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Keywords:

  • diffusion;
  • relative humidity;
  • specific humidity;
  • bimodal

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The AC model
  5. 3. Moist Brownian parcels
  6. 4. Steady solution with ‘resetting’ at y = 0
  7. 5. An example
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgements
  11. Appendix. Normalization as a differential boundary condition
  12. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The AC model
  5. 3. Moist Brownian parcels
  6. 4. Steady solution with ‘resetting’ at y = 0
  7. 5. An example
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgements
  11. Appendix. Normalization as a differential boundary condition
  12. 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

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

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

  • equation image(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

  • equation image(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 χ2q 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

  • equation image(3)

where τc is the time-scale associated with condensation, qs(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 τc [RIGHTWARDS ARROW] 0. In the rapid-condensation limit, the details in �� are unimportant and condensation can instead be represented as a rule enforcing subsaturation:

  • equation image(4)

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

  • equation image(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 qs(x). Thus, in the absence of sources (Sukhatme and Pierrehumbert, 2006, pers. comm.),

  • equation image(6)

where

  • equation image(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 qs(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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The AC model
  5. 3. Moist Brownian parcels
  6. 4. Steady solution with ‘resetting’ at y = 0
  7. 5. An example
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgements
  11. Appendix. Normalization as a differential boundary condition
  12. 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 qs depends only on the meridional coordinate y in the domain 0 < y < L, and that qs decreases monotonically from qmax at y = 0 to qmin 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.

thumbnail image

Figure 1. In the rapid-condensation limit the accessible domain in the (q,y) plane is the shaded region shown above. The model assumes that qs(y) decreases monotonically from qmax = qs(0) to qmin = qs(L). The functions qs(y) and ys(q) are inverses of each other: qs[ys(q)] = q.

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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:

  • equation image(8)

where W(t) is a Weiner process, κb 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 dqdy is equal to

  • equation image(9)

where N is the total number of parcels.

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

  • equation image(10)

with the normalization

  • equation image(11)

In the limit of rapid condensation, as qminq(y,t) ≤ qs(y), the normalization takes the form

  • equation image(12)

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

  • equation image(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:

  • equation image(14)

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

  • equation image(15)

Note that the left of (15) is consistent with the global normalization in (11). In the limit τc [RIGHTWARDS ARROW] 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The AC model
  5. 3. Moist Brownian parcels
  6. 4. Steady solution with ‘resetting’ at y = 0
  7. 5. An example
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgements
  11. Appendix. Normalization as a differential boundary condition
  12. 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

  • equation image(16)

For example, completely saturating the parcels at y = 0 corresponds to the choice Φ(q) = δ(qqmax). 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 τMoist(x). In effect, by resetting only at y = 0 we are considering the extreme case in which τMoist(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 = qmin, 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(qqmin), 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 equation image, with the immediate solution

  • equation image(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(qqmin). This ‘dry spike’ contains parcels created by encounters with the northern boundary at y = L. These considerations refine (17) to

  • equation image(18)

The term L−2δ(qqmin) 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

  • equation image(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 = ys(q) to re-write (19) as

  • equation image(20)

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

  • equation image(21)

where Λ(q) is the cumulative distribution

  • equation image(22)

Assembling the pieces, we obtain the PDF

  • equation image(23)

The expression above applies only within the shaded region in Figure 1 where q < qs(y); in the supersaturated region q > qs(y), P(q,y) = 0. The PDF is 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–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

  • equation image(24)

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

  • equation image(25)

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

  • equation image(26)

which is consistent with (12).

Independent of model details, such as the specification of qs(y) and Λ(q), the dry spike δ(qqmin) 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 qqmin, while the other half constitute the dry spike with q = qmin.

4.3. Averages

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

  • equation image(27)

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

  • equation image(28)

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

  • equation image(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 qmin to qmax, we have

  • equation image(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

  • equation image(31)

as

  • equation image(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 κb 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 τc [RIGHTWARDS ARROW] 0, the mean condensation rate is

  • equation image(33)

Although equation image, the mean condensation 〈��〉 in (33) is independent of τc. 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 = qs(y) in Figure 1.

