Lake models are increasingly being incorporated into global and
regional climate and numerical weather prediction systems. Lakes interact
with their surroundings through flux exchange at their bottom sediments and
with the atmosphere at the surface, and these linkages must be well
represented in fully coupled prognostic systems in order to completely
elucidate the role of lakes in the climate system. In this study schemes for
the inclusion of wind sheltering and sediment heat flux simple enough to be
included in any 1-D lake model are presented. Example
simulations with the Canadian Small Lake Model show improvements in surface-wind-driven mixing and temperature in summer and a reduction of the bias in
the change in heat content under ice compared with a published simulation
based on an earlier version of the model.
Introduction
The surface roughness of a small lake or reservoir nearly always contrasts
sharply with that of its terrestrial surroundings and will thus be
associated with a different atmospheric boundary layer than would be found
on shore. The new boundary layer does not develop instantaneously over the
entire lake: as the atmospheric flow encounters a sudden change (generally a
decrease) in roughness at the shoreline, an internal boundary layer (IBL)
develops with a transition region whose properties, including key lake
mixing parameters like wind stress, vary with fetch. In fact, for
sufficiently small lakes this transition region might occupy the entire
surface area: wind stress varies with downstream distance and an equilibrium
boundary layer never forms. In addition, for elongated lakes and reservoirs,
the net response to wind forcing will vary depending on whether the wind is
along or across the primary axis. The impact of fetch-varying surface
wind speed due to the presence of IBLs on lake evaporation (e.g. Venäläinen et al., 1998) and gas flux (e.g. Kwan and Taylor, 1994) has been considered, but the
effect of a sudden change in aerodynamic roughness on lake mixing does not
appear to have been thoroughly examined in the literature.
On the other hand, the relationship between lake area and mixing has been
investigated. Mazumder and Taylor (1994) related both lake size and water
clarity to epilimnion depth and found linear relationships between
epilimnion depth and log fetch (where fetch is taken as the square root of
lake surface area) for different transparency classes. Fee et al. (1996)
also investigated the effects of lake size and water clarity on mixed layer
depths and found that lake size was the more important factor for
determining mixed layer depth in a set of Canadian shield lakes, though
transparency modulated this response for smaller lakes. The influence of
fetch on epilimnion depth cannot be unambiguously attributed to variations
in surface stress, since lake size influences seiching and thus shear
production of turbulence at the thermocline, which can also contribute to
mixed layer deepening (e.g. Gorham and Boyce, 1989), but these studies do hint at
the possible importance of IBLs and fetch-varying surface stress for lake
hydrodynamics. Most 1-D lake models assume that turbulent flux
exchange with the atmosphere (heat and moisture as well as momentum) takes
place under equilibrium conditions such as is assumed, for example, when
employing Monin–Obukhov similarity theory.
Another physical process, particularly relevant for ice-covered lakes, is
the sediment heat flux. This flux arises because lake sediment, especially
in shallow littoral zones, warms under the influence of penetrating
radiation (as well as thermal contact with warm water) during summer and
releases this heat during the ice-cover season. Rizk et al. (2014) report results
from numerous studies that found sediment heat fluxes of the order of a few
Wm-2, generally larger during early winter and tapering off as winter
progresses. This is insignificant compared with other energy fluxes during
the open water season and probably within the observational uncertainty of
meteorological forcing (not to mention the uncertainty in process
parameterization) in numerical modelling studies. During the ice-cover
season, however, energy fluxes into the lake are small, and this source can
become important for many applications. For example, sediment heat fluxes
have been linked to basin-scale circulations in ice-covered lakes (Kirillin
et al., 2012; Rizk et al., 2014), which can strongly influence the thermal structure and
distribution of dissolved oxygen and nutrients in deep waters. The impact of
sediment heat flux on surface conditions such as ice thickness and ice
phenology is generally assumed small, except perhaps for very shallow lakes,
though this has not been thoroughly examined in the literature.
The Canadian Small Lake Model (CSLM; MacKay, 2012; MacKay et al., 2017) is a
1-D thermodynamic lake scheme developed for coupling within global
and regional climate and numerical weather prediction systems. This model
computes the complete nonlinear surface energy balance in a thin layer (5 cm) and then solves the heat equation on a uniform finite-difference grid
throughout the column. Shortwave radiation extinction follows Beer's law
for both visible and near-infrared bands. A diurnal surface mixed layer
is simulated based on an integrated turbulent kinetic energy (TKE) approach,
with a variety of well-known sources and sinks of TKE parameterized. The
seasonal thermocline arises naturally as a result of the daily excursions of
the surface mixed layer. Congelation (i.e. black) ice forms when the energy
balance in a layer is sufficiently negative to cool it below 0 ∘C. Snow
(i.e. white) ice forms when the weight of the overlying snowpack is sufficient
to crack the ice and allow lake water to flood a layer of snow. The snow
itself is represented with the complete snowpack parameterization component
of the Canadian Land Surface Scheme (Verseghy and MacKay, 2017). Both
fractional ice cover and fractional snow on ice are permitted. The
current formulation of the model does not account for wind sheltering or
sediment heat fluxes. To begin to address these shortcomings, this study
proposes two new schemes which are simple and flexible enough to be easily
incorporated into any 1-D lake model. Below these schemes are fully
developed from theoretical considerations and some example CSLM simulations
demonstrating their impacts are presented.
