The advection-condensation model and water vapour PDFspordlabs.ucsd.edu/wryoung/reprintPDFs/WaterVapourPDF1.pdf · 2017-04-15 · The advection-condensation model and water vapour

Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–13 (2010)

The advection-condensation model and water vapourPDFs

∗Jai Sukhatmea William R. Youngb

a Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore 560012, Indiab Scripps Institute of Oceanography, University of California at San Diego, San Diego, CA 92093-0213, USA.

∗Correspondence to: [email protected]

The statistically steady humidity distribution resulting from aninteraction of advection, modeled as an uncorrelated randomwalk of moist parcels on an isentropic surface, and a vapoursink, modeled as immediate condensation whenever the specifichumidity exceeds a specified saturation humidity, is explored withtheory and simulation. A source supplies moisture at the deep-tropical southern boundary of the domain, and the saturationhumidity is specified as a monotonically decreasing functionof distance from the boundary. The boundary source balancesthe interior condensation sink, so that a stationary spatiallyinhomogeneous humidity distribution emerges. An exact solutionof the Fokker-Planck equation delivers a simple expression forthe resulting probability density function (PDF) of the watervapour field and also of the relative humidity. This solutionagrees completely with a numerical simulation of the process,and the humidity PDF exhibits several features of interest, suchas bimodality close to the source and unimodality further fromthe source. The PDFs of specific and relative humidity are broadand non-Gaussian. The domain averaged relative humidity PDFis bimodal with distinct moist and dry peaks, a feature which weshow agrees with middleworld isentropic PDFs derived from theERA interim dataset.Copyright c© 2010 Royal Meteorological Society

Key Words: advection, diffusion, condensation, probability density function, relative humidity,

specific humidity, water vapor, bimodal

Received . . .

Citation: . . .

1. Introduction

Water vapour plays a significant role in diverseproblems pertaining to the dynamics of the Earth’sclimate (Pierrehumbert 2002; Schneider, O’Gormanand Levine 2010). In particular, with regards toits radiative effects, estimating the amount of watervapour in the upper troposphere is crucial (Held andSoden 2000; Spencer and Braswell 1997). Significantefforts — especially directed towards the subtropical

troposphere — have yielded a key insight intothe nonlocal nature of processes that control thedistribution of water vapour (Yang and Pierrehumbert1994; Sherwood 1996; Salathe and Hartman 1997;Pierrehumbert 1998; Pierrehumbert and Roca 1998;Galewsky, Sobel and Held 2005; Brogniez, Roca andPicon 2009).

The resultant framework for understanding theevolution of water vapour, or generally any condensable

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2 J. Sukhatme and W. R. Young

substance in a fluid dynamical setting, is known as theadvection-condensation (AC) model. The recent reviewby Sherwood et. al (2009) provides an introduction,while the book chapter by Pierrehumbert, Brogniezand Roca (2005) (hereafter, PBR) provides a thoroughoverview of the AC model.

Further, as elaborated on by Spencer and Braswell(1997) and PBR, in addition to the amount, theprobability density function (PDF) of the water vapourfield has a strong influence on the longwave coolingof the atmosphere. Explicit calculations illustratingthe effect of the PDF on the outgoing longwaveradiation (OLR) are given by Zhang, Soden and Mapes(2003). The importance of the PDF stems from theapproximately logarithmic dependence of the changein OLR to the specific humidity of the water vapourin the domain. Further discussion of this logarithmicdependence which results from a spectral broadeningof absorption peaks can be found in the text byPierrehumbert (2010).

Following Soden and Bretherton (1993), therehave been numerous efforts that examine water vapourPDFs in the tropical and subtropical troposphere. Aprincipal feature noted by these studies is the non-normality of the PDFs. A range of distributions,from bimodality in the deep tropics (Brown andZhang 1997; Zhang, Soden and Mapes 2003; Moteand Frey 2006; Luo, Kley and Johnson 2007), tounimodal PDFs with a roughly lognormal or power-lawform in the subtropics have been documented (Sodenand Bretherton 1993; Sherwood, Kursinski and Read2006). Ryoo, Igusa and Waugh (2009) have emphasizedthat a salient and baffling feature of water vapourPDFs is their spatial inhomogeneity. Models basedon the statistics of subsidence drying and random re-moistening of parcels reproduce many of the observedfeatures of the humidity PDFs, provided that oneconsiders the model parameters to be functions ofposition determined to match observations (Sherwood,Kursinski and Read 2006; Ryoo, Igusa and Waugh2009).

One of the main achievements of this paper isan exact solution for the water vapour PDF thatcharacterizes the statistically steady state emergingfrom an interaction of advection, condensation anda spatially localized moisture source. In particular,we consider moist parcels advected by a white-noisestochastic velocity field. Each parcel carries a specifichumidity and experiences condensation whenever andwherever the specific humidity exceeds a specifiedsaturation profile. The parcel humidity is reset tosaturation by a moist source at the southern boundaryof the domain. In the ultimate statistically steady state,the southern vapour source is balanced by condensationthroughout the interior of the domain. In contrast toearlier considerations of the AC model with white-noiseadvection (O’Gorman and Schneider 2006), we considerstatistically steady states, rather than the initial valueproblem, and we obtain the full water-vapour PDFrather than the mean humidity.

The AC model is described in more detail insections 2 and 3, and in section 4 we solve the Fokker-Planck equation to obtain the PDF of the specific

humidity. The utility of the PDF is demonstrated bycalculating the average moisture flux and condensation.In section 5 we focus on a particular saturationprofile — the standard exponential model — andexhibit analytical solutions for the specific and relativehumidity PDFs; these are shown to be in very goodagreement with PDFs estimated from a Monte Carlosimulation. In section 6 we analyse daily isentropicrelative humidity data from the ERA interim product,and show that the PDFs derived from this datasetagree with principal features predicted by our idealizedmodel. A discussion of the PDFs, with implications forthe atmospheric radiation budget and the present-daydistribution of water vapour, followed by a conclusionsection 7 ends the paper.

