The effect of coherent stirring on the advection-condensation of water vapour.

Atmospheric water vapour is an essential ingredient of weather and climate. The key features of its distribution can be represented by kinematic models which treat it as a passive scalar advected by a prescribed flow and reacting through condensation. Condensation acts as a sink that maintains specific humidity below a prescribed, space-dependent saturation value. To investigate how the interplay between large-scale advection, small-scale turbulence and condensation controls moisture distribution, we develop simple kinematic models which combine a single circulating flow with a Brownian-motion representation of turbulence. We first study the drying mechanism of a water-vapour anomaly released inside a vortex at an initial time. Next, we consider a cellular flow with a moisture source at a boundary. The statistically steady state attained shows features reminiscent of the Hadley cell such as boundary layers, a region of intense precipitation and a relative humidity minimum. Explicit results provide a detailed characterization of these features in the limit of strong flow.

Introduction

Liquid water evaporates from land and ocean into the atmosphere. The interaction between the subsequent transport and condensation of this evaporated water gives rise to intriguing distributions of water vapour in the atmosphere: for example, persistent relative humidity minima are observed in the subtropics [1,2], and bimodal distributions have been reported in the tropics [3]. Knowledge of the full distribution of atmospheric humidity is crucial for understanding the Earth’s energy balance and climate. This is because the absorption of outgoing long-wave radiation by water vapour increases nonlinearly (roughly logarithmically) with specific humidity [4]. The atmospheric moisture distribution and transport are also closely linked to global and regional precipitation patterns which have high social and economic impacts [5].

A framework to explain the key features of the atmospheric humidity distribution is the advection–condensation model [1,6]. In this model, a moist air parcel is transported through the atmosphere’s saturation humidity field and condensation occurs when its humidity exceeds the local saturation value. The excessive water is rained out of the system. As a result, the humidity at a particular location is equal to the minimum saturation value the air parcel has encountered since leaving the moisture source. Critically, all complex cloud-scale microphysics and molecular diffusion are excluded from this model [1]. Research over the last several decades has demonstrated the value of the idea of advection–condensation. Brewer in 1949 was able to deduce the existence of a general circulation in the stratosphere from water-vapour distribution measurement [7]. More recently, many studies have reconstructed humidity fields in the troposphere [8-12] and the stratosphere [13] by simulating particle trajectories using observed wind fields.

The success in numerical and observational studies has led to theoretical investigations of the advection–condensation model in idealized settings. A continuum formulation of the model, with the water-vapour distribution represented by a coarse-grained field [2,14], is prone to produce overly saturated air [6]. Here, we employ a Lagrangian particle formulation. A few previous works have taken this approach, starting with Pierrehumbert et al. [6], who considered an ensemble of moist air parcels undergoing Brownian motion and condensation in one dimension. Among other results, they obtained analytically the time-dependent probability distribution function (PDF) of the local specific humidity when initially saturated parcels are allowed to dry in the absence of a moisture source—stochastic drying. The stochastic drying problem where the parcel velocity has a finite correlation time was solved by O’Gorman & Schneider [14]. Sukhatme & Young [15] studied Brownian parcels forced by a moisture source located at one end of a bounded one-dimensional domain and derived an exact solution for the water-vapour PDF of the resulting statistically steady state. A generalization of this steady-state problem to the case of time-correlated parcel velocity was considered by Beucler [16]. All these studies employ a one-dimensional Lagrangian velocity with no spatial correlation to mimic turbulent motions. However, analyses of observational data [17,18] and idealized simulations [19] demonstrate that synoptic-scale eddies play an important role in atmospheric transport. Pauluis et al. [20] have also shown that the global moisture circulation can be viewed as a single overturning cell in moist isentropic coordinates. Roughly speaking, water vapour evaporated into the planetary boundary layer is drawn towards the tropics where it is transported upward. Large-scale advection then carries the moisture from the tropical upper troposphere to other regions where the air subsides [1].

In this paper, we aim to gain insight into the effects of coherent stirring on the transport and distribution of water vapour. We consider a two-dimensional advection–condensation system where the velocity of an air parcel consists of a large-scale circulation and a small-scale stochastic component. We use this idealized model to investigate how the large- and small-scale velocities interact to produce the resulting humidity distribution and answer questions such as: How does the large-scale circulation create an area of low relative humidity? How does the precipitation pattern change with the strength of the circulation?

Following the presentation of our model in §2, we investigate in §3 the drying of a moisture patch in the presence of a single vortex and no moisture source. The drying process consists of an initial fast advective stage and a later slow stochastic stage. In the limit of strong circulation, we obtain an analytical expression for the decay of the mean moisture in the system. In §4, we consider a cellular circulation in a bounded domain with a moisture source at the bottom boundary. This set-up roughly resembles the Hadley cell [21]. We discuss the general features of the statistically steady humidity distribution and their dependence on the circulation strength. In the strong circulation limit, we derive an expression for the specific humidity PDF from which diagnostics such as evaporation rate and precipitation rate are obtained. Section 5 concludes the paper.

