Numerical Simulation of Katabatic Flow with Changing Slope Angle

Craig M. Smith College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Eric D. Skyllingstad College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Abstract

A large eddy simulation (LES) model and the Advanced Regional Prediction System (ARPS) model, which does not resolve turbulent eddies, are used to study the effect of a slope angle decrease on the structure of katabatic slope flows. For a simple, uniform angle slope, simulations from both models produce turbulence kinetic energy and momentum budgets that are in good overall agreement. Simulations of a compound angle slope are compared to a uniform angle slope to demonstrate how a changing slope angle can strongly affect the strength of katabatic flows. Both ARPS and the LES model show that slopes with a steep upper slope followed by a shallower lower slope (concave shape) generate a rapid acceleration on the upper slope followed by a transition to a slower evolving structure characterized by an elevated jet over the lower slope. In contrast, the case with uniform slope (having the same total height change) yields a more uniform flow profile with stronger winds at the slope bottom. Higher average slope in the uniform slope angle case generates greater gravitational potential energy, which is converted to kinetic energy at the bottom of the slope. Analysis of the total energy budget of slope flows indicates a consistent structure where potential energy generated at the top of the slope is transported downslope and converted into kinetic energy near the slope base.

Corresponding author address: Craig Smith, COAS, 104 COAS Admin. Bldg., Oregon State University, Corvallis, OR 97331. Email: csmith@coas.oregonstate.edu

Abstract

A large eddy simulation (LES) model and the Advanced Regional Prediction System (ARPS) model, which does not resolve turbulent eddies, are used to study the effect of a slope angle decrease on the structure of katabatic slope flows. For a simple, uniform angle slope, simulations from both models produce turbulence kinetic energy and momentum budgets that are in good overall agreement. Simulations of a compound angle slope are compared to a uniform angle slope to demonstrate how a changing slope angle can strongly affect the strength of katabatic flows. Both ARPS and the LES model show that slopes with a steep upper slope followed by a shallower lower slope (concave shape) generate a rapid acceleration on the upper slope followed by a transition to a slower evolving structure characterized by an elevated jet over the lower slope. In contrast, the case with uniform slope (having the same total height change) yields a more uniform flow profile with stronger winds at the slope bottom. Higher average slope in the uniform slope angle case generates greater gravitational potential energy, which is converted to kinetic energy at the bottom of the slope. Analysis of the total energy budget of slope flows indicates a consistent structure where potential energy generated at the top of the slope is transported downslope and converted into kinetic energy near the slope base.

Corresponding author address: Craig Smith, COAS, 104 COAS Admin. Bldg., Oregon State University, Corvallis, OR 97331. Email: csmith@coas.oregonstate.edu

1. Introduction

Atmospheric conditions in mountain basins are controlled by a number of processes generated both locally and by synoptic-scale weather systems. When synoptic conditions are relatively weak, for example during anticyclone periods, basin circulations often follow a diurnal cycle with upslope flow during the day when surrounding mountains are heated, and downslope flows in the nighttime when the same slopes are cooled. Upslope flows typically “mix out” or combine with the surrounding boundary layer growth and lose their identity as daytime heating progresses. Downslope flows, on the other hand, are by nature relatively shallow and can persist for many hours after sunset. Downslope or katabatic flows transport dense air into mountain basins and contribute to stagnant conditions during the winter season by filling basins with relatively cold air, enhancing the strong stratification generated by local radiative cooling.

Katabatic flows can also occur over glaciers. To gain a better understanding of feedback mechanisms between the cryosphere and global climate, many recent observational studies have focused on katabatic flows over ice sheets and glaciers (e.g., Smeets et al. 1998; Oerlemans and Grisogono 2002, and references therein). An overview of some of these studies can be found in Oerlemans (1998). On some glaciers, katabatic flows dominate surface climatology (e.g., Renfrew and Anderson 2002).

Past theoretical and modeling studies of katabatic flows have generally examined simple slope cases with uniform slope angles. Most field observations also have concentrated on locations with only small variations in slope angle (e.g., Horst and Doran 1986). As a result, our understanding of how slope angle decreases affect the dynamics of slope flows is rather incomplete. We address this issue in this study by applying two modeling systems; a large eddy simulation (LES) model capable of resolving finescale turbulent motions and a mesoscale model [the Advanced Regional Prediction System (ARPS)] more appropriate for simulating a larger-scale mountain-basin circulation system. Our goals in using these two models are to first assess how well the ARPS model compares with the LES in simulating realistic slope flow behavior, and then to examine the dynamics and energetics of slope flows for cases with an abrupt slope angle decrease. Use of LES provides a means of testing the turbulence parameterizations commonly used in mesoscale models such as ARPS; previous simulations using LES compared favorably with observations of uniform slope flow mean quantities such as flow velocity and temperature structure (Skyllingstad 2003).

Here we compare the ARPS model with LES to determine if the flow transition produced by a slope angle decrease can be realistically simulated in a mesoscale model. Observations suggest that slope flows can develop features similar to hydraulic flow, for example, jump conditions or turbulent transitions (Monti et al. 2002). With changing slope angle, rapid deceleration of katabatic flows may generate increased turbulent mixing and slope flow structure that is very different from the simple scenario examined in past studies (e.g., Manins and Sawford 1979; Skyllingstad 2003). One of the reasons field studies have avoided slopes that have a change in slope angle is because of the transitions that occur in katabatic flows over these slopes. Flow features in these transitions are hard to explain using limited observations from towers and tethered instruments. With a numerical model, we can use idealized slopes with a single slope angle decrease allowing detailed examination of the changes in slope flow structure and energetics. The paper is structured as follows. A description of the LES and ARPS models is presented in section 2 along with an outline of the experiments performed in our study. Results are presented next in section 3 for a comparison between the LES and ARPS models for a simple, uniform slope. In section 4 we present a comparison between the LES and ARPS models for a compound slope. In section 5, simple and compound slope flows modeled with the ARPS model are contrasted and described in terms of the total energy budget. Summary and conclusions are given in section 6.

