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Journal of Advances in Modeling Earth Systems
RESEARCH ARTICLE Transient aspects of the Hadley circulation forced by an
10.1002/2016MS000837 idealized off-equatorial ITCZ
Key Points: Alex O. Gonzalez1,2 , Gabriela Mora Rojas3, Wayne H. Schubert1 , and Richard K. Taft1
 Transient diabatic heating in the ITCZ
forces balanced and transient 1Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado, USA, 2Now at Joint Institute for
behavior in the southern and
northern Hadley cells Regional Earth System Science and Engineering, University of California, Los Angeles, California, USA,
3Department of
 When the ITCZ intensifies within 3 Atmospheric Physics, Oceanography, and Planetary Science, University of Costa Rica, San Pedro Montes de Oca,
days, equatorially trapped inertia- Costa Rica
gravity waves have significantly large
amplitudes
 The mass in the Hadley cells oscillate Abstract This paper presents analytical solutions of large-scale, zonally symmetric overturning circula-
on 1–3 day timescales for typical ITCZ
diabatic heating vertical modes tions in the tropical free troposphere forced by transient diabatic heating in the off-equatorial intertropical
convergence zone (ITCZ). The dynamics are discussed in the context of the time-dependent meridional
Correspondence to: circulation equation arising in an equatorial b-plane model. The solutions of these differential equations con-
A. O. Gonzalez, tain terms for the slow, quasi-balanced part of the response and terms for the transient, zonally symmetric,
alex.o.gonzalez@jpl.nasa.gov inertia-gravity wave part of the response. When the off-equatorial (north of the equator) ITCZ diabatic heat-
ing is switched on at various rates, both parts of the response reveal a basic asymmetry between the south-
Citation: ern and northern hemispheres, with the southern hemisphere side containing most of the quasi-balanced
Gonzalez, A. O., G. Mora Rojas,
W. H. Schubert, and R. K. Taft (2017), compensating subsidence and transient inertia-gravity wave activity. The inertia-gravity waves travel in
Transient aspects of the Hadley wave packets that bounce off a spectrum of turning latitudes and are analyzed in the context of an average
circulation forced by an idealized conservation law approach. These traveling wave packets cause the mass flux in the southern and northern
off-equatorial ITCZ, J. Adv. Model.
Earth Syst., 9, 668–690, doi:10.1002/ Hadley cells to pulsate on timescales of about 1, 2, and 3 days for diabatic heating of the external, first inter-
2016MS000837. nal, and second internal vertical modes, respectively. The spectral characteristics of the vertical motion in the
ITCZ and subsidence regions are slightly more complicated and depend on ITCZ location.
Received 15 OCT 2016
Accepted 15 FEB 2017
Accepted article online 20 FEB 2017
Published online 24 MAR 2017
1. Introduction
A typical, boreal summer 6.7 lm water vapor image of the eastern Pacific from the GOES West satellite is
shown in Figure 1. Under clear-sky conditions, the 6.7 lm channel is sensitive to the vertically averaged
humidity in the 200–500 hPa layer, so the dark blue areas on either side of the intertropical convergence
zone (ITCZ) indicate regions of low humidity in the upper troposphere and hence regions of enhanced sub-
sidence in the downward branches of the southern hemisphere and northern hemisphere Hadley cells. The
complete explanation of atmospheric water vapor distributions can be quite complicated and involve sev-
eral different physical processes, such as the stretching and folding processes associated with the Rossby
wave pattern just east of Hawaii in Figure 1. For detailed discussions of tropical moisture distributions,
including trajectory analysis and the concept of ‘‘time since last condensation,’’ see Sun and Lindzen [1993],
Soden and Fu [1995], Salathe and Hartmann [1997], Pierrehumbert [1998], Pierrehumbert and Roca [1998],
Galewsky et al. [2005], Sherwood et al. [2006], Cau et al. [2007], and Schreck et al. [2013]. In spite of the intrica-
cies involved in comprehensive explanations of tropical water vapor distributions, the explanation of the
water vapor distribution in the eastern Pacific is simpler than in many other areas during much of the year.
An important part of the explanation lies in the large-scale balanced dynamics of the Hadley cells, with the
VC 2017. The Authors. southern hemisphere Hadley cell having larger meridional extent and larger overturning mass flux [Oort
This is an open access article under the and Rasmusson, 1970; Lindzen and Hou, 1988; Hack et al., 1989].
terms of the Creative Commons
Attribution-NonCommercial-NoDerivs The tropical Hadley circulation is often thought of as a slowly evolving, zonally symmetric phenomenon in
License, which permits use and which the divergent part of the flow (i.e., the meridional and vertical components) is diagnostically deter-
distribution in any medium, provided mined via an elliptic equation, while the rotational part of the flow (i.e., the geostrophically balanced zonal
the original work is properly cited, the
component) evolves as the potential vorticity (PV) in the ITCZ develops in response to diabatic and frictional
use is non-commercial and no
modifications or adaptations are forcing [Hack et al., 1989; Hack and Schubert, 1990; Schubert et al., 1991; Gonzalez and Mora Rojas, 2014]. The
made. PV distribution then naturally evolves into a structure in which the meridional gradient of PV has both signs.
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 668
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
Figure 1. NOAA GOES West satellite water vapor image (6.7 lm) at 06 UTC 25 June 2013. This image is typical of the eastern Pacific during
the boreal summer when the ITCZ is located at 108N–158N. The dark blue areas on either side of the ITCZ indicate regions of low humidity
in the upper troposphere, and hence regions of enhanced subsidence in the downward branches of the southern hemisphere and
northern hemisphere Hadley cells.
Such PV structures have the distinct possibility of supporting combined barotropic/baroclinic instability, which
leads to a breakdown of the zonally symmetric structure [Nieto Ferreira and Schubert, 1997;Wang and Magnus-
dottir, 2005; Magnusdottir and Wang, 2008]. In this study, we relax the assumption of geostrophic balance of
the zonal flow so that the meridional winds associated with the zonally symmetric Hadley circulation can
evolve in a way that allows for the meridional propagation of equatorially trapped inertia-gravity waves. These
large-scale inertia-gravity waves are mainly forced by transient convection in the ITCZ.
Inertia-gravity waves in the tropics have been widely studied, but not in the context of the large-scale ITCZ
and Hadley circulation system. Takayabu [1994] showed evidence that tropospheric inertia-gravity waves,
often called ‘‘two-day waves,’’ are important for large-scale tropical dynamics, especially in conjunction with
the Madden-Julian oscillation. These inertia-gravity waves have been classified as convectively coupled
n5 1 waves, and they mainly propagate westward, but also have a nonnegligible meridional component
[Haertel and Kiladis, 2004]. Wunsch and Gill [1976] also found observational evidence of n5 1 and n5 2
inertia-gravity waves in sea level and surface meridional wind data over the central Pacific Ocean, with spec-
tral peaks at 4–5 day timescales. Similar timescales have also been observed in regards to stratospheric
inertia-gravity waves forced by tropical convection, as discussed in Tsuda et al. [1994], Karoly et al. [1996],
Evan and Alexander [2008], and Evan et al. [2012]. In particular, two-day oscillations associated with
inertia-gravity waves emanating from tropical convection have been observed in the stratosphere [Evan
and Alexander, 2008; Evan et al., 2012]. The two-day stratospheric inertia-gravity waves travel mainly in the
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 669
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
east-west direction, but do have a meridional component as well [Evan et al., 2012]. It is possible that the
tropical free troposphere contains a considerable amount of equatorially trapped inertia-gravity wave activi-
ty associated with the ITCZ and Hadley circulation system, but their contribution to the large-scale flow may
be difficult to discern in observations and reanalyses.
It is also difficult to correctly initialize inertia-gravity waves because of their small temporal and spatial scales
[Charney, 1955; Daley, 1981]. Therefore, idealized models often filter out inertia-gravity waves [Gill, 1980; Chao,
1987; Schubert et al., 2009]. Our understanding of transient inertia-gravity waves in the tropics has been lim-
ited by a lack of observations, especially over the oceans where the Hadley circulation is strongest. Reanalyses
can be used to help our understanding, although they can have problems resolving small-scale inertia-gravity
waves sufficiently as they are constrained by their coarse horizontal and vertical resolution. Therefore, ideal-
ized modeling of inertia-gravity waves can be useful for improving our understanding of inertia-gravity waves.
