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  • View in gallery

    TRMM 3B42 composites of July in the Northern Hemisphere and January in the Southern Hemisphere, for (top) ratio of diurnal harmonic amplitude to monthly mean precipitation and (bottom) diurnal harmonic phase expressed as LST of maximum precipitation. In the top map, ratios greater than unity are colored white (where 24-h Fourier amplitude for the diurnal cycle exceeds the mean monthly value). The few grid points where no precipitation occurs during the study period are colored gray. In the bottom map, areas where either monthly mean precipitation or its diurnal cycle is weak are colored gray (using masking criteria described in the text). Black dots in both maps locate grid points selected for later time series analysis.

  • View in gallery

    Amplitude ratios, as in the top map in Fig. 1, for AMIP simulations by the four highest-resolution CMIP5 models: (top left) 0.5° and (top right) 0.25° GFDL model and (bottom left) high- and (bottom right) super-high-resolution MRI model.

  • View in gallery

    As in Fig. 2, but for phase, with definitions and thresholds for masking (gray colors) matching the bottom map in Fig. 1.

  • View in gallery

    Composite diurnal time series from TRMM 3B42 observations (black dots) and from the four highest-resolution CMIP5 models (colored dots), at the eight sample points shown in the maps above. Models are color coded as described in the figure legend. Observed time points are fit by (dashed lines) the diurnal harmonic only and by (solid lines) the combination of diurnal and semidiurnal harmonics, with error bars showing standard error of the mean.

  • View in gallery

    Harmonic dial plots of the amplitude and phase of Fourier components, after vector averaging over land and ocean areas separately, for TRMM 3B42 observations (black lines and dots), for the four highest-resolution CMIP5 models color coded as in Fig. 4 (colored lines and dots), and for the other 17 CMIP5–AMIP models with only July results shown for clarity (gray dots). For TRMM and the highest-resolution models, solid lines mark January results and dashed lines mark July results.

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Metrics for the Diurnal Cycle of Precipitation: Toward Routine Benchmarks for Climate Models

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  • 1 Program for Climate Model Diagnosis and Intercomparison, Lawrence Livermore National Laboratory, Livermore, California
  • 2 University at Albany, State University of New York, Albany, New York
  • 3 National Center for Atmospheric Research, Boulder, Colorado
  • 4 International Research Institute for Climate and Society, Columbia University, New York, New York
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Abstract

Metrics are proposed—that is, a few summary statistics that condense large amounts of data from observations or model simulations—encapsulating the diurnal cycle of precipitation. Vector area averaging of Fourier amplitude and phase produces useful information in a reasonably small number of harmonic dial plots, a procedure familiar from atmospheric tide research. The metrics cover most of the globe but down-weight high-latitude wintertime ocean areas where baroclinic waves are most prominent. This enables intercomparison of a large number of climate models with observations and with each other. The diurnal cycle of precipitation has features not encountered in typical climate model intercomparisons, notably the absence of meaningful “average model” results that can be displayed in a single two-dimensional map. Displaying one map per model guides development of the metrics proposed here by making it clear that land and ocean areas must be averaged separately, but interpreting maps from all models becomes problematic as the size of a multimodel ensemble increases.

Global diurnal metrics provide quick comparisons with observations and among models, using the most recent version of the Coupled Model Intercomparison Project (CMIP). This includes, for the first time in CMIP, spatial resolutions comparable to global satellite observations. Consistent with earlier studies of resolution versus parameterization of the diurnal cycle, the longstanding tendency of models to produce rainfall too early in the day persists in the high-resolution simulations, as expected if the error is due to subgrid-scale physics.

