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Balanced Contribution to the Intensification of a Tropical Cyclone Simulated
in TCM4: Outer-Core Spinup Process*
HIRONORI FUDEYASU1 AND YUQING WANG
International Pacific Research Center, School of Ocean and Earth Science and Technology,
University of Hawaii at Manoa, Honolulu, Hawaii(Manuscript received 5 April 2010, in final form 17 November 2010)
ABSTRACTThe balanced contribution to the intensification of a tropical cyclone simulated in the three-dimensional,
nonhydrostatic, full-physics tropical cyclone model version 4 (TCM4), in particular the spinup of the outercore circulation, is investigated by solving the Sawyer–Eliassen equation and by computing terms in theazimuthal-mean tangential wind tendency equation. Results demonstrate that the azimuthal-mean secondary
circulation (radial and vertical circulation) and the spinup of the midtropospheric outer-core circulation in thesimulated tropical cyclone are well captured by balance dynamics. The midtropospheric inflow develops in
response to diabatic heating in mid–upper-tropospheric stratiform (anvil) clouds outside the eyewall in active
spiral rainbands and transports absolute angular momentum inward to spin up the outer-core circulation.
Although the azimuthal-mean diabatic heating rate in the eyewall is the largest, its contribution to radialwinds and thus the spinup of outer-core circulation in the middle troposphere is rather weak. This is because
the high inertial stability in the inner-core region resists the radial inflow in the middle troposphere, limiting
the inward transport of absolute angular momentum. The result thus suggests that diabatic heating in spiralrainbands is the key to the continued growth of the storm-scale circulation.

1. Introduction
To first order, the primary circulation of a strong tropical cyclone can be considered as a warm-cored, quasiaxisymmetric vortex in gradient wind and hydrostaticbalance. As a tropical cyclone evolves slowly while its
primary circulation remains in gradient wind and hydrostatic balance, the secondary circulation (radial and vertical circulation) can be considered as a result of the
response to both diabatic heating and momentum forcing, including surface friction. The secondary circulationtransports high absolute angular momentum inward tospin up the tropical cyclone primary circulation. This
spinup process can be well described by the Sawyer–* School of Ocean and Earth Science Technology Publication
Number 8060 and International Pacific Research Center Publication Number 741.
1
Current affiliation: Faculty of Education and Human Sciences,
Yokohama National University, Yokohama, Japan.

Corresponding author address: Dr. H. Fudeyasu, Yokohama
National University, 79-1 Tokiwadai, Hodogaya-ku, Yokohama,
Kanagawa, Japan, 240-8501.
E-mail: fude@ynu.ac.jp
DOI: 10.1175/2010JAS3523.1
Ó 2011 American Meteorological Society

Eliassen (SE) equation following the classic work ofEliassen (1952).
The general form of the SE equation is a two-dimensional
diagnostic equation describing the azimuthal-mean
secondary circulation forced by 1) axisymmetric diabatic
heating, 2) eddy heat flux, 3) axisymmetric surface friction and subgrid-scale vertical mixing (frictional forcing),
and 4) eddy momentum flux. Many theoretical studies
have applied the SE equation to calculate the secondary
circulation in response to idealized forcing mechanisms,
such as point sources of diabatic heating and frictional
forcing, excluding eddy heat and momentum fluxes(e.g., Willoughby 1979; Shapiro and Willoughby 1982;
Schubert and Hack 1982; Holland and Merrill 1984; Hackand Schubert 1986; Pendergrass and Willoughby 2009). In
these studies, diabatic heating is generally assumed tooccur in a convective ring of the eyewall (EW) and isgiven as a prescribed function of radius and height (e.g.,
Pendergrass and Willoughby 2009). The solution of theSE equation in response to prescribed axisymmetriceyewall-like heating produces an updraft through the locus of heat source with the lower-tropospheric inflow andupper-tropospheric outflow, consistent with the axisymmetric conceptual model of a tropical cyclone.

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Many aspects of a tropical cyclone can be explained
based on the SE equation or often alternatively termedbalance dynamics. In particular, contribution by balance
dynamics to the intensification of a tropical cyclone haslong been studied as a basic aspect of tropical cyclone
dynamics in not only theoretical studies but also diagnostic
studies from observations (e.g., Molinari and Vollaro
1990; Molinari et al. 1993) and full or simplified-physics
model simulations (e.g., Persing et al. 2002; Möller andShapiro 2002; Hendricks et al. 2004; Montgomery et al.
2006; Bui et al. 2009).
Persing et al. (2002) and Möller and Shapiro (2002)
diagnosed the secondary circulation using the SE equation and discussed the intensity change of Hurricane Opal
(1995). They showed that, with all forcing terms derived
from the full-physics model output, the SE equation solution captured the model secondary circulation quite
well. Möller and Shapiro (2002) found that, althoughcontributions by diabatic heating and frictional forcing totropical cyclone intensification were much greater, theeddy processes also contributed to the rapid intensificationof Hurricane Opal to some degree. Hendricks et al. (2004)
and Montgomery et al. (2006) demonstrated that thesecondary circulation during the tropical cyclone genesisstage in a cloud-resolving model simulation could be well
explained by balance dynamics in the context of the SE
equation. Bui et al. (2009) investigated the extent to which
the secondary circulation and intensification of a simulated tropical cyclone can be explained by balance dynamics. Their results further confirmed that the solutionof the SE equation with all forcings from a simplifiedphysics model simulation well captured the secondary
circulation of a simulated tropical cyclone, except in theboundary layer, where the unbalanced flow becomessignificant. They also showed that inflow above theboundary layer in response to diabatic heating transports
absolute angular momentum inward, spinning up the primary circulation in regions more than 150-km radius or in
the outer-core region (between radii 18 and 2.58 latitude)
as defined by Weatherford and Gray (1988a,b). This result thus demonstrated that, in addition to the storm
intensity, the intensification of the primary circulationin the outer-core region can also be explained by balance dynamics.
Pendergrass and Willoughby (2009) found that thebalanced response of the secondary circulation is sensitive
to the radial location of heat source and the inertial stability of the vortex itself. In the inertially stable regions,the secondary circulation is confined horizontally close to
the heat source. Because the inertial stability is generallyhigh in the eyewall region, diabatic heating in the eyewall
is expected to force relatively weak midtropospheric inflow in the inner-core region, contributing little to the431

