a. Linear model

As in previous studies (e.g., Robinson et al. 2008) a semianalytical solution is obtained to the dry, linearized, two-dimensional (2D) Boussinesq system of equations. Although 2D is a major limitation that limits the real-world applicability of these results, it enables straightforward physical interpretations and thus provides a basis for understanding more complicated flows. The Coriolis force is neglected and the background flow is assumed to be steady, which renders this analysis most applicable to synoptically quiescent tropical or high-Rossby-number midlatitude flows. We begin by presenting a form of the dry, nonlinear, 2D Boussinesq system that represents the basis for all linear and nonlinear analysis to follow:

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where is the total wind velocity, decomposed into a mean [, where is a zonal background wind] and perturbation component. In addition, P = p/ρ₀ is the density-normalized pressure perturbation (relative to the hydrostatically balanced basic state) and p and ρ₀ are the pressure perturbation and reference density, b = gθ′/θ₀ is the buoyancy and θ′ and θ₀ are the perturbation and reference potential temperature, Q is a prescribed heating function, and N is the Brunt–Väisälä frequency. The time-dependent variables are u, w, b, and P. To represent dissipation, Rayleigh damping terms are added to the right-hand sides of each prognostic equation, all with the coefficient α = 2 × 10⁻⁵ s⁻¹, which is similar to that inferred experimentally and used in previous linear studies (e.g., Stevens et al. 2002; Jiang 2012a). By expanding (1) and removing all products of perturbation terms, we obtain the linearized equations for a given layer (where the layer number is denoted by the subscript i):

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The diurnally varying surface-based heat source Q is based on CT05, who considered a steady, exponentially decaying heating function of the form

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where is a Gaussian terrain profile with a peak of h_m, a half-width of a_m, and a centerpoint of x_c; D is an e-folding heating decay scale; and , where is a heating rate (K s⁻¹) and T₀ is a reference temperature. Because the heating function is horizontally invariant over flat terrain, the only source of baroclinicity is variations in terrain height. To the accuracy of the linear approximation, the differential heating owing to elevated terrain is given by

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where Ω = 2π day⁻¹ is the earth’s angular speed. This expression, which is henceforth used in (6), is identical to that used by CT05 except for its inclusion of diurnal variability (the time-varying exponential).

Although this formulation contains important features that were previously neglected, it still fails to capture many real-world complexities. First, it does not take into account diurnal changes in boundary layer stability—it only isolates the dynamical impacts of diurnal heating and cooling within a steady background flow. Similarly, the assumption of an exponential heating function (9) with a fixed vertical structure differs from reality, where the effective heating depth shrinks in the stable nocturnal boundary layer. Despite these simplifications, this formulation allows for an attractively straightforward analysis that, as will be seen, does offer useful insights into more complex flows.

With some arithmetic, the above system of equations may be combined into a single partial differential equation for w_i:

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Applying the Fourier transform in the x and time dimensions, w in (10) may be represented as , where k is the horizontal wavenumber and ω the angular frequency. Because only the diurnal temporal mode is considered (i.e., ω = Ω), this may be simplified to (e.g., Jiang 2012a). Adopting this representation, (10) may be written as an ordinary differential equation

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where σ = Ω − Uk − iα and . We solve (11) for a two-layer atmosphere (layers 0 and 1) with stabilities N₀ and N₁, separated by an interface at z = H₀ (shown schematically in Fig. 1). This configuration can accommodate a uniform stability profile as in previous studies (e.g., CT05) or a more realistic weakly stratified boundary layer beneath a stable free troposphere [as in Jiang (2012b)].

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

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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The linear model is solved by applying four boundary conditions: a free-slip lower surface, an upper radiation condition, and matching of displacement and pressure at z = H₀ (see appendix for details). It is represented on a discretized x-periodic domain with a length of L_x = 400 km and a grid spacing of Δx_lin = 0.2a_m. These values of domain length and horizontal resolution are large enough to ensure numerical robustness yet small enough to allow for efficient solutions. Because the Fourier transform is not applied in the vertical, the solution does not depend on the vertical grid settings. Thus the domain depth L_z is set to be just high enough to contain the dynamics of interest, and the vertical grid spacing is Δz_lin = 50 m to adequately resolve the low-level flow.

b. The heat-engine framework

Renno and Ingersoll (1996) and Souza et al. (2000) developed a theory for thermally driven topographic circulations based on an analogy with thermodynamic heat engines, which was refined and applied to the mountain heating problem by Tian and Parker (2003). This theory is briefly reviewed here and applied to the flows under consideration in subsequent sections. The approach integrates Bernoulli’s equation and the first law of thermodynamics around the closed loop of a steady-state convective circulation. This allows the strength of a convective updraft W_t to be related to the total convective available potential energy available for mechanical work (TCAPE) by the following expression:

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where μ is a dimensionless coefficient of mechanical dissipation. In a dry convective boundary layer, TCAPE ≈ ηc_pΔT, where η is the thermodynamic efficiency of the heat engine, c_p is the specific heat of dry air at constant pressure, and ΔT is the mean horizontal temperature difference between the updrafts and downdrafts, or the “nonadiabatic” temperature difference between the heated terrain and the surrounding plain. In our experiments, the maximum ΔT may be estimated by integrating the positive phase of the surface heating function [from 0600 to 1800 local time (LT), or t = −π/2Ω to t = π/2Ω]:

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Assuming that the system acquires its heat from the surface and dissipates it at the top of the convective boundary layer, the thermodynamic efficiency may be written

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(Souza et al. 2000), where T_h is the temperature of the hot reservoir (the heated surface) and T_c is the temperature of the cold reservoir (the boundary layer top). To obtain μ we use the formulation from Tian and Parker (2003), which assumes that scale of the dominant convective circulation is dictated by the geometry of the surface heat source and the boundary layer depth:

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Substituting (13)–(15) into (12), we obtain

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Because the terrain is idealized as a flat heat source, no provision is made for the adiabatic temperature difference between the mountain base and peak, but it may be easily incorporated if needed (Tian and Parker 2003). Note that although the diurnally varying situations considered herein are obviously not in steady state, this theory is still reasonable if the time scale for an air parcel to loop through the circulation is much shorter than the diurnal time scale. This condition is only satisfied in the more strongly forced cases with more vigorous convective circulations; thus, as will be seen, the theory tends to poorly represent weakly forced cases. Note also that because this derivation assumes a closed circulation in a convective boundary layer, we only apply it to cases with .

c. Idealized numerical simulations

To critically assess the validity of the linear approximation and to extend our examination to the nonlinear regime, we perform idealized numerical simulations with the Bryan cloud model (cm1), version 14 (Bryan and Fritsch 2002). This model solves the primitive moist atmospheric equations using a split time step procedure to maintain stability of acoustic modes. On the large time step, time integration is performed with a third-order Runge–Kutta scheme. Ten 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 applied in the horizontal, explicit sixth-order horizontal diffusion is added to minimize spurious grid-scale waves. The domain dimensions are L_x = 400 km and L_z = 12 km and the grid resolution is Δx_sim = 500 m and Δz_sim = 100 m. Boundary conditions are open in x and closed in z with a free-slip lower surface and a Rayleigh damper over the uppermost 4 km to absorb upward-propagating gravity waves. The only physical parameterization used in the simulations is a 1.5-order TKE-based subgrid-turbulence scheme. The runs are “dry” in that they contain no water vapor.

The simulations are performed in double precision to diminish the amplitude of numerical roundoff error, which can trigger instabilities in weakly unstable flows like some of those considered here. Although turbulent eddies are physically realistic, they mask the thermal circulations of interest and thus cause the numerical and analytical solutions to grossly differ. By suppressing their growth and allowing model diffusion and/or subgrid turbulence to carry out the mixing, we may directly compare the two models on equal footing. The simulated circulations are thus best interpreted as time-averaged fields within a turbulent flow. However, their reliance on subgrid-scale (rather than resolved) turbulence and numerical diffusion may underestimate the degree of boundary layer mixing as well as the entrainment across the boundary layer top. Because both effects tend to weaken the horizontal buoyancy gradients that drive thermal circulations, the numerical solutions obtained herein may tend to overestimate the strength of these circulations.

Except for a single simulation that verifies the validity of the localized heating function in (9) for representing elevated heating, all experiments use flat terrain with this heating function. This is done for consistency with the linear model and because it isolates the thermal response to localized heating, which is the focus of this study, in the absence of any mechanical forcing. Another benefit of this approach is that the results are generalizable to any mechanism of differential surface heating (such as land surface heterogeneities).

The model initialization must be handled with some care to reasonably match the linear solutions. 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 (or with a uniform basic-state flow) imposes differences from the linear solution that persist indefinitely. To overcome this problem, the model was started from rest but the 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 48 h. By allowing the flow dynamics to develop gradually, the numerical-model solutions were able to fall into line with the corresponding linear solutions. All simulations are initialized at 0600 LT so that t = π/2Ω and Q = 0 in (9).

a. Vertical velocity scale

We begin by estimating a characteristic magnitude of the thermally induced updraft velocity W_t. Although such a scaling is straightforward for a uniform atmosphere, it can become much more challenging in multilayer flows where the dynamics of a given layer are influenced by interactions with surrounding layers. However, if one restricts consideration to two basic yet highly relevant situations—a uniformly stratified atmosphere (N₀ = N₁) and a neutrally stratified boundary layer overlaid by a stable free troposphere (N₀ = 0)—simple yet accurate scalings do become possible.

