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  • View in gallery

    Slope-following coordinate system. The downslope (x) and slope-normal (z) coordinates are inclined at an angle to the horizontal and vertical coordinates, respectively. The motion is independent of the cross-slope (y; into page) coordinate.

  • View in gallery

    Schematic of the diurnally varying diffusivity K(t) and surface buoyancy functions considered in section 4. The diffusivity varies as a step function with daytime value and nighttime value . The surface buoyancy varies as a sawtooth function that increases from a minimum at sunrise (t = 0) to a maximum at time , a few hours before sunset (time ).

  • View in gallery

    Time dependence of (left) b and (right) υ on the slope (z = 0) computed using a progressively larger number of terms (see legends) in the truncated series form of (3.17) for experiment BH. As more terms are included, the variables approach their appropriate slope distributions: no slip for υ, and sawtooth function for b, with and .

  • View in gallery

    Wind speed (m s−1) as a function of height and time over a 24-h period in experiment BH. Sunrise is at t = 0 s. Sunset is at tset = 43 200 s. The surface buoyancy peaks at tmax = 32 400 s.

  • View in gallery

    Postsunset evolution of (left) and (right) profiles in experiment BH. Curves are shown at 1.5-h intervals starting from sunset. Labeled curves correspond to times (line a), (line b), (line c), and (line d).

  • View in gallery

    Evolution of the wind hodographs in experiment BH over a 24-h period at different heights. Curves are plotted (left) at low levels in 100-m increments up to the jet maximum and (right) at and above the jet maximum in 500-m increments. The times of sunset, sunrise, and peak surface buoyancy are indicated on the hodographs. The blue diamond indicates the geostrophic wind.

  • View in gallery

    As in Fig. 4, but for b (m s−2).

  • View in gallery

    As in Fig. 5, but for b(z) profiles.

  • View in gallery

    Peak υ as a function of slope angle for (solid curve) and (dashed curve). Other parameters are as in experiment BH.

  • View in gallery

    Wind speed (m s−1) as a function of height and time over a 24-h period for an intense jet associated with strong surface buoyancy and a strong geostrophic wind . Other parameters are as in experiment BH. Sunrise is at t = 0 s. Sunset is at tset = 43 200 s. The surface buoyancy peaks at tmax = 32 400 s.

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A Unified Theory for the Great Plains Nocturnal Low-Level Jet

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  • 1 School of Meteorology, and Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma
  • | 2 School of Meteorology, University of Oklahoma, Norman, Oklahoma
  • | 3 Department of Atmospheric Science, University of Wyoming, Laramie, Wyoming
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Abstract

A theory is presented for the Great Plains low-level jet in which the jet emerges in the sloping atmospheric boundary layer as the nocturnal phase of an oscillation arising from diurnal variations in turbulent diffusivity (Blackadar mechanism) and surface buoyancy (Holton mechanism). The governing equations are the equations of motion, mass conservation, and thermal energy for a stably stratified fluid in the Boussinesq approximation. Attention is restricted to remote (far above slope) geostrophic winds that blow along the terrain isoheights (southerly for the Great Plains). Diurnally periodic solutions are obtained analytically with diffusivities that vary as piecewise constant functions of time and slope buoyancies that vary as piecewise linear functions of time. The solution is controlled by 11 parameters: slope angle, Coriolis parameter, free-atmosphere Brunt–Väisälä frequency, free-atmosphere geostrophic wind, radiative damping parameter, day and night diffusivities, maximum and minimum surface buoyancies, and times of maximum surface buoyancy and sunset. The Holton mechanism, by itself, results in relatively weak wind maxima but produces strong jets when paired with the Blackadar mechanism. Jets with both Blackadar and Holton mechanisms operating are shown to be broadly consistent with observations and climatological analyses. Jets strengthen with increasing geostrophic wind, maximum surface buoyancy, and day-to-night ratio of the diffusivities and weaken with increasing Brunt–Väisälä frequency and magnitude of minimum slope buoyancy (greater nighttime cooling). Peak winds are maximized for slope angles characteristic of the Great Plains.

Corresponding author address: Alan Shapiro, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Room 5900, Norman, OK 73072. E-mail: ashapiro@ou.edu

Abstract

A theory is presented for the Great Plains low-level jet in which the jet emerges in the sloping atmospheric boundary layer as the nocturnal phase of an oscillation arising from diurnal variations in turbulent diffusivity (Blackadar mechanism) and surface buoyancy (Holton mechanism). The governing equations are the equations of motion, mass conservation, and thermal energy for a stably stratified fluid in the Boussinesq approximation. Attention is restricted to remote (far above slope) geostrophic winds that blow along the terrain isoheights (southerly for the Great Plains). Diurnally periodic solutions are obtained analytically with diffusivities that vary as piecewise constant functions of time and slope buoyancies that vary as piecewise linear functions of time. The solution is controlled by 11 parameters: slope angle, Coriolis parameter, free-atmosphere Brunt–Väisälä frequency, free-atmosphere geostrophic wind, radiative damping parameter, day and night diffusivities, maximum and minimum surface buoyancies, and times of maximum surface buoyancy and sunset. The Holton mechanism, by itself, results in relatively weak wind maxima but produces strong jets when paired with the Blackadar mechanism. Jets with both Blackadar and Holton mechanisms operating are shown to be broadly consistent with observations and climatological analyses. Jets strengthen with increasing geostrophic wind, maximum surface buoyancy, and day-to-night ratio of the diffusivities and weaken with increasing Brunt–Väisälä frequency and magnitude of minimum slope buoyancy (greater nighttime cooling). Peak winds are maximized for slope angles characteristic of the Great Plains.

