momentum balance and eliassen–palm flux on moist

Momentum Balance and Eliassen–Palm Flux on Moist Isentropic Surfaces

RAY YAMADA

Courant Institute of Mathematical Sciences, New York University, New York, New York

OLIVIER PAULUIS

Courant Institute of Mathematical Sciences, New York University, New York, New York, and Center for

Prototype Climate Modeling, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

(Manuscript received 4 August 2015, in final form 29 October 2015)

ABSTRACT

Previous formulations for the zonally averagedmomentumbudget andEliassen–Palm (EP) flux diagnostics

do not adequately account for moist dynamics, since air parcels are not differentiated by their moisture

content when averages are taken. The difficulty in formulating the momentum budget in moist coordinates

lies in the fact that they are generally not invertible with height. Here, a conditional-averaging approach is

used to derive a weak formulation of the momentum budget and EP flux in terms of a general vertical

coordinate that is not assumed to be invertible. The generalized equation reduces to the typical mass-

weighted zonal-mean momentum equation for invertible vertical coordinates.

The weak formulation is applied here to study the momentum budget on moist isentropes. Recent studies

have shown that the meridional mass transport in the midlatitudes is twice as strong onmoist isentropes as on

dry isentropes. It is shown here that this implies a similar increase in the EP flux between the dry and moist

frameworks. Physically, the increase in momentum exchange is tied to an enhancement of the form drag

associated with the horizontal structure of midlatitude eddies, where the poleward flow of moist air is located

in regions of strong eastward pressure gradient.

1. Introduction

High-energy air parcels rise in the tropics and flow

poleward along the upper branch of the meridional

overturning circulation. From the conservation of axial

angular momentum, these air parcels would eventually

attain unrealistically high zonal velocities in the mid-

latitudes if it were not for the downward momentum

transport induced by the form drag associated with

baroclinic eddies. The transformed Eulerian-mean

(TEM) equations, derived by Andrews and McIntyre

(1976), successfully capture these interactions between

eddies and the mean flow. In this framework, the mean

meridional circulation is represented by the residual

circulation, which consists of a single thermally direct

cell in each hemisphere. It approximates the Lagrangian-

mean circulation and is predominantly eddy driven in the

midlatitudes. The Coriolis torque on the residual circu-

lation acts to accelerate the zonal wind in the upper

troposphere. In steady state, it is balanced by the

Eliassen–Palm (EP) flux divergence, which represents

the form drag due to the eddies (e.g., Andrews et al.

1987; Vallis 2006).

The EP flux has become a standard diagnostic tool for

studying wave–mean flow interactions (Edmon et al.

1980). Andrews and McIntyre (1976) generalize the

original EP relation (Eliassen and Palm 1961), showing

that the EP flux divergence vanishes when the waves are

assumed to be steady, conservative, and of small am-

plitude. Under such conditions, it follows from the TEM

equations that the waves cannot induce changes in the

mean flow—a statement of the nonacceleration theorem

(Charney and Drazin 1961)—and thus a divergent EP

flux provides a measure of how wave transience and

dissipation affect the zonal-mean circulation. The non-

acceleration result can be arrived at more readily if

quasigeostrophic scaling is imposed (Edmon et al. 1980).

In this case, the total eddy forcing of the mean state is

accounted for by the EP flux divergence, which acts as a

Corresponding author address: Ray Yamada, Courant Institute

of Mathematical Sciences, New York University, 251 Mercer St.,

New York, NY 10012.

E-mail: [email protected]

MARCH 2016 YAMADA AND PAULU I S 1293

DOI: 10.1175/JAS-D-15-0229.1

� 2016 American Meteorological Society

mailto:[email protected]

zonal forcing. The generalized Lagrangian-mean for-

mulation in Andrews and McIntyre (1978a,b) general-

izes the nonacceleration theorem to finite-amplitude

waves, but is difficult to use in practice as it involves

particle displacements. More recently, Nakamura and

Zhu (2010) extended the nonacceleration result to finite

amplitude in pressure coordinates by using a hybrid

Eulerian–Lagrangian approach.

Following the works of Andrews andMcIntyre (1976)

and Edmon et al. (1980), subsequent studies utilized

isentropic averaging (i.e., averaging on surfaces of con-

stant potential temperature u) to make further advances

to the TEMdescription of the large-scale circulation. On

the one hand, isentropic averaging was pursued to gen-

eralize the EP flux diagnostic to finite-amplitude waves

and nongeostrophic flows in an Eulerian framework.

Andrews (1983) derived a finite-amplitude nondiver-

gent EP flux from the isentropic primitive equations,

assuming nonacceleration conditions a priori. Tung

(1986) relaxed the assumption of nonacceleration con-

ditions and used mass-weighted averages to derive an

isentropic EP flux that includes diabatic effects. Iwasaki

(1989) formulated a similar finite-amplitude nongeo-

strophic EP flux using isentropic zonal-mean pressure as

the vertical coordinate instead.

Meanwhile, isentropic averaging was applied to study

the mean meridional circulation (e.g., Dutton 1976;

Townsend and Johnson 1985; Johnson 1989). In isen-

tropic coordinates, poleward fluxes of warm air are

separated from equatorward fluxes of cool air, and so a

single-celled circulation emerges naturally when the

mass flux is averaged on isentropes. The isentropic cir-

culation is qualitatively similar to the TEM residual

circulation, and the two can be shown to be equivalent to

leading order in wave amplitude (McIntosh and

McDougall 1996; Pauluis et al. 2011). Moreover, the is-

entropic circulation offers several advantages over the

TEM circulation: (i) no small wave amplitude or qua-

sigeostrophic scaling assumptions are made; (ii) the

streamlines of isentropic circulation close at the lower

boundary, while those of the TEM circulation do not

(Held and Schneider 1999; Tanaka et al. 2004; Lalibertéet al. 2013); and (iii) the vertical motion in the isentropic

streamfunction directly reflects diabatic heating.

The isentropic EP flux and circulation are dynamically

tied through zonal momentum balance, as is the case

with the TEM EP flux and residual circulation. The is-

entropic EP flux divergence is related to the eddy flux of

Ertel potential vorticity (PV; Tung 1986). Since eddies

tend to mix PV downgradient (equatorward) along is-

entropic layers that do not intersect the surface (e.g.,

Yang et al. 1990), it can be shown (Held and Schneider

1999) that eddies drive a thermally direct isentropic

circulation. In the steady-state momentum equation, the

EP flux divergence balances the Coriolis force on the

isentropic circulation. The isentropic formalism gives

the EP flux a clear physical relationship to form drag,

which is expressed in terms of correlations between is-

entropic pressure anomalies and the isentropic slope

(Andrews 1983; Juckes et al. 1994).

The aforementioned studies outline some of the sig-

nificant theoretical advances in our understanding of the

momentum cycle, wave–mean flow interactions, and the

general circulation. These momentum budget analyses

have been limited, however, to dry dynamics with a

single exception (Stone and Salustri 1984). Air parcels

with the same value of potential temperature but dif-

fering moisture contents are not distinguished when

averages are taken. Recently, Pauluis et al. (2008,

hereafter PCK08) and Pauluis et al. (2010, hereafter

PCK10) compared the dry and moist isentropic

streamfunctions, where the latter is computed by aver-

aging the meridional mass transport on surfaces of

constant equivalent potential temperature ue rather than

potential temperature.1 Like the dry isentropic circula-

tion, the moist isentropic streamfunction consists of a

single direct cell in each hemisphere, but in the mid-

latitudes, the total mass transport of the moist isentropic

circulation is about twice as large as that of the dry is-

entropic circulation. The additional mass flux on moist

isentropes arises from a low-level flow of warm,moist air

that is advected poleward by bothmidlatitude baroclinic

eddies and subtropical stationary eddies (Pauluis et al.

2011; Shaw and Pauluis 2012). Air parcels within this

moist branch of the circulation rise within the storm

tracks, precipitate, and enter the upper branch of the

circulation, where they cool radiatively and eventually

subside into the low-level return flow of the circulation.

The dry isentropic diagnostic fails to clearly differentiate

the low-level poleward moist branch from the equator-

ward return flow, since the two branches consist of air

masses with similar values of potential temperature. This

leads to partial cancellation in the mass transport when

averaged on potential temperature surfaces. In contrast,

the moist branch has a similar value of equivalent po-

tential temperature as the upper branch of the circulation,

which allows for its clear separation from the low-level

return flow when using the moist isentropic diagnostic.

The additional eddy mass flux associated with the

moist branch has significant implications for the zonal

momentum cycle. The enhanced mass flux on moist

1 Czaja and Marshall (2006) had computed a similar version of

the dry and moist isentropic streamfunction, in which they used

surfaces of constant dry and moist static energy instead.

1294 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

isentropes implies a stronger Coriolis torque on the

zonal flow than that previously indicated by dry diag-

nostics. For momentum balance, the form drag and EP

flux on moist isentropes must increase as well. However,

there are currently no available formulations of the

momentum budget on moist isentropes. A major chal-

lenge with moist isentropes is that, unlike potential tem-

perature, equivalent potential temperature is not invertible

with height, especially at low latitudes (Xu and Emanuel

1989). This prevents the primitive equations from being

transformed onto moist isentropes via a simple change of

variables, and so zonal averages at constant equivalent

potential temperature cannot be taken in the usual way.

