Latent heat fluxes during stably stratified conditions
Stephan de Roode
with contributions from Fred Bosveld and Reinder Ronda
Turbulent transport: Turbulent kinetic energy E > 0
For very stable conditions shear generation of TKE is not sufficient(?)
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'w'pE'w
zz
V'w'v
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U'w'u''w
g
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E
buoyancy shear production turbulent transport dissipation
TKE production requires
Write
and if KH = KM then TKE production for the following criterion
Bulk critical Richardson number
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1VU
zg
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Stable boundary layers
Cabauw data 2000-2006
Data selection:
weak winds: Utot (z=10 m) < 3 m/s
clear skies: LWnet,sfc > 40 W/m2
nighttime: SWnet,sfc = 0 W/m2
7.1% of all data points satisfy these criteria
Stable boundary layers
Cabauw data 2000-2006
Further data selection:
Latent, sensible and ground heat flow at 5 cm
are available
weak winds: Utot (z=10 m) < 3 m/s
clear skies: LWnet,sfc > 40 W/m2
nighttime: SWnet,sfc = 0 W/m2
3.5% of all data points
Surface energy balance during the night
-G = SW + LW + H + LE
G = energy flow into the ground
SW = net shortwave radiation
LW = net longwave radiation
H = sensible heat flux
LE = latent heat flux (evaporation)
night-time
Low wind speeds during stable nights:
turbulent fluxes H and LE become very small.
Cabauw Monthly Mean Surface Energy Balance results
for stable boundary layers
month N10mins G|10 cm G|05 cm LWnet H LE
Jan 536 -6.6 -9.7 50.4 -4.7 -1.4
Feb 751 -6.6 -10.0 54.9 -6.8 -0.1
Mar 1478 -3.0 -8.4 52.1 -5.9 0.4
Apr 1620 -0.9 -7.8 49.6 -6.2 0.8
May 1291 0.3 -7.4 48.5 -6.5 0.3
Jun 867 -0.0 -8.0 48.0 -7.9 2.1
Jul 798 0.0 -7.3 46.5 -8.7 1.1
Aug 1333 -1.8 -7.4 46.9 -9.8 1.5
Sep 1557 -3.9 -9.9 46.5 -8.5 -0.3
Oct 1098 -3.9 -7.6 47.7 -9.8 -0.9
Nov 723 -7.1 -10.2 48.4 -6.0 -2.1
Dec 1122 -8.7 -11.9 50.7 -5.7 -1.4
[W/m2]
Is it possible to close the surface energy balance from observations?
Dew formation in Wageningen
Results based on modeling and observations
from Jacobs et al. (2006)
Percentage of monthly dew nights in Wageningen
Energy equivalence of dew formation
Evaporation of 1 kg (= 1mm) of water:
2.5·106 J
In Wageningen monthly mean dew formation:
3.5 mm/month = 0.12 mm/day
Rough estimation: Assume that dew formation takes place during 12 hours
(half a day) then the typical heat production due to dew formation amounts
2.5·106 x 0.12 / 12 / 3600 = 7 W/m2
A few examples from Cabauw
• Select clear nights with low wind speeds
• Select negative humidity tendencies
• In the examples that will be shown the observed latent heat flux ~ 0 W/m2
Moisture tendencies: weak wind velocities
z=2m
z=10m
z=80m
z=200m
qsat,sfc
Moisture tendencies: weak wind velocities
z=2m
z=10m
z=80m
z=200m
qsat,sfc
Moisture tendencies: weak wind velocities
z=2mz=10m
z=80m
z=200m
qsat,sfc
Moisture tendencies: very weak wind velocities
z=2mz=10m
z=80m
z=200m
qsat,sfc
Moisture tendencies: very weak wind velocities
z=2m
z=10m
z=80mz=200m
qsat,sfc
Add selection criterion: humidity tendency < 0.02 g/kg/hour at
z=20m
Monthly mean humidity tendencies
z=2m
z=10m
z=80m
z=200m
Conclusions from observations
• Specific humidity in the lower part of the atmosphere follows surface
saturation specific humidity rather well
• It is difficult to assess the magnitude of the large-scale horizontal advection
of moisture
Diagnose the latent heat flux from the humidity budget equation
x
qU
z
'q'w
t
q
ii
Latent heat fluxes diagnosed from the humidity budget equation
Monthly mean values
I
II
I. Neglect large-scale advection term
II. Large-scale (ls) tendency correction.
Assume that the tendency at 200 m is
representative for the ls-tendency at every
height.
Can we possibly explain transport for stable conditions?
Prognostic equations
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Turbulent potential energy TPE
Total turbulent energy TTE (Zilitinkevich, 2007)
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z/
''g
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1ETPETKETTE
v
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z/
g'w'pTTE'w
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U'w'u
t
TTE
v00
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shear production turbulent transport total dissipation
Prognostic equations
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pvv
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Stable <0 always >0 ! transport dissipation radiation
Unstable > 0
Large eddy simulation
• Initial (very) stable lapse rate:
• No fluxes at the surface, no moisture present
• No horizontal winds: U=V=0.001 m/s
• nr of grid points Nx=Ny=64 , Nz=80
• x=y=z=5m
• sinusoidal initial perturbation at layers between 50 and 150 m
• amplitude of perturbation AMPL =0.5 K
What will happen?