Corresponding author address: Alan Shapiro, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Room 5900, Norman, OK 73072. E-mail: ashapiro@ou.edu

1. Introduction

The nocturnal low-level jet (LLJ) is a low-level maximum in the boundary layer wind profile common to the Great Plains of the United States (Bonner 1968; Mitchell et al. 1995; Stensrud 1996; Whiteman et al. 1997; Arritt et al. 1997; Song et al. 2005; Walters et al. 2008) and other places worldwide (Sládkovič and Kanter 1977; Stensrud 1996; Beyrich et al. 1997; Rife et al. 2010; Fiedler et al. 2013). Typically LLJs begin to develop around sunset in fair weather conditions, reach peak intensity a few hours after midnight, and dissipate with the onset of daytime convective mixing. Peak LLJ winds are generally supergeostrophic, with wind vectors that turn anticyclonically through the night. Data from Doppler sodars, lidars, and other high-resolution observational platforms indicate that peak LLJ winds are often found within 500 m of the ground (Whiteman et al. 1997; Banta et al. 2002; Banta 2008; Song et al. 2005; Conangla and Cuxart 2006; Baas et al. 2009; Werth et al. 2011; Kallistratova and Kouznetsov 2012; Hu et al. 2013; Klein et al. 2014). Early analyses of pibal and rawinsonde data over the Great Plains of the United States revealed that LLJs in that region can be hundreds of kilometers wide (Means 1954; Hoecker 1963; Bonner 1968; Bonner et al. 1968) and up to 1000 km long (Bonner 1968; Bonner et al. 1968). The maximum winds in the majority of Great Plains LLJs are southerly; the southerly jets occur roughly twice as often in the warm season as in the cold season (Bonner 1968; Whiteman et al. 1997; Song et al. 2005; Walters et al. 2008). In Bonner’s (1968) 2-yr climatological analysis, Great Plains LLJs occurred most frequently over Kansas and Oklahoma in a corridor roughly straddling the 98°W meridian. However, there is much interannual variability in the jet characteristics (Song et al. 2005; Walters et al. 2008), and the peak frequency from a longer-term (40 yr) analysis (Walters et al. 2008) was located farther south than in the Bonner (1968) study, in southern and southwestern Texas.

LLJs have significant economic and public health and safety impacts. LLJs promote deep convection and beneficial (as well as hazardous) heavy rain events through moisture transport and lifting (Means 1954; Pitchford and London 1962; Maddox 1980, 1983; Cotton et al. 1989; Stensrud 1996; Higgins et al. 1997; Arritt et al. 1997; Trier et al. 2006, 2014; French and Parker 2010). They transport pollutants hundreds of miles over the course of a night and mix elevated pollutants down to the surface (Zunckel et al. 1996; Banta et al. 1998; Solomon et al. 2000; Mao and Talbot 2004; Darby et al. 2006; Bao et al. 2008; Klein et al. 2014; Delgado et al. 2014). They also transport fungi, pollen, spores, and insects, including allergens, agricultural pests, and plant and animal pathogens (Drake and Farrow 1988; Wolf et al. 1990; Westbrook and Isard 1999; Isard and Gage 2001; Zhu et al. 2006; Westbrook 2008). LLJs are an important wind resource for the wind energy industry (Cosack et al. 2007; Storm et al. 2009; Emeis 2013, 2014; Banta et al. 2013).

There is considerable interest in characterizing and improving the representation of LLJs in numerical weather prediction. Storm et al. (2009) found that the Weather Research and Forecasting (WRF) Model run with a variety of boundary layer and radiation parameterizations underestimated LLJ strength and overestimated LLJ height in two southern Great Plains test cases. Steeneveld et al. (2008) reported similar difficulties in experiments with three regional models. Werth et al. (2011) found that the life cycles of southern Great Plains LLJs were generally well predicted using a regional model but that the modeled jets were again placed too high. Mirocha et al. (2016) found large discrepancies between lidar observations and WRF Model predictions of near-surface winds in northern Great Plains LLJs. Increasing the model spatial resolution did little to improve the wind speed forecasts and sometimes even degraded the results. In a comparison of wind profiles from the North American Regional Reanalysis (NARR) and rawinsonde observations, Walters et al. (2014) noted that the NARR underestimated LLJ frequencies and urged caution in the use of NARR LLJ winds in LLJ case studies and numerical model validations. The problems identified in some of these studies were attributed, in part, to deficiencies in parameterizations of turbulent exchange in the nocturnal stable boundary layer. Indeed, understanding and modeling boundary layers under stably stratified conditions is notoriously difficult (Mahrt 1998, 1999; Derbyshire 1999; Mironov and Fedorovich 2010; Fernando and Weil 2010; Holtslag et al. 2013; Sandu et al. 2013; Steeneveld 2014).

In addition to the current poor state of knowledge of nocturnal stable boundary layers, there are also remaining uncertainties in understanding the physical mechanisms of the development of Great Plains LLJs. We now discuss the main theories underpinning this phenomenon.

Concerning the geographical preference of the Great Plains LLJ, Wexler (1961) proposed that the strong southerly time-mean current originates from blocking of easterly trade winds by the Rocky Mountains in a manner similar to the westward intensification of oceanic currents along eastern seaboards. Terrain versus no-terrain experiments performed in a regional model (Pan et al. 2004) and a general circulation model (Ting and Wang 2006; Jiang et al. 2007) suggested that topography is essential for maintaining a strong southerly time-mean flow over the Great Plains during the summer. However, Parish and Oolman (2010) found that the strong southerly mean flow over the Great Plains in their nonhydrostatic mesoscale simulations of summertime LLJs originates from the mean heating of the gentle slope of the Great Plains rather than from mechanical blocking by the Rocky Mountains.

Blackadar (1957, hereafter B57) and Buajitti and Blackadar (1957, hereafter BB57) described the LLJ as an inertial oscillation (IO) in the atmospheric boundary layer resulting from the sudden disruption of the Ekman balance near sunset when turbulent (frictional) stresses are rapidly shut down.1 The IO-theory predictions of low-altitude winds that turn anticyclonically during the night and reach peak intensity in the early morning have been amply verified in the literature. However, the B57 IO theory cannot explain how the peak winds in some LLJs exceed the geostrophic values by more than 100%.2 Additionally, B57 proposed a close association between the height of the wind maximum and the top of the nocturnal surface inversion layer. While there are some confirmations of such an association (B57; Coulter 1981; Baas et al. 2009; Werth et al. 2011; Kallistratova and Kouznetsov 2012), the many studies where this association was not found (Hoecker 1963; Bonner 1968; Mahrt et al. 1979; Brook 1985; Whiteman et al. 1997; Andreas et al. 2000; Milionis and Davies 2002) suggest that such a link may not be straightforward.

Holton (1967, hereafter H67) studied the response of the atmosphere to a diurnally heated/cooled slope without provision for a time dependence in the turbulent friction (i.e., without the B57BB57 frictional relaxation). The analysis was restricted to cases where the free-atmosphere geostrophic wind flowed parallel to terrain isoheights. The governing equations were the one-dimensional (1D; in slope-normal coordinate) Boussinesq equations of motion and thermal energy equation for a viscous stably stratified fluid. The H67 solutions described baroclinically generated wind oscillations in the atmospheric boundary layer, but the phase of the oscillations was not captured correctly, and the wind profiles were not as jetlike as in observed LLJs.