Stone and Salustri (1984) modify the quasigeostrophic

TEM residual circulation and EP flux to account for the

eddy transport of both sensible and latent heat, but they

assume amoist static stability that is vertically stratified and

latitudinally independent. Their formulation is also con-

strained by the same assumptions that go into the TEM

equations, and so the accuracy of their method near the

surface is not reliable. Their results do, however, show that

when large-scale condensation is accounted for, there is a

dramatic increase in the EP flux and residual circulation in

the subtropics and midlatitudes, as compared to the origi-

nal TEM description.

The goal of this work is twofold. First, we generalize

the isentropic EP flux diagnostics (Andrews 1983; Tung

1986; Iwasaki 1989) to allow for a general vertical co-

ordinate h that may be noninvertible. The generalized

framework is discussed in section 2 and involves condi-

tionally averaging the primitive equations based on the

value of h in the zonal–height plane. It is consistent with

the conventional mass-weighted zonal-mean equations

that are obtained when the vertical coordinate is in-

vertible. Second, we apply this new method in section 3

to evaluate the momentum budget on dry and moist

isentropes using the Modern-Era Retrospective Analy-

sis for Research and Applications (MERRA) reanalysis

from 1979 to 2012. The moist isentropic EP flux is shown

to have amuch largermagnitude and greater subtropical

extent than the dry isentropic EP flux. These differences

are explained by analyzing the zonal structure of synoptic

eddies in section 4. In section 5, we discuss the decompo-

sition of the EP flux into orographic and atmospheric form

drags, which leads to a more familiar expression for the EP

flux. Conclusions are given in section 6.

2. Weak formulation of the zonally averagedprimitive equations in generalized verticalcoordinates

In this section, we formulate the zonally averaged

mass-weighted primitive equations on constant-h surfaces,

where h is a general vertical coordinate that is not as-

sumed to be invertible with height. We derive a gener-

alized EP flux and EP theorem and give a clear physical

interpretation of the EP flux and its relationship to form

drag. These results are analogous to those derived in

Andrews (1983) and Tung (1986) in u coordinates but

have been generalized to hold on arbitrary h surfaces.

Deriving these results requires the use of distributions,

since traditional zonal averaging cannot be applied

along noninvertible h surfaces. For clarity, we begin

with the primitive equations in Cartesian coordinates (x,

y, z), but one could also derive the results from

spherical–pressure coordinates (appendix B):

›r

›t1= � (rv)5 0, (1a)

›u

›t1 v � =u52

1

r

›p

›x1 f y1D , (1b)

›p

›z52rg , (1c)

›h

›t1 v � =h5 _h, and (1d)

›C

›t1 v � =C5 _C . (1e)

The above equations are the continuity equation, zonal

momentum equation, hydrostatic relationship, and

general evolution equations for h and a tracer C, re-

spectively. The meridional momentum and temperature

equations are not explicitly written out, but they can be

considered as special cases of the tracer equation. The

velocity vector is denoted by v5 (u, y,w) andD denotes

frictional drag. The notation is otherwise standard.

The domain is assumed to be a periodic, zonal channel

[0,L]3 [y1, y2]3 [zsfc(x, y), ‘) that is bounded between

two fixed latitudes y1 and y2. The surface height is de-

noted by zsfc. The velocity normal to the surface and

along the latitudinal boundaries is taken to be zero. We

will use the symbol D to refer to the domain [0, L] 3[zsfc, ‘) within the x–z plane at fixed latitude. If X is a

subset of the domain, then ›(X) will denote the

boundary of the set X.

a. Weak formulation of the mass-weighted zonalaverage

When h is invertible with height, the primitive equa-

tions can be transformed from (x, y, z) to (x, y, h) co-

ordinates. Zonal averages (denoted by a bar) can be

taken in the usual way at constant value of h:

f ( y,h, t)51

L

ðL0

f (x, y,h, t) dx . (2)


When h is not invertible with height, this expression for

the zonal mean breaks down since f becomes multi-

valued at a given value of h. To work around this, we

use a conditional averaging approach similar to that

used in Pauluis and Mrowiec (2013) for studying the

upward mass transport in moist convection and also to

that in PCK10 for computing the meridional mass

transport on dry and moist isentropes.2 Here, we con-

sider the distribution of f over h at each latitude y and

time t, where the distribution is computed, in principle,

by integrating f in the x–z plane over those regions in

which h takes on a designated value h0:

hf i(y,h0, t)

51

L

ðL0

ð‘zsfc(x,y)

f (x, y, z, t)d[h02h(x, y, z, t)] dz dx .

(3)

The Dirac delta function d(x) is given by the derivative

of the Heaviside functionH(x) so (3) can be rewritten as

h f i5 limDh/ 0

1

2LDh

ððjh2h0j,Dh

f (x, y, z, t) dz dx . (4)

Hence, h f i involves integrating f over the x–z plane in

the regions in which h is within Dh of h0 in the limit as

Dh goes to zero. In practice, a small but finite Dh is used

to approximate h f i. We will refer to h f i as the delta

distribution of f on h, or simply the delta distribution of

f, when h is understood from context. The units of h f iare the units of f multiplied by meters per units of h.

We also define the following Heaviside distribu-

tion h�iH:

hf iH(y,h

0, t)

51

L

ðL0

ð‘zsfc(x,y)

f (x, y, z, t)H[h02h(x, y, z, t)] dz dx .

(5)

The delta distribution is given by the derivative of the

Heaviside distribution with respect to h0: hf i5 ›h0hf iH .

Although the delta distribution (3) defines a gener-

alized average on noninvertible h surfaces, hfi is not

generally equal to f even for invertible h. Instead, it can

be shown that for invertible h,

hf (x, y, z, t)i5 f (x, y,h, t)

��›z›h�� . (6)

The derivation is given in appendixA. The factor j›z/›hjis the Jacobian of transformation that relates the volume

element in (x, y, z) space to that in (x, y, h) space (i.e.,

dxdydz 5 j›z/›hjdxdydh). Hence, when f is chosen to

be a quantity per unit volume, then the delta distribution

of f is equal to the zonalmean of f on constant h surfaces.

In particular, mass-weighted zonal averages of f are

preserved by the delta distribution:

hrf (x, y, z, t)i5 rf (x, y,h, t)

��›z›h��5 r

hf (x, y,h, t) , (7)

where rh 5 rj›z/›hj is the density in (x, y, h) coordinates.

For (6) to hold on h surfaces that intersect the ground, we

must assume that f 5 0 below ground. This falls out natu-

rally for mass-weighted averages (7) if we adopt the com-

mon convention of setting the density to zero below ground

(Lorenz 1955). With this convention, mass-weighted aver-

ages contain no contribution from the portions along the

isentropes that are below ground.While such underground

conventions are necessarywhen taking zonalmeans in (x, y,

h) coordinates, the formulation for the delta distribution

(3) only involves integrating above the surface so that un-

derground conventions are not needed.

For invertible h, the conventional mass-weighted

zonal mean of f will be denoted by

~f (y,h, t)5rhf

rh

. (8)

For general, possibly noninvertible, h we define a weak

formulation of the mass-weighted zonal average of f on

h surfaces as follows:

f (y,h, t)5hrf ihri . (9)

Let f*(x, y, z, t)5 f (x, y, z, t)2 f [y, h(x, y, z, t), t] de-

note the departure of f from f . From (7), it follows im-

mediately that for invertible h the two definitions (8)

and (9) are consistent; that is, f 5 ~f . The weak form

of the mass-weighted average allows for the typical

manipulations of averages (e.g., bf*5 0,bf 5 f , andbfg5 f g1 df*g*), while the delta distribution h�i by itself

does not.

b. Weak form of the zonally averaged primitiveequations

The weak form of the zonally averaged primitive

equations can be derived by first rewriting (1) in flux

2 The meridional mass transport on a moist isentrope can be

computed as Lhryidue, which is a special case of the delta distri-

bution (3) with f 5 ry and h 5 ue. The moist isentropic stream-

function is given byC(y, ue0, t)5LhryiH, which is equivalent to theexpression given in PCK08 and PCK10 before time averaging.


form (so that mass-weighted averages can be taken) and

then taking the delta distribution (3) of the equations.

The weak forms of the zonally averaged continuity,

zonal momentum, hydrostatic balance, and tracer

equation are given as follows:

›

›thri1 ›

›y(hriy)1 ›

›h0

(hrib_h)5 0, (10a)

›

›t(hriu)1 ›

›y(hricuy)1 ›

›h0

(hricu _h)52

�›p

›x

�1 f hriy1 hriD , (10b)�

›p

›z

�52hrig, and (10c)

›

›t(hriC)1 ›

›y(hricCy)1 ›

›h0

(hricC _h)5 hrib_C . (10d)

In deriving these equations, we only need to consider

the manipulation of the tracer equation, since the

other equations are all just special cases of it [e.g.,

C5 1 with _C5 0 for the continuity equation and C5 u

with _C being the rhs of (1b) for the zonal momentum

equation]. The derivation of the tracer equation is

given in appendix A and is similar to the derivation for

the moist isentropic continuity equation given in

Pauluis and Mrowiec (2013). The equations formu-

lated from spherical–pressure coordinates are given in

appendix B.

