a. Physics

Equation (12), the nonlinear impedance relation for form-preserving disturbances in pressure, has numerous implications for the wind perturbations in gravity waves, wake lows, solitary waves, and other pressure disturbances. Rearranging Eq. (5), one obtains

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The three terms on the rhs of Eq. (13) represent the three processes responsible for the acceleration of the local wind. The first term is simply the pressure gradient force. The second and third terms on the rhs of Eq. (13) represent the advection of perturbation momentum by the background wind and by the perturbation wind, respectively.

Considering the nonlinear term [the third term on the rhs of Eq. (13)], one may note that it is similar to the linear advection term, but it indicates the advection of momentum by the perturbation wind rather than by U. In the case of an approaching pressure fall (rise), u′ is negative (positive) and ∂u′/∂x is positive (negative). In either case, the nonlinear term −u′∂u′/∂x > 0 produces a positive effect on ∂u′/∂t and on u′. In other words, the nonlinear term reduces the magnitude of the wind perturbation in a pressure fall (where u′ < 0) and increases the magnitude of the wind perturbation in a pressure ridge (where u′ > 0). As will be shown in section 4, a parcel of air in a pressure trough (with u′ < 0, or in the opposite direction of the motion of the disturbance) moves through the disturbance more quickly than a parcel in a ridge (with u′ > 0, in the same direction as the motion of the disturbance). The longer parcel residence times in the pressure ridge, associated with nonlinear effects, allow for larger magnitude wind perturbations in a pressure ridge than in a pressure trough, given the same magnitude pressure perturbation and the same intrinsic speed. Once the pressure trough (ridge) has passed by a location, u′ is still negative (positive), but at that time the direction of the pressure gradient force has changed, so ∂u′/∂x is negative (positive). Here, in both cases, the nonlinear term −u′∂u′/∂x < 0 produces a negative effect on ∂u′/∂t, canceling out the extra positive effect on u′ ahead of the trough or ridge and allowing the wind perturbation to return to zero once p′ = 0.

b. Idealized examples

The new, nonlinear IR [Eq. (12)], using only the negative root (as discussed in section 2), is now applied to two types of idealized pressure disturbances, and the results are compared to the classical IR. The first pair of disturbances (one trough, one ridge) are sinusoidal in pressure and are represented by p′ = A cos(kx − ωt). These approximate ducted gravity waves, and initially p′ = 0. The amplitude is 1 hPa, the propagation speed (relative to the ground) is c = 40 m s⁻¹ (in the positive x direction, or eastward), and the component of the background wind in the direction of the disturbance motion is U = 15 m s⁻¹, producing an intrinsic propagation speed (relative to the mean wind) of c − U = 25 m s⁻¹. The period is 2000 s, and the density is assumed to be 1.2 kg m⁻³. First of all, it may be shown that p′ < ½ρ(c − U)² throughout both disturbances, so the usual case of the nonlinear impedance relation applies. Figure 1 shows the sinusoidal disturbances in pressure, and the resulting wind perturbations predicted by the classical and the nonlinear impedance relations at the point x = 0.

The second pair of disturbances represents a linear pressure rise and a linear pressure fall, consistent with a front or density current. The pressure change occurs over 500 s, representing a quarter of a wavelength of a sawtooth wave. The amplitude and intrinsic speed of the disturbances, and the density, are the same as in the sinusoidal disturbances described above. The pressure perturbations and the wind perturbations predicted by both impedance relations are shown in Fig. 2.