Denotes Open Access content.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-15-0664.s1.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Curt Covey, Program for Climate Model Diagnosis and Intercomparison, Lawrence Livermore National Laboratory, LLNL Mail Code L-103, 7000 East Ave., Livermore, CA 94550. E-mail: covey1@llnl.gov

Abstract

Metrics are proposed—that is, a few summary statistics that condense large amounts of data from observations or model simulations—encapsulating the diurnal cycle of precipitation. Vector area averaging of Fourier amplitude and phase produces useful information in a reasonably small number of harmonic dial plots, a procedure familiar from atmospheric tide research. The metrics cover most of the globe but down-weight high-latitude wintertime ocean areas where baroclinic waves are most prominent. This enables intercomparison of a large number of climate models with observations and with each other. The diurnal cycle of precipitation has features not encountered in typical climate model intercomparisons, notably the absence of meaningful “average model” results that can be displayed in a single two-dimensional map. Displaying one map per model guides development of the metrics proposed here by making it clear that land and ocean areas must be averaged separately, but interpreting maps from all models becomes problematic as the size of a multimodel ensemble increases.

Global diurnal metrics provide quick comparisons with observations and among models, using the most recent version of the Coupled Model Intercomparison Project (CMIP). This includes, for the first time in CMIP, spatial resolutions comparable to global satellite observations. Consistent with earlier studies of resolution versus parameterization of the diurnal cycle, the longstanding tendency of models to produce rainfall too early in the day persists in the high-resolution simulations, as expected if the error is due to subgrid-scale physics.

Denotes Open Access content.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-15-0664.s1.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Curt Covey, Program for Climate Model Diagnosis and Intercomparison, Lawrence Livermore National Laboratory, LLNL Mail Code L-103, 7000 East Ave., Livermore, CA 94550. E-mail: covey1@llnl.gov

1. Introduction

The purpose of this paper is to propose standard metrics for evaluating the diurnal timing of precipitation in atmospheric general circulation models (GCMs). By “metrics” we mean a few numbers that condense large amounts of output into summary statistics (Gleckler et al. 2008). Such metrics are useful in assessing overly frequent occurrence of light rainfall (Trenberth et al. 2003; Dai 2006; Stephens et al. 2010) and tropical precipitation occurring too early in the day (Dai and Trenberth 2004; Dai 2006; Dai et al. 2007; Kikuchi and Wang 2008). These problems impair fundamental understanding of how climate emerges from the statistics of weather and also have important practical consequences because the frequency and intensity of precipitation impact human and natural systems.

Straightforward Fourier analysis provides both spatial structures and time series at selected locations, sampling data from a subset of models. Vector averaging over large areas then provides the summary metrics. As noted below, more elaborate approaches exist but are less appropriate for applying metrics to a large multimodel ensemble.

2. Data sources and algorithms

Simulations come from phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012), including models with sufficient horizontal resolution to be comparable to satellite observations of precipitation. Table 1 shows that our two highest-resolution models (GFDL HiRAM-C360 and MRI-AGCM3.2S) have grid spacings as fine as or finer than 0.25° latitude/longitude, the resolution of commonly used observational datasets. Each of these two models is paired with a lower-resolution version (GFDL HiRAM-C180 and MRI-AGCM3.2H). These four models only provide output for the subset of CMIP in which sea surface temperatures (SSTs) and sea ice concentrations are matched to observations for the period 1979–2008. In many ways, such Atmospheric Model Intercomparison Project (AMIP) boundary conditions give GCMs their best opportunity for accurate climate simulation, though not necessarily for coupled variations (Wang et al. 2005), as the SSTs do not respond to surface forcing. Also, as the specified SSTs do not contain diurnal variations, the diurnal cycle over oceans is damped (Dai and Trenberth 2004; Covey et al. 2011), although this effect is not significant in our case, as shown below.

Table 1.

CMIP5 3-hourly precipitation from AMIP forcing.a (Expansions of model name acronyms are available online at http://www.ametsoc.org/PubsAcronymList.)

Table 1.

