The “paleo calendar effect” is a common expression for
the impact that changes in the length of months or seasons over time,
related to changes in the eccentricity of Earth's orbit and precession, have
on the analysis or summarization of climate-model output. This effect can
have significant implications for paleoclimate analyses. In particular,
using a “fixed-length” definition of months (i.e., defined by a fixed
number of days), as opposed to a “fixed-angular” definition (i.e., defined
by a fixed number of degrees of the Earth's orbit), leads to comparisons of
data from different positions along the Earth's orbit when comparing paleo
with modern simulations. This effect can impart characteristic spatial
patterns or signals in comparisons of time-slice simulations that otherwise
might be interpreted in terms of specific paleoclimatic mechanisms, and we
provide examples for 6, 97, 116, and 127 ka. The calendar effect is
exacerbated in transient climate simulations in which, in addition to spatial
or map-pattern effects, it can influence the apparent timing of extrema in
individual time series and the characterization of phase relationships among
series. We outline an approach for adjusting paleo simulations that have
been summarized using a modern fixed-length definition of months and that
can also be used for summarizing and comparing data archived as daily data.
We describe the implementation of this approach in a set of Fortran 90
programs and modules (PaleoCalAdjust v1.0).
Introduction
In paleoclimate analyses, there are generally two ways of defining months or
seasons (or any other portion of the year): (1) a “fixed-length”
definition, wherein, for example, months are defined by a fixed number of days
(typically the number of days in the months of the modern Gregorian
calendar), and (2) a “fixed-angular” definition, wherein, again for example,
months are defined by a fixed number of degrees of the Earth's orbit.
Variations in the Earth's orbit over time will have different effects on
fixed-length versus fixed-angular months: fixed-length months will contain
the same number of days through time, but the arc of the Earth's orbit
traversed during that interval will vary over time, while fixed-angular
months will each sweep out the same arc of the Earth's orbit through time,
but the number of days they contain will vary over time. The issue for
paleoclimate analyses is that, using a fixed-length definition of months,
comparisons of paleo simulations for different time periods may incorporate
data from different positions along the Earth's orbit for a particular
month, which can produce patterns in data–model and model–model comparisons
that mimic observed paleoclimatic changes.
This paleo calendar effect arises from a consequence of Kepler's (1609)
second law of planetary motion: Earth moves faster along its elliptical
orbit near perihelion and slower near aphelion. Because the time of year of
perihelion and aphelion varies over time, the length of time that it takes the
Earth to traverse one-quarter (90∘) or 1/12 (30∘) of
its orbit (a nominal season or month) also varies, so months or seasons
are shorter near perihelion and longer near aphelion. For example, a 30 or
90∘ portion of the orbit will be traversed in a shorter period of time
when the Earth is near perihelion (because it is moving faster along its
orbit) and a longer period when it is near aphelion. Likewise, a 30 or
90 d interval will define a longer orbital arc near perihelion and a
shorter one near aphelion. When examining present-day and paleo simulations,
summarizing data using a fixed-length definition of a particular month (e.g.,
31 d of a 365 d year), as opposed to a fixed-angular definition (e.g.,
31.0 d × (360.0∘/ 365.25 d) degrees of orbit, where 365.25 is the
number of days in a year), will therefore result in comparing conditions
that prevailed as the Earth traversed different portions of its orbit (e.g.,
Kutzbach and Gallimore, 1988; Joussaume and Braconnot, 1997). Consequently,
comparisons of, for example, present-day and paleoclimatic simulations that
use the same fixed-length calendar (e.g., a present-day calendar definition
of January as 31 d long) will include two components of change, one
consisting of the actual model-simulated climate change between the
present-day and paleo time period and a second arising simply from the
difference in the angular portion of the orbit defined by 31 d at present
as opposed to 31 d at the paleo time period.
This impact of the calendar effect on the analysis of paleoclimatic
simulations and their comparison with present-day or “control” simulations
is well known and not trivial (e.g., Kutzbach and Gallimore, 1988; Joussaume
and Braconnot, 1997). The effect is large, spatially variable, and can
produce apparent map patterns that might otherwise be interpreted as
evidence of, for example, latitudinal amplification or damping of
temperature changes, the development of continent–ocean temperature
contrasts, interhemispheric contrasts (the “bipolar seesaw”), changes in
the latitude of the intertropical convergence zone (ITCZ), variations in the
strength of the global monsoon, and other effects (see examples in Sect. 3.1 to 3.3).
In transient climate-model simulations, time series of data aggregated using
a fixed-length modern calendar, as opposed to an appropriately changing one,
can differ not only in the overall shape of long-term trends in the series,
but also in variations in the timing of, for example, Holocene “thermal
maxima” which, depending on the time of year, can be on the order of
several thousand years. The impact arises not only from the orbitally
controlled changes in insolation amount and the length of months or seasons,
but also from the advancement or delay in the starting and ending days of
months or seasons relative to the solstices. Even if daily data are
available, the calendar effect must still be considered when summarizing
those data by months or seasons or when calculating climatic indices such
as the mean temperature of the warmest or coldest calendar month – values
that are often used for comparisons with paleoclimatic observations (e.g.,
Harrison et al., 2014, 2016; see Kageyama et al., 2018, for a further
discussion). As will be discussed further below (Sect. 3.1), the calendar
effect must be considered not only in data–model comparisons, but also in
model-only intercomparisons. It is also the case that the calendar effect
can have a small impact on annual average values because the first day of
the first month of the year may fall in the previous year, and the last day
of the last month of the year may fall in the next year.
Various approaches have been proposed for incorporating the calendar effect
or “adjusting” monthly values in analyses of paleoclimatic simulations
(e.g., Pollard and Reusch, 2002; Timm et al., 2008; Chen et al., 2011).
Despite this work, the calendar effect is generally ignored, so our
motivation here is to provide an adjustment method that is relatively simple
and can be applied generally to CMIP-formatted (https://esgf-node.llnl.gov/projects/cmip5/, last access: 11 August 2019) files, such as those
distributed by the Paleoclimate Modelling Intercomparison Project (PMIP;
Kageyama et al., 2018). Our approach (broadly similar to Pollard and Reusch,
2002) involves (1) determining the appropriate fixed-angular month lengths
for a paleo experiment (e.g., Kepler, 1609; Kutzbach and Gallimore, 1988),
(2) interpolating the data to a daily time step using a mean-preserving
interpolation method (e.g., Epstein, 1991), and then (3) averaging or
accumulating the interpolated daily data using the appropriate (paleo) month
starting and ending days, thereby explicitly incorporating the changing
month lengths. In cases in which daily data are available (e.g., in CMIP5–PMIP3
“day” files), only the third step is necessary. This approach is
implemented in a set of Fortran 90 programs and modules (PaleoCalAdjust v1.0, described below). With a suitable program code “wrapper” file, the
approach can also be applied to transient simulations (e.g., Liu et al.,
2009; Ivanovic et al., 2016).
In the following discussion, we describe (a) the calendar effect on month
lengths and their beginning, middle, and ending days over the past 150 kyr;
(b) the spatial patterns of the calendar effect on temperature and
precipitation rate for several key times (6, 97, 116, and 127 ka); and (c) the methods that can be used to calculate month lengths (on various
calendars) and to “calendar adjust” monthly or daily paleo model output to
an appropriate paleo calendar.
Month-length variations
The fixed-angular length of months as they vary over time can be calculated
using the algorithm in Appendix A of Kutzbach and Gallimore (1988) or via
Kepler's equation (Curtis, 2014), which we use here and which is described
in detail in Sect. 4. The algorithms yield the length of time (in
real-number or fractional days) required to traverse a given number of
degrees of celestial (as opposed to geographical) longitude starting from
the vernal equinox, the common “origin” for orbital calculations (see
Joussaume and Braconnot, 1997, for a discussion), or from the changing time of
year of perihelion. We use the Kepler-equation approach to calculate the
month-length values that are plotted in Figs. 1–5, and the specific values
plotted are provided in the code repository in the folder
/data/figure_data/month_length_plots/ (see the “Code and data availability” section).
