## Introduction

Bluestein and Crawford (1997) concluded that computing gradients of wind and other quantities is complicated by sloping terrain. They used adjustments suggested by Schaefer (1973) to take the terrain into account. Schaefer (1973) noted that when horizontal divergence is computed over nonhorizontal surfaces, extra terms appear in the equations. The extra terms can be estimated and are probably important enough that they should not be ignored. Computations made from observations taken over flat terrain are straightforward, but most observations of interest are made over land surfaces with a certain amount of irregular topography.

This issue arose in the Upper Missouri River Basin Pilot Project (UMRBPP), which gathered data during the period of 5 April through 5 May 1999 over the Black Hills of South Dakota and Wyoming. Observations around a small watershed were the focus of the project, with the objective to refine the understanding of the water budgets in the watershed. Rawinsondes were launched from three sites, and a wind profiler and radiometer (water vapor) installation made up a fourth site enclosing the watershed area. The measurements were made in mountainous terrain with topography relief of about 1 km. A scheme was developed to compute the gradients of the observations over the watershed.

## Analysis

*q*represent a general coordinate (

*x*or

*y*); then some quantity

*A*may be represented to first order as

*q,*

*z*) and (

*q*+ Δ

*q,*

*z*+ Δ

*z*), on the terrain

*E*at elevations

*z*=

*E*(

*q*) and

*z*+ Δ

*z*=

*E*(

*q*+ Δ

*q*). Substituting this into Eq. (1), rearranging, and dividing by Δ

*q*results in

*q*becomes very small is

Equation (3) is Schaefer's (1973) Eq. (1) and Wigley's (1964) Eq. (11). The left-hand side is the gradient along *q,* which can be measured on the terrain surface. The first term on the right-hand side is the value that is desired. The last quantity on the right side involves the unknown gradient of *A* with respect to *z* and the slope of the terrain. Schaefer (1973) suggests that the gradient of *A* in the vertical may be estimated from a sounding. However, soundings do not measure *A* on the surface of the terrain, but rather they measure the variation of *A* above the surface. Equation (3) has two unknowns—the gradients of *A* with respect to *q* and *z.* It is identical to Eq. (1) for all practical purposes. Equation (1) or (2) can be used with observational data, which are taken at finite intervals, rather than at infinitesimal intervals as suggested by Eq. (3).

*u*= 0 on the left at the surface increasing linearly to 2

*u*

_{1}at 2 km above the surface. At the top of the hill the flow is 2

*u*

_{1}at all levels, decreasing to ⅔

*u*

_{1}on the right. The mass flux (assumed density is 1 kg m

^{−3}) is constant at 2000

*u*

_{1}kg m

^{−1}s

^{−1}across the hill. Three observations of the wind may be made along the surface to determine the gradients. After choosing the left side of the hill as the basis point, the following two equations can be constructed from Eqs. (1) or (2):

*u,*and 2 times Eq. (5) is subtracted from (4) to obtain the horizontal gradient. Clearly the gradient of

*u*with respect to

*z*is 20/9

*u*

_{1}km

^{−1}and the gradient of

*u*with respect to

*x*is −1/45

*u*

_{1}km

^{−1}. Schaefer's (1973) approach would produce different results depending on the value used for the vertical shear in

*u.*On the left side of the hill, the shear is

*u*

_{1}km

^{−1}. Schaefer's method gives

*z.*Adding an observation upstream from the left foot of the hill (

*u*= 0 at the surface) and one downstream on the plateau [

*u*= (2/3)

*u*

_{1}] will result in a set of four equations that can be solved for the gradients. The results are:

*x*and

*y*directions. With this in mind, a three-dimensional Taylor series expansion in

*x,*

*y,*and

*z*may be written (neglecting higher-order terms):

