www.sciencemag.org/content/347/6221/540/suppl/DC1

Supplementary Materials for

Constrained work output of the moist atmospheric heat engine in a warming climate

F. Laliberté,* J. Zika, L. Mudryk, P. J. Kushner, J. Kjellsson, K. Döös

*Corresponding author. E-mail: frederic.laliberte@utoronto.ca

Published 30 January 2015, Science 347, 540 (2015) DOI: 10.1126/science.1257103

This PDF file includes:

Materials and Methods Supplementary Text Figs. S1 to S5 References

Materials and Methods

Data

From the CESM 1.02 coupled climate model with the CAM4 dynamical core, we use six hourly

surface pressure ps (time staggered, instantaneous), temperature T (instantaneous), specific wa-

ter vapor qv (instantaneous), zonal velocity u (time averaged) and meridional velocity v (time

averaged) on hybrid-pressure model levels on the output A-grid at 2.5◦×1.9◦ lon-lat. The ve-

locities are first multiplied by the model level thickness ∆p to convert them to mass fluxes and

are then remapped to the cell edges using a first-order conservative remapping (converting the

A-grid to a C-grid).

From the MERRA reanalysis, we use six hourly layer edge pressure pe (time staggered,

instantaneous), temperature T (instantaneous, analysis time), specific water vapor qv (instan-

taneous, analysis time), zonal mass fluxes u∆p (time averaged), meridional mass fluxes v∆p

(time averaged) on hybrid-pressure model levels on the native C-grid at 0.67◦×0.5◦ lon-lat (ob-

tained from datasets MAI3NECHM, MAI6NVANA, MAT3NVCHM).

Using data on a C-grid should allow us to evaluate the continuity equation accurately, but

the remapping step applied to the CESM data (necessary because the output is on an A-grid)

and the data assimilation step in MERRA (33) lead to small errors in continuity. In subsequent

attempts to quantify energy fluxes using the data, these small errors can become significant. In

order to minimize their influence on our results we use a common procedure (34,35) to compute

a barotropic correction to the horizontal winds and therefore ensure continuity. Thanks to this

correction the vertical velocity ω, recovered by integrating the continuity equation, will vanish

both at the surface and at the top of the atmosphere. The barotropic correction is small except

in regions with large ω. For MERRA, the recovered ω is very close to the vertical mass flux

found in dataset MAT3NECHM. Furthermore, with this correction the data analyzed from both

1

CESM and MERRA will conserve mass, thus making it is possible to compare quantitatively

their energy fluxes in thermodynamic diagrams.

In this work, we use the specific water vapor qv to approximate the specific humidity qT .

This approximation ensures that our method is portable to typical model outputs. The moist

entropy s is computed at the center node of the C-grid for both CESM and MERRA using

the moist entropy provided by the TEOS-10 (36), implemented by the pyteos_air package

(available on pypi) and described below in the Supplementary Text section. This formulation

includes the effect of the latent heat of fusion and is close to other formulations found in the

literature (21). The annual-mean and zonal-mean s, qT and p of s, qT and p are computed from

the six hourly data on model levels.

Thermodynamic Transform Method

In the atmosphere, the moist entropy s and the specific humidity qT satisfy ∂ts + v · ∇s = s

and ∂tqT + v · ∇qT = qT , where s and qT are their respective sources and sinks. The pres-

sure velocity ω satisfies ∂tp + v · ∇p = ω and these equations can be binned at each time t

by computing s(s, qT , p, t) = [(∂ts + v · ∇s)δ(s − s′)δ(qT − q′T )δ(p − p′)], qT (s, qT , p, t) =

[(∂tqT + v · ∇qT )δ(s− s′)δ(qT − q′T )δ(p− p′)] and ω(s, qT , p, t) = [(∂tp+v·∇p)δ(s−s′)δ(qT−

q′T )], where [·] =∫ 2π

0

∫ π/2−π/2

∫∞0ρ(·) cosφdzdφdλ is a mass-weighted integral over the whole at-

mosphere with ρ representing the density, δ(·) the Dirac delta function, z the height from the

surface, λ the longitude and φ the latitude. The primed variables s′, q′T and p′ represent the

moist entropy, specific humidity and pressure coordinates in the physical (z, φ, λ) space and the

unprimed variables s, qT and p represent the same variables in the thermodynamic transform

phase space. It is shown in the Supplementary Text section that the thermodynamic transform

preserves the continuity equation in thermodynamic space and that it is invariant if other ther-

modynamic variables are used instead of s, qT and p.

