## 1. Introduction

The meteorology of orographic precipitation was summarized by Sawyer (1956), p. 375) as being composed of three factors:

“First, there are the large-scale synoptic factors which determine the characteristics of the air mass which crosses the hills, its wind-speed and direction, its stability, and its humidity. Second, there is the dynamics of the air motion over and around the hill or hills with which we are concerned; this determines to what depth and through what layers the air mass is lifted. Thirdly, there is the microphysics of the cloud and rain, which determines whether the water which is condensed as cloud will reach the ground as rain or snow or whether it will be merely re-evaporated on the leeward side.”

The present study is concerned with the second factor as it addresses the dynamics of airflow in the observationally important case where the air mass crossing a hill is both saturated and nearly neutrally stable.

Although the significance of saturated moist neutral conditions for orographic precipitation has been debated in the literature since the 1940s (Smith 1979, 178–183), there has been very little work on this topic from a theoretical standpoint. This paucity of work is most likely due to the conceptual difficulty associated with the nonlinearity of the processes associated with latent heat release. Even in the simple case where there is no rainfall, it has long been noted that latent heating affects the dynamics only where the air is saturated, but it is such a large effect that it can render the atmosphere statically neutral or unstable locally even if it were stable to unsaturated displacements. In trying to understand the orographic-flow modification for an initially saturated flow, it is not possible to know a priori where the air will be saturated since the absence or presence of saturated regions depends on the flow itself, which is what one is trying to predict. Barcilon et al. (1979) carried out an analytical calculation of moist stable nonrotating flow in the case of a small-amplitude two-dimensional ridge, maintaining in a limited fashion the nonlinearity associated with the process of the local switching between saturated air (moist stability) and unsaturated air (dry stability) showing that the latent heating decreases wave drag.

As analytical calculations yield rather limited results, it is understandable that almost all subsequent work on the effects of moisture on orographically modified airflow has been done through numerical modeling. The first such studies solving the nonhydrostatic primitive equations, including moisture variables and a simple microphysical parameterization, were those of Durran and Klemp (1982a, 1983). These papers were concerned with cases in which the effects of moisture produce relatively modest reductions of the stability of the unsaturated profile. Both in the context of nonrotating mountain waves and lee waves, these papers showed that latent heating decreases the effective static stability and hence increases the effective vertical wavelength (since it is proportional to *U*/*N*, where *N*^{2} is the static stability and *U* is the ambient flow speed) of the waves and may thereby disrupt the conditions required for strong mountain and lee waves. Conditions were chosen such that no precipitation was expected or produced in the simulations. Systematic numerical studies of the rainfall produced with rotating saturated stable flow over a two-dimensional ridge have only recently been carried out (Colle 2004).

Miglietta and Buzzi (2001) have recently reported on simulations of nonrotating, moist stably stratified flows past idealized isolated three-dimensional obstacles. Their experiments showed that latent heating significantly changes the flow response with respect to that of the dry case. This change occurs especially in the range characterized by values of the nondimensional mountain height *h _{m}N*/

*U*(

*h*is the mountain height) in the vicinity of a bifurcation between different flow regimes. In particular, nonlinear flow features, such as wave breaking and flow splitting, occur at higher values of

_{m}*h*/

_{m}N*U*than those at which they occur for dry flows.

Systematic numerical studies of the rainfall produced with conditionally unstable flow over a two-dimensional ridge are also currently in progress (Kirshbaum and Durran 2004; Fuhrer and Schär 2002). In the latter case, the vertical motion (which leads to condensation and precipitation), although initiated by forced orographic lifting, will have an intensity dominated by the degree of convective instability rather than that implied by the orographic lifting itself. Moist unstable flows over a ridge therefore represent an entirely different category and hence are outside the scope of the present study, which is the case of moist neutral flow over a two-dimensional ridge.

Conditions of moist neutrality in association with orographic precipitation have been observed from the earliest times (e.g., Douglas and Glasspoole 1947; Sawyer, 1956). Recent observations of such conditions in connection with Alpine precipitation (Doswell et al. 1998; Buzzi and Foschini 2000; Rotunno and Ferretti 2003) have produced renewed interest in the idea that such a condition is quite common and, moreover, a basic ingredient in nonconvective flood-producing events. For example several nearly moist neutral soundings have also been observed during the Mesoscale Alpine Program (MAP) Intensive Observation Periods (IOPs). Figure 1 taken from Rotunno and Ferretti (2003), their Fig. 7a) shows the nearly saturated moist neutral state of the atmosphere approaching the Alps during MAP/IOP2b. The basic idea is that, although the upslope-forced updrafts in neutral flow are normally weaker (0.1–1 m s^{−1}) than convective updrafts, they may, however, be maintained for long periods by slowly moving synoptic-scale systems. Moreover numerical sensitivity studies have shown (Buzzi et al. 1998; Ferretti et al. 2000) the critical importance of latent heating in allowing moist flow to surmount the 3-km-tall barrier presented by the Alps to the oncoming flow and hence in allowing the condensation of large amounts of water vapor. Moist neutrality is hence of interest since it represents the limiting case in which no gravitational resistance is offered to air as it tries to surmount a barrier.

