obtain a better physical insight as to where the differences originate.

To do so, we will be utilizing the concept of PV and, more specifically, the PV perturbations that are generated by the convective heating. This approach is similar to the one used by NG03 to explain the weakening of their symmetric vortex, and it is therefore adapted here also for consistency and for a direct comparison.

In the presence of the diabatic heating rate and in the absence of friction, the equation governing the PV is given by1

 
formula

where ω is the 3D vorticity vector and is the diabatic heating rate. Decomposing each variable into a mean state and an associated perturbation term, the prognostic equation for the PV perturbation q′ can be written as

 
formula

At the initial time when the thermal anomaly is introduced, ω′ is zero and Eq. (14) simplifies to

 
formula

Taking into account the absence of a symmetric secondary circulation and that the mean state is considered to be azimuthal invariant, Eq. (15) reduces to

 
formula

Equation (16) states that the structure and the magnitude of the generated PV perturbation is mainly determined by the baroclinicity of the symmetric vortex (vertical wind shear of the mean azimuthal winds), the radial shear of the mean azimuthal winds, and finally, the radial and vertical gradients of the diabatic heating rate. For surface vortices forced by thermal anomalies having their largest magnitudes away from the surface, Eq. (16) implies that if the first (second) term on the rhs of Eq. (16) dominates, then the resultant PV perturbation will have a dipole pattern in the radial (vertical) direction. In both NG03 and Nolan et al. (2007), the vortices under consideration had large vertical wind shear, thus leading to the generation of a dipole PV perturbation in the radial direction [see Fig. 18c of NG03 and Fig. 11a of Nolan et al. (2007)]. The crucial point made by NG03 as to how a thermal anomaly can result in the deceleration of the symmetric vortex originates from the fact that, in the case of a pure harmonic thermal anomaly and in the presence of a highly baroclinic vortex, two concentric sets of PV perturbation will be generated on either side of the maximum thermal anomaly in the radial direction. For a monopolar vortex, the inner PV perturbation ring will rotate faster (since it is located closer to the vortex center), and it therefore may become phase locked with the outer PV perturbation ring. During this phase of evolution, the two PV perturbation rings will help each other to grow (PV perturbation is amplifying) by extracting energy from the symmetric vortex. This energy extraction by the amplifying PV perturbation is the main reason for the deceleration of the vortex. Therefore, the key question to ask here is as follows: Is the dominant generated PV perturbation in the present study amplified with time or does it follows a damping behavior? As previously shown from the wave activity diagnostics, the response of the symmetric vortex to the localized convective forcing is dominated by the rotational component of pseudomomentum (a quadratic term in terms of the PV perturbation) that follows a damping behavior (see Figs. 3b and 4b). As such, different results with NG03 and Nolan et al. (2007) should be expected.

f. Sensitivity tests on the baroclinicity of the basic-state vortex and the radial and vertical gradients of the thermal forcing

The previous section revealed the main difference between the present study and the work by NG03 and Nolan et al. (2007). In our study, the convectively induced PV perturbation follows a damping behavior and acts to accelerate the symmetric vortex through the process of wave–mean flow interaction, while in the latter studies, the generated PV perturbation amplifies with time, thus leading to a deceleration of the symmetric vortex. It also emphasizes the importance of the baroclinicity of the basic-state vortex, as well as the importance of the radial and vertical gradients of the thermal forcing to the structure and magnitude of the generated PV perturbations. In this section, sensitivity tests are designed to assess how vortices with different vertical structures and how thermal sources with different radial and vertical gradients can alter the generated PV perturbations. For each test, the time evolution of the maximum amplitudes of the PV perturbations will be monitored to address the possible implications to the intensification process of the symmetric vortex.

