A Nonlinear Impedance Relation for the Surface Winds in Pressure Disturbances

Timothy A. Coleman Department of Atmospheric Science, University of Alabama in Huntsville, Huntsville, Alabama

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Kevin R. Knupp Department of Atmospheric Science, University of Alabama in Huntsville, Huntsville, Alabama

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Abstract

The “impedance relation” between the wind perturbation within an ageostrophic atmospheric disturbance and its pressure perturbation and intrinsic propagation speed has been in use for decades. The correlation between wind and pressure perturbation was established through this relation. However, a simple Lagrangian model of an air parcel traversing an idealized sinusoidal wave in the pressure field indicates that the impedance relation produces significant errors. Examination of the nonlinearized horizontal momentum equation with a sinusoidal disturbance in pressure reveals an additional nonlinear term in the impedance relation, not previously included.

In this paper, the impedance relation is rederived, with the solution being the original equation with the addition of the nonlinear term. The new equation is then evaluated against the Lagrangian model of an air parcel traversing an idealized gravity wave, as well as three observed cases. It is shown that the new impedance relation is indeed more accurate in predicting wind perturbations in disturbances based on pressure perturbations and intrinsic speed than the accepted equation. Implications for determination of the intrinsic phase speed of a disturbance when pressure and wind perturbations are known (another widely used application of the impedance relation) are also discussed.

Corresponding author address: Dr. Timothy A. Coleman, Department of Atmospheric Science, University of Alabama in Huntsville, NSSTC, 320 Sparkman Drive, Huntsville, AL 35805. Email: coleman@nsstc.uah.edu

Abstract

The “impedance relation” between the wind perturbation within an ageostrophic atmospheric disturbance and its pressure perturbation and intrinsic propagation speed has been in use for decades. The correlation between wind and pressure perturbation was established through this relation. However, a simple Lagrangian model of an air parcel traversing an idealized sinusoidal wave in the pressure field indicates that the impedance relation produces significant errors. Examination of the nonlinearized horizontal momentum equation with a sinusoidal disturbance in pressure reveals an additional nonlinear term in the impedance relation, not previously included.

In this paper, the impedance relation is rederived, with the solution being the original equation with the addition of the nonlinear term. The new equation is then evaluated against the Lagrangian model of an air parcel traversing an idealized gravity wave, as well as three observed cases. It is shown that the new impedance relation is indeed more accurate in predicting wind perturbations in disturbances based on pressure perturbations and intrinsic speed than the accepted equation. Implications for determination of the intrinsic phase speed of a disturbance when pressure and wind perturbations are known (another widely used application of the impedance relation) are also discussed.

Corresponding author address: Dr. Timothy A. Coleman, Department of Atmospheric Science, University of Alabama in Huntsville, NSSTC, 320 Sparkman Drive, Huntsville, AL 35805. Email: coleman@nsstc.uah.edu

1. Introduction

The classical “impedance relation,” relating the wind perturbation u′ within a traveling atmospheric disturbance to its pressure perturbation p′ and intrinsic propagation speed cU, where c is the ground-relative phase speed and U is the component of the ambient wind in the direction of disturbance motion, has been in use for decades (e.g., Gossard and Munk 1954; Gossard and Hooke 1975). This relation may be written as
i1520-0469-67-10-3409-e1
where ρ is density. The correlation between wind perturbations and pressure perturbations in disturbances such as gravity waves (e.g., Coleman and Knupp 2009; Trexler and Koch 2000; Bosart and Sanders 1986) is consistent with this relationship. The classical impedance relation may be derived by linearizing the horizontal momentum equation at the surface (where w = 0), and then integrating this equation over time in a propagating, form-preserving disturbance in pressure. The impedance relation has also been widely used to determine the theoretical speed of a pressure disturbance (such as a gravity wave), given the pressure and wind perturbations (e.g., Gossard and Sweezy 1974; Gossard and Munk 1954).