Figure 1 shows a minor difference in u′ predicted by the linear and nonlinear IRs. As mentioned above, the addition of the nonlinear term in the derivation of the IR causes a positive effect on ∂u′/∂t as either a pressure rise or pressure fall approaches a location, meaning that u′ is more positive (or less negative) using the nonlinear IR. This implies smaller magnitude perturbations in the trough (Fig. 1a) and larger magnitude perturbations in the ridge (Fig. 1b), consistent with the discussion in section 3a. For these fairly small-magnitude (1 hPa) pressure disturbances, the maximum magnitude of u′ predicted by the classical impedance relation in the trough is 3.33 m s⁻¹, while that predicted by the nonlinear impedance relation is 3.14 m s⁻¹; in the ridge, the maximum magnitudes of u′ are 3.33 and 3.59 m s⁻¹, respectively. The decrease in the magnitude of u′ in the trough (when comparing nonlinear to linear predicted values) is 5.7%, while the increase in the ridge is 7.8%. Notice also that the value of u′ predicted by both relations returns to zero once p′ = 0, also consistent with section 3a.

Figure 2 shows a similar analysis for linear pressure changes, with the u′ predicted using the nonlinear impedance relation being greater than that predicted by the linear impedance relation (less negative in the pressure fall, more positive in the pressure rise). The maximum magnitudes of u′ predicted by the two IRs in these sawtooth cases are exactly the same as those predicted for the sinusoidal cases. This makes mathematical sense, in that both IRs (except in the special case outlined in section 2, which does not apply here) simply relate u′ to p′, with no regard for the shape of the changes in p′. The nonlinear term simply causes |p′u′⁻¹| to be smaller in pressure ridges and larger in pressure troughs than predicted by the linear IR. It should also be noted that despite the changes in magnitude of |p′u′⁻¹|, p′ and u′ are still correlated in troughs and ridges, with the maximum |u′| occurring at the same time as the maximum |p′|, and the sign of p′ always being the same as the sign of u′—in other words, p′u′ > 0 as long as c − U > 0.

In the special case mentioned above and outlined in section 2, involving a slow-moving, large-amplitude pressure ridge, where p′ > ½ρ(c − U)², Eq. (12) still accurately represents the relationship between p′ and u′. However, at the point where the above inequality becomes true, the special case of the nonlinear impedance relation becomes valid, and the winds adjust to the positive root of Eq. (12). These cases are discussed in section 4.

c. A note on determining the motion of pressure disturbances

As mentioned in section 1, many have also used the IR to calculate the theoretical c − U of a pressure disturbance, given p′ and u′. Dividing both sides of Eq. (11) by ρu′ and rearranging terms yields the nonlinear IR as an expression for determining intrinsic speed:

View Expanded

The second term on the rhs of Eq. (14) is present because of the nonlinear term in Eq. (11). Without the second term on the rhs, Eq. (14) is simply the linear IR [Eq. (1)], written in terms of intrinsic speed. The effect of the nonlinear term in Eq. (14) is fairly straightforward. Since p′u′ > 0, the first term on the rhs of Eq. (14) is always positive. In the case of a pressure ridge, the addition of the positive u′/2 partially compensates for the smaller |p′u′⁻¹| than that predicted by the linear IR (see section 3b). Similarly, in a pressure trough, the addition of the negative u′/2 partially compensates for the larger |p′u′⁻¹|. In any event, Eq. (14) will also provide an accurate calculation for c − U, given the accurate, nonlinear IR-predicted p′ and u′.

a. Development of a Lagrangian model

The Lagrangian numerical model is fairly simple, applying the irrotational, inviscid horizontal equation of motion [Eq. (3)] at z = 0 (where υ = 0 and w = 0) to a parcel of air traversing a form-preserving, traveling disturbance in pressure that is a function p(x, t). We fix U, c, A (the maximum magnitude of p′), and wave period T, and an air parcel enters the disturbance at the point where p′ = 0, at the front edge of the disturbance at t = 0 s. The parcel enters the disturbance at U, so that u′ = 0. The model time step is 1 s. At each time step n (n = 0, 1, 2, 3, 4, …), the pressure perturbation p′_n is determined using p(x, t), and the horizontal pressure gradient acceleration (dp/dx)_n is also determined and used to determine the horizontal acceleration of the parcel (du/dt)_n at that time step. The position of the parcel at each time step x_n is determined from