Observations come from the 3-hourly, 0.25° latitude–longitude resolution Tropical Rainfall Measuring Mission (TRMM) 3B42 dataset coarchived with CMIP (Teixeira et al. 2014). This is the most common source for global high-time-frequency observations. It includes data from a wide variety of sources in addition to the TRMM satellite and covers over 75% of the globe (Huffman et al. 2007). Simulations and observations overlap during 1998–2008, but we omit 1998 to minimize interannual variance (Fig. S1 in supplemental material). All data used are publically available (http://cmip-pcmdi.llnl.gov/cmip5).

Python-based software (Williams et al. 2013) employed to study atmospheric tides in surface-pressure fields (Covey et al. 2011) performs Fourier analysis in time over the 10 Januaries and 10 Julys spanning 1999–2008. This is equivalent to first averaging the time series into a composite 24-h day and then applying Fourier analysis (Covey and Gleckler 2014). We consider only the first two harmonic components: diurnal (24 h) and semidiurnal (12 h) periods. Although the daily cycle of rainfall is incompletely described by two sinusoids, most simulated and observed global data available—including all data in this study—exist at 3-hourly time resolution. Thus, the semidiurnal harmonic, with only four time points per cycle, already has large uncertainties, while the terdiurnal harmonic (8 h) is close to the limiting Nyquist period (6 h) and would add little real information.

An alternative procedure identifies minimum and maximum points in the composite diurnal cycle and computes empirical orthogonal functions (EOFs; Kikuchi and Wang 2008; Pritchard and Somerville 2009a,b; Wang et al. 2011). Since EOFs are eigenvectors of either the covariance or correlation matrix (which are symmetric and positive definite), the EOF procedure is mathematically guaranteed to produce an orthogonal set of normal modes, each associated with a positive eigenvalue that may be interpreted as the fractional variance associated with the corresponding mode. But from a physical point of view, some variations are more suitable for EOF analysis than others. Real-valued EOFs make each mode of variation a standing wave: any location is either in phase or 180° out of phase with any other location. Thus, EOFs are a natural way to study phenomena like the large-scale El Niño–La Niña “seesaw” oscillation of warm surface water between the western and eastern Pacific Oceans, but they are less natural diagnostics for other processes. This limitation may be overcome in various ways, including combined analysis of different EOFs [see Wang et al. (2011, their Fig. 2) for the diurnal cycle; Sperber and Kim (2012) for the Madden–Julian oscillation] or complex EOFs created from two-component vectors (e.g., Trenberth et al. 2000). Elaborated EOF analyses, however, seem less appropriate than straightforward Fourier analysis for first-look metrics.

3. Guidelines for metrics

Figure 1 presents the diurnal harmonic amplitude and phase from the observations. Since warm seasons have similar dynamics, Fig. 1 combines Northern Hemisphere July and Southern Hemisphere January in both maps. The resulting discontinuity at the equator is small. (Corresponding cold season maps provide little additional information because coherent diurnal variation outside the tropics is small; cf. Figs. S2–S5 in the supplemental material.) The diurnal amplitude map (Fig. 1, top) is normalized by the monthly mean (similar to Dai et al. 2007), since otherwise an imprint of the mean precipitation dominates. Grid points with amplitudes greater than monthly mean values (ratios >1, colored white) appear prominently in dry regions because of occasional showers occurring for one time step. The Dirac delta function would give a ratio of exactly 2 for each Fourier component; in our results, very dry regions give ratios close to 2 (Fig. S10 in the supplemental material). Wet regions generally give ratios ≤1, although exceptions occur (e.g., in Florida) as explained below in our discussion of individual gridpoint results.

Fig. 1.
Fig. 1.

TRMM 3B42 composites of July in the Northern Hemisphere and January in the Southern Hemisphere, for (top) ratio of diurnal harmonic amplitude to monthly mean precipitation and (bottom) diurnal harmonic phase expressed as LST of maximum precipitation. In the top map, ratios greater than unity are colored white (where 24-h Fourier amplitude for the diurnal cycle exceeds the mean monthly value). The few grid points where no precipitation occurs during the study period are colored gray. In the bottom map, areas where either monthly mean precipitation or its diurnal cycle is weak are colored gray (using masking criteria described in the text). Black dots in both maps locate grid points selected for later time series analysis.