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

5. An example

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

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

  • equation image(34)

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

  • equation image(35)

This form of qs has been employed previously in AC studies (Pierrehumbert et al., 2005; O'Gorman and Schneider, 2006).

With Φ(q) and qs(y) in (34) and (35), the steady solution of the Fokker–Planck equation is

  • equation image(36)

and away from the southern boundary

  • equation image(37)

The solution (37) can be non-dimensionalized with

  • equation image(38)

and P(q,y) = qmaxLP(q,y). This scaling identifies the fundamental non-dimensional control parameter α, and the scaled PDF is

  • equation image(39)

with

  • equation image(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. 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 = 0.5. Indeed, for y > 0.5 the α = 4 PDF falls monotonically from equation image. In the case with α = 1, the PDF increases with increasing q at all values of y. Therefore, in combination with equation image, the PDF is always bimodal if α < 2.

thumbnail image

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. The major axes correspond to α = 4; there is a transition from a bimodal to a unimodal PDF at y = 2 = 0.5. Larger y have smaller ranges of permissible q, and hence the curves are ordered such that y increases as we move to the left. The transition curve at y = 2 = 0.5 is shown by the dark line in the plot. The inset corresponds to α = 1. The curves are ordered as in the main plot. In this case, counting the dry spike at qmin, the PDF is bimodal at all values of y. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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(qqmin), corresponds to a singularity at qmin after which the PDF falls off, only to rise again to a secondary moist peak as q [RIGHTWARDS ARROW] qs(y). Further from the vapour source (i.e. for y > 2) the PDF is unimodal, with a dry spike at qmin followed by a gradual monotonic decay to a minimum at q = qs(y).

We proceed to discuss the implications of the solution in (37) using dimensional variables. The non-dimensional versions of all formulae are obtained with L [RIGHTWARDS ARROW] 1, α [RIGHTWARDS ARROW] α and equation image.

5.1. Statistics in a strip y1< y < y2

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 ∈ [y1,y2]. Integrating (37) over such a strip, and denoting this by P12(q), we have

  • equation image(41)

where the strip function is

  • equation image(42)

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

thumbnail image

Figure 3. Comparison between P12(q,y) from Monte Carlo with the smooth portion of the analytic result in (41). In this illustration α = 5. Crosses, open circles and stars are the Monte Carlo PDFs while the smooth solid curves are the analytical expression. Top panel: the bimodal PDFs for y < 2. Middle panel: the PDFs for larger y which show the transition from a bimodal to a unimodal distribution. Bottom panel: the unimodal PDFs far from the source, i.e. for y > 2. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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The effect of the strip function s12 is to clip the secondary peak at q = qs, and this clipping becomes more extreme if y2y1 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 y1 [RIGHTWARDS ARROW] 0 and y2 [RIGHTWARDS ARROW] 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

  • equation image(43)

where q ∈ [qmin,qmax]. 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 = 105 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, qi(0) = qmin. The parcels perform an uncorrelated random walk in y, and parcel i is reflected back into the domain if yi> L or yi < 0. The forcing �� is implemented by setting qi = qmax when a parcel is reflected at y = 0. Rapid condensation is implemented by enforcing qi = min[qi,qs(yi)] after each random displacement.

