Download Sources and sinks of entropy in the atmosphere

Sources and Sinks of Entropy in the Atmosphere
1
David J. Raymond
Corresponding author: D. J. Raymond, Physics Department and Geophysical Research Center,
New Mexico Tech, Socorro, NM 87801, USA. ([email protected])
1
Department of Physics and Geophysical
Research Center, New Mexico Tech,
Socorro, New Mexico, USA
This article has been accepted for publication and undergone full peer review but has not been
through the copyediting, typesetting, pagination and proofreading process which may lead to
differences between this version and the Version of Record. Please cite this article as an
‘Accepted Article’, doi: 10.1002/2013MS000247
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Abstract.
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Moist entropy is generally treated as a conserved variable in
atmospheric models in the absence of external heat and moisture sources such
as radiation and surface fluxes. However, both irreversible generation and
non-advective transport of entropy occur in the atmosphere, with many of
these associated with moisture and precipitation. These entropy sources and
sinks are needed to balance the global entropy budget. Existing work on calculating the irreversible generation of entropy and non-advective transport
is extended to include the major effects of ice.
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1. Introduction
Johnson [1997] showed the importance of getting the entropy budget right in climate
models. In particular, he pointed out potential problems arising from the numerical
generation of entropy in these models. He noted that such extra entropy generation may
be responsible for the cold bias that most climate models show. This numerical entropy
source is hard to avoid with the normal choice of dependent variables in such models.
Peixoto et al. [1991], Goody [2000], and Warner [2005] made estimates of the natural
generation of entropy in the atmosphere, while Pauluis [2000], Pauluis et al. [2000], and
Pauluis and Held [2002a,b] evaluated the production of entropy in a numerical model of
convection. Bister and Emanuel [1998] showed that entropy generation by dissipative
heating is important in determining the maximum potential intensity of tropical cyclones.
As Emanuel et al. [1994] and Goody [2000] point out, thinking of latent heat release as a
source of heat external to the atmosphere, as is done by Peixoto et al. [1991], is probably
not a good idea, as the generation of latent heat is a process tightly integrated with
atmospheric motions. This implies that the use of moist entropy, where latent heating is
an internal process, is preferable to the use of dry entropy, where latent heating constitutes
an external entropy source. Consistent with this view, the term “entropy” is here taken
to mean “moist entropy”.
The specific entropy of air parcels (often in the guise of equivalent potential temperature) is generally considered to be conserved in the absence of external heat sources.
However, entropy can be created by real irreversible generation processes as well as by
spurious numerical processes in models. It also can be transported by molecular and eddy
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diffusive processes. Textbook examples of irreversible entropy generation are the rapid
expansion and compression of an ideal gas. Such violent processes are generally only important in the atmosphere when shock waves occur, a phenomenon not associated with
meteorological flows. Important irreversible processes generating entropy in the atmosphere include mechanical dissipation in turbulence, heating due to work done by falling
hydrometeors, evaporation of precipitation in dry air, and mixing of dry and moist air,
among others.
The form of the entropy source in the atmosphere depends on what is included as
part of an “atmospheric parcel”. This issue becomes rather complex when various forms
of condensate and precipitation exist in the atmosphere. Pauluis [2000] and Pauluis
and Held [2002a,b] include all water substance whether in the form of vapor, advected
condensate (small particles), or precipitating condensate (large particles). We propose
instead to include dry air plus water vapor and advected condensate (known as “total
cloud water”) as part of the parcel and consider precipitation to be external. This makes
it easier to implement numerically the entropy governing equation, as the advection of
falling precipitation does not enter. Of course, interactions between the air parcel and
the precipitation must then be considered. One advantage of this approach is that the
entropy inherent in precipitation that falls to the surface is lost to the atmosphere and
should not be counted as part of the atmospheric entropy, though of course it is part of
the overall entropy of the earth system.
As noted above, if the problem of numerical generation of entropy by numerical models
is not overcome, then careful evaluation of the natural sources of entropy is for naught.
One solution to this problem is to write the numerical governing equations in a form that
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explicitly conserves entropy in the absence of non-conservative physical processes. This
can be achieved by making the entropy and total cloud water primary dependent variables
with the governing equations for them written in flux form. One disadvantage of this is
that the entropy and total cloud water must be inverted for temperature and water vapor
mixing ratio. The iterative calculations needed to accomplish this can be computationally
intensive, but this problem can be minimized by generating look-up tables at the beginning
of the computation. Another important issue is the simultaneous conservation of entropy
and energy. Obtaining correct budgets for both constitutes a difficult challenge. If this is
impossible, then a decision has to be made as to which quantity it is more important to
conserve.
