1. Introduction

Mountains perturb incident airflow by mechanically lifting or deflecting it and by generating differential surface heating that drives thermal circulations. The associated vertical motions control local cloud patterns, cumulus convection, and boundary layer aerosol distributions (e.g., Banta 1990). Because mountain-forced ascent is at best partially resolved in weather and climate models, it can only be properly represented with the aid of subgrid parameterization. However, to our knowledge, no parameterization schemes consider the impacts of subgrid-scale terrain circulations on the initiation of deep convection. Common mass flux convection schemes like those of Gregory and Rowntree (1990) and Kain (2004) treat convective “triggering” by adding a thermodynamic perturbation to low-level air parcels that may depend on resolved, but not subgrid, vertical motion. This neglect of subgrid-scale ascent may contribute to biases in the representation of orographic convection in large-scale models (e.g., Schwitalla et al. 2008). As a step toward addressing this shortcoming, we endeavor to improve the quantitative understanding of mountain-forced vertical motion.

To briefly review some essential concepts relating to mountain-forced ascent, we first consider the simple case of dry, two-dimensional (2D), mechanically forced (nondiabatically heated) airflow with uniform winds and stability over an isolated mountain. The dynamics of such flows are largely controlled by the nondimensional mountain height (alternatively called the inverse Froude number) and width , where N is a characteristic Brunt–Väisälä frequency, h_m and a_x are the mountain height and width in the along-flow (here x) direction, and is a characteristic wind speed (e.g., Smith 1989; Lin 2007). While M_M indicates the extent to which the flow can be represented by linear theory, indicates the extent to which the flow is hydrostatic. Within the confines of linear theory, for the flow is hydrostatic with vertically propagating gravity waves directly above the mountain, while for it is nonhydrostatic with evanescent waves above the mountain and/or downwind-propagating gravity waves.

For M_M ≪ 1, the magnitude of the mountain-induced flow perturbations is far less than that of the mean wind and the flow is well described by linear theory. Incident air freely ascends the mountain and the surface-based ascent depends on the product of and the terrain slope. As M_M increases toward unity, the magnitude of mountain-induced flow perturbations becomes similar to that of the mean wind, rendering nonlinearities important. Strongly nonlinear flows (M_M ≳ 1) feature increasingly complicated dynamics, including gravity wave breaking and/or upstream blocking (e.g., Smith 1979). The transition between unblocked and blocked flow generally occurs at M_M at or slightly below unity. For hydrostatic flow over a bell-shaped mountain, Miles and Huppert (1969) found it to occur more precisely at M_M = 0.85. In general, blocked 2D flows exhibit significantly smaller vertical displacements than unblocked flows. In the absence of rotation, these flows may develop an upstream-propagating bore originating at the foot of the mountain that leaves behind a wake of decelerated fluid (Pierrehumbert and Wyman 1985).

In three dimensions (3D), the flow is afforded another degree of freedom to split upstream of, and detour around, the mountain, which tends to weaken or eliminate the upstream-propagating bore from 2D. Moreover, hydrostatic gravity wave updrafts are no longer localized over the mountain; they extend downwind in a V-shaped pattern with a vertex over the lee (e.g., Lin 2007). The impacts of finite-length cross-flow terrain may be viewed as a function of the horizontal aspect ratio r = a_y/a_x, where a_y is the cross-flow length. As described by Smith (1989) and Lin (2007), for r ≪ 1 both wave breaking and flow splitting are suppressed by the narrow cross-flow terrain, while for r ≫ 1 the flow dynamics approach the 2D solution discussed above. As r increases between these extremes, the flow transitions from linear to nonlinear at progressively smaller values of M_M. As in 2D, the linear flows are unblocked, with windward ascent controlled by the product of and the terrain slope. By contrast, the blocked flows exhibit complex leeside flow patterns including a turbulent wake and possibly lee vortices (Smolarkiewicz and Rotunno 1989; Epifanio and Durran 2001), with surface-based updrafts forming upwind in decelerating air, along the lee slope in reversed wake flow, and downwind as split low-level airstreams converge. As M_M → ∞, the solution (at least in theory) transitions to a potential flow regime where displacements are purely horizontal and vertical motion ceases.

Thermally forced orographic flows also represent an important meteorological problem, particularly for applications such as boundary layer aerosol venting and warm-season convection initiation (e.g., Banta 1990; Wulfmeyer et al. 2008). Studies of the combined mechanical and thermal forcing problem include the work of Raymond (1972), who solved Long’s nonlinear equations for steady 2D airflow over a heated ridge and found that diabatic heating (cooling) tends to weaken (strengthen) mechanically forced wave breaking. Smith and Lin (1982) used linear theory to investigate the feedbacks of cloud latent heating on flow over a bell-shaped 2D ridge, and found an out-of-phase relationship between the mechanically and thermally forced gravity wave updrafts. A similar out-of-phase relationship between mechanical and thermal updrafts (the latter forced by elevated heating) was found in the linear solutions of Crook and Tucker (2005, hereafter CT05). Using a combination of linear theory and nonlinear simulations, Reisner and Smolarkiewicz (1994, hereafter RS94) found that in blocked flows with M_M > 1, localized diabatic heating over the terrain enhanced the windward ascent, shifting the flow into a more unblocked regime. Tian and Parker (2003), who studied the combined mechanical and thermal forcing problem using 2D numerical simulations, found (among other things) that a thermodynamic heat-engine model based on Souza et al. (2000) accurately diagnosed simulated thermally forced updrafts.

Compared to previous theoretical studies of steady mountain thermal forcing in a uniformly stratified atmosphere, Kirshbaum (2013, hereafter K13) considered the more realistic case of diurnal thermal forcing in a two-layer atmosphere consisting of a neutral boundary layer and a stable free troposphere. Using linear theory, they separated thermally forced mountain flows into three regimes based on the dominant terms in the thermodynamic budget. They also derived a nondimensional thermal forcing amplitude M_T analogous to M_M and analyzed both linear and nonlinear thermally forced flows. Their analysis suggested that most meteorologically significant thermally forced mountain flows are highly nonlinear (M_T ≳ 1). In such flows, a positive feedback loop driven by nonlinear horizontal advection rapidly contracts and intensifies the circulation (as in semigeostrophic frontogenesis) before nonlinear vertical advection emerges to curtail its growth. The narrow but intense thermally forced updrafts produced by this process are highly effective at initiating deep convection under weak wind conditions. Consistent with Tian and Parker (2003), the vertical motions in those nonlinear flows were predicted reasonably well by heat-engine theory.

