### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

The statistically steady humidity distribution resulting from an interaction of advection, modelled as an uncorrelated random walk of moist parcels on an isentropic surface, and a vapour sink, modelled as immediate condensation whenever the specific humidity exceeds a specified saturation humidity, is explored with theory and simulation. A source supplies moisture at the deep-tropical southern boundary of the domain and the saturation humidity is specified as a monotonically decreasing function of distance from the boundary. The boundary source balances the interior condensation sink, so that a stationary spatially inhomogeneous humidity distribution emerges. An exact solution of the Fokker–Planck equation delivers a simple expression for the resulting probability density function (PDF) of the water-vapour field and also the relative humidity. This solution agrees completely with a numerical simulation of the process, and the humidity PDF exhibits several features of interest, such as bimodality close to the source and unimodality further from the source. The PDFs of specific and relative humidity are broad and non-Gaussian. The domain-averaged relative humidity PDF is bimodal with distinct moist and dry peaks, a feature which we show agrees with middleworld isentropic PDFs derived from the ERA interim dataset. Copyright © 2011 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

Water vapour plays a significant role in diverse problems pertaining to the dynamics of the Earth's climate (Pierrehumbert, 2002; Schneider *et al.*, 2010). In particular, with regards to its radiative effects, estimating the amount of water vapour in the upper troposphere is crucial (Spencer and Braswell, 1997; Held and Soden, 2000). Significant efforts–especially directed towards the subtropical troposphere–have yielded a key insight into the non-local nature of processes that control the distribution of water vapour (Yang and Pierrehumbert, 1994; Sherwood, 1996; Salathe and Hartman, 1997; Pierrehumbert, 1998; Pierrehumbert and Roca, 1998; Galewsky *et al.*, 2005; Brogniez *et al.*, 2009).

The resultant framework for understanding the evolution of water vapour, or generally any condensable substance in a fluid dynamical setting, is known as the advection–condensation (AC) model. The recent review by Sherwood (1996)-rev provides an introduction, while the book chapter by Pierrehumbert *et al.* (2005) (hereafter PBR) provides a thorough overview of the AC model.

Further, as elaborated on by Spencer and Braswell (1997) and PBR, in addition to the amount, the probability density function (PDF) of the water-vapour field has a strong influence on the long-wave cooling of the atmosphere. Explicit calculations illustrating the effect of the PDF on the outgoing long-wave radiation (OLR) are given by Zhang *et al.* (2003). The importance of the PDF stems from the approximately logarithmic dependence of the change in OLR to the specific humidity of the water vapour in the domain. Further discussion of this logarithmic dependence, which results from a spectral broadening of absorption peaks, can be found in the text by Pierrehumbert (2010).

Following Soden and Bretherton (1993), there have been numerous efforts that examine water-vapour PDFs in the tropical and subtropical troposphere. A principal feature noted by these studies is the non-normality of the PDFs. A range of distributions, from bimodality in the deep Tropics (Brown and Zhang, 1997; Zhang *et al.*, 2003; Mote and Frey, 2006; Luo *et al.*, 2007) to unimodal PDFs with a roughly log-normal or power-law form in the subtropics have been documented (Soden and Bretherton, 1993; Sherwood *et al.*, 2006). Ryoo *et al.* (2009) have emphasized that a salient and baffling feature of water-vapour PDFs is their spatial inhomogeneity. Models based on the statistics of subsidence drying and random re-moistening of parcels reproduce many of the observed features of the humidity PDFs, provided that one considers the model parameters to be functions of position determined to match observations (Ryoo *et al.*, 2009; Sherwood *et al.*, 2006).

One of the main achievements of this article is an exact solution for the water-vapour PDF that characterizes the statistically steady state emerging from an interaction of advection, condensation and a spatially localized moisture source. In particular, we consider moist parcels advected by a white-noise stochastic velocity field. Each parcel carries a specific humidity and experiences condensation whenever and wherever the specific humidity exceeds a specified saturation profile. The parcel humidity is reset to saturation by a moist source at the southern boundary of the domain. In the ultimate statistically steady state, the southern vapour source is balanced by condensation throughout the interior of the domain. In contrast to earlier considerations of the AC model with white-noise advection (O'Gorman and Schneider, 2006), we consider statistically steady states rather than the initial value problem and we obtain the full water-vapour PDF rather than the mean humidity.

