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  • View in gallery

    The log of the global mean variance of tracer mixing ratio from simulations initialized 17 Oct 1991 at 350, 460, 850, and 1900 K. The e-folding times at each level were calculated using a linear least squares fit shown by the dashed lines.

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    Scatterplots of TrEL for two tracer advection runs, initialized separately on 17 Oct 1991 and 1 Jun 1998. Both runs were initialized at 350 K with mixing ratio equal to sine(latitude) and were advected with Met Office winds. The normalization operator [Eq. (3)] was applied to each time step.

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    (a) Lambert equal area plots of TrEL at 850 K covering the Northern Hemisphere for 16, 20, and 26 Dec. The outer edge of each circle is the equator. Row 1 shows spectral scheme T96 resolution, Met Office winds. Row 2 shows van Leer scheme, n = 5 resolution, Met Office winds. Row 3 shows van Leer scheme, n = 7 resolution, Met Office winds. Row 4 shows van Leer scheme, n = 5 resolution, GEOS-DAS 24-hourly winds. Row 5 shows van Leer scheme, n = 5 resolution, GEOS-DAS 6-hourly winds

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    (b) Same as (a) but covering the Southern Hemisphere

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    The rms difference for each of the sensitivity tests (see Table 1) calculated at 850 K over the month of Dec 1998 and in 2° equivalent latitude bins.

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    Lambert equal area plots of TrEL at 460 K over the Northern Hemisphere for 1 Mar–10 May 1994 at 10-day intervals. The outer edge of each circle is the equator. The 60° TrEL contour is highlighted in black

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    Same as Fig. 5, but for PVEL

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    Scatterplots between TrEL and PVEL at 460 K for 1 Mar–10 May 1994 at 10-day intervals. The plots cover the range from 45° to 90° equivalent latitude. The correlation coefficient for data in this range is provided above each frame.

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    (a) Lambert equal area map of PVEL at 460 K for 11 Mar 2000. Overlaid are locations of POAM (black dots) and ER-2 (blue dots) measurements along with black lines for the inner (solid) and middle (dashed) vortex edge. (b) Ozone vs PVEL for 11 Mar 2000. Only data contained within a 10° bin centered at 460 K are included here. Lines mark inner (solid) and middle (dashed) vortex edge. (c), (d) Same as (a) and (b) but using TrEL. The sloping dashed line in (d) is a linear least squares fit to the POAM data from 65° to 90°.

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    Column ozone from earth probe TOMS (left), TrEL reconstruction (middle), and PVEL reconstruction (right) for 30 Nov–2 Dec 1999. The TrEL reconstruction was produced with the van Leer model at n = 6 resolution (approximately 120 km) in order to more closely match the resolution of the TOMS data. Correlation coefficients for TrEL/TOMS and PVEL/TOMS are provided above each plot.

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    Scatterplots for TOMS level 2 data vs TrEL and PVEL reconstructions on 30 Nov 1999 over the Northern Hemisphere. The reconstructed data are interpolated to the TOMS level 2 data points (in space and time) to account for the asynoptic TOMS sampling. The TOMS points are subsampled by a factor of 10. Column ozone is calculated from the proxy using the method described in Allen and Nakamura (2002).

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Tracer Equivalent Latitude: A Diagnostic Tool for Isentropic Transport Studies

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  • 1 Remote Sensing Division, Naval Research Laboratory, Washington, D.C
  • | 2 Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois
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Abstract

Area equivalent latitude based on potential vorticity (PV) is a widely used diagnostic for isentropic transport in the stratosphere and upper troposphere. Here, an alternate method for calculating equivalent latitude is explored, namely, a numerical synthesis of a PV-like tracer from a long-term integration of the advection–diffusion equation on isentropic surfaces. It is found that the tracer equivalent latitude (TrEL) behaves much like the traditional PV equivalent latitude (PVEL) despite the simplified governing physics; this is evidenced by examining the kinematics of the Arctic lower stratospheric vortex. Yet in some cases TrEL performs markedly better as a coordinate for long-lived trace species such as ozone. These instances include analysis of lower stratospheric ozone during the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment (SOLVE) campaign and three-dimensional reconstruction of total column ozone during November–December 1999 from fitted ozone-equivalent latitude relationship. It is argued that the improvement is due to the tracer being free from the diagnostic errors and certain diabatic processes that affect PV. The sensitivity of TrEL to spatial and temporal resolution, advection scheme, and driving winds is also examined.

Corresponding author address: Dr. Douglas Allen, Naval Research Laboratory, Code 7227, 4555 Overlook Avenue SW, Washington, DC 20375. Email: drallen@nrl.navy.mil

Abstract

Area equivalent latitude based on potential vorticity (PV) is a widely used diagnostic for isentropic transport in the stratosphere and upper troposphere. Here, an alternate method for calculating equivalent latitude is explored, namely, a numerical synthesis of a PV-like tracer from a long-term integration of the advection–diffusion equation on isentropic surfaces. It is found that the tracer equivalent latitude (TrEL) behaves much like the traditional PV equivalent latitude (PVEL) despite the simplified governing physics; this is evidenced by examining the kinematics of the Arctic lower stratospheric vortex. Yet in some cases TrEL performs markedly better as a coordinate for long-lived trace species such as ozone. These instances include analysis of lower stratospheric ozone during the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment (SOLVE) campaign and three-dimensional reconstruction of total column ozone during November–December 1999 from fitted ozone-equivalent latitude relationship. It is argued that the improvement is due to the tracer being free from the diagnostic errors and certain diabatic processes that affect PV. The sensitivity of TrEL to spatial and temporal resolution, advection scheme, and driving winds is also examined.