As an extension to K13, who focused exclusively on mountain thermal forcing, we consider the combined problem of mechanical and thermal orographic forcing, using two-layer models of diurnally forced mountain flows. Our specific objectives are (i) to analytically quantify boundary layer orographic ascent in both stable and convective boundary layers and (ii) to analyze the detailed interactions between thermal and mechanical flow dynamics, particularly the impact of mechanical blocking on the flow response to elevated mountain heating. We address these objectives through a combination of linear theory, a nonlinear thermodynamic heat-engine model, and a fully nonlinear atmospheric model. Section 2 describes the theoretical framework, including scalings for mechanical and thermal boundary layer ascent, parameters distinguishing different flow regimes, and the derivation of nonlinearity parameters. Section 3 describes the numerical model and section 4 the experimental setup. Section 5 presents the results, including direct comparisons between the numerical simulations and the theoretical scalings, along with physical interpretation. Section 6 presents a discussion of the results and section 7 provides the conclusions.

2. Analytical models

The theory presented herein closely follows that of K13, with some exceptions. To limit redundancy, we restate only the essential and novel aspects of the analysis and refer the reader to K13 for further details. For reference, all of the parameters used in the theory, scalings, and numerical simulations are defined in Table 1.

Table 1.

Definition of various parameters used in the analytical and numerical modeling sections.

View Table

a. Linear theory

As in K13, we consider a two-layer model (Fig. 1) with Brunt–Väisälä frequencies N₀ in the lower layer and N₁ in the upper layer, separated by an interface at z = H₀. This two-layer configuration enables consideration of both stable and convective boundary layers. For added realism we extend their 2D linear, Boussinesq analysis to 3D. The atmosphere is considered to be dry and nonrotating, which renders this analysis valid for equatorial regions and high Rossby number flows at other latitudes. The governing equations for each layer i are thus

View Expanded

View Expanded

View Expanded

where u = (u, υ, w) is the three-dimensional wind perturbation; is a uniform background zonal wind; ϕ = p′/ρ₀ is the density-normalized pressure perturbation (or kinetic pressure) relative to the hydrostatically balanced basic state; b = gθ′/θ₀ is the buoyancy; is the potential temperature perturbation from the hydrostatically balanced basic-state profile ; ρ₀ = 1 kg m⁻³ and θ₀ = 300 K are constant reference density and potential temperature, respectively; is the Brunt–Väisälä frequency; α is a Rayleigh damping or Newtonian cooling coefficient; and Q = Q(x, y, z, t) is a prescribed heating function. Combining (1)–(3) into a single partial differential equation for w_i gives

View Expanded

where is the horizontal Laplacian. To prescribe Q, we follow a similar procedure as in CT05. We begin with a diurnally varying function of the form Q = Q₀e^iΩte^−(z−h)/D, where is the heating amplitude, g is the gravitational acceleration, is the heating rate, Ω is the diurnal frequency, h = h(x, y) is the terrain height, and D is the vertical heating scale. Given that the heating function is horizontally uniform (or barotropic) everywhere except over the terrain, we subtract off the contribution from flat terrain to isolate the elevated heating component:

View Expanded

To the accuracy of the linear approximation, this expression may be simplified to

View Expanded

which effectively treats the elevated mountain heating as an isolated heat source over flat terrain. We use a simple Gaussian function for the terrain height:

View Expanded

where x = y = 0 is at the mountain centerpoint and a_x and a_y are the mountain widths in the x and y directions, respectively.

View Full Size

Schematic diagram of the two-layer model. All symbols are defined in the text.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

1) Linear scaling of vertical motion

If one is interested in roughly estimating the maximum low-level vertical velocity, a full solution to the linear system of equations is unnecessary. For unblocked, purely mechanical flow, we can scale the free-slip lower boundary condition as

View Expanded

Because fluid motions are highly nonlinear and often turbulent for blocked flow, no linear scaling for W_M is available in that regime. However, as noted by Ólafsson and Bougeault (1996) and others, the vertical displacement may still be scaled as . For thermally forced flow, we can readily scale (4) to estimate a characteristic vertical velocity W_T,lin in the lower layer:

View Expanded

where L_x1 is the dominant wavelength of the terrain feature; L_x2 and L_y are the zonal and meridional scales of buoyancy variations (and horizontal convergence), respectively; and H is a characteristic height scale. The physical interpretation of, and assignments for, these length scales is provided in section 2a(3).

2) Nonlinearity parameters

For mechanically forced flow, the degree of nonlinearity is determined by M_M (e.g., Epifanio and Durran 2001). K13 derived an analogous M_T for heated mountains using the weakly nonlinear approximation, whereby second-order nonlinear quantities may be approximated as products of linear first-order perturbations. To determine the degree of nonlinearity, the magnitudes of these terms are compared to that of the linear time-tendency or advection terms. The choice of linear reference term depends on the ratio ; if this ratio is greater (smaller) than unity, the time-tendency (advection) term dominates. Through substitution of relevant scaling parameters, M_T may be expressed as

View Expanded

3) Flow regimes

As discussed in section 1, two flow regimes (unblocked and blocked) exist for mechanically forced flow depending on the value of M_M. For thermally forced flow, at least three basic flow regimes exist. These can be identified from the linearized thermodynamic equation [see (2)], where Q may be balanced by three basic processes (or some combination thereof). In the “growth–decay” (GD) regime, where , the heating term is balanced by the time-tendency and/or dissipation terms. In the “ventilation” (VE) regime, where , the heating is balanced by the background wind advection. In the “stratification” (ST) regime, where , the heating is balanced by the stability term. Based on these dominant balances, (9) can be simplified as follows for each regime (assuming further that H ≪ [L_x1, L_x2, L_y]):

View Expanded

As shown by K13, the structure and amplitude of the flow response fundamentally differs in each regime: the GD regime produces a strong surface-based updraft directly over the crest, the VE regime produces a weaker longitudinal ascent band downwind, and the ST regime produces much weaker gravity wave updrafts. Because these regime classifications are based on linear theory, however, they do not capture the full complexity of real flows. For example, K13 showed that within the GD regime, nonlinearities can change the evolution, intensity, and some parameter sensitivities of the flow response. The impact of such nonlinearities on each flow regime will be considered in subsequent sections.

We now return to the scaling of W_T,lin in (9), which is facilitated by the above regime classifications. K13 assigned L_y = ∞ (for 2D) and considered a single zonal length scale L_x = L_x1 = L_x2. For the GD and VE regimes, they set L_x = a_x and H = H₀/2, and for the ST regime they set , where and one of the two scales is tied to the scale of the heating function (L_x = a_x or H = D). The latter scaling is dictated by the homogeneous form of (4), which provides a natural ratio between the horizontal and vertical scales. Although this scaling produced reasonable agreement with the 2D simulations in K13, its performance degraded in our 3D simulations. This prompted us to rethink the length-scale assignments.