The AC model is described in more detail in sections 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 by calculating the average moisture flux and condensation. In section 5 we focus on a particular saturation profile–the standard exponential model–and exhibit analytical solutions for the specific and relative humidity PDFs; these are shown to be in very good agreement with PDFs estimated from a Monte Carlo simulation. In section 6 we analyze daily isentropic relative humidity data from the ERA interim product, and show that the PDFs derived from this dataset agree with the principal features predicted by our idealized model. A discussion of the PDFs, with implications for the atmospheric radiation budget and the present-day distribution of water vapour, followed by a conclusion in section 7 ends the article.

### 2. The AC model

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

The central quantity in the AC model is the specific humidity, defined by

- (1)

where **x** = (x,y) is the position on an isentropic surface. Our main focus here is the extratropical vapour distribution, which is determined in large part by the wandering of air parcels on isentropic surfaces and the temperature changes associated with latitudinal excursions.

Following PBR, the AC model is

- (2)

Above, �� is the vapour source, �� is the condensation sink and **u**(**x**,t) is the velocity on an isentropic surface. In the absence of sources and sinks, *Dq/Dt* = 0, i.e. the specific humidity is a materially conserved quantity. The AC formalism does not consider the diffusive homogenization of water vapour, e.g. via the addition of *χ*∇^{2}*q* on the right of (2). This limitation of the model is discussed further in section 6.

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

The condensation sink on the right of (2) might be modelled with

- (3)

where *τ*_{c} is the time-scale associated with condensation, *q*_{s}(**x**) is the prescribed saturation specific humidity and *H*(*x*) denotes the Heaviside step function, which is zero if *x <* 0 and unity if *x >* 0. Observed atmospheric supersaturations rarely exceed a few per cent and therefore the sink �� is extremely effective (Wallace and Hobbs, 1977). In other words, we are concerned with the rapid-condensation limit *τ*_{c} 0. In the rapid-condensation limit, the details in �� are unimportant and condensation can instead be represented as a rule enforcing subsaturation:

- (4)

Therefore *q*(**x**,t) ≤ *q*_{s}(**x**) and there is an upper bound on the specific humidity at every **x**. Globally, the maximum tolerable specific humidity is

- (5)

If the flow is ergodic then a wandering parcel explores the entire domain and eventually encounters the driest regions with the smallest value of *q*_{s}(**x**). Thus, in the absence of sources (Sukhatme and Pierrehumbert, 2006, pers. comm.),

- (6)

where

- (7)

On the other hand, without advection the source will eventually saturate *q*(**x**,t) at every point **x** where ��*≠*0. Therefore, interesting statistically steady states are expected only if there is both a source �� and advection **u**(**x**,t).

The problem is to determine the evolution and the statistics of the specific humidity *q*(**x**,t), given the velocity field **u**(**x**,t) and the saturation profile *q*_{s}(**x**). Due to its physical significance, the principal statistical feature we focus here on is the probability density function (PDF) of *q*. Quantities such as the mean moisture flux and the mean condensation rate are also crucial, and have been examined in the AC framework (O'Gorman and Schneider, 2006; O'Gorman *et al.*, 2011). These mean quantities can be deduced from the PDF of *q* via integration over *q*.

### 3. Moist Brownian parcels

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

The simplest representation of random transport is obtained by taking the velocity **u**(**x**,t) in (2) as having rapid temporal decorrelation so that the motion of an air parcel is Brownian. Then the spatial evolution of an ensemble of parcels, each carrying its own individual value of *q*, is described by the diffusion equation. We also assume that *q*_{s} depends only on the meridional coordinate *y* in the domain 0 *< y < L*, and that *q*_{s} decreases monotonically from *q*_{max} at *y* = 0 to *q*_{min} at *y* = *L*. Each parcel is located at a point in the configuration space (*q,y*). Because of the condensation rule (4), the ensemble of parcels is confined to the shaded region of configuration space illustrated in Figure 1.

