The basis of the moisture tagging technique is to replicate for moisture tracers the prognostic equation for total moisture: ∂qn∂t=-v⋅∇qn+νq⋅∇2qn+∂qn∂tPBL+∂qn∂tmicrophysics+∂qn∂tconvection, where qn refers to the different moisture species considered, namely water vapor, cloud water, rain water, snow, ice and graupel. The first two terms on the right-hand side of Eq. () represent the tendencies due to advection and molecular diffusion, respectively, and the others correspond to tendencies resulting from parameterized turbulent transport (planetary boundary layer – PBL scheme), microphysics and convection. The latter three terms account for subgrid physical processes affecting atmospheric moisture, such as phase changes and precipitation, or redistribution by convection and turbulent diffusion.

To replicate Eq. (), six new variables tqn are created corresponding to the tracers of the different moisture species: tvapor, tcloud, train, tsnow, tice and tgraupel. We note that in earlier studies, only water vapor was tagged (tvapor); hence, the name is WVT method. Perhaps this denomination is no longer accurate when tagging all six moisture species, and more properly the technique should be referred to as simply the moisture tracers (MTs) method; we will keep in the text, however, the common WVT term, as it is already well established in the literature.

The general form of the prognostic equation for WVTs is totally analogous to Eq. (), just replacing qn by tqn. The Eulerian form of this equation and the fact that it is solved simultaneously with Eq. () are the reasons for the method to be classified as “online” Eulerian. One could think that since the prognostic equations for WVTs and total moisture have the same form, it would suffice with repeating the calculations performed for total moisture species for the tracer species, just changing initial or boundary conditions. However, this is not the case, since the behavior of the tagged moisture is not independent from that of total moisture. In other words, the tagged moisture does not evolve as if it was completely on its own. A very simple example of this is saturation conditions and phase changes, which would hardly occur if only tagged moisture were considered. When an air parcel saturates, it does so in regards to its total moisture content, independently of whether its tracer moisture content is high or low. Similarly, since it is total moisture that determines the thermodynamical setting for turbulence and convection, primary and derived variables in the basis of the parameterizations of those processes, such as virtual temperature, dew point, profile instability, convective available potential energy (CAPE), level of free convection, eddy diffusivity and many more, must be computed using total moisture, even when calculations are performed for tagged moisture tendencies. Therefore, the prognostic equations for tracer moisture must be solved coupled to the governing equations of the model, i.e., “online”, although tracer variables do not appear elsewhere and hence do not have an effect on the model's dynamics in any way.

Thus, for the implementation of WVTs into WRF, three fundamental parameterizations of the model, such as the turbulence (PBL) scheme, microphysics and convection, must be modified for calculating the associated tracer moisture tendencies, as discussed above. Conversely, advection and diffusion routines can be simply called for tracers in the same way as for total moisture or any other scalar, since in these processes tracer moisture can indeed be treated independently from total moisture. We note that it is important to use an advection numerical scheme that is positive definite, conserves mass and minimizes numerical diffusion, in order to limit numerical errors in WVT calculations. Both total moisture and tagged moisture must use the same scheme. All other components of the model remain unchanged, since they do not affect moisture dynamics directly.

Of the several scheme options available, we have altered for moisture tagging the Yonsei University YSU; PBL scheme, the WRF Single-Moment 6-class WSM6; microphysics scheme and the Kain–Fritsch convective parameterization. These schemes have been selected because they are some of the most commonly used and show a reliable performance in numerous situations.

The validation simulation for the newly implemented moisture tagging tool is performed with the WRF model version 3.8.1 for the duration of 1 month (November 2014) and with a domain D1 of 20 km horizontal resolution and 35 vertical levels (Fig. ). Initial and boundary conditions, updated every 6 h, were obtained from the National Centers for Environmental Prediction (NCEP) Final (FNL) Operational Model Global Tropospheric Analyses, available at 1∘ resolution . In addition to the YSU PBL, WSM6 microphysics and Kain–Fritsch convective parameterizations that we adapted to calculate the corresponding tracer tendencies (as described in Sect. 2), in the simulations, we also use the Noah land surface model Noah LSM; and the Rapid Radiative Transfer Model RRTM; and Dudhia schemes for long and short wave radiation, respectively. Moisture and tracer advection are calculated with the fifth-order weighted essentially non-oscillatory WENO; scheme with a positive definite limiter. Spectral nudging of waves longer than around 1000 km is activated to avoid distortion of the large-scale circulation within the regional model domain due to the interaction between the model's solution and the lateral boundary conditions .