The main limitation of K13’s scaling is that it admitted only one zonal length scale while the physical problem contains two. The first (L_x1) relates to background-wind advection over the mountain heat source and the second (L_x2) relates to the streamwise structure of diabatic buoyancy anomalies. Both scales must be considered for consistency with linear and numerical solutions over the full parameter space. Given that advection scales with the terrain wavelength, we select L_x1 = 4a_x (approximately two half-widths on either face). Based on empirical tests, smaller (larger) L_x1 produced unrealistically strong (weak) sensitivity of W_T,lin to in (9). The second zonal scale (L_x2) depends on the flow regime. For the GD and ST regimes, where the buoyancy anomalies are largely trapped over the mountain, we set L_x2 = a_x, which is most consistent with the structure of thermal circulations in linear and numerical solutions (as will be seen). For the VE regime, however, L_x2 is controlled by background wind ventilation, which creates diurnally alternating longitudinal warm–cold bands of length .

The length scales in the y and z directions are straightforward. For the former, where there is no background wind, the buoyancy anomalies simply scale as L_y = a_y. For the latter, H = H₀ in the GD and VE regimes where the response is dictated by the boundary layer depth, and H = D in the ST regime. Unlike the length scales used by K13, these scalings are dictated purely by the diurnal heating function, H₀, and , which is attractively simple and produces the best empirical agreement with the numerical simulations.

b. Nonlinear theory

The application of thermodynamic heat-engine theory to mountain thermal circulations, which was proposed by Renno and Ingersoll (1996) and Souza et al. (2000), and advanced by Tian and Parker (2003), is briefly reviewed here. It is based on the integration of Bernoulli’s equation and the first law of thermodynamics around the closed loop of a steady-state convective circulation, which, after all of the exact differentials vanish, gives

View Expanded

where T is temperature, s is entropy, f is the frictional force, and l is a unit displacement vector locally tangent to the circulation streamline. Thus, in steady state, the net heat input is balanced by the frictional dissipation, the latter of which may be expressed as

View Expanded

where μ is a coefficient of mechanical dissipation and W_T,nonlin is the circulation speed. The net heat input is c_pΔT_sfc, where c_p is the specific heat of dry air at constant pressure and ΔT_sfc is the “nonadiabatic” temperature difference between the mountain crest and the surrounding plain. Solving for W_T,nonlin gives

View Expanded

where η is the thermodynamic efficiency. For a Carnot cycle in a convective boundary layer, η may be approximated as

View Expanded

where T_h and T_c are the temperatures of the hot and cold reservoirs at the surface and the top of the neutral boundary layer, respectively. In addition, μ may be estimated using the convective boundary layer scaling of Tian and Parker (2003):

View Expanded

where L_T represents the horizontal scale of the circulation, defined as the distance between the main updraft and downdraft. For a circulation trapped over the mountain in the GD regime, L_T = min(a_x, a_y) so that its strength is controlled by the strongest horizontal buoyancy gradient. For the VE regime we set L_T = a_y because the downwind response is characterized by a longitudinal updraft separated by a_y from two descent branches on either side. We do not apply the heat-engine scaling to the ST regime because the associated gravity waves are not closed circulations.

For the GD regime we estimate ΔT_sfc as the integral of Q over the positive half of the heating cycle, while for the VE regime we integrate the maximum heating rate over the advective time scale :

View Expanded

Substituting (17), (16), and (15) into (14) gives¹

View Expanded

Because this theory assumes a steady-state circulation, it is most applicable when the time scale for an air parcel to loop through the circulation [~2(H + a_x)/W_T] is much shorter than the diurnal time scale (2π/Ω). This condition is only satisfied in the more strongly forced cases with more vigorous convective circulations, so (18) tends to overestimate the updraft strength in weakly forced cases. This nonlinear heat-engine scaling thus provides a useful complement to the linear scaling, which is only accurate for weakly forced, quasi-linear flows.

3. Numerical simulations

We perform idealized numerical simulations with version 14 of the Bryan Cloud Model (cm1; Bryan and Fritsch 2002), which solves the primitive moist atmospheric equations using a split-time-step procedure to maintain the stability of acoustic modes. On the large time step, time integration is performed with a third-order Runge–Kutta scheme. Eight small time steps are performed for each large time step. Horizontal (vertical) advection uses a centered sixth-order scheme (a fifth-order scheme with implicit diffusion). Because no implicit diffusion is used in the horizontal, explicit sixth-order horizontal diffusion is added (with a filter coefficient of 0.48) to diminish spurious poorly resolved waves. The only physical parameterization used in the simulations is a 1.5-order TKE-based subgrid-turbulence scheme, with the turbulent kinetic energy initialized to zero over the domain. For consistency with the theoretical models, the simulations are dry (no water vapor) and nonrotating (no Coriolis force).

The standard domain configuration uses dimensions of G_x = 300 km, G_y = 100 km, and G_z = 12 km, with the mountain centered in the upstream half of the domain (x₀ = 75 km and y₀ = 50 km) to allow thermal circulations to fully develop downwind. The horizontal grid is regular with a uniform spacing of Δx = Δy = 500 m, while the vertical grid uses a stretched, terrain-following coordinate with a spacing of Δz = 100 m from 0 to 4 km, Δz = 400 m from 8 to 12 km, and a linear increase of Δz in-between, giving 66 levels. Boundary conditions are open in x and y and closed in z with a free-slip lower surface and a Rayleigh damper over the uppermost 4 km to absorb vertically propagating gravity waves.

To directly compare the simulations with linear theory, the model initialization must be handled with some care. Because the linear flow dynamics are periodic in time and state-variable perturbations are always present over the flow volume, starting the model from rest imposes differences from the linear solution that persist indefinitely. To overcome this problem, the model was started from rest but the thermal forcing was linearly increased from zero to its full amplitude over the first 24 h of integration, after which the simulation was integrated for an additional 24 h. By allowing the flow dynamics to develop gradually, the numerical model solutions agreed more closely with the corresponding theoretical predictions. All simulations are initialized at 0600 local time (LT), at which time t = π/2Ω and Re(Q) = 0 in (6).