As mentioned above there has been very little modeling/theoretical work on orographic-flow modification in the limit of moist neutral conditions. There is the single paper by Sarker (1967) who solved analytically the (linear) Taylor–Goldstein equation with zero moist stability (but with a nonconstant wind shear profile) with application to orographic rainfall over the western Ghats (India). This approach appears to be valid as long as the atmosphere remains saturated both upstream and downstream of the ridge in question. Rotunno and Ferretti (2001), their section 4c) pointed out that, for the most part, the effects of moisture have been understood as a qualitative extension of the known results for orographic-flow modification in the absence of latent heating by a simple reduction of static stability. However Rotunno and Ferretti (2001) also observe that taking the static stability to zero would be problematic for an interpretation based on a simple extension of the results from dry dynamics since the available evidence indicates that orographic flows tend not to remain everywhere saturated. Rotunno and Ferretti (2001) performed idealized numerical calculations for a saturated nearly neutral (constant equivalent potential temperature) ambient flow, including for the first time the effects of rotation, to help explain results from their full-physics numerical simulations of a case of intense Alpine rainfall.

In the present paper, we report on our attempts to study the case of nonrotating orographic-flow modification over a two-dimensional ridge under conditions of saturated moist neutrality. The plan of the present paper is as follows. First, we analyze the conditions that must be imposed in order to obtain a moist neutral atmospheric vertical profile. Section 3 describes the model characteristics and the numerical setup of the experiments to be reported. Section 4 presents the numerical solutions for saturated nearly moist neutral flow past a small-amplitude bell-shaped mountain with comparison against linear theory. Section 5 is a study of the numerical solutions for large mountain heights; several novel features emerge and are analyzed. Section 6 summarizes the results.

## 2. Creation of a moist neutral initial state

*N*, which according to Lalas and Einaudi (1974) iswhere

_{m}*z*is the vertical coordinate,

*g*the gravitational acceleration,

*T*the atmospheric temperature,

*γ*the saturated adiabatic lapse rate;

_{m}*L*is the latent heat of vaporization,

_{υ}*R*is the ideal gas constant for dry air,

_{d}*q*is the water vapor saturation mixing ratio, and

_{s}*q*is the total water mixing ratio. Equivalent expressions have been calculated by Durran and Klemp (1982b), hereafter DK), Emanuel (1994) and Richiardone and Giusti (2001). We note that the Clausius–Clapeyron equation for the saturation vapor pressure

_{w}*e*(

_{s}*T*) has been used in the derivation of (2.1). Moreover

*γ*is as yet unknown and must be determined through the first law of thermodynamics. Since different numerical models may use different

_{m}*e*(

_{s}*T*) (e.g., based on empirical formulas), the first thing to be done here is to develop a formula analogous to (2.1) for the Weather Research and Forecasting (WRF) model (described in section 3) used in the present study.

*e*(

_{s}*T*) we derive in appendix A the expressionwhere all definitions may be found in appendix A. To make further progress one needs to determine

*d*ln

*θ*/

*dz*|

_{p}in terms of environmental variables. To do this we use the approximate form of the first law of thermodynamics used in the WRF model:Using (2.3) and expressing

*q*=

_{s}*q*(ln

_{s}*θ*,

*π*) to compute

*d*ln

*θ*/

*dz*|

_{p}, the final expression for

*N*

^{2}

_{m}after some manipulation becomeswithAll variables in (2.4)–(2.5) refer to the environment. To calculate Γ, we use the definition of

*q*[=

_{s}*ϵe*/(

_{s}*p*−

*e*)] and take(Wexler's formula, which is the one used in the Kessler scheme adopted here). With the approximations

_{s}*q*

_{s}≈

*ϵ*

*e*

_{s}/

*p*,

*q*

_{s}≪

*ϵ*,

*q*

_{w}≪ 1 and with

*e*(

_{s}*T*) given by the Clausius–Clapeyron equation, it is easy to show thatand that (2.4) with (2.5) reduces to DK's Eq. (36).

^{1}

Now that all variables in (2.4) are environmental variables, we are in a position to ask what vertical distribution of those variables corresponds to a moist neutral (*N*^{2}_{m} = 0) atmosphere. The term involving Γ in (2.4) is proportional to the model's version of the lapse rate of equivalent potential temperature *θ _{e}*; as was observed by Lalas and Einaudi (1974), p. 323), an atmosphere with

*θ*(

_{e}*z*) = const is, in fact, slightly stable since the second term in (2.4) tends to be positive as

*q*(

_{w}*z*) ≈

*q*(

_{s}*z*) and the latter decreases with height. By the same reasoning, for conditions of constant ambient cloud water, an atmosphere with

*N*

^{2}

_{m}= 0, must have

*θ*decreasing with height. [For this case of constant cloud water, we note that Eq. (10) of Richiardone and Giusti (2001) is an exact analytic expression for the temperature lapse rate −

_{e}*dT*/

*dz*corresponding to the state

*N*

^{2}

_{m}= 0.] For the sake of simplicity we study here initial states in which

*q*is a constant and

_{c}*N*

^{2}

_{m}(

*z*) is prescribed.

