Physical mechanisms of the thermally driven cross-basin circulation

Transcription

1 Quarterly Journalof the Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: , April 24 DOI:.2/qj.295 Physical mechanisms of the thermally driven cross-basin circulation Manuela Lehner* and C. David Whiteman Department of Atmospheric Sciences, University of Utah, Salt Lake City, UT, USA *Correspondence to: M. Lehner, Department of Atmospheric Sciences, University of Utah, 35 S 46 East Rm 89, Salt Lake City, UT 842-, USA. manuela.lehner@utah.edu The physical mechanisms responsible for the formation of a thermally driven crossbasin circulation in a basin with asymmetric heating of opposite mountain sidewalls are investigated. A large-eddy simulation is performed with the Weather Research and Forecasting model for an idealized basin based on the topography of Arizona s Meteor Crater. The individual components of the horizontal momentum and thermodynamic balance equations are analyzed to determine their respective contributions in forcing the cross-basin circulation. A cross-basin pressure gradient, with higher pressure on the less irradiated sidewall, leads to the development of a cross-basin flow near the basin floor. A weak opposing return flow develops above this cross-basin flow as a result of a reversed cross-basin pressure gradient. The reversed cross-basin pressure gradient is caused by cold-air advection by upslope winds in the stable morning atmosphere on the sunlit sidewall and warm-air advection by downslope winds on the still shaded sidewall, as this reverses the cross-basin temperature gradient, producing a higher temperature on the less sunlit sidewall. Key Words: basin meteorology; large-eddy simulation; METCRAX; return flow; thermally driven winds; WRF Received 24 October 22; Revised 2 May 23; Accepted 2 May 23; Published online in Wiley Online Library 26 July 23. Introduction Thermally driven winds are a regularly occurring phenomenon in complex terrain under synoptically undisturbed conditions. Whiteman (2) distinguishes three thermal wind regimes at the scale of a single valley: (i) along-valley flows resulting from a pressure difference between the valley and the plain; (ii) slope flows resulting from heating or cooling of an inclined surface; and (iii) cross-valley winds resulting from asymmetric irradiation of opposing sidewalls. Few studies have dealt with cross-valley winds in comparison with slope flows and alongvalley flows. Early observations of thermally driven cross-valley winds were made, for example, in valleys in Tyrol, Austria (Moll, 935); in the Columbia River Valley, Canada (Gleeson, 95); in the Kananaskis Valley, Canada (MacHattie, 968); in the Dischma Valley, Switzerland (Hennemuth and Schmidt, 985; Hennemuth, 986; Urfer-Henneberger, 97); and in the Brush Creek Valley, Colorado (Whiteman, 989). In a recent study, Lehner et al. (2) used data from the Meteor Crater experiment (METCRAX: Whiteman et al., 28) field campaign to describe the diurnal cycle of the surface cross-basin flow and its relation to horizontal temperature and pressure gradients in the closed and almost circular basin of Arizona s Meteor Crater. Thermally driven cross-valley flows are a result of asymmetric irradiation. As shown by Lehner et al. (2), a horizontal temperature gradient forms across the valley, which is accompanied by a pressure gradient with higher pressure on the less irradiated, i.e. colder, sidewall. Cross-valley flows form in response to the pressure gradient. Simulations by Lehner and Whiteman (22), who used an idealized basin topography based on the Meteor Crater to study the impact of background winds and basin size on the thermal cross-basin circulation, showed the presence of a return flow (RF) of opposing direction on top of the cross-basin flow (CBF). Lehner et al. (2) also described a case of an RF (elevated crossbasin flow in their terminology) that occurred in the Meteor Crater in the morning from the sunlit to the shaded sidewall. In this case, the RF was collocated with an elevated inversion layer, in agreement with the conceptual model by Vergeiner and Dreiseitl (987), who postulated that the mass flux in the upslope-flow layer is proportional to the stability in the valley atmosphere, so that in the presence of an elevated inversion layer the mass flux is reduced, resulting in a flow away from the slope due to mass conservation. A simple calculation of the mass fluxes in the upslope-flow layer and the RF layer for the Meteor Crater case also agreed with this conceptual model (Lehner et al., 2). In this study we use a semi-idealized Weather Research and Forecasting (WRF) simulation to investigate the physical mechanisms that produce the cross-basin (or equivalently cross-valley) circulation. The vertical structure of the crossbasin circulation is shown and the respective key mechanisms responsible for the formation of the individual layers are identified, based on an analysis of the horizontal momentum and thermodynamic balance equations. A conceptual model is developed to explain the formation of the return branch (i.e. the RF) of the cross-basin circulation in the absence of vertical changes in atmospheric stability. c 23 Royal Meteorological Society

2 896 M. Lehner and C. D. Whiteman North south distance (km) East west distance (km) Height (m) w3 wu w6 c2 w c e wl flr el Distance (km) c e6 eu e3 Figure. Idealized model topography (white contour lines), Meteor Crater topography (black contour lines) and difference (m) between Meteor Crater topography and idealized model topography (grey shading). The contour interval for the model topography and the Meteor Crater topography is 5 m. The location of lowest elevation in the Meteor Crater basin is aligned with the centre of the idealized model topography. West east cross-section through the axisymmetric model topography (bold line) together with west east and south north cross-sections through Arizona s Meteor Crater (solid lines). Grey dots indicate grid points with time-series output along the centre (c 2), west slope (w 3) and east slope (e 3) lines and black crosses indicate five METCRAX instrumented-tower measurement sites along a west east line through the crater, used for comparison. This figure is available in colour online at wileyonlinelibrary.com/journal/qj 2. Model set-up The simulation was performed with the Advanced Research WRF (ARW) version 3 (Skamarock and Klemp, 28; Skamarock et al., 28). The model topography is an