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**7.
Modelling the behaviour of water **

**Abstract
**

J. C. I. Dooge

Scientists in many disciplines are concerned with the movement of water. One of the results of their varying viewpoints is that their models relate to a specific length scale and a particular time scale. The mathematical models at the different scales are not only dissimilar but in certain cases the assumptions made at one scale may actually contradict the assumptions made at another scale. In certain cases, microscale phenomena may be successfully parametrized at a macroscale, but the procedure is rarely simple. The transition from physically based models of hydrological processes to conceptual models at the catchment scale has been systematically studied in recent years. The transition from catchment models to general circulation models presents even more formidable problems.

The main links between the land surface and the atmosphere are formulated on the basis of the energy balance the heat balance, and the moisture balance. A key problem is the reduction of estimated potential evapotranspiration, which is determined by atmospheric variables, to actual transpiration. To handle this coupling on the macroscale it is necessary to parametrize the physical processes of soil moisture movement, plant water uptake, and transpiration. Since forests constitute about one third of the land surface cover and about two thirds of the biomass production, the successful parametrization of these processes for the forested condition is essential if accurate representation of the fluxes of water from land surface is to be incorporated in general circulation models.

Macroscale water balance models may be (a) purely empirical (in which the relationship is only statistical), (b) phenomenological (in which primary behavioural characteristics, such as storage or flux between storages, are modelled by conceptual elements), or (c) dynamic (in which the equations expressing hydrological processes at the microscale are parametrized).

The coupling of soil moisture,
vegetation, and climate is still more complex. In this case also
one can distinguish between empirical, phenomenological, and
dynamic models. It is desirable to establish, if possible, the
long-term equilibrium relationships between these factors for
large homogeneous areas on the basis of a parametrization of the
physical processes in which the data requirements will be
relatively compact and readily observable. Success in this task
would pave the way for tackling the problems of spatial
variability and temporal variability. A large-scale (104 km^{-2})
experiment to study the hydrology/atmospheric circulation
interface is currently being planned, under the aegis of the
World Climate Research Programme.

**Modelling
the behaviour of water**

**Scales and Parametrization**

Each scientific discipline has not only its own distinctive jargon but also a set of one or more preferred scales at which mathematical models are constructed. These preferred scales seem to reflect well-spaced discrete states along the natural continua of space and time that can be isolated and subjected to conceptual analysis. The models based on such an analysis and tested by observations at this preferred scale cease to be useful for predictive purposes once we move more than one or two orders of magnitude away from the optimum space and time scales for this particular analysis. If the discrete scales, for which a verified analysis is available, are close enough together so that regions of application overlap, then the two modes of conceptual analysis reinforce one another. If, however, the phenomena of interest fall within a gap between two discrete modes of analysis, then prediction may become quite unreliable. Faced with such a situation, we are obliged to follow one of three courses: (a) to attempt to parametrize those processes at the microscale that affect behaviour at the mesoscale, which is the scale of interest in terms of the processes and parameters at the mesoscale; (b) to attempt to disaggregate the conceptualization at the macroscale to allow for processes and for parameter variations at the mesoscale; of interest; or (c) to postulate and seek to verify new laws of behaviour appropriate to phenomena at the mesoscale; of interest.

The first attempts at a scientific conceptualization of nature were at the human scale, at which intuition is fairly reliable. As science developed other levels became the basis for conceptual analysis. Figure I is one representation of the principal levels of analysis (Klemes 1983). Other values might be preferred by other authors, but the general picture can be accepted. Klemes (1983) comments that physical science did not expand regularly upwards and downwards from the human scale. Thus, analysis at the atomic level preceded analysis at the molecular level and analysis of the solar system preceded the analysis of global phenomena.