*A*in

*x,*

*y,*and

*z*), so four measurements of

*A*are needed to solve for the unknowns. As before, assume that measurements

*A*

_{0}(

*x*

_{0},

*y*

_{0},

*z*

_{0}),

*A*

_{1}(

*x*

_{1},

*y*

_{1},

*z*

_{1}),

*A*

_{2}(

*x*

_{2},

*y*

_{2},

*z*

_{2}), and

*A*

_{3}(

*x*

_{3},

*y*

_{3},

*z*

_{3}) have been made. Use

*A*

_{0}as the starting point to define the delta values

*x*direction), the gradients in the

*y*direction will be poorly resolved. Substantial departures in the vertical from a planar surface similarly facilitate resolution of the gradients in

*z.*

*u*and

*υ*components of the wind can then be resolved, and the gradients of the wind in three dimensions can be computed using Eq. (9). The horizontal divergence

*D*is then calculated directly from the gradients:

## Error analysis

*x,*

*y,*

*z*). The sides of the rectangle are

*x*and

*y*units long. Without loss of generality, the sites 1 and 2 may be assumed to be at the same elevation as site 0. Recall that three nonlinear points determine a plane. Thus the four observation sites are in two distinct planes, with site 3 on high ground. If the site positions are known exactly, then Eq. (9) gives

*x*

_{3}and

*y*

_{3}are

*x*

_{1}and

*y*

_{2}with some uncertainty built in. So the determinant is

*x*

_{3}and

*y*

_{3}include error terms that are multiplied by the respective

*z*error terms. Thus, to first order, the determinant simplifies to

*x*

_{1},

*y*

_{2}, and

*z*

_{3}have errors that have not been accounted for as yet. The following includes these errors to first order:

*z*s are on the order of a meter for general data. The implication is that

*z,*the height of the high ground, should be on the order of 100 m to reduce the errors to a few percent in the inverse matrix. Latitude and longitude are usually used as the location coordinates for meteorological sites. These are typically given to the nearest 0.01°. This implies a precision of about 0.5 km. The distance between two sites is then known to about 1 km, which implies that the horizontal distances should be on the order of 100 km or so to reduce errors to a few percent.

In the case of the data presented here, taken from UMRBPP, the latitude and longitude are given to an accuracy of 0.001° and the vertical coordinate to 1 m. The observation sites are about 30 km apart, and the vertical distance is 500 m. Thus, position errors are very small and sum to a few percent. The errors in calculating the divergence should not exceed a few percent. The precision reported in the wind speeds is to the nearest 0.1 m s^{−1}. The divergence errors due to wind speed error are on the order of 5 × 10^{−6} s^{−1}.

## Example of application to surface pressure calculations

*p*

*z*

^{−6}

*z*

^{2}

*p*between the observation points:

## Example of application to divergence calculations from upper-air observations

The same approach can be applied to compute gradients and divergence values from upper-air observations, provided the observations are selected to represent noncoplanar “layers.” Figure 3 shows a contour plot of the Black Hills with the four UMRBPP observation sites depicted. Details about the site locations are given in Table 3. Note the high ground directly north and west of Custer (the southernmost site), which was the site with the wind profiler and microwave radiometer. The profiler made measurements continuously and recorded an observation 2 times per hour. The Custer Crossing and Four Corners sites had the National Center for Atmospheric Research cross-chain long-range navigation atmospheric sounding system (CLASS) rawinsonde system; Four Corners, the highest site, is west of the highest terrain in the Black Hills. The watershed of interest is west of Rapid City (the easternmost site), the location of the NWS rawinsonde data.

Figures 4 and 5 show the observed wind components (*u* and *υ*) from the soundings for 1800 UTC 22 April 1999. Each site had similar wind structure above the ground. The *u* component increases with height to about 2.5 km above mean sea level (MSL) in Fig. 4 then becomes variable but somewhat weaker with increasing height. The *υ* component is weaker but has a similar structure (Fig. 5). Note that the winds are from the east-northeast, so there were upslope winds on the eastern slopes of the Black Hills. Easterly winds were observed for the previous 12 h and the following 12 h of this observation. Precipitation in the form of light snow was observed over the watershed during the period from 1200 UTC April 22 through 0000 UTC April 23, so one might expect to find convergence at some level.