Variations over the past 150 kyr in the beginning and
ending days of fixed-angular months for a 365 d noleap calendar, shown
for 1 kyr intervals beginning at 0 ka (1950 CE). The left side of each
horizontal bar shows the beginning day, while the right side shows the ending
day of a particular month for each 1 kyr interval. The month-length
“anomalies” or differences from the present day are shown by shading, with
individual paleo months that are shorter than those at present indicated by
green shades and those that are longer indicated by blue shades. The day
that perihelion occurs for each 1 kyr interval is indicated by a magenta
dot, and the overall pattern of month-length anomalies can be seen to follow
the day of perihelion. The figure shows that the changing month lengths move
the beginning, middle, and ending days of each month (as well as the
beginning and ending days of the year). The day of the Northern Hemisphere
summer solstice is indicated by a black diamond on the x axes.
Variations in the difference (in days) between the
mid-month day of each month and the day of the June solstice. Months that
are shifted closer to the June solstice are indicated by orange hues, while
those that are farther away are indicated by blue. As in Fig. 1, variations
over the past 150 kyr in the beginning and ending days of fixed-angular
months for a 365 d noleap calendar are shown for 1 kyr intervals
beginning at 0 ka (1950 CE). The left side of each horizontal bar shows the
beginning day, while the right side shows the ending day of a particular
month for each 1 kyr interval. Variations in the beginning and ending days
of individual months can be seen to track the climatic precession parameter
(e⋅sinω, where e is eccentricity and ω is the
longitude of perihelion measured from the vernal equinox, an index of
Earth's distance from the Sun at the summer solstice), which is plotted at
the right side of the figure (red dots). (Note that the inverse of the
climatic precession parameter is plotted for easier comparison.) The day of
the Northern Hemisphere summer solstice is indicated by a black diamond on
the x axes.
Calendar effects on insolation at 45∘ N. The
differences plotted show the values of average daily insolation at mid-month
days identified using the appropriate fixed-angular paleo calendar minus
those using the fixed-length definition of present-day months, with orange
hues showing positive differences and purple hues negative differences. As
in Fig. 1, variations over the past 150 kyr in the beginning and ending days
of fixed-angular months for a 365 d noleap calendar are shown for 1 kyr
intervals beginning at 0 ka (1950 CE). The left side of each horizontal bar
shows the beginning day, while the right side shows the ending day of a
particular month for each 1 kyr interval.
Calendar effects on insolation at the Equator. The
differences plotted show the values of average daily insolation at mid-month
days identified using the appropriate fixed-angular paleo calendar minus
those using the fixed-length definition of present-day months, with orange
hues showing positive differences and purple hues negative differences. As
in Fig. 1, variations over the past 150 kyr in the beginning and ending days
of fixed-angular months for a 365 d noleap calendar are shown for 1 kyr
intervals beginning at 0 ka (1950 CE). The left side of each horizontal bar
shows the beginning day, while the right side shows the ending day of a
particular month for each 1 kyr interval.
Calendar effects on insolation at 45∘ S. The
differences plotted show the values of average daily insolation at mid-month
days identified using the appropriate fixed-angular paleo calendar minus
those using the fixed-length definition of present-day months, with orange
hues showing positive differences and purple hues negative differences. As
in Fig. 1, variations over the past 150 kyr in the beginning and ending days
of fixed-angular months for a 365 d noleap calendar are shown for 1 kyr
intervals beginning at 0 ka (1950 CE). The left side of each horizontal bar
shows the beginning day, while the right side shows the ending day of a
particular month for each 1 kyr interval.
The beginnings and ends of each fixed-angular month in a 365 d noleap
calendar are shown at 1 kyr intervals for the past 150 kyr in Fig. 1,
calculated using the approach described in Sect. 4.2–4.5 below. (See Sect. 4.4.1 of the NetCDF Climate and Forecast Metadata Conventions (http://cfconventions.org, last access: 11 August 2019) for a discussion of
climate-model output calendar types.) The month-length anomalies (i.e.,
long-term differences between paleo and present month lengths, with present
defined as 1950 CE) are shown in color, with (paleo) months that are shorter
than those at present in green shades and months that are longer than those
at present in blue shades. Not only do the lengths of fixed-angular months
vary over time, but so do their middle, beginning, and ending days (Fig. 2),
with mid-month days that are closer to the June solstice indicated in orange
and those that are farther from the June solstice in blue. The variations in
month length (Fig. 1) obviously track the changing time of year of
perihelion, while the beginning and ending day anomalies reflect the
climatic precession parameter (Fig. 2). The shift in the beginning, middle,
or end of individual months relative to the solstices ultimately controls
the average or mid-month daily insolation at different latitudes (Figs. 3–5).
Figure 2 essentially maps the systematic displacement of the stack of
horizontal bars for individual months, which reflects the changes during the
year of the beginning and end of each month. Using 15 ka as an example,
perihelion occurs on day 111.87 (relative to 1 January), and consequently
the months between March and August are shorter than present (Fig. 1). That
effect in turn moves the beginning, middle, and ending day of the months
between April and December earlier in the year (Fig. 2). July therefore
begins a little over 5 d earlier than at present, i.e., closer within
the year to the June solstice. June is likewise displaced earlier in the
year, with the beginning of the month 3.41 d farther from the June
solstice and the end a similar number of days closer to the June solstice
than at present. Thus, the calendar effect arises more from shifts in the
timing (beginning, middle, and end) of the months than from changes in their
lengths.
The calendar effect is illustrated below for four times: 6 and 127 ka are
the target times for the planned warm-interval midHolocene and lig127k CMIP6–PMIP4 (Coupled
Model Intercomparison Project Phase 6–Paleoclimate Modelling Intercomparison
Project Phase 4) simulations (Otto-Bliesner et al., 2017) and illustrate the
calendar effects when perihelion occurs in the boreal summer or autumn (Fig. 6); 116 ka is the time of a proposed sensitivity experiment for the onset of
glaciation (Otto-Bliesner et al., 2017) and illustrates the calendar effect
when perihelion occurs in boreal winter; and 97 ka was chosen to illustrate
an orbital configuration not represented by the other times (i.e., one with
boreal spring months occurring closer to the June solstice).
Orbital parameter variations at 1 kyr intervals over the
past 150 kyr for obliquity, climatic precession, eccentricity, and day of
perihelion (relative to 1 January). Climatic precession is calculated as
e⋅sinω, where e is eccentricity and ω is the longitude of
perihelion measured from the vernal equinox.
At 6 ka, perihelion occurred in September (Fig. 6), and the months from May
through October were shorter than today (Fig. 1), with the greatest
differences in August (1.65 d shorter than present). This contraction of
month lengths moved the middle of all of the months from April through
December closer to the June solstice (Fig. 2), with the greatest difference
in November (5.01 d closer to the June solstice and therefore 5.01 d farther
from the December solstice). At 127 ka, perihelion was in late June, and the
months April through September were shorter than today (Fig. 1), with the
greatest difference in July (3.19 d shorter than present). As at 6 ka,
the shorter boreal summer months at 127 ka move the middle of the months
between July and December closer to the June solstice (Fig. 2), with the
greatest difference in September and October (12.80 and 12.70 d closer,
respectively). At both 6 and 127 ka, the longer boreal winter months begin
and end earlier in the year, placing the middle of January 3.38 (6 ka) and
4.35 (127 ka) days farther from the June solstice than at present. As can be
noted in Figs. 1 and 2, 127 ka does not represent a simple amplification of
6 ka conditions. Although broadly similar in having shorter late boreal
summer and autumn months that begin earlier in the year (and hence closer to
the June solstice), the two times are only similar in the relative
differences from the present in month length and beginning and ending days.
At 116 ka, perihelion was in late December, and consequently the months from
October through March were shorter than present (Fig. 1). This has the main
effect of moving the middle of the months July through December farther from
the June solstice (with a maximum in September of 5.80 d; Fig. 2),
somewhat opposite to the pattern at 6 and 127 ka. At 97 ka, perihelion
occurred in mid-November, between its occurrence in September at 6 ka and
December at 116 ka (Fig. 1). The impact on month length and mid-month timing
is complicated, with the mid-month days of January through March and July
through October occurring farther from the June solstice (Fig. 2).