Fetch-varying wind stress and mixing in small lakes
In the analysis that follows we consider for simplicity the case of neutral
atmospheric stability with no surface heat or vapour flux. Under steady-state conditions over horizontally homogeneous surfaces we expect the wind
profile to be given by the familiar logarithmic form:
u‾(z)=u∗klnzz0,
where u‾ is the mean wind profile, k is the von Karman constant,
z0 is the surface roughness, and u∗ is the surface friction
velocity of the air, which is related to surface stress by τ=ρu∗2, where ρ is the density of air. Thus upstream of a
shoreline over a (likely vegetated or urbanized) terrestrial landscape we
find (adopting the notation of Jensen, 1978)
u‾(z)=u-klnzz-,
and far downstream of the shoreline under equilibrium conditions over a very
large lake we expect
u‾(z)=u+klnzz+,
where the - and + subscripts refer to upstream and far downstream quantities
respectively. In what follows the ratio of the upstream-to-downstream
roughness lengths, characterized by
M=lnz-z+,
is an important parameter governing the response of the surface stress due
to the sudden change in surface roughness at the shoreline. Over lakes the
roughness elements are provided by surface waves, and thus roughness is not
static but rather a function of wind speed; nevertheless, we shall consider
it O10-3m in what follows. Common terrestrial surfaces
are forest (z-=1.0m nominally), shrubland (z-=10-1m nominally),
and grassland (z-=10-2m nominally), which yield M=6.9, 4.6, and 2.3 respectively.
The development of IBLs due to discontinuities in surface roughness over
rigid surfaces has been studied for many years. This might not have been the
case: as pointed out by Jensen (1978), in the planetary boundary layer the
downstream equilibrium surface stress (τ+=ρu+2) under
realistic conditions is different from the upstream surface stress (τ-=ρu-2) by at the very most a factor of 3. If the
transition from upstream to downstream conditions were simply monotonic,
then Jensen speculates that there would be little interest in the
development of IBLs. For most purposes a simple interpolation rule could be
developed to distribute wind stress over a surface, if one were to bother at
all. However, experience shows that this transition is not so simple. For
atmospheric flow from a rough to a smooth surface, both experimental and
theoretical results indicate that surface stress drops suddenly across the
roughness transition before asymptotically approaching its new equilibrium
value (e.g. Garratt, 1990). On the other hand, for flow from a smooth surface to
a rough surface, the surface stress initially “overshoots” the final
equilibrium value before slowly asymptoting to the final value, though this
situation seems less relevant for our purposes as most lakes are situated
within environments that are usually rougher than the water surface.
Theoretical IBL depths (a) and surface stress (b) as a function of
fetch for the rough-to-smooth transition case of Bradley (1968). Solid
curves – Panofsky and Townsend (1964) approach; dashed curves –
Jensen (1978) approach with A=1 (thin dash) and A=2 (thick dash) as indicated.
Error bars in (b) represent the range of observed values from Bradley
(1968), and surface stress values are normalized by the upstream value.
Panofsky and Townsend (1964) developed a simple, approximate analytical
model describing this phenomenon, and Bradley (1968) showed that this model
qualitatively captures the observed behaviour over the surfaces he examined,
though he did find that the observed transition region was smaller than
predicted. Panofsky and Townsend solve for a surface stress parameter given
by (again adopting the notation of Jensen, 1978)
S=u--u0/u-,
where u0=u0(x) is the surface friction velocity (of air) and is a
function of fetch, x, with x= 0 representing the location of the roughness
discontinuity (in our case the shoreline). Panofsky and Townsend argue that
S can be approximated by
S≈Mlndz+-1,
where d is the depth of the IBL. They develop a relation between boundary
layer depth d and fetch which can be solved iteratively and substituted into
Eq. (2) to solve for S as a function of fetch. From Eq. (1) we can then
evaluate the ratio of the over-lake wind stress to the upstream terrestrial
surface wind stress as
τ0τ-=1-S2.
Jensen (1978) suggests a different approach to this problem. He argues that the
depth of the IBL can be solved from
dz+lndz+-1=Axz+-1,
for some constant A, and that the wind stress ratio becomes
τ0τ-=1-S′2,
where
S′=Mlndz+.