The weak formulation (10) provides the advantage of

formulating the zonally averaged momentum and tracer

budgets for a noninvertible choice of vertical co-

ordinate. When h is invertible, the weak formulation

reduces to the same set of mass-weighted zonally aver-

aged equations that would result when the vertical co-

ordinate is first transformed to h before averaging. For

example, when h is chosen to be u, (10) reduces to the

mass-weighted zonally averaged isentropic primitive

equations (e.g., Gallimore and Johnson 1981). In this

case, mass-weighted averages can be rewritten in terms

of the usual mass-weighted average (8); for example,

hriy5 ru~y. The pressure gradient can be written in terms

of the gradient of the Montgomery streamfunction

h›p/›xi5 rug›c/›x, where c 5 cpT 1 gz.

c. Generalized EP flux and form drag decomposition

The zonal momentum (10b) can be rewritten in terms

of a generalized Eliassen–Palm flux vector F, as follows:

›

›t(hriu)1 ›

›y(hriuy)1 ›

›h0

(hriub_h)5= � F1 f hriy1 hriD , (11)

where

F5 (Fy,F

h)5

�2hridu*y*,2hri du* _h*2�›p

›x

�H

�(12)

and

= � F5›

›yFy1

›

›h0

Fh0.

The momentum flux terms in (10b) were separated into

their mean and eddy components. The EP flux vector

contains the eddy momentum fluxes and a pressure gra-

dient term, where we have used the fact that h›p/›xi5›h0

h›p/›xiH .The EP flux divergence physically describes the eddy

momentum transfer into an infinitesimal h layer lying

in the x–z plane. This layer describes the region of fluid

lying within Dh of some fixed value, h0, and has width

Dy in the latitudinal direction. Andrews (1983) and

Tung (1986) provide a similar physical interpretation of

= � F for u layers. But unlike u, h is not assumed to be

invertible, and so an h0 contour could consist of disjoint

pieces with vertical folds and loops. Figure 1 shows

three general cases: (i) loop, (ii) zonal ring, and

(iii) surface-intersecting contour, where the h0 contour

is drawn as a thick solid line and the infinitesimal h layer

is denoted by the surrounding thin dashed lines. More

general setups can be broken up into combinations of

cases (i)–(iii).

The first term in the horizontal and vertical compo-

nents of = � F are the along-h and across-h components

of the eddy momentum flux convergence. Below, we

show that the pressure gradient term in the vertical

component of = � F describes the form drag (i.e., the

average pressure exerted in the zonal direction on an

infinitesimal h layer per unit h) that arises from both

topography and the surrounding atmosphere. When the

topography is flat (i.e., no zonal asymmetries in the

surface height), the only contribution to form drag

comes from atmospheric eddies.

A general form of theEP theorem follows from (11): if

the atmosphere is frictionless and adiabatic in the sense

that _h vanishes, then a necessary condition for steady

flow is that the = � F must vanish. This can be shown as

follows. Under the assumptions stated above, the con-

tinuity (10a) implies that ›y(hriy)5 0. Furthermore,

since y vanishes on the meridional boundaries and hri isnonzero this implies that y5 0; that is, there is no mean

circulation on h surfaces. Then from (11),= � Fmust also

vanish. The EP theorem stated above holds for finite-

amplitude waves and nongeostrophic flows. Its state-

ment is analogous to the versions given in Andrews


(1983) and Tung (1986), but is now generalized for

arbitrary h coordinates.

We now give a physical interpretation of the pressure

gradient terms, 2h›p/›xi and 2h›p/›xiH, as form drags.

Consider the finite atmospheric layer in the x–z plane that is

bounded by the h0 contour and indicated by the stippled

region in Fig. 1 (i.e., this layer describes the subset

fh, h0g). The formdrag described by the term2h›p/›xiHis the average pressure exerted in the zonal direction on the

finite layer fh , h0g:

2

�›p

›x

�H

521

L

ððh,h0

›p

›xdz dx52

1

L

þ›(h,h0)

p dz , (13)

where Green’s theorem was used in the last step. The

increment dz along the boundary contour describes the

effective surface area (per unit y) on which the pressure

acts zonally. The boundary of the finite layer is illus-

trated in Fig. 1 by thick lines with arrows indicating the

direction of integration. The pressure along the bound-

ary of the layer can be separated into its surface (along

gsfc) and atmospheric (along gatm) components, as in-

dicated in Fig. 1:

21

L

þ›(h,h0)

p dz521

L

ðgsfc

p dz1

ðgatm

pdz

!.

We then separate the atmospheric component into its

mean and eddy parts, so that

2

�›p

›x

�H

521

L

ðgsfc

p dz1 p

ðh5h0

dz1

ðh5h0

p*dz

!,

(14)

where p(y, h0, t) was pulled out of the integral because

›(h , h0) 5 fh 5 h0g on gatm. The mean pressure p

describes the average pressure along the h0 contour and

p* is the deviation from thatmean. The formdrag exerted

on a finite layer therefore consists of the zonal pressure

exerted from the surface (i.e., an orographic form drag)

and from the surrounding atmospheric layers.

The pressure gradient term,2h›p/›xi52›h0h›p/›xiH ,

then describes the form drag acting on the infinitesimal

layer of air lying between h0 2 Dh and h0 1 Dh. This isillustrated in Fig. 1 in which 2h›p/›xi involves the re-

gion (between the thin dashed lines) given by the dif-

ference between two finite layers, fh , h0 2 Dhg and

fh, h01Dhg, in the limit as Dh goes to zero (4). When

the h0 contour does not intersect the surface, as in cases

(i) and (ii), or when there is flat topography (such that dz

along gsfc is zero) the orographic form drag vanishes.

From (14), the mean component of the atmospheric

form drag is just 2L21›h0(pDz), where Dz is the height

displacement between the endpoints of the h0 contour.

It also vanishes when the layer does not intersect the

surface or when there is flat topography, since Dz is zeroin these cases. Thus, without topography, the only con-

tribution to form drag comes from atmospheric eddies.

The form drag description of the pressure gradient

given here generalizes the discussion in Andrews (1983)

and Tung (1986) for isentropic layers to noninvertible

h layers.Additionally, our treatment of surface-intersecting

contours is handled differently. Tung (1986) does not

consider this case, while in Andrews (1983), surface-

intersecting isentropes are extended ‘‘just under the

surface’’ by a convention developed in Lorenz (1955), in

which the pressure along underground portions of

isentropes is taken to be the pressure at the surface. This

FIG. 1. A schematic showing three general types of constant-h contours (fh 5 h0g) that can occur: (left) case (i) loop, (center) case

(ii) zonal ring, and (right) case (iii) surface-intersecting contour. The gray-shaded region near the surface represents topography. The

finite layer fh, h0g is indicated by the stippled region. The boundary of the finite layer is outlined by the thick lines with arrows drawn

such that the curve is positively oriented. Thick solid lines indicate the atmospheric component of the boundary on which h5 h0, denoted

gatm, while thick dashed lines indicate the remainder of the boundary, including the surface component gsfc. In case (ii), integration over

the sidewalls of the boundary cancels from periodicity. The thin, dashed lines surrounding gatm indicate the infinitesimal layer fh0 2Dh , h , h0 1 Dhg.


convention is mathematically convenient but has no

physical basis, and the explanation for the form drag

contribution from subterranean isentropes is unclear.

Here, underground conventions are avoided altogether

and the pressure gradient is only integrated at or above

the surface. As a result, the decomposition (14) provides

a clear and physically meaningful separation of the form

drag into its orographic and atmospheric components. In

section 5, we will discuss the form drag terms in (14) in

more detail and derive alternative expressions for them,

which will give the vertical EP flux its familiar form.

In computing the EP flux (12) in practice, we use

p0 5 p2 p (i.e., deviation from the zonally averaged

mean state) instead of p. This does not affect the pres-

sure gradient since ›p/›x 5 ›p0/›x; however, the use of

p0 becomes important in the decomposition (14) for re-

moving large cancellations between the orographic and

mean atmospheric form drags that arise from the hy-

drostatic pressure.

3. Zonal momentum budget on dry and moistisentropes

In this section, we compute the terms in the zonal

momentum budget (11) in both dry and moist isentropic

coordinates (i.e., choosing h 5 u and h 5 ue, re-

spectively) using MERRA data from 1979 to 2012

(Rienecker et al. 2011). The MERRA data are output

eight times daily on a 1.258 3 1.258 latitude–longitudegrid at 42 pressure levels. Only the lowest 31 pressure

levels, which lie between 1000 and 10hPa, are used,

since we only consider isentropes which lie between 240

and 360K. The tendency terms, _u and _ue, are computed

diagnostically from the data using finite differences in

space and time. Delta distributions (4) are computed

using a finite layer width of Dh 5 1.3K.

