ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-17-8177-2017The boundary condition for vertical
velocity and its interdependence with surface gas exchangeKowalskiAndrew S.andyk@ugr.esDepartmento de Física Aplicada, Universidad de Granada, Granada, 18071, SpainInstituto Interuniversitario de Investigación del Sistema Tierra
en Andalucía, Centro Andaluz de Medio Ambiente (IISTA-CEAMA), Granada, 18071, SpainAndrew S. Kowalski (andyk@ugr.es)5July201717138177818725February201723March201724May201729May2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/17/8177/2017/acp-17-8177-2017.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/17/8177/2017/acp-17-8177-2017.pdf
The law of conservation
of linear momentum is applied to surface gas exchanges, employing scale
analysis to diagnose the vertical velocity (w) in the boundary layer. Net
upward momentum in the surface layer is forced by evaporation (E) and
defines non-zero vertical motion, with a magnitude defined by the ratio of
E to the air density, as w=Eρ. This is true even right down
at the surface where the boundary condition is w|0=Eρ|0 (where w|0 and ρ|0 represent the vertical velocity
and density of air at the surface). This Stefan flow velocity implies upward
transport of a non-diffusive nature that is a general feature of the
troposphere but is of particular importance at the surface, where it assists
molecular diffusion with upward gas migration (of H2O, for example)
but opposes that of downward-diffusing species like CO2 during
daytime. The definition of flux–gradient relationships (eddy diffusivities)
requires rectification to exclude non-diffusive transport, which does not
depend on scalar gradients. At the microscopic scale, the role of
non-diffusive transport in the process of evaporation from inside a narrow
tube – with vapour transport into an overlying, horizontal airstream – was
described long ago in classical mechanics and is routinely accounted for by
chemical engineers, but has been neglected by scientists studying stomatal
conductance. Correctly accounting for non-diffusive transport through
stomata, which can appreciably reduce net CO2 transport and
marginally boost that of water vapour, should improve characterisations of
ecosystem and plant functioning.
Introduction
The vertical velocity (w) is a key variable in the
atmospheric sciences, and its precise diagnosis is essential for numerous
applications in meteorology. Above the boundary layer, the weather is largely
determined by adiabatic adjustments to vertical motion that is slight
compared to horizontal winds. Closer to the surface, even a tiny w can
result in relevant transport; for example, in a typical boundary layer –
with representative temperature (T=298K), pressure (p=101325Pa) and CO2 mass fraction
(607 mgkg-1; a molar ratio of about 400 ppm) – just
61 µms-1 of average vertical velocity is needed to waft a
biologically significant 44 µgCO2m-2s-1 (a
CO2 molar flux density of 1 µmolm-2s-1). Modern
anemometry cannot resolve such a minuscule airflow (Lee, 1998), and generally
w is immensurable at many scales so that it must be derived from other
variables (Holton, 1992). Such diagnostic estimation is traditional in
synoptic meteorology, but has been developed less rigorously near the surface
boundary.
The characterisation of boundary conditions for state and flow variables, in
order to enable atmospheric modelling at larger scales, is a fundamental goal
of micrometeorology. Since w is air velocity, its boundary condition w|0 describes the surface-normal or vertical motion of the gas molecules
found closest to the surface (at some height z|0, very nearly but not
exactly zero). The Navier–Stokes equations, when applied to the lower
atmosphere, are particularly sensitive to the conditions specified at the
boundary (Katul et al., 2004), and this lends great importance to w|0 in
the context of dynamic modelling. Nevertheless, until now w|0 has
received inadequate attention in boundary-layer meteorology.
Micrometeorologists have made presuppositions regarding w|0 without
formal justification and in contradiction to deductions from classical
mechanics. The traditional hypothesis about near-surface winds is that they
flow parallel to underlying terrain (Kaimal and Finnigan, 1994; Wilczak et
al., 2001) and vanish at the surface (Arya, 1988), implying w|0=0. This
assumption underlies many derivations and abets the prevailing belief that
vertical exchanges are accomplished purely by molecular diffusion within a
millimetre of the surface (Foken, 2008) or purely by turbulent diffusion at
heights of metres or more within the atmospheric boundary layer. However,
such a premise is inconsistent with net surface gas exchange
(predominantly evaporative), which implies Stefan flow with a mean velocity
component normal to the surface. Net mass transfer across a surface results
in a velocity component normal to the surface, and an associated
non-diffusive flux in the direction of mass transfer (Kreith et al., 1999).
The existence and relevance of Stefan flow – first derived and described in
the 19th century – is certain. Indeed, engineers necessarily account for its
role in heat and mass transfer (Abramzon and Sirignano, 1989) when precisely
controlling industrial processes that include phase change, such as
combustion. For these reasons, it is to be expected that a more accurate
means of estimating w|0 for the atmospheric boundary layer can be
achieved by rigorous examination of known surface flux densities in the light
of physical laws.
The remaining sections of this work aim to diagnose a defensible lower
boundary condition for the vertical velocity (w|0) and to interpret its
significance. Section 2 presents the theory and illustrates types of mass
transport and heat exchange in fluids via an example from the liquid phase.
In Sect. 3, an analytical framework is established and conservation of linear
momentum is applied to derive w|0 from published magnitudes of surface
gas exchanges, demonstrating that it is directly proportional to the
evaporative flux density (E), consistent with the findings of Stefan. The
derived vertical velocity is seen to be relevant in defining the mechanisms
of gas transport, which is not accomplished by diffusion alone – even at the
surface interface. Section 4 highlights the need to rectify flux–gradient
relationships by taking into account the non-diffusive component of
transport; this includes boundary-layer similarity theory and physiological
descriptions of stomatal conductance. Thus, the implications of these
analyses are broad and interdisciplinary.
Theory
The objective of this section is to establish the theoretical bases for the
analyses and interpretations that follow. It opens with a list of symbols
(Table 1) along with the meaning and SI units of each variable
represented, and finishes with a summary of the most salient points regarding
physical laws and transport mechanisms to be recalled in Sect. 3.
List of symbols, with their meanings and units.
