We follow and use phase-lagged correlations between adjacent levels of the tape recorder signal to estimate an effective vertical transport velocity, a method previously introduced by and recently used in modified form by . The earlier studies used large sample sizes (∼1 year) to compute the correlations. These sample sizes tend to highlight interannual variability (such as due to the QBO) over seasonal variability. Here, we modify this method to parse out shorter-duration variability. First, we obtain correlation coefficients between daily data at consecutive levels. The data at the higher level are then shifted in 1-day increments up to 14 months to find the largest correlation coefficient. A strong correlation between the data at the lower level and the shifted data at the higher level is assumed to follow the tape recorder. The effective transport vertical velocity, assigned to midpoints between levels and time steps, is simply the distance between the levels divided by the time shift associated with the largest correlation coefficient. We consider effective transport velocities in both pressure and isentropic coordinates. Vertical velocities in pressure coordinates will be presented as log–pressure velocities to give the more often used unit of mm s-1, using a constant scale height of 7 km

A more appropriate scale height for the tropical lowermost stratosphere would be 6 km (H=RT0/g, where R is the gas constant for dry air, T0 is a reference temperature, and g is the acceleration due to gravity). However, we opt for 7 km as this is the most commonly used scale height in the expression for log–pressure coordinates e.g.,.

Instead of using a large (∼365 days) sample size for computing the correlation coefficients , we have found that a sample size of ∼180 days is capable of parsing out the seasonal cycle of effective transport velocity. Further, unlike , we retain high correlations that occur at lags of less than 1 month. However, lags of less than 7 days are omitted because they produce unrealistic and temporally unvarying speeds with low correlations. Our modified phase-lagged correlation method was tested on a synthetic tape recorder signal with varying advection scenarios. The results show that the method is more likely to underestimate by 0.05 hPa day-1 below 60 hPa and more likely to overestimate by 0.05 hPa day-1 above 60 hPa. Small vertical velocities in the middle stratosphere and rapid water vapor changes in time are not fully identified (e.g., in May when the signal goes from dry to moist). Overall, the method appears to successfully capture the seasonality and magnitude of the transport.

We emphasize that this lag correlation method based on the observed tape recorder signal results in an effective (vertical) transport velocity. When mixing has negligible influence on the signal, this velocity may be assumed to be approximately equal to the residual vertical velocity . However, especially in the lowermost tropical stratosphere, the effects of horizontal and vertical mixing may be significant. Vertical mixing will cause the signal to spread between two levels while reducing the time lag for maximum correlation and therefore increase the inferred velocity. The influence of horizontal mixing dilutes the tape recorder signal , but this depends on the horizontal background structure that varies seasonally.

Estimates of the effects of vertical and horizontal mixing on the tape recorder signal may be obtained by simulating this signal with a 1-D transport model: ∂tχ‾=-ω‾*∂pχ‾+∂p(Kp∂pχ‾)-αp(χ‾-χ‾ML)+S. Here, χ‾ is the water vapor mixing ratio, ω‾* is the residual vertical velocity, Kp is the vertical diffusivity in pressure coordinates, αp is the horizontal dilution rate in pressure coordinates, χ‾ML is the midlatitude reference value of χ‾, and the overbars represent the zonal mean. χ‾ML is obtained from the actual (seasonally varying) water vapor for each dataset, averaged over 30 to 60∘ latitude. We have tested an alternative latitude range for χ‾ML of 15 to 45∘ and found no significant changes to our results. S is a chemical source–sink term. We set S to zero because we are only interested in the tape recorder below the level of methane oxidation, which becomes important above ∼ 40 hPa . This also neglects cloud formation or evaporation just above the tropopause, as well as a potential contribution due to dehydration at the local cold point tropopause. We will discuss this potential drawback in detail in Sect. 6. Our model is similar to the one used in , except that it uses pressure coordinates (we also use a potential temperature coordinate version; see below).