We first consider the EP flux divergence and the

Coriolis term. In the TEM and dry isentropic frame-

works, these terms are well known (e.g., Vallis 2006,

section 12.4) to make up the dominant balance in the

midlatitude momentum budget (11):

f hriy’2= � F . (15)

The EP flux divergence is largest in its vertical compo-

nent, which is mostly due to the downward transfer of

momentum by form drag. This implies from (12) that, at

least in the dry diagnostics = � F ’ 2h›p/›xi, and so the

leading-order relation (15) is essentially geostrophic

balance in the isentropic zonal mean. In contrast, the

zonally averaged circulation on pressure surfaces does

not capture the leading-order geostrophic flow in the

midlatitudes because the zonal mean of the meridional

geostrophic velocity is zero. Consequently, the isobarically

averagedmomentumbudget only reflects the higher-order

balance, which relates the Coriolis force by the ageo-

strophic flow, the eddy momentum fluxes, and surface

friction (e.g., Peixoto and Oort 1992, section 14.5).

The isentropic-mean circulation y is important in this

discussion and can be visualized by its streamfunction.

The dry (moist) isentropic streamfunction represents

the meridional mass flux across a given latitude for air

parcels with a value of u (ue) less than u0 (ue0). It is

computed as

C(f,h0)5

2pa cosf

t

ðt0

hrpyi

Hdt ,

where h denotes u (ue) in the dry (moist) case, t is the

time period over which the circulation is averaged, rp 5g21 is the density in pressure coordinates, and the

spherical–pressure coordinate form of h�iH is given in

appendix B. Figure 2 shows the 1979–2012 annual mean

of the dry and moist isentropic streamfunctions (Figs. 2a

and 2b, respectively). The dashed (solid) contours rep-

resent clockwise (counterclockwise) rotation. In Figs. 2a,b,

the contours are drawn at every 2.5 3 1010 kg s21,

omitting the zero contour. The black lines denote the

median value of zonal-mean surface u and ue, respec-

tively. PCK10 give a detailed analysis of the dry and moist

isentropic streamfunctions. Here we list a few important

features:

(i) In both the dry and the moist case, the stream-

function consists of a single, direct cell in each

hemisphere.

(ii) The cells tilt downward with latitude because air

parcels cool radiatively as they travel poleward and

gain heat from surface fluxes as they return from

higher latitudes. This tilt is steeper in the moist

case, since ue includes surface evaporation as well

as sensible heat fluxes. Additionally, the radiative

cooling of u is partially compensated by latent heat

release.

(iii) The dry circulation has two distinct cores: a tropical

Hadley circulation and an eddy-induced midlati-

tude circulation. Cool, air parcels subside in sub-

tropical latitudes near the edge of the Hadley cell,

while in themidlatitudes, air parcels rise in the storm

track from the latent heat released in precipitation.

In the moist circulation the cooling occurs at nearly

the same rate in the subtropics and midlatitudes.

(iv) The streamlines of the return flow tend to lie below

themedian surface value of u and ue, indicating that

the equatorward branch of the circulation consists

largely of cold-air outbreaks (Held and Schneider

1999; Laliberté et al. 2013).


(v) In the midlatitudes, the total mass transport is

around twice as large when averaged on moist,

instead of dry, isentropes (PCK08; PCK10). Pauluis

et al. show that this difference can be attributed to a

low-level flow of moist air that is advected pole-

ward from the subtropics by midlatitude eddies.

Figure 3 shows the EP flux vectors (arrows) and di-

vergence (shading) computed on dry (Fig. 3, left) and

moist (Fig. 3, right) isentropes for the 1979–2012 annual

(Figs. 3a,b), December–February (DJF) (Figs. 3b,d),

and June–August (JJA) (Figs. 3e,f) means. The color

scale for all the panels is the same. The streamfunctions

are drawn to show which isentropes correspond to the

poleward and equatorward branches of the circulation

at a given latitude.

The dry isentropic EP flux qualitatively resembles its

quasigeostrophic TEM analog. In the time mean, the

midlatitude EP flux has a large vertical component, a

characteristic feature of growing baroclinic waves (Edmon

et al. 1980), and exhibits a primarily equatorward tilt in the

upper troposphere. TheEP flux divergence (convergence)

is strongest near the lower boundary (upper troposphere)

and indicates the downward transfer of momentum

through form drag, which is necessary to maintain the

surface westerlies (e.g., Vallis 2006, chapter 12). These

features are present in both hemispheres and for all

seasons, but the EP flux and circulation are much more

intense in the winter hemisphere than in the summer

hemisphere. The main difference between the dry is-

entropic and TEM formulations of the EP flux lies near

the surface, where the TEM version has a stream-

function that does not close and unrealistically large

values of the EP flux (Held and Schneider 1999; Tanaka

et al. 2004).

At high-ue (poleward branch) levels, the moist EP

flux convergence extends well into the subtropics, to

around 208 in both hemispheres and for all seasons. In

comparison, the dry EP flux convergence becomes

much weaker equatorward of around 408. The moist EP

flux also has a significantly stronger convergence maxi-

mum at high-ue levels. For the annual mean, the con-

vergence maximum is 1.5 (2.2) times stronger in the

Northern (Southern) Hemisphere as compared to the

dry case. In the winter, the convergence maximum is

around twice as large with a factor of 2.0 (2.2) for the

NH DJF (SH JJA). The factors are even greater in the

summer, 2.4 (NH JJA) and 2.6 (SH DJF), because

the moist EP flux and circulation still have a strong

signal in the summer, whereas the dry EP flux and

circulation do not.

At low-ue (equatorward branch) levels, the differ-

ences between the dry and moist EP flux divergence are

similar to those in the EP flux convergence at high-uelevels but are not as distinct. With the exception of the

summertime (NH JJA and SH DJF), the moist EP flux

divergence extends only slightly farther into the sub-

tropics. Its maximum is greater than the dry case by a

FIG. 2. The 1979–2012 annual-mean climatology of the (a) dry and (b) moist isentropic streamfunctions. Dashed

(solid) contours represent clockwise(counterclockwise) rotation. The contours are drawn at every 2.53 1010 kg s21,

omitting the zero contour. The black lines denote the median value of zonal-mean surface u and ue.


factor of 1.1 (1.1) for the Northern (Southern) Hemi-

sphere in the annual mean and by a factor of 1.1 (1.1) in

the winter and 1.3 (1.1) in the summer.

Stone and Salustri (1984) found qualitatively similar

results (i.e., an enhancement and subtropical extension

of the EP flux) in the Northern Hemisphere. Their moist

EP flux is derived from the quasigeostrophic TEM for-

mulation that has been modified to account for large-

scale condensation but assumes a vertically stratified

moist static stability. They find that the inclusion of

condensation effects increases the TEM EP flux con-

vergence and divergence maxima by around 2.5 times in

the annual mean. In contrast, the maxima in our results

do not increase by such a large factor; we find only a 1.5

(1.1) times increase in the convergence (divergence)

maximum for the Northern Hemisphere annual mean.

This quantitative difference between their results and

ours may arise from the inaccuracy of the TEM formu-

lation, which overestimates the value of the EP flux and

residual circulation near the surface (Held and Schneider

1999; Tanaka et al. 2004). In their results, both the con-

vergence and divergence maxima occur near the surface

(their Fig. 6) at latitudes where the residual stream-

function does not close at the surface (their Fig. 8).

The enhancement and subtropical extension of the EP

flux reveals a greater and more extensive momentum

exchange between moist isentropes than between dry

isentropes. These differences are consistent with the

difference in the mass transport, discussed in point (v)

above. Dynamically, a larger poleward mass transport

onmoist isentropes implies a stronger Coriolis torque on

the zonal flow. The steady-state balance (15) can only

hold if there is also an increase in the eddy momentum

transfer between moist isentropes.

FIG. 3. The 1979–2012 (a),(b) annual-, (c),(d) DJF-, and (e),(f) JJA-mean climatologies of the EP flux vectors

(arrows) and divergence (shading) computed on (left) dry and (right) moist isentropes. The streamfunctions and

median surface value of zonal-mean surface u and ue are drawn as in Fig. 2, but the contours for the streamfunctions

are drawn at every 5.0 3 1010 kg s21 starting from 62.5 3 1010 kg s21. The low-level EP flux divergence is over-

saturated in the Southern Hemisphere for all seasons. The maximum values attained in (a)–(f) are [6.3, 6.9, 6.9, 7.7,

7.6, 8.5] 3 1022 PaK21, respectively.


Figure 4 shows that this is indeed the case. Figures

4a,b show the annual mean of the Coriolis force on the

isentropic-mean circulation f hriy for the dry (Fig. 4a)

and moist (Fig. 4b) cases. Similarly, Figs. 4c,d show the

isentropic-mean pressure gradient h›p/›xi which is the

form drag up to a minus sign (section 2). The plots in

Figs. 4a,b and Figs. 4c,d are nearly identical, showing

that (i) where there is an increase in the Coriolis force

from dry to moist isentropes, there is also an increase in

the form drag and (ii) both the dry and moist isentropic-

mean flows are nearly geostrophic. Figures 4e,f show the

Coriolis force on the ageostrophic circulation, which is

given by the difference between Figs. 4a,b and Figs. 4c,d.