In this study, we consider only the flow in the inviscid interior (i.e., above the 900 hPa isobaric surface) fol-
lowing the methods of Gonzalez and Mora Rojas [2014]. Our analysis involves solving a partial differential
equation in (y, z, t), with appropriate boundary and initial conditions. As described in section 3, the first step
to solving this system involves the application of a vertical transform that converts the original partial differ-
ential equation in (y, z, t) into a system of partial differential equations in (y, t). Gonzalez and Mora Rojas
[2014] proceed by using evanescent basis functions, or Green’s functions, to solve the slowly forced version
of this equation. The Green’s function approach yields the most physical insight into the quasi-balanced
meridional flow and the fundamental asymmetry between the northern hemisphere and southern hemi-
sphere Hadley cells for an ITCZ centered off of the equator, which is what is typically observed in nature.
We use oscillatory basis functions, or the Hermite transform approach, to solve this equation for transient
forcings. This approach yields the most physical insight into the transient aspects of the flow and, in particu-
lar, how zonally symmetric inertia-gravity waves can be emitted due to transient convection in the ITCZ.
The paper is organized in the following way. In section 2, the primitive equation model is presented and the
associated time-dependent meridional circulation equation is derived. A vertical transform is performed in sec-
tion 3, and section 4 introduces a Hermite transform in y that converts the set of equations in (y, t) into a set of
ordinary differential equations in t. In sections 5 and 6, we discuss the free tropospheric response to transient dia-
batic forcings in the ITCZ with vertical structures comprised solely of one vertical mode. An analysis of inertia-
gravity wave packet properties is given in section 7. Some concluding remarks are presented in section 8.
2. Model Equations
In order to gain insight into the transient aspects of the Hadley circulation, we consider zonally symmetric
motions in a stratified, compressible atmosphere on the equatorial b-plane. We use z5H ln ðp0=pÞ as the
vertical coordinate, where p05900 hPa, T05293 K, R5 287 J K
21 kg21, and g5 9.81 m s22 so that H5RT0=g
 8572 m. We consider the case of weak horizontal flow and weak baroclinicity, so that the vð@u=@yÞ and
wð@u=@zÞ terms in the zonal momentum equation, the vð@v=@yÞ and wð@v=@zÞ terms in the meridional
momentum equation, and the vð@T=@yÞ term in the thermodynamic equation can be neglected. With these
simplifications, the governing equations are
@u
2byv50; (1)
@t
@v @U
1byu1 50; (2)
@t @y
@U g
5 T ; (3)
@z T0
@v @w w
1 2 50; (4)
@y @z H
@T T0 Q
1 N2w5 ; (5)
@t g cp
where u is the zonal velocity, v the meridional velocity, w the log-pressure vertical velocity, U the perturba-
tion geopotential, T the perturbation temperature, N the constant buoyancy frequency, Q the diabatic heat-
ing, and by the Coriolis parameter.
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 670
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
Our goal is to derive a time-dependent meridional circulation equation. This is a partial differential equation
in (y, z, t) that can be solved analytically for a variety of forcings. We begin the derivation by eliminating u
between (1) and (2), yielding   
@ @U @2

1 1b2y22 v50: (6)@y @t @t
Similarly, from the elimination of T between (3) and (5) we obtain
@ @U gQ
1N2w5 : (7)
@z @t cpT0
The elimination of ð@U=@tÞ between (6) and(7) then yields
2 @w @
2
2 2 @v g @QN 2 1b y 5 : (8)
@y @t2 @z cpT0 @y
Equations (4) and (8) can be regarded as a closed system in v and w. One way of proceeding from this sys-
tem is to make use of (4) to express the meridional circulation components vðy; z; tÞ and wðy; z; tÞ in terms
of the stream function wðy; z; tÞ by
2z=H @w @we v52 and e2z=Hw5 ; (9)
@z @y
and then to use (9) in (8) to obtain a single equation in w. This procedure yields the time-dependent meri-
dional circulation equation that is given below in (10). Assuming that v ! 0 as y ! 61 and that w vanishes
at the top boundary (z5zT ), we obtain the boundary conditions given below in (12) and (13). Following
Gonzalez and Mora Rojas [2014], we assume that the actual vertical velocity (i.e., the physical height vertical
velocity) is specified at the lower isobaric surface z5 0 (i.e., the top of the boundary layer) for the lower
boundary condition. Concerning the initial conditions, we assume that the meridional circulation and its
tendency both vanish at t5 0.
In summary, the time-dependent meridiona l circulation  !
equation is
@2 2 2 @
2ŵ ŵ 2 @
2ŵ g @Q̂
1b y 2 1N 5 ; (10)
@t2 @z2 4H2 @y2 cpT0 @y
where
ŵðy; z; tÞ 5wðy; z; tÞ e z=2H;
(11)
Q̂ðy; z; tÞ 5Qðy; z; tÞ e2z=2H;
with boundary conditions
ŵ ! 0 as y ! 61; (12)
 ŵ50 !at z5zT ; (13)
@2
 2
1b2y2
@ŵ ŵ @ ŵ @W
2 2 1g 2 5g at z50; (14)@t @z 2H @y @y
where Wðy; tÞ is the physical height vertical velocity at z5 0, and with initial conditions
@ŵ
ŵ50 and 50 at t50: (15)
@t
The use of (11) simplifies the time-dependent meridional circulation equation (10) by eliminating the ez=H
factors. Note that the diabatic forcing appears through the right-hand side of the interior equation (10)
while the frictional forcing appears through the right-hand side of the lower boundary condition (14).
The first step in solving the time-dependent meridional circulation problem (10)–(15) involves a vertical
transform of (10), which is presented in section 3. This vertical transform converts (10) from a partial differ-
ential equation in (y, z, t) to a system of partial differential equations in (y, t), as shown in (28). There are two
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 671
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
methods that can be used to proceed from (28). The first method involves using a Green’s function
approach that has evanescent basis functions; this was explored in Gonzalez and Mora Rojas [2014]. The
second method involves a Hermite transform that contains oscillatory basis functions as is presented in
section 4.
3. Vertical Transform of the Meridional Circulation Equation
We seek solutions of (10)–(15) via the vertical transform pair [Fulton and Schubert, 1985; Gonzalez and
Mora Rojas, 2014]
X1
ŵðy; z; tÞ5 ŵmðy; tÞZmðzÞ; (16)
ð m50
N2 zT
ŵmðy; tÞ5 ŵðy; z; tÞZmðzÞ dz1ŵðy; 0; tÞZmð0Þ: (17)g 0
In other words, the stream function ŵðy; z; tÞ is represented in terms of a series of vertical structure func-
tions ZmðzÞ, with the coefficients ŵmðy; tÞ given by equation (17). The last term in (17) may seem unfamiliar;
it arises from the lower boundary condition (14). The vertical structure functions ZmðzÞ are solutions of the
Sturm-Liouville eigenvalue problem
d2Zm Zm N2Zm
dz2
2 52 ; (18)
4H2 ghm
Zm50 at z5zT ; (19)
dZm Zm Zm
2 52 at z50; (20)
dz 2H hm
with eigenvalues (or equivalent depths) denoted by hm. Since the eigenvalue hm appears in both the differ-
ential equation (18) and the lower boundary condition (20), this Sturm-Liouville problem is slightly more
general than those usually treated in standard texts. A discussion of the transform pair (16) and (17) is given
in Gonzalez and Mora Rojas [2014, Appendices], along with a proof that hm > 0, a derivation of the solutions
ZmðzÞ, as well as an analysis of the completeness of the set ZmðzÞ for m50; 1; 2;   , and a discussion of the
associated orthonormality relation.