The purpose of this paper is to lay the groundwork for numerical models of the atmosphere that treat the entropy accurately. Aside from taking entropy as a primary
dependent variable as discussed above, the sources of entropy in the atmosphere must be
accurately evaluated. Pauluis [2000] and Pauluis and Held [2002a,b] have taken initial
steps in this direction. We use a somewhat more general technique for evaluating these
sources to extend their results. Certain classical textbooks in the area of non-equilibrium
thermodynamics are not clear in their presentation of irreversible entropy generation. In
particular, the texts by De Groot [1951], De Groot and Mazur [1962], and Yourgrau et al.
[1982] can mislead the unwary. (See Landau and Lifschitz [1959] and Prigogine [1967] for
alternate treatments.) We attempt to clarify these derivations, as they are pertinent to
this paper.
Section 2 presents the atmospheric governing equations for the case in which entropy
is a primary dependent variable. Care is taken to account for the fact the atmosphere
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is an open system with respect to water substance. In section 3 we present an accurate
definition of entropy that takes into account the existence of the ice phase. This definition
is a generalization of that presented by Emanuel [1994]. Section 4 presents the derivation
of entropy sources due to heating, phase changes, diffusion, and the sources and sinks of
water vapor. Section 5 compares our results with those of others and the conclusions are
presented in section 6.
2. Governing equations
The specific entropy s is defined here as the entropy of a parcel divided by the mass of
dry air in the parcel, which may be written
s = sD + rV sV + rL sL + rI sI
(1)
where sD is the specific entropy of dry air, sV , sL , and sI are the specific entropies of
vapor, advected liquid water, and advected ice condensate, while rV , rL , and rI are the
mixing ratios of these quantities. Since the system for which the entropy is defined consists
of components that move with the dry air component, the governing equation takes the
simple form
ds
= XE
dt
(2)
where d( )/dt represents the usual material derivative of fluid dynamics and XE represents
the reversible and irreversible sources of entropy per unit mass of dry air.
Defining specific entropy relative to the mass of dry air is consistent with writing the
mass continuity equation for dry air alone, thus avoiding the necessity of including water
substance source terms in this equation:
dρD
+ ρD ∇ · v = 0
dt
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(3)
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where ρD is the density of dry air and v is its velocity, assumed to be the same as that of
the water vapor and advected condensate.
The momentum equation for the dry air component of the atmosphere in this formulation is
dv ∇p − ∇ · (2KD)
+
dt
ρ (1 + rT )
( D
)
1 + rT + rR
+g
k + 2Ω × v = 0
1 + rT
(4)
where rT = rV + rL + rI is the total cloud water mixing ratio, rR is the mixing ratio of
precipitation, p is the total pressure, K is the dynamic eddy mixing coefficient (i.e., it
includes the density), k is the vertical unit vector, g is the acceleration of gravity, Ω is
the rotation vector of the earth, and
1
Dij =
2
(
∂vi
∂vj
+
∂xj ∂xi
)
(5)
is the strain rate. (This, of course, assumes that the turbulence can be represented by
a flux-gradient relationship with an eddy viscosity, however derived. There are many
schemes of varying complexity to determine K, which we do not go into here.) The
precipitation is assumed to be falling at its terminal fall velocity with respect to the
air. Transient situations in which this is not true are assumed to be unimportant. It is
sufficient to focus on the motion of the dry air component, as the vapor and advected
condensate are assumed to move with the dry air.
The total cloud water mixing ratio obeys the equation
drT
1
=
∇ · (K∇rT ) + E − P
dt
ρD
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(6)
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where E is the evaporation rate of precipitation per unit mass of dry air and P is the
formation rate of precipitation. The corresponding equation for precipitation is
drP
1
=
∇ · (ρD wT k + K∇rP ) + P − E
dt
ρD
(7)
where only a single category of precipitation with terminal fall speed wT is included for
simplicity.
The flux forms of (2), (4), (6), and (7) can be obtained in the standard manner by
multiplying each by ρD and invoking the mass continuity equation (3).
3. Entropy definition
The specific entropy for dry air in terms of temperature T and dry air partial pressure
pD is
sD = CP D ln(T /TF ) − RD ln(pD /pR ) + sRD
(8)
where CP D is the specific heat of dry air at constant pressure, the gas constant for air
is RD = R/mD where mD is the molecular weight of dry air and R is the universal gas
constant, and sRD is the constant reference entropy for dry air. The reference temperature
has been taken to be the freezing point TF . The reference pressure for dry air pR is taken
to be 1000 hPa.