2. The AC model

The central quantity in the AC model is the specifichumidity, defined by

q(x, t) def=mass of water vapour in a parcel

total mass of the parcel, (1)

where x = (x, y) is position on an isentropic surface.Our main focus here is the extratropical vapourdistribution, which is determined in large part by thewandering of air parcels on isentropic surfaces andthe temperature changes associated with latitudinalexcursions.

Following PBR, the AC model is:

∂tq + u·∇q = S − C . (2)

Above, S is the vapour source, C is the condensationsink and u(x, t) is velocity on an isentropic surface. Inthe absence of sources and sinks, Dq/Dt = 0, i.e. thespecific humidity is a materially conserved quantity.The AC formalism does not consider the diffusivehomogenization of water vapour e.g., via the additionof χ∇2q on the right of (2). This limitation of the modelis discussed further in section 6.

Latent heat release and radiative cooling result insignificant cross-isentropic motion in the troposphere,and we have ignored this process in formulating a two-dimensional isentropic model in (2). We emphasize thatthe AC formulation is not inherently limited to two-dimensions or to motion on isentropic surfaces. But forease of exposition, and simplicity, isentropic motion isa useful starting point.

The condensation sink on the right of (2) mightbe modeled with

C = τ−1c [q(x, t)− qs(x)] H [q(x, t)− qs(x)] , (3)

where τc is the timescale associated with condensation,qs(x) is the prescribed saturation specific humidity;H(x) denotes the Heaviside step function, which is zeroif the x < 0 and one if x > 0. Observed atmosphericsupersaturations rarely exceed a few percent andtherefore the sink C is extremely effective (Wallaceand Hobbs 1977). In other words, we are concerned

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Advection-Condensation 3

with the rapid-condensation limit τc → 0. In the rapid-condensation limit, the details in C are unimportantand condensation can instead be represented as a ruleenforcing subsaturation:

q(x, t)→ min [q(x, t), qs(x)] . (4)

Therefore, q(x, t) ≤ qs(x) and there is an upper boundon the specific humidity at every x. Globally, themaximum tolerable specific humidity is

qmaxdef= max

∀xqs(x) . (5)

If the flow is ergodic then a wandering parcelexplores the entire domain and eventually encountersthe dryest regions, with the smallest value of qs(x).Thus in the absence of sources (Sukhatme andPierrehumbert 2006)

limt→∞

q(x, t) = qmin , (6)

whereqmin

def= min∀x

qs(x) . (7)

On the other hand, without advection, the source willeventually saturate q(x, t) at every point x where S 6=0. Therefore, interesting statistically steady states areexpected only if there is both a source S and advectionu(x, t).

The problem is to determine the evolution andthe statistics of the specific humidity q(x, t), given thevelocity field u(x, t) and the saturation profile qs(x).Due to its physical significance, the principal statisticalfeature we focus here on is the probability densityfunction (PDF) of q. Quantities such as the meanmoisture flux and the mean condensation rate are alsocrucial, and have been examined in the AC framework(O’Gorman and Schneider 2006; O’Gorman, Lamquinand Schneider 2011). These mean quantities can bededuced from PDF of q via integration over q.

3. Moist brownian parcels

The simplest representation of random transport isobtained by taking the velocity u(x, t) in (2) as havingrapid temporal decorrelation so that the motion of anair parcel is brownian. Then the spatial evolution of anensemble of parcels, each carrying its own individualvalue of q, is described by the diffusion equation. Wealso assume that qs depends only on the meridionalcoordinate y in the domain 0 < y < L, and that qsdecreases monotonically from qmax at y = 0 to qmin

at y = L. Each parcel is located by a point in theconfiguration space (q, y). Because of the condensationrule (4), the ensemble of parcels is confined to theshaded region of configuration space illustrated inFigure 1.

The AC model is then expressed as a set of coupledstochastic differential equations governing the positionY (t) and specific humidity Q(t) of a moist brownian

Figure 1. In the rapid-condensation limit the accessible domainin the (q, y) plane is the shaded region shown above. The modelassumes that qs(y) decreases monotonically from qmax = qs(0) toqmin = qs(L). The functions qs(y) and ys(q) are inverses of eachother: qs(ys(q)) = q.

parcel

dY (t) =√

2κb dW (t),dQ(t) = {S [Y (t)]− C [Y (t), Q(t)]} dt (8)

where W (t) is a Weiner process, κb is the browniandiffusivity associated with the random walk, and Sand C are the source and condensation functions in (2).The numerical simulations described below amount toEuler-Maruyama discretization of (8). The midlatitudebaroclinic eddies responsible for transport have non-trivial spatio-temporal correlations (Sukhatme 2005;O’Gorman and Schneider 2006). Thus the white noiseadvection used in (8) and in earlier AC studies is anidealization.

The statistical properties of the ensemble of moistbrownian parcels are characterized by a PDF P (q, y, t),such that the expected number of parcels in dq dy isequal to

N P (q, y, t) dq dy , (9)

where N is total number of parcels.