4. Experimental design

The parameter space of the combined mechanical and thermal forcing problem is too large to comprehensively investigate in a single study. Given the 11 control parameters (h_m, a_x, a_y, , N₀, N₁, Q₀, D, H₀, Ω, and α), there are 10 nondimensional parameters that can be formulated to describe the problem, 9 of which are independent. Systematically varying all of these parameters is a practical challenge, particularly in light of the substantial computational cost of the 3D numerical simulations. We thus choose to vary a few key dimensional parameters over realistic ranges and evaluate their impacts on nondimensional scaling parameters and the simulated flow dynamics. The fixed dimensional parameters, which are listed in Table 2, are mostly the same as those in the “baseline” simulations of K13. To focus on smaller obstacles that are poorly resolved in large-scale models, an axisymmetric mountain with a Gaussian width of a_x = a_y = 5 km is chosen. The boundary layer depth and heating decay scale (H₀ and D) are both assigned a standard value of 1 km. The upper-layer Brunt–Väisälä frequency (N₁ = 0.013 s⁻¹) is characteristic for the tropical free troposphere. The damping–cooling coefficient (α = 5.0 × 10⁻⁵ s⁻¹) is loosely based on observational estimates (e.g., Stevens et al. 2002) but is set relatively large for application to the highly turbulent convective boundary layers of the GD regime. The linear scalings are largely insensitive to α in the VE and ST regimes.

Table 2.

Fixed dimensional parameters in the experiments.

View Table

To cover a broad swath of the relevant parameter space, we consider four variable parameters: , h_m, N₀, and . As shown in Table 3, we consider six suites of experiments, each with six cases sampling a range of (0–5 m s⁻¹ in steps of 1 m s⁻¹). We omitted larger values of because at those values the thermal response weakens and stretches increasingly far downwind, requiring substantially larger computational domains to capture it. Each suite is defined by its values of h_m, , and N₀; for example, the first suite with h_m = 1 m, K h⁻¹, and N₀ = 0 is named H1-T01-N0. The suites are designed to sample different flow regimes and degrees of nonlinearity, as quantified by the nondimensional parameters and regime classifications in Table 3. These nondimensional parameters include the ratio of thermal to mechanical updraft strengths (W_T,lin/W_M), the two nonlinearity parameters M_M and M_T, and the inverse thermal Froude number . As shown by Lin and Smith (1986) and Raymond and Rotunno (1989), influences the gravity wave response to diabatic heating in a stratified atmosphere.

Table 3.

List of all of the experiments along with dimensional parameters and nondimensional scaling parameters.

View Table

The naming convention for individual simulations appends the wind speed to the suite name. For example, the name H1-T01-N0-U0 corresponds to a simulation from the H1-T01-N0 suite with .

1. H1-T01-N0, linear (GD and VE);

2. H1-T01-N013, linear (ST);

3. H500-T01-N0, linear mechanical, partially nonlinear thermal (GD and VE);

4. H500-T01-N013, nonlinear mechanical, linear thermal (ST);

5. H500-T1-N0, linear mechanical, fully nonlinear thermal (GD and VE); and

6. H500-T1-N013, nonlinear mechanical, weakly linear thermal (ST).

The most relevant suite for meteorological applications is H500-T1-N0, which considers a moderately sized mountain, strong thermal forcing, and a neutrally stratified lower layer (representing a convective boundary layer under strong surface heating). Although the flow regimes are similar between the H500-T01-N013 and H500-T1-N013 suites, our consideration of both provides a more systematic test of the scalings and investigation of the interactions between mechanical and thermal circulations. For simplicity, the mountain is always shorter than the boundary layer depth (i.e., h_m < H₀) so that we can cleanly distinguish the flow regimes and degrees of nonlinearity, which are quantified using a single-layer framework. We defer the h_m > H₀ scenario, which is meteorologically important but significantly more complex, to future work.

To better interpret the simulated interactions between mechanical and thermal forcings, we conduct three numerical simulations for each case, one with combined mechanical and thermal forcing [using the nonlinear heating function in (5)], one with purely mechanical forcing (the “mechanical simulation”), and one with purely thermal forcing (the “thermal simulation”). In the mechanical simulations, the thermal forcing is easily eliminated by setting . In the thermal simulations, the mechanical forcing is eliminated by removing the terrain but retaining h in the thermal forcing function. However, a complication arises in that the area formerly occupied by the mountain becomes part of the atmosphere. If the heating was applied in the exact same region as in cases with terrain, it would be elevated above a mountain-shaped region of unheated air, which would create an unrealistic elevated circulation. To avoid that outcome, we simply reapply (5) over a flat lower boundary and place the heating lower in the atmosphere.

In two simulations (H500-T01-N0-U0 and H500-T1-N0-U0), the lack of a mean wind or background stability leads to very strong nocturnal downslope (or katabatic) flow. The relatively narrow y dimension of the default domain (see section 3) allows these outward-propagating density currents to easily reach the y boundaries and reflect back deep into the domain interior. To diminish these spurious boundary reflections, we change the horizontal domain size of these simulations to G_x = G_y = 200 km, with the mountain centered at the domain centerpoint. In the other simulations these perturbations are either too weak or carried sufficiently far downwind by the background flow to have only modest impacts on the circulations of interest.

5. Results

a. Qualitative comparisons

To illustrate the varying flow dynamics among the experiments, we begin with a qualitative comparison of the simulated flow fields. Due to the large number of cases under consideration, we show only eight simulations (the and members from the H1-T01 and H500-T1 suites), which together capture most of the key features. Figure 2 shows quasi-horizontal cross sections of w at the midpoint of the boundary layer [z = (h + H)/2], along with wind vectors from the lowest model level, for each of these eight simulations. Each case is displayed near the time at which the simulated w reaches its maximum; this occurs at midday or in the early afternoon for all cases except the H1-T01-N0 simulations (Figs. 2a,b), where it occurs significantly later. As discussed by Rotunno (1983) and K13, this delay is a natural consequence of the linear balances in the governing equations, which imply a 12-h lag between Q and w for (for the H1-T01-N0-U0 case in the GD regime), and a 6-h lag between Q and w when and N₀ = 0 (for the H1-T01-N0-U3 case in the VE regime). For the H1-T01-N013 simulations, as well as for all the cases with h_m = 500 m, the strong boundary layer stability and/or the emergence of nonlinearities all but eliminates the phase difference between Q and w.