The advection–condensation model

Consider an ensemble of moist air parcels passively advected by a velocity field in a two-dimensional domain. When the specific humidity Q of an air parcel at position =(X,Y) exceeds the local value of the saturation specific humidity qs(), the excessive moisture condenses and precipitates out of the system. To a very good approximation, qs is proportional to the saturation vapour pressure, which varies with temperature according to the Clausius–Clapeyron relation [22]. Assuming that the temperature is independent of x and decreases linearly with y, qs decays exponentially in y [6]. Thus for the rest of this paper, we take for some constant α>0.

Our goal is to investigate the effect of a large-scale circulation on the distribution of moisture. To this end, the prescribed velocity in our model is composed of a deterministic part =(u,v) representing large-scale coherent motions and a stochastic, δ-correlated in time (white noise) component which mimics the small-scale random transport of the air parcels. Hence, the Lagrangian formulation of our advection–condensation model takes the form of a set of stochastic differential equations for the random variables (X,Y,Q), The Brownian motion of the parcel is modelled via the Wiener processes W1(t) and W2(t) with diffusivity κ. represents a moisture source. Generally, the condensation sink is given by where τc is the condensation time scale and denotes the Heaviside step function. Following previous studies [6,14,15], we take the rapid condensation limit . Effectively, this means that Q is reset to qs(Y) whenever the former exceeds the latter, The specific form of and (u,v) will be given in the following sections when we consider initial-value and steady-state problems.

The two-dimensional model described above can represent an isentropic surface in the mid-troposphere with x the distance in the east–west direction and y the distance from the Equator. The present set-up can also be considered as a crude model for moisture transport by an overturning circulation in the free troposphere. Then x represents the latitude or longitude and y is the altitude. Generally, the typical length scales in the x- and y-directions are different. Here, it is understood that X and Y have been scaled by their respective typical length scales. For simplicity, the re-scaled diffusivities in the two directions are assumed equal.

Initial-value problem

Let us consider a patch of initially saturated air in an unbounded domain with no moisture source, . Condensation may occur when individual air parcels move in the y-direction, hence reducing the total moisture content in the system. We are interested in how a vortex, taken to be the solid-body rotation with constant Ω, added to the random motion of the air parcels modifies the drying process. Later, we shall see that the present set-up is relevant to the emergence of a dry zone in the forced problem considered in §4. With typical length α−1 set by qs in (2.1) and typical velocity Ωα−1, the inverse Péclet number measures the importance of random motion relative to the circulation.

Figure 1 shows a typical Monte Carlo simulation of (2.2) using the Euler–Maruyama method [23] for an ensemble of 107 air parcels. The simulation parameters are Ω=5, κ=10−1, and qmax=0.1. This gives ϵ≈0.01≪1, so this case is in the fast circulation limit. The value of α mimics the situation in the troposphere where the saturation specific humidity varies by several orders of magnitude with altitude as well as between the tropical and polar regions [1]. The parcels are initially distributed evenly over a circular area centred at the origin with radius R=5π (figure 1a). We are interested in a large patch R≫α−1. Generally, we find that the drying process consists of a fast advective stage and a slow stochastic stage. We discuss these two stages in the following sections.

Figure 1.

(a–c) Monte Carlo simulation of the stochastic advection–condensation model (2.2) for the initial-value problem in §3. The position of air parcels is shown at different times with a colour scale that indicates the specific humidity carried by each parcel. The values of the simulation parameters are given in the text. (Online version in colour.)

Advective drying

Initially, at t=0, all parcels are saturated and have Q(0)=qs[Y (0)]. At t=0+, the air parcels start to move in the counterclockwise sense along the circular streamlines of (u,v) with small random fluctuations induced by the Brownian motion. The parcels that move in the −y direction are entering regions where qs(Y)>Q, thus no condensation occurs and Q remains constant. On the other hand, for parcels moving in the +y direction along a streamline of radius r, condensation starts immediately. These parcels continue to lose water vapour as condensation goes on until they reach (X,Y)≈(0,r) and Q≈qs(r)—the minimum of qs on the streamline. By the time every parcel has made one complete revolution and a large amount of moisture has been lost: the moisture distribution becomes more or less axisymmetric with for each parcel (figure 1b). The rapid initial drying is best exhibited by the decay of the global specific humidity defined as where the sum is over all N air parcels. Assuming all parcels have the same air mass, is simply the ratio of the total moisture mass to the total air mass in the system. During the advective drying stage, drops rapidly from its initial value at t=0 to at t≈t. Further drying from this time on relies on the Brownian motion of the parcels and corresponds to the slow stochastic drying phase. Figure 2a shows this transition for different Ω including the case without a vortex (Ω=0); other simulation parameters are the same as in figure 1.