The first-order impact of the calendar effect can be gauged by comparing (at
a particular latitude) daily insolation values for mid-month days determined
using the appropriate paleo calendar (which assumes fixed-angular
definitions of months) with insolation values for mid-month days using the
present-day calendar (which assumes fixed-length definitions of months).
Using the example of 45∘ N, at 6 ka the shorter (than
present) and earlier (relative to the June solstice) months of September
through November had insolation values over 10 W m-2 (12.67, 15.59,
and 10.38 W m-2, respectively) greater for mid-month days defined
using the fixed-angular paleo calendar in comparison with values determined
using the fixed-length present-day calendar (Fig. 3), and at 127 ka, the
differences exceeded 35 W m-2 for the months of August through
October (41.27, 49.74, and 38.66 W m-2, respectively). These
positive insolation differences were accompanied by negative differences
from January through June. At first glance, it may seem counterintuitive
that the calendar effects that yield positive differences in mid-month
insolation are not balanced by negative insolation differences, as is the
case with the month-length differences. However, the calendar effects on
insolation include both the month-length differences and long-term
insolation differences themselves (Figs. 7–9), which are not symmetrical
within the year, so the calendar effects do not “cancel out” within
the year.
Long-term differences in mid-month average daily
insolation relative to present (0 ka or 1950 CE) at 45∘ N for a
fixed-angular calendar. As in Fig. 1, variations over the past 150 kyr in
the beginning and ending days of fixed-angular months for a 365 d noleap
calendar are shown for 1 kyr intervals beginning at 0 ka (1950 CE). The left
side of each horizontal bar shows the beginning day, while the right side
shows the ending day of a particular month for each 1 kyr interval.
Long-term differences in mid-month average daily
insolation relative to present (0 ka or 1950 CE) at the Equator for a
fixed-angular calendar. As in Fig. 1, variations over the past 150 kyr in
the beginning and ending days of fixed-angular months for a 365 d noleap
calendar are shown for 1 kyr intervals beginning at 0 ka (1950 CE). The left
side of each horizontal bar shows the beginning day, while the right side
shows the ending day of a particular month for each 1 kyr interval.
Long-term differences in mid-month average daily
insolation relative to present (0 ka or 1950 CE) at 45∘ S for a
fixed-angular calendar. As in Fig. 1, variations over the past 150 kyr in
the beginning and ending days of fixed-angular months for a 365 d noleap
calendar are shown for 1 kyr intervals beginning at 0 ka (1950 CE). The left
side of each horizontal bar shows the beginning day, while the right side
shows the ending day of a particular month for each 1 kyr interval.
At 116 ka, the later occurring months of September and October had negative
differences in mid-month insolation that exceeded 10 W m-2 (-14.80 and -15.23 W m-2, respectively; Fig. 3). For regions
where surface temperatures are strongly tied to insolation with little lag,
such as the interiors of the northern continents, these calendar effects on
insolation will be directly reflected by the calendar effects on
temperatures. By moving the beginning, middle, and end of individual months
(and seasons) closer to or farther from the solstices, the “apparent
temperature” of those intervals will be affected (i.e., months or seasons
that start or end closer to the summer solstice will be warmer). The
calendar effect on insolation varies strongly with latitude, with the sign
of the difference broadly reversing in the Southern Hemisphere (Figs. 3–5).
Figures 3 to 5 show the calendar effect on insolation at three different
latitudes (which are longitudinally uniform, and hence not much would be
gained from mapping them), and that effect can be thought of as being
compounded by the month-length effects superimposed on the time-varying
insolation. The amplitude of the calendar effect on insolation in December
at 45∘ N (Fig. 3) only occasionally exceeds the range between
-2.0 and +2.0 W m-2 because it is winter in the Northern
Hemisphere and insolation in general is low. Likewise, the calendar effects
on insolation at 45∘ S (Fig. 5) are quite muted in June, which is
winter in the Southern Hemisphere.
Impact of the calendar effect
Past demonstrations of the calendar effect have used “real” paleoclimatic
simulations, so the climate patterns being used in these demonstrations
include both the calendar effect and the long-term mean differences in
climate between the experiment and control simulations. Comparison of Figs. 3
and 7 clearly shows, however, that the variations over time in insolation
and in the calendar effect are not identical, so the use of an actual
paleoclimatic experiment (e.g., for 6 or 127 ka) to illustrate the
calendar effect will inevitably be confounded by the climatic response to
changes in insolation (and other boundary conditions). The impact on the
analysis of paleoclimatic simulations of the calendar effect can
alternatively be assessed by assuming that the long-term mean difference in
climate (also referred to as the experiment minus the control “anomaly”) is
zero everywhere, illustrating the “pure” calendar effect. Pseudo-daily
interpolated values (or actual daily output, if available) of present-day
monthly data can then simply be reaggregated using an appropriate paleo
calendar and compared with the present-day data. (The pseudo-daily values
used here were obtained by interpolating monthly data to a daily time step
using the monthly-mean-preserving algorithm described below.)
The pure calendar effect is demonstrated here using present-day monthly
long-term mean (1981–2010) values of near-surface air temperature (tas) from
the Climate Forecast System Reanalysis (CFSR; Saha et al., 2010; https://esgf.nccs.nasa.gov/projects/ana4mips/, last access: 11 August 2019) and monthly precipitation
rate (precip) from the CPC Merged Analysis of Precipitation (CMAP; Xie and Arkin,
1997; https://www.esrl.noaa.gov/psd/data/gridded/data.cmap.html, last access: 11 August 2019) (Fig. 10).
These data were chosen because they are global in extent and are of
reasonably high spatial resolution. The long-term mean values of both data
sets follow an implied 365 d noleap calendar.
Present-day (1981–2010 CE) long-term mean values of
monthly near-surface air temperature (tas) from the Climate Forecast System
Reanalysis (CFSR), the mean temperatures of the warmest (MTWA) and coldest
(MTCO) months and their differences from the same data, and precipitation
rate (precip) from the CPC Merged Analysis of Precipitation (CMAP).
If it is assumed that there is no long-term mean difference between a
present-day and paleo simulation (by adopting the present-day data as the
simulated paleo data), then the unadjusted present-day data can be compared
with present-day data adjusted to the appropriate paleo month lengths. The
calendar-adjusted minus unadjusted differences will therefore reveal the
inverse of the built-in calendar-effect “signal” in the unadjusted data,
which might readily be interpreted in terms of some specific paleoclimatic
mechanisms while instead being a data analytical artifact. Positive values
on the maps (Figs. 11–13) indicate, for example, where temperatures would be
higher or precipitation greater if a fixed-angular calendar were used to
summarize the paleo data.
Calendar effects on near-surface air temperature for 6 ka (a), 97 ka (b), 127 ka (c), and 116 ka (d). The maps show the patterns of month-length adjusted average
temperatures minus the unadjusted values using 1981–2010 long-term averages
of CFSR tas values, with positive differences (indicating that the adjusted data
would be warmer than unadjusted data) in red hues and negative differences
in blue.
Calendar effects on the mean near-surface air
temperatures of the warmest (MTWA) and coldest (MTCO) months and their
differences (an index of seasonality) for 6, 97, 116, and 127 ka
(top to bottom row). The maps show the patterns of month-length adjusted
average temperatures minus the unadjusted values for MTWA and MTCO using
1981–2010 long-term averages of CFSR tas values, with positive differences
(indicating that the adjusted data would be warmer than unadjusted data) in
red hues and negative differences in blue.
Calendar effects on precipitation rate for 6 ka (a), 97 ka (b), 127 ka (c), and 116 ka (d).
The maps show the patterns of month-length adjusted precipitation rate minus
the unadjusted values using 1981–2010 long-term averages of CMAP precip values,
with positive differences (indicating that the adjusted data would be wetter
than unadjusted data) in blue hues and negative differences in brown.