Before examining the results of these models for forest, shrubland, and
grassland environments, we first revisit the experiments of Bradley (1968)
in order to gain some insight into the models. Bradley examined the flow
transition from a rough surface (created with a mesh of wire spikes: z-=2.5×10-3 m) to a smooth surface (runway tarmac embedded with sand
and small pebbles: z+=2.0×10-5 m), and vice versa, in an experiment at
an airfield in New South Wales, Australia, yielding M=4.8 –
approximately the value for our shrubland environment. IBL depths and stress
ratios τ0/τ- as a function of fetch are indicated in
Fig. 1 for the Panofsky–Townsend model (solid) and Jensen (A=1 thin
dash; A=2 thick dash) models. Jensen (1978) reports that for the smooth-to-rough transition case, setting A=1 reproduces the observed Bradley data
very well and yields results that are nearly identical to the theoretical
approach of Rao et al. (1974) which is based on a second-order turbulence
closure scheme. However, Fig. 1 suggests that for the rough-to-smooth case,
setting A=2 may be equally appropriate based on the limited data from
Bradley (the maximum fetch was only about 12 m). In either case the results from
Jensen appear to better represent the observed data than does the Panofsky–Townsend approach. In addition, the free parameter A could be adjusted
should more observed data become available, giving this approach some
appeal.
The final downstream equilibrium surface stress τ+ (or
equivalently u+) is not discussed by Panofsky and Townsend, but Jensen
finds that
τ+τ-=1-MlnRo2,
where Ro is the downstream surface Rossby number given by
Ro=Gfz+.G is the geostrophic wind speed and f is the Coriolis parameter. Thus for a
given value of Ro, Eq. (6) can be used to recast results of the Panofsky–Townsend and Jensen models in terms of the approach to equilibrium of
surface stress (τ0/τ+) by simply dividing Eqs. (3) or
(5) by Eq. (6). For midlatitude lakes, we take f=10-4 s-1
and G=10 ms-1. Jensen's primary interest was the planetary boundary,
and he considered equilibrium achieved when the IBL filled the entire PBL.
As indicated below, this is not an appropriate equilibrium value even for
midlatitude lakes, and fails entirely in the tropics.
Results for lakes within forest (M=6.9; black curves), shrubland (M=4.6;
blue curves), and grassland (M=2.3; red curves) environments are shown in
Fig. 2. Figure 2a highlights the fact that IBL depth is not a function of M in
the Jensen model (dashed curves), though it is obviously a strong function of
A. Observed IBL depth data would help constrain the value of A and would be
much easier to measure than surface stress – especially over open water for
a sufficient range in fetch. Figure 2b shows that surface stress values are
far from the equilibria proposed by Jensen (thin horizontal lines), but for
all three surface types the surface stress approaches an asymptotic value at
fetches around 5 km or less.
Theoretical IBL depths (a) and surface stress (b) as a function of
fetch over water for a rough-to-smooth transition (offshore flow) for
forested landscapes (black curves, M=6.9), shrubland (blue curves,
M=4.6), and grassland (red curves, M=2.3). Solid curves – Panofsky and
Townsend (1964) approach; dashed curves – Jensen (1978) approach with A=1 (thin dashed) and A=2 (thick dashed). Surface stress values are
normalized by the upstream value. Thin horizontal lines represent
equilibrium downstream values proposed by Jensen (1978) for downstream
surface Rossby number Ro=108.
The problem at hand is to estimate a mean surface wind stress value for
small lakes, given that they are embedded within terrestrial environments of
vastly different surface roughness and will thus be subject to an internal
boundary layer that is almost certainly not in equilibrium with the lake
surface. Figure 2 shows that the actual surface stress τ0 is a
function of fetch and always less than the equilibrium value τ+.
We have chosen the Jensen model with A=2 and a land cover of forest with
M=6.9. A reasonable approach to estimate a lake-averaged surface wind
stress is then the following.
Iteratively solve Eq. (4) for IBL depth d.
Solve Eq. (5) for the stress ratio τ0/τ-.
Compute the average stress ratio over the mean fetch of the lake. The mean
fetch can be taken as the square root of the surface area. This is exact for
circular lakes but becomes progressively inaccurate as lakes become
elongated and wind direction becomes important.
Divide this mean value by asymptotic value τ+/τ-. This is easily read from Fig. 2b or can be computed in the model by
setting a threshold (e.g. 1 %) which the change in the stress ratio with fetch
does not exceed. This yields the required stress reduction factor τ0/τ+, i.e. the reduction in mean surface stress the lake
experiences compared to the equilibrium surface layer estimate based on
Monin–Obukhov similarity theory.
It is worth pointing out that Vickers and Mahrt (1997) found that surface
drag decreased with fetch, because younger, growing waves are associated with larger
surface drag than older waves (unless they are breaking). In reality, both
processes may be occurring, but it is clear that for the smallest lakes the
mechanism described here will dominate. Including the impact of wave state
on the results reported here is certainly beyond the scope of the present
study. Stepanenko et al. (2014) have proposed that excessive drag in the LAKE 1-D model
for a simulation of Lake Valkea–Kotinen, a 4.1 ha boreal lake in
southern Finland, can be compensated for by partitioning atmospheric
momentum flux between wave development and surface currents. Our results
provide an alternative (or perhaps additional) explanation. Fetch in this
lake varies from about 100 to 400 m, and the lake is surrounded by forest.