idealized, rotationally symmetric basin, which is based on the topography of Arizona s Meteor Crater and is surrounded by flat terrain (Figure ). The basin has a diameter of 2 m at the rim and 5 m at the floor, a depth of 7 m from the floor to the basin rim and a rim that extends 4 m above the surrounding plain. The model domain is km 3 with a horizontal grid spacing of 5 m and 5 vertical grid points. The vertical grid spacing is 4.5 m near the surface and is stretched to 625 m at the domain top. 3 model levels are located in the lowest 2 m above ground level (AGL) and 27 model levels are located within the lowest 5 m AGL. At 5 m AGL, the grid spacing is 23 m. The exact heights of the model levels vary slightly within the domain, since WRF uses terrain-following pressure coordinates in the vertical. The simulation is run for a 7 h period from 3 to 2 local time (LT: UTC 7 h) on 23 October with a.5 s time step. Temperature and humidity fields are initialized as horizontally homogeneous with observations taken during the METCRAX field campaign at 3 LT on 23 October 26, a clear-sky night with typical synoptically undisturbed conditions at the Meteor Crater. The model sounding consists of data from a tethersonde that was flown from the centre of the Meteor Crater to a height of 235 m and is complemented above this height by data from a rawinsonde launched approximately 5 km to the northwest of the crater. Wind speed is initialized at m s to focus on the thermally driven wind circulation. The model is run in large-eddy simulation (LES) configuration using the model s.5-order turbulent kinetic energy (TKE) subgrid-scale scheme (Deardorff, 98) to parametrize the effects of small-scale, unresolved turbulent motions on the flow. An anisotropic diffusion option was chosen, which calculates separate horizontal and vertical length-scales to account for the grid anisotropy near the surface ( x/ z 3.5). Short-wave and long-wave radiation are parametrized using the Mesoscale Model version 5 (MM5) short-wave scheme (Dudhia, 989) and the Rapid Radiative Transfer Model (for global circulation models RRTMG: Mlawer et al.., 997) long-wave scheme, respectively. The short-wave scheme accounts for topographic shading, i.e. shadows cast by the surrounding topography, and self-shading, i.e. whether a sloping surface faces towards or away from the Sun. The Noah Land Surface Model (Chen and Dudhia, 2) is used, together with the Eta surface-layer scheme (Janjić, 994), which is based on Monin Obukhov theory, to calculate surface fluxes. The land-surface model has four soil layers, which were initialized at.2,.4,.6 and.8 K warmer than the surface skin temperature, from top to bottom. Soil moisture was set to.5 m 3 m 3. The Coriolis force is neglected because of the small model domain. Periodic boundary conditions are applied at the lateral boundaries and a Rayleigh damping layer is applied to the top 5 km. A sixth-order numerical diffusion scheme (Knievel et al., 27) is used to dampen 2 x waves. In addition to the three-dimensional fields of standard meteorological variables, time series are output at every time step for 72 grid points (c 2, n 3, e 3, s 3 and w 3) shown in Figure. Individual terms of the horizontal momentum and thermodynamic equations were output in the time series, together with standard meteorological variables and other auxiliary variables necessary for the analysis. All of the analysis using this time-series output is based on 3 min averages (Figures 3 6 and 9 2). Times indicated in the figures and the text refer to the end of the averaging period and are thus representative of the 3 min period prior to the time mentioned. The 3 min averaging period was chosen to reduce fluctuations from convective turbulence. A comparison with a shorter 5 min averaging time showed that the results are not affected by the longer averaging period. Geopotential heights have been interpolated from w grid points (WRF uses a staggered Arakawa C grid) to mass grid points. Wind components and their tendencies are not interpolated to mass grid points from the u and v grid points unless stated otherwise and are located half a horizontal grid-point distance (25 m) to the west (u) andsouth(v) of the respective mass grid points. 3. Diurnal evolution of the basin atmosphere Local sunrise occurs first at about 75 LT at the basin rim and the upper northwest sidewall. It then propagates down the sidewall towards the southeast (Figure 2). The difference in incoming solar radiation between the north and south sidewalls is thus positive in the morning and remains positive throughout the day, except for the difference between the grid points n3 s3 located at the rim top, which is negative although weak (not shown). Between the east and west sidewalls, the difference is negative during the morning but becomes positive in the afternoon after approximately 23 LT. The difference e3 w3 is again of opposite direction but weak. The difference in sensible heat flux between opposite sidewalls follows the difference in radiation, with positive differences between the north and south sidewalls throughout the day and negative differences between the east and west sidewalls during the morning, which then change sign at 3 LT, i.e. about half an hour after the radiation difference changes sign. Only at the rim grid points (e3 w3 and n3 s3) is the heat flux difference reversed, although weak (not shown). c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

3 Thermal Cross-Basin Circulation LT (f) 3 LT.3.2. m s m s 9 LT (g) 4 LT u v nsl esl ssl wsl.3.2. (c) LT.3.2. (h) 5 LT.3.2. (d) LT.3.2. (i) 6 LT.3.2. (e) 2 LT.3.2. (j) 7 LT.3.2. m s 5 R (W m 2 ) θ (K) R (W m 2 ) θ (K) 2 2 Figure 2. Overview of the basin atmosphere between 8 and 7 LT. Left panels: plan view of short-wave incoming radiation and horizontal wind vectors at the first model level. Middle panels: vertical profiles of potential temperature (black line, bottom axis) and u and v wind components (grey lines, top axis) in the centre of the basin. Right panels: slope wind components, i.e. u on the east and west slopes and v on the north and south slopes. Wind components are interpolated to mass grid points and wind vectors are plotted at every third grid point; all data are instantaneous values. At 8 LT, the greater part of the basin is still shaded and wind speeds in the confined cold-air pool within the basin are close to m s (Figure 2). One hour later, the shadow has retreated to the southeast sidewall and surface