Distributions of Climate Variables

3.0.1 The Components of the Climate System.

The climate system is composed of all processes that directly or indirectly control the atmospheric environment of humans and ecosystems. The main components of the system are the hydro- and thermodynamic states of the atmosphere and the ocean. Sea ice affects the exchange of heat, momentum and fresh water between oceans and atmosphere. On longer time scales, the shelf ice and the land ice become relevant since these components are able to store and release large quantities of fresh water. The atmosphere, ocean, and land surface are interconnected by means of the hydrological cycle on a number of time scales. Precipitation falls on the land where it affects land surface properties such as albedo and heat capacity. Some of this precipitation evaporates into the atmosphere, and some flows to the ocean as runoff. Fresh water flux into the ocean by means of precipitation and runoff, and out of the ocean through evaporation, affects ocean variability, which in turn feeds back on atmospheric variability.

Changes in the chemical composition of the atmosphere also impact the climate system because the concentration of carbon dioxide, ozone, or other radiatively active gases affects the radiative balance of the atmosphere. These concentrations are controlled by the state of the atmosphere and the ocean, as well as the biospheric and anthropogenic sinks and sources of these chemicals. Clearly the components of the climate system cannot be defined exhaustively, since it is not a closed system in a strict sense.

In the following sections we describe several atmospheric, oceanic, cryospheric (ice and snow) and hydrologic variables.1 The choice of the variables is subjective and biased towards those that are most easily observed. Shea et al. [348] list addresses of atmospheric and oceanographic data centres in the US, and give an overview of easily accessible atmospheric and oceanographic data sets at the National Center for Atmospheric Research (NCAR).

1 Biospheric variables are beyond the scope of this text.

3.0.2 The Law of Large Numbers and Climate Time Scales. The instantaneous values or daily accumulations of many climate variables have skewed distributions. On the other hand, averages or accumulations taken over long periods tend to be 'near normal' because of the Central Limit Theorem [2.7.5].

3.0.3 Length and Time Scales. Two terms often used in climate research are time scale and length scale. Although these terms are vaguely defined, thinking about the temporal and spatial resolution needed to describe a phenomenon accurately will help us to select suitable variables for study and to find suitable approximations of the governing equations (see Pedlosky's book [310] on geophysical fluid dynamics).

A length scale is a characteristic length that is representative of the spatial variations relevant to the process under investigation. For instance, if this process is an extratropical storm, then its length scale may be taken as its diameter or as the distance between a pressure minimum and the closest pressure maximum. The length scale of a wind sea2 may be the distance between a wave crest and a wave valley, or between two consecutive crests.

The term 'time scale' is defined similarly. Time scales are representative of the duration of the phenomenon of interest and the greater environment. For example, extratropical storms dissipate within a few days of cyclogenesis, so suitable time scales range from about an hour to perhaps two days. Convective storms (thunder storms), on the other hand, occur on much shorter spatial (up to tens of kilometres) and temporal scales (minutes to several hours). In both cases, the time scale gives an indication of the 'memory' of the process. A statistical measure of 'memory' is the decorrelation time described in Section 17.1. The decorrelation time for the mean sea-level pressure (SLP) is typically three to five days in

2 The part of the ocean wave field that is in dynamical contact with the wind.

the extratropics, while that for convection is on the order of half a day.

We choose variables that describe the variation on the length and time scales of interest that are relevant to a problem. For example, to study extratropical cyclones, the divergence should be chosen rather than the velocity potential.3 When we observe the process, we sample it at spatial and temporal increments that resolve the length and time scales.

An interesting feature of the climate system is that the length and times scales of climate variability are often related. Processes with long length scales have long time scales, and short time scales are associated with short length scales. This fact is illustrated for the atmosphere and the ocean in Figure 3.1. However, this rule is far from precise. Atmospheric tides are an example of a process with large spatial scales and short temporal scales.

3.1 Atmospheric Variables

3.1.1 Significant Variables. A myriad of variables can be used to monitor the physical state of the atmosphere; so understandably the following list of variables commonly used in climate research is not at all exhaustive.

Local climate is often monitored with station data: temperature (daily minimum and maximum, daily mean), precipitation (5 min, 10 min, hourly and daily amounts, monthly amount, number of wet days per month), pressure, humidity, cloudiness, sunshine, and wind (various time averaging intervals).

The large-scale climate is generally described with gridded data, such as: sea-level pressure, geopotential height, temperature, the vector wind, stream function and velocity potential, vorticity and divergence, relative humidity, and outgoing long-wave radiation. Some of these are based on observations (e.g., temperature) while others are derived quantities (e.g., vorticity).

The main problem with time series from station data is that the data are often not homogeneous; they exhibit trends or sudden jumps in the mean or variance that are caused by changes in the physical environment of the observing site, in the observing equipment, of the observing procedures and time, and of the responsible personnel (see Figure 1.9).