*z*to

*p*and hence allows (2.4) be solved as a first-order nonlinear differential equation that determines

*T*(

*p*) given the surface temperature. The finite-difference form of (2.4) [using (2.6)] relating variables at pressure level

*j*+ 1 to those at the lower pressure level

*j*is

The solution is considered converged when the difference between the lhs and the rhs is less than an imposed accuracy error. This error is chosen to be equal to 10^{−6}, which is the limit of our computer single-precision accuracy. Since we are limited to single-precision accuracy with the current version of the WRF model, we found that we were only able to produce with confidence profiles with *N*^{2}_{m} approaching 10^{−6} s^{−2}. In the current study we will consider a two-layer atmosphere with *N*^{2}_{m} = 3 × 10^{−6} s^{−2} in the troposphere and an isothermal stratosphere where *N*^{2}_{m} = 4 × 10^{−4} s^{−2}. Smaller values of *N*^{2}_{m} in the troposphere produced solutions that appeared to be convectively unstable. Hence, the value of *N*^{2}_{m} we choose for the troposphere is small enough to reproduce nearly neutral conditions in the troposphere but large enough to avoid problems related to the limited computer accuracy. Figure 2 shows the corresponding thermodynamic profile (the thick line) used for the majority of simulations in this study. Also shown in Fig. 2 is a profile with *θ _{e}* = const [i.e., a profile calculated by setting to zero the first term in (2.4)] in the troposphere (the thick dashed line); according to (2.4) this profile corresponds to moist stability of

*N*

^{2}

_{m}≈ 1.5 × 10

^{−5}s

^{−2}. Since it is difficult from an observational point of view to measure such fine distinctions in moist stability we will also discuss solutions for the case with

*θ*= const.

_{e}## 3. Model characteristics and numerical setup

*a*equal to 10 km. The latter is chosen with the recognition that for much smaller values of

*a*, the typical time scale for the microphysical processes to produce rain that falls to the ground (roughly 20 min—Sawyer's third factor, see the introduction) is larger than the typical advective time scale (

*a*/

*U*) and so for

*a*≪ 10 km orographic effects on rainfall would be much harder to detect. On the other hand, for

*a*much larger than 10 km one would need to consider effects of the earth's rotation, which we have neglected here to focus attention on the simplest nontrivial problem.

To achieve the above-stated objective we have carried out numerical integrations using the WRF model (Michalakes et al. 2001; Skamarock et al. 2001; more information available online at www.wrf-model.org). WRF solves the fully compressible, nonhydrostatic three-dimensional equations of fluid motion. In the present application, the height-based, terrain-following-coordinate version of the model has been chosen (WRF version 1.3). We have configured the model to solve the equations on the domain shown in Fig. 3; the chosen domain is 800 km wide (in the *x* direction) and 20 km deep (in the *z* direction) with corresponding grid resolutions of Δ*x* = 2 km and Δ*z* = 0.25 km. The WRF model uses a split-explicit technique to advance the equations in time (Klemp and Wilhelmson 1978); in the present application we have chosen a time step of 20 s for the advection terms and a 2-s time step for the fast-moving acoustic modes. We have also selected the third-order Runge–Kutta technique option for the time integration, the fifth-order scheme for horizontal advection, and the third-order scheme for vertical advection following the suggestion of Wicker and Skamarock (2002). The simulations are done with a second-order filter in the *x* and *z* directions with filter coefficients of 3000 and 3 m^{2} s^{−1}, respectively.

A wave radiation condition is used at the horizontal boundaries. At the upper boundary the vertical velocity is set to zero and a damping layer is used in the uppermost 5 km to absorb downward-reflected wave energy. Along the lower surface (3.1) there is no normal flow and the stress is set to zero.

The simulations are initialized using the saturated sounding shown in Fig. 2 and a constant wind *U* = 10 m s^{−1}. In the control case, we prescribed a small uniform initial cloud water content *q _{c}* (=0.05 g kg

^{−1}), but as described in section 5, we performed experiments with other values. The mountain is introduced impulsively at

*t*= 0 into the basic flow and produces a potential flow condition at

*t*= 0

^{+}, which subsequently evolves with time as buoyancy effects come into play.

As noted above, ice microphysics is neglected for simplicity and accordingly we have used the widely known Kessler parameterization (see, e.g., Emanuel 1994, p. 296) in all simulations reported here. The moist nearly neutral case considered here revealed some oddities that required minor modifications of the Kessler scheme (see appendix B).

## 4. Small-amplitude mountains

In the present section, we consider the solutions for flows over small-amplitude mountains. Steady-state solutions are first obtained in the numerical integrations and then we show how those solutions are well described by linear theory.

### a. Numerical integrations

In this first example we choose *h _{m}* = 50 m and, as mentioned above,

*q*= 0.05 g kg

_{c}^{−1}. Figure 4a shows the vertical velocity field

*w*after the flow has reached a steady state. The picture here is basically as one might expect as the

*w*field indicates there is the maximum upward displacement at the hill crest. At first glance

*w*appears as it would in potential flow (maximum at the surface and decreasing in magnitude with height); however, Fig. 4b shows that the horizontal perturbation velocity

*u*does not conform to that description. We will describe below how these flow features are consistent with the slightly positive

*N*implied by the sounding (Fig. 2).

_{m}The solution shown in Fig. 4 is deceptively simple. When the hill is put into motion at *t* = 0, the initial potential flow implies a downward motion on the lee side, which, if the initial *q _{c}* were not sufficiently large, would desaturate the air and hence produce a zone of locally high stability and the solution would dramatically change in character. We will encounter and analyze this effect in the following section describing solutions for large-amplitude mountains. For clarity of exposition, we restrict attention in this section to the relatively simple case with flow everywhere saturated as shown in Fig. 4.