spinup of the outer-core circulation. This suggests that diabatic heating in the eyewall region enhanced the primarycirculation mainly in the inner-core region and thus thestorm intensity, whereas heat sources outside the eyewall could contribute the changes in primary circulationin the outer-core region and thus the size of a tropical
cyclone. This indeed has been studied based on sensitivity experiments in idealized numerical simulations by
Wang (2009), Hill and Lackmann (2009), and Xu andWang (2010a). These studies show that diabatic heating
in both inner and outer spiral rainbands contributes tothe size increase of tropical cyclones. Therefore, diabatic
heating outside the eyewall may contribute significantly tothe intensification of the primary circulation in the outercore region, where the inertial stability is much weaker
than that in the eyewall. Although Bui et al. (2009) demonstrated the importance of diabatic heating to the spinupof the outer-core circulation above the inflow boundary
layer they did not evaluate the relative contributions byheat source in different regions in their simulated storm.
The objective of this study is to extend Bui et al.’s (2009)
work to examine forcings to the spinup of the outer-coreprimary circulation of a simulated tropical cyclone in afull-physics model under idealized conditions by solving
the SE equation and by computing terms in the azimuthalmean tangential wind equation. The SE equation is a linear partial differential equation and thus, for suitableboundary conditions, solutions to different forcings are
additive. This allows us to evaluate contribution of heatsource in different regions to the spinup of the outer-corecirculation of the simulated tropical cyclone. The restof the paper is organized as follows: the next section briefly
describes the tropical cyclone model version 4 (TCM4)
experiment and the SE equation. Section 3 validates theSE equation as a tool to diagnose the secondary circulation in the simulated tropical cyclone. Contributionsof different forcing mechanisms to the spinup of the outercore circulation in the simulated tropical cyclone are
evaluated based on the azimuthal-mean tangential wind
tendency equation in section 4. The main findings are
summarized and discussed in the last section.

2. Methodology
a. TCM4 experiment
TCM4 is a fully compressible, quadruply nested, nonhydrostatic tropical cyclone model. The model physics
include an E–« turbulence closure scheme for subgridscale vertical turbulent mixing (Langland and Liou 1996);
a modified Monin–Obukhov scheme for the surface flux
calculation (Fairall et al. 2003); explicit treatment of
mixed-phase cloud microphysics (Wang 2001); fourthorder horizontal diffusion with deformation-dependent

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diffusion coefficient for all prognostic variables, except forthat related to the mass conservation equation; a Newtonian cooling term added to the potential temperatureequation to mimic radiative cooling as used in Rotunno
and Emanuel (1987); and dissipative heating due to molecular friction, which is included by adding the turbulent
kinetic energy dissipation rate « into the thermodynamic
equation. Because no large-scale environmental flow is
included in this study, convection is mainly active within
about a radius of 300 km from the storm center and thus iscovered by the finest innermost domain. Therefore, cumulus parameterization is not used in this study. A complete description of TCM4 and its recent applications tothe studies of tropical cyclone inner-core dynamics can befound in Wang (2007, 2008a,b, 2009), Wang and Xu(2010), and Xu and Wang (2010a,b).
The model has 32 vertical levels with relatively highresolution both in the lower troposphere and near the
tropopause, and it has its lower boundary at a flat surface
with the unperturbed surface pressure of 1010 hPa and its
top at about 38 km. The four meshes are quadruply nested with two-way interactive nesting and with the innermeshes automatically moving to follow the model storm(Wang 2001, 2002). The outermost mesh has 251 3 251
grid points with a horizontal resolution of 67.5 km. The
three inner meshes have 109 3 109, 127 3 127, and 271 3
271 grid points with horizontal resolutions of 22.5, 7.5, and2.5 km, respectively.
Following the experimental design of Wang (2008a,b),
the model was initialized with an axisymmetric cyclonicvortex in a quiescent environment over the ocean with aconstant SST of 298C and was run on an f plane of 188N.
The initial thermodynamic structure of the unperturbed
model atmosphere is defined as the western Pacific clearsky environment given in Gray et al. (1975). The initial
cyclonic vortex has a maximum tangential wind speed of
20 m s21 at the surface and at a radius of 80 km anddecreases sinusoidally with pressure to vanish at 100 hPa.
The mass and thermodynamic fields are obtained bysolving the nonlinear balance equation as described in
Wang (2001). Because our focus is on the detail structureof the developed tropical cyclone, the results in the
innermost mesh with 15-min intervals after the 20-h spinup period are used in the diagnostic and analyses discussed below. Thus, the TCM4 output with high temporaland spatial simulations allows us to study contributions
to the intensification of the storm-scale circulation byheating source in different regions in the simulated storm.

b. The Sawyer–Eliassen equation
With the Boussinesq approximation, the SE equationin the radius–pseudoheight (r–Z) coordinates (Hendricks
et al. 2004; Montgomery et al. 2006) can be written as

VOLUME 68› A ›cB ›c› C ›cB ›c1
1
1›r r ›r
r ›Z›Z r ›Z
r ›r
5

›(jF)
›Q
1
,
›Z
›r

(1)

where r is radius, c is the azimuthal-mean transverse
streamfunction explained by y r 5 (1/r)(›c/›Z), w 5(1/r)(›c/›r), y r is the azimuthal-mean radial wind, and w
is the azimuthal-mean vertical velocity. The term Z is thepseudoheight defined as Z 5 [1  ( p/p0 )Rd /Cp ](Cp u0 /g),
where p is pressure, p0 is a reference pressure (1000 hPa),
Rd is the gas constant for dry air, Cp is the specific heatof dry air at constant pressure, g is the acceleration due
to gravity, and u0 is a reference potential temperature(300 K). The overbar denotes the azimuthal mean at aconstant pseudoheight about the circulation center. The
left-hand side of Eq. (1) includes three parameters defined as
A 5 N2 5B 5 jg ›u, static stability;
u0 ›Z

