a. Budget technique

The first is the budget technique. In this approach, the entrainment rate is obtained by estimating all the sources and sinks of a tracer species, S(x, y, z, t) = S(z) + s(x, y, z, t), where S is the mean concentration at height z in the PBL and s a fluctuation in the mean. One of the terms is the entrainment flux (ws)_zi, where the overbar denotes an average over an area (or along a line) that is large (or long) enough to give a statistically significant value. As shown by Russell et al. (1999), assuming a discrete jump over an infinitesimal layer at the top of the PBL, δS, the entrainment velocity at the top of the PBL is

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where δS = S_zi+ − {S}, S_zi+ is the concentration just above the PBL and Q_s is the chemical source or sink of S. The curly braces { } denote an average throughout the PBL over the time period of measurement and z_i is the average depth of the PBL over the period of measurement. Russell et al. (1999) extended this approach to a two-layer system in which entrainment occurs in both directions across the top of the PBL, and the second layer, which is intermittently turbulent, extends up to a second interface, across which entrainment occurs only from the free atmosphere into the second layer. Boers et al. (1999) and Bretherton et al. (1995) also discuss applications and limitations of this technique. The other two techniques are described in the next two sections. Further details on implementing these two techniques are given by Lenschow (1996).

b. Flux/δS approach

The second is a direct measurement technique where a conserved (on a timescale of several hours or more) tracer is used to obtain flux profiles through a well-mixed PBL. The entrainment velocity is calculated from the ratio of the flux extrapolated to the top of the PBL (ws)_zi to the mean difference in tracer concentration across the top (Lilly 1968):

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This is the approach used by Kawa and Pearson (1989a). Comparing their two estimates for each flight (one using total water q_t and the other ozone as tracers), we see that in most cases they are in reasonable agreement—the average difference is 0.0007 m s⁻¹. However, both ozone and q_t have problems for this application. They both have significant concentrations in the overlying free atmosphere, which tend to vary significantly both horizontally and vertically because of the absence of mixing between decoupled layers that have diverse origins and the intermittency of events that produce vertical transport. Ozone in particular is generated mostly in the stratosphere and removed mostly at or near the earth’s surface so that the concentration just above the PBL is typically greater than within the PBL (e.g., Kawa and Pearson 1989b). The result is that the jump across the top of the PBL typically shows considerable horizontal variability. In addition, it has only a small sink at the surface [a deposition velocity, w_d ≡ −(wO^′₃)₀/O₃ ≃ 3 × 10⁻⁴ m s⁻¹, where (wO^′₃)₀ is the surface flux of ozone and O₃ is the mean concentration at some reference height in the surface layer; Kawa and Pearson (1989b)]. Another problem with using q_t is loss of liquid water from the cloud layer through drizzle.

The ideal characteristics of a tracer species would be 1) a horizontally homogeneous surface source; 2) a lifetime of 10⁴ < τ < 10⁶ s, short enough to ensure a large jump across the PBL top and minimize horizontal variations in concentration above the PBL and yet with a long lifetime as compared to the turbulence mixing time in the PBL z_i/w∗, where w∗ is the convective velocity scale; and 3) negligibly soluble in clouds. Two examples of naturally occurring scalars that have these characteristics are CH₃SCH₃ (dimethyl sulfide; DMS) and CH₃I (methyl iodide). In addition, of course, it is necessary to be able to measure fluxes of these species—preferably via the eddy-correlation technique (Lenschow 1995), which requires fast-response (≥5 Hz) sensors. Although eddy-correlation fluxes of DMS have not yet been measured directly, Hills et al. (1998) have developed a prototype fluorine-induced chemiluminescence DMS sensor that has sufficient sensitivity and time response for eddy flux measurement. In the absence of direct flux measurements, we do not have good spatial definition of DMS emission; however, it is closely related to oceanic concentration of DMS, which has been found to be reasonably uniform on a scale of hundreds of kilometers (e.g., Bates et al. 1987). No technique is currently available to measure eddy flux of CH₃I.

Straightforward eddy correlation is not the only possibility for measuring the entrainment flux. Other techniques exist that do not require such fast-response measurements (Lenschow 1995).

• Conditional sampling—collecting air samples in more than one reservoir according to some property of the vertical velocity. Eddy accumulation denotes collecting samples at a rate proportional to the instantaneous vertical velocity in two reservoirs—one for positive w and the other for negative w. The flux is proportional to the difference in concentration between the two reservoirs. A “relaxed” version of eddy accumulation collects samples at a constant rate in either of two reservoirs according to whether w is positive or negative.

• Intermittent or disjunct sampling—grabbing a sample quickly, then measuring the tracer concentration more leisurely. High-frequency response is maintained by the fast sampling. If the measurement is made once per integral scale, which is typically about 1 s for aircraft measurements, the random error due to finite sample length is not appreciably increased (Lenschow et al. 1994). Cooper and Shertz (1995) describe an airborne system that can be used for both eddy accumulation and intermittent sampling.