Corresponding author address: Dr. Douglas Allen, Naval Research Laboratory, Code 7227, 4555 Overlook Avenue SW, Washington, DC 20375. Email: drallen@nrl.navy.mil

1. Introduction

Daily isentropic maps of Ertel potential vorticity (PV) are very useful for analyzing the motion of air in the upper troposphere and stratosphere (Hoskins et al. 1985). This owes to the fact that PV is approximately conserved under adiabatic motion on synoptic timescales and that it maintains sufficient gradients on the isentropic surface. In fact, its monotonic pole-to-pole gradients allow PV to be used as a marker of “latitude.” The PV area equivalent latitude (PVEL) may be defined as
i1520-0469-60-2-287-e1
where A = A(q, θ, t) is the area in which PV is less than q on a particular isentropic surface with potential temperature θ at time t, and a is the radius of the earth. This would be the limiting latitude if A were reshaped into a pole-centered circle. To the extent that isentropic winds are divergence free and that PV is materially conserved, ϕq is also conserved. Since ϕq itself is moving with the air, effects of adiabatic transport can be removed when a quantity is measured in the (ϕq, θ) coordinate.

The use of PV or PVEL as a coordinate first entered the scene through the analysis of stratospheric sudden warming and associated mixing (McIntyre 1980; Dunkerton et al. 1981; McIntyre and Palmer 1983; Butchart and Remsberg 1986). Later their utility has been expanded greatly. Exploiting the correlations between PV and mixing ratios of long-lived trace constituents [see Danielsen (1968) for PV–ozone correlation], Schoeberl et al. (1989, 1992) and Lait et al. (1990) successfully reconstructed synoptic maps of constituents from incomplete observation. PVEL fields have further been incorporated into studies of polar vortex processes, three-dimensional model initialization, and isentropic cross-tropopause tracer exchange (e.g., Manney et al. 1999b; Lary et al. 1995; Seo and Bowman 2001).

There are several limitations, however, to using PV or PVEL as a horizontal coordinate. First, PV is constructed from a rather complicated mathematical manipulation of observed or assimilated winds and temperature (Newman et al. 1989). In doing so, analysis errors are convoluted, causing PV to have a noisy appearance. Although the noise does not affect appreciably the PV–PVEL relationship q(ϕq) for a given θ and t, errors in PVEL at a given geographical location and time will be amplified when PV gradients are weak, since
i1520-0469-60-2-287-e2
where δϕq and δq denote errors in PVEL and PV, respectively. In addition, the quality of winds and temperature degrades with altitude, making PV considerably noisier in the upper stratosphere (Manney et al. 1996). Furthermore, the resolution of PV is limited by the resolution of the wind and temperature fields. Finally, PV is not conserved isentropically beyond the timescale in which diabatic effects are negligible, or where mechanical forcing is significant. The forms of local sources and sinks are complicated, and it is difficult to distinguish them from diagnostic errors (i.e., the analysis errors in winds and temperature that propagate into the PV calculation).

Attempts have been made to “improve” the effective resolution of PV through trajectory calculations. The domain fill (DF) approach (O'Neill et al. 1994) and the reverse domain fill (RDF) approach (Sutton et al. 1994; Schoeberl and Newman 1995) have proved particularly useful in generating realistic small-scale features. However, Fairlie et al. (1997), in an analysis of nitrous oxide data along ER-2 flight tracks, showed that although the RDF calculation produced increased small-scale structure compared to analyzed PV, the RDF product showed no statistically significant improvement over analyzed PV in forecasting tracer structure. They argued that small-scale noise in PV and errors in advecting winds adversely affect the small-scale structure of trajectory mapped fields. The contour advection with surgery (CAS) technique has also been adopted for routinely for generating high-resolution PV-like fields (Waugh and Plumb 1994; Norton 1994). Methven and Hoskins (1999) showed that errors in the PV analyses used for initialization strongly limit the smallest scale for believable features in this approach. They conclude that even for very high-resolution winds, only filaments wider than 50 km are believable. Recent work by Dragani et al. (2002) examined the predictive skill of both the RDF and CAS techniques in simulating ozone structure along aircraft flight tracks. They found that when the measurements clearly show evidence of filamentary tracer structure these techniques can improve the statistical agreement between simulated and measured tracer profiles over that obtained using analyzed PV.

This paper investigates an alternate way of producing high-resolution equivalent latitude maps for tracer analysis, based on a numerical synthesis of a PV-like tracer by solving the advection–diffusion problem on isentropic surfaces (Haynes and Shuckburgh 2000a; Allen and Nakamura 2001; Lee et al. 2001). Although the initial value problem approach is computationally more intensive than the instantaneous calculation of PVEL, the obtained tracer equivalent latitude (TrEL) has several advantages over PVEL. First, since the tracer is predicted explicitly, diagnostic errors that result from the PV calculation are avoided altogether. Second, TrEL is unaffected by nonconservative processes (diabatic heating and friction) that affect PVEL, making it a better tracer of adiabatic motion under certain circumstances, as we will show later. Third, TrEL can be calculated at arbitrarily high resolution, whereas PVEL is limited to the resolution of the analyzed winds and temperatures (although RDF techniques effectively enhance the resolution of PV). Furthermore, unlike the trajectory calculations, the technique we propose here allows a long-term integration without reinitialization, and as will be shown, the effects of initial condition decay rather quickly. Thus, by continuously running the advection–diffusion model forward in time, TrEL can be updated accurately and economically.

This paper is outlined as follows. Section 2 explains how TrEL is calculated. The subtle role of numerical diffusion to remove the dependence on initial conditions is discussed. Section 3 tests the sensitivity of TrEL to four factors in the advection model: spatial and temporal resolution, advection scheme, and input wind fields. Section 4 demonstrates the utility and advantage of TrEL over PVEL in diagnosing stratospheric tracer transport within three common applications: tracer kinematics during Arctic vortex breakup, analysis of ozone in the Arctic vortex region, and three-dimensional ozone reconstruction. Section 5 provides a summary and discussion.

2. Calculation of TrEL

a. The advection–diffusion problem

The solution procedure for the advection–diffusion problem on a sphere follows Allen and Nakamura (2001). Briefly, the tracer is initialized on each isentropic surface with “mixing ratio” equal to the sine of latitude and is advected using the streamfunction (i.e., nondivergent wind only), which is spectrally derived from assimilated winds on isentropic surfaces. Unless otherwise stated, we use the Met Office assimilated winds (Swinbank and O'Neill 1994) to drive a finite volume model on an icosahedral triangular grid with the van Leer flux limiter (van Leer 1977; Peyret and Taylor 1983; Putti et al. 1990; Edouard et al. 1996) at a mean node separation of 240 km [n = 5, where the number of triangles covering the sphere varies as 20 × 4n, see Allen and Nakamura (2001) for details]. Sensitivity of TrEL to the input winds and model characteristics will be examined in section 3. The tracer mixing ratio predicted from the model is converted to a regular longitude–latitude grid and is saved once per day at 1200 UTC.