For vapor we have a similar equation
sV = CP V ln(T /TF ) − RV ln(pV /eSF ) + sRV
(9)
where pV is the partial pressure of vapor, CP V is the specific heat of water vapor at
constant pressure, RV = R/mV is the gas constant for water vapor where mV is the
molecular weight of water, and sRV is the constant reference entropy for water vapor.
The reference pressure for water vapor is eSF , the saturation vapor pressure at freezing.
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For condensate, assumed to be incompressible, the specific entropy is
sC = CC ln(T /TF ) + sRC
(10)
where CC is its specific heat and sRC is the constant reference entropy for the condensate.
This equation applies to both ice (C → I) and liquid water (C → L) with appropriate
choice of constants.
Following convention, the reference entropies for dry air and liquid condensate are set to
zero, sRD = sRL = 0. Equilibrium between a saturated atmosphere and liquid condensate
then requires that
sRV =
LL (TF )
TF
(11)
where LL (T ) is the latent heat of condensation. For ice, the reference entropy in (10) is
negative as the result of equilibrium between liquid water and ice at freezing,
sRC = sRI = −
LF
TF
(12)
where LF is the latent heat of freezing at the freezing point.
Putting these results together and defining separate liquid and ice condensate components results in a seamless expression for the specific entropy valid both above and below
freezing,
s = (CP D + rV CP V + rL CL + rI CI ) ln(T /TF )
− RD ln(pD /pR ) − rV RV ln(pV /eSF )
+
LL (TF )rV − LF rI
,
TF
(13)
where CL and CI are the respective specific heats of liquid and ice, assumed constant.
(Note that the assumption of constancy for CI is not a good approximation. However,
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the vapor pressure is so low below freezing that this approximation is unlikely to have
significant consequences in most circumstances.) Since this is an equilibrium equation,
rL = 0 below the freezing point and rI = 0 above the freezing point. Furthermore,
rV equals the saturation mixing ratio over liquid if rL is non-zero, and that for ice if
rI is non-zero. One could rewrite the vapor pressure for water vapor as the saturation
vapor pressure times the relative humidity, as is done by Emanuel [1994]. This has some
conceptual advantages, but (13) is simpler to calculate numerically.
For later reference, we present the final equations for the specific entropies of vapor,
liquid, and ice:
sV = CP V ln(T /TF ) − RV ln(pV /eSF ) + LL (TF )/TF ,
(14)
sL = CL ln(T /TF ),
(15)
sI = CI ln(T /TF ) − LF /TF .
(16)
We also need the chemical potentials µ of liquid and ice condensate. Using the enthalpy
equation hC = CC (T − TF ) − LC (TF ) + CP V TF and the general relation µ = h − sT , we
find the chemical potentials of liquid and ice to be
µL = CL (T − TF ) − CL T ln(T /TF ) − LL (TF ) + CP V TF ,
(17)
and
µI = CI (T − TF ) − CI T ln(T /TF )
+ T LF /TF − LI (TF ) + CP V TF .
(18)
Recognizing that LF = LI (TF ) − LL (TF ), we easily verify that µL = µI at T = TF , as
required by liquid-ice equilibrium at freezing.
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4. Entropy sources and sinks
We now consider the form of the specific entropy source term XE . By “source”, we
mean all non-advective contributions to the Eulerian entropy tendency, including (1)
non-advective transport, which moves entropy around but is not associated with a net integrated source; (2) entropy entering and exiting the atmosphere due to radiation, surface
fluxes, and interaction with precipitation, which is considered to be outside the atmospheric system; and (3) the irreversible generation of entropy.
The bases for the understanding of entropy sources and sinks within a parcel are the
Gibbs equation and the open system version of the first law of thermodynamics (Landau
and Lifschitz [1959]; Prigogine [1967]). The Gibbs equation comes from the assumption
that the entropy S of a parcel is a function of its internal energy E, its volume V , and
the masses of the individual components Mi , resulting in
(
)
∑ ( ∂S )
∂S
dS =
dE +
dV +
dMi
∂V
∂M
i
i
∑ µi dMi
dE pdV
=
+
−
T
T
T
i
∂S
∂E
)
(
(19)
where the thermodynamic definitions of temperature T , pressure p, and the chemical
potential of each component µi are used.
For internal transformations such as phase changes, the total mass of the atmospheric
parcel does not change. However, for transfers of mass into and out of the system, it does.