Although neither the H67 slope theory nor the B57BB57 IO theory is generally sufficient to explain observations of Great Plains LLJs, the physical mechanisms underlying these theories are plausible, and it has long been speculated that both can be important in the development of Great Plains LLJs. The question of dominant mechanism is not without some controversy, however. For instance, on the basis of two-dimensional numerical modeling of the boundary layer diurnal cycle over gently sloping terrain, McNider and Pielke (1981) and Savijärvi (1991) reach opposite conclusions. A slope effect plays “a dominant role in the evolution of the diurnal wind structure” in the former study but “has only a small guiding effect” in the latter study. These disparate conclusions may be an outcome of seasonal differences: Savijärvi (1991) used composite data from 21 cases in March–June 1967, while McNider and Pielke (1981) used data typical of midsummer. The diurnal range of the surface geostrophic wind (representing thermal forcing of the slope) was 2.5 times larger in the latter study, while the free-atmosphere geostrophic wind was 1.6 times larger in the former study. In an analysis of general circulation model output, Jiang et al. (2007) concluded that both the IO and buoyancy-associated slope flow mechanisms were equally important in establishing the phase and amplitude of the modeled LLJs. Based on results from a mesoscale numerical model and an idealized inviscid analysis, Parish and Oolman (2010) concluded that the nocturnal wind maxima in their summertime Great Plains LLJs resulted from inertial oscillations arising from frictional decoupling. However, they also found that diabatic heating of the slope strengthened the late-afternoon low-level geostrophic wind—a background field on which the frictional decoupling acted—which then intensified the nocturnal wind maximum.

Early studies that sought to combine the H67 and B57BB57 mechanisms into a single framework considered the 1D (in true vertical coordinates) equations of motion but did not make provision for a thermal energy equation [Bonner and Paegle (1970) and references therein]. Rather, a diurnally warming/cooling atmospheric boundary layer was explicitly specified by imposing a corresponding geostrophic wind profile.3 The eddy viscosity and geostrophic wind were specified as sums of time-mean and diurnally varying components, with the diurnally varying part of the geostrophic wind decreasing exponentially with height. When appropriately tuned, the Bonner and Paegle (1970) model [and its extension by Paegle and Rasch (1973)] produced results that were in reasonable agreement with observations. More recently, Du and Rotunno (2014) simplified the Bonner and Paegle (1970) model by setting the stress-divergence term proportional to the velocity vector, resulting in a zero-dimensional (no height dependence) model. Such a model cannot predict the vertical structure of a jet but can be used to estimate jet strength and timing. The amplitude and phase of the modeled LLJs were in good agreement with NARR data, particularly when NARR-derived latitudinal variations in the amplitudes of the mean and diurnally varying components of the pressure gradient force were specified.

Shapiro and Fedorovich (2009) treated the combined H67 and B57/BB57 problem as an inviscid postsunset initial-value problem, essentially generalizing the B57 formulation to include a thermal energy equation for a stably stratified fluid overlying the sloping Great Plains. The wind and buoyancy variables were specified through initial (sunset) conditions, and the subsequent motion was an inertia–gravity oscillation. Stronger initial buoyancies played an analogous role in strengthening the LLJ as stronger initial ageostrophic winds did in the IO theory. The initial buoyancy was obtained in terms of slope angle, free-atmosphere stratification, distance beneath the capping inversion, and parameters characterizing the residual layer and capping inversion. The free-atmosphere stratification played a dual role: it attenuated the upslope and downslope motions (as had been noted by H67 and others) but was also associated with increased initial buoyancy, which tended to increase the oscillation amplitude.

In this study, we present a unified theory for the Great Plains LLJ in which the jet appears in the nighttime phase of oscillations arising from diurnal cycles of turbulent mixing (Blackadar mechanism) and heating/cooling of the slope (Holton mechanism). As in H67, the equations of motion are supplemented with a thermal energy equation. The buoyancy evolves in accord with the coupled governing equations, unlike the thermal field proxy (geostrophic wind or equivalent) in Bonner and Paegle (1970), Paegle and Rasch (1973), Parish and Oolman (2010), and Du and Rotunno (2014), which is specified.

In section 2, we formulate our problem and show how a special linear transformation simplifies the governing equations. An analytical solution for periodic motions is derived in section 3. The case where the diffusivities vary as piecewise constant functions of time and the surface buoyancy varies as a piecewise linear function of time is explored in section 4. A summary follows in section 5.

2. Problem formulation and governing equations

An inclined planar surface (slope) is subjected to a diurnally periodic but spatially uniform thermal forcing. Far above the slope, a spatially and temporally uniform pressure gradient drives a geostrophic wind that blows parallel to the terrain isoheights (as in H67). For a Great Plains analysis, this free-atmosphere geostrophic wind is considered southerly. As in H67, Parish and Oolman (2010), and Du and Rotunno (2014), the model variables are independent of the cross-slope coordinate. We parameterize the turbulent transfer of heat and momentum using an eddy viscosity approach in which the diffusivities are diurnally periodic. Attention is restricted to periodic flows; transients associated with any particular initial state are not investigated.

We work with a slope-following right-handed Cartesian coordinate system (Fig. 1) with x axis pointing downslope, y axis pointing across the slope, and z axis directed perpendicular to the slope. The slope is inclined at an angle to the true horizontal coordinate x*. For a Great Plains analysis, the x* axis points eastward (higher terrain toward the west). The true vertical coordinate (normal to x*) is denoted by z*. The unit vectors in the x, , z, and directions are denoted by , , , and , respectively.

Fig. 1.
Fig. 1.

Slope-following coordinate system. The downslope (x) and slope-normal (z) coordinates are inclined at an angle to the horizontal and vertical coordinates, respectively. The motion is independent of the cross-slope (y; into page) coordinate.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

a. Governing equations

The governing equations are the equations of motion, mass conservation, and thermal energy under the Boussinesq approximation. Given the along-slope homogeneity of the forcings, the x- and y-velocity components (u and υ, respectively) are constant on xy planes, and the mass conservation equation (incompressibility condition) together with the impermeability condition imply that the slope-normal (z) velocity component w is identically zero. The remaining governing equations4 can then be written as
e2.1
e2.2
e2.3
e2.4
where (2.1) and (2.2) are the downslope and cross-slope equations of motion, respectively, (2.3) is the quasi-hydrostatic equation (Mahrt 1982), and (2.4) is the thermal energy equation. Here, is a kinematic pressure perturbation, is buoyancy, p is pressure, is potential temperature, is the potential temperature in the free atmosphere, is the free-atmosphere Brunt–Väisälä frequency (considered constant), and a subscript “0” denotes a constant reference value of the corresponding thermodynamic variable. The quantity is the pressure profile in the free atmosphere at a prescribed location . Since , the free-atmosphere geostrophic balance must be supported by rather than by P. For simplicity, we refer to ( is the angular velocity of Earth’s rotation) as the Coriolis parameter since its value differs insignificantly from that of the true Coriolis parameter on the Great Plains, where the angle between and is ~0.1°. We treat f as constant.