• Mixed-layer gradient or variance—using mean concentration measurements from a minimum of three levels through the mixed layer or variance measurements from a minimum of two levels to solve for the surface and entrainment fluxes using the top–down—bottom–up (TD–BU) gradient or variance relations, respectively, first proposed by Wyngaard and Brost (1984), and later refined by Moeng and Wyngaard (1984, 1989) using large-eddy numerical simulations (LES). The gradient relations can be integrated to solve for the surface and entrainment fluxes in terms of the two concentration difference measurements (Davis 1992; Davis et al. 1994), while the variance relations can be applied directly to the measured variances to obtain the entrainment velocity (e.g., Davis et al. 1997). The variance technique technique is, however, sensitive to possible mesoscale contributions to the variance, and hence to the long-wavelength cutoff used in estimating the variance.

1) DMS as a tracer

Examples of the use of DMS mean profiles are presented by Russell et al. (1999) and Lenschow et al. (1999). Because of the ∼1–2-day (10⁵ to 2 × 10⁵ s) lifetime of DMS, air above the PBL is likely to have negligible concentration of DMS. This is particularly true of the air overlying the stratus regions associated with the summertime subtropical anticyclone over the major ocean basins since the stably stratified subsiding flow has likely not been in contact with PBL air for many days previously. In this case, the budget equation for a chemically reactive species in a horizontally homogenous PBL reduces to

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where τ is the species decay time. Integrating from the surface to the mean top of the PBL z_i, and assuming steady-state conditions,

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Equation (4) can be solved for the flux at the top, (ws)_zi = −W_eδS. For DMS, we assume that S_zi+ = 0, so that δS ≃ −S. Therefore,

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For (ws)₀ = 2 pptv m s⁻¹, (≃6 μmole m⁻² day⁻¹), W_e = 0.005 m s⁻¹, τ = 2 days, and z_i = 10³ m, we obtain from (5) S ≃ 185 pptv; and from (4), (ws)_zi ≃ 0.93 pptv m s⁻¹. That is, the flux at z_i is typically about half the surface flux, which means that the entrainment velocity should be relatively well defined and easy to estimate.

We also note that the budget Eq. (3) can be integrated between two levels Δz = z₂ − z₁ and solved for the reactivity time constant τ,

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Kawa and Pearson (1989b) used this technique to estimate the lifetime of ozone (10–32 days) in the PBL during DYCOMS.

Finally, DMS has great potential as a tracer for addressing other problems. For example, deep convection provides a conduit for transporting PBL air to the upper troposphere on a timescale of an hour. Measuring DMS in the outflow regions of such convective events over the ocean could provide a means for estimating the amount of PBL air brought up to these levels. Another example is its use as a tracer for measuring TD–BU functions in the atmosphere. Thus far, these functions have been estimated from LES and laboratory convection tank experiments (Piper et al. 1995), but not in the atmosphere. Combining DMS with other species that have quite different behavior in the PBL provides the necessary contrast in structure that both top–down and bottom–up functions can be solved for simultaneously. Ozone provides a good contrast since it has a surface sink and typically a large negative entrainment flux.

2) Estimate of the sampling error

The error variance for a scalar flux measured over a finite sample length L is given by (Lenschow and Kristensen 1985)

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where ℓ_w is the integral length scale of the vertical velocity, and σ²_w and σ²_s are the variances of w and a scalar s. Lenschow and Stankov (1986) found empirically that

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where z∗ = z/z_i. We can estimate σ_s from the TD–BU formulation obtained by Moeng and Wyngaard (1984, 1989) using large-eddy numerical simulations that relate mixed-layer variance of a scalar to surface and entrainment fluxes,

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where f_b ≃ z^−0.9_*, f_t ≃ 3.1(1 − z∗)^−3/2, and f_bt ≃ 1.5 are the dimensionless bottom–up and top–down variance functions, and the TD–BU covariance function, respectively. Solving for the normalized scalar variance we obtain

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where α = (ws)₀/(ws)_zi.

This normalized variance is plotted in Fig. 1. Using the above case as an example, α = 2.2, and we let the convective velocity scale w∗ = 1 m s⁻¹ and z∗ = 0.5. The result is σ²_sw²_*/(ws)²_zi ≃ 24. This gives σ_s ≃ 4.5 pptv. The relative error for a 200-km flight path with σ_w = 0.7 m s⁻¹ is σ_F/(ws)_zi ⩽ 0.20. Thus, the random error variance due to a finite sample length for flux measurement of DMS can be made small enough that it is possible to obtain estimates of W_e with reasonable accuracy.

c. Divergence technique

From the continuity equation for an incompressible fluid,

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where A is the area enclosed by the integration path, w̃ is the area-averaged vertical velocity within the integration path, and υ_n is the horizontal velocity component normal to the path of integration. The most efficient aircraft flight path for this measurement is a circle since 1) a circle has the largest enclosed area of any closed curve of a given circumference (e.g., 27% more area than a square of the same perimeter) and 2) no time need be wasted in turns since the rate of turning is slow enough that there is no reduction in accuracy as may occur during a sharp turn. For a circular flight path of radius R,

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where V_n is the mean normal wind component averaged around the closed flight track. For constant divergence from the surface to height z, the mean vertical velocity at z is

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This vertical velocity estimate assumes that the horizontal velocity field is constant with time in the coordinate system of the measurement throughout the measurement period. We can most closely realize this assumption by flying the closed path in a Lagrangian framework—that is, flying the aircraft in a circular pattern where the center of the circle drifts with the mean wind.