To calculate TrEL from the saved mixing ratios, we first calculate the area for each grid box. We sort the mixing ratios and areas into one-dimensional arrays ordered by increasing mixing ratio. Then, for each point in this array we calculate the cumulative area over the regions in which the mixing ratio is less than or equal to that for the given point. This area A is converted to equivalent latitude using Eq. (1). The one-dimensional TrEL array is then reformed back to the regular two-dimensional grid. This method efficiently produces equivalent latitude at the highest possible resolution. We produce PVEL using the same procedure with Met Office PV fields on isentropic surfaces.

b. Global homogenization and tracer normalization

The subgrid diffusion, though implicit in our finite volume model, slowly homogenizes the tracer. The rate at which this occurs varies with height due to varying production rates of small scales (stretching rates). To estimate the timescale at which the gradients diminish, the global mean tracer variance was calculated for each day of a multiyear run initialized 17 October 1991. Figure 1 shows that the log of this variance decays nearly linearly with time. The e-folding decay times were calculated to be 114, 343, 257, and 113 days at 350, 460, 850, and 1900 K (approximately 12, 20, 30, and 50 km), respectively. The diminishing gradients do not affect the tracer shape and hence TrEL, but eventually the mixing ratios become comparable to the machine precision, at which point the finite-volume approximation is subject to significant truncation errors.

In their tracer simulations, Haynes and Shuckburgh (2000a) avoided facing this problem by limiting the integration to 4 months, but they suggested that runs could probably be done on timescales of a year without needing reinitialization. For longer runs, Shuckburgh et al. (2001) used a weak relaxation toward the initial sine(latitude) profile with a timescale of 120 days.

In this study the truncation errors are avoided by normalizing the tracer at each time step using the following linear operator:
i1520-0469-60-2-287-e3
where q is the tracer mixing ratio and qmin and qmax are the minimum and maximum values of q. This operator stretches the mixing ratio to maintain maximum and minimum values of ±1.0 but does not affect TrEL. We found that the two TrEL fields at 350 K (initialized on 17 October 1991) calculated with and without the normalization are virtually identical until late 1998, when the unnormalized tracer starts to be contaminated by truncation errors, but the normalized tracer continues to evolve smoothly.

c. Spinup period

Since TrEL is calculated as a solution of an initial value problem, its dependence on initial conditions is of critical interest. The numerical diffusion again plays a subtle but important role by dissipating the memory of the initial conditions in TrEL.

To demonstrate this, we compare the evolution of two tracers at 350 K that are initially out of equilibrium, but are subjected to the same flow field. The first tracer is initialized with sine(latitude) on 17 October 1991 and advected to the end of year 1998 using Met Office winds. The second tracer is initialized with sine(latitude) on 1 June 1998 and is also advected to the end of 1998. Figure 2 shows scatterplots of the first run (TrELoct) versus the second run (TrELjun) at 40-day intervals. Initially (1 June 1998), there is a significant degree of scatter at all equivalent latitudes due to zonal asymmetries in TrELoct. By 11 July the scatter has decreased sharply except for a band from 10°–30°N. The scatter in this band persists into August, while that outside the band reduces dramatically. By 18 December there is virtually no scatter between the two fields; the tracers are nearly perfectly aligned.

It is evident that the numerical diffusion is responsible for the loss of scatter, because two tracer fields simply advected by the same wind will not change their mutual relationship. The scatter diminishes when the difference of the two solutions decays faster than the solutions themselves. A typical scenario for this would be an anisotropic flow in which shear dispersion assists the difference to cascade into smaller scales and hence to be dissipated fast by diffusion, whereas each solution retains similar large-scale variance in the cross-stream direction (Rhines and Young 1983).

Figure 2 suggests that the required length of the spinup period, the time it takes for TrEL to lose the memory of initial conditions, varies with flow conditions. We have thus repeated this analysis for different isentropic levels and different seasons for initialization. In most cases examined, the memory of initial conditions is found to decay on timescale of tens of days, an order of magnitude less than the global homogenization timescale. However, when the runs are initialized in a dynamically active period (e.g., Northern Hemisphere winter in the middle stratosphere), the tracer difference can actually grow before it starts to decay. At 350 and 460 K, the estimated e-folding decay time of the rms difference between two initially decorrelated tracers is around 10–50 days, consistent with Haynes and Shuckburgh (2000b) who estimated a spinup time of 30 days at 350 K in their simulated tracer field. A significantly longer spinup may be necessary for higher altitudes, depending on the timing of initialization. However, given advection over a complete annual cycle, our calculations show that removal of initial conditions will be complete at all levels. Note also that the obtained TrEL distribution is insensitive to the initial latitudinal profile [here sine(latitude)], as demonstrated by Haynes and Shuckburgh (2000a). If a reduced spinup time is desired, one could try initializing the tracer using analyzed PV, as in Lee et al. (2001, 2002). However, in this study, since we are directly comparing TrEL with PVEL, we do not want the TrEL field to be influenced by the analyzed PV.

3. Sensitivity tests

Since TrEL is numerically synthesized, it is important to evaluate its sensitivity to the model settings such as temporal and spatial resolution, input wind field, and advection scheme. To perform such tests we choose a dynamically active time period, December 1998 in the middle stratosphere (850 K). This period has strong mixing in the southern high latitudes due to the breakup of the Antarctic polar vortex. It also has strong activity in the northern high latitudes due to a major warming event (Manney et al. 1999a). Any model sensitivity of TrEL should be apparent during this active period, and the resulting differences are expected to be toward the upper limit of the possible errors.

