## Introduction

Snow cover is frequently heterogeneous on length scales too small to be
resolved by a general circulation model (GCM) grid, introducing marked sub-grid
heterogeneities in land-surface characteristics and fluxes. A high-resolution
two-dimensional (2-D) boundary-layer model is used in this study to model
turbulent fluxes of heat and moisture over heterogeneous snow cover. The
performance of a “tile” model, which parameterizes gridbox-average surface
fluxes as weighted averages of fluxes over snow-covered and snow-free regions,
is assessed in comparison with the boundary-layer model. Using the tile model
to allow for heterogeneous snow cover in a single-column version of the Hadley
Centre GCM is found to have a large impact on the partitioning of available
energy into latent and sensible heat fluxes.

## The Boundary-Layer Model

The boundary-layer model used here is described by Wood and Mason (1991). Velocity components, potential
temperatures and specific humidities are found as solutions of the Boussinesq
equations with first-order turbulence closure on a 2-D grid that has a
horizontal spacing of 31.25 m and 20 vertical levels (five in the lowest 10 m)
extending up to 5000 m. The numerical scheme used is second-order accurate, and
energy conserving. Vertical fluxes of heat and moisture are set to zero at the
upper boundary, and the flow is driven by a constant horizontal-pressure
gradient. Periodic boundary conditions are imposed at the left-hand and
righthand edges of the model domain, which is 1 km wide.

Parameterizations of surface radiation, sensible heat and moisture fluxes have
been added to the boundary-layer model (Essery,
in press). Sensible heat fluxes (*H*) and moisture
fluxes *(E)* are proportional to differences in potential
temperature *(θ)* and specific humidity *(q)*
between the surface and the lowest model level (at height *z* =
0.25 m), divided by appropriate resistances;

and

where *θ*_{0}
is the surface potential temperature,
*q*_{sat}(T_{0}) is the saturation
humidity at surface temperature *T*_{0}, and
*r*_{s}
is a surface resistance for moisture transfer (zero for saturated
surfaces such as snow). The aerodynamic resistance,
*r*_{a}, increases with increasing atmospheric
stability, decreasing surface roughness (characterized by roughness length
*z*_{0}
) and decreasing windshear, all of which suppress turbulent
transport.

Given downward fluxes of solar and longwave radiation *(SW*_{↓}
and *LW*_{↓}
) and assuming unit longwave emissivity, the net radiation at a point
on the surface is

where *α* is the surface albedo, and *σ* is the
Stefan Boltz-mann constant. The net radiation is partitioned into sensible,
latent, ground and snowmelt heat fluxes. Surface temperatures are found by
inverting the surface-energy balance,

The sign convention used is that *R* and *G* are
positive downward, and *H* and *E* are positive
upward. *H*_{sm}
is the snow-melt heat flux required to ensure that the calculated
snow-surface temperature does not exceed 0°C, *G* is the ground
heat flux (assumed to be negligible) and λ is taken to be the latent heat of
sublimation at snow-covered points or the latent heat of vaporization at
snow-free points.

The boundary-layer model was run with surface parameters, shown in Table 1, chosen to represent forest, grass
and snow-covered grass. The snow is given the same roughness length as the
grass (roughness lengths for deep, continuous snow covers are generally much
lower) but has higher albedo and zero surface resistance, whereas the forest
has the same albedo as the grass but larger roughness length and surface
resistance. It should be noted that the drag at model levels that would lie
within the forest canopy is not explicitly represented, and the modelled wind
profile is only valid at heights above the canopy.

Table 1. Surface parameters for forest, grass and snow-covered grass

Figure 1a shows heat fluxes across a
surface with 500 m fetches of forest and grass, homogeneous downward radiation
fluxes *(SW*_{↓}
= 400 W m^{−2} and LW_{↓} = 300 W m^{−2}) and
a 10 m s^{−1} geostrophic wind blowing from left to right. The sensible
heat flux is higher and the latent heat flux is lower over the forest, but both
are upward everywhere, and the energy available to be partitioned into latent
and sensible heat fluxes is nearly homogeneous across the surface. Replacing
the grass by snow gives similar latent heat fluxes, as shown in Figure 1b, but the high albedo of the snow
gives a lower available energy, requiring the sensible heat flux to change
direction from upward over the forest to downward over the snow. Heat is
advected from the warm forest to the cold snow, and a shallow, stable internal
boundary layer forms over the snow. The stable layer is capped by an inversion
in the local temperature profile, which reaches a height of 13 m at the
downwind edge of the snow patch in this case. Advection over heterogeneous snow
has been studied by Liston (1995) using a
similar modelling strategy.

Fig. 1. Surface sensible and latent heat fluxes for (a) forest and gray, and
(b) forest and snow. Solid lines are from the boundary-layer model and
dashed lines are from the tile model.