Defining ĥ5ð2NHÞ2=g  4314 m, corresponding to our choice N51:231022 s21, we can organize the solu-
tions of equations (18–20) into two cases: hm > ĥ (Case 1), which results in evanescent behavior of the
eigenfunction; and 0 < hm < ĥ (Case 2), which results in oscillatory behavior of the eigenfunction. For Case
1 there is only one eigenvalue, denoted by the external mode h0. For Case 2 there is an infinite set of eigen-
values, denoted by the internal modes h(1; h2;   . In summary, the corresponding eigenfunctions are
A0sinh ½j0ð12z=zT Þ if m50;
ZmðzÞ5 (21)
Amsin ½jmð12z=zT Þ if m  1;
where j0 and jm (m  1) are related to the eigenvalues hm through the formula given below as equation
(26). The normalization factor for the external mode m5 0 is
 2    1N zT sinh ðj0Þcosh ðj0Þ 22A05 21 1sinh 2ðj0Þ ; (22)2g j0
and the normalization factors for the internal modes m  1 are
 2    1N zT sin ðjmÞcos ðj 22A 1 mÞm5 2 1sin 2ðjmÞ : (23)2g jm
As can be shown by substituting z5zT into (21), the eigenfunctions satisfy the upper boundary condition
(19). Through application of the lower boundary condition (20), it can be shown that j0 is the solution of
the transcendental equation
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 672
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
j0
Table 1. Information About the First Five Vertical Modesa tanh ðj0Þ5 ; (24)ðzT=ĥÞ½12ĥ=ð2HÞ2ð2Hj =z Þ20 T 
m h (m) c (m s21) bm m m (km)
0 7095 263.8 3394 while the jm are the solutions of the transcendental
1 229.5 47.45 1440 equation
2 61.36 24.53 1035
3 27.63 16.46 848.1
4 15.61 12.38 735.2 tan ð jj mmÞ5 2 : (25)
aThe spectra of equivalent depths hm, gravity wave speeds ðzT=ĥÞ½12ĥ=ð2HÞ1ð2Hjm=zT Þ 
c 5ðgh Þ1=2m m , and equatorial Rossby lengths bm5ðc 1=2m=bÞ for
the five values of m listed in the left column, where the values After the transcendental equations (24) and (25) are
have been computed from (26) using zT513 km, N51:231022
s21, and H5 8572 m. solved, the equivalent depths hm can be obtained
from
><8 h i21ĥ 12ð2Hj =z Þ2 if m50;
hm5>: h
0 T i (26)
2 21ĥ 11ð2Hjm=zT Þ if m  1:
The first five eigenvalues hm (m50; 1; 2; 3; 4) are listed in the second column of Table 1, while the
corresponding eigenfunctions ZmðzÞ are shown in Gonzalez and Mora Rojas [2014, Figure 2]. Note that
the dependence of the normalization factors Am on m is weak because jm  mp, making the sin ðjmÞ terms
in equation (23) negligible, leading to Am  ½2g=ðN2zT Þ1=2  3:2. The solution of equation (24) is
j0  0:4747.
To take the vertical transform of equation (10), we first multiply it by ZmðzÞ and integrate over z from 0 to zT .
The integral involving the second-order vertic"al de rivative te!rm in(10) is then inte#grated by parts twice, yieldingz2 T@ 2 2 @ŵ ŵ dZm Z1b y mZ 2 2ŵ 2
@t2 m ð@z 2H dz 2H 0
@2 z
 
T 2
2 2 d Zm Zb y ŵ m1 1 2 dz (27)
@t2 ð dz202 z ð4H
2
@ T g @ zT
1N2 ŵZ
y2 m
dz5 Q̂Z dz:
@ 0 cpT
m
0 @y 0
To simplify (27), first use (18) in the second line and then use (13) and (19) to show that the upper boundary
term in the first line vanishes. To evaluate the lower boundary term in the first line, we use (14) to eliminate
½ð@ŵ=@zÞ2ðŵ=2HÞ and then group the resulting terms with the third line of (27). Similarly, we use (20) to
eliminate ½ðdZm=dzÞ2ðZm=2HÞ and then group the resulting Zm=hm term with the second line of (27).
Making use of (17), this procedure then simpl ifies (27) to!
@2ŵ @2m y
2 @F
2gh mm 2 4 ŵm52ghm ; (28)@t2 @y2 b @ym
with boundary conditions
ŵm ! 0 as y ! 61; (29)
and with initial conditions
@ŵ
ŵm50 and
m 50 at t50; (30)
@t
where the equatorial Rossby length is defined by b 5ðc =bÞ1=2m m . The forcing term Fmðy; tÞ on the right-
hand side of (28) is given by  !
ð Þ gQ̂mðy; tÞ ð gQ̂ðy; 0; tÞFm y; t 5 1 W y; tÞ2 Zmð0Þ; (31)cpT0N2 cpT0N2
where
2 ðN zT
Q̂mðy; tÞ5 Q̂ðy; z; tÞZmðzÞ dz1Q̂ðy; 0; tÞZmð0Þ: (32)g 0
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 673
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
Note that the bm definition of Rossby length is conpvffieffiffinient when working with Hermite functions Hmn ðyÞ
that we use in the next section, while the bm5bm= 2 definition of equatorial Rossby length is convenient
when working with parabolic cylinder functions DnðyÞ [Gonzalez and Mora Rojas, 2014].
4. Solution via Hermite Transforms
The solution of (28)–(32) is now constructed by using Hermite transform methods. The Hermite transform
pair for the stream function is X1
ŵmðy; tÞ5 ŵ mmnðtÞHn ðyÞ; (33)
ð n50
1 1
ŵmnðtÞ5  ŵmðy; tÞH
m
n ðyÞ dy; (34)bm 21
where the meridional structure functionsHm	n ðyÞ ar
e related to the Hermite polynomials H ðy=b

n mÞ by
21 2
Hmn ð
1 2 1 
yÞ5 p22nn! H ðy=b Þ e22ðy=bmÞn m : (35)
Since the Hermite polynomials satisfy the recurrence relation Hn11ðxÞ5 2xHnðxÞ22nHn21ðxÞ, it can be
shown that the meridional structure functionsHmn ðyÞ satisfy the recurrence relation 1 
2 2 	 
1
Hmn11ð Þ
y
y 5 m Hn ð Þ
n 2
y 2 Hm
n11 b n11 n21
ðyÞ: (36)
m
1 1 2
We first compute Hm0 ðyÞ from H0ðxÞ51, obtaining Hm0 ðyÞ5p24e22ðy=b

mÞ . All succeeding Hermite functions
can be computed using the recurrence relation (36), with the understanding that the last term in (36) van-
ishes when n5 0. Computing Hmn via its recurrence relation is much preferable to computing Hn via its
recurrence relation and then computing Hmn by evaluation of the right-hand side of (35), because the for-
mer method avoids explicit calculation of the factor 2nn! for large n. Plots of Hmn ðyÞ for m5 0, 1, 2 and n50;
1; 2; 3; 4 are shown in the three panels of Figure 2.
The meridional structure functions satisfy the!second-orde r equat!ion
d2 y2
2 Hm4 n ðyÞ
2n11
52 m
dy2  2
Hn ðyÞ; (37)
bm bm
so that Hmn ðyÞ is an eigenfunction of the operator that appears in parentheses on the left-hand side of (28).
This eigenfunction property makes the transform pair (33) and (34) convenient for the solution of (28). Note
that solutions of (37) transition from oscillatory to evanescent when ~y 56bmn m ð2n11Þ
1=2, which we define
as the turning (or critical) latitudes [Wunsch and Gill, 1976]. In Table 2, we display the turning latitudes for ver-
tical modesm50; 1; 2; 3; 4 and meridional modes n50; 1; 2; 3; 4.
Another convenient property of thðe meridional structure f(unctions H
m
n ðyÞ is that they satisfy the orthonor-
mality relation 1 b n05n;
Hmn ðyÞHm
m
n0 ðyÞ dy5 (38)
21 0 n0 ¼6 n:
Note that (34) can be obtained through multiplication of (33) by Hmn0 ðyÞ, followed by integration over y and
use of (38).
To take the meridional transform of (28), first multiply it by Hmn ðyÞ and integrate over y. The integral involv-
ing the second-order y-derivative term in (28) is then integrated by parts twice, making use of the boundary
conditions (30), to yield ð ð  !
@2 1 1 2 2
ŵmð Þ mð
d y
y; t Hn yÞ dy2gh ŵ ðy; tÞ 2 HmðyÞ dy@t2 m m 2 4 n21 2ð1 dy bm (39)1 @Fmðy; tÞ
52ghm H
m
n ðyÞ dy:
21 @y
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 674
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
To simplify (39) we first use (37) in the
integrand of the second integral. We
then make use of (34) to simplify (39) to
the second-order ordinary differential
equation
d2ŵmn 1m2
0
mnŵmn52ghmFmn; (40)dt2
with the initial conditions
dŵ
ŵmn50 and
mn 50 at t50; (41)
dt
where the inertia-gravity wave frequen-
cy mmn is given by
1
m 1=2mn5  ½ghmð2n11Þ ; (42)bm
and the forcinðg by
1 10 ð Þ @FF t mðy; tÞ5 mmn b
Hn ðyÞ dy:
m 21 @y
(43)
The inertia-gravity wave frequencies for
the first five vertical wave numbers
Figure 2. Plots of Hmn ðyÞ for m5 0, 1, 2 and n50; 1; 2; 3; 4. Note that, as n (m50; 1; 2; 3; 4) are shown as a function
increases (for a given m), the width of the oscillatory region of Hmn ðyÞ increases as of meridional mode n in Figure 3. Val-
n1=2, so the magnitude ofHmn ðyÞ in the oscillatory region decreases as n21=4 in ues of the c21 parameter in the switch-
order to satisfy the normalization imposed by (38).
on function TðtÞ, presented in the next
section, are plotted in the four horizon-
tal dashed lines, c2153; 6; 12; 24 h. Notice how c21524 h does not intersect with any of the m50; 1; 2; 3; 4
inertia-gravity wave frequencies. As will be seen later, when diabatic heating is switched on at this slow
rate, inertia-gravity wave activity is minimal and the transient solutions are approximately equal to the bal-
anced solutions derived in Gonzalez and Mora Rojas [2014]. In the next section, we solve (40) and (41) for a
particular forcing.