View Full Size

Plan-view cross sections of vertical velocity at z = (H + h)/2 (filled contours) with wind vectors (arrows) and terrain contours (thin lines with 100-m interval) from selected simulations: (a) H1-T01-N0-U0, (b) H1-T01-N0-U3, (c) H1-T01-N013-U0, (d) H1-T1-N013-U3, (e) H500-T1-N0-U0, (f) H500-T1-N0-U3, (g) H500-T1-N013-U0, and (h) H500-T1-N013-U3. The time for each case (upper-right corner) corresponds to that at which the maximum w occurred. The maximum vertical velocity in the plane of the diagram w_max (m s⁻¹) is shown in the lower-left corner.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

The presence of background winds and/or boundary layer stability dramatically changes the properties of the low-level updrafts. All cases with develop an updraft at the mountain centerpoint, fed by converging low-level flow (Fig. 2, left). In these cases, the updrafts are much weaker for N₀ = N₁ than for N₀ = 0 due to the inhibition of vertical motion by the background stability. For , the position of the main updraft is very sensitive to N₀; for N₀ = 0, it forms an elongated band extending downwind (Figs. 2b,f), while for N₀ = N₁ it forms directly over the crest (Figs. 2d,h). Unlike the cases with , the maximum updraft magnitudes for the cases with are dominated by mechanical rather than thermal forcings, so they exhibit less sensitivity to N₀. The main updraft apparent in the H1-T01-N013-U3 simulation is the lowest ascending branch of a mountain wave (Fig. 2d) and that in the H500-T1-N013-U3 simulation is a breaking mountain wave, downwind of which lies a decelerated wake with weakly reversed flow along its centerline (Fig. 2h). Comparing the H1-T01 simulations (Figs. 2a–d) with the H500-T1 simulations (Figs. 2e,f), the main updrafts become narrower and more concentrated as the flow becomes more nonlinear.

Figure 3 presents vertical cross sections of w, θ, and the horizontal wind vectors along the mountain centerline (y = 0) for the same eight simulations. For , a deep updraft forms directly over the mountain (Fig. 3, left). This updraft reaches its maximum intensity near the center of the boundary layer for N₀ = 0 (Figs. 3a,e) and near the surface for N₀ = N₁ (Figs. 3c,g), where its exponentially decaying shape closely matches that of Q. In the latter case, the dominant response to the diabatic heating is local ascent in proportion to the local diurnal-heating amplitude, which gives rise to gravity wave oscillations with a frequency Ω. Again, the updrafts become narrower and more intense with increasing nonlinearity, which is most evident in the H500-T1-N0-U0 case as the vigorous central core induces 200–300 m of ascent at the mixed-layer top (Fig. 3e).

View Full Size

Vertical cross sections of vertical velocity along the y centerline (filled contours) with wind vectors (arrows) and potential temperature contours (thin lines with 1-K interval) from selected simulations: (a) H1-T01-N0-U0, (b) H1-T01-N0-U3, (c) H1-T01-N013-U0, (d) H1-T01-N013-U3, (e) H500-T1-N0-U0, (f) H500-T1-N0-U3, (g) H500-T1-N013-U0, and (h) H500-T1-N013-U3. The time for each case (upper-right corner) corresponds to that at which the maximum w occurred. The reference vertical velocity in the plane of the diagram w_max (m s⁻¹) is shown in the upper-right corner (below the time).

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

For the simulations with N₀ = 0, the mechanically forced updrafts form an evanescent pattern directly over the mountain while the thermally forced updrafts form a longitudinal band extending downwind (Figs. 3b,f). In contrast to CT05, who concluded that the mechanical and thermal responses interfere destructively, these two responses are spatially separated and do not interfere. By contrast, for the H1-T01-N013-U3 simulation (Fig. 3d), which is in the ST regime like the flows considered by CT05, the mechanical and thermal responses do interfere destructively: thermally forced ascent over the lee slope coincides with mechanical descent. However, this cancellation is imperceptible because the mechanical response is over 100 times stronger than the thermal response (Table 3). More noticeable cancellation would require an unrealistically large . The most complex response occurs in the H500-T1-N013-U3 case (Fig. 3h), where, in addition to the breaking gravity wave directly above the mountain, two other regions of ascent develop: a weak updraft at x/a ≈ 1.25 caused by reversed flow ascending the lee slope and a stronger longitudinal updraft band downwind (4 < x/a < 9) resulting from leeside convergence of blocked airstreams that split and separated around the barrier (this surface-based updraft is too shallow to appear in Fig. 2h).

The locations of the thermally forced updrafts in Figs. 2 and 3 are broadly consistent with the values in Table 3. As found by Lin and Smith (1986), favors updrafts forming directly over the heat source while favors updrafts forming downwind. For all cases with , and the updraft forms directly over the crest (e.g., Fig. 2a). Similarly, for cases with and a stable boundary layer (N₀ = N₁), and the updrafts form just downwind of the crest. Although these thermal updrafts are masked by the dominant mechanical downdrafts in Figs. 2d and 2h, they are apparent in the thermal H1-T1-N013-U3 simulation (not shown). Only in the cases with and a neutral boundary layer (where ) does the thermal updraft extend well downwind of the crest (e.g., Figs. 2b,f).

b. Performance of the scalings

For each suite of experiments, we calculate linear scalings for W_M and W_T,lin, along with nonlinear heat-engine scalings for W_T,nonlin (the latter restricted to cases with N₀ = 0). These are directly compared to the corresponding numerically simulated values of w_max, defined as the maximum w within the region |x| ≤ 2L_x2, |y| ≤ 2L_y, and z ≤ 2H over the final 24 h of the simulation. Results are presented for all experiments, including the combined simulations (with orography and elevated heating), the pure mechanical simulations (with orography and zero heating), and the pure thermal simulations (with heating and flattened orography).

1) Quasi-linear simulations

For a suite of experiments to be termed quasi linear, M_M and M_T must both be subunity for all six cases. By design, this only holds for the H1-T01-N0 and H1-T01-N013 suites (Table 3). For the H1-T01-N0 suite, the linear scalings of W_T,lin and W_M accurately predict the maximum vertical motions within the corresponding thermal and mechanical simulations (Fig. 4a). As increases, the thermal scaling captures the decrease in w_max due to the effects of downwind heat ventilation, while the mechanical scaling captures the increase in w_max due to stronger forced ascent. The thermal scaling also captures the break in the slope of w_max at , where the flow transitions from the GD to the VE regime. In contrast to the linear scalings, the heat-engine (or Carnot) scaling overestimates w_max by about two orders of magnitude. As mentioned in section 2b, this follows from its steadiness assumption, which is strongly violated in these weakly forced cases. Because interference between the thermal and mechanical responses is minimal when N₀ = 0, w_max for this suite simply takes on the value of the stronger pure response. It is nearly identical to that from the thermal simulations for and that from the mechanical simulations for . This is consistent with the linear prediction that the W_M curve crosses the W_T,lin curve at (Fig. 4a).