We therefore separate mass changes into two parts,
dMi = dNi + dOi
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(20)
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with the dNi referring to internal transformations and the dOi indicating masses entering
and exiting the system. The sum of the dNi is zero, but this condition does not hold for
the dOi .
The open system version of the first law is
dE = dQ − dW +
∑
ei dOi
(21)
i
where the heat added to the test parcel is dQ, the work done by it is dW , and the specific
internal energies of the various components are ei . The existence of the last term in
(21) simply indicates that mass entering or exiting the system carries energy with it.
Substituting (20) and (21) into (19) results in
T dS = dQ − dW + pdV
−
∑
µi dNi +
∑
i
(ei − µi ) dOi
(22)
i
We assume that the volume change for an open system dV = dVX +dVM is either a result
of expansion and compression of the atmospheric parcel at constant mass (dVX ) or due
to the addition or removal of mass from the parcel at constant pressure and temperature
(dVM ). The work done by the parcel is therefore dW = pdVX , since adding mass to
the parcel without changing its temperature and pressure changes the parcel’s volume,
but does no work. This is because no actual expansion occurs; in essence we are simply
re-defining the boundary of the (open) parcel to include the added mass. For this process,
pdVM =
∑
∑
d (pi VM ) =
i
=
i
∑
∑
d (Mi Ri T )
i
Ri T dOi =
(hi − ei )dOi
(23)
i
where the ideal gas law is used, with pi being the partial pressure of the ith component,
hi = ei + Ri T is its specific enthalpy, and where it is recognized that dMi = dOi in this
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case. (This also works for the condensed components, which can be considered to be ideal
gases with very massive molecules, resulting in vanishingly small partial pressure and gas
constant.)
Further insight comes from noting that
dW = pdVX = pdV − pdVM
= pdV −
∑
(hi − ei )dOi .
(24)
i
Recalling that the chemical potential can be written in terms of the specific enthalpy hi ,
the specific entropy si , and the temperature as µi = hi − si T , substitution of (24) into
(22) yields
1
dS =
T
(
dQ −
∑
)
µi dNi
+
∑
i
si dOi .
(25)
i
Therefore, the entropy of the parcel can change by adding heat, by undergoing nonequilibrium phase transformations or chemical reactions, and by adding or subtracting
mass. For multiple phases in equilibrium, the chemical potentials µi are all equal, meaning
that the second term on the right side of (25) vanishes in equilibrium, since
∑
dNi = 0.
Dividing (25) by MD dt, the mass of dry air in the parcel and the time differential results
in an expression for the entropy source term XE :
( )
∑ ( dri )
q
1∑
dri
XE = −
µi
+
si
T
T i
dt N
dt O
i
(26)
where q is the heating rate per unit mass of dry air, ri is the mixing ratio of the ith component, the subscript N indicates an internal phase change or chemical transformation,
and the subscript O indicates the addition or removal of a component from the parcel.
We split XE into three parts corresponding to the three terms in (26),
XE = XQ + XN + XO ,
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(27)
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and discuss each below.
4.1. Internal phase transformations
Since the atmospheric parcel considered here does not include precipitation, internal
phase transitions are those between vapor, advected liquid, and advected ice. Advected
vapor and condensate tend to be very close to equilibrium. However, small liquid cloud
droplets can be lifted significant distances above the freezing level before freezing occurs.
We neglect this effect for advected condensate, as we assume that all phase transitions
within the atmospheric system (i.e., excluding precipitation) occur in equilibrium, and
therefore generate no entropy. It seems likely in any case that the entropy generation due
to this effect is minor compared to other sources of entropy. Thus, we set XN = 0.
Since precipitation is outside of the atmospheric system, non-equilibrium effects in the
evaporation and melting of precipitation are treated differently. These effects are described
below.
4.2. Entropy sources due to heating
The entropy source due to heating is
XQ =
∑
i
XQi =
∑ qi
Ti
i
(28)
where the qi are the heating rates per unit mass of dry air from various mechanisms
and the Ti are the temperatures at which the respective heat sources act. The heating
has five main components, heating due to radiation qR , large-scale heat conduction qC ,
heat conduction on the microscale near precipitation particles qM , viscous dissipation of
kinetic energy qV , and transfer of sensible heat to and from precipitation particles qP .
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The corresponding entropy source terms are
XQ = XQR + XQC + XQM + XQV + XQP .
(29)
Evaluation of qR and XQR require a radiative transfer model and are not considered further
here. The term XQM due to microscale heat conduction is discussed later in conjunction
with the microscale vapor diffusion term XOM .