Monthly temperature
The impacts of using the appropriate calendar to summarize the data (as
opposed to not) are large, often exceeding 1 ∘C in absolute value
(Fig. 11). The effects are spatially variable and are not simple functions
of latitude, as might initially be expected, because the effect increases
with the amplitude of the annual cycle (which has a substantial longitudinal
component) for temperature regimes that are in phase with the annual cycle
of insolation. For temperature regimes that are out of phase with
insolation, the calendar-adjusted minus unadjusted values would be negative
and largest when the temperature variations were exactly out of phase. (If
there were no annual cycle, i.e., if a climate variable remained constant
over the course of a year, the calendar effect would be zero.) The
interaction between the annual cycle and the direct calendar effect on
insolation produces patterns of the overall calendar effect that happen to
resemble some of the large-scale responses that are frequently found in
climate simulations, both past and future, such as high-latitude
amplification or damping, continent–ocean contrasts, interhemispheric
contrasts, and changes in the seasonality of temperature (see Izumi et al.,
2013). Because the month-length calculations use the Northern Hemisphere
vernal equinox as a fixed origin for the location of Earth along its orbit,
the effects seem to be small during the months surrounding the equinox (i.e.,
February through April; Fig. 11), and indeed the selection of a different
origin would produce different apparent effects (see Joussaume and
Braconnot, 1997; Sect. 2.1). However, the selection of a different origin
would not change the relative (to present) length of time it would take
Earth to transit any particular angular segment of its orbit.
At 6 ka, the largest calendar effects on temperature can be observed over
the Northern Hemisphere continents for the months from September through
December (Fig. 11), consistent with the earlier beginning of these months
(Fig. 2) and the direct calendar effect on insolation at 45∘ N
(Fig. 3). For example, in the interior of the northern continents, as well
as North Africa, temperature is in phase with insolation, and therefore the
calendar effect on insolation (Fig. 3), which produces strongly positive
differences from August through November, is reflected by the calendar
effect on temperature. Over the northern oceans, temperature is broadly in
phase with insolation but with a lag, which reduces the magnitude of the
effect and gives rise to an apparent land–ocean contrast that otherwise
might be interpreted in terms of some particular paleoclimatic mechanism.
The calendar effect on temperature from January through March produces
negative calendar-adjusted minus unadjusted values in the northern
continental interiors (Fig. 11), which is also consistent with the calendar
effect on insolation. In the Southern Hemisphere at 6 ka, the calendar
effects on temperature produce generally negative differences, which is
consistent with the calendar effects on mid-month insolation at
45∘ S (Fig. 5) that produce generally negative differences
throughout the year, particularly during the months of August through
November. Like the continent–ocean contrast in the Northern Hemisphere,
the Northern Hemisphere–Southern Hemisphere contrast in the calendar
effect on temperature could also be interpreted in terms of one or another
of the mechanisms thought to be responsible for interhemispheric temperature
contrasts.
At 127 ka, the calendar effect on temperature is broadly similar to that at
6 ka over the months from September through March, but it differs in sign from
April through July and in magnitude in August (Fig. 11). These patterns are
also consistent with the direct calendar effects on insolation. At 127 ka,
the calendar effect on insolation produces strongly positive differences in
the Northern Hemisphere earlier in the northern summer than at 6 ka (Fig. 3), while at 45∘ S the calendar effect on insolation produces
strongly negative differences in July and persists that way through November
(Fig. 5). At 116 ka, perihelion occurs in late December in comparison to
late June at 127 ka (Figs. 1 and 6), and not surprisingly the calendar
effect on temperature is nearly the inverse of that at 127 ka (Fig. 11).
This pattern has important implications for paleoclimatic studies because
in addition to all of the changes in the forcing and the paleoclimatic
responses accompanying the transition out of the last interglacial, the
possibility that some of the apparent simulated changes between 127 and 116 ka may be an artifact of data analysis procedures cannot be discounted.
At 97 ka, a time selected to illustrate a different orbital configuration
(i.e., one with boreal spring months occurring closer to the June solstice)
than the similar (6 and 127 ka) or contrasting (127 and 116 ka)
configurations, the calendar effect on temperature in the Northern
Hemisphere (Fig. 11) shows a switch from positive differences in the early
boreal summer (May and June) to negative in the late summer (August and
September). This switch is again consistent with the direct calendar effect
on insolation (Fig. 3). Like the other times, these spatial variations in
the calendar effect could easily be interpreted in terms of one kind of
paleoclimatic mechanism or another.
The generally larger calendar effect on temperature over the continents than
over the oceans implicates the amplitude of the seasonal cycle in the size
of the effect. This situation suggests that even in model-only
intercomparisons (and even in the unlikely case that all models involved in
an intercomparison use the same calendar) the calendar effect could be
present because the amplitude of the seasonal cycle is dependent on model
spatial resolution (and its influence on model orography).
Mean temperature of the warmest and coldest months
Although the calendar effects on monthly mean temperature show some
subcontinental-scale variability, the overall patterns are of relatively
large spatial scales and are interpretable in terms of the direct orbital
effects on month lengths and insolation. The calendar effects on the mean
temperature of the warmest (MTWA) and coldest (MTCO) calendar months (and
their differences) are much more spatially variable (Fig. 12). This
variability arises in large part because of the way these variables are
usually defined (e.g., as the mean temperature of the warmest or coldest
conventionally defined month as opposed to the temperature of the warmest
or coldest 30 d interval), but also because the calendar adjustment can
result in a change in the specific month that is warmest or coldest. These
effects are compounded when calculating seasonality (as MTWA minus MTCO).
Other definitions of the warmest and coldest month are possible, such as the
warmest consecutive 30 d period during the year (e.g., Caley et al., 2014),
and such definitions will not be susceptible to the calendar effect. In
practice, however, paleoclimatic reconstructions based on calibrations or
forward-model simulations routinely use conventional calendar-month
definitions of the warmest and coldest months and of seasonality (Bartlein
et al., 2011; Harrison et al., 2014), and often only monthly output from
paleoclimatic simulations is available, necessitating consistent definitions
when summarizing model output.
In the particular set of example times chosen here, the magnitudes of the
calendar effects are also smaller than those of individual months because,
as it happens, the calendar effects in January and February (typically
the coldest months in the Northern Hemisphere) and July and August (typically
the warmest months in the Northern Hemisphere) are not large. There are also
some surprising patterns. The inverse relationship between the calendar
effects at 116 and 127 ka that might be expected from inspection of the
monthly effects (Fig. 11) are not present, while the calendar effects on
MTCO and MTWA at 97 and 116 ka tend to resemble one another (Fig. 12).
Across the four example times, there is an indistinct but still noticeable
pattern in reduced seasonality (MTWA minus MTCO) between the adjusted and
unadjusted values, which like the other patterns described above could tempt
interpretation in terms of some specific climatic mechanisms.
Monthly precipitation
In contrast to the large spatial-scale patterns of the calendar effect on
temperature, the patterns of the calendar effect on precipitation rate are
much more complex, showing both continental-scale patterns (like those for
temperature) and smaller-scale patterns that are apparently related to
precipitation associated with the ITCZ and regional and global monsoons
(Fig. 13). The continental-scale patterns are evident in the calendar
effects at 6 and 127 ka, particularly in the months from September through
November (Fig. 13), when it can also be noted (especially over the
midlatitude continents in both hemispheres) that there is a positive
association with the calendar effect on temperature. This association is
simply related to similarities in the shapes of the annual cycles of those
variables and not to some kind of more elaborate thermodynamic constraint.
At 116 ka, as for temperature, the large-scale calendar-effect patterns
appear to be nearly the inverse of those at 127 ka. The smaller-scale kind
of pattern is well illustrated at 127 ka in the tropical North Atlantic,
sub-Saharan Africa, and south Asia. There, negative calendar-adjusted minus
unadjusted values can be noted for June through August, giving way to
positive differences from September through November, and the same
transition appears inversely at 116 ka. Another example can be found in the
South Pacific Convergence Zone in austral spring and early summer (September
through November) at 6 and 127 ka, when generally positive differences
between calendar-adjusted and unadjusted values in July and August give way
to negative differences from September through December. This second kind of
pattern, most evident in the subtropics, is not mirrored by the calendar
effects on temperature.