5. Transient Hadley Circulations Forced by a Switch-On of ITCZ Convection
Consider the response to a forcing for which the Ekman pumping and the diabatic heating at z5 0 are
related by
ð Þ gQ̂ðy; 0; tÞW y; t 5 ; (44)
c 2pT0N
so that (31) simplifies to
ð Þ gQ̂mðy; tÞFm y; t 5 : (45)cpT N20
This relation between the Ekman pumping and diabatic heating is assumed in order to use the vertical
structure of a single vertical mode to represent the vertical structure of the prescribed diabatic heating. For
example, the vertical structure of Z1ðzÞ is not equal to zero at z5 0 therefore there must be a small nonzero
vertical velocity at z5 0. This simplification can be avoided if the prescribed diabatic heating were equal
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zero at z5 0. However, more than one ver-
Table 2. Information About the Turning Latitudesa
tical heating mode would be required to
m n5 0 n5 1 n5 2 n5 3 n5 4
represent this behavior. For brevity, we
0 3,395 5,880 7,590 8,981 10,184 proceed using only single vertical modes.
1 1,440 2,494 3,230 3,809 4,319
2 1,035 1,793 2,315 2,739 3,106 Assume that Q̂ðy; z; tÞ vanishes everywhere
3 848.1 1,469 1,896 2,244 2,544
4 735.3 1,274 1,644 1,945 2,206 except in the latitudinal range y1 < y < y2,
where y
a 1 and y2 are constants that specify theThe turning latitudes in the units of km for m50; 1; 2; 3; 4 and n50; 1; 2;
3; 4 using the formula ~y 56b 1=2mn mð2n11Þ . Note how the turning latitudes
south and north boundaries of the ITCZ. Within
increase as the meridional mode n increases and the vertical mode m this ITCZ region, thediabatic heating is assumed
decreases. to be independent of y and to be smoothly
8X switched on to a steady state value, i.e.,>< 1 Q~mZmðzÞ if y1 < y < y2;
Q̂ðy; z; tÞ5TðtÞ:>m50 (46)
0 otherwise;
where the constants Q~m specify the projection of the vertical structure of Q̂ðy; z; tÞ onto the vertical modes,
and where the time dependence is given by
TðtÞ512ð11ctÞe2ct; (47)
with the constant c specifying the sharpness of the switch-on function TðtÞ. Figure 4 displays four TðtÞ
curves for the particular values c2153; 6; 12; 24 h.
Substituting (46) into (32), and then using the orth(onormality relation associated with ZmðzÞ, we obtain
Q~
ð Þ ð Þ m
if y1 < y < y2;
Q̂m y; t 5T t (48)
0 otherwise:
Use of (45) and (47) in (43) now yieðlds
gHm0 ðy Þ y11 @Q̂ ðy; tÞ gHmðy Þ ðy21 @Q̂
F ðtÞ n 1 m dy n 2 mðy; tÞmn 5  1  dy5TðtÞF2 y 2 y mn; (49)cpT0N bm y @ @12 cpT0N bm y22
where
gQ~m  
F mmn5  Hn ðy1Þ2H
m
c 2 n
ðy2Þ ; (50)
pT0N bm
and where we have made use of the fact that the narrow integral of ð@Q̂m=@yÞ across y1 is Q~mTðtÞ and the
narrow integral across y is 2Q~2 mTðtÞ.
The final equality in (49) can now be used in the right-hand side of (40), and the complete solution can be writ-
ten as the sum of the homogeneous solution and a particular solution. As can be checked by direct substitution
into (40), the solution satisfying t(he initial condit!ions (41) is  !
ðm2 2c2Þc2ð Þ mn ð 2c
3m
ŵmn t 5Wmn 2 cos m tÞ
mn
2 sin ðm tÞ
ð 2 2Þ mn ð 2 2Þ2 mnmmn1c  mmn1c  ) (51)
m2mn13c
2 m2 2ct
112 1ct mn
e
;
m2mn1c
2 m2mn1c
2
where Wmn52ghmF 2mn=mmn. In summary, the solution of the original meridional circulation problem is
obtained by combining equations (11), (16), andX(33X), yielding1 1
wðy; z; tÞ5e2z=2H ŵmnðtÞHmn ðyÞZmðzÞ; (52)
m50 n50
where ŵmnðtÞ is given by equation (51). Plots of the stream function can be constructed by first calculating
Fmn from (50), then calculating ŵmnðtÞ from (51), and finally calculating wðy; z; tÞ from (52).
Note that when c  mmn, the solution (51) simplifies considerably since the coefficients of the cos ðmmntÞ
and sin ðmmntÞ terms become much smaller than unity, while the second line in (51) approaches TðtÞ. Then,
the spectral space solution (51) simplifies to
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ðbÞ
ŵmnðtÞ5WmnTðtÞ (53)
so that the physical space solution
(52) becomes
X1
wðbÞðy; z; tÞ5e2z=2H
X m501 (54)ðbÞ
ŵ ðtÞHmmn n ðyÞZmðzÞ;
n50
where the superscript (b) indicates the
balanced (or filtered) solution. Since
the time dependence on the right-
hand side of (53) is TðtÞ, the wðbÞðy; z;
tÞ field develops in lock-step with the
forcing, i.e., there is no time delay
between the forcing and the
response, no matter how far one is
from the forcing. Since this represents
‘‘action at a distance,’’ it should be
regarded as a filtered approximation
Figure 3. Plots of mmn, computed from (42), for m50; 1; 2; 3; and 4 and of the actual dynamics, valid only in
n50; 1; . . . ; 15. The four horizontal dashed lines indicate the values of c corre- the case of a ‘‘slowly varying forcing.’’
sponding to the four switch-on functions TðtÞ presented in (47) and plotted in
Figure 4. To better understand how slow the
forcing needs to be, Figure 3 includes
horizontal dashed lines for the four values of c used in Figure 4. As an example, for m5 1 and c21524 h the
condition c  mmn holds for essentially all n, while for m5 1 and c2153 h the condition does not hold for
the smaller values of n. Thus, for the external mode and the first two internal modes, the c21524 h dashed
line in Figure 3 would correspond to a forcing that is probably slow enough for the filtered approximation
to be reasonably accurate, but the c2153 h dashed line would correspond to a forcing that excites a nonne-
gligible inertia-gravity wave response, especially for the higher internal modes. This hypothesis will be con-
firmed by the examples shown in the next section.
6. Examples Using Single Vertical Mode Diabatic Heating
6.1. Diabatic Heating of the m51 Mode
In this section, we present examples forced by diabatic heating in an ITCZ centered off of the equator with
a simplified single vertical mode structure. We begin with a diabatic heating of the first internal mode
m5 1, given by
>8<  ~ ð5 K d21Þ 500 kmQm > if m51;5: y22y1 (55)cp
0 if m ¼6 1;
where Q~m has been normalized in such a way that the horizontally integrated forcing ðy22y1ÞQ~m is fixed.
Figures 5 and 6 show isolines of wðy; z; tÞ and contour shading of Qðy; z; tÞe2z=H=cp at t512; 36; 60; 84 h
and t5108; 132; 156; 180 h for the switch on rates c2153 h and c21524 h, respectively. The w field is com-
puted from (52) using the parameter choices z 513 km, N51:231022 s21T , and ðy1; y2Þ5ð500; 1000Þ km.
The choice of c2153 h corresponds to localized ITCZs while c2156 h is more likely in zonally elongated
ITCZs. When c21 equals 12 or 24 h, transient activity in the Hadley circulation decreases substantially, as
demonstrated in Figure 6. This can be explained by the w solutions in (51) approaching the balanced solu-
tions in (53) as c21 increases. The spectral space solution (51) can be considered as the sum of two parts.