The methodology followed to validate WVTs is analogous to that used previously by and , and it is based on tagging moisture from all possible sources, so that if the method were exact, the difference between tracer and total moisture should be zero. In other words, let Sn (with n=1,2,3…) be a set of moisture sources covering all possible atmospheric moisture sources, tqSn the total moisture (the sum of all moisture species) from each source Sn, and q the total moisture; then, the absolute error (εa) of the method can be written as εatq(x,y,η,t)=∑ntqSn(x,y,η,t)-q(x,y,η,t), or in terms of precipitable water, integrating Eq. () in the vertical yields εaTTPW(x,y,t)=∑nTTPWSn(x,y,t)-TPW(x,y,t), where TTPWSn refers to the total precipitable water coming from source Sn and TPW is the total precipitable water simulated by the model. Similarly, for precipitation, εaTP(x,y,t)=∑nTPSn(x,y,t)-P(x,y,t), where TPSn corresponds to the precipitation from source Sn and P is the total precipitation produced by the model. Equation () can also be applied to any particular type of precipitation, such as rain, snow or graupel, individually.

Here, we have divided the possible moisture sources into five (S1,…,S5) sources, three of them two-dimensional (Fig. a) and two three-dimensional (Fig. b). The two-dimensional source regions cover all evaporative sources within the domain, namely sea, land and lakes, whereas the three-dimensional sources tag incoming moisture from the lateral boundaries and the moisture contained in the full atmospheric volume of the domain at initial time. For the latter purpose, the three-dimensional source “INITIAL” (Fig. b) is activated only at the first time step of the simulation. The “BOUNDARY” source (Fig. b) is a wall-like volume encompassing the relaxation zone where lateral boundary conditions are applied, along the domain's outer edges. To prevent moisture from evaporative or initial condition sources to be counted twice, this boundary volume becomes a sink for tracers of these other origins; that is, tagged moisture species from other sources are set to zero when they enter BOUNDARY. All possible atmospheric moisture sources are covered by the aforementioned five sources, and therefore Eqs. (), () and () should be fulfilled at all times with zero error if the method were perfectly accurate.

To provide insights on the temporal evolution of the error, we follow the statistical treatment of , based on the calculation of the mean (ME) and standard deviation (SD) of the error at each point in time, that can be written as (following the notation used previously) ME=1N∑i=1NεaαiSD=∑i=1N(εaαi-ME)2N, where N is the number of grid cells in the domain and α can correspond to TP (total tracer precipitation or rain, snow or graupel separately) or TTPW (tracer total precipitable water).

An alternative statistical treatment, which is very visual and can be used as a second test of the reliability of the method, is that of , based on computing the relative contribution of each moisture source to total precipitable water, total precipitation or to each type of precipitation (rain, snow or graupel) separately. The calculation returns the relative error of the mean values of those variables at each instant in time. For example, let P‾ be the mean total precipitation; then, the contribution (in %) of each source Sn is FTPSn=100⋅TP‾SnP‾, where TP‾Sn represents the mean total precipitation from source Sn. Then, if the method were perfectly accurate, the sum of all contributions (∑nFTPSn) should equal 100 %. The degree of deviation of this sum with respect to the latter value yields the relative error (εr) of the mean tracer precipitation: εrTP‾=∑n100⋅TP‾SnP‾-100=100⋅∑nTP‾Sn-P‾P‾.

The above equation can be applied not only to total precipitation but also to any particular type of precipitation or to total precipitable water. Finally, we note that the concept of relative error of the mean variables should not be confused with the mean relative error, which would be expressed, following the notation used in the equation above, as εrTP‾=1N∑i=1N100⋅∑nTPSn-PPi. This last variable will also be used during the validation treatment shown below.

As mentioned earlier, the validation experiment is a month-long simulation for November 2014 over North America. Figure  shows the results obtained in this simulation for total precipitation from each of the five analyzed sources (TPS1, TPS2, …) depicted in Fig.  and the total sum of precipitation from all sources (∑nTPSn). The relaxation zone along the boundaries is excluded in these figures. The largest contribution to total precipitation is from external advection into the domain, and in the eastern half of it, also from sea evaporation. Lake evaporation is locally important around the Great Lakes and in Canada, where most smaller lakes in the grid are located. Evapotranspiration over land is not very relevant in the month of November and neither is its contribution to precipitation. Moisture present at initial time precipitates significantly only toward the eastern boundary of the domain in the downwind direction of the dominant westerly flow.