winds in the basin are from the southeast, i.e. away from the shaded and thus colder sidewall. Upslope winds are present on the north and west sidewalls, whereas a downslope flow has developed on the east sidewall; slope winds on the south sidewall are still close to ms. During the following morning hours, incoming radiation increases and the areas of strongest and weakest irradiation move to the north and south, respectively, as the Sun moves across the sky (Figure 2(c) (e)). Potential-temperature profiles indicate mixing of the basin atmosphere. At LT, a weak and shallow inversion layer is still present below m AGL. By LT, the stratification in the centre of the basin is neutral above a shallow super-adiabatic layer. The overall direction of the surface winds shifts, together with the radiation gradient, to a more southerly direction. Slope-wind speeds increase and the downslope flow on the east sidewall turns to an upslope direction, starting in the upper part of the slope and propagating downward. Slope winds on the south sidewall show a similar evolution to those on the east sidewall, with an initial downslope flow that then turns upward in the upper part. However, in contrast to the east sidewall, katabatic flows continue near the basin floor on the south sidewall throughout the day. During the early afternoon hours (Figure 2(e) (g)), the surface wind directions become increasingly variable. By 5 LT, the surface wind direction is again relatively consistent throughout the basin, with the wind field directed from the southwest to the northeast, towards what is now the most strongly irradiated sidewall. The shadow on the southwest sidewall propagates down the slope in the late afternoon until approximately 73 LT, when sunset occurs last on the basin rim and the upper northeast sidewall. Slope wind speeds decrease after 4 LT. They turn to a downslope direction on the west sidewall around 5 LT and on the east and north sidewalls near the basin floor around 7 LT. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

4 898 M. Lehner and C. D. Whiteman R (W m 2 ) 5 wu, w8 wl, w5 flr, c el, e5 eu, e7 H (W m 2 ) 2 2 T ( o C) 3 2 dir (deg) Figure 3. Comparison of WRF model results with METCRAX observational data. From top to bottom, rows show time series of short-wave incoming radiation, sensible heat flux, temperature, wind speed and wind direction, respectively, at five sites along a west east line through the crater and at the five model grid points closest in elevation to the METCRAX sites. The black dashed line shows the model simulation, the black solid line observations from 23 October (used for model initialization) and the grey lines observations from 2, 9 and 22 October. Model data are at the first model level, i.e. 8 m AGL. METCRAX temperature and wind measurements are at 5 m AGL and heat flux at 2 m AGL. Temperature data at eu and el were corrected for overheating due to non-aspirated radiation shields; for details see Lehner et al. (2). METCRAX data are 3 min means of 5 min averaged data. Vertical lines are sunrise and sunset times at the respective model grid points. Model outputs are compared with observations from the METCRAX field campaign in Figure 3 for short-wave incoming radiation, heat flux, temperature and wind speed and direction. The comparisons are made between five observational sites that were deployed along a west east line through the Meteor Crater (wu, wl, flr, el and eu) and model data from the nearest grid points. In addition to observations from 23 October (black solid line), which was used for model initialization, observations are plotted for three other days (grey lines) to evaluate the representativeness of the results for clear-sky conditions. The model calculates the incoming solar radiation reasonably well. Sunrise seems to occur slightly later than in the Meteor Crater, which is probably due to local variations in the slope angle. At the lower sites wl, flr and el, WRF underestimates the sensible heat flux, particularly during the morning and early afternoon hours. Further up the west sidewall (wu), on the other hand, the modelled heat flux agrees well with observations. Unfortunately, no heat-flux measurements are available for eu. The model also reproduces the diurnal evolution of the temperature on 23 October at all five sites except for the relatively steep increase in the morning. Temperature curves from other days can deviate more strongly because of different initial temperatures and stabilities. When comparing wind speed and direction, it must be remembered that the model is initialized with m s and that there is no synoptic forcing. Overall, the model agrees well with the wind observations. Before sunrise, winds are mostly close to zero, i.e. no perceptible downslope flow is present. With sunrise on the west sidewall, upslope winds develop along this slope, which last until local sunset. On the basin floor an easterly CBF develops at the time of local sunrise on the west sidewall, i.e. at the time when the east west radiation difference is first established. In the evening, the westerly CBF lasts beyond the time of sunset on the east sidewall. On the east sidewall, downslope wind speeds also increase following local sunrise on the west sidewall. The weak downslope flow lasts beyond local sunrise. In the observations, this downslope flow changes somewhat earlier to an upslope direction than it does in the model simulations, particularly at eu. Afterwards, upslope winds also continue until after local sunset. 