Overall, the magnitude and spatial patterns of the calendar effects on
temperature and precipitation (Figs. 11 and 13) resemble those in the
paleoclimatic simulations and observations that we attempt to explain in
mechanistic terms (Harrison et al., 2016). Depending on the sign of the
effect, neglecting to account for the calendar effects could spuriously
amplify some signals in long-term mean differences between the experiment
and control simulations, while damping others.
Calendar effects and transient experiments
Calendar effects must also be considered in the analysis of transient
climate-model simulations (even if those data are available on the daily
time step). This can be illustrated for a variety of variables and regions
using data from the TraCE-21ka transient simulations (Liu et al., 2009;
https://www.earthsystemgrid.org/project/trace.html, last access: 11 August 2019). The series
plotted in Fig. 14 are area averages for individual months on a yearly time
step, with 100-year (window half-width) locally weighted regression curves
added to emphasize century-timescale variations. The original yearly
time-step data were aggregated using a perpetual noleap (365 d)
calendar (using the present-day month lengths for all years). The gray and
black curves in Fig. 14 show these unadjusted “original” values, while the
colored curves show month-length adjusted values (i.e., pseudo-daily
interpolated values, reaggregated using the appropriate paleo fixed-angular
calendar). Area averages were calculated for ice-free land points.
Time series of original and month-length adjusted annual
area-weighted averages of TraCE-21ka data (Liu et al., 2009), expressed as
differences from the 1961–1989 long-term mean for (a–c) 2 m air temperature,
(d) precipitation rate, and (e–f) precipitation minus evaporation (P-E).
The original or unadjusted data are plotted in gray and black, and the
adjusted data are in color. The area averages are grid-cell area-weighted
values for land grid points in each region, and the smoother curves are
locally weighted regression curves with a window half-width of 100 years.
The regions are defined as (a) 15 to 75∘ N and -170 to
60∘ E, (b) 10 to 50∘ S and 110 to 160∘ E, (c) global ice-free land area, (d) 0 to 30∘ N and -30 to 120∘ E, (e) 5 to 17∘ N and -5 to 30∘ E, and
(f) 31 to 43∘ N and -5 to 30∘ E.
Figure 14a shows area-weighted averages for 2 m air temperature for a region
that spans 15 to 75∘ N and -170 to 60∘ E, the region
used by Marsicek et al. (2018) to discuss Holocene temperature trends in
simulations and reconstructions. The largest differences between
month-length adjusted values and unadjusted values occur in October between
14 and 6 ka, when perihelion occurred during the northern summer months.
October month lengths during this interval were generally within 1 d of
those at present (Fig. 1), but the generally shorter months from April
through September resulted in Octobers beginning up to 10 d earlier in
the calendar than at present, i.e., closer in time to the boreal summer
solstice (Fig. 2). The calendar-effect-adjusted October values therefore
average up to 4 ∘C higher than the unadjusted values during this
interval (Fig. 14a), consistent with the direct calendar effects on
insolation at 45∘ N (Fig. 3). The calendar effect also changes the
shape of the temporal trends in the data, particularly during the Holocene.
October temperatures in the unadjusted data showed a generally increasing
trend over the Holocene (i.e., since 11.7 ka), reaching a maximum around 3
ka, comparable with present-day values, while the adjusted data reached
levels consistently above present-day values by 7.5 ka. The unadjusted
October temperature data could be described as reaching a “Holocene thermal
maximum” only in the late Holocene (i.e., after 4 ka), while the adjusted
data display more of a mid-Holocene maximum. As is the case with the mapped
assessments of the pure calendar effect, the differences between
unadjusted and adjusted time series are of the kind that could be
interpreted in terms of various hypothetical mechanisms. For example, the
calendar-effect adjustment advances the time of the occurrence of a Holocene
thermal maximum in October by about 3 kyr for North America and Europe.
As in North America and Europe, the adjusted temperature trends in Australia
(10 to 50∘ S and 110 to 160∘ E) (Fig. 14b) are
consistent with the direct calendar effects on insolation (i.e., for
45∘ S; Fig. 5). The difference between adjusted and unadjusted
values are again largest in October between 14 and 6 ka, but the difference
is the inverse of that for the North America and Europe region because the
annual cycle of temperature for Australia is inversely related to the annual
cycle of the insolation anomalies (Fig. 9) and therefore to the direct calendar
effects on insolation (Fig. 5). Again, the shapes of the Holocene trends in
the adjusted and unadjusted data are noticeably different. In the Australia
(Fig. 14b) and North America and Europe (Fig. 14a) examples, relatively
large areas are being averaged, and the calendar effect becomes more
apparent as the size of the area decreases. Notably, the effect does not
completely disappear at the largest scales, i.e., for area-weighted averages
for the globe (for ice-free land grid cells) (Fig. 14c). The differences are
smaller but still discernible.
In the Northern Hemisphere (African–Asian) monsoon region (0 to
30∘ N and -30 to 120∘ E), the calendar effects on
precipitation rate are similar to those on temperature in the midlatitudes
because the annual cycle of precipitation is roughly in phase with that of
insolation (Fig. 7). There is little effect in the winter and spring but a
substantial effect in summer and autumn over the interval from 17 to
about 3 ka (Fig. 14d). The calendar effect reverses sign between July and
August (when the month-length adjusted precipitation rate values are less
than the unadjusted ones) and September and October (when the adjusted
values are greater than the unadjusted ones). In July, the timing of
relative maxima and minima in the two data sets is similar, while in
October, in particular, the Holocene precipitation maximum is several
thousand years earlier in the adjusted data than in the unadjusted data.
The time-series expression of the latitudinally reversing calendar effect on
precipitation rate evident in Fig. 13 (e.g., July vs. October at 127 ka) can
be illustrated by comparing precipitation or precipitation minus evaporation
(P-E) for the North African (sub-Saharan) monsoon region (5 to 17∘ N
and -5 to 30∘ E) with the Mediterranean region (31 to
43∘ N and -5 to 30∘ E) (Fig. 14e and f). The
differences between the adjusted and unadjusted data in the North African
region (Fig. 14e) parallel that of the larger monsoon region (Fig. 14d). The
Mediterranean region, which is characteristically moister in winter and
drier in summer, shows the reverse pattern: when the calendar-adjusted minus
unadjusted P-E difference is positive in the monsoon region, it is negative in
the Mediterranean region. Dipoles are frequently observed in climatic data,
both present-day and paleo, and are usually interpreted in terms of
broad-scale circulation changes in the atmosphere or ocean. This example
illustrates that they could also be artifacts of the calendar effect. Such
changes in the timing of extrema could also influence the interpretation of
phase relationships among simulated time series and time series of potential
forcing (Joussaume and Braconnot, 1997; Timm et al., 2008; Chen et al.,
2011).
There are other interesting patterns in the monthly time series from the
transient simulations, some of which are amplified by the calendar effect
and others damped. The monthly time series suggest that the traditional
meteorological seasons (i.e., December–February, March–May, June–August,
September–November) are not necessarily the optimal way to aggregate
data – the September time series in Fig. 14 often look like they are more
similar to, and should be grouped with, July and August rather than with October
and November, the traditional other (northern) autumnal months. Figure 14a
(North America and Europe), for example, suggests that the July through
November time series are similar in their overall trends, and even more so
for the adjusted data (in pink and red). Similarly, months that appear
highly correlated over some intervals (e.g., July and June global
temperatures from the Last Glacial Maximum to the Holocene) become decoupled at other times.
The impacts of the calendar effect on temporal trends in transient
simulations (Fig. 14), when compounded by the spatial effects (Figs. 11–13),
make it even more likely that spurious climatic mechanisms could be inferred in
analyzing transient simulations rather than in the simpler time-slice simulations.
Summary
Several observations can be made about the calendar effect and its
potential role in the interpretation of paleoclimatic simulations and
comparisons with observations.