4.2.1. Large-scale heat conduction
For heat conduction, ρD qC equals minus the divergence of the molecular heat flux
ρD qC = ∇ · (κ∇T )
(30)
where κ is the thermal conductivity of air. Thus, the entropy source due to heat conduction is
XQC =
∇ · (κ∇T )
,
ρD T
(31)
where we have assumed that heat from this source is added to the atmosphere at the
ambient atmospheric temperature. Equation (31) can be written
ρD XQC = ∇ · (κ ln T ) +
κ|∇T |2
,
T2
(32)
where the entropy source per unit volume ρD XQC consists of minus the divergence of
a molecular entropy flux −κ∇T and a positive definite quantity proportional to the
temperature gradient squared. The first term represents lossless molecular transport of
entropy while the second represents the irreversible generation of entropy in molecular
conduction.
Heat conduction in laminar flows in the atmosphere provides a negligible source of
entropy. However turbulence sends variance in temperature T down-scale where the corresponding temperature variance is dissipated by heat conduction. We thus replace the
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molecular heat flux −κ∇T by the turbulent heat flux −CP D K∇T where K is the dynamic
eddy mixing coefficient. Thus, in the presence of turbulence,
XQC =
CP D ∇ · (K∇T )
.
ρD T
(33)
This has the same properties as the molecular case, resulting in an irreversible entropy
source plus the divergence of the eddy flux of entropy.
It would be tempting to replace ∇T by ∇θ for the eddy flux case, where θ is the
potential temperature, since the potential temperature is conserved in eddy motions on
scales much greater than the inner scale. However, eddy viscosity differs from molecular
viscosity in that some of the kinetic energy is transformed into potential energy and vice
versa. This energy does not ultimately contribute to the irreversible generation of entropy.
Furthermore, replacement of ∇T by ∇θ results in a form for (33) that cannot be reduced
to a positive definite source plus the divergence of a flux, indicating that this replacement
is inappropriate. That said, (33) constitutes no more than a consistent educated guess
for the form of the entropy source in the case of turbulence representable by an eddy
viscosity. A deeper analysis is needed to determine whether this guess is correct.
4.2.2. Viscous dissipation
Heating due to viscous dissipation in the atmosphere has two sources, the end point of
the turbulent energy cascade and dissipation of work done on the atmosphere by falling
precipitation. The source of kinetic energy per unit volume due to the the turbulent
energy cascade in an essentially incompressible fluid is
v · ∇ · (KD) = ∇ · (Kv · D) − 2K|D|2
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(34)
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where v is the velocity, D is the strain rate given by (5), and KD is the Reynolds stress.
The first term on the right side of this equation simply moves kinetic energy around in
the fluid without dissipating it. However, minus the second term is the heat source due to
viscous dissipation, which should appear as a heat source in the entropy source equation.
The entropy source due to turbulent dissipation and falling precipitation together takes
the form
XQV =
2K|D|2 + gρD rP wT
ρD T
(35)
(Landau and Lifschitz [1959], p 54; Pauluis et al. [2000]) where g is the acceleration
of gravity, rP is the mixing ratio of precipitation, and wT is the terminal fall speed
of hydrometeors. Note that both terms in this equation represent irreversible entropy
generation and are positive definite.
4.2.3. Heat transfer to precipitation
We now consider the transfer of heat to the atmosphere that is required to keep the
precipitation at the wet bulb temperature TW as it warms while falling into warmer air,
possibly melting at the freezing level. For simplicity, we assume that melting occurs
instantaneously as precipitation particles cross the freezing level. The heat flow toward
hydrometeors associated with evaporation is treated separately.
The precipitation-related heating term takes the form
[
]
∂TW
qP = rP (vz − wT ) −CC
+ LF δ(z − zF )
∂z
(36)
where vz is the vertical component of air velocity, the condensate specific heat CC is that
appropriate to ice or liquid depending on the temperature, and zF is the elevation where
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TW = TF . The corresponding atmospheric entropy source term is
XQP =
qp
.
TW
(37)
In most cases qP , and hence XQP , are negative, since heat is flowing from the atmosphere
into the precipitation as it falls into warmer air. However, if a hydrometeor is being
carried upward in an updraft, then qp could be positive as the hydrometeor cools. Since
precipitation is external to the atmospheric system, this term results in the transfer of
entropy in and out (mostly out) of the atmosphere as precipitation warms in its fall to
the surface.
4.3. Mass sources and sinks
Mass sources and sinks of vapor and condensate come from three processes, large-scale
diffusion, the formation and evaporation of precipitation, and microscale diffusion near
evaporating hydrometeors:
XO = XOD + XOP + XOM .
