the impact of gradient wind imbalance on tropical cyclone

The Impact of Gradient Wind Imbalance on Tropical Cyclone Intensificationwithin Ooyama’s Three-Layer Model

THOMAS FRISIUS AND MARGUERITE LEE

Meteorological Institute, University of Hamburg, Hamburg, Germany

(Manuscript received 3 November 2015, in final form 8 June 2016)

ABSTRACT

This paper addresses the validity of the gradient wind balance approximation during the intensification

phase of a tropical cyclone in Ooyama’s three-layer model. For this purpose, the sensitivity to various model

modifications is examined, given by the inclusion of (i) unbalanced dynamics in the free atmosphere, (ii)

unbalanced dynamics in the slab boundary layer, (iii) a height-parameterized boundary layermodel, and (iv) a

rigid lid. The most rapid intensification occurs when the model employs the unbalanced slab boundary layer,

while the simulation with the balanced boundary layer reveals the slowest intensification. The simulation with

the realistic height-parameterized boundary layer model exhibits an intensification rate that lies in between.

Intensification is induced by a convective ring in all experiments, but a distinct contraction of the radius of

maximum gradient wind only takes place with unbalanced boundary layer dynamics. In all experiments the

rigid lid and the balance approximation for the free atmosphere have no crucial impact on intensification,

and a linear stability analysis cannot explain the found sensitivity to intensification. Most likely the nonlinear

momentum advection term plays an important role in the boundary layer. It is found on the basis of a di-

agnostic radial mass flux equation that the source term for latent heat provides the largest contribution to

intensification and contraction. Furthermore, it turns out that the position of the convective ring inside or

outside of the radius of maximum gradient wind (RMGW) is of vital importance for intensification and most

likely explains the large impact of boundary layer imbalance.

1. Introduction

The mechanism for tropical cyclone intensification is

still controversially debated [for a review, see Smith and

Montgomery (2015)], although many numerical non-

linear models can capture this phenomenon properly.

The numerical axisymmetric model developed by Ooyama

(1969, hereafter O69) was one of the first to reveal tropical

cyclone intensification. The model is based on the hy-

drostatic Boussinesq equations and has three layers of

uniform density, where the lowest one is a slab boundary

layer. This simple model conception enables a better

understanding of the intensification process and has the

advantage of a high numerical efficiency. However, the

Ooyama model includes the balance approximation—

that is, the tangential wind underlies the gradient wind

balance—and it is questionable if this assumption is

justified during the intensification stage, when the rate at

which pressure falls becomes large in magnitude. In the

present study we will analyze the impact of this assump-

tion by relaxing the balance assumption within Ooyama’s

model. Indeed, K.V. Ooyama (1968, unpublished manu-

script)1 presented results of a modified three-layer model

that includes an unbalanced boundary layer. He found

more realistic solutions than with the original model (see

also Smith and Montgomery 2008). An unbalanced non-

axisymmetric Ooyama model was also developed by

Schecter and Dunkerton (2009) and compared to a cloud-

resolving model by Schecter (2011). These studies in-

vestigated the sensitivity of tropical cyclone formation and

maximum intensity with respect to various model param-

eters but did not address the impact of gradient wind

imbalance.

Another aimof this study is to advance understanding of

the intensificationmechanism. The balance approximation

has the advantage that the cause of tangential wind rise can

be revealed by the Sawyer–Eliassen equation (Shapiro and

Corresponding author address: Thomas Frisius, CliSAP, Uni-

versität Hamburg, Grindelberg 5, D-20144 Hamburg, Germany.

E-mail: [email protected]

1 ‘‘Numerical Simulation of Tropical Cyclones with an Axi-

symmetric Model.’’

SEPTEMBER 2016 FR I S IU S AND LEE 3659

DOI: 10.1175/JAS-D-15-0336.1

� 2016 American Meteorological SocietyUnauthenticated | Downloaded 12/25/21 04:19 AM UTC

mailto:[email protected]

Willoughby 1982; Bui et al. 2009). This diagnostic equation

results from the time derivative of the thermal wind bal-

ance equation. In the Ooyama model a simpler diagnostic

equation can be derived because of the three-layer formu-

lation. We will see that time differentiation of the gradient

wind balance equation yields a single ordinary differential

equation for the radial mass flux when a rigid lid is

assumed. Shapiro and Willoughby (1982) also analyzed

tropical cyclone intensification, and they found that a

single point source for heat can induce contraction of the

wind maximum and intensification. We will see that this

convective ring contraction scenario also does take place

in the Ooyama model and that the findings by Shapiro

and Willoughby (1982) are indeed relevant. However,

less clear is the role of the boundary layer. The boundary

layer supplies the necessary latent energy that is released

in the contracting convective ring. Smith andMontgomery

(2008) found that the gradient wind imbalance has a large

impact on the wind profiles in a steady-state slab boundary

layer model of a tropical cyclone. It is likely that this also

has an effect on the strength and position of the convec-

tive ring fed by boundary layer air. Therefore, the balance

assumption in the boundary layer could sensitively influ-

ence the intensification rate. On the other hand, Kepert

(2010a,b) and Williams (2015) found by comparing the

slab boundary layer model with a height-resolving bound-

ary layer model that the former tends to overestimate su-

pergradient winds and vertical velocities. Kepert (2010b)

suggested using a height-parameterized boundary layer

model instead, which can be coupled to theOoyamamodel

indeed, as we will demonstrate in this paper.

The paper is organized as follows. Section 2 contains

the model description and the outline of the simulations.

In section 3, results of the performed simulations are

presented and the differences due to the balance ap-

proximation are identified. Section 4 clarifies the in-

tensification mechanism by analyzing the linearized

equations and evaluating the diagnostic equation for the

induced secondary circulation. Finally, the conclusions

are summarized in section 5.

2. Description of the model

Ooyama’s model includes three layers lying upon each

other, which is sketched in Fig. 1. The interface between

the middle and the upper layers is a free material surface,

while the interface between the lower andmiddle layers is

fixed but permeable. The lowermost layer forms the

boundary layer where microturbulent exchange with the

ocean surface is relevant. Convection can pervade all in-

terfaces, and it gives rise to the vertical mass fluxes Qb,1,

Qb,2, and Q1;2. The density of the two lower layers is r0,

while that of the upper layer takes the value �r0 with �, 1.

In the following subsections the governing equations, the

physical parameterization schemes, the rigid-lid modifi-

cation, Kepert’s height-parameterized boundary layer

model, and the performed experiments are outlined.

a. Governing equations of the free-surface Ooyamamodel

The governing equations of the free-surface Ooyama

model are as follows:

d1

�›u

j

›t1 u

j

›uj

›r

�2

�f 1

yj

r

�yj

52›P

j

›r1 d

1(D

y,uj1D

h,uj), j5 1; 2, (1)

›yj

›t1 z

juj5D

y,yj1D

h,yj, j5 1; 2, (2)

d2

�d3

›ub

›t1u

b

›ub

›r

�2

�f 1

yb

r

�yb

52›P

1

›r1 d

2(D

y,ub1D

h,ub1D

s,ub), (3)

d2d3

›yb

›t1 z

bub5 d

2(D

y,yb1D

h,yb)1D

s,yb, (4)

›ue,b

›t1 u

b

›ue,b

›r5 (D

y,ue,b1D

h,ue,b1D

s,ue,b), (5)

›h1

›t11

r

›

›r(ru

1h1)5Q

b,12Q

1,b2Q

1;2, (6)

›h2

›t1

1

r

›

›r(ru

2h2)5Q

b,2/�1Q

1;2/�, (7)

w52hb

r

›rub

›r, (8)

FIG. 1. Schematic showing the design of Ooyama’s

three-layer model.

3660 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

Unauthenticated | Downloaded 12/25/21 04:19 AM UTC

P15 g(h

12H

1)1 �g(h

22H

2), and (9)

P25 g(h

12H

1)1 g(h

22H

2) , (10)

where r denotes the radius, t the time, u the radial wind,

y the tangential wind, w the vertical velocity at the top of

the boundary layer, h the layer depth, H the mean layer

depth, P the kinematic pressure anomaly, ue the equiva-

lent potential temperature, f the Coriolis parameter,

z5 f 1 r21›(ry)/›r the absolute vorticity, Qj,k the vertical

mass flux from layer j into layer k2, and g the gravity ac-

celeration. The indices b, 1, and 2 denote the boundary

layer, middle layer, and upper layer, respectively. The

terms Dy,X , Dh,X , and Ds,X describe the tendencies of

quantity X as a result of vertical exchange between the

layers, horizontal mixing, and surface fluxes, respectively.

The switches d1, d2, and d3 include additional terms that

were absent in the original formulation by O69. With

d1 5 1, the balance approximation for the free atmosphere

(layers 1 and 2) is switched off. Switch d2 5 1 includes an

unbalanced slab boundary layer model. Setting d3 5 0 in

the case d2 5 1 removes the local time derivatives of the

boundary layer momentum equations so that the bound-

ary layer model becomes diagnostic as in Smith and

Montgomery (2008). With this switch, we can estimate the

importance of the boundary layer adjustment time scale for

intensification. For d3 5 1, the boundary layer model also

includes the effect of local wind change in the momentum

budget. For d1 5 d2 5 d3 5 0, the originalOoyamamodel is

recovered, while for d1 5 d2 5 d3 5 1, the model formula-

tion corresponds to that by Schecter andDunkerton (2009).

b. Parameterization of irreversible physical processes

The Ooyama model comprises parameterization

schemes for updrafts, surface fluxes, and vertical and

horizontal diffusion. The updraft parameterization yields

the mass fluxes between the various layers. It is assumed

that the upward mass fluxes are proportional to the up-

ward boundary layer mass flux Qb 5 (w1 jwj)/2 so that

Q1;2

51

2[(h2 1)1 j(h2 1)j]Q

b, (11)

Qb,1

51

2[(12h)1 j(12h)j]Q

b, and (12)

Qb,2

5Qb2Q

b,1, (13)

where h denotes the so-called entrainment parameter.