Testing shows that solution patterns are independent of the vertical resolution of the model in the troposphere, but are quite dependent on it in the stratosphere where the intensity of the perturbation doubles for a case with twice the vertical resolution. Apparently, the small vertical wavelength of the stable waves in the stratosphere (< 3 km), due to the large Brunt–Väisälä frequency, requires a high resolution to be reproduced correctly. Other sensitivity experiments have been undertaken with twice the horizontal resolution and show that flow patterns do not change significantly.

The velocity pattern, in particular the *u* component, is quite sensitive to small changes in *N _{m}*. We have observed that for values of

*N*

^{2}

_{m}even just a little different from 3 × 10

^{−6}s

^{−2}, the vertical profile of

*u*over the ridge may change considerably, exhibiting, for example, negative values near the hill top. We show below that this sensitivity is consistent with the linear theory.

### b. Linear theory

*N*in the troposphere and a much larger stability

_{m}*N*in the stratosphere. Such calculations are standard (see, e.g., Klemp and Lilly 1975) and only the results will be given here. The linear solution for the lower layer of a two-layer atmosphere for a single Fourier component [

_{s}

*u**k, z*),

*w**k, z*)]exp(

*ikx*) isandwhere

*n*=

*N*

^{2}

_{s}/

*U*

^{2}−

*k*

^{2}

*m*=

*k*

^{2}−

*N*

^{2}

_{m}/

*U*

^{2}

*H*is the depth of the troposphere, and

*k*is the wavenumber.

*N*is large so that

_{s}*n*≈

*N*/

_{s}*U*. On the other hand the Fourier components of the assumed topography (3.1) may have significant amplitude at wavenumbers

*k*<

*N*/

_{m}*U*. [If

*k*<

*N*/

_{m}*U*,

*m*is imaginary and cosh(

*ix*) → cos

*x*and sinh(

*ix*) →

*i*sin

*x*in (4.1)–(4.2).] Hence we have the situation where

*n*/|

*m*| ≫ 1, which may be interpreted as indicating that the strong change in stability at the tropopause acts to reflect mountain-produced waves back into the troposphere. Taking

*n*/|

*m*| ≫ 1, (4.1)–(4.2) simplify toand

*m*were real for all

*k*, then inspection of (4.3) shows that

*u*would always decay with height from a maximum at

*z*= 0. Since

*u*tends increase with height in the troposphere, we are led to examine the case where

*k*<

*N*/

_{m}*U*; for simplicity consider the extreme case

*m*=

*iN*/

_{m}*U*so that (4.3)–(4.4) imply thatandWith

*N*

^{2}

_{m}= 3.0 × 10

^{−6}s

^{−2},

*H*= 11.7 km, and

*U*= 10 m s

^{−1}, (4.5) shows that the

*u*component increases with height (dashed line in the inset in Fig. 4b), while (4.6) shows that the

*w*component is maximum close to the ground and decays with height (dashed line in the inset in Fig. 4a). For both wind components, (4.5)–(4.6) imply that the phase lines are vertical as in a neutral flow pattern; however in this case the vertical phase lines are an indication of the interference between upward- and downward-propagating waves, rather than being an indication of potential flow. Included in the insets in Figs. 4a,b are curves corresponding to

*N*

^{2}

_{m}= 1.7 × 10

^{−6}s

^{−2}showing that slight changes in

*N*imply significant shifts in the vertical profile of

_{m}*u*, consistent with our experience from the numerical solutions mentioned above.

## 5. Large-amplitude mountains

Entering the regime of large-amplitude mountains, the present simulations of moist nearly neutral flow exhibit several responses that have no analog in simulations of dry stably stratified atmospheres. Out of the many experiments that we have done with *h _{m}* ranging from 50 to 3000 m, we have attempted to discuss here in detail only the few that most clearly illustrate the unique features of the orographic-flow modification in the regime under consideration. As discussed in the introduction, the major uncertainty in the present problem is that associated with whether and where the air remains saturated. In this section we discuss cases where desaturation occurs and examine its consequences.

The impulsive introduction of the mountain at *t* = 0 generates at *t* = 0^{+} a potential flow with initial vertical displacements *O*(*h _{m}*). In the previous section we discussed the case of small-amplitude mountains where the displacements are so small that a small amount of initial cloud water can prevent the appearance of zones of unsaturated air. Examination of the initial sounding (Fig. 2) in the lower to middle troposphere indicates that a downward vertical displacement of an air parcel by 500 m would require roughly 1 g kg

^{−1}of evaporated cloud water to maintain parcel saturation. Since the latter amount is close to the usual 1 g kg

^{−1}threshold for the autoconversion of cloud water

*q*to rainwater

_{c}*q*in the Kessler scheme (Emanuel 1994, his Table 10.1), for mountains with

_{r}*h*

_{m}≳ 500 m, it becomes increasingly more difficult to maintain saturation during the initial adjustment of the airflow.

In the first sequence of simulations discussed below, we present for continuity with the previous section simulations for *h _{m}* = 700 and 2000 m using the same small initial

*q*(=0.05 g kg

_{c}^{−1}). The autoconversion to rain was withheld for the first 5 hours so that the effects of the initial adjustment could be clearly isolated. Desaturation occurs shortly after

*t*= 0 and has some remarkable consequences, which are described in section 5a. In section 5b we describe simulations with greater initial

*q*(=0.50 g kg

_{c}^{−1}) in combination with other strategies that attempt to avoid the desaturation originating with the potential-flow initial condition.

