## 1. Introduction

The definition of the gradient wind invoked here is the following paraphrase of Dutton’s (1986, p. 311) textbook definition: The gradient wind is a horizontal wind having the same direction as the geostrophic wind but with a magnitude consistent with a balance of three forces: the pressure gradient force, the Coriolis force, and the centrifugal force arising from the curvature of a parcel trajectory. In the usual order of presentation to students, the atmospheric dynamical equations are derived, subjected to scale analysis for large-scale mid- and upper-tropospheric flow, and simplified to explain the concept of the geostrophic wind (e.g., Dutton 1986). The geostrophic wind arises from a balance of the Coriolis and pressure gradient forces. The gradient wind is then introduced as a refinement of the geostrophic wind to obtain a better approximation for frictionless large-scale horizontal flow (e.g., Dutton 1986).

From the perspective of the gradient wind as a refinement of the geostrophic wind, it may seem mandatory that the parcel trajectories should depend on the geostrophic streamfunction. Then the trajectory curvature would necessarily result from parcels following the geostrophic streamfunction, even as it changes with time. Yet, using parcel curvature derived from an actual wind does not violate the nominal definition of the gradient wind. It is in consideration of this ambiguity that the author proposes the categorization of gradient wind into two types discussed herein. Haltiner and Martin (1957, p. 183) recognize two types of gradient wind. Their terminology characterizes a horizontal flow as being either “gradient at a point or over an extended path.” This article revisits the notion of two ways of obtaining the gradient wind, proposes new terminology, and explores the different types of gradient wind by way of descriptive and quantitative examples.

The concept of the gradient wind is a familiar pedagogical tool found in many textbooks on atmospheric dynamics. The properties of the solutions to the gradient wind quadratic equation have been exhaustively examined (e.g., Fultz 1991). Beyond the classroom in research, the gradient wind is sometimes used as a background or reference state in analytic or numerical modeling (e.g., Emanuel 1986; Hodyss and Nolan 2007). In practical applications, the gradient wind may serve as a better estimate of a wind field associated with a mass field than does the geostrophic wind (e.g., Dutton 1986, p. 311; Holton 2004, p. 65).

The goals of the analysis presented here are to draw distinctions among valid methods of computing the gradient wind and to characterize basic differences, both from a computational point of view and how the different gradient wind magnitudes compare to the actual wind as a benchmark. Accomplishing this requires some examination of the fundamental mathematics behind the gradient wind. With reference to the mathematics, the proposed new terminology is offered. In the following, the second section presents a concise background for the mathematical aspects of this analysis, introduces the new terminology, and then derives and discusses a formula for comparing gradient wind and actual wind magnitudes. The third section applies the new terminology, offering both qualitative and quantitative examples of the gradient wind types in comparison to actual winds. The final section presents summarizing material and concluding remarks.

## 2. Mathematical analysis

### a. Background

*V*is the wind (parcel) speed,

^{ 1 }and

*t*is the independent variable time. Equation (1) is often expressed in terms of the radii of curvature, which are the reciprocals of the curvature parameters. The second term on the rhs of (1) accounts for the change with time of the streamlines. Curvature is positive (negative) for a cyclonically moving parcel and negative (positive) for the anticyclonic case in the Northern (Southern) Hemisphere. It should be noted that (1) is general and applies to any horizontal wind field.

The infinitesimal displacement of a parcel moving along a curved trajectory occurs in a local horizontal tangent plane and, in the parcel frame of reference, is represented mathematically as an infinitesimally small rotation of a natural coordinate system. The orthogonal natural coordinate system is (*τ*, *η*, *κ*, *t*), for which unit vector **
τ
** is along the horizontal velocity vector

**V**, unit vector

**is to the left of**

*η***, unit vector**

*τ***is along the local vertical, and**

*κ**t*is the time coordinate. The choice of vertical coordinate determines the formulation expressing the pressure gradient force. In this article, selecting the vertical coordinate to be pressure allows the horizontal pressure gradient force to be expressed as the gradient of the geopotential (

*g*is the acceleration of gravity and

*z*is the height of the isobaric surface above mean sea level).

