# PALM-SLUrb Module Reference#

## Overview#

PALM-SLUrb (PALM's Single-Layer Urban surface model) is a physically based surface energy balance model and solver for non-resolved scale urban surfaces for the PALM model system. The model can be used to parametrize online surface fluxes from urban surfaces in PALM for simulations where individual buildings cannot be resolved at the selected grid resolution. Its purpose is to provide a relatively simple yet physically based method to model urban surface fluxes in PALM in cases where resolving urban canopies in detail is not a priority (mesoscale studies etc). The fluxes are computed online during the simulation, with two-way online coupling to the atmospheric (LES/RANS) simulation.

This technical documentation provides a brief overview of the model. A more comprehensive model description is currently under preparation and will be available publicly as a peer-reviewed journal article in due course. The model is currently considered **experimental**, and the users are recommended to carefully check and evaluate the results in their use case.

### Main features#

- Energy balance scheme for generalised urban morphology, modelling radiative fluxes, as well as sensible, latent and conductive heat fluxes.
- Prognostic equation for street canyon air temperature and humidity.
- Transfer model for energy and momentum based on a resistance network.
- Two-way online coupling with the atmospheric LES/RANS simulation.
- Parametrised shading and reflections of long and shortwave radiation within the urban canopy with coupling to atmospheric radiation models.
- Partial transmission of shortwave radiation through window layers.
- Moist physical processes: evaporation, dewfall, precipitation interception and runoff.
- Possibility for anisotropic street canyons (specified street orientation).
- Mixed urban–natural tiles are realised by aggregating fluxes from SLUrb and PALM’s land surface model (PALM-LSM), weighted by urban fraction.
- Input of user-defined surface parameters through a netCDF I/O interface.
- Instantaneous and temporally averaged output of model variables and a range of additional diagnostic quantities through the netCDF I/O interface.

## Physical model description#

SLUrb is a resistance-based single-canyon model based on the physical formulation of the Town Energy Balance (TEB) model (Masson V., 2000 and subsequent papers, see References). However, no codebase is shared between SLUrb and TEB. Numerical schemes are similar to those of PALM's land surface model (LSM) and urban surface model (USM). The atmosphere-surface coupling scheme is specifically developed for PALM. Mixed urban-natural tiles are realised using aggregated fluxes from SLUrb and LSM, weighted by urban fraction.

The urban surface is modelled with singular street canyon. Subsequently, the SLUrb uses six prognostic surfaces: roof, wall A, wall B, window A, window B and road. Walls and windows A and B are located on opposite sides on the street canyon. Walls (and windows) A and B recieve different amount of direct solar radiation depending on the street canyon orientation (input by the user) and time of day. Alternatively, the user may opt to use isotropic street canyons where walls A and B are represented by one singular wall integrated over all solar angles. In addition to the prognostic equations for surfaces, SLUrb has an internal street canyon model to model the atmospheric conditions within it.

As SLUrb is a single-layer model, the thermodynamical quantities for walls, windows and street canyon air are representative of those at canyon half-height (\(H_{avg}/2\)).

Fig. 1: An overview of the physical processes included in PALM-SLUrb, where \(\tau\) represents parametrised momentum fluxes, \(L\) longwave radiative, \(H\) sensible heat, \(LE\) latent heat and \(G\) conductive heat fluxes respectively. The modelled resistance network is illustrated with zigzag lines. The surfaces are illustrated with the default four material layers

### Heat fluxes#

Generally in the model, sensible (\(H\)) and latent heat (\(LE\)) fluxes between given points \(A\) and \(B\) are computed using bulk transfer equations

where \(r_{ah}\) is the aerodynamic resistance for heat between \(A\) and \(B\).

For vertical fluxes (roofs, roads, canyon-atmosphere fluxes), resistances based on Monin-Obukhov similarity theory (MOST) are used. The aerodynamic resistances for heat are modelled as

where \(z_{MO}\) is the half-height of first grid level (\(0.5\Delta z\)) and \(L\) is the Obukhov length. Based on a theoretical relationship between the roughness lengths for momentum and heat and an experimental fit to model data by Kanda et al. (2007), \(z_{0,h}\) computed as

where \(\nu=1.46\times 10^{-5}\) m\(^2\)s\(^{-1}\) is the kinematic viscosity of air. Alternatively, this may be fixed to a prescribed constant value by the user. The Obukhov length is computed using Newton-Raphson iteration based on bulk Richardson number (modified version of Maronga et al., 2020).