SymbolVariable representedSI unitsTensor orderGeneral variable representations ξAn arbitrary magnitude (can represent any scalar variable)Depends on ξ0 (scalar)ξiThe magnitude of arbitrary variable ξ for gas species iDepends on ξ0 (scalar)∇ξThe spatial gradient in arbitrary variable ξDepends on ξ1 (vector)ξ|0The lower boundary condition for arbitrary variable ξDepends on ξ0 (scalar)Specific variable representations Δx, ΔyHorizontal dimensions of an analytical volumem0 (scalars)δzVertical dimension (thickness) of an analytical volumem0 (scalar)EEvaporative flux density across a horizontal surfacekgm-2s-10 (component)esSaturation vapour pressurePa0 (scalar)fMass fractionNon-dimensional0 (scalar)FiVertical flux density of gas species ikgm-2s-10 (component)Fi,nonNon-diffusive component of Fikgm-2s-10 (component)iIndex for counting gas species (as in Table 2)–0 (scalar)KMolecular diffusivitym2s-10 (scalar)LAILeaf area indexNon-dimensional0 (scalar)pPressurePa0 (scalar)qSpecific humidityNon-dimensional0 (scalar)ρAir densitykgm-30 (scalar)σStomatal fraction of leaf areaNon-dimensional0 (scalar)TAir temperatureK0 (scalar)tTimes0 (scalar)t0Initial instant of a case scenarios0 (scalar)teqEquilibrium instant of a case scenarios0 (scalar)tfFinal instant of a case scenarios0 (scalar)vAir velocityms-11 (vector)wVertical component of vms-10 (component)WUEWater use efficiencyNon-dimensional0 (scalar)zHeight above the surfacem0 (component)Relevant scientific lawsThe law of conservation of linear momentum
The principle of conservation of momentum is most fundamental in physics,
more so than even Newton's first law (Giancoli, 1984). It defines the momentum
of a system of particles as the sum of the momenta of the individual
components and establishes that this quantity is conserved in the absence of
a net external force. Accordingly, in atmospheric dynamics (Finnigan, 2009) a
system may be defined as the N component gas species comprising a
particular mass of air, with a net vertical flux density of
wρ=∑i=1Nwiρi.
In Eq. (1), w and ρ represent the vertical velocity and density of
air, respectively, while wi and ρi are the properties of
component i, the species flux density of which is wiρi. For this
species i, total transport wiρi can be attributed to mechanisms
that are diffusive (if wi≠w), non-diffusive (if w≠0) or more
generally a combination of these two types of transport. Dividing Eq. (1) by
the net air density defines the system's vertical velocity as a weighted
average of its components (Kowalski, 2012), where the weighting factors are
the species' densities.
The zeroth law of thermodynamics
The zeroth law establishes the
temperature as the variable whose differences determine the possibility
for heat exchange between thermodynamic systems. For two systems in thermal
contact, if they have the same temperature then they are in thermodynamic
equilibrium and therefore exchange no heat. If their temperatures differ,
then heat will be transferred from the system with the higher temperature to
that with the lower temperature. Heat transfer by molecular conduction
depends on gradients in the temperature; in compressible fluids like air,
however, turbulent diffusion can occur without thermal contact and yet bring
about heat transfer as determined by gradients in the potential temperature
(Kowalski and Argüeso, 2011), accounting for any work done/received
during the expansion/compression associated with vertical motions.
Fick's first law of diffusion
Molecular diffusion has no effect on the net fluid momentum, but “randomly”
redistributes fluid components and can cause different species to migrate in
different directions, according to component scalar gradients. Regrettably,
scientific literature contains inconsistencies regarding the scalar gradient
that determines diffusion in the gas phase (Kowalski and Argüeso,
2011). The proper form of Fick's first law for diffusion in the vertical
direction is
Fi,M=-ρK∂fi∂z,
where Fi,M is the vertical flux density of species i due to
molecular diffusion, which is proportional to the vertical gradient in that
species' mass fraction (fi; Bird et al., 2002), and z is height. Also
relevant are the fluid density (ρ) and molecular diffusivity (K).
However, ρ must not be included in the derivative in Eq. (2), unless for
the trivial case in which it is constant (in an incompressible fluid); in
compressible media, gradients in gas density can arise, with no direct
relevance to diffusion, due to gradients in pressure or temperature as
described by the ideal gas law. It is relevant to note that Adolf Fick
arrived at this law, not by experimentation, but rather by analogy with
Fourier's law for heat conduction (Bird et al., 2002). By the same analogy,
the product of the diffusivity with the scalar gradient in Eq. (2) yields a
kinematic flux, which requires multiplication by the fluid density in order
to yield the flux density of interest.
Fluxes due to molecular diffusion are referenced to the motion of the fluid's
centre of mass or “mixture velocity” (Bird et al., 2002). The simplest
example for describing this is that of binary diffusion, where only two species
compose the fluid, as in the traditional meteorological breakdown of air into
components known as dry air and water vapour. In the case of static
diffusion, the fluid velocity is zero and the mass flux of one gas species
(water vapour) counterbalances that of the other (dry air). When diffusion
occurs in a dynamic fluid (non-zero velocity), then overall transport must be
characterised as the sum of diffusive and non-diffusive components.
Turbulent diffusion is analogous to molecular diffusion in the sense that
fluid components are randomly redistributed, with different species migrating
as a function of gradients in their mass fractions. The primary difference is
that eddies rather than molecular motions are responsible for mixing, and the
eddy diffusivity (the value of K in Eq. 2, describing K-theory; Stull, 1988) is a property
of the flow rather than the fluid. The Reynolds number describes the relative
importance of molecular and turbulent diffusion, which are otherwise
indistinct with respect to the analyses that follow, and will simply be
grouped and referred to as “diffusive transport”.
Transport processes
In this section, two case studies from the liquid phase will help to identify
and define non-diffusive and diffusive types of transport, as well as their
scalar source/sink determinants. Let us consider the case of freshwater
(35×10-5 mass fraction of salt) with constant temperature and
composition flowing through a tube into the bottom of a pool (Fig. 1) of
salinity specified according to the two case scenarios defined below.