The Hadley, Ferrel, and polar cells can be clearly

identified. The Coriolis force acts to accelerate the zonal

flow in the upper branches of the direct cells and de-

celerate the winds in the upper branch of the indirect

Ferrel cell. The signs of the Coriolis force are opposite in

the lower branches of the cells. Notice that the color

scale is smaller by a factor of 10, and so in the extra-

tropics, the ageostrophic Coriolis term is around an or-

der Rossby number smaller. The separation between the

dominant balance and higher-order terms is greater in

the moist case than the dry case.

The other higher-order, steady-state terms in (11)

include the momentum flux convergence terms [of

which there are four different components: isentropic

mean, 2›y(hriuy); cross-isentropic mean, 2›h0(hriub_h);

FIG. 4. The 1979–2012 annual-mean climatology of the leading-order terms in the zonal momentum budget (11)

computed on (left) dry and (right) moist isentropes: (a),(b) Coriolis force on the isentropic-mean circulation f hriyand (c),(d) isentropic-mean pressure gradient h›p/›xi. (e),(f) The Coriolis force on the ageostrophic circulation,

which is given by the difference between (a),(b) and (c),(d), respectively. The color scale for (a)–(d) is 10 times that

for (e),(f). Oversaturation tends to occur near the surface and is greatest in (e),(f), with some values reaching

around 2–3 times the color scale maximum. The magenta and black lines are as in Fig. 3.


isentropic eddy, 2›y(hridu*y*); and cross-isentropic

eddy, 2›h0(hri du* _h*)], the metric term from spherical

coordinates (B4b), and the surface drag, hriD. The

isentropic eddy momentum flux convergence shown

in Figs. 5a,b has a similar structure as that when

computed on isobars. Rossby waves transfer easterly

momentum away from the midlatitude jet, which

results in momentum convergence in the jet region

and divergence on the flanks of the jet (e.g., Vallis

2006, chapter 12). The wave propagation is strongest

in the upper troposphere, where there is little mois-

ture, and so the dry and moist computations are quite

similar.

The cross-isentropic eddy momentum flux conver-

gence (Figs. 5c,d) is noticeably stronger in the sub-

tropics on moist isentropes. There is a vertical dipole

structure in each hemisphere, which indicates there is

an easterly momentum transfer to higher-ue isentropes

near the surface. This low-level eddy momentum flux is

not seen in the dry case and may be related to surface

evaporation. From Dalton’s evaporation law (Peixoto

and Oort 1992, section 10.7.3), the evaporation is pro-

portional to the surface wind speed multiplied by the

saturation deficit of the air. Since the mean surface

winds are easterly at low latitudes, an easterly wind

anomaly would trigger stronger evaporation than a

westerly wind anomaly.

The residual (Figs. 5e,f) is computed by isolating

hrpiD on the right-hand side of (B4b) and then sum-

ming all the terms on the left-hand side of the equa-

tion. The residual mainly comprises the surface drag

that is needed to balance the Coriolis force on the

ageostrophic circulation (Figs. 4e,f) and is positive

(negative) in the regions of surface easterlies (west-

erlies). We do not discuss the mean momentum flux

convergence terms nor the metric term here, but their

structures follow intuitively from the structure of the

isentropic-mean winds.

FIG. 5. Eddy components of the momentum flux convergence: (a),(b) isentropic, 2›y(hridu*y*); (c),(d) cross

isentropic,2›h0(hri du* _h*), plotted for the 1979–2012 annual mean on (left) dry and (right) moist isentropes. (e),(f)

The residual, explained in the text. As in Fig. 4, oversaturation occurs near the surface in (e),(f) with some values

reaching around 2–3 times the color scale maximum.


4. Physical description of EP flux enhancement onmoist isentropes

To physically explain why the moist EP flux is

stronger and has a greater subtropical extent than the

dry EP flux, we now turn to analyzing a case study of

synoptic eddies. Figure 6 shows a cyclone–anticyclone

pair over the South Atlantic Ocean on 1200 UTC

6 August 2000. Figure 6a shows in shading the geo-

potential height anomaly at 700 hPa. The centers lie

within a latitude band between 308 and 458S and are

centered about 158W and 108E, respectively. To the

east of the cyclone, warm moist air is being advected

poleward (the flow velocity is shown with arrows). The

boundary between the warm subtropical and cooler

extratropical air masses is roughly delineated by the

300-K u contour (black) and 310-K ue contour (ma-

genta). The warm front appears sharper when ue is used

instead of u. This difference reflects the lower-level

transport of water vapor by the eddies. Figure 6b shows

the relative humidity in which there is a channel-like

flow of water vapor, often referred to as an ‘‘atmo-

spheric river’’ (Newell et al. 1992), from the tropics to

the warm front. The 310-K ue contour closely follows

the sharp front between moist and dry air and pro-

trudes past the 300-K u contour near 108W.

FIG. 6. A case study of synoptic eddies over the South Atlantic Ocean at 1200 UTC 6 Aug 2000 to illustrate the

enhancement of momentum exchange between moist isentropes as compared to dry isentropes. (a) Geopotential

height anomaly at 700 hPa. The 300-K u (black) and 310-K ue (magenta) contours are shown. The flow velocity is

depicted by the arrows. The dotted black lines mark the latitude circles at 308 and 408S for reference. (c) Longitude–pressure profile of the geopotential height anomaly at 408S. The 305-K u (black) and ue (magenta) contours are

shown. The gray contours show the geostrophic meridional wind at every 10m s21, omitting the zero contour and

with equatorward (poleward) wind denoted by solid (dashed) contours. (e) As in (c), but at 308S and showing the

310-K u and ue contours. (b),(d),(f) As in (a),(c),(e), but for the relative humidity.


Vertical cross sections of the height anomaly and rela-

tive humidity are shown in shading at 408S (Figs. 6c,d) and

308S (Figs. 6e,f). The latitude circles are marked for ref-

erence in Figs. 6a and 6b by the dotted line. The geo-

strophic velocity is shown in gray contours with poleward

(equatorward) wind denoted by dashed (solid) lines. The

305-K (310K) u and ue contours are shown in Figs. 6c,d for

408S (Figs. 6e,f for 308S). These contours roughly divide

the troposphere into upper- and lower-isentropic layers.

At 408S, the poleward advection of warm air is indicated

by the dip in both the u and ue contours on the eastern side

of the cyclone. The warm front is again sharper (i.e., the

contours dip further) when observed using ue instead of u.

This is because the poleward flow of warm air from the

tropics has a high moisture content, as can be seen by the

region of high relative humidity (Fig. 6d) that coincides

with the poleward geostrophic winds.

The difference in the zonal structure of dry and moist

isentropes associated with typical midlatitude eddies, such

as the one shown in Fig. 6, can explain the larger EP flux

observed onmoist isentropes. In Fig. 6c, there is a clear bias

for poleward geostrophic velocity in the upper-isentropic

layer and equatorward velocity in the lower layer, as is

expected for a thermally direct eddy-driven circulation. To

leading order, the EP flux divergence is proportional to the

geostrophic velocity (i.e., = � F ’ 2h›z/›xi } hygi), sincef, 0 in the Southern Hemisphere. Hence, there is EP flux

convergence and divergence in the upper and lower layers,

respectively. The convergence is stronger in the moist case,

since the moist upper layer includes the poleward flows of

both the low-levelmoist air, aswell as the upper-levelwarm

air. Physically, the poleward geostrophic wind lies in a re-

gion of strong eastward height gradient, which explains the

westward acceleration on the upper layer.

Further equatorward at 308S (Figs. 6e,f), the sharp

moisture front is still clearly defined by the 310-K uecontour. Thus, even at subtropical latitudes away from

the storm centers, midlatitude eddies can induce a large

momentum exchange between moist isentropic layers.

In contrast, there is only a weak zonal asymmetry in the

temperature structure, as indicated by the nearly hori-

zontal 310-K u contour, which shows that the eddies

have little effect on themomentum transfer between dry

isentropic layers in the subtropics.

A summary schematic is shown in Fig. 7. The hori-

zontal (Fig. 7, top) and vertical (Fig. 7, bottom) profiles

are shown for a Northern Hemisphere midlatitude eddy

system, consisting of three centers—high, low, and high.

In the top panel, a contour of constant pressure is shown

in green. The geostrophic flow is depicted by the black

lines and arrows. The advection of warm and cold air by

the cyclone create zonal perturbations in the 300-K

contours of u (orange) and ue (magenta). The ue

contour is marked by an asymmetry, in which high-ue air

penetrates farther poleward than high-u air, while the

cold protrusions of low-u and low-ue air are comparable.

This is because the poleward branch of the moist cir-

culation includes the contribution from low-level moist

air, while the return flow is fairly dry.

In the vertical profile, the 300-K isentrope divides the at-

mosphere into upper- and lower-isentropic layers. The

asymmetry in the ue contour is also observed, inwhich it dips

lower than the u contour in the regionofmoisture advection.