The variations in eccentricity and perihelion over time are large enough to
produce differences in the length of (fixed-angular) months that are as
large as 4 or 5 d, as well as differences in the beginning and ending
times of months on the order of 10 d or more (Fig. 1).
These month-length and beginning and ending date differences are large
enough to have noticeable impacts on the location in time of a fixed-length
month relative to the solstices and hence on the insolation receipt during
that interval (Figs. 2 through 5). The average insolation (and its
difference from the present) during a fixed-length month will thus include the
effects of the orbital variations on insolation and the changing month
length.
However, such insolation effects are not offset by the changing insolation
itself but instead can be reinforced or damped (Figs. 7 through 9). (In
other words, orbitally related variations in insolation do not take care
of the calendar definition issue.)
The pure calendar effects on temperature and precipitation (illustrated
by comparing adjusted and non-adjusted data assuming no climate change;
Figs. 11–13) are large, spatially variable, and could easily be mistaken
for real paleoclimatic differences (from the present).
The impact of the calendar effect on transient simulations is also large
(Fig. 14), affecting the timing and phasing of maxima and minima, which,
when combined with the spatial impacts of the calendar effect, makes transient
simulations even more prone to misinterpretation.
PaleoCalAdjust v1.0
The approach we describe here for adjusting model output reported either as
monthly data (using fixed-length definitions of months) or as daily data to
reflect the calendar effect (i.e., to make month-length adjustments) has two
fundamental steps: (1) pseudo-daily interpolation of the monthly data on a
fixed-month-length calendar (which, when actual daily data are available, is
not necessary), followed by (2) aggregation of those daily data to
fixed-angular months defined for the particular time of the simulations. The
second step obviously requires the calculation of the beginning and ending
days of each month as they vary over (geological) time, which in turn
depends on the orbital parameters. The definition of the beginning and
ending days of a month in a leap-year, Gregorian, or proleptic
Gregorian calendar (http://cfconventions.org) additionally
depends on the timing of the (northern) vernal equinox, which varies from
year to year. Here we describe the pseudo-daily interpolation method first,
followed by a discussion of the month-length calculations. Then we describe
the calendar-adjustment program, along with a few demonstration programs
that exercise some of the individual procedures. All of the programs,
written in Fortran 90, are available (see the “Code and data availability” section).
Pseudo-daily interpolation
The first step in adjusting monthly time-step model output to reflect the
calendar effect is to interpolate the monthly data (either long-term means
or time-series data) to pseudo-daily values (a step that is not required if
the data are daily time-step values). It turns out that the most common way
of producing pseudo-daily values, linear interpolation between monthly
means, is not mean preserving; the monthly (or annual) means of the
interpolated daily values will generally not match the original monthly
values. An alternative approach, and the one we use here, is the
mean-preserving “harmonic” interpolation method of Epstein (1991), which
is easy to implement and performs the same function as the parabolic-spline
interpolation method of Pollard and Reusch (2002). As is also the case with
Pollard and Reusch's (2002) method, Epstein's (1991) approach can
occasionally produce overshoots that are physically impossible, as can
happen in the application of the method to variables like precipitation,
which may have monthly values that alternate between zero and nonzero
values. For practical reasons, variables like precipitation are therefore
“clamped” at zero, which can introduce small differences between the
annual and monthly means of the original and interpolated data, and we
illustrate a pathological case of this below.
Pseudo-daily interpolated temperature (a, b) and
precipitation (c, d) for some representative locations: (a, c) Madison, Wisconsin, USA, (b) Australia, and (d) the Indian Ocean. The
original monthly mean data are shown by the black dots and stepped curves
(black lines), daily values linearly interpolated between the monthly mean
values are shown in blue, and daily values using the mean-preserving
approach of Epstein (1991) are shown in red. The annual interpolation error
(or the difference between the annual average calculated using the original
data and the pseudo-daily interpolated data) is given for the
mean-preserving approach in each case. The interpolated data for this figure
were generated using the program demo_01_pseudo_daily_interp.f90.
The linear and mean-preserving interpolation methods can be compared using
the Climate Forecast System Reanalysis (CFSR) near-surface air temperature
and CPC Merged Analysis of Precipitation (CMAP) 1981–2010 long-term mean
data (Fig. 15). A typical example for temperature appears in Fig. 15a for a
grid point near Madison, Wisconsin (USA). The difference between the annual
mean values of the interpolated data for the two approaches is small and
similar (ca. 2.0×10-6), but the difference between the
original monthly means and the monthly mean of the linearly interpolated
daily values can exceed 0.8 ∘C in some months (e.g., December).
(The differences from the original monthly means for the mean-preserving
interpolation method are less than 1.0×10-3∘C
for every month in Fig. 15a.) Figure 15b shows an example for a grid point in
Australia, where again the difference between the original monthly means and
the monthly means of the linearly interpolated daily values is not
negligible (i.e., 0.4 ∘C). Similar results hold for precipitation
(Fig. 15c), whereby the difference can exceed 0.1 mm d-1. Like other
harmonic-based approaches, the Epstein (1991) approach can create
interpolated curves that are wavy (see Pollard and Reusch, 2002, for
a discussion), but these effects are small enough to not be practically
important in nearly all cases. The pathological case for precipitation is
shown in Fig. 15d at a grid point in the Indian Ocean. Here, the difference
between an original monthly mean value and one calculated using the
mean-preserving interpolation method reaches -0.12 mm d-1 in
March and April, but the differences between the original monthly means and
the monthly means of the linearly interpolated daily values are nearly 3
times larger.
The map patterns of the interpolation errors (the monthly mean values
recalculated using the linear or mean-preserving pseudo-daily interpolated
values minus the original values) appear in Fig. 16. (Note the differing
scales for the linear interpolation errors and the
mean-preserving interpolation errors.) The linear interpolation errors are
quite large, with absolute values exceeding 1 ∘C and 1 mm d-1, and have distinct seasonal and spatial patterns:
underpredictions of Northern Hemisphere temperature in summer (and
overpredictions in winter), underpredictions of precipitation in the wet
season (e.g., southern Asia in July), and overpredictions in the dry season
(southern Asia in May). The magnitude and patterns of these effects again
rival those we attempt to infer or interpret in the paleo record. The
mean-preserving interpolation errors for temperature are very small and
show only vague spatial patterns (note the differing scales). The errors for
precipitation are also quite small but can be locally larger, as in the
pathological case illustrated above. However, the map patterns of the
interpolation errors strongly suggest that those cases are not practically
important.
Pseudo-daily interpolation errors for CFSR near-surface
air temperature (a, c) and CMAP precipitation rate (b, d). (a, b) The interpolation errors, or the
differences between the original monthly mean values and the monthly mean
values recalculated from linear interpolation of pseudo-daily values. (c, d) The interpolation errors for mean-preserving
(Epstein, 1991) interpolation. The errors for linear interpolation of the
temperature data (∘C) range from -1.20851 to 1.29904, with a
mean of 0.05664 and standard deviation of 0.16129 (over all months and
grid points), while those for mean-preserving interpolation range from
-0.00002 to 0.00050, with a mean of -0.0061 and standard deviation of
0.00007. The errors for linear interpolation of the precipitation data (mm d-1) range from -1.10617 to 1.40968, with a mean of 0.00087 and
standard deviation of 0.11851, while those for mean-preserving interpolation
range from -0.00002 to 0.00383, with a mean of 0.00001 and standard
deviation of 0.00163.
The mean-preserving interpolation method is implemented in the Fortran 90
module named pseudo_daily_interp_subs.f90. The subroutine hdaily(…)
manages the interpolation, first getting the harmonic coefficients (Eq. 6 of
Epstein, 1991) using the subroutine named harmonic_coeffs(…) and then applying these coefficients in the
subroutine xdhat(…) to get the interpolated values.
