Local-scale advection of energy from warm snow-free surfaces to cold snow-covered surfaces is an important component of the energy balance during snow-cover depletion. Unfortunately, this process is difficult to quantify in one-dimensional snowmelt models. This paper proposes a simple sensible and latent heat advection model for snowmelt situations that can be readily coupled to one-dimensional energy balance snowmelt models. An existing advection parameterization was coupled to a conceptual frozen soil infiltration surface water retention model to estimate the areal average sensible and latent heat advection contributions to snowmelt. The proposed model compared well with observations of latent and sensible heat advection, providing confidence in the process parameterizations and the assumptions applied. Snow-covered area observations from unmanned aerial vehicle imagery were used to update and evaluate the scaling properties of snow patch area distribution and lengths. Model dynamics and snowmelt implications were explored within an idealized modelling experiment, by coupling to a one-dimensional energy balance snowmelt model. Dry, snow-free surfaces were associated with advection of dry air that compensated for positive sensible heat advection fluxes and so limited the net influence of advection on snowmelt. Latent and sensible heat advection fluxes both contributed positive fluxes to snow when snow-free surfaces were wet and enhanced net advection contributions to snowmelt. The increased net advection fluxes from wet surfaces typically develop towards the end of snowmelt and offset decreases in the one-dimensional areal average melt energy that declines with snow-covered area. The new model can be readily incorporated into existing one-dimensional snowmelt hydrology and land surface scheme models and will foster improvements in snowmelt understanding and predictions.

Sensible and latent turbulent heat fluxes contributing to snowmelt are complicated during snow-covered area (SCA) depletion by the lateral redistribution of energy from snow-free surfaces to snow. Unfortunately, many calculations of the snow surface energy balance have largely been limited to one-dimensional model frameworks (Brun et al., 1989; Gray and Landine, 1988; Jordan, 1991; Lehning et al., 1999; Marks et al., 1999) which simulate melt at points without considering variations in SCA. Despite the sophistication of these methods, they have neglected local-scale advection of energy.

The differences in surface energetics between snow-covered and snow-free areas lead to a heterogeneous distribution of surface temperature and near-surface water vapour. These horizontal gradients drive a lateral exchange of heat (sensible heat advection) and water vapour (latent heat advection when considering the induced condensation or sublimation) over the leading edge of a snow patch. Advection contributions to snowmelt are not negligible as sensible heat advection has been estimated to account for up to 55 % of the snowmelt energy balance (Granger and Male, 1978), resulting in areal melt rates being greatest when snow cover is between 40 % and 60 % (Shook, 1995; Marsh et al., 1997). Advection is very challenging to directly observe due to the dynamic nature of snow-cover ablation. Direct observations of the melt implications of advection have utilized repeat terrestrial laser scanning to identify and quantify larger melt rates on the leading edge of snow patches (Mott et al., 2011). The development of internal boundary layers as air flows over heterogenous surfaces provides an alternate approach to measure the advection energy flux directly (Garratt, 1990). Measurements of these internal boundary layers across snow surface transitions reflect established power laws of boundary layer height (Shook, 1995; Granger et al., 2006) and can quantify the latent and sensible heat advection through boundary layer integration (Granger et al., 2002, 2006; Harder et al., 2017). In contrast to these findings the formation of boundary layers has also been identified as a potential cause of atmospheric decoupling of the atmosphere from the snow surface, leading to the suppression of sensible heat advection in the Alps (Mott et al., 2013, 2016, 2017). The reader is referred to Fujita et al. (2010), Granger et al.(2002, 2006), Mott et al. (2013, 2016, 2017) and Sauter and Galos (2016) for discussions on the complexities of boundary layer development over snow during advection situations and how this may influence overall energy exchange. Due to the complexity of the process and difficulties in observation, modelling has been the focus of much more work on this topic.

To model advection to snow patches, Weisman (1977) applied
mixing length theory, implicitly accounting for both latent heat advection
(LE

The major obstacle to the development of an energy balance model for calculating melt quantities is the lack of reliable methods for evaluating the sensible heat flux. A priority research need is the development of “bulk methodologies” for calculating this term, especially for patchy, snow-cover conditions.

