Supplementary Electronic Material for
NUMERICAL STUDY OF THE URBAN GEOMETRICAL REPRESENTATION IMPACT
IN A SURFACE ENERGY BUDGET MODEL
in the journal Environmental Fluid Mechanics
Antonio Cantelli1, Paolo Monti1, Giovanni Leuzzi1
1
Department of Civil and Environmental Engineering, “Sapienza” Università di Roma, Via
Eudossiana 18, 00184 Roma, Italy.
email: paolo.monti@uniroma1.it
In what follows the equations involved in the infrared budget for both the L1 and L2 schemes are
reported. For the equations that remain unchanged here and in the paper, the numbering is preserved
in order to make easier the use of this material. For both L1 and L2 schemes, the trapping of longwave radiation by the canyon surfaces is computed following the one re-emission formulation as
reported in [1].
Infrared radiation for the L1 scheme
Considering the urban canyon surfaces numbering introduced in the paper (see Fig. 2b of the
paper), for the L1 scheme the net infrared radiation of wall 2 is:
1
L∗2 = ψ2 ε2 ε3 σT34 + (1 − 2ψ2 )ε2 ε4 σT44 + ψ2 ε2 L ↓ −ε2 σT24 + ε2 σψ2 (1 − ε3 )(1 − ψ3 ) 2 (ε2 T24 +
ε4 T44 ) + ε2 σ(1 − 2ψ2 )(1 − ε4 )[(1 − 2ψ4 )ε2 T24 + ψ4 ε3 T34 ] + ε2 L ↓ [ψ3 ψ2 (1 − ε3 ) +
ψ4 (1 − 2ψ2 )(1 − ε4 )]
(9)
where  is the Stefan-Boltzmann constant. ϵi , Ti and ψi indicate, respectively, the infrared
emissivity, the temperature and the sky view factor of the i-th surface. L ↓ is the incoming infrared
radiation. The meaning of the terms of Eq. (9) is reported below.
ψ2 ε2 ε3 σT34 : radiation emitted by the road and seen and absorbed by wall 2.
(1 − 2ψ2 )ε2 ε4 σT44 : radiation emitted by wall 4 and seen and absorbed by wall 2.
ψ2 ε2 L ↓ : incoming sky radiation seen and absorbed by wall 2.
ε2 σT24 : infrared radiation emitted by wall 2.
1
ε2 σψ2 (1 − ε3 )(1 − ψ3 ) (ε2 T24 + ε4 T44 ) : infrared radiations emitted by walls 2 and 4, received
2
and absorbed and reflected by the road and finally seen and absorbed by wall 2.
ε2 σ(1 − 2ψ2 )(1 − ε4 )[(1 − 2ψ4 )ε2 T24 + ψ4 ε3 T34 ] : contribution of radiations emitted by the road
and wall 2, received and reflected by wall 4, and finally seen and absorbed by wall 2.
ε2 L ↓ [ψ3 ψ2 (1 − ε3 ) + ψ4 (1 − 2ψ2 )(1 − ε4 )] : contribution of incoming sky infrared radiation
portions (L ↓) seen and reflected by wall 4 and by the road, and received by wall 2.
The net infrared radiation of the road, L∗3 , reads:
1
1
L∗3 = (1 − ψ3 )ε3 σ 2 [ε4 T44 + ε2 T24 ] + ψ3 ε3 L ↓ −ε3 σT34 + ε3 σ(1 − ε2 )(1 − ψ3 ) 2 [ψ2 ε3 T34 +
1
(0.5 − ψ2 )ε4 T44 ] + ε3 σ(1 − ε4 )(1 − ψ3 ) [ψ4 ε3 T34 + (0.5 − ψ4 )ε2 T24 ] + (1 − ψ3 )ε3 L ↓
2
1
2
[ψ2 (1 − ε2 ) + ψ4 (1 − ε4 )]
(10)
where:
1
(1 − ψ3 )ε3 σ [ε4 T44 + ε2 T24 ] : radiations emitted by walls 2 and 4 and received and absorbed by
2
the road.
ψ3 ε3 L ↓ : incoming infrared radiation received by the road.
ε3 σT34 : radiation emitted by the road.
1
ε3 σ(1 − ε2 )(1 − ψ3 ) 2 [ψ2 ε3 T34 + (0.5 − ψ2 )ε4 T44 ] : contribution of radiations emitted by wall 4
and the road, received and reflected by wall 2, and finally absorbed by the road.