Month-length calculations
The calculation of the length and the beginning, middle, and ending (real number
or fractional) days of each month at a particular time is based on an
approach for calculating orbital position as a function of time using
Kepler's equation:
M=E-ε⋅sinE,
where M is the angular position along a circular orbit (referred to by
astronomers as the mean anomaly), ε is eccentricity, and
E is the eccentric anomaly (Curtis, 2014; Eq. 3.14). Given the angular
position of the orbiting body (Earth) along the elliptical orbit, θ
(the “true anomaly”), E can be found using the following expression
(Curtis, 2014; Eq. 3.13b):
E=2⋅tan-11-ε/1+ε0.5tanθ/2.
Substituting E into Eq. (1) gives us M, and then the time since
perihelion is given by
t=M/2πT,
where T is the orbital period (i.e., the length of the year) (Curtis,
2014; Eq. 3.15).
This expression can be used to determine the traverse time or
time of flight of individual days or of segments of the orbit equivalent
to the fixed-angular definition of months or seasons. Doing so involves
determining the traverse times between the vernal equinox and perihelion,
between the vernal equinox and 1 January (set at the appropriate number of
degrees prior to the vernal equinox for a particular calendar), and the
angle between perihelion and 1 January and using these values to translate
time since perihelion to time since 1 January. The true anomaly
angles along the elliptical orbit (θ) are determined using the
present-day (e.g., 1950 CE) definitions of the months in different
calendars (e.g., January is defined as having 30, 31, and 31 d in
calendars with a 360, 365, or 366 d year, respectively). For example,
January in a 365 d year is defined as the arc or “month angle” between
0.0 and 31.0 d × (360.0∘/ 365.0 d). Note that when perihelion is
in the Northern Hemisphere winter, the arc may begin after 1 January as a
consequence of the occurrence of shorter winter months, and when perihelion
is in the Northern Hemisphere summer, the arc may begin before 1 January as a
consequence of longer winter months (Fig. 1).
We also implemented the approximation approach described by Kutzbach and
Gallimore (1988; Appendix A) for calculating month lengths. There were no
practical differences between their approach and our implementation of
Kepler's equation based on the Curtis (2014) approach.
Application of this algorithm requires as input eccentricity and the
longitude of perihelion (∘) relative to the vernal equinox, and the
generalization of the approach to other calendars, such as the proleptic
Gregorian calendar (that includes leap years; http://cfconventions.org), also requires the (real number or fractional)
day of the vernal equinox. To calculate the orbital parameters using the
Berger (1978) solution and the timing of the (northern) vernal equinox (as
well as insolation itself), we adapted a set of programs provided by the
National Aeronautics and Space Administration (NASA) Goddard Institute for
Space Studies (GISS) (now available at https://web.archive.org/web/20150920211936/http://data.giss.nasa.gov/ar5/solar.html, last access: 11 August 2019).
Simulation ages and simulation years
Inspection shows that different climate models employ different starting
dates in their output files for both present-day (piControl) and paleo (e.g.,
midHolocene) simulations (https://esgf-node.llnl.gov/projects/cmip5/). For
models that use a noleap (constant 365 d year) calendar, such as CCSM4
(Otto-Bliesner, 2014), the starting date is not an issue, but for MPI-ESM-P
(Jungclaus et al., 2012), which uses a proleptic Gregorian calendar, or
CNRM-CM5 (Sénési et al., 2014), which uses a “standard” (i.e., mixed
Julian–Gregorian) calendar, as examples, the specific starting date
influences the date of the vernal equinox through the occurrence of
individual leap years. For example, in the CMIP5–PMIP4 midHolocene simulations, output
from MPI-ESM-P starts in 1850 CE and that from CNRM-CM5 in 2050 CE (and it
can be verified that leap years in those output files occur in a fashion
consistent with the “modern” calendar). Consequently, we need to make a
distinction between two notions of time here: (1) the simulation age,
expressed in (negative) years BP (i.e., before 1950 CE), and (2) the
simulation year, expressed in years CE. The simulation age controls the
orbital parameter values, while the simulation year, along with the
specification of the CF-compliant calendar attribute (http://cfconventions.org), controls the date and time of the vernal
equinox.
Month-length programs and subprograms
Month lengths are calculated in the subroutine get_month_lengths(…) (contained in the Fortran 90
module named month_length_subs.f90), which in
turn calls the subroutine monlen(…) to get real-type month
lengths for a particular simulation age and year. (The subroutine
get_month_lengths(…) can be
exercised to produce tables of month lengths, beginning, middle, and ending
days of the kind used to produce Figs. 1–5 and 7–9 using a driver program
named month_length.f90.) The subroutine get_month_lengths(…) uses two other modules,
GISS_orbpar_subs.f90 and GISS_srevents_subs.f90 (based on programs originally downloaded
from GISS; now available at https://web.archive.org/web/20150920211936/http://data.giss.nasa.gov/ar5/solar.html),
to get the orbital parameters and vernal equinox dates.
The specific tasks involved in the calculation of either a single year's set
of month lengths, or a series of month lengths for multiple years, include
the following steps, implemented in get_month_lengths(…):
generate a set of “target” dates based on the simulation ages and
simulation years;
obtain the orbital parameters for 0 ka (1950 CE), which will be used to
adjust the calculated month-length values to the conventional definition of
months for 1950 CE as the reference year; and
obtain the present-day (i.e., 1950 CE) month lengths (along with the
beginning, middle, and ending days relative to 1 January) for the appropriate
calendar using the subroutine monlen(…).
Then loop over the simulation ages and simulation years, and for each
combination do the following:
obtain the orbital parameters for each simulation age using the subroutine
GISS_orbpars(…);
calculate real-type month lengths (along with the beginning, middle, and
ending days relative to 1 January) for the appropriate calendar using
monlen(…);
adjust (using the subroutine adjust_to_ref_length(…)) those month-length values to the
reference year (e.g., 1950 CE) and its conventional set of month-length
definitions so that, for example, January will have 31 d, February 28 or
29 d, etc., in that reference year;
further adjust the month-length values to ensure that the individual monthly
values will sum exactly to the year length in days using the subroutine
adjust_to_yeartot(…);
convert real-type month lengths to integers using the subroutine
integer_monlen(…) (these integer values are not
used anywhere but may be useful in conceptualizing the pattern of
month-length variations over time);
get integer-valued beginning, middle, and ending days for each month; and
determine the mid-March day using the subroutine GISS_srevents(…) to get the vernal equinox date for calendars in which it varies.
Month-length tables and time series
Tables and time series of month lengths, beginning, middle, and ending days,
and dates of the vernal equinox can be calculated using the program
month_length.f90. This program reads an “info file”
(month_length_info.csv) consisting of an
identifying output file name prefix, the calendar type, the beginning and
ending simulation age (in years BP), the age step, the beginning
simulation year (in years CE), and the number of simulation years. Note that
in the approach described above, orbital parameters are calculated once per
year (step 4 in Sect. 4.4) and are assumed to apply for the whole year.
This assumption can lead to small differences (ranging from -0.000863 to
0.000787 d over the past 22 kyr with a mean of -0.00000389 d) in
the ending day of one year and the beginning day of the next.
Paleo calendar adjustment
The objective of the principal calendar-adjustment program
cal_adjust_PMIP.f90 is to read and clone CMIP–PMIP-formatted netCDF files, replacing the original monthly or
daily data with calendar-adjusted data, i.e., data aggregated using a
fixed-angular calendar appropriate for a particular paleo experiment. In the
case of monthly input data, either climatological long-term means or monthly
time series, the data are first interpolated to a daily time step and then
reaggregated to monthly time-step mean values using an appropriate paleo
calendar. In the case of daily input data, the interpolation step is
obviously unneeded, so the data are simply aggregated to the monthly
time step. In both cases, new time-coordinate variables are created
(consistent with the paleo calendar), and all other dimension information,
coordinate variables, and global attributes are copied and augmented by
other attribute data that indicate that the data have been adjusted. The
reading and rewriting of the netCDF file is handled by subroutines in a
module named CMIP_netCDF_subs.f90, and various
modules and subprograms for month-length calculations described above are
also used here. Additional details regarding the model code can be found in
the README.md file in the code repository folder /f90.