2

At six hours intervals, s(s, qT , p, t), qT (s, qT , p, t) and ω(s, qT , p, t) are obtained using a

technique that requires C-grid data and exact mass conservation (27). We first define a thermo-

dynamic phase-space rectangular grid with indices (i, j, k): (s0, s0 + ∆s, . . . , s0 + i∆s, . . .)×

(qv0, qv0 + ∆qT , . . . , qv0 + j∆qT , . . .) × (p0, p0 + ∆p, . . . , p0 + k∆p, . . .). We initialize s, qT

and ω on this grid to 0. We pick a grid box edge in the Eulerian space that has mass flux m with

thermodynamical states (s−, q−T , p−) upstream and (s+, q+T , p

+) downstream. We then find the

indices (i−, j−, k−) and (i+, j+, k+) of the upstream and downstream states in the phase-space

so that i−∆s ≤ s−−so < (i−+1)∆s and similarly for the other indices. We compute the num-

ber of phase-space grid boxes to be crossed for each coordinate: ∆i = i+ − i−, and similarly

for ∆j,∆k. Then, we start recording the mass flux in phase-space. We begin by finding the

maximum among |∆i|, |∆j|, |∆k|. Assuming that it is |∆i|, we would add m to s(s, qT , p, t) at

(i− + sgn(∆i)/2, j−, k−) and find the maximum among |∆i| − 1, |∆j|, |∆k|. Assuming that it

is |∆j|, we would add m to qT (s, qT , p, t) at (i− + sgn(∆i), j− + sgn(∆j)/2, k−). We repeat

this procedure |∆i| + |∆j| + |∆k| times in total and for all grid box edges in Eulerian space.

In order to numerically estimate the time-derivative, a similar procedure is used except that the

mass flux is replaced by the grid box mass divided by a time interval of six hours. The upstream

variables are taken as the current Eulerian grid box thermodynamic state and the downstream

variables are taken as the thermodynamic state six hours later.

We calculate monthly mean values of s(s, qT , p, t) from 6-hourly data, apply a twelve month

running mean and project the result onto the (s,T) plane to obtain s(s, T ). We use the same

procedure to obtain qT (qT , µ) from qT (s, qT , p, t).

3

Supplementary Text

Moist thermodynamicsAbove the freezing point

When temperatures are above the freezing point, we use the equation of state defined by the wet

air Gibbs function gAW (eq. 7.1, 19). The wet air entropy sAW and wet air specific volume vAW

are obtained through the appropriate derivatives of gAW (eq. 7.10-11, 19). The dry air content

in wet air wA is defined as the ratio of dry air mass to wet air mass (dry air plus water vapor

plus liquid water), and the dry air content in humid air A is defined as the ratio of dry air mass

to humid air mass (dry air plus water vapor). The wet air enthalpy hAW can then be computed

using a Legendre Transform (eq. 7.13, 19).

The first law of thermodynamics for wet air with pressure P and temperature T then reads

dhAW = µAWdwA +wA

A

(µAV − µAW

)dA+ TdsAW + vAWdP.