2. Model introduction and setup

a. Large eddy simulation model

Experiments were performed using a modified version of the LES model described in Skyllingstad (2003). This model is based on the Deardorff (1980) LES equation set, with the subgrid-scale model described by Ducros et al. (1996). In the first set of experiments (L14, see Table 1), we concentrated on a constant slope angle and used the rotated domain applied in Skyllingstad (2003), which effectively transforms the model into terrain following coordinates, with x, y, and z coordinates in the downslope, cross-slope, and slope-normal directions, respectively. Simulations were conducted using a narrow channel domain with open boundaries at the base of the slope and along the top of the model. A closed boundary was prescribed at the top of the slope, which is nearly equivalent to simulating a symmetric mountain ridge with no cross-ridge flow. The domain size was set to 3072 × 128 grid points in the horizontal and 56 grid points in the vertical with grid spacing of 2.5 m.

For the second set of LES experiments (LC) we modeled a compound-angled slope with a Cartesian model domain, as shown schematically in Fig. 1. Top and lateral boundary conditions in these experiments were the same as in the rotated case. However, terrain was simulated directly using filled grid cells and a shaved cell topography representation taken from Steppler et al. (2002) to approximate the sloping lower boundary. Using this approach, the z axis is not normal to the slope as in the rotated domain experiments, but is in the same direction as the gravitational force. Surface cooling was spread over the first two grid cells in the shaved-cell technique to avoid discontinuities produced by the changing grid volume at the slope surface. The domain size in these experiments was set to 2560 × 100 grid points in the x and y directions, and 290 points in the z direction with grid spacing of 3 m.

b. Mesoscale model

The mesoscale model used in our experiments is ARPS described in Xue et al. (2000). ARPS is a nonhydrostatic, compressible model that utilizes a terrain-following coordinate system with a stretched vertical grid. Turbulence parameterization is provided by the Deardorff (1980) level-1.5 closure scheme. Here, we used the model in a two-dimensional framework, representing a section across an infinite ridgeline. The model domain (Fig. 2) was 32 km × 0.6 km × 4 km in the x, y, and z directions, respectively. Grid spacing was 100 m × 100 m in the horizontal with 320 grid points in the x direction and 6 grid points in the y direction. The terrain-following vertical coordinate was stretched to allow for a very fine grid size of 5 m near the surface where resolution is critical, expanding to 50 m well above the surface where flow variations were small. Eighty vertical levels were used spanning a total distance of 4 km with rigid top and bottom boundary conditions. Periodic boundary conditions were used in the y direction, while in the x direction the boundary conditions were open (radiation). Surface heat and momentum flux were set using the same constant heat flux and roughness length as used in the LES cases. Radiation and moisture effects were not considered in our experiments.

c. Slope geometry

For the constant, uniform slope case (A14) presented in section 3, the slope angle was set to 14°, with a slope length of 7.7 km. The simple (one slope angle) and compound (two slope angles) slope comparisons (AS and LC/AC, respectively) presented in sections 4 and 5 utilized a linear terrain profile to focus specifically on the effect of a slope angle decrease on katabatic flows. An xz cross section of the model right-hand side slope for the combined and simple slopes is shown in Fig. 1. For the purposes of this study, the simple slope is constrained to have the same horizontal distance and height as the combined slope. The lower- and upper-slope angles of the combined slope were set to α1 = 1.6°, and α2 = 11.6°, yielding an equivalent simple slope angle of αs = 6.5°. The total height of the combined and simple slope was set to 822 m, giving a total horizontal run for the two slope scenarios of 7.3 km.

All simulations were forced with a constant surface cooling rate of 30 W m−2 and a surface roughness length of 0.01 m. The atmosphere was initialized at rest with a base-state potential temperature, θ0, of 18°C. Simulations were conducted for 1 h of model time, by which time temperature and velocity profiles at any point along the slope were not changing with time, and the katabatic flows had reached a steady state. A summary of the numerical experiments can be found in Table 1.

3. Uniform, constant slope intercomparisons

Our first set of experiments center on a simple slope with uniform slope angle. These cases are used to compare the basic flow properties from the ARPS (A14) and LES (L14) model simulations and examine how the turbulence parameterization in the ARPS model affects the flow structure.

a. Comparison between ARPS and LES

We begin our analysis by showing a cross section for the rotated LES and ARPS models of downslope velocity and potential temperature deficit, θ′ = θ(x, z) − θ̂(z), where θ(x, z) is the potential temperature and θ̂(z) represents the free-air potential temperature away from the slope at the same height, z, above sea level (Fig. 3). Overall, the two simulations produce flows that are very similar in both magnitude and physical shape. Both cases produce a near linear growth in the katabatic layer depth as a function of the downslope distance and velocities that are similar in magnitude for a given location. Results from the LES display more variability, representing turbulent eddies that are resolved in the LES model and not resolved in ARPS. Potential temperature deficit in the LES results has a maximum near the middle of the slope, whereas the ARPS value is at a maximum closer to the top of the slope.