4. Cross-basin circulation At the basin centre, both u and v components start to increase from ms after 8 LT, first at lower levels and slightly later at higher levels (Figure 4), i.e. shortly after sunrise on the northwest sidewall, when part of the basin is still shaded (Figure 2). Winds weaken again in the evening at about 73 LT and then oscillate around m s. Wind speeds are strongest near the surface (c 3) and mostly increasing until 3 LT. An easterly wind component is continuously present at c and c2 before 23 LT and at c3 from LT, which then changes to a westerly direction in the early afternoon. The southerly component at c 3 lasts throughout the day until 73 LT, starting at (c2 3) or before (c) 9 LT. The wind direction of the morning southwesterly CBF at c 3 changes continuously to a southwesterly direction in the late afternoon (Figure 5). The wind direction at c4 is also from the southeast during part of the morning hours but is relatively weak. From approximately 83 LT, a northwesterly RF occurs at c3 7 (Figures 4 and 5). A more detailed view of the horizontal wind components in the morning before 2 LT is shown in Figure 6 for the centre grid points below the basin rim, together with selected forcing terms, which will be discussed later. Wind speeds in the RF are weak compared with the CBF, with magnitudes of the order of. (Figures 4 6). The RF layer moves upward in time as the CBF layer grows underneath. For instance, c4 is within the RF layer from 83 9 LT, but then becomes part of the CBF layer. The RF layer grows to c7 at 93 LT, although it is extremely weak and occurs only at 93 LT in the u component at this height. The top of the RF is capped by a secondary southeasterly CBF that starts at progressively later times at increasing heights (Figures 4 and 6). At c, easterly and southerly wind components start to increase continuously at 83 LT. At c2, the increase starts only at LT c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

5 Thermal Cross-Basin Circulation 899 c c2 c3 c4 c5 c6 c7 c8 c9 c c c2 c3 c4 c5 c6 c7 c8 c9 c2 (c) u (m s ) (d) (e) (f) v (m s ) Figure 4. Time series (3 min averages) of (c) u and (d) (f) v wind components at c 2. See Figure for the location of grid points c 2. dir ( ) c c2 c3 c4 c5 c6 c Figure 5. Time series (3 min averages) of wind speed and direction at c 7. The respective u and v components are shown in Figure 4. (Figure 4). Although the southerly flow near or above the top of the basin stops almost concurrently at levels c 8 at 3 LT, the easterly flow lasts slightly longer until 23 LT (c 6) or 2 LT (c7 2). The three-layer structure of the cross-basin circulation at the basin centre between 8 and LT can also be seen in Figure 2 (c). Vertical cross-sections through the basin at 93 LT in Figure 7 and (f) show the spatial extent of the three-layer structure. Southeasterly wind components in the lowest 5 m above the basin floor indicate the location of the CBF, with maximum speeds near the surface. A westerly RF component is located above the CBF. It extends over the entire basin width except for the upslope-flow layer on the west sidewall. The easterly component of the secondary southeasterly flow layer near the top of the basin shows its highest wind speeds over the basin centre, decreasing towards the sidewalls. Winds at the basin centre (Figure 4) become relatively constant with height and strongly variable after approximately 2 LT, except for the three or four lowest grid points, as the basin atmosphere becomes increasingly well mixed in the afternoon. The u components are also relatively weak during the afternoon hours, when the east west radiation difference is small, while the v components at c 3 are comparatively strong from the south. Between about 5 and 7 LT, the u component at c 3 is from the west while the v component is still from the south, thus producing a well-developed southwesterly CBF (Figure 5). CBF speeds are similar to those of the morning with ms at c and.5 m s at c2 and c3. The CBF breaks down at about 7 LT with a strong decrease in wind speed, particularly at c and c2. The duration of the CBF is thus shorter in the evening than it is in the morning. This is due to turbulent motions in the unstable basin atmosphere before 5 LT, which inhibit the formation of a constant CBF. The heat flux difference between the east and west sidewalls is also weaker in the afternoon than it is in the morning and the period with higher heat flux on the east sidewall in the afternoon is shorter than the period with higher heat flux on the west sidewall in the morning (not shown). Above c3, wind speeds are relatively weak in the late afternoon, staying mostly below.5 m s, and are relatively constant with height, particularly above c (Figure 4). Overall, even though the magnitudes of the cross-basin radiation gradients in the morning and late afternoon are similar, the cross-basin circulation in the afternoon differs from the cross-basin circulation in the morning due to differences in atmospheric stability, with a stable atmosphere in the morning and an unstable atmosphere in the afternoon. Previous studies have shown that an elevated cross-valley flow forms in the presence of an elevated inversion layer or a surface inhomogeneity (e.g. Vergeiner and Dreiseitl, 987; Shapiro and Fedorovich, 27; Gohm et al., 29; Lehner and Gohm, 2; Lehner et al., 2). For example, at transitions to more stably stratified layers, the along-slope mass flux in the slope wind layer is reduced, producing a cross-valley flow directed away from the slope (Vergeiner and Dreiseitl, 987). Our simulation was initialized with a relatively constant stability within the basin (see e.g. the potential temperature profile in Figure 2). Diurnal heating, however, produces a shallow mixed layer in the morning that is topped by the elevated remnant of the nocturnal inversion. A comparison of wind and potential temperature profiles shows that the top of the CBF layer is, with very few exceptions, not collocated with the top of the mixed layer (not shown), suggesting that another mechanism is at work in this case. Slope winds along the basin sidewalls couple the basin atmosphere to the atmosphere aloft so that mass is not conserved within the cross-basin circulation. This is confirmed by an estimate of the volume flow in the CBF and RF layers through west east and south north cross-sections through the basin c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

6 9 M. Lehner and C. D. Whiteman u/v ( ) PGF (.3 m s 2 ) p (2 Pa) T ( K) West east direction South north direction c e3 w3 c n3 s3 c e w c n s c9 e w c9 n s c8 e w c8 n s c7 e w c7 n s c6 e9 w9 c6 n9 s9 c5 e9 w9 c5 n9 s9 c4 e8 w8 c4 n8 s8 c3 e7 w7 c3 n7 s7 c2 e6 w6 c2 n6 s6 c e4 w4 c n4 s Figure 6. Time series (3 min averages) of (left) west east and (right) south north wind components at c and their respective forcing terms between 73 and 2 LT: pressure-gradient force PGF at c and the pressure and temperature differences p and T between opposing sidewalls. Differences between sidewalls are calculated between (left) east and west sidewalls and (right) north and south sidewalls using grid points on the sidewalls located at approximately the same elevation as the respective centre grid point. The time series have been normalized by the values indicated in parentheses in the figure legend. This figure is available in colour online at wileyonlinelibrary.com/journal/qj centre, calculated between 