## Structure of the convective mixed layer

The goal of this research is to describe the mean wind speed (*M*) and potential temperature (*θ*) profiles within the bottom fifth of the convective boundary layer. Figure 1a identifies layers in the convective mixed layer (ML), using wind speed as an example.

Starting from the top, the free atmosphere is where surface friction is not felt, and the actual wind speed is nearly equal to the geostrophic or gradient wind speed *G.* This layer is above the boundary layer. Next is the entrainment zone, a region of subadiabatic temperature profiles, overshooting thermals, intermittent turbulence, and wind shear (Deardorff et al. 1980).

Further down is a region where wind speed and direction are nearly uniform with height, *z.* This is the uniform layer (UL), where the wind speed is *M*_{UL} and the potential temperature is *θ*_{UL}. The wind is subgeostrophic because thermals communicate surface drag information via nonlocal transport. Traditionally, the term“mixed layer” is reserved for the whole turbulent region between the surface and the average ML top, *z*_{i}; hence, we distinguish between the ML and the UL to avoid confusion.

Between the surface and the UL is a radix layer (RxL) of depth *z*_{RM}. Like the roots of a tree, the radix (Latin for “root”) layer is where smaller plumes merge into large-diameter ML thermals. It contains the classic surface layer as a subdomain (Fig 1a), as will be explained below. Winds in this layer are zero near the ground, and smoothly increase to become tangent to *M*_{UL} at the RxL top.

Within the bottom of the RxL is the classic surface layer (SL), the nearly constant flux region where Monin–Obukhov similarity theory applies. The SL is the region where predominantly mechanically generated turbulence within the wall shear flow produces a nearly logarithmic wind profile. Again, we distinguish between the RxL and the SL to avoid confusion because we retain the traditional definition that the SL is that region where Monin–Obukhov similarity applies. Although not the subject of this paper, Zilitinkevich (1994) further subdivides the traditional surface layer into (from bottom up) a mechanical turbulence layer, an alternative turbulence layer, and a free convection layer, where the top two layers combine to form his convective and mechanical layer.

Radix-layer definitions also can be made for potential temperature, as sketched in Fig. 1b. The depth of the SL and RxL for temperature can be different than for momentum. For potential temperature, we denote these depths as *z*_{Sθ} and *z*_{Rθ}, respectively.

Based on these traditional and new definitions, we see that the ML (order of a couple kilometers thick) contains the RxL (hundreds of meters thick) as a subdomain, and the RxL contains the SL (tens of meters thick) as a subdomain. Hence, there is a superposition of layers in the bottom fifth of the convective boundary layer. This is an alternative view to the three-layer model of Garratt et al. (1982), where their surface layer merges directly into the uniform layer.

Because the proposed profile equations will be expressed as a dimensionless similarity theory, section 2 starts by reviewing similarity hypotheses for the bottom of the boundary layer. In section 3 similarity equations for the radix layer are proposed and calibrated against field data. In section 4 there is a first attempt to identify parameters that control the radix-layer depth. The value of similarity wind profiles over similarity shear profiles is discussed in section 5.

## Similarity hypotheses

### Surface layer

Most SL similarity theories are based on the following premises:

flux is approximately uniform with height (constant flux ± 10%);

turbulence consist of “small eddies,” causing local transport;

turbulence is predominantly generated mechanically by shear flow near the ground, with minor modifications for static stability; and

feedback exists between the mean flow and the dominant eddies.

*z*

_{i}should not be relevant. The third premise implies that surface roughness length

*z*

_{0}is important. The fourth premise is discussed later.

Most classic SL similarity equations are strongly dependent on the surface but are virtually independent of factors higher in the ML. Typically lacking is dependence on ML depth, temperature within the UL, winds within the UL, and geostrophic wind speed. For this reason, we cannot expect the SL equations to merge smoothly into the UL because no information about the UL is included in those equations. Panofsky (1978) points out that convective-matching-layer and free-convection-layer formulations (Priestley 1955; Kaimal et al. 1976) fail near the bottom of the UL, where the shear and potential-temperature gradient approach zero.