Considering only flow within the tube (at point 1), whether laminar or
turbulent, it clearly realises non-diffusive transport of salt, since the
salt has no particular behaviour with respect to the fluid, but simply goes
with the flow. There are no scalar gradients within the tube, and so there is
neither diffusion nor advection. Let us now describe diffusive transport
processes within the pool (at point 2), and the nature (whether absolute or
relative) of the relevant fluid properties whose
gradients determine them by
defining sources/sinks, using two illustrative case scenarios.
A pool of water being fed from below by a tube. The points indicate
water (1) in the tube and (2) in the pool. The arrow indicates the direction
of flow.
The temperature is constant in time and space, but other
characteristics of the two case scenarios are chosen to elucidate the
relationship between diffusive transport processes and scalar gradients:
Due to surface evaporation that balances the mass input from the tube,
the pool mass is constant. The water remains isothermal by surface
heating, which supplies the (latent) energy for evaporation. Initially
(t0) the pool has zero salt mass, but salinity increases constantly,
equalling that of the tube water at some moment (teq) and rising
by another 2 orders of magnitude to reach that of sea water (35×10-3) by the end of the scenario (tf). This case is of
interest from both (a) salt/solute and (b) thermodynamic points of view:
In solute terms, the tube represents a source of (absolute) salt to
the pool, but not always of (relative) salinity. Initially (t0), the water
from the tube is more saline than that in the pool, such that non-diffusive
and diffusive transport processes operate in tandem to transport salt from
the tube upward into the pool; at this moment, the tube is a source of
salinity. Salinity advection, defined as the negative of the inner product of
two vectors (the velocity with the salinity gradient, with opposite signs),
is then positive. Ultimately, however (at t>teq), the water in
the pool is more saline than that entering from the tube, such that
non-diffusive and diffusive salt transport are in opposite directions. Then
the tube dilutes the pool and is a salinity sink, but still a salt source.
Salinity advection at tf is negative. The pool continues to gain
salinity after teq, despite the diluting effects of the tube, due
to the concentrating effects of evaporation, which is the ultimate source of
salinity. This distinction matters because the gradients that drive advection
and diffusion are those in salinity, a relative (not absolute) salt measure.
At tf, the diffusive salinity fluxes are oriented against the
flow within the pool (downward, and radially inward towards the diluting
tube, despite its being a net salt source). By contrast, non-diffusive
transport always goes with the flow and accounts for continued upward and
outward salt transport, increasing the salt content at the surface.
Although thermodynamically trivial – with no heat exchanges
whatsoever within the water as determined by the zeroth law – this case
nonetheless illustrates the nature of the scalars that determine heat
transfer by advection and diffusion (conduction). The heat content of the
pool decreases as it becomes more and more saline, due to the inferior heat
capacity of saltwater versus freshwater. Similarly, salt diffusion/advection
is initially upward/positive but ultimately downward/negative, yet the
corresponding implications regarding heat content fluxes say nothing about
the transfer of heat. The point here is that the dynamics of the heat content
must not be interpreted in terms of heat fluxes, which was done by Finnigan
et al. (2003). For this reason, meteorologists correctly define “temperature
advection” (Holton, 1992) based on the thermodynamic relevance of gradients in the variable singled
out by the zeroth law.
Let us now specify that the water in the pool has the same
(freshwater) salinity as that coming from the tube (35×10-5). If
we furthermore remove both surface evaporation and heating from scenario (1),
then the temperature remains constant and the salinity corresponds uniformly
to that of freshwater, but the pool accumulates mass. In this case, there are
convergences in the non-diffusive transports of water, salt and heat content:
fluxes into the pool are positive, while fluxes out are null. However, there
are no gradients in temperature or salinity, and so there is neither
diffusion nor advection in this scenario. The pool does gain volume (depth)
but this is only because the fluid under consideration is incompressible. By
contrast, for the gas phase, accumulation of absolute quantities – such as
air and trace constituent mass and heat content – can occur in a constant
volume context (e.g. at a point in an Eulerian fluid specification) due to
convergent, non-diffusive transport that defines compression. In the pool,
diffusion and advection are clearly null because they are determined by
gradients in the relative trace gas amount, the mass fraction, which is a
variable of essential utility for the gas phase because it is immune to the
effects of compression.
An advection-diffusion synopsis
The analyses that follow rely on the succeeding key points drawn from
Sects. 2.1 and 2.2. Advection and diffusion depend on gradients in scalars
with relative rather than absolute natures. In incompressible thermodynamics,
the relevant gradients are those in the temperature and not the heat content.
For trace constituents, the relevant scalar is the mass fraction (e.g.
salinity) and not the species density. Advection and diffusion are otherwise
physically very distinct. Like non-diffusive transport, diffusion is a vector
whose vertical component is of particular interest in the context of
surface–atmosphere exchange. By contrast, advection is a scalar; for some
arbitrary quantity ξ, it is defined as the negative of the inner product
v⋅∇ξ, where v is the fluid velocity and
∇ is the gradient operator. Thus advection, unlike diffusion, is
not a form of transport, but rather a consequence of differential transport.
The scenarios depicted above correspond to the incompressible case (liquid).
When the effects of compressibility are irrelevant, it can be convenient to
add the incompressible form of the continuity equation (∇⋅v=0) to advection yielding -∇⋅ξv, the convergence
of a kinematic flux. This is called the “flux form” of advection. For a
compressible medium such as the atmosphere, however, if ξ is taken to
represent some “absolute fluid property such as the (gas) density”
(Finnigan et al., 2003), then the transformation of advection into flux form
cannot be justified (Kowalski and Argüeso, 2011), since using the
incompressible form of the continuity equation leads to unacceptable errors
in conservation equations for boundary-layer control volumes (Kowalski and
Serrano-Ortiz, 2007). By contrast, the expression of advection in flux form
can be valid if the scalar ξ is carefully chosen for its immunity to the
effects of compression, as is the case for the mass fraction. These
generalisations regarding the nature of transport by non-diffusive and
diffusive mechanisms, and also the nature of advection, will now be applied
to the case of vertical transport very near the surface and the mechanisms
that participate in surface exchange, after first deriving the boundary
condition w|0.