73 and 2 LT (Figure 8). Assuming that air density is constant within the basin, the volume flow is linearly proportional to the mass flow. For the calculation of the volume flow, the CBF layer is defined along south north and west east cross-sections through the basin centre as the layer above the surface with an easterly or southerly wind component, respectively. The RF is defined accordingly as the layer with a westerly or northerly component directly above the CBF layer. Uncertainties arise from the calculation of the area of the grid boxes adjacent to the topography, which is not exact. Wind fields were interpolated to a Cartesian grid, so that grid boxes near the slopes are intersected by the topography. Through both west east and south north cross-sections, the volume flow in the CBF layer is larger than the volume flow in the RF layer before LT, when the cross-basin circulation is best developed. This means that the cross-basin circulation is not closed within the basin but that part of the volume flow in the CBF layer is transported above the RF layer by the slope winds along the basin sidewalls. Ignoring the flow through the grid boxes intersected by the topography, which contain most of the flow in the slope-wind layer, results in a reduction of the difference between the volume flow in CBF and RF layers (not shown). The volume flows through the CBF c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

7 Thermal Cross-Basin Circulation 9 Geopotential height (m) Geopotential height (m) θ (contour interval:.2 K) 3 2 ( θ/ t) vmix 3 2 (c) W E distance (km) (d) u, w ( θ/ t) adv W E distance (km) u (m s ) θ/ t (K s ) Geopotential height (m) 3 2 θ (contour interval:.2 K) (e) Distance (km) u, w (f) Distance (km) Figure 7. West east cross-sections of potential temperature (contour interval.2 K), west east (colour) and vertical (black contour lines) wind components, (c) potential temperature tendencies from vertical turbulent mixing and (d) potential temperature advection at 93 LT, together with vertical cross-sections in the approximate CBF direction (25 ) of (e) potential temperature and (f) the wind component along the cross-section. Northwest is to the left in (e) and (f). Contour lines for w in and (f) are drawn at.5,.3,.2,.and.5 m s (dashed lines); at. m s (bold lines); and at.5,.,.2,.3 and.5 m s (solid lines). This figure is available in colour online at wileyonlinelibrary.com/journal/qj V flow (x 4 m 3 s ) CBF (E to W) CBF (S to N) RF (W to E) RF (N to S) E S W N V flow (x 4 m 3 s ) Figure 8. Time series of the volume flow in the CBF and RF layers through west east and south north cross-sections through the basin centre and the total easterly and westerly volume flow through the south north cross-section and the total northerly and southerly volume flow through the west east cross-section. The volume flow was calculated from min instantaneous model fields. and RF layers are compared with the total volume flows through the west east and south north cross-sections below 7 m (i.e. approximate rim height) in Figure 8. It shows that the total easterly and southerly volume flows through the cross-sections, which are not restricted to the CBF layer, are larger than the total westerly and northerly flows, independent of whether the grid boxes intersected by the topography are counted (Figure 8) or not (not shown). 5. Analysis of the momentum and thermodynamic balance equations 5.. Momentum balance equation The horizontal momentum equation can be written as dv = PGF + MIX + DIFF6 + DAMP. () dt The total derivative on the left-hand side is dv dt = v t + u v x + v v y + w v z, (2) where v is horizontal velocity (on a Cartesian plane), t is time and x, y and z are distances in the west east, south north and vertical directions, respectively. The individual terms on the right-hand side of () are the pressure-gradient force PGF, subgrid-scale turbulent mixing MIX, sixth-order diffusion DIFF6 to dampen 2 x waves and Rayleigh damping DAMP. Contributions from Rayleigh damping are neglected in the following analysis, since the damping is only active in the top 7 km and thus does not influence the area of interest near the surface. The analysis is based on the terms calculated by the model, which were output at every time step. In the centre of the basin, PGF is the determining term during most of the morning period, except for the lowest c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

8 92 M. Lehner and C. D. Whiteman c9 c7 c5 c3 c c9 c7 c5 c3 c c9 c7 c5 c3 c c9 c7 c5 c3 c PGF MIX DIFF Figure 9. Forcing-term vectors (3 min averages) of the horizontal momentum equation at every second model level in the centre of the basin between 83 and 2 LT. North is at the top of the figure. The vector magnitude is indicated in the figure legend. levels near the surface, for which the contributions from sixthorder diffusion and subgrid-scale turbulent mixing are relatively large (Figure 9). Subgrid-scale vertical turbulent mixing plays an important role near the surface because of the surface momentum flux but quickly decreases with height. In addition, subgrid-scale horizontal turbulent mixing and sixth-order diffusion are higher at c 4 than at higher levels and sixth-order diffusion in particular can reach relatively large values near the surface. In the morning, PGF is mostly directed towards the northwest at the centre grid points near the surface (Figure 9). The direction of the pressure gradient is thus in agreement with the cross-basin radiation and heat-flux gradients, with higher pressure to the southeast. The sign of PGF also agrees with the easterly and southerly CBF components at the centre, confirming that the CBF is driven by the cross-basin pressure gradient. A reversed PGF, i.e. a PGF directed towards the southeast, occurs at several model levels during the morning hours, for example at c5 8 at 9 LT. In the east west direction, a reversed PGF, i.e. a positive u tendency, occurs at c3 some time before LT, in agreement with the timing of the RF at c3 7 (Figure 9). In the north south direction, the RF that forms in the morning before LT is even weaker than in the east west direction. However, the timing agrees again, with short periods of negative PGF at c 3 before 83 LT and at c5 2 some time between sunrise and LT. The secondary southeasterly flow at higher levels within and above the basin seems to be initialized by a combination of a weak PGF (Figure 9) and advection (not shown). PGF and advection also start