The chemical potential µAW is obtained by taking the wA-derivative of gAW keeping A, T and

P fixed (eq. 7.4], 19)

µAW =gAV − gW

A= µAV +

µV − gWA

,

where gAV is the Gibbs function of humid air (eq. 5.13, 19), gW is the Gibbs function of liquid

water and A is the ratio of dry air mass to humid air mass (dry air plus water vapor). The

chemical potential µAV is equal to the difference between the relative chemical potentials of dry

air µA and water vapor µV (eq. 5.14-15, 19). We can then rewrite the first law as

dhAW = µAdwA − gWdwA + (µV − gW)d(wA(1/A− 1)) + TdsAW + vAWdP. (1)

The water vapor content in wet air is given by wA(1/A − 1). In the absence of change in

chemical composition (dwA = 0 and d(wA(1/A − 1)) = 0) this equation reduces to the first

4

law of thermodynamics for a closed system, for which every term carries a well-established

physical meaning.

To establish some physical insight we look at the contribution that each of the remaining

terms make to the steam cycle per unit dry air (10). We first convert the extensive quantities

per mass of wet air in eq. (1) to extensive quantities per mass of dry air. The first law becomes

d(hAW/wA) = gWd(1/wA) + (µV − gW)d(1/A) + Td(sAW/wA) + (vAW/wA)dP. (2)

The steam cycle is a Carnot cycle where heat exchanges with the reservoirs are entirely mediated

by moisture exchanges:

1. The first step is to add a small amount of liquid water to the air parcel isothermally at input

temperature Tin and isobarically at input pressure Pin. This increases 1/wA by ∆(1/wA)

and leaves A unchanged, leading to a change in enthalpy ∆(hAW/wA) = gW∆(1/wA) +

TinsWin ∆(1/wA), where sW is the entropy of liquid water (eq. 3.4 with SA = 0, 19).

2. Since the air parcel is unsaturated, the added liquid water will evaporate and expand at

constant hAW and constant 1/wA with 1/A increasing by ∆(1/wA). This process is irre-

versible because it is physically impossible to convert the water vapor back to liquid water

without bringing the air parcel to saturation first. Adding water vapor to unsaturated air

therefore results in the production of irreversible entropy (Section 4, 10). This can be seen

from the first law, 0 = (µV−gW)in∆(1/wA)+Tin∆(sAW/wA)evap+(vAWin /wA)(∆P )evap,

by noting that below saturation (µV − gW)in < 0, making the first term of the right hand

side negative. This term thus quantifies the moistening inefficiencies related to the addi-

tion of water vapor to unsaturated air, and when divided by Tin it quantifies how much

entropy is generated by these inefficiencies. As a rule of thumb, the lower the relative

humidity, the more negative (µV − gW)in is.

5

The air parcel is then cooled along a reversible adiabat until it becomes saturated and all

the water vapor that was evaporated has condensed back into liquid water.

3. This liquid water is then removed isothermally and the air parcel compressed so that

the decrease in enthalpy equals the enthalpy input in Step 1, and the decrease in en-

tropy equals the increase in entropy from the two previous steps. The first law then

implies −∆hAW = −gWout∆(1/wA) − Tout∆(sAW/wA) + (vAWout /wA)(∆P )out, where

∆(sAW/wA) = sWin ∆(1/wA) + ∆(sAW/wA)evap. Once again, the first term on the right

hand side quantifies the energy difference between dry air and liquid water that is not

accounted for by the change in entropy.

4. Finally, the air parcel is brought back adiabatically and at constant chemical composition

to the input temperature Tin and input pressure Pin, thus closing the steam cycle.

Adding the changes in enthalpy from the first three steps yields

W/wA = (gWin − gWout)∆(1/wA) + (µV − gW)in∆(1/wA) + (Tin − Tout)∆(sAW/wA),

where W/wA = −(vAWin (∆P )evap + vAWout (∆P )out)/wA is the work per unit of dry air produced

by the cycle. The first term on the right hand side is generally positive and it quantifies the in-

creased enthalpy transport enabled by the water flux between the input and output temperatures

(Appendix B, 10). The second term quantifies the work reduction from the irreversible entropy

production associated with the addition of water vapor into unsaturated air, as mentioned in the

description of the cycle.

