INTRODUCTION - Hydrology: An Introduction - page 9

1.4.3 Spatial scale and parameterization

General approach
All natural flow phenomena are governed by the principles of conservation of mass,
momentum and energy, which can be expressed by a number of equations to provide a
mathematical description of what goes on. However, because there are normally more
dependent variables than available conservation equations, in order to close the system,
additional relationships must be introduced. These closure relationships, also called
parameterizations , relate some of the variables with each other to describe certain specific
physical mechanisms; the mathematical form of these relationships, and the values of
the material constants or parameters are usually based on experimentation.
A second point is that any physical phenomenon must be considered at a given scale;
this scale is the available (depending on the data) or chosen (depending on the objectives
of the study) resolution. It will become clear later on in this topic that, while the funda-
mental conservation equations remain unaffected by the scale at which the phenomenon
is being considered, most closure relationships in them are quite sensitive to scale.
Indeed, a parameterization can be considered as a mathematical means of describing
the subresolution (or microscale) processes of the phenomenon, in terms of resolvable
scale (or macroscale) variables; these macroscale variables are the ones, which can be
treated explicitly in the analysis or for which measured records are obtainable. Thus,
the details of the microscale mechanisms are not considered explicitly, but their statis-
tical effect is formulated mathematically by a parameterization in terms of macroscale
variables.
All this suggests that a sound criterion to distinguish, in principle at least, one approach
from another, may be the spatial scale at which the internal mechanisms are parame-
terized. For example, Newton's equation for viscous shear stress (see Equation (1.12)
below) is a parameterization in terms of variables typically at the millimeter to centimeter
scale; however, it reflects momentum exchanges at molecular scales, which are orders
of magnitude smaller. The hydraulic conductivity is a parameter at the so-called Darcy
scale (see Chapter 8), namely a scale somewhere intermediate between the Newtonian
viscosity (or Navier-Stokes) scale for water and air inside the soil pores, on the one hand,
and the field scales for infiltration and drainage, on the other. Several spatial scales are
illustrated with the corresponding characteristic temporal scales in Figure 1.2 for some
general types of water transport processes as they have been considered in hydrologic
studies.

On the land surfaces of the Earth, the catchment or river basin sizes appear as scales of
central importance. The terms basin, catchment, watershed and drainage area are roughly
synonymous and are often used interchangeably. A basin can be defined as all of the upstream
area, which contributes to the open channel flow at a given point along a river. The size of the
basin depends on the selection of the point in the river system under consideration. Usually
this point is taken, where the river flows into a large water body, such as a lake or the ocean,
or where it changes its name as a tributary into a larger river. However, a basin or catchment
can also defined by any point along the river, where the river flow is being measured. Basins
are delineated naturally by the land surface topography, and topographic ridges are usually