The first part consists of the oscillatory terms cos ðmmntÞ and sin ðmmntÞ; the second part contains the evanes-
cent term with the e2ct factor. Also, the term outside of the brackets happens to be the term in the steady
state limit, Wmn. For large times (i.e., ct  1), the second part is negligible and the oscillatory terms
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represent inertia-gravity waves that have
propagated far from any confined region
of forcing [Salby and Garcia, 1987; Garcia
et al., 1987].
The evanescent part of w changes very lit-
tle after approximately 12 h in both Fig-
ures 5 and 6 while the oscillatory part of
w continues to be active long after the
forcing has been fully switched on, as
demonstrated in Figure 5. When c2153 h,
the southern and northern Hadley cells
Figure 4. Plots of the switch-on function TðtÞ for the four choices c2153; 6; are almost symmetric in size and magni-
12; and 24 h. The ‘‘filtered solutions’’ are valid for the ‘‘slow switch-on’’ cases,
i.e., for large values of c21. tude at t5 12 h, but by t5 36 h, the
northern cell, which is typically thermally
direct, has become thermally indirect. At
t5 60 h, the northern cell is both wider
and stronger than the southern cell, and the opposite occurs at t5 84 h. In addition to this transient activity
there is an inherent asymmetry between the Hadley cells in a time averaged sense due to the anisotropy of
the inertial stability [Hack et al., 1989; Gonzalez and Mora Rojas, 2014]. This is even more clear in Figure 6,
where the transient activity is not excited as effectively, implying that the zonal winds in the Hadley cells
are essentially in geostrophic balance. Thus, we can refer to the Hadley circulation as a balanced Hadley cir-
culation when the ITCZ diabatic heating is switched on slowly [Gonzalez and Mora Rojas, 2014]. When the
diabatic heating in the ITCZ is switched on rapidly, there are both balanced and transient aspects to the
Hadley circulation, where the transient activity is in the form of equatorially trapped inertia-gravity waves.
Since we have assumed a vertical structure composed of the single m5 1 internal mode, the vertical struc-
ture of wðy; z; tÞ is simply e2z=2HZ1ðzÞ. Therefore, we analyze w where it maximizes in the vertical, at
z5 5.7 km. Figure 7 illustrates w as a function of time at z5 5.7 km and at the southern edge of the ITCZ
(y5 y1, southern Hadley cell) in the blue curves, northern edge of the ITCZ (y5 y2, northern Hadley cell) in
the red curves, and the black curves represent the total mass flux, wðy2Þ2wðy1Þ. All curves are computed
using the same parameter values as in Figure 5 and c2153; 6; 12; 24 h in the four curves for each color.
Note that the total mass flux, given by wðy2Þ2wðy1Þ, is strictly positive because wðy1Þ < 0 and wðy2Þ is either
positive or near zero. The blue and red curves in Figure 7 show that both the southern and northern Hadley
cells have similar oscillatory behavior but are out of phase, with the southern cell peaking in intensity about
20–30 h after the northern cell peaks in intensity. This lag occurs because the inertia-gravity waves that
travel northward from the ITCZ reach their turning latitudes before the inertia-gravity waves traveling
southward from the ITCZ. This behavior causes the northern cell strength to fluctuate significantly with
time; it can be as strong as the southern cell and it can also disappear, as seen in Figures 5 and 7. The total
mass flux in the ITCZ, shown in the black curves, also contains high frequency behavior due to the time lag
between the two cells. Note that the time average asymmetry between the southern and northern cells is
approximately 2 to 1 for all ITCZ switch on rates, similar to the balanced solutions shown in Gonzalez and
Mora Rojas [2014].
Figure 8 illustrates the normalized power of w at the southern and northern edge of the ITCZ represented
by the color shading and black line contours, respectively, as a function of frequency in h21 and central
ITCZ location ðy11y2Þ=2 for a 500 km wide ITCZ diabatic heating and for c2156 h. The power spectrum is
computed by first removing the time mean of w, then performing a Hann window, and computing a dis-
crete fast Fourier transform on the resulting time series. Note that the power in the shading and the black
contours is normalized by the maximum power in the southern Hadley cell. The amplitude of the inertia-
gravity waves in both Hadley cells maximize at an approximate timescale of 50 h and have their largest ampli-
tude when the ITCZ is centered off of the equator. More specifically, the inertia-gravity wave amplitude peaks
when ðy11y2Þ=2  1150 km and 950 km in the southern and northern cell, respectively. These ITCZ locations
nearly coincide with the ITCZ location where there is the maximum asymmetry between the southern and
northern Hadley cells [Hack et al., 1989; Gonzalez and Mora Rojas, 2014]. Also, the inertia-gravity waves in the
southern cell are about twice as large in amplitude as those in the northern cell, similar to the time average
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asymmetry between the two cells. Note
that the northern cell also contains
high frequency variability when c2153
h (secondary peak near the 30 h time-
scale, not shown) because it is located
farther away from the equator than the
southern cell where the low n inertia-
gravity waves have smaller amplitudes,
as implied from the Hermite functions
Hmn ðyÞ in Figure 2. When the ITCZ dia-
batic heating is switched on at the rate
c21512 h (not shown), the inertia-
gravity wave activity decreases signifi-
cantly, but variability is still largest near
the 50 h timescale. It is not until c215
24 h that we see the 50 h variability
associated with inertia-gravity waves
becomes negligible.
Although we have focused on solu-
tions for w thus far, we emphasize
that the horizontal structure of v is the
same as w. The ð@u=@tÞ field is also
very similar to v, with an additional by
factor, as seen from (1). The w,
ð@T=@tÞ, and ð@U=@tÞ fields have
quite different horizontal structures
than the w and v fields due to the
meridional derivative of w in (9).
Figure 9 shows the ‘‘full solution’’ of w,
the ‘‘evanescent part’’ of w, and the
‘‘oscillatory part’’ of w at z5 7.6 km
(where it maximizes in the vertical)
using the same parameters as Figure 8.
Figure 5. Contoured wðy; z; tÞ and shaded Qðy; z; tÞe2z=H=cp fields for ðy1; y2Þ5 Recall that the spectral space solution
ð500; 1000Þ km, m5 1, and c2153 h at t512; 36; 60; 84 h. The contour interval
for wðy; z; tÞ is 400 m2 s21, the maximum (magnitude) of wðy; z; tÞ is 4156 m2 s21, (51) can be considered as the sum of
and the zero line is omitted. The Qðy; z; tÞe2z=H=cp shade interval is 0.5 K d21, and two parts, with the first part containing
the maximum (magnitude) of the diabatic heating is 3.496 K d21. the oscillatory terms cos ðmmntÞ and sin
ðmmntÞ and the second part containing
the evanescent term with the e2ct factor. Similar to the w field, the evanescent w solutions in the middle
panel change very little after approximately 12 h while the inertia-gravity wave w solutions in the bottom
panel continue to be active long after the forcing has been fully switched on. The oscillatory terms illustrate
the equatorially trapped inertia-gravity wave activity in both Hadley cells, however, both the oscillatory and
evanescent solutions are larger in the southern cell. Also, the irregular pulsation of the southern and north-
ern cells occurs because the waves that emanate from the north edge of the ITCZ bounce off of their turn-
ing latitudes before those traveling from the south edge of the ITCZ bounce off of their turning latitudes. It
is easier to trace individual paths, or rays, of the inertia-gravity wave packets in the w field than the w field,
especially in the subsidence regions, as we discuss in more depth in section 7.
The ‘‘oscillatory’’ and ‘‘evanescent’’ parts of w are nonzero at t5 0, but their sum is zero, as seen in the ‘‘full
solution’’ of w. This is similar to the classic f-plane geostrophic adjustment problem where the oscillatory
and evanescent parts of the height field are initially nonzero, but their sum equals zero. In the classic geo-
strophic adjustment problem, the inertia-gravity waves satisfy f-plane dynamics rather than equatorial b-
plane dynamics, so that the waves do not bounce back toward their source [Gill, 1982, section 7.3]. Another
interesting feature of Figure 9 is that the top two panels reveal essentially time-independent spatial
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oscillations in the evanescent part and
in the full solution for w, especially in
the subsiding regions (blue shading)
on either side of the ITCZ. We do not
regard these as physically significant,
but rather as Gibbs phenomenon oscil-
lations associated with the use of the
Hermite representation in y, combined
with the assumption (46) that the dia-
batic heating is discontinuous at the
edges of the ITCZ. Interestingly, while
the assumption of discontinuous dia-
batic heating leads to a very compact
representation of the solution in terms
of Green’s functions [Gonzalez and
Mora Rojas, 2014], it leads to this Gibbs
phenomenon when the solution repre-
sentation is in terms of the Hermite
series in y.