However, a simple Lagrangian model of an air parcel traversing an idealized disturbance in the pressure field indicates that the classical impedance relation (IR) has errors that in some cases may be significant. Examination of the nonlinear horizontal momentum equation, applied to a propagating, form-preserving disturbance in pressure, reveals an additional nonlinear term in the IR, not previously included. In general, the classical (linear) relation overestimates the magnitude of the wind perturbation in a traveling disturbance associated with a temporary (or sometimes semipermanent) decrease in pressure, such as a pressure trough. The linear relation underestimates the magnitude in a traveling pressure ridge (or rise in pressure). Both of these errors are associated with the advection of perturbation momentum by the perturbation wind; this is ignored in the linear relation. In a propagating pressure fall, the perturbation momentum causes an air parcel to traverse the disturbance more quickly than linear dynamics would predict, allowing the parcel less residence time in the perturbation pressure gradient associated with the disturbance. In a pressure rise, an air parcel traverses the disturbance more slowly, allowing more residence time in the pressure gradient. In extreme cases, referred to in this paper as “special cases,” a parcel starting from rest may never even reach the pressure ridge, as the pressure gradient accelerates the parcel away from the ridge. These nonlinear effects may be significant in the cases of gravity waves, in addition to wake lows (e.g., Fujita 1955; Johnson and Hamilton 1988; Stumpf et al. 1991; Vescio and Johnson 1992; Haertel and Johnson 2000), density currents (e.g., Simpson 1997), atmospheric bores (e.g., Rottman and Simpson 1989; Koch et al. 1991; Knupp 2006), solitary waves (e.g., Christie et al. 1978), and fronts.

In this paper, the IR is rederived including the nonlinear term in the horizontal momentum equation. The solution is similar to the classical impedance relation but includes a quadratic term. The new nonlinear IR and the classical IR are compared with the results of the Lagrangian model mentioned above. It is shown that the new IR is indeed more accurate than the accepted linear equation in predicting perturbation winds in propagating pressure disturbances.

The nonlinear IR is developed mathematically in section 2. The physical applications of the new relation are discussed in section 3. The Lagrangian parcel model is described in section 4, and the results of its application to various types of idealized pressure disturbances are compared to the classical and the nonlinear IR. A discussion and conclusions are presented in section 5.