solved for annual mean parameters by defining the tape recorder as a wave solution and inverse solving for advection, diffusion (vertical mixing), and dilution (horizontal mixing). Although they tested their model on synthetic data, the solutions from this approach are restricted because they rely on the tape recorder fitting a perfect wave at each level, which may be problematic in the presence of mixing. The most severe restriction, however, comes from using an annual mean value for the residual vertical velocity. It is by now well established that the strength of residual tropical upwelling undergoes a significant seasonal cycle, with smaller values in boreal summer e.g.,. The vertical transport due to residual vertical advection alone slows down significantly during boreal summer, enhancing the relative importance of mixing to total transport, particularly in this season. Assuming annual mean values for the transport parameters essentially underestimates the contribution due to mixing. We therefore introduce seasonality in these parameters by prescribing reductions and enhancements of 50 % over the course of the seasonal cycle. The 50 % value results in realistic variations for vertical advection, which corresponds to estimates in the literature e.g.,.

Seasonal variations in the mixing strengths are less well constrained by past studies; for these we use the same value of 50 %. We tested other seasonal cycle amplitudes for the MLS dataset in pressure coordinates by keeping a 50 % amplitude in two of the transport parameters while varying the third (ranging from 0 to 100 %). Overall, we find that the seasonality in the transport parameters has little influence on the tape recorder as long as the amplitude remains below 75 % for the vertical transport parameters. Small improvements in the simulated tape recorder signal can be obtained by making slight adjustments to the seasonal cycle amplitude in vertical advection or vertical mixing. In comparison, the tape recorder signal is hardly affected by changes in the seasonal cycle amplitude of the horizontal mixing strength. Given the mostly insignificant changes in the simulated tape recorder signal for modest changes in seasonal cycle amplitudes, we stick to the 50 % values in this study for simplicity.

We further remove the perfect wave restriction by running a parameter sweep with varying strengths of each transport. Control values for the annual mean solutions (denoted by the subscript “ctrl”) are taken to be the solutions obtained by , including their vertical structure. Transport strengths are varied from 0 to 10 times their control value. Apart from these modifications, our model carries the same assumptions as discussed by . It assumes that tropical air is horizontally well mixed within the latitude bounds (here, 10∘ S to 10∘ N) and is notably different, though not completely isolated, from midlatitude air. The vertical eddy water vapor flux in the full water vapor budget can be represented as instantaneous diffusion acting on the vertical gradient of water vapor (ω′χ′‾≃-Kp∂pχ‾, with K as a positive constant; further discussed below). Horizontal mixing by midlatitude air is modeled by a linear relaxation process (dilution) in which tropical air is relaxed towards χ‾ML with rate αp. This last assumption represents a crude approximation; horizontal mixing in the lowermost stratosphere is generally a more complex process . Note that by taking χ‾ML from the actual seasonally varying data, we expect to better represent influences due to the monsoon circulations, e.g., during boreal summer e.g.,.

We prescribe the seasonal cycle of advection (ω‾∗) as a sine wave that peaks during boreal winter (on 1 January) when the meridional circulation is strongest according to observations. Vertical diffusion (K) is prescribed with the same seasonality, i.e., strongest during boreal winter, which is consistent with the results in , and when convective influence on the TTL is strongest . While observational estimates of vertical mixing and its seasonal cycle are sparse to nonexistent, this seasonal cycle can be considered a plausible first guess and is found not to have a strong influence on our results (see below). The seasonal cycle of horizontal mixing αp is opposite from that of vertical advection and vertical mixing. Horizontal mixing is prescribed to maximize during boreal summer (on 1 July) when the subtropical mixing barrier (jet) is relatively weak . We tested the model with seasonality in vertical advection only, which resulted in a somewhat lower performance with respect to its ability to reproduce the observations; however, the main qualitative features of our results to be presented in Sect. 4 are not affected.