We begin by nondimensionalizing the state variables through the following assignments: , , , , , and , where the variables with tilde symbols are nondimensional and of (1). Substituting these relations into (10) and cancelling (1) terms, we obtain the following expression for W_t:

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Care is required in selecting the horizontal (L) and vertical (H) length scales. For the continously stratified case (N₀ = N₁), the solution consists primarily of vertically propagating gravity waves. Following Rotunno (1983), the governing partial differential equation in (10) implies a natural aspect ratio of . Since the prescribed horizontal (a_m) and vertical (D) forcing scales are unlikely to obey this relationship, only one of them can control the response. To identify the controlling scale, we first select L* = a_m and calculate the associated natural vertical scale (H* = a_m|σ_L|/N₀). For H* > D, we set H = H* to select the larger of the two relevant vertical scales, along with L = a_m. For H* < D, we set H = D and L = DN₀/|σ_L| to again choose the larger scale and enforce the natural aspect ratio. In the latter case, because |σ_L| is a function of L, the equation is implicit and an iterative procedure is required to solve for both |σ_L| and L.

For the neutral boundary layer (N₀ = 0), the two-layer response does not conform to the natural aspect ratio of either layer alone. A mixed response develops in the vertical, with the dominant signal confined to the boundary layer and a weak extension into the stable free troposphere. An appropriate choice for the vertical scale is H = H₀/2, midway through the boundary layer where the strongest vertical motions are found. In the horizontal, where the circulation extent is dominated by the applied forcing scale, we set L = a_m.

b. Validity of linear theory

The linear approximation is only valid if the magnitudes of the linear terms in (4)–(7) are much larger than the (unwritten) nonlinear terms. To this end, we estimate the magnitude of the nonlinear advection terms in (5) and compare them to that of the dominant linear terms. This approach is valid under the weakly linear approximation, whereby the first-order perturbations obtained from the linear solution may be used to estimate the second-order nonlinear terms. The nonlinear horizontal advection of w may be estimated as

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where a scaling of (7) was used to estimate the magnitude of the perturbation u-wind. The specific choice of the nonlinear advection term in this equation (i.e., zonal advection versus vertical advection) is unimportant because both terms share the same characteristic magnitude. To estimate the magnitude of the linear terms, we first must identify which term on the left-hand side of (5) is dominant. This depends on ; when the linear advection term dominates and when the time-tendency term dominates. This leads to two nonlinearity parameters (M_t) for the two cases:

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c. Flow regimes

Based on (6) and (17), three distinct flow regimes naturally emerge, the first being the “growth–decay” (GD) regime, where the heating term in (6) is balanced by the time-tendency and/or dissipation terms. One can show that this regime prevails when . The two other basic regimes are the “ventilation” (VE) regime , where the heating is balanced by the linear advection term, and the “stratification” (ST) regime , where the heating is balanced by the stability term [the first on the right-hand side of (6)]. Based on these dominant balances, (17) can be simplified to give the following W_t estimates for each regime:

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where we have also assumed that H/L ≪ 1. In all three regimes W_t depends directly on Q₀h_m/D, which determines the low-level baroclinicity. It also increases with H² in the GD and VE regimes for two reasons: (i) based on the hydrostatic form of (6), the accumulation of buoyancy in a neutral layer gives rise to pressure perturbations (and hence horizontal wind perturbations U) that scale with H and (ii) from (7), W_t ~ UH/L, hence W_t ~ H². In the GD regime W_t is inversely dependent on (Ω + α)² and L², the former being the inverse time scale for perturbation growth and the latter controlling the horizontal buoyancy gradients that drive the circulation. In the VE regime, W_t depends inversely on , which causes downwind ventilation of heat and thus weakens the buoyancy anomalies over the heat source. Finally, in the ST regime, W_t is inversely dependent on , which controls the fluid resistance to vertical motion.

To provide a tangible comparison of the updraft magnitudes and linearity within these different flow regimes, Table 1 lists six different situations distinguished by their values of and N₀ for a short (h_m = 100 m) and narrow (a_m = 5 km) mountain under moderate heating . Other parameters include H₀ = 1 km (for cases with N₀ = 0) and T₀ = 300 K. The regime classification, along with W_t and M_t, is provided for each case. Updrafts are the strongest, and the flow the most nonlinear, for a neutral boundary layer with (the GD regime). This is because neither stable ascent nor wind ventilation exists to diminish the growth of buoyancy over the heat source, so the response attains a large amplitude. The updrafts are weaker and the flow more linear in the ST regime, owing to the effectiveness of stable ascent at balancing the imposed heating. The other linear term capable of effectively balancing the heating is horizontal advection by the background wind, which strongly reduces M_t in the VE regime. Thus the linear approximation is more valid in the ST and VE regimes than in the GD regime, where most real-world flows easily exceed the nonlinearity threshold. Nonetheless, the analysis in section 4 demonstrates that even in nonlinear cases, the linear theory is still useful for physically interpreting some (but not all) of the basic parameter sensitivities.