Examination of Fig. 2 suggests that surface stress (or equivalently the
surface drag coefficient) is only a fraction of the equilibrium value:
perhaps between 25 % and 50 %, depending on the model chosen.
We have applied this technique to a simulation of the CSLM for a small
boreal lake (L239 at the Experimental Lakes Area, Canada) for 2013–2014
that has been described in MacKay et al. (2017). In that study the primary focus
was on the winter season, though the simulation actually began in July. The
first few days of this simulation (17–25 July 2013) are illustrated in
Fig. 3, where observations are indicated with black curves, the original
simulation with blue curves, and a modified simulation with red curves. Both
19 and 22 July were relatively windy days (Fig. 3a), with winds
generally from the west or north-west (not shown) yielding a maximum fetch
around 500 m. Based on the technique outlined above, we compute a drag
reduction factor of 0.5 in the modified simulation. The impact has been an
improved simulation of surface (i.e. 0.5 m depth) temperature (Fig. 3b) caused by
a reduction in simulated depth of turbulent mixing (Fig. 3c), especially
during or shortly following the wind events. A few observed and simulated
temperature profiles during this period are shown in Fig. 3d, e.
Observations suggest that between 19 and 20 July the epilimnion cooled from
24.7 to 23.4 ∘C and deepened from 3.0 to 3.8 m. On 19 July
both simulations and the observations show (Fig. 3d) identical epilimnion
temperatures and depths (the simulations were initialized just 2 days
earlier). However, by 20 July the original simulation (blue curves) indicates
excessive deepening and cooling (4.5 m, 22.5 ∘C) compared with the
modified simulation (4.0 m, 23.2 ∘C). Note that by 23 July (Fig. 3e),
even though both simulations now show good agreement with the observed
epilimnion temperature, an error of 0.5 m in the original simulated
epilimnion depth persists.
Observed 2 m wind speed (a), observed and simulated 0.5 m water
temperatures (b), and simulated diurnal turbulent mixed layer depths (c) for
17–25 July 2013. Temperature profiles (midnight) for (d) 19 July (dash) and
20 July (solid); and (e) 21 July (dash) and 23 July (solid). Standard simulation
results (blue curves) taken from MacKay et al. (2017). Modified simulation results
(red curves) have a surface drag coefficient reduced by 50 %. Observed
curves are black.
With this approach we must keep in mind a number of caveats. Very large
changes in roughness or very dense canopies are strictly speaking not
consistently handled as no allowance has been made for changes in the zero
plane displacement. Recent wind tunnel (Markfort et al., 2014), large-eddy
simulation (Schlegel et al., 2015), and field (Detto et al., 2008) experiments all show
complex vertical and horizontal structure in the Reynolds stress resulting
from streamline displacement and the turbulent wake at a canopy edge.
Markfort et al. (2014) showed that flow separation and a shallow recirculating zone
can develop in the immediate lee of the forest edge, with flow reattachment
some distance downstream – reminiscent of the well-known circulation
pattern for a backward-facing step (e.g. Driver and Seegmiller, 1985). If one
accounts for this reattachment distance (i.e. the origin in Fig. 2 is at the
point of reattachment rather than the shoreline), then the empirical model
for surface stress of Markfort et al. (2014) (their Fig. 16) appears similar to that
proposed here. However, both studies consider only neutrally stratified
boundary layer conditions. Unstable conditions such as would occur in autumn
and early winter before ice-on would almost certainly have an impact on
the turbulent canopy wake and the distribution of Reynolds stress. This is
currently an active area of research. In addition, unlike for rigid surfaces,
steady wind over open water will generally lead to an increase in surface
roughness with time. Note that computing the surface drag coefficient based
on lake surface roughness values of 10-2 and 10-4 m changes the
stress reduction factors to 0.59 and 0.45 respectively, similar to the factor
of 0.5 proposed here. All of these issues could likely be incorporated into
the theory at the expense of increasing complexity; nevertheless, this
approach should at least qualitatively describe the importance of IBLs and
fetch-varying wind stress over small lakes and appears a suitable first step.
Sediment heat flux
Currently the CSLM employs an adiabatic boundary condition at the lake bottom.
In the recent simulation of L239 mentioned above, it was found that during a
100 d ice-covered period (2013–2014) the simulated change in lake heat
content corresponded to a mean bias of about -2.5 W m2 compared to
observations (MacKay et al., 2017). The authors noted that this was of the same
order of magnitude as the wintertime sediment heat flux found in a number of
previous studies and suggested this warranted further investigation. Here we
test whether the inclusion of a simple sediment heat flux scheme can
ameliorate this bias.
A simple scheme for the storage and subsequent flux of heat from lake
sediments can be constructed by considering a sediment slab of fixed
thickness, and uniform thermal conductivity and volumetric heat capacity.
