2.6.1 Conceptualisation of the real world
Numerical modelling provides a useful tool to quantitatively understand how an a system behaves under different constraints. A model can be used to explain observations, reconstruct past or predict futures scenarios to input
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parameters and boundary constraints. By comparing model results to ob-servations, missing processes can be identified. Where field observations are absent, the sensitivity of different input parameters can be found that drive a system.
However, it is important to notice that a model in environmental science is an abstraction and simplification of the real world. Some processes may be neglected, the geometry simplified, the material properties homogenised, and the forcing factors simplified.
2.6.2 Initial conditions and boundary conditions
Initial and boundary conditions of a model define the extent of the model domain. The boundary conditions represent processes or values at the border of the spatial domain of the model. The initial conditions represent values of the model domain at t=t0 (Allen et al., 1988; Wainwright , 2004). Model results are usually very sensitive to the choice of the initial and boundary conditions and are discussed in this section.
2.6.2.1 Initial condition
The initial conditions specify a solution to the problem domain at t=t0. Commonly, a steady state problem is assumed for the initial condition (e.g.
McKenzie et al., 2007; Bense et al., 2009). For systems with a long response time, the initial conditions should be calculated early enough in order for any relevant past changes to be included in the model. For example, the hydrogeology of a previously glaciated area may still be influenced by past conditions, and therefore setting the initial conditions to present day in order to model future responses may neglect past glacial influences. In addition, Bense et al. (2012) use an initial condition of an intra-permafrost talik, for which a transient model has been used.
2.6.2.2 Boundary conditions
The choice of boundary conditions of any model is crucial and determines what happens at the edge of the model domain and how they change over time (Wainwright , 2004). At the boundary, either a value to the solution can be specified, also known as Dirichlet boundary condition, or the normal derivative of the equation, the Neumann boundary condition can be given.
The Dirichlet boundary condition specifies a value at the boundary; for heat flow this can be the ground surface temperature at the top boundary,
or for Darcy flow the hydraulic head distribution over the model domain.
The value specified at the boundary can be a function of length along the boundary and time.
The Neumann boundary specifies a flux into or out from the model do-main. Considering a Laplace equation:
∇ · ∇u = 0 (2.21)
here for two dimensions, the Divergence Theorem states that the flux outward of a vector field is equal to the volume integral of the equation (Kreyszig et al., 2011) as follows:
where u is a variable, A the area of the domain, S the surface, and n the normal vector. n · ∇u is the gradient at the boundary and describes the Neumann boundary condition (PDE Solutions, 2006).
A flux boundary can either be equal to zero, when no fluxes enters or leaves the system, be a set value over the entire side, as for example a heat flux defined at the base of the model, or be defined with an equation, as for example regions with high groundwater discharge whose temperature is defined by the component of advective heat flow normal to the surface.
How the initial and boundaries are implemented for each model scenario, will be discussed in each chapter separately.
2.6.3 Model geometry
The implementation of the model can be done in different coordinate systems:
1D, 2D Cartesian, 2D radial, and 3D. Each of the geometries have their advantages and disadvantages for modelling different systems.
In this section, the model geometries for a lake, surrounded by bedrock is evaluated. In order to model the real world lake geometry, the lake geometry is measured and its coordinates imported into a 3D geometry. When concep-tualised and simplified, the lake geometry could be simplified to a circular or elliptical lake geometry and modelled in 3D. However, in FlexPDE, using coupled heat and fluid flow including phase change in 3D is computation-ally very expensive. Therefore, the model has to be implemented into 2D geometry. Here the choice is either with cylindrical coordinates or 2D Carte-sian coordinates. Cylindrical coordinates would be ideal to model a circular
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Table 2.1: Definition of lake geometries compared in Figure 2.3. A lake modelled in 3D is defined by its major axis a, and the minor axis b. In a 2D model, the lake size is defined by its major axis, the radius.
lake. However, in cylindrical coordinates heat conduction only models can be done, as fluid flow could only flow radially either into or from the lake.
Thus, to incorporate a flow from one side to the other side of the model domain, 2D Cartesian coordinates have to be used. However, assigning the thermal boundaries for a lake in 2D is difficult, as 2D models a cross section through the lake and is essentially a river. In 2D Cartesian, the thermal disturbance of a lake is only influenced from the sides of the model and the depth, whereas in 2D cylindrical or 3D, the thermal disturbance is influenced by the x-y plain and the depth. Therefore, the thermal disturbance is larger if modelled in 2D Cartesian than for 2D radial or 3D.
In order to find the relative importance of the lake geometry, different lake geometries modelled in a steady state are compared. Compared geometries are a circular lake, elongated lakes, and square lakes modelled in 3D and 2D radial and 2D Cartesian lake (Table 2.1).
Figure 2.3a compares the talik distributions, where the permafrost bound-ary is defined as 95% ice saturation (pi = Θi/n), under elliptical lakes with different proportions of major radius a [m] versus minor radius b [m]. For a circular lake modelled in 3D, there is no through talik modelled, whereas for an oval lake with a/b = 2 a through talik exists for the set boundary conditions. For a longer major radius, the permafrost boundary becomes shallower, and approaches the one of a river or an infinite long lake.
Figure 2.3b compares a talik under a 2D Cartesian lake and a 3D infinite rectangle, or river and demonstrates a good match. The mesh of the 3D simulation is coarser, resulting in a less smooth permafrost boundary than for the 2D simulation. In addition, Figure 2.3b compares a circular lake modelled in 3D and 2D radial coordinates and shows that the 2D talik is deeper than the 3D talik. This is possibly due to numerical instability and the coarser mesh of the 3D model.
a)
b)
Figure 2.3: a) Comparison of permafrost geometry under lakes with different elongation. b) Comparison of permafrost geometry under lakes for a 2D and 3D model domain. The permafrost boundary represented is defined as 95% ice saturation.