1) Sampling errors

Two approaches to measuring the divergence through the PBL are to 1) fly a series of closed (circular) flight paths at several levels in the PBL and 2) obtain vertical soundings of the normal wind component at various points on the periphery of the area under consideration, for example, from dropwindsondes launched from an aircraft flying above the PBL or rawindsondes launched from a series of stations surrounding the area. The error variance of a variable υ (here we drop the subscript from υ_n, but have in mind still the horizontal velocity component normal to the flight path) is defined as

σ²υ, Lυ_Lυ²(16)

where L is the sample length and 〈 〉 denotes an ensemble average. For an ergodic variable measured by, for example, an aircraft, (16) can be reduced, for L ≫ ℓ_υx, to (Lenschow et al. 1994)

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where ℓ_υx is the integral length scale of υ along the flight path and σ²_υ is the variance of υ. The integral scale along the flight path is defined as

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where ρ_υx(ξ) is the autocorrelation function, which for L ≫ ℓ_υx, can be written as

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Lenschow and Stankov (1986) found that the integral scale for horizontal velocity components measured in the horizontal plane is

_υxz_i(20)

For typical marine stratocumulus, we assume σ²_υ ≃ 1 m² s⁻² and z_i = 1000 m. For a circular flight path of radius R = 29 km (at an airplane speed of 100 m s⁻¹, this takes one half-hour per revolution),

σ⁻¹(21)

We also note that eight revolutions would reduce this number to 0.025 m s⁻¹. Thus the standard deviation due to sampling length for eight revolutions is less than the value specified in (15).

The second approach, using soundings, suffers from large sampling errors unless an inordinate number of soundings is obtained. If the distance between soundings Δ ≫ ℓ_υx, then from the results of Lenschow et al. (1994), assuming vertical homogeneity the random error variance based on a set of instantaneous independent measurements over the horizontal distance L becomes

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Each sounding, however, may give more than one independent estimate of υ_n as it passes through the PBL. We can incorporate this by modifying (22) to read

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where n is the number of independent estimates of υ_n obtained per sounding.

We can estimate the sampling error of a set of soundings in the following way. The autocorrelation function between horizontal velocity components along the vertical direction, again assuming ergodicity, can be written as

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Admittedly, the assumption of ergodicity is less tenable in the vertical than in the horizontal since the velocity field undergoes changes both near the surface and near the top of the PBL. However, observations (e.g., Lenschow et al. 1980) show that through the bulk of the mixed layer, statistics of the horizontal velocity components vary only slightly with height.

The autocorrelation function can also be expressed in terms of the Fourier transform of the coherence function coh(k, ζ) and the spectrum of υ_n along the aircraft track, given by Φ_υ(k) (Panofsky and Dutton 1984). If we assume that the phase angle between the horizontal velocity components at different levels in the PBL is negligible (i.e., that the turbulent eddy structures are not tilted in the vertical),

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We assume a simple power spectrum of the form (Mann and Lenschow 1994)

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Lenschow and Kristensen (1988) proposed the following formulation for the coherence function:

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where the constant b = 0.7468. Davis (1992) estimated the coherence of w from two vertically stacked aircraft flying in formation (the experiment is described by Lenschow and Kristensen 1988). He found that for a_z ≃ 0.25, Eq. (27) is a reasonable fit to the coherence of w along the vertical axis, using the integral scale for w in the horizontal plane obtained by Lenschow and Stankov (1986), which is given by (8). In the middle of the mixed layer, ℓ_w/z_i ≃ 0.17. We assume that we can replace ℓ_υ in (27) with the midmixed-layer value of ℓ_w. This would be strictly true if the turbulence were isotropic; even for vertically “squashed” turbulence it is probably true within a factor of about 2 (Kristensen et al. 1983). Here we are interested only in an approximate estimate of the correlation function, so substituting ℓ_w/z_i = 0.17 along with (26) and (27) into (25) and numerically integrating, we obtain the autocorrelation function in terms of the normalized separation distance ζ/z_i, which is plotted in Fig. 2.

Integrating the vertical autocorrelation function (25) over the normalized separation distance, the integral scale for υ in the vertical direction is ℓ_υz/z_i ≃ 0.34, about 75% of ℓ_υx/z_i. Thus, based on this very approximate analysis there is no evidence for any strong asymmetry in the horizontal velocity structure through the mixed layer. The result indicates that in principle it is possible to obtain about three statistically independent estimates of υ per sounding through the PBL. [This also means that three aircraft flying coincident closed flight paths simultaneously (one near the surface, one in the middle, and the other near the top of the PBL) will make nearly independent measurements of υ_n, and that remote measurements of a vertical cross section of the normal component of the horizontal wind by, e.g., an airborne Doppler lidar or radar will approximately triple the sample size of an in situ measurement.]