The AC model is then expressed as a set of coupled stochastic differential equations governing the position *Y*(*t*) and specific humidity *Q*(*t*) of a moist Brownian parcel:

- (8)

where *W*(*t*) is a Weiner process, *κ*_{b} is the Brownian diffusivity associated with the random walk and �� and �� are the source and condensation functions in (2). The numerical simulations described below amount to Euler–Maruyama discretization of (8). The midlatitude baroclinic eddies responsible for transport have non-trivial spatio-temporal correlations (Sukhatme, 2005; O'Gorman and Schneider, 2006). Thus the white-noise advection used in (8) and in earlier AC studies is an idealization.

The statistical properties of the ensemble of moist Brownian parcels are characterized by a PDF *P*(*q,y,t*), such that the expected number of parcels in d*q*d*y* is equal to

- (9)

where *N* is the total number of parcels.

The evolution of *P*(*q,y,t*) is governed by the Fokker–Planck equation

- (10)

with the normalization

- (11)

In the limit of rapid condensation, as *q*_{min} ≤ *q*(*y,t*) ≤ *q*_{s}(*y*), the normalization takes the form

- (12)

Integrating *P*(*q,y,t*) over *q* produces the marginal density

- (13)

Because the Brownian motion of a parcel is independent of the specific humidity it carries, we expect that integration of (10) over *q* will show that *p*(*y,t*) satisfies the diffusion equation, and indeed this result is immediate:

- (14)

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

- (15)

Note that the left of (15) is consistent with the global normalization in (11). In the limit *τ*_{c} 0, the steady, marginal normalization condition (15) applies at every *y*, and for all models of �� and ��. In other words, in a statistical steady state the normalization condition (15) is equivalent to enforcing the rapid-condensation rule in (5). The normalization (15) figures prominently in the following solution of the Fokker–Planck equation. Similar ‘variable limit’ normalization conditions are encountered in the solution of integrate-and-fire neuron models (Fusi and Mattia, 1998).

### 4. Steady solution with ‘resetting’ at *y* = 0

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

Now we adopt a ‘resetting’ model for the vapour source ��: on encountering the southern boundary at *y* = 0, parcels are reflected back into the domain with a new value of *q* chosen from a specified probability density function Φ(*q*), with the normalization

- (16)

For example, completely saturating the parcels at *y* = 0 corresponds to the choice Φ(*q*) = *δ*(*q* − *q*_{max}). This is a simple representation of tropical moistening. The technique of resetting boundary conditions has been employed in a previous study of AC on isentropic surfaces (Yang and Pierrehumbert, 1994), as well as in advection–diffusion of passive scalars (Pierrehumbert, 1994; Neufeld *et al.*, 2000).

We emphasize that resetting the humidity only at *y* = 0 is a strong model assumption that distinguishes this work from the AC models developed by Sherwood *et al.* (2006) and Ryoo *et al.* (2009). The statistical model developed in those works assumes that random resetting occurs throughout the domain as an exponential or gamma stochastic process with a characteristic time-scale *τ*_{Moist}(**x**). In effect, by resetting only at *y* = 0 we are considering the extreme case in which *τ*_{Moist}(**x**) = ∞ except at *y* = 0. Physically, the picture we have in mind is that parcels experience moistening when they encounter convective regions that 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 AC model along with the ‘re-setting throughout the domain’ scenario considered by Sherwood *et al.* (2006) and Ryoo *et al.* (2009).

An important consequence of resetting only at *y* = 0 is that the driest parcels, which are created at *y* = *L* with *q* = *q*_{min}, maintain their extreme dryness till they eventually strike *y* = 0. The solution below requires explicit consideration of these driest parcels by inclusion of a component *δ*(*q* − *q*_{min}), referred to as the ‘dry spike’, in the PDF.