Two main sets of sensitivity experiments are configured, starting with a more barotropic symmetric vortex (experiment I) and a highly baroclinic symmetric vortex (experiment II). The horizontal grid mesh, grid spacing, number of vertical levels, and the constant f are the same as in the control experiment. The initial ζ0 radial profile is prescribed utilizing Eq. (1), in which {ζm, Ca} = {5 × 10−4 s−1, 100 km}. The velocity profile is extended into the vertical in a similar manner as in Nolan et al. (2007) by utilizing a function of the form

 
formula

Here z0 indicates the height of the maximum velocity, Hd defines the depth of the barotropic part of the vortex, and α2 is a constant. For experiment I {z0, Hd, α2} = {0 km, 8 km, 1.5}, and for experiment II {z0, Hd, α2} = {0 km, 5 km, 4}. Figure 12 shows the radial and vertical structure of the symmetric tangential winds for experiment I (barotropic vortex; Fig. 12a), and for experiment II (highly baroclinic vortex; Fig. 12b). The maximum azimuthal velocity for both vortices is close to 16 m s−1 (Fig. 12c; black line), which is typical for a tropical storm vortex. The vorticity profile again resembles a monopolar vortex (Fig. 12c; gray line) and therefore satisfies the necessary Rayleigh condition for linear stability.

Fig. 12.

Radius–height sections of the tangential winds (m s−1) of the basic-state vortex used in (a) experiment I and (b) experiment II. (c) Surface tangential winds (left axis; black), and surface relative vorticity (right axis; gray) of the basic-state vortex used in experiments I and II.

Fig. 12.

Radius–height sections of the tangential winds (m s−1) of the basic-state vortex used in (a) experiment I and (b) experiment II. (c) Surface tangential winds (left axis; black), and surface relative vorticity (right axis; gray) of the basic-state vortex used in experiments I and II.

For each experiment, two simulations will be conducted in which the symmetric vortex will be forced by a thermal anomaly having large vertical gradients (LVG) and a thermal anomaly with large radial gradients (LRG). Similar to the control experiment, the thermal forcing is taken to be a localized single pulse of a double-cluster anomaly specified again by Eq. (3). The parameters , rcb, δr, and zcb are the same as in the control experiment. The major differences reside in the parameters β1 and β2, the modification of which can result in sharp radial and/or sharp vertical gradients. For the LVG, {δz, β1, β2} = {1.8 km, 2, 6}, and for the LRG, {δz, β1, β2} = {3 km, 6, 2}. Figure 13 shows a vertical cross section of the double-cluster thermal anomaly taken along the center of each blob for LVG (Fig. 13a) and LRG (Fig. 13b). It is pointed out that an additional final experiment in which the forcing resembles a quadruple cluster will be also presented.

Fig. 13.

Vertical cross section of the double-cluster thermal anomalies used to force the simulations in experiments I and II. Thermal anomaly with (a) LVG and (b) LRG.

Fig. 13.

Vertical cross section of the double-cluster thermal anomalies used to force the simulations in experiments I and II. Thermal anomaly with (a) LVG and (b) LRG.

1) Experiments I and II

As previously mentioned, experiment I deals with a barotropic basic-state vortex. Figure 14 shows the convectively induced PV perturbation, taken at t = 2 h for both LVG (Figs. 14a,c) and LRG (Figs. 14b,d). Note that the horizontal cross sections (Figs. 14a,b) are taken at approximately the level at which the maximum PV perturbation occurs, and the vertical cross sections (Figs. 14c,d) are taken along the black dashed line. There are a few important things to note. First, the generated positive PV perturbations in the presence of a barotropic basic-state vortex are of the same order of magnitude, regardless of the gradients of the thermal forcing. Second, for the LVG case, the PV perturbations are dominated by positive values while the negative PV perturbations are one order of magnitude smaller at the level of the maximum PV perturbation (Fig. 14a). In contrast, for LRG the PV perturbation is dominated by a dipole PV pattern in the radial direction in which both positive and negative values are of the same order of magnitude (Fig. 14b). This is to be expected if one considers that, in this simulation, the second term on the rhs of Eq. (16) is dominant. From the vertical cross sections, it is clear that the axis of the PV dipole is more vertical (horizontal) for the LVG (LRG) case, consistent again with Eq. (16). Third, the maximum convectively induced PV perturbations are realized at different vertical levels for the two different cases. For the LVG it occurs near 2.5 km, while for the LRG, it occurs near 6.2 km.