This situation is illustrated in Fig. 2, where the abscissa has been normalized according to the Businger (1971)–Dyer (1974) similarity theory. In this normalization, all the data will collapse to a single curve regardless of static stability in those regions where SL similarity theory is valid. While Fig. 2a shows that SL similarity works well in the bottom 40 m of the ML for the Minnesota dataset (to be described in more detail later), Fig. 2b shows that SL theory is less successful higher in the RxL and in the UL. Namely, the classic surface layer does not extend up to the base of the UL for these data, resulting in data points that do not collapse onto a single curve.

*z*/

*L*is defined as

*L*is the Obukhov length (negative for a statically unstable BL),

*k*is von Kármán’s constant (approximately 0.4), and

*u*∗ is the friction velocity. This expression can be integrated to yield the mean wind profile,

*z*

_{0}is the aerodynamic roughness length and

*z*′ is a dummy of integration. The function

*ϕ*

_{m}varies away from unity as the static stability varies from neutral; however, the functional form of

*ϕ*

_{m}is not known from first principles.

*ϕ*

_{m}have been suggested for the statically unstable surface layer. These forms can be broadly classified into two groups. One group is the modified logarithmic:

*α, β, γ,*and

*δ*are arbitrary empirical constants that usually differ from author to author (e.g., Businger et al. 1971; Dyer 1974; Dyer and Bradley 1982; Högström 1988; Frenzen and Vogel 1992).

*a, b,*and

*c*are arbitrary constants that also differ from author to author (e.g., Swinbank 1968; Zilitinkevich and Chalikov 1968; Foken and Skeib 1983; Kader and Yaglom 1990). Both broad groups are a function of

*z*/

*L,*which implies that both tacitly assume that the dominant generation of turbulence is mechanical, formed by wall shear (Stull 1997).

Some SL similarity equations have been proposed to apply higher into the ML. For example, Fig. 3 compares the SL similarity relationships proposed by Kader and Perepelkin (1989) and Sorbjan (1986, hereafter ZS) to the Minnesota data of run 7C1. Also plotted is an extension of the Businger–Dyer profile equations above the surface layer for comparison. Of the three relationships plotted in Fig. 3, the proposal by Sorbjan appears to work the best above the top of the SL; however, even it has substantial errors as shown in appendix C for all Minnesota runs. The Sorbjan relationship will be discussed in more detail in section 5. The difficulties of most SL theories at heights above the top of the traditional SL provide motivation for the definition of an RxL and suggest that classic profile-matching methods (e.g., Rossby similarity) linking SL profiles with UL profiles are probably not justified.

In closing this brief review of SL similarity, we return to the fourth premise listed at the start of this section. That feedback premise has the following interpretation for flow very near the bottom boundary. Turbulence transports momentum, momentum-flux divergence alters the mean-wind profile, and shear in the mean-wind profile generates small-eddy turbulence. The feedback loop is closed, at least for shear-driven surface layers. This is a fundamental, but infrequently discussed, premise underlying SL similarity theory.

### Radix layer

In the nearly free-convective ML, such feedback is broken. Turbulence still transports momentum, and momentum-flux divergence alters the mean wind profile. However, the mean wind profile does *not* generate large-eddy turbulence. Instead, surface heating generates the large, coherent, thermal structures. Because the thermal structures have a length scale proportional to the ML depth, we can infer that the wind profile in the radix layer must also be a function of *z*_{i}.

According to convective transport theory (CTT) for surface fluxes (Stull 1994), the UL is independent of the surface layer. Namely, the UL interacts with the surface directly via the large thermal structures, making it independent of the small-eddy SL and of the roughness length (Fig. 4, after Stull 1994). However, the SL in that previous paper, here better identified as the radix layer, must depend on both roughness length and UL characteristics if it is to become tangent to the UL profile. This also suggests that the RxL is a function of *z*_{i}.