Subsequent models have had variable complexity. Marsh and Pomeroy (1996) estimated areal averageThere remains a pressing need for an approach that can estimate areal
average

The methodology to address the research objectives is briefly outlined here.
A conceptual and quantitative model framework extended the Granger et
al. (2002) advection model, hereafter referred to as the extended GM2002, to
also consider LE

Horizontal gradients of scalar properties are a first-order control on the
advection flux. For snowmelt the gradients are conceptualized as snow-free
surfaces upwind of a transition to a snow-covered surface. During melt
periods upwind snow-free surfaces are typically comprised of dry soil and/or
ponded water which correspond to warm dry and/or warm moist near-surface air
properties, respectively. In contrast snow is commonly assumed to be

To scale any estimate of fetch length advection to an areal average
representation the geometric properties and extent of exchange are needed.
Over the course of melt, SCA declines from completely snow-covered to
snow-free conditions, with the intermediate periods defined by a
heterogeneous blend of both. Conceptually the advection of energy to snow
therefore is bounded by the areas of snow-free and snow-covered surfaces
that constrain energy transfer. Initial advection contributions to melt are
dominated by energy advecting from emerging snow-free patches to the
surrounding snow (Fig. 2a). The total amount of energy advected will be
limited by the smaller snow-free surface source area available to exchange
energy; all energy entrained by air movement across isolated snow-free
patches will be completely advected to the surrounding snow surfaces. At the
end of snowmelt, snow patches remain in a snow-free domain, and some energy
is advected from the warm surrounding snow-free surface to isolated snow
patches (Fig. 2b). The amount of energy advected is limited by the smaller
snow surface area available to exchange energy. When the snow surface is the
most heterogeneous, with a complex mixture of snow and snow-free patches,
advection occurs between isolated snow-free patches, surrounding snow cover,
snow-free surfaces, and isolated snow patches at the same time. Conceptually
there will be a gradual transition from isolated snow-free patch to isolated
snow patch advection constraints. Marsh and
Pomeroy (1996) and Shook et al. (1993b) found that magnitude of the snowmelt
advection flux will be greatest when SCA is 40 %–60 % and this range was
used to bound the transition of advection constraints. The advection
mechanism transitions over the course of the melt and was conceptually
related to SCA by a fractional source (

Conceptual model of advection dynamics for

During snowmelt, meltwater may infiltrate into the frozen soil and any
excess will pond prior to and during the runoff phase; these interactions
will influence the near-surface humidity of the snow-free surface. Thus,
LE

Parameterizations for extended GM2002.

Granger et al. (2002) developed a simplified approach
to estimate the advection over a surface transition from boundary layer
integration. Advected energy,

The humidity of the air at the surface interface is rarely observed but is
needed to quantify the LE

To estimate

To relate

Conceptual water area–volume relationship diagram where a cross section of land surface microtopography (brown is soil and blue is water) is assumed to follow a sinusoidal profile.

The SCA constrains the overall exchange of energy between the snow surface
and the atmosphere. Essery and Pomeroy (2004) developed an
SCA parameterization from the closed form fit to the parametric SCA
curve produced by homogeneous melt of a log-normal SWE distribution,

Perimeter–area relationships and patch area distributions of snow and
snow-free patches show fractal characteristics that can be exploited to
simplify the representation of snow-cover geometry needed to calculate
advection. There are two commonly used scaling relationships. From
application of Korcak's law by Shook et al. (1993a) the
fraction of snow patches greater than a given area,

The relationships of Eqs. (10) and (11) were exploited to develop a
probability distribution of

Probability of patch size occurrence and its transformation to
fractional area patch sizes for a range in patch sizes from 1 to
1000 m

Using the above-described parameterizations of

The coefficients for the snow-cover geometry relationships are based on
oblique terrestrial photography or aerial photography with coarse resolution
and limited temporal sampling (Shook et al., 1993b).
Recent advances in UAV technologies provide a tool to re-evaluate these
relationships with georectified high-resolution imagery. During the 2015 and
2016 snowmelt seasons, 0.035 m