For this section we initialize all runs on 1 December 1998 at 850 K using the tracer mixing ratio from a multiyear run, which has been integrated since 17 October 1991. We then advect the tracer until 31 December 1998, varying the parameter in question. Table 1 lists the model parameters used for each of the sensitivity tests.

a. Advection scheme (Met Office T96 versus Met Office n5)

We first test the sensitivity to the numerical scheme used to solve the advection–diffusion equation. Previous studies have compared tracers advected by different isentropic advection schemes. Most relevant to this analysis, Shuckburgh et al. (2001) compared results from a spectral model and the Single Layer Isentropic Model of Chemistry and Transport (SLIMCAT) model (Chipperfield 1999; Hansen and Chipperfield 1999), which uses the Prather advection scheme (Prather 1986). They showed very good agreement between normalized equivalent lengths (Nakamura 1996) calculated from the two tracer fields. Allen and Nakamura (2001) similarly compared equivalent lengths calculated from the van Leer and spectral models. They too found very good agreement between the two schemes at similar resolutions. Since equivalent lengths are equal for two tracers with identical equivalent latitude distributions (Allen et al. 1999b, appendix), these observations suggest that equivalent latitude distribution is indeed similar among tracers produced from different schemes. Here we confirm this by comparing TrEL obtained using a spectral transform model (T96 resolution with a constant second-order diffusion of 1.25 × 105 m2 s−2) and the van Leer model with n = 5, as described in Allen and Nakamura (2001).

Figure 3a shows Northern Hemisphere synoptic plots of TrEL for 16, 20, and 26 December 1998 for all of the runs described in Table 1. The top two rows of Fig. 3a show the spectral and van Leer model results. On 16 December, the vortex, indicated by high values of TrEL, is pushed off the pole and deformed into a “comma” shape. A large anticyclone (“Aleutian high”) is present, centered over northwestern Canada. A tongue of low TrEL is observed over Japan, encroaching on the higher latitudes. These features are similar in the two runs, but there are minor differences. For example, some of the high TrEL contours in the spectral run show a wavy pattern, indicative of minor numerical noise.

On 20 December the vortex is pushed farther off the pole with the tail stretched and thinned. Both runs show a similar vortex shape and a similar tongue of low TrEL that has moved poleward crossing over Alaska, the Arctic region, and into Asia. By 26 December the vortex has evolved into three distinct lobes. The spectral model shows more structure than the van Leer model; for example, the swirling in the upper lobe is more distinct in the spectral model. This indicates that the spectral model at T96 has a higher effective horizontal resolution than the van Leer code with n = 5. Comparing the spectral results with the van Leer model at n = 7 (Fig. 3a, row 3), one sees that the finescale features are indeed resolved by the van Leer code at higher resolution.

The results for the Southern Hemisphere are provided in Fig. 3b. This period follows the breakup of the Antarctic vortex, which is accompanied by vigorous mixing of vortex and midlatitude air. The TrEL distributions on 16, 20, and 26 December are very complicated, with the lowest TrEL air (the vortex remnant) contorted into an irregular mass displaced from the pole. Spectral and van Leer results (top two rows of Fig. 3b) are very similar for this case. One noticeable deviation is the scalloping that occurs at very high TrEL values in the spectral run, again indicative of minor numerical noise.

A plot of the monthly averaged rms difference between the two runs is shown in Fig. 4 (green line) for the entire globe. The difference is smaller in the Tropics than in the extratropics, likely due to less mixing there, and is slightly larger in the Southern than in the Northern Hemisphere. The difference is less than 5° everywhere except at the equivalent South Pole. To understand the implication of these differences, consider a TrEL field that has a contour displaced from the control run by a normal distance δr. The equivalent latitude difference between the two, δϕe, will be large where the gradient in TrEL is large, since
δϕeδrϕe
Thus, if the position of the tracer contour is off by δr, the above relation gives a rough estimate of difference in TrEL. Large δϕe does not necessarily mean large δr—it can simply reflect large gradient of ϕe. If contours of TrEL are evenly spaced, as in the zonally symmetric case, then the relationship between δr and δϕe is simply the spacing of (geographic) latitude contours, 111 km for each degree, so that a 5° rms difference indicates mean contour displacements of 555 km. Using this rough rule of thumb, we can translate equivalent latitude errors into displacement errors. Since 555 km is just over twice the horizontal resolution of the van Leer code (for n = 5), we do not consider rms errors on the order of 5° to be significant.

b. Spatial resolution (Met Office n5 versus Met Office n7)

In the stratosphere and upper troposphere small-scale structures in the passive tracer emerge primarily from smooth large-scale flow. Thus the details of tracer structure are rather insensitive to the spatial resolution of the wind fields (Waugh and Plumb 1994; Methven and Hoskins 1999). However, they are sensitive to the spatial resolution of the tracer itself. Allen and Namaura (2001) examined the equivalent length calculated from tracer fields with resolutions of n = 5, 6, and 7. They found that the magnitude of the equivalent length increases with model resolution as finer-scale features in the tracer field become resolved, although the overall large-scale structure of the equivalent length remains qualitatively the same.

Synoptic plots for the van Leer runs at n = 5 and n = 7 (240- and 60-km resolution) for the Northern Hemisphere are shown in Fig. 3a, rows 2 and 3. On 16 December, the two fields show similar vortex structure and similar tongue of low TrEL air over Japan. Inside the Aleutian high, the n = 7 run shows more swirling structure than n = 5. The n = 7 run also shows finescale filaments outside the vortex that cannot be resolved at n = 5. By 20 December the vortex shapes of the two runs are still similar, and both show a tongue of low TrEL air. Yet the n = 7 run shows a sharper structure for this tongue and lower values of TrEL. By 26 December differences are significantly enhanced as the n = 7 run shows considerably more structure, particularly in the swirling of high TrEL air in the three lobes.