Gas components comprising the system to be examined, and their
masses.
The analysis will focus on a system defined as a mixture of gas molecules of
different species, and its momentum will be examined. The system's mass is
defined (Table 2) by gas components in a ratio that corresponds quite closely
to that of the atmosphere (Wallace and Hobbs, 2006) but updated to more
closely reflect actual atmospheric composition. At a representative ambient
temperature (T=298K) and pressure (p=101325Pa),
the many millions of molecules forming this system occupy a volume of
10-15m3 with 70 % relative humidity. The system geometry
will be specified in four different ways, according to the different spatial
scales for which w|0 is to be described:
At the synoptic scale, the volume occupied by the system is a lamina
of depth δz∼10-27m, bounded above and below by
constant geopotential surfaces, with horizontal dimensions (Δx and
Δy) of the order of 106m. The fact that δz is
thinner than the dimension of a molecule does not matter at all when
classifying any and all molecules with centres of mass (points, with neither
size nor dimension) occupying the lamina as belonging to the volume.
At the micrometeorological scale, the volume overlies a flat surface
and is shaped as a rectangular lamina of depth δz∼10-21m, with horizontal dimensions (Δx and Δy)
of 103m.
At the leaf scale, the volume is a rectangular lamina of depth
δz∼10-11m, with horizontal dimensions (Δx and
Δy) of 10-2m.
At the microscopic scale of plant stomata, the volume is a cube with
Δx=Δy=δz=10-5m. For the purpose of
transitioning between the leaf and microscopic scales, plant pores are
assumed to occupy a stomatal fraction σ of the leaf surface and yet
accomplish all gas exchange. The remaining fraction (1-σ) is
occupied by a cuticular surface and its gas exchange is assumed to be null
(Jones, 1983).
Independent of scale, the base height z|0 of the volume is the lowest
for which only air – and neither ocean wave nor land surface element –
occupies the volume. The land/ocean/leaf surface will be assumed to be static
(i.e. its vertical velocity is zero), impenetrable to the wind (explicitly
neglecting ventilation of air-filled pore space), smooth, level and uniform,
all for the sake of simplicity. The temporal framework for the analysis is
instantaneous, with no need to choose between Eulerian and Lagrangian fluid
specifications.
The direction of momentum transport to be examined is vertical, meaning
perpendicular to constant geopotential surfaces and therefore to the
underlying surface. At the stomatal scale, the stoma to be examined is
situated on the upper side of a flat, horizontal leaf; water vapour exiting
the stomatal aperture during transpiration therefore has a positive vertical
velocity. These analyses can be generalized to sloping surfaces and/or
stomata on the underside of leaves, simply by referring to the
surface-normal rather than vertical velocity. Hereinafter, however,
the term “vertical” will be employed for conciseness.
The vertical velocity at the surface boundary
Knowledge regarding surface exchange (gas flux densities) has advanced to the
point where the boundary condition for the vertical velocity (w|0) can
be estimated from conservation of linear momentum – applying Eq. (1) to the
system defined in Table 2 – and vastly simplified to a simple function of
the evaporation rate (E). The species flux densities (wiρi) within
the system represent the surface exchanges of the corresponding gas species
(i). Scale analysis of surface gas exchange magnitudes, published from
investigations at a particularly well-equipped forest site in Finland
(Table 3), reveals that for the water vapour species (i=4), the flux
density (E=w4ρ4) is several orders of magnitude larger than both
the flux density of any dry air component species and even the net flux
density of dry air. Such dominance by water vapour exchanges is
representative of most surfaces worldwide. This is especially so because the
two largest dry air component fluxes are opposed, with
photosynthetic/respiratory CO2 uptake/emission largely offset by
O2 emission/uptake (Gu, 2013). Hence, following tradition in
micrometeorology (Webb et al., 1980), surface exchange of dry air can be
neglected, allowing the elimination from Eq. (1) of all species flux
densities except for that of water vapour (H2O; i=4). Therefore,
net air transfer across the surface can be approximated very accurately as
w|0ρ|0=w4|0ρ4|0=E,
where w4|0 and ρ4|0 are the H2O species velocity and
density at the surface. Equation (3) states that, at the surface, the net
vertical flux density of air is equal to the net vertical flux density of
water vapour, which is the evaporation rate. Solving this for w|0 allows
estimation of the lower boundary condition for the vertical velocity as
w|0=Eρ|0.
The representative evaporation rate prescribed in Table 3 and vertical
velocity resulting from Eq. (4) are valid for most of the scales defined
above. Thus, the boundary condition w|0 is valid for the synoptic scale
(notwithstanding vertical motion aloft, such as subsidence), for the
micrometeorological scale and even for the leaf scale. In the context of
scale analysis, leaves may be approximated as having equal area as the
underlying surface (i.e. a unit leaf area index or LAI = 1), and equal
evaporation rates as the surface in general. This latter assumption does not
neglect soil evaporation, but only excludes the possibility that it
would dominate leaf evaporation by 1 order of magnitude. Thus, it will be assumed here that the
assumed evaporation rate and derived vertical velocities are equally valid at
synoptic (a), micrometeorological (b) and leaf (c) scales. The order of
magnitude is different, however, at the microscopic (d) scale. To show this,
it will be assumed here that all leaf evaporation (or transpiration) occurs
through the small fraction of the leaf that is stomatal (σ), such that
both the stomatal evaporative flux density and the lower boundary condition
for the vertical velocity (w|0) are a factor 1/σ greater than
those at larger scales. Independent of scale, Eq. (4) states that, for a
positive evaporation rate, the boundary condition for the vertical velocity
is non-zero and upward.
The first six air components by their surface exchange scale
magnitude, and the net exchange of air as the sum of these flux densities.
Representative surface exchanges are taken from the Finnish boreal forest
site (Suni et al., 2003; Aaltonen et al., 2011). The O2 exchange rate
assumes 1:1 stoichiometry with CO2.