›y t
, baroclinicity;›ZandC 5 jh, inertial stability,

(2)

(3)
(4)

where yt is the azimuthal-mean tangential wind, u is theazimuthal-mean potential temperature, h 5 f 0 1 z is
the azimuthal-mean absolute vertical vorticity, z is theazimuthal-mean relative vertical vorticity, f0 is the planetary vorticity at the cyclone center (188N), andj 5 f 0 1 2y t /r is the vortex inertial parameter of theswirling flow. Terms on the right-hand side of Eq. (1) arethose due to heat source and momentum forcing,
Q5



g
›u9
›u9 _
y9r
 w9
1 u , heating source, and
u0
›r
›Z
(5)

F 5 y9r z9  w9›y9t
1 F sg , momentum forcing,›Z(6)where the prime denotes the deviation (eddy) from theazimuthal mean. In Eq. (5), the first two terms represent
the azimuthal-mean eddy radial and vertical fluxes of theasymmetric potential temperature and u_ is the azimuthalmean diabatic heating rate. Terms in Eq. (6) represent theazimuthal-mean eddy radial and vertical fluxes of theasymmetric tangential wind and the azimuthal-mean frictional forcing F sg , including subgrid-scale vertical turbulent
mixing and surface friction. TCM4 provides a complete
dynamically consistent dataset, including diabatic heating u_
and friction F sg with high temporal and spatial resolutions.

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Given the structure of the vortex, thus the parameters
A, B, and C defined in Eqs. (2)–(4) and the distribution ofheating source and momentum forcing in Eqs. (5) and (6),the solution of Eq. (1) will give the secondary circulation
if the elliptical discriminant condition A  C 2 B2 . 0 is
satisfied everywhere in the vortex and proper boundary
conditions are specified (Shapiro and Willoughby 1982).
Although the discriminant derived from TCM4 output ispositive throughout the middle–lower troposphere, there
are small negative values in the upper-tropospheric outflow (see Fig. 4a), similar to those in Hendricks et al.
(2004). In our calculations, if the absolute vorticity
h 5 f 0 1 z , 0.5 3 f 0 , we set h 5 0.5 3 f 0 to allow thediscriminant to be satisfied. This does not affect the global
solution to the SE equation as we tested (Fudeyasu et al.
2010). The boundary conditions for the SE equation are
C 5 0 at r 5 0 km, Z 5 0 km, and Ztop and ›C/›r 5 0 as
r 5 ‘, where Ztop is the top of the computational domain,which is 0–18 km in the vertical with 500-m grid intervaland 0–300 km in the radial direction with 2.5-km grid
spacing. To match the SE equation calculation, all fields
of TCM4 output are first interpolated into the radius–
pseudoheight coordinates.

3. Diagnostics of the secondary circulation
Figure 1 shows the time–radius cross section of theazimuthal-mean tangential wind averaged in the middle
troposphere between 3- and 9-km heights. The simulated
tropical cyclone intensified rapidly until 48 h with the
maximum azimuthal-mean tangential wind increased toabout 50 m s21 and the radius of maximum wind decreased to about 30 km. This intensification was followed
by a quasi-steady evolution. In the outer-core region,1 theazimuthal-mean tangential wind strengthened intermittently in the simulation, implying a corresponding increasein the storm-scale size. Three episodes of the increase in
the outer-core azimuthal-mean tangential wind occurred
during 32–48, 56–78, and 86–96 h, respectively, corresponding to the episodes of active spiral rainbands as will
be discussed later. This indicates that the spinup of theouter-core circulation in the middle troposphere could beclosely related to the activity of spiral rainbands in thesimulated storm.
Figure 2 shows the surface rain rate, its azimuthal mean,
the east–west cross section of radar reflectivity across the1
Here we define the inner core as the region including the eye,eyewall, and inner rainbands within about a 70-km radius, whereas
the outer core is defined as the region immediately outside theinner-core region including the outer rainbands and the associatedanvil clouds in the outflow layer in the upper troposphere between
radii of 70 and 210 km.

433

FIG. 1. Time–radius cross section of the azimuthal-mean tangential wind (m s21) averaged between 3- and 9-km heights. Contour
interval is 2 m s21.

tropical cyclone center, and the azimuthal-mean radarreflectivity, all averaged between 71 and 72 h of simulation, during which the storm-scale size increased (Fig. 1).
The surface rainfall field (Fig. 2a) displays a quasiaxisymmetric pattern in the inner-core region and shows
little rainfall in the eye, which is surrounded by a closed
heavy rainfall ring in the eyewall between radii of 15 and
30 km. Immediately outside the eyewall are the wellorganized fine inner spiral rainbands, mainly within a radius of around 70 km. Outside the inner core, convection
is loosely organized in two major outer spiral rainbands. The azimuthal-mean rainfall rate shows two peaks(Fig. 2b): one over 80 mm h21, at about 20-km radius in
the eyewall, and the other at about 40 mm h21, near the
40-km radius associated with inner rainbands (IB).
The eyewall is characterized by high reflectivity
throughout the troposphere (Fig. 2c). The weak reflectivity in the eye indicates the existence of hub clouds, andthe high reflectivity immediately outside the eyewall is
associated with inner spiral rainbands. The relatively highreflectivity between radii of 90 and 110 km to the west of
the storm center is associated with an outer rainband(OB; Fig. 2a). Weak reflectivity in the middle–upper
troposphere extending radially outward up to a radius of
180 km indicates the existence of anvil (stratiform) clouds
in the outflow layer. The azimuthal-mean radar reflectivity
shows high reflectivity in the eyewall and inner rainband
and relatively weak reflectivity in the outer rainband
and anvil clouds (Fig. 2d), consistent with observations(e.g., Barnes et al. 1983; Marks and Houze 1987; Markset al. 2008).
The axisymmetric structure and some other dynamicalparameters of the simulated storm averaged between 71
and 72 h are shown in Figs. 3 and 4. The storm develops astrong inflow in the boundary layer, an outflow in the434