Table 1.

Regime classifications and scaled thermal and mechanical updraft velocities (W_t and W_m) from (17) for a few different combinations of and N₀. In all cases h_m = 100 m, a_m = 5 km, D = H₀ = 1 km, and , where and T₀ = 300 K is a reference temperature. The linearity is determined by application of (19).

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d. Mechanical forcing

In addition to inducing thermal circulations, mountains force mechanical ascent when the background cross-barrier winds are nonzero. For taller obstacles, strong interactions between mechanical and thermal forcing may occur, such as when flow splitting around a massif creates a decelerated wake region that gives rise to an amplified thermal circulation (e.g., Jury and Chiao 2011). However, because mechanical forcing is not a focus of this paper, a thorough analysis of such complex interactions is deferred to future work.

As a starting point, we restrict our analysis to the linear approximation—for which the thermal and mechanical responses formally decouple—and offer some preliminary insights into the relative strengths of the thermal and mechanical responses. The mechanical-lifting strength may be estimated from the free-slip lower boundary condition as . Values of W_m are compared to W_t for the six cases in Table 1. The ratio of thermal to mechanical lifting is then given by

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As before, one may simplify this ratio for the three regimes identified in section 3c:

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This suggests that the thermal response tends to dominate when the cross-barrier winds and boundary layer stratification are weak, which is consistent with the numerical findings of Tian and Parker (2003), or when the heating rate and boundary layer depth are large. For the VE regime, the above scaling is identical to that derived by previous authors (Smith and Lin 1982; CT05). However, (22) reveals that those studies only addressed a fraction of the relevant parameter space.

a. Examples of linear solutions

Linear-model solutions for the “baseline” case of h_m = 1 m, a_m = 5 km, , N₀ = 0 s⁻¹, N₁ = 0.013 s⁻¹, H₀ = D = 1 km, and are shown in Fig. 3 alongside corresponding fields from the numerical solution. With its short terrain and weak forcing, this case falls well within the linear regime defined by (20). However, the linear and numerical model solutions are not identical. This is primarily due to the damping term in the linear system, which, at least in this case, is unrealistically large. Although one can reduce α to produce a nearly identical match between these two solutions, it would degrade the comparison for other cases, particularly those with stronger thermal forcing. Thus we hold α fixed and accept some modest disagreements between the two models.

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Comparison of linear and numerical model solutions for the baseline case of h_m = 1 m, a_m = 5 km, , N₀ = 0 s⁻¹, H₀ = D = 1 km, and . Filled grayscale contours denote positive vertical motion; dashed lines denote negative vertical motion (using the same contour scale as in the color bar). Thin black contours are isentropes at a 1-K interval. The arrows show perturbation wind vectors. The velocity reference scales in (g) and (h) also apply to (a)–(f).

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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The vertical-motion patterns in Fig. 3 generally differ from the steady-state solutions obtained by CT05 (e.g., their Fig. 4) that took the form of upstream-tilted vertically propagating gravity waves. These differences arise from our consideration of time variability, weak background winds, and a neutrally stratified boundary layer, all of which are common characteristics of thermally forced flows but were previously neglected. As in Fig. 2, the boundary layer response at 0000 LT is a solenoidal circulation with an updraft over the terrain centerpoint surrounded by two symmetric downdrafts. Strong horizontal convergence is apparent at the updraft bottom, with divergence at the boundary layer top. The peak updraft speeds at that time are similar to the W_t estimates from (17).

Surprisingly, both models generate their strongest downdrafts at around noon and their strongest updrafts at around midnight, or about 12 h after these features normally develop in reality. As discussed by Rotunno (1983), this is due to the dominance of the time-tendency terms in (5) and (6), which together imply up to a 12-h lag between the Q maximum at 1200 LT and the peak kinematic response. This is seen by solving (5) and (6) for the idealized case of (i.e., the parcel approximation in a wind-free, neutral, inviscid flow), which gives w = −Q/Ω². Hence, at least for this simple case, the phases of w and Q differ by 180°. In reality, the emergence of other terms in these equations, namely nonlinearities and turbulent dissipation, reduces the delay between the heating cycle and the kinematic response. This is reflected by Fig. 3, where the stronger effective dissipation in the linear model shifts the diurnal cycle of vertical motion forward by about 2 h. When α is increased to 1.0 × 10⁻⁴ s⁻¹ the timing of the maximum updraft shifts forward all the way to 1400 LT (not shown).