GasTypical mass flux, FiCorresponding molar fluxSourcei(mgm-2s-1)(mmolm-2s-1)H2O362Suni et al. (2003)4CO2-0.088-0.002Suni et al. (2003)5O20.0640.002Gu (2013)2CH4-0.000032-0.000002Aaltonen et al. (2011)6O30.0000096-0.0000002Suni et al. (2003)8N2O0.000000880.00000002Aaltonen et al. (2011)7Air35.98–This study–
Given that the surface boundary is static, it may well be asked why there is
a non-zero boundary condition for the vertical velocity of air. The answer is
that evaporation induces a pressure gradient force that pushes air away from
the surface. Evaporation into air increments the water vapour pressure and
thereby the total pressure, according to Dalton's law. If evaporation were to
proceed until equilibrium is achieved, the pressure added by evaporation would
correspond to the saturation vapour pressure (es; Fig. 2). Its
temperature dependency has been quantified empirically and is described by
the Clausius–Clapeyron relation. It is this evaporation-induced pressure
gradient force that pushes the manometer in Fig. 2 to its new position, and
similarly drives winds away from the surface.
Although this upward air propulsion occurs at the surface, air velocities are
generally upward throughout the boundary layer in a climatological context.
Indeed, the dominant role of water vapour in determining the net vertical
momentum of air is a general feature of the troposphere. In the context of
the hydrological cycle, water vapour is transported from the surface where it
has an evaporative source, to further aloft where clouds develop via
processes that act as water vapour sinks: condensation and vapour deposition
onto ice crystals (or ice nuclei). In terms of total water, upward transport
in the gas phase is offset, over the long term, by downward transport in
liquid and solid phases (e.g. rain and snow); unlike the water vapour flux,
however, precipitation does not directly define air motion. It is true that
downward water vapour transport occurs during dewfall – with surface
condensation, as described by Eq. (4) with a negative evaporation rate
(E< 0) – but this plays a minor role in the global water balance.
Generally, the relative magnitudes of gas exchanges used for the scale
analysis in Table 3 are representative throughout most of the troposphere,
with upward water vapour flux densities dominating those of other gases in
the vertical direction. In the surface layer, sometimes termed the “constant
flux layer” (Dyer and Hicks, 1970), Eq. (4) can be extrapolated away from
the surface under steady-state conditions to yield
w=Eρ.
Illustration of evaporation incrementing air pressure. Chamber air
evolves from (a) dry air initially at atmospheric pressure to
(b) moist air at a pressure that has risen by the partial pressure
of water vapour, ultimately at equilibrium (saturation vapour pressure,
es). The force generated by evaporation propels the mercury in
the manometer from its initial position.
Mechanisms of gas transport at the surface
Non-zero vertical momentum in the lower atmosphere and right at the surface
boundary – dominated by the flux density of water vapour and generally
upward due to evaporation – means that diffusion is not the lone relevant
transport mechanism that participates in surface exchange, as has been
generally supposed. This is true for all atmospheric constituents, not
only for water vapour; over an evaporating surface, any molecule undergoing
collisions with its neighbours does not experience a random walk (a
characteristic of static diffusion), but rather tends to be swept upward with
the flow. The upward air current similarly wafts aerosol particles, although
these may move downwards if their fall velocities exceed the upward air
motion. The upward flow velocity is rather small – just
31 µms-1 for the conditions specified above and the
evaporation rate of Table 3, according to Eqs. (4) and (5). It does not
exclude the possibility of diffusive transport in any direction, but does
imply a relevant, non-diffusive component of transport for any gas, with a
magnitude that is not related to the scalar gradient of that gas.
The non-diffusive flux density of species i can be expressed as
Fi,non=wρi,
and when substituting for w from Eq. (5) this becomes
Fi,non=Efi,
that is, the product of the
evaporation rate and the species mass fraction. Examination of its magnitude
near the surface for different gases will now show that, while this is often
small in comparison with the diffusive component, it is not negligible in
every case, depending on the magnitudes of the mass fraction and surface
exchange for the gas considered.
Interpreting decomposed transport is simplest when examining a gas whose
surface exchange is very well known, such as the null value for inert Argon
(Ar) that constitutes ca. 1.3 % of dry atmospheric mass (Wallace and
Hobbs, 2006). Considering the state variables defined by Table 2 and the
evaporation rate of Table 3, Eq. (7) indicates
458 µgm-2s-1 (a molar flux density of
11.6 µmolm-2s-1) of upward, non-diffusive Ar transport
(F3,non) . To comprehend this, it helps to recall that the
constant addition of H2O dilutes dry air at the surface and promotes
its downward diffusion. For a null net flux of inert Ar to exist, downward
diffusion of this dry air component must exactly cancel the upward
non-diffusive transport, and therefore it is 458 µgm-2s-1
for the state and evaporative conditions specified above. These opposing
non-diffusive and diffusive Ar transport processes are quite analogous to
case scenario 1 of Sect. 2.2, at the instant tf when the fluid
emitted into the pool has a diluting effect. Such dual transport mechanisms
are also relevant for vital gases, with different transport directions and
degrees of relevance, depending on the density and flux density of the gas in
question.
For H2O, the two types of gas transport mechanisms operate in tandem,
with the non-diffusive component contributing a fraction of upward
H2O transport that, according to Eq. (7), is exactly the water vapour
mass fraction or specific humidity (Wallace and Hobbs, 2006)
q≡f4≡ρ4ρ.
This is just 2 % for the state conditions previously specified, but can
approach 5 % for very warm evaporating surfaces and/or high-altitude
environments. The breakdown of H2O transport into diffusive and
non-diffusive components is analogous to case scenario 1 of Sect. 2.2 at an
instant prior to teq when the fluid introduced to the pool is
highly concentrated in comparison with the fluid already in the pool. In any
case, non-diffusive H2O transport is generally secondary to diffusive
transport, but its neglect in an ecophysiological context can lead to larger
relative errors, as will be shown in Sect. 4.