(38)
4.3.1. Large-scale diffusive source of entropy
The molecular diffusion of condensate particles (i.e., Brownian motion) is much weaker
than the diffusion of vapor, due to the large masses of these particles. However, eddy
motions produce eddy diffusive fluxes of vapor and advected condensate that are quite
similar. Thus, from (26), we have
XOD =
∑
i
(
si
dri
dt
)
=
O
1 ∑
si ∇ · (K∇ri ) ,
ρD V,L,I
(39)
where (dri /dt)O is the eddy diffusive time tendency of the ith component and the sum is
over vapor, advected liquid, and advected ice. Note that diffusion of the dry air component
does not occur since the parcel is considered to move with the dry air. As with the case
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of heat conduction, the diffusive flow of vapor away from an evaporating hydrometeor is
considered separately.
Equation (39) may be written
ρD XOD = ∇ ·
∑
V,L,I
(Ksi ∇ri ) − K
∑
∇si · ∇ri .
(40)
V,L,I
The first term on the right in this equation is minus the divergence of the entropy flux due
to diffusion of the components of water substance. The second term should, in principle,
be a positive definite entropy source. For the water vapor component, the gradient of
entropy is
∇sV = CP V ∇ ln T − RV ∇ ln pD − RV ∇ ln rV
where rV ∝ pV /pD is used.
(41)
In the original derivation of the Gibbs equation (see
23), the pressure and temperature are assumed to be approximately constant, whence
∇T, ∇pD ≈ 0. In this approximation, −∇sV · ∇rV ≈ RV ∇ ln rV · ∇rV , which is positive
definite. For liquid and ice, Ri → 0, and the entropy generation by condensate mixing is
approximately zero. Thus, making approximations consistently results in positive-definite
entropy generation due to mixing of water substance components. In a more accurate formulation, water vapor fluxes proportional to the gradients in T and pD would need to be
taken into account.
4.3.2. Formation and evaporation of precipitation
The evaporation of precipitation in an unsaturated environment is a non-equilibrium
process. However, a microscopic view of this process reveals that the evaporation from the
hydrometeor into the thin layer adjacent to its surface is essentially reversible, since this
layer is very close to saturation, with a temperature nearly equal to the temperature of the
hydrometeor. The true irreversible entropy source in this case arises from a combination
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of molecular heat conduction toward the hydrometeor and water vapor diffusion away
from it.
In what follows, we assume that the hydrometeor takes on the wet bulb temperature
TW of the atmosphere, i.e., it acts analogously to a wet bulb thermometer. Transients
in temperature due to the fact that a hydrometeor is generally encountering increasing
atmospheric temperatures as it falls are ignored. This simplifying assumption is valid for
small hydrometeors, such as those occurring in stratiform rain, and poor for very large
particles such as hail. In principle TW could be replaced by the actual temperature of the
precipitation, as obtained, for instance, from a full energy budget of the falling particles.
Precipitation formation occurs formally in the present context by the aggregation of
advected condensate or its accretion onto existing precipitation. In both cases, advected
condensate leaves the atmospheric system and the entropy sink due to this process is
sC (T )P where P is the formation rate of precipitation mass per unit mass of dry air.
Evaporation is the reverse process, but it occurs at the wet bulb temperature at the
surface of hydrometeor, with an entropy source due to the transfer of condensate back
into the atmospheric system equal to sC (TW )E where E is the evaporation rate of precipitation mass per unit mass of dry air. The specific entropy of condensate is used rather
than that of vapor, because we formally account for the evaporation after the condensate to be evaporated leaves the hydrometeor and enters the atmospheric system, thus
cooling the atmosphere rather than the condensate. Since the evaporation then occurs
in the microlayer next to the drop where the air is saturated, it takes place under nearly
equilibrium conditions, thus producing no additional entropy. Putting the precipitation
generation and evaporation terms together, the entropy source due to the mass of advected
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condensate transferred to and from precipitation is given by
XOP = sC (TW )E − sC (T )P.
(42)
This is another term representing the flow of entropy between the atmosphere and precipitation as it forms and evaporates.
4.3.3. Microscale diffusion near hydrometeors
We now consider the small scale diffusion of heat toward an evaporating hydrometeor
and the diffusion of vapor away from it. Both processes can be significant sources of
entropy.