K/km 10z
Turbulent kinetic energy (TKE) - Turbulent potential energy (TPE)
TKE
TPE (buoyancy variance term)
● TKE generation by TPE, their sum is not conserved (dissipation)
● Rapid oscillations (but not Brunt-Vaissala frequency)
● vertical integral buoyancy flux > 0 for d/dt TKE > 0 and vice versa
Lorenz, Available potential energy and the maintenance of the
general circulation, Tellus., 1955
Lorenz showed that available potential energy (APE) is given
approximately by the volume integral over the entire atmosphere of
the variance of potential temperature on isobaric surfaces:
As the potential temperature is conserved for adiabatic processes,
and as kinetic energy is produced, the enthalpy (cpT) of the
atmosphere should decrease (see also Holton 1992)
dV
'
V
1APE
2
2
Temperature evolution
t=0
t=1200 s
Enthalpy change
enthalpy
PKE
● Enthalpy in phase with PKE and TKE
TKE
Conclusions
● Surface energy balance does not close for Cabauw for strong stable conditions
Observations suggests the observed latent heat flux is underestimated
● Concept of 'total turbulent energy' is just a smart combination of the variance
equations for E and the virtual potential temperature. Note that Nieuwstadt also
uses these equations and gets identical solutions as from a simple TKE equation
(Baas et al., 2007)
● However, note that buoyancy fluctuations plane will trigger vertical motions
● The radiation term might be very important in generating temperature variance
Can we measure small latent heat fluxes?
Write the flux according to an updraft downdraft decomposition:
with = updraft fraction
wu (wd) = updraft (downdraft) vertical velocity
qu (qd) = updraft (downdraft) specific humidity
qqwqqww1'q'w uududu
Can we measure small latent heat fluxes?
Measure 7 W/m2:
Examples wu (m/s) qt' (g/kg)
1 0.0028
0.1 0.028
0.001 0.28
Li-Cor Li7500 RMS noise ± 0.0033 g/kg
m/sg/kg 108.2L
7'q'w 3
v
g/kg w
108.2
w
'q'wqq'q
u
3
uut
Soil heat flux
Soil heat flux: soil heat flux plates.
The six plates are burried at the three vertices of an equilateral triangle with sides of 2 m at depths of 0.05 and 0.10 m. The measurements are averaged over the three plates at each depth.
To obtain the surface soil heat flux a Fourier decompostion method is used.
The instruments are manufactured by TNO-Delft. Type: WS31S, measuring principle: thermo-pile, diameter 0.11 m, thickness 5 mm, sensitive surface: central square of 25*25 mm2. Thermal conductivity of the sensor 0.2-0.3 W/m/K.
Temperature tendencies during stable, clear nights:
Monthly mean values [K/hour]
Month 2m 10m 20m 40m 80m 140 m 200m
Jan -0.874 -0.772 -0.710 -0.470 -0.214 -0.117 -0.048
Feb -0.714 -0.652 -0.610 -0.449 -0.322 -0.135 -0.053
Mar -0.871 -0.691 -0.679 -0.523 -0.336 -0.190 -0.113
Apr -0.775 -0.594 -0.492 -0.396 -0.287 -0.181 -0.163
May -0.833 -0.668 -0.618 -0.430 -0.282 -0.188 -0.166
Jun -0.820 -0.719 -0.651 -0.513 -0.406 -0.264 -0.218
Jul -0.966 -0.755 -0.695 -0.562 -0.363 -0.295 -0.254
Aug -0.883 -0.787 -0.752 -0.618 -0.443 -0.275 -0.118
Sep -0.843 -0.746 -0.694 -0.567 -0.380 -0.241 -0.165
Oct -0.815 -0.744 -0.717 -0.597 -0.380 -0.197 -0.141
Nov -0.941 -0.720 -0.696 -0.526 -0.316 -0.149 -0.059
Dec -0.692 -0.650 -0.658 -0.410 -0.204 -0.106 -0.035
Humidity tendencies during stable, clear nights:
Monthly mean values [g/kg/hour]
Month 2m 10m 20m 40m 80m 140 m 200m
Jan -0.176 -0.160 -0.167 -0.098 -0.050 0.002 -0.007
Feb -0.148 -0.130 -0.150 -0.093 -0.069 -0.044 -0.043
Mar -0.167 -0.147 -0.173 -0.104 -0.073 -0.052 -0.066
Apr -0.181 -0.189 -0.242 -0.146 -0.104 -0.089 -0.064
May -0.244 -0.194 -0.271 -0.158 -0.095 -0.074 -0.031
Jun -0.282 -0.241 -0.272 -0.186 -0.118 -0.104 -0.025
Jul -0.372 -0.283 -0.358 -0.204 -0.096 -0.051 -0.080
Aug -0.334 -0.272 -0.333 -0.191 -0.073 -0.086 -0.064
Sep -0.297 -0.274 -0.323 -0.152 -0.144 -0.110 -0.070
Oct -0.283 -0.231 -0.252 -0.171 -0.125 -0.103 -0.085
Nov -0.248 -0.197 -0.218 -0.134 -0.086 -0.062 -0.077
Dec -0.147 -0.139 -0.170 -0.092 -0.067 -0.057 -0.041
Humidity budget equation
I. Neglect large-scale advection term
II. Assume 200 m tendency is due to horizontal advection
month LE_I LE_II (advection correction)
Jan -7.32 -6.30
Feb -10.07 -4.15
Mar -11.64 -2.46
Apr -16.16 -7.22
May -15.37 -11.07
Jun -18.37 -14.83
Jul -18.43 -7.38
Aug -17.97 -9.13
Sep -20.52 -10.82
Oct -19.24 -7.44
Nov -14.49 -3.74
Dec -10.71 -5.02