For CO2, which usually migrates downward during evaporative
conditions because of photosynthetic uptake, upward transport of a
non-diffusive nature is even more relevant, opposing the downward flux due to
diffusion. To see this, let us examine the typical gas transport magnitudes
of Table 2 and the atmospheric state conditions specified above. According to
Eq. (7), non-diffusive CO2 transport (F5non) is then
21.5 µgm-2s-1 (a molar flux density of
0.49 µmolm-2s-1) in the upward direction, requiring
that downward CO2 diffusion be 109.5 µgm-2s-1
in order to yield 88 µgm-2s-1 of net surface uptake; if
not accounting for the non-diffusive resistance to net transport, the
CO2 diffusivity would be underestimated by ca. 20 %. The case of
CO2 uptake is not analogous to any pool/tube scenario in Fig. 1.
However, different conditions with equal evaporation (E=36mgm-2s-1) and CO2emission in the amount of
21.5 µgm-2s-1 (by respiration, for example) would
correspond to the case of zero CO2 diffusion (as at the instant
teq), since the CO2 mass fractions of both the atmosphere
and the gas mixture emitted by the surface are identical. Viewed in the
traditional diffusion-only paradigm, such a situation involving a net flux
but no gradient (F3=F3,non) would require a physically
absurd infinite diffusivity. At this same evaporation rate, but with lower
CO2 emission, diffusion of CO2 would be downward, towards the
surface which is a source of CO2 but a sink of the CO2 mass
fraction (analogous to salinity in case scenario 1 of Sect. 2.2 at some
instant between teq and tf when the fluid emitted to
the pool has a diluting effect). Whatever the direction of net CO2
transport, these case examples demonstrate the need for sometimes substantial
rectifications to flux–gradient relationships whether expressed as a
conductance, resistance, deposition velocity or eddy diffusivity
(K-theory) when correctly
accounting for non-diffusive transport.
Discussion
Relevant transport of a non-diffusive nature implies a need to revise the
basis of flux–gradient theory, both in the boundary layer and also at smaller
scales regarding gas transfer through plant pores. One of the key goals of
micrometeorology has been the derivation of the vertical transports of mass,
heat, and momentum from profiles of wind speeds and scalar variables in the
boundary layer (Businger et al., 1971). The analyses above elucidate how
gradients relate to only the diffusive components of such exchanges.
Therefore, non-diffusive flux components must be subtracted out in order to
characterise turbulent transport in terms of eddy diffusivities, a key goal
of Monin–Obukhov similarity theory (Obukhov, 1971). Perhaps more important
is the need to distinguish between non-diffusive and diffusive transport
mechanisms prior to assessing molecular diffusivities (conductances).
When Eq. (3) is applied at the stomatal apertures where virtually all plant
gas exchanges occur, it is revealed that jets of air escape from these pores
during transpiration. In the context of the scale analysis begun in
Sect. 3.2, it is appropriate to note that even fully open stomata occupy just
1 % of leaf area (Jones, 1983), leaving 99 % cuticular and inert with
regard to vital gas exchanges (σ=0.01). As noted in Sect. 3.2, this
means that for the microscopic scale (d; Sect. 3.1) of the stomatal aperture,
both the local evaporative flux density (E) and therefore the lower
boundary condition for the vertical velocity (w|0) predicted by Eq. (4)
are 2 orders of magnitude greater
than the 31 µms-1 estimated above. In other words, a typical
average airspeed exiting a stomatal aperture is 3.1 mms-1. For
non-turbulent flow through a cylindrical tube/aperture (i.e. Poiseuille
flow), the velocity at the core of such an air current is twice as large. If
a characteristic timescale is defined for air blowing through stomata as the
ratio of a typical stomatal aperture diameter (ca. 6 µm) to this
core velocity, it is found to be of the order of 10 ms, illustrating
that air is expulsed from plants in the form of “stomatal
jets”. Non-diffusive gas transport by such airflow exiting stomata – assisting with water vapour egress but inhibiting CO2 ingress – has been previously conceived.
The concept of net motion and consequent non-diffusive transport out of
stomata is not new, but has been disregarded by plant ecologists. Parkinson
and Penman (1970) put forth that the massive water vapour flux from
transpiration implies an outbound air current as a background against which
diffusion operates. Regrettably, however, their interpretation has largely
been forgotten, having been refuted in an analysis (Jarman, 1974) that
incorrectly assumed “no net flow of air” – disregarding conservation of
momentum – and yet seems to have gained acceptance (von Caemmerer and
Farquhar, 1981). Similarly, Leuning (1983) recognised the relevance of
non-diffusive transport and furthermore identified excess pressure inside the
stomatal cavity as the impetus for the outward airstream (which he termed
“viscous flow”), but had little impact on the mainstream characterisation
of stomatal conductance. Rather, important aspects of ecophysiology continue
to hinge upon the assumption that diffusion alone transports vital gases
through plant pores, disregarding both the above-mentioned studies and more
importantly the fact that gas transport mechanisms through such apertures
were accurately described by one of the great physicists of the 19th century.
Because Josef Stefan substantially helped to establish the fundamentals of
classical physics, his name is often mentioned in the same breath as
Boltzmann (regarding blackbody radiation) and Maxwell (for diffusion).
However, his work in the latter regard has been broadly ignored by scientists
studying gas exchanges through plant stomata. Stefan's study of evaporation
from the interior of a narrow, vertical cylinder with vapour transport into
an overlying, horizontal stream of air is of particular relevance to the
discipline of ecophysiology. He determined that this is not a problem of
static diffusion, but rather includes an element of non-diffusive transport
due to a mean velocity in the direction of the vapour flux, induced by
evaporation and now commonly known as Stefan flow. Engineers know this
history and refer to such a scenario as a Stefan tube (Lienhard IV and
Lienhard V, 2000), and routinely reckon transport by Stefan flow in addition
to that caused by diffusion. Such accounting is necessary for precise control
in industrial applications such as combustion and is described in many
chemical engineering texts (Kreith et al., 1999; Lienhard IV and Lienhard V,
2000; Bird et al., 2002). The phenomenon of transpiration through a stoma is
a reasonable proxy for a Stefan tube, the main difference being that
evaporation in the Stefan tube depletes the pool of evaporating liquid, whose
surface therefore recedes downward. By contrast, the evaporating water in the
stomatal cavity is continually replenished by vascular flow from within the
plant. If anything, this reinforces the magnitude of the upward vertical air
velocity, in contrast to the Stefan tube, and is consistent with that derived
from momentum conservation in Eqs. (4) and (5).