The entropy generation from the diffusive flow of vapor from the saturated microlayer
next to evaporating hydrometeors to the free atmosphere is
XOM =E [sV (T, pV ) − sV (TW , pS )]
[
(
)
(
)]
T
pV
=E CP V ln
− RV ln
TW
pS (TW )
(43)
where pV is the vapor pressure in the free atmosphere and pS (TW ) is the saturation vapor
pressure at the wet bulb temperature.
An additional source of entropy in this case comes from the microscale heat conduction
toward the hydrometeors needed to balance the evaporative cooling adjacent to the surface
of the drop. This takes the form
(
XQM = ELC (TW )
1
1
−
TW
T
)
(44)
since the heat flow is down the temperature gradient from the ambient temperature T to
the surface temperature of the drop TW . The latent heat of evaporation LC becomes the
liquid form above the freezing level and the ice form below.
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Equations (43) and (44) may be combined into a single, simplified equation. Invoking
the minor approximation that ln(T /TW ) = (T − TW )/T and using the Clausius-Clapeyron
equation
(
pS (T ) = eSF
TF
T
)(CC −CP V )/RV
(
exp
LC (TF ) LC (T )
−
RV TF
RV T
)
(45)
where eSF is the saturation vapor pressure at freezing, the result is
[
XOM + XQM = E −RV ln
(
pV
pS (T )
)
(
+ CP V ln
T
TW
)]
.
(46)
This represents irreversible entropy generation and is positive-definite since pV ≤ pS (T )
and TW ≤ T .
4.4. Heat and moisture fluxes from the ocean surface
The analysis of the entropy created by sensible and latent heat fluxes from the ocean
surface differs from that for evaporating precipitation in that the surface itself is assumed
to provide the heat driving the evaporation. There is thus no flow of heat to the surface
needed to evaporate the surface condensate. The evaporation produces a source of water
vapor in a thin layer of the atmosphere adjacent to the surface with entropy per unit mass
equal to sV [TS , pS (TS )], where TS is the temperature of the surface, pS (TS ) is the saturation
vapor pressure at the surface temperature, and T and pV are the temperature and water
vapor pressure in the boundary layer. In analogy with the diffusion of water vapor away
from evaporating precipitation, the entropy generated per unit mass as the vapor diffuses
into the free boundary layer is sV (T, pV ) − sV [TS , pS (TS )]. Summing these results in a
net entropy generation per unit mass due to evaporation of sV (T, pV ). Assuming an
evaporation rate per unit area per unit time of FV and a sensible heat flux out of the
ocean surface of FS , the combined entropy source due to surface heat and moisture fluxes
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is
FE = FV sV (T, pV ) + FS /T.
(47)
Land surfaces aren’t necessarily saturated and they require a slightly different treatment.
5. Comparison with other results
5.1. Pauluis and Held
Pauluis [2000] and Pauluis and Held [2002a,b] derived equations for the irreversible
generation of entropy in a moist atmosphere. Most of their results had to do with generally
neglected heat sources such as turbulent dissipation, dissipative work done by falling
raindrops, and the divergence of the heat flux due to molecular conduction. However,
they also showed that the diffusion of water vapor in the atmosphere is a significant source
of irreversible entropy generation. Their expression in our notation for the irreversible
generation of entropy by the diffusive transfer of a mass of water vapor dO from parcel 1
to parcel 2 is
dSirr = RV dO ln(pV 1 /pV 2 )
(48)
where RV is the gas constant for water vapor and pV 1 and pV 2 are the vapor pressures of
water vapor in parcel 1 and parcel 2.
The change in entropy in a parcel according to our formulation is given by (25). Neglecting heating and phase changes, this equation reduces to dS = sV dO for the diffusive
transfer of a mass dO of water vapor into a parcel. For two parcels in which transfer is
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from parcel 1 to parcel 2, the net irreversible entropy source is
dSirr = dS2 + dS1
= sV 2 dO2 + sV 1 dO1
= (sV 2 − sV 1 )dO
= [CP V ln(T2 /T1 ) + RV ln(pV 1 /pV 2 )] dO
(49)
where dO ≡ dO2 = −dO1 , T1 and T2 are the temperatures in the two parcels, and CP V is
the specific heat of water vapor at constant pressure.
Note that our result is equivalent to that of Pauluis and Held if the temperatures of the
two parcels are the same. However, since our results do not take into account all of the
effects of temperature gradients in this case, our results are not necessarily more accurate
than those of Pauluis and Held.