to increase at progressively later times at increasing elevation, in agreement with the upward-growing layer of southeasterly wind. The northwest southeast pressure gradient at this elevation is caused by stronger subsidence over the southeast half of the basin (see e.g. the cross-sections at 93 LT in Figure 7 and (f)), which seems to be related to stronger heating of the southeast rim and the resulting vertical motions. The stronger heating of the southeast rim is reflected in a tilting of the isentropes, with a potential temperature gradient from northwest to southeast (Figure 7 and (e)). The unobstructed exposure of the outer southeast sidewall to solar irradiation produces a large sensible heat flux (included in the vertical turbulent mixing in Figure 7(c); a more detailed discussion of the potential temperature tendency terms follows in section 5.3) and a strong upslope flow (Figure 7) that separates from the surface at the rim in an upward motion, which advects warmer surface-air upwards (Figure 7(d)). When analyzing the budget equations, it has to be kept in mind, however, that changes in the signs of the total tendencies do not necessarily result in a change of direction of the respective wind component. For example, an easterly wind component decreases but does not necessarily change to a westerly direction if the u tendency becomes positive. Between 5 and 6 LT, PGF is mostly positive for both u and v components at c 3 (not shown). At higher levels, however, momentum-budget terms are difficult to interpret. For example, although there is a short period of northeasterly flow above c9 from LT (Figure 4), only the PGF in the north south direction has the correct sign to produce a northerly flow component, whereas the sign of the PGF in the east west direction does not agree with the easterly flow component. There is thus no clear indication that this flow is part of the thermal cross-basin circulation. The flow could be part of turbulent motions in the still neutral layer above the increasingly stable layer near the surface. The horizontal wind field at higher model levels does indeed indicate more turbulent motions than at the lowest levels at this time (not shown). Because the cross-basin circulation is not as well developed in the evening as it is in the morning, further analysis focuses only on the morning situation The pressure gradient The model PGF is calculated between two adjacent grid points. To relate the PGF in the centre of the basin to heating of the sidewalls, we first determine how representative this local PGF is of the cross-basin PGF, which we calculate between pairs of grid points on opposite sidewalls. Minor differences (of the order of 2 m) can occur in the height of the model levels on the opposing sidewalls (e.g. e4 and w4) because of the model s vertical pressure coordinate. For this analysis, however, it is assumed that the grid points are located at the same height and that the pressure difference reflects the horizontal pressure gradient. Correlation coefficients between the local PGF at the centre grid points and the PGF calculated between the east and west sidewalls and between the north and south sidewalls for the entire day between sunrise and sunset are relatively poor away from the lowest levels (Table ). For the comparison, the PGF between the sidewalls was linearly interpolated to the respective height of the model levels at the basin centre. It is not surprising that correlation decreases with height, considering the increasing distance between the opposing sidewalls. While the distance between e and w is 2 x = m, the distance between e3 and w3 is 26 x = 3 m. In addition, the pressure at the first model level above the ground, which is used for the calculation of the PGF between opposing sidewalls, is influenced by boundary-layer processes occurring near the slope surface. In the north south direction, the local PGF oscillates mostly around m s 2 (c5 8) or even becomes negative at c9 (see Figure 6 for the time before 2 LT), whereas the PGF between the sidewalls is mostly positive throughout the day, in agreement with stronger solar irradiation on the north sidewall (negative p in Figure 6). The PGF between the sidewalls is generally stronger than the local PGF in the morning and evening. The large differences between the local PGF and the PGF between opposite sidewalls develop mainly in the late morning, which is reflected in the higher correlation coefficients for the period c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

9 Thermal Cross-Basin Circulation 93 Table. Correlation coefficients between the model s local PGF calculated between adjacent grid points and the PGF calculated between opposing sidewalls for unaveraged data. West East South North LT 75 LT LT 75 LT c c c c c c c c c c c LT, particularly below c8 (Table ). This suggests that, at least in the early morning, the cross-basin PGF is representative of the local PGF in the centre of the basin, which forces the cross-basin circulation. A detailed look at the pressure and temperature differences between the sidewalls shows an almost constantly positive pressure difference and negative temperature difference between e 6 and w 6 (elevation compares with c 2) between 8 and 23 4 LT (exact times vary among the grid points; Figure and (c)). Similarly, a constantly negative pressure difference and positive temperature difference occur in the north south direction between n 7 and s 7 (elevation compares with c 3) before 63 7 LT (Figure and (d)). At higher altitudes, the negative pressure difference between the north and south sidewalls and the positive pressure difference between the east and west sidewalls is preceded or interrupted by a short period of positive or negative difference, respectively (Figure 6). A reversed pressure difference occurs in the east west direction between e7 3 and w7 3 (elevation compares with c3 ) as well as in the north south direction between n8 3 and s8 3 (elevation compares with c4 ), in approximate agreement with the height of the RF, starting first at lower altitudes and progressing upward (83 9 LT at e8 w8 and n8 s8, 9 LT at e w and n s). Similarly, the pressure difference returns first to its normal direction at the lower altitude, in agreement with the upward movement of the RF layer (Figure 6). A reversed temperature difference occurs in the east west direction between e7 3 and w7 3 and in the north south direction between n8 3 and s8 3 (n s is close to K) during