CTT identifies a buoyancy velocity scale, *w*_{B} = [(*g*/*T*_{υ})Δ*θ*_{υs}*z*_{i}]^{1/2}, that is valid for free convection. It is proportional to the Deardorff velocity, *w*∗ = *g*/*T*_{υ})*w*′*θ*^{′}_{υs}*z*_{i}]^{1/3},*g* is gravitational acceleration, subscript *s* denotes a surface value, *T*_{υ} is average virtual temperature, Δ*θ*_{υs} is the virtual potential temperature difference between the surface skin and UL, and *w*′*θ*^{′}_{υs}

## Radix-layer wind and temperature profiles

### Profile equations in the radix layer

For the purpose of the similarity analysis in the RxL/ML system, we hypothesize here that RxL depths (*z*_{RM} and *z*_{Rθ}) are the relevant height scales and that the winds and temperatures in the UL (*M*_{UL} and *θ*_{UL}) are the relevant velocity and temperature scales. After testing several candidate functions, the following empirical profile equations were selected for further study based on their close agreement with the data.

*A*

_{1}and

*A*

_{2}are empirical constants and

*θ*

_{0}is the potential temperature of the air near the surface.

Note that both sets of equations above satisfy the desired constraint that the partial derivative of the parameters (*M* or *θ*) with respect to *z* is zero at the top of the radix layer, *z*_{R}. Both sets of the equations also show that the mean profiles and the vertical gradients are continuous and smoothly merge at the top of the RxL. Namely, the RxL profiles are tangent to the UL at a finite height, as observed, rather than asymptotically approaching the UL at infinite height, or rather than crossing the UL at an arbitrary matching height (Panofsky 1978).

### Calibration against Minnesota data

To locate the top of the RxL, vertically contiguous mean (time averaged) wind speed and temperature data are needed from near the surface through the interior of the ML. Many field experiment datasets do not satisfy this requirement because of the artificial discontinuity created when time-averaged surface-layer data from masts or towers are combined with instantaneous rawinsonde observations above (Clarke and Brooke 1979). Also, many wind profiler systems do not give robust wind-speed measurements below 100–200 m.

However, published data from the 1973 Minnesota field experiment (Izumi and Caughey 1976) are satisfactory. The instrument platforms included a combination of a 32-m tower and sensors at fixed locations on the cable of a tethered balloon. Similar averaging times of 75 min were used at all heights from near the surface through the midmixed layer, yielding profiles of wind and temperature that are self-consistent, smooth, and contiguous. There were 11 datasets (“runs”) that were obtained during the Minnesota experiment. The site was a flat, recently harvested, and plowed farm square-mile section (1.609 km on a side) with no vegetation close by, a roughness length of *z*_{0} ≈ 2.4 mm, elevation of 255 m above sea level, at location 48°34′N, 96°51′W. A uniform fetch of 10 km existed to the north, which was the predominant wind direction. A more detailed description of the site, instruments, and experimental procedures can be found in Izumi and Caughey (1976) and Kaimal et al. (1976). Appendix A (Table A1) lists key dates, times, and scales for the 11 runs. Figure 5 shows the range of wind speeds and potential temperature profiles present during the runs.

A nonlinear regression (Bevington 1969; Press et al. 1992) is used to determine best-fit parameters *M*_{UL}, *z*_{RM}, and *A*_{1} for wind; and *θ*_{0}, *θ*_{UL}, *z*_{Rθ}, and *A*_{2} for potential temperature, for each dataset from the Minnesota experiment. An iterative process is used to minimize the sum of squared deviations between the regression equation (RxL and UL taken together) and the data. Best-fit estimates of the parameters are listed in appendix B for each of the 11 datasets.

### Universal constants

Using all the Minnesota runs, parameter *A*_{1} = 0.095 ± 0.011 (mean plus or minus standard deviation) for wind speed, and *A*_{2} = 0.104 ± 0.009 for potential temperature (see appendix B). One wind dataset (run 5A1) did not have a UL in wind speed and, hence, was excluded from the wind speed analysis of this study because no top to the RxL could be found. The relatively small variations in *A* over the wide range of wind speeds (Fig. 5a), potential temperatures (Fig. 5b), and ML depths suggest that they might be universal. We will make the assumption of universality. Instead of averaging the previously calculated *A* values to find the universal value, we rearrange the profile equations into a dimensionless form and apply the nonlinear regression to the full set of 10 or 11 runs grouped together as one run. This allows us to find the best-fit *A* values for a superset of data consisting of the whole Minnesota experiment. This is a more stringent test because it demands similarity simultaneously over a wide variety of wind and depth regimes. That is, while any single dataset might cluster tightly around the curve, the superset is more likely to have scatter (i.e., unexplained variance) if the theory was poor.