### a. Initial q_{c} = 0.05 g kg^{−1}

Figure 5 shows aspects of the numerical solution for the case *h _{m}* = 700 m at

*t*= 5 h when the flow in the vicinity of the mountain is nearly steady. The most striking feature of the solution is the midlevel zone of unsaturated air that extends both upstream and downstream of the mountain (Fig. 5d). As discussed above, the initial adjustment of the airflow to the forward motion of the obstacle requires leeside downward motion that leads to a local desaturation of the air; subsequently however, the zone of unsaturated air propagates both upstream and downstream. The structure of the flow response (Figs. 5a,c) is quite different from that of the linear solution (Figs. 4a,b) and is obviously related to the drastic change in static stability associated with the unsaturated layer that develops in the solution (Fig. 5d). Since autoconversion of cloud water to rainwater has been withheld during the first 5 hours, effects leading to desaturated conditions may all be traced back to the evolution of the flow from its potential-flow initial condition. [To gain confidence that the above-described response is not a model artifact we repeated this experiment with different momentum-advection (the second-order scheme both for horizontal and vertical advection) and time-integration schemes (the second-order Runge–Kutta), and found that the upwind-propagating disturbance appeared in all of them.]

*b*that, under saturated moist neutral conditions, becomeswhere

*H*(

*ζ*) is the Heavyside function,

*ζ*(

*x*,

*z*,

*t*) is the vertical displacement, and the dry stability is

*N*

^{2}

_{d}≈ 10

^{−4}s

^{−2}. Equation (5.1) indicates that

*b*= 0 for

*ζ*> 0 (saturated regions) and

*b*= −

*N*

^{2}

_{d}

*ζ*for

*ζ*< 0 (unsaturated regions). Referring back to Fig. 5, one can see that over the mountain there is a zone of

*b*> 0 corresponding roughly to the region of descent; at lower levels, where saturation is maintained,

*b*≈ 0. As a result of this desaturation of the middle troposphere, in effect a strong jump develops in static stability near

*z*= 4 km that, as discussed in section 4b, acts to reflect wave energy below; the patterns of

*u*and

*w*below

*z*= 4 km are consistent with this interpretation. Air parcels in the unsaturated layer above the mountain (4 <

*z*< 8 km) descend, experience strong warming, and subsequently experience strong positive vertical motion (Fig. 5a) and resaturation (Fig. 5d).

*ζ >*0 with

*b*= 0 upwind, and

*ζ*< 0 with

*b*> 0 downwind. The local maximum in

*b*produces initially a dipole in the

*y*component of vorticity

*η*through the equation:(see, e.g., Rotunno et al. 1988). The implied counterclockwise circulation (

*u*,

*w*) on the left side of the positive buoyancy anomaly has downward motion in the saturated air; this downward motion is apparently enough to desaturate the air through (5.1) and produces a leftward extension of the positive buoyancy anomaly. Thereafter a steadily propagating signal is established so that

*∂*→ −

_{t}η*cη*, where

_{x}*c*(< 0) is the speed of propagation. Including the effect of the mean wind it is easy to show thatso

*η*∼ −

*b*. Analysis of this signal as it occurs in the present series of simulations indicates that it moves at the group velocity

*U*−

*N*/

_{d}d*π*for hydrostatic internal gravity waves trapped in a duct of vertical extent

*d*corresponding to the unsaturated layer (e.g., as shown in Fig. 5d). To illustrate these points we consider, in Fig. 6,

*q*(

_{c}*x*),

*w*(

*x*), and

*b*(

*x*) at

*z*= 4 km for

*t*= 1, 2, and 3 h. Figure 6a shows that the zone where

*q*= 0 corresponds to the maximum in

_{c}*b*; since

*η*≈ −

*w*, it is clear that

_{x}*η*∼ −

*b*, as argued above. As time goes by, Figs. 6b,c show how the upwind-propagating signal emerges and leaves behind a stationary disturbance over the mountain.

Simulations with much higher mountains produced further unanticipated results. Figure 7 shows the numerical solution in the case with *h _{m}* = 2000 m at

*t*= 19 h. The solution in this regime shows no evidence of the upstream-propagating disturbance discussed above. In this case there is a larger amount of cloud water produced by the greater vertical displacement; the buoyancy anomaly produced in the initial leeside descent and desaturation process is apparently not strong enough to produce a circulation with sufficient strength to desaturate the air upwind of it. Thus, the upstream solution is everywhere saturated and becomes quasi stationary after just a few hours. To be a little more quantitative we show in Table 1 an estimate of maximum upslope wind

*w*

_{up}=

*Uh*/

_{m}*a*and the maximum buoyancy anomaly at

*z*= 4 km and at

*t*= 1 h (as in Fig. 6a) versus mountain height

*h*. This table shows that for

_{m}*h*> 1500 m,

_{m}*b*

_{max}effectively reaches a limit, while

*w*

_{up}continues to increase. We speculate that this limit for

*b*

_{max}is related to the fact that the downwind parcel displacements

*ζ*are limited for large values of

*N*/

_{d}h_{m}*U*and thus through (5.1), so is

*b*

_{max}.