Figure 2.

Decay with time of the global specific humidity defined in (3.5) for the initial-value problem in §3. (a) Transition from advective to stochastic drying for different values of angular velocity Ω. (b) Monte Carlo simulation results (circles) for Ω=5 (ϵ≈0.01) are compared with the theoretical prediction in (3.22) (solid line) and with the long-time asymptotic formula (3.23) (dashed line). The inset shows the initial rapid decay, with the dotted line indicating the value at the end of the advective drying stage, in (3.6). (Online version in colour.)

Stochastic drying

In the stochastic drying phase, an air parcel on a streamline of radius r1 that wanders onto another streamline of radius r2>r1 is being quickly advected into the region of y≈r2 with lower saturation specific humidity. Rapid condensation within this region reduces the specific humidity of the parcel from Q=qs(r1) to Q=qs(r2). Our primary goal in this section is to calculate the resulting PDF of Q. Following previous work [6,14], this is achieved by considering the maximum excursion statistics of an air parcel.

Define the maximum excursion (in the y-direction) at time t of an air parcel as Because of rapid condensation (2.4), the specific humidity of an air parcel at time t is the minimum qs it encounters during the time interval [0,t]. Since qs decreases monotonically with y, this implies that the random variables Q and Λ are related by We first derive the equation satisfied by the cumulative distribution function C(λ|;t) of Λ for an air parcel located at at time t. Suppose there is an absorbing barrier at y=λ. We follow a parcel backward in time according to (2.2a) and (2.2b) and remove it from the system if its trajectory (t) hits the absorbing barrier at some t>0. It then follows that where , and denote, respectively, the probability and expectation of event E1 conditioned upon event E2, and χ is the indicator function Since backward trajectories are equivalent to forward trajectories under a reversal of , (3.9) gives and it follows that C(λ|;t) satisfies the backward Kolmogorov equation [24,25], with the boundary condition C(λ|;t)=0 at y=λ. The initial condition is and we adopt the convention for the Heaviside function.

We now solve (3.12) perturbatively for C in the fast flow limit ϵ≪1. Non-dimensionalizing using , and , then suppressing the hats, (3.12) becomes Adopting polar coordinates, we expand where the powers of ϵ1/2 turn out to be required for matching with a boundary layer around r=λ. At the leading order ϵ−1, we find that which means C0=C0(λ|r;t) is constant along streamlines. Hence, at the lowest order, the moisture distribution is axisymmetrized by , as described in §3a. The next-order solution is similarly axisymmetric, C1=C1(λ|r;t). At O(ϵ0), we obtain For the solid-body rotation (3.1), ⋅∇C2=Ω∂C2. Hence, averaging (3.16) over θ eliminates the term involving C2, leading to the one-dimensional heat equation for C0 (in dimensional variables) For fast circulation, the boundary and initial conditions of C imply C0(λ|r;t)=0 for r≥λ and . The solution C0 obtained in this manner has discontinuous derivatives at t=0 and r=λ. These are smoothed out in boundary layers: a boundary layer in time of size O(ϵ) matches with the advective drying solution described in §3a; a boundary layer around r=λ of size O(ϵ1/2) where radial diffusion is important ensures a smooth transition between the positive values of C for r<λ and zero values for r≥λ. The details of the solution within the boundary layers are unimportant for C0 outside and we do not consider them further. Solving (3.17) for C0, we obtain the PDF of the maximum excursion for a parcel landing at r at time t, for r<λ. Here, J0 and J1 are the zeroth and first-order Bessel functions of the first kind, and z is the nth zero of J0. Using (3.8), we finally have the leading-order PDF of the specific humidity for a parcel arriving at position r at time t, for , where , and .

Using parameters matching those of figure 1, figure 3a plots (3.18) and (3.19) for r=π/2 at different times t. At early times, most air parcels have not moved far from their initial position. So a parcel landing at r is most probably coming from the vicinity of r, implying its maximum excursion is either equal to or only slightly larger than r, hence its specific humidity is equal to or slightly less than qs(r). As time goes by, more and more parcels have visited places with small qs and undergone condensation before arriving at r. Thus, the peak of P(λ|r,t) shifts to larger λ while that of P(q|r,t) shifts to smaller q.

Figure 3.