#### 4.1. Solution of the Fokker–Planck equation

In the rapid-condensation limit, the accessible part of the (*q,y*) space is the shaded area in Figure 1, where �� = 0. With the resetting boundary condition, however, the source �� is also zero within the shaded region. Thus the steady Fokker–Planck equation in the shaded region of Figure 1 collapses to , with the immediate solution

- (17)

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

Corresponding to the driest parcels in the domain, *B*(*q*) in (17) must contain a component proportional to *δ*(*q* − *q*_{min}). This ‘dry spike’ contains parcels created by encounters with the northern boundary at *y* = *L*. These considerations refine (17) to

- (18)

The term *L*^{−2}*δ*(*q* − *q*_{min}) in (18) is obtained by applying the normalization condition (15) at *y* = *L*.

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

- (19)

Inspecting (19), we see that it is possible to solve for *F*(*q*) by using *q*, rather than *y*, as the independent variable. That is, following the notation in Figure 1, we use *y* = *y*_{s}(*q*) to re-write (19) as

- (20)

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

- (21)

where Λ(*q*) is the cumulative distribution

- (22)

Assembling the pieces, we obtain the PDF

- (23)

The expression above applies only within the shaded region in Figure 1 where *q < q*_{s}(*y*); in the supersaturated region *q > q*_{s}(*y*), *P*(*q,y*) = 0. The PDF is discontinuous at the saturation boundary *q* = *q*_{s}(*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–Planck equation is solved exactly by (23). The normalization constraint in (15), and the use of *q* as the independent variable produces this relatively simple solution. In fact, the normalization constraint (15) can also be regarded as a differential boundary condition. Manipulating this boundary condition provides an alternate route to the solution, outlined in the Appendix.

#### 4.2. The global PDF

The global PDF of specific humidity is the marginal density

- (24)

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

- (25)

A comforting check on (25) is that *g*(*q*) satisfies the normalization

- (26)

which is consistent with (12).

Independent of model details, such as the specification of *q*_{s}(*y*) and Λ(*q*), the dry spike *δ*(*q* − *q*_{min}) in (25), contains exactly half of the parcels in the ensemble. Professor P. O'Gorman noted that this feature can be understood on the basis of symmetry: exactly half the parcels have more recently encountered the moist southern boundary than the dry northern boundary. The former half have *q* ≥ *q*_{min}, while the other half constitute the dry spike with *q* = *q*_{min}.

#### 4.3. Averages

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

- (27)

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

- (28)

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

- (29)

#### 4.4. Average humidity and condensation rate

To understand the transport of moisture in this model and the associated mean condensation rate, we return to (10) and momentarily retreat from the rapid-condensation limit*. Then, multiplying (10) by *q* and integrating from *q*_{min} to *q*_{max}, we have

- (30)

where the condensation sink �� is given by the model in (3). Using this expression for �� and integrating the left of (30) by parts, we obtain the mean condensation rate defined by

- (31)

as

- (32)

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

This argument shows that the flux of moisture has the same diffusivity *κ*_{b} as a Brownian parcel. As explained by O'Gorman and Schneider (2006), this is a special property of the Brownian limit: if the velocity has a non-zero integral time-scale (e.g. as in the Ornstein–Ulenbeck process) then the diffusivity of moisture will be systematically less than the random-walk diffusivity. We emphasize that while the moisture flux is diffusive, even with a Brownian advecting flow, the condensation sink cannot be adequately represented by a bulk diffusivity. This inadequacy of a mean-field representation of condensation is clearly demonstrated in PBR.

In the rapid condensation limit, the right of (32) can be computed by putting *f*(*q*) = *q* in (29). Thus if *τ*_{c} 0, the mean condensation rate is

- (33)

Although , the mean condensation 〈��〉 in (33) is independent of *τ*_{c}. This singular limit is achieved because the non-zero probability of supersaturation is confined to a boundary layer, lying just above the saturation boundary *q* = *q*_{s}(*y*) in Figure 1.