Vertical velocity fields of linear solutions for the six cases from Table 1 are compared in Fig. 4, where the display times are chosen separately for each case to capture their mature responses to the daytime heating. A rich spectrum of dynamical responses arises depending on the choices of governing parameters. For zero wind, a vertically decaying plume of ascent develops with a maximum just above the surface in the stable case (ST1; Fig. 4b). This is surrounded by two vertically tilted downdraft beams, forming a gravity wave circulation that transports perturbation energy large distances from the heat source. This response differs from the solenoidal circulation in the neutral case (GD1; Fig. 4a), where the perturbations are stronger but more localized at the heat source. The differences in the timing of these responses relate to their different thermal inertia: whereas strong buoyancy anomalies are sustained for several hours after the heating ceases in the GD1 case, they are rapidly weakened by stable ascent in the ST1 case. When the stability term dominates over the time-tendency terms in (6), the vertical motion becomes in phase with the heating function and the strongest updraft occurs at noon.

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As in Fig. 3, but for six different linear model solutions using the parameter settings in Table 1. For reference, the name of each case follows the convention in Table 1 and the time is shown in the upper-right corner. The times are chosen to capture the thermal response in a mature state over the 1200–0000 LT period. Thin black contours are isentropes at a 1-K interval.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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For the neutral boundary layer, modest winds of are sufficient to dramatically weaken the central updraft and displace it downwind (Fig. 4c). This is consistent with the cloud-resolving simulations of Kirshbaum (2011), where background winds of 1.5 m s⁻¹ strongly weakened boundary layer updrafts and suppressed the initiation of deep convection. Stronger winds also cause the emergence of upstream-tilted, vertically propagating gravity waves, which become better defined for larger N₀ and (Figs. 4c–f). These waves, which owe their existence to the interaction between ambient winds and a stationary heat source, are similar to those found in recent studies of sea-breeze dynamics (Qian et al. 2009; Jiang 2012b). As in CT05, the gravity wave response is characterized by low-level descent upstream of the terrain, where the flow accelerates into the low pressure center, and ascent downwind. Because the exponential heating function extends into the stable troposphere, elevated waves form even in the neutral cases. However, these are not visible in the zero-wind case (Fig. 4a) because they are masked by the dominant boundary layer circulation. In all of the cases with nonzero winds, the thermal inertia is diminished by horizontal advection and/or stable ascent/descent, so the diurnal updrafts are in phase with the heating cycle.

b. Parameter sensitivities

Experiments covering a broad range of parameter space are conducted to examine the environmental and terrain-related sensitivities of thermally forced updrafts. For all other parameters equal to the baseline case (a_m = 5 km, , D = 1 km, and H₀ = 1 km), Fig. 5 shows that the updraft magnitude depends nearly linearly on the terrain height h_m (for ) and the surface heating rate (for h_m = 1 m), in agreement with (17). The W_t values from both the linear scaling and the numerical simulations are presented, the latter of which is taken as the maximum simulated w over the region |x| ≤ 5a_m and z ≤ 2H over the full diurnal cycle. Because of their close agreement with the linear scalings (and for the sake of clarity), the corresponding W_t values from the full linear solutions are omitted. Results are compared for both a neutral boundary layer and a stable boundary layer with N₀ = 0.013 s⁻¹, indicating a dramatic (10²–10³-fold) enhancement of updraft velocity in the neutral case. A vertical dashed line is added to indicate the threshold values of and h_m at which M_t = 0.25 in (19), beyond which nonlinearities figure prominently in the numerical model solution. This threshold, which varies according to regime (see section 3c), is reached at much smaller values of h_m and in the neutral case (for the stable boundary layer in Fig. 5a, this threshold falls outside the axis limits). The agreement between the simulated and scaled W_t is nearly exact in the linear regime and very good into the nonlinear regime.

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Sensitivities of updraft velocity to (a) heating rate (for a fixed value of h_m = 1 m) and (b) mountain height (for a fixed value of ). Results for a neutral (N₀ = 0) and stable (N₀ = 0.013 s⁻¹) boundary layer are shown by black and gray markers, respectively. Nonlinearity thresholds of M_t = 0.25 are shown for by vertical dashed lines, adhering to the same color scale as the markers.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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Also shown in Fig. 5 are the corresponding W_t predictions from the heat-engine theory in (16), for the neutral cases only. These grossly overestimate W_t under weak forcing but closely match the simulations within the nonlinear regime. The poor performance in the linear regime is likely linked to the violation of the steady-state assumption in (16). With a characteristic updraft strength of W_t ≤ 0.1 m s⁻¹, the time required for an air parcel to complete a full cycle through the convective circulation is over 12 h, which renders the steady-state assumption invalid. However, because meteorologically significant circulations usually exceed this strength, this overprediction in the linear regime is not a major concern.