Such a scheme is simpler than some existing multilayer sediment models
(Stepanenko et al., 2016), but has the appeal of not requiring many additional
levels to be carried in global weather and climate simulations. Mean
temperature in the slab evolves due to thermal and radiative fluxes at the
lake water–sediment interface in such a way as to conserve energy. The
boundary condition at the base of the slab is isothermal, rather than
adiabatic as is sometimes assumed (e.g. Stepanenko et al., 2016), which places a
constraint on the minimum slab thickness. Note that for terrestrial (i.e. soil or
rock) surfaces the diurnal temperature wave is believed to penetrate about 1 m and the annual temperature wave penetrates about 20 m (e.g. Carslaw and
Jaeger, 1959), below which geothermal heating acts to maintain a constant
temperature gradient. A number of studies have found that temperature
oscillations in lake sediments are substantially damped after only a few meters,
below which temperatures are essentially isothermal (e.g. Likens and Johnson,
1969; Tsay et al., 1992). Here we consider a slab thickness of 10 m, with thermal
properties consistent with sand. A layer of pure sand has a thermal
conductivity of about 2.5 W K-1 m-1 (about 4 times larger than
that for liquid water) and a volumetric heat capacity of about 2.13×106 J m-3 K-1 (about half that of liquid water), and these
values are adopted here. Values for clay would be similar, though sediments
with significant organic matter would differ. The system is solved by
asserting the continuity of both temperature and heat flux at the water–sediment interface (i.e. option 1 in Stepanenko et al., 2016) and a lower boundary
condition temperature of 6.0 ∘C.
The difficulty with applying such an approach here is that the CSLM does not
include lake bathymetry information, and lake bottom is everywhere assumed
to be at the mean lake depth. In fact, many lakes have extensive shallow
littoral zones in which we would expect much larger shortwave (SW)
insolation, and thus sediment heat flux, than in more “bathtub”-shaped
lakes which nevertheless have the same mean depth. Here we propose a scheme
to compute the net sediment SW insolation based on minimal bathymetric data
– maximum and mean depth only.
In what follows, all lower-case variables are dimensionless – scaled by
appropriate length and depth scales for the lake in question. Figure 4a shows
a family of one-parameter theoretical “hypsographs” given by
y=xs,
where x is the (dimensionless) radial distance from the lake center, y is the
(dimensionless) height from the lake bottom at maximum depth, and s is a
shape factor. Theoretical, axially symmetric lakes are formed by rotating
these curves around the line x=0. Thus for s=1 the lake is conical, for
values of s less than 1 the lake takes on a “birdbath” shape, while for
values of s greater than 1 it is more “bathtub” shaped. In the limit of
very large s the lake becomes a right circular cylinder (the default shape
assumed in CSLM). Birdbath lakes have extensive shallow littoral zones whose
sediment would absorb much more SW radiation than bathtub lakes of the same
surface area.
(a) Theoretical “hypsographs” for idealized axially symmetric
lakes with different shape factors s indicated; (b) schematic to aid in
determining the volume of an axially symmetric lake; (c) normalized observed
hypsograph (blue) and theoretical estimate (red) for L239. See text for
details.
Reference to Fig. 4b shows that the volume of revolution of a thin disk at
height y is
dv=πx2dy,
so that the total volume is simply
v=∫01πx2dy=∫01πy2/sdy=πss+2.
The dimensional volume is thus
V=πss+2L2Hmax,
where Hmax is the maximum depth and L is the radius of a circular lake
with the same surface area as the lake in question. We can determine the
value for s by equating this volume to the actual volume of the lake, or
equivalently,
πss+2L2Hmax=πL2H‾,
where H‾ is the mean lake depth. This yields
s=2H‾Hmax1-H‾Hmax.
For L239 we have H‾=11.0 m and Hmax=30.0 m, so that s=1.16.
Figure 4c compares the actual (normalized) hypsograph for L239 compared with
our theoretical curve.
Schematic to aid in the estimation of sediment shortwave insolation
for an axially symmetric lake with a given shape factor s<1.
The task now is to estimate the mean absorbed SW radiation at the
sediment–water interface given this lake shape profile. It is clear from
Fig. 5 that if i0 is the intensity of SW radiation reaching the lake
surface (per unit area), then the intensity reaching the sediment at radial
distance x is given by
isedx=i0cosθe-β1-xs,
where β is the (dimensionless) shortwave extinction (i.e. β=β^Hmax where β^ is the dimensional extinction) and
1-xs is the depth at x. Consider the surface area of a thin ring of radius x and width dl. Clearly
da=2πxdl,
so that the total radiation reaching this elemental surface is
isedxda=i0cosθe-β1-xs2πxdl=2πi0xe-β1-xsdx.
The total insolation of the lake sediment surface is found by integrating
over all x, and the mean sediment insolation per unit lake surface area is
found by dividing this integral by the lake surface area. Recalling that the
dimensionless lake surface area is simply π, we get
i‾sed=1π∫012πi0xe-β1-xsdx=i02e-β∫01xeβxsdx.