The evolution of P (q, y, t) is governed by theFokker-Planck equation

∂tP + ∂q[(S − C)P ] = κb∂2yP , (10)

with the normalization,∫ L

0

∫ qmax

qmin

P (q, y, t) dq dy = 1 . (11)

In the limit of rapid condensation, as qmin ≤q(y, t) ≤ qs(y), the normalization takes the form∫ L

0

∫ qs(y)

qmin

P (q, y, t) dq dy = 1 . (12)

Integrating P (q, y, t) over q produces the marginaldensity

p(y, t) def=∫ qs(y)

qmin

P (q, y, t) dq . (13)




Because the brownian motion of a parcel is independentof the specific humidity it carries, we expect thatintegration of (10) over q will show that p(y, t)satisfies the diffusion equation, and indeed this resultis immediate

∂tp = κb∂2yp . (14)

If parcels are reflected back into the domain at y = 0and L then the solution of (14) is limt→∞ p(y, t) = L−1.Thus if we seek long-time equilibrium solutions of theFokker-Planck equation (10), then we require

L−1 =∫ qs(y)

qmin

P (q, y) dq . (15)

Note that the left of (15) is consistent with the globalnormalization in (11). In the limit τc → 0, the steady,marginal normalization condition (15) applies at everyy, and for all models of S and C. In other words, in astatistical steady state the normalization condition (15)is equivalent to enforcing the rapid-condensation rulein (5). The normalization (15) figures prominently inthe following solution of the Fokker-Planck equation.Similar “variable limit” normalization conditions areencountered in the solution of integrate-and-fire neuronmodels (Fusi and Mattia 1998).

4. Steady solution with “resetting” at y = 0

Now we adopt a “resetting” model for the vapoursource S: on encountering the southern boundary aty = 0, parcels are reflected back into the domain witha new value of q chosen from a specified probabilitydensity function Φ(q), with the normalization∫ qmax

qmin

Φ(q) dq = 1 . (16)

For example, completely saturating the parcels aty = 0 corresponds to the choice Φ(q) = δ(q − qmax).This is a simple representation of tropical moistening.Resetting boundary conditions have been employed ina previous study of AC on isentropic surfaces (Yangand Pierrehumbert 1994), as well as in advection-diffusion of passive scalars (Pierrehumbert 1994;Neufeld, Haynes and Picard 2000).

We emphasize that resetting the humidity only aty = 0 is a strong model assumption that distinguishesthis work from the AC models developed by Sherwood,Kursinski and Read (2006) and Ryoo, Igusa and Waugh(2009). The statistical model developed in those worksassumes that random resetting occurs throughout thedomain as an exponential or gamma stochastic processwith a characteristic time scale τMoist(x). In effect,by resetting only at y = 0, we are considering theextreme case in which τMoist(x) =∞, except at y =0. Physically, the picture we have in mind is thatparcels experience moistening when they encounterconvective regions which are restricted to the tropics,and therefore y is the distance from this saturated zone.In fact, this serves as another limiting case for the ACmodel along with the re-setting throughout the domainscenario considered by Sherwood, Kursinski and Read(2006) and Ryoo, Igusa and Waugh (2009).

An important consequence of resetting only aty = 0 is that the dryest parcels, which are created aty = L with q = qmin, maintain their extreme drynesstill they eventually strike y = 0. The solution belowrequires explicit consideration of these dryest parcelsby inclusion of a component δ(q − qmin), referred to asthe “dry-spike”, in the PDF.

4.1. Solution of the Fokker–Planck equation

In the rapid-condensation limit, the accessible part ofthe (q, y)-space is the shaded area in Figure 1, whereC = 0. But with the resetting boundary condition, thesource S is also zero within the shaded region. Thusthe steady Fokker-Planck equation in the shaded regionof Figure 1 collapses to ∂2

yP = 0, with the immediatesolution

P (q, y) =Φ(q)L

+ yB(q) . (17)

The first term on the right of (17) is determinedby the resetting condition and by application of thenormalization in (15) at y = 0.

Corresponding to the dryest parcels in the domain,B(q) in (17) must contain a component proportional toδ(q − qmin). This “dry spike” contains parcels createdby encounters with the northern boundary at y = L.These considerations refine (17) to

P (q, y) =Φ(q)L

+[δ(q − qmin)

L2+ F (q)

]y . (18)

The term L−2δ(q − qmin) in (18) is obtained byapplying the normalization condition (15) at y = L.

In (18), F (q) is the smooth part of the PDF, whichis determined by substituting into the normalization(15). This requirement leads to

L

y

∫ qmax

qs(y)

Φ(q′) dq′ = 1 + L2

∫ qs(y)

qmin

F (q′) dq′ . (19)

Inspecting (19), we see that it is possible to solve forF (q) by using q, rather than y, as the independentvariable. That is, following the notation in Figure 1,we use y = ys(q) to re-write (19) as

L

ys(q)

∫ qmax

q

Φ(q′) dq′ = 1 + L2

∫ q

qmin

F (q′) dq′ . (20)

The derivative with respect to q then determines F (q),i.e.

LF (q) =d

dq

Λ(q)ys(q)

, (21)

where Λ(q) is the cumulative distribution

Λ(q) def=∫ qmax

q

Φ(q′) dq′ . (22)

Assembling the pieces, we obtain the PDF

P (q, y) = − 1L

dΛdq

+y

L

[δ(q − qmin)

L+

d

dq

Λ(q)ys(q)

].

(23)




The expression above applies only within theshaded region in Figure 1 where q < qs(y); in thesupersaturated region q > qs(y), P (q, y) = 0. The PDFis discontinuous at the saturation boundary q = qs(y).This completes the solution for the steady-state Fokker-Planck equation with resetting forcing at y = 0.

In summary, in the rapid-condensation limit,and with brownian motion, the steady Fokker-Planckequation is solved exactly by (23). The normalizationconstraint in (15), and the use q as the independentvariable, produces this relatively simple solution.In fact, the normalization constraint (15) can alsobe regarded as a differential boundary condition.Manipulating this boundary condition provides analternate route to the solution outlined in AppendixA.