Equating the error variance from continuous sampling (17) to that from instantaneous measurements (23) with the number of independent samples per sounding, n = 3, we obtain Δ ≃ 6ℓ_υx. That is, a sounding is required about every 3 km for a 1-km-deep PBL to match the sampling error from a continuous horizontal velocity measurement.

From (11), to measure W_e, we also need to measure dz_i/dt. For a 4-h period, to measure W_e to 0.001 m s⁻¹, the change in z_i must be measured to Δz_i ≃ 15 m. This accuracy is possible to achieve using either a downward-pointing lidar on a traverse over the area of interest or sufficient penetrations through the top of the cloud layer. We can estimate the requisite number of penetrations in the following way. Lenschow (1990) reports the standard deviation of cloud-top height measured by lidar for two cases in the California stratocumulus to be σ_z = 13 and 26 m. If we assume conservatively that σ_z ⩽ 30 m, then if z_i is normally distributed, the error for N independent estimates of z_i is σ_z/N. Thus, in general, a minimum of about four independent penetrations through the cloud top should be sufficient for estimating z_i to 15-m accuracy for California-type stratocumulus. In certain situations, however, estimating the change in PBL height may become more problematic since for σ_z = 60 m, the number of penetrations increases to 16.

2) Vorticity

The same technique discussed above can be used for measuring the circulation or the vertical component of vorticity within the closed flight path by substituting the velocity component parallel to the flight track υ_t (positive for counterclockwise rotation) for υ_n in (11),

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This may be useful also as an independent estimate of divergence, since the equation for the vertical component of absolute vorticity, neglecting the tilting and solenoidal terms and terms involving the horizontal momentum transport by turbulence, can be written as (e.g., Holton 1972)

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where f is the Coriolis parameter. Normally, f ≫ ζ_υ. Therefore, if we assume steady state and a linear decrease in the stress term with height to zero at z_i, integrate vertically, and approximate the stress term by a drag formulation, we obtain

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where C_D is the drag coefficient and U_m is the mean horizontal wind speed at the height for which the drag coefficient applies.

Ching (1975) applied this approach to measurements of rawinsonde wind profiles from ships obtained during the Barbados Oceanographic and Meteorological Experiment to estimate the drag coefficient. Here we point out that for the idealized situation indicated above, the divergence is approximately related to the vorticity by the relation

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Strong divergence occurs with anticyclonic circulation;in the Northern Hemisphere, positive divergence implies negative vorticity. For typical values in a nonequatorial PBL, (31) indicates that the magnitude of the vorticity is typically from several times to an order of magnitude larger than divergence and thus can be measured more accurately.

3) Instrumental errors

We now consider errors in the mean divergence measurement from the perspective of likely errors in the airborne sensors. The actual horizontal wind measured by an aircraft is obtained from the difference between the velocity of the airplane in a geographic reference frame [measured, e.g., by an inertial navigation system (INS) with long-term corrections provided by the Global Positioning System (GPS)] and the velocity of the air in the airplane coordinate system (obtained, e.g., from differential pressure measurements between ports on a nose radome). Here we are interested only in the component normal to the airplane flight track υ_n. Errors in the airplane velocity measurement (from INS updated with GPS) tend to be independent of aircraft heading. Therefore, in considering their impact, we consider the simple case of a horizontal velocity field that is constant in direction but increases linearly in the x direction, U = U₀ + U₁(x/R), as shown in Fig. 3. (Since the flight path is assumed to be circular, the wind direction we choose is arbitrary.) The normal wind component in polar coordinates is given by

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The divergence for this case is

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Assuming constant divergence through the PBL and a required measurement accuracy of W_zi = 0.001 m s⁻¹, for R = 29 km (30-min revolution), and z_i = 1000 m, the required accuracy is U₁ = 0.03 m s⁻¹, which is equivalent to ∼54 m per revolution. Currently available GPSs can measure to about 100 m absolute accuracy, which therefore seems marginal for estimating the horizontal airplane velocity averaged over a circle with R = 29 km. We point out, however, that higher-accuracy GPS information (which GPS is inherently capable of providing with the approval of the controlling governmental agency) would then be essential.

The next step is to consider errors in angle measurement. Probably the simplest error to consider is an error in the angle measurement of the normal wind component of a constant value ε, which might be due to, for example, a misalignment between the INS and the wind-sensing instrumentation, which gives the measured divergence

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Thus, for a relative error of ⩽10%, ε ⩽ 26° per revolution, so that an error from this source is not significant.

However, if we introduce a time-dependent error, or drift in the angular misalignment of ε(t) = αθ, then for small α, it can be shown that, by setting sinαθ ≃ αθ and cosαθ ≃ 1,

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We let U₀ = 5 m s⁻¹. Then for a relative error of ⩽10%, ε ⩽ 0.33° per revolution. This is a much more significant source of error and is comparable to the accuracy of the INS measurement of true heading, which is the most likely source of such a drift.