3 The divergence is approximately the second derivative of the velocity potential and is sensitive to small scale features, as in extratropical storms. The velocity potential, on the other hand, displays planetary scale divergent, as in the large tropical overturning of the Hadley circulation.

More examples, for instance of the complicating effect of observation time on the daily temperature, the snow cover derived from satellite data, and the effect of lifting a precipitation gauge from the 1 m level to the 2 m level, are described in a review paper by Karl, Quayle, and Groisman [213].

Jones [201] discusses, in some detail, the problems in long time series of precipitation, temperature and other atmospheric data, and lists many relevant papers.

Gridded data have the advantage that they represent the full spatial distribution. However, in data sparse areas, the gridded value may be more representative of the forecast models and interpolation schemes that are used to do the objective analysis4 than they are of the state of the climate system. Unfortunately, when used for diagnostic purposes, it is impossible to distinguish between observed and interpolated, or guessed, information. Difficulties also arise because most gridded data are a byproduct of numerical weather forecasting and therefore affected by changes in the forecast and analysis systems.5 Such changes are made almost continually in an effort to improve forecast skill by incorporating the latest research and data sources and exploiting the latest computing hardware (see, e.g., Trenberth and Olsen [370], or Lambert [240] [241]).

Finally, we note in passing that climate model output is not generally affected by the kinds of problems described above, although it too can have its own idiosyncrasies (see, e.g., Zwiers [444]). However, simulations that are constrained by observations in some way can be affected.6

4 Objective analysis is used to initialize numerical weather forecasting models. Most weather forecasting centres reinitialize their forecasting models every six hours. Typically, the objective analysis system adjusts the latest six-hour forecast by comparing it with station, upper air, satellite, airline, and ship reports gathered during a six-hour window centred on the forecast time. The adjusted forecast becomes the initial condition for the next six-hour numerical forecast. Objective analysis systems are the source of most gridded data used in climate research. See Thiebaux and Pedder [362] or Daley [98] for comprehensive descriptions of objective analysis.

5 Re-analysis projects (Kalnay et al. [210]) have done much to ameliorate this problem. These projects re-analysed archived observational data using a fixed analysis system. Note that re-analysis data are still affected by changes in, for example, the kind of observing systems used (e.g., many different satellite based sensors have been 'flown' for various lengths of time) or the distribution and number of surface stations.

6 Examples include 'AMIP' (Atmospheric Model Intercom-parison Project; see Gates [137]) and 'C20C' (Climate of the Twentieth Century; see Folland and Rowell [123]) simulations. Sea-surface temperature and sea-ice extent are prescribed from observations in both cases, and thus the models are forced with data that are affected by observing system changes.

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Figure 3.1: Length and time scales in the atmosphere and ocean. After [390].

5.00

.01 .05 .1 .2 .5 .8 .9 .95 .99 Cumulative Probability

Figure 3.2: Empirical distribution functions of the amount of precipitation, summed over a day, a week, a month, or a year, at West Glacier, Montana, USA. The amounts have been normalized by the respective means, and are plotted on a probability scale so that a normal distribution appears as a straight line. For further explanations see [3.1.3]. From Lettenmaier [252].

3.1.2 Precipitation. Precipitation, in the form of rain or snow, is an extremely important climate variable: for the atmosphere, precipitation indicates the release of latent heat somewhere in the air column; for the ocean, precipitation represents a source of fresh water; on land, precipitation is the source of the hydrological cycle; for ecology, precipitation represents an important external controlling factor.

There are two different dynamical processes that yield precipitation. One is convection, which is the means by which the atmosphere deals with vertically unstable conditions. Thus, convection depends mostly on the local thermodynamic conditions. Convective rain is often connected with short durations and high rain rates. The other precipitation producing process is large-scale uplift of the air, which is associated with the large-scale circulation of the troposphere. Large-scale rain takes place over longer periods but is generally less intense than convective rain. Sansom and Thomson [338] and Bell and Suhasini [40] have proposed interesting approaches for the representation of rain-rate distributions, or the duration of rain-events, as a sum of two distributions: one representing the large-scale rain and the other the convective rain.

There are a number of relevant parameters that characterize the precipitation statistics at a location.

• The statistics of the amount of precipitation depend on the accumulation time, as demonstrated in Figure 3.2. The curves, which are empirical distribution functions of accumulated precipitation, are plotted so that a normal distribution appears as a straight line. For shorter accumulation times, such as days and weeks, the curves are markedly concave with medians (at probability 0.5) that are less than the mean (normalized precipitation = 1), indicating that these accumulations are not normally distributed. For the annual accumulation, the probability plot is a perfect straight line with coinciding mean and median. Thus, for long accumulation times the distribution is normal. Figure 3.2 is a practical demonstration of the Central Limit Theorem.