2. The nonlinear impedance relation

We begin by defining a form-preserving, two-dimensional disturbance in the pressure field, traveling in the positive x direction (eastward) at the surface (z = 0), given by the function
i1520-0469-67-10-3409-e2
If, for example, the disturbance were sinusoidal, it could be described by p′ = A cos[k(xct)] = A cos(kxωt), where A is the amplitude, k is the wavenumber of the disturbance, and ω is the angular frequency. It should be noted that for periodic disturbances, c = ωk−1 (e.g., Pedlosky 2003). The inviscid horizontal equation of motion in the x direction, ignoring Coriolis forces (which is appropriate for the mesoscale disturbances being considered herein), may be written as
i1520-0469-67-10-3409-e3
The total x component of u is composed of U (which is constant in x and t, implying that ∂U/∂x = 0 and ∂U/∂t = 0) and u′ (i.e., u = U + u′). Expanding the total derivative and rearranging terms,
i1520-0469-67-10-3409-e4
Note that terms containing υ′ were also neglected, since we are considering a two-dimensional disturbance. We will assume that w′ = 0 at the surface. It should be noted here that the assumption that the third and fourth terms on the left-hand side (lhs) of Eq. (4) are equal to zero is generally applicable not only at the surface but also in the lower parts of the boundary layer (e.g., in the lowest 75 m AGL). Scale analysis, even with very large vertical wind shear (10−1 s−1), and with significant horizontal divergence O(10−3 s−1) producing average w′ (through continuity) over the lowest 75 m AGL O(10−2 m s−1), indicates that the third and fourth terms on the lhs of Eq. (4) are O(10−3 m s−2), while the second term on the lhs (the horizontal advection of momentum) is O(10−2 m s−2), an order of magnitude larger than the vertical advection terms. A further study of the terms in Eq. (4) is presented in a case study in section 5b. Therefore, Eq. (4) may be approximated well by
i1520-0469-67-10-3409-e5
Note that we are keeping the horizontal nonlinear term [the third term on the lhs of Eq. (5)]. Now, the following transformations are valid for a form-preserving, constant amplitude disturbance:
i1520-0469-67-10-3409-e6
i1520-0469-67-10-3409-e7
Substituting Eq. (6) into Eq. (5) and rearranging terms yields
i1520-0469-67-10-3409-e8
Multiplying both sides of Eq. (8) by dt and integrating definitely from t = 0 s to t = t, assuming that p′ = 0 and u′ = 0 at t = 0 s, we get
i1520-0469-67-10-3409-e9
The first term on the right-hand side (rhs) of Eq. (8) may be integrated using Eq. (7), as follows:
i1520-0469-67-10-3409-eq1
The second term may be integrated fairly simply:
i1520-0469-67-10-3409-eq2
The third term is integrated as
i1520-0469-67-10-3409-eq3
Therefore, the solution to Eq. (9) may be written as
i1520-0469-67-10-3409-e10
Solving Eq. (10) for p′ yields
i1520-0469-67-10-3409-e11
The nonlinear term [the third term on the rhs of Eq. (11)] is present because the nonlinear advection term was maintained in Eq. (5) and therefore in Eqs. (8), (9), and (10). If one linearizes Eq. (11) by removing the third term on the rhs and then solves for u′, the classical linear impedance relation [Eq. (1)] is found:
i1520-0469-67-10-3409-eq4
The quadratic formula may is used to solve Eq. (11) explicitly for u′, yielding the new impedance relation:
i1520-0469-67-10-3409-e12

It should be noted that in most cases the negative root of Eq. (12) is the only root that makes physical sense. As mentioned above, p′ = 0 and u′ = 0 at t = 0 s. The positive root implies that u′ = 2(cU) ≠ 0; this is clearly not the case at t = 0 s. Cases using the negative root only are referred to in this study as “usual cases.” There are, however, cases of intense, slow-moving pressure ridges where the positive root applies well after t = 0 s. Parcels may change their direction of motion (relative to the moving disturbance) and retrace part of the ridge. This produces one set of (p′, u′) values as the parcel first enters the front of the ridge at and immediately after t = 0 s (corresponding to the negative root), and a different set of (p′, u′) values as it attempts to retrace the front of the ridge (corresponding to the positive root). These cases are discussed more thoroughly in section 4 and are referred to as special cases.

In addition, the expression under the radical in Eq. (12) is never negative for a pressure trough, since p′ < 0. However, in the case of a slow-moving, large-amplitude pressure ridge, where p′ > ½ρ(cU)2, Eq. (12) produces imaginary solutions for u′. It is shown numerically in section 4 that the expression under the radical in Eq. (12) constrains the motion of an air parcel that starts from rest and enters a pressure ridge. The parcel changes direction relative to the propagating disturbance, as mentioned above, as p′ becomes sufficiently large to cause an imaginary u′—that is, where p′ = ½ρ(cU)2—and the expression under the radical equals zero, implying that u′ = cU. At that point, the parcel is moving at exactly the same intrinsic speed as the disturbance itself. The parcel then retraces the front of the ridge. This is the special case defined above, and even though it is consistent with the nonlinear impedance relation, it is most easily illustrated using a numerical model. It should be emphasized that that the new, nonlinear IR developed herein is valid physically and mathematically at all times and points.

3. Physical interpretation using usual cases

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
i1520-0469-67-10-3409-e13
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 termu′∂u′/∂x > 0 produces a positive effect onu′/∂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(kxωt). 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 cU = 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(cU)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 |pu−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 |pu−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, pu′ > 0 as long as cU > 0.