Figure 10 shows the time evolution of
the vertical log-pressure velocity w as a
function of time at z5 7.6 km and at
y5 0 km and y5 1500 km in the
southern and northern Hadley cells in
the blue and red curves, respectively,
using the same parameter values as in
Figure 7. These meridional locations
can be thought of as representative of
the subsidence regions in the southern
and northern Hadley cells. The tempo-
ral evolution of the w field in the center
of the ITCZ is not shown but essentially
has the same behavior as the total
Figure 6. The same as Figure 5, but using c21524 h and t5108; 132; 156; 180 h. mass flux in the ITCZ shown in Figure
The maximum (magnitude) of wðy; z; tÞ is 2818 m2 s21. 7. There is a significant amount of tran-
sient activity in both subsidence
regions in Figure 10. Subsidence in the southern cell is typically larger than the subsidence in the northern
cell, as expected, but their vertical motion ratio is much smaller than their mass flux ratio. Both Hadley cells
experience brief periods when their vertical motion transitions from subsidence to weak ascent, especially
in the northern cell. Inertia-gravity wave activity is concentrated at a higher frequency than the 50 h time-
scale. Also, there seems to be significant transient activity at a secondary timescale.
Figure 11 illustrates the normalized power associated with w at z5 7.6 km and at y5 0 km (southern cell,
colored shading) and y5 1500 km (northern cell, black contours) as a function of frequency in h21 and cen-
tral ITCZ location ðy11y2Þ=2 for a 500 km wide ITCZ using the same parameters as Figure 8. Note that the
power in the colored shading and the black contours is normalized by the maximum power in the southern
Hadley cell. Transient convection in the ITCZ excites a range of frequencies in the w field subsidence
regions, with the largest amplitude inertia-gravity waves peaking near 30 and 50 h periods. The 30 h time-
scale is most dominant when the ITCZ is close to the equator while the 50 h timescale is preferred when
the ITCZ is far from the equator. More specifically, the southern Hadley cell is dominated by 30 h oscillations
when the ITCZ is centered 0–7.58 off of the equator and 50 h oscillations when the ITCZ is centered 12.5–
208 off of the equator. On the other hand, the northern Hadley cell is dominated by 30 h oscillations when
the ITCZ is 0–2.58 off of the equator and 50 h oscillations when the ITCZ is 2.5–208 off of the equator. When
the ITCZ is centered 7.5–12.58 off of the equator, the southern cell experiences oscillations at both 30 and
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Figure 7. The time evolution of w as a function of time at z5 5.7 km and at the southern edge of the ITCZ (y5 y1, southern Hadley cell) in
the blue curves, the northern edge of the ITCZ (y5 y2, northern Hadley cell) in the red curves, and the black curves represent the total
mass flux (wðy2Þ2wðy1Þ) for the four ITCZ diabatic heating switch on rates c2153; 6; 12; 24 h. All other parameters are the same as those
in Figure 5.
50 h timescales. The vertical motion in the ITCZ, although not shown explicitly, is mainly dominated by 50 h
oscillations when c2156 h and by 30 h oscillations when c2153 h.
6.2. Diabatic Heating of Other Modes
In this section, we examine the solutions for experiments where the vertical structure of the diabatic heat-
ing is composed of individual modes other than the m5 1 internal mode. It has been shown that diabatic
heating in the ITCZ tends to contain contributions mainly from the m5 0 external mode and the m5 1 and
2 internal modes [Fulton and Schubert, 1985]. In particular, regions such as the Atlantic and eastern Pacific
tend to have strong bimodal (m5 1, 2) variability in the vertical structure of diabatic heating [Zhang and
Hagos, 2009].
In Figure 12 we display w as a function of
time at z5 5.7 km and at the southern
and northern edge of the ITCZ in the blue
and red curve, respectively, and the total
mass flux, wðy2Þ2wðy1Þ, in the black curve
using the same parameter values as in
Figure 8, except for an ITCZ diabatic heat-
ing of the m5 2 internal mode. The ideas
we have postulated thus far for the m5 1
diabatic forcing apply to the m5 0 (not
shown) and m5 2 cases in that there is
significant pulsating of the southern and
northern Hadley cells due to equatorially
trapped inertia-gravity waves. The pulsat-
ing for the m5 2 diabatic heating leads
to slower-moving inertia-gravity wave
Figure 8. The normalized power associated with w at the southern and north-
ern edge of the ITCZ represented by the color shading and black line con- packets than the m5 1 case, with the
tours, respectively, as a function of frequency in h21 and central ITCZ location amplitude of inertia-gravity waves in w
ðy11y2Þ=2 in km for the ITCZ diabatic heating switch on rate c2156 h. All
other parameters are the same as those in Figure 5. The shading and isoline peaking at a period of approximately
contour interval is 0.1. 70 h, while the pulsating for the m5 0
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Figure 9. Solutions of w(y, t) at z5 7.6 km: the ‘‘full solution’’ of w (top), the ‘‘evanescent part’’ of w (middle), and the ‘‘oscillatory part’’ of
w (bottom) using the same parameters used in Figure 8.
diabatic heating leads to faster propagating inertia-gravity wave packets than the m5 1 case, correspond-
ing to a period of approximately 20 h. These results can be explained by the internal gravity wave speed
decreasing as a function of vertical wave number m even though the turning latitude decreases as m
increases. This implies that the wave packets take longer to reach their critical latitudes as the vertical struc-
ture of diabatic heating becomes more complex (i.e., involving higher internal modes), leading to an excita-
tion of lower frequencies of the entire tropical belt. In addition to the w field, these ideas apply to the
transient behavior in the w field, e.g., the southern Hadley cell is dominated by 40 h oscillations when the
ITCZ is centered 0–7.58 off of the equator and 70 h oscillations when the ITCZ is centered 12.5–208 off of the
equator. On the other hand, the northern Hadley cell is dominated by 40 h oscillations when the ITCZ is 0–
2.58 off of the equator and 70 h oscillations when the ITCZ is 2.5–208 off of the equator. A possible explana-
tion for this behavior is that as the forcing involves higher vertical wave numbers, the asymmetry between
the southern and northern cells increases, as discussed in Gonzalez and Mora Rojas [2014].
When diabatic heating in the ITCZ is convectively coupled to equatorial waves, the static stability is effec-
tively smaller [Wheeler and Kiladis, 1999]. We can simulate this in our idealized model by decreasing the
value of N, in which case the equatorially trapped wave packets travel slower. For an ITCZ diabatic heating
of the m5 0, 1, 2 vertical modes in the region ðy1; y2Þ5 ð500; 1000Þ km with the new value of N5 0.06 s21,
the peak transient activity in w occurs at approximately 70, 90, and 115 h timescales, respectively. Therefore,
it is possible that inertia-gravity wave activity associated with the Hadley cells may occur at slightly different
timescales in observations. We performed a preliminary spectral analysis of the meridional wind field during
the months of July and August using the YOTC reanalysis at various locations in the eastern Pacific ITCZ and
found there to be prominent peaks at 3–5 day timescales (not shown). This result is promising; however, we
leave a more in-depth analysis to future work.
In concluding this section it is interesting to note that, as t becomes large, TðtÞ ! 1 and the evanescent
part of the forced divergent circulation (v, w) comes into steady state. However, as can be seen from (1) and
(5), the zonal flow and the temperature continue to evolve. In fact, as discussed by Gonzalez and Mora Rojas
[2014], these fields evolve in such a way that the associated potential vorticity field develops local extrema
in the ITCZ, leading to a zonal flow that satisfies the Charney-Stern necessary condition for combined
barotropic-baroclinic instability [Charney and Stern, 1962]. Thus, one should not expect the evolving zonal
flow to remain zonally symmetric for more than approximately a week [Nieto Ferreira and Schubert, 1997;
Wang and Magnusdottir, 2005; Magnusdottir and Wang, 2008].
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Figure 10. The time evolution of w at z5 7.6 km and in the two subsidence regions: the southern cell (at y5 0 km) in the blue curves and
northern cell (at y5 1500 km) in the red curves. The four lines for each subsidence region represent the ITCZ diabatic heating switch on
rates c2153; 6; 12; 24 h. All other parameters are the same as those in Figure 7.