This gives themoist isentropic layers a larger vertical surface

area on which pressure acts in the zonal direction (13). The

westward phase lag of the isentropes relative to the pressure

anomaly creates a net westward pressure force acting on the

upper-isentropic layer and an equal and opposite force on

the lower-isentropic layer. Hence, the form drag induced by

atmospheric eddies acts todecelerate (accelerate) theupper-

isentropic (lower isentropic) layer.

5. Formal decomposition of the pressure gradientinto form drags

In section 2, we gave a physically motivated de-

composition (14) of the pressure gradient term h›p/›xiHinto the sum of the orographic form drag and the mean

and eddy components of atmospheric form drag on the

finite layer fh, h0g. We will refer to these components

of form drag [i.e., the three terms on the rhs of (14)] as

Mo, Mm, and Me, respectively, so that

2

�›p

›x

�H

5Mo1M

m1M

e. (16)

Here, we will formallymake this decomposition and find

alternative expressions for Mo, Mm, and Me to those

given in (14) that are computationally simple and have

the familiar forms of the orographic form drag and EP

flux. Additionally, we will show that the form drag in-

duced by atmospheric eddiesMe is the dominant means

by which momentum is transferred between dry and

between moist isentropic layers.

Using the product rule to incorporate the Heaviside

function inside the derivative, the pressure gradient can

be decomposed as follows:

2

�›p

›x

�H

521

L

ðL0

ð‘zsfc

›

›x[pH(h

02h)]1p

›h

›xd(h

02h)dzdx

521

L

ððD

›

›x[pH(h

02h)]dzdx2

�p›h

›x

�(17)

521

L

þ›DpH(h

02h)dz2p

�›h

›x

�2hribp*�1

r

›h

›x

�*.

(18)


The first term in (17) was transformed into the first term

in (18) by usingGreen’s theorem, and the second term in

(17) was split into the last two terms in (18) by using the

mean-eddy decomposition:

2

�p›h

›x

�52hribp�1

r

›h

›x

�52p

�›h

›x

�2 hribp*�1

r

›h

›x

�*.

The three terms in (18) are equivalent to Mo, Mm, and Me,

respectively, and upon furthermanipulation can bewritten as

Mo52

1

L

ðL0

psfcH(h

02h

sfc)›z

sfc

›xdx , (19a)

Mm52

1

Lp�

n

i51

[zsfc(x

2i21, y)2 z

sfc(x

2i, y)], and (19b)

Me52hribp*�1

r

›h

›x

�*. (19c)

The details are given in appendix A. In (19b), the points

x2i21 (x2i) represent the left (right) endpoints of the

h0 contour(s) that intersect the surface, where there are

n disjoint contours.

The orographic form drag,Mo, describes the pressure

along the surface acting in the zonal direction on the

finite layer fh , h0g. The orographic form drag on an

infinitesimal layer, or h0 contour, is given by

›Mo

›h0

521

L

ðL0

psfcd(h

02h

sfc)›z

sfc

›xdx (20)

521

L�i

psfc(x

i, y, t)

›zsfc

›x(x

i, y)

��›hsfc

›x(x

i, y, t)

��21

,

(21)

where the xi denote the x positions of the points of in-

tersection between the surface and h0 contours in the

x–z plane. The factor j›hsfc/›xj21 arises since the delta

function is composed with h0 2 hsfc which has an x de-

pendence. The pressure on the h0 contour exerted by

topography is determined by the surface pressure at its

endpoints and depends on the geometric factors ›zsfc/›x

FIG. 7. A schematic diagram to illustrate the form drag on dry and moist isentropic layers

associated with midlatitude eddies. (top) Longitude–latitude and (bottom) longitude–height

profiles are shown for a Northern Hemisphere midlatitude eddy system, consisting of three

centers: high, low, and high. The 300-K u and ue contours are shown in orange and magenta,

respectively. In (top), the streamlines of the flow are depicted by black lines and arrows. A

representative pressure contour is shown in green. In (bottom), the pressure anomalies are

shown by green contours, with solid (dashed) lines denoting positive (negative) values.


and j›hsfc/›xj21. These factors describe the effective

surface area for pressure acting in the zonal direction

and the thickness of the h0 contour, respectively. If the

topography is flat, then ›zsfc/›x is zero and there is no

orographic form drag. The expression (21) is similar to

that given in isentropic coordinates (Koh and Plumb

2004). In height coordinates (h 5 z), the geometric

factor is simply 61, since layers of constant height are

horizontal (Peixoto and Oort 1992, section 11.1).

From (19b), it is clear that the mean component of the

atmospheric form drag Mm can only be nonzero in the

presence of topography. But unlikeMo, it does not directly

involve surface pressure and should not be confused as an

orographic form drag. Rather, it arises from the fact that

surface-intersecting h0 contours do not form closed con-

tours and can therefore have a net height displacement

between their endpoints. And so, Mm describes the mean

atmospheric pressure p along the boundary of the finite

layer fh, h0g acting on the net vertical surface area of thelayer that is exposed to the atmosphere.

When topographic effects are small, the form drag

arising from atmospheric eddiesMemakes up the largest

contribution to the total form drag in (16). The vertical

component of the EP flux is approximately

Fh’2hri du* _h*2 hribp*�1

r

›h

›x

�*.

With h 5 0, Me describes the correlation between the

pressure and isentropic-slope anomalies and has a sim-

ilar form to the dry isentropic EP flux in Andrews (1983)

and Tung (1986).

Isentropic mass-weighted means and the isentropic EP

flux do not depend on whether they are computed from

height or pressure coordinates, but the individual terms

within the form drag decomposition (16) do depend on

the choice of coordinate when the isentropes intersect the

surface (appendix B). In pressure coordinates (denoted

by superscript p), the orographic form dragM(p)o involves

the surface height zsfc rather than the surface pressure,

and the mean and eddy components of atmospheric form

drag, M(p)m and M(p)

e , are formulated in terms of the is-

entropic mass-weightedmean and eddy heights, z and z*,

rather than those for pressure (B8).

Figure 8 shows the decomposition of the isentropic-

mean pressure gradient h›p/›xi into the orographic

(›h0M(p)

o ; Figs. 8a,b), mean (›h0M(p)

m ; Figs. 8c,d), and

eddy (›h0M(p)

e ; Figs. 8e,f) atmospheric form drags on dry

(Fig. 8, left) and moist (Fig. 8, right) isentropes. The

orographic and mean atmospheric form drags are shown

for their DJF mean, while the eddy atmospheric form

drag is shown for its annual mean. The eddy form drag is

nearly identical to the EP flux divergence shown in

Figs. 3a,b. It makes up the dominant contribution to the

pressure gradient term and provides the downward

transfer of momentum needed for geostrophic balance.

In the previous section, we showed how this form drag

arises from the zonally asymmetric structure of isen-

tropes induced by atmospheric eddies.

The orographic and mean form drags are of higher

order and are mostly localized near the surface. Here we

explain their observed features in the NH midlatitudes

during DJF. These features are also present, but weaker,

in the annual mean. Near the surface around 458N, in

both the dry and moist cases, the orographic and mean

form drags form a vertical dipole, centered around 280K,

with acceleration below and deceleration above. In con-

trast, the classic description of orographic form drag on

isobars involves a deceleration of the surface flow, similar

to friction (Peixoto and Oort 1992, their Fig. 11.12). This

difference can be explained by the temperature structure

associated with large-scale mountain ranges.

Figure 9 shows the 20 January 2000 daily average of the

geopotential height and potential temperature fields. The

top panel shows the horizontal distribution of the geo-

potential height at 500hPa. There are troughs to the east of

the major mountain ranges—Rockies, Alps, and Tibetan

Plateau—and ridges to theirwest. Similar zonal asymmetries

appear in the January climatology of the height field (not

shown). A simple potential vorticity conservation argument

(Holton 2004) shows that westerly wind over mountain

ranges tends to form troughs on their leeward side.

The middle and bottom panels of Fig. 9 show the ver-

tical profile of the potential temperature and geopotential

height anomaly at 458N, respectively. The three distinct

mountain ranges shaded in gray are, fromwest to east, the

Rockies, Alps, andMongolian plateau. The troughs to the

east of the mountain ranges bring about a northerly ad-

vection of cold air on the leeward side of the mountains.

This can be seen in the temperature profile by the domes

of cold air that form just to the east of the mountains. The

280-K contours are highlighted in blue. The temperature

280K lies near the center of the vertical dipoles in ›h0M(p)

o

and ›h0M(p)

m at 458N, and is where there is a positive

maximum in M(p)o and M(p)

m , as can be inferred from the

sign of the dipole. The positive signs ofM(p)o andM(p)

m can

be explained using (B8a) and (B8b). Here,M(p)o describes

the orographic form drag imparted along the surface un-

der the cold domes marked by the blue contours. Its

dominant contribution comes from the eastern slope of

the mountains, where ›psfc/›x is largest and positive and

the surface height anomaly z0sfc is also positive. From

(B8a), M(p)o must then be positive. The sign of M(p)

m is

determined by the sign of the average height anomaly

along the 280-K contours and the sign of the difference in

surface pressure between endpoints (left minus right) of


the contours (B8b). The average height anomaly is neg-

ative (i.e., bz0 , 0), as can be inferred from the bottom

panel of Fig. 9 in which the 280-K contours extend mostly

through the troughs on the leeward side of the mountains.