For vertical surfaces (walls and windows), a parametrisation by Mills (1993) is used:

where \(U_{can,eff}\) is canyon effective wind speed (see below).

The heat fluxes from the roof are directly aggregated into urban-averaged heat fluxes based on building plan area fraction, while the fluxes from walls, windows and roads are first aggregated into a street canyon model.

### Radiative fluxes#

SLUrb interfaces with several of PALM's radiation schemes for incoming and outgoing shortwave and longwave radiation (see radiation_scheme). Currently, `clear-sky`

, `external`

, `constant`

and `rrtmg`

are supported.

For shortwave radiation balance in anisotropic street canyons, analytical solution of net radiative flux on surfaces by Lemonsu et al. (2012) after infinite reflections within the street canyon is used. For isotropic canyons, the original parametrization by V. Masson (2000) is used, with the addition of windows. Window transmissivity is modelled as in the PALM-USM model.

The longwave radiation interactions are modelled after Johnson et al. (1991), with first order reflections resolved within the street canyon. The first order reflections contribute around 5% of the total LW budget, while higher order reflections would contribute only <0.5%. The interaction coefficients are computed during initialization and stored in memory in order to save computation time during simulation.

### Street canyon model#

Unlike TEB, which uses quasi-equilibrium model for modelling the temperature and humidity within the street canyon, SLUrb uses additional prognostic equations for \(T_{can}\) and \(q_{can}\). This change to the model formulation is done to stabilize the solution even in highly turbulent flows, where the conditions at the first atmospheric grid level may rapidly change due to resolved-scale turbulence.

In principle, this means that the \(T_{can}\) and \(q_{can}\) tendencies are computed using net sensible and latent heat fluxes applied on total canyon air volume, assuming the street canyon air is fully mixed. For temperature tendency, this is

where \(H_{surf}\) and \(H_{can}\) are the sensible heat fluxes from the canyon surfaces and between canyon air and the atmosphere respectively. Remembering that the fluxes are given per unit area, \(H_{surf}\) is computed as

As SLUrb doesn't consider condensation or evaporation of liquid water on vertical surfaces, \(q_{can}\) tendency is computed as

The prognostic equations for \(T_{can}\) and \(q_{can}\) are then solved using the same Runge-Kutta 3rd order time integration as for the atmospheric model.

A parametrized street canyon wind speed \(U_{can}\) is used to compute the aerodynamic resistances on the surfaces within the street canyon. Currently, the user is provided an option to select from three different exponential formulations.

To address the mixing enhancement due to mechanical and convective turbulence, an effective canyon wind speed is used in the resistance model

where \(u_*\) and \(w_*\) are the friction velocity and the convective velocity scale within the street canyon, respectively.

### Liquid water reservoirs#

SLUrb considers evaporation, dewfall, precipitation interception and runoff of liquid water for horizontal surfaces, namely roofs and roads. The prognostic equation for the liquid water reservoirs \(m_{*,liq}\) on roofs and roads is written as

where \(P\) is the precipitation rate and \(R\) is runoff. The liquid water column is restricted to 1 mm, with excess water removed from the model as runoff. The water reservoir is rarely distributed evenly on the surface, especially when only a small amount of water is stored on the surfece. As evaporation can only occur on the liquid water - air boundary, it has a dependency on liquid water coverage on the surface (following V. Masson, 2000):

where

For dewfall and interception, the whole surface area is used as they are not limited by the water coverage on surface.

### Energy balance linearization#

For the numerical solution of the energy balance on surfaces, the energy balance linearised in time:

where

where \(T_{\star,1,p}\) is the prognostic surface temperature of the surface, \(\Delta t\) is the time step, \(S^\updownarrow\) and \(L^\updownarrow\) are the net surface shortwave and longwave radiation, \(\varepsilon_{eff}\) the effective emissivity of the surface (including the effect of backreflection from other street canyon surfaces), \(C_{R,1}\) is the outermost layer (total) heat capacity, \(d_1\) and \(d_2\) are the thickness of first and second material layers, \(f_H=\frac{\rho c_p}{r_\star}\), \(\Pi\) is the Exner function, \(\theta\) the potential temperature in the adjacent air (canyon air for walls and roads or first atmospheric level for roofs), \(q\) the water vapour mixing ratio in the adjacent air, \(q_s\) the saturation water vapour mixing ratio at the surface,, \(T_{\star,1}\) the first subsurface layer temperature and \(\Lambda_{1}=\frac{d_1 + d_2}{\left(d_1/\lambda_1\right) + \left( d_2 / \lambda_2\right) }\) the heat conductivity between layers 1 and 2.