On the other hand, the solution behavior downstream is rather complex. On the lee side close to the ground the wind is strong (32 m s^{−1}), and the atmosphere resembles the classical structure observed in dry downslope windstorm conditions (see, e.g., Durran and Klemp 1987). We also observe the formation of a transient rain cell associated with a hydraulic-jump-like feature downwind in this case. Analysis indicates that the presence of nonnegligible amounts of rainwater in this case leads to local regions of *N*^{2}_{m} < 0. The reasons for this can be understood by making reference to the definition of *N*^{2}_{m}(2.4). As discussed in section 2, (2.4) is composed of two terms, the first proportional to *dθ _{e}*/

*dz*and the second to −

*dq*/

_{w}*dz*; in the initial state (Fig. 2), the first term is designed to be negative so that it would mostly offset the necessarily positive second term. Figure 8 shows several typical rain cells along with

*N*

^{2}

_{m}and the constituent terms in (2.4); the rain cells are clearly associated with regions where

*N*

^{2}

_{m}< 0 (Fig. 8b). The contributions from the two terms in (2.4) show that, while the first term continues to be negative (Fig. 8c), the offsetting influence of the second term is reduced (Fig. 8d) and instability results. In fact, it follows from the conservation of

*θ*and

_{e}*q*[assuming Γ,

_{w}*T*constant and neglecting terms

*O*(

*q*

^{−1}

_{w})] that

*dN*

^{2}

_{m}/

*dt*≈ 0 if

*N*

^{2}

_{m}≈ 0 initially. However, once there is rainfall

*q*is no longer conserved,

_{w}*dN*

^{2}

_{m}/

*dt*≠ 0 and instability may develop.

### b. Initial q_{c} = 0.50 g kg^{−1}

Considering that all the above-described behavior can be traced back to the early time adjustment from the initially saturated conditions in Fig. 2, one may wonder if such behavior would result if the model atmosphere were to be somehow maintained in a saturated state. As described above, simply putting in more initial cloud water is not enough to prevent desaturation for mountains much higher than ∼ 500 m. Through trial and error we found that for an initial *q _{c}* = 0.50 g kg

^{−1}with

*h*= 700 m, saturated conditions could be maintained as the flow evolved from the potential-flow initial condition. However, with the larger initial

_{m}*q*, the threshold for autoconversion to rainwater (1 g kg

_{c}^{−1}) is readily reached over the mountain. To separate clearly the effects of rainfall from these associated with the initial adjustment, we set the autoconversion coefficient

*c*

_{1}(see appendix B) to zero for the first 10 hours of simulation; after

*t*= 10 h the autoconversion coefficient was restored to its proper value. We note here that in the first simulations we did with

*q*= 0.50 g kg

_{c}^{−1}the appearance of rain after

*t*= 14 h rapidly led to unsaturated conditions everywhere in the domain. This odd behavior necessitated a minor change in the Kessler microphysics, which we describe in appendix B.

Figure 9 shows the solution at *t* = 10 h, just before the autoconversion is switched on. The solution in essence is identical to that obtained in the small-amplitude-mountain limit (Fig. 4), which, as noted in section 4b, is well described by linear theory. After *t* = 10 h, Fig. 10 shows that the solution changes dramatically as the autoconversion allows the formation of rainwater, which then falls to ground and so is no longer available for reevaporation in the leeside descending motion. Figure 10 shows that the basic features identified for the cases with small initial *q _{c}* = 0.05 g kg

^{−1}are still present (upstream-propagating region of subsaturated air, downstream unstable cells) and hence are not peculiar to the evolution away from the initial potential flow. Performing the same experiment with the higher mountain

*h*= 2000 m leads to a similar conclusion in that a flow much like that obtained in Fig. 7 results.

_{m}### c. Initial q_{c} = 0.50 g kg^{−1}, *θ*_{e}(z) = const

Although the observational motivation for the present study is based on soundings like the one shown in Fig. 1, it is difficult to say whether that sounding is best characterized by *θ _{e}*(

*z*) = const, or rather the idealized sounding in Fig. 2 where

*θ*(

_{e}*z*) decreases slightly with

*z*. To cover the range of possibilities, we show in this subsection the solution with

*θ*(

_{e}*z*) = const (the dashed line in Fig. 2) with initial

*q*= 0.50 g kg

_{c}^{−1}in the case where

*h*= 700 m. The autoconversion of rain is withheld until

_{m}*t*= 5 h. Figure 11 shows that the solution at

*t*= 11 h is similar to that described in Fig. 5 (

*q*= 0.05 g kg

_{c}^{−1}) in that there is an apparently trapped wave below a subsaturated zone at mid levels that propagates upstream. Absent are the downstream cells evident in Fig. 10 (

*q*= 0.50 g kg

_{c}^{−1}); this is consistent with the explanation embodied in Fig. 8, which traces the development of instability to the initial profile in which

*θ*(

_{e}*z*) decreases with

*z*.

### d. Summary

A summary of the main results for all the simulations performed for this study is presented in Table 2. Generally speaking the simulations behaved according to the linear theory for mountain heights less than 250 m, as no rain was produced and no desaturated layers formed. For mountain heights greater than 2000 m, the atmosphere upwind remained saturated, and large rain rates (which varied in proportion with mountain height) were produced on the windward slope. For intermediate mountain heights the solution behavior was more complex involving a subtle balance among the tendencies that determine the formation of the desaturated layer. In all these cases, a desaturated layer formed but with varying size and ability to propagate upstream; these layers had a strong effect on the rain rate since contributions to condensation in the affected layers were essentially zero. The last two columns show the accumulated rain total and that due to cells downstream; the downstream cells apparently can make a significant contribution to the total under some parameter settings. Finally, the main effect of having *θ _{e}* = const was the elimination of the downstream instability.