Fig. 14.

Experiment I. (left) PV perturbations (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) generated at t = 2 h for the LVG case: (a) horizontal cross section at about 2.5-km height and (c) radius–height cross section taken along the black dashed line. (right) PV perturbations (PVU) generated at t = 2 h for the LRG case: (b) horizontal cross section at about 6.2-km height and (d) radius–height cross section taken along the black dashed line. In all panels, positive contours are red and negative contours are blue. The contour interval is 0.5 PVU (lowest positive contour shown is 0.5 PVU, lowest negative contour shown −3 PVU).

Fig. 14.

Experiment I. (left) PV perturbations (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) generated at t = 2 h for the LVG case: (a) horizontal cross section at about 2.5-km height and (c) radius–height cross section taken along the black dashed line. (right) PV perturbations (PVU) generated at t = 2 h for the LRG case: (b) horizontal cross section at about 6.2-km height and (d) radius–height cross section taken along the black dashed line. In all panels, positive contours are red and negative contours are blue. The contour interval is 0.5 PVU (lowest positive contour shown is 0.5 PVU, lowest negative contour shown −3 PVU).

Figure 15 shows the time evolution of the maximum amplitudes of the PV perturbations for the whole time duration of the simulations. Of interest to note is that, in both simulations, the PV perturbations follow an overall damping behavior. For the LVG case in which the PV pattern has a more vertical axis and in which the positive values are dominant, such damping behavior is to be expected and can be easily understood in terms of axisymmetrization by the differential rotation. For the LRG case in which the PV dipole is one step closer to that of NG03 and Nolan et al. (2007) (PV dipole in the radial direction), such damping behavior might be considered at first to be surprising. However, upon careful inspection, one can justify why damping should be again expected. In NG03, the initial thermal anomaly was in the form of a pure harmonic asymmetry. As such, the resultant PV perturbation resembled two “rings” of PV perturbation with alternative positive and negative values in the azimuthal direction. Therefore, phase lock between “same sign” inner and outer asymmetries, which in turn may imply amplification, was possible. Different from NG03, the thermal anomaly in the present study resembles a localized asymmetry, and thus, such phase lock between the inner and outer convectively induced PV perturbations is not possible. This is the main reason why, in the LRG case, no amplification of the PV perturbations is seen.

Fig. 15.

Experiment I. Time evolution of the maximum amplitudes of PV perturbations (PVU) for (a) LVG and (b) LRG cases.

Fig. 15.

Experiment I. Time evolution of the maximum amplitudes of PV perturbations (PVU) for (a) LVG and (b) LRG cases.

As a reminder, experiment II deals with a highly baroclinic basic-state vortex. Figure 16 again shows the resultant PV perturbations after 2 h of simulation, for both LVG (Figs. 16a,c) and LRG (Figs. 16b,d). The overall pattern of the generated PV perturbation, as well as the level of occurrence of the maximum magnitudes, closely resembles that obtained from the previous experiment. One major difference resides in the magnitudes of the PV perturbations obtained in the LRG case, which are more than double those obtained from the LRG case in experiment I. Taking into account the quite large vertical wind shear of the basic-state vortex here, such differences are not surprising [see second term on the rhs of Eq. (16)]. Finally, and consistent with the findings of experiment I, the time evolution of the maximum amplitude of the convectively induced PV perturbation shows again a damping behavior (Fig. 17).

Fig. 16.

As in Fig. 14, but for Experiment II.

Fig. 16.

As in Fig. 14, but for Experiment II.

Fig. 17.

As in Fig. 15, but for Experiment II.

Fig. 17.

As in Fig. 15, but for Experiment II.

2) Experiment with a quadruple-cluster thermal forcing

Experiments I and II suggests that for a localized convective forcing, regardless of the baroclinicity of the basic-state vortex, and regardless of the radial and vertical gradients of the convective bursts, amplification of the induced PV asymmetries should not be expected. This, in turn, implies that a deceleration of the basic-state vortex should not be expected either.