4.2. The global PDF

The global PDF of specific humidity is the marginaldensity

g(q) def=∫ ys(q)

0

P (q, y) dy . (24)

The integral on the right can be evaluated using thesolution in (23), and one finds

g(q) =12δ(q − qmin)− 1

2ddq

(ys(q)Λ(q)) . (25)

A comforting check on (25) is that g(q) satisfies thenormalization ∫ qmax

qmin

g(q) dq = 1 , (26)

which is consistent with (12).

Independent of model details, such as thespecification of qs(y) and Λ(q), the dry spike, δ(q −qmin) in (25), contains exactly half of the parcelsin the ensemble. Prof. P. O’Gorman noted that thisfeature can be understood on the basis of symmetry:exactly half the parcels have more recently encounteredthe moist southern boundary than the dry northernboundary. The former half have q ≥ qmin, while theother half constitute the dry spike with q = qmin.

4.3. Averages

Knowledge of the stationary PDF allows us to evaluatethe average of any function of q, say f(q), as

〈f〉(y) def= L

∫ qs(y)

qmin

f(q)P (q, y) dq . (27)

Using the solution for P (q, y) in (23), one can showthat the average defined above is

〈f〉 = f(qs)Λ(qs)−∫ qs

qmin

f(q)dΛdq

dq − y∫ qs

qmin

dfdq

Λ(q)ys(q)

dq .

(28)Taking the derivative of (28) with respect to y weobtain, after some remarkable cancellations,

d〈f〉dy

= −∫ qs

qmin

dfdq

Λ(q)ys(q)

dq . (29)

4.4. Average humidity and condensation rate

To understand the transport of moisture in this model,and the associated mean condensation rate, we returnto (10) and momentarily retreat from the rapid-condensation limit∗. Then, multiplying (10) by q andintegrating from qmin to qmax, we have

−∫ qmax

qs

q ∂q(CP ) dq = κbd2〈q〉dy2

, (30)

where the condensation sink C is given by the modelin (3). Using this expression for C and integrating theleft of (30) by parts, we obtain the mean condensationrate, defined by

〈C〉 def= τ−1c

∫ qmax

qs

(q − qs)P (q, y) dq , (31)

as

〈C〉 = κbd2〈q〉dy2

. (32)

The right hand side of (32) is the convergence of themoisture flux, which balances the mean condensation〈C〉 on the left.

This argument shows that the flux of moisture hasthe same diffusivity κb as that of a brownian parcel.As explained by O’Gorman and Schneider (2006), thisis a special property of the brownian limit: if thevelocity has a nonzero integral time scale (e.g., as inthe Ornstein-Ulenbeck process) then the diffusivity ofmoisture will be systematically less than the random-walk diffusivity. We emphasize that while the moistureflux is diffusive, even with a Brownian advectingflow, the condensation sink cannot be adequatelyrepresented by a bulk diffusivity. This inadequacy ofa mean-field representation of condensation is clearlydemonstrated in PBR.

In the rapid condensation limit, the right of (32)can be computed by putting f(q) = q in (29). Thus ifτc → 0, the mean condensation rate is

〈C〉 = −κby

dqsdy

Λ (qs(y)) . (33)

Although C ∝ τ−1c →∞, the mean condensation 〈C〉 in

(33) is independent of τc. This singular limit is achievedbecause the nonzero probability of supersaturation isconfined to a boundary layer, lying just above thesaturation boundary q = qs(y) in Figure 1.

If the resetting PDF Φ(q) is non-singular at qmax,then 〈C〉 in (33) is finite as y → 0, despite the factory−1. Specifically, one can show that limy→0 Λ(qs)/y =−Φ(qmax)(dqs/dy).

5. An example

The solution from the previous section is most simplyillustrated by supposing that the resetting produces

∗We continue using resetting forcing, so that S = 0 when y > 0.




complete saturation at y = 0,

Φ(q) = δ(q − qmax) , (34)

so that Λ(q) = 1. For the saturation humidity we usethe exponential model

qs(y) = qmax e−αy . (35)

This form of qs has been employed previously inAC studies (Pierrehumbert, Brogniez and Roca 2005;O’Gorman and Schneider 2006).

With Φ(q) and qs(y) in (34) and (35), the steadysolution of the Fokker-Planck equation is

P (q, 0) =δ(q − qmax)

L, (36)

and away from the southern boundary

P (q, 0 < y) =y

L

[δ(q − qmin)

L+

α

q log2(q/qmax)

].

(37)The solution (37) can be non-dimensionalized with

α∗def= αL , q∗

def= q/qmax , y∗def= y/L , (38)

and P∗(q∗, y∗) = qmaxLP (q, y). This scaling identifiesthe fundamental non-dimensional control parameterα∗, and the scaled PDF is

P∗(q∗, 0 < y∗) = y∗

[δ(q∗ − ε) +

α∗

q∗ log2(q∗)

], (39)

withε

def=qmin

qmax= e−α∗ . (40)

At the southern boundary P∗(q∗, 0) = δ(q∗ − 1). Themain point is that the shape of P∗(q∗, y∗) is controlledby a single parameter α∗.

There is an interesting change in the structure ofP∗ at α∗ = 2: P∗ is bimodal for all y∗ if α∗ < 2; if α∗ > 2then P∗ is bimodal only in the region 0 < y∗ < 2/α∗.This is illustrated in Figure 2 which shows the smoothportion of P∗ with α∗ = 4 and α∗ = 1. The PDF withα∗ = 4 changes from a bimodal to a unimodal form asone crosses y∗ = 2/α∗ = 0.5. Indeed, for y∗ > 0.5 theα∗ = 4 PDF falls monotonically from q∗ = ε. In thecase with α∗ = 1, the PDF increases with increasingq∗ at all values of y∗. Therefore, in combination withδ(q∗ − ε), the PDF is always bimodal if α∗ < 2.