View Full Size

Comparison of theoretical scalings of vertical motion with numerical simulations for the (a) H1-T01-N0 and (b) H1-T01-N013 suites of experiments. The theoretical scalings are shown by solid lines and the simulation results are shown by symbols.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

Close agreement is also found between the linear scalings and the numerical simulations for the H1-T01-N013 suite (Fig. 4b). Whereas the ascent in the mechanical simulation again scales with , that in the thermal simulation exhibits no sensitivity to it. This is because in the ST regime, the thermal response is dominated by stable ascent directly over the terrain. Rather than being carried downwind, the internal energy gained over the mountain is quickly converted to kinetic energy and radiated away by gravity waves. In the combined-forcing simulations, the mechanical response is dominant everywhere except for , where it is nonexistent. The dominance of mechanical forcing in this suite arises from the strong boundary layer stability, which greatly suppresses the thermal response.

2) Nonlinear simulations

Nonlinearities emerge when h_m is increased to 500 m, which is reflected by the strong overestimation of updraft strength by the linear thermal scaling for in the H500-T01-N0 suite (Fig. 5a). Table 3 shows that M_T is well above unity in these cases but, as it decreases toward unity for , the scaling performs much better. The degradation of this scaling for M_T > 1 arises from nonlinear advection, which strengthens and contracts the circulation early in the day but ultimately constrains it by ventilating heat vertically (as shown by K13). This causes the simulated w_max to fall well below the linear predictions, which assume that the circulation gains strength throughout the day. In contrast to the H1-T01-N0 suite, the heat-engine scaling outperforms the linear thermal scaling, providing accurate estimates of w_max in all of the thermal simulations. This marked improvement is due to the stronger mountain forcing, which produces circulations with turnover time scales [O(10⁴ s)] similar in magnitude to the diurnal time scale (86 400 s).

View Full Size

As in Fig. 4, but for the (a) H500-T01-N0 and (b) H500-T01-N013 suites of experiments.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

Because M_M = 0 in the H500-T01-N0 suite, the linear W_M scaling remains extremely accurate. Moreover, as for the H1-T01-N0 suite, the large horizontal separation between the thermal and mechanical responses renders their interference minimal, so that w_max for the combined-forcing cases again takes on the value of the stronger pure response. Thus, w_max mostly aligns with the pure thermal response for and the pure mechanical response for . The one exception to this trend is for , where w_max for the combined simulation is significantly weaker than for the thermal simulation. This discrepancy will be addressed shortly.

For the H500-T01-N013 suite, the thermal response is linear while the mechanical response becomes nonlinear (Table 3). As in the H1-T01-N013 suite, the linear W_T,lin scaling accurately predicts w_max for the thermal simulations (Fig. 5b). However, the nonlinearity of the mechanical response causes a degradation of the linear w_M scaling, manifested as a substantial underprediction of updraft strength. The nonlinear response consists of low-level breaking gravity waves (see, e.g., Fig. 3h) containing powerful elevated updrafts that strengthen with . In the combined simulation, w_max is dominated by these mechanically forced updrafts, which are one to three orders of magnitude stronger than the thermal updrafts. Although the linear mechanical scaling fails to accurately predict w_max in this suite, the maximum vertical displacement still scales straightforwardly as . This is reflected by Fig. 3h, where the vertical displacements of the isentropes over the crest and downwind of the terrain (around 200 m) are consistent with m for the H500-T1-N013-U3 simulation.

For the H500-T1-N0 suite, the combination of a tall mountain and strong diabatic heating enhances the nonlinearity of the thermal response, as M_T > 1 for all values of (Table 3). The linear scaling of W_T,lin fails more dramatically than before, as it grossly overestimates w_max as well as its downward trend with increasing (Fig. 6a). Again, this is likely due to the increasing role of nonlinear vertical advection in restraining the circulation. Because the weak wind cases are more nonlinear, this process commences earlier and creates larger departures from the linear predictions. As expected, the heat-engine W_T,nonlin scaling agrees more closely with the thermal simulations than the linear scalings. Because the thermal response dominates the mechanical response in this suite, W_T,nonlin is also consistent with w_max for the combined simulations.

View Full Size

As in Fig. 4, but for the (a) H500-T1-N0 and (b) H500-T1-N013 suites of experiments.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

Similar to the H500-T01-N013 suite, the H500-T1-N013 suite is governed by a nonlinear mechanical response (Table 3 and Fig. 6b). However, the thermal forcing is much stronger, which renders M_T large enough for nonlinearities to become nontrivial. These effects are qualitatively apparent in Figs. 3c and 3g, where the vertical plume of ascent breaks down into four vertically aligned cells in the H500-T1-N013-U0 case. This more cellular response, which is also present in the thermal H500-T1-N013-U0 simulation but absent in the less strongly forced H500-T01-N013-U0 case (not shown), likely arises due to nonlinear horizontal thermal advection. As shown by K13, a positive feedback loop may be established where this advection squeezes isentropes together to drive a stronger thermal circulation that, in turn, strengthens the thermal advection. The enhanced updrafts at the core of the circulation overshoot their level of neutral buoyancy to create a secondary gravity wave. The emergence of this nonlinearity likely explains the significant underpredictions of w_max by the linear thermal scaling at small . Despite their increased strength, these updrafts are likely still too weak (~0.1 m s⁻¹) to carry out significant vertical advection. This leaves the enhancements due to nonlinear horizontal advection as the principal nonlinear effect.

Interestingly, in all of the simulations with h_m = 500 m and , the thermal response is significantly stronger than the combined response (Figs. 5 and 6). The origin of this discrepancy depends on N₀. For N₀ = 0, the boundary layer depth directly over the mountain is effectively halved because the surface rises up to H₀/2. Because the strength of the thermal response scales with H in (18) or H² in (11), this implies a 50%–75% decrease in w_max. For N₀ = N₁, the nonlinear advection is very sensitive to the presence of the orography. Whereas the convergent low-level flow at the surface moves purely horizontally in the thermal simulation, it travels upslope in the combined simulation. The resistance to this ascent by the background stability diminishes the strength of the low-level convergence at the crest and, in turn, the strength of the main updraft.

c. Interactions between mechanical and thermal responses

Of the many flows under consideration herein, only a few exhibit significant interaction between their thermal and mechanical responses. For the linear H1-T01-N0 and H1-T01-N013 suites, there is at most weak interference between these responses (as described by CT05) but no nonlinear interaction. Moreover, for the H500-T01-N0 and H500-T1-N0 suites, the evanescent mechanical wave response is confined to the mountain while the thermal response extends downwind, leaving them too far apart to interact. Furthermore, although the mechanical response in the H500-T01-N013 suite is nonlinear, the thermal response is too weak to allow for significant interaction. Only in the H500-T1-N013 suite, where the mechanical response is nonlinear and the thermal forcing is strong, does such interaction occur. This is suggested by Figs. 4–6, where only the H500-T1-N013-U2–H500-T1-N013-U4 cases exhibit slightly larger w_max in their combined-forcing simulations than in the corresponding mechanical or thermal simulations (Fig. 6b). Although these 5%–10% enhancements in w_max are barely detectable on the logarithmic ordinate, the overall flow dynamics in these cases are actually strongly modified by the diabatic heating, which renders them worthy of some attention.