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The parcel’s velocity at each time step u_n is then calculated through

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Finally, the perturbation velocity at each time step is determined to be u′_n = u_n − U. The above describes a Lagrangian model in which a single parcel is followed. The pressure perturbation and pressure gradient acceleration (both functions of x and t) are determined at each point, and then the velocity, allowing determination of the parcel’s position at the following time step. At this new position and time, the new p′ and acceleration are determined, and so on. Since U is constant, u′ = u − U allows determination of the perturbation velocity, and this value is compared to the linear and nonlinear impedance relations.

b. Examples of usual cases

The two sinusoidal pressure disturbances described in section 3b, one being a ridge and one a trough, are now simulated using the Lagrangian parcel model described above, and the results are compared to those predicted by the linear and the nonlinear IRs. Figure 3 is a plot of parcel p′, parcel acceleration Du′/Dt, and u′ for a parcel initially moving at the speed of the background wind and encountering a propagating wave ridge at t = 0 s. Figure 3 shows that the parcel reaches the maximum p′ at t = 881 s. At that time, dp′/dx = Du/Dt = 0, u reaches a local maximum of 18.59 m s⁻¹, and u′ = 3.59 m s⁻¹. By t = 882 s, the pressure gradient acceleration has reversed, and the parcel decelerates until t = 1762 s, when it exits the pressure ridge with its velocity returning to u = 15 m s⁻¹ (u′ = 0). Figure 4 shows a similar plot for the simulation of a parcel encountering a wave trough. The parcel encounters acceleration in the negative x direction (toward the center of the trough) until it reaches the minimum p′ at t = 741 s, where u has decreased to 11.87 m s⁻¹, implying that u′ = −3.13 m s⁻¹. The parcel then accelerates back into the positive x direction until p′ = 0 and u′ = 0 at t = 1483 s. Consistent with section 3, the parcel residence time in the pressure ridge is longer (1762 s) than in the pressure trough (1483 s). Also, the magnitude of the maximum wind perturbation is larger in the wave ridge than in the wave trough.

The simulations illustrated in Figs. 3 and 4 indicate maximum wind perturbations that are consistent with the nonlinear IR and deviate from the classical linear IR. Solving the linear IR [Eq. (1)] for u′ at the time of the largest magnitude p′ in the ridge and trough yields 3.33 and −3.33 m s⁻¹, respectively. These solutions have an RMS error of 0.23 m s⁻¹ relative to the Lagrangian model. The nonlinear IR [Eq. (12)] at the largest magnitude p′ yields u′ of 3.59 and −3.14 m s⁻¹, respectively. These solutions’ RMS error is only 0.01 m s⁻¹ when compared to the Lagrangian model. These example cases, and other simulations (in usual cases) at varying amplitudes and intrinsic speeds (not shown), indicate that the nonlinear IR is a more accurate indicator of the perturbation winds in a propagating pressure disturbance than the linear IR.

c. Special case

We now examine a Lagrangian simulation for a large-amplitude, sufficiently slow-moving pressure ridge that in some parts of the pressure disturbance p′ > ½ρ(c − U)². These are special cases as defined in section 2, where the nonlinear impedance relation [Eq. (12)] contains imaginary solutions for u′. [Recall that no pressure troughs satisfy the inequality p′ > ½ρ(c − U)², since p′ < 0.] In this simulation, the amplitude is 5 hPa, c = 15 m s⁻¹, U = −5 m s⁻¹ (so c − U = 20 m s⁻¹), and the period is 2000 s. Figure 5, like Figs. 3 and 4, shows parcel p′ and Du′/Dt, as well as u′, for a parcel initially moving at the speed of the background headwind (U = −5 m s⁻¹) encountering the wave ridge at t = 0 s. Given the amplitude of the ridge, the parcel quickly encounters a large pressure gradient and a resulting large acceleration in the positive x direction of almost 0.1 m s⁻². Since the acceleration is proportional to the pressure gradient ∂p′/∂x, which is maximized for a sinusoidal wave in the pressure field at points where p′ = 0, the acceleration decreases somewhat as the parcel moves toward the center of the wave ridge.