If the resetting PDF Φ(*q*) is non-singular at *q*_{max}, then 〈��〉 in (33) is finite as *y* 0, despite the factor *y*^{−1}. Specifically, one can show that lim_{y0}Λ(*q*_{s})*/y* = −Φ(*q*_{max})(d*q*_{s}/d*y*).

### 5. An example

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

The solution from the previous section is most simply illustrated by supposing that the resetting produces complete saturation at *y* = 0,

- (34)

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

- (35)

This form of *q*_{s} has been employed previously in AC studies (Pierrehumbert *et al.*, 2005; O'Gorman and Schneider, 2006).

With Φ(*q*) and *q*_{s}(*y*) in (34) and (35), the steady solution of the Fokker–Planck equation is

- (36)

and away from the southern boundary

- (37)

The solution (37) can be non-dimensionalized with

- (38)

and *P*_{∗}(*q*_{∗}*,y*_{∗}) = *q*_{max}*LP*(*q,y*). This scaling identifies the fundamental non-dimensional control parameter *α*_{∗}, and the scaled PDF is

- (39)

with

- (40)

At the southern boundary *P*_{∗}(*q*_{∗},0) = *δ*(*q*_{∗} − 1). The main point is that the shape of *P*_{∗}(*q*_{∗}*,y*_{∗}) is controlled by a single parameter *α*_{∗}.

There is an interesting change in the structure of *P*_{∗} at *α*_{∗} = 2: *P*_{∗} is bimodal for all *y*_{∗} if *α*_{∗} < 2; if *α*_{∗} > 2 then *P*_{∗} is bimodal only in the region 0 *< y*_{∗} < 2*/α*_{∗}. This is illustrated in Figure 2, which shows the smooth portion of *P*_{∗} with *α*_{∗} = 4 and *α*_{∗} = 1. The PDF with *α*_{∗} = 4 changes from a bimodal to a unimodal form as one crosses *y*_{∗} = 2*/α*_{∗} = 0.5. Indeed, for *y*_{∗} > 0.5 the *α*_{∗} = 4 PDF falls monotonically from . In the case with *α*_{∗} = 1, the PDF increases with increasing *q*_{∗} at all values of *y*_{∗}. Therefore, in combination with , the PDF is always bimodal if *α*_{∗} < 2.

On a middleworld isentrope, the saturation mixing ratio usually changes by more than an order of magnitude as one spans the entire midlatitudes, hence from (35) we expect that *α*_{∗} > 2 is the relevant atmospheric case. Therefore, close to the vapour source the PDF is bimodal. One mode, the dry spike *δ*(*q* − *q*_{min}), corresponds to a singularity at *q*_{min} after which the PDF falls off, only to rise again to a secondary moist peak as *q* *q*_{s}(*y*). Further from the vapour source (i.e. for *y*_{∗} > 2*/α*_{∗}) the PDF is unimodal, with a dry spike at *q*_{min} followed by a gradual monotonic decay to a minimum at *q* = *q*_{s}(*y*).

#### 5.1. Statistics in a strip *y*_{1}*< y < y*_{2}

The expressions in (36) and (37) provide the PDF at a specified *y*. From an observational and numerical view, recording the PDF at a particular *y* can be difficult. It is more likely that one obtains an estimate of the PDF over a strip, i.e. for *y* ∈ [*y*_{1}*,y*_{2}]. Integrating (37) over such a strip, and denoting this by *P*_{12}(*q*), we have

- (41)

where the strip function is

- (42)

The slightly complicated form of *s*_{12}(*q*) results because when *q*_{s}(*y*_{2}) *< q < q*_{s}(*y*_{1}) the upper limit of the integral is *y*_{s}(*q*) *< y*_{2}. We compare *P*_{12} above with the results of a Monte Carlo simulation in Figure 3.

The effect of the strip function *s*_{12} is to clip the secondary peak at *q* = *q*_{s}, and this clipping becomes more extreme if *y*_{2} − *y*_{1} is made larger. Thus, aggregating measurements of *q* from different spatial locations will obscure the bimodal structure of *P*(*q,y*).