According to Eq. (), for the absolute error to be zero at each point, the result in Fig. f should exactly match the total precipitation calculated by the model, shown in Fig. a. The values of this error (i.e., the differences between the results of Fig. f and Fig. a) are depicted in Fig. b. The maximum deviations between the sum of the precipitation coming from the five considered sources and the total precipitation calculated by the model occur over the sea, near the domain's edges, and hover around -3 mm. These values correspond to very low relative errors (Fig. c), since the cumulative precipitation in these areas during the month of November is very high, often exceeding 300 mm. In most regions, however, the absolute error is clearly less than 1 mm, close to zero for the most part and thus very small, even in the relative sense. Neglecting cells where the total monthly precipitation is less than 1 mm to avoid arithmetical problems, the area-averaged value of the relative error (Eq. ) is -0.17 %, with a standard deviation of 0.20 %. The maximum relative error found at any point is only -3.73 % in areas of the US desert southwest with low accumulated precipitation during the simulated month of November.

Figure  shows, at 3 h intervals, the ME and SD for the three precipitation types, rain (Fig. a), snow (Fig. c) and graupel (Fig. e), throughout the monthly period of simulation. Values of the mean error are very close to zero at all times, with small standard deviations of about 0.05 mm day-1 for rain, 0.01 mm day-1 for snow and 0.005 mm day-1 for graupel, indicating that the compensations between positive and negative errors are not very relevant. As expected, the error is larger for the domain-wide most abundant precipitation types (rain and snow, in this order) and smaller for the most residual type of precipitation (graupel). found mean errors very close to zero for precipitation, as in our case, but comparatively much larger standard deviations of about 0.2 mm day-1 (∼5%). In addition, Fig.  shows the relative contribution of each considered moisture source to area-averaged rain (Fig. b), snow (Fig. d) and graupel (Fig. f). Moisture initially present in the domain's atmospheric columns only contributes to any precipitation type during approximately the first week of simulation. Rain is roughly about 40 % of sea evaporation origin and 60 % from moisture influxes from the lateral boundaries, with these values oscillating throughout the month. In comparison with rain, snow and graupel have a stronger contribution from external moisture advection and also from land evapotranspiration and lake evaporation, and a much smaller fraction of sea evaporation input. As these figures are cumulative diagrams, the upper line (which separates the white zone from the color zone), indicates the combined contribution of all sources to precipitation. The deviation of this line from 100 % represents the relative error of mean domain precipitation (Eq. ), which, as it is apparent, is very small for all three precipitation types and at all times. Further discussion will follow later in this section.

Validation results in terms of total precipitable water are presented in the diagrams of Fig. , which are similar to those in Fig.  for precipitation. In this case, the mean error (Fig. a) takes values around -0.01 mm, whereas the standard deviation is about 0.1 mm, which are very small numbers. To contextualize these results, we refer again to , who show a mean error around -0.5 mm (∼2%) and standard deviation of about 0.5 mm.

Finally, Fig.  shows in more detail the time evolution of the relative error of mean domain precipitation of all three types, as well as of mean domain TPW. This corresponds to the deviation from 100 % in the cumulative values in figures Figs. b, d, f and b, as discussed previously. Numbers are similar for the three precipitation types and do not exceed ±0.4 %. On average, about 0.2% of precipitation is not associated with any of the five considered moisture sources; i.e., the mean domain relative error is around -0.2%. For TPW, errors are even smaller. In this case, the deviation of the sum of contributions from all sources from 100 %, is roughly -0.1 % (Fig. ), which means that only 0.1 % of TPW is not traceable. found, at first, errors that were around 10 % for TPW, and later this value was improved to 1–2 % . Finally, we note that during the simulation period (1 month) there is no increasing trend in these errors, which attests to the method's stability.

Both the small absolute and relative values of the analyzed error measures in this section, together with the lack of trends in the errors, demonstrate the high accuracy and soundness of the method. Finally, with regard to the causes of these inaccuracies, most likely, they are largely caused by numerical errors derived from the very large moisture tracer gradients that occur in some regions of the domain, for example, in the separation region between the BOUNDARY source (Fig. b) and the interior of the domain. These sharp transitions can induce small errors in the advection scheme and also stronger numerical diffusion than for full moisture. In addition, other errors, such as rounding errors or small inaccuracies in the water budget, contribute secondarily.