Mass of layer 1will be entrained into the updraft forh. 1

and transformed into mass of layer 2 having a lower

density. Then deep convection takes place, while h# 1

yields detrainment, which is characteristic for shallow

convection. The entrainment parameter h is a function of

the vertical thermal and moisture stratification, namely,

h5 11ue,b

2 ue,2*

ue,2*2 u

e,1

, (14)

where ue* is the saturation equivalent potential temper-

ature. A constant value is assumed for ue,1, while ue,2*

results from the approximation

ue,2* 5 u

e,2* 1 a(P

22P

1) , (15)

where the overbar denotes the ambient value and a is a

thermodynamic constant.

The upward boundary layer mass flux Qb,1 must be

compensated by a downward mass flux Q1,b in order to

conserve the mass of the boundary layer. This is ensured

by setting

Q1,b

521

2(w2 jwj) . (16)

There is no downwardmass flux from layer 2 to layer 1 in

the original Ooyama model.

Note that this parameterization differs from conven-

tional convective parameterization schemes since it does

not include downdrafts. Indeed, Ooyama’s scheme is

rather valid for a single convective cloud or convective

ring than for an ensemble of many convective elements

that usually include downdrafts. Therefore, a cloud-scale

model resolution does not disagree with this scheme.

The tendencies due to surface fluxes are parameter-

ized as follows:

Ds,ub

52C

D0

hb

(11 0:12Vb)V

bub, (17)

Ds,yb

52C

D0

hb

(11 0:12Vb)V

byb, and (18)

Ds,ue,b

5C

E0

hb

(11 0:12Vb)V

b(u

e,s* 2 u

e,b), (19)

whereCD0 andCE0 denote theminimum surface transfer

coefficients for momentum and enthalpy, respectively.

Furthermore,Vb 5 (d2u2b 1 y2b)

1/2 is the wind speed in the

boundary layer and ue,s* the saturation equivalent po-

tential temperature at the sea surface, which is a func-

tion of pressure and is given by

ue,s* 5 u

e,s* 2 bP

1, (20)

where b is another thermodynamic constant. Since we

make use of a slab boundary layer model, the surface

transfer is regulated by the depth-averaged wind instead

2 Strictly speaking, Qj,k represents mass fluxes divided by refer-

ence density r0.



of the conventional 10-m wind. This obvious deficiency

is eliminated in Kepert’s height-parameterized bound-

ary layer model, which will be described below.

For horizontal exchange, a simple diffusion scheme is

applied: that is,

Dh,uj

51

hjr2

›

›r

�nhhjr3›

›r

�uj

r

��, j5 b, 1; 2, (21)

Dh,yj

51

hjr2

›

›r

�nhhjr3›

›r

�yj

r

��, j5 b, 1; 2, and (22)

Dh,ue,b

51

r

›

›r

�nhr›

›r(u

e,b)

�, (23)

where nh denotes the horizontal diffusion coefficient.

Vertical exchange is related to vertical mass fluxes,

causing the following tendencies:

Dy,ub

5Q1,b

u12 u

b

hb

, (24)

Dy,yb

5Q1,b

y12 y

b

hb

, (25)

Dy,u1

5Qb,1

ub2 u

1

h1

, (26)

Dy,y1

5Qb,1

yb2 y

1

h1

, (27)

Dy,u2

5Qb,2

ub2 u

2

�h2

1Q1;2

u12 u

2

�h2

, (28)

Dy,y2

5Qb,2

yb2 y

2

�h2

1Q1;2

y12 y

2

�h2

, and (29)

Due,b

5Q1,b

ue,1

2 ue,b

hb

. (30)

O69 also includes shearing stress at the interface be-

tween the layers. However, we found a negligible impact

of these stresses in the simulations performed, and,

therefore, we left this process out here.

c. Rigid-lid assumption

Some of the governing equations have to be modified

when a rigid lid is assumed. This assumption has the

consequence that

h11 h

25H

11H

2dH . (31)

Therefore, we obtain for the pressure anomalies

P15P

l1 (12 �)g(h

12H

1) and (32)

P25P

l, (33)

where Pl is the kinematic pressure anomaly at the rigid

lid. These equations replace Eqs. (9) and (10). Because of

the enforced volume conservation of the free atmosphere,

the radial velocity u2 becomes a function of u1, ub, and

h1, namely,

u252(u

1h11 u

bhb)/(H2 h

1) . (34)

With the rigid-lid assumption, mass conservation does

not hold anymore when convection is included, since the

expansion due to latent heat release cannot be consis-

tent with the volume conservation enforced by the rigid

lid. Then the model loses some mass in the course of

time. However, the artificial mass loss is rather small,

since only the fraction 12 � of the total upper layer mass

gain is removed, and � is usually close to 1.

The rigid-lid assumption has the advantage that the

radial flow of the free-atmosphere layer can be de-

termined by a single diagnostic equation in the balanced

case (d1 5 0). The time derivative of the gradient wind

balance equation for the middle layer becomes the fol-

lowing by substituting Eqs. (2), (32), and (6):�f 1 2

y1

r

�(z

1u12D

y,y12D

h,y1)

52›2P

l

›r›t2 g(12 �)

›

›r

�1

r

›C1

›r1Q

b,12Q

1,b2Q

1;2

�,

(35)

where C1 52rh1u1 denotes the inward radial volume flux

in the middle layer. The contribution of the lid pressure

tendency can be evaluated by the gradient wind balance

equation for the upper layer, and the result leads to the

followingdiagnostic equation for the inwardvolumefluxC1:

r›

›r

�1

r

›C1

›r

�2 SC

15B

A1B

B1B

C1B

D1B

E1B

F,

(36)

in which

S51

g(12 �)�2

j51

�f 1 2

yj

r

�z1

hj

(37)

is a factor measuring the inertial stability, and the vari-

ous source terms are as follows:

BA52r

›

›r(Q

b,12Q

1,b), (38)

BB5 r

›

›r(Q

1;2), (39)

BC5

r

g(12 �)

�f 1 2

y1

r

�D

y,y1, (40)

BD52

r

g(12 �)

�f 1 2

y2

r

�D

y,y2, (41)

BE52�

2

j51

(21) jr

g(12 �)

�f 1 2

yj

r

�D

h,yj, and (42)



BF5

1

g(12 �)

�f 1 2

y2

r

�z2ru

b. (43)

The source termBA results from the mass fluxes between

the middle layer and the boundary layer due to frictional

convergence or divergence, while BB is associated with

latent heat release due to deep convection. The source

terms BC and BD refer to vertical flux of tangential mo-

mentum from the boundary layer into the middle layer

and from the two lower layers into the upper layer, re-

spectively. The source term BE arises as a result of hori-

zontal diffusion, and BF is the effect of radial angular

momentum advection in the upper layer due to frictional

convergence. The diagnostic Eq. (36) forms the equivalent

to the Sawyer–Eliassen equation that applies to a vertically

continuousmodel. In the latter equation, however, vertical

and horizontal momentum advection do not appear as

source terms. These diagnostic equations can help in un-

derstanding tropical cyclone intensification, since they di-

agnose the inflow in the free atmosphere, which is required

for the spinup of the vortex. The linear nature of Eq. (36)

facilitates the assignment of the various source terms to

associated tangential wind tendencies resulting from in-

ward angular momentum advection.

In the rigid-lid case, the solution method for the free-

atmosphere equations is as follows. First, the gradientwind

balance equations are integrated inward from the outer

boundary to the center to obtain the lid pressure anomaly

Pl and the middle-layer depth h1. The values PL 5 0 and

h1 5H1 are assumed at the outer boundary for the in-

tegration. Then the diagnostic Eq. (36) is solved, and the

resulting radial velocity is used for the time integration of

the prognostic tangential wind equation [see Eq. (2)].

d. Kepert’s height-parameterized boundarylayer model

The slab boundary layer model used in the present

study has some shortcomings, which were demonstrated

by Kepert (2010a,b), and Williams (2015). They found

by comparing the slab boundary model with a height-

resolving model that the inflow has too large an ampli-

tude and that the departure from gradient wind balance

is overestimated as a result of excessive surface drag.