On a middleworld isentrope, the saturation mixingratio usually changes by more than an order ofmagnitude as one spans the entire midlatitudes, hencefrom (35), we expect that α∗ > 2 is the relevantatmospheric case. Therefore, close to the vapour sourcethe PDF is bimodal. One mode, the dry spike δ(q −qmin), corresponds to a singularity at qmin, after whichthe PDF falls off, only to rise again to a secondarymoist peak as q → qs(y). Further from the vapoursource (i.e. for y∗ > 2/α∗) the PDF is unimodal witha dry spike at qmin followed by a gradual monotonicdecay to a minimum at q = qs(y).

We proceed to discuss the implications of thesolution in (37) using dimensional variables. The non-dimensional versions of all formulas are obtained withL→ 1, α→ α∗ and qmin → ε.

Figure 2. The behaviour of the smooth portion of P∗(q∗, y∗)for two choices of α∗; we do not show the dry spike at qmin. Themajor axes correspond to α∗ = 4; there is a transition from abimodal to unimodal PDF at y∗ = 2/α∗ = 0.5. Larger y∗’s havesmaller ranges of permissible q∗, and hence the curves are orderedsuch that y∗ increases as we move to the left. The transitioncurve at y∗ = 2/α∗ = 0.5 is shown by the dark line in the plot.The inset corresponds to α∗ = 1. The curves are ordered as inthe main plot. In this case, counting the dry spike at qmin, thePDF is bimodal at all values of y∗.

5.1. Statistics in a strip y1 < y < y2

The expression in (36) and (37) provides the PDF at aspecified y. From an observational and numerical view,recording the PDF at a particular y can be difficult. Itis more likely that one obtains an estimate of the PDFover a strip, i.e. for y ∈ [y1, y2]. Integrating (37) oversuch a strip, and denoting this by P12(q), we have

P12(q) =∫ y2

y1

P (q, y) dy ,

=[δ(q − qmin)

2L2+

α

2Lq log2(q/qmax)

]s12(q) ,

(41)

where the strip function is

s12(q) =

{y22 − y2

1 , if qmin < q < qs(y2);y2s(q)− y2

1 , if qs(y2) < q < qs(y1).(42)

The slightly complicated form of s12(q) is because whenqs(y2) < q < qs(y1), the upper limit of the integral isys(q) < y2. We compare P12 above with the results ofa Monte Carlo simulation in Figure 3.

The effect of the strip function s12 is to clipthe secondary peak at q = qs, and this clippingbecomes more extreme if y2 − y1 is made larger. Thus,aggregating measurements of q from different spatiallocations will obscure the bimodal structure of P (q, y).

5.2. The global PDF of q

In the extreme case y1 → 0 and y2 → L, we obtain theglobal PDF of the specific humidity, denoted by g(q),




Figure 3. Comparison between P12(q, y) from Monte Carlowith the smooth portion of the analytic result in (41). In thisillustration α∗ = 5. Crosses, open circles and stars are the Monte-Carlo PDFs while the smooth solid curves are the analyticalexpression. Top panel: The bimodal PDFs for y∗ < 2/α∗. Middlepanel: The PDFs for larger y∗ which show the transition from abimodal to a unimodal distribution. Bottom panel: The unimodalPDFs far from the source, i.e. for y∗ > 2/α∗.

in (24). Using (25), the global PDF in this example is

g(q) =12δ(q − qmin) +

12αLq

, (43)

where q ∈ [qmin, qmax]. The smooth part of the globalPDF is g(q) ∼ q−1.

5.3. Monte Carlo Simulations

To test the analytic solution above, we proceed to aMonte Carlo simulation of our system. Typically oursimulations use N = 105 parcels which are released inthe domain y ∈ [0, L]. We have used α = 1, L = 5 whichgives α∗ = 5. This implies a change of approximatelytwo orders of magnitude in the saturation specifichumidity across the domain. The y-coordinate of theparcels are initially uniformly distributed and aretagged with the minimum saturated specific humidity,i.e. on parcel i, qi(0) = qmin. The parcels perform anuncorrelated random walk in y, and parcel i reflectedback into the domain if yi > L or if yi < 0. Theforcing S is implemented by setting qi = qmax whena parcel is reflected at y = 0. Rapid condensation isimplemented by enforcing qi = min[qi, qs(yi)] after eachrandom displacement.

To approximate brownian motion, the root meansquare step length of the random walk, `, should be assmall as computationally feasible — we typically used` = 2× 10−3L. We specify the brownian diffusivityas κb = 0.5, and then the time between steps, τ , iscomputed via

κb =`2

2τ. (44)

The simulation runs till P (q, y) attains a stationaryform and then data is collected in several strips forcomparison with (41).

The results of the simulations are shown inFigure 3. The top panel of this figure shows acomparison between the smooth portion of thetheoretical expression in (41) and the numericallyestimated PDF for y∗ < 2/α∗. There is good agreementbetween the Monte-Carlo simulation and the Fokker-Planck solution. Some minor discrepancies are evidentas q → qmin; this is anticipated as we have suppressedthe numerical spike at q = qmin (indeed, the theoreticalPDF is given by a δ-function at this location). Themiddle panel of Figure 3 shows the PDF at largery, specifically around the region where the PDFtransitions from a bimodal to a unimodal form. Thebottom panel of Figure 3 shows the PDF’s far from thesource, where the PDFs have a single peak at q = qmin,followed by decay and termination at q = qs(y). Finally,the first panel of Figure 4 shows the global PDF g(q)in (43). As per the log-log inset, the global PDF closelyfollows the result in (43), with a spike at q = qmin

followed by an algebraic tail g(q) ∼ q−1.