The influence of the advection model upon snowmelt dynamics was explored
with two approaches. The first approach is a scenario and sensitivity
analysis where inputs are fixed and a selection of process parameterizations
are employed to illustrate the relationship between

Example of snow-cover geometry scaling properties, exceedance
faction versus patch area (

To explore the dynamics of modelled advection contributions several
scenarios were implemented with the model. The first scenario (No Advection)
constitutes a baseline for a typical one-dimensional model that assumes no
advection, the second (dry surface) includes advection from a warm dry surface, the third (wet surface) includes advection from a warm wet surface,
and the fourth (dry to wet surface) includes advection from a warm surface
that transitions from dry to wet as a function of the
INF–

The sensitivity of SLHAM to

Input variables for scenario analysis of SLHAM dynamics.

Conditions controlling advection processes are not constant over snowmelt;
therefore SLHAM was coupled with a one-dimensional snowmelt model (SSAM) to
estimate the role of advection contributions over a snowmelt season.
Briefly, SSAM describes the relationships between shortwave, longwave and
turbulent exchanges between a snow surface underlying exposed crop stubble
and the atmosphere. The surface energy balance was coupled to a single-layer
snow model to estimate snowmelt. A slight modification of SSAM, or any
one-dimensional model that computes areal average snowmelt, is needed to
include advection. The energy terms of one-dimensional energy balance models
are represented as flux densities (W m

The SSAM, SSAM–SLHAM and EBSM simulations were driven by common observed
meteorological data, parameters and initial conditions obtained from
intensive field campaigns at a research site near Rosthern, Saskatchewan,
Canada (52.69

The extended GM2002 proposed here was tested using advection estimates from
vertical air temperature and water vapour profiles as reported in Harder et al. (2017); the results are summarized in
Table 3. The model slightly overestimated

Model parameters, estimates and observations for evaluation of the extended GM2002.

Differences exist between the originally reported parameters and those found
from the analysis of UAV imagery (mean coefficients summarized in Table 4).
Early work applying fractal geometry to natural phenomena (Mandelbrot, 1975,
1982) discusses the Korcak exponent as a fractal dimension. More recent work
suggests that the Korcak law describing the area–frequency relationship is
not a fractal relationship but rather a mathematically similar, but distinct,
scaling law (Imre and Novotn, 2016). Therefore, the

Updated mean snow-cover geometry parameters.

Time series of fitted

Time series of fitted

The dynamics of the various scenarios are expressed through visualizations
of SWE depletion (Fig. 8) and magnitudes of the

Modelled snow water equivalent depletion for various advection scenarios.

Latent heat (green), sensible heat (red) and net (blue) advection components for the SLHAM scenarios plotted with snow-covered area (black).

Sensitivity of snow water equivalent and snow-covered area depletion, ponded water fraction, sensible heat advection, latent heat advection and net advection with respect to variation in water surface temperature.

The influence of the input variables on the SLHAM model is evaluated through
a sensitivity analysis (Fig. 10). It is apparent from the variability in
SWE depletion that the

Sensitivity to any variable is only expressed towards the end of the
snowmelt, when SWE

The scenario analysis demonstrates the melt response to variations in surface wetness but actual snowmelt situations have forcings that vary diurnally and with meteorological conditions. Snowmelt simulations with three models of varying complexity provide insight into the implications of process representation. SSAM and SSAM–SLHAM show considerable improvement when compared to EBSM (Fig. 11 and Table 5). The SSAM simulation is by itself a significant improvement upon EBSM for SWE prediction during melt. The addition of SLHAM does not change the SWE simulation performance appreciably but does increase the physical realism of the model with its more complete surface energy balance. The SSAM–SLHAM simulations including advection, relative to SSAM simulations without advection, led to lower areal average melt rates in 2015 and higher rates in 2016. Lower wind speeds in 2015 led to lower advection contributions than 2016, which had relatively higher wind speeds. The comparison of the simulated melt with snow survey SWE observations showed that the differences are minimal (Fig. 11 and Table 5). While the SSAM–SLHAM simulations do not change melt rates or total amount of energy, the sources of energy driving snowmelt do change. Early melt displays no differences as SCA remains relatively homogenous. Differences appear due to decreases in the turbulent radiation fluxes, with a decrease in the SCA exchange surface due to increases in advection fluxes with increasing horizontal scalar gradients and surface heterogeneity. The cumulative net energy from advection for these two seasons contributed energy to melt 4 and 5 mm of SWE in 2015 and 2016, respectively (Fig. 12). The advection energy contribution represents 6.5 % and 10.6 % of total snowmelt in 2015 and 2016, respectively.