Furthermore, the vertical and horizontal wind compo-

nents can display unnaturally large oscillations for cer-

tain gradient wind profiles. Kepert (2010b) proposed a

height-parameterized boundary layer model that re-

solves these issues. In this model, the vertical profiles of

the boundary layer wind components are prescribed by

the Ekman spiral as a function of height z and are given

by the following:

u5 [ub(r, t)2 ~y

b(r, t)] cos

�zd

�e2z/d

1 [ub(r, t)1 ~y

b(r, t)] sin

�zd

�e2z/d, and (44)

y2 y15 [~y

b(r, t)2 u

b(r, t)] sin

�zd

�e2z/d

1 [ub(r, t)1 ~y

b(r, t)] cos

�zd

�e2z/d , (45)

where d denotes the height scale. Kepert (2010b) found

that 2.5 3 d approximately yields the boundary layer

height hb. The vertical averages of these profiles

becomes

1

hb

ðhb0

u dz’1

hb

ð‘0

u dz5d

hb

ub

and (46)

1

hb

ðhb0

y2 y1dz’

1

hb

ð‘0

y2 y1dz5

d

hb

~yb. (47)

Therefore, ub and yb 5 ~yb 1 y1 are not the vertically av-

eraged wind components, since hb/d’ 2:5. These com-

ponents rather yield the scale of the boundary layer

winds. The equations for the height-parameterized

model become

›ub

›t1

3ub

4

›ub

›r2

ub

4

›~yb

›r2~yb

4

›ub

›r1~yb

4

›~yb

›r2 f y

b2

y21 1 2~yby1

r2

u2b 1 2u

b~yb1 3~y 2

b

4r

5Q1,b

u12 u

b

d1D

h,ub2›P

1

›r2

CD0

d(11 0:12V

s)V

sus, (48)

›yb

›t1

ub

4

›ub

›r1

3ub

4

›~yb

›r2~yb

4

›ub

›r2~yb

4

›~yb

›r1u

b

›y1

›r1 fu

b1

uby1

r1

u2b 1 2u

b~yb2 ~y 2

b

4r

5Q1,b

y12 y

b

d1D

h,yb2

CD0

d(11 0:12V

s)V

sys, and (49)

›ue,b

›t1

d

hb

ub

›ue,b

›r5Q

1,b

ue,1

2 ue,b

hb

1Dh,ue,b

1C

E0

hb

(11 0:12Vs)V

s(u

e,s* 2 u

e,b), (50)



where us, ys, and Vs respectively denote the radial

wind, tangential wind, and wind speed at the surface

(z 5 0) deduced from Eqs. (44) and (45). Using these

surface wind values instead of vertically averaged

values yields a more realistic surface transfer param-

eterization. Most terms result by vertically averaging

the terms of the height-dependent axisymmetric mo-

mentum and temperature equations for the given

vertical wind profiles, but the vertically averaged

vertical advection term has been simplified as in the

original slab boundary layer model. The vertical ve-

locity at the top of the boundary layer becomes

approximately

w521

r

ðhb0

›ru

›rdz’2

d

r

›rub

›r. (51)

Note that Kepert’s height-parameterized boundary

layer model deviates from the slab boundary layer

model in terms of (i) different expressions for the inertia

forces, (ii) the use of surface wind for the surface

transfer parameterization, and (iii) a 2.5-times-smaller

height scale in the momentum equations. Kepert

(2010b) also included a radial variation of boundary

layer depth, which is omitted here. Furthermore, there

is a small inconsistency in the boundary condition, since

the direction of turbulent stresses becomes discontinu-

ous immediately above the surface (Kepert 2010b).

However, the height-parameterized boundary layer

model still produces more realistic results compared to

the slab boundary layer model.

e. Numerical method and experimental design

For the numerical solution, a stretched staggered grid

has been introduced, as described by Frisius (2015). The

grid has an inner part and an outer part with gridpoint

distances of 250 and 2500m, respectively. Between these

regions there is a smooth transition zone that is centered

between the first and second quarter of all grid points.

For time integration, a leapfrog scheme including a

Robert–Asselin time filter is applied. The lateral

boundary is located at r 5 4647.08 km where the radial

velocity and all horizontal diffusive fluxes vanish. These

boundary conditions also apply at the center axis where

the tangential wind becomes zero in addition. In the

balanced case (d1 5 0), the equations for the upper two

layers are solved as described in O69. To solve the

steady unbalanced boundary equations (d2 5 1 and

d3 5 0), the boundary layer model is integrated sepa-

rately in time at each time step until the solution attains

approximately a steady state.

The initial fields for all simulations are identical to

those of case A in O69. The parameters were also taken

from O69 (Table 1), except for the interfacial friction

coefficient, which is set to zero in the present study. The

density ratio �5 0:9 appears too small for the tropo-

sphere. However, DeMaria and Pickle (1988) found that

the Ooyama model can alternatively be interpreted in

terms of compressible isentropic layers. Then the chosen

� value agrees indeed with a typical tropospheric strati-

fication. Table 2 shows the designation and configura-

tion of the various experiments. Experiment REF

corresponds to the original model experiment per-

formed by O69. Experiments BALBL and UNBALBL

include unbalanced dynamics in the free atmosphere

and in the boundary layer, respectively, while experi-

ment UNBAL adopts unbalanced dynamics in both the

boundary layer and free atmosphere. Experiment SBL

has unbalanced dynamics in the boundary layer, but

local time tendencies for momentum are switched off

so that equations for a steady-state boundary layer

are solved (d3 5 0). Experiment KEPERTBL includes

Kepert’s height-parameterized boundary layer model

but is identical to UNBALBL otherwise. Experiments

REF_RIGID, UNBALBL_RIGID, SBL_RIGID and

KEPERTBL_RIGID employ a rigid lid and conform

in other respects with REF, UNBALBL, SBL and

KEPERTBL, respectively.

TABLE 1. Values of the model parameters.

Parameter Value

� 0.9

hb 1000m

H1, H2 5000m

f 0:53 1024 s21

ue,1 332K

ue,2* 342K

ue,s* 372K

a 0.001K s2m22

b 0.0002K s2m22

nh 1000m2 s21

CD0 0.005

CE0 0.005

TABLE 2. Specification of the various experiments. In experi-

ments KEPERTBL and KEPERTBL_RIGID, boundary layer

equations described in section 2d were adopted.

Expt Rigid lid Switches

REF No d1 5 0, d2 5 0, d3 5 0

UNBAL No d1 5 1, d2 5 1, d3 5 1

UNBALBL No d1 5 0, d2 5 1, d3 5 1

BALBL No d1 5 1, d2 5 0, d3 5 0

SBL No d1 5 0, d2 5 1, d3 5 0

KEPERTBL No d1 5 0, d2 5 1, d3 5 1

REF_RIGID Yes d1 5 0, d2 5 0, d3 5 0

UNBALBL_RIGID Yes d1 5 0, d2 5 1, d3 5 1

SBL_RIGID Yes d1 5 0, d2 5 1, d3 5 0

KEPERTBL_RIGID Yes d1 5 0, d2 5 1, d3 5 1



3. Results

Before discussing all experiments, the results of the

reference experiment REF are compared with those of

O69, which should be similar. However, some differ-

ences may arise because of the different numerical

scheme, model resolution, and boundary conditions.

a. Reference experiment

Figure 2 shows the maximum tangential wind of the

middle layer and the radiuswhere it is located. The former

is taken as a measure for intensity while the latter con-

stitutes the radius ofmaximumgradientwind (RMGW) in

the runs based on the balanced free atmosphere (d1 5 0).

The present simulation reveals a similar result compared

to that obtained by O69 (see his Fig. 4). Nevertheless, the

maximal tangential wind peaks at 64.9ms21 instead of

58ms21, as found by O69, and the final radius of maxi-

mum tangential wind becomes 158km, which is much

larger than in the simulation by O69. Figure 3 presents

radial profiles of various model variables at time t5 81h,

when the intensification rate is close to its maximum. This

figure should be compared to the middle panel of Fig. 5 in

O69. The profiles for the displayed tangential and radial

wind components are very similar inmagnitude and shape

to those found by O69. However, the peak vertical ve-

locity of 1.7ms21 is about 0.5ms21 larger than in the

original Ooyama model. This could possibly explain the

higher peak intensity, as the vertical velocity w is pro-

portional to the latent heat release in the developing

eyewall. The shape of the profile for the entrainment

parameterh resembles that displayed byO69, except near

the center axis, where O69 found somewhat smaller

values. This parameter is closely related to the convective

available potential energy (CAPE). Therefore, CAPE is

minimal close to the developing eyewall, and it increases

toward the center as well as outwards. Three reasons can

be responsible for the differences found between the

simulations. First, the grid spacing is reduced by a factor of

20 in the inner part of the model compared to that in the

original Ooyama model. Second, the model domain of

O69 has only a radius of 1000km, which is less than a

quarter of the domain size used here. Furthermore, O69

treated the lateral wall differently by allowing fluxes

across the boundary. Third, O69 did not make use of

horizontal diffusion in the prognostic equation for the

equivalent potential temperature in the boundary layer.

We also made a simulation with uniform grid spacing of

5km, no horizontal diffusion in Eq. (5), and a lateral wall

at r 5 1000km. In this case, the tangential wind y1 only

reaches a maximum of 45ms21, and the RMGWdoes not

expand in the decay phase. Furthermore, the profiles of

vertical velocity w and entrainment parameter h exhibit

gridpoint scale oscillations. However, the value of h at t581h is now at the axis (r5 0) very similar to that found by

FIG. 2. Maximum of tangential wind y1 (solid line) and the

radius of maximal y1 (dashed line) as a function of time for the

experiment REF.

FIG. 3. Radial profiles at t 5 81 h for the experiment REF:

(a) Tangential wind y1 (solid line), inward boundary layer wind

2ub (dotted line), and tangential wind y2 (dashed line); (b) vertical

velocity w (solid line) and entrainment parameter h (dashed line).