Non-diffusive transport by Stefan flow has implications for defining key
physiological parameters, which are more significant
than the percentages of CO2 and water vapour transport calculated
above. Plant physiologists have postulated that stomata act to maximise the
ratio of carbon gain to water loss (Cowan and Farquahar, 1977) or water use
efficiency (WUE), an ecosystem trait that constrains global biogeochemical
cycles (Keenan et al., 2013). In formulating this parameter, presuming
molecular diffusion to be the lone transport mechanism, the water vapour
conductance is usually taken as 1.6 times that of CO2 (Beer et
al., 2009), based on the ratio of their diffusivities – the inverse of the
square root of the ratio of their molecular masses, according to Graham's
law. Such an assumption underlies the very concept of stomatal control
(Jones, 1983), but neglects the role of non-diffusive transport for both
gases. Net momentum exiting stomata both expedites water vapour egress and
retards CO2 ingress, versus the case of static diffusion, in each
case acting to reduce the WUE. Importantly, water vapour transport by
stomatal jets depends not only on physiology but also physically on the state
variable q, according to Eq. (8). Consistent with the determinants of q,
as the temperature of a (saturated) stomatal environment increases, even for
a constant stomatal aperture, the WUE is reduced, wresting some control over
gas exchange rates from the plant. Perhaps equally importantly, opposition to
CO2 uptake by stomatal jets also should be considered when modelling
the most fundamental of biological processes, namely photosynthesis.
Accurate modelling of primary production in plants may require a fuller
description of stomatal transport mechanisms, including non-diffusive
expulsion by jets. The partial pressure of CO2 inside the stomata is
a key input parameter for the classic photosynthesis model (Farquhar et
al., 1980), but is never directly measured. Rather, it must be inferred from
gas exchange measurements and assumptions about the relative conductance of
water vapour and CO2, as described above. The amendment of such
calculations to account for non-diffusive transport of both CO2 and
H2O should help to improve the accuracy of physiological models.
As a final note regarding ecophysiology, studies of plant functioning
conducted using alternative gas environments should be interpreted with care.
Stomatal responses to humidity variations have been studied in several plant
species using the He:O2 gas mixture termed “helox”
(Mott and Parkhurst, 1991). In the context of conservation of linear
momentum, it is relevant that the effective molecular weight of helox is just
29 % that of dry air. Under equal
conditions of temperature and pressure, helox has far less density, and so
during transpiration both w|0 from Eq. (4) and the non-diffusive
component of stomatal transport from Eq. (7) are 3.5 times greater than in
air. The validity of helox for characterising natural plant functioning is
thus dubious due to its low inertia versus that of air.
Conclusions
Evaporation (E) is the dominant surface gas exchange, and
forces net upward momentum in the surface layer such that the lower boundary
condition for the vertical velocity is w|0=Eρ|0,
where ρ|0 is the air density at the surface. This non-zero vertical
velocity describes Stefan flow and implies gas exchange of a non-diffusive
nature, which must be extracted from the net transport of any gas prior to
relating that gas's resultant diffusive transport component to scalar
gradients, as in the Monin–Obukhov similarity theory. Such a correction of the
flux–gradient theory is of particular importance for descriptions of gas exchange
through plant stomata, which should be amended to account for non-diffusive
transport by stomatal jets, which help to expel water vapour but hinder the
ingress of CO2.
No data sets were used in this article.
The author declares that he has no conflict of interest.
Acknowledgements
This work is dedicated, with fondness and great esteem, to the memory of Ray
Leuning whose insights led to substantive improvements both in this work and
broadly in the science of surface gas exchanges. Investigation into this
matter was funded by Spanish national project GEISpain
(CGL2014-52838-C2-1-R). The author thanks Penélope Serrano-Ortiz, Enrique
Pérez Sánchez-Cañete, Óscar Pérez-Priego, Sonia Chamizo,
Ana López-Ballesteros, Russell L. Scott, and Jorge Pérez-Quezada and
anonymous reviewers for bibliographical guidance, comments and criticisms
that helped to clarify the manuscript.
Edited by: Armin Sorooshian
Reviewed by: Werner Eugster and two anonymous referees
ReferencesAaltonen, H., Pumpanen, J., Pihlatie, M., Hakola, H., Hellén, H.,
Kulmala, L., Vesala, T., and Bäck, J.: Boreal pine forest floor biogenic
volatile organic compound emissions peak in early summer and autumn, Agr.
Forest Meteorol., 151, 682–691, 10.1016/j.agrformet.2010.12.010, 2011.
Abramzon, B. and Sirignano, W. A.: Droplet vaporization model for spray
compustion calculations, Int. J. Heat Mass Tran., 32, 1605–1618, 1989.
Arya, S. P.: Introduction to Micrometeorology, Academic Press, San Diego,
307 pp., 1988.Beer, C., Ciais, P., Reichstein, M., Baldocchi, D., Law, B. E., Papale, D.,
Soussana, J.-F., Ammann, C., Buchmann, N., Frank, D., Gianelle, D., Janssens,
I. A., Knohl, A., Köstner, B., Moors, E., Roupsard, O., Verbeeck, H.,
Vesala, T., Williams, C. A., and Wohlfahrt, G.: Temporal and among-site
variability of inherent water use efficiency at the ecosystem level, Global
Biogeochem. Cy., 23, GB2018, 10.1029/2008GB003233, 2009.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N.: Transport Phenomena, John
Wiley & Sons, Cambridge, 2002.
Businger, J. A., Wyngaard, J. C., Izumi, Y., and Bradley, E. F.: Flux-profile
relationships in the atmospheric surface layer, J. Atmos. Sci., 28, 181–189,
1971.
Cowan, I. R. and Farquahar, G. D.: Stomatal function in relation to leaf
metabolism and environment, Sym. Soc. Exp. Biol., 31, 471–505, 1977.