The sensitivities of W_t to other governing parameters are shown in Fig. 6 for a weakly heated bump (h_m = 1 m, ) (left panels) and a strongly heated hill (h_m = 100 m, ) (right panels). With some exceptions, the updraft velocities in the neutral boundary layer are multiple orders of magnitude stronger than those in the stable boundary layer, which reinforces the strong sensitivity of W_t to N₀. The boundary layer stability also controls the sensitivity to background winds—whereas W_t sharply decreases with increasing in the neutral case, it is insensitive to for the stable case (Figs. 6a,b). The neutral boundary layer transitions from the GD into the VE regime for small values of , causing W_t to become inversely proportional to and thus decrease rapidly with increasing in (20). By contrast, the stable cases lie uniformly within the ST regime, which from (20) implies no sensitivity to . One shortcoming of the linear scaling is that it overpredicts W_t in the neutral boundary layer for larger , likely because of its neglect of free-tropospheric gravity waves that extend downward into the boundary layer (e.g., Fig. 4e). Nonetheless, it still matches the simulations to within a factor of 2. Although the scaling performance worsens in the nonlinear regime (the neutral cases in Fig. 6b), it still captures the trend for W_t to decrease rapidly with increasing background winds. This suggests that the downwind advection of heat, a process reasonably well described by linear theory, is the principal mechanism for the rapid weakening of the updrafts for increasing .

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Sensitivities of updraft velocity to (a),(b) background cross-barrier wind speed, (c),(d) terrain width, (e),(f) heating depth, and (g),(h) boundary layer depth, for a neutral (N₀ = 0, black) and stable (N₀ = 0.013 s⁻¹, gray) boundary layer. Left panels correspond to a short mountain and weak heating rate (h_m = 1 m and ), and right panels correspond to a taller hill and strong heating rate (h_m = 100 m and ). Nonlinearity thresholds of M_t = 0.25 are shown for the neutral cases by vertical dashed lines. These lines are not drawn for the stable cases, which all fall within the linear regime.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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Similarly, the neutral and stable boundary layers exhibit contrasting linear sensitivities to a_m: the former sharply decreases with increasing a_m but the latter is largely insensitive to it (Fig. 6c). In the neutral boundary layer, W_t is controlled by the amplitudes of horizontal buoyancy gradients, which, all else being equal, strengthen as the terrain narrows. By contrast, for a stable boundary layer the vertical motion is governed primarily by the stability and the heating-amplitude Q₀h_m/D [see (20)], which is invariant among these cases. Although the linear scaling performs extremely well in the linear regime, it performs poorly in the nonlinear regime where the simulations exhibit virtually no sensitivity to a_m (Fig. 6d). As will be discussed in section 5, this is due to nonlinear advection, which profoundly modifies the evolution of the convective circulation. Compared to the linear scaling, the heat-engine framework provides a much better estimate of W_t in the nonlinear regime.

The updrafts tend to strengthen with decreasing D and increasing H₀, both in the linear and nonlinear regimes (Figs. 6e–h). The linear scaling performs extremely well in the linear regime but, like the heat-engine model, it tends to overestimate W_t up to fourfold in the nonlinear regime. The linear scaling and heat-engine model show slightly different sensitivities to D and H₀, with the former overestimating the slope of the simulated trend and the latter underestimating it. The general overestimation of W_t by the heat-engine model may not be a flaw with the theory but the result of the assumption that ΔT equals the integrated surface-based heating in (13). This is surely an overestimate—advection and diffusion both diminish ΔT by transporting heat away from the terrain. Nonetheless, Figs. 5–6 illustrate that the theory generally performs well in the nonlinear GD regime. Thus it serves as a useful complement to the linear theory, which performs best in the VE and ST regimes.