Moisture distribution for the initial-value problem. (a) Theoretical PDFs for the maximum excursion Λ and the specific humidity Q of a parcel at r=π/2 and different times t. To ensure convergence in the tails, these PDFs are calculated from (3.18) and (3.19) by including the first 50 terms of the series. The same parameter values as those in the Monte Carlo simulation of figure 1 are used in the formulae. (b) Profile of the ensemble mean specific humidity 〈Q〉 at different times t. The curves are theoretical predictions calculated using (3.20) and the first 10 terms in expression (3.19) for P(q|r,t). The crosses are the results from Monte Carlo simulations. The saturation specific humidity profile qs(r) is also shown. (Online version in colour.)

We now compare predictions of our theory with results from the Monte Carlo simulation described in figure 1, which has ϵ≈0.01. The fast circulation limit () assumed in the theory means that the moisture field is axisymmetrized instantaneously at t=0+. However, it always takes a finite amount of time, namely t=2πϵ (see the dimensional (3.3)), for that to happen in a simulation with small but finite ϵ. We will, therefore, compare theoretical prediction at time t with the corresponding numerical results at time t+t.

We first look at the spatial profile of the mean specific humidity Figure 3b compares simulation results for 〈Q〉 at different times with the theoretical prediction calculated from (3.19). The numerical estimate of 〈Q〉 at a given r is obtained by averaging the specific humidity Q over all the parcels located within a thin annulus of radii r±δ with δ=0.05. Note that our theoretical prediction assumes the parcels are initially distributed uniformly across the (x,y)-plane while the Monte Carlo simulation initializes parcels inside a circle of radius R only. However, since R≫α−1 and ϵ≪1, the parcels that are not sampled make a negligible contribution to the statistics. We find reasonable agreement between the theoretical and numerical results with the largest discrepancy near r=0. This is due to the lack of data points and the deviation from the fast circulation limit near r=0 (recall ||=Ωr).

We can also predict the decay of the global specific humidity , defined in (3.5), for a patch of initially saturated parcels. The circles in figure 2b show measured in the simulation. For N parcels distributed uniformly in the (x,y)-plane with number density ρ=N/(πR2), the expectation value of can be calculated as Performing the spatial integration, we obtain In contrast with P, P or 〈Q〉 in figure 3, (3.22) for is dominated by the first term, which we plot as a solid line in figure 2b. We see that the theory is in good agreement with the Monte Carlo simulation. The long-time decay of can be found from (3.22) using Laplace’s method as detailed in appendix A. The result, also plotted in figure 2b, is

A general incompressible flow

In this section, we outline an extension of the above calculation to arbitrary flows with closed streamlines. A motivation for this extension is that the transport of moisture in mid-latitudes is primarily along moist isentropic surfaces. Such transport is driven by large-scale baroclinic eddies and can roughly be modelled by a wavy velocity field in a periodic channel, which our extension covers.

The main idea is to generalize the polar coordinates (r,θ) used for axisymmetric flows to the pair (ψ,τ), where ψ is the value of the streamfunction and τ is the elapsed time along a streamline defined by where l is the arclength and the integral is along a streamline. The advective phase of the drying reduces the humidity of air parcels initially located on a streamline ψ to Q=qs(y), where y denotes the maximum value of y along the streamline. To analyse the later phase of stochastic drying, we need to consider the backward Kolmogorov equation (3.13) for C(λ|ψ,τ;t). To leading order this reduces to which implies that C0=C0(λ|ψ;t). Introducing this into (3.16) and averaging over τ yields (see [26] and appendix A of [27] for details). This heat-like equation, which reduces to (3.17) for a solid-body rotation, can be solved (numerically in general) with the initial condition C0(λ|ψ;0)=1 if y<λ and 0 otherwise. The PDFs P(λ|ψ,t) and P(q|ψ,t) follow.

Steady-state problem

Water vapour condensed and precipitated out of the atmosphere is replenished by evaporation of liquid water from the oceans and land. As mentioned in §1, the large-scale cycling of atmospheric water can sometimes be viewed as taking place inside a single overturning cell [1,20]. One question that naturally arises is: how does the moisture distribution within the cell change with the strength and other properties of the circulation? Here, we investigate this within the context of the advection–condensation paradigm.

As a simple representation of an overturning cell, we consider the velocity (u,v)=(−∂ψ,∂ψ) given by the streamfunction in a bounded domain [0,πL]×[0,πL] with reflective boundary condition; see figure 4. Recall that x and y are re-scaled to have the same typical length L as discussed near the end of §2. The evaporation source is modelled as a boundary condition at y=0: the specific humidity Q of air parcels hitting (and reflecting on) the bottom boundary is reset to qmax, the saturation value there [6,15]. The saturation profile is given by (2.1). From here on, we fix qmax=qs(0)=1 and qmin≡qs(πL)=0.01. The fate of a moist parcel under the action of large-scale circulation (4.1), Brownian motion and condensation is then governed by (2.2). If we interpret x as the meridional direction and y as the altitude, this set-up resembles the Hadley cell.