Equation (2) is valid in either the Northern or Southern Hemisphere. In crossing the equator from one hemisphere to the other, both *K* and *f* change algebraic signs. The geopotential gradient term also changes sign for a symmetrical configuration reflected across the equator in the opposite hemisphere. Consider a midlatitude geopotential trough in the Northern Hemisphere with westerly flow at the base and

Actual wind data are needed to justify the choice of the anomalous solution as exemplified in the approach of Mogil and Holle (1972). This is a point that deserves particular emphasis. For anticyclonic curvature, both the normal and anomalous solutions for the gradient wind magnitude exceed the geostrophic wind (Holton 2004, p. 67). Therefore, there is no basis for determining whether or not to choose the anomalous solution without looking beyond the geostrophic wind to the actual wind. If the anomalous gradient wind magnitude is a better match to the actual wind speed, then it is appropriate to choose the anomalous solution.

The streamline curvature *direction* of the gradient wind is clearly defined to be identical to that of the geostrophic wind, the *magnitude* of the gradient wind, computed using (4), is dependent on the values of

The fact that the choice between two wind fields may exist for the curvature calculation provides the basis for the proposed terminology, *contour* versus *natural* gradient winds. If the streamlines of the geostrophic wind are used in (4), the result is a contour gradient wind. If the streamlines of the actual wind are used in (4), the result is a natural gradient wind. If

### b. Assessment of supergradient winds

^{ 2 }Thus, for cross-contour flow by the actual wind, these last two parenthetically enclosed terms of (5) evaluate to a positive number rather than zero. This positive value is subtracted, diminishing any supergradient wind effect that may arise from the first two terms on the rhs of (5).

The effect of the first two terms on the rhs of (5) is examined in the following. Table 1 applies for steady contour, nonsteady contour, or steady natural gradient winds. Nonsteady natural gradient winds are treated as a separate case below. The last column of Table 1 states the condition that does not contradict

Relative values for

## 3. Demonstrating the new terminology

### a. Schematic illustration

Figure 1 presents schematically a collection of possible trajectories occurring for different assumed conditions for contour or natural gradient winds. Figure 1 depicts an idealized large-scale mid- or upper-tropospheric trough occurring well away from the equator in the Northern Hemisphere. The trajectory assessment in Fig. 1 applies at the instant the parcel passes through the identified point circumscribed by the thick circle. If steady flow conditions are assumed for the contour gradient wind, then only the first term on the rhs of (1) imparts curvature and the parcel follows the contour of the geostrophic streamfunction. This steady contour (SC) gradient wind trajectory is identified by SC in Fig. 1 and is coincident with the geopotential contour labeled

The steady natural (SN) gradient wind is drawn somewhat arbitrarily and labeled SN in Fig. 1. If the large-scale mid- or upper-tropospheric trough depicted in Fig. 1 is well away from the equator, then “the geostrophic wind approximates the true horizontal velocity to within 10%–15%” (Holton 2004, p. 40). For this reason the SN trajectory is depicted so that the instantaneous streamline of the actual wind is not much different from the instantaneous geostrophic streamfunction

Figure 1 is intended primarily to illustrate different trajectory paths possible for steady and nonsteady contour and natural gradient winds in an idealized schematic form. Figure 1 also serves to demonstrate the influence of Blaton’s equation for the nonsteady cases. Although Fig. 1 is not a quantitative depiction, the relative magnitudes of the gradient wind at the identified point can be inferred by examining the curvature of the trajectories. In order of SC, SN, NSC, and NSN, the curvatures become less cyclonic, and the centrifugal force (directed to the right in Fig. 1) in the gradient wind balance becomes less for each successive trajectory in that order. Therefore, the gradient wind magnitude must increase in the progression of SC, SN, NSC, and NSN because an increased Coriolis force (also directed to the right) is needed to balance the same pressure gradient force (directed to the left). The sequence of curvature change depicted in Fig. 1 applies only to the particular case shown and will vary in general. For example, if the motion were retrogressive in Fig. 1 as mentioned earlier, the nonsteady gradient wind trajectories would become more cyclonically curved and the corresponding gradient wind magnitudes would be less.