## The Tile Model

GCMs require parametefizations to calculate gridbox-average fluxes given
gridbox-average temperatures, humidities and windspeeds at a reference height
in the atmosphere. Local fluxes, however, depend non-linearly on local vertical
gradients, and average fluxes are not simply related to average gradients over
heterogeneous surfaces. It has been suggested that surface heterogeneity can be
represented by gathering distinct surface types within a gridbox into
homogeneous “tiles” and calculating fluxes separately over each tile (Avissar and Pielke, 1989; Claussen, 1991).

A tile model, using the same surface-flux parameterizations as the
boundary-layer model and driven by area-average data from the boundary-layer
model at a height of 19 m (typical of the lowest atmospheric level in a GCM,
but too high to resolve the shallow, stable layer over the snow), has been
assessed in comparison with the boundary-layer model by Essery (in press). As shown by Blyth and others (1993), a tile model can work well for
heterogeneous vegetation. The dashed lines on Figure 1a show fluxes calculated for forest and grass tiles. The
tile model does not represent advective effects at the edges of the forest and
grass patches, but it does give very good values for average fluxes in this
case (Table 2). Figure 1b shows that the performance of the tile model is
degraded for forest and snow as a result of the large variation in stabilily
across the surface; the magnitudes of the fluxes over the snow tile are
underestimated, giving an overestimate of the average sensible heat flux and an
underestimate of the average latent heat flux (Table 2). Nevertheless, the tile model captures the change in
stability between forest and snow, and still estimates the average fluxes to
within about 10%. A slight improvement in the partitioning of the available
energy over the snow tile can be achieved by using windspeeds on a separate
reference height from temperatures and humidities (Essery, in press).

Table 2. Average sensible and latent heat fluxes (W m^{−2}) from the
boundary-layer model and the tile model

As a first step towards a GCM implementation, a tile model of heterogeneous
snow cover has been tested in a single-column version of the Hadley Centre GCM.
Unlike the boundary-layer model, the GCM uses a soil model to calculate
ground-heat fluxes. Figure 2 shows surface
temp-eratures and fluxes during one day for a model gridbox with 50%
snow-covered grass; the snow-free fraction is forested, as in the
boundary-layer model. The forest (dotted lines) absorbs much more radiation and
has a higher temperature than the snow (dashed lines). Latent heat fluxes from
the snow and the forest are comparable, although the evaporation peaks later in
the day over the forest than over the snow. The sensible heat flux over the
snow is small and downward throughout the day, but there is a large, upward
flux from the forest around midday that dominates the gridbox-average sensible
heat flux (thin solid line). The ground heat flux is very small (less than 3 W
m^{−2}), when averaged over 24 hours, but gives a fairly strong
warming for the forest fraction during the day and cooling at night.

Fig. 2. Gridbox-average surface temperatures and fluxes obtained using a tile
model (thin lines) and effective surface parameters (thick lines) in a
single-column version of the Hadley Centre GCM. Dotted and dashed
lines show values for forest and snow tiles respectively.

Most current GCM land-surface schemes do not calculate separate fluxes for
snow-covered and snow-free fractions of a gridbox — CLASS (Verseghy, 1991) is an exception — but
instead use “effective” surface parameters to relate gridbox-average fluxes to
gridbox-average gradient. Thick lines on Figure
2 show results obtained from the single-column model with the
following parameter choices. Assuming the snow to be uniformly distributed
across the gridbox. the surface resistance is set to zero. Simply using the
average albedo (α = 0.5) gives the correct average net surface shortwave
radiation. The roughness length is set to the log average of the local
roughness lengths (*z*_{0}
= 0.1 m), the effective value predicted by linear theory for small
variations in roughness. The effective parameter model gives nearly the same
surface temperature and net radiation as the tile model, but the partitioning
of the available energy is changed; the peak latent heat flux is higher and the
peak sensible heat flux is lower.

## Conclusions

Although the tile model performs less well in situations with large sub-grid
variations of stability, it still gives reasonable estimates of average and
local fluxes over heterogeneous snow cover. However, the tile model has only
been assessed in comparison with a boundary-layer model that uses the same
surface flux parameterizations, and tile model fluxes have not been compared
with fluxes measured over real surfaces.

The explicit representation of heterogeneities in snow cover is likely to have
large impacts on GCM surface fluxes. For a complete parameterization of
heterogeneous snow-cover, the problem of determining a closure relating
fractional snow coverage to surface characteristics and average snow depth
remains.

## Acknowledgements

N. Wood supplied the boundary-layer model. P. Rowntree, J. King and an
anonymous referee made a number of useful comments in reviewing this paper.
This work was supported by the U.K. Department of Environment.

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