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

  • equation image(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. There is good agreement between the Monte Carlo simulation and the Fokker–Planck solution. Some minor discrepancies are evident as q [RIGHTWARDS ARROW] qmin; this is anticipated, as we have suppressed the numerical spike at q = qmin (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 = qmin followed by decay and termination at q = qs(y). Finally, the first panel of Figure 4 shows the global PDFg(q) in (43). As per the log–log inset, the global PDF closely follows the result in (43), with a spike at q = qmin followed by an algebraic tail g(q) ∼ q−1.

thumbnail image

Figure 4. Top panel: the global PDF of the specific humidity g(q) from Monte Carlo simulations. The inset shows a log–log plot, 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 numerical simulation (dotted circles). This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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

  • equation image(45)

where 〈q〉 is the mean specific humidity and κb 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 qs(y) in (35), the average gradient of specific humidity is obtained as

  • equation image(46)

where equation image 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.

thumbnail image

Figure 5. A comparison of the flux 〈ℱ〉 measured in the Monte Carlo simulation with the theoretical result using (45) and (46). This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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

  • equation image(47)

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

  • equation image(48)

where r ∈ [rmin(y),1], equation image is defined in (40) and

  • equation image(49)

A comparison between (48) and the Monte Carlo simulation is shown in Figure 6. The main mismatch we see is for r [RIGHTWARDS ARROW] 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 rmin(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 rmin(y) (suppressed in the plot), after which the PDF gradually levels off as r [RIGHTWARDS ARROW] 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.

thumbnail image

Figure 6. Comparison between Q12(r,y) from Monte Carlo with the smooth portion of the analytic result in (48). In this illustration α = 5 and y2y1 = 0.02L. Crosses, open circles and stars are the Monte Carlo PDFs while the smooth solid curves are the analytical expression. Top panel: the bimodal PDFs for y < 2 (the curve in the middle corresponds to y = 0.08). Middle panel: the PDFs for larger y, which show the transition from a bimodal to a unimodal distribution. Bottom panel: the unimodal PDFs far from the source, i.e. for y > 2. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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5.6. The global RH PDF

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

  • equation image(50)

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

  • equation image(51)

As a check, one can verify that on integration over the range of RH, that is equation imager ≤ 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 [RIGHTWARDS ARROW] 1.

6. Discussion

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

6.1. Radiative effects of water vapour

In the Introduction, we referred to the importance of the vapour PDF in understanding the effect of humidity on the OLR. In fact, the effect of humidity on OLR was quantified by Zhang et al. (2003) in terms of the PDF of the RH field. Their numerical experiments, employing a full-fledged radiation code, showed that a domain with a bimodal RH PDF allowed for significantly larger cooling than one with a unimodal distribution (holding the total and mean the same in the two cases). In the present context, we see that the PDF of the RH over the domain, i.e. the lower panel of Figure 4, is bimodal. Therefore, along with regions of moderate RH, the domain consists of subsets with very small RH (the first peak in the aforementioned plot) and subsets that are nearly saturated (the second peak). A clearer picture is seen in the upper panel of Figure 7, which shows the average specific humidity and RH over the domain, i.e. 〈q〉(y) and 〈r〉(y) respectively. While the specific humidity decays monotonically from the source, the RH has a pronounced mid-domain minimum representing a pool of ‘relatively’ dry air flanked on either side by nearly saturated regions.

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Figure 7. Upper panel: average profiles of the specific humidity and RH computed from the PDFs in (37) and (48) respectively. Lower panel: log〈q〉(y) and 〈log(q)〉(y) as computed using (28). The inset shows the difference 〈log(q)〉(y) − log〈q〉(y) and demonstrates the increase in the OLR due to fluctuations in q at every y. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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As described in PBR, the presence of water vapour affects the OLR in an approximately logarithmic fashion, i.e. OLR ∝ −log(q). Therefore, the importance of knowing the PDF of q at every y is illustrated by comparing log(〈q〉) with 〈log(q)〉: it is the latter quantity that accounts for fluctuations in the specific humidity at every location and requires a knowledge of P(q,y). As expected (see for example the discussion in PBR) and as shown in the lower panel of Figure 7 (especially the inset, which shows the difference between the two estimates), the fluctuations in the specific humidity field increase the OLR compared with 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 upper panel of Figure 4 is unimodal, while the global PDF of RH in the lower panel of Figure 4 is bimodal. Thus, according to our model, RH bimodality is so robust that it survives the mixing of PDFs from diverse locations into the global PDFh(r) in (51).