Although Fig. 1 applies to the Northern Hemisphere, a simple exercise can transform it into a Southern Hemisphere depiction: place an inverted display of Fig. 1 facing a mirror and view the reflected image. The arrow labeled “NORTH” is pointing south. The translation of the trough is still west to east (from left to right). The turning of the geostrophic wind vector associated with the translation of the trough is counterclockwise rather than clockwise. In the Southern Hemisphere, cyclonic curvature is negative; so, with

### b. Example gradient wind calculations

To show quantitative examples, the four species of gradient wind are computed using the solution to (3) or (4) and displayed in Fig. 2 as differences between the actual wind magnitude and the gradient wind magnitude. The calculations for Fig. 2 use forecast fields from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS). The forecast projections are 24 and 27 h corresponding to forecasts of upper-tropospheric flow having good agreement to verifying analyses. The calculations are carried out using the average of model output fields at 24 and 27 h. Specific date information is given in the Fig. 2 caption. Time tendency calculations needed for the nonsteady contour gradient winds are based on the difference between the 27- and 24-h forecast fields. The GFS data are at 1° × 1° horizontal resolution on a simple cylindrical projection. Since the gradient wind applies to large-scale flows and to ensure aesthetically appealing displays, the data are filtered (the *e*-folding wavelength is seven grid spaces for a filter having a Gaussian distribution of weights) so that the numerical values of gradients are more typical of those obtained for coarser spatial resolution with larger separation of grid points. The General Meteorological Package (GEMPAK) described by desJardins et al. (1991) was invoked for gridded data computation and graphical displays. The GEMPAK programs and parameter settings used for the computations and graphics are provided in a UNIX shell context in the supplemental material.

Differences between actual wind magnitude and that of the gradient wind (^{−1}, shaded, color bar at top) are shown for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level. The data are 1° × 1° latitude–longitude filtered GFS forecast grids, averages of the 24- and 27-h forecasts from initial time 1200 UTC 27 Jan 2013, with the valid time at the interval midpoint (1330 UTC 28 Jan 2013). Averaged height contours (dam, thin, solid) are shown. Averaged forecast actual winds are shown by wind barbs (m s^{−1}, half barb is 2.5, full barb is 5, flag is 25). Nonshaded areas are where the gradient wind calculation fails. The zero contour for the differences is drawn as a thick solid line.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

Differences between actual wind magnitude and that of the gradient wind (^{−1}, shaded, color bar at top) are shown for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level. The data are 1° × 1° latitude–longitude filtered GFS forecast grids, averages of the 24- and 27-h forecasts from initial time 1200 UTC 27 Jan 2013, with the valid time at the interval midpoint (1330 UTC 28 Jan 2013). Averaged height contours (dam, thin, solid) are shown. Averaged forecast actual winds are shown by wind barbs (m s^{−1}, half barb is 2.5, full barb is 5, flag is 25). Nonshaded areas are where the gradient wind calculation fails. The zero contour for the differences is drawn as a thick solid line.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

Differences between actual wind magnitude and that of the gradient wind (^{−1}, shaded, color bar at top) are shown for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level. The data are 1° × 1° latitude–longitude filtered GFS forecast grids, averages of the 24- and 27-h forecasts from initial time 1200 UTC 27 Jan 2013, with the valid time at the interval midpoint (1330 UTC 28 Jan 2013). Averaged height contours (dam, thin, solid) are shown. Averaged forecast actual winds are shown by wind barbs (m s^{−1}, half barb is 2.5, full barb is 5, flag is 25). Nonshaded areas are where the gradient wind calculation fails. The zero contour for the differences is drawn as a thick solid line.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

*ζ*in the natural coordinate framework (Dutton 1986, p. 337):

**V**is the actual wind. The

The other practical aspects of the calculations for Fig. 2 are the time averaging and smoothing mentioned previously in discussing the source of data. Both help to prevent small-scale spatial fluctuations in curvature, especially in the regions of transition from cyclonic to anticyclonic curvature or vice versa. Smoothing reduces noisy small-scale variations in the magnitude of the geostrophic wind. The time averaging and smoothing act to eliminate any spurious influences of gravity waves associated with topography or geostrophic adjustment (Holton 2004) occurring on a scale smaller than that typically associated with midlatitude gradient wind balance.

Figure 2 is designed to reveal with little more than a glance that the various kinds of gradient wind are in fact different as evidenced by the varying color shaded patterns of supergradient (pink to brown) and subgradient (green to purple) flow. The gradient wind magnitude itself is obtained by subtracting the shaded difference value from the actual wind magnitude shown by the wind barbs. Super- and subgradient wind speeds are evident for both steady and nonsteady contour gradient winds as well as for the steady natural gradient wind. Only subgradient flow is found for the nonsteady natural gradient wind, except for small isolated areas where the linear solution using (9) is required. Similar small isolated areas requiring (9) also exist for the other types of gradient wind and are shown as scattered dark shaded regions in Fig. 3.