More detailed comparisons between the LES and ARPS simulations can be made using horizontally averaged profiles. Horizontal averaging for Figs. 4 –7 is performed between 6.2 and 7.3 km in the downslope direction. For the LES model, we also average in the cross-flow direction to remove turbulent perturbations. As shown in Fig. 4, vertical profiles of downslope velocity and potential temperature deficit indicate differences between the two simulations that are not as obvious in the cross-section plots. Below 20-m height, ARPS displays a much smaller temperature deficit and a slightly weaker downslope wind maximum. The location of the downslope jet is also slightly higher in the ARPS simulation. Some of the near-surface differences may be a result of the finer resolution used in the LES model; 2.5 m in the LES versus ∼5 m in the ARPS case. Coarser resolution in ARPS automatically forces increased vertical mixing, which transports cooler air away from the surface region. The net effect is a more rapid vertical mixing of potential temperature deficit, which slightly increases the flow at higher heights in the ARPS model as shown by the stronger winds above ∼20 m.

b. Momentum budget analysis

The mean momentum equation in the downslope direction, neglecting storage and the Coriolis term, may be written as in Mahrt (1982):
i1520-0493-133-11-3065-e1
where h is the depth of the flow, and the pressure gradient term has been rewritten using the hydrostatic balance so that
i1520-0493-133-11-3065-eq1
where ρ0 is the base-state density, and p is the pressure. The layer average potential temperature is defined as
i1520-0493-133-11-3065-e2
where H is the height of surface of the slope and θ0 represents a constant, reference potential temperature. Turbulent flux divergence is expressed as an eddy viscosity flux:
i1520-0493-133-11-3065-eq2
where Km is the turbulent eddy coefficient, u and w represent the downslope and slope-normal velocity components, and primed quantities represent turbulent velocity fluctuations as in Eq. (3). We have calculated the downslope momentum budgets by decomposing the horizontal and vertical momentum budgets into downslope, s, and slope-normal, n, coordinates. Following Mahrt (1982), terms in (1) are grouped as follows. Term I is the sum of the buoyancy and pressure terms, including the katabatic acceleration, which is the primary driving force for the flow. Term II is the thermal wind term, which represents a retarding factor due the downslope increase in stability. For the flows presented here, the thermal wind term is much smaller than the katabatic acceleration term, and can be ignored. Term III is the sum of turbulent mixing and surface drag, which is included as a boundary condition. For the LES model, this term includes both grid-scale and subgrid-scale mixing. Term IV is the sum of the downslope and slope-normal advection terms, and represents the horizontal advection of lower momentum from upslope.

The momentum balance for the downslope flows examined here is among buoyancy, advection, and mixing. Katabatic flows having this balance of forcing are commonly referred to as “shooting” flows (Mahrt 1982). The relative magnitude of the individual terms in the momentum balance varies with height as shown by plots of the momentum budget from ARPS and the LES (Fig. 5). As might be expected given the similarity of the velocity fields, the momentum budget terms display good agreement between the two models. Near the surface, where potential temperature deficits are greatest, the buoyancy force (term I) is large in magnitude, and is driving the flow downhill, while vertical mixing and drag (term II) acts to slow down the flow. Advection (term III) becomes more important in the momentum budget away from the slope, while mixing and drag is the dominant retarding mechanism near the surface.

Away from the surface, the buoyancy term is greater in magnitude in the ARPS model than in the LES model, due to the greater temperature perturbation in the ARPS model between ∼5- and 100-m height. In the LES momentum budget, advection becomes more important than mixing and drag at around 10 m, near the height of the jet, whereas, in the ARPS model, advection becomes more important than mixing and drag at 20 m, just above the height of the jet. This may be related to the turbulence scheme in ARPS, given that the mixing term in the ARPS momentum budget between 10 and 30 m in height is rather large compared to that of the LES.

c. Turbulence kinetic energy budget analysis

Turbulence strength in stable boundary layers is critical for determining the exchange of heat and momentum between the surface and overlying ambient air. For katabatic flows turbulence has an important role in deepening the flow and controlling the strength of the downslope jet. The presence of turbulence in the LES results is evident in the velocity cross section (Fig. 3), which shows significant small-scale variations. In contrast, ARPS flow fields are smooth because turbulent mixing is parameterized using an average turbulence kinetic energy (TKE) budget equation. Because turbulence is directly simulated in the LES, it provides a unique tool for examining the accuracy of the ARPS turbulence parameterization. In this section, we focus on TKE fields and budget from both models and relate TKE to the katabatic flow properties described above.

We can define turbulence velocity fields in the LES model by considering perturbations in the flow that are not related to the mean downslope flow variations:
i1520-0493-133-11-3065-e3
where ψ is velocity components or potential temperature, prime represents the turbulent variation, and the overbar represents an average in the cross-slope direction. TKE is then defined as
i1520-0493-133-11-3065-eq3
Profiles of TKE from the LES and ARPS models (Fig. 6), show similarities in shape, but large differences at specific heights. Both models predict a sharp peak in TKE just above the surface, a minimum at the level of the jet, and a broader maximum in the shear zone above the jet. Overall the profiles have a similar shape, however TKE in the ARPS model is greater in magnitude and is shifted upward relative to the LES profile.
Factors controlling TKE in the two models can be diagnosed using the TKE budget equation:
i1520-0493-133-11-3065-e4
where e represents TKE, Uj represent the average velocity components, ε is the dissipation of turbulence, and primed quantities are as defined above for the LES model. Terms in (4) are defined as TKE storage (I), horizontal advection (II), buoyant production/destruction (III), shear production (IV), turbulent transport (V), pressure transport (VI), and dissipation (VII). Primed quantities in the ARPS model are parameterized; for the shear production term (IV):
i1520-0493-133-11-3065-e5
and the pressure and turbulent transport terms (V and VI) are treated as diffusive processes as in Deardorff (1980):
i1520-0493-133-11-3065-e6
Dissipation in ARPS is parameterized using
i1520-0493-133-11-3065-e7
where C is 3.9 at the surface and 0.93 elsewhere and l = Δs = (ΔxΔyΔz)1/3 for neutral and unstable cases and
i1520-0493-133-11-3065-e8
for stable cases, where N = |(g/θ)(∂θ/∂z)|1/2 is the absolute value of the buoyancy frequency. Dissipation in the LES is calculated from the velocity fields using
i1520-0493-133-11-3065-e9