Snow water equivalent simulation for EBSM (red line), SSAM (green line) and SSAM–SLHAM (blue line) with respect to snow survey mean (black points) and 95 % percentile sampling confidence interval (black lines).

Error metrics of snow water equivalent simulation versus snow survey observations for EBSM, SSAM and SSAM–SLHAM models.

An unappreciated dynamic of local-scale advection during snowmelt is that
LE

Cumulative sensible (red), latent (green) and net (blue) advection terms in terms of energy (MJ: left) and equivalent melted snow water equivalent (mm SWE: right axis).

The advection fluxes may also be of opposite sign to the sensible
(

Cumulative energy from sensible, latent and net exchange for 2015 and 2016 snowmelt simulations with (SSAM–SLHAM) and without (SSAM) advection.

The simulation of snowmelt with, and without, advection gave minimal
differences in the resulting SWE simulation. This demonstrates system
insensitivity to processes that on their own appear to be important. This
may explain why EBSM, like many other physically based snowmelt models
(Jordan, 1991; Lehning et al., 1999; Marks et
al., 1998), does not accommodate heterogeneous snow cover yet successfully
simulates SWE depletion. In EBSM the simulation of an areal average albedo
rather than a snow albedo performed relatively well in simulating SWE
(Fig. 11) without considering SCA depletion or advection controls. The
modelling challenges of

Understanding the implications of land-use and climate changes on variables
beyond SWE are needed to fully inform coupled modelling of land–atmosphere
and radiation feedbacks between land surface and numerical weather or
climate models. The framework presented explicitly considers advection and
scales it with SCA,

The SLHAM framework replaces the large uncertainty deriving from physically
unrealistic albedo parametrizations (Gray and Landine, 1987; Raleigh et al.,
2016) and ignored SCA dynamics (Essery and Pomeroy, 2004) with a more
physically realistic framework. The individual process parametrizations still
have uncertainties that need to be constrained. The advection versus patch
length parametrization of GM2002 lacks inclusion of surface roughness
differences and the valid bounds of the parametrizations need clarification.
Observations of stable atmospheric profiles over snow patches (Fujita et al.,
2010; Mott et al., 2015, 2016; Shook and Gray, 1997) complicate energy
exchange. The goal of this simple model was to develop an easy-to-implement
advection framework with stability represented by the Weisman (1977)
stability parameters. Future work will need to revaluate the stability
assumptions of Granger et al. (2002) and Weisman (1977) or devise more
appropriate schemes to account for the stability influence. The SCA model of
Essery and Pomeroy (2004) is challenged by exposure of vegetation in shallow
snow. The conceptual surface water ponding model developed in this work
requires field observations or further parameterizations to accurately
quantify the relevant variables. The transition of advection mechanism from
snow-free sources to snow patch sources uses a conceptualized relationship to
SCA. A targeted field campaign is needed to assess the validity of the
conceptualized

To date, the development of easily implementable and appropriate models to
estimate the advection of

The data and code discussed in this paper are available through the corresponding author, Phillip Harder (phillip.harder@usask.ca).

Funding was provided by the Natural Sciences and Engineering Research Council of Canada through discovery grants, research tools and Instruments and the Changing Cold Regions Network and the Canada Research Chairs programme. Field and technical assistance from Bruce Johnson, Chris Marsh, Kevin Shook and Michael Schirmer is gratefully acknowledged. This work would not have been possible without the cooperation of Nathan Janzen and Robert Regehr, the farmers who accommodated the intensive field campaigns. Edited by: Sean Carey Reviewed by: Rebecca Mott and Richard L. H. Essery