O69. Using a lateral wall at r 5 1750km yields a time

evolution of maximum y1 and RMGW that is very similar

to the result of O69. From these results, we can presume

that both the different boundary conditions and grid

spacing are responsible for differences in the evolution of

intensity and the RMGW. On the other hand, horizontal

diffusion in the thermodynamic equation causes a higher

h value at the vortex axis. The reason why O69 did not

detect gridpoint-scale oscillations remains unclear. It is

likely that the different gridpoint discretization schemes

are responsible for the deviations.3 We also performed a

simulation at doubled gridpoint spacing andwith a smaller

model domain. In both cases, we found only very small

differences in the results. Therefore, the positions of the

lateral wall and the radial gridpoint resolution appear

suitable for the present simulation. However, this was

possibly not the case in the original simulation by O69.

b. Sensitivity experiments

In this subsection we compare the outcome of all

performed simulations. Figure 4a shows the time evo-

lution of the maximum tangential wind of layer 1. Ob-

viously, large differences arise between the various

developments. The peak intensity varies between 58 and

80ms21, and the time of maximum intensification rate

ranges from 1.5 to 4 days. The intensification phase can

be assigned to four experiment groups. The most rapid

intensification takes place when the model includes

the full unbalanced boundary layer model (experi-

ments UNBAL, UNBALBL, and UNBALBL_RIGID),

while a somewhat slower intensification is found for the

model configurations neglecting the local momentum

tendencies in the boundary layer model (experiments

SBL and SBL_RIGID). The tangential wind in experi-

ments KEPERTBL andKEPERTBL_RIGID intensifies

in turn less rapidly compared to SBL and SBL_RIGID.

The slowest growth emerges in the experiments based on

the balanced boundary layer model (REF, BALB, and

BALB_RIGID). In these simulations, intensification is

slower, and a decay of vortex intensity takes place

eventually in contrast to the other experiments. The cases

UNBAL, UNBALBL, and UNBALBL_RIGID reveal a

contraction of the radius of maximum y1, after which this

radius increases again very slowly (see Fig. 4b). This

behavior is similar to that typically observed in more

sophisticated tropical cyclone models (e.g., Hausman

et al. 2006; Hill and Lackmann 2009; Xu and Wang

2010; Persing et al. 2013; Frisius 2015; Smith et al.

2015; Stern et al. 2015). The developments in the

experiments SBL, SBL_RIGID, KEPERTBL, and

KEPERTBL_RIGID are qualitatively similar, but the

contraction begins several hours later and is slower. In

Stern et al. (2015), it is noted that the vortex contraction

stops well before the intensity reaches its maximum.

This is also fulfilled in the abovementioned experi-

ments, although after contraction only little further in-

tensification happens. The experimentsREF,BALB, and

BALB_RIGID reveal only a weak contraction and an

untypically fast outward migration of the wind maximum

afterward.

Figure 5 displays radius–time diagrams of tangential

wind y1 and vertical velocity w for selected experiments.

The diagrams for UNBALBL and SBL_RIGID are very

similar to UNBAL and SBL, respectively, and have,

therefore, not been displayed in Fig. 5. Also in this figure

the differences and commonalities suggest the subdivision

FIG. 4. (a) Maximum middle-layer tangential wind as a function

of time for the experiments REF (red solid curve), UNBAL (green

solid curve), UNBALBL (blue solid curve), SBL (violet solid

curve), BALBL (light blue solid curve), and KEPERTBL (black

solid curve). The corresponding experiments assuming a rigid lid

are displayed by dashed curves. (b) As in (a), but the radius of

maximum y1 is shown.

3 The discretization of the prognostic equation for boundary

layer equivalent potential temperature was not described in O69.



FIG. 5. Radius–time diagrams showing tangential wind y1 (shading; m s21) and vertical velocity w (white contours; m s21) for the

experiments (a) REF, (b) UNBAL, (c) BALBL, (d) SBL, (e) KEPERTBL, (f) REF_RIGID, (g) UNBALBL_RIGID, and

(h) KEPERTBL_RIGID. The contour interval for vertical velocity is 3m s21 in (b),(d), and (g) and 1m s21 in (a),(c),(e),(f), and (h).



into the four experiment groups. All unbalanced

boundary layer experiments exhibit the single convec-

tive ring contraction scenario, as suggested by Shapiro

and Willoughby (1982). The convective ring develops

already at the beginning of the intensification phase just

inside of the tangential wind maximum. Then the solu-

tion of the Sawyer–Eliassen equation reveals both con-

traction and intensification as a result of the latent heat

release (Shapiro and Willoughby 1982). Frisius (2006)

also detected intensification by a single convective ring

in an axisymmetric nonhydrostatic cloud model. Such a

scenario may even be seen in more realistic 3D models

when the azimuthally averaged vertical velocity is ana-

lyzed (e.g., Braun et al. 2006; Persing et al. 2013). The

single convective ring scenario could be disturbed by

further convection outside the developing eyewall as

a consequence of latent cooling in downdrafts (Wang

2002a; Frisius and Hasselbeck 2009). Furthermore,

Wang (2002b) found that vortex Rossby waves can lead

to outward-propagating spiral rainbands and eyewall

breakdown. However, the effects of latent cooling and

asymmetries like vortex Rossby waves are not consid-

ered in the Ooyama model, and, therefore, further

convective cells do not develop in our experiments.

In the experimentsUNBAL,UNBALBL_RIGID, and

SBL the vertical velocityw takes values of up to 18ms21,

which appear unrealistically large. On the other hand, w

is much smaller and more realistic in experiments

KEPERTBL andKEPERTBL_RIGID.Hence, Kepert’s

height-parameterized model obviously corrects an un-

warranted feature of the slab boundary model in this

context. The experiments based on a balanced boundary

layer (REF, BALB, and BALB_RIGID) reveal a single

convective ring too, but it does not contract significantly

and is weaker. It is always located outside of the tan-

gential wind maximum. This could be the reason why

contraction and intensification are much weaker in these

experiments. The results are consistent with Heng and

Wang (2016), who found in a nonhydrostatic tropical

cyclone model with a prescribed heating and the Sawyer–

Eliassen equation that the balance approximation is

fulfilled quite well above the boundary layer but that

unbalanced boundary layer processes may enhance eye-

wall contraction and produce more realistic boundary

layer structures. At last, we can draw the conclusion that

the balance approximation above the boundary layer and

the rigid-lid assumption may have some effect on the

peak intensity but are not very crucial for the intensifi-

cation phase. Therefore, we only consider the rigid-lid

experiments in the remainder of this study.

Figure 6 shows radial profiles of the boundary layer

wind components at the time of maximum intensi-

fication for the three experiments REF_RIGID,

FIG. 6. Radial profiles of tangential boundary layer wind yb (solid

curve), radial boundary layer wind ub (dashed curve), vertical wind

w (dotted curve), and tangential wind y1 (dotted–dashed curve)

at the time of maximum intensification for the experiments

(a) REF_RIGID, (b) UNBALBL_RIGID, and (c) KEPERTBL_

RIGID. Note that y1 is not displayed in (a), since it is identical to ybin this case. Furthermore, (c) shows the surface wind components

ys and us as thin solid and dashed curves, respectively.



UNBALBL_RIGID, and KEPERTBL_RIGID. Note

that the tangential boundary layer wind in REF_

RIGID is identical to the gradient wind of the middle

model layer. In this simulation, the maximum inflow is

found at a radius of more than 50 km, while the

RMGW is located at a significantly smaller radius. This

has the consequence that the maximum vertical ve-

locity also lies at a radius larger than the RMGW. The

maximum inflow velocity of 36m s21 is very large, and

the resulting inflow angle of about 408 appears un-

realistically high (e.g., Frank 1977). In contrast, the

simulations UNBALBL_RIGID and KEPERTBL_

RIGID reveal a lower inflow angle, and the radius of

maximum inflow is close to the RMGW. Therefore, the

vertical wind peaks inside of this maximum. Further-

more, tangential winds become supergradient beneath

the developing eyewall in UNBALBL_RIGID and

also in KEPERTBL_RIGID but with a smaller am-

plitude in the latter. Experiment KEPERTBL_RIGID

has smaller horizontal wind shear inside of the tangential

wind maximum, and the peak vertical velocity is also

much smaller in comparison with UNBALBL_RIGID.

Boundary layer profiles similar to KEPERTBL_RIGID

were also observed in more complex cloud-resolving

models [e.g., Fig. 11a of Schecter (2011)]. We can also

clearly see weaker tangential winds at the surface, with

the consequence that the supergradient wind nearly

vanishes there, which is consistent with results found in

multilevel models [e.g., Fig. 10 of Bryan and Rotunno

(2009)]. The radial surface wind becomes weaker too at

most radii, but the reduction is smaller, and the radial

surface wind minimum is located more inward. Ooyama

found similar results in his unpublished work. In the

modified experiment conforming to experiment SBL, he

detected in comparison with the balanced model faster

intensification, RMGW contraction, a shift of the updraft

toward the center, and less expansion of theRMGWafter

the intensification phase. The smaller surface wind speed

considered in KEPERTBL_RIGID can explain the

slower intensification compared to UNBALBL_RIGID.

Indeed, the surface wind intensification rate might be

similar to that observed in the balanced boundary

layer simulation REF_RIGID. Therefore, the balanced

boundary layer model has, compared to a realistic model,

two deficiencies: (i) it overestimates the radius of eyewall

convection, and (ii) it overestimates the near-surface

wind speed. The experiments show that these two ef-

fects have opposing impacts. Profiles of UNBALBL_

RIGID and SBL_RIGID are almost identical at the time

of maximum intensification (not shown). Consequently,

the consideration of the local time tendency of boundary

layer momentum only enhances the amplification rate

but does not influence the radial wind structure. A

plausible candidate for explaining the differences to

REF_RIGID is the momentum advection term ub›ub/›r

in Eq. (3). It leads to higher inertia and more inward

intrusion of the inflow beyond the RMGW, where su-

pergradient winds and an updraft arise [for more dis-

cussion, see Frisius et al. (2013)]. This updraft lying

closer to the center provokes further contraction and

intensification, as will be shown in the next section.