Eddy viscosity in ARPS is calculated using Km = max(0.1e1/2l, βΔs), where β is a small number (e.g., 1 × 10−6). In the LES model, Km is calculated using a subgrid-scale parameterization based on Ducros et al. (1996).

Plots from the LES and ARPS models of the primary terms in the TKE budget are shown in Fig. 7. In katabatic flows, TKE is produced almost entirely through shear production (term IV). Velocity gradients are highest near the surface and just above and below the jet height, hence shear production is highest in these areas. At the jet height, where velocity gradients become small, there is a local minimum of shear production of TKE. The minimum of TKE with height is lower in the LES model than in ARPS, since the velocity jet is lower in the LES model than in ARPS. Since shear-produced eddies tend to be small and locally dissipated, the dissipation term (VII) is large where shear production is large, near the surface and above and below the height of the jet. The average buoyancy term (III) is almost always negative in katabatic flows, because stable stratification acts to suppress turbulence. Near the surface, where stratification is highest, term III is greatest in magnitude. The advection and mixing term includes the advection of TKE by the mean flow (II), and turbulent and pressure transport (V and VI), and acts to redistribute TKE. This term is positive near the jet height and becomes strongly negative just below the jet height, indicating that turbulent eddies transport TKE from areas of high TKE, where it is produced, to areas of lower TKE. The turbulent transport term is important in preventing decoupling of the flow at the height of the jet, where there is no shear production of TKE (Denby 1999).

One possible explanation for the higher TKE values in the ARPS simulation could be overestimation of l as defined in (8). Turbulent length scale in the Deardorff (1980) scheme has a large influence on TKE production and dissipation, which are both directly related to l. Overestimation of l could cause decreased dissipation, allowing for a greater equilibrium TKE value and stronger vertical mixing in the ARPS solutions.

4. Compound-angled slopes

In our next set of experiments we expand on the simple, uniform slope problem by considering the effect of a decrease in slope angle on katabatic flows. For these cases, the LES model was run in a nonrotated model domain using the shaved-cell approach discussed in section 2. For ARPS, a configuration similar to the uniform slope cases was applied, but with the ridge having a decrease in slope angle at a horizontal distance equal to the LES scenario as shown in Fig. 1.

Cross-section plots of downslope velocity and potential temperature deficit from the LES (LC) and ARPS (AC) compound-angled simulations are shown in Fig. 8. Overall, the two models predict similar slope flow behavior. Over the steep angle portion of the slope, both models indicate a nearly linear increase in the slope flow depth with the flow maximum velocity located just above the surface. At the juncture point between the large- and small-angle slope sections, the flow depth increases rapidly over a distance of ∼500 m. At the same time, the height of the maximum velocity moves upward about 5–10 m. The flow structure below the slope transition reestablishes a new equilibrium with decreased growth in the slope flow depth and more rapid cooling of the near-surface temperatures. Near-surface downslope velocity in this part of the flow decreases, producing a more distinct, detached jet structure between 10- and 15-m height.

Quantitative differences between the two simulations are most apparent when analyzing profiles from the two models (Fig. 9) taken from the three different points along the slope indicated in Figs. 1 and 8. In general, the LES slope flow is slightly faster and deeper than the ARPS case, with significant flow differences depending on the slope location. Near the top of the slope in the high-angle region (A), the height of maximum velocity is about the same in both simulations. Moving to the lower-angle portion of the slope (C), the LES model predicts a jet maximum located at ∼20-m height, versus ∼10 m for the ARPS model. Potential temperature in the two simulations is consistent with the velocity fields; the LES case shows cooler air above ∼10-m height in comparison with the ARPS case, suggesting stronger upward turbulent flux of cool surface air.

Vertical profiles shown in Fig. 9 indicate that the Cartesian coordinate version of the LES model simulates more mixing for slope flows in comparison with the rotated coordinate system LES results presented in section 3. Greater mixing of cool air in the LES detaches the slope flow jet from the frictional drag of the slope surface. Consequently, the LES slope flow accelerates more rapidly, which generates more turbulence and greater vertical heat flux. Stronger vertical mixing in the LES simulation may be a result of the implicit, two grid-cell cooling that is applied over the shaved-cell orography. On average, this adds about a 1/2 grid cell (2 m) to the flow depth. A second reason for the increased flow depth in the LES case could be the stair-step pattern of the resolved topography used in the Steppler et al. (2002) shaved-cell approach. Small discontinuities in the surface boundary conditions are produced using this method and can add more energy to near-surface turbulent velocities. Differences can also be attributed to the parameterized turbulence in the ARPS model versus the resolved eddy mixing generated in the LES as suggested in section 3.