Citation: Journal of Climate 29, 12; 10.1175/JCLI-D-15-0664.1

The phase map (bottom) explicitly masks out areas where the diurnal cycle is weak and hence phase is ill defined (gray colors). Our criteria for a weak diurnal cycle are that either monthly mean precipitation <0.75 mm day−1, about 25% of the global average, or that the amplitude/monthly mean ratio <25%. This masking allows visualization of the main features that metrics should encapsulate. It has the additional benefit of removing high-latitude wintertime ocean areas where traveling synoptic storms (baroclinic waves) alias into the diurnal signal. Some aliasing remains after masking, appearing as north–south stripes in phase maps (e.g., in Fig. S5). This problem has come up in earlier work (e.g., Fig. 5d of Dai et al. 2007). It appears more prominently in our graphics because we display higher resolution. As shown in the next section, however, our proposed summary metrics are not significantly affected by these spurious phase stripes.

General conclusions from the maps are insensitive to the choice of thresholds for data masking. Dai et al. (2007) mask out areas where daily mean precipitation is less than 0.1 mm day−1 and find 1) diurnal amplitudes 30%–100% of the mean over most land areas in summer and 10%–30% over most ocean areas and 2) diurnal maxima in the afternoon to evening hours over land and midnight to early morning over ocean. These conclusions also emerge from inspection of both Fig. 1 and corresponding maps for the semidiurnal harmonic (Figs. S6–S9 in the supplemental material).

Mapping Fourier components at satellite resolution visualizes observations that have been noted in regional studies and in the EOF approach. Most prominent is a summertime eastward propagation of rainfall in the center of North America (Tripoli and Cotton 1989; Dai et al. 1999; Jiang et al. 2006) that appears as north-to-south stripes in the phase map. Also apparent are phase discontinuities between land and ocean areas associated with the sea breeze (Dai 2001; Pritchard and Somerville 2009b). Although regional phase propagation is not represented in the large-scale average metrics we propose below, the very sharp distinction between land and ocean areas in Fig. 1 makes it clear that averages must be taken separately over each.

For CMIP’s highest-resolution GCMs, diurnal amplitudes are in fair agreement with observations (Figs. 2 and S2), but corresponding phases give warm season rainfall too early in the day, at least over land (Figs. 3 and S3). The semidiurnal harmonic exhibits similar errors (Figs. S6 and S7). Prior work revealed this problem in earlier, lower-resolution models (see above) as well as in higher resolutions, up to the point where explicitly simulated convection can replace subgrid-scale parameterizations (Sato et al. 2009; Dirmeyer et al. 2012). An important caveat is that TRMM 3B42 phase values are biased late by up to 3 h because the 3B42 algorithm includes outgoing longwave radiation (OLR) and thus includes high cold clouds, which peak after surface precipitation (Dai et al. 2007; Kikuchi and Wang 2008). This bias is larger than CMIP time-sampling errors (Berg et al. 2014; Zhou et al. 2015). Nevertheless, the problem with too-early climate model rainfall also appears in surface in situ observations, for example, in the North American Great Plains, as discussed below in conjunction with Fig. 4. We show in conjunction with Fig. 5 that average CMIP bias exceeds the TRMM observational uncertainty.

Fig. 2.
Fig. 2.

Amplitude ratios, as in the top map in Fig. 1, for AMIP simulations by the four highest-resolution CMIP5 models: (top left) 0.5° and (top right) 0.25° GFDL model and (bottom left) high- and (bottom right) super-high-resolution MRI model.

Citation: Journal of Climate 29, 12; 10.1175/JCLI-D-15-0664.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for phase, with definitions and thresholds for masking (gray colors) matching the bottom map in Fig. 1.

Citation: Journal of Climate 29, 12; 10.1175/JCLI-D-15-0664.1

Fig. 4.
Fig. 4.