Another consideration with a circular flight path is the magnitude of the roll angle ϕ that results from a constant-rate turn. This can be easily estimated by equating the centrifugal force mU²_a/R, where U_a is the true airspeed and m the mass of the aircraft, to the component of the lift force that is opposite to the centrifugal force, mg tanϕ, where g is the gravitational acceleration. Solving for the roll angle,

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For U_a = 100 m s⁻¹ and R = 29 km (equivalent to a 30-min flight time around the circle), ϕ ≃ 2°, which would have negligible effect on the accuracy of the air velocity measurement.

Finally, we consider errors in the measurement of the velocity of the air with respect to the aircraft. For divergence measurement, this is very nearly the horizontal component of the air velocity normal to the longitudinal axis of the aircraft (or the lateral component) since the difference between the aircraft heading and track angle is small. Currently, a standard technique for obtaining this measurement is to use pressure difference measurements from ports on a radome mounted on the nose of the aircraft to calculate the flow angle, called sideslip, β. The normal air velocity component with respect to the aircraft is then ≃βU_a. Certainly the absolute accuracy of this measurement is greater than 0.006 m s⁻¹, primarily since the system is calibrated by flight maneuvers, which probably cannot be used to achieve absolute accuracies better than a few tenths of a meter per second because of spatial and temporal variability in the wind field. However, the effect of a mean offset in β can probably be mostly cancelled out by flying circles in opposite directions. The remaining error is difficult to quantify and remains a question. A similar result holds for the effect of the true airspeed, which is also obtained from a differential pressure measurement (corrected for air density), on the longitudinal component of air velocity, and thus on the vorticity measurement.

An alternative approach for air motion measurement is a side-looking Doppler laser system, sometimes called a velocimeter. A forward-looking laser velocimeter was successfully flown on the NCAR Sabreliner (Keeler et al. 1987), and design criteria for a more complex scanning laser velocimeter are discussed by Kristensen and Lenschow (1987). The advantages of this approach are 1) the measurement volume can be displaced several meters away from the aircraft where the flow is not perturbed by the aircraft and thus flight maneuvers are not required to calibrate the sensor, 2) Doppler shift is inherently an absolute measure of velocity with no hysteresis or environmental sensitivities, and 3) cloud droplets should not affect the accuracy. This is an attractive option that offers the potential of taking full advantage of the recent improvement in measurement accuracy of the airplane velocity through GPS. Currently, air motion measurement accuracy is the limiting factor in wind measurement from aircraft. Improved wind measurement would have application to many other research problems as well, such as measuring mesoscale wind fields, aerodynamic transfer coefficients, and mean advection.

d. Optimal flight track

Summarizing the results of the previous sections into a reasonable scenario for an observational study, the requirements for an optimal flight pattern to measure entrainment velocity in a marine stratocumulus regime are as follows.

• The three independent techniques for estimating W_e should be simultaneously implemented with a single aircraft flight pattern.

• The flight pattern should be as close to optimal as possible for all the techniques.

• The flight pattern should permit correlative measurements essential to a complete documentation of the processes involved in generating entrainment.

a. ACE-1 case study

The Aerosol Characterization Experiment (ACE-1), carried out primarily over the ocean south of Australia during November–December 1995 (Bates et al. 1999), afforded an opportunity to test the techniques proposed here for measuring entrainment, divergence, and vorticity. One objective of ACE-1 was to estimate budgets of trace atmospheric constituents in the PBL over time periods of a day or more using the NCAR C-130 aircraft. This required a series of flights in the same air mass whose trajectory was labeled by means of constant-level balloons released from a ship. These sets of flights are called Lagrangian experiments. Strictly speaking, the trajectory is not truly Lagrangian, since the balloons do not stay in the same air mass, but rather stay at approximately the same altitude. However, because of efficient mixing, the PBL normally has a fairly uniform velocity, and the trajectory should be close to the trajectory of the mean PBL flow. An important term in the constituent PBL budgets is the contribution of entrainment through the top of the PBL, for which the circular flight patterns, centered on a balloon’s position as determined by GPS, were employed. Russell et al. (1999) have already used the C-130 measurements during the second set of Lagrangian measurements (flights 24–26) to compare entrainment rates calculated by all three techniques, and Wang et al. (1999a) have compared the flux and divergence techniques for the first Lagrangian period (flights 18–20).

To illustrate the calculation of divergence and vorticity for these circular flight patterns, we focus on flight 18 of ACE-1, the first flight of the first Lagrangian experiment. The NCAR C-130 flew southwest of Tasmania with the first circle centered near 44.8°S, 143.5°E. Over a 6-h period on 1 December 1995, the C-130 completed two sets of circles while drifting with the ambient wind so that the C-130 sampled nearly the same air mass throughout the flight. Each set consisted of five circles at various levels in the lower troposphere, with one in the free troposphere, in a quasi-Lagrangian frame of reference as estimated from an approximately constant-level balloon. The pressure altitude and flight track are shown in Fig. 4, with the circle number indicated on the pressure altitude plot.