a. Physical characteristics

Some essential differences between the linear and nonlinear flow responses are shown in Fig. 7, which compares w and b fields for the case with h_m = 100 m and a_m = 5 km (termed the “HM100-AM5” case) at 0600 and 1000 LT. At sunrise (0600 LT), the linear solution contains a strong downdraft at the terrain centerpoint surrounded by two updrafts (Fig. 7a), which is identical to the baseline case (Fig. 3a) except for a larger magnitude. The numerical model also has a central downdraft at that time, but its magnitude falls below the minimum contour interval (Fig. 7b). These circulations are accompanied by radically different b signatures; whereas the linear model has a cold anomaly extending vertically through the boundary layer, the numerical model has a thin, surface-based cold layer spread over a broad area (Figs. 7c,d). In the latter, nocturnal cooling above the terrain leads to a downward plunging of cold air and the formation of two density currents that transport negatively buoyant air outward (not shown). Despite the very idealized nature of this experiment, this is a realistic effect that resembles katabatic winds or coastal land breezes (e.g., Mahrt 1982). Between 0600 and 1000 LT the boundary layer downdraft strengthens in the linear model as cold air persists over the terrain, reflecting the high thermal inertia of this case (Figs. 7e and g). By contrast, the numerically simulated downdraft transitions into a compact updraft over the terrain centerpoint, which coincides with a relatively narrow buoyancy anomaly (Figs. 7f and h). Again, the narrowness of this updraft is consistent with real thermal circulations, the central updrafts of which collapse into sharp zones of concentrated ascent (e.g., Kirshbaum 2011; Barthlott et al. 2011).

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Comparison of linear and numerical model solutions for the HM100-AM5 case. At 0600 LT, solutions for w are provided from the (a) linear and (b) numerical models, followed by b for the (c) linear and (d) numerical models. Similarly, 1000 LT solutions for w are provided from the (e) linear and (f) numerical models, followed by b from the (g) linear and (h) numerical models. Positive values are in filled grayscale contours and negative values are shown by dashed lines (using the same contour interval). Thin black contours are isentropes at a 1-K interval. Wind velocity vectors are overlaid as arrows.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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The sensitivity of the thermal circulations to a_m is illustrated by Fig. 8, which compares time series of relevant quantities for two cases in the nonlinear GD regime: the HM100-AM5 case (again, with a_m = 5 km) and the HM100-AM20 case (with a_m = 20 km). Four quantities are calculated within an analysis box covering : (i) the averaged buoyancy b_avg, (ii) the minimum surface pressure , (iii) the maximum vertical velocity w_max, and (iv) the averaged horizontal vorticity η_avg. Both b_avg and are largely insensitive to a_m in the linear solutions, with the maximum of the former and minimum of the latter coinciding at around 1700 LT (Figs. 8a and c). By contrast, the numerical model solutions indicate a major phase shift between the two cases, with the maximum b_avg (and minimum ) falling at around 1130 LT in the HM100-AM5 case and at around 1600 LT in the HM100-AM20 case (Figs. 8b and d). Moreover, the peak magnitudes of the simulated b_avg and are significantly lower in the HM100-AM5 case than in the corresponding linear solution or the HM100-AM20 case. The linear model predicts both circulations reaching their maxima at 2200 LST, with much larger w_max and η_avg for the HM100-AM5 case (Figs. 8e and g). By contrast, in the numerical solutions w_max and η_avg have similar peak magnitudes but differ mainly in phase (Figs. 8f and h).

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Time series of various quantities, calculated over the region for the HM100-AM5 (black lines) and HM100-AM20 (gray lines) cases from the (left) linear and (right) numerical models: (a),(b) mean buoyancy, (c),(d) minimum pressure, (e),(f) maximum vertical velocity, and (g),(h) averaged vorticity.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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b. Vorticity analysis

To conceptually explain the substantial differences between the linear and nonlinear responses in Figs. 7–8, we examine the horizontal vorticity and the processes that control it. Taking the curl of the nonlinear Boussinesq momentum equation in (1) gives

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where we have set for application to the GD regime. The first term on the right-hand side of (23) is the nonlinear vorticity advection, the second is the horizontal buoyancy gradient, and the third is damping. Although the simplified Boussinesq representation of η in (23) differs from that in the fully compressible numerical model, it still provides useful and straightforward insight into the behavior of the simulated flow.

Figure 9 provides a snapshot of the advection and buoyancy gradient terms of (23) during the growth phase of the HM100-AM5 simulation at 1000 LT (because the damping term cannot strengthen the circulation, we do not analyze it in detail). The buoyancy gradient term clearly increases (decreases) η for x > 0 (x < 0), which tends to strengthen the overall convective circulation (Fig. 9a). The advection terms have a more complex structure, with a strong couplet straddling x = 0 that locally sharpens the circulation and secondary features farther from the center (Fig. 9b). The broad evolution of these terms is shown by time series for the HM100-AM5 and HM100-AM20 cases in Fig. 10, which are again averaged over . Whereas the buoyancy gradient monotonically increases the vorticity for several hours in both cases, the nonlinear advection generally tends to weaken it. Although the couplet at x = 0 in Fig. 9b amplifies the vorticity locally, it evidently does not strengthen the overall mountain-scale solenoidal circulation because of cancellations with other features within the analysis box. Thus, the buoyancy gradient term drives the overall circulation, with the nonlinear advection acting primarily to tighten the central updraft into a narrow core.