As a final test, it is of interest to examine the extent to which the current conclusions may be modified in the case of a quadruple thermal cluster anomaly. Figure 18 shows the horizontal structure of the resultant PV perturbations (Figs. 18a,c) and the time evolution of their maximum amplitude (Figs. 18b,d) for both the barotropic basic-state vortex (Figs. 18a,b) and the highly baroclinic basic-state vortex (Figs. 18c,d), forced by a quadruple single pulse with LRG. A quadruple convective cluster with LVG leads to the formation of a distinct quadruple pattern of positive PV perturbations that undergoes an axisymmetrization. This is to be expected, and it is therefore not shown. Consistent again with our previous statements, the convectively induced PV perturbation follows a damping behavior. Taking into account that observed convective bursts resemble more closely the thermal forcing used in the present study, rather than a pure harmonic asymmetry such the one specified in NG03, suggests that the current results are more likely to represent a more realistic impact of the convective forcing to the intensification process.

Fig. 18.

(left) PV perturbations (PVU) generated at t = 2 h by a quadruple-cluster thermal forcing with LVG in (a) the presence of the barotropic basic-state vortex and (c) the presence of the highly baroclinic vortex [positive contours (red), negative contours (blue); contour interval 0.5 PVU]. (right) Time evolution of the maximum amplitudes of PV perturbations (PVU) for (b) the barotropic vortex and (d) the highly baroclinic vortex.

Fig. 18.

(left) PV perturbations (PVU) generated at t = 2 h by a quadruple-cluster thermal forcing with LVG in (a) the presence of the barotropic basic-state vortex and (c) the presence of the highly baroclinic vortex [positive contours (red), negative contours (blue); contour interval 0.5 PVU]. (right) Time evolution of the maximum amplitudes of PV perturbations (PVU) for (b) the barotropic vortex and (d) the highly baroclinic vortex.

5. Summary and conclusions

Even though it is now generally accepted that hurricanes are mainly axisymmetric systems, the underlying dynamics associated with the hurricane inner-core asymmetries and their contribution to intensity and structural changes are not well understood. One school of thought relates the convectively forced VRWs and the subsequent axisymmetrization process induced by the differential wind speeds of the background vortex to the intensification of an incipient weak vortex. Such conclusions are based in a number of earlier works that study the evolution of initially imposed vorticity perturbations on the symmetric flow, presumed to be the end result of adjustment to convective heating. In contrast, another school studying the evolution of asymmetric temperature and asymmetric diabatic heat sources (intended to represent one step closer to reality) concluded that purely asymmetric heat sources will generally lead to the weakening of the symmetric vortex.

In this study, a dry but thermally forced version of WRF is used and a series of numerical experiments are designed to revisit first the role of asymmetric convection to the intensification of a weak vortex from the perspective of wave radiation and wave–mean flow interactions (control experiment).

In the control experiment, the forcing is a single pulse that takes the shape of a localized double-cluster thermal anomaly with moderate magnitudes. With the aid of the conservation law of pseudomomentum, it is found that the response of the initial symmetric vortex to the asymmetric heating is dominated by the radiation of a sheared VRW that follows a damping behavior due to the axisymmetrization process. From the EP flux diagnostics, it is shown that the damped sheared VRW transfers angular momentum radially inward and acts to accelerate the symmetric flow close to the initial RMW. Sensitivity tests performed by changing the structure as well as the magnitude of the initial heating confirm the robustness of the results and emphasize further the significance of the wave–mean flow interaction mechanism to the intensification process. Finally, the azimuthally averaged eddy kinetic energy budget calculations performed complement our findings and show how the axisymmetrization process acts to increase the kinetic energy of the symmetric flow.