However, by t = 246 s, the ground-relative speed of the parcel has increased from its initial value of −5 to 15 m s⁻¹, implying that the wind perturbation of the parcel is u′ = 20 m s⁻¹, which is equal to the intrinsic speed of the disturbance. At this point the parcel is moving at the same speed as the wave ridge itself, and any further acceleration in the positive x direction results in the parcel moving faster than the wave and away from the center of the wave ridge. This is reflected in Fig. 5, which shows the parcel reaching a maximum p′ of 2.4 hPa at t = 246 s, then experiencing decreasing p′ after that point. In other words, after t = 246 s, the parcel is retracing the front of the pressure ridge. The parcel continues to accelerate until t = 493 s, when it completely outruns the pressure disturbance, experiencing p′ = 0, and no additional pressure gradient acceleration. After t = 493 s, the parcel continues to move eastward at a constant speed, with u = 35 m s⁻¹ and u′ = 40 m s⁻¹.

Figure 6 illustrates the position and speed of the air parcel relative to the eastward-propagating disturbance at various times. At t = 0 s, the front of the wave ridge is just arriving at the parcel’s location, but the parcel has yet to be influenced by the wave ridge, so its δx = 0, and u = U = −5 m s⁻¹ (westward motion). At t = 58 s, the parcel has moved slightly to the west (δx = −0.15 km) because of its initial westward background motion. However, the eastward-directed pressure gradient acceleration associated with the pressure ridge has eliminated the westward wind: u′ = 5 m s⁻¹, u = 0. Thus, for a brief moment, the parcel is at rest relative to the ground. The eastward acceleration continues, and the parcel then begins to move eastward. At t = 150 s, the parcel has accelerated to u = 7.59 m s⁻¹ and u′ = 12.59 m s⁻¹ and has moved eastward past its initial location (δx = 0.2 km). At t = 246 s, a mathematically significant point (as shown below), the parcel has accelerated to a wind perturbation of u′ = 20 m s⁻¹. This means that the parcel is moving at a speed of 20 m s⁻¹ relative to the background wind, the same as the intrinsic speed of the propagating wave ridge (c − U = 20 m s⁻¹). At this point, relative to the moving atmosphere, the air parcel and the wave ridge are moving at exactly the same speed, and with continued eastward acceleration, the parcel will shortly begin to move faster than the wave ridge. Therefore, the pressure perturbation of the air parcel due to the wave ridge at that point, p′ = 2.41 hPa, is the maximum p′ that a parcel in the background flow, entering the ridge at the very front, can reach. So, after t = 246 s, the parcel will begin to retrace the front of the ridge. At t = 400 s, the parcel p′ due to the wave ridge has already decreased to 1.53 hPa, but its speed has increased to u = 27 m s⁻¹. At t = 493 s, the parcel has reached the front of the wave ridge again, so p′ = 0. It has reached a speed of u = 35 m s⁻¹ (u′ = 40 m s⁻¹); note that this u′ = 2(c − U). From this point on, the parcel is away from the influence of the wave ridge and its pressure gradient acceleration, so it will theoretically continue to move at a constant speed, moving farther ahead of the wave ridge. Note that at t = 600 s and t = 700 s, u = 35 m s⁻¹. The parcel continues to pull away from the wave ridge, also. At t = 600 s, δx = 11.2 km, and the parcel is 2.2 km ahead of the front of the ridge. At t = 700 s, δx = 14.7 km, and the parcel is 4.2 km ahead of the front of the ridge.

As shown in section 5, in the real atmosphere, mixing and viscous forces have a large negative effect on u′ once the parcel begins to outrun the pressure disturbance. Actually, the parcel u′ seems to settle between c − U and 2(c − U). However, the fact remains, in practice and in theory, that the wind perturbation in these special cases becomes independent of the amplitude of the disturbance and depends only on the intrinsic speed of the disturbance.