The example application experiment is run for 4.5 days (17:00–00:00 to 21:00–12:00 UTC November) in a D2 domain nested within the validation simulation and encompassing the Great Lakes region with a horizontal resolution of 5 km and the same 35 vertical levels as the parent domain D1 (Fig. ). Tracer moisture from the parent domain can feed the nested domain through its lateral boundaries, which are not set to zero. The simulation serves also as an example of the versatility of the tagging tool. The physics settings in this experiment are identical to those in the validation simulation, except for spectral nudging and the convective parameterization, which are turned off.

Figure  shows the general synoptic situation for the selected case, in terms of surface pressure and 850 hPa temperature (Fig. a) along with 500 hPa geopotential height and temperature (Fig. b), both at 12:00 UTC on 18 November 2014. The situation is that which is typically associated with Great Lake-effect snowstorms: a deep trough with low temperatures aloft over the region, causing intense west–northwest winds at lower levels across the Great Lakes and very cold air advection. The lakes were mostly ice free at this time, with temperatures between 0 and 8 ∘C, the warmest in Lake Erie (Fig. b), contrasting markedly with the below -15 ∘C values at 850 hPa. The topography of the area is also shown in Fig. , with the highest terrain east of Lake Erie.

Figure  shows the daily evolution, from 17 to 20 November 2014, of the precipitable water originating from evaporation in the lakes and the 10 m wind at 12:00 UTC. Paired panels depict the percentage of total precipitable water that those amounts represent, together with 850 hPa temperature. At 12:00 UTC on 17 November, a short wave trough was pushing past the region. Winds ahead of the associated front were still from the south over lakes Erie and Ontario, with moderately low temperatures above -6 ∘C at 850 hPa; however, behind the trough, a very cold air mass was already in place over lakes Superior, Michigan and Huron, where winds had already veered and were at this time from the west–northwest direction. The enhancement of evaporation from the lakes is already apparent at this time, with precipitable water plumes from lakes Superior and Huron with values around 2–3 mm, which represent a contribution of 20–30 % of the total. After frontal passage, the next day, winds increase in intensity and change direction to the west–northwest, and the cold air settles in with temperatures around -16 ∘C at 850 hPa. The arrival of the cold and dry air mass, together with the wind intensity rise, augment evaporation fluxes from the surface of the lakes, so that the precipitable water with this origin practically doubles with respect to the previous day, increasing the lake moisture contribution to about 30–60 % of the total. The highest values are attained in plumes aligned with the main wind direction that originate from open waters and extend leeward of the lakes. The cold air stays in place for the next days and lake water evaporation values remain high; however, the direction of the moisture plumes from this source vary as wind changes due to the approach of another short wave trough, turning more toward the north as the flow becomes southerly on 19 November and again westward of the lakes when winds turn in this direction on 20 November. In the areas where the 850 hPa temperatures remain below about -15 ∘C during the short wave passage, plumes of moisture from the lakes still develop, with an input of lake moisture above 30 % of total content.

The previous results suggest that the lakes' contribution to atmospheric moisture in the region is very significant for this event, and we assess next whether this is also the case for precipitation. Observed snowfall totals for the period between 17 November at 06:00 UTC and 21 November at 06:00 UTC (Fig. a, from NOAA's National Snow Analyses data; ) were very high, with peak values close to 100 mm in the Buffalo, NY, area, to the lee of Lake Erie, and with other pockets of over 60 mm of snow water equivalent accumulations on the leeward shores of lakes Huron and Ontario, where orographic lifting from the existing hills further enhances precipitation. Figure b shows model results for the same period, which are in very good agreement with the observations, in amounts and distribution. This is particularly true for the aforementioned areas of highest snowfall totals.

The part of precipitation originating from lake evaporation during the same 4-day period is shown in Fig. , in terms of absolute (Fig. a) and relative (Fig. b) values to total accumulations. The role of the lakes as moisture sources is very relevant. In general, in all regions immediately downwind of the Great Lakes, water vapor with this origin accounts for more than 30 % of precipitation. The areas where the contribution of lake water vapor fluxes to precipitation is largest coincide with the locations of maximum snowfall totals, to the lee of lakes Huron, Erie and Ontario. Here, more than 50 % percent of the snow water equivalent has its source in lake evaporation, which attests to the fundamental role of lake moisture in producing the observed localized extreme accumulations during these events. In regions further from the lakes, the pattern of total precipitation and that of precipitation originating from lake evaporation lose correlation.