The integrated downslope heat flux produced by the two models is very similar as shown by the vertically integrated potential temperature deficit at each point along the slope (Fig. 10). Over the steep angle slope (between 0- and 3500-m horizontal distance), both models show a relatively small increase in temperature deficit in comparison with the lower, small angle portion of the slope. Downslope velocity over the upper slope is strongest near the surface and advects air rapidly down the slope. When the slope angle decreases, the jet height increases and the near-surface flow speed decreases. As a consequence, surface cooling acts on the slope flow for a longer period of time in comparison with the upper slope, causing a more rapid increase in the integrated flux.

Differences in the slope flow structure between the upper and lower slope agree with the analytical slope flow model presented in Oerlemans and Grisogono (2002), which is based on theory proposed by Prandtl (1942), but with variable vertical mixing coefficients. The basis of this model is an assumed balance between the downslope buoyancy forcing and vertical momentum flux generated by surface friction, and a heat budget dominated by vertical mixing and downslope transport of heat. Oerlemans and Grisogono (2002) show that the height of the velocity maximum is inversely proportional to the slope angle and that the maximum flow speed is proportional to the flow temperature deficit. Here, the increase in the maximum flow height agrees with theory, however the flow speed does not always increase in proportion to the temperature deficit (see Fig. 10), suggesting that the variable mixing coefficients employed by Oerlemans and Grisogono (2002) could use modification to allow for changing stratification.

5. Comparison between the uniform and compound angle slopes (ARPS)

Our final set of experiments focuses on a comparison of uniform (AS) and compound angled (AC) slope cases, each having the same total slope height. Two slope configurations were considered (Fig. 1) using ARPS, the first duplicating the compound slope described above, and the second applying a uniform slope with an identical initial height and slope length as the compound angled slope (yielding a slope angle of 6.5°).

Cross-section plots from the two simulations are presented in Fig. 11. A number of distinct flow features are produced in the two scenarios. First, we note that except near the top of the slope, the uniform slope generates a stronger jet in comparison with the compound slope case. Slope flow depth, however, is greater for the compound case over the same slope region. Temperature deficit in the uniform slope case is less than the compound slope case, especially over the lower portion of the slope where the low angle slope in the compound case limits buoyant forcing of the flow. Weaker winds in the compound angle case limit the transport of cold air down the slope, leading to a greater temperature deficit.

Because surface cooling rates are held constant, buoyancy deficits in the flow are largely controlled by flow velocity and slope angle. This has important implications for the momentum budget of the flow, which was described earlier using Eq. (1). Previous studies, such as Nappo and Rao (1987), indicate that the bulk velocity magnitude of katabatic flows is a strong function of slope angle as indicated by term I of (1). Therefore, in the compound slope angle case it is reasonable to expect a flow transition when going from the high- to low-angle sections of the slope as shown by Fig. 11 (assuming the flow is near equilibrium). Forces acting on air parcels moving through the slope angle transition change suddenly, with term I decreasing rapidly. To maintain balance between the momentum budget terms, the flow can either lose momentum via vertical mixing and surface drag (increasing term III) or undergo a reduction in the advection magnitude (term IV). Figure 11 suggests that both of these mechanisms are active in the compound slope flow case; the depth of the flow increases more rapidly and the strength of the near-surface jet decreases so that the surface drag is smaller. Reduced surface velocity also decreases the downslope advection term (IV) downstream from the angle decrease.

Plotting the vertically averaged mixing and drag term (III) normalized by buoyancy (I) provides a more direct example of this effect (Fig. 12). For most of the slope flow, the vertical mixing and drag term balances about 70% of the buoyancy forcing. As the flow moves over the slope angle decrease, the buoyancy term decreases rapidly to the point where term III is larger than term I. Adjustment occurs as the flow depth increases and the near-surface velocity decreases between x = 3500 and 4500 m. Slower downslope velocities also increase the potential temperature deficit because the air has more time to cool before advecting downslope.

Effects of slope angle decreases are made clearer by examining the total energy distribution of the flow system. In slope flows, the source of total kinetic energy is gravitational or buoyant potential energy defined here for a vertical height, z, of the slope flow as
i1520-0493-133-11-3065-e10
Similarily, we define the total kinetic energy as
i1520-0493-133-11-3065-e11
We also define a vertically integrated potential energy, PET and kinetic energy KET as
i1520-0493-133-11-3065-eq4
where the integration goes from the surface of the slope through the depth of the flow. Katabatic flows are generated when buoyant potential energy is added to the system through surface cooling and converted into kinetic energy. Kinetic energy is lost primarily via surface drag and dissipation of turbulence.

Plots of PET and KET from the uniform and compound slope ARPS case (Fig. 13) demonstrate how the total energy is partitioned along the slope. Near the top of the slope in the uniform case, PET climbs rapidly to a maximum while KET slowly increases from the initial state of no motion. Rapid growth of PET is produced by the relatively high altitude of the slope and the limited amount of slope available for the flow to accelerate. For an equilibrium slope flow in a neutral atmosphere forced by surface cooling, the downhill flux of heat at any point along the slope must equal the total heat lost to the surface along the slope above the point. Near the top of the slope, the downslope velocities are relatively weak, consequently the temperature deficit must be large to yield a downhill heat flux equal to the total surface heat loss.

Moving down the uniform slope, PET decreases as KET increases, so that the flow eventually has an equal energy partition at x = ∼6500 m. Figure 13 shows that PET decreases more rapidly than the increase in KET in the lower half of the slope. Energy loss from surface drag and turbulence dissipation prevents KET from balancing the decrease in PET. At the bottom of the slope, KET reaches a maximum that is about one-third of the maximum PET value on the upper slope, indicating that about 33% of the available PET is converted to KET for a 6.5° slope.