Mathematically, the above simulation is consistent with the nonlinear impedance relation [Eq. (12)] and provides excellent insight into the special case scenario when the positive root becomes the correct root of Eq. (12) for predicting u′. First of all, once p′ > ½ρ(c − U)², Eq. (12) produces imaginary values for u′. Allowing p′ = ½ρ(c − U)², one can determine the p′ above which imaginary solutions occur. In this case, that occurs where p′ = 2.40 hPa, within 0.01 hPa of the maximum p′ reached by the parcel in the numerical simulation at t = 246 s. So, if we seek real values of u′ (those that are physically valid), the nonlinear impedance relation, through the terms under the radical in the numerator, places a constraint on the p′ a parcel entering the front of a positive pressure disturbance can attain (i.e., how far into the ridge a parcel can move). It should also be noted that at this maximum value of parcel p′, since p′ = ½ρ(c − U)², the expression under the radical in Eq. (12) equals zero, and division by density ρ shows that u′ = c − U. This was also the case in the above numerical simulation, as u′ = c − U = 20 m s⁻¹ when the maximum p′ of 2.41 hPa was reached.

The negative root of Eq. (12) is required as the parcel enters the wave ridge and remains the correct one until the parcel reaches its maximum p′ (at t = 246 s), when it begins to move faster than the pressure ridge. Since the parcel is still on the leading side of the pressure ridge, a positive pressure gradient acceleration causes a continued increase in u′ as the parcel retraces the front of the ridge. This implies that u′ is greater at each p′ as the parcel retraces the ridge than it was at that same p′ the first time, when the parcel was entering the ridge. The term under the radical in Eq. (12) must be greater than or equal to zero at all times to yield real solutions for u′, and since there are only two roots to a quadratic equation, the positive root of the nonlinear impedance relation must be used for a parcel as it retraces the front of the ridge. Therefore, the correct root of the nonlinear IR to be used changes from the negative root to the positive root at the exact point when the parcel’s perturbation velocity is equal to the intrinsic speed of the propagating ridge.

Figure 7 shows the u′ of the parcel as a function of time and allows comparison to the negative and positive roots of the nonlinear impedance relation. During the time period 0 < t < 229 s, the simulated u′ is predicted very well by the negative root, with the RMS error of the negative root being only 0.17 m s⁻¹ during this time period. During the 34-s time period defined by 229 < t < 264 s, the numerical simulation indicates that the parcel reaches a p′ that is slightly greater than ½ρ(c − U)², so the nonlinear impedance relation produces imaginary values for u′ during this short time period when the parcel is near its maximum p′. However, the positive root of the nonlinear impedance relation begins to accurately predict u′ again at t = 264 s and all times after that (see Fig. 7), with the RMS error of the positive root during the time period 264 < t < 500 s being 0.23 m s⁻¹. Therefore, the nonlinear impedance relation, when the appropriate roots are used, predicts u′ as a function of p′ in a pressure disturbance very accurately (except during the very short time period when imaginary solutions were produced), with RMS errors of only 1%. It is also of note that once the parcel exits the ridge at t = 493 s and p′ = 0, the positive root of Eq. (12) predicts that u′ will reach a maximum value of 2(c − U). Since c − U = 20 m s⁻¹, the maximum predicted u′ is 40 m s⁻¹, which is the same as the maximum wind perturbation in the simulation. It should be stressed again here that once a parcel begins to outrun the pressure ridge, turbulent mixing and viscous forces rapidly damp any further increases in u′.