We also use the 1-D transport model in isentropic coordinates. This has the advantage that the representation of horizontal mixing becomes more realistic. This process is driven by Rossby wave breaking and takes place approximately along isentropes in the real atmosphere. Furthermore, vertical mixing is partially an adiabatic process (e.g., if driven by small-scale gravity wave breaking) and is therefore expected to contribute less to transport through isentropic surfaces. On the other hand, comparison to our datasets is more straightforward in pressure coordinates, so additional insight may be gained by comparing the two coordinate systems. In isentropic coordinates the model may be written as ∂tχ‾∗=-Q‾∗∂θχ‾∗+σ‾-1∂θ(σ‾Kθ∂θχ‾∗)-αθ(χ‾∗-χ‾ML∗)+S, where Q is the diabatic heating rate and σ is isentropic (mass) density (also often referred to as thickness). The overbars with asterisks denote the mass-weighted zonal averages (e.g., χ‾∗≡σχ‾/σ‾).

As measures of the model's performance in simulating the tape recorder, we analyze the amplitude, phase, and annual mean of the water vapor mixing ratio at 80 hPa and 400 K for each parameter combination. The phase is obtained using simple Fourier analysis, while the amplitude is obtained simply from the minimum and maximum values. We introduce a score (out of 100 %; see equation below) that is a function of the multiplying factors (a,b,c) on the control values of the residual vertical velocity or the diabatic heating rate, vertical diffusivity, and the horizontal dilution rate. For example, in pressure coordinates, the factors (a,b,c) determine the values of (ω‾∗,Kp,αp)=(aω‾ctrl∗,bKp,ctrl,cαp,ctrl). Generally, we find that the strengths of vertical advection and horizontal mixing are not independent, and their variations result in similar structures (i.e., a=c with the high-scoring combination). This is perhaps not surprising as both are a function of subtropical Rossby wave breaking e.g.,. To highlight that typically a=c, we denote the combined effects of vertical advection and horizontal mixing by G with the control value Gctrl for a=c=1. There are rare cases in which the original values for vertical advection and horizontal mixing must be multiplied by different factors to create the highest score (i.e., where a≠c). In these cases, Gctrl represents a. For example, if an optimal solution requires (a,b,c)=(1,1,3), then 1×Gctrl corresponds to a=1 and c=3, while 2×Gctrl corresponds to a=2 and c=6, and so on. These rare cases will be discussed separately.

The score as a function of (a,b) (assuming c=a) is score(a,b)=1001+|As-Ar||Ar|+|ϕs-ϕr||ϕr|+|χs-χr||χr|, where A is the amplitude, ϕ is the phase, and χ is the water vapor mixing ratio. The subscripts “s” and “r” refer to the synthetic and real tape recorder signals, respectively.

Both MLS and HALOE show seasonal variations in effective vertical transport velocity in the TTL and the lower stratosphere, with stronger upwelling during boreal winter (Fig. ). Boreal winter upward transport is 2–3 times stronger than during summer in these datasets. A finer vertical resolution in HALOE results in more consistent vertical spreading of this seasonality up to ∼50 hPa. The difference in the depth of the signal may be due to our method of underestimating small speeds, which may be more pronounced in MLS with its coarser vertical resolution. Seasonal variations are qualitatively similar in ERA-i, but velocities are 2–4 times greater than in MLS and HALOE.

Figure  highlights that the inferred effective vertical transport velocity is not necessarily the same as the residual upward velocity. In ERA-i, the effective vertical transport velocity is about 4 times larger than the residual vertical velocity at 80 hPa, which points to the role of vertical and/or horizontal mixing in transport just above the tropical tropopause (and amplified dispersion due to data assimilation; see below). The MLS- and HALOE-derived transport velocities are of similar magnitude as the residual circulation velocity in ERA-i. Taking into account that ERA-i's residual vertical velocity seems biased high near the tropical tropopause , this indicates that effective vertical transport is stronger than by the residual circulation alone. The double-minimum structure between April and August in MLS effective transport velocity is likely the result of noisy data around the transition between the wet and the dry part of the signal.