A different behavior is noted in the compound angle case where PET increases more slowly at the top of the slope (Fig. 13). This is because the slope height is decreasing more rapidly (since the upper slope angle for the combined slope is greater than the simple slope angle) and KET is growing faster in the compound angle slope flow. By the time the flow reaches the decrease in slope angle, it has almost twice the KET as in the uniform slope angle case. However, PET is much less in the compound case, which greatly affects the growth in KET downslope from the transition. Interestingly, PET in the transition zone downslope from the angle decrease increases slightly after having fallen from a peak value of ∼300 at x = ∼1500 m. The increase in PE results from the added vertical mixing of potential temperature, noted above, that occurs when the flow decelerates. More significant near-surface cooling in the lower angle slope region also adds to PET. Because PET decreases more rapidly in the compound slope case, KET is unable to increase significantly downslope from the angle transition. In contrast, the uniform slope case KET continues to increase, reaching a value about 50% larger at the slope bottom than the compound angle case.

Conversion of PE to KE in slope flows can be examined by deriving a budget equation for the slope PE from the perturbation potential temperature equation:
i1520-0493-133-11-3065-e12
where is the turbulent heat flux. By assuming that the flow is in equilibrium [i.e., (∂θ′/∂t) = 0], ignoring transport by subgrid turbulence, and multiplying by (g/θo)z, where θo is the average background potential temperature and z is the average Cartesian height of the slope flow above the valley floor, we get
i1520-0493-133-11-3065-e13
where Hf is the surface heat flux, ρ is the average density, Cp is the atmospheric heat content at constant pressure, and h is the depth of the slope flow. Using the chain rule yields
i1520-0493-133-11-3065-eq5
Finally we use (∂z/∂s) = −sinα, (∂z/∂n) = cosα, substitute our definition of PE from (10), and depth average to obtain the PE budget equation:
i1520-0493-133-11-3065-e14
where (∂z/∂s) = −sinα 〈〉 represents the vertical average over a flow depth of h. This equation represents a balance of vertically averaged PE along the average slope flow height, z.

Plots of the three terms in (14) are presented in Fig. 14, representing (I) the transport of PE, (II) conversion of PE to KE, and (III) the local change in PE from the surface heat flux (for simplicity, we ignore the conversion term associated with the vertical velocity, w). Analysis of the PE terms is consistent with the KET and PET presented in Fig. 13. Surface cooling (Hf < 0) is always increasing PE, while the conversion term is always removing PE. The transport term (I) in the PE budget exhibits interesting behavior. Near the top of the slope the transport term reduces the local PE, indicating that much of the cold air generated in this region of the slope is transported downhill without immediately increasing KE. The transport term changes sign midway down the slope and begins to increase local PE. At this point along the slope, cold air transported downhill begins to add significantly to the KE budget and eventually exceeds the contribution of the surface heat flux term. Careful examination of Fig. 14 indicates that the three terms in (14) do not exactly balance, especially near the top of the slope. Most of the imbalance can be explained by the difficulty in converting from the terrain-following coordinate system to the slope-following coordinates used in (14). Errors are also introduced by not considering the turbulent flux terms.

The PE budget analysis suggests a conceptual model for the energy budget of katabatic flows. Near the top of the slope, the flow is unable to convert buoyant PE into KE fast enough to balance the increase from surface cooling. Excess PE generated at the top of the slope is transported by the flow and released farther down the slope where velocities are large enough to increase the conversion term.

6. Conclusions

Experiments are performed using ARPS and a LES model aimed at understanding how decreases in slope angle affect the structure and evolution of katabatic slope flows. By comparing results from the ARPS and LES models, we show that one common turbulence parameterization method (Deardorff 1980) can yield results similar to LES, which directly resolves turbulent fluxes. Analysis of the momentum and turbulence kinetic energy budget from the two models demonstrates that the models are consistent in predicting both the mean fields and higher-order terms such as the vertical momentum flux.

Simulations of a compound angle slope show how changing slope angle strongly affects the strength of katabatic flows. Both models show that slopes with a steep upper slope followed by a more shallow lower slope (concave shape) generate a rapid acceleration on the upper slope followed by a transition to a slower evolving structure characterized by an elevated jet over the lower slope. In contrast, a case with uniform slope having the same total height change yielded a more uniform slope flow profile with stronger winds at the slope bottom. Elevation of the jet in the compound slope angle case is in agreement with Prandtl model results presented in Oerlemans and Grisogono (2002). Less available potential energy in the compound angle case dramatically decreases the flow kinetic energy in comparison with the uniform slope example. Analysis of the total energy budget of the slope flows indicates a consistent structure where a significant fraction of the potential energy generated at the top of the slope was transported downslope and converted into kinetic energy near the slope base.

Acknowledgments

This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC03-76SF00098. This work was supported by the DOE Office of Biological and Environmental Research Environmental Meteorology Program, Grant DE-FG03-99ER62840. The authors also wish to acknowledge the help of the anonymous reviewers who contributed greatly to the manuscript.

REFERENCES

  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18 , 495527.

  • Denby, B., 1999: Second-order modeling of turbulence in katabatic flows. Bound.-Layer Meteor., 92 , 67100.

  • Ducros, F., P. Comte, and M. Lesieur, 1996: Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech., 326 , 137.

    • Search Google Scholar
    • Export Citation
  • Horst, T. W., and J. C. Doran, 1986: Nocturnal drainage flow on simple slopes. Bound.-Layer Meteor., 34 , 263286.