The “universal” *A* values are computed using nonlinear least squares analysis, as before, between *M*/*M*_{UL} versus *z*/*z*_{RM} for the 10 valid wind runs taken together as one large dataset. Similarly, nonlinear regression is used for (*θ* − *θ*_{UL})/(*θ*_{0} − *θ*_{UL}) versus *z*/*z*_{Rθ} using data from all 11 temperature runs taken together as one large dataset. The *A* values are determined such that *M*/*M*_{UL} approaches one as *z*/*z*_{RM} approaches one, and (*θ* − *θ*_{UL})/(*θ*_{0} − *θ*_{UL}) approaches zero as *z*/*z*_{Rθ} approaches one.

The resulting best-fit values are *A*_{1} = 0.0959 ± 0.011 (mean plus or minus standard deviation) for wind and *A*_{2} = 0.101 ± 0.009 for temperature. The correlation coefficient between observed and computed nondimensional equations is *r* = 0.992 for wind speed and *r* = 0.986 for potential temperature. Given these fixed values for *A*_{1} and *A*_{2}, the other parameters in the wind and temperature equations found using nonlinear regression are listed in Table 1 for each of the Minnesota runs. Plots of dimensionless wind and temperature profiles using these universal *A* values are shown in Figs. 6 and 7.

The parameters listed in Table 1 are not expected to be universal, but they can be functions of external forcings such as geostrophic wind speed *G,* boundary effects such as *z*_{i} and *z*_{0}, and scaling variables such as *w*_{B}. Most similarity theories incorporate such locally varying nonuniversal parameters, such as the way the Obukhov length is incorporated in SL similarity theory. Note from Table 1 that wind speeds varied by roughly a factor of 2, and RxL depths by a factor of 3, yet the data from each of the 10 wind datasets collapsed into a single similarity curve in Fig. 6. Analogous similarity behavior is evident in Fig. 7 for the 11 temperature datasets. For both wind and temperature, the RxL depth exhibits very little correlation with either classic measures of surface-layer thickness, namely, the Obukhov length *L,* or 10% of the mixed-layer depth *z*_{i} (see Table 1). This latter result differs from the findings of Brutsaert and Sugita (1992) for a different dataset, although their results are for the surface-layer depth rather than the RxL depth.

There are noticeable differences between RxL depths between Table 1 and the tables in appendix B. Because of the gradual approach of the RxL profile to the UL profile, the RxL height is somewhat sensitive to the *A* parameters and to scatter in the data. Because of this sensitivity, we looked for a better depth scale, such as an integral scale or a half-width height. To date, a better depth scale has not been found. However, the shape of the profile is relatively insensitive to the RxL depth, which means the former sensitivity is not very important.

## Similarity scaling

What controls RxL depth? From Table 1 it is obvious that RxL depth varies from run to run. It probably also varies from location to location, although the Minnesota case study alone is insufficient to address dependency on roughness length.

In this section we attempt to parameterize RxL depth as a function of constraints and external forcings for the Minnesota case study. These factors include both SL variables such as *z*_{0} and surface buoyancy flux, but they also include ML variables such as *z*_{i}. A new field program called Boundary Layer Experiment 1996 (BLX96) was recently conducted to study roughness dependency of RxL characteristics (Stull et al. 1997). Results from this new study will be reported later in a separate paper.

*w*

_{B}was as defined earlier and

*c*is an empirical constant. An expression equivalent to (9) is possible using the Deardorff velocity

*w*∗ instead of the buoyancy velocity

*w*

_{B}because the two are linearly related by

*w*∗ ≈ 0.08

*w*

_{B}(Stull 1994). The ratio

*w*

_{B}/

*M*

_{UL}is the square root of the “mixed-layer Richardson number” (Stull 1994). The roughness-length dependence in (9) is tentative, based on theoretical expectations.