Character of curvature (anticyclonic light shaded, cyclonic not shaded) for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level, corresponding to the gradient winds depicted in Fig. 2. Dark shaded areas denote regions of near zero curvature requiring the application of (9) in the calculation of gradient winds displayed in Fig. 2.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

Character of curvature (anticyclonic light shaded, cyclonic not shaded) for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level, corresponding to the gradient winds depicted in Fig. 2. Dark shaded areas denote regions of near zero curvature requiring the application of (9) in the calculation of gradient winds displayed in Fig. 2.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

Character of curvature (anticyclonic light shaded, cyclonic not shaded) for (a) steady contour, (b) nonsteady contour, (c) steady natural, and (d) nonsteady natural gradient winds at the 250-hPa level, corresponding to the gradient winds depicted in Fig. 2. Dark shaded areas denote regions of near zero curvature requiring the application of (9) in the calculation of gradient winds displayed in Fig. 2.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00088.1

In Fig. 3, the regions of anticyclonic curvature are lightly shaded in gray. Figure 3 is provided to facilitate a deeper discussion of the patterns seen in Fig. 2. The curvature of the actual wind is shown in Fig. 3d. In general, there is agreement among the different curvature calculations for cyclonic curvature (nonshaded areas in Fig. 3) associated with the trough over the western continental United States (CONUS). The trough off the eastern coast of the CONUS is also an area of general agreement on cyclonic curvature. Areas of anticyclonic curvature over the northwestern CONUS and over the eastern CONUS and southeastern Canada are also generally evident in all four panels of Fig. 3. However, an area of disagreement between the character of curvature of the nonsteady natural gradient wind (also the curvature of the actual wind) in Fig. 3d and that of the other gradient winds is apparent over the lower (southern) Mississippi Valley (LMV). The curvature of the actual wind (Fig. 3d) is anticyclonic over the LMV region, except for a narrow strip of cyclonic curvature extending from southern Louisiana southward into the central Gulf of Mexico where actual wind barbs suggest a shortwave trough. Figures 3a–c show cyclonic curvature over the LMV. By the second row of Table 1, the LMV area should be quite susceptible to supergradient winds for all species of gradient winds except the nonsteady natural gradient wind because the actual wind curvature is anticyclonic and the gradient wind curvature is cyclonic. The presence of supergradient winds over the LMV is confirmed in Figs. 2a–c.

Example Lagrangian gradient wind calculations for a 3-h segment of the cold conveyor belt trajectory from Table 3 of Whitaker et al. (1988). The successive positions numbered in the first column are the ending points for each subsegment and are separated by 1 h. The entries for the vector wind (parcel velocity) ^{−1}. The parcel curvature ^{−1}. The gradient wind speed ^{−1}. Latitude and longitude are expressed in degrees, with longitude negative west of the Greenwich meridian. The average geostrophic wind speed for each hourly segment is used in the gradient wind computation.

## 4. Summary and conclusions

While the gradient wind is precisely defined with respect to its direction, the magnitude of the gradient wind is not so precisely defined owing to its dependence on the curvature of a parcel trajectory, which may follow the geostrophic wind or the actual wind. Thus, the nominal definition of the gradient wind leaves ambiguous the method of computing the requisite radius of curvature. This article defines new terminology for two types of gradient wind based upon how the parcel curvature is computed: contour gradient winds are computed using the curvature of the geostrophic streamfunction. Natural gradient winds are computed using the curvature of the streamlines of the actual wind or an observed parcel trajectory. If the geostrophic streamfunction or streamlines are not changing in time, the corresponding type of gradient wind is steady; otherwise, the gradient wind type is nonsteady. The kinds of gradient wind are summarized in Table 3, the contents of which reflect the findings discussed in the previous sections of this article. In the last column of the first two rows of Table 3, the disadvantage related to justification of the anomalous solution for the contour gradient wind assumes that no actual wind is available.

Summary of types of gradient wind calculations with advantages and disadvantages inferred from this article.