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FIG. 2. (a) Plane view of rain rate (mm h21), (b) radial distribution of the azimuthal-mean rain rate (mm h21),(c) vertical (pseudoheight) cross section of radar reflectivity (dBZ) along the east–west direction across the storm
center, and (d) the azimuthal-mean radar reflectivity (dBZ), all hourly averaged between 71 and 72 h of simulation.
Regions with rain rate greater than 1 mm h21 are shaded in (a). Regions with radar reflectivity higher than 15 and 3
dBZ are heavily and lightly shaded in (c) and (d), respectively.

upper troposphere (Fig. 3a), strong updrafts in the eyewall
and inner rainband, and descent in the eye (Fig. 3b). In
addition to the inflow in the boundary layer, a relatively weak inflow occurs outside the eyewall in the middle troposphere between 3- and 9-km heights (Fig. 3a).
Tangential winds show the maximum over 70 m s21, justabove the surface at a radius of about 20 km under the
eyewall (Fig. 3c), The storm has a warm core in the
middle–upper troposphere with the maximum temperature anomaly over 18 K (Fig. 3d). The absolute verticalvorticity shows an off-center maximum inside the eyewall
throughout the troposphere (Fig. 4a). The surface of absolute angular momentum defined as L 5 ry t 1 f 0 (r2 /2)
is upright in the eyewall in the middle–lower tropospherebut tilts outward with height in the upper troposphereand outside the eyewall (Fig. 4b). The inertial stability

parameter is high in the inner-core region (larger than
1 3 1026 s22) but decreases rapidly outward and becomesless than 1 3 1028 s22 outside a radius of 60 km in theouter-core region (Fig. 4c).
Because the SE equation is based on the assumptionof gradient wind balance, it is necessary to see to what
extent the simulated tropical cyclone remains in thequasi-balanced state. Figure 4d shows the radial–height
distribution of agradient tangential winds, defined asy a 5 y t  y g , where y g is gradient wind calculated fromthe azimuthal-mean perturbation pressure field in TCM4.
Positive agradient winds, namely supergradient winds,appear in the boundary layer under the eyewall becauseof surface friction, consistent with previous studies (e.g.,
Kepert and Wang 2001; Smith and Montgomery 2008;
Smith et al. 2009). Relatively large positive and negative

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435

FIG. 3. The vertical structure of the simulated storm in pseudoheight coordinates. The azimuthal-mean (a) radialwind (m s21), (b) vertical velocity (1021 m s21), (c) tangential wind (m s21), and (d) temperature anomaly (K), allaveraged between 71 and 72 h of simulation. Contours are at 0 (thick), 61, 62, 65, 610, and 615 m s21, and regionswith values greater than 1 m s21 (less than 21 m s21) are heavily (lightly) shaded in (a). Contours are at 0 (thick),62, 65, 68, 611, 614, 617, 620, and 623 3 1021 m s21, and regions with values larger than 2 3 1021 m s21 (8 31021 m s21) are lightly (heavily) shaded in (b). Contour interval is 10 m s21 in (c). Contour interval is 3 K, andregions with values larger than 9 K (18 K) are lightly (heavily) shaded in (d).

agradient winds occur in the upper-tropospheric outflow
layer mainly because of the eddy effects, similar to thoseshown in Möller and Shapiro (2002). The agradient winds
are overall small compared with the gradient winds, inparticular in the middle troposphere below 10-km height
above the inflow boundary layer, consistent with Bui et al.
(2009). Therefore, the SE equation is applicable for the
purpose of this study.
Figures 5a,b show the radial wind and vertical velocity
obtained from the SE equation with all forcing terms in
Eqs. (5) and (6). The SE equation solution captures thesecondary circulation simulated in TCM4 well (Figs.
3a,b). To validate quantitatively, we show in Figs. 5c,d thenormalized differences between the radial wind andvertical velocity derived from the SE equation and thosefrom TCM4, defined as D 5 jSE 2 TCM4j/jTCM4j. Notethat regions with radial wind less than 1 m s21 and vertical velocity less than 0.2 m s21 in TCM4 are not shownin Figs. 5c,d. The D values in vertical velocity are small(less than 0.2) in the inner-core region, and those in radialwind are small (less than 0.3), except in the boundary
layer and in the upper-tropospheric outflow layer. The
D values larger than 0.4 are shown in regions where relatively strong agradient winds occur (Fig. 4d). Therefore,
the azimuthal-mean secondary circulation simulated in
TCM4 can be well explained by balance dynamics, inparticular in the middle–lower troposphere, except in theboundary layer.

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FIG. 4. The vertical cross section of the azimuthal-mean (a) absolute vertical vorticity (1024 s21), (b) absolute
angular momentum (106 m2 s21), (c) inertial stability (1028 s22), and (d) agradient winds (m s21), all averaged
between 71 and 72 h of simulation. Contours are at 0, 60.5, 5, 10, 15, 20, 30, 40, 50, 60, and 70 3 1024 s21, and regionswith values greater than 5 3 1024 s21 are heavily shaded and regions with values less than 0.5 3 f0 are lightly shaded in(a). Contour interval is 0.5 3 106 m2 s21 in (b). Contours are at 0.5, 1, 10, 100, 500, 1000, and 5000 3 1028 s22, andregions with values larger than 1 3 1028 s22 (100 3 1028 s22) are lightly (heavily) shaded in (c). Contour interval is
1 m s21, and regions with values greater than 2 m s21 (less than 22 m s21) are heavily (lightly) shaded in (d).