However, the slab boundary model overestimates

the overshoot by inertia (Kepert 2010b). The height-

parameterized boundary layer model corrects this

overestimation to a high degree, but the inertia of the

inflow can still be important, since supergradient winds

also appear in KEPERTBL_RIGID.

4. Analysis of the intensification processes

O69 found conditional instability of the second kind

(CISK; Charney and Eliassen 1964) in his model by

linearizing the equations with respect to a conditionally

unstable atmosphere at rest. Growth due to CISK in-

creases with decreasing perturbation size. This property

is known as the ‘‘ultraviolet catastrophe’’ (Montgomery

et al. 2006) and leads to the consequence that only

horizontal diffusion can prevent shrinking of the per-

turbation to arbitrary small scales.4 However, the nu-

merical results are not inconsistent with such a scenario

indeed, since the width of the convective ring is close to

the scale of the grid and only remains finite because of

the inclusion of horizontal diffusion. The effect of hor-

izontal diffusion on the updraft profile in the in-

tensification phase for experiment UNBALBL_RIGID

is displayed in Fig. 7. Obviously, the width and radius

increases significantly with increasing horizontal diffu-

sion, while themaximum updraft velocity decreases. For

the low-diffusion case (nh 5 200m2 s21), a doubled grid

resolution was actually necessary to resolve the updraft

properly. This result is qualitatively consistent with

CISK, except for the difference that the updraft does not

develop at the vortex axis. On the other hand, Eliassen

(1971) showed that the updraft forms at a finite radius

when a quadratic drag law is used that is more similar to

that employed in the Ooyama model than a linear drag

law. Therefore, some aspects of intensification could

possibly be understood in terms of the CISK theory.

This possibility will be checked in the following.

4 It has to be noted that Charney and Eliassen (1964) found in

their CISKmodel rather uniform growth rates for disturbance sizes

smaller than about 100 km. Therefore, arbitrarily small distur-

bances grow only slightly faster than those having a scale of

about 100 km.



a. Linear stability analysis

The linear instability analysis has been performed in a

similar way to O69. We premise a basic-state atmosphere

that is at rest and conditionally unstable. Furthermore, the

surface drag coefficient remains finite in the linear model

by assuming a gustiness velocity Vg 5 10ms21 so that

Ds,ub

52C

D0

hb

Vgu0b, D

s,yb52

CD0

hb

Vgy0b , (52)

where the prime denotes the linear perturbation. The

linearized model also includes latent cooling for w, 0

besides latent heating for w. 0. Then the model may

reveal very unrealistic states after a certain time period.

However, linear instability models are only valid for a

limited duration anyhow. With these assumptions, the

linearized equations in the rigid-lid case read as follows:

f y0j 5›P 0

j

›r, j5 1; 2, (53)

›y0j›t

52fu0j, j5 1; 2, (54)

d2d3

›u0b

›t5 f y0b 2

›P01

›r2 d

2Ku0

b , (55)

d2d3

›y0b›t

52fu0b 2Ky0b , (56)

›h01

›t52

H1

r

›

›r(ru0

1)2 (h2 1)w0 , (57)

w0 52hb

r

›

›r(ru0

b) , (58)

P01 5P0

2 1 (12 �)gh01, and (59)

u02 52u0

1 2hb

H2

u0b , (60)

where K5CD0Vg/hb. By assuming exponentially grow-

ing perturbations of the form F(r, t)5Fr(r)est, one can

derive a single equation in P 01r:

s(12R2D)P 01r

52[11 2R2(h2 1)D]

3

�f 2

2

hb

H2

d2d3s1K

f 2 1 d2d3(s2 1 2sK)1 d

2K2

�P 01r, (61)

where s denotes the growth rate, D the horizontal

Laplace operator, and R5 f[g/2(12 �)][H1/f2]g1/2 the

internal Rossby radius. For solving this equation, it is

appropriate to assume a radial structure of the form

P 01r(r)5

cP1J0(kr) , (62)

where J0(kr) is the zeroth-order Bessel function of the

first kind, and k is the radial wavenumber. Inserting this

solution yields the following cubic equation:

d2d3(s3 1 2Ks2)1 ( f 2 1 d

2K2)s

2 f 2hb

H2

�h2 12 j2

2j2 1 1

�(d

2d3s1K)5 0, (63)

where j5 1/(ffiffiffi2

pkR) has been introduced in O69 as a

nondimensional measure for the horizontal scale of the

disturbance. There is only one real root in the unstable

case, since all factors in front of the exponential growth

rates are positive for reasonable h and hb. Figure 8 shows

the resulting growth rate as a function of j for the three

casesREF_RIGID (d2 5 0, d3 5 0), UNBALBL_RIGID

(d2 5 1, d3 5 1), and SBL_RIGID (d2 5 1, d3 5 0). In all

cases, the largest growth rate results at j5 0, and in-

stability arises only for j, 1: that is, k. 1/(ffiffiffi2

pR). The

FIG. 8. Nondimensional growth rate s/f as a function of

nondimensional perturbation size j for the cases REF_RIGID

(d2 5 0, d3 5 0), UNBALBL_RIGID (d2 5 1, d3 5 1), and SBL_

RIGID (d2 5 1, d3 5 0).

FIG. 7. Radial profiles of vertical wind for the experiment

UNBALBL_RIGID at the time of maximum intensification for

various horizontal diffusion coefficients.



unstable modes have an updraft with the radius 2:4048/k

in the center, and the growth rate s increases with de-

creasing updraft diameter. The largest growth rates occur

for d2 5 1 and d3 5 1 (UNBALBL_RIGID). Therefore,

the inclusion of the local time tendencies ›u0b/›t and

›y0b/›t in UNBALBL_RIGID enhances the instability as

in the nonlinear experiments (see Fig. 4a). Equation (63)

simplifies drastically in the case d3 5 0 and has the fol-

lowing solution:

s5hb

H2

K

11 d2K2/f 2

h2 12 j2

2j2 1 1. (64)

For d2 5 0, this formula is very similar to that obtained by

O69 [see his Eq. (8.10)]. In his formula the additional

summand (12 �)j4 shows up in the denominator of the

second factor, but it has a small impact since 12 � � 1.

For d2 5 1, the growth rate becomes smaller than in the

case d2 5 0. Therefore, the violation of the balance as-

sumption in the steady boundary layer has a damping

effect in the linearized system. Thus, only nonlinear terms

in the boundary layer model can explain the larger in-

tensification rate observed in the nonlinear experi-

ments SBL_RIGID in comparison to REF_RIGID (see

Fig. 4a). This outcome does not appear surprising, since

the abovementioned radial momentum advection term

is not part of the linear model, and it is, therefore, un-

derstandable that it cannot explain the unbalanced

intensification phase properly. Furthermore, representa-

tion of a convective ring in terms of a Fourier–Bessel

series does not exhibit the ring contraction when each

coefficient growswith the corresponding growth rate (not

shown). Consequently, the linear model cannot explain

the contraction of the vortex either.

Further simulations have been performed in which the

vertically averaged radial momentum advection term

2u›u/›r has been set to zero in the boundary layer to

test if this term plays an important role for the inten-

sification. The experiments called SBLMOD_RIGIDand

KEPERTBLMOD_RIGID are identical to SBL_RIGID

and KEPERTBL_RIGID, respectively, except for this

modification. Figure 9 shows the maximum tangential

wind y1 and the RMGW for these additional simulations

as a function of time. The results for the unmodified runs

are also displayed for comparison. Obviously, the devel-

opment for SBLMOD_RIGID significantly differs from

that for SBL_RIGID. Both intensity and intensification

rate take much lower values than for SBL_RIGID. Fur-

thermore, the contraction of the RMGW already stops at

37km compared to the minimum RMGW of 12km ob-

served in SBL_RIGID. The final intensity is even lower

than in REF_RIGID. The omission of the radial mo-

mentum advection term has a visibly smaller effect in

Kepert’s height-parameterized boundary layer model, as

can be seen in Fig. 9b. This brings again intomind that the

slab boundary model overestimates the role of radial

momentum advection. We also performed simulations in

which both nonlinear advection terms 2u›u/›r and

2u›(y2 y1)/›r are omitted in the boundary layer and the

time tendencies are retained. In the case of a steady slab

boundary, these additional modifications have little ef-

fect, but neglecting 2u›(y2 y1)/›r reduces the intensifi-

cation and contraction by another significant amount in

Kepert’s height-parameterized boundary layer model.

These results demonstrate the vital role of nonlinear

momentum advection for intensification in the Ooyama

model. Possibly, nonlinear momentum advection in the

boundary layer could also explain some aspects of the

finite-amplitude nature of tropical cyclogenesis.5

b. Evaluation of the middle-layer momentum budget

The source termsBA,BB,BC,BD,BE, andBF [see Eqs.

(38)–(43)] cause a radial middle-layer flow, which in turn

FIG. 9. As in Fig. 2, but for (a) SBLMOD_RIGID and (b)

KEPERTBLMOD_RIGID (see text). The thin curves in (a) and

(b) show for comparison the results of the experiments SBL_

RIGID and KEPERTBL_RIGID, respectively.