The rms difference between the two runs averaged over December 1998 is shown with a black line on Fig. 4. The differences are generally less than 5° in the NH and increase to over 10° at the equivalent South Pole. The larger rms differences in the Southern Hemisphere are due to the vigorous mixing that occurs at 850 K during the breakup of the Antarctic vortex. Synoptic plots for the Southern Hemisphere are provided in Fig. 3b, rows 2 and 3. Here the enhanced structure in the high-resolution run is obvious for all three days. Although the general pattern is the same, the filamentary structures are much sharper in the n = 7 run than the n = 5 run. In this rapid stirring regime, small differences in tracer fields tend to be amplified in TrEL via the enhanced |∇ϕe| in Eq. (4).

c. Temporal resolution [Data Assimilation Office (DAO) 24 h versus DAO 6 h]

The sensitivity of high-resolution advection calculations to temporal resolution of the winds was examined by Waugh and Plumb (1994). They showed that in contour advection simulations, significantly different structures can be generated when using 6-, 18-, and 24-hourly winds. Methven and Hoskins (1999) further showed that although contour stretching rates are insensitive to spatial truncation in the wind fields, they are rather sensitive to the temporal truncation. Knudsen et al. (2001) compared Lagrangian parcel trajectories calculated with 24-hourly winds and 12-hourly winds with balloon flight tracks and showed that the former gives substantially larger errors. Although individual trajectories and contour advection simulations show sensitivity to temporal resolutions, it is not clear how the Eulerian calculation of tracer transport with diffusion will respond. Here we test the sensitivity by comparing TrEL calculations for December 1998 using 6- and 24-hourly winds from a test system of the finite volume version of the Goddard Earth Observing System-Data Assimilation System (GEOS-DAS; DAO 2000). For both temporal truncations, the winds are linearly interpolated to the same time step (8 min).

Synoptic plots of the resulting TrEL for the Northern Hemisphere are shown in Fig. 3a, rows 4 and 5. For all three days, the fields show very similar structure, even after the vortex has completely broken up on 26 December. Synoptic plots for the Southern Hemisphere (Fig. 3b, rows 4 and 5) are also similar. However, there are subtle differences, such as in the position and orientation of the lowest equivalent latitudes (the black regions). The rms differences for December in Fig. 4 (blue line) are less than 4° throughout the Northern Hemisphere. In the Southern Hemisphere, the differences are somewhat larger, peaking near 10° at the equivalent South Pole.

d. Wind field (Met Office n5 versus DAO n5)

To test the sensitivity of TrEL to the input wind field, we ran the van Leer model at n = 5 over the month of December 1998 separately using Met Office and GEOS-DAS winds. Coy and Swinbank (1997) compared in detail assimilated winds and temperatures from these two analyses, which revealed, on the whole, mutually consistent pictures of the large-scale circulation. Both also showed similar PV structure in the vicinity of the lower stratospheric polar vortex. However, small-scale differences were seen to occur, particularly in the PV fields near the pole. Manney et al. (1996) similarly showed differences in small scales between PV fields calculated from the Met Office assimilation and the National Centers for Environmental Prediction (NCEP) analyses. Comparing individual trajectories arising from different wind fields, various authors have shown that under certain flow conditions differences of over 1000 km can occur even on a timescale of only a few days (e.g., Allen et al. 1999a; Knudsen et al. 2001).

Synoptic plots of the two runs for the Northern Hemisphere are shown in Fig. 3a, rows 2 and 4. For 16 and 20 December, the shape of the vortex is similar for both runs. On 20 December, the vortex core is smaller for the GEOS-DAS run and the stretched tail is broader. By 26 December, very large differences develop. Rather than a triple-lobed structure, the GEOS-DAS run shows multiple isolated regions of high TrEL air. These regions are connected by long thin arcs, which are not seen in the Met Office n = 5 run, but are more evident in the Met Office n = 7 run (Fig. 3a, row 3). The low TrEL in the Aleutian high region, pronounced in the Met Office run, is hardly discernible in the GEOS-DAS run.

Results for the Southern Hemisphere (Fig. 3b, rows 2 and 4) also show significant difference in TrEL produced from the Met Office and GEOS-DAS winds. For example, on 26 December, the Met Office run shows the equivalent South Pole over Antarctica, while the GEOS-DAS run positions the equivalent pole near Tasmania. The position and orientation of several other significant features, such as the large tongues of low equivalent latitude, are quite different between the two runs, resulting in large rms differences.

The rms difference between the two runs is shown with a solid red line on Fig. 4. The differences are everywhere larger than those from any of the previous sensitivity tests. In the Northern Hemisphere, differences range from 7°–11°, while in the Southern Hemisphere, differences increase to over 30° at the equivalent South Pole. There is also a local maximum in the rms difference centered near 40°S. These results suggest that TrEL is very sensitive to errors in the wind fields, particularly in regions where mixing is fast. One should therefore use the TrEL field with some caution during these periods, unless it can be independently discerned that one of the wind fields is of much better quality than the other. This problem is not unique to TrEL, however. The dashed red line in Fig. 4 shows the rms difference between the Met Office and GEOS-DAS PVEL for this time period. It follows nearly the same pattern as the TrEL difference, showing that the sensitivities of TrEL and PVEL to wind fields are very similar.

4. TrEL versus PVEL

To the extent that PV approximately obeys the same advection–diffusion equation that governs the numerical tracer and to the extent that PV maintains variance at large scales, we expect PV and the tracer to share similar geometry (and hence equivalent latitude) after the spinup period: PV can be thought of as another tracer, “initialized” a long time ago, to which the numerical tracer asymptotes in the manner described in section 2c.

In practice, in addition to diagnostic errors in PV and model sensitivity of TrEL, differences arise between TrEL and PVEL because of differences in the governing physics. Apart from the small numerical diffusion, the tracer is solely governed by advection due to the nondivergent part of isentropic winds, whereas PV is additionally affected by 1) advection due to the divergent part of isentropic winds, 2) various nonadiabatic effects including cross-isentropic advection and “sources,” and 3) frictional effects [Andrews et al. 1987, their Eq. (3.8.5)]. In the stratosphere, frictional effects should be completely negligible. Nakamura (2001) showed that at 350 K, in the lowermost stratosphere, the divergent part of the wind was less than 10% in root-mean-square of the total wind in the extratropics. Therefore, the difference between TrEL and PVEL is likely generated, apart from the diagnostic errors, primarily by nonadiabatic effects.