As shown by the w cross section from the mechanical H500-T1-N013-U3 simulation (or, more succinctly, the H500-T0-N013-U3 simulation) at z = (h + H)/4 and 1400 LT in Fig. 7a, the strongest mechanically forced updraft forms over the lee slope, giving way to a decelerated wake region. This updraft is the merger of the lowest ascending branch of the mountain wave and a secondary updraft arising from weakly reversed leeside upslope flow. A second prominent updraft also lies downwind—a longitudinal band arising from the convergence of the two airstreams that split upstream and separated around the lateral edges of the mountain. Because they are based at the surface where the moist instability is often the largest, such leeside convergence bands are often capable of effectively initiating deep convection (e.g., Mass 1981; Houze 1993; Cosma et al. 2002).

View Full Size

The impact of diabatic heating in the H500-T1-N013-U3 simulation: (a) w, (c) b, and (e) p′ from the mechanical (or unheated) simulation (named H500-T0-N013-U3), and the differences in the same three fields—(b) Δw, (d) Δb, and (f) Δp—between the heated H500-T1-N013-U3 simulation and the H500-T0-N013-U3 simulation. Similarly, the vectors in (a),(c),(e) represent absolute surface wind velocities v while those in (b),(d),(f) represent the differences between the two simulations Δv.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0287.1

• Download Figure
• Download figure as PowerPoint slide

Surface heating in the H500-T1-N013-U3 simulation acts to strengthen both updrafts, but not at the same height. Whereas the updraft over the crest is enhanced the most at z/H ≈ 1, the leeside convergence band is enhanced near the surface at z/H ≈ 0.25. The slight (6%) enhancement of the former is associated with a larger-amplitude mountain wave, which extends deep into the troposphere (not shown). As explained by Reisner and Smolarkiewicz (1994), this results from lower pressure over the high terrain induced by the heating, which draws more air up the windward slope and strengthens the wave activity above the crest.

A more pronounced enhancement of the leeside convergence band is shown by the w difference Δw between the heated H500-T1-N013-U3 simulation and the unheated H500-T0-N013-U3 simulation at z = (h + H)/4 in Fig. 7b. To explain this enhancement, we present the surface b and p′ fields for the unheated H500-T0-N013-U3 simulation in Figs. 7c and 7e. Positive b and negative p′ are found in the wake, where blocked low-level flow is replaced by potentially warmer upper-level air, leading to a hydrostatic pressure minimum. When diabatic heating is applied, the wake buoyancy (pressure) increases (decreases) further, as shown by the surface buoyancy (Δb) and pressure (Δp) differences between the two simulations (Figs. 7d,f). The decreased wake pressure in the heated case, which arises as a hydrostatic response to diabatic heating deposited in the wake, accelerates the flow toward the wake center, increasing both the reversed zonal flow along the centerline and the meridional convergence downwind. Another prominent feature in Fig. 7d is the negative b perturbation surrounding the wake, which results from a contraction of the wake boundary in the heated case due to its lower internal pressure. The warm air that was previously in the outer wake is replaced by ambient flow, yielding a negative Δb.

Importantly, the strong flow reversal over the lee slope in Figs. 2h and 7b is promoted by the decelerated wake caused by mechanical blocking. In the absence of such a wake, the cross-barrier flow may actually accelerate over the windward slope (due to lower pressure over the crest), then return to its ambient speed in the lee, without any flow reversal (CT05). By protecting it from the ambient winds, the wake thus provides a favorable environment for thermally driven reversed flow up the lee slope. As with the aforementioned leeside convergence band, this surface-based upslope flow can effectively initiate moist convection. Recent observational and numerical studies show a tendency for surface-based convergence zones, and convection initiation, to form over the lee slope under moderate cross-barrier winds (e.g., Hagen et al. 2011; Kirshbaum 2011; Soderholm et al. 2014).

As a first step toward quantifying the impact of mountain heating on wake dynamics, we use the linearized, hydrostatic, inviscid Boussinesq momentum equations to analyze the perturbations induced by the thermal forcing:

View Expanded

View Expanded

View Expanded

where Δϕ, Δu, and Δυ are the flow perturbations purely associated with the mountain heating (which, in the confines of linear theory, can be separated from those associated with mechanical forcing). Although these linearized equations cannot describe the nonlinear dynamics of mountain wave breaking, they can still provide useful insights into the quasi-linear thermal response.

First, consider (21), which can be used to relate the diabatic heating gained by the flow in traversing the mountain to the vertical pressure gradient. Following the parcel arguments of Reisner and Smolarkiewicz (1994), this heating may be roughly approximated as the integral of Q along a flow path crossing the ridge centerline (y = 0):

View Expanded

where the latter expression is obtained through substitution of the steady (Ω = 0) form of (5), followed by integration of the Gaussian argument h(x, 0). Because this trajectory absorbs the maximum diurnal heating over the full mountain profile, it represents an upper bound on Δb. In reality, most of the wake air traverses only a small portion of the mountain, and not necessarily at the time of peak heating. The surface pressure perturbation is then estimated by integrating (22) over the atmospheric depth to give

View Expanded

For h_m = 500 m and , we obtain Δϕ_sfc = 13.4 J kg⁻¹, which, as expected, is significantly larger than Δp in Fig. 7f (although the units are different, the magnitudes are similar because ρ₀ = 1 kg m⁻³). Substitution of this value into scaled versions of (19) and (20) gives characteristic thermally forced horizontal flow perturbations of Δu_sfc = Δυ_sfc ~ 4 m s⁻¹ (assuming the scales of variability are equal in x and y).

Although the above analysis overestimates the differences between the H500-T1-N013-U3 and H500-T0-N013-U3 simulations, it illustrates that simple, hydrostatic arguments can account for the enhanced leeside updrafts associated with diabatic surface heating. The flow on the downwind side of the wake, which in the unheated case was nearly stagnant, is more strongly reversed in the heated case, with an easterly perturbation of about 2 m s⁻¹ that terminates as it ascends the lee slope (Fig. 7b). Similarly, the air streaming around the wake acquires a stronger meridional component in the heated case, with |υ| increasing by 1–2 m s⁻¹ on either side of the convergence band. This enhances the horizontal convergence arising from the collision of these airstreams, which significantly enhances the vertical motion at the upwind edge of the leeside convergence band.