Composite diurnal time series from TRMM 3B42 observations (black dots) and from the four highest-resolution CMIP5 models (colored dots), at the eight sample points shown in the maps above. Models are color coded as described in the figure legend. Observed time points are fit by (dashed lines) the diurnal harmonic only and by (solid lines) the combination of diurnal and semidiurnal harmonics, with error bars showing standard error of the mean.

Citation: Journal of Climate 29, 12; 10.1175/JCLI-D-15-0664.1

Fig. 5.
Fig. 5.

Harmonic dial plots of the amplitude and phase of Fourier components, after vector averaging over land and ocean areas separately, for TRMM 3B42 observations (black lines and dots), for the four highest-resolution CMIP5 models color coded as in Fig. 4 (colored lines and dots), and for the other 17 CMIP5–AMIP models with only July results shown for clarity (gray dots). For TRMM and the highest-resolution models, solid lines mark January results and dashed lines mark July results.

Citation: Journal of Climate 29, 12; 10.1175/JCLI-D-15-0664.1

A noteworthy feature of Figs. 2 and 3 is a close resemblance of MRI-AGCM3.2S and MRI-AGCM3.2H, suggesting that subgrid-scale “column physics” rather than explicitly resolved dynamics is responsible for model errors. GFDL HiRAM-C360 and GFDL HiRAM-C180 also resemble each other, except for diurnal harmonic phase in Australia and some coastal zones, where the C360 version is nearly 180° out of phase with the C180 version. Since column physics is identical except for tuning parameters in the GFDL HiRAM versions archived in CMIP5 (S.-J. Lin 2015, personal communication), their differences probably involve explicitly resolved dynamics. It may be relevant that GFDL HiRAM uses a cubed-sphere grid that inevitably generates discontinuities (Putman and Lin 2007).

Figures 13 come from Fourier analysis at each individual grid point. To sample this process, we choose 7 of the 10 points in Fig. 4 of Dai et al. (2007) plus the Southern Great Plains site of the Atmospheric Radiation Measurement program (ARM SGP). These eight points are marked by black dots in Figs. 13; results for each are shown in Fig. 4. Observed composite diurnal variations are fitted rather well by only the first two harmonic components (black dots and lines) wherever a coherent diurnal cycle exists. Observed rainfall at the Florida and Yangtze River valley locations peaks in late afternoon and evening hours, typical of warm land areas (Figs. 4a,b). Northeast Argentina and the ARM SGP site, both in areas of zonal propagation (Fig. 1), have broad maxima (Figs. 4c,d). The Pacific warm pool and Indian Ocean points experience mainly 24-h cycles of low amplitude with peaks occurring in early morning, typical of the tropical oceans (Figs. 4e,f). The final two grid points in Fig. 4 exemplify higher-latitude ocean areas with no obvious diurnal cycle (Figs. 4g,h).

The Florida grid point (Fig. 4a) has a particularly simple diurnal cycle with very little rainfall in the morning and one distinct peak in late afternoon–early evening. Therefore, Fourier analysis gives a ratio of harmonic amplitude to mean value of 1.3, lying in between 1 (for a pure sine wave) and 2 (for a delta function). The same results are obtained for other wet locations when rainfall occurs at restricted times of day (Fig. S10).

We have added ±1σ error bars to the observed 3-hourly values plotted in Fig. 4, with σ defined as standard error of the composite mean, that is, we divided the standard deviation of the data points contributing to each local standard time (LST) by . This underestimates the true uncertainty by assuming that each of the 310 days of a given month during 1999–2008 is independent. In spite of this underestimate, the error bars are rather large, not only at locations where one must question the significance of Fourier analysis (Figs. 4g,h) but also in locations with coherent diurnal cycles. Applying a rigorous statistical test could therefore suppress physically important results, for example, the well-established morning rainfall peaks observed in the tropical Pacific and Indian Oceans (Figs. 4e and 4f; compare Figs. 4f and 4g in Dai et al. 2007) and the peak around midnight at the ARM SGP site (Fig. 4c). Therefore, we retain the less severe constraints of the masking procedure defined above.