5 Substantial intensification within 10 days occurs in UNBALBL_

RIGID only when the initial tangential wind maximum is above

3m s21 (not shown).



induces a tendency in tangential wind y1 through angular

momentum advection. Therefore, at least one of these

source terms should be responsible for intensification,

because the resulting radial flow should be inward to

produce a local increase of angular momentum. How-

ever, vertical transport of boundary layer air could also

contribute to a local increase of angular momentum,

but it appears rather unlikely that this process solely

accounts for intensification. Figure 10 displays the var-

ious contributions to the tendency of tangential wind y1as a function of radius and time for the experiment

UNBALBL_RIGID. As expected, latent heat (source

term BB) release provides the major contribution to

intensification and RMGW contraction. During the

contraction phase, it has a distinct and narrowmaximum

inside the RMGW. The small horizontal extension of

the response can be explained by the high inertial sta-

bility inside the RMGW where the maximum vertical

mass flux occurs (see Fig. 5). At the end of the con-

traction phase, nonnegligible contributions due to fric-

tional convergence (source term BA) and upper-layer

gradient wind change (source term BD 1BF) become

apparent. The former arises because of downward mo-

tion immediately inside of the eyewall leading to inward

flow in the middle layer. This descent could be an arti-

fact of the simple slab boundary layer model, since it

emerges with a much smaller amplitude in KEPERTBL_

RIGID (see Fig. 6). A corresponding upward flux from

the boundary layer into the middle layer does not ap-

pear, since the entrainment parameter h is larger than 1

below the eyewall, and, therefore, the updraft at the top

of the boundary layer transports air directly into the

upper layer [see Eqs. (11)–(13)]. The upper-layer gradi-

ent wind change contributes to the y1 tendency, since an

alteration of the radial pressure gradient in the upper

layer also modifies the radial pressure gradient in the

middle layer, leading to additional inflow. A relevant

process for upper-layer gradient wind intensification is

given by the vertical momentum flux Dy,y2 . However, a

large portion of this intensification is compensated by

outward advection of angularmomentum so that finally a

positive contribution remains, which is considerably

smaller than that of latent heat release. The impact of

upward momentum flux (source term BC) and diffusion

(source term BE) on midlevel inflow is negligible. Nev-

ertheless, both these processes also have a direct con-

tribution to the tangential momentum budget. Diffusion

mainly dampens intensification but also supports the

contraction of the RMGW. The direct tendency due to

upwardmomentum flux appears to be small compared to

latent heat release. All the displayed tendencies add up

to the total tendency (shown in Fig. 10h). The pattern

of the total tendency resembles that due to latent heat

release, but the radial extension of the maximum is

somewhat larger, which is mainly a consequence of hor-

izontal diffusion.

Figure 11 shows the middle-layer momentum budget

for experiment KEPERTBL_RIGID. In this experi-

ment, the entrainment parameter h remains above 1

during the complete simulation. Therefore, no shallow

convection and no momentum transport from the

boundary layer to the middle layer takes place. Conse-

quently, the tendencies due to upward momentum flux

vanish identically. The tendency due to frictional con-

vergence attains much smaller values, presumably be-

cause of the smaller vertical velocity at the top of the

boundary layer. The tendency due to horizontal diffusion

induces, in contrast to UNBALBL_RIGID, a spinup of

the tangential wind inside the RMGW but with a small

amplitude and after the vortex has already terminated its

contraction. The other tendency terms resemble quali-

tatively those of the experiment UNBALBL_RIGID.

However, the tendencies have a significantly smaller

amplitude, which explains the slower intensification in

KEPERTBL_RIGID.

Figure 12 displays the contributions to the tendencies

of y1 for the experiment REF_RIGID. Note that up-

wardmomentum flux out of the boundary layer does not

change the middle-layer tangential wind because in

REF_RIGID it is assumed that yb 5 y1. In contrast to

UNBALBL_RIGID, the tendency due to latent heat

release spreads over a larger radial range. This hap-

pens because the updraft has a larger width than in

UNBALBL_RIGID and is located outside of the

RMGW (see Fig. 5f), where a smaller inertial stability

occurs. Furthermore, the maximum tendency due to

heating is about a factor of 10 smaller compared to

UNBALBL_RIGID. Although the total heating might

be similar to UNBALBL_RIGID, the smaller peak

value can explain the smaller intensification rate in

REF_RIGID, since the radial gradient of latent heating

and not the total heating induces the gradient wind

increase [cf. Eq. (39)]. Frictional convergence only

dampens intensification in REF_RIGID, and its magni-

tude is very small. The upper-layer gradient wind change

contributes with a large amount to the midlevel inflow

in the mature stage. The radial flow induced by diffu-

sion also supports intensification in the mature stage,

but the resulting tendencies are very small. The direct

tendency due to diffusion is negative near the RMGW,

and, therefore, it mainly dampens intensification. After

maximum contraction, the RMGW migrates outward

rapidly. This happens because latent heat release pro-

duces a positive tangential tendency outside of the

RMGW, while frictional convergence decreases tan-

gential wind inside of the RMGW.



FIG. 10. Radius–time diagrams for experiment UNBALBL_RIGID, showing tendencies of gradient wind y1stemming from (a) source term BA (frictional convergence), (b) source term BB (latent heat release), (c) source

termBC (upward momentum flux), (d) source termBD 1BF (upper-layer gradient wind change), (e) source term

BE (diffusion), (f) direct effect of upward momentum flux Dy,y1, (g) direct effect of diffusion Dy,y1, and (h) total

tendency ›y1/›t. The contour interval is displayed in the upper-right corner of each panel. Negative isolines are

dashed, and the thick solid line displays the position of RMGW.



Comparison of Figs. 10 and 11 with Fig. 12 manifests

large qualitative and quantitative differences in the in-

tensification dynamics. In UNBALBL_RIGID and

KEPERTBL_RIGID the tendencies are mostly con-

centrated in a narrow ring inside of the RMGW, while in

REF_RIGID the tendency profiles are smoother and

sometimes are maximized outside the RMGW. The

reason for the different behavior of REF_RIGID is the

neglect of gradient wind imbalance in the boundary layer

model. The consequence is that the maximum ascent out

of the boundary layer occurs outside of the RMGWwith a

smooth radial profile (see Fig. 6a). This leads via latent

heat release to a gradient wind intensification with little

contraction and whose profile is also smooth.

To substantiate this conclusion further, steady-state

solutions of the unbalanced slab boundary layer model

and of Kepert’s height-parameterized boundary layer

model have been calculated using the gradient wind field

FIG. 11. Radius–time diagrams for experiment KEPERTBL_RIGID showing tendencies of gradient wind y1stemming from (a) source term BA (frictional convergence), (b) source term BB (latent heat release), (c) source

termBD 1BF (upper-layer gradient wind change), (d) source termBE (diffusion), (e) direct effect of diffusionDy,y1 ,

and (f) total tendency ›y1/›t. The contour interval is displayed in the upper-right corner of each panel. Negative

isolines are dashed, and the thick solid line displays the position of RMGW.



of REF_RIGID at t 5 99 h. The results are shown in

Fig. 13, which should be compared with Fig. 6a. As ex-

pected, the boundary layer wind profiles differ greatly to

those of REF_RIGID. For the slab boundary layer

model, the vertical wind maximum arises far inside of

the RMGW, where strong supergradient winds are

found. The radial inflow also maximizes inside of the

RMGW and takes smaller values compared to the bal-

anced boundary layer calculation in REF_RIGID. The

steady-state solution of Kepert’s height-parameterized

boundary layer model also reveals an inward shift of the

vertical wind maximum and a decrease of the radial in-

flow velocity. However, the maximum vertical velocity

and the supergradient wind appear to be much smaller

than in the slab boundary layer model. Such differences

have already been noted and are consistent with the

findings of section 3. An inward shift of the frictional

updraft location was also found by Kepert (2013) with a

height-resolving boundary layer model when compared

with a linearized one. Furthermore, he demonstrated

that the linearized version yields vertical wind profiles

that resemble those found with the balanced slab

FIG. 12. As in Fig. 11, but for the experiment REF_RIGID.



boundary layer model of O69. Running the model by

using the gradient wind of REF_RIGID at t 5 99 h as

initial condition for d2 5 d3 5 1 would lead to immediate

contraction and rapid intensification. These results

support our conclusion on the importance of the non-

linear advection terms and show that it is of importance

for intensificationwhether the convective heating occurs

inside or outside of the RMGW.

5. Conclusions

In this study we have investigated the impact of the

balance assumption for the intensification of a tropical

cyclone in Ooyama’s three-layer model. Furthermore,

the effects of a rigid lid, the neglect of local time de-

rivatives in the boundary layer momentum equations,

and the inclusion of Kepert’s height-parameterized

boundary layer model have also been examined. We

found that the balance approximation in the two upper

layers and the rigid-lid assumption are of minor impor-

tance for the intensification phase, while the balance

approximation in the boundary layer has a significant

impact. With a balanced boundary layer (as given in ex-

periment REF_RIGID), the tropical cyclone intensifies

much more slowly than in the simulation employing

an unbalanced boundary layer (as given in experiment

UNBALBL_RIGID). Furthermore, the vortex contracts

only slightly during intensification, and the RMGW

increases dramatically after the intensity has maxi-

mized in REF_RIGID. In contrast, the simulation

UNBALBL_RIGID reveals a substantial vortex con-

traction with little RMGW variation after the intensi-

fication stage. In both experiments intensification is

accompanied by the occurrence of a convective ring in the

vicinity of the RMGW. However, in REF_RIGID the con-

vective ring is located outside and in UNBALBL_RIGID

inside the RMGW. We suggest that this discrepancy is

crucial for the different intensification rates. Because of

the higher inertial stability, latent heating inside the

RMGW leads to a larger and tighter tangential wind ten-

dency than in the case with heating outside the RMGW.