Interpolation and (re)aggregation
The pseudo-daily interpolation and (re)aggregation are done using two
subroutines, mon_to_day_ts(…) and day_to_mon_ts(…), in the module calendar_effects_subs.f90. The pseudo-daily interpolation is done a
year at a time, creating slight discontinuities between one year and the
next in the case of transient or multiyear “snapshot” simulations. The
subroutine mon_to_day_ts(…) has options for smoothing those discontinuities and
restoring the long-term mean of the interpolated daily data to that of the
original monthly data.
The (re)aggregation of the daily data is also done a year at a time by
collecting the daily data for a particular year and “padding” it at the
beginning and end with data from the previous and following year if
available, as in transient or multiyear simulations (to accommodate the
fact that under some orbital configurations the first day of the current
year may occur in the previous year or the last day in the following year;
Fig. 1). For example, at 6 ka, the changes in the shape of the orbit and the
consequently longer months from January through March (32.5, 29.5, and 32.4 d, respectively) displace the beginning of January 4 d into the
previous year, with the last day of December consequently falling just
before day 361 in a 365 d year. In the case of long-term mean
“climatological” data (Aclim data; see Sect. 5.2), the padding is done
with the ending and beginning days of the single year of pseudo-daily data.
The calculation of monthly means is done by calculating weighted averages of
the days that overlap a particular month as defined by the (real number
or fractional) beginning and ending days of that month (from the subroutine
get_month_lengths(…)). Each whole
day in that interval gets a weight of 1.0, and each partial day gets a
weight proportional to its part of a whole day. It should be noted that in
transient simulations, annual averages, constructed either by averaging
actual or pseudo-daily data (or by month-length weighted averages), will
differ from the unadjusted data.
Processing individual netCDF files
The cal_adjust_PMIP.f90 program reads an
info file that provides file and variable details and can handle
CMIP6–PMIP4-formatted files (https://pcmdi.llnl.gov/CMIP6/Guide/modelers.html#5-model-output-requirements, last access: 11 August 2019)
as they become available. The fields in the info file include (for each
netCDF file) the activity (PMIP3 or PMIP4), the variable (e.g.,
tas, pr), the realm-plus-time-frequency type (e.g., Amon,
Aclim), the model name, the experiment name (e.g.,
midHolocene), the ensemble member (e.g., r1i1p1), the grid label (for
PMIP4 files), and the simulation year beginning date and ending date (as a
YYYYMM or YYYYMMDD string). An input file name suffix field is also read
(which is usually blank but is “-clim” for Aclim-type files), as is an
output file name suffix field (e.g., _cal_adj), which is added to the output file name to indicate that it has been
modified from the original. The info file also contains the simulation age
beginning and end (in years BP), the increment between simulation ages
(usually 1 in the application here), the beginning simulation year (years
CE), the number of simulation years, and the paths to the source and
adjusted files. This information could also be obtained by parsing the netCDF
file names and reading the calendar attribute and time-coordinate variables,
but that would add to the complexity of the program.
The output netCDF files have the string _cal_adj appended to the end of the file name. In the case of monthly time
series (e.g., Amon) or long-term means (e.g., Aclim) the file names
are otherwise the same as the input data. In the case of the daily input
data, with “day” as the realm-plus-time-frequency string, that string
is changed to Amon2.
The adjustment of a file using cal_adjust_PMIP.f90 includes the following steps:
read the info file, construct various file names, and allocate month-length
variables;
generate month lengths using the subroutine get_month_lengths(…); and
open input and output netCDF files. For each file,
redefine the time-coordinate variable as appropriate using the subroutines
new_time_day(…) and
new_time_month(…) in the module
CMIP_netCDF_subs.f90;
create the new netCDF file, copy the dimension and global attributes from
the input file using the subroutine copy_dims_and_glatts(…), and define the output variable using
the subroutine define_outvar(…);
get the input variable to be adjusted;
for each model grid point, get calendar-adjusted values as described above
using the subroutines mon_to_day_ts(…) and day_to_mon_ts(…); and
write out the adjusted data, and close the output file.
Further examples
Five other main programs that serve as “drivers” for some of the
subroutines or that demonstrate particular aspects of procedures used here
are included in the GitHub repository for the programs (https://github.com/pjbartlein/PaleoCalAdjust, last access: 11 August 2019).
GISS_orbpar_driver.f90 and GISS_srevents_driver.f90 are the main programs that call the subroutines
GISS_orbpars(…) and GISS_srevents(…) to produce tables of orbital parameters and “solar
events” like the dates of equinoxes, solstices, and perihelion and aphelion.
demo_01_pseudo_daily_interp.f90 is the main program that demonstrates linear and
mean-preserving pseudo-daily interpolation.
demo_02_adjust_1yr.f90 is the main
program that demonstrates the paleo calendar adjustment of a single year's
data.
demo_03_adjust_TraCE_ts.f90 is the main program that demonstrates the adjustment
of a 22 040-year-long time series of monthly TraCE-21ka data.
Summary
As has been done previously (e.g., Kutzbach and Otto-Bliesner, 1982; Kutzbach
and Gallimore, 1988; Joussaume and Braconnot, 1997; Pollard and Reusch,
2002; Timm et al., 2008; Chen et al., 2011; Kageyama et al., 2018), we have
described the substantial impacts of the paleo calendar effect on the
analysis of climate-model simulations and provide what we hope is a
straightforward way of making adjustments that incorporate the effect. At
some point in the course of the development of protocols for model
intercomparisons and comparisons of model-simulated data with observed
paleoclimatic data, such adjustments will become unnecessary when model
output is archived at daily (and sub-daily) intervals and when
paleoclimatic reconstructions are no longer tied to conventionally defined
monthly and seasonal climate variables but instead use more biologically or
physically based variables such as growing degree days or plant-available
moisture. The interval between previous calls to include consideration of
the calendar effect in paleoclimate analyses has ranged between 3 and
9 years over the past nearly 4 decades, with a median interval of 6
years. The size and impact of the calendar effect warrant its consideration
in the analysis of paleo simulations, and we hope that by providing a
relatively easy-to-implement method, that will become the case.
Code and data availability
The Fortran 90 source code (main programs and modules), example data sets,
and the data used to construct the figures (v1.0d) are available from Zenodo
(https://zenodo.org/, last access: 11 August 2019) at 10.5281/zenodo.1478824 (Bartlein and Shafer, 2019) and from GitHub (https://github.com/pjbartlein/PaleoCalAdjust). CMIP5 data are available at: https://esgf-node.llnl.gov/projects/cmip5/. Climate Forecast System Reanalysis (CFSR) temperature data are available at https://esgf.nccs.nasa.gov/projects/ana4mips/, and CPC Merged Analysis of Precipitation (CMAP) monthly precipitation rate (precip) data are available at https://www.esrl.noaa.gov (last access: 11 August 2019). TraCE-21ka transient climate-simulation data are available at https://www.earthsystemgrid.org/project/trace.html.
Author contributions
PJB designed the study, developed the Fortran 90 programs, and wrote the first draft of the paper. Both authors contributed to the final version of the text.
Competing interests
The authors declare that they have no conflict of interest.
Disclaimer
Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.
Acknowledgements
We thank Jay Alder, Martin Claussen, and Anne Dallmeyer for their comments on earlier versions of the text. This
publication is a contribution to PMIP4. TraCE-21ka was made possible by the
DOE INCITE computing program and supported by NCAR, the NSF P2C2 program,
and the DOE Abrupt Change and EaSM programs. CMAP precipitation data were
provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their website at https://www.esrl.noaa.gov/psd/ (last access: 11 August 2019). CFSR near-surface
air-temperature data were obtained from https://esgf.nccs.nasa.gov/projects/ana4mips/ (for the original source see
http://cfs.ncep.noaa.gov, last access: 11 August 2019). Maps were prepared using NCL, the
NCAR Command Language (UCAR/NCAR/CISL/TDD, 2017). Sarah L. Shafer
was supported by the U.S. Geological Survey Land Change Science Program.
Review statement
This paper was edited by Didier Roche and reviewed by three anonymous referees.
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