The measured winds were interpolated into 25 levels beginning at the surface and extending to 3000 m MSL at each site. The resulting (nonplanar) layers are thickest at Rapid City and thinnest at Four Corners. The interpolated data were used to compute the divergence values shown in Fig. 6; two computations are shown. The Rapid City and Four Corners sites are nearly on an east–west line (Fig. 3) and so can be used to compute the gradient of *u* according to Eq. (2). The Custer and Custer Crossing sites are nearly on a north–south line and can be used to approximate the gradient of *υ.* This has been done to get the “terrain-neglected” approximation. The “terrain-included” curve was computed using Eq. (9), with Rapid City as the base station. Note that this computation used data from all four sites to get the gradients of *u* and *υ* below 3000 m MSL.

In principle, there should be more valid information in the terrain-included curve. There is divergence from the surface to 2400 m MSL in both computations. Above 2400 m MSL, there is a layer of convergence about 400 m thick in the terrain-included curve, whereas the terrain-neglected approximation remains divergent. In this particular case, the terrain-included analysis shows enhanced divergence at the surface and convergence above 2400 m MSL that does not appear in the straightforward terrain-neglected analysis. The uncertainties in the results should increase near the upper boundary as the layers become more nearly planar; some indications of this effect appear in Fig. 6.

## Conclusions

As seen in the synthetic example (section 2), topography complicates the computation of horizontal gradients. Using the maximum vertical shear, Schaefer's (1973) scheme results in some divergence whereas the scheme proposed herein does not. Based on the constant mass flux in the example, the correct answer is thought to be no divergence. Schaefer's scheme gives reasonable results when applied to the surface pressure in the example of section 4. In this case, the vertical gradient of the pressure varies smoothly and a sounding in proximity to the surface data gives an approximate result. However, the proposed scheme gives a more accurate result for interpolation purposes and a horizontal pressure gradient that is closer to the horizontal pressure gradient calculated from the mean sea level pressures. Results from Schaefer's (1973) scheme depend greatly on the accuracy of the vertical gradient that is used.

The example of divergence in complex terrain in section 5 makes use of Schaefer's (1973) scheme difficult. The winds (Figs. 4 and 5) have different shears at different sites. The sites are at markedly different altitudes, so choosing a value of shear is complicated and it would have to be computed numerically. The proposed scheme, by using more data points, is able to resolve a value for both the horizontal and vertical gradients that is accurate to first order. In addition, resolving the solution into meridional and latitudinal gradients is part of the scheme. Schaefer's (1973) scheme, using two points, will give a gradient in the direction defined by the two points. At least one additional point is needed to resolve the orthogonal direction, and then additional computations may be needed to resolve the results into meridional and latitudinal components.

The proposed scheme, based on a Taylor series expansion, may be enhanced to get higher-order derivatives if more data sites are available. There are six second-order terms in the Taylor expansion, requiring 10 observation sites to solve the equations for all of the terms, which greatly complicates the problem. The sites should be either on a 3 × 3 grid with one extra site somewhere or, perhaps, a triangle with four sites per side and one in the center.

## Acknowledgments

This research was supported by the National Aeronautics and Space Administration as part of the Upper Missouri River Basin Pilot Project under Grant NAG8-1447.

## REFERENCES

Bluestein, H. B. and T. M. Crawford. 1997. Mesoscale dynamics of the near-dryline environment: Analysis of data from COPS-91.

*Mon. Wea. Rev*125:2161–2175.Schaefer, J. T. 1973. On the computation of the surface divergence field.

*J. Appl. Meteor*12:546–547.Taylor, A. E. 1955.

*Advanced Calculus*. Blaisdell Publishing, 786 pp.Wigley, T. M. L. 1964. Approximations in the use of pressure coordinates.

*Tellus*16:26–31.

Sounding sites

Soundings—00 UTC 18 Apr 2001

Site locations