The work produced by a steam cycle per unit wet air based on eq. (1) for an input of−∆wA

of liquid water at the input temperature and pressure is

W = (gWin − gWout)∆wA + (µV − gW)in∆wA − (µAin − µAout)∆wA + (Tin − Tout)∆sAW, (3)

6

where W = −(vAWin (∆P )evap + vAWout (∆P )out) is the work per unit of wet air produced by the

cycle and ∆sAW = (sAV − sWin )(∆wA/A) + ∆sAWevap is the change of entropy per unit of wet air.

In this case, the first and second terms on the right hand side have the same meaning as in the

steam cycle per unit dry air whereas the third term quantifies the enthalpy transport associated

with the dry air flux between the input and output temperatures that is required to keep the wet

air mass constant along the cycle.

At or below the freezing point

Below the freezing point, the previous discussion holds by replacing the thermodynamic func-

tions for water by their equivalent for ice. At the freezing point, water vapor, liquid water and

ice can co-exist and a special treatment is necessary (19). This added complexity does not affect

the conclusions of the above discussion.

Implementation

The TEOS-10 (36) implementation of moist thermodynamics used here does not allow us to set

wA and A independently. Below saturation, wA must be equal to A, and the first law reduces to

dhAV =((µA − gC)− (µV − gC)

)dA+ TdsAV + vAVdP.

where hAV, sAV and vAV are the enthalpy, entropy (eq. 5.10-11, 19) and specific volume of

humid air, respectively. Here gC is the Gibbs function of the condensate. It is equal to gW above

the freezing point and to gI the Gibbs function of ice, below. Above saturation, A must equal

the saturation value AsatAW(T, P ) obtained from the equilibrium equation µV − gC = 0 (eq. 7.5,

19) and the first law reduces to (eq. 7.15, 19)

dhAW =(µA − gC − (µV − gW)

)dwA + TdsAW + vAWdP.

Note that in both saturation cases the first law of thermodynamics can be written as

dh = µd(1− qT ) + Tds+ αdp, (4)

7

where h = hAV or hAW, s = sAV or sAW and α = vAV or vAW below or above saturation,

respectively. Here, qT represents the specific humidity, defined as the ratio of the mass of

moisture (vapor plus condensate) to the mass of wet air (dry air plus water vapor or dry air

plus water vapor plus condensate). The chemical potential µ equals µA − gC − (µV − gW) in

both saturation cases. As discussed in our presentation of the steam cycle per unit wet air, the

term −(µV − gW)d(1 − qT )/T quantifies the irreversible entropy production associated with

the addition of water vapor to unsaturated air while the term (µA − gC)d(1− qT ) quantifies the

enthalpy changes associated with the changing air composition (replacing dry air by moisture

and vice versa).

Theory for Thermodynamic Transform method

The transformed quantities s(s, qT , p, t) and qT (s, qT , p, t) represent the sources / sinks of moist

entropy and specific water vapor while ω(s, q, p, t) represents the pressure change in a purely

thermodynamic space. Assuming that the climate is steady, when averaged over a sufficiently

long time, we will show that the phase space divergence of these three quantities vanishes :

limτ→∞

1

τ

∫ τ

0

(∂ss(s, qT , p, t) + ∂qT qT (s, qT , p, t) + ∂pω(s, qT , p, t))dt = 0.

Moreover, this property does not depend on the choice of (s, qT , p) as our three thermodynamic

coordinates and holds for any change of variable to another thermodynamic state variable. For

example, if we change variables from pressure p to temperature T the following quantities will

all be non-divergent under the same condition:

s(s, qT , T, t) =

∫ ∞0

s(s, qT , p, t)δ(T − T (s, qT , p))dp,

qT (s, qT , T, t) =

∫ ∞0

qT (s, qT , p, t)δ(T − T (s, qT , p))dp,

T (s, qT , T, t) =

∫ ∞0

(∂sT s(s, qT , p, t)+∂qTT qT (s, qT , p, t)+∂pTω(s, qT , p, t))δ(T−T (s, qT , p))dp.

8

The same will hold if T is replaced by µ in the previous equations.