This has analytic solutions in terms of the gamma and incomplete gamma
functions:
i‾sed=2i0e-β1s-β-2sΓ2s-Γ2s,-β;s>0.
Note that for s=0 (i.e. a birdbath of zero depth everywhere except at the
center) the solution Eq. () does not apply, but Eq. () can be integrated
trivially to get
i‾seds=0=2i0e-β∫01xeβdx=i0,
which is the correct limit. On the other hand, for very large s our lake
becomes cylindrical and Eq. () can also be solved trivially. Expanding the
exponential in a Taylor series, we find
i‾sed,∞=s→∞lim2i0e-β∫01xeβxsdx=2i0e-βs→∞lim∫01x1+βxs+(βxs)22!+…dx=2i0e-βs→∞lim12x2+βs+2xs+2+β22(2s+2)x2s+2+…01=i0e-β,
which is again the correct limit.
For L239 we have found s=1.16 and the (dimensionless) extinction is β=27, so that Eq. () or Eq. () becomes
i‾sed=0.06i0.
In other words, of the net surface SW radiation on L239, about 6 % reaches
the sediments, leaving 94 % to be absorbed by the lake water on a lake-wide
average. In the standard simulation assuming the entire lake is at the mean
depth (11 m), the sediment insolation is only
i‾sed=5.0×10-5i0,
3 orders of magnitude less. The approach taken here is to simply reduce
the incoming net SW insolation at the lake surface by 6 % and apply this
energy flux directly at the lake water–sediment interface. In this way we
estimate the mean sediment insolation for any lake whose surface insolation
is known, along with extinction, mean and maximum depths. For example,
Table 1 lists sediment SW insolation fractions for a variety of shape factors s for lakes with the same (dimensionless) extinction as L239.
Sediment shortwave insolation fractions for lakes with
dimensionless shortwave extinction β=27 and various shape
factors s.
si‾sed/i00.10.430.50.131.00.072.00.0410.00.01
The above approach was used in two new simulations of L239 for 2013–2014
(Fig. 6). In the standard simulation for this period (MacKay et al., 2017) the lake
bottom was adiabatic. In order to isolate the impact of geothermal heating
alone, we relaxed the adiabatic boundary condition as described above, but set
the sediment SW insolation fraction to 0.0 (experiment X1; Fig. 6 blue
curves). The second experiment (experiment X2) is identical to the first,
except we set the sediment SW insolation fraction to the proposed value of
0.06 (Fig. 6, red curves). The simulated ice thicknesses (Fig. 6a) are
virtually identical in the two experiments, though both the sediment
temperatures (dotted curves) and lowest water layer temperatures (solid
curves) differ even well before ice-on (Fig. 6b). Note that the sediment
heat flux is close to zero or negative (i.e. into the water column) throughout
these simulations, with values never exceeding a few W m-2 (Fig. 6b,
dashed curves, right-hand scale). During the ice-cover season the geothermal
component is significant, generally about half of the total.
Finally, the impact of the new sediment heat flux scheme on lake thermal
structure during the ice-cover period is illustrated in Fig. 7. Ice
phenology and thickness for the original and modified simulations are
virtually identical (Fig. 7a). Temperature profiles in the original
experiment show no sign of water column warming (Fig. 7b) – in fact, a
general cooling trend is evident in the top 5 m until ablation begins (after
snow cover is gone – not shown). The observations on the other hand show
clear signs of warming throughout most of the column under ice (Fig. 7c).
Results from the sediment heat flux experiment (X2) more closely resemble
this pattern (Fig. 7d), though there are obvious deficiencies in the
simulation, including a more stratified region for several meters below the
ice as well as a general warm bias in deeper waters and a too strongly
stratified surface layer near and following ice-off.
(a) Simulated ice thickness; (b) simulated sediment temperature
(dotted), lowest-level water temperature (solid), and sediment heat flux
(dashed – right-hand scale). Simulated results are from experiments X1
(blue) and X2 (red). Zero sediment heat flux (dashed black curve) is also
indicated in (b). The period shown covers 19 July 2013–25 May 2014.
Simulated ice thickness (a), and simulated (b)(d) and observed (c) temperature profiles for 1 November 2013–20 May 2014. The standard
simulation (MacKay et al., 2017) is shown in (b) and the blue curve in (a).
Simulation X2 is shown in (d) and the red curve in (a).
The change in total heat content under ice is, however, improved. MacKay et al. (2017)
found that between 26 November 2013 and 6 March 2014 (100 d) the
simulated 1–10 m water column lost 7.26×106 (J m-2)
corresponding to a mean heat flux of -0.84 (W m-2), whereas observations
suggested warming corresponding to +1.66 (W m-2). Including sediment
heat flux improves this. Geothermal heating alone (X1) brings the mean heat
flux into the column up to 0.08 (W m-2), while including the radiative
forcing (X2) yields 1.05 (W m-2) for a net bias of -0.61 (W m-2).