Figure 4.

Steady-state problem. (a) Schematics showing the streamlines (4.1) and the moisture source at the bottom boundary. (b and c) Snapshots from a Monte Carlo simulation of (2.2) with ϵ=10−2 showing the positions of the moist parcels; the colour scales indicate the value of (b) and the relative humidity Q/qs(Y) (c). (Online version in colour.)

Figure 4 shows a snapshot of the statistically steady state attained in a Monte Carlo simulation of the system. The domain is initially saturated. For all simulations presented in this section, we use 106 parcels. We focus on situations when the circulation is strong, with the inverse Péclet number Our simulations show that there are generally three distinct regions inside the cell:

(i) The source boundary layer. Since the circulation is tangential to the boundary, it is by means of the small-scale Brownian motion that the parcels hit the bottom boundary and moisture is injected into the domain. When the vertical random motion of the recently saturated parcels near the source is balanced by the sweeping (towards x=0) of the circulation, a boundary layer of high humidity is formed at y=0. Interestingly, as can be seen in figure 4, mixed inside this layer of mostly wet parcels are parcels with Q≈qmin that subside from aloft. This results in a bimodal local PDF P(q|x,y) for the specific humidity inside this boundary layer; see figure 5a.

Figure 5.

(a) Conditional PDF P(q|x,y) of specific humidity inside the source boundary layer at (x,y)=(L/2,L/200) for ϵ=3×10−2. The PDF consists of a continuous part (solid line) and a dry spike indicated by an arrow at q=qmin. (b) As in (a) but for (x,y)=(L/200,L/2) inside the condensation boundary layer. The dashed line marks the value of qs(L/2). (c) Mean relative humidity (RH) for parcels in the neighbourhood (within a distance of L/80) of the cell centre as a function of time t. For small ϵ, its long-time value RH approaches the minimum attainable value RH (dotted line). The inset shows the difference RH RH scaling as ϵ1/2. (Online version in colour.)

(ii) The condensation boundary layer. Advected by the circulation, the wet parcels with Q≈qmax in the source boundary layer converge towards a narrow region near x=0 before moving upward. The water vapour in these parcels then quickly condenses as qs decreases, keeping the relative humidity Q/qs(Y)≈1 (figure 4). Such a region of intense precipitation is reminiscent of the Intertropical Convergence Zone. Figure 5b shows a typical PDF of Q inside this boundary layer. Similar to (i), the dry parcels brought in by the Brownian motion give P(q|x,y) two peaks at q=qmin and q=qs(y).

(iii) The dry interior. The bulk interior (as well as the top boundary and the descending arm) of the cell is mainly occupied by parcels with Q≈qmin, creating a patch of relative humidity minimum [2] about the cell centre. This is because parcels that pick up moisture from the source are quickly advected by the circulation around the periphery, leaving the interior largely oblivious of the source. The upshot is the inner region losing its moisture through advective and stochastic drying (§3). Figure 5c shows the decrease of relative humidity at the centre of the cell with time. The equilibrium mean specific humidity inside the dry patch is maintained slightly above qmin by moisture mixing in [28] from the condensation and source boundary layers via Brownian motion.

Through the interplay between coherent stirring and small-scale random motion, our idealized model develops the interesting features of boundary layers and relative humidity minimum. This is in contrast with a one-dimensional system of Brownian parcels [15]. With the qualitative picture described above in mind, we examine quantitatively how the strength of the circulation controls the system in the next sections.

Water-vapour probability distribution function in the fast circulation limit

We first derive the steady joint PDF P(q,x,y) governing the equilibrium statistics of the parcel position and specific humidity in (2.2). The steady Fokker–Planck equation satisfied by P(q,x,y) is Following Sukhatme & Young [15], rapid condensation (2.4) implies that we only need to consider (4.3) in a region of the (q,x,y)-space where qmin≤q≤qs(y). Within this region, . We are interested in the limit of fast circulation. Thus, upon re-scaling , and suppressing the hats, we consider with ϵ≪1. From (4.4), it follows [15] that the marginal PDF This simply means that the number density of parcels is uniform over the entire cell. The boundary conditions are no-flux at all edges except the bottom one, at which P(q,x,0)=π−2δ(q−qmax) representing the source. We also know that P(q,x,π)=π−2δ(q−qmin) because when a parcel hits y=π (and then subsides) it has a probability 1 that Q=qmin. This idealization at the top edge coupled with a localized boundary source implies that P(q,x,y) generally contains a singular dry spike [15] at q=qmin, as exemplified in figure 5, in addition to a continuous part F(q,x,y):

