Before describing the hydrological cycle, we will pause here to consider the very notions of “system” and “complexity.” According to Bertalanfy (1968) :
“A system can be defined as a complex of elements in interaction. By interaction, we understand that the elements p are connected by relationships R, so that the behavior of an element p in R differs from its behavior in another relationship R'.” This definition applies perfectly to the hydrological cycle, given that it is made up of several elements (the different reservoirs) and that these elements interact and exchange materials (water). Moreover, the behavior of water is specific to the reservoir it occupies and to other particular conditions.
Thus, we can attribute to each element piof the system a measure of flux Qi, and
then define the general dynamic of the water cycle by a system of differential equations using relationships between the derivatives of mathematical equations:.
(2.2)
Even without solving these equations, it is possible to draw certain conclusions about the behavior of the system by studying the equations that describe the dynamic. For example, we can observe whether the system under study is relatively stable or unstable. It is important to know if a disturbance to the system modifies its behavior permanently or whether the system can rapidly compensate for the disturbance.
In the case of the hydrological cycle, the traditional formulation is to express the total stock as a function of precipitation, flow, and evapotranspiration. This relationship, which we will look at in greater detail in the following sections, is called the water budget equation. Then we can show that a disturbance to such a system is of little importance and the system will quickly revert to its equilibrium state. This type of reasoning could lead us to refuse to acknowledge climatic changes or their impacts on the total water cycle. But if we acknowledge the role of these disturbances, we can deduce that the traditional model of the water balance equation is too simplistic a representation to be real or even credible. In that case, we need to reformulate the
d d d d Q t f Q Q Q Q t f Q Q Q n n n 1 1 1 2 1 1 2 = =
⎧
⎨
⎪
⎪
⎩
⎪
⎪
( , , ..., ) .... ( , , ..., )model to include more complex relationships between the various reservoirs in the water cycle, and that takes into account, for example a model of climate change. This is what the meteorologist E. N. Lorenz did in 1963, proposing a model comprising only three linear differential equations. With this model, it was possible to shown that a seemingly minor disturbance in the system could grow over time by a factor of about
e10.
Considering the foregoing, we can come to of number of important conclusions. First, the water cycle is a complex system. By complex, we mean that the system contains information that is difficult to obtain (Ruelle, 1997). Secondly, no unique method is adequate for representing the water cycle, either with a verbal description (a qualitative explanation) or mathematically (with a system of equations). There are a number of possible explanations of the water cycle, depending on the level of detail available. At this point, we revisit a concept that we touched upon in the first chapter and will raise again in the following pages: the concept of scale. Any description that we attempt of a particular phenomenon or process is always closely linked to the scale we adopt, whether spatial or temporal. The choice of scale is determined in part by the scientist’s search for precision, but also depends on the degree of variability of the element being studied. For example, a single atmospheric reservoir suffices to study the carbon cycle, while a regional analysis is generally necessary when studying the sulfur cycle because there are such fluctuating concentrations of sulfuric products in the air. Thirdly, given the complexity of the phenomena being considered, we can conclude that there is not a single water cycle but several water cycles that are closely related to other cycles of energy and matter. This is another reason why scale is so important. A description of the water cycle on the global scale actually incorporates other internal cycles, while the vector properties of water means that the water cycle also incorporates the cycles of energy and matter we call associated cycles.
For the purpose of illustration, we can describe the water cycle at the level of plants by studying the interface between soil, vegetation and the atmosphere. Precipitation is an essential factor in plant growth, and plants in turn contribute water vapor to the atmosphere by means of transpiration. The water they release absorbs some radiation, which influences plant growth. Meanwhile, this absorption leads to the formation of clouds, which modify temperature and pressure fields, which modify the wind velocity field, causing thunderstorms and other precipitation phenomena. This example shows the importance of actions and reactions within a part of the water cycle (which itself constitutes a separate cycle) on a given scale. Add to this the fact that there is obviously a fundamental difference between this water cycle and the water cycle considered on a global scale. On the scale of the Earth, we can consider the system as a closed system because the total quantity of water does not change. But otherwise, as soon as we change the scale of study, the water cycle becomes more and more complex because it is no longer a closed system, but an open system interacting with its environment.
One final remark before we look at particular representations of the water cycle: the scale of study is equally important when examining the causes of water exchange, or put more simply, the mechanisms by which water moves through the natural world. These movements are determined by solar thermal energy as well as by gravity, solar and lunar attraction, atmospheric pressure, intermolecular forces, chemical and nuclear
reactions, biological activities, and finally by human activities. Because the earth’s surface is heated unequally, thermal energy from the sun causes air to circulate in the atmosphere. The force of gravity is responsible for the phenomena of precipitation, in- filtration, runoff, and convection currents. Solar and lunar attraction produce marine tides and currents. Differences in atmospheric pressure cause horizontal displacements of the air. The resulting winds are responsible for the movements of the surface layers in lakes and oceans. Intermolecular forces in the soil affect capillary phenomena and viscosity and so influence the flow rate. Water is also one of the components of many organic and inorganic chemical reactions. Another type of transformation of water is the physiological process that occurs in animal organisms. Finally, humans intervene directly in the processes of water movement and transformation. Our actions can lead to improved water management, but can also cause many problems, especially when we disrupt the hydrological cycle, either quantitatively or qualitatively.