We obtain s(s, T ) and qT (qT , µ) using the following projection:

s(s, T ) =

∫ ∞0

∫ ∞0

s(s, qT , p, t)δ(T − T (s, qT , p))dpdqT ,

qT (qT , µ) =

∫ ∞−∞

∫ ∞0

qT (s, qT , p, t)δ(µ− µ(s, qT , p))dpds,

where (·) represents an annual mean.

To prove those assertions, we transform the mass of the atmosphere M to our thermody-

namical space:

M(s, qT , p, t) = [δ(s− s′))δ(qT − q′T )δ(p− p′)],

where the operator [·] represents a mass-weighted integral over the whole atmosphere and where

primed quantities are functions of spatial and temporal co-ordinates. Taking the phase-space

divergence of s(s, qT , p, t), qT (s, qT , p, t) and ω(s, qT , p, t), we obtain:

∂ss+ ∂qT qT + ∂pω =

[dδ(s− s′)

dtδ(qT − q′T )δ(p− p′)

]+ . . .[

δ(s− s′)dδ(qT − q′T )

dtδ(p− p′)

]+ . . .[

δ(s− s′)δ(qT − q′T )dδ(p− p′)

dt

]= . . .[

d

dt(δ(s− s′)δ(qT − q′T )δ(p− p′))

]= ∂tM(s, qT , p, t). (5)

This equation averages to 0 over a sufficiently long time if the climate is steady in the sense that

M(s, qT , p, t) can have at most a sub linear trend in time, thus proving the first assertion.

To prove the second assertion, we define

M(s, qT , T, t) =

∫ ∞0

M(s, qT , p, t)δ(T − T (s, qT , p))dp,

and we observe that

∂T T (s, qT , T, t) = −∂ss(s, qT , T, t)− ∂qT qT (s, qT , T, t) + ∂tM(s, qT , T, t),

9

by integration by parts. The transformation therefore leads to a non-divergent thermodynamic

transform under the same conditions.

10

Fig. S1Supplementary Figures!

� � � � !! ! ! ! � ! � ! ! !!!

gC � µV

A B

Same as Fig. 1 C, D but using gC − µV instead of µ. This diagram depicts only the moistening

inefficiencies related to the addition of water vapor to unsaturated air and the associated produc-

tion of irreversible entropy. At a fixed pressure level and fixed qT , larger gC−µV correspond to

lower relative humidity.

11

Fig. S2

� !Figure 3 with the dry air flux removed from Q_moist!!!!

A

B

C

D

As in Fig. 2, with Qirrmoist obtained using the cycles in Fig. S1.

12

Fig. S3!

� � �

� � !

� � !!Work diagrams!!!!!

A B

DC

Thermodynamic diagrams for years 1981-2012 of CESM (A, B) and MERRA (C, D). (A,C):

α-p diagram. Ψwork =∫∞αω(p, α′)dα′ in colour shading and grey contours (-125, -255, -375

Sv). Axes are oriented so that the lower left corner is closest to typical tropical surface α-p

values and the upper left corner is closest to typical tropical upper tropospheric α-p values.

For plotting purposes, instead of using α we are using the virtual potential temperature θv =

α(p/Rd)(pref/p)Rd/cp , with pref = 1000 hPa, Rd = 287.06 Jkg−1K−1 and cp = 1003 Jkg−1K−1.

Thin black curves indicate where Ψwork = −1Sv. They indicate the small-magnitude cutoff over

which W was computed to avoid float-point errors. (B,D) the contribution to W per pressure

level, computed from A, C by the integral∫∞0

Ψworkdα.

13

Fig. S4!

� � � !

� � !!!Change in work diagrams!!!!!

A B

(A): Response of Ψwork for CESM (2076-2098 minus 1981-2012, colour shading). Grey con-

tours indicate the 1981-2012 streamlines (Fig. S2A). (B): Corresponding response of∫∞0

Ψworkdα.

14

Fig. S5

� !!Figure 4 with the dry air flux removed from Q_moist!!!

A

B

As in Fig. 4, with Qirrmoist as in Fig. S3 and Wmod = Qtotal − Qirr

moist.

15

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