Figure 8 shows this regression. The dimensionless slope is *c* = 21.4 × 10^{3} and the correlation is *r* = 0.545, which means that only about 30% of the variance in the data is explained by the regression line. It is clear that additional work needs to be done to identify the factors that control the RxL depth.

The RxL depth for temperature, *z*_{Rθ}, shows even more scatter when plotted (Fig. 9) against the same parameters as for *z*_{RM}. Also, there was very little correlation (*r* = 0.145) between *z*_{RM} and *z*_{Rθ}. This might be expected because the wind speed is very highly constrained between zero at the surface and the geostrophic wind aloft, but the temperature profile floats as both the surface skin temperature and the UL temperature increase during the day.

## Wind profiles versus shear profiles

During this work, it became apparent that inaccuracies of wind profile relationships are hidden when those profiles are expressed in terms of wind shear. However, when those profiles are integrated with height to get the actual wind speeds, the errors are revealed and can accumulate to cause substantial discrepancies between the observed wind speed and the parameterized profile. This is unfortunate because for many practical purposes, such as wind loads on structures, pollutant transport, and wind power generation, it is the speed and not the shear that is needed.

One might argue that this is an unfair test because integrating up from a small wind near the surface, such as from zero wind at the roughness height, might amplify small initial errors. To examine this argument, one can recompute the height integration, but starting at different heights. This process is repeated for each data point in the observed wind profile, generating a set of curves such that each curve exactly passes through one point. If the profile similarity theory is valid, then all of the curves should lie nearly on top of each other.

Examples of the integrated ZS wind-speed profiles are shown in Fig. 11, for runs 3A1 and 6A1. Most of the curves do not lie on top of each other. Furthermore, the direction of the error is not consistent: run 3A1 has less shear than the integrated ZS curves, while run 6A1 has more shear. We repeated this exercise for each of the Minnesota runs and found magnitudes of ZS wind-speed errors of roughly ±3.0 m s^{−1} for runs 2A1, 2A2; ±1.0 m s^{−1} for runs 3A1 and 3A2; ±0.7 m s^{−1} for runs 6A1, 6A2, 6B1, and 7C2; and ±0.3 m s^{−1} for runs 7C1 and 7D1. This compares to wind-speed errors of ±0.3 m s^{−1} or less for all runs using RxL Eq. (5), as already plotted in Fig. 6.

Thus, it appears that wind speed gives a more sensitive test of the accuracy of a similarity relationship than does shear. We recommend that future proposals for similarity relationships be tested in their integrated form, such as wind speed profiles.

## Summary and recommendations

Equations (7) and (8) describe similarity relationships for wind speed and potential temperature within the whole radix layer. The radix layer is identified as the region between the surface and the base of the uniform portion of the convective boundary layer. The top of the radix layer (i.e., the base of the uniform layer) is usually well above the top of the classic surface layer.

Using data from the 1973 Minnesota experiment, the resulting best-fit value for the exponents in Eqs. (7) and (8) are *A*_{1} = 0.0959 ± 0.011 for wind and *A*_{2} = 0.101 ± 0.009 for temperature. These exponents are relatively constant for a wide variety of wind speeds and surface heating, suggesting that they might be universal. The correlation coefficient between observed and parameterized profiles is *r* = 0.992 for wind speed and *r* = 0.986 for potential temperature. This suggestion of universality of equations and exponents should be considered tentative until independent verification tests are published and wind-speed ranges of validity are identified.

The radix-layer depth differs from run to run, analogous to run-to-run variations of surface-layer scales such as the Obukhov length. The radix-layer depth for wind is greater than that for temperature, and there is little correlation between the two. An attempt was made to parameterize the radix-layer depth for wind as a function of the mixed-layer Richardson number; however, the fit was poor, suggesting that more work is needed.

The profile equations proposed here are applicable in the range of 1 m < *z* < 0.7 *z*_{i}. Thus, they span the previously little-studied region above the top of the surface layer and below the base of the uniform layer, where classic Monin–Obukhov similarity theory performs poorly. Potential applications of better wind speed profiles include wind turbine electrical generation, wind loads on buildings and bridges, and air pollutant transport.