**FIG. 1. Scales for conceptual
analysis**

Even if we take the levels of figure I as marking significant scientific divisions, we must take account of lesser peaks in the spectrum of significant scales when tackling the particular problem of the behaviour of water. The present author (Dooge 1983' has suggested as significant length and time scales in the various approaches to water movement as shown in figure 2. The mathematical models appropriate to these different levels of analysis may be quite different in character and when compared may seem to contradict one another.

Atmospheric phenomena also span a wide range of space and time scales, as is shown in figure 3 (Smagorinsky 1974). The amount of kinetic energy involved in molecular diffusion and in the internal sound waves is negligible compared to that involved in the central part of the figure, where turbulent flow phenomena predominate. The fact that these turbulent phenomena lie about a line corresponding to the five-thirds turbulence law emphasizes both the linkage between space scales and time scales and also the importance of seeking relationships at a scale appropriate to the phenomena of interest.

The area outlined by a dotted line in the upper right-hand corner of figure 3 represents the scales typically resolved by general circulation models (GCMs). The phenomena outside this area, which affect the macroscale processes modelled by the GCM, must be taken into account by some form of parametrization (GARP 1972, 1974).

There is a similar range of scale in the mathematical models used in hydrology. To link the models based on hydrologic physics with the models based on atmospheric physics one needs a number of successive parametrizations: from "point" scale, to field scale, to catchment scale, to regional scale, to GCM grid scale (Dooge 1982). Before seeking to span this wide range it is instructive to review the basis of the equations of hydrologic physics, which are themselves the result of parametrization from a still smaller scale formulation of physical laws.

**FIG. 2. Scales in water movement**

**FIG. 3. Space and time scales of
atmospheric phenomena **

**From Water Molecule to Continuum Point**

The modelling of the general circulation of the atmosphere of this planet would be less complicated if there were no oceans and if the transformation of water from one phase (vapour, liquid, solid) to another in the hydrological cycle did not occur. Of the other inner planets of our solar system, the hydrosphere of Mars is largely frozen, that of Venus is largely in vapour form, while Mercury has no hydrosphere. The fact that the earth's hydrosphere is largely liquid is due to the fact that the evolution of the earth's oceans and atmosphere resulted in an average surface temperature of about 15°C, which, at atmospheric pressure, is within the liquid range of water. On Mars the evolution of even a slight atmosphere was sufficient to produce equilibrium at - 54°C, at which water occurs in solid form. In the case of Venus the initial temperature was high enough to produce a runaway greenhouse effect (Goody and Walker 1972), resulting in a surface temperature of 457°C and a gaseous hydrosphere.

The fact that water is liquid at 15°C is, however, seen to be anomalous if one compares water with the other Group VI anhydrides (Dooge 1983). On the basis of the position of oxygen in the periodic table one would expect oxygen anhydride (i.e. water) to melt at about - 100°C and to boil at about - 80°C. The fact that this expectation is not realized is only one of the anomalies in the physical properties of water. The anomalously high values of a number of its physical properties (latent heat of vaporization, specific heat, thermal conductivity, surface tension, dielectric constant, minimum water volume at 4°C) are all significant in land surface processes and in the general circulation of the atmosphere.

The physical chemist explains these anomalous properties of water on the basis of the polarity of the water molecule and of the consequent hydrogen bonding between water molecules. The extra energy required to overcome hydrogen bonding as well as covalent bonding accounts for the high melting point, the high boiling point, the high latent heat, and the high surface tension. The essential assumption made in the models that successfully predict the actual properties of water is that the water molecule is highly polar, that is non-isotropic. Without this assumption we would predict that the earth's hydrosphere would all occur as water vapour.

When we come to predict the movement of liquid water, as distinct from its occurrence, we take a completely different view. The molecular structure of water is ignored and water is assumed to be a fluid continuum. The continuum is assumed to be a Newtonian fluid, one in which the local shear tensor is a linear function of the local deformation tensor. This can be written as

(1)

where t _{ij} is the second order local
shear tensor at a continuum point, D_{rs} is the second
order local deformation tensor, and C_{ijrs} is a fourth
order tensor whose 81 elements are functions only of the
thermodynamic variables of mass density (r ) and
thermodynamic temperature (q ). Since the two second order tensors are
symmetric, the linking fourth order tensor will also be symmetric
and hence only 36 of its elements represent independent
parameters. It can be appreciated that the prediction of the
movement of a fluid whose constitutive equation has 36
independent parameters would present some difficulty.