These results are summarized in Table 2.
Observed and simulated change in 1–10 m heat content from 26 November 2013 to 6 March 2014.
ΔQ1-10mBiasExperiment(W m-2)(W m-2)Observations1.66–MacKay et al. (2017)-0.84-2.50X10.08-1.58X21.05-0.61
A disadvantage of the method described here is that all sediment heat flux
into the lake water takes place at the mean lake depth (which is the lowest
model level). Circulation processes that redistribute heat are clearly
active in the observations (Fig. 7c), but are not represented in the model.
In fact, the only reason the simulated heat is redistributed vertically is
that the bottom simulated temperatures are near the temperature of maximum
density (4 ∘C), so that warming of these waters produced convection. But
temperatures in the bottom half of the simulated lake are clearly biased
warm during ice cover, and free convection is likely not taking place in
reality. So while the change in total heat content under ice has improved
with the new scheme, work remains for parameterizing mixing processes to
appropriately redistribute heat emanating from the sediments.
Conclusions
Lakes are increasingly being recognized as important components of the land
surface, and 1-D modelling schemes are currently being developed for
inclusion in a number of climate and numerical weather prediction systems
around the world. Lakes interact with their environment through flux
exchange with their bottom sediments and at the surface with the atmosphere,
and even a perfect lake model will perform poorly in a prognostic sense if
these linkages are represented poorly. However, these are challenging areas
of research: high-quality data in lake sediments (e.g. temperature, thermal
properties) are difficult to achieve at regional or global scales, and the
process understanding itself of turbulent exchange under non-equilibrium
conditions, ubiquitous for all but the largest of boreal lakes, is lacking.
This study contributes to this discussion with two simple schemes that begin
to represent these linkages in simple 1-D “bathtub”-like models like the CSLM.
Terrestrial landscapes play a role in lake hydrodynamics through the sudden
drop in aerodynamic roughness that the atmosphere encounters at the
shoreline of virtually any boreal lake. While some models do make
adjustments to surface drag to account for various processes (or for
“tuning”), none to our knowledge factors in the terrestrial roughness in a
quantitative way. Here we propose a straightforward scheme based on earlier
IBL research that begins to address this. When applied to a simulation of
the CSLM we have found an improvement in the near-surface temperature
following two wind events a few days apart that resulted from a more
realistically simulated diurnal mixed layer depth.
It is well known that shallow lakes or lakes with extensive shallow littoral
zones may be subject to sediment heating that is significant for many
applications, and several models have attempted to account for this process.
While it is trivial to add a sediment layer beneath a lake in order to
absorb and subsequently release heat, problems arise when the lake model
itself does not represent bathymetry. Since a significant fraction of the
sediment heat content can arise from SW insolation at the sediment–water
interface, lake models that assume uniform depth (i.e. constant depth set equal
to the lake mean depth) would in most cases receive almost no radiative
forcing at the sediment unless the lake was exceptionally shallow or clear.
To account for this a scheme is proposed to estimate the actual sediment
insolation by approximating any given lake as an axially symmetric lake with
a shape factor determined from mean and maximum depth data only. When
applied in a simulation of the CSLM we have found an improvement in the
change in lake heat content under ice cover over a 100 d period. Issues
remain regarding the distribution of the sediment heating throughout the
water column, as currently all heating takes place at the lake mean depth.
Code and data availability
Code for the CSLM and data used for its forcing and evaluation are available at 10.5281/zenodo.2554524 (MacKay, 2019).
Author contributions
MDM performed all tasks associated with this paper, including
developing the mathematical framework, coding and running the numerical
model, analyzing the simulations, and writing the manuscript.
Competing interests
The author declares that there is no conflict of interest.
Special issue statement
This article is part of the special issue “Modelling lakes in the climate system (GMD/HESS inter-journal SI)”. It is a result of the 5th workshop on “Parameterization of Lakes in Numerical Weather Prediction and Climate Modelling”, Berlin, Germany, 16–19 October 2017.
Acknowledgements
Mike Rennie provided temperature profile observations under ice cover for
L239. As always, I am indebted to the IISD-ELA staff, who have provided
ongoing and invaluable support throughout this research programme.
Review statement
This paper was edited by Qiang Wang and reviewed by two anonymous referees.
References
Bradley, E. F.: A micrometeorological study of velocity profiles and surface
drag in the region modified by a change in surface roughness, Q. J. Roy.
Meteorol. Soc., 94, 361–379, 1968.
Carslaw, H. S. and Jaeger, J. C.: Conduction of Heat in Solids, Oxford Science
Publications, Oxford, England, 1959.
Detto, M., Katul, G. G., Siqueira, M., Juang, J.-Y., and Stoy, P.: The structure
of turbulence near a tall forest edge: the backward-facing step flow analogy
revisited, Ecol. Appl., 18, 1420–1435, 2008.
Driver, D. M. and Seegmiller, H. L.: Features of a reattaching turbulent shear
layer in divergent channel flow, AIAA, 23, 163–171, 1985.