The ARM data are particularly important because they are ground based and in excellent agreement with TRMM 3B42 (cf. Fig. 1b in Jiang et al. 2006). At this location, occasional storms produce high rainfall rates while most time bins record zero rain. In statistical terms, large skewness moves the standard deviation far out beyond the mean value because the distribution is more Poisson-like than Gaussian (Fig. S11 in the supplemental material). Uncertainty in average amplitude and phase, however, is smaller than the uncertainty at each LST and location. Also, the central limit theorem implies that statistics become more Gaussian as more space–time points are averaged together. Accordingly, the summary metrics presented in Fig. 5 below are fairly robust.

In contrast to observations, modeled rainfall (colored dots) in Fig. 4 is quite noisy. Often its pattern is poorly fit by low-order Fourier harmonics (or other simple curves), implying that Fourier amplitudes and phases at single grid points are not ideal for its decomposition. Florida and the Yangtze River valley sample the widespread land areas where observed precipitation peaks in late afternoon–early evening, and the models rain too early in the day, as noted above. At other points, however, a systematic difference between model simulations and observations is not evident. The implication is that model output needs to be averaged over more than one grid point, either “by eye” in visual inspection of maps, or more systematically as proposed below.

4. Proposed metrics

Careful inspection of several maps per model is impractical in a large multimodel ensemble. Zonal averaging provides some insight if land and ocean areas are each homogeneous and averaged separately (see top panels in Fig. 17 of Dai 2006). Here we propose an extension of this procedure to area averages. The “harmonic dial” diagram originally developed in atmospheric tide research—treating the amplitude and phase of each harmonic component as a two-dimensional vector—provides the most appropriate display graphic (see section 1.2B in Chapman and Lindzen 1970). Averaging separately over all land and all ocean is a reasonable first step because the observed phase is roughly uniform (apart from areas of phase propagation).

In this approach, adding vectors during the averaging process automatically down-weights areas where the diurnal cycle is weak. There is no need to choose arbitrary thresholds for data masking. In any case the choice of thresholds or filtering makes little difference. Table 2 compares averages computed using no filtering or masking of any kind with analogous averages using the same filtering procedure as in Figs. 1 and 3. Averaged phases with and without filtering are virtually identical. Averaging over the spurious phase stripes that arise from synoptic time-scale aliasing (e.g., appearing in the bottom panel of Fig. 1) leads to cancellation of vector sums because the phases run through one or more 360 cycles; hence, this aliasing does not affect the end results. Averaged amplitudes are increased by the filtering, because it masks out shorter vectors, but the inflation factor is roughly uniform and thus intercomparison of amplitudes from different data sources is not meaningfully affected. In short, our proposed metrics are not sensitive to masking strategy. Therefore, the simplest and best strategy is to use no masking at all when computing area averages.

Table 2.

Vector-averaged spatial means without and with masking: amplitude (mm day−1), phase (LST) (masked results in curly brackets). Obs4MIPs stands for Observations for Model Intercomparison Projects.

Table 2.

Figure 5 shows resulting plots for TRMM 3B42 observations and for all models in Table 1. No masking or filtering is used in producing this figure. Each pair of maps (like Fig. 1) has become a pair of points (one for January and one for July) on each of four harmonic dials representing diurnal versus semidiurnal components and land versus ocean averaging. Land areas (Fig. 5, top panels) exhibit considerable scatter among the models, but both the diurnal and semidiurnal components from most models have vector-averaged lengths comparable to observations. A notable exception is GFDL HiRAM-C360 (blue lines and points). In this model, the out-of-phase spatial variation noted above produces a weak mean amplitude. This problem arises occasionally in the harmonic dial approach, for example, in Fig. 5 of the tide study of de Argandoña et al. (2010).