a. Usual case—Gravity wave, 7 March 2008

On 7 March 2008, a gravity wave ridge moved across central Alabama, producing high winds and reports of wind damage (see http://www.alabamawx.com/?p=5964). The wave disturbance was primarily a single wave of elevation, producing positive wind perturbations. The convergence ahead of the wave ridge produced a thin band of intense precipitation, as shown in the radar reflectivity field from the Birmingham, Alabama (KBMX), Weather Surveillance Radar-1988 Doppler (WSR-88D) (Fig. 8). The disturbance was moving northeastward, or from 240°, at 24 m s⁻¹, so positive values of u appear in the Doppler velocity measurements in Fig. 8 as inbound (negative) velocities. Surface data from Calera, Alabama (KEET), where the wave passed around 1100 UTC, are shown in Fig. 9a. The amplitude of the pressure ridge was approximately 4 hPa, and the maximum pressure was achieved in only 13 min. Given the speed of wave propagation, this implies a pressure gradient of 21.7 hPa (100 km)⁻¹, and a geostrophic wind speed of 179 m s⁻¹. Therefore, this disturbance could never come close to achieving geostrophic balance, and its wind perturbations are better described using the momentum equation in its direction of motion, and consequently the impedance relations. It should also be noted that, given light northeasterly background winds, U = −2 m s⁻¹, so the intrinsic speed of the wave ridge was c − U = 26 m s⁻¹.

The component of the wind in the direction of wave propagation, given by the maximum wind gust each minute, is plotted in Fig. 9a, and perturbations in u show excellent correlation with p′. The maximum u was 17.8 m s⁻¹ according to 1-min surface gusts, so the maximum u′ (u′ = u − U) was 19.8 m s⁻¹. A vertical profile of u, derived using radial velocity data at 1050 UTC at azimuth 240° and range 5 km from the KBMX radar (parallel to wave propagation and at the range of maximum wind perturbations at that time), is shown in Fig. 9b. It indicates a maximum u of 21.8 m s⁻¹ at 75 m AGL, or u′ = 23.8 m s⁻¹. This is fairly consistent with the maximum gust, and averaging the two provides an observed maximum u′ of 21.8 m s⁻¹. Given the observed p′ and intrinsic speed, p′ < ½ρ(c − U)², so the expression under the radical in Eq. (12) will yield only positive roots even though p′ > 0, making this case “usual.” The atmospheric density was 1.21 kg m⁻³, and the linear IR [Eq. (1)] predicts a maximum u′ = 12.7 m s⁻¹, while the nonlinear IR [Eq. (12)] predicts u′ = 22.1 m s⁻¹. In this case, as expected, the linear IR is underestimating u′ in the ridge, by 42%. The nonlinear IR is very accurate in its prediction of u′, with an error of only 1.3%.

b. Usual case—Wake low, 20 December 2007

An intense wake low event affected the Birmingham, Alabama, metropolitan area between 2200 UTC 20 December 2007 and 0000 UTC 21 December 2007. The high winds associated with the wake low caused tree and power line damage across the area (Storm Data). Radar imagery from the KBMX WSR-88D radar (Fig. 10) illustrates the southeasterly flow into the southeast-moving wake low that was located near the back edge of the rain area associated with a mesoscale convective system (MCS) over central Alabama. In Fig. 11, surface data from Birmingham, Alabama (KBHM), show a large pressure fall of 7.8 hPa in slightly over one hour, again indicating ageostrophy. There was a synoptic-scale pressure trend, about −0.5 hPa h⁻¹, imposed on the surface data, implying that the largest magnitude p′ = −5.9 hPa. The disturbance was propagating southeastward at 10 m s⁻¹, and surface data indicate that U = −2 m s⁻¹, so the intrinsic speed of propagation was 12 m s⁻¹. Using the maximum 1-min surface wind gust, the maximum u′ = −14.1 m s⁻¹. However, peak gusts at slightly elevated stations in downtown Birmingham, Inverness, and Cullman, Alabama (see http://www.alabamawx.com/?p=4606), were much higher, with these stations reporting an average maximum u′ = −20.9 m s⁻¹, with a variation of only 0.5 m s⁻¹ among the three stations. The elevations of these three stations average approximately 75 m above the surrounding terrain (similar to the case in section 5a), so they may provide a better representation of maximum u′ without the effects of friction.