In the special case mentioned above and outlined in section 2, involving a slow-moving, large-amplitude pressure ridge, where p′ > ½ρ(cU)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 cU 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:
i1520-0469-67-10-3409-e14
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 pu′ > 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 |pu−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 |pu−1|. In any event, Eq. (14) will also provide an accurate calculation for cU, given the accurate, nonlinear IR-predicted p′ and u′.

4. Numerical simulations

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 pn 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 xn is determined from
i1520-0469-67-10-3409-e15
The parcel’s velocity at each time step un is then calculated through
i1520-0469-67-10-3409-e16
Finally, the perturbation velocity at each time step is determined to be un = unU. 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′ = uU 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(cU)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(cU)2, since p′ < 0.] In this simulation, the amplitude is 5 hPa, c = 15 m s−1, U = −5 m s−1 (so cU = 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 (cU = 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(cU). 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 cU and 2(cU). 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(cU)2, Eq. (12) produces imaginary values for u′. Allowing p′ = ½ρ(cU)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(cU)2, the expression under the radical in Eq. (12) equals zero, and division by density ρ shows that u′ = cU. This was also the case in the above numerical simulation, as u′ = cU = 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(cU)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 cU = 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′.

5. Case studies

Three case studies of disturbances in pressure, associated with highly ageostrophic flow in the direction of motion of the disturbance, are examined below. We predict u′ (the component of the wind perturbation in the direction of disturbance motion) for each using the linear IR and the new, nonlinear IR and then compared to the observed u′. Determination of the ageostrophically forced u′ at a given location is fairly straightforward in weak disturbances, as the wind adjusts slowly and friction plays a fairly insignificant role. However, in higher-amplitude disturbances, the determination of u′ is slightly more difficult, given the rapid increase in wind speed and frictional effects on measured surface wind speeds. In these cases, maximum u′ during surface wind gusts, and/or radar measurements of u′ up to 75 m AGL, provide a more accurate measure of the maximum u′ than the maximum 2-min average surface wind does. There is no systematic bias introduced by using this method to compare the linear and nonlinear IRs, since the nonlinear IR predicts lower-magnitude u′ than the linear IR in pressure troughs, and higher-magnitude u′ in pressure ridges.

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 cU = 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′ = uU) 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(cU)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 uin 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 uu′/∂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, |uu′/∂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, cU = 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(cU)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(cU)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′ = cU = 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 cU and 2(cU). 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(cU).

6. Discussion and conclusions

A nonlinear impedance relation between perturbation pressure and wind is developed by including the effects of nonlinear advection of perturbation momentum by the perturbation wind. Solving the new equation for u′ in terms of p′ shows that u′ is a quadratic. In “usual cases,” consisting of all pressure troughs and any pressure ridges in which p′ < ½ρ(cU)2, the following effects occur: In a propagating pressure fall, the perturbation momentum causes an air parcel to traverse the disturbance more quickly than linear dynamics would predict, producing smaller parcel residence time in the disturbance and therefore smaller wind perturbations than those predicted using linear theory. In a pressure rise, the perturbation momentum causes an air parcel to remain in the disturbance longer, producing larger wind perturbations.

In “special cases” (slow-moving, large-amplitude pressure ridges), the nonlinear impedance relation, when applied near the maximum pressure perturbation, yields an imaginary solution for u′. In these cases, a parcel entering the pressure ridge will never reach the pressure maximum, as the pressure gradient combined with the positive advection of momentum causes a parcel retrace part of the ridge. Since u′ is always real, the expression under the radical in Eq. (12) predicts the point (in terms of p′) along the front of a pressure ridge where a parcel begins to retrace the front of the ridge.

