## 1. Introduction

The nocturnal low-level jet over the Great Plains of the United States has been a noted feature of its warm-season climatology since the 1950s [see the comprehensive literature review in Shapiro et al. (2016), hereafter SFR]. Observations of the horizontal wind vector

In addition to the qualitative difference in the turning of the surface hodograph between models with semislip and no-slip lower-boundary conditions, there are quantitative differences in the wind profiles above the surface. To compare the present results with those of SFR, and to explore efficiently the parameter space, we will present most of the results in a nondimensional framework in which the SFR no-slip solutions emerge in the limit as the surface-drag coefficient becomes large.

In section 2, the SFR model is summarized and extended to have a semislip lower-boundary condition. As only numerical solutions are available to us, we first check the present numerical model results against the analytical solutions reported in SFR. A direct comparison of the latter against a semislip solution for the same set of external parameters shows counterclockwise turning of the surface wind is a result of the semislip condition. Quantitative differences noted in the hodographs above the surface motivate the dimensional analysis of section 3, where several important nondimensional parameters are identified. Results for the maximum jet strength, its height, and its time of occurrence are tabulated as a function of the most important nondimensional input parameters of the present simple model. A summary is given in section 4.

## 2. The model

### a. Governing equations

*b*is the buoyancy,

*N*is the Brunt–Väisälä frequency,

*f*is the Coriolis parameter,

*δ*is the radiative-damping coefficient, and

*α*represents the rotation of the coordinate system such that the

*x*(east–west) axis aligns with the sloping plain, which is assumed to be invariant in the

*y*(north–south) direction (see Fig. 1 of SFR). The upper-boundary conditions are

*d*is within the surface layer. In SFR, the no-slip conditionis used instead of (6); we will show below that the solutions using (6) approach the no-slip solutions as

The different lower-boundary conditions, (6) and (7), warrant a few words of explanation. In the case of (7) (no slip), the eddy viscosity defines the shear stress (4) at all heights, including *d* is several meters above the actual lower boundary at

Following SFR, we adopt a square wave for the time variation of ^{1}

The SFR model contains the two most popular mechanisms proposed to explain the low-level jet in particular, and the diurnal wind oscillation in general, over the Great Plains. With

For the numerical solution of (1)–(6), the *z* derivatives are discretized using centered differences with

### b. Numerical solutions

To test the present numerical solution, we use (7) and set the input parameters as specified in Table 1 of SFR (^{2}

Figure 1b shows the hodographs from numerical solutions using the semislip condition (6) with *υ*). However, at the 10-m level, the hodograph behaviors of the no-slip and semislip cases are qualitatively different. In the no-slip case at

Figure 2 shows the variation with height of the hodograph in the semislip case as the counterclockwise turning changes to clockwise turning in the lowest 100 m. The length scale of the variation with height is roughly the nighttime Ekman depth *f* (not shown).

The profiles in Fig. 3 illustrate the sunset transition *υ* *υ* increasing until roughly *u* and *υ* are qualitatively the same at each level. In the semislip case (Fig. 3b), the profile evolution after sunset in the layer extending from roughly *υ* immediately to roughly constant value, while

Figure 4 shows the budget terms for the no-slip case at *u* equation abruptly decreases, and *u* decelerates since

In the semislip case at *u* in contrast with the no-slip case (cf. Fig. 4a). Similarly, Fig. 6b shows a sharp deceleration of *υ* at sunset, in contrast with the no-slip case (cf. Fig. 4b). During the evening hours, Fig. 6a indicates the buoyancy term reverses sign and leads to a weak increase of *u*, while Fig. 6b shows that *υ* is nearly constant, consistent with Fig. 1b. At sunrise, the increase in both stress components in the interior leads to a recoupling of the interior and surface stresses and the restoration of the daytime near-Ekman balance.

In contrast with the no-slip case, the balance of terms in the semislip case is qualitatively different at higher levels. Figure 7 shows that at sunset, the acceleration due to the change in the stress profile is qualitatively consistent with the no-slip case at either level (Figs. 4, 5). And, as shown in Fig. 1b, the hodograph at

The sunset transition for the semislip case is illustrated in detail in Fig. 8. Just before sunset *z* over the lowest 100 m. However, at sunset *υ* decrease, which, in turn, leads to a smoother variation of the stress profiles (blue dashed lines) with height.

Before leaving this section, we note that both the Blackadar and Holton mechanisms are needed to produce something resembling a counterclockwise-rotating hodograph at the surface. [The pure Blackadar (no slope with time-varying eddy viscosity) solution at the surface goes to a single point at night since there is no katabatic wind, while the pure Holton (slope with eddy viscosity constant in time) solution has clockwise turning at all levels since there is never a discontinuity in the vertical stress profile.] This result is consistent with Bluestein et al. (2018), in which both the observational analysis (their Fig. 5) and the time-averaged full-physics model simulations (their Fig. 12) indicate that counterclockwise turning with time of the surface hodographs only occurred on the gently sloping terrain roughly west of a line from Oklahoma City, Oklahoma, to Wichita, Kansas.

## 3. Dimensional analysis and parameter dependence

In their section 4d, SFR describe the main parameter dependence of their analytical solutions in terms of the maximum strength of the low-level jet, its height, and its time of occurrence. With the semislip condition (6), *d* enter the list of external parameters that could influence the solution. Given the already-long list of external parameters, we are motivated to seek a reduced set of nondimensional parameters that most influence the solution.