Dyer, A. J. and Hicks, B. B.: Flux-gradient relationships in the constant
flux layer, Q. J. Roy. Meteor. Soc., 96, 713–721, 1970.Farquhar, G. D., von Caemmerer, S., and Berry, J. A.: A biochemical model of
photosynthetic CO2 assimilation in leaves of C3 species,
Planta, 149, 78–90, 1980.
Finnigan, J. J.: Response to comment by Dr. A.S. Kowalski on “The storage
term in eddy flux calculations”, Agr. Forest Meteorol., 149, 725–729, 2009.
Finnigan, J. J., Clement, R., Malhi, Y., Leuning, R., and Cleugh, H. A.: A
re-evaluation of long-term flux measurement techniques. Part I: Averaging and
coordinate rotation, Bound.-Lay. Meteorol., 107, 1–48, 2003.
Foken, T.: Micrometeorology, Springer-Verlag, Berlin, 306 pp., 2008.
Giancoli, D. C.: General Physics, Prentice-Hall, Englewood Cliffs, NJ, 1984.Gu, L.: An eddy covariance theory of using O2 to CO2 exchange
ratio to constrain measurements of net ecosystem exchange of any gas species,
Agr. Forest Meteorol., 176, 104–110, 10.1016/j.agrformet.2013.03.012,
2013.
Holton, J. R.: An Introduction to Dynamic Meteorology, Academic Press, San
Diego, 511 pp., 1992.
Jarman, P. D.: The diffusin of carbon dioxide and water vapour through
stomata, J. Exp. Bot., 25, 927–936, 1974.
Jones, H. G.: Plants and microclimate: a quantitative approach to
environmental plant physiology, Cambridge University Press, New York,
323 pp., 1983.
Kaimal, J. C. and Finnigan, J. J.: Atmospheric Boundary Layer Flows Their
Measurement and Structure, Oxford University Press, New York, 242 pp., 1994.
Katul, G., Cava, D., Poggi, D., Albertson, J., and Mahrt, L.: Stationarity,
homogeneity, and ergodicity in canopy turbulence, in: Handbook of
Micrometeorology, edited by: Lee, X., Massman, W., and Law, B., Kluwer
Academic, New York, 29, 161–180, 2004.Keenan, T. F., Hollinger, D. Y., Bohrer, G., Dragoni, D., Munger, J. W.,
Schmid, H. P., and Richardson, A. D.: Increase in forest water-use efficiency
as atmospheric carbon dioxide concentrations rise, Nature, 499, 324–327,
10.1038/nature12291, 2013.
Kowalski, A. S.: Exact averaging of atmospheric state and flow variables,
J. Atmos. Sci., 69, 1750–1757, 2012.
Kowalski, A. S. and Argüeso, D.: Scalar arguments of the mathematical
functions defining molecular and turbulent transport of heat and mass in
compressible fluids, Tellus B, 63, 1059–1066, 2011.Kowalski, A. S. and Serrano-Ortiz, P.: On the relationship between the eddy
covariance, the turbulent flux, and surface exchange for a trace gas such as
CO2, Bound.-Lay. Meteorol., 124, 129–141, 2007.
Kreith, F., Boehm, R. F., Raithby, G. D., Hollands, K. G. T., Suryanarayana,
N. V., Carey, V. P., Chen, J. C., Lior, N., Shah, R. K., Bell, K. J., Moffat,
R. J., Mills, A. F., Bergles, A. E., Swanson, L. W., Antonetti, V. W.,
Irvine, T. F., and Capobianchi, M.: Heat and Mass transfer, in: Mechanical
Engineering Handbook, edited by: Kreith, F., CRC Press LLC, Boca Raton,
4-1–4-287, 1999.
Lee, X.: On micrometeorological observations of surface-air exchange over
tall vegetation, Agr. Forest Meteorol., 91, 39–49, 1998.Leuning, R.: Transport of gases into leaves, Plant Cell Environ., 6,
181–194, 10.1111/1365-3040.ep11587617, 1983.
Lienhard IV, J. H. and Lienhard V, J. H.: A heat transfer textbook, J. H.
Lienhard V, Cambridge, MA, USA, 2000.
Mott, K. A. and Parkhurst, D. F.: Stomatal responses to humidity in air and
helox, Plant Cell Environ., 14, 509–515, 1991.Obukhov, A. M.: Turbulence in an atmosphere with a non-uniform temperature,
Bound.-Lay. Meteorol., 2, 7–29, 1971.
Parkinson, K. J. and Penman, H. L.: A possible source of error in the
estimation of stomatal resistance, J. Exp. Bot., 21, 405–409, 1970.Stull, R. B. (Ed.): An introduction to boundary layer meteorology, in:
Atmospheric and Oceanographic Sciences Library, Kluwer Academic Publishers,
The Netherlands, 13, 670 pp., 10.1007/978-94-009-3027-8, 1988.
Suni, T., Rinne, J., Reissell, A., Altimir, N., Keronen, P., Rannik, Ü.,
Dal Maso, M., Kulmala, M., and Vesala, T.: Long-term measurements of surface
fluxes above a Scots pine forest in Hyytiälä, southern Finland,
1996–2001, Boreal Environ. Res., 8, 287–301, 2003.von Caemmerer, S. and Farquhar, G. D.: Some relationships between the
biochemistry of photosynthesis and the gas exchange of leaves, Planta, 153,
376–387, 10.1007/bf00384257, 1981.
Wallace, J. M. and Hobbs, P. V.: Atmospheric science: an introductory survey,
International Geophysics, edited by: Dmowska, R., Hartmann, D., and Rossby,
H. T., Academic Press, Amsterdam, 483 pp., 2006.
Webb, E. K., Pearman, G. I., and Leuning, R.: Correction of flux measurements
for density effects due to heat and water vapour transfer, Q. J. Roy. Meteor.
Soc., 106, 85–100, 1980.
Wilczak, J. M., Oncley, S. P., and Stage, S. A.: Sonic anemometer tilt
correction algorithms, Bound.-Lay. Meteorol., 99, 127–150, 2001.