This case is automatically usual because p′ < 0. In this case, the linear IR predicts a maximum u′ = −41 m s⁻¹, which is much too large (the error is 96%). However, the nonlinear IR predicts a maximum u′ = −21.6 m s⁻¹, having an error of only 3%. This is consistent with the fact that once nonlinear processes are included, an air parcel traverses a pressure trough much more quickly than it traverses a pressure ridge of the same magnitude and intrinsic speed.

This case also helps verify the scale analysis used to obtain Eq. (5). The three terms containing advection of momentum in Eq. (4) may be combined and written as u∂u′/∂x + w′∂u/∂z, since u = u′ + U, and ∂U/∂x = 0. Using a combination of Doppler radar velocity data and surface data, and assuming two-dimensional Boussinesq continuity, it may be shown that, even in this case with extremely large vertical shear, |u∂u′/∂x| = 17.5 × 10⁻³ m s⁻² and |w′∂u/∂z| = 4.8 × 10⁻³ m s⁻². The horizontal momentum advection term is 3–5 times larger than the combined vertical momentum advection terms. This is in general agreement with the scale analysis in section 2, where the vertical momentum advection is neglected. The effect on u′ associated with vertical momentum advection is, in general, quite small and is beyond the scope of this study.

c. Special case—Cold front, 30 January 2008

On 30 January 2008, a strong cold front moved through Alabama, producing widespread wind damage (Storm Data) and wind gust observations up to 29 m s⁻¹ (see http://www.alabamawx.com/?p=5251). Such winds are quite unusual for the passage of a cold front without any significant deep convection. However, as shown in Fig. 12 [surface data from the University of Alabama in Huntsville (UAH) Mobile Integrated Profiling System (MIPS) berm the amplitude of the surface pressure increase was significant, with an increase in pressure of 2.2 hPa in only 4 min, indicating a highly ageostrophic disturbance that should be well described using the IR. The front was moving at a ground-relative speed of 21 m s⁻¹; given U = 2.5 m s⁻¹, c − U = 18.5 m s⁻¹. The atmospheric density was 1.18 kg m⁻³, and the linear IR predicts a maximum u′ = 9.9 m s⁻¹, or a maximum u of only 12.4 m s⁻¹. However, Fig. 12 shows that the maximum wind speed was near 20 m s⁻¹. Therefore, the observed u′ = 17.5 m s⁻¹, and the linear IR has an error of 76%.

In this case, p′ > ½ρ(c − U)², so solving the nonlinear IR (12) directly for u′ using the observed maximum p′ of 2.2 hPa yields an imaginary root. This represents a special case of the nonlinear IR. Setting p′ = ½ρ(c − U)² to find the p′ at which the expression under the radical in the nonlinear IR becomes zero shows that the maximum pressure a parcel entering the cold front may achieve is p′ = 2.02 hPa. Placing this p′ into the nonlinear IR yields u′ = c − U = 18.5 m s⁻¹, implying an observed wind u = 21 m s⁻¹, which is very close to the observed maximum wind speed u = 20 m s⁻¹. As suggested in section 4, viscous forces and mixing force a parcel in special cases to achieve a maximum u′ somewhere between c − U and 2(c − U). In this case, this would imply that the maximum u′ should be between 18.5 and 37 m s⁻¹. These estimates are fairly good in this case, with the maximum observed u′ at UAH being 17.5 m s⁻¹, and the maximum observed u′ in northern Alabama being 26.5 m s⁻¹. In this case, the nonlinear advection of perturbation momentum kept air parcels under the influence of the large pressure gradient force for longer than the actual period of the disturbance, producing maximum perturbation winds that were much higher than those predicted by the linear IR, and better predicted by the nonlinear IR, except for some overestimation of winds by the frictionless nonlinear IR once the parcel outruns the front, where theory predicts u′ = 2(c − U).