Fee, E.,
Hecky, R., Kasian, S., and Cruikshank, D.: Effects of lake size, water clarity,
and climatic variability on mixing depths in Canadian Shield lakes, Limnol.
Oceanogr., 41, 912–920, 1996.
Garratt, J.: The internal boundary layer – a review, Bound.-Lay. Meteorol., 50,
171–203, 1990.
Gorham, E. and Boyce, F.: Influence of lake surface area and depth upon
thermal stratification and the depth of the summer thermocline, J. Great
Lakes Res., 15, 233–245, 1989.
Jensen, N. O.: Change of surface roughness and the planetary boundary layer,
Q. J. Roy. Meteorol. Soc., 104, 351–356, 1978.Kirillin , G., Leppäranta, M., Terzhevik, A., Granin, N.,
Bernhardt, J., Engelhardt, C., Efremova, T., Golosov, S.,
Palshin, N., Sherstyankin, P., Zdorovennova, G., and Zdorovennov, R.: Physics of seasonally ice – covered lakes: a review,
Aquat. Sci., 74, 659–682, 10.1007/s00027-012-0279-y, 2012.
Kwan, J. and Taylor, P.: On gas fluxes from small lakes and ponds,
Bound.-Lay. Meteorol., 68, 339-356, 1994.Likens, G. E. and Johnson, N. M.: Measurement and analysis of the annual heat
budget for the sediments of two Wisconsin lakes, Limnol. Oceanogr., 14,
115–135, 1969.
MacKay, M. D.: A process – oriented small lake scheme for coupled climate
modeling applications, J. Hydrometeorol., 13, 1911–1924,
10.1175/JHM-D-11-0116.1, 2012.MacKay, M. D.: CSLM2.0, Zenodo, 10.5281/zenodo.2554524, 2019.MacKay, M. D., Verseghy, D. L., Fortin, V., and Rennie, M. D.: Wintertime
simulations of a boreal lake with the Canadian Small Lake Model, J.
Hydrometeorol., 18, 2143–2160, 10.1175/JHM-D-16-0268.1, 2017.Markfort, C. D., Porté-Agel, F., and Stefan, H. G.: Canopy – wake dynamics
and wind sheltering effects on Earth surface fluxes, Environ. Fluid Mech.,
14, 663–697, 10.1007/s10652-013-9313-4, 2014.
Mazumder, A. and
Taylor, W.: Thermal structure of lakes varying in size and water clarity,
Limnol. Oceanogr., 39, 968–976, 1994.
Panofsky, H. and Townsend, A.: Change of terrain roughness and the wind
profile, Q. J. Roy. Meteorol. Soc., 90, 147–155, 1964.
Rao, K., Wyngaard, J., and Coté, O.: The structure of the two –
dimensional internal boundary layer over a sudden change of surface
roughness, J. Atmos. Sci., 31, 738–746, 1974.Rizk, W., Kirillin, G., and Lepparanta, M.: Basin – scale circulation and heat
fluxes in ice – covered lakes, Limnol. Oceanogr., 59, 445–464,
10.4319/lo.2014.59.2.0445, 2014.
Schlegel, F., Stiller, J., Bienert, A., Maas, H.-G., Queck, R., and Bernhofer, C.:
Large-eddy simulation study of the effects on flow of a heterogeneous forest
at sub-tree resolution, Bound.-Lay. Meteorol., 154, 27–65, 2015.Stepanenko, V., Jöhnk, K. D., Machulskaya, E.,
Perroud, M., Subin, Z., Nordbo, A., Mammarella, I., and Mironov, D.: Simulation of surface energy fluxes and
stratification of a small boreal lake by a set of one – dimensional models,
Tellus A, 66, 21389, 10.3402/tellusa.v66.21389, 2014.Stepanenko, V., Mammarella, I., Ojala, A., Miettinen, H., Lykosov, V., and Vesala, T.: LAKE 2.0: a model for temperature, methane, carbon dioxide and oxygen dynamics in lakes, Geosci. Model Dev., 9, 1977–2006, 10.5194/gmd-9-1977-2016, 2016.
Tsay, T.-K., Ruggaber, G. J., Effler, S. W., and Driscoll, C. T.: Thermal
stratification modeling of lakes with sediment heat flux, J. Hydraul.
Eng., 118, 407–419, 1992.
Venäläinen, A., Heikinheimo, M., and Tourula, T.: Latent heat flux
from small sheltered lakes, Bound.-Lay. Meteorol., 86, 355–377, 1998.Verseghy, D. L. and MacKay, M. D.: Offline implementation and evaluation of the
Canadian Small lake Model with the Canadian Land Surface Scheme over Western
Canada, J. Hydrometeorol., 18, 1563–1582, 10.1175/JHM-D-16-0272.1, 2017.
Vickers, D. and Mahrt, L.: Fetch limited drag coefficients, Bound.-Lay.
Meteorol., 85, 53–79, 1997.