• The number ofrainy days per month is often independent of the amount of precipitation.

• The time between any two rainfall events, or between two rainy days, is the interarrival time.

Lettenmaier [252] deals with the distribution aspects of precipitation and offers many references to relevant publications.

3.1.3 Probability Plots—a Diversion. Diagrams such as Figure 3.2 are called probability plots, a type of display we discuss in more detail here.

The diagram is a plot of the empirical distribution function, rotated so the possible outcomes y lie on the vertical axis, and the estimated cumulative probabilities p(y) = Fy(y) lie on the horizontal axis.7 Alternatively, if we consider p the independent variable on the horizontal axis, then y = Fy (p) is scaled by the vertical axis. For reasons outlined below, the variable p is re-scaled by x = F—l(p) with some chosen distribution function Fx. The horizontal axis is then plotted with a linear scale in x. The p-labels (which are given on a nonlinear scale) are retained. Thus, Figure 3.2 shows the function x ^ Fy [ Fx(x)]. If fy = Fx, the function is the identity and the graph is the straight line (x, x). The probability plot is therefore a handy visual tool that can be used to check whether the observed random variable Y has the postulated distribution FX.

7 The 'hat' notation, as in Fy(y), is used throughout this book to identify functions and parameters that are estimated.

Maximum Temperature

Figure 3.3: Frequency distribution of daily maximum temperature in °F at Napoleon (North Dakota, USA) derived from daily observations from 1900 to 1986. From Nese [291].

Maximum Temperature

Figure 3.3: Frequency distribution of daily maximum temperature in °F at Napoleon (North Dakota, USA) derived from daily observations from 1900 to 1986. From Nese [291].

When the observed and postulated random variables both belong to a 'location-scale' family of distributions, such as the normal family, a straight line is also obtained when Y and X have different means and variances. In particular, if a random variable X has zero mean and unit variance such that Fy(y) = FX(y-p), then y = F-l( p) = p + a F-l( p) = p + a x.

The line has the intercept p at x = 0 and a slope of a.

When we think Y has a normal distribution, the reference distribution Fx is the standard normal distribution. S-shaped probability plots indicate that the data come from a distribution with wider or narrower tails than the normal distribution. Probability plots with curvature all of one sign, as in Figure 3.2, indicate skewness. Other location scale families include the log-normal, exponential, Gumbel, and Weibull distributions.

3.1.4 Temperature. Generally, temperature is approximately normally distributed, particularly if averaged over a significant amount of time in the troposphere. However, daily values of near-surface temperature can have more complicated distributions.

The frequency distribution of daily maximum temperature at Napoleon (North Dakota) for 1900 to 1986 (Figure 3.3, Nese [291]) provides another interesting example.

• The distribution is skewed with a wide left hand tail. Cold temperature extremes apparently occur over a broad range (causing the long negative tail) whereas warm extremes are more tightly clustered.

• The distribution has two marked maxima at 35 °F and at 75 °F. This bimodality might be due to the interference of the annual cycle: summer and winter conditions are more stationary than the 'transient' spring and autumn seasons, so the two peaks may represent the summer and winter modes. The summer peak is taller than the winter peak because summer weather is less variable than winter weather. Also, the peak near the freezing point of 33 °F might reflect energy absorption by melting snow.

• There is a marked preference for temperatures ending with the digits 0 and 5. Nese [291] also found that the digits 2 and 8 were overrepresented. This is an example of psychology interfering with science.

Averages of daily mean air temperature8 are also sometimes markedly non-normal, as is the case in Hamburg (Germany) in January and February. Weather in Hamburg is usually controlled by a westerly wind regime, which advects clouds and maritime air from the Atlantic Ocean. In this weather regime temperatures hover near the median. However, the westerly flow is blocked intermittently when a high pressure 'blocking' regime prevails. In this case, the temperature is primarily controlled by the local radiative balance. The absence of clouds and the frequent presence of snow cover cause the temperatures to drop significantly due to radiative cooling. Thus, daily temperatures are sometimes very low, but they usually vary moderately about the mean of -0.4 °C. Strong positive temperature deviations from the mean occur rarely. This behaviour is reflected in the empirical distribution function of the winter mean anomalies (Figure 3.4): the minimum two-month mean temperature in the 1901-80 record is -8.2°C, the maximum is +3.2 °C, while the median is +0.2 °C.

The distribution function in Figure 3.4 is not well approximated by a normal distribution. It is markedly skewed (with an estimated skewness of -1.3 and an estimated third L-moment of -2.86). The degree of convergence towards the normal

8'Daily means' are supposed to represent diurnal averages. In practice, they are obtained by averaging a small number of regularly spaced observations taken over the 24 hours of each day, or, more often, as the mean of the daily maximum and minimum temperatures.

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