Equations of the form (4.4) have been widely studied in different areas such as magnetohydrodynamics [29], transport in convective rolls [30,31] and two-dimensional vortex condensate [32]. As , it is well known that boundary layers of thickness ϵ1/2 form around the periphery. Following standard procedures, we introduce the von Mises transformation [33] where is the integral of the speed along the cell boundary. The speed is parametrized by the arclength l and we choose l=0 at (π,0) so that γ=0 at (π,0), 2 at (0,0), 4 at (0,π) and 6 at (π,π). The variables ψ and γ track the variation of P across and along streamlines, respectively. Inside the boundary layers, ∂∼ϵ−1/2 and ∂∼O(1) as advection along streamlines balances diffusion (of probability) across streamlines. We let and substitute into (4.4), then we have, to leading order in ϵ, On the other hand, in the cell interior outside the boundary layers as well as the corner regions, the Laplacian term in (4.4) is negligible to leading order [29,32]. Thus, we have In the following, we derive P0 for different regions of the cell.

Cell interior, boundary layers at x=π and y=π

The amplitude of the dry spike in (4.6) drops sharply from β=1 at y=π to β≈1/2 over a distance of y∼O(ϵ1/2). Inside this top boundary layer, let . We then see that rapid condensation restricts the specific humidity to lie within Since Q=qmin to leading order for all parcels, we choose not to resolve the separation between the dry spike and the smooth contribution F(q,x,y) to P(q,x,y) and we take

We now turn to the cell interior. Equation (4.10) implies that P0=P0(q,ψ) is constant along streamlines. To derive P0, we consider (4.4) at order ϵ: ⋅∇P2=∇2P0. Integrating this equation along streamlines (the same calculation that leads to (3.26)) gives the solvability condition for P2, The circulation Γ increases monotonically from Γ(0)=−8 at the boundary to Γ(1)=0 at the centre [27]. Hence, we conclude that, for finite ∂P0, P0 must be independent of ψ. By matching to the boundary layer as , we see that P0 in the cell interior is also given by (4.12).

Inside the boundary layer near x=π (where ), (4.12) provides the ‘initial’ condition (at γ=6) for (4.9) because (4.10) in the corner regions ensures P0 joins smoothly across neighbouring boundary layers. With zero-flux at the boundary and matching to the interior solution (4.12) as , it follows that P0 is once again given by (4.12).

Source boundary layer

We have seen in figure 5a an example of the bimodality of extreme high and low specific humidity in the source boundary layer near y=0 (where ). From the discussion in the previous section, we know that the dry parcels flowing in from upstream and from the interior have qmin≤QQ=qmin exactly, G is a leading-order approximation encompassing the range of Q in (4.11). From (4.9), G satisfies the heat equation The initial condition at γ=0 is obtained by joining the boundary layer upstream via the corner region at x=π. The source at the bottom edge and matching to the interior as give the boundary conditions. Thus, and the solution is

Condensation boundary layer

The condensation layer near x=0 (where ) is the region of concentrated precipitation in the model. Figure 5b shows a typical bimodal distribution of Q in this layer. Dry parcels in (4.11) once again contribute to the peak near qmin. The peak at qs(y) has an O(ϵ) width extended towards q0) are able to random walk downward against to reach y. We estimate the maximum y1 by balancing upward advection and downward Brownian motion: (in dimensional variables) for some time τ1. This leads to y1∼κU∼ϵL, which implies that these parcels have Q≈qs(y)−O(ϵ) due to rapid condensation. (This is consistent with the leading-order equation (4.9), which neglects along-flow diffusion.) The solution can, therefore, be written as where . The function G in the range 2<γ<4 satisfies the heat equation (4.15) with initial and boundary conditions with obtained from (4.17). The solution is given by Note that the integral term above tends to zero as or as expected.

Surface evaporation, boundary-layer ventilation and vertical flux

Equipped with the joint PDF P0(q,x,y), we now study the transport of moisture from the source to the upper part of the domain. The mean specific humidity at position (x,y) is given by the conditional expectation with the conditional probability density . The steady Fokker–Planck equation (4.3) implies the balance for y>0. Apart from a factor of constant air density, is the mean moisture mass condensed per unit time per unit area. By integrating (4.22) over the region above a given y and applying the divergence theorem, we find that the net upward transport of moisture mass across height y per unit time is proportional to We refer to Φ(y) as the vertical moisture flux.

The surface evaporation rate, i.e. the rate at which moisture is introduced by the source at y=0, is given by Φ(0). The idealization of Brownian small-scale motion leads to air parcels continuously picking up and losing moisture by bouncing on and off the bottom edge multiple times in quick succession. This results in an infinite Φ(0), although much of this moisture is quickly lost in the immediate vicinity of y=0 [15]. Thus, for the present model, we focus on a more relevant measure of moisture input. We define the net surface evaporation rate Enet to be the surface moisture flux attributed only to the dry air parcels, specifically parcels with Q

Figure 6.