  • Mahrt, L., 1982: Momentum balance of gravity flows. J. Atmos. Sci., 39 , 27012711.

  • Manins, P. C., and B. L. Sawford, 1979: A model of katabatic winds. J. Atmos. Sci., 36 , 619630.

  • Monti, P., H. J. S. Fernando, M. Princevac, W. C. Chan, T. A. Kowalewski, and E. R. Pardyjak, 2002: Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci., 59 , 25132534.

    • Search Google Scholar
    • Export Citation
  • Nappo, C. J., and S. Rao, 1987: A model study of pure katabtic flows. Tellus, 39A , 6171.

  • Oerlemans, J., 1998: The atmospheric boundary layer over melting glaciers. Clear and Cloudy Boundary Layers, A. A. M. Holtslag and P. G. Duynkerke, Eds., Royal Netherlands Academy of Arts and Sciences, 129–153.

    • Search Google Scholar
    • Export Citation
  • Oerlemans, J., and B. Grisogono, 2002: Glacier winds and parameterisation of the related surface heat fluxes. Tellus, 54A , 440462.

  • Prandtl, L., 1942: Führer Durch die Strömunglehre. Braunschweig, F. Vieweg & sohn, 382 pp. [Published in English in 1952 as Essentials of Fluid Dynamics, Hafner, 452 pp.].

    • Search Google Scholar
    • Export Citation
  • Renfrew, I. A., and P. S. Anderson, 2002: The surface climatology of an ordinary katabatic wind regime in Coats Land, Antarctica. Tellus, 54A , 463484.

    • Search Google Scholar
    • Export Citation
  • Skyllingstad, E. D., 2003: Large eddy simulation of katabatic flows. Bound.-Layer Meteor., 106 , 217243.

  • Smeets, P., P. Duynkerke, and H. Vugts, 1998: Turbulence characteristics of the stable boundary layer over a mid-latitude glacier. Part I: A combination of katabatic and large scale forcing. Bound.-Layer Meteor., 87 , 117145.

    • Search Google Scholar
    • Export Citation
  • Steppler, J., H-W. Bitzer, M. Minotte, and L. Bonaventura, 2002: Nonhydrostatic atmospheric modeling using a z-coordinate representation. Mon. Wea. Rev., 130 , 21432149.

    • Search Google Scholar
    • Export Citation
  • Xue, M., K. Droegemeier, and V. Wong, 2000: The Advanced Regional Prediction System (ARPS)—A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteor. Atmos. Phys., 75 , 161193.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Cross section of a portion of the model domain for the simple/compound runs, showing locations where vertical profiles will be shown.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 2.
Fig. 2.

Three-dimensional view of ARPS model domain and ridgeline.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 3.
Fig. 3.

Cross sections (xz) of (a) downslope velocity, u (m s−1), from the LES L14 and ARPS A14 simulations and (b) potential temperature deficit, Δθ (°C), from the LES and ARPS simulations. Plots are at 1 h of simulation time.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 4.
Fig. 4.

Vertical profiles of (a) downslope velocity, u (m s−1), from the LES L14 (solid) and ARPS A14 (dashed) simulations and (b) potential temperature deficit, Δθ (°C), from the LES (solid) and ARPS (dashed) simulations.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 5.
Fig. 5.

Horizontally averaged downslope momentum budget terms from (a) the LES L14 and (b) the ARPS A14 simulations.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 6.
Fig. 6.

TKE profiles for the LES L14 (solid) and ARPS A14 (dashed) simulations.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 7.
Fig. 7.

Horizontally averaged TKE budget profiles as a function of height for the (a) LES L14 and (b) ARPS A14 simulations. Terms shown are as follows: shear production (SP), dissipation (D), buoyant destruction (BD), advection and mixing (AM), and the sum of all terms (S).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 8.
Fig. 8.

Same as in Fig. 3, but (a) from the LES LC and ARPS AC simulations and (b) from the LES and ARPS simulations. The arrow indicates the location of the slope angle decrease. Locations A, B, and C indicate the location of profiles shown in Fig. 9.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 9.
Fig. 9.

Vertical profiles of (a) downslope velocity, u (m s−1), and (b) potential temperature deficit, Δθ (°C), at the locations marked in Fig. 2, for the compound slope angle LES LC (solid) and ARPS AC (dashed) simulations.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 10.
Fig. 10.

Vertically integrated potential temperature deficit vs downslope distance for the compound angle LES LC (solid) and ARPS AC (dashed) simulations. The arrow indicates the location of the slope angle decrease.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 11.
Fig. 11.

Same as in Fig. 3, but from the compound AC and uniform AS simulations. Thearrow indicates the location of the slope angle decrease. Results are from the ARPS model.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 12.
Fig. 12.

Vertically averaged normalized mixing and drag vs downslope distance for the compound AC (solid) and simple AS (dashed) simulations.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 13.
Fig. 13.

Vertically integrated (a) buoyant potential energy and (b) kinetic energy vs downslope distance for the compound angle AC (solid) and uniform angle AS (dashed) simulations. The arrow indicates the location of the slope angle decrease.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Fig. 14.
Fig. 14.

Vertically averaged terms from the potential energy budget equation [Eq. (14)] for the uniform slope angle (AS) simulation. Terms are defined in the text.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR2982.1

Table 1.

Names, model used, slope angle, and section where presented in text of experimental cases studied.

Table 1.
Save
  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18 , 495527.

  • Denby, B., 1999: Second-order modeling of turbulence in katabatic flows. Bound.-Layer Meteor., 92 , 67100.