The above results demonstrate that the SE equation
can be used to examine contributions of different forcing
processes to the secondary circulation in the simulated
tropical cyclone. Here, three calculations using the SE
equation are performed: each calculation only includes
one forcing term, namely the axisymmetric diabatic heating [the last term in Eq. (5)], the axisymmetric frictional
forcing [the last term in Eq. (6)], or the eddy terms associated with the azimuthal-mean eddy radial and verticalfluxes of asymmetric angular momentum or the asymmetric potential temperature [the first two terms in Eqs. (5)and (6)]. Results from these calculations (Fig. 6) indicatethat the secondary circulation in the middle–lower troposphere above the boundary layer is mainly a balanced

response to the axisymmetric diabatic heating, consistent
with previous studies (Möller and Shapiro 2002; Bui et al.
2009). The shallow outflow immediately above the inflow
boundary layer is forced mainly by the frictional forcing.
Contributions to the secondary circulation by theazimuthal-mean diabatic heating in different regions inthe storm can be examined using the SE equation as
well. Based on the radar reflectivity in Figs. 2c,d and theazimuthal-mean diabatic heating rate in Fig. 7a, we simply
divide the azimuthal-mean diabatic heating rate into theheating rates in the eyewall within a radius of about
30 km, in inner rainbands between radii of about 30
and 70 km, and in outer rainbands and the mid–uppertropospheric anvil clouds beyond 70 km (Figs. 7b–d,

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437

FIG. 5. The azimuthal-mean (a) radial wind (m s21) and (b) vertical velocity (1021 m s21) derived from the SE
equation, (c) the normalized difference D in radial wind, and (d) the normalized difference D in vertical velocity, allaveraged between 71 and 72 h of simulation, as in Figs. 3a,b. Regions with D values greater than 0.01 are shaded in (c)and (d).

respectively). Figure 8 shows the secondary circulation in
response to each heating source. In response to the narrow eyewall heating are a weak subsidence in the eye andan in–up–out circulation, namely a boundary layer inflow,
strong eyewall updraft, an upper-tropospheric outflow,
and a weak midtropospheric inflow just outside the eyewall (Figs. 8a,b), very similar to that shown in previous
studies with specified eyewall-like heating (Shapiro andWilloughby 1982; Pendergrass and Willoughby 2009).
The response of the secondary circulation to heating ininner rainbands is very similar to that in the eyewall, except for the midtropospheric inflow (Figs. 8c,d). Although
the azimuthal-mean diabatic heating rate in inner rainbands is only about 50% of diabatic heating in the eyewall(Figs. 7b,c), the midtropospheric inflow extends farther
radially outward to beyond 200 km from the storm center(Fig. 8c). Diabatic heating in inner rainbands shows two

peaks in the vertical: one in the upper troposphere and one
in the middle troposphere (Fig. 7c). Additional calculation with heating either below or above 6-km heightindicates that heating in the middle–upper troposphere
contributes much more to that in the midtroposphericinflow than heating in the middle–lower troposphere (notshown). The heating rate in outer rainbands and in theupper-tropospheric outflow layer between the radii of 70
and 150 km is generally less than 20% of that in theeyewall and is dominated by heating in the anvil clouds(Figs. 7b,d). It forces an inflow in the middle troposphereand an outflow in the upper troposphere in the outer-core
region with quite weak azimuthal-mean vertical motion(Figs. 8e,f). The results thus suggest that the midtropospheric inflow in the outer-core region is mainly inducedby mid–upper-tropospheric heating in both inner andouter rainbands.

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FIG. 6. (left) The azimuthal-mean radial wind (m s21) and (right) the vertical velocity (1021 m s21) derived fromthe SE equation with the azimuthal-mean (a),(b) diabatic heating forcing; (c),(d) frictional forcing; and (e),(f) eddy
heating and momentum forcing, all averaged between 71 and 72 h of simulation, as in Figs. 3a,b.

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439

FIG. 7. The azimuthal-mean diabatic heating rate (1023 K s21) of (a) total; (b) in the EW; (c) in IB; and (d) in OB
and the mid–upper-tropospheric anvil clouds, all averaged between 71 and 72 h of simulation. Contours are at
0 (thick), 0.1, 1, 2, 3, 5, 7, and 10 3 1023 K s21, and regions with values greater than 0.1 3 1023 K s21 are shaded.

4. Contributions to the spinup of outer-corecirculation
To quantify the contribution by balance dynamics to the
spinup of the primary circulation in the outer-core regionof the simulated storm, we calculated the individual
terms contributing to the azimuthal-mean tangential
wind tendency. The tangential wind tendency equationin cylindrical coordinates following the moving storm
can be written as (Xu and Wang 2010a,b)
›y t
›y
›y9
5 y r h  y9r h9  w t  w9 t 1 F sg ,
›t
›z
›z

(7)

where t is time and z is physical height. The five terms on
the right-hand side are radial advection of the azimuthalmean absolute angular momentum by the azimuthal-mean

radial wind (mean radial advection), radial advection by
eddy processes (eddy radial advection), vertical advectionof azimuthal-mean tangential wind by the azimuthal-meanvertical motion (mean vertical advection), vertical advection by eddy processes (eddy vertical advection), andsurface friction and subgrid-scale vertical mixing (friction
and vertical mixing), respectively. To be consistent withdiscussions in the previous section, the analysis results are
displayed in r–Z coordinates.
Figure 9 shows the analysis results averaged between
71 and 72 h of simulation, the same period as discussed in
section 3. The time tendency of azimuthal-mean tangential wind estimated from the tangential wind tendency equation [Eq. (7); Fig. 9b] shows a distribution
very similar to that simulated in TCM4 (Fig. 9a), although
the former looks noisy because of the large variability ofcontributions by eddy processes (Fig. 9e). The difference

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FIG. 8. (left) The azimuthal-mean radial wind (m s21) and (right) the vertical velocity (1021 m s21) derived fromthe SE equation with the azimuthal-mean diabatic heating sources in (a),(b) the EW (Fig. 7b); (c),(d) the IB (Fig. 7c);and (e),(f) the OB and the mid–upper-tropospheric anvil clouds (Fig. 7d), all averaged between 71 and 72 h ofsimulation, as in Figs. 3a,b.