A detailed account of the dynamics and thermodynamics of the PBL for flight 18 can be found in Wang et al. (1999a) hereafter W99a; Wang et al. 1999b. To summarize, the Lagrangian began in an air mass behind a weakening front. The flight logs note a complex PBL structure with decoupling occurring at the top of a surface-based turbulently mixed layer and two layers of cloud (broken stratocumulus and scattered cumulus). A change in horizontal wind of over 10 m s⁻¹ was observed across the lower troposphere below the free atmosphere. Details of the synoptic meteorology and its impact on the Lagrangian trajectory can be found in Businger et al. (1999) and Siems et al. (1999; hereafter S99).

The set of divergence and vorticity estimates obtained for each circle is shown in Figs. 5a and 5b, where the solid dots indicate clockwise and the squares counterclockwise circles. Here, and in subsequent figures and tables showing ACE-1 data, we have reversed the sign of the vorticity so that the results conform to what would be expected in the Northern Hemisphere. Numbers next to the points indicate the circle number. We calculated means and standard deviations of the divergence and vorticity for flight 18, excluding circles 5 and 10, which were above the PBL. The standard deviations (the values below preceded by ±) were obtained by fitting a least squares linear fit separately to the set of four clockwise and counterclockwise divergence and vorticity measurements versus static pressure, then calculating the standard deviations of the remaining fluctuations. These standard deviations were then used to calculate the standard deviations of a set of eight estimates, using the assumption that the ratio of the sampled mean to the sampled variance follows a Student t-distribution (e.g., Bendat and Piersol 1971).

The resulting divergence is 5.1 ± 5.4 × 10⁻⁶ s⁻¹. In this case, even the sign of the mean value has little significance; however, it is consistent with our expectation of a positive divergence in a postfrontal environment as the cold air from higher latitudes subsides behind the front. The average vorticity for the same set of circles is 1.9 ± 0.7 × 10⁻⁵ s⁻¹ (Fig. 5b). In this case, the sign is significant; a clockwise (cyclonic) rotation, which is also consistent for airflow just behind a front in the Southern Hemisphere, but in disagreement with Eq. (31), which predicts counterclockwise rotation, likely because the terms neglected in obtaining (31)—that is, the time rate-of-change, tilting, and solenoidal terms—are significant in a situation such as this close to a front.

From (13), the normal wind component averaged around the closed flight track calculated from the estimated divergence error is V_n ≃ ±0.075 m s⁻¹. This is about three times the theoretical sampling error estimated earlier for a set of eight circles. In the next section, we examine some reasons for the measured error being larger than predicted for this case.

b. Sensitivity to circle closure and mesoscale variability

As discussed earlier, neglecting instrument errors, the C-130 measurements should be sufficiently accurate to estimate the divergence and vorticity. There are, however, some further aspects specific to this case that need to be considered. First, we consider a particular circle from flight 18. Figure 6 shows the first circle from this flight plotted with the trajectory corrected for the mean wind measured by the aircraft. It would be highly unlikely for the circle to be precisely closed; however, the integrals in Eqs. (28) and (12) are still valid provided we can reasonably estimate a “closure point” near the end of the circle that minimizes any extraneous contribution to the closed integrals from overlap between the beginning and the end of the circle. Furthermore, (28) and (12) are valid for any closed curve. The criterion we used here for “closing” the circle was to integrate around the circle to the point of minimum distance from the starting point. The shortest distance we achieved was 13 m; in a few extreme cases during ACE-1 circles had gaps between starting and ending of over 1000 m. Figure 7 shows the running calculation of the divergence and vorticity for our example, which was within 175 m, or 0.1% of the total circumference, of closure.

To test the sensitivity of the calculations to the error in closure, we consider small changes to the length of the flight track. Table 1 shows divergences and vorticities as a function of time for this circle. At an aircraft speed of 100 m s⁻¹, a 10-s error leads to a distance error of 1 km. Table 1 illustrates that for this case, the relative change in divergence with closure distance is larger than for vorticity, even though the actual change in vorticity is larger. This is a reflection of the order of magnitude larger amplitude of the vorticity. In this particular case, Fig. 8, which zooms in on the last few minutes of Fig. 7, shows that the normal wind component passes through zero at about 1744 UTC, which minimizes the closure error. We also see that divergence and vorticity are smoothly varying functions of time, so that it is feasible to estimate circle closure with considerable accuracy up to a few hundred meters. For this example, if the circle is closed to within about 3 s, or 300 m, the contribution of nonclosure to the measurement error of divergence and vorticity is negligible.

We note that the divergence sensitivity can be minimized by orienting the aircraft along the wind direction at the beginning of the circle so that there will be little contribution to the flux divergence. At the same time, this maximizes the component along the flight path so that the vorticity will be maximally sensitive to the circle closure. The results indicate that adding a few seconds flight time at the end to overlap the starting point is useful.