7. Analysis of Inertia-Gravity Wave Packets
When the intensity of ITCZ convection fluctuates, inertia-gravity wave packets are emitted toward the north
and south, as we have discussed in the previous section. The movement of these wave packets depends
critically on the waveguide effect, i.e., the effect by which the variable Coriolis parameter traps the inertia-
gravity wave energy in the equatorial region. We can understand this process via a variety of approaches,
including the following three: (i) asymptotic results obtained from the exact solution (52); (ii) average con-
servation law approach [Whitham, 1965a]; (iii) variational approach [Whitham, 1965b, 1974, section 11.7].
Approaches (ii) and (iii) have the advantage that the exact solution of the problem is not required, i.e., the
mathematical apparatus of Hermite transforms can be bypassed. Although the variational approach is the
most general, approach (ii) is perhaps
more easily understood and is presented
in this section. For readers familiar with
variational methods, approach (iii) is dis-
cussed in Appendix A.
The discussion of the average conserva-
tion law approach begins with the
unforced version of the horizontal struc-
ture equation (28), written in the form
wtt2c
2wyy1b
2y2w50; (56)
where, for simplicity, we have dropped
the subscript m and the hat on w. The
homogeneous equation (56) essentially
describes the propagation of zonally sym-
Figure 11. The normalized power associated with w in the southern and metric inertia-gravity waves in the equa-
northern cell represented by the color shading and black line contours, torial waveguide after they have been
respectively, as a function of frequency in h21 and central ITCZ location ðy11 excited in the ITCZ, although this should
y2Þ=2 in km for the ITCZ diabatic heating switch on rate c2156 h. All other
parameters are the same as those in Figure 8. The shading and isoline contour be regarded as an approximation since a
interval is 0.1. wave packet that is turned back through
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the ITCZ could experience an addition-
al forcing not accounted for in (56). If
the factor b2y2 in (56) were replaced
by a constant f 2, the resulting equa-
tion would be the classic f-plane geo-
strophic adjustment equation (i.e., the
Klein-Gordon equation) studied by
Cahn [1945]. Thus, the transient ITCZ
adjustment problem is essentially a
classic geostrophic adjustment prob-
lem, but with the interesting added
effects produced by a variable Coriolis
parameter.
Multiplying (56) by wt, we obtain the
energyconservation relation
@ 1	 

w2t1c
2w21b2y2w2
@t 2 y
@ 	 
 (57)
1 2c2wywt 50:@y
Similarly, multiplying (56) by 2wy , we
obtain the pseudomomentum conser-
vation relation
@ 	 

 2w w	 @t y t@ 1 
2 2 2 2 2 2 2 2
Figure 12. The top figure shows the time evolution of the stream function w at 1 wt1c wy2b y w 52b yw :
z5 5.7 km and at the southern and northern edge of the ITCZ in the blue and red @y 2
curve, respectively, and the black curve represents the total mass flux, (58)
wðy2Þ2wðy1Þ, for an m5 2 ITCZ diabatic heating with the switch on rate c2156 h.
The bottom figure illustrates the normalized power associated with w at The term on the right-hand side of (58)
z5 5.7 km and at the southern and northern edge of the ITCZ in the color shad-
1 arises because the inertia-gravitying and black line contours, respectively, as a function of frequency in h2 and
ITCZ central location ðy11y2Þ=2 in km. All other parameters are the same as in waves propagate through a spatially
Figure 8. nonuniform medium, with the nonuni-
formity due to the y-variation of the
Coriolis parameter. An analogous term does not appear on the right-hand side of (57) because, although
the medium is spatially nonuniform, it is not time-dependent.
We now search for solutions that have t	he form of a
slowly varying wave train, i.e.,
wðy; tÞ5 Real Aðy; tÞeihðy;tÞ 5aðy; tÞcos ðhðy; tÞ1gðy; tÞÞ; (59)
where hðy; tÞ is the phase, aðy; tÞ5jAðy; tÞj is the amplitude, and gðy; tÞ5arg Aðy; tÞ is the phase shift.
Differentiation of (59) yields
wt52ðht1gtÞa sin ðh1gÞ1at cos ðh1gÞ;
(60)
wy52ðhy1gyÞa sin ðh1gÞ1ay cos ðh1gÞ:
Now assume that a(y, t) and gðy; tÞ are slowly varying so that the terms involving at; gt; ay; gy can be
neglected. Application of this approximation to (60) yields
wt5xasin ðh1gÞ;
(61)
wy52‘asin ðh1gÞ;
where
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 684
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
ð Þ @hðy; tÞ ð Þ @hðy; tÞ‘ y; t 5 and x y; t 52 (62)
@y @t
are the local meridional wave number and the local frequency, respectively. When these approximations
are used in (56), we obtain the inertia-gravity wave dispersion relation
x25c2‘21b2y2: (63)
In contrast to (42), equation (63) is a local dispersion relation relating the local frequency xðy; tÞ, the local
wave number ‘ðy; tÞ, and the latitude y. For our linear problem, the amplitude a(y, t) does not appear in the
local dispersion relation.
While the analytical solution (52) gives the complete structure of the w field, the present ‘‘slowly varying
wave train analysis’’ gives a more macroscopic view of the w field. In this spirit, it is natural to apply a spatial
running mean operator ð Þ to the conservation equations (57) and (58). The spatial interval of this running
mean operator is small compared to the scale of variation of ‘ðy; tÞ; xðy; tÞ, and a(y, t), but it includes one
or more cycles of the phase h. Application of this averaging operator to (57) and (58) results in
@ 1	 
 @ 	 

w2t 1c
2w2y1b
2y2w2 1 2c2wywt 50; (64)@t 2
	 
 
@y 
@ @ 1	 

2w 2ywt 1 wt 1c
2w2y2b
2y2w2 52b2yw2 : (65)
@t @y 2
Our goal is to use the average conservation equations (64) and (65), together with the local dispersion rela-
tion (63), to obtain governing equations for the slowly varying fields ‘ðy; tÞ; xðy; tÞ, and a(y, t). Since the
mean values of sin 2ðh1gÞ and cos 2ðh1gÞ over one oscillation are both equal to 1/2, we obtain from (61)
the following relations for the mean values appearing in (64) and (65)
1 1
w2 2 2 2 2 2t 5 x a ; w 5 ‘ a ;2 y 2
(66)
1 1
w25 a2; 2w w 5 ‘xa2:
2 y t 2
Using (66) in (64) and (65) we obtain the average conservation equations in the forms
@    
x2a2
@
1 c2‘xa2 50; (67)
@t @y
@    
‘xa2
@
1 c2‘2a2 52b2ya2; (68)
@t @y
where we have used the dispersion relation (63) in both (67) and (68). The energy conservation equation
(67) can also be written as
@E @ðcgEÞ
1 50; (69)
@t @y
where the energy density Eðy; tÞ and the group velocity cgðy; tÞ are given by
1 c2‘
E5 x2a2 and c
2 g
5 : (70)
x
Using (69) and the dispersion relation (63), it can be shown that the conservation equation (68) leads to
@x @x
1c 50: (71)
@t g @y
According to (69), the total energy in the area between two group lines remains fixed.
Since the characteristic forms of (69) and (71) are
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 685
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
dE @cg
52 E>
dt @y =
9
> dy; on 5cdt g; (72)dx50
dt
the frequency x is invariant along a group line, while the energy density E decays or grows along a group
line due to the divergence or convergence of the group lines. Another form of (72) is
da 1 @c 9g
52 a>
dt 2 @y => dy c
2ðx22b2y2Þ1=2
; on 56 : (73)dx dt x50
dt
Note that the equations for x and y are decoupled from the equation for a.
Now we would like to plot ray trajectories for choices of x. Using the local dispersion relation (63), we can
write ðdy=dtÞ5cg in the form
 dy  c dt56 : (74)
x22b2
1=2
y2 x
Integration of (74) yields the solution
     
yð x by bctt;x; y0; cÞ5 sin sin21 0 6 : (75)b x x
For given values of x, y0, and c, (75) describes two ray trajectories, one starting at y0 and initially moving
northward, and the other also starting at y0 but initially moving southward. Plots of yðt;x; y0; cÞ are shown
in Figure 13 for c5 47.45 m s21 (the m5 1 vertical mode), y05500 km and y051000 km, and the two fre-
quencies x5ð3 hÞ21 and x5ð12 hÞ21. The blue and red curves in Figure 13 represent the ray trajectories
that initially move southward and northward, respectively. These frequencies are chosen to correspond
with the switch-on rates of c21 that we analyzed previously. Note that low frequencies, such as x5ð12 hÞ21
and x5ð24 hÞ21, do not yield ray trajectories for y0 poleward of a particular meridional location (e.g., y05
1000 km for x5ð12 hÞ21, in Figure 13). This is because these trajectories are beyond their turning latitudes.