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FIG. 9. The time tendency in the azimuthal-mean tangential wind (1024 m s22) derived from (a) TCM4, (b) the sum of all terms in thetangential wind tendency equation, (c) mean radial advection, (d) mean vertical advection, (e) eddy radial and vertical advections, (f)friction, and (g) the difference in the tendency derived from TCM4 and from the tangential wind tendency equation, all averaged between
71 and 72 h of simulation. Contours are at 60.5, 62, 65, 610, 630, and 650 3 1024 m s22, and regions with values greater than 0.5 31024 m s22 (less than 20.5 3 1024 m s22) are heavily (lightly) shaded in (a)–(f). Regions with values greater than 1 3 1024 m s22 (4 31024 m s22) are heavily (lightly) shaded in (g). Solid box indicates the midtropospheric outer-core region.

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FIG. 10. Contributions to the time tendency in the azimuthal-mean tangential wind (1024 m s22)
by terms in the tangential wind tendency equation, averaged in a volume the midtroposphericouter-core region (the boxed area in the annulus shown in Fig. 9) and between 71 and 72 h ofsimulation. (a) All terms are derived from TCM4. Mean radial and vertical advections derived
from the diagnosed winds by the SE equation (b) with full forcing, diabatic heating, eddy
advection, and frictional forcing and (c) with diabatic heating in the EW, IB, and OB and mid–
upper-tropospheric anvil clouds. Filled boxes indicate the mean radial advection, open boxes
indicate the mean vertical advection, and gray boxes indicate other terms in the tangential
wind tendency equation.

between the time tendency of the simulated and estimated azimuthal-mean tangential winds is small (less than
1 3 1024 m s22) in the outer-core region throughout thetroposphere and is large (greater than 4 3 1024 m s22) inthe eyewall region (Fig. 9g). Significant positive tendencies appear in the upper part in the eyewall andoutside the eyewall in the middle troposphere, whereas
weak negative tendencies occur in the middle–lower partin the eyewall and in the upper part of the outwardly tilted
inner rainbands.
Changes in the azimuthal-mean tangential wind
above the boundary layer in the eyewall and inner rainbands result mainly from the upward and outward advection of the azimuthal-mean tangential wind by theazimuthal-mean secondary circulation (Figs. 9c,d). Surface friction and vertical mixing contribute to the spindown of tangential wind in the boundary layer (Fig. 9f).
Note that surface friction has an indirect effect in spinningup the wind by allowing air parcels to have larger inward
radial displacements (Smith et al. 2009). Eddy processes

contribute to the spinup of tangential wind inside theeyewall and the overall spindown of tangential wind outside the eyewall in the middle–lower troposphere (Fig. 9e).
The mean radial advection contributes to the spinup oftangential wind in the midtropospheric outer-core region(the boxed region in Fig. 9) with a small positive contribution by the mean vertical advection and a small negative
contribution by the eddy processes. To quantify contributions to the azimuthal-mean tangential wind tendency
in the midtropospheric outer-core region, in Fig. 10a weshow tendencies on the right-hand side of the azimuthalmean tangential wind tendency equation [Eq. (7)], averaged in a volume in the midtropospheric outer-core region(the boxed area in the annulus in Fig. 9). The simulated
and estimated tendencies agree quantitatively well (0.66 31024 versus 0.71 3 1024 m s22). The maximum contribution to the spinup of tangential wind is by the mean
radial advection (96% of estimated tendency), and thesecond contribution is by the mean vertical advection(32%). The eddy processes contribute negatively (total

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FIG. 11. As in Figs. 9c,d, but for the time tendency in the azimuthal-mean tangential wind(1024 m s22) resulting from (left) the mean radial advection and (right) the mean verticaladvection derived from the diagnosed winds by the SE equation with the azimuthal-mean diabatic heating in (a),(b) total; (c),(d) the EW; (e),(f) the IB; and (g),(h) the OB and mid–uppertropospheric anvil clouds.

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FIG. 12. The azimuthal-mean (a),(c) radial wind (m s21) and (b),(d) vertical velocity (1021 m s21) derived from(a),(b) TCM4 and (c),(d) the SE equation, (e) diabatic heating rate (1023 K s21), and (f) time tendency in theazimuthal-mean tangential wind (1024 m s22) derived from TCM4, all averaged between 79 and 80 h of simulation.
Contours are at 0 (thick), 61, 62, 65, 610, and 615 m s21, and regions with values greater than 1 m s21 (less than
21 m s21) are heavily (lightly) shaded in (a),(c). Contours are at 0 (thick), 62, 65, 68, 611, 614, 617, 620, and623 3 1021 m s21, and regions with values greater than 2 3 1021 m s21 (8 3 1021 m s21) are lightly (heavily) shadedin (b),(d). Contours are at 0 (thick), 60.1, 1, 2, 3, 5, 7, and 10 3 1023 K s21, and regions with values greater than 0.1 31023 K s21 are shaded in (e). Contours are at 60.5, 62, 65, 610, 630, and 650 3 1024 m s22, and regions withvalues greater than 0.5 3 1024 m s22 (less than 20.5 3 1024 m s22) are heavily (lightly) shaded in (f).

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FIG. 13. Time–radius cross section of the azimuthal-mean diabatic heating rate (K s21) averaged between 3- and 12-km heights.
Contours are at 0 (thick), 1, 2, 3, and 4 3 1023 K s21, and regionswith values greater than 1 3 1023 K s21 are shaded.