As far as we are aware, bimodal RH PDFs that have been documented in literature are mostly derived from satellite data over the deep Tropics (Brown and Zhang, 1997; Zhang et al., 2003; Mote and Frey, 2006; Luo et al., 2007).

Exceptions are the three-dimensional and isentropic model-based AC experiments by PBR and Yang and Pierrehumbert (1994) respectively: in figure 16 in PBR and figure 6 of Yang and Pierrehumbert (1994) the midlatitude RH PDF is clearly bimodal.

On the other hand, the stochastic drying models developed by Sherwood et al. (2006) and Ryoo et al. (2009) predict only unimodal RH PDFs. Indeed, Sherwood et al. (2006) concluded that a broad and unimodal RH density might be regarded as a general outcome of the AC model. This conclusion is inconsistent with our implementation of the AC model, which results in bimodal RH PDFs. We discuss this point further in section 6.3.

Because our main focus is humidity fields on middleworld isentropic surfaces, we proceed with a data comparison by analyzing the RH on the 330 K isentropic surface as presented in the ERA interim product. The data were obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) web portal (http://data-portal.ecmwf.int/data/d/interim_daily/). We analyze daily data for the year 2008 given at a horizontal resolution of 1.5° × 1.5° and at 24 vertical levels from 1000–175 mbar. As the RH data are only available at isobaric levels, we interpolate to generate RH fields on the 330 K isentrope. Further, as we are interested in the PDFs from the midlatitudinal portion of the isentropic surface, we collect data between 15° and 60° in both hemispheres.

A reviewer noted that the 330 K isentrope intersects the tropopause by 60°. To alleviate concerns that these bimodal PDFs result from sampling distinct air masses separated by a mixing barrier, we have also constructed PDFs by considering data only from the troposphere, for example, from 15° and 35° in both hemispheres, and verified that these PDFs display the same qualitative form as those in Figure 8.

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Figure 8. The midlatitudinal RH PDF on the 330 K isentrope as derived from the ERA interim data product for the individual months of 2008. Upper panel: PDFs from the two hemispheres plotted separately (ordering: from back to front January–December). Lower panel: lines with dots are the PDFs with both hemispheres taken together, and the foremost bold line with star symbols is the mean of these monthly curves. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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As seen in Figure 8, which shows the PDFs from the different months of 2008 (PDFs from the individual hemispheres are plotted in the upper panel and a composite from both hemispheres is shown in the lower panel), there are considerable differences in the RH PDFs between the two hemispheres. For example, the dry peak in the PDFs is more prominent in each hemisphere's winter season. As these differences are not of primary interest to us we do not dwell on them here. There are a few values in the midlatitudes where the ERA dataset shows supersaturation; these have been set to 100% in the present calculation. Naturally, this raises the secondary peak but does not not alter the form of the PDF. Quite clearly, the PDF does vary from month to month but its main features are robust throughout the year. More importantly, we observe that the ERA interim data PDF, in all months, is bimodal with distinct dry and moist peaks, which conforms with the results of the AC model in section 5.

6.3. Special features of the AC model in this article

In reconsidering the different ingredients in our idealized implementation of the AC model, we emphasize that there are at least four important assumptions affecting our conclusions:

  • isentropic transport is modelled as Brownian motion;

  • parcels are re-moistened only at y = 0;

  • the saturation humidity is qs(y) = qmaxeαy;

  • the model neglects diffusive processes.

The first assumption, which is equivalent to saying that the Lagrangian velocity correlation time is zero, is crucial for the derivation of the Fokker–Planck equation, and thus for all the results in this article. Moreover, following O'Gorman and Schneider (2006), we emphasize that the important condensation–diffusion balance in (32) also relies crucially on the assumption of Brownian motion. With non-zero velocity correlation time it might still be possible to argue that mean condensation 〈��〉 is balanced by diffusive convergence, that is

  • equation image(52)

However, the humidity diffusivity κq will be systematically lower than the Brownian parcel diffusivity κb. Understanding the relation between κq and κb is beyond our scope here.