  • Ducros, F., P. Comte, and M. Lesieur, 1996: Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech., 326 , 137.

    • Search Google Scholar
    • Export Citation
  • Horst, T. W., and J. C. Doran, 1986: Nocturnal drainage flow on simple slopes. Bound.-Layer Meteor., 34 , 263286.

  • Mahrt, L., 1982: Momentum balance of gravity flows. J. Atmos. Sci., 39 , 27012711.

  • Manins, P. C., and B. L. Sawford, 1979: A model of katabatic winds. J. Atmos. Sci., 36 , 619630.

  • Monti, P., H. J. S. Fernando, M. Princevac, W. C. Chan, T. A. Kowalewski, and E. R. Pardyjak, 2002: Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci., 59 , 25132534.

    • Search Google Scholar
    • Export Citation
  • Nappo, C. J., and S. Rao, 1987: A model study of pure katabtic flows. Tellus, 39A , 6171.

  • Oerlemans, J., 1998: The atmospheric boundary layer over melting glaciers. Clear and Cloudy Boundary Layers, A. A. M. Holtslag and P. G. Duynkerke, Eds., Royal Netherlands Academy of Arts and Sciences, 129–153.

    • Search Google Scholar
    • Export Citation
  • Oerlemans, J., and B. Grisogono, 2002: Glacier winds and parameterisation of the related surface heat fluxes. Tellus, 54A , 440462.

  • Prandtl, L., 1942: Führer Durch die Strömunglehre. Braunschweig, F. Vieweg & sohn, 382 pp. [Published in English in 1952 as Essentials of Fluid Dynamics, Hafner, 452 pp.].

    • Search Google Scholar
    • Export Citation
  • Renfrew, I. A., and P. S. Anderson, 2002: The surface climatology of an ordinary katabatic wind regime in Coats Land, Antarctica. Tellus, 54A , 463484.

    • Search Google Scholar
    • Export Citation
  • Skyllingstad, E. D., 2003: Large eddy simulation of katabatic flows. Bound.-Layer Meteor., 106 , 217243.

  • Smeets, P., P. Duynkerke, and H. Vugts, 1998: Turbulence characteristics of the stable boundary layer over a mid-latitude glacier. Part I: A combination of katabatic and large scale forcing. Bound.-Layer Meteor., 87 , 117145.

    • Search Google Scholar
    • Export Citation
  • Steppler, J., H-W. Bitzer, M. Minotte, and L. Bonaventura, 2002: Nonhydrostatic atmospheric modeling using a z-coordinate representation. Mon. Wea. Rev., 130 , 21432149.

    • Search Google Scholar
    • Export Citation
  • Xue, M., K. Droegemeier, and V. Wong, 2000: The Advanced Regional Prediction System (ARPS)—A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteor. Atmos. Phys., 75 , 161193.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Cross section of a portion of the model domain for the simple/compound runs, showing locations where vertical profiles will be shown.

  • Fig. 2.

    Three-dimensional view of ARPS model domain and ridgeline.

  • Fig. 3.

    Cross sections (xz) of (a) downslope velocity, u (m s−1), from the LES L14 and ARPS A14 simulations and (b) potential temperature deficit, Δθ (°C), from the LES and ARPS simulations. Plots are at 1 h of simulation time.

  • Fig. 4.

    Vertical profiles of (a) downslope velocity, u (m s−1), from the LES L14 (solid) and ARPS A14 (dashed) simulations and (b) potential temperature deficit, Δθ (°C), from the LES (solid) and ARPS (dashed) simulations.

  • Fig. 5.

    Horizontally averaged downslope momentum budget terms from (a) the LES L14 and (b) the ARPS A14 simulations.

  • Fig. 6.

    TKE profiles for the LES L14 (solid) and ARPS A14 (dashed) simulations.

  • Fig. 7.

    Horizontally averaged TKE budget profiles as a function of height for the (a) LES L14 and (b) ARPS A14 simulations. Terms shown are as follows: shear production (SP), dissipation (D), buoyant destruction (BD), advection and mixing (AM), and the sum of all terms (S).

  • Fig. 8.

    Same as in Fig. 3, but (a) from the LES LC and ARPS AC simulations and (b) from the LES and ARPS simulations. The arrow indicates the location of the slope angle decrease. Locations A, B, and C indicate the location of profiles shown in Fig. 9.

  • Fig. 9.

    Vertical profiles of (a) downslope velocity, u (m s−1), and (b) potential temperature deficit, Δθ (°C), at the locations marked in Fig. 2, for the compound slope angle LES LC (solid) and ARPS AC (dashed) simulations.

  • Fig. 10.

    Vertically integrated potential temperature deficit vs downslope distance for the compound angle LES LC (solid) and ARPS AC (dashed) simulations. The arrow indicates the location of the slope angle decrease.

  • Fig. 11.

    Same as in Fig. 3, but from the compound AC and uniform AS simulations. Thearrow indicates the location of the slope angle decrease. Results are from the ARPS model.

  • Fig. 12.

    Vertically averaged normalized mixing and drag vs downslope distance for the compound AC (solid) and simple AS (dashed) simulations.

  • Fig. 13.

    Vertically integrated (a) buoyant potential energy and (b) kinetic energy vs downslope distance for the compound angle AC (solid) and uniform angle AS (dashed) simulations. The arrow indicates the location of the slope angle decrease.

  • Fig. 14.

    Vertically averaged terms from the potential energy budget equation [Eq. (14)] for the uniform slope angle (AS) simulation. Terms are defined in the text.

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