### e. Discussion

The flow patterns obtained here for the case with *h _{m}* = 2000 m basically look similar to those from previous studies (e.g., Figs. 14a,b of Rotunno and Ferretti 2001) and from observations of orographic rain (Fig. 12 of Medina and Houze 2003). Moreover the numerical solutions for small-amplitude mountains conform to expectations based on linear theory as demonstrated in section 4. However, the solution for intermediate mountains (

*h*= 500–1000 m) has the odd and unexpected feature of an upstream-propagating disturbance that acts to desaturate the initially saturated sounding. Since the authors are unaware of any observations showing such behavior, it is natural to conclude that something is missing from our idealization of the upwind conditions in orographic-rain scenarios. Attempts to avert desaturation with large amounts of initial

_{m}*q*are ultimately thwarted by the microphysical parameterization that converts cloud water to rainwater at a threshold far below that required to maintain saturation. Our alternative attempt at initialization using

_{c}*q*below the threshold and also withholding autoconversion until an everywhere-saturated steady state is achieved, also failed to prevent the formation of an upwind propagation of unsaturated air, once autoconversion was reactivated. Pending evidence to the contrary we conclude that in actual cases of orographic rain over intermediate-sized mountains, there must be some large-scale lifting that opposes the tendency toward desaturation identified in the present idealized study. Such a conclusion is consistent with some of the earliest studies of orographic influences on rainfall over relatively low hills (Browning et al. 1974).

_{c}## 6. Conclusions

The presence of nearly moist neutral lapse rates in cases of heavy orographic rain has been noted since the 1940s (Douglas and Glasspoole 1947) and identified by Sawyer (1956) as one of the conditions favorable for orographic enhancement of rainfall, as it allows for the complete lifting of oncoming air by the obstacle. In spite of its observational and theoretical importance, no numerical study has attempted to analyze the moist neutral regime for orographic-flow modification in the simplified two-dimensional context.

In this first systematic study of orographic-flow modification under moist neutral conditions, we identified a number of obstacles ranging from the definition of a moist stability consistent with the model equations and the accuracy requirements for its computation (section 2 and appendix A) to the odd behavior of the standard Kessler parameterization for the conditions under study here (appendix B).

The very nature of the strong nonlinearity inherent in the internal switching between weak stability in saturated zones and strong stability in unsaturated zones makes a strong microphysical sensitivity likely. The Kessler scheme used in the present study, for all its previously noted shortcomings, is at least well understood (Emanuel 1994). Hence the present study is to be regarded as a first attempt, with the understanding that studies using more refined microphysical schemes will be necessary to truly assess the range of orographic-flow responses identified in the present study.

Those responses can be summarized as falling into three categories:

- Small-mountain regime (
*h*= 0–250 m): If a mountain is small enough, it is always possible to add a small amount of initial cloud water to prevent the appearance of unsaturated regions. The numerical solutions are well described by linear theory using the moist stability in place of the dry stability. A novel feature of the latter is that the moist tropospheric stability used here is much smaller than that of the stratosphere so that wave reflection off the tropopause plays a prominent role._{m} - High-mountain regime (
*h*> 1500 m): For tall mountains, the numerical solutions indicate that the upwind atmosphere remains saturated; in the unsaturated lower troposphere downstream, features associated with downslope windstorms characterize the solution; for the case of a nearly neutral state, convective cells form as a consequence of locally generated zones of moist instability._{m} - Intermediate-mountain regime (
*h*= 250–1500 m): The flow in this regime is characterized by an upwind-propagating disturbance that desaturates the initial sounding. The presence of this unsaturated layer of high stability at mid level produces a flow consistent with that expected in the case of a two-layer dry troposphere with a neutral layer surmounted by a stable layer. Experiments done in the course of the study show that for even smaller amounts of initial_{m}*q*the upwind-propagating signal can occur for smaller mountains._{c}

This theoretical study is confined to the case of nonrotating flow past two-dimensional mountains. We are currently performing simulations including the effect of the Coriolis force and will be investigating the flow response for three-dimensional obstacles. Motivated in part by the present work, Ralph et al. (2005) have just completed a study of thermodynamic conditions observed in two recent field campaigns upwind of California's coastal orography. They found that the atmosphere below approximately 3 km was essentially moist neutral (by the precise definition discussed in section 2) for cases of heavy orographic rain.

## Acknowledgments

We thank Joe Klemp, Jimy Dudhia, and Bill Skamarock for their help with the WRF model simulations and George Bryan for helping us identify and solve the problem discussed in appendix B. The stay of M. Miglietta in Boulder was supported in part by the CNR Short-Term Mobility Program and in part by NCAR.