Given the differences between PVEL and TrEL, an important question is which serves better as a coordinate of long-lived chemical tracers; that is, which mimics the isentropic kinematics of stratospheric constituents more faithfully? The numerical tracer is free from diagnostic errors and nonadiabatic sources that affect PV, but it also neglects cross-isentropic advection that can influence the true chemical tracers. Thus the answer to the above question is not obvious a priori.

To address this, we shall compare TrEL and PVEL in stratospheric flows and examine their utilities for the transport analysis and reconstruction of long-lived trace constituents such as ozone. Here TrEL is based on the van Leer calculation driven by the Met Office winds, while PVEL is based on the same Met Office assimilation. This ensures that both quantities are compared under identical flow conditions.

a. Breakup of the Arctic polar vortex

The first comparison concerns the final breakup of the Arctic lower stratospheric vortex. This is the period in which isentropic winds become quite weak and the timescale of nonadiabatic processes can be comparable to the advection timescale. Indeed, during this period, PV and chemical tracers are known to be decorrelated (Hess 1991). It is thus of interest to compare PVEL and TrEL for the same period. We take March–May 1994 as a case study. Eight synoptic plots of Northern Hemisphere TrEL at 460 K (Fig. 5) can be compared directly with similar plots of PVEL (Fig. 6). The plots are at 10-day intervals, starting on 1 March 1994.

Overall, the shapes of the TrEL and PVEL contours in the polar region are remarkably similar during March. For example, the 60° contours (highlighted in black) both start out nearly circular on 1 March and are similarly distorted by planetary wave activity during the rest of the month. Scatterplots of PVEL versus TrEL (Fig. 7) show tight correlation during March in the 60°–75° range, and the correlation coefficient covering 45°–90°N is greater than 0.85.

There are, however, some noticeable differences between the two fields during March. First, PVEL shows a larger degree of patchiness than TrEL, particularly in the middle latitudes. We believe that this is due to inaccuracy in winds and temperature at small scales, combined with the weak background PV gradients [see Eq. (2)]. Next, in the Tropics, TrEL tends to show more structure than PVEL. In fact, there appears little hint of PVEL being advected in the Tropics; for example, large tongues of low TrEL are observed on 31 March, whereas PVEL shows more zonally symmetric structure. Finally, within the vortex core on 1 March, PVEL has a local minimum that is absent in TrEL. This feature in PVEL has a certain ubiquity (see section 4b) and appears to be of physical origin. The difference contributes to enhanced PVEL/TrEL scatter poleward of 75° (see Fig. 7).

The bottom rows in Figs. 5 and 6 show the conditions occurring as the Arctic polar vortex breaks up. In the first two plots, 10 and 20 April, TrEL and PVEL still show similar vortex structure as it separates into two distinct lobes. There is some “scalloping” in the PV field on 20 April in the lower lobe. By 30 April and 10 May, however, the spatial structures of the two fields rapidly diverge as the structure of the vortex remnant increases in complexity. Correspondingly the bottom row of Fig. 7 shows rapidly increasing scatter with time, with the correlation coefficient dropping from 0.75 on 10 April to 0.23 on 10 May. The rapid decorrelation is similar to the one observed by Hess (1991) between PV and chemical tracers. Hess attributed the decorrelation to the nonadiabatic sources that affect only PV. In the present case, cross-isentropic advection is also neglected for the numerical tracer (unlike PV or true chemical tracers), which therefore counts as another potential cause of the decorrelation. However, these effects are not readily discernible from Figs. 5 and 6 since the small-scale PVEL features at this stage are likely subject to significant diagnostic errors. (We also note that, during periods of rapid stirring and mixing, TrEL is expected to have a higher sensitivity to the spatial resolution of the van Leer model; see section 3.) The comparison between TrEL and PVEL in the summer months (not shown here) is very similar to that on 10 May. The PVEL structure is nearly zonally symmetric in the Tropics and rather noisy in the extratropics, while the TrEL fields show highly irregular, but smooth features throughout the Northern Hemisphere, qualitatively similar to that observed on 10 May. The correlation coefficients from May–August are less than 0.5, reaching a minimum of 0.11 around the middle of July. More work is necessary to quantify the causes of the breakdown in correlation between TrEL and PVEL in the summer lower stratosphere.

b. Analysis of POAM and ER-2 ozone during SOLVE

To highlight how the differences between PVEL and TrEL may affect interpretation of transport and chemistry, we examine UV-Absorption Ozone Photometer (Proffitt et al. 1989) and Polar Ozone and Aerosol Measurements (POAM) III (Lucke et al. 1999) ozone data taken during the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment (SOLVE) 2000 campaign. During two deployments the ER-2 aircraft made multiple flights out of Kiruna, Sweden, which provided quality data inside the core of the Arctic polar vortex in the lower stratosphere. Here we focus on data from 11 March. Following Lumpe et al. (2002) we interpolate equivalent latitude (both TrEL and PVEL) to the locations of the POAM and ER-2 measurements for this day. To limit the vertical range we only include data within a 10-K potential temperature bin centered at 460 K.

Figure 8 shows synoptic plots of (Fig. 8a) PVEL and (Fig. 8c) TrEL at 460 K overlaid with the measurement locations of the POAM and ER-2 data along with scatterplots of (Fig. 8b) ozone versus PVEL and (Fig. 8d) ozone versus TrEL. POAM data from a 3-day period centered on the flight date are included in the analysis. The synoptic plots show that both POAM and ER-2 made measurements inside and outside the vortex edge [the solid and dashed lines indicate the inner and middle vortex edges calculated from the PV data using the Nash et al. (1996) criteria]. Both PVEL and TrEL show similar placement of the vortex edge along the ER-2 flight path. However, within the inner vortex edge, PVEL shows two local minima; values within these minima lie in between the inner and middle edge values. TrEL, on the other hand, increases monotonically toward the vortex core.