The ray trajectories shown in Figure 13 turn back toward the equator at their turning latitudes, 64000 and
6900 km. Recall that we introduced turning latitudes in the previous section, but because we have aban-
doned the idea of meridional modes n, the turning latitudes seen in Figure 13 do not correspond to any
particular meridional mode n. The solutions presented in the last section are summed over all meridional
modes, therefore they contain information about all of the meridional modes, but with varying amplitudes
wmnðy; tÞ. When the wave packets reach their turning latitude, their local meridional wave number crosses
‘50, and the analysis breaks down [Wunsch and Gill, 1976]. Plots of y(t) and ‘ðtÞ for m5 0, 2 using the same
two meridional wave numbers ‘ as in Figure 13 are similar to the plots shown in Figure 13, with particles
traveling faster for m5 0 and slower for m5 2 along their ray trajectories due to c increasing when m5 0
and c decreasing when m5 2 (not shown), as we would expect from the results shown in section 6. Also,
the ray trajectories spread out over time, in general agreement with (72).
8. Concluding Remarks
To understand the transient dynamics of meridional overturning circulations, a zonally symmetric model on
the equatorial b-plane has been formulated and its associated meridional circulation equation has been
derived. This meridional circulation equation is a partial differential equation in (y, z, t). It contains two types
of forcing: (1) horizontal variation of the interior diabatic heating; (2) Ekman pumping at the top of the
boundary layer. Since the problem is linear, the meridional circulations attributable to these two forcing
effects can be treated separately, and then the resulting flows can simply be added together to obtain the
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 686
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
total response. In this study we focus on
the free tropospheric response to off-
equatorial transient diabatic heating in the
ITCZ.
The meridional circulation equation has
been solved analytically by first performing
a vertical transform that converts the partial
differential equation in (y, z, t) into a system
of partial differential equations in (y, t) for
the meridional structures of all the vertical
modes m. These partial differential equa-
tions have been solved via both the Green’s
function approach (evanescent basis func-
tions) in Gonzalez and Mora Rojas [2014]
and the Hermite transform approach (oscil-
latory basis functions) in this study. These
two approaches yield two different mathe-
matical representations of the same physi-
cal solution; for understanding the transient
behavior of the Hadley cells, it is advanta-
geous to solve the equations using Hermite
functions. The solutions suggest that the
Hadley cells contain inertia-gravity wave
packets that emanate from the ITCZ and
Figure 13. Plots of (75) for m5 1 and the switch-on rates c2153 h and bounce off a spectrum of turning latitudes
c21512 h. The blue and red curves represent the ray trajectories that ini- when convection in the ITCZ is temporally
tially move southward and northward, respectively. Note how the rays turn
at particular latitudes, approximately 64000 km for c2153 h and approxi- evolving. These equatorially trapped wave
mately 6900 km for c21512 h. packets cause the mass flux associated with
the Hadley cells to pulsate with periods of
about 1, 2, and 3 days for the m5 0, 1, 2
vertical modes while the vertical motion in the ITCZ and subsidence regions are slightly more complicated
and depend on ITCZ location, e.g., transient activity in the southern Hadley cell peaks at timescales of
approximately 30 h and 2 days for ITCZs 0–7.5 and 12.5–208 off of the equator, respectively, for ITCZ dia-
batic heating of the m5 1 vertical mode. When the m5 1 ITCZ is centered 7.5–12.58 off of the equator, the
southern cell experiences oscillations at both 30 and 50 h timescales. When the forcing is switched on
slowly (e.g., about 80% switched on after 3 days), the transient behavior decreases significantly and the sol-
utions are similar to the balanced results shown in Gonzalez and Mora Rojas [2014].
There have been a number of studies that have explored inertia-gravity waves forced by tropical convec-
tion, but not in the context of the ITCZ and the Hadley circulation. In regions such as the Pacific Ocean, the
dynamics are nearly zonally symmetric; therefore, the analytical solutions derived can provide realistic
insight into the dynamics. In other regions, such as the Indian Ocean, the assumption of zonal symmetry is
not often met; therefore, the inertia-gravity waves emanating from tropical convection likely behave differ-
ently. We expect that these inertia-gravity waves travel in both the zonal and meridional directions, possibly
bearing resemblance to the two-day inertia-gravity waves discussed in Takayabu [1994] and Haertel and
Kiladis [2004]. In closing, we emphasize that the tropical atmosphere may contain a considerable amount of
inertia-gravity wave activity, but its contribution to the large-scale flow may be difficult to discern in obser-
vational analyses. Therefore, this theoretical work should serve as motivation for future observational work
on inertia-gravity waves in the tropics.
Appendix A: Variational Approach
As a complement to the analysis in section 7, we now discuss the variational approach to understanding
the inertia-gravity wave aspects of the transient solutions. The argument again begins with (56), which is
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 687
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
the unforced version of the horizontal structure equation (28). We first note that (56) has the equivalent
variational formulation ð ð
d Lðwt;wy ;w; yÞ dt dy50; (A1)
where the Lagrangian is given by
Lðwt;wy;w; yÞ
1 2 1 2 2 15 wt2 c wy2 b
2y2w2: (A2)
2 2 2
To confirm the equivalence of the variational formulation (A1) and (A2) with the partial differential equation
(56), note that
ð ð
0 5d Lðwt;wy ;wÞ dt dy
ð ð "     #
@L @w @L @w @L
5 d 1 d 1 dw dt dy
@w @t @w @y @w (A3)
ð ð " t   y ! #
@ @L @ @L @L
5 2 2 1 dw dt dy;
@t @wt @y @wy @w
where the last line has been obtained through integr ation!by parts in both t and y. It follows from (A3) that 
@ @L @ @L @L
1 2 50: (A4)
@t @wt @y @wy @w
Differentiating (A2) we obtain
@L @L @L
5wt; 52c
2wy ; 52b
2y2w; (A5)
@wt @wy @w
which, when substituted into (A4), leads directly to (56).
In order to study slowly varying wave trains of the form w 	 a cos ðh1gÞ,Whitham [1965b, 1974, section 11.7]
introduced the concept of an ‘‘average vaðriaðtional principle,’’ which is analogous to (A1) and takes the form
d Lð2ht; hy ; a; yÞ dt dy50; (A6)
where the local frequency and wave number are given by x52ht and ‘5hy , and where the average
Lagrangian is given by
 
Lðx; ‘; a; yÞ 15 x22c2‘22b2y2 a2: (A7)
4
The average variational principleð(Að6) can then be written as
0 5d Lð2ht; hy ; a; yÞ dt dyð ð      
@L @h @L @h @L
5 d 1 d 1 da dt dy (A8)
ð ð @ht @t @hy @y @a 
@ @L @ @L @L
5 2 2 dh1 da dt dy;
@t @ht @y @hy @a
where the last line has again been obtained through integration by parts in both t and y. Independent varia-
tions da and dh respectively yield
GONZALEZ ET AL. TRANSIENT HADLEY CIRCULATION 688
Journal of Advances inModeling Earth Systems 10.1002/2016MS000837
@L
50; (A9)
 @a  
@ @L @ @L
2 50: (A10)
@t @x @y @‘
Differentiating (A7) we obtain
@L 1
5 xa2
@L 1 @L 1
; 52 c2‘a2; 5 ðx22c2‘22b2y2Þa; (A11)
@x 2 @‘ 2 @a 2
which, when substituted into (A9) and (A10), lead directly to the local dispersion relation
x25c2‘21b2y2; (A12)
and to the wave action equation    
@ E @ E
1 cg 50; (A13)
@t x @y x
where E5 1 22x a
2 is the energy density, E=x is the wave action, and the group velocity is given
by cg5c2‘=x. The dispersion relation links the local meridional wave number ‘ðy; tÞ and the local frequency
xðy; tÞ of the nonuniform wave train propagating in a medium with variable Coriolis parameter by.
From the relations x52ht and ‘5hy , it follows that ‘t1xy50. Then, since the local dispersion relation
yields xt5cg‘t , we conclude that
@x @x
1cg 50: (A14)
@t @y
Note that (A14) is identical to (71) and that (A13) and (A14) can be combined to obtain (69). In other words,
the average conservation law approach of section 7 yields the same results as the variational approach.
However, the variational approach gives a more direct route to the wave action principle (A13).
In section 7 and this appendix we have limited our discussion to the zonally symmetric case. Interesting dis-
cussions of ray theory for the zonally asymmetric case can be found in Blandford [1966] and Ripa [1994].
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