Equations (2.1)(2.4) comprise a nearly exact set of governing equations for 1D Boussinesq flow, with most of the missing terms vanishing identically because of spatial homogeneity (in x) rather than being neglected. For instance, the momentum advection terms in the three equations of motion are identically zero. The only thermal advection term to survive in (2.4) is the advection of the free-atmosphere potential temperature by the true vertical (z*) velocity component (which is ). All x-derivative diffusion terms vanish in (2.1), (2.2), and (2.4)

As in the katabatic wind studies of Egger (1985) and Mo (2013), we include a radiative damping (Newtonian heating/cooling) term in the thermal energy equation, with the damping parameter held constant. The reciprocal parameter is the damping time scale. A radiative damping term was also included in H67 but was based on the difference between temperature and a specified diurnally varying radiative equilibrium temperature.

As in H67, the turbulent heat and momentum exchanges are parameterized through height-invariant turbulent diffusion coefficients (diffusivities) , considered equal for momentum and heat (i.e., the turbulent Prandtl number is unity). However, while H67 also took K to be temporally constant, our K is diurnally periodic. Bonner and Paegle (1970) also took their turbulent momentum exchange coefficient to be diurnally periodic and height invariant.

It can be noted that estimated K(z) profiles in Ekman layers typically vary slowly in height and attain a local maximum at low or midlevels of the boundary layer (O’Brien 1970; Stull 1988; Grisogono 1995; Jeričević and Večenaj 2009). Steady-state solutions of the Ekman equations associated with such K have been obtained analytically by Brown (1974), Nieuwstadt (1983), Miles (1994), Grisogono (1995), Tan (2001), and others. In contrast, in stable boundary layers featuring a pronounced low-level wind maximum, as in nocturnal low-level jets or katabatic flows, height variations in K are of finer scale, and K(z) can exhibit multiple extrema (McNider and Pielke 1981; Cuxart and Jiménez 2007; Axelsen and van Dop 2009). The shape of K profiles in sloping boundary layers with jets therefore varies considerably over a diurnal cycle. Such K are nonseparable functions of z and t, which render the governing equations insoluble by traditional analytical methods. Accordingly, for analytical tractability, we proceed with height-invariant K. Reasonable results obtained with this simplified approach might suggest that it is more important to account for the rapid and drastic temporal changes of K attending the early evening and morning transitions than to account for the vertical variations of K.

We have chosen a turbulent Prandtl number of unity primarily for mathematical expediency: it is the only choice for which the governing equations reduce to the simpler forms derived in section 2b. However, Wilson (2012) obtained qualitatively reasonable eddy viscosity model simulations of day 33 of the Wangara experiment using that value. A value of unity is also consistent with estimates reported in stable conditions (nighttime phase of our analysis). For weakly stable conditions, Howell and Sun (1999) show Prandtl numbers scattered around unity, with no strong dependence on the stability parameter, while Schumann and Gerz (1995) estimate Prandtl numbers between 0.8 and 1.2. Cuxart and Jiménez’s (2007) estimates of the Prandtl number in their nocturnal LLJ LES using the Stable Atmospheric Boundary Layer Experiment in Spain-1998 (SABLES-98) data are generally between 0.8 and 1. However, under unstable conditions (daytime phase of our analysis), Prandtl numbers in the surface layer are generally smaller than 1—about 0.3–0.4 for strongly unstable regimes (Businger et al. 1971; Gibson and Launder 1978; results in those studies shown for the inverse Prandtl number)—suggesting that during the day our analysis likely underestimates the mixing of heat relative to momentum at low levels. Prandtl numbers are generally not reported for the mixed layer because of measurement difficulties and the large uncertainty in estimates of vertical gradients of wind and temperature when these gradients are small.

On the slope we impose the no-slip conditions
e2.5
and specify the surface buoyancy
e2.6
where is diurnally periodic. Far above the slope we take , in which case (2.4) indicates that u must also vanish. Thus, the wind far above the slope blows parallel to the topographic height contours. In view of (2.1), this remote wind is in a geostrophic balance. These remote conditions can be written as
e2.7a
e2.7b
For brevity, we refer to as the geostrophic wind even though it actually is the geostrophic wind far above the slope (i.e., free-atmosphere geostrophic wind).

Since b is independent of x, taking of (2.3) yields , the z-integral of which is , where the function of integration F is, at most, a function of x and t. In view of (2.7b), . Since is constant, F is constant. Thus, the along-slope perturbation pressure gradient is spatially and temporally constant; it is imposed on the boundary layer from aloft.

In terms of the ageostrophic wind components and u (which is ageostrophic since the x-component geostrophic wind is zero), (2.1) and (2.2) become
e2.8
e2.9
with . Equations (2.4), (2.8), and (2.9) comprise a closed system. Equation (2.3) was used to establish constancy of the along-slope pressure gradient throughout the boundary layer but is now no longer needed. It can be seen from (2.8) that positive buoyancy plays an analogous role in forcing upslope flow as a negative ageostrophic wind. During the late afternoon, at the low levels where the LLJ will eventually develop, the buoyancy is positive and the wind is subgeostrophic, implying that the ageostrophic wind is northerly (i.e., negative). The effects of buoyancy and ageostrophic wind during the early evening transition are thus additive and set the stage for the initiation of a stronger inertia–gravity oscillation. This joint effect is described more fully in Shapiro and Fedorovich (2009).

Although our analysis is restricted to constant values of the remote geostrophic wind , a time-dependent function would not violate the 1D model restriction and could, in principle, be incorporated into a revised analysis. Had we made provision for such a function, (2.9) would include an inhomogeneous term . The two equations for Q derived in the next section would then also be inhomogeneous.

b. Uncoupling the governing equations

Although our governing equations can be cast as a single sixth-order partial differential equation (as in H67), a special linear transformation can be found that reduces the governing equations to two much simpler equations. Our approach parallels Gutman and Malbakhov’s (1964) analysis for 1D katabatic flows but is more involved in the present case because of our inclusion of a radiative damping term.