If the single assumption is now
made that the fluid is isotropic, then the number of independent
parameters in the fourth order tensor C_{ijrs} is reduced
from 36 to 2. Thus, for an isotropic Newtonian fluid we have

(2)

where d _{ij} is the Kronecker delta, D_{rr}
is the local divergence, and the two parameters l and µ are
functions only of the thermodynamic variables. If the Newtonian
fluid is incompressible, then the divergence is zero and there is
only one parameter, the dynamic viscosity µ

For an incompressible fluid, the three equations for the conservation of linear momentum can (with the aid of tensor notation and the summation convention of repeated indices) be written as

(3)

where u_{i}, is the i
component of the velocity, F_{i} is the i component of
the body force, and t _{ij} the i component of the
surface force on the plane whose normal is in the j direction.
Substitution from the incompressible form of equation (2) into
equation (3) gives the Navier-Stokes equations

(4)

which form the basis for the study of the motion of viscous fluids.

Hydrology has been defined as the
study of the occurrence and movement of water on our planet. When
we look to the fundamental sciences we find that (a) to predict
correctly the occurrence of liquid water on earth we must assume
water to be nonisotropic and (b) to describe the movement of
water adequately we must assume it to be isotropic. This apparent
contradiction should not deter us from using either type of model
in the solution of appropriate problems. A thorough understanding
of the range of validity of the Navier-Stokes equations must
await a complete parametrization from the non-isotropic molecular
scale (10^{-10} m) to the isotropic continuum scale (10^{-5}
m). Meanwhile, these equations can be used for problems in which
predictions based on the equations have been empirically verified
and for problems using parameter values determined by laboratory
methods. The lesson to be learnt is that problems of differing
scale can be successfully tackled by the use of apparently
conflicting approaches.

**From Fluid Mechanics to Physical Hydrology**

Before attempting to incorporate or parametrize the equations of physical hydrology in general circulation models, it is necessary to realize that these hydrologic equations themselves involve parametrization at a macroscale of the basic microscale equations of fluid mechanics such as the Navier-Stokes equations. The most important hydrologic processes for climate modelling are evaporation from various types of surface, infiltration of the surface, percolation through the unsaturated zone resulting in groundwater outflow, and free surface flow over the land surface and in the drainage network. The hydrologic equations used to describe these processes all involve parametrization of the basic Navier-Stokes equations.

In both the atmospheric boundary
layer in which evaporated moisture is transported and the free
surface flow in the drainage network the flow conditions are
turbulent. Under these conditions the Navier-Stokes equations
still apply at any given continuum point, but they cease to be
useful because of the extreme complexity of the turbulent flow
pattern. Accordingly, the velocity components are expressed as
the sum of a mean value () and fluctuations from this mean (u'_{i}).
If the Navier-Stokes equation given by equation (4) is then
averaged over a time scale that is large with respect to the time
scale of any deviations from the mean flow, we obtain the
Reynolds equation of motion:

(5)

Though the Reynolds equation differs from the original Navier-Stokes equation only in respect of the third term, which involves the cross product of the turbulent fluctuations, the difference is an important one. In virtually all cases of turbulent flow the viscous stresses represented by the third term on the right-hand side of equation (5) are negligible compared with the Reynolds "stresses" represented by the new third term on the left-hand side of the equation. In order to solve the Reynolds equations at the new macroscale, it is necessary to provide a closure for the problem by expressing the Reynolds stresses that are generated at the microscale in terms of the macroscale mean variables of flow (or by adding further equations with all but one variable in common). Models proposed for this purpose over the past 100 years use such concepts as eddy viscosity, mixing length, kinetic energy of turbulence, and so forth (Rodi 1980; Brutsaert 1982).