For subsurface layers, a regular heat equation is solved. The bottom boundary condition is either a indoor temperature set by the user (roofs, walls, windows) or a deep soil temperature, set by the user as well (roads).

In the prognostic equation for street canyon air temperature, the street canyon air - atmosphere exchange is linearized similarly. However, the heat fluxes gathered from the surfaces are not linearized with respect to the street canyon air temperature but rather entered directly to the prognostic equation to ensure consistent fluxes within the model.

### Urban aggregation#

The momentum transport between the urban surface and the atmosphere is modelled as

where

By default, the urban roughness length for momentum is parametrised as \(z_{0,m}=h/10\), but the user is recommended to provide a gridded input of pre-computed values based on e.g. formulations by Macdonald (1998), Kanda (2007), or Kent (2017). Obukhov length \(L_{urb}\) for complete urban surface is computed using ensemble temperature of roof and canyon air.

For \(H\) and \(LE\), area-weighted fluxes from roofs and canyon air are used, with addition of optional user input of \(H_{ext}\) and \(LE_{ext}\) from external sources (e.g. industry, HVAC exhaust):

After being weighted using urban fraction \(f_{urb}\), the fluxes enter the PALM's prognostic equations for \(u_i\), \(\theta\), \(q\) and \(e\) on the first grid point above topography together with the natural fluxes as computed by PALM-LSM. SLUrb provides effective albedo and emissivity as well as radiative surface temperature for the PALM's radiation schemes (clear-sky, external, constant and RRTMG) to ensure consistent representation of radiative fluxes throughout the model. These are also used by the radiative transfer model (RTM) when computing reflections and shading due to resolved topography.

The energy balance closure in SLUrb is represented by the following equation

where (A) is a sum of the net atmosphere-surface fluxes, (B) storage term of canyon air, and (C), (D), (E), and (F) are the storage and bottom boundary flux terms for roofs, walls, windows and roads respectively, and (G) the evolution of liquid water storages including intercepted precipitation and runoff. After expansion, positive terms represent the inflow of energy to the system whereas the negative terms correspond to the outflow of energy.

## References#

Johnson, G. T., T. R. Oke, T. J. Lyons, D. G. Steyn, I.D. Watson and J. A. Voogt (1991): Simulation of surface urban heat islands under ‘IDEAL’ conditions at night part 1: Theory and tests against field data, Bound.-Lay. Meteorol, 56, 275-294

Kanda, M., M. Kanega, T. Kawai, R. Moriwaki, and H. Sugawara (2007): Roughness Lengths for Momentum and Heat Derived from Outdoor Urban Scale Models, J. Appl. Meteorol. Climatol., 46(7), 1067-1079

Krayenhoff, E. S., J. A. Voogt (2007): A microscale three-dimensional urban energy balance model for studying surface temperatures, Bound.-Lay. Meteorol, 123, 433-461

Lemonsu, A., C. S. B. Grimmond, and V. Masson (2004): Modeling the Surface Energy Balance of the Core of an Old Mediterranean City: Marseille, J. Appl. Meteorol., 43(2), 312-327

Lemonsu, A., V. Masson, L. Shashua-Bar, E. Erell, and D. Pearlmutter (2012): Inclusion of vegetation in the Town Energy Balance model for modeling urban green areas, Geosci. Model Dev., 5, 1377-1393

Masson, V. (2000): A Physically-Based Scheme for the Urban Energy Budget in Atmospheric Models, Bound.-Lay. Meteorol, 94, 357-397

Masson, V., C. S. B. Grimmond, and T. R. Oke (2002): Evaluation of the Town Energy Balance (TEB) Scheme with Direct Measurements from Dry Districts in Two Cities, J. Appl. Meteor. Climatol., 41, 1011–1026

Pigeon G., K. Zibouche, B. Bueno, J. Le Bras, and V. Masson (2014): Improving the capabilities of the Town Energy Balance model with up-to-date building energy simulation algorithms: an application to a set of representative buildings in Paris. Energy and Build., 76, 1–14

Rowley F. B., and W.A. Eckley (1932): Surface coefficients as affected by wind direction. ASHRAE Trans., 38, 33–46