The different equivalent latitude distributions within the vortex will produce different ozone&sol=uivalent latitude scatterplots. For the PVEL analysis (Fig. 8b), the ozone from 75°–80° is multivalued, with clusters of points centered near 1.3 and 2.0 ppmv. This is because the local minima cause measurements taken near the vortex edge and in the core to be lumped together in PVEL coordinates. However, in the TrEL analysis (Fig. 8d), the ozone decreases monotonically (with some scatter) from the vortex edge to the (equivalent) pole. The POAM data from 65°–90° cluster around the dashed line, showing the nearly linear decrease of ozone with increasing TrEL. Whereas the multivalued PVEL analysis can be taken (mistakenly) as indicative of isolated photochemical ozone loss, such interpretation would not be made with the monotonic TrEL analysis. The local PV minimum that causes the confusion still remains on the following date, 12 March (plots not shown), again causing a multivalued ozone distribution near PVEL = 75°–80°, whereas the TrEL analysis suggests that ozone decreases monotonically from 65° to the vortex center.

In fact, pockets of reduced PV in the lower stratospheric vortex are found throughout the SOLVE period. It is unclear whether they are caused by errors in the diagnostic or assimilation procedures or whether they have a physical origin, such as the meridional variation in the nonadiabatic sources of PV and/or cross-isentropic advection. However, the latter could equally affect ozone in the presence of strong vertical gradients. Although a more careful analysis will be necessary to completely discern the causes of the PV minima, the fact that ozone mixing ratio lines up better with TrEL than PVEL suggests that TrEL may be a better coordinate for ozone, at least for the Arctic winter in the lower stratosphere.

The TrEL-based interpretation of the data sheds light on the large differences above 400 K between the vortex-averaged ozone calculated for this date from the ER-2 and POAM data shown in Fig. 6 of Lumpe et al. (2002). According to the TrEL analysis, the average of the ER-2 data is expected to be smaller, since the ER-2 measurements were taken only at very high equivalent latitudes, while the POAM data coverage spans the entire vortex.

c. Three-dimensional ozone reconstruction

To further compare the utility of TrEL and PVEL as coordinates of transport, we examine the reconstruction of the three-dimensional ozone field using the procedure described in Randall et al. (2002) and Allen and Nakamura (2002), in which the relationship between equivalent latitude and ozone on isentropic surfaces is used to create a proxy ozone field. To this end, satellite observations of ozone mixing ratio from POAM III, Halogen Occultation Experiment (HALOE, Russell et al. 1993), and SAGE II (Cunnold et al. 1989) are correlated separately with PVEL and TrEL on 16 isentropic surfaces from 320–1900 K. Here we use two months of ozone data, November and December 1999, which allows nearly complete global coverage. The median ozone value is calculated at each level from the ozone measurements contained within 10° equivalent latitude bins. Using this equivalent latitude–ozone relationship and the spatiotemporal distribution of equivalent latitude during the 2-month period, the daily 3D proxy distribution of ozone from 320–1900 K is reconstructed. The reconstructed column below 320 K is computed from the Total Ozone Mapping Spectrometer (TOMS) standard ozone profiles (McPeters et al. 1998).

Figure 9 compares the earth probe TOMS level 3 (gridded) column ozone for 30 November–2 December 1999 with the reconstructed column using TrEL and PVEL. This period covers a record-low column ozone event over Europe (Allaart et al. 2000; Hood et al. 2001). Both reconstructions are able to capture the salient features in the TOMS data, particularly the “mini-hole,”the high ozone ridge in the middle latitudes, and the “tongue” of higher column ozone that extends westward from 180° to 45°W longitude. The TrEL proxy does a somewhat better job at tracking the extent and evolution of the mini-hole, while the PVEL proxy is able to better reproduce the high ozone over the northern Pacific on 1 and 2 December. The TrEL field is generally smoother than PVEL, consistent with the smoother structure seen in the synoptic plots in Figs. 5 and 6.

To gauge the proximities of the TrEL and PVEL reconstructions to TOMS, we calculate a correlation coefficient between TOMS total ozone and its reconstruction for all available points in the Northern Hemisphere. The correlation coefficients are high (>0.93) for both proxies, but the TrEL proxy coefficient is consistently higher than the PVEL proxy. Some of the differences in Fig. 9 reflect the asynoptic nature of the TOMS observations, while the proxy maps are for snapshots in time (1200 UTC). To account for the asynoptic sampling we also interpolated the two proxies to the time and location of the TOMS level 2 (along track) data for 30 November 1999. Figure 10 shows scatterplots of TrEL and PVEL proxies versus TOMS. The TrEL plot shows slightly less scatter than the PVEL plot and the standard deviation from TOMS is 14.7 DU for TrEL compared with 17.2 DU for PVEL. Since about 70% of the total mass of ozone resides in layers between 10 and 30 km, the better reproduction of TOMS with TrEL likely reflects better correlation between TrEL and ozone in the lower stratosphere. Indeed, the correlation with ozone degrades both for TrEL and PVEL in the upper stratosphere, where ozone is no longer materially conserved due to fast photochemistry.

5. Summary and conclusions

This paper describes an alternate method for calculating equivalent latitude using a global isentropic advection simulation. During the spinup stage, numerical diffusion assists TrEL to lose the memory of initial conditions. The timescale for this process is less than 2 months at 350 and 460 K, but it can be longer in the middle to upper stratosphere, particularly when the winter polar vortex breaks up. After a complete annual cycle, there is virtually no dependence on the initial conditions at any level, so the resulting tracer field, and calculated TrEL, can be considered a function of the flow. Without forcing, the tracer homogenizes with an e-folding time on the order of hundreds of days. Hence, eventually the machine precision limits the accuracy of the advection calculation. This problem can be easily mended with a simple normalization applied to the tracer field. Thus the TrEL technique removes the main difficulty with the trajectory calculations, namely, the need for frequent reinitialization. However, it is important to note that the tracer calculations performed here will be subject to many of the same errors of trajectory calculations, particularly analysis errors in the advecting winds. As with RDF calculations, small-scale features such as filaments ejected from the polar vortex by Rossby wave breaking may be realistically produced by the tracer calculation, but the exact structure and location may be wrong (e.g., Fairlie et al. 1997). Errors in the tracer will also reflect the numerics of the advection scheme and the subgrid diffusion.

Sensitivity tests show, perhaps not surprisingly, that the TrEL calculation is much more sensitive to the choice of wind field than it is to either spatial or temporal resolution or choice of advection scheme: the diagnostic is only as good as the input winds. However, it is important to note that PVEL has the same problem, since PVEL calculated from different meteorological analyses also show differences on the same order.