#### 5.2. The global PDF of *q*

In the extreme case *y*_{1} 0 and *y*_{2} *L*, we obtain the global PDF of the specific humidity, denoted by *g*(*q*), in (24). Using (25), the global PDF in this example is

- (43)

where *q* ∈ [*q*_{min}*,q*_{max}]. The smooth part of the global PDF is *g*(*q*) ∼ *q*^{−1}.

#### 5.3. Monte Carlo simulations

To test the analytic solution above, we proceed to a Monte Carlo simulation of our system. Typically our simulations use *N* = 10^{5} parcels which are released in the domain *y* ∈ [0*,L*]. We have used *α* = 1*,L* = 5 which gives *α*_{∗} = 5. This implies a change of approximately two orders of magnitude in the saturation specific humidity across the domain. The *y*-coordinates of the parcels are initially uniformly distributed and are tagged with the minimum saturated specific humidity, i.e. on parcel *i*, *q*_{i}(0) = *q*_{min}. The parcels perform an uncorrelated random walk in *y*, and parcel *i* is reflected back into the domain if *y*_{i}*> L* or *y*_{i} < 0. The forcing �� is implemented by setting *q*_{i} = *q*_{max} when a parcel is reflected at *y* = 0. Rapid condensation is implemented by enforcing *q*_{i} = min[*q*_{i}*,q*_{s}(*y*_{i})] after each random displacement.

To approximate Brownian motion, the root-mean-square step length of the random walk, *ℓ*, should be as small as is computationally feasible–we typically used *ℓ* = 2 × 10^{−3}*L*. We specify the Brownian diffusivity as *κ*_{b} = 0.5, and then the time between steps, *τ*, is computed via

- (44)

The simulation runs till *P*(*q,y*) attains a stationary form and then data are collected in several strips for comparison with (41).

The results of the simulations are shown in Figure 3. The top panel of this figure shows a comparison between the smooth portion of the theoretical expression in (41) and the numerically estimated PDF for *y*_{∗} < 2*/α*_{∗}. There is good agreement between the Monte Carlo simulation and the Fokker–Planck solution. Some minor discrepancies are evident as *q* *q*_{min}; this is anticipated, as we have suppressed the numerical spike at *q* = *q*_{min} (indeed, the theoretical PDF is given by a *δ*-function at this location). The middle panel of Figure 3 shows the PDF at larger *y*, specifically around the region where the PDF undergoes transition from a bimodal to a unimodal form. The bottom panel of Figure 3 shows the PDFs far from the source, where they have a single peak at *q* = *q*_{min} followed by decay and termination at *q* = *q*_{s}(*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 closely follows the result in (43), with a spike at *q* = *q*_{min} followed by an algebraic tail *g*(*q*) ∼ *q*^{−1}.

#### 5.4. Transport and condensation

To understand the water-vapour distribution, it is important to quantify the transport of moisture. In the context of the AC model, it was shown by O'Gorman and Schneider (2006) that Brownian motion results in the mean moisture flux

- (45)

where 〈*q*〉 is the mean specific humidity and *κ*_{b} is the Brownian diffusivity. Then, as in (32), in a statistical steady state the convergence of 〈ℱ〉 balances the mean condensation 〈��〉 at every *y*.

Using *f*(*q*) = *q* in (29) and *q*_{s}(*y*) in (35), the average gradient of specific humidity is obtained as

- (46)

where is the exponential integral. The moisture flux 〈ℱ〉 can also be diagnosed from the simulation and thus (45) is tested, with agreement shown in Figure 5.