of 226%), whereas contribution by surface friction andvertical mixing is negligible (22%).
Because the midtropospheric outer-core region can bewell explained by balance dynamics (Fig. 5), we estimatedcontributions by the total and the individual heating
sources to the mean radial and vertical advections ofazimuthal-mean tangential wind based on the diagnosed
secondary circulation from the SE equation (Figs. 10b,c).
The mean radial and vertical advections of azimuthalmean tangential wind based on SE equation solution withall forcing processes are very close to the budget directlyfrom TCM4 output (Fig. 10a), except for a slight overestimate for the mean radial advection (0.68 3 1024
versus 0.75 3 1024 m s22) and an slight underestimate forthe mean vertical advection (0.23 3 1024 versus 0.16 31024 m s22). Consistent with the balanced contribution to
the secondary circulation as discussed in section 3 (Fig. 6),
contribution by the azimuthal-mean diabatic heating to
the mean radial and vertical advections of azimuthalmean tangential wind is the largest (Fig. 10b). Contribution by the azimuthal-mean frictional forcing to thetangential wind tendency is small and negative because of
the outflow just above the inflow boundary layer forcedby frictional forcing (Fig. 6c), whereas contribution by theeddy processes is negligible (Figs. 6e,f). Diabatic heating in the eyewall contributes little to the spinup of tangential wind in the midtropospheric outer-core region(Figs. 10c, 11), whereas heating in both inner and outerrainbands contributes largely, consistent with the response of the secondary circulation to heating in different
regions discussed in section 3. Note that the mean verticaladvection, the second largest contribution to the tangential
wind tendency, is mostly forced by diabatic heating inouter rainbands. Therefore, the intensification of theazimuthal-mean tangential winds in the midtropospheric445

outer-core region results mostly from the mid–uppertropospheric diabatic heating in both inner and outerrainbands.
Immediately after the active period of rainbands between 71 and 72 h of simulation as discussed above, aninactive period appears between 79 and 80 h (see Fig. 13).
During this period, the azimuthal-mean diabatic heating
rate is characterized by heating in the eyewall between
radii of 20 and 30 km and heating in the outer rainbands around 180-km radius is dominated by mid–uppertropospheric anvil clouds (Fig. 12e). The mean radial/vertical winds diagnosed from the SE equation (Figs. 12c,d)are qualitatively consistent with those simulated in TCM4
(Figs. 12a,b). Weak inflow appears near the 8-km height
below the upper-tropospheric heating region, whereas
weak outflow immediately above the inflow boundary
layer extends outward from the inner-core region. As aresult, the spinup of the azimuthal-mean tangential windin the midtropospheric outer-core region (Fig. 12f) ismuch smaller than that between 71 and 72 h (Fig. 9a).
Figure 13 shows the time–radius cross section of theazimuthal-mean diabatic heating rate averaged between
3- and 12-km heights, and Fig. 14f shows the time evolution of the diabatic heating rate averaged between radii of
0 and 30, 30 and 70, and 70 and 150 km, respectively. The
eyewall is featured by high heating rate after its contraction in the first 36 h. Diabatic heating between radii of 30
and 70 km shows some temporal variations, indicating thequasi-periodic development of inner rainbands in thesimulated storm (not shown). Three episodes of the elevated diabatic heating rate initiated at about a 70-km radius during 30–44, 54–74, and 82–96 h correspond to threeoutward-propagating outer rainbands that appeared outside about a 70-km radius (Fig. 13) and the outward
expansions in the azimuthal-mean tangential wind (Fig. 1).
The azimuthal-mean diabatic heating rate associated withouter rainbands and mid–upper-tropospheric anvil clouds
between radii of 70 and 150 km is much smaller than thateither in the eyewall or in inner rainbands.
Figures 14a–e show the evolution of the tangential wind
tendency in the midtropospheric outer-core region andcontributions by mean radial, vertical, and eddy advections in the tangential wind tendency equation [Eq. (7)].
The evolution of the tangential wind tendency estimatedfrom the tangential wind tendency equation [Eq. (7), Fig.
14b] is quantitatively very close to that directly obtained
from TCM4 output (Fig. 14a). The variability in the mean
radial advection mainly contributes the evolution of theazimuthal-mean tangential wind tendency (Fig. 14c),
whose evolution is related to the three episodes ofoutward-propagating outer rainbands during 30–44, 54–
74, and 82–96 h (Fig. 13). Accompanied with the three
episodes are positive in the mean vertical advection

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FIG. 14. Temporal evolutions of (a)–(e), (g)–(j) time tendency of the azimuthal-mean tangential wind (1024 m s22)in the midtropospheric outer-core region (radii of 70–210 km and heights of 3–9 km, the box shown in Fig. 9) and (f)the azimuthal-mean diabatic heating (1023 K s21) averaged between 3- and 12-km heights and between radii of 0 and
30 (dashed), 30 and 70 (solid), and 70 and 150 (dotted) km, respectively. The evolutions derived from (a) TCM4; (b)the sum of all terms in the tangential wind tendency equation; (c) mean radial advection; (d) mean vertical advection;(e) eddy radial and vertical advections; and (g),(i),(j) mean radial advection and (h) mean vertical advection based on
the diagnosed winds by the SE equation with (g),(h) total diabatic heating and (i),(j) diabatic heating between (i)radii of 0 and 30 km (dashed) and (j) between radii of 30 and 70 (solid) and 70 and 150 km (dotted), respectively.

associated with the activity of outer rainbands (Fig. 14d).
Overall, the eddy associated with outer rainbands spins
down the midtropospheric outer-core circulation, partly
offsetting the spinup by the mean radial/vertical advections (Fig. 14e). Therefore, three processes associated withouter rainbands may contribute to the spinup of the midtropospheric outer-core region: overall positive contributions by the mean radial and vertical advections anda negative contribution by eddy advections.
Figures 14g–j show the mean radial and vertical advections based on the diagnosed secondary circulation

derived from the SE equation. The evolution of the mean
radial and vertical advections obtained from the SE equation with total diabatic heating (Figs. 14g,h) is quantitatively very close to that estimated from the tangential
wind tendency equation [Eq. (7), Figs. 14c,d]. Diabatic
heating between radii of 30 and 70 km and between
radii of 70 and 150 km in both inner and outer rainbands contributes the evolution of the mean radial advection (Figs. 14i,j), whereas the evolution of the mean
vertical advection is consistent with the evolution of
diabatic heating between radii of 70 and 150 km in outer

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447

FIG. 15. Schematic diagram showing the azimuthal-mean secondary circulation forced by the azimuthal-meandiabatic heating in the EW, IB, and mid–upper-tropospheric anvil clouds and surface friction. Regions with significant diabatic heating are shaded. Arrows indicate azimuthal-mean secondary circulation. Dashed boxes indicate theregion of the midtropospheric outer-core region. The outer-core spinup process is associated with the midtropospheric inflow in response to the mid–upper-tropospheric diabatic heating outside the eyewall.

rainbands (not shown). These further demonstrate that
diabatic heating in both inner and outer rainbands isimportant for the spinup of the midtropospheric circulation in the outer-core region.