We showed that for stratospheric applications, the behavior of TrEL is comparable to that of PVEL, but the former tends to be less noisy than the latter. Arguably this owes to the fact that the numerical tracer is devoid of the diagnostic errors that plague PV: TrEL is rather insensitive to the small-scale details in the advecting wind, whereas small-scale errors in wind does affect PVEL. However, differences also stem from the governing physics absent for the tracer, most notably nonadiabatic effects, when their timescale is comparable to that for isentropic advection. Interestingly, when PVEL and TrEL differ, the latter sometimes serves better as a coordinate for ozone (or any long-lived chemical constituent). For example, in the analysis of ER-2 and POAM ozone during the SOLVE campaign, PVEL was shown to have a local minimum in the vortex that confused the analysis, whereas no such problem existed for TrEL, which increased monotonically toward the vortex center. It appears for this case that the nonadiabatic sources that affect only PVEL outweigh cross-isentropic advection neglected in TrEL, making the latter behave more closely to the chemical tracers.

To the extent that TrEL correlates with a chemical constituent better than PVEL does, one expects the former to do a better job when used to reconstruct the constituent. Indeed, in a three-dimensional reconstruction of ozone TrEL was shown to produce column amounts that match slightly better with TOMS observation than the conventional PVEL reconstruction techniques.

Since TrEL is relatively easy to compute from assimilated winds and a stable advection scheme, and since its spatial resolution can be made arbitrarily higher, it can be a viable alternative to PVEL for problems related to isentropic advection. Further applications will be presented in forthcoming papers.

Acknowledgments

We thank R. Bevilacqua, G. Nedoluha, and three anonymous reviewers for helpful comments on this manuscript, L. Coy for providing the GEOS-DAS data, and M. Fromm for arranging the SAGE data for easy handling. We also appreciate the efforts of the Goddard Space Flight Center (GSFC) Data Assimilation Office and the Met Office in producing the assimilated data products. The SAGE and HALOE data were obtained from the Langley Radiation and Aerosols Branch of the NASA Langley Research Center. The Atmospheric Chemistry and Dynamics Branch at GSFC provided access to the TOMS and Met Office data. The POAM III instrument is sponsored by the Office of Naval Research. Support for DRA comes from the Naval Research Laboratory and from the NASA Atmospheric Chemistry Modeling and Analysis Program. NN acknowledges support from the NSF Grant ATM9980676.

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Fig. 1.
Fig. 1.

The log of the global mean variance of tracer mixing ratio from simulations initialized 17 Oct 1991 at 350, 460, 850, and 1900 K. The e-folding times at each level were calculated using a linear least squares fit shown by the dashed lines.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 2.
Fig. 2.

Scatterplots of TrEL for two tracer advection runs, initialized separately on 17 Oct 1991 and 1 Jun 1998. Both runs were initialized at 350 K with mixing ratio equal to sine(latitude) and were advected with Met Office winds. The normalization operator [Eq. (3)] was applied to each time step.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Lambert equal area plots of TrEL at 850 K covering the Northern Hemisphere for 16, 20, and 26 Dec. The outer edge of each circle is the equator. Row 1 shows spectral scheme T96 resolution, Met Office winds. Row 2 shows van Leer scheme, n = 5 resolution, Met Office winds. Row 3 shows van Leer scheme, n = 7 resolution, Met Office winds. Row 4 shows van Leer scheme, n = 5 resolution, GEOS-DAS 24-hourly winds. Row 5 shows van Leer scheme, n = 5 resolution, GEOS-DAS 6-hourly winds

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 3.
Fig. 3.

(b) Same as (a) but covering the Southern Hemisphere

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 4.
Fig. 4.

The rms difference for each of the sensitivity tests (see Table 1) calculated at 850 K over the month of Dec 1998 and in 2° equivalent latitude bins.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 5.
Fig. 5.

Lambert equal area plots of TrEL at 460 K over the Northern Hemisphere for 1 Mar–10 May 1994 at 10-day intervals. The outer edge of each circle is the equator. The 60° TrEL contour is highlighted in black

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 6.
Fig. 6.

Same as Fig. 5, but for PVEL

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 7.
Fig. 7.

Scatterplots between TrEL and PVEL at 460 K for 1 Mar–10 May 1994 at 10-day intervals. The plots cover the range from 45° to 90° equivalent latitude. The correlation coefficient for data in this range is provided above each frame.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 8.
Fig. 8.

(a) Lambert equal area map of PVEL at 460 K for 11 Mar 2000. Overlaid are locations of POAM (black dots) and ER-2 (blue dots) measurements along with black lines for the inner (solid) and middle (dashed) vortex edge. (b) Ozone vs PVEL for 11 Mar 2000. Only data contained within a 10° bin centered at 460 K are included here. Lines mark inner (solid) and middle (dashed) vortex edge. (c), (d) Same as (a) and (b) but using TrEL. The sloping dashed line in (d) is a linear least squares fit to the POAM data from 65° to 90°.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 9.
Fig. 9.

Column ozone from earth probe TOMS (left), TrEL reconstruction (middle), and PVEL reconstruction (right) for 30 Nov–2 Dec 1999. The TrEL reconstruction was produced with the van Leer model at n = 6 resolution (approximately 120 km) in order to more closely match the resolution of the TOMS data. Correlation coefficients for TrEL/TOMS and PVEL/TOMS are provided above each plot.

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Fig. 10.
Fig. 10.

Scatterplots for TOMS level 2 data vs TrEL and PVEL reconstructions on 30 Nov 1999 over the Northern Hemisphere. The reconstructed data are interpolated to the TOMS level 2 data points (in space and time) to account for the asynoptic TOMS sampling. The TOMS points are subsampled by a factor of 10. Column ozone is calculated from the proxy using the method described in Allen and Nakamura (2002).

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0287:TELADT>2.0.CO;2

Table 1. 

Parameters used for the runs in each of the four sensitivity tests. Parameters include advection scheme, temporal truncation of the winds, horizontal resolution, and wind field. DAO (UKM) refers to the GEOS-DAS (Met Office) assimilation

Table 1. 
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