the morning, mostly before LT. The signs of the cross-basin pressure and temperature differences thus agree with the directions of the CBF and RF. This means that not only the CBF but also the RF is produced by a thermal gradient, with higher temperatures on the southeast, less irradiated sidewall. Figure 6 also indicates that there is a time lag between the reversed temperature and pressure differences between e8 and w8 and between n8 and s8, with the temperature difference preceding the pressure difference. However, compensatory downslope flows in the layer above the upslope flow on the northwest sidewall (Figure 7 and (f)) may also contribute to the horizontal pressure gradient and to the formation of the RF Thermodynamic balance equation The thermodynamic equation can be written as θ t = ADV + RAD + HMIX + VMIX + DIFF6 + DAMP, (3) where the individual terms are advection ADV, radiation RAD, horizontal and vertical subgrid-scale turbulent mixing HMIX and VMIX, sixth-order diffusion DIFF6 and Rayleigh damping DAMP. No cumulus parametrization is run and microphysics are turned off, i.e. only water vapour is taken into account. Because of the dry atmosphere, neglecting phase changes is not expected to affect the results. Rayleigh damping is again neglected in the following analysis and subgrid-scale turbulent mixing is separated into a horizontal component HMIX and a vertical component VMIX. Radiation and sixth-order diffusion terms are generally small and are therefore not shown in Figure. Radiation is positive during the day until sunset and has about the same order of magnitude as the total θ tendency or is even smaller. The sixth-order diffusion term is also relatively small, except for the morning hours at upper levels, particularly on the east and south sidewalls (e 3 and s 3), where it opposes advection and reaches similar magnitudes (not shown). Vertical turbulent mixing VMIX near the surface is strongly determined by the sensible heat flux from the surface. It is thus positive throughout the day and generally increases with elevation on the north and west sidewalls in the morning. Horizontal turbulent mixing HMIX is generally negative and produces a cooling of the near-sidewall air, thus reducing the temperature contrast between the air close to the sidewall and the air farther away. e w e4 w4 e7 w7 e w e3 w3 n s n4 s4 n7 s7 n s n3 s3 p (Pa) 2 E W N S 2 (c) (d) T (K) E W N S Figure. Time series (3 min averages) of, pressure difference and (c), (d) temperature difference between, (c) the east and west sidewalls and, (d) the north and south sidewalls. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

10 94 M. Lehner and C. D. Whiteman total ADV VMIX HMIX.. w e.. w9 e9. θ/ t (K s ).. w7 e7. w5 e5.. w3 e3.. w e Figure. Time series (3 min averages) of potential temperature tendency terms at grid points on the west (left) and east (right) sidewalls. Vertical lines indicate the times of local sunrise at the respective grid points. The advection term is mostly negative at all sidewalls (i.e. cold-air advection) during the morning (Figure ). On the more strongly irradiated north and west sidewalls, ADV decreases from Ks to relatively strong negative values after local sunrise, together with an increase in VMIX. Cold-air advection reaches values of. K s at w8, countering the heating from radiation and surface sensible heat flux. The spatial extent of the cold-air advection on the west sidewall is also shown in Figure 7(d). On the east and south sidewalls, on the other hand, the advection term is slightly positive (i.e. warm-air advection) between sunrise on the opposite sidewalls and local sunrise. Since VMIX starts to increase only after local sunrise, ADV leads to positive heating rates on the shaded sidewalls. The warm-air advection on the shaded sidewalls is caused by the downslope flow, which advects potentially warmer air from above in the stable basin atmosphere, whereas the cold-air advection on the irradiated sidewalls is caused by the upslope flow, which advects potentially colder air from below. This interpretation is also supported by the fact that potential-temperature advection is dominated by along-slope advection, particularly during the morning hours. Correlation coefficients between total advection and along-slope advection, which was calculated using centred along-slope θ differences from unaveraged.5 s model output, are mostly.8 on the west, east and north sidewalls for the period 75 LT. At or shortly after the time of local sunrise on the east and south sidewalls, potential-temperature advection changes sign from positive to negative, countering the heating from surface sensible heat flux (VMIX) and radiation. Katabatic winds on the east sidewall, however, continue until after LT at e (up to 3 LT at e 2). Only at e 2 does the slope wind direction change at the same time or before the change from warm-air advection to cold-air advection (not shown). On the south sidewall, downslope winds continue throughout the day, except for s 3. There is some indication that the reversed advection is produced by a locally unstable stratification along the slope. Local sunrise on the southeast sidewall occurs first at the basin floor, from where the shadow line moves up the slope. This means that the lower slope is heated earlier than the upper slope, thus reversing the vertical θ gradient locally along the sidewall so that downslope winds advect colder air from above. On the east sidewall at e 9, the local along-slope θ gradient calculated between two adjacent grid points becomes negative or close to neutral at about the same time as θ advection changes sign (not shown). On the south sidewall, however, the timing is not as clear as on the east sidewall. The difference in total θ tendency between opposing sidewalls oscillates around K s in the morning, including positive differences between the east and west sidewalls at all levels, i.e. less warming on the west, more strongly irradiated, sidewall and, similarly, negative values between the north and south sidewalls (Figure 2). Such reversed heating periods occur, for example, between e 8 and w 8 some time between 9 and 3 LT and between e7 and w7 some time before 9 LT. As discussed earlier, no reversed temperature gradient occurs between the lowest sidewall grid points e 6 and w 6 or between n 6 and s 6. An early negative difference in total θ tendency occurs between e2 6 and w2 6 and a positive difference c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