The residual vertical velocity is about 25 % weaker in GEOS CCM (the green lines in Fig. ) than in ERA-i, albeit with identical seasonality. Its effective vertical transport velocity, however, only shows very little seasonal variation and is significantly smaller than in ERA-i, allowing GEOS CCM to be in closer agreement with MLS and HALOE during boreal winter. The large difference between the effective transport transport velocities in GEOS CCM and ERA-i (up to 4 times larger in ERA-i) suggests that excessive vertical dispersion due to data assimilation dominates in ERA-i. Mixing appears to have a stronger influence on transport in boreal summer in GEOS CCM (cf. the difference between the green dashed and solid lines in Fig. ).

Figure  shows that a range of combinations that slightly vary G, but more so K, result in high-scoring simulations of observed water vapor at 80 hPa for both MLS and HALOE. Moderately high scores may be achieved by using the control value for vertical mixing (Kctrl), but this requires increases in vertical advection and horizontal mixing by more than 50 % of their control values. Using Gctrl, on the other hand, a near-perfect score (near or above 90 %) results from increasing K by a factor of 4 for MLS or 5.5 for HALOE. The strength of vertical advection may be considered better constrained from past studies e.g.,, while the strength of vertical mixing remains more ambiguous. Closer inspection of the individual tape recorder characteristics shows that high-scoring (above 90 %) simulations of its amplitude alone require at least 3×Kctrl. High-scoring simulations of only its phase, on the other hand, are more sensitive to the strength of vertical advection and to allowing transport seasonality (particularly vertical advection).

Figure  compares the tape recorder signal between our simple model and the satellite observations using the transport combination (1×Gctrl,4×Kctrl) for MLS and (1×Gctrl,5.5×Kctrl) for HALOE (the white stars in Fig. ). The time series at 80 hPa (bottom panels) further shows that our parameter settings better capture the observed seasonal water vapor evolution than the control setting, although the seasonal cycle amplitude is still somewhat underestimated. Note that the solution much better captures the HALOE time series (on which it was based).

Inspecting the individual transport contributions to the time tendency of water vapor for MLS (Fig. ) shows that vertical advection and vertical mixing play equally significant roles in forming the tape recorder signal at 80 hPa. Vertical mixing plays a particularly large role during late summer/early fall. The transport contributions based on HALOE are very similar (see Fig. S2 in the Supplement).

Horizontal mixing generally plays a small role, except during boreal spring. Note that the tendency due to horizontal mixing is a function of both the dilution rate αp and the background meridional gradient (∼χ‾-χ‾ML). The latter becomes small during boreal summer, when meridional mass transport maximizes, and this explains why the tendency due to horizontal mixing maximizes during boreal spring. This is consistent with the results in ; their Fig. 1a shows from trajectory calculations that the difference in the water vapor mixing ratio between in-mixed extratropical air and tropical air is the largest during boreal spring and becomes negligible during boreal summer. Nevertheless, the strongest signature in the tropical mean water vapor will be found when the accumulated tendency is the strongest; this happens during late boreal summer when the tendency (the green line in Fig. ) crosses through zero.

High-scoring simulations of the ERA-i tape recorder signal in pressure coordinates require much greater amounts of vertical mixing than for the MLS or HALOE observations (Fig. ). Based on all parameter combinations tested, vertical mixing needs to be at least an order of magnitude larger than the control values. We found that high-scoring solutions also require strongly enhanced horizontal mixing (multiple times its control value), whereas vertical advection may remain unchanged from its control value (small changes in it require large changes in both types of mixing to compensate). Gctrl therefore corresponds to c=3a in the transport combinations for ERA-i, and the x axis in Fig. b only extends to 3×Gctrl because changes in transport strength by more than 1 order of magnitude have not been examined.

The 1-D model results imply that eddy transport in the lowermost stratosphere is strongly enhanced in ERA-i compared to observations, especially in the vertical. Amplified vertical advection alone does not result in very improved tape recorder simulations. In fact, even reduced vertical advection may easily be compensated for by slight further amplifications of the vertical and horizontal mixing. The enhanced eddy mixing in ERA-i is likely a result of spurious dispersion due to data assimilation , but could also result from diffusive numerical schemes. In the case of the GEOS CCM, a transport combination more similar to MLS or HALOE produces the highest scores: enhanced vertical mixing with vertical advection and horizontal mixing near their control values (not shown). This difference between the free-running model and the reanalyses further points to excessive dispersion due to data assimilation in ERA-i. We also note that our simulations of the GEOS CCM tape recorder signal are not very sensitive to changes in vertical mixing strength, which is likely due to its small vertical water vapor gradient so that vertical diffusion remains small.