While the above results complement the findings of one school of thought, they contradict the findings of the other. With the aid of the governing equation for the PV perturbation, the underlying physical reasons why disagreements related to the role of asymmetric convection in the intensification process exist (intensification versus weakening) are addressed. In the present dry framework (with friction being neglected), four main parameters that may significantly alter the structure and the magnitude of the convectively induced PV perturbations are recognized. These include the baroclinicity of the basic-state vortex, the radial gradients of the thermal forcing, the vertical gradients of the thermal forcing, and the horizontal shape of the forcing. Two sets of sensitivity experiments are designed to further address how the baroclinicity and the gradients (radial and vertical) of the thermal forcing may affect the generated PV perturbation and in turn the intensification of the incipient vortex (experiments I and II). More specifically, experiment I (II) deals with a barotropic (highly baroclinic) vortex, and two individual simulations are performed in which the thermal forcing is characterized by large vertical gradients and large radial gradients.

From the PV perturbation governing equation, it is realized that in the presence of a barotropic basic-state vortex and for a thermal forcing with large vertical gradients, the resultant PV perturbation will be characterized by a dipole pattern (positive and negative PV2) in the vertical direction. In such configuration, the PV perturbation can be subjected directly to an axisymmetrization process, thus following a damping behavior. In the presence of a highly baroclinic vortex and for a thermal forcing with large radial gradients, the resultant PV perturbation will be characterized by a dipole pattern in the radial direction (inward and outward of the maximum thermal forcing). Depending on the horizontal shape of the thermal forcing (localized versus harmonic perturbation), the resultant PV perturbation can follow an amplifying behavior, extracting energy from the basic-state vortex, thereby leading to a weakening of the vortex. This, however, may be expected to occur when the thermal forcing has a pure harmonic shape. Such forcing will induce two concentric sets of PV perturbations with alternative signs in the azimuthal direction. Same-sign inner and outer PV anomalies may then possibly come into phase, amplifying each other at the expense of the energy of the basic-state vortex.

In both experiments (I and II) in which the thermal forcing takes the shape of a localized double-blob thermal anomaly, regardless of the baroclinicity and the gradients (both radial and vertical) of the forcing, the resultant PV perturbation followed a damping behavior, and no amplification that can justify weakening of the basic-state vortex was observed. An additional sensitivity test in which the forcing resembles a quadruple cluster with large radial gradients left the above conclusions unaltered. Taking into account that the shape of observed convective bursts is more likely to resemble the shape of the forcing used in the present study, rather than having a pure harmonic shape, suggests that the present findings may be more realistic than previous studies.

Acknowledgments

This research is sponsored by Natural Science and Engineering Research Council of Canada (NSERC) and Hydro-Quebec through the IRC program.

APPENDIX A

Hydrostatic Primitive Equations in Cylindrical and Isentropic Coordinates

Here we want to provide the governing equations for a hydrostatic flow in the cylindrical and isentropic framework:

 
formula
 
formula
 
formula
 
formula
 
formula

Here u and υ represent the radial and tangential wind component, respectively, and is the diabatic heating rate and equivalently the vertical velocity in isentropic coordinates. The variables p, σ, f, pa, R, Cp, g, M, and Fr and Fλ denote the pressure, isentropic density, Coriolis parameter, ambient surface pressure, gas constant of dry air, specific heat at constant pressure, gravitational acceleration, Montgomery streamfunction (M = CpT + ϕ, where T is the temperature and ϕ is the geopotential), and body forces per unit mass, respectively.

APPENDIX B

Derivation of the Azimuthal-Mean Eddy Kinetic Energy Equation in Cylindrical Coordinates