The vast majority of models, however, produce coherent spatial averages over land that reveal systematic phase errors consistent with Figs. 13. The difference between TRMM and the models, 6–10 h in the dominant diurnal component, is much greater than either the 2–3-h observational error estimated by Dai et al. (2007) or the 3-h observational error estimated by Kikuchi and Wang (2008). For the weaker semidiurnal component, the difference between TRMM and the models is roughly 3 h on average, perhaps within observational uncertainty. Still, model rainfall peaks systematically earlier than the observations. In both observations and models the semidiurnal component sharpens the diurnal peak, for example, TRMM has a diurnal maximum around 1900 LST reinforced by semidiurnal maxima around 1600 LST.

For ocean areas the length scale in Fig. 5 (bottom panels) is expanded to accommodate considerably smaller amplitudes. For the diurnal component, nearly all models have amplitudes roughly comparable to observations but peak times systematically earlier. These oceanic diurnal results from the AMIP-type runs are consistent with those from the coupled model simulations (Dai 2006; Flato et al. 2013). It seems that the removal of the SST diurnal cycle has little effect on the mean diurnal cycle of oceanic precipitation. The semidiurnal component over oceans is small in both models and observations, consistent with Fig. 4.

5. Future extensions

The metrics proposed here would be only a first step in quantifying the diurnal cycle. Vector means could also be computed for smaller scales than the all-land and all-ocean areas shown in Fig. 5, giving more information about spatial variations. The GFDL HiRAM-C360 problem arising from unwanted phase cancellation (Fig. 5a) could be “solved” by ad hoc division of land into coastal areas versus interior areas, but this seems arbitrary. A physically based geographical division suggested by Kikuchi and Wang (2008, their Fig. 10) might be more informative, but it would not solve the GFDL HiRAM-C360 problem, nor would the geographical regions necessarily remain fixed when the climate changes.

In addition to area-mean bias, vector averaging can be extended to other familiar statistics by simply replacing the multiplication of numbers with the scalar (dot) product of vectors. For example, the spatial variance of a vector field x becomes , where the angle brackets denote area means and the primes denote differences from area means, for example, . The resulting algebra expresses the centered mean-square difference (i.e., the difference after removing the area-mean bias of each field) exactly as for scalar fields in a Taylor (2001) diagram. The mean-square difference between two vector fields x and y becomes , where Rxy is their correlation . In this equation, the scalar product on the right-hand side is the squared difference of vector means, given by constructing arrows that join observed with model points in Fig. 5. The remaining terms on the right form the centered mean-square difference.

Taylor diagrams reasonably summarize model behavior if the bias is small compared with the centered root-mean-square (RMS) error. This is true for monthly mean precipitation but not true for its diurnal cycle. The substantial phase discrepancies between models and observations discussed above are coherent across wide areas. Thus, Taylor diagrams constructed from vector statistics would probably not be meaningful for the diurnal cycle of precipitation, although one could compute results analogous to Fig. 5 for any region. For example, dividing North America into a few domains with different character would raise the question of how much the phase can be allowed to vary within a domain before it becomes significantly different. Here some form of classification or EOF analysis might apply, but a key point is that one can reduce the dimensionality many fold.

Finally, we note that precipitation is only one of 23 fields archived at 3-hourly frequency in CMIP5. This database remains a largely unexplored source of information about the diurnal cycle of near-surface fields. Relationships between different fields may aid satellite retrieval algorithms, as well as climate model development.

Acknowledgments

We thank Karl Taylor for mathematical consultation, the Working Group on Coupled Modelling for CMIP planning, and the modelers (Table 1) for providing output. We also thank our colleagues Shaocheng Xie and Chengzhu Zhang for useful discussion of the relationship between ARM and TRMM observations. Work was performed under auspices of the DOE Office of Science by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, at the National Center for Atmospheric Research, and at Columbia University and the State University of New York. We acknowledge funding from NSF Grants AGS-1353740 and AGS-1331375, DOE Awards DE-SC0012602 and DE-SC0012711, and NOAA Award NA15OAR4310086.

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