A Lagrangian model is developed to track the movement of a parcel as it traverses a sinusoidal pressure disturbance. This model verifies the results predicted by the nonlinear impedance relation in usual cases very well, showing that the magnitudes of the wind perturbations are smaller in troughs than they are in ridges, and parcel residence times are longer in ridges than in troughs. The numerical simulation of the special case disturbance is very illustrative of the process that occurs in these disturbances and, in combination with the usual cases, verifies the correctness of the nonlinear impedance relation. The nonlinear impedance relation remains valid for simulated disturbances because a parcel will never reach a p′ such that imaginary values of u′ are predicted by Eq. (12). When the parcel reaches the front edge of the pressure disturbance, the simulation indicates that u′ = 2(cU), consistent with the positive root and p′ = 0. Here, the linear IR would completely fail. Therefore, it may be stated that the maximum uin all pressure troughs, and any pressure ridges in which p′ < ½ρ(cU)2, is provided by the nonlinear IR applied to the maximum p′. Otherwise, the maximum u′ = 2(c − U). Also, the nonlinear IR clearly provides a better estimate of u′ in any pressure disturbance.

Three case studies were also used to demonstrate the improved accuracy in predicting u′ with the nonlinear IR versus the linear IR. In the first case, a gravity wave ridge, the observed u′ was much larger (by 42%) than that predicted by the linear IR, while the nonlinear IR was almost exactly correct in its prediction of the maximum u′. In the second case, a wake low, the linear IR predicted a maximum u′ with a magnitude almost twice as large as that observed. However, the nonlinear IR predicted a much smaller magnitude u′ with an error of only 3%. The third case involved a cold front special case. The linear IR greatly underestimated the maximum u′. Calculating the maximum p′ a parcel could attain and allowing the parcel to retrace the front of the ridge, and considering friction, the nonlinear IR implied a range of maximum u′ values between cU and 2(cU), or between 18.5 and 37 m s−1. The maximum observed u′ associated with the cold front was 26.5 m s−1, well within this range.

The implications of special cases of the nonlinear IR may be important. Relatively large-amplitude pressure ridges or pressure rises may produce strong winds, and these winds become more dependent on the intrinsic speed of the disturbance than on the amplitude, with a maximum theoretical value of 2(cU). This nonlinear process may help explain the rear-to-front feeder flow often found in density currents (e.g., Simpson 1997). Also, the high winds associated with some rapidly moving squall lines and strong cold fronts may in part be related to the rapid movement of these features. This is especially the case for weakening squall lines and cold pool density currents and decaying gravity waves. In these phenomena, there may be a period of time when the pressure perturbation and associated pressure gradient force no longer support high winds, but air parcels having experienced the nonlinear advection of perturbation momentum may continue to move at or near the intrinsic speed 2(cU) for a while, producing high winds simply related to the motion of the feature and not its intensity. These types of applications require further study, including observations and numerical simulations of high winds associated with some squall lines and density currents long after they have weakened dynamically and have weak pressure perturbations.

Acknowledgments

The authors wish to thank Dr. Carmen Nappo for his review of this manuscript. His comments and suggestions greatly improved the manuscript. The authors also wish to thank the additional anonymous reviewers, whose comments also improved the manuscript. This research was funded by grants from the National Oceanic and Atmospheric Administration (NOAA Grant NA07OAR4600493) and the National Science Foundation (NSF Grant ATM0533596).

REFERENCES

  • Bosart, L. F., and F. Sanders, 1986: Mesoscale structure in the megalopolitan snowstorm of 11–12 February 1983. Part III: A large-amplitude gravity wave. J. Atmos. Sci., 43 , 924939.

    • Search Google Scholar
    • Export Citation
  • Christie, D. R., K. J. Muirhead, and A. L. Hales, 1978: On solitary waves in the atmosphere. J. Atmos. Sci., 35 , 805825.

  • Coleman, T. A., and K. R. Knupp, 2009: Factors affecting surface wind speeds in gravity waves and wake lows. Wea. Forecasting, 24 , 16641679.