#### 5.5. The relative humidity (RH)

It is the immense subsaturation of the troposphere that makes the study of water vapour an interesting and challenging problem (Spencer and Braswell, 1997; Pierrehumbert *et al.*, 2005). This subsaturation is most starkly revealed by examining the RH, which is

- (47)

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

- (48)

where *r* ∈ [*r*_{min}(*y*),1], is defined in (40) and

- (49)

A comparison between (48) and the Monte Carlo simulation is shown in Figure 6. The main mismatch we see is for *r* 1, where the numerical PDFs taper off while expression (48) continues to rise till *r* = 1. This discrepancy is due to the collection of numerical data in a strip (as before), while (48) provides an expression for the PDF at a particular location. Note that the numerical PDFs show a spike at *r*_{min}(*y*)–shown only in the first panel–which agrees with the shifting *δ*-function in (48). Bimodality of the PDF at small *y* is evident, while further from the source there is a peak at *r*_{min}(*y*) (suppressed in the plot), after which the PDF gradually levels off as *r* 1. The transition from a bimodal to a unimodal PDF, as well as the PDFs far from the source, are shown in the second and third panels of Figure 6 respectively.

#### 5.6. The global RH PDF

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

- (50)

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

- (51)

As a check, one can verify that on integration over the range of RH, that is ≤ *r* ≤ 1, the global PDF in (51) normalizes to unity. The lower panel of Figure 4 shows a comparison between the Monte Carlo global RH PDF and (51). There is a good match between the theoretical prediction (50) and the simulation. The global PDF exhibits a mix of distinct features of the PDFs at different *y*, e.g. *h*(*r*) is bimodal with a primary peak at small RH followed by a decay at intermediate RH and then a secondary ‘moist peak’ as *r* 1.

### 7. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. The AC model
- 3. Moist Brownian parcels
- 4. Steady solution with ‘resetting’ at
*y* = 0 - 5. An example
- 6. Discussion
- 7. Conclusions
- Acknowledgements
- Appendix. Normalization as a differential boundary condition
- References

We have studied the AC model forced by resetting the specific humidity at the southern boundary of the model isentrope. If the flow can be approximated by Brownian motion (i.e. a very rapidly decorrelating velocity), then the PDF of the vapour field is governed by the Fokker–Planck equation, which can be solved in the limit of rapid condensation. The solution is illustrated for a saturation profile that decays exponentially along the isentrope in the meridional direction. Analysis of this solution and supporting Monte Carlo simulations documents a non-trivial statistical steady state in which the southern moisture source is balanced by condensation throughout the interior of the domain. ‘Non-trivial’ refers to the fact that the steady state is far from the limiting states of complete saturation or complete dryness, and instead is characterized by a spatially inhomogeneous PDF. This non-uniformity manifests itself in the co-existence of very dry and nearly saturated regions within the domain. We find this to be encouraging, as a strongly inhomogeneous steady state is reminiscent of the atmosphere. In fact, the bimodality of the global RH PDF predicted by the model is supported by midlatitudinal middleworld isentropic RH PDFs from the ERA interim product.

An important property of our solution is its spatial structure: near the resetting source (i.e. for small *y*) the PDF is bimodal with a peak at *q*_{min}, a rapid decay followed by a rise to a second peak as *q* *q*_{s}(*y*) ≈ *q*_{max}. Further from the source (i.e. for large *y*) the PDF is unimodal and terminates at *q* = *q*_{s}(*y*) ≪ *q*_{max}. To facilitate a comparison with numerical experiments and observations we also derived an expression for the PDF as estimated over a strip of non-zero width. Extending the strip to cover the domain, we obtained the domain-averaged PDF, which in the case of an exponential saturation profile has a *q*^{−1} form. These features are in very good agreement with numerical simulations employing a large number of diffusively moving parcels. We also demonstrated the utility of the PDF in estimating mean field observables such as the moisture flux and condensation rate.

Though this work deals with an idealized model, we believe the analysis afforded in this setting provides a useful understanding of advection–condensation and the equilibria supported by the interaction of these processes with a steadily maintained source of moisture. Indeed, with a proper identification of the saturation profile and the resetting protocol, our methodology can be applied wherever the AC model finds use. Also, we argue that the algorithm outlined here is of practical interest in that, given a saturation profile from a general circulation model, one can then predict the form of the PDF of the vapour and relative humidity field. In turn, this can be compared with the PDFs generated directly from the model itself, thus providing a method to validate the fidelity of moisture evolution in a general circulation model.