This can explain the faster growth and contraction in

UNBALBL_RIGID compared to REF_RIGID. An en-

hancement of intensification with increasing inertial sta-

bilitywas already foundbySchubert andHack (1982) in an

idealized analytical solution of the Sawyer–Eliassen

equation. This result hints at the importance of non-

linearity in tropical cyclone intensification, and the mech-

anismwas further elaborated byHack andSchubert (1986)

on the basis of nonlinear primitive equation and balanced

models. However, they assumed a time-independent heat

source, and, therefore, they excluded a possible feedback

of inertial stability on heating. This could stem from a

modification of the boundary layer inflow and the as-

sociated vertical mass fluxes. The results based on

Ooyama’s three-layer model show that the increasing

inflow in the unbalanced boundary layer supplies the

eyewall with enough moist air so that heating is main-

tained or even increased at larger inertial stability.

Eventually, the intensification stops because the fric-

tional dissipation rate increases at a much faster rate

than the energy input rate from the ocean (Wang 2012).

The simulations with neglected time derivatives (as

given in experiment SBL_RIGID) and with Kepert’s

height-parameterized boundary layer model (as given in

experiment KEPERTBL_RIGID) exhibit vortex con-

traction like in UNBALBL_RIGID, but the intensifica-

tion rates are smaller. Experiment KEPERTBL_RIGID

has more realistic wind profiles, and the smaller intensi-

fication rate in this experiment partially results because

the surface flux parameterization includes surface winds

FIG. 13. As in Fig. 6a, but the calculated boundary layer wind

profiles have been determined by finding the steady-state solutions

of (a) the unbalanced slab boundary layer model and (b) Kepert’s

height-parameterized boundary layer model.



instead of vertically averaged winds, as in the slab bound-

ary layer model. Furthermore, the radial overshoot is

also reduced in the height-parameterized boundary

layer model (see Fig. 13) so that convection forms in a

less inertially stable environment, where a diminished

intensification rate results.

The linear instability analysis of the governing equa-

tions does not reveal an enhancement of the in-

tensification rate by the relaxation of the balance

assumption in the boundary layer, and it cannot ex-

plain the contraction of the intensifying vortex either.

Therefore, nonlinear terms appear to enhance intensi-

fication in the Ooyama model as a result of gradient

wind imbalance. The radial advection of radial mo-

mentum likely supports the intrusion of boundary layer

air inside of the RMGW, where the eyewall develops. A

simulation without this term in the slab boundary layer

model exhibits a much slower development with lit-

tle contraction. The reason for faster intensification

and contraction in UNBALBL_RIGID compared to

REF_RIGID has been found by analyzing the various

terms in the tangential wind equation of the middle

model layer. In both experiments latent heat release is

the main driver for intensification. Gradient wind

intensification by upward momentum flux from the

middle layer to the upper layer also contributes to in-

tensification in the middle layer in the later stages of

the development. The frictionally induced downdraft

inside the eyewall yields another contribution in

UNBALBL_RIGID. However, the crucial difference

between both experiments is the radial scale of the wind

tendency. In UNBALBL_RIGID the radial extent is

much smaller than in REF_RIGID because the eyewall

emerges in the inertially stable region inside of the

RMGW, while in REF_RIGID it arises outside of the

RMGW, where weak inertial stability dominates.

Although the intensification rate is smaller, experi-

ment KEPERTBL_RIGID reveals an intensification

mechanism resembling that of UNBALBL_RIGID.

Therefore, a more realistic representation of bound-

ary layer dynamics still supports the finding that gra-

dient wind imbalance in the boundary layer is of

importance for the intensification scenario and vortex

contraction.

These results suggest the following intensification

mechanism in Ooyama’s three-layer model. First, an

incipient vortex generates inflow, and the inflow over-

shoots the RMGW as a result of its inertia. The over-

shoot causes an updraft inside the RMGW, which

would occur outside the RMGW in the balanced and

the linearized boundary layer model [as found by

Kepert (2013)]. Then moist convective instability re-

leased by Ekman pumping generates a convective ring,

and the entrainment of air above the boundary layer

intensifies the tangential wind because of angular mo-

mentum import in the middle layer. Finally, the re-

sulting increase of gradient wind enhances the inflow

with more inward intrusion, which causes an inward

migration of the eyewall and further intensification.

Fluxes of latent heat from the sea surface are, of course,

necessary in this feedback loop to maintain the con-

vective instability near the eyewall radius. This picture

has some similarity with the CISK theory, since there is

also a cooperation of a large-scale vortex with con-

vection, and the latent heating depends on frictional

convergence in the boundary layer. However, the CISK

theory refers to an ensemble of individual convection

cells, but in the present model convection only appears

in the form of a single convective ring. Obviously,

convection is in the nonlinear Ooyama model only

triggered at the position where maximum frictional

convergence appears. Furthermore, the linear CISK

model by Charney and Eliassen (1964) predicts, like

the linearized Ooyama model, a maximum heating at

the vortex center, while in the nonlinear Ooyama

model it appears at a finite radius where the contracting

convective ring is located. This difference is in agree-

ment with the finding by Eliassen (1971) that a qua-

dratic drag law yields a boundary layer updraft

maximum at a finite radius. Ooyama (1982) already

appreciated the role of nonlinearities in his co-

operation intensification theory, but he did not con-

sider gradient wind imbalance in the boundary layer,

although he found previously in his unpublished study

that it enhances intensification. The frictionally in-

duced intensification mechanism is also consistent with

the study ofWang andXu (2010), who found in a cloud-

resolving model that inward boundary layer enthalpy

transport is important to the energy balance in the

eyewall.

The present study suggests that gradient wind imbal-

ance must be taken into account for a proper un-

derstanding of the intensification process. However, one

may ask why some balanced models already provide a

reasonable picture for intensification and vortex con-

traction. For example, Emanuel (1989) developed a

simple balanced hurricanemodel that reveals a behavior

as in the unbalanced case discussed here. On the other

hand, Emanuel (1989) applied a convective parameter-

ization scheme inwhich the convectivemass flux depends

directly on buoyancy and is independent of moisture

convergence in the boundary layer. Therefore, the

placement of the developing eyewall follows other rules

in his model than in the Ooyama model. A possible

drawback of the Ooyama model is the nonexistence of

ordinary conditional instability like that found by Lilly



(1960). Convective instability can take place without

frictional convergence in the boundary layer only if h

becomes unrealistically large. This is unlike the situation

in the vertically continuous atmosphere, where both

kinds of conditional instability occur simultaneously, if at

all (Fraedrich and McBride 1995). Possibly, convective

ring formation can result from ordinary conditional in-

stability in a more realistic model. This could potentially

explain why intensification and contraction arise without

surface drag in one of the axisymmetric model experi-

ments by Craig and Gray (1996). Such an outcome does

not invalidate the importance of unbalanced dynamics,

since ordinary convection cannot evolve in a balanced

model. Nevertheless, to reveal the relevance of the results

of the present study and the possible limitations of the

Ooyama model, it is necessary to investigate further the

intensification mechanism in more complex models.

Acknowledgments. This work was supported by the

Deutsche Forschungsgemeinschaft (DFG) within the

following individual research grant: The role of con-

vective available potential energy for tropical cyclone

intensification FR 1678/2-1. The first author also grate-

fully acknowledges support by the Cluster of Excellence

CliSAP (EXC177) funded by the DFG. We thank M. T.

Montgomery for the hint that K. V. Ooyama already

indicated the importance of unbalanced boundary layer

dynamics for intensification in an unpublished manu-

script that was kindly provided to us. Furthermore, we

thank three anonymous reviewers for their valuable

comments.

REFERENCES

Braun, S. A., M. T. Montgomery, and Z. Pu, 2006: High-resolution

simulation of Hurricane Bonnie (1998). Part I: The organiza-

tion of eyewall vertical motion. J. Atmos. Sci., 63, 19–42,

doi:10.1175/JAS3598.1.

Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical

model for the maximum intensity of tropical cyclones.

J. Atmos. Sci., 66, 3042–3060, doi:10.1175/2009JAS3038.1.

Bui, H. H., R. K. Smith, M. T. Montgomery, and J. Peng, 2009:

Balanced and unbalanced aspects of tropical cyclone in-

tensification. Quart. J. Roy. Meteor. Soc., 135, 1715–1731,

doi:10.1002/qj.502.

Charney, J. G., and A. Eliassen, 1964: On the growth of the hur-

ricane depression. J. Atmos. Sci., 21, 68–75, doi:10.1175/

1520-0469(1964)021,0068:OTGOTH.2.0.CO;2.

Craig, G. C., and S. L. Gray, 1996: CISK or WISHE as the

mechanism for tropical cyclone intensification. J. Atmos.

Sci., 53, 3528–3540, doi:10.1175/1520-0469(1996)053,3528:

COWATM.2.0.CO;2.

DeMaria, M., and J. D. Pickle, 1988: A simplified system of

equations for simulation of tropical cyclones. J. Atmos. Sci.,

45, 1542–1554, doi:10.1175/1520-0469(1988)045,1542:

ASSOEF.2.0.CO;2.

Eliassen, A., 1971: On the Ekman layer in a circular vortex.