5.4. Transport and condensation

To understand the water vapour distribution, it isimportant to quantify the transport of moisture. In thecontext of the AC model, it was shown by O’Gormanand Schneider (2006) that brownian motion results inthe mean moisture flux

〈F〉 = −κbd〈q〉dy

, (45)




Figure 4. Top panel: The global PDF of the specific humidityg(q) from Monte Carlo simulations. The inset shows a log-logplot which reveals a spike at qmin followed by an algebraic q−1

tail as predicted by (43). Bottom panel: The analytical curve h(r)from (51) (solid line) and the global RH PDF from the numericalsimulation (dotted circles).

where 〈q〉 is the mean specific humidity and κb is thebrownian diffusivity. Then, as in (32), in a statisticalsteady state the convergence of 〈F〉 balances the meancondensation 〈C〉 at every y.

Using f(q) = q in (29), and qs(y) in (35), theaverage gradient of specific humidity is obtained as

d〈q〉dy

= αqmax [E(αL)− E(αy)] , (46)

where E(x) def=∫∞x

(e−t/t) dt is the exponential integral.The moisture flux 〈F〉 can also be diagnosed fromthe simulation and thus (45) is tested with agreementshown in Figure 5.

5.5. The Relative Humidity (RH)

It is the immense subsaturation of the tropospherethat makes the study of water vapour an interestingand challenging problem (Spencer and Braswell

Figure 5. A comparison of the flux 〈F〉 measured in the MonteCarlo simulation with the theoretical result using (45) and (46).

1997; Pierrehumbert, Brogniez and Roca 2005). Thissubsaturation is most starkly revealed by examiningthe RH, which is

rdef=

q

qs(y). (47)

Converting the PDF in (37) to a PDF for r, denotedby Q(r, y), gives (for y > 0)

Q(r, y) =y

L

[δ(r − rmin)

L+

α

r log2(εr/rmin)

], (48)

where r ∈ [rmin(y), 1], ε is defined (40), and

rmin(y) = εeαy . (49)

A comparison between (48) and the Monte Carlosimulation is shown in Figure 6. The main mismatch wesee is for r → 1, where the numerical PDF’s taper offwhile expression (48) continues to rise till r = 1. Thisdiscrepancy is due to the collection of numerical datain a strip (as before), while (48) provides an expressionfor the PDF at a particular location. Note that thenumerical PDF’s show a spike at rmin(y) — shownonly in the first panel — which agrees with the shiftingδ-function in (48). Bimodality of the PDF at smally is evident, while farther from the source there is apeak at rmin(y) (suppressed in the plot), after whichthe PDF gradually levels off as r → 1. The transitionfrom a bimodal to a unimodal PDF, as well as thePDFs far from the source, are shown in the secondand third panels of Figure 6 respectively.

5.6. The global RH PDF

The global PDF, h(r), of RH is

h(r) def=∫ log(r/ε)/α

0

Q(r, y)dy. (50)

Evaluating (50) using Q(r, y) in (48) gives

h(r) =1

αLrlog[

αL

log(1/r)

]. (51)




Figure 6. Comparison between Q12(r, y) from Monte Carlowith the smooth portion of the analytic result in (48). In thisillustration α∗ = 5 and y2 − y1 = 0.02L. Crosses, open circlesand stars are the Monte-Carlo PDFs while the smooth solidcurves are the analytical expression. Top panel: The bimodalPDFs for y∗ < 2/α∗ (the curve in the middle corresponds toy∗ = 0.08). Middle panel: The PDFs for larger y∗ which show thetransition from a bimodal to a unimodal distribution. Bottompanel: The unimodal PDFs far from the source, i.e. for y∗ >2/α∗.

As a check one can verify that on integration over therange of RH, that is ε ≤ r ≤ 1, the global PDF in (51)normalizes to unity. The lower panel of Figure 4 showsa comparison between the Monte Carlo global RH PDFand (51). There is a good match between the theoreticalprediction (50) and the simulation. The global PDFexhibits a mix of the distinct features of the PDFs atdifferent y e.g., h(r) is bimodal with a primary peakat small RH, followed by a decay at intermediate RH,and then a secondary “moist peak” as r → 1.

6. Discussion

6.1. Radiative effects of water vapour

In the introduction, we referred to the importanceof the vapour PDF in understanding the effect ofhumidity on the OLR. In fact, the effect of humidityon OLR was quantified by Zhang, Soden and Mapes(2003) in terms of the PDF of the RH field.Their numerical experiments, employing a full-fledgedradiation code, showed that a domain with a bimodalRH PDF allowed for significantly larger cooling thanone with a unimodal distribution (holding the totaland mean the same in the two cases). In the presentcontext, we see that the PDF of the RH over thedomain, i.e. the lower panel of Figure 4, is bimodal.Therefore, along with regions of moderate RH, thedomain consists of subsets with very small RH (the firstpeak in the aforementioned plot) and those that arenearly saturated (the second peak). A clearer pictureis seen in the upper panel of Figure 7 which showsthe average specific humidity and RH over the domain,i.e. 〈q〉(y) and 〈r〉(y) respectively. While the specifichumidity decays monotonically from the source, the RHhas a pronounced mid-domain minimum representinga pool of “relatively” dry air which is flanked on eitherside by nearly saturated regions.

As described in PBR, the presence of water vapouraffects the OLR in an approximately logarithmicfashion, i.e. OLR∝ − log(q). Therefore, the importanceof knowing the PDF of q at every y is illustratedby comparing log(〈q〉) with 〈log(q)〉 — it is the latterquantity that accounts for fluctuations in the specifichumidity at every location and requires a knowledge ofP (q, y). As expected (see for example the discussion inPBR), and as is shown in the lower panel of Figure7†(especially the inset, which shows the differencebetween the two estimates) the fluctuations in thespecific humidity field increase the OLR as comparedto the estimate based only on the mean profile 〈q〉(y).