    • Search Google Scholar
    • Export Citation
  • Fujita, T. T., 1955: Results of detailed synoptic studies of squall lines. Tellus, 7 , 405436.

  • Gossard, E., and W. Munk, 1954: On gravity waves in the atmosphere. J. Meteor., 11 , 259269.

  • Gossard, E., and W. Sweezy, 1974: Dispersion and spectra of gravity waves in the atmosphere. J. Atmos. Sci., 31 , 15401548.

  • Gossard, E., and W. H. Hooke, 1975: Waves in the Atmosphere. Elsevier, 456 pp.

  • Haertel, P. T., and R. H. Johnson, 2000: The linear dynamics of squall line mesohighs and wake lows. J. Atmos. Sci., 57 , 93107.

  • Johnson, R. H., and P. J. Hamilton, 1988: The relationship of surface pressure features to the precipitation and airflow structure of an intense midlatitude squall line. Mon. Wea. Rev., 116 , 14441473.

    • Search Google Scholar
    • Export Citation
  • Koch, S. E., P. B. Dorian, R. Ferrare, S. H. Melfi, W. C. Skillman, and D. Whiteman, 1991: Structure of an internal bore and dissipating gravity current as revealed by Raman lidar. Mon. Wea. Rev., 119 , 857887.

    • Search Google Scholar
    • Export Citation
  • Knupp, K. R., 2006: Observational analysis of a gust front to bore to solitary wave transition within an evolving nocturnal boundary layer. J. Atmos. Sci., 63 , 20162035.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 2003: Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics. Springer-Verlag, 260 pp.

  • Rottman, J. W., and J. E. Simpson, 1989: The formation of internal bores in the atmosphere: A laboratory model. Quart. J. Roy. Meteor. Soc., 115 , 941963.

    • Search Google Scholar
    • Export Citation
  • Simpson, J. E., 1997: Gravity Currents in the Environment and the Laboratory. 2nd ed. Cambridge University Press, 244 pp.

  • Stumpf, G. J., R. H. Johnson, and B. F. Smull, 1991: The wake low in a midlatitude mesoscale convective system having complex convective organization. Mon. Wea. Rev., 119 , 134158.

    • Search Google Scholar
    • Export Citation
  • Trexler, C. M., and S. E. Koch, 2000: The life cycle of a mesoscale gravity wave as observed by a network of Doppler wind profilers. Mon. Wea. Rev., 128 , 24232446.

    • Search Google Scholar
    • Export Citation
  • Vescio, M. D., and R. H. Johnson, 1992: The surface-wind response to transient mesoscale pressure fields associated with squall lines. Mon. Wea. Rev., 120 , 18371850.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Plot of p′ (hPa, dark solid curve), and the associated u′ predicted by the nonlinear IR (m s−1, gray dashed curve) and by the linear impedance relation (m s−1, gray solid curve), associated with the passage of (a) a sinusoidal pressure trough and (b) a sinusoidal pressure ridge. The disturbances have amplitude 1 hPa, period 2000 s, and an intrinsic speed of 25 m s−1.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 2.
Fig. 2.

As in Fig. 1, but for a linear pressure (a) fall and (b) rise.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 3.
Fig. 3.

The change with time of the parcel p′ (hPa, dark solid curve), u′ (light solid curve), and Du′/Dt (10−3 m s−2). The maximum u′ = 3.59 m s−1.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for a 1-hPa wave trough. The minimum u′ is −3.13 m s−1.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for a 5-hPa amplitude pressure ridge. Note that the parcel never reaches the ridge itself, attaining a maximum p′ = 2.4 hPa.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 6.
Fig. 6.

Plot of p′ (y axis) as a function of x at various times for the Lagrangian parcel numerical simulation (hPa, dark curve), the x position of the air parcel at each time (gray vertical line segment), and c of the parcel (vectors).

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 7.
Fig. 7.