We recommend that profile similarity relationships be presented and tested as wind-speed profiles, rather than as shear profiles. The latter tend to hide or disguise errors and give a false picture of the accuracy of the relationship.

Datasets for independent validation are scarce because of the requirements for time- or space-averaged, not instantaneous, observations from the surface into the midmixed layer. To test the above equations, a new field program (Boundary Layer Experiment 1996) was conducted in July and August 1996 in the central United States (Stull et al. 1997). Special vertical zigzag flight patterns in the University of Wyoming King Air aircraft were flown to measure the mean profiles of wind speed and potential temperature. Results from this new validation will be forthcoming.

## Acknowledgments

This work was initiated while both authors were at the University of Wisconsin. The following funding agencies are gratefully acknowledged for their support: U.S. National Science Foundation (NSF ATM-9411467), the U.S. Department of Energy (ARM Grant DE-FG02-9ZER61361), the Canadian Atmospheric Environment Service (AES subventions), and the Canadian Natural Sciences and Engineering Research Council (NSERC operating grants).

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## APPENDIX A

### Scaling Variables for Individual Minnesota Runs

Dates, times, and boundary layer scaling variables for the Minnesota field experiment. Friction velocity is *u**, Deardorff velocity is *w**, buoyancy velocity is *w*_{B}, mixed-layer depth is *z*_{i}, Obukhov length is *L,* surface kinematic heat flux is *Q*_{0}, surface-layer temperature scale is *T** = −*Q*_{0}/*u**, and mixed-layer temperature scale is *θ** = *Q*_{0}/*w**. Central Daylight Time (CDT) = UTC − 5 h. The buoyancy velocity *w*_{B} was calculated from its definition (see section 2b) using Δ*θ*_{υs} equal to *θ*_{υUL} − *θ*_{υ0} and was not calculated from *w**. Also *L* was calculated using a von Kármán constant of 0.4.

## APPENDIX B

### Nonlinear Regression Parameters for Individual Minnesota Runs

Best-fit estimates of the parameters in Eq. (5) for wind speed in the combined radix and uniform layers. Here, *M*_{UL} is wind speed in the uniform layer, *z*_{RM} is the radix-layer depth for wind, and *A*_{1} is the exponent parameter.

Best-fit estimates of the parameters in Eq. (6) for potential temperature in the combined radix and uniform layers. Here, *θ*_{UL} and *θ*_{0} are the potential temperatures in the uniform layer and in the air adjacent to the surface, respectively; *z*_{Rθ} is the radix–layer depth for potential temperature; and *A*_{2} is the exponent parameter for potential temperature.

## APPENDIX C

### Comparison of Several Wind-Speed Profile Relationships Against Each Individual Minnesota Run

(*Continued*)

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

(*Continued*)

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

(*Continued*)

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Idealized profiles showing components of the convective mixed layer for (a) wind *M* and (b) potential temperature *θ.* Here, *G* is geostrophic wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Idealized profiles showing components of the convective mixed layer for (a) wind *M* and (b) potential temperature *θ.* Here, *G* is geostrophic wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Idealized profiles showing components of the convective mixed layer for (a) wind *M* and (b) potential temperature *θ.* Here, *G* is geostrophic wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind profiles for all 11 runs of the Minnesota campaign. Abscissa is normalized using surface-layer similarity, where *z*_{0} is aerodynamic roughness length, *k* = 0.4 is von Kármán’s constant, and *ψ*_{m} is the integrated wind profile stability–correction function of Businger and Dyer: (a) within the surface layer and (b) within the radix and uniform layers.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind profiles for all 11 runs of the Minnesota campaign. Abscissa is normalized using surface-layer similarity, where *z*_{0} is aerodynamic roughness length, *k* = 0.4 is von Kármán’s constant, and *ψ*_{m} is the integrated wind profile stability–correction function of Businger and Dyer: (a) within the surface layer and (b) within the radix and uniform layers.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind profiles for all 11 runs of the Minnesota campaign. Abscissa is normalized using surface-layer similarity, where *z*_{0} is aerodynamic roughness length, *k* = 0.4 is von Kármán’s constant, and *ψ*_{m} is the integrated wind profile stability–correction function of Businger and Dyer: (a) within the surface layer and (b) within the radix and uniform layers.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind observations from the Minnesota field experiment run 7C1 compared to three surface-layer models [KP = Kader and Perepelkin 1989; ZS = Sorbjan 1986; BD = Businger–Dyer (Businger et al. 1971)]: (a) above and (b) within the traditional SL. See appendix C for plots of the other runs.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind observations from the Minnesota field experiment run 7C1 compared to three surface-layer models [KP = Kader and Perepelkin 1989; ZS = Sorbjan 1986; BD = Businger–Dyer (Businger et al. 1971)]: (a) above and (b) within the traditional SL. See appendix C for plots of the other runs.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind observations from the Minnesota field experiment run 7C1 compared to three surface-layer models [KP = Kader and Perepelkin 1989; ZS = Sorbjan 1986; BD = Businger–Dyer (Businger et al. 1971)]: (a) above and (b) within the traditional SL. See appendix C for plots of the other runs.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Potential temperature (left) and wind (right) superposition of transport processes (center) imposed on the background states (a) and (d), by large thermals (b) and (e) in the uniform layer, and smaller shear-driven eddies (c) and (f) in the radix layer (after Stull 1994).