5. Conclusions and discussion
a. Conclusions
We have examined contributions by diabatic heating
in different regions to the spinup of the outer-core circulation in a tropical cyclone simulated in the threedimensional, nonhydrostatic, full-physics model TCM4
by solving the SE equation and by computing terms in
the azimuthal-mean tangential wind tendency equation.
The azimuthal-mean secondary circulation (radial andvertical circulation) and the spinup of the midtroposphericouter-core circulation in the simulated tropical cyclone are
well captured by the SE equation, namely by balance
dynamics. The azimuthal-mean tangential wind tendency
equation shows that the mean radial and vertical advections that are mainly driven by diabatic heating in spiralrainbands and in the upper-tropospheric anvil clouds
outside the eyewall contributed the spinup of the primarycirculation in the midtropospheric outer-core region immediately outside the inner core.
Figure 15 provides a synthesis of the outer-core spinup
process based on our diagnostic analyses. The azimuthalmean diabatic heating outside the eyewall in the middle–
upper troposphere drives radial inflow in the outer-core
region in the middle troposphere. The radial inflow brings
about high absolute angular momentum inward to intensify the primary circulation in the midtroposphericouter-core region. Although the diabatic heating rate in

the eyewall is the largest, its contribution to the spinup of
the midtropospheric outer-core primary circulation is
generally much less than diabatic heating outside theeyewall because of the high inertial stability in the innercore region. Diabatic heating in outer rainbands contributes positively to the spinup of the outer-core primarycirculation by the mean vertical and radial transport ofangular momentum and negatively by the eddy advections. The contribution by surface friction to thetangential wind tendency is negligible above the inflow
boundary layer, whereas the outflow just above the inflow
boundary layer in response to surface friction contributes
negatively to the outer-core primary circulation above
the boundary layer.

b. Discussion
This study demonstrates the important contributionof diabatic heating outside the eyewall to the secondary
circulation and the spinup of the outer-core primarycirculation of a simulated tropical cyclone. Recently,
Smith et al. (2009) and Bui et al. (2009) showed that theincrease in the outer-core circulation above the inflow
boundary layer results from the radial convergence of theabsolute angular momentum. We have extended theirwork to examine the contributions of diabatic heating
in different regions in the storm to the spinup of theouter-core circulation. We show that diabatic heating
in both inner and outer rainbands, in particular that in
the mid–upper-tropospheric anvil clouds, plays a dominant role.
Weatherford and Gray (1988a,b) showed the independency between the inner-core intensity (storm intensity)and the outer-core strength (winds outside the eyewall).
Our azimuthal-mean tangential wind tendency equation

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shows that the inner-core strength can be explained by
the mean vertical advection of azimuthal-mean tangential
wind above the boundary layer and radial advection of
azimuthal-mean tangential wind within the boundary layer(Fig. 9). Diabatic heating in the eyewall contributes to theinner-core strength and thus the storm intensity. Diabatic
heating in the middle–upper troposphere in both innerand outer rainbands contributes largely to the spinupof the outer-core strength, whereas it contributes little tothe inner-core strength or storm intensity (Fig. 15).
Wang (2009) evaluated the effect of diabatic heating inouter spiral rainbands on the storm structure and intensity
through sensitivity experiments by artificially modifying
the heating rate in the region outside a radius of about
90 km. He found that diabatic heating in outer rainbands
limits the storm intensity while it increases the storm innercore size. In this study, we have further quantified the
spinup process of the outer-core circulation by heating inouter rainbands based on balance dynamics. The results
confirm that diabatic heating outside the eyewall (both ininner and outer spiral rainbands) contributes predominantly to the spinup of the outer-core circulation andthus the increase in storm-scale size. Our finding also
supports the results of Hill and Lackmann (2009), whoshowed that the tropical cyclone size in their cloudresolving simulation was sensitive to the environmental
relative humidity. In a relatively moist environment, thestorm developed considerable precipitation (and thus diabatic heating) outside the core (in spiral rainbands), accompanied by significant outward expansion of the windfield and increase in storm size.
We also demonstrated that balance dynamics can capture the detailed features of the secondary circulation
simulated in a full-physics model, including its temporal
evolution. Therefore, balance dynamics can be applied
to the understanding of structure and intensity changesof the tropical cyclone simulated in high-resolution
nonhydrostatic models as a diagnostic tool, consistent
with the conclusions of Bui et al. (2009). In particular,balance dynamics can help estimate potential contributions of a heating source in different radial range anddifferent vertical layers to the intensification and sizechange of the simulated tropical cyclone. This study isa first step toward the understanding of the spinup
process of the outer-core circulation in a tropical cyclone simulated in a full-physics model under idealized
conditions based on balance dynamics. The findings arevalid only for a quasi-axisymmetric storm, because thesimulation was performed on an f plane in a quiescentenvironment. With any environmental forcing, thestorm may develop considerable asymmetric structure. Inthat case, the eddy processes enhanced by the environmental forcing may have considerable contribution to the

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spinup of the outer-core circulation. In addition, the outercore strength and storm size should be significantly
affected by environmental synoptic conditions, which
may influence activity of spiral rainbands and modify
the distribution of diabatic heating in the tropical cyclone
circulation (Merrill 1984; Holland and Merrill 1984;
Weatherford and Gray 1988b; Cocks and Gray 2002).
We plan to include the effect of environmental flow in
a future study.
Acknowledgments. This study has been supported inpart by NSF Grant ATM-0754039 and in part by the
JAMSTEC, NASA, and NOAA through their sponsorships of the International Pacific Research Center (IPRC)
in the School of Ocean and Earth Science and Technology(SOEST) at the University of Hawaii.
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