The second assumption of a localized vapour source probably explains the difference between our conclusions and those of Sherwood et al. (2006) regarding bimodality of the RH PDFs. This point could be examined within the Fokker–Planck framework by modelling the source �� in (10) as a re-setting, q [RIGHTWARDS ARROW] qs(y), occurring at a rate τMoist(y) throughout the domain. We do not pursue this generalization here.

The third assumption, that the saturation humidity varies exponentially with y, is a consequence of the Clausius–Clapeyron relation and the assumption that the temperature gradient on an isentrope is uniform. We have examined alternative models of qs(y), e.g. taking qs(y) as a linear or parabolic function of y while ensuring qs(y) is still monotonically decreasing qmax at y = 0. The main qualitative features of these solutions shown in Figures 3, 4 (lower panel) and 6 are unchanged. The only PDF that is sensitive to qs(y) is the global specific humidity PDF, g(q), shown in the upper panel of Figure 4. Thus, compared with the q−1 algebraic decay in the exponential qs, a linear (parabolic) saturation qs(y) yields a uniform (growing) PDF as q [RIGHTWARDS ARROW] qmax. These differences can be traced to the form of the function ys(q) in the analytic expression for g(q) provided by (25).

The fourth assumption above is also crucial to the AC model. Haynes and Anglade (1997) estimate that in the upper troposphere and lower stratosphere the ‘mix-down’ time for synoptic-scale fluid parcels to be sheared and strained to molecular-diffusive scales is a few weeks. This time-scale is comparable to the mean residence time of water in the free troposphere (Quante and Matthias, 2006). Thus molecular diffusion between fluid parcels may have an impact on the water-vapour field, and in particular on its PDF. This important open question cannot be addressed till the advection–condensation model is extended to the advection–diffusion–condensation model.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The AC model
  5. 3. Moist Brownian parcels
  6. 4. Steady solution with ‘resetting’ at y = 0
  7. 5. An example
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgements
  11. Appendix. Normalization as a differential boundary condition
  12. 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 qmin, a rapid decay followed by a rise to a second peak as q [RIGHTWARDS ARROW] qs(y) ≈ qmax. Further from the source (i.e. for large y) the PDF is unimodal and terminates at q = qs(y) ≪ qmax. 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.

From the PDF of the specific humidity, we deduced the PDF of the RH. Once again, the PDFs near the source are bimodal with a peak at the minimum RH for a given y, i.e. at rmin(y), followed by a decay at intermediate RH values and a rapid rise as r [RIGHTWARDS ARROW] 1. The bimodality gives way to a unimodal distribution at large distances from the source, and here the PDF is controlled by the δ-function at rmin(y), which levels off to a constant as r [RIGHTWARDS ARROW] 1. The domain-averaged RH PDF displays a mix of these distinct features; in particular–even though the global specific humidity PDF is unimodal–the global RH PDF is bimodal with a primary peak at small RH followed by a decay at intermediate RH and then a secondary peak as r [RIGHTWARDS ARROW] 1.

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.

Appendix. Normalization as a differential boundary condition

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

Here we show that the normalization constraint obtained in the limit of rapid condensation can be interpreted as a differential boundary condition. Specifically, we consider this interpretation for the resetting problem and pick up the story at (18) where

  • equation image(A1)

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

  • equation image(A2)

Differentiating with respect to y, Leibnitz's rule gives

  • equation image(A3)

A second differentiation yields

  • equation image(A4)

On specifying qs(y), we see that (A4) is an ordinary differential equation for the function F(q). Indeed, (A4) is the interpretation of the normalization constraint as a differential boundary condition. To see the utility of (A4) in solving for F(q), consider qs(y) = eαy. This yields

  • equation image(A5)

and hence F(q) = α/[Lqlog2(q)], which agrees with the earlier result (37).

References

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

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

  • Both curves start at zero as we have chosen qmax = 1; in general the value at y = 0 is log(qmax).