The horizontal momentum equations and the mass continuity can be written in cylindrical coordinates as

 
formula
 
formula
 
formula

Here w is the vertical velocity and ρ is the density that is considered to be constant. The kinetic energy, azimuthal-mean kinetic energy, and eddy kinetic energy are defined respectively as

 
formula

Multiplying Eq. (B1) by u and Eq. (B2) by υ and then adding the resulting equations gives

 
formula

Utilizing mass continuity, Eq. (B5) can be rewritten as

 
formula

Azimuthally averaging the horizontal momentum equations and then following a similar procedure as above, the prognostic equation for can be obtained:

 
formula

By subtracting Eq. (B7) from Eq. (B6) and then azimuthally averaging the resulting equation, one can obtain the prognostic equation for the azimuthally averaged eddy kinetic energy , given by

 
formula

REFERENCES

REFERENCES
Anthes
,
R. A.
,
1972
:
Development of asymmetries in a three-dimensional numerical model of the tropical cyclone
.
Mon. Wea. Rev.
,
100
,
461
476
, doi:.
Enagonio
,
J.
, and
M. T.
Montgomery
,
2001
:
Tropical cyclogenesis via convectively forced vortex Rossby in a shallow water primitive equation model
.
J. Atmos. Sci.
,
58
,
685
706
, doi:.
Hendricks
,
E. A.
,
M. T.
Montgomery
, and
C. A.
Davis
,
2004
:
The role of “vortical” hot towers in the formation of Tropical Cyclone Diana (1984)
.
J. Atmos. Sci.
,
61
,
1209
1232
, doi:.
Melander
,
M. V.
,
J. C.
McWilliams
, and
N. J.
Zabusky
,
1987
:
Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation
.
J. Fluid Mech.
,
178
,
137
159
, doi:.
Möller
,
J. D.
, and
M. T.
Montgomery
,
1999
:
Vortex Rossby waves and hurricane intensification in a barotropic model
.
J. Atmos. Sci.
,
56
,
1674
1687
, doi:.
Möller
,
J. D.
, and
M. T.
Montgomery
,
2000
:
Tropical cyclone evolution via potential vorticity anomalies in a three-dimensional balance model
.
J. Atmos. Sci.
,
57
,
3366
3387
, doi:.
Montgomery
,
M. T.
, and
R. J.
Kallenbach
,
1997
:
A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes
.
Quart. J. Roy. Meteor. Soc.
,
123
,
435
465
, doi:.
Montgomery
,
M. T.
, and
J.
Enagonio
,
1998
:
Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model
.
J. Atmos. Sci.
,
55
,
3176
3207
, doi:.
Montgomery
,
M. T.
,
M. E.
Nicholls
,
T. A.
Cram
, and
A. B.
Saunders
,
2006
:
A vortical hot tower route to tropical cyclogenesis
.
J. Atmos. Sci.
,
63
,
355
386
, doi:.
Nolan
,
D. S.
, and
M. T.
Montgomery
,
2002
:
Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability, and evolution
.
J. Atmos. Sci.
,
59
,
2989
3020
, doi:.
Nolan
,
D. S.
, and
L. D.
Grasso
,
2003
:
Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part II: Symmetric response and nonlinear simulations
.
J. Atmos. Sci.
,
60
,
2717
2745
, doi:.
Nolan
,
D. S.
,
M. T.
Montgomery
, and
L. D.
Grasso
,
2001
:
The wavenumber-one instability and trochoidal motion of hurricane-like vortices
.
J. Atmos. Sci.
,
58
,
3243
3270
, doi:.
Nolan
,
D. S.
,
Y.
Moon
, and
D. P.
Stern
,
2007
:
Tropical cyclone intensification from asymmetric convection: energetics and efficiency
.
J. Atmos. Sci.
,
64
,
3377
3405
, doi:.
Shapiro
,
L. J.
,
2000
:
Potential vorticity asymmetries and tropical cyclone evolution in a moist three-layer model
.
J. Atmos. Sci.
,
57
,
3645
3662
, doi:.
Wang
,
Y.
,
2002
:
Vortex Rossby waves in a numerically simulated tropical cyclone. Part I: Overall structure, potential vorticity, and kinetic energy budgets
.
J. Atmos. Sci.
,
59
,
1213
1238
, doi:.

Footnotes

1

Note that NG03 made use of the definition of PV rather than the prognostic PV equation.

2

A dipole PV pattern is to be expected, assuming that the largest heating magnitudes occur away from the surface.