1
ε3 σ(1 − ε4 )(1 − ψ3 ) 2 [ψ4 ε3 T34 + (0.5 − ψ4 )ε2 T24 ] : contribution of radiations emitted by wall 2
and the road, received and reflected by wall 4, and finally absorbed by the road.
1
(1 − ψ3 )ε3 L ↓ [ψ2 (1 − ε2 ) + ψ4 (1 − ε4 )] : contribution of incoming infrared radiation (L ↓)
2
seen and reflected by walls 4 and 2, and received by the road.
Similarly to wall 2, the net infrared radiation term for wall 4, L∗4 , reads:
1
L∗4 = ψ4 ε4 ε3 σT34 + (1 − 2ψ4 )ε2 ε4 σT24 + ψ4 ε4 L ↓ −ε4 σT44 + ε4 σψ4 (1 − ε3 )(1 − ψ3 ) 2 (ε2 T24 +
ε4 T44 ) + ε4 σ(1 − 2ψ4 )(1 − ε2 )[(1 − 2ψ2 )ε4 T44 + ψ2 ε3 T34 ] + ε4 L ↓ [ψ3 ψ4 (1 − ε3 ) +
ψ2 (1 − 2ψ4 )(1 − ε2 )]
(11)
The terms in L∗4 are analogous to the corresponding terms in L∗2 , with the differences of the indices
due to geometrical reasons in mind.
Infrared radiation for the L2 scheme
Equations (33-35) regulate the infrared budget for the L2 scheme and are analogous to Eqs. (9-11).
Two main differences between the two equations sets are present. The first one is related to the
utilization in Eqs. (33-35) of the surface-view factors instead of the sky-view factors (see the
paper). The second one is due to the presences of the edge terms associated to crossroads. For this
ed
ed
reason only the edge terms Led
2 , L3 and L4 , which appear respectively in Eqs. (33), (34) and (35),
are described here in details. Led
2 reads:
4
4
4
4
4
4
Led
2 = ε2 εw ψ2,4u σ(T4u + T4l ) + ε2 εw ψ2,2eu σ(T2eu + T4el ) + ε2 ε3 σ[ψ2,3u (T3u + T3l ) +
4
4
4
4
ψ2,cr−u (Tcr−u
+ Tcr−l
) + ψ2,3eu (T3eu
+ T3el
)]
(36)
It must be observed that in Eq. (34), as in Eqs. (33-35), new indices are needed to indicate the
surface view factors and the temperatures. These indices refer to the canyon surfaces located
beyond the intersection (see Fig. 3a in the paper). The meaning of the terms of Eq. (36) is as
follows:
4
4
ε2 εw ψ2,4u σ(T4u
+ T4l
) : infrared radiations emitted by 4u and 4l walls and received by wall 2.
4
4
ε2 εw ψ2,2eu σ(T2eu
+ T4el
) : infrared radiations emitted by 2eu and 4el walls and received by wall 2.
4
4
4
4
4
4
ε2 ε3 σ[ψ2,3u (T3u
+ T3l
) + ψ2,cr−u (Tcr−u
+ Tcr−l
) + ψ2,3eu (T3eu
+ T3el
)]
:
infrared
radiations
emitted by 3u and 3l roads, by upper and lower crosses, by 3eu and 3el roads and received by
wall 2.
The edge term, Led
3 , which appears in Eq. (34), reads:
4
4
4
4
4
4
4
4
Led
3 = ε3 εw ψ3,2u σ(T4u + T2u + T4l + T2l ) + ε3 εw ψ3,2eu σ(T2eu + T4el + T4wl + T2wu )
(37)
where:
4
4
4
4
ε3 εw ψ3,2u σ(T4u
+ T2u
+ T4l
+ T2l
) : infrared radiation emitted by walls 4u, 4l, 2u, 2l and seen by
the road.
4
4
4
4
ε3 εw ψ3,2eu σ(T2eu
+ T4el
+ T4wl
+ T2wu
) : infrared radiation emitted by walls 2eu, 4el, 4wl, 2wu
and seen by the road.