There is also a difference between the Pauluis-Held results and our equation (46) for
the generation of entropy by evaporating precipitation. In addition to −RV ln[pV /pS (T )]
in the Pauluis and Held papers, we have the term CP V ln(T /TW ). However, this is much
smaller than the first term since typically ln(T /TW ) ≪ 1 whereas − ln[pV /pS (T )] can
become quite large for low relative humidities. The difference probably has to do with our
assumption that the droplet temperatures are equal to the wet bulb temperature rather
than the actual temperature.
Pauluis [2000] and Pauluis and Held [2002a,b] have no entropy source corresponding to
water substance entering and exiting the atmospheric system, as represented by (42), as
they consider precipitation to be part of the system. The corresponding physical effects
appear as precipitation falls to the surface.
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5.2. Classical textbooks
Landau and Lifschitz [1959] and Prigogine [1967] explicitly take into account the fact
that isothermal diffusion transfers the enthalpy as well as the mass of the diffused component (see page 221). Without this enthalpy transfer, the specific entropy si in (25) would
be replaced by −µi /T . The irreversible generation of entropy by diffusion given by (49)
would be unaffected, since −µV /T differs from sV by only an additive constant that would
cancel in the calculation of dSirr . However, the entropy source due to precipitation evaporation given in (42) (approximating TW by T ) would be given in terms of the chemical
potential of the condensate µC rather than its specific entropy,
XOP = −
µC (E − P)
T
(incorrect),
(50)
which is seriously incorrect. (The author was first alerted to this problem by the disastrous
effect of (50) on a numerical model of convection!)
Certain well-known texts (e.g., de Groot [1951], de Groot and Mazur [1962], Yourgrau et
al. [1982]) do not make it clear that the enthalpy transport associated with mass transport
into a parcel (by diffusive or other mechanisms) must be included in the calculation of
the irreversible generation of entropy. In the case of mass transfer by diffusion of an
ideal gas, the effects of the error on irreversible entropy generation vanish by accident, as
noted above. However, if the specific enthalpy of the substance being diffused does not
vary linearly with absolute temperature, the effects of the error reappear. Furthermore, if
material actually enters or exits the system under consideration (as with the conversion
of total cloud water to and from precipitation in our case), the results are disastrously
wrong, as shown above.
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6. Conclusions
Summarizing our analysis, the total atmospheric entropy source (exclusive of radiative
effects) is
1
XE =
ρD
[
]
CP D ∇ · (K∇T ) ∑
+
si ∇ · (K∇ri )
T
V,L,I
2K|D|2 + gρD rP wT
ρD T [
]
rP (vz − wT )
∂TW
+
−CC
+ LF δ(z − zF )
TW
∂z
+
+ EsC (TW ) − PsC (T )
[
(
)
(
)]
pV
T
+ E −RV ln
+ CP V ln
.
pS (T )
TW
(51)
Line by line, we have:
1. molecular processes of heat conduction and diffusion of water substance, augmented
by the down-scale transfer of variance to molecular scales by turbulence;
2. production of heat by viscous dissipation augmented by turbulence and the work
done by falling precipitation on the atmosphere;
3. heat transfer to falling precipitation required to drive its temperature toward the
wet bulb temperature of the air; also included is the heat transfer required to melt frozen
precipitation as it descends through the freezing level;
4. transfer of entropy to and from the atmospheric microlayer surrounding hydrometeors by the formation and evaporation of precipitation;
5. entropy source associated with the microscale diffusion of water vapor away from
and the microscale heat conduction toward evaporating hydrometeors.
In addition, entropy can enter the atmosphere from surface fluxes given by (47).
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Equation (51) represents the total non-advective source of entropy, including irreversible
entropy generation, flow of entropy to and from precipitation and the surface, and nonadvective transports. It is conventional to separate these quantities in textbooks, but
from the point of view of numerical simulations, the total source (which may be negative
in places) is what is needed. The results as presented are also specific to the case in which
turbulent fluxes are represented by an eddy viscosity, though no restriction is placed on
how this eddy viscosity is determined.
Our results extend the theoretical development of Pauluis [2000] and Pauluis and Held
[2002a,b] in a number of respects. In particular, the difference between the atmospheric
temperature and that of precipitation particles is accounted for in the present work and
the most important effects of the ice phase are included. It remains to incorporate these
results in a model that exhibits no numerical generation of entropy. This is feasible, with
certain other compromises, given the development in section 2. The model of Raymond
and Zeng [2005] conserves the integrated equivalent potential temperature explicitly, so
only a small modification of that model is required to achieve that goal.
Acknowledgments. Thanks are due to Kerry Emanuel and Olivier Pauluis for penetrating reviews. Thanks also to Sharon Sessions for useful discussions. This work was
supported by National Science Foundation Grant 1021049.
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