6.2. Bimodality of the RH PDF

It is interesting that the global PDF of q in the upperpanel of Figure 4 is unimodal, while the global PDFof RH in the lower panel of Figure 4 is bimodal. Thus,according to our model, RH bimodality is so robust

†Both curves start at zero as we have chosen qmax = 1, in generalthe value at y = 0 is log(qmax).




Figure 7. Upper panel: The average profiles of the specifichumidity and RH computed from the PDFs in (37) and (48)respectively. Lower panel: log〈q〉(y) and 〈log(q)〉(y) as computedusing (28). The inset shows the difference 〈log(q)〉(y)− log〈q〉(y)and demonstrates the increase in the OLR due to fluctuations inq at every y.

that it survives the mixing of PDFs from diverselocations into the global PDF h(r) in (51).

As far as we are aware, bimodal RH PDFs thathave been documented in literature are mostly derivedfrom satellite data over the deep tropics (Brown andZhang 1997; Zhang, Soden and Mapes 2003; Mote andFrey 2006; Luo, Kley and Johnson 2007). Exceptionsare the three-dimensional and isentropic model basedAC experiments by PBR and Yang and Pierrehumbert(1994) respectively: in Figure 16 in PBR and in Figure6 of Yang and Pierrehumbert (1994) the midlatitudeRH PDF is clearly bimodal.

On the other hand, the stochastic drying modelsdeveloped by Sherwood, Kursinski and Read (2006)and Ryoo, Igusa and Waugh (2009) predict onlyunimodal RH PDF’s. Indeed, Sherwood, Kursinski andRead (2006) concluded that a broad and unimodalRH density might be regarded as a general outcomeof the AC model. This conclusion is inconsistent withour implementation of the AC model, which results inbimodal RH PDFs. We discuss this point further insection 6.3.

Because our main focus is humidity fields onmiddleworld isentropic surfaces, we proceed with adata comparison by analyzing the RH on the 330Kisentropic surface as presented in the ERA interimproduct. The data was obtained from the ECMWFweb portal (http://data-portal.ecmwf.int/data/d/interim_daily/). We analyse daily data for theyear 2008 given at a horizontal resolution of 1.5◦ ×1.5◦ and at 24 vertical levels from 1000 mbar to 175mbar. As the RH data is only available on isobariclevels, we interpolate to generate RH fields on the330K isentrope. Further, as we are interested in thePDFs from the midlatitudinal portion of the isentropicsurface, we collect data between 15◦ and 60◦ in boththe hemispheres.

Figure 8. The midlatitudinal RH PDF on the 330K isentropeas derived from the ERA interim data product for the individualmonths of 2008. Upper panel: PDFs from the two hemispheresplotted separately (ordering: January — back to December —front). Lower panel: The lines with dots are the PDFs with bothhemispheres taken together, and the foremost bold line with starsymbols is the mean of these monthly curves.

A reviewer noted that the 330K isentropeintersects the tropopause by 60◦. To alleviate concernsthat these bimodal PDFs result from sampling distinctair masses separated by a mixing barrier, we have alsoconstructed PDFs by considering data only from thetroposphere, for example, from 15◦ and 35◦ in bothhemispheres, and verified that these PDFs display thesame qualitative form as those in Figure 8.

As seen in Figure 8 — which shows the PDFsfrom the different months of 2008 (PDFs from theindividual hemispheres are plotted in the upper panel,and a composite from both the hemispheres is shownin the lower panel) — there are considerable differencesin the RH PDFs between the two hemispheres. Forexample, the dry peak in the PDFs is more prominentin each hemisphere’s winter season. As these differencesare not of primary interest to us we do not dwell onthem here. There are a few values in the midlatitudeswhere the ERA dataset shows supersaturation, thesehave been set to 100% in the present calculation.



http://data-portal.ecmwf.int/data/d/interim_daily/

http://data-portal.ecmwf.int/data/d/interim_daily/


Naturally, this raises the secondary peak but does notnot alter the form of the PDF. Quite clearly, the PDFdoes vary from month to month but its main featuresare robust throughout the year. More importantly, weobserve that the ERA interim data PDF, in all months,is bimodal with distinct dry and moist peaks, whichconforms with the results of the AC model in section5.

6.3. Special features of the AC model in this paper

In reconsidering the different ingredients in ouridealized implementation of the AC model, weemphasize that there are at least four importantassumptions affecting our conclusions:

• Isentropic transport is modeled as brownianmotion;

• Parcels are re-moistened only at y = 0;• The saturation humidity is qs(y) = qmaxe−αy.• The model neglects diffusive processes.

The first assumption, which is equivalent to sayingthat the lagrangian velocity correlation time is zero,is crucial for the derivation of the Fokker-Planckequation, and thus for all of the results in thispaper. Moreover, following O’Gorman and Schneider(2006), we emphasize that the important condensation-diffusion balance in (32) also relies crucially onthe assumption of brownian motion. With nonzerovelocity correlation time it might still possible to arguethat mean condensation 〈C〉 is balanced by diffusiveconvergence, that is

〈C〉 = κqd2〈q〉dy2

. (52)

But the humidity diffusivity κq will be systematicallyless than the brownian parcel diffusivity κb. Under-standing the relation between κq and κb is beyond ourscope here.

The second assumption of a localized vapoursource probably explains the difference between ourconclusions and those of Sherwood, Kursinski andRead (2006) regarding bimodality of the RH PDFs.This point could be examined within the Fokker-Planck framework by modeling the source S in(10) as a re-setting, q → qs(y), occurring at a rateτMoist(y) throughout the domain. We do not pursuethis generalization here.