For the edge term, Led
4 :
4
4
4
4
4
4
Led
4 = ε4 εw ψ4,2u σ(T2u + T2l ) + ε4 εw ψ4,2wu σ(T2wu + T4wl ) + ε4 ε3 σ[ψ4,3u (T3u + T3l ) +
4
4
4
4
) + ψ4,3wu (T3wu
)]
ψ4,cr−u (Tcr−u
+ Tcr−l
+ T3wl
(38)
the contributions are analogous to the corresponding terms in the Led
2 equation, taking into account
the differences of the indices due to geometrical reasons. The last three equations are written
considering the geometrical simplifications adopted in the L2 scheme, that produce the following
identity among edge surface-view factors: ψi,ju = ψi,jl , ψi,jeu = ψi,jel , ψi,jwu = ψi,jwl, where
indices i,j=2-4 (where i≠j) indicate surfaces 2, 3 and 4. For the sake of simplicity, it is assumed that
all the wall-surfaces beyond the edges share the same emissivity, εw .
In L2 scheme the net infrared radiation terms must be calculated for both the crossroad surfaces
located at the edge of the canyon. At the crossroads, the infrared radiation balance is derived
employing the same approach used for the main UC, in which the net infrared radiation is
calculated from the incoming and outgoing infrared radiations. For the i-th crossroad (where
indexes i=u,l indicate the upper and lower crosses, respectively) the net radiation term is:
4
cr−i
L∗3cr−i = (1 − 8ψ∞
cr−i,2 )ε3 L ↓ −ε3 σTcr−i + L3
(41)
The meaning of the terms of Eq. (41) is:
(1 − 8ψ∞
cr−i,2 )ε3 L ↓ : incoming infrared radiation received by the crossroad.
4
ε3 σTcr−i
: outgoing infrared radiation emitted by the i-th crossroad surface.
Lcr−i
: infrared radiation coming from the crossroad edges.
3
Here ψ∞
cr−i,2 is the wall view factor of the crossroad assuming UCs of infinite length, while Tcr−i is
the crossroad surface temperature. Lcr−i
3 , which takes into account the contribution of the surfaces
facing the crossroads, reads:
4
4
4
4
4
4
) + ψ4,2 ψcr−i,2 (1 − εw )ε3 σ(T24 +
Lcr−i
= ψcr−i,2 εw ε3 σ(T24 + T2ei
+T2wi
+T2i
+T44 +T4ei
+T4wi
+T4i
3
4
4
4
4
4
4
)
T2ei
+T2wi
+T2i
+T44 +T4ei
+T4wi
+T4i
where:
(40)
4
4
4
4
4
4
) : infrared radiation emitted by the eight
ψcr−i,2 εw ε3 σ(T24 + T2ei
+T2wi
+T2i
+T44 +T4ei
+T4wi
+T4i
walls and seen by the crossroad cr-i.
4
4
4
4
4
4
) : infrared radiation reflected by
ψ4,2 ψcr−i,2 (1 − εw )ε3 σ(T24 + T2ei
+T2wi
+T2i
+T44 +T4ei
+T4wi
+T4i
facing walls (respectively 2 and 4, 2i and 4i, 2ei and 4ei, 2wi and 4wi) and seen by crossroad cr-i.
As showed in the paper, L2 scheme can be simplified. Referring to Fig. 3b of the paper, the
equations that describe the edge terms in the infrared budget (Eqs. 36-38) can be rewritten in the
simplified L2 scheme as:
2
4
4
4
4
4
Led
2 = 2(ψ2,2u + ψ2,2eu )ε2 σ(T2 + T4 ) + 2ε2 [ε3 σ(ψ2,3u T3 + ψcr,2 Tcr + ψ2,3eu T3 )]
4
4
Led
3 = 2(ψ3,2u + ψ3,2eu )ε3 ε2 σ(T2 + T4 )
ed
Led
4 = L2
Similarly, Eq. (42) can be simplified as follows:
4
4
4
4
Lcr
3 = 4ψcr,2 ε2 ε3 σ(T2 + T4 ) + 4ψ4,2 ψcr,2 (1 − ε2 )ε3 σ(T2 +T4 )
Note the change of notation with respect to Eq. (42) for the crossroad surfaces (superscript and
subscript “cr”). Due to geometrical simplification, it is irrelevant to make a distinction between
upper and lower crossroad intersection, i.e. “cr-u” and “cr-l“. Unlike Eq. (42), Eq. (41) does not
change significantly, except for the substitution of the index “cr-i” with “cr”.
References
[1] Masson V (2000) A physically-based scheme for the urban energy budget in atmospheric
models Bound-Lay Meteorol 94:357–397.