The surface pressure difference is generally negative as a

result of the elevation difference between the endpoints,

and so from (B8b) M(p)m is positive. The terms M(p)

o and

M(p)m attain their maxima near 280K because above this

temperature, the isentropes begin to pass over the

mountains, while below it the isentropes emanate from

the mountains at lower elevations.

6. Conclusions

The quasigeostrophic TEM equations have become a

textbook standard (e.g., Holton 2004; Vallis 2006) for

their mathematical simplicity and clear exposition of the

effect of eddies on the large-scale circulation. The TEM

framework shows how eddies drive a direct meridional

circulation in the midlatitudes. The EP flux divergence

contains the explicit eddy forcing of the mean flow.

Physically, it describes a downward transfer of mo-

mentum through form drag and explains how eddies drive

the surface westerlies. Isentropic EP flux diagnostics

(Andrews 1983; Tung 1986; Andrews et al. 1987; Iwasaki

1989) were developed to extend the original TEM results

to finite-amplitude waves and nongeostrophic flows. De-

spite these advances in our understanding of wave–mean

flow interactions, the existing descriptions of the large-

scale circulation have largely neglected the role of moist

processes because of the difficulty in dealing with non-

invertible moist coordinates. One exception is the work

FIG. 8. The different form drags are plotted on (left) dry and (right) moist isentropes: (a),(b) orographic, ›h0M(p)

o ;

(c),(d) mean atmospheric, ›h0M(p)

m ; and (e),(f) eddy atmospheric, ›h0M(p)

e . Panels (a)–(d) show the DJF-mean is-

entropic streamfunctions, while (e),(f) show the annual mean.


of Stone and Salustri (1984), who modify the quasigeo-

strophic TEM formulation to include the effects of

large-scale condensation, but require the choice of a

vertically stratified and latitude-independent moist

static stability parameter. PCK08 and PCK10 showed

that the midlatitude circulation is twice as large when

the mass flux is separated by its value of ue instead of u.

This implies a much larger Coriolis acceleration on the

residual circulation and greater momentum exchange by

midlatitude eddies than explained by dry diagnostics.

In this paper, we established the necessary framework

for studying the momentum budget on moist isentropes

and computed the diagnostic results using MERRA re-

analysis data (1979–2012). The issue of noninvertibility is

handled by formulating aweak coordinate transformation,

which involves conditionally averaging the mass-weighted

primitive equations on the value of a general vertical co-

ordinate h in the zonal–height plane. The generalized EP

flux describes the eddymomentum transfer into an h layer

from both eddymomentum fluxes and form drag. A finite-

amplitude nongeostrophic EP theorem holds in which the

EP flux divergence vanishes for steady flow in a frictionless

and adiabatic ( _h5 0) atmosphere. When the vertical co-

ordinate is invertible, the weak formulation reduces to the

usual set of mass-weighted zonal-mean equations that

could have alternatively been derived by first changing

variables and then taking the zonal mean. The formulation

does not rely on the use of underground conventions

(Lorenz 1955) and isentropes that intersect the surface give

rise to form drag terms with clear physical interpretations.

The leading-order balance in the momentum budget,

when computed on both dry and moist isentropes, is be-

tween the Coriolis force on the geostrophic flow and the

EP flux divergence. As compared to the dry EP flux, the

moist EP flux is much stronger in the midlatitudes and

extends well into the subtropics. Stone and Salustri (1984)

found a qualitatively similar increase between the dry and

moist TEMEPfluxes, but the increase they found ismuch

larger than the increase found here between the dry and

moist isentropic EP fluxes. This quantitative discrepancy

likely arises from the inaccuracy of the TEM formulation

near the surface, which overestimates the value of the

residual streamfunction and EP flux (Held and Schneider

1999; Tanaka et al. 2004).

The difference between the dry and moist EP fluxes can

be traced back to the moist branch of the circulation

(PCK08; PCK10), which consists of a low-level poleward

flow ofmoist air that rises in the storm track through latent

heat release. The moist branch is associated with an ad-

ditional momentum exchange and energy transport that is

FIG. 9. The 20 Jan 2000 daily average of (top) geopotential height at 500 hPa over the NH extratropics, (middle)

potential temperature at 458N, and (bottom) geopotential height anomaly at 458N. The latitude circle at 458N is

delineated in (top) by the dotted horizontal line. In (middle) and (bottom), the 280-K contour is highlighted in blue.

The contour intervals are 0.1 km, 5K, and 50m in (top), (middle), and (bottom), respectively.


not captured on dry isentropes. The low-level moist air

flows poleward through regions of strong eastward pres-

sure gradients associated with midlatitude eddies. Its high

angular momentum is thereby transferred through form

drag to isentropic layers with lower values of ue but com-

parable values of u. Themoist branch is also closely tied to

the poleward latent heat transport. It can be shown that the

moist isentropic circulation accounts for the total moist

static energy transport, whereas the dry isentropic circu-

lation only reflects the total dry static energy transport

(Döös andNilsson 2011; Yamada andPauluis 2015). In the

midlatitudes, the moist static energy transport is around

double the dry static energy transport, and the difference

between the two is given by the latent heat transport

(Peixoto and Oort 1992; Trenberth and Stepaniak 2003).

Moist isentropes better partition the poleward and

equatorward flows in the midlatitudes but have the draw-

back of losing height information as a result of their non-

invertibility.Despite the increase in the vertical component

of the EP flux on moist isentropes, there is no corre-

sponding increase in the horizontal component of the EP

flux (Fig. 5), as one might expect based on baroclinic life

cycle studies (e.g., Edmon et al. 1980). Since moist isen-

tropic layers can extend from the surface to the upper

troposphere, the additionalmomentumexchange observed

betweenmoist isentropic layersmay be the result of lateral,

rather than downward, momentum exchange.

Another open question for future research is finding a

relationship between the moist EP flux presented here

and a moist PV. In the dry TEM framework, the EP flux

divergence is equal to the quasigeostrophic eddy PV flux

(e.g., Edmon et al. 1980). This characterizes the eddy

forcing of the mean flow in terms of mixing, since PV is

materially conserved and the flux is downgradient in the

time average. Tung (1986) derived a nongeostrophic

relationship between an isentropic EP flux divergence

and the eddy flux of Ertel PV. This particular relation-

ship is not formulated in terms of mass-weighted aver-

ages and, therefore, does not have a generalization

within the framework presented here. Furthermore,

formulating a moist PV brings its own challenges. A

simple moist generalization of the Ertel PV by replacing

uwith ue does not carry over the fundamental properties

from dry theory; namely, (i) this quantity is not mate-

rially conserved even for a moist adiabatic atmosphere

and (ii) it does not have an invertibility principle

(Schubert et al. 2001).

Czaja and Marshall (2006), PCK08, and PCK10 first

used moist coordinates to study the global meridional

mass and energy transports. In this study,moist isentropic

averaging was applied to the full momentum budget,

which led to the formulation of amoist EP flux. Theweak

formulation presented here is not limited to the mo-

mentum equation, moist isentropes, or to the global cir-

culation. The tracer (10d) allows for other large-scale

tracer budgets to be computed in general vertical co-

ordinates. Schneider et al. (2006) studied the hydrological

cycle by computing zonal-mean water vapor fields and

fluxes on dry isentropes. Likewise, the weak formulation

could be used to compute the moisture budget on dry or

moist isentropes directly from data output in pressure

coordinates. A similar approach to moist isentropic av-

eraging has also found useful application in moist con-

vection (Pauluis and Mrowiec 2013; Mrowiec et al. 2015)

and hurricanes (Mrowiec et al. 2016).

Acknowledgments. We thank three anonymous re-

viewers for their thoughtful comments on an earlier

version of this manuscript. This work was supported by

the National Science Foundation under Grant AGS-

0944058 and the NYUAbuDhabi Institute under Grant

G1102. The MERRA dataset was provided by NASA’s

Global Modeling and Assimilation Office.

APPENDIX A

Derivations

a. Equivalence of f and ~f when h is invertible

Here we derive (6), which was used in section 2 to

show that f and ~f are equivalent for invertibleh. Assume

h increases monotonically with height from hsfc to ‘,where hsfc(x, y, t) 5 h[x, y, zsfc(x, y), t]. Then,

hf (x, y, z, t)i(y,h0, t)5

›

›h0

(1

L

ðL0

ð‘zsfc(x,y)

f [x, y,h(x, y, z, t), t]H[h02h(x, y, z, t)] dz dx

)

5›

›h0

"1

L

ðL0

ð‘hsfc

f (x, y,h, t)H(h02h)

��›z›h��dhdx

#

51

L

ðL0

f (x, y,h0, t)

��›z›h�� dx

5 f (x, y,h0, t)j›z/›hj

h5h0.


Wemust assume f5 0 below ground for the third equality

to hold [i.e., for those points at whichh0, hsfc, we have f(x,

y, h0, t)5 0, which makes no contribution to the integral].

b. Weak form of zonally averaged tracer equation

Using the continuity (1a), the tracer equation can be

written in flux form:

›

›t(rC)1= � (rCv)5 r _C . (A1)

We then take the Heaviside distribution (5) of (A1).