(a) Surface evaporation rate Enet and boundary-layer ventilation Qvent defined by (4.26): the Monte Carlo estimates of Enet (crosses) and Qvent (circles) are compared with the asymptotic predictions Enet/(UL)∝ϵ1/2 in (4.24) (dash-dotted line) and ΦFT(ϵ1/2L) from (4.25) (dashed line). (b) Vertical moisture flux Φ(y): Monte Carlo results (symbols) are compared with the asymptotic prediction (4.25). (Online version in colour.)

We are also interested in the moisture flux outside the source boundary layer. In this ‘free troposphere’ of the model, (4.23) is dominated by the first term. Using P0 for the cell interior (4.12) and for the condensation boundary layer (4.18) in (4.23), we find (see appendix B for details) Figure 6b plots the scaled Φ(y) from several Monte Carlo simulations (see also appendix C) with different ϵ together with the prediction ΦFT(y). The collapse of all the data onto the theoretical curve for y≫ϵ1/2L verifies the prediction. The position where the numerical results start to deviate from the theory indicates that the thickness of the source boundary layer is of order ϵ1/2L.

The value of the moisture flux at the top of the planetary boundary layer Qvent is of particular importance for atmospheric moisture transport as it represents the amount of moisture ventilated from the boundary layer [34]. Figure 6a demonstrates the good agreement between measured from simulations and the prediction ΦFT(ϵ1/2L). Like Enet, Qvent increases with U. In fact, Qvent≈Enet for small ϵ, showing that the large-scale circulation acts like a conveyor belt: air parcels enter the source boundary layer at one end and travel within the layer to the other end where they exit, carrying with them almost all the moisture they pick up from the surface source.

Surface precipitation rate

As wet parcels emerge from the source boundary layer and move upward into regions of low saturation qs(y), condensation occurs. We assume that all condensed moisture becomes precipitation. With y interpreted as altitude and assuming precipitation falls vertically, we can consider the distribution R(x) of the surface precipitation rate. R(x) measured from Monte Carlo simulations (as described in appendix C) can have a significant contribution RBL(x) from the frequent condensation near y=0 induced by the Brownian small-scale motion described below (4.23). We generally find that RBL depends very weakly on x. So we instead consider the net surface precipitation rate Figure 7a shows Rnet normalized by U from simulations of different ϵ. When the circulation strength increases, the precipitation rate increases and the distribution of precipitation becomes more localized around x=0 (when κ is held fixed), in line with the boundary-layer thickness scaling like ϵ1/2.

Figure 7.

(a) Distribution of the net surface precipitation rate Rnet in (4.27) for different ϵ≡κ/(UL). Simulation results (symbols) are compared with the prediction (4.30) (solid lines). Precipitation increases and becomes more localized as the circulation strength U increases. (b) Mean relative humidity profile at x=πL/2 for different ϵ compared with the limiting behaviour qmin/qs(y) (dotted line). Inset: global specific humidity: the time-averaged of (3.5) in the simulation (circles) is compared with the prediction (4.32) (dashed line). (Online version in colour.)

We can calculate the leading-order Rnet(x) by considering transport inside the condensation boundary layer. Vertical transport is dominated by v, so we are in the ballistic limit studied by O’Gorman & Schneider [14]. Only parcels with Q(t)=qs[Y (t)] contribute to precipitation at time t. The amount of condensation from one such parcel between time t and t+Δt is The mean condensation rate at position (x,y) is thus where the conditional PDF is obtained from (4.18). We derive Rnet(x) by integration over y to find with G given by (4.20). Good agreement between this prediction and numerical results is seen in figure 7a.

Relative humidity and global specific humidity

When evaporation balances condensation, the system reaches a statistically steady state and the mean moisture distribution has a steady profile. The snapshot in figure 4 confirms that the relative humidity in the centre of the cell is near its minimum qmin/qs(πL/2); the inset in figure 5c shows that it decreases towards this minimum as . Here, we take a closer look by plotting in figure 7b the mean relative humidity as a function of y along a fixed x=πL/2. These profiles are obtained from Monte Carlo simulations by averaging Q/qs(Y) over parcels as well as over time. Figure 7b shows that the relative humidity decreases from y=0, reaches a minimum, then increases with y, approximately as qmin/qs(y). When the circulation strength U increases with κ fixed, the minimum relative humidity decreases and its location shifts towards y=0, or the direction of increasing qs.