6. Discussion

The first objective of this study was to analytically quantify boundary layer ascent over heated mountains, for both stable and convective boundary layers. To this end, linear and nonlinear scalings were derived in section 2 and critically evaluated in section 5. The appropriate choice of scaling depends on the degree of nonlinearity of the flow, as quantified by the nondimensional forcing amplitudes M_M (for mechanical forcing) and M_T (for thermal forcing), and the thermal flow regime (GD, VE, or ST). The well-performing scalings included the linear mechanical scaling for M_M < 1, the linear thermal scaling for M_T < 1 and M_M < 1 (for the all three thermal flow regimes), and the nonlinear thermal scaling for M_T > 1 and M_M < 1 (for the GD and VE regimes). Thus, the theory provides a means of predicting boundary layer orographic ascent for a broad range of situations. Although no vertical-velocity scaling was proposed for nonlinear mechanical flows with M_M ≳ 1, a simple vertical-displacement scaling estimated the mechanically forced ascent reasonably well, consistent with previous numerical studies (e.g., Ólafsson and Bougeault 1996).

The second objective was to quantify and interpret the impact of nonlinear interactions between the thermal and mechanical responses on boundary layer ascent. While such interactions were modest in most experiments, noticeable interactions did occur over taller, strongly heated mountains under moderate background winds (M_M > 1 and M_T ~ 1), for which the flows were blocked by the terrain and a lee wake developed. In those cases, the diabatic heating caused a hydrostatic lowering of the wake pressure, which enhanced the flow reversal along the wake centerline and strengthened the leeside convergence band downwind. This diurnal wake-flow reversal is consistent with observations and simulations of trade wind flow over the island of Hawaii (e.g., Chen and Nash 1994; Yang and Chen 2008). However, while lee flow reversal over Hawaii is typically attributed to the land–sea contrast, the present experiments suggest that it can also be produced by elevated heating in continental regions.

Our findings show promise for the parameterization of orographically forced vertical motion in large-scale models. The vertical motion was reasonably well approximated by at least one of the scalings for all cases with M_M < 1. Of particular relevance is the case of a neutral boundary layer (N₀ = 0), which commonly develops under strong solar heating. This situation, which has been neglected in most previous theoretical studies of heated mountains (e.g., Smith and Lin 1982; Reisner and Smolarkiewicz 1994; Crook and Tucker 2005), is treated herein with a two-layer model. Because of their accuracy and computational efficiency, these scalings may ultimately help to improve the representation of subgrid orographic ascent in large-scale models. However, because of the many idealizations contained in the present experiments, further evaluation of the scalings in more challenging experiments is required before we pursue that application.

Some of the most important idealizations to be relaxed in future studies include (i) the free-slip lower boundary, (ii) the neglect of the Coriolis force, (iii) the exclusion of realistic turbulence, and (iv) the simplified diurnal heating function in (5), which lacks the diurnally asymmetric sensible heat fluxes and complex vertical heating structures of reality. We will also consider the case of a mountain whose crest protrudes above the convective boundary layer, which is common over taller mountains in the morning when the boundary layer is shallow (e.g., Banta 1990; Tian and Parker 2003). Depending on the mountain width, a local mixed layer may develop over the mountain with different properties than that over the surrounding plains. This presents a particularly challenging situation that may require new theoretical approaches to quantify.

7. Summary and conclusions

We have quantified the strength of boundary layer updrafts produced by combined mechanical and thermal forcing over heated mountains using theory and numerical simulation. Based on the linearized, Boussinesq equations of motion as well as a nonlinear thermodynamic heat-engine model, we obtained theoretical scalings for updraft strength and directly compared them to those produced by idealized numerical simulations. The experiments were characterized by varying background wind speeds , boundary layer stabilities N₀, mountain heights h_m, and diurnal heating amplitudes . Different combinations of these parameters allowed for the sampling of various flow regimes, mechanical and thermal forcing amplitudes, and degrees of nonlinearity. The nonlinearity parameters for mechanical forcing M_M and for thermal forcing M_T were used to distinguish linear from nonlinear flows. To isolate the flow responses to mechanical and thermal forcing, simulations were performed with each forcing applied separately as well as with both forcings combined.

For the quasi-linear simulations where both M_M and M_T were less than unity, the linear scalings accurately predicted the updraft strength. The mechanical updrafts scaled simply as the product of and the terrain slope. The thermal response depended on both N₀ and . For a neutral boundary layer (N₀ = 0), it consisted of a strong updraft centered over the crest for that transitioned to longitudinal updraft bands downwind as increased. Stronger background winds also weakened the updraft by distributing the diabatic surface heating over a longer downwind swath. For a stable boundary layer (N₀ > 0), the thermal response consisted of vertically propagating gravity waves over the obstacle with relatively weak updrafts that were virtually independent of .

Many of the linear results carry over to the nonlinear regime, including the tendency for the mechanical response to strengthen with increasing and the thermal response to weaken with increasing N₀ or . For a neutral boundary layer and nonlinear thermal forcing (M_T > 1), the thermal response was qualitatively similar to that in the linear cases. A concentrated updraft over the crest transitioned to an elongated band downwind as increased. However, nonlinearities caused the updrafts to collapse into a narrower core, which ultimately restrained the circulation strength due to vertical heat ventilation [as described in Kirshbaum (2013)]. The nonlinear heat-engine scaling outperformed the linear scaling in these cases and remained accurate even as M_T exceeded 100. Similarly, the linear mechanical scaling failed in all cases with M_M > 1, where it was outperformed by a simple displacement scaling. Consistent with previous studies, the nonlinear mechanical response exhibited upstream blocking, breaking gravity waves above the crest, a wake with reversed flow along its centerline, and a leeside convergence band downwind.

Because the theoretical scalings treated the mechanical and thermal responses separately, they did not address the interactions between these responses. Analysis of the numerical simulations suggested that these interactions were generally weak due to the physical separation between the two responses and/or the dominance of one response over the other. However, they were significant in some experiments with taller mountains, for which M_M > 1 and the flow was blocked by the terrain. In those cases, the diabatic surface heating lowered the pressure in the lee, which strengthened the flow reversal within the mountain wake as well as the leeside convergence band downwind. As both of these surface-based updrafts can effectively initiate deep convection (e.g., Hagen et al. 2011; Mass 1981), elevated heating may thus significantly enhance the potential for convection initiation in blocked orographic flows. The impacts of diabatic heating on the flow dynamics in these cases were roughly quantified using simple parcel arguments. Future research will test the scalings in more realistic environments, with an eye toward the improved representation of mountain convection in large-scale models.