In theory, we could have also calculated the divergence (and vorticity) from semicircles that begin and end with the lateral (longitudinal) winds perpendicular to the airplane heading. Beginning and end points of the semicircle calculations, however, are hard to estimate. Combining this with the increased sampling error means that we would have little confidence in the results.

Figures 7 and 8 also give some indication of how sensitive the measurements are to mesoscale variability in the wind field. The winds in flight 18 are characteristic of a relatively turbulent marine PBL containing mesoscale variations. In Fig. 9, the autocorrelation functions and the integral of the autocorrelation functions are shown for the circle from flight 18 shown in Fig. 7. Before calculating the autocorrelations, the zero mode plus the first three harmonics of the Fourier transforms of the time series are removed from each component to eliminate modulation of the time series by the circular flight path. An inverse transform is then performed on the Fourier spectra to obtain the autocorrelation functions. The maxima in the integrals are estimates of the integral scales of both the normal (lateral) and the along-flightpath (longitudinal) wind components (Lenschow and Stankov 1986). We see that the values of the maxima are 13 (∼1.3 km) and 29 s (∼2.9 km), respectively. We note that these values are a small fraction of the distance around the circle, so that the effect of aircraft turning should be minimal. The ratio is 0.46, which is not far off from their predicted ratio in isotropic turbulence of 0.5 (e.g., Batchelor 1953)—of course, this does not imply that the turbulence is isotropic.

For this case, W99b estimated the depth of the PBL at about 900 m. Lenschow and Stankov (1986) used the average of the longitudinal and lateral integral scales in their analysis, and obtained ℓ_υx/z_i = 0.45. Here we obtain ℓ_υx/z_i ∼ 2.3, which is about five times what they obtained. We suspect that in this case there is significant mesoscale variability, which can sometimes occur in the horizontal wind (e.g., if the C-130 flew through or under a patch of cumulus clouds, or across an ocean temperature discontinuity), which means that the sampling error in this example is larger than that predicted by (17). Lenschow and Stankov (1986) obtained their data from a more unstable PBL, which consequently likely had less mesoscale variability. The boundary layer in flights 13 and 19 (Table 2), for example, had such large mesoscale variability that we have little confidence in their estimated divergence and vorticity. Thus far we have not been able to determine the extent to which systematic errors in the air velocity measurements contribute to the observed error. This could be accomplished by carrying out a more focused flight program, as described in our conclusions.

The estimates for flight 18 are an example of a consistent set of measurements. Often the estimates from the ACE-1 flights are not as easily interpreted, both because of more complicated synoptic situations and because the flight patterns were not optimal for application of this technique. Since the measurements are generally not sufficiently accurate to easily detect changes with height, we have not looked into what measurement accuracy might be possible for changes with height. In this case, however, the vorticity shows a consistent increase with height of about 4 × 10⁻⁸ m⁻¹ s⁻¹, which seems significant. The difference between the clockwise and counterclockwise circles is due to biases in the differential pressure measurement for sideslip (divergence) and the Pitot-static pressure measurement for airspeed (vorticity), and illustrates the importance of flying in both directions.

Generally, the vorticity profiles showed considerably less relative scatter than the divergence, which is consistent with the vorticity being typically as much as an order of magnitude larger than the divergence (31). It is also possible that the assumption of constant divergence across the PBL is invalid. Divergences produced from numerical models often have gradients across the lower troposphere (e.g. W99a). If the divergence does vary across the PBL, this could have serious implications for many numerical studies of the PBL. These models are known to be highly sensitive to the prescribed divergence (Schubert et al. 1979). We note that an appropriately designed experiment using circular flight tracks may be able to address whether there is a random or systematic variability in the divergence and vorticity in the lower troposphere.

c. Summary of ACE-1 measurements

Since ACE-1 was the first field experiment to extensively use circular flight paths for measuring divergence and vorticity in the PBL, we summarize the results in Table 2. Only flights with four or more PBL circles are included, and a few flights were rejected because the set of circles had problems (e.g., one clockwise and three counter-clockwise circles, or circles not in a Lagrangian framework). Circles from the free troposphere were not included.

Ideally, we would like to compare the C-130 measurements with an independent measure of the divergence and vorticity taken on the same scale. Unfortunately, that is not possible because of the remote location and the Lagrangian approach to the measurements. Our alternative is to compare with synoptic-scale calculations based on the global analyses used to initialize numerical weather prediction (NWP) models. These are the values often used in numerical models of the PBL. From the horizontal wind analyses, it is straightforward to derive the divergence and vorticity (we note again that the sign of vorticity is changed so that Southern Hemisphere results conform to what is expected in the Northern Hemisphere):

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Table 2 also contains vorticity and divergence calculations from the wind fields of the Australian Bureau of Meteorology Global Assimilation Prognosis (GASP) NWP model (Bourke et al. 1995). There is little agreement between the two techniques, especially for the divergence, as only one flight shows good agreement for the divergence. The vorticity shows some agreement in six of the nine flights. Furthermore, the average vorticity over the nine flights of −1.8 × 10⁻⁵ s⁻¹ is in reasonable agreement with the NWP model average of −2.2 × 10⁻⁵ s⁻¹. However, the ratio of the average measured divergence to the average vorticity is about −0.6—considerably smaller in magnitude than that predicted from (31). We anticipate the better agreement for the vorticity as it is also a more robust measurement for calculations based on the global NWP. Equation (39) typically leads to a small difference between two relatively large numbers, while the vorticity (38) is typically a summation, which results in a smaller error.