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## APPENDIX A

### Derivation of N2m for Arbitrary es(T)

*w*is the vertical velocity,

*p*the pressure, and

*ρ*the total density, which may be expressed aswhere

_{m}*ρ*is the density of dry air; ice is neglected in this study for simplicity. Next we express

_{d}*p*and

*ρ*, respectively, as the sum of an initial undisturbed state variable, which is a function of

_{d}*z*only (denoted by an overbar), and of a perturbation term (denoted with a prime). (This is done to follow the WRF model in which the basic-state variables are defined only for a dry atmosphere, and the moisture is considered as a perturbation to the basic state.) Using the hydrostatic equation [(A.1) with the lhs = 0] to relate the basic-state pressure to the basic-state dry density, we write (A.1) asThe last term on the rhs is formally the buoyancy, which due to the WRF-defined dry base state, is balanced by the pressure gradient. A more meaningful definition of the buoyancy is obtained by subtracting its initial value from the last term on the rhs and then dividing by

*ρ*

_{d}to arrive at the expression for the buoyancy:

*N*:where the subscripts

_{m}*p*and

*e*denote the values for the parcel and for the environment, respectively. Substituting the equation of state,(where

*ϵ*=

*R*/

_{d}*R*and

_{υ}*R*is the gas constant for water vapor) into (A.5), we arrive at DK's (10):Assuming that a displaced parcel maintains its total liquid waterand that the parcel pressure adjusts instantaneously to that of the environment, (A.7) becomeswhere

_{υ}*θ*is the potential temperature

*T*/

*π*[

*π*= (

*p*/

*p*

_{0})

*R*/

_{d}*c*and

_{p}*c*is the heat capacity of dry air]. At this point we depart from DK's derivation to allow for arbitrary formulas

_{p}*ε*

_{s}(

*T*) [or equivalently,

*q*(

_{s}*T*,

*p*)]. Since

*q*is a state variable,

_{s}*q*=

_{s}*q*(ln

_{s}*θ*,

*π*); recalling that

*p*is assumed the same for both parcel and environment, (A.8) may be written as

## APPENDIX B

### A Modification to Kessler Microphysics

*q*> 0.2 g kg

_{c}

^{−}^{1}, the numerical solutions exhibited the odd feature of producing rainfall close to the mountain but then spreading far away from the mountain, and thus depleting all of the initial

*q*after a few hours. To better understand this behavior, we analyzed a simple one-dimensional model, based on the equations that represent the microphysical processes in the Kessler microphysics parameterization (Klemp and Wilhelmson 1978). The equations for the evolution of the cloud water mixing ratio

_{c}*q*and of the rain mixing ratio

_{c}*q*are, respectively,The terms

_{r}*A*and

_{r}*C*represent respectively the rates of autoconversion and of accretion of rain and are represented by the formulas:andwhere

_{r}*c*

_{1}= 0.001 s

^{−}^{1},

*c*

_{2}= 0.001 g g

^{−}^{1},

*c*

_{3}= 2.2 s

^{−}^{1},

*c*

_{4}= 0.875. The coefficient

*K*is the diffusion coefficient associated with the filter. To emulate the conditions encountered in our numerical simulations, we examine the evolution of an initial condition in which

_{h}*q*is a bell-shaped distribution, with

_{c}*q*greater than the threshold

_{c}*c*

_{2}in a limited zone in the middle of the domain and with

*q*constant and different from 0 in the rest of the channel. Since

_{c}*q*is larger than

_{c}*c*

_{2}, the rainwater

*q*is generated by autoconversion in the middle of the channel, reducing

_{r}*q*and increasing

_{c}*q*there. The accretion

_{r}*C*is then different from 0 and produces further decreases of

_{r}*q*and increases of

_{c}*q*. As a consequence, the gradient of

_{r}*q*grows and becomes large enough to diffuse the rain progressively away from the center, allowing for more rain to be produced there by accretion. In this way, the combination of diffusion and of accretion progressively makes the rain spread throughout the channel and at the same time reduces the cloud everywhere. After a few hours, all the cloud water is removed since it has been changed into rainwater. This effect was not noticed in simulations with small initial

_{r}*q*because in those cases

_{c}*A*and

_{r}*C*were too small to have any significant effect.

_{r}To correct this problem, a threshold has been included into the accretion term so that this term is activated only when the rain produced by autoconversion is greater than 0.0001 g kg^{−}^{1}. This modification has been included in the WRF model Kessler microphysics for the simulations presented herein.

Estimate of maximum upslope wind *w*_{up} = *Uh _{m}/a* (in m s

^{−1}, second column) and maximum buoyancy anomaly

*b*

_{max}at

*z*= 4 km and

*t*= 1 h (in m s

^{−2}, third column) vs mountain height

*h*(in m, first column)

_{m}Summary of the results from the series of numerical simulations reported in this paper. The first column contains the name of the simulation, indicated by the mountain height *h*, followed by the value of *q _{c}*. The

*θ*-constant profile is denoted with a “

_{e}*θ*” at the end of the name. The next five columns describe some characteristics of the simulations at

_{e}*t*= 10 h (except for the experiment

*h*700

*qc*.5, for which the descriptions apply at

*t*= 15 h); they are, respectively, the upstream propagation of the desaturated region, the presence of downstream convective instability, the intensity of the upstream maximum rainfall rate

*R*

_{max}(in mm h

^{−1}), evaluated at the surface, and its distance

*X*

_{max}from the mountain peak (in km), and the surface maximum of the horizontal velocity perturbation (

*u*

_{max}in m s

^{−1}). The last two columns show the accumulated precipitation at the surface (in mm) integrated in the domain (

*R*

_{tot}) and in the region downstream (

*x*> 40 km) affected by convective cells (

*R*

_{down}), respectively, during the first 10 h of the simulations

^{}

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

^{1}

Durran and Klemp (1982b, p. 2157) state that their Eq. (36) was derived using DK's Eq. (21), which as noted by Kirshbaum and Durran (2004), is not correct; the reduction of (2.4)–(2.5) to DK's Eq. (36) shows, however, that DK's (36) is correct and, in fact, can be derived without using Durran and Klemp's (1982b) Eq. (21).