J. Meteor. Soc. Japan, 49, 784–789.

Emanuel, K. A., 1989: The finite-amplitude nature of tropical cy-

clogenesis. J. Atmos. Sci., 46, 3431–3456, doi:10.1175/

1520-0469(1989)046,3431:TFANOT.2.0.CO;2.

Fraedrich, K., and J. L. McBride, 1995: Large-scale convective

instability revisited. J. Atmos. Sci., 52, 1914–1923, doi:10.1175/

1520-0469(1995)052,1914:LSCIR.2.0.CO;2.

Frank, W. M., 1977: The storm structure and energetics of the trop-

ical cyclone. I: Storm structure.Mon.Wea. Rev., 105, 1119–1135,

doi:10.1175/1520-0493(1977)105,1119:TSAEOT.2.0.CO;2.

Frisius, T., 2006: Axisymmetric tropical cyclogenesis via a

single convective ring. 27th Conf. on Hurricanes and

Tropical Meteorology, Monterey, CA, Amer. Meteor. Soc.,

14B.7. [Available online at https://ams.confex.com/ams/

27Hurricanes/webprogram/Paper108695.html.]

——, 2015: What controls the size of a tropical cyclone? In-

vestigations with an axisymmetric model. Quart. J. Roy. Me-

teor. Soc., 141, 2457–2470, doi:10.1002/qj.2537.——, and T. Hasselbeck, 2009: The effect of latent cooling pro-

cesses in tropical cyclone simulations. Quart. J. Roy. Meteor.

Soc., 135, 1732–1749, doi:10.1002/qj.495.

——, D. Schönemann, and J. Vigh, 2013: The impact of gradient

wind imbalance on potential intensity of tropical cyclones in

an unbalanced slab boundary layer model. J. Atmos. Sci., 70,

1874–1890, doi:10.1175/JAS-D-12-0160.1.

Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of

atmospheric vortices to heating by organized cumulus

convection. J. Atmos. Sci., 43, 1559–1573, doi:10.1175/

1520-0469(1986)043,1559:NROAVT.2.0.CO;2.

Hausman, S. A., K. V. Ooyama, and W. H. Schubert, 2006: Po-

tential vorticity structure of simulated hurricanes. J. Atmos.

Sci., 63, 87–108, doi:10.1175/JAS3601.1.

Heng, J., and Y. Wang, 2016: Nonlinear response of a tropical cy-

clone vortex to prescribed eyewall heating with and without

surface friction in TCM4: Implications for tropical cyclone

intensification. J. Atmos. Sci., 73, 1315–1333, doi:10.1175/

JAS-D-15-0164.1.

Hill, K. A., andG.M. Lackmann, 2009: Influence of environmental

humidity on tropical cyclone size. Mon. Wea. Rev., 137, 3294–

3315, doi:10.1175/2009MWR2679.1.

Kepert, J. D., 2010a: Slab- and height-resolving models of the

tropical cyclone boundary layer. Part I: Comparing the simu-

lations. Quart. J. Roy. Meteor. Soc., 136, 1686–1699,

doi:10.1002/qj.667.

——, 2010b: Slab- and height-resolving models of the tropical cyclone

boundary layer. Part II: Why the simulations differ.Quart. J. Roy.

Meteor. Soc., 136, 1700–1711, doi:10.1002/qj.685.

——, 2013: How does the boundary layer contribute to eyewall

replacement cycles in axisymmetric tropical cyclones?

J. Atmos. Sci., 70, 2808–2830, doi:10.1175/JAS-D-13-046.1.

Lilly, D. K., 1960: On the theory of disturbances in a conditionally

unstable atmosphere. Mon. Wea. Rev., 88, 1–17, doi:10.1175/

1520-0493(1960)088,0001:OTTODI.2.0.CO;2.

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:10.1175/

JAS3604.1.

Ooyama, K. V., 1969: Numerical simulation of the life cycle of

tropical cyclones. J. Atmos. Sci., 26, 3–40, doi:10.1175/

1520-0469(1969)026,0003:NSOTLC.2.0.CO;2.

——, 1982: Conceptual evolution of the theory and modelling of

the tropical cyclone. J. Meteor. Soc. Japan, 60, 369–379.



http://dx.doi.org/10.1175/JAS3598.1

http://dx.doi.org/10.1175/2009JAS3038.1

http://dx.doi.org/10.1002/qj.502

http://dx.doi.org/10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1996)053<3528:COWATM>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1996)053<3528:COWATM>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1988)045<1542:ASSOEF>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1988)045<1542:ASSOEF>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1989)046<3431:TFANOT>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1989)046<3431:TFANOT>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1995)052<1914:LSCIR>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1995)052<1914:LSCIR>2.0.CO;2

http://dx.doi.org/10.1175/1520-0493(1977)105<1119:TSAEOT>2.0.CO;2

https://ams.confex.com/ams/27Hurricanes/webprogram/Paper108695.html

https://ams.confex.com/ams/27Hurricanes/webprogram/Paper108695.html



http://dx.doi.org/10.1175/JAS-D-12-0160.1

http://dx.doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2




http://dx.doi.org/10.1175/2009MWR2679.1




http://dx.doi.org/10.1175/1520-0493(1960)088<0001:OTTODI>2.0.CO;2

http://dx.doi.org/10.1175/1520-0493(1960)088<0001:OTTODI>2.0.CO;2



http://dx.doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2

Persing, J., M. T. Montgomery, J. C. McWilliams, and R. K. Smith,

2013: Asymmetric and axisymmetric dynamics of tropical

cyclones. Atmos. Chem. Phys., 13, 12 299–12 341, doi:10.5194/

acp-13-12299-2013.

Schecter, D. A., 2011: Evaluation of a reduced model for in-

vestigating hurricane formation from turbulence. Quart.

J. Roy. Meteor. Soc., 137, 155–178, doi:10.1002/qj.729.

——, and T. J. Dunkerton, 2009: Hurricane formation in diabatic

Ekman turbulence. Quart. J. Roy. Meteor. Soc., 135, 823–838,

doi:10.1002/qj.405.

Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical

cyclone development. J. Atmos. Sci., 39, 1687–1597,

doi:10.1175/1520-0469(1982)039,1687:ISATCD.2.0.CO;2.

Shapiro, L. J., andH. E.Willoughby, 1982: The response of balanced

hurricanes to local sources of heat and momentum. J. Atmos.

Sci., 39, 378–394, doi:10.1175/1520-0469(1982)039,0378:

TROBHT.2.0.CO;2.

Smith, R. K., and M. T. Montgomery, 2008: Balanced boundary

layers used in hurricane models. Quart. J. Roy. Meteor. Soc.,

134, 1385–1395, doi:10.1002/qj.296.

——, and ——, 2015: Toward clarity on understanding tropical cy-

clone intensification. J. Atmos. Sci., 72, 3020–3031, doi:10.1175/

JAS-D-15-0017.1.

——, G. Kilroy, and M. T. Montgomery, 2015: Why do model

tropical cyclones intensify more rapidly at low latitudes?


Stern,D. P., J. L.Vigh,D. S.Nolan, andF.Zhang, 2015:Revisiting the

relationship between eyewall contraction and intensification.


Wang, Y., 2002a: An explicit simulation of tropical cyclones

with a triply nested movable mesh primitive equation model:

TCM3. Part II: Model refinements and sensitivity to cloud

microphysics parameterization. Mon. Wea. Rev., 130, 3022–

3036, doi:10.1175/1520-0493(2002)130,3022:AESOTC.2.0.

CO;2.

——, 2002b: Vortex Rossby waves in a numerically simulated

tropical cyclone. Part II: The role in tropical cyclone struc-

ture and intensity change. J. Atmos. Sci., 59, 1239–1262,

doi:10.1175/1520-0469(2002)059,1239:VRWIAN.2.0.CO;2.

——, 2012: Recent research progress on tropical cyclone structure

and intensity.Trop. CycloneRes. Rev., 1, 254–275, doi:10.6057/

2012TCRR02.05.

——, and J. Xu, 2010: Energy production, frictional dissipation, and

maximum intensity of a numerically simulated tropical cyclone.

J. Atmos. Sci., 67, 97–116, doi:10.1175/2009JAS3143.1.

Williams, G., 2015: The effects of vortex structure and vortex trans-

lation on the tropical cyclone boundary layer wind field. J. Adv.

Model. Earth Syst., 7, 188–214, doi:10.1002/2013MS000299.

Xu, J., and Y. Wang, 2010: Sensitivity of the simulated tropical

cyclone inner-core size to the initial vortex size. Mon. Wea.

Rev., 138, 4135–4157, doi:10.1175/2010MWR3335.1.



http://dx.doi.org/10.5194/acp-13-12299-2013

http://dx.doi.org/10.5194/acp-13-12299-2013



http://dx.doi.org/10.1175/1520-0469(1982)039<1687:ISATCD>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2






http://dx.doi.org/10.1175/1520-0493(2002)130<3022:AESOTC>2.0.CO;2

http://dx.doi.org/10.1175/1520-0493(2002)130<3022:AESOTC>2.0.CO;2

http://dx.doi.org/10.1175/1520-0469(2002)059<1239:VRWIAN>2.0.CO;2

http://dx.doi.org/10.6057/2012TCRR02.05

http://dx.doi.org/10.6057/2012TCRR02.05

http://dx.doi.org/10.1175/2009JAS3143.1

http://dx.doi.org/10.1002/2013MS000299

http://dx.doi.org/10.1175/2010MWR3335.1

the impact of gradient wind imbalance on tropical cyclone

Documents