Taking sinα × (2.4) + k × (2.8) + l × (2.9), where k and l are constants, produces
e2.10
where . We seek k and l such that the sum of the undifferentiated terms in (2.10) is proportional to Q (, where is a constant of proportionality). In other words, if k and l (and ) are chosen such that
e2.11
then (2.10) reduces to . If then (2.11) is satisfied by
e2.12
Eliminating l and from (2.12) in favor of k yields the cubic equation
e2.13
where .
Equation (2.13) is solved in appendix B. One root is real, and two roots ( and ) form a complex conjugate pair: that is, , where the asterisk denotes complex conjugation. Corresponding to are real values of and l from (2.12), labeled and l1. The associated Q is
e2.14
For , (2.12) yields generally complex and , and we obtain a complex Q with the real and imaginary parts, respectively:
e2.15
e2.16
Since u, υ, and b are completely specified by and (shown below), it suffices to work with and (equivalently, we could work with and ). The governing equations thus reduce to the two uncoupled parabolic equations
e2.17
subject to boundary conditions based on (2.5), (2.6) and (2.7a):
e2.18a
e2.18b
Inverting (2.14)(2.16) yields expressions for u, υ, and b in terms of and :
e2.19
e2.20
and
e2.21
where
e2.22
In the next section, we obtain periodic solutions for and .

3. Periodic solutions

We seek diurnally periodic solutions for over a 24-h interval from sunrise, at , until the next sunrise, at . Periodicity is achieved by setting
e3.1
The solution for can be obtained from the 24-h solution through the relation , where , and M is the day number.

a. Nonexistence of periodic solutions for

If there is no radiative damping , then (B7) yields , and (2.13) yields , which violates the prerequisite condition for (2.13). We must therefore revisit the case (though there is no such difficulty for ). Setting and reduces (2.10) to the diffusion equation
e3.2
for . This is essentially one of the two uncoupled equations obtained by Gutman and Malbakhov (1964). The boundary conditions for are
e3.3
e3.4
Averaging (3.2) over the 24-h interval and using (3.1) yields
e3.5
where an overbar denotes a 24-h average. Equation (3.5) integrates to
e3.6
where A and B are constants. In view of (3.4), , and so for all z. However, since is specified at through and [the latter specified via (3.3) through choices for , N, f, , and ], only vanishes at for particular (artificial) arrangements of the governing parameters. Alternatively, if A is chosen so that (3.3) is satisfied, then (3.4) is violated. We conclude that diurnally periodic solutions are not possible for arbitrary governing parameters.

To shed light on this result, consider a simple initial value problem consisting of (3.2) with temporally constant K, , initial state , remote condition , and a surface buoyancy that is suddenly imposed at and thereafter maintained such that is a nonzero constant for t > 0. This problem is equivalent to 1D heat conduction in a semi-infinite solid whose boundary is suddenly subjected to a constant temperature perturbation. The analytical solution (Carslaw and Jaeger 1959; Shapiro and Fedorovich 2013), describes the continual vertical growth of a thermal boundary layer. The layer depth becomes infinite as . Similar behavior is found in 1D katabatic flows with provision for the Coriolis force but without radiative damping (Gutman and Malbakhov 1964; Lykosov and Gutman 1972; Egger 1985; Stiperski et al. 2007; Shapiro and Fedorovich 2008) and in 1D models of oceanic flows over sloping seabeds (Garrett 1991; MacCready and Rhines 1991, 1993; Garrett et al. 1993). We speculate that the tendency of the 1D boundary layer to deepen is also a feature of the differential equations for our low-level jet problem (with ); any tendency of the solutions to oscillate would be conflated with an inexorable deepening of the boundary layer, at least for the buoyancy and cross-slope-flow variables.

b. Periodic solutions for

In the 1D katabatic flow problem considered by Egger (1985), provision for radiative damping led to steady-state solutions that satisfied both the surface and remote conditions; continual slope-normal growth of thermal and cross-slope momentum boundary layers no longer occurred. The H67 equations also included a radiative damping term and did admit periodic solutions. We anticipate that this will also be the case for our analysis.

We seek the solution of (2.17) using separation of variables. With in the form
e3.7
(2.17) becomes
e3.8
from which follow
e3.9
e3.10
where is a constant. The general solutions of (3.9) and (3.10) are
e3.11
e3.12
where , , and are constants. To satisfy (2.18b), or must be zero, and we can write
e3.13
where the plus-or-minus sign choice ensures satisfaction of (2.18b).
Applying the periodicity condition in (3.12) yields
e3.14
where
e3.15
Writing the left-hand side of (3.14) as , where m is an integer, we find that and thus obtain a distinct for each m:
e3.16
The solution (3.13), generalized to include summation over m, appears as
e3.17
e3.18
e3.19
Last, we determine so that the slope conditions (2.18a) are satisfied. Unfortunately, since K is a function of time, a standard Fourier series approach will not lead to explicit formulas for . Instead, we must derive orthogonality relations for the functions. Toward that end, we consider the time derivatives of (3.18):
e3.20a
e3.20b
Adding × (3.20a) to × (3.20b) leads to
e3.21
which integrates over the 24-h interval to . For , the integral must vanish, and therefore and are orthogonal over the 24-h interval with respect to the weighting function K(t). For , the integral is evaluated as . We thus obtain the orthogonality relations
e3.22
where is the Kronecker delta. The solution procedure now parallels the usual Fourier approach. Applying (2.18a) for in (3.17) leads to
e3.23
Multiplying (3.23) by , integrating the resulting equation over the 24-h interval, and making use of (3.22) and (3.18), yields
e3.24

4. Examples: Piecewise constant , piecewise linear

a. Analytical solution

We prescribe K(t) and to be broadly representative of the diurnal cycle on clear days and simple enough to facilitate evaluation of (3.24). The diffusivity is assigned a constant daytime value from sunrise until sunset and a constant nighttime value from sunset until the next sunrise:
e4.1
The decrease in turbulent diffusivity from day to night is quantified through
e4.2
The surface buoyancy is specified as the piecewise linear (sawtooth) function
e4.3
where , is the buoyancy minimum, which occurs at sunrise, and is the buoyancy maximum, which occurs a few hours before sunset, at time . Sawtooth functions provide reasonable representations of the diurnal temperature/buoyancy cycle on clear days (Sanders 1975; Reicosky et al. 1989; Sadler and Schroll 1997). Schematics of K(t) and are given in Fig. 2.
Fig. 2.
Fig. 2.