11 Thermal Cross-Basin Circulation 95 ( θ/ t) ( 3 K s ).5 e3 w3 e5 w5 e9 w9 e3 w n3 s3 n5 s5 n9 s9 n3 s Figure 2. Time series (3 min averages) of the difference in total potential temperature tendency between the east and west sidewalls and the north and south sidewalls. between n3 6 and s3 6 prior to the reversed heating periods. The difference in VMIX thus seems to dominate the difference in ADV initially. This may be a result of the shorter time period between local sunrise on opposing sidewalls or of the slope angle, i.e. in the steep upper part the vertical component of advection along the slope is larger, whereas in the lower part horizontal advection dominates. Although cooling due to horizontal turbulent mixing is generally weaker than cooling due to cold-air advection (Figure ), it still contributes to the stronger cooling on the more irradiated sidewalls. HMIX is generally stronger on the north and west sidewalls than on the east and south sidewalls, particularly at higher elevations, i.e. grid points 7 3 from the basin centre (the difference is very small at lower elevations). This difference in HMIX can be at least partially attributed to the stronger temperature differences between the surface and the free basin atmosphere on the more irradiated sidewalls. A secondary contribution may be caused by stronger turbulence on the north and west sidewalls. In the morning, along-slope wind speeds are higher along the more irradiated sidewalls, with values of u m s on the west sidewall and v.5 3 m s on the north sidewall compared with u. m s on the east sidewall and v.5 m s on the south sidewall. Measurements on the sidewalls of Arizona s Meteor Crater during four clear-sky mornings also indicate cold-air advection along the slope on the west sidewall after local sunrise and contemporaneous warm-air advection on the east sidewall (Figure 3 (d)). Along-slope advection was calculated based on the temperature difference between the upper and lower towers on the respective sidewall (see Figure for their locations). Comparing the magnitude of the observed advection with the modelled values in Figure shows good agreement. Cold-air advection on the west sidewall is of the order of. K s and warm air-air advection on the east sidewall is much lower. The temperature difference between the two pairs of instrumented sites, however, does not reverse before local sunrise on the east sidewall (Figure 3(e) (f)). 6. Summary and conclusion The thermal cross-basin circulation in an idealized basin based on the topography of Arizona s Meteor Crater was simulated with WRF. The modelled horizontal momentum and thermodynamic budgets were analyzed to investigate the physical mechanisms contributing to the formation of the cross-basin circulation caused by asymmetric irradiation. The vertical structure of the cross-basin circulation is summarized in Figure 4. We could identify three different mechanisms that lead to the formation of cross-basin flows either from or towards the more sunlit sidewall, with all three mechanisms being a result of the asymmetric irradiation. () A southeasterly cross-basin flow (CBF) develops in the morning above the basin floor, from the less irradiated sidewall towards the more irradiated one, as the result of a horizontal pressure gradient, with higher pressure on the less irradiated and thus colder sidewall. (2) Above this cross-basin flow a weak opposing return flow (RF) develops towards the less irradiated sidewall. Our analysis suggests that differential temperature advection by the slope winds is responsible for a reversal in the crossbasin temperature and thus pressure gradient, leading to the formation of the RF. (3) Above the RF, a secondary southeasterly flow occurs that grows upward with time. Stronger irradiation on the outer southeast sidewall causes higher potential temperature near the southeast rim and strong vertical rising motions. Compensatory subsidence is stronger over the southeast half of the basin, resulting in a horizontal pressure gradient. The mechanisms outlined above and summarized in Figure 4 are based on an analysis of the cross-basin circulation in the centre of the basin away from the basin sidewalls, where the resulting flow is mostly two-dimensional. The flow close to the basin sidewalls, however, is likely influenced by the three-dimensional topography, including flow convergence or divergence along the basin sidewalls. Even away from the complicating factors near the sidewalls, the analysis showed that the thermal crossbasin circulations resulting from asymmetric irradiation are part of a highly complex system, even in such simple topography as an idealized, rotationally symmetric basin without synoptic influences. In valleys, the cross-valley circulation can thus be expected to be even more complicated when additional alongvalley winds are present. The findings for the cross-basin circulation, however, are equally applicable to valleys in the absence of along-valley winds, e.g. during the up-valley downvalley wind transition periods or in very long valleys with a horizontal valley floor far enough away from the valley exit, similar to the idealized valley topography discussed by Serafin and Zardi (2). The near-surface branch of the cross-basin circulation, which is directed towards the more irradiated sidewall, forms as a result of the higher pressure on the less irradiated sidewall compared with the pressure on the more irradiated sidewall, as documented previously in the literature (e.g. Gleeson, 95; Hennemuth, 986; Lehner et al., 2). However, this explains only one part of the whole circulation system, as slope winds on the mountain sidewalls influence and interact with the cross-basin circulation, indicating that the CBF cannot be treated completely independently from the slope-wind system. Since the formation mechanisms of the CBF and the slope flows are different, we think, however, that they should be considered as two individual phenomena. The CBF may influence the slope wind system, as the example of the developing downslope wind on the east sidewall after local sunrise on the west sidewall suggests. This indicates that in some cases the local surface energy balance may be insufficient to explain the reversal of slope winds in the morning and evening. Conversely, the RF is a result of the influence of slope winds on the cross-basin circulation. Based on an analysis of the horizontal c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)