This amounts to multiplying (A1) by H(h0 2 h) to get

›

›t(rC)H(h

02h)1= � (rCv)H(h

02h)5 r _CH(h

02h)

(A2)

and then integrating this expression. The derivatives can

be taken outside the integral by moving the Heaviside

function inside the derivatives. This can be done by us-

ing the product rule and the following identities:

›

›t[rCH(h

02h)]5

›

›t(rC)H(h

02h)

2 rCd(h02h)( _h2 v � =h) and

= � [rCvH(h02h)]5= � (rCv)H(h

02h)

2 rCd(h02h)v � =h ,

where we have used the fact that the derivative of

the Heaviside function is the delta function and

(1e) in the first identity. And so (A2) can be written

as

›

›t[rCH(h

02h)]1= � [rCvH(h

02h)]

1›

›h0

[rC _hH(h02h)]5 r _CH(h

02h) . (A3)

Integrating (A3) over the domain D, one immediately

obtains ›thrCiH for the first term, ›h0hrpC _hiH for the

third term, and hr _CiH for the rhs. For the integral over

the divergence term,

1

L

ðL0

ð‘0

= � [rCvH(h02h)]H(z2 z

sfc) dz dx5

1

L

ðL0

ð‘0

= � [rCvH(h02h)H(z2 z

sfc)] dz dx

21

L

ðL0

ð‘0

rCH(h02h)d(z2 z

sfc)v � =(z2 z

sfc) dz dx

5›

›y

�1

L

ðL0

ð‘0

rCyH(h02h)H(z2 z

sfc) dz dx

�2

1

L

ðL0

rCH[h02h]v

sfc� n dx .

We first make the integration limits constant by bringing

H(z 2 zsfc) inside the integral. The integral is then re-

written using the product rule as the integral over a di-

vergence term minus an integral along the surface. The

integral over the x- and z-derivative terms in the divergence

are zero from their boundary conditions, and only the

y-derivative term remains. The last line is just ›yhrCyiH ,since the velocity normal to the surface, vsfc � n, is zero.So the integral of (A3) becomes

›

›thrCi

H1

›

›yhrCyi

H1

›

›h0

hrC _hiH5 hr _Ci

H, (A4)

fromwhich the tracer (10d) follows by taking the derivative

with respect toh0. Towrite the expression in terms ofmass-

weighted averages, we use the fact that hrf i5 hrif .c. Showing the equivalence of Mo, Mm, and Me in

(19) and (14)

Here we want to show that the expressions for Mo,

Mm, andMe given in (19) are equal to the three terms in

the pressure gradient decomposition (14), respectively.

We do this by showing that (i) the decomposition (18) is

equal term by term to the decomposition (14) and

(ii) the three terms in (18) are equal to the expressions

for Mo, Mm, and Me given in (19).

As forMo, it follows immediately that the first term in

(18) is equivalent to the first term in (14), since the

contour integral around ›D only consists of the contri-

bution from the bottom boundary because the left and

right boundaries cancel by periodicity and pressure de-

cays to zero at infinity. The bottom boundary can be

parameterized by x, and so (19a) follows from the first

term in (18).

As for Mm, we start by using Green’s theorem on the

second term in (18),

�›h

›x

�5

›

›h0

1

L

ððh,h0

›h

›xdz dx

!5

›

›h0

1

L

þ›(h,h0)

hdz

!.

(A5)


To evaluate this integral, again consider the three general

cases for the shape of the h0 contour: (i) loop, (ii) zonal

ring, and (iii) surface-intersecting contour (Fig. 1). In the

case of the loop, h equals h0 along the boundary contour

›(h, h0) and so (A5) is zero since the contour is closed. In

the case of the zonal ring, the line integrals over the left-

and right-side boundaries cancel out. The top boundary

fh 5 h0g is closed by zonal periodicity and evaluates to

zero as in the case of the loop. The integral along the

surface is independent of h0, so (A5) is again zero.

The only contribution toMm then comes from surface-

intersecting contours. In this case, the left and right

endpoints of the contour, x1 and x2 respectively, depend

on h0. However, the term associated with the change in

the endpoints with respect to h0 appears for both the gatmand gsfc integrals with opposite signs and, therefore, has

no contribution to the total contour integral. The in-

tegrand along gsfc has no h0 dependence. Along gatm,

h equals h0, but since gatm does not form a closed contour

(A5) is in general nonzero. Thus, the only contribution to

the contour integral arises from the gatm segment and is

related to the change in surface height between the

endpoints of the h0 contour. In this case, (A5) becomes

L21fzsfc[x1(h0), y]2 zsfc[x2(h0), y]g, from which the

equivalence of Mm as given in (14) and (18) follows im-

mediately. In the general case, when there are multiple

h0 contours intersecting the surface,Mm is given by (19b).

Since the first two terms in the decompositions (14)

and (18) are equivalent, it follows that their third terms

must also be equal. So Me is given by (19c).

APPENDIX B

Weak Formulation of the Primitive Equations inSpherical–Pressure Coordinates

Equations (1) in spherical–pressure coordinates (e.g.,

chapter 3 of Peixoto and Oort 1992) become

1

a cosf

›u

›l1

1

a cosf

›

›f(cosfy)1

›v

›p5 0, (B1a)

du

dt5

tanf

auy2

1

a cosf

›F

›l1 f y1D , (B1b)

2r›F

›p5 r

pg5 1, (B1c)

dh

dt5 _h, and (B1d)

dC

dt5 _C , (B1e)

whereF5 gz is the geopotential; rp5 g21 is the density

in pressure coordinates; the components of velocity are

u 5 a cosfdl/dt, y 5 adf/dt, and v 5 dp/dt; and the

advective derivative is given by

d

dt5

›

›t1

u

a cosf

›

›l1

y

a

›

›f1v

›

›p.

The Heaviside and delta distribution operators are de-

fined as

hfiH(f,h

0, t)5

1

2p

ð2p0

ðpsfc(l,f,t)0

f (l,f, p, t)H(h02h) dp dl

(B2)

and

hf i(f,h0, t)5

›

›h0

hf iH, (B3)

where the surface pressure psfc depends on time. The

units of hfi are the units of f multiplied by pascals per

units of h. The mass-weighted average is defined as

f 5 hrpf i/hrpi.The weak form of the equations can be derived

analogously to the Cartesian case. The equations are

summarized as follows:

›

›thr

pi1 1

a cosf

›

›f(hr

piy cosf)1 ›

›h0

(hrpib_h)5 0,

(B4a)

›

›t(hr

piu)1 1

a cosf

›

›f(hr

piuy cosf)1 ›

›h0

(hrpiub_h)

5= � F1 hrpi tanf

acuy1 f hr

piy1 hr

piD ,

(B4b)�2r

›F

›p

�5 h1i, and (B4c)

›

›t(hr

piC)1 1

a cosf

›

›f(hr

picCy cosf)

1›

›h0

(hrpicC _h)5 hr

pib_C , (B4d)

where the generalized EP flux vector F is defined as

F5 (Ff,F

h0)5

�2hr

pidu*y*,2hr

pi du* _h*2 1

a cosf

�›z

›l

�H

�(B5)

and

= � F51

a cosf

›

›f(F

fcosf)1

›

›h0

Fh0. (B6)


In the momentum equation, the geopotential gradient

term can be decomposed into orographic and mean and

eddy atmospheric form drags. The derivation is similar

to that in section 5:

21

a cosf

�›z0

›l

�H

5M(p)o 1M(p)

m 1M(p)e , (B7)

where

M(p)o 5

1

L

ðL0

z0sfcH(h02h

sfc)›p

sfc

›xdx , (B8a)

M(p)m 52bz0� 1

a cosf

›h

›l

�5 bz0 1

2pa cosf�n

i51

psfc[l

2i21(h

0),f, t]

2 psfc[l

2i(h

0),f, t], and (B8b)

M(p)e 52hr

pibz0* 1

rpa cosf

›h

›l

!*. (B8c)

The points l2i21 (l2i) represent the left and right end-

points of the h0 contour, which intersects topography.

The height deviation z0 5 z2 z is used instead of z to

remove large cancellations betweenM(p)o andM(p)

m in the

form drag decomposition, but the use of z0 does not af-fect the EP flux.

If we let h�i(z)H denote the Heaviside distribution

(B2) in spherical–height coordinates (l, f, z), then it is

easy to show that hrpf (l,f,p, t)iH 5 hrf (l,f, z, t)i(z)H .

Similarly, h›z/›liH 5 h›p/›li(z)H , and so the weak form

of the equations does not depend on whether we start

with the primitive equations in height or pressure

coordinates. However, the individual terms in the

form drag decomposition can differ depending on the

choice of coordinate. Going back to (14) and using

integration by parts, this difference can be attributed

to a boundary term:ðg

p dz5pzj10 2ðg

z dp ,

where pzj10 denotes the difference in pz between the

endpoints of g. If g is closed then this difference vanishes.

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momentum balance and eliassen–palm flux on moist

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