For the simulation illustrated in Fig. 6, (a) the simulated u′ (m s−1, gray circles), and the u′ predicted by the negative root (m s−1, heavy dashed) and the positive root (m s−1, solid) of the nonlinear IR, and (b) the p′ (hPa) attained by a parcel entering the front of the disturbance, as a function of t.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 8.
Fig. 8.

(a) Reflectivity (dBZ) and (b) Doppler radial velocity (m s−1) from KBMX WSR-88D at 0.4° elevation at 1031 UTC 7 Mar 2008. Negative radial velocities are toward the radar; positive radial velocities are away from the radar. The radar is located at the origin.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 9.
Fig. 9.

(a) Time series of surface observations of u (m s−1, gray dots) and MSL pressure (hPa, solid curve) at KEET, on 7 Mar 2008, and (b) BMX WSR-88D radar-indicated vertical profile of u (m s−1) at 1050 UTC 7 Mar 2008, along azimuth 240° at a range of 5 km (near the center of the maximum wind perturbations due to the wave at that time).

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 10.
Fig. 10.

As in Fig. 8, but for the BMX radar at 2227 UTC 20 Dec 2007.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 11.
Fig. 11.

(a) Time series of surface observations of u (m s−1, gray dots) and (b) MSL pressure (hPa, solid curve) at KBHM, on 20–21 Dec 2007.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Fig. 12.
Fig. 12.

Surface observations of p (hPa, solid dark curve) and u (m s−1, gray dots) at the UAH MIPS surface berm location in Huntsville, AL, on 30 Jan 2008.

Citation: Journal of the Atmospheric Sciences 67, 10; 10.1175/2010JAS3457.1

Save
  • Bosart, L. F., and F. Sanders, 1986: Mesoscale structure in the megalopolitan snowstorm of 11–12 February 1983. Part III: A large-amplitude gravity wave. J. Atmos. Sci., 43 , 924939.

    • Search Google Scholar
    • Export Citation
  • Christie, D. R., K. J. Muirhead, and A. L. Hales, 1978: On solitary waves in the atmosphere. J. Atmos. Sci., 35 , 805825.

  • Coleman, T. A., and K. R. Knupp, 2009: Factors affecting surface wind speeds in gravity waves and wake lows. Wea. Forecasting, 24 , 16641679.

    • Search Google Scholar
    • Export Citation
  • Fujita, T. T., 1955: Results of detailed synoptic studies of squall lines. Tellus, 7 , 405436.

  • Gossard, E., and W. Munk, 1954: On gravity waves in the atmosphere. J. Meteor., 11 , 259269.

  • Gossard, E., and W. Sweezy, 1974: Dispersion and spectra of gravity waves in the atmosphere. J. Atmos. Sci., 31 , 15401548.

  • Gossard, E., and W. H. Hooke, 1975: Waves in the Atmosphere. Elsevier, 456 pp.

  • Haertel, P. T., and R. H. Johnson, 2000: The linear dynamics of squall line mesohighs and wake lows. J. Atmos. Sci., 57 , 93107.

  • Johnson, R. H., and P. J. Hamilton, 1988: The relationship of surface pressure features to the precipitation and airflow structure of an intense midlatitude squall line. Mon. Wea. Rev., 116 , 14441473.

    • Search Google Scholar
    • Export Citation
  • Koch, S. E., P. B. Dorian, R. Ferrare, S. H. Melfi, W. C. Skillman, and D. Whiteman, 1991: Structure of an internal bore and dissipating gravity current as revealed by Raman lidar. Mon. Wea. Rev., 119 , 857887.

    • Search Google Scholar
    • Export Citation
  • Knupp, K. R., 2006: Observational analysis of a gust front to bore to solitary wave transition within an evolving nocturnal boundary layer. J. Atmos. Sci., 63 , 20162035.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 2003: Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics. Springer-Verlag, 260 pp.