In isentropic coordinates, the effective vertical transport velocity corresponds to diabatic heating. Figure  shows this diabatic effective vertical transport velocity for MLS and ERA-i using the phase-lagged correlation method as before. The averages (zonally and in time) in isentropic coordinates are appropriately obtained by applying mass weighting, which is implicit in pressure coordinates. The seasonal cycles of diabatic heating rates thus obtained are similar between MLS and ERA-i, with maxima in the lowermost stratosphere during boreal winter, as expected. The maximum diabatic heating from MLS is ∼1 K day-1. That from ERA-i is 4–5 times larger and located slightly higher (at 410 K versus 390 K for MLS, perhaps related to temperature differences in this region). The enhanced diabatic heating in ERA-i compared to MLS is consistent with and , who found longwave cloud radiative heating rates above 200 hPa to be larger in ERA-i compared to other reanalyses and a detailed radiative transfer model. note that water vapor contents, and therefore the treatment of convective anvil clouds in ERA-i, could partially explain the anomalous heating rates. ERA-i has also been found to exhibit clear-sky radiative heating rates that are ∼40 % too large . However, the discrepancy between MLS and ERA-i is likely also due to excessive vertical and horizontal dispersion, as discussed in the previous section.

The difference between the effective vertical transport velocities and the contributions due to vertical and horizontal mixing may be better understood by considering the zonal mean tracer evolution equation, which in pressure coordinates reads (written in Cartesian coordinates and neglecting sources and sinks for simplicity) ∂tχ‾+ω‾∂pχ‾+v‾∂yχ‾=-∂yv′χ′‾-∂pω′χ′‾. Here, v is the meridional velocity and the primes denote deviations from the zonal mean (denoted by the overbars as before). The effective vertical transport velocity (ωeff) results formally from establishing that ∂tχ‾+ωeff∂pχ‾=0. Hence, ωeff=ω‾+(∂pχ‾)-1(v‾∂yχ‾+∂yv′χ′‾+∂pω′χ′‾). This shows how both horizontal and vertical eddy fluxes (∼ mixing) lead to differences between ω‾ and ωeff (note that in the residual, transformed Eulerian mean form, horizontal mixing is partially included in ω‾∗; more precisely, the part that is aligned with the meridional eddy heat flux , so in that form it is primarily the vertical mixing that creates differences between ω‾ and ωeff). Horizontal advection may cause an additional difference but is generally small in the deep tropics.

In isentropic coordinates, the corresponding zonal mean tracer evolution equation reads ∂tχ‾∗+Q‾∗∂θχ‾∗+v‾∗∂yχ‾∗=-σ‾-1∂yv^σχ^‾-σ‾-1∂θQ^σχ^‾. Here, the hats denote deviations from the mass-weighted zonal mean (e.g., χ^≡χ-χ‾∗). This is a slightly modified version of that given in , formulated here for the mass-weighted tracer mixing ratio. The effective vertical transport velocity in this case (Qeff) results from ∂tχ‾∗+Qeff∂θχ‾∗=0. Hence, Qeff=Q‾∗+(∂θχ‾∗)-1v‾∗∂yχ‾∗+σ‾-1∂yv^σχ^‾+σ‾-1∂θQ^σχ^‾. In this case, assuming quasi-adiabatic mixing processes (Q^≈0, e.g., due to Rossby and gravity waves in the horizontal and vertical direction, respectively) and neglecting horizontal advection, horizontal mixing is the primary process that leads to differences between Q‾∗ and Qeff.

Our estimates of Qeff from MLS agree roughly with diabatic heating rate estimates in the TTL and the lower stratosphere e.g.,, indicating that horizontal mixing does not play a big role in the observed tape recorder signal cf.. However, the difference between Q‾∗ and Qeff is substantial in ERA-i, indicating excessive horizontal dispersion in the lowermost stratosphere.