The gradient wind concept is generally accepted as an important teaching tool. To get some idea as to the prevalence of the utility of the gradient wind beyond the classroom, the author conducted a search for the phrase “gradient wind” in the abstracts of articles published in all American Meteorological Society (AMS) journals after 2006. The articles found were examined to determine the type of gradient wind used. If the determination of type was possible and decisive, the article is included in Table 4. Two noticeable aspects of Table 4 are the predominance of the steady contour gradient wind and its frequent application to tropical cyclones.

Recent AMS articles mentioning “gradient wind” in the abstract and for which the type of gradient wind could be determined decisively. The entries are organized chronologically from earliest at top to most recent at bottom.

The prevalence of the steady contour gradient wind concept in tropical cyclone research has a long history. The application of gradient wind balance appears in early work on the topic of tropical cyclones and is justified with reference to scale analysis (e.g., Charney and Eliassen 1964). From early composite data studies (e.g., Gray and Shea 1973) and modeling studies (e.g., Emanuel 1986) to the contemporary work on tropical cyclones, the steady contour gradient wind continues to play an important role. A relatively recent article by Schwendike and Kepert (2008) gives an excellent justification for and description of applying the steady contour gradient wind to tropical cyclones. Given the preponderance of the steady contour gradient wind in the tropical cyclone arena, findings of supergradient winds are ubiquitous (e.g., Gray and Shea 1973; Bryan and Rotunno 2009; Frisius et al. 2013) as would be expected from the analysis and results presented in the previous sections of this article.

The geostrophic wind results from applying scale analysis to large-scale midtropospheric flow (Dutton 1986). The gradient wind, as a refinement of the geostrophic wind, is similarly applicable to large-scale flow. Yet, in many cases the gradient wind is applied on much smaller scales. For example, fine temporal and spatial resolution of Lagrangian trajectory data results in more accurate and applicable gradient wind estimates for parcels. Tropical cyclones are not large-scale phenomena from the midlatitude perspective; yet, the gradient wind is routinely applied to tropical cyclones (Table 4). In the opinion of the author, the large-scale context for the gradient wind is not necessary so long as the application of the gradient balance is justified in general by scale analysis or demonstrated to be valid for specific cases using real data.

In conclusion, the gradient wind remains an important concept of utility in teaching, as well as in research and practical applications (Table 4). If four people are given the same dataset with unqualified instructions to compute the gradient wind, they could return with four different values, all valid from the standpoint of the definition of the gradient wind. The primary goal of this article is to propose simple terminology that clearly distinguishes among the possible magnitudes of the gradient wind based on the method of computing the trajectory curvature for the gradient wind calculation. *Contour* gradient wind parcels follow trajectories based on the geostrophic streamfunction. *Natural* gradient wind parcels follow trajectories based on the actual wind field. In the case of steady flow of either type, the trajectories are streamlines; otherwise, Blaton’s Eq. (1) must be included as in (4) to account for the local time tendency of the respective wind direction. The nonsteady natural gradient wind may be computed using the curvature obtained from successive parcel positions in the case of Lagrangian data or by computing the curvature directly applying (8) as described in section 3. The nonsteady natural gradient wind magnitude provides an upper bound for the actual wind speed.

## Acknowledgments

The insights of Dr. Daniel Keyser in an informal review of the draft manuscript were very valuable and much appreciated. The author is grateful to Dr. Richard Rotunno, whose perusal of preliminary notes and informal review of a draft version resulted in many helpful suggestions. Reviews by Drs. David Novak and Wallace Hogsett yielded helpful suggestions for clarifying the presentation. Formal reviews by five anonymous reviewers and Dr. David Schultz were very helpful, resulting in improved and expanded content vis-à-vis the original manuscript. The encouragement and advice of Dr. Louis Uccellini are much appreciated. The National Centers for Environmental Prediction provided funding for this publication. Any opinions expressed in this article are solely those of the author and do not necessarily represent the position of the funding entity.

## APPENDIX

### Demonstration of an Anticyclonic Curvature Limitation

This demonstration shows that a supergradient wind speed is not possible if the gradient wind magnitude is that for the nonsteady natural case. Referring to (6) in the main text, which applies only to the nonsteady natural gradient wind, if

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^{1}

Some textbooks (e.g., Holton 2004) give the angle in Blaton’s equation as the mathematical angular direction, positive in the *counterclockwise* direction.

^{2}

This article assumes all flows are baric with positive values of the geostrophic wind (Holton 2004, p. 67).