It is interesting to assess the dependence of the total water-vapour content, as estimated by the global specific humidity in (3.5), on the circulation strength. The theoretical prediction for is given by the expectation value Clearly, as ϵ→0 since the area of the source and condensation boundary layers (where Q≠qmin) tends to 0 with ϵ. The leading-order correction is controlled by the solution near the corner of the domain at (x,y)=(0,0), where the moisture content is at its largest. A computation detailed in appendix D gives with the appearance of a logarithmic factor that can be traced to the streamline geometry near the corner. Thus, the total moisture increases with the diffusivity κ, that is, with the intensity of small-scale turbulence, and decreases as the strength of the large-scale circulation U increases. The inset in figure 7b confirms this result.

Discussion and conclusion

Motivated by the importance of synoptic-scale moisture transport in the atmosphere, we have studied two idealized problems based on the advection–condensation paradigm. The key element in both cases is that the advecting velocity has a large-scale coherent component in addition to small-scale white noise. The analytically tractable models introduced here capture some of the essential processes that control the large-scale dynamics of atmospheric water vapour, enabling us to examine the three-way interaction between large-scale advection, small-scale turbulence and moisture condensation.

We first study in §3 the drying of a patch of initially saturated air and show how the action of a vortex speeds up the process. We predict the long-time decay of total moisture from the statistics of maximum excursion. The drying mechanism in this initial-value problem is responsible for the creation of a dry zone in the steady-state problem discussed in §4.

For the steady-state problem, we consider the single overturning cell (4.1) on the (x,y)-plane with a moisture source at the boundary y=0. This can be interpreted as a large-scale circulating flow on an isentropic surface if we take x and y as the zonal and meridional directions, respectively. Alternatively, this set-up is a crude representation of the Hadley cell if we interpret x as latitude and y as altitude. This simple model produces some interesting features reminiscent of the atmosphere. First, a boundary layer near the source is formed as a result of the balance between large-scale and small-scale motions. This layer roughly mimics the atmospheric boundary layer whose role in moisture transport has been investigated using idealized simulations with full physics [34]. There is another boundary layer along the rising arm of the cell near x=0—the tropics of the model—where intense precipitation occurs. Second, we find that the moisture distributions inside both boundary layers are bimodal. The dry peak is a consequence of the subsidence of parcels with low humidity originated from the top of the cell. Satellite measurements indeed show that the PDF of relative humidity over the whole tropics is bimodal although the PDF within a subregion could be unimodal [35]. Finally, the coherent stirring in the model produces a region of low relative humidity about the centre of the cell. A similar dry area in the subtropics is observed in the zonal-temporal-averaged relative humidity obtained from satellite measurements [1] and reanalysis data [2]. The importance of these subtropical dry zones lies in their large influence on the radiation budget [6] and the high sensitivity of such influence to water-vapour feedback [36]. Using an idealized model, O’Gorman et al. [2] show a strong correlation between the position of the relative humidity minimum and the inflection point of the saturation profile. In our model, the minimum is located at the edge of the source boundary layer, at an altitude of a few times , independent of the details of the saturation profile.

There is a continuous interest in how climatological and seasonal variations in the strength and width of the Hadley cell [37,38] affect rainfall patterns. Some analysis associates the increase in tropical precipitation with the intensification of the Hadley circulation [39]. Increasing the strength of the circulation U in our model does increase the amount of moisture injected into the system through surface evaporation, with the specific scaling in the limit of strong circulation. This is balanced by a larger moisture flux and higher precipitation rate. The precipitation becomes more concentrated around x=0, with an extent that scales like ; as a result the local precipitation intensity increases like U. An increase in circulation strength also leads to a drier atmosphere with humidity values that are only substantially larger than qmin in the increasingly small source and condensation boundary layers. Interestingly, the net moisture input (4.24), and as a consequence the total condensation above the source boundary layer, and the total moisture (4.32) depend only on qmin and qmax rather than on the full saturation profile qs(y), which only affects the spatial distribution of rainfall. It is known that changes in global-mean evaporation and precipitation with surface temperature are strongly constrained energetically [40]. How well our simple qualitative conclusions apply to more complete models of the atmosphere remains to be assessed.

Previous work using simplified one-dimensional models has established the Lagrangian formulation of the advection–condensation paradigm as a promising strategy to investigate atmospheric water vapour. The present study provides a step forward in this direction through the analysis of a stochastic Lagrangian model that incorporates the dynamics of a two-dimensional large-scale circulation. An important extension in the future is to include the effects of latent heat by making temperature a dynamical variable and the saturation profile temperature dependent. As it is often difficult to untangle the many interacting processes in full general circulation model simulations, idealized models such as the one introduced here can help to reveal the role of specific processes in controlling the distribution of water vapour in the atmosphere.