The NWP-based estimates suffer from additional problems. First, they are particularly sensitive to the initial position and time. Changing the height of the calculations from 500 to 1000 m leads to a change of as much as 3 × 10⁻⁶ s⁻¹ in the divergence in some flights. European Centre for Medium-Range Weather Forecasts (ECMWF)-based divergences also varied strongly with altitude for the ACE-1 Lagrangians, as shown in W99a. These divergences may also vary in time as noted by Bretherton et al. (1995) for ECMWF-based calculations for the ASTEX Lagrangians.

Second, the difference in scales between the in situ measurements and the NWP measurements may also be a problem. At best, measurements based on global NWP datasets are smoothed to at least the resolution of the model, typically on the order of 1°. In this region of the globe, however, observations are sparser. The global analyses usually apply only to synoptic-scale variations, which means that smaller-scale structures, such as fronts, are underresolved. Flights 13, 18, and 19 were conducted in the neighborhood of fronts and have poor agreement in vorticity calculations in Table 2. If the divergence or vorticity has a mesoscale variability, we would not be able to see it in the NWP-based calculations. This explains why the in situ calculations of divergence can be an order of magnitude greater than the synoptic-scale values. We also note that there is little advantage in using smaller-scale, nested models to derive the divergence and vorticity since these models still depend on the global analysis for boundary conditions and initialization in this region of the globe.

We can average over some of the temporal and spatial variability by considering the divergence and vorticity observed on numerical trajectories. Here we used the change in altitude of a trajectory to obtain the divergence and the rotation of a cluster of trajectories to obtain the local vorticity. Following S, we consider numerical trajectories calculated from three independent global NWP models: the Australian Bureau of Meteorology GASP model, the National Oceanic and Atmospheric Administration (NOAA) aviation model and the NOAA medium-range forecast model. The trajectories were produced by the HYSPLIT code (Draxler 1991); full details of these calculations are found in S99. For this paper, numerical trajectories were initialized at the center of the first circle for each flight at an elevation of 500 m and allowed to run for 6 h. Trajectories were based on the full three-dimensional winds.

Table 3 shows that the vorticity is reasonably consistent among the different models but the divergence is not. S99 found that the numerical horizontal motions of the trajectories were also consistent among the models. Errors typically were 5%–15% of the total trajectory length. The vertical motion of their trajectories, however, varied among the models so that there was little confidence in the divergence derived from global NWP models.

Finally we illustrate the calculation of the entrainment rate W_e using the divergence from a particular ACE-1 case. Boers et al. (1998) have already derived the divergences for flight 12, which they described as the best example of a classic well-mixed, cloud-topped marine PBL among the ACE-1 flights. They used three different methods to calculate the entrainment rates including Eq. (11). They also discuss the uncertainty in each technique and sources of potential error. First Boers et al. calculated the entrainment rate from the evolution of the thermodynamic budgets of the PBL (Bretherton et al. 1995). Using the liquid water potential temperature, they obtain W_e = 0.004 m s⁻¹; using the total water mixing ratio, W_e = 0.006 m s⁻¹. These calculations require estimates of surface fluxes, radiative exchange through the PBL and precipitation, all of which have inherent uncertainties. They also assume simple averages of quantities across the PBL, which sometimes may be inadequate because of the wind shear.

Next Boers et al. calculated W_e from (2) using ozone as a tracer and obtained W_e = 0.0033 m s⁻¹. However, there may be large uncertainties associated with both the jump across the inversion and the flux at cloud top (Bretherton et al. 1995). Finally, Boers et al. calculated the entrainment directly from Eq. (2). They found virtually no change in z_i, which was ∼1300 m over the course of the flight. Using global NWP data they found a divergence of 2.7 × 10⁻⁶ s⁻¹; thus W_e = 0.0035 m s⁻¹. In addition to the uncertainty in the divergence, the change in the inversion height was based on C-130 soundings. This method has a considerable uncertainty due to variations in cloud-top height. Examining the lidar legs for these flights confirms that the height of the inversion was unchanging, although it seemed to be closer to 1100 m. In Boers et al., this last technique is no more accurate than either of the first two. The divergence they use, 2.7 × 10⁻⁶ s⁻¹, is roughly three times greater than the GASP values we have in Table 3 because of using different levels for their calculations. Using the circles to calculate the divergence (Table 2) leads to a divergence of 5.1 × 10⁻⁶ s⁻¹. This is approximately twice that calculated by Boers et al., so that our entrainment velocity would be twice what they derived.