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Potential temperature (left) and wind (right) superposition of transport processes (center) imposed on the background states (a) and (d), by large thermals (b) and (e) in the uniform layer, and smaller shear-driven eddies (c) and (f) in the radix layer (after Stull 1994).

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Potential temperature (left) and wind (right) superposition of transport processes (center) imposed on the background states (a) and (d), by large thermals (b) and (e) in the uniform layer, and smaller shear-driven eddies (c) and (f) in the radix layer (after Stull 1994).

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Raw (a) wind speeds and (b) potential temperatures for all Minnesota runs, showing the range of conditions studied here.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Raw (a) wind speeds and (b) potential temperatures for all Minnesota runs, showing the range of conditions studied here.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Raw (a) wind speeds and (b) potential temperatures for all Minnesota runs, showing the range of conditions studied here.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 10 wind datasets collapse to a common similarity wind profile when radix-layer depth *z*_{RM} is used to make height *z* dimensionless, and uniform-layer wind *M*_{UL} is used to make wind speed *M* dimensionless. Empirical Eq. (7) is plotted as the solid line, using a “universal” value of *A*_{1} = 0.0959. Note that the wind speed Eq. (7) does become zero at the surface, even though there are no data low enough to show it.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 10 wind datasets collapse to a common similarity wind profile when radix-layer depth *z*_{RM} is used to make height *z* dimensionless, and uniform-layer wind *M*_{UL} is used to make wind speed *M* dimensionless. Empirical Eq. (7) is plotted as the solid line, using a “universal” value of *A*_{1} = 0.0959. Note that the wind speed Eq. (7) does become zero at the surface, even though there are no data low enough to show it.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 10 wind datasets collapse to a common similarity wind profile when radix-layer depth *z*_{RM} is used to make height *z* dimensionless, and uniform-layer wind *M*_{UL} is used to make wind speed *M* dimensionless. Empirical Eq. (7) is plotted as the solid line, using a “universal” value of *A*_{1} = 0.0959. Note that the wind speed Eq. (7) does become zero at the surface, even though there are no data low enough to show it.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 11 potential-temperature datasets collapse to a common similarity profile when radix-layer depth *z*_{Rθ} is used to make height *z* dimensionless, and temperature difference between the surface skin and the uniform layer (*θ*_{0} − *θ*_{UL}) is used to make potential temperature *θ* dimensionless. Empirical Eq. (8) is plotted as the solid line, using a universal value of *A*_{2} = 0.101.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 11 potential-temperature datasets collapse to a common similarity profile when radix-layer depth *z*_{Rθ} is used to make height *z* dimensionless, and temperature difference between the surface skin and the uniform layer (*θ*_{0} − *θ*_{UL}) is used to make potential temperature *θ* dimensionless. Empirical Eq. (8) is plotted as the solid line, using a universal value of *A*_{2} = 0.101.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Data (open circles) from the 11 potential-temperature datasets collapse to a common similarity profile when radix-layer depth *z*_{Rθ} is used to make height *z* dimensionless, and temperature difference between the surface skin and the uniform layer (*θ*_{0} − *θ*_{UL}) is used to make potential temperature *θ* dimensionless. Empirical Eq. (8) is plotted as the solid line, using a universal value of *A*_{2} = 0.101.