  • Rottman, J. W., and J. E. Simpson, 1989: The formation of internal bores in the atmosphere: A laboratory model. Quart. J. Roy. Meteor. Soc., 115 , 941963.

    • Search Google Scholar
    • Export Citation
  • Simpson, J. E., 1997: Gravity Currents in the Environment and the Laboratory. 2nd ed. Cambridge University Press, 244 pp.

  • Stumpf, G. J., R. H. Johnson, and B. F. Smull, 1991: The wake low in a midlatitude mesoscale convective system having complex convective organization. Mon. Wea. Rev., 119 , 134158.

    • Search Google Scholar
    • Export Citation
  • Trexler, C. M., and S. E. Koch, 2000: The life cycle of a mesoscale gravity wave as observed by a network of Doppler wind profilers. Mon. Wea. Rev., 128 , 24232446.

    • Search Google Scholar
    • Export Citation
  • Vescio, M. D., and R. H. Johnson, 1992: The surface-wind response to transient mesoscale pressure fields associated with squall lines. Mon. Wea. Rev., 120 , 18371850.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Plot of p′ (hPa, dark solid curve), and the associated u′ predicted by the nonlinear IR (m s−1, gray dashed curve) and by the linear impedance relation (m s−1, gray solid curve), associated with the passage of (a) a sinusoidal pressure trough and (b) a sinusoidal pressure ridge. The disturbances have amplitude 1 hPa, period 2000 s, and an intrinsic speed of 25 m s−1.

  • Fig. 2.

    As in Fig. 1, but for a linear pressure (a) fall and (b) rise.

  • Fig. 3.

    The change with time of the parcel p′ (hPa, dark solid curve), u′ (light solid curve), and Du′/Dt (10−3 m s−2). The maximum u′ = 3.59 m s−1.

  • Fig. 4.

    As in Fig. 3, but for a 1-hPa wave trough. The minimum u′ is −3.13 m s−1.

  • Fig. 5.

    As in Fig. 3, but for a 5-hPa amplitude pressure ridge. Note that the parcel never reaches the ridge itself, attaining a maximum p′ = 2.4 hPa.

  • Fig. 6.

    Plot of p′ (y axis) as a function of x at various times for the Lagrangian parcel numerical simulation (hPa, dark curve), the x position of the air parcel at each time (gray vertical line segment), and c of the parcel (vectors).

  • Fig. 7.

    For the simulation illustrated in Fig. 6, (a) the simulated u′ (m s−1, gray circles), and the u′ predicted by the negative root (m s−1, heavy dashed) and the positive root (m s−1, solid) of the nonlinear IR, and (b) the p′ (hPa) attained by a parcel entering the front of the disturbance, as a function of t.

  • Fig. 8.

    (a) Reflectivity (dBZ) and (b) Doppler radial velocity (m s−1) from KBMX WSR-88D at 0.4° elevation at 1031 UTC 7 Mar 2008. Negative radial velocities are toward the radar; positive radial velocities are away from the radar. The radar is located at the origin.

  • Fig. 9.

    (a) Time series of surface observations of u (m s−1, gray dots) and MSL pressure (hPa, solid curve) at KEET, on 7 Mar 2008, and (b) BMX WSR-88D radar-indicated vertical profile of u (m s−1) at 1050 UTC 7 Mar 2008, along azimuth 240° at a range of 5 km (near the center of the maximum wind perturbations due to the wave at that time).

  • Fig. 10.

    As in Fig. 8, but for the BMX radar at 2227 UTC 20 Dec 2007.

  • Fig. 11.

    (a) Time series of surface observations of u (m s−1, gray dots) and (b) MSL pressure (hPa, solid curve) at KBHM, on 20–21 Dec 2007.

  • Fig. 12.

    Surface observations of p (hPa, solid dark curve) and u (m s−1, gray dots) at the UAH MIPS surface berm location in Huntsville, AL, on 30 Jan 2008.

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