Further insight into the role of vertical mixing may be obtained by comparing the effective vertical transport velocities in pressure and isentropic coordinates. Specifically, an approximate expression relating their difference to the vertical eddy tracer flux may be derived (outlined in “Appendix A”): ω′χ′‾≈ωeff-Qeff(∂pθ‾)-1(∂θ¯χ‾)2∂θ¯θ¯χ‾. The factor outside the square brackets involves derivatives of the mean tracer mixing ratio with respect to the mean potential temperature, where both means are taken in pressure coordinates. This expression suggests that differences between the effective vertical transport velocities in the pressure versus isentropic coordinates are directly related to vertical mixing.

Figure  shows the vertical profiles of this approximate vertical eddy flux of water vapor for DJF and JJA. The flux is predominantly negative in the lowermost stratosphere (in pressure coordinates), indicating the expected upward eddy transport from high to low background concentrations in height coordinates. This may serve as a sanity check that the above approximation gives physically reasonable results. The vertical gradient of the shown eddy flux (∂pω′χ′‾) confirms that vertical mixing contributes approximately 10-3 to 10-2 ppmv day-1 to the overall water vapor tendency just above the tropical tropopause. However, the tendencies resulting from these eddy flux estimates only agree to within a factor of 10 with those derived from our 1-D model (cf. Fig. ) and during DJF even have the opposite sign. This indicates that the approximations going into our eddy flux estimate at best provide qualitative results, although uncertainties also exist with our 1-D model results. Nevertheless, given the lack of observational estimates of vertical eddy tracer fluxes on a zonal mean scale, our approach, which at its heart takes advantage of comparing tracer evolutions in pressure and isentropic coordinates, may prove useful when applied to future higher resolution datasets.

In Sect. 4.2, we found that a successful simulation of the water vapor tape recorder signal in pressure coordinates requires strongly enhanced values for vertical mixing. A different story emerges when simulating the tape recorder signal in isentropic coordinates. In this case, the original transport parameters obtained in and translated into isentropic coordinates cf. also lead to a successful simulation matching the observations (with a score of ∼90 %, shown in Fig. ). The corresponding time tendencies at 400 K (roughly corresponding to 80 hPa), shown in Fig. , reveal that the total tendency is explained almost entirely by the contributions due to vertical advection (∼ diabatic heating; the red line) and horizontal mixing (green), with the former dominating throughout Northern Hemisphere (NH) winter and the latter dominating through NH spring and early summer.

Figure a shows the scores for a range of parameter combinations at 400 K for MLS, similar to Fig. . The range of high-scoring solutions is narrower than in pressure coordinates. Other than the reference or control set of parameters (a=b=c=1), we also find maximum scores for two other cases. The first case requires zero vertical mixing (b=K=0), while vertical advection and horizontal mixing are at their control values (a=c=1). The other case requires vertical mixing to remain at its control value (b=K=1), while vertical advection and horizontal mixing are reduced by half of their control values (a=c=0.5). Overall, vertical mixing plays a smaller role in isentropic coordinates compared to pressure coordinates. This is expected based on the assumption that vertical mixing takes place quasi-adiabatically (see the discussion in the previous section).

Simulating the ERA-i tape recorder in isentropic coordinates increases in all three transports (Fig. b). A factor of 2–3 increase in vertical advection and horizontal mixing compared to the control values, together with an increase by at least a factor of 4 in vertical mixing, leads to maximum scores (>90 %). The increase in vertical advection points once more to biases in the diabatic heating rates in ERA-i (presumably due to longwave cloud radiative biases in the TTL, as stated earlier). The increase in vertical mixing indicates excessive dispersion even in isentropic coordinates. We have found, however, that large changes in vertical mixing strength only lead to small changes in the simulated tape recorder signal (the vertical gradients in the score distribution in Fig. b are much smaller than the horizontal gradients), indicating that it is not very sensitive to this transport contribution.