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Illustration of the dominant terms contributing to the averaged vorticity tendency in (23), for the HM100-AM5 simulation at 1000 LT: (a) the negative horizontal buoyancy gradient and (b) the nonlinear advection. Positive values are shown by filled grayscale contours and negative values are shown by dashed lines (with the same contour interval). Wind velocity vectors are overlaid as arrows.

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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Time series of the dominant terms contributing to the averaged vorticity tendency in (23), for the HM100-AM5 (black lines) and HM100-AM20 (gray lines) simulations: (a) the negative horizontal buoyancy gradient and (b) the nonlinear advection. These quantities are calculated over .

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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The main sensitivity to a_m in Fig. 10 is that |∂b/∂x| initially increases faster but reaches its apex much earlier in the HM100-AM5 case (around 1000 LT) than in the HM100-AM20 case, where it slowly increases to a maximum at around 1430 LT. Because the buoyancy gradient driving the latter circulation is maintained for substantially longer, the circulation ultimately becomes more vigorous. This is consistent with the evolution of the four quantities in Fig. 8. Whereas the magnitude of each quantity initially grows faster in the HM100-AM5 case, it eventually becomes larger in the HM100-AM20 case owing to its extended growth phase. In addition, Fig. 10b indicates that nonlinear momentum advection substantially weakens the circulation in the HM100-AM5 case between 1000 and 1800 LT (Fig. 10b). A similar trend is apparent in the HM100-AM20 case over 1400–2100 LT, but with a significantly lower amplitude.

Given the central importance of the horizontal buoyancy gradient for driving the circulation, one must understand the factors that control it to interpret the behavior of thermally forced flows. To this end, we analyze the time derivative of −∂b/∂x, which is given by

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The first two terms on the right-hand side are related to the nonlinear buoyancy advection, the third is the horizontal gradient of the heating function, and the fourth is damping. The evolution of the first three terms (again averaged over ) is compared for the HM100-AM5 and HM100-AM20 cases in Fig. 11. As the convective circulations gain strength under diurnal heating, the horizontal advection term (u∂b/∂x)_x transitions from positive to negative (Fig. 11a). Henceforth, this term possesses the same sign as, and attains similar magnitudes to, the diurnal heating term (−∂Q/∂x) (Fig. 11c). It enhances the horizontal buoyancy gradients by squeezing b contours together, which is reflected by the b evolution over 0930–1230 LT in Fig. 12. From 0930 to 1030 LT the main buoyancy anomaly transitions from a cone into a mushroom shape, with a relatively narrow central core (Figs. 12a,b and Fig. 7h). The associated increase in |∂b/∂x| forces |η_avg| to increase, which, in turn, contracts the anomaly further. This positive feedback continues until the vertical advection term (w∂b/∂z)_x sharply increases to oppose further growth of the circulation (Fig. 11b).

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Time series of the dominant terms contributing to the −∂b/∂x tendency in (24), for the HM100-AM5 (black lines) and HM100-AM20 (gray lines) simulations: (a) the zonal buoyancy advection, (b) the vertical buoyancy advection, and (c) the horizontal gradient of the heating function Q. These quantities are calculated over .

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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Evolution of the convective circulation and central buoyancy anomaly in the HM100-AM5 case, in hourly intervals: (a) 0930, (b) 1030, (c) 1130, and (d) 1230 LT. Positive values of b are shown by filled grayscale contours and negative values by dashed lines (with the same contour interval). Velocity vectors shown by arrows, all using the same reference scale as that shown in (a).

Citation: Journal of the Atmospheric Sciences 70, 6; 10.1175/JAS-D-12-0199.1

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The emergence of the vertical buoyancy advection term occurs earlier (1000 LT) in the HM100-AM5 case than in the HM100-AM20 case (1430 LT) because its stronger initial buoyancy gradients promote faster development of the circulation. At this time, a sharp increase in heat ventilation by the main updraft core transports the warmest air from the surface to the upper boundary layer (Figs. 12b,c). This air then spreads laterally, entrains free-tropospheric air to warm further, and begins to subside within the convective downdrafts. A large thermal anomaly is apparent by 1230 LT with two broad, symmetric lobes centered at (Fig. 12d). With the buoyancy spread over a relatively large area, the overall convective circulation undergoes a broadening and weakening, which is reflected by the slow decrease in η_avg and w_max from 1100 to 1800 LT in Figs. 8f and h. The same process occurs in the HM100-AM20 case after 1430 LT case, once the circulation is sufficiently strong for vertical momentum advection to displace the main surface-based buoyancy anomaly.