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Linear regression of dimensionless RxL depth *z*_{RM} for wind against a dimensionless group (*w*_{B}/*M*_{UL}) that is proportional to the square root of the mixed-layer Richardson number (Stull 1994), where *z*_{0} is the aerodynamic roughness length, *w*_{B} is the buoyancy velocity, and *M*_{UL} is the wind speed in the uniform layer.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Linear regression of dimensionless RxL depth *z*_{RM} for wind against a dimensionless group (*w*_{B}/*M*_{UL}) that is proportional to the square root of the mixed-layer Richardson number (Stull 1994), where *z*_{0} is the aerodynamic roughness length, *w*_{B} is the buoyancy velocity, and *M*_{UL} is the wind speed in the uniform layer.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Linear regression of dimensionless RxL depth *z*_{RM} for wind against a dimensionless group (*w*_{B}/*M*_{UL}) that is proportional to the square root of the mixed-layer Richardson number (Stull 1994), where *z*_{0} is the aerodynamic roughness length, *w*_{B} is the buoyancy velocity, and *M*_{UL} is the wind speed in the uniform layer.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Dimensionless potential temperature RxL depth *z*_{Rθ} shows little correlation when plotted against the RxL dimensionless group that was successful for wind.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Dimensionless potential temperature RxL depth *z*_{Rθ} shows little correlation when plotted against the RxL dimensionless group that was successful for wind.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Dimensionless potential temperature RxL depth *z*_{Rθ} shows little correlation when plotted against the RxL dimensionless group that was successful for wind.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Comparison of wind shear profiles from Sorbjan (ZS) and from Eq. (10, radix) for typical conditions at Minnesota (*z*_{i} = 2000 m, *z*_{RM} = 200 m, *L* = −20 m, *M*_{UL} = 10 m s^{−1}, and *w*∗ = 2 m s^{−1}). The differences are subtle but significant when integrated over height to get wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Comparison of wind shear profiles from Sorbjan (ZS) and from Eq. (10, radix) for typical conditions at Minnesota (*z*_{i} = 2000 m, *z*_{RM} = 200 m, *L* = −20 m, *M*_{UL} = 10 m s^{−1}, and *w*∗ = 2 m s^{−1}). The differences are subtle but significant when integrated over height to get wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Comparison of wind shear profiles from Sorbjan (ZS) and from Eq. (10, radix) for typical conditions at Minnesota (*z*_{i} = 2000 m, *z*_{RM} = 200 m, *L* = −20 m, *M*_{UL} = 10 m s^{−1}, and *w*∗ = 2 m s^{−1}). The differences are subtle but significant when integrated over height to get wind speed.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind speed *M* profiles observed (data points) and found by integrating the Sorbjan (1986) profile relationship (curves) up and down from each data point.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind speed *M* profiles observed (data points) and found by integrating the Sorbjan (1986) profile relationship (curves) up and down from each data point.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Wind speed *M* profiles observed (data points) and found by integrating the Sorbjan (1986) profile relationship (curves) up and down from each data point.

Citation: Journal of Applied Meteorology 37, 6; 10.1175/1520-0450(1998)037<0545:WATPIT>2.0.CO;2

Best-fit estimates of parameters in the dimensionless wind, Eq. (7), for *A*_{1} = 0.0959, and in the dimensionless potential temperature, Eq. (8), for *A*_{2} = 0.101. Also shown for comparison with the radix-layer depths for momentum *z*_{RM} and temperature *z*_{Rθ} are the Obukhov length *L,* and 10% of the mixed-layer depth *z*_{i}, which are often used as depth scales for the classic surface layer. Here, UL is the uniform layer, and *M*_{UL} and *θ*_{UL} are the wind speed and potential temperature in the UL. The surface potential temperature based on the best-fit profile is *θ*_{0}.