Monday, 24 March 2014

Re-entering the Matrix

Whilst I won't be donning a pair of black glasses and a large trench coat to enter a parallel universe via a phone line, I have returned to the world of matrices after a 10 year hiatus.

It has been a while since my last post, and after giving myself a kick up the backside I am back to it. After a period of reading (note to self update online bibliography) focused on urban surface models of varying degrees of complexity and how they treat vegetation (if at all), I have started the first modelling phase. After discussions with the Prof. during a period of disillusionment (first of many I am sure over the next few years) it was decided that it is time to get practical and put into practice some of the topics I have been reading about.

So as a first exercise I am focusing on modelling radiation exchange within a 2D urban canyon using the method of Sparrow and Cess (1970) as applied by Harman et al. (2004), in which shapefactors are determined for a chosen canyon geometry and used to calculate the exchange of diffuse radiation between surfaces. This scheme was chosen as it is shown by Harman et al. (2004) to be more accurate than two commonly utilised approximations to determine the net radiation balance of a canyon used in a number of models and the scheme is utilised (in a simplified form) within the MORUSES model (Porson et al. 2010) that I plan to adapt to include urban vegetation.

The 2D urban canyon alluded to earlier has four facets (roof, road, wall 1 and wall 2), with the road having width, W, the building walls having height H and the canyon is of infinite length (Figure 1). The canyon geometry is then described by the height to width ratio of the canyon (H/W) which from inference identifies the magnitude of the surface area within it for radiative transfer and emission (radiation trapping). In this modelling exercise the surfaces considered are the road, two walls and the sky (s) at the top of the walls which forms a closed system. The roof is treated separately as a flat surface and not considered in this initial exercise.
Figure 1. Schematic of urban canyon system (Harman et al. 2004)

The aims of the modelling exercise are to recreate the results of Harman et al (2004), to look deeper at the underlying geometry and mathematics used in determining the solution, as a starting point in the coding of my model, and to begin to consider the impact of additional facets such as trees from a coding and theoretical prospective.

So where do matrices come into this I hear you say? It is possible to describe the exchange of radiation within the above defined closed system by a set of three equations (see below) that consider the emitted (Sigma), total incoming (Lambda) and total outgoing (B) radiation for each facet (i) and the other facets (j) within the system and the net radiative balance (Q).
Equations 6, 7 and 8 from Harman et al. (2004)

The solutions for the 3 equations can be determined using a matrix that considers the interactions between the 4 facets (sky, road, wall1, wall2) in terms of calculated shapefactors (Fij), facet emissivity (epsilon) and the identity matrix (deltaij ,not shown). From this matrix the outgoing radiant density for each facet (B) is then determined using a set of 4 simultaneous equations solved using Gaussian Elimination (a.k.a. LU Factorization). Results to follow...

References
Harman, I. N., Best, M. J., and Belcher, S. E.: 2004, Radiative exchange in an urban street canyon. Boundary Layer Meteorology, 110, 301-316.

Porson, A., Clark, P. A., Harman, I. N., Best, M. J., and Belcher, S. E.: 2010, Implementation of a new urban energy budget scheme in the MetUM. Part I: Description and idealized simulations. Quarterly Journal of the Royal Meteorological Society, 136, 1514-1529.

Sparrow, E. M., and Cess, R. D.: 1970, Radiation Heat Transfer, Chapters 3-4, Appendices A & B, Thermal Science Series, Brooks/Cole, pp.75-136, 300-313.