### a. Dimensional analysis

*α*and

*δ*has very little influence on the solutions, and thus, we neglect it from this point forward. Furthermore, by meteorological convention,

*d*and

*δ*, (8) indicates there are eight dimensional parameters. Since our equations involve two dimensions (length and time), dimensional analysis indicates that the eight dimensional parameters in (8) may be reduced to six dimensionless parameters; these six plus the dimensionless parameters

*α*and

*α*and

### b. Nondimensional governing equations

*ϕ*is the latitude. Since

*ω*is a constant, the scale factor for time is the same for all experiments. In the computations to follow, we will keep

### c. Solution dependence on the nondimensional external parameters

Each entry in Table 1 shows numerical solutions for ^{3} to those in Table 1 at

Each entry shows *ϕ* (°),

A few general features of the solution for *ϕ* for any value of *ϕ* dependence of the amplitude and phase and Shapiro and Fedorovich [2010, their (7)] for the *ϕ* dependence of the vertical scale}. An increase in ^{4} The effect of the semislip condition (finite

Turning to the case with *ϕ*. As in the case with

As in Table 1, but for the case

Keeping *α*, ^{−1}, ^{−2} and ^{−2}; we infer that ^{−2} and ^{−2} and find ^{−1}. Since ^{−1} ^{−1}, which is consistent with SFR’s result. The semislip results shown in Table 3 show the same general tendencies with respect to the no-slip cases, as seen in Table 1 (

As in Table 1, but for the case

The present calculations reproduce the dependence of *α*, as described in Fig. 9 of SFR (peaking at *α*). Finally, increasing

## 4. Conclusions

Surface hodographs over the Great Plains exhibit anomalous counterclockwise turning over the diurnal cycle, which is opposite of the expected clockwise turning based on higher-level observations and theory; mesoscale-model forecasts over the continental United States from the warm season exhibit the same behavior (Bluestein et al. 2018). Analysis of the mesoscale-model forecasts in Bluestein et al. (2018) revealed that at sunset and sunrise, sharp vertical gradients in the stress profiles occur due to the continuous action of the surface stress and the decay and growth of interior stress due to the diurnal cycle of turbulence. The present paper extends the simple one-dimensional model of Shapiro et al. (2016) to include a semislip (instead of a no-slip) lower-boundary condition and is able to qualitatively reproduce the anomalous counterclockwise turning over the diurnal cycle in the surface layer while retaining the expected clockwise turning at higher levels.

The semislip lower boundary condition, in addition to its effect on the direction of turning in the surface layer, also produces quantitative changes in the solutions at all levels. As the number of potentially important input parameters in the extended SFR model is large, we reduced that number by looking for the most important dependence of the solution on several dimensionless input parameters. In summary, the nondimensional parameters that influence the no-slip or semislip solutions to the SFR model are given by (19). A novel result is the identification of the control parameter *υ* is supergeostrophic include the nature of the underlying vegetation and the wetness of the soil, which could affect the buoyancy. Since low-level vertical shear may be increased as *υ* increases in speed, one would expect the intensity of convective storms that might form would also increase as the soil wetness decreases or if vegetation becomes sparser.

## Acknowledgments

H. Bluestein is supported by NSF Grant AGS-1560945. We thank Prof. Yu Du (Sun Yat-sen University, China) for his comments on the first draft of this paper.

## REFERENCES

Blackadar, A. K., 1957: Boundary layer wind maxima and their significance for the growth of nocturnal inversions.

,*Bull. Amer. Meteor. Soc.***38**, 283–290.Bluestein, H. B., G. S. Romine, R. Rotunno, D. W. Reif, and C. C. Weiss, 2018: On the anomalous counterclockwise turning of the surface wind with time in the plains of the United States.

,*Mon. Wea. Rev.***146**, 467–484, https://doi.org/10.1175/MWR-D-17-0297.1.Bonner, W. D., and J. Paegle, 1970: Diurnal variations in boundary layer winds over the south-central United States in summer.

,*Mon. Wea. Rev.***98**, 735–744, https://doi.org/10.1175/1520-0493(1970)098<0735:DVIBLW>2.3.CO;2.Du, Y., and R. Rotunno, 2014: A simple analytical model of the nocturnal low-level jet over the Great Plains of the United States.

,*J. Atmos. Sci.***71**, 3674–3683, https://doi.org/10.1175/JAS-D-14-0060.1.Glickman, T., Ed., 2000:

*Glossary of Meteorology.*2nd ed. Amer. Meteor. Soc., 855 pp., http://glossary.ametsoc.org/.Holton, J. R., 1967: The diurnal boundary layer wind oscillation over sloping terrain.

,*Tellus***19A**, 199–205, https://doi.org/10.1111/j.2153-3490.1967.tb01473.x.Lumley, J. L., and H. A. Panofsky, 1964:

*The Structure of Atmospheric Turbulence.*Wiley, 239 pp.Rotunno, R., 1983: On the linear theory of the land and sea breeze.

,*J. Atmos. Sci.***40**, 1999–2009, https://doi.org/10.1175/1520-0469(1983)040<1999:OTLTOT>2.0.CO;2.Shapiro, A., and E. Fedorovich, 2010: Analytical description of a nocturnal low-level jet.

,*Quart. J. Roy. Meteor. Soc.***136**, 1255–1262, https://doi.org/10.1002/qj.628.Shapiro, A., E. Fedorovich, and S. Rahimi, 2016: A unified theory for the Great Plains nocturnal low-level jet.

,*J. Atmos. Sci.***73**, 3037–3057, https://doi.org/10.1175/JAS-D-15-0307.1.

^{1}

Otherwise, the ratio of the periods of surface temperature increase to decrease would present yet another input parameter.

^{2}

Note that the noon icon in Fig. 1 denotes

^{3}

With the SFR value of

^{4}

Note that these dimensional values will depend on