Schematic of the diurnally varying diffusivity K(t) and surface buoyancy functions considered in section 4. The diffusivity varies as a step function with daytime value and nighttime value . The surface buoyancy varies as a sawtooth function that increases from a minimum at sunrise (t = 0) to a maximum at time , a few hours before sunset (time ).

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

With the above specifications, in (3.19) is evaluated as
e4.4
where
e4.5
and in (3.24) is evaluated as
e4.6
where
e4.7
e4.8

Use of these analytical expressions obviates the need for numerical integration in the evaluation of (3.17). The resulting solution is independent of the spatial and temporal resolution of the grid on which it is evaluated, but grid resolution should be considered when graphing or taking finite differences of the solution.

Although we have not proved that the form a complete set, we can show that the lower conditions (2.5) and (2.6) can be recovered to any desired accuracy by including a sufficiently large number of terms in the finite series approximation of (3.17) with given by (4.6). An example is shown in Fig. 3.

Fig. 3.
Fig. 3.

Time dependence of (left) b and (right) υ on the slope (z = 0) computed using a progressively larger number of terms (see legends) in the truncated series form of (3.17) for experiment BH. As more terms are included, the variables approach their appropriate slope distributions: no slip for υ, and sawtooth function for b, with and .

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

b. Reference experiment BH

In each experiment, the analytical solution was evaluated for one 24-h period at 10-min intervals with a grid spacing of . The series in (3.17) were truncated at . A reference experiment BH (B for Blackadar mechanism and H for Holton mechanism) was designed to provide a baseline description of fair weather warm-season diurnal cycles over the sloping portion of the southern Great Plains (e.g., in western Oklahoma), where both Blackadar and Holton mechanisms are present. The parameters in BH are also the default parameters in all of the other experiments.

In BH we take and , values that are appropriate for western Oklahoma (36.4°N, 99.4°W). The slope is also close to the 1/400 slope used in H67. We take based on the approximate times of sunrise [0630 central standard time (CST)] and sunset (1830 CST) in late September at the chosen location (U.S. Naval Observatory 2016). From an observed diurnal range of surface geostrophic winds (described in appendix A), we calculate a diurnal range of surface buoyancy of . We split this range equally between a peak of at (3 h before sunset) and a sunrise minimum of but will consider unequal magnitudes of and in the sensitivity experiments. At sunset the diffusivity drops from to , values that are within the range of published estimates of K for the daytime convective atmospheric boundary layer and the nocturnal stable boundary layer, respectively, cited in Shapiro and Fedorovich (2010). We adopt typical free-atmosphere values for N (=0.01 s−1) and (=10 m s−1). The radiative damping parameter is the same as in Egger (1985). These parameters are summarized in Table 1.

Table 1.

Parameters in reference experiment BH. Times are in hours after sunrise. Sunrise is at ~0630 CST.

Table 1.

Figures 48 show that BH provides a qualitatively reasonable depiction of the diurnal cycle of the wind and buoyancy in the atmospheric boundary layer, including the development and breakup of both a nocturnal stable boundary layer and a low-level jet. The low-level wind speeds shown in Fig. 4 are subgeostrophic and relatively steady during most of the afternoon (15 000 s until sunset) but thereafter increase rapidly in strength and become supergeostrophic. After sunset, the height of the wind maximum descends until ~65 000 s and thereafter remains at ~480 m for several hours. The peak speed of ~21 m s−1 puts the modeled jet (barely) into the strongest category (category 3) of the Bonner (1968), Whiteman et al. (1997), and Song et al. (2005) classification systems. The peak speed occurs at t = ~73 800 s, 3 h after midnight (CST). Figure 5 shows the evolution of υ(z) from a broad relatively weak late-afternoon profile to the graceful jetlike shape characteristic of LLJs. At the time of the speed maximum (roughly the time of curve d in Fig. 5), the peak speed is almost entirely associated with the υ wind component5 (this is also evident from the 500-m hodograph curve in Fig. 6). Figure 5 also shows the postsunset development of upslope winds, with a peak magnitude of ~10 m s−1 reached approximately 3 h after sunset. The hodographs in Fig. 6 show an anticyclonic turning of the wind vectors from sunset until sunrise. During this period the hodographs are approximately semicircles reminiscent of IOs, although the curves become increasingly asymmetrical at lower levels. The shortness of the hodograph segments between the times of peak surface buoyancy and sunset further indicate that the late-afternoon winds are in a near-steady state. Figures 7 and 8 show the development of a shallow nocturnal stable boundary layer, with negative buoyancies extending up to ~200 m at the time of the speed maximum. Shortly after sunrise, mixing spreads the negatively buoyant air upward and dilutes the cold layer. By late morning, the boundary layer is dominated by positive buoyancy.

Fig. 4.
Fig. 4.

Wind speed (m s−1) as a function of height and time over a 24-h period in experiment BH. Sunrise is at t = 0 s. Sunset is at tset = 43 200 s. The surface buoyancy peaks at tmax = 32 400 s.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

Fig. 5.
Fig. 5.

Postsunset evolution of (left) and (right) profiles in experiment BH. Curves are shown at 1.5-h intervals starting from sunset. Labeled curves correspond to times (line a), (line b), (line c), and (line d).

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

Fig. 6.
Fig. 6.

Evolution of the wind hodographs in experiment BH over a 24-h period at different heights. Curves are plotted (left) at low levels in 100-m increments up to the jet maximum and (right) at and above the jet maximum in 500-m increments. The times of sunset, sunrise, and peak surface buoyancy are indicated on the hodographs. The blue diamond indicates the geostrophic wind.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

Fig. 7.
Fig. 7.

As in Fig. 4, but for b (m s−2).

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

Fig. 8.
Fig. 8.

As in Fig. 5, but for b(z) profiles.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

c. Pure Blackadar- and Holton-mechanism experiments

Experiments were conducted to gauge the separate effects of the Blackadar and Holton mechanisms. The Blackadar mechanism was simulated by turning off the Holton mechanism (setting , which decouples the wind and buoyancy fields), and the Holton mechanism was simulated by turning off the Blackadar mechanism (setting ).6 We focus on the peak υ wind component (which, as noted, differed from the wind speed by less than ), and the height and time at which this peak υ is attained. The results are summarized in Table 2. In Blackadar-mechanism experiment B, is weaker than in BH. In a second Blackadar-mechanism experiment , a 50% increase in yields a corresponding ~50% increase in over that in B, with essentially no change in or . Holton-mechanism experiments were conducted using (weak-mixing experiment ), (moderate-mixing experiment H), and (strong-mixing experiment ). The corresponding values were remarkably similar to each other and were significantly less than in B and BH. However, despite the similarity of the values, varied by an order of magnitude, from 320 m in to 3160 m in . In a final Holton-mechanism experiment with moderate mixing (, as in H) and larger surface buoyancy (, 50% larger than in H), only increased by (~10%) over in H. In short, the Holton mechanism produced wind maxima that were notably weaker than those produced by the Blackadar mechanism. We also note that the in the Blackadar-mechanism experiments exceeded those in the Holton-mechanism experiments by more than 3 h and exceeded in BH by 0.5 h. The relative strengths and timings of the wind maxima in these experiments are in qualitative agreement with the results of Du and Rotunno (2014, Fig. 5).