In the case of turbulent flow as described above, the Reynolds smoothing represents a parametrization of the Navier-Stokes with respect to time. In the case of the equations for subsurface flow, the Navier-Stokes equations are parametrized with respect to space. For subsurface flow the velocities are usually low enough and the characteristic linear dimensions small enough for the viscous forces to dominate the inertial forces sufficiently to prevent the onset of turbulence. In addition, the accelerations on the lefthand side of equation (4) are considered negligible compared with the forces on the right-hand side, so that the Navier-Stokes equation is simplified to

(6)

If we now integrate over the cross-section of an individual pore, we obtain for the mean pore velocity

(7)

where L is a length scale characterizing the cross-section of the pore space and o is a dimensionless number that depends on the shape of the cross-section (and in the case of unsaturated flow on the capillary potential).

A second integration must now be performed over a representative cross-sectional area of the porous medium in order to obtain the face velocity

(8)

where K_{j} is the
intrinsic permeability of the porous medium in the i direction.
In the case of unsaturated flow, the permeability will be a
function of the capillary potential.

It is clear that the equations used in physical hydrology involve the parametrization of the Navier-Stokes equations, which themselves ignore the structure of liquid water. This indicates that models based on these macroscale equations must be verified at the scale of interest before they can be applied with confidence to problems at that scale. Even if they are verified in certain cases, it must be remembered that the best established laws of physics are transcended under certain conditions (Wheeler 1973).

**From Physical Hydrology to Climate Modelling**

There has been an increasing interest in recent years in the parametrization of land surface processes in general circulation models (Eagleson 1982b). In principle the GCM will be able to provide values of net solar and infra-red radiation, air temperature and specific humidity, and precipitation in the form of rain and snow. The modeller will require, through coupling with a hydrologic model, an estimate of the variation in time of the moisture flux and the heat flux from the land surface into the atmosphere. The key hydrologic variable involved is the actual evaporation into the atmsophere, which depends not only on the atmospheric conditions but on the state of the soil moisture (including the storage in the unsaturated zone and the level of the water-table) and on the nature of the vegetal cover.

The relationship between rainfall and runoff is highly non-linear (Dooge 1982). Models based on the threshold concept of infiltration will predict zero direct runoff as long as the rate of precipitation is less than the rate of potential infiltration, which is determined by the state of the soil moisture.

Models based on the contributing area concept predict a varying percentage of runoff during the course of a storm as the water table approaches the surface over an increasing area during rainfall and over a decreasing area during the recession. No matter which type of model is adopted to explain and to predict direct storm runoff, the rate of infiltration will vary throughout a catchment area and the non-linearity involved will make direct parametrization very difficult.

Studies of the variation of soil properties and of infiltration for areas between 10 and 150 ha indicate variations of one or two orders of magnitude that cannot be related to soil characteristics (Dooge 1982). It is not clear whether the solution to the problem of parametrization—of soil moisture even on a catchment scale of 10 km by 10 km—will be found through direct parametrization of the processes described by the equations of physical hydrology or through the development at catchment scale of reliable conceptual models and the systematic analysis of the parameter values that optimize the fit of model predictions to hydrologic records.

Even if hydrologists are successful in parametrizing physical equations to a catchment size of 100 km2, there remains the final problem of parametrization from this catchment scale to a (GCM scale in which the areas involved would be between 104 and 105 km2. At these scales, not only do the spatial variability of the land surface and the soil properties create problems but there is the added problem of the inhomogeneity of the precipitation input, which in many climates has a characteristic scale of the order of 10 km. Whether parametrization or direct conceptual modelling is used, there is clearly a need to consider whether meaningful experiments can be carried out on a catchment of a GCM scale. This topic will be resumed at the end of this review paper.