The third assumption, that the saturationhumidity varies exponentially with y, is a consequenceof the Clausius-Clapeyron relation and the assumptionthat the temperature gradient on a isentrope isuniform. We have examined alternative models of qs(y),e.g., taking qs(y) as a linear or a parabolic function ofy while ensuring qs(y) is still monotonically decreasingqmax at y = 0. The main qualitative features of thesesolutions shown in Figures 3, the lower panel of Figure 4and in Figure 6 are unchanged. The only PDF which issensitive to qs(y) is the global specific humidity PDF,g(q) shown in the upper panel of Figure 4. Thus, ascompared to the q−1 algebraic decay in the exponential

qs, a linear (parabolic) saturation qs(y) yields a uniform(growing) PDF as q → qmax. These differences can betraced to the form of the function ys(q) in the analyticexpression for g(q) provided by (25).

The fourth assumption above is also crucial tothe AC model. Haynes and Anglade (1997) estimatethat in the upper troposphere and lower stratospherethe “mix-down” time for synoptic-scale fluid parcels tobe sheared and strained to molecular-diffusive scalesis a few weeks. This time scale is comparable to themean residence time of water in the free troposphere(Quante and Matthias 2006). Thus molecular diffusionbetween fluid parcels may have an impact on thewater vapour field, and in particular on its PDF.This important open question cannot be addressed tillthe advection-condensation model is extended to theadvection-diffusion-condensation model.

7. Conclusions

We have studied the AC model forced by resettingthe specific humidity at the southern boundary of themodel isentrope. If the flow can be approximated bybrownian motion (i.e., a very rapidly de-correlatingvelocity), then the PDF of the vapor field is governedby the Fokker-Planck equation, which can be solvedin the limit of rapid condensation. The solutionis illustrated for a saturation profile which decaysexponentially along the isentrope in the meridionaldirection. Analysis of this solution, and supportingMonte Carlo simulations, document a non-trivialstatistical steady state in which the southern moisturesource is balanced by condensation throughout theinterior of the domain. “Non-trivial” refers to thefact that the steady state is far from the limitingstates of complete saturation or complete dryness, andinstead is characterized by a spatially inhomogenousPDF. This non-uniformity manifests itself in the co-existence of very dry and nearly saturated regionswithin the domain. We find this to be encouraging as astrongly inhomogeneous steady state is reminiscent ofthe atmosphere. In fact, the bimodality of the globalRH PDF predicted by the model is supported bymidlatitudinal middleworld isentropic RH PDFs fromthe ERA interim product.

An important property of our solution is its spatialstructure: near the resetting source (i.e. for small y)the PDF is bimodal with a peak at qmin, a rapid decayfollowed by a rise to a second peak as q → qs(y) ≈ qmax.Futher from the source (i.e. for large y) the PDFis unimodal and terminates at q = qs(y)� qmax. Tofacilitate a comparison with numerical experimentsand observations we also derived an expression forthe PDF as estimated over a strip of non-zero width.Extending the strip to cover the domain, we obtainedthe domain averaged PDF which in the case of anexponential saturation profile has a q−1-form. Thesefeatures are in very good agreement with numericalsimulations employing a large number of diffusivelymoving parcels. We also demonstrated the utility ofthe PDF in estimating mean field observables such asthe moisture flux and condensation rate.




From the PDF of the specific humidity we deducedthe PDF of the RH. Once again, the PDFs near thesource are bimodal with a peak at the minimum RHfor a given y, i.e. at rmin(y), followed by a decay atintermediate RH values and a rapid rise as r → 1. Thebimo a unimodal distribution at large distances fromthe source, and here the PDF is controlled by the δ-function at rmin(y) which levels off to a constant asr → 1. The domain averaged RH PDF displays a mixof these distinct features, in particular — even thoughthe global specific humidity PDF is unimodal — theglobal RH PDF is bimodal with a primary peak atsmall RH, followed by a decay at intermediate RH, andthen a secondary peak as r → 1.

Though this work deals with an idealized model,we believe the analysis afforded in this setting providesa useful understanding of advection-condensationand the equilibria supported by the interaction ofthese processes with a steadily maintained sourceof moisture. Indeed, with a proper identification ofthe saturation profile and the resetting protocol, ourmethodology can be applied wherever the AC modelfinds use. Also, we argue that the algorithm outlinedhere is of practical interest in that, given a saturationprofile from a general circulation model, one can thenpredict the form of the PDF of the vapour and relativehumidity field. In turn, this can be compared with thePDFs generated directly from the model itself, thusproviding a method to validate the fidelity of moistureevolution in a general circulation model.

Acknowledgement

WRY is supported by the NSF under GrantNo. OCE07-26320. We thank Paul O’Gorman for auseful discussion of the AC model.

A. Normalization as a differential boundarycondition

Here we show that the normalization constraintobtained in the limit of rapid condensation canbe interpreted as differential boundary condition.Specifically, we consider this interpretation for theresetting problem and pick up the story at (18) where

P (q, y) =Φ(q)L

+[δ(q − qmin)

L2+ F (q)

]y . (53)

Applying the normalization (15), for y ∈ (0, L) weobtain

y

L2+ y

∫ qs(y)

qmin

F (q) dq =1L. (54)

Differentiating with respect to y, Leibnitz’s rule gives

1L2

+∫ qs(y)

qmin

F (q)dq + yF [qs(y)]dqsdy

= 0 . (55)

A second differentiation yields,

2F [qs(y)]dqsdy

+ ydFdy

(dqsdy

)2

+ yF [qs(y)]d2qsdy2

= 0 .

(56)

On specifying qs(y), we see that (56) is an ordinarydifferential equation for the function F (q). Indeed, (56)is the interpretation of the normalization constraint asa differential boundary condition. To see the utility of(56) in solving for F (q), consider qs(y) = e−αy, thisyields

[2 + log(x)]F (x) + x log(x)dF (x)dx

= 0. (57)

and hence, F (q) = α/(Lq log2(q)) which agrees withthe earlier result (37).

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