Table 2.

Characteristics of the modeled LLJ in a set of Blackadar-mechanism and Holton-mechanism experiments. Parameters are as in experiment BH (Table 1), except where noted. The peak υ () is found at height at time . Times are in hours after sunrise. Sunrise is at ~0630 CST.

Table 2.

d. Exploring the parameter space

We now examine the sensitivity of the solution to the governing parameters. Table 3 summarizes experiments in which one parameter is varied and the others are fixed at their reference values. The experiment names are of the form , where represents the varied parameter, and the plus-or-minus symbol indicates that the magnitude of that parameter is larger (+) or smaller (−) than in BH.

Table 3.

Sensitivity of modeled LLJ characteristics to the governing parameters. In each experiment, one parameter is varied from its value in experiment BH (Table 1). Times are in hours after sunrise. Sunrise is at ~0630 CST.

Table 3.

The evolution of the jet heights in these experiments was remarkably similar to that in BH: the wind maxima descended rapidly after sunset, leveled off to a near-constant height by the time of peak jet intensity, and stayed close to that height for the rest of the night. In most experiments, was in a narrow range between 440 and 520 m. A notable exception was in the nighttime diffusivity experiments, where a large diffusivity (in ) yielded a large (900 m), and a small diffusivity (in ) yielded a small (240 m). In light of these results, the tendencies noted in Steeneveld et al. (2008), Storm et al. (2009), and Werth et al. (2011) for their regional model-simulated jets to be too deep suggests that the parameterized mixing in those simulations may have been too aggressive at night.

The time was relatively insensitive to most of the parameters, generally being well within an hour of in BH. An exception was, not surprisingly, in experiments and , where shifting later or earlier by 2 h shifted later or earlier by ~2 h. Also, in and , decreased with increasing latitude, in qualitative accord with the IO and inertia–gravity oscillation theories. More specifically, the ratio of the inertial period in to that in (~0.753) was close to the ratio of the time intervals in those experiments (~0.786). An even better agreement with the latter ratio was obtained using the ratio of the inertia–gravity oscillation period (Shapiro and Fedorovich 2009) in to that in (~0.772).

There are many interesting aspects to the jet strength sensitivities:

  1. The value of was very sensitive to (cf. and ). A strong dependence of jet strength on geostrophic wind is a hallmark of the IO theory (implied in Fig. 10 of B57 and explicit in the analytical solution of Shapiro and Fedorovich 2010). The importance of a strong mean background wind (or geostrophic wind) to jet strength has also been recognized by Wexler (1961), Arritt et al. (1997), Zhong et al. (1996), Jiang et al. (2007), and Parish and Oolman (2010).
  2. Larger values of were also associated with larger values of . Recall that in the pure Holton-mechanism experiments, increasing to only produced a increase in ( vs H). Since the same increase in now increases by ( vs BH), we see that the interaction of the Holton mechanism with the Blackadar mechanism is cooperative, at least with respect to the role of daytime heating.
  3. Although the solution is less sensitive to than it is to , increasing the magnitude of (greater nighttime cooling) actually decreases ( vs ). This is consistent with Parish and Oolman’s (2010) finding that nocturnal cooling is inimical to the LLJ. It should be borne in mind, however, that in real atmospheric boundary layers, nocturnal cooling also affects the level of turbulence, and it is somewhat artificial to vary the diffusivities independently of the buoyancy parameters.
  4. Consistent with the B57BB57 theory, the jet winds intensify with an increasing day-to-night ratio of diffusivities (decreasing ), associated with either an increase in the daytime diffusivity (in ) or decrease in the nighttime diffusivity (in ).
  5. As in H67 and Shapiro and Fedorovich (2009), the jet strengthens as the stratification weakens ( vs ). In further experiments (Fig. 9), unphysically large values of are obtained in the (unrealistic) case of a neutrally stratified free atmosphere .
  6. There is a slight increase in as the lag between the time of peak surface buoyancy and sunset is decreased, either through increasing (in ) or decreasing (in ).
  7. Although provision for radiative damping was essential to obtain purely periodic solutions (section 3), these periodic solutions exhibit little sensitivity to the actual value of the damping parameter; varying by an order of magnitude changed by less than (cf. and ). A lack of sensitivity to was also reported in Egger’s (1985) katabatic flow study.
  8. Further experiments (Fig. 9) reveal a strong dependence of on slope angle. For a realistic free-atmosphere stratification , increases from (its value in B) at to a local maximum of ~21.6 m s−1 for between and . Similar optimal values of were obtained in Shapiro and Fedorovich (2009). The optimal value is in qualitative agreement with climatological studies (Bonner 1968; Walters et al. 2008), though with the preferred longitudes of LLJ occurrence in those studies associated with slope angles somewhat lower (terrain farther east) than indicated here. It should be kept in mind, however, that while we fixed at its value in BH, the summer-mean geostrophic wind—which affects jet strength—undergoes significant spatial variation across the Great Plains (Jiang et al. 2007; Pu and Dickinson 2014; Du and Rotunno 2014).
Fig. 9.
Fig. 9.

Peak υ as a function of slope angle for (solid curve) and (dashed curve). Other parameters are as in experiment BH.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

In a final experiment, we increased the values of two parameters that individually had a large impact on jet strength. Setting and produced an intense jet, with (Fig. 10). LLJ winds of this magnitude, though infrequent, do occur over the southern Great Plains (Bonner et al. 1968, their Fig. 2; Whiteman et al. 1997, their Fig. 11; Song et al. 2005, their Fig. 2).

Fig. 10.
Fig. 10.

Wind speed (m s−1) as a function of height and time over a 24-h period for an intense jet associated with strong surface buoyancy and a strong geostrophic wind . Other parameters are as in experiment BH. Sunrise is at t = 0 s. Sunset is at tset = 43 200 s. The surface buoyancy peaks at tmax = 32 400 s.

Citation: Journal of the Atmospheric Sciences 73, 8; 10.1175/JAS-D-15-0307.1

5. Summary and future work

Our theory for the Great Plains nocturnal LLJ combines the Blackadar mechanism for IOs arising from a sudden decrease of turbulent mixing at sunset, with the Holton mechanism for an oscillation arising from the diurnal heating/cooling of a sloping surface. Periodic solutions of the 1D Boussinesq equations of motion and thermal energy for a stably stratified fluid were obtained analytically for the combined mechanisms, with the separate mechanisms studied as particular cases. In the context of our 1D model, provision for a radiative damping term (Newtonian heating/cooling) in the thermal energy equation was essential for periodic solutions to exist. The model jets exhibited little sensitivity to the actual value of the damping parameter.

A reference experiment in which both mechanisms were operating provided a baseline description of the fair-weather warm-season diurnal cycle over the sloping portion of the southern Great Plains, including the emergence of a strong LLJ in the nocturnal phase of the diurnal cycle. The strength, timing, and vertical structure of the analytical LLJ were in good qualitative agreement with typical LLJ observations over the southern Great Plains.

Experiments were conducted with the Holton and Blackadar mechanisms operating separately. The Holton mechanism tended to produce relatively weak jets. The Blackadar mechanism produced jets that were roughly 50% stronger than in the Holton-mechanism experiments. In a combined Holton- and Blackadar-mechanism experiment (the reference experiment), the peak winds increased by ~25% over those in the Blackadar-mechanism experiment. These strongly supergeostrophic winds (marginally) exceeded the theoretical 100% supergeostrophic limit for the Blackadar mechanism.

Sensitivity tests with the 11 governing parameters showed that the height of the wind maxima evolved in remarkably similar fashion for all parameter combinations: a rapid descent after sunset, followed by a leveling off to a near-constant value through much of the night. This equilibrium height was relatively insensitive to all parameters except the nighttime diffusivity, larger values of which were associated with higher jets. Jet wind speeds increased with increasing geostrophic wind, increasing maximum surface buoyancy, and increasing day-to-night ratio of the diffusivities. The peak winds were maximized for small slope angles characteristic of the Great Plains, although with a small westward bias relative to climatologies. Peak speeds decreased with increasing free-atmosphere stratification and increasing magnitude of minimum slope buoyancy (strength of nocturnal cooling).

We plan to test the unified LLJ model predictions against observations from the Plains Elevated Convection at Night (PECAN) field project, run from 1 June–15 July 2015. The PECAN campaign included the deployment of fixed and mobile Doppler radars, Doppler lidars, radiosonde systems, experimental profiling sensors, and research aircraft over a domain extending from northern Oklahoma through south-central Nebraska. Wind and thermodynamic profiles through many LLJs were obtained during the course of the project.

Finally, we note several possible extensions of our 1D theory. First, as discussed in section 2a, a time-dependent free-atmosphere geostrophic wind can, in principle, be incorporated into a revised 1D analysis. In addition, it may be possible to make provision for a free-atmosphere geostrophic wind that varies linearly with height. This would extend the theory to cases where background (nonterrain associated) synoptic-scale baroclinicity is important. It also appears likely that the theory can be revised to take surface roughness into account. To accomplish this, the surface boundary conditions would need to be reformulated as normal gradient conditions using bulk parameterizations for the momentum and heat fluxes (Stull 1988), although to keep the analysis tractable these conditions would have to be linearized. The surface drag coefficient and surface heat exchange coefficient would then appear as input parameters.

Acknowledgments

We thank the anonymous referees and Joshua Gebauer for their constructive comments and suggestions for extending the theory. This research was supported by the National Science Foundation under Grant AGS-1359698.

APPENDIX A

Geostrophic Wind, Thermal Wind, and Buoyancy in a Sloping Atmospheric Boundary Layer

In our 1D theory of the sloping atmospheric boundary layer, all variables are independent of the cross-slope coordinate, and the geostrophic wind blows parallel to the terrain isoheights. We assume that approaches a constant value far above the slope but do not otherwise specify its spatial (height) or temporal dependencies. As we now show, the evolving profile can be simply related to the buoyancy field.

By definition, is related to the true horizontal component of the perturbation pressure gradient by
ea.1
where we have approximated the true Coriolis parameter by (see section 2a). With the help of Fig. 1, we find that
ea.2
Application of (2.3) in (A.2) yields
ea.3
where is a suitable approximation for the Great Plains. Since b is independent of x, and is independent of x and z (see section 2a), (A.3) shows that —and thus —are, at most, functions of z and t. Furthermore, since is independent of x and z, and far above the slope,
ea.4
Applying (A.4) in (A.3) and substituting the resulting expression into (A.1) then yields
ea.5

To guide the specification of the surface buoyancy parameters in our experiments, we used (A.5) together with data from Sangster’s (1967) analysis of the surface geostrophic wind (which was primarily southerly) for June 1966 over a line from Oklahoma City (φ = 35.5°N) to Amarillo, Texas. Using Sangster’s estimates of 18 knots (1 kt = 0.51 m s−1)A1 for the mean diurnal range of the southerly surface geostrophic wind and 1/530 for the average slope of that region, (A.5) yields a diurnal range of for the surface buoyancy changes.

To get an equation for the thermal wind , take the z* derivative of (A.5) and use . This latter relation follows from the elimination of between the equations for the x* and z* derivatives of , and . We obtain
ea.6
In view of (A.5) and (A.6), one may consider the thermal structure of this sloping boundary layer in terms of the buoyancy, geostrophic wind, or thermal wind.

APPENDIX B

Solution of (2.13)

We solve (2.13) using standard formulas for cubic equations (e.g., Abramowitz and Stegun 1964, pg. 17). Write (2.13) as
eb.1
eb.2
and define
eb.3
eb.4
eb.5
and
eb.6
The roots of (2.13) can then be written as
eb.7
eb.8
eb.9
For a typical free-atmosphere value of N , Great Plains slope , and latitudes spanning the Great Plains across North America , ranges from to (and obviously ). For damping parameters in a range from 0.1 to 1 day−1 that includes values estimated by Kuo (1973) and the value 0.2 day−1 adopted by Egger (1985), ranges from to , so by one to four orders of magnitude. In view of that inequality, we see from (B.5) that , while a comparison of (B.5) with (B.4) shows that . Thus, is positive, while is negative. Taking the 1/3 power of those factors then produces a real and positive and a real and negative , which we write asB1
eb.10

Since and are real, (B.7)(B.9) indicate that is real, while and form a complex conjugate pair.

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