Energy balance and warming
Ned Bair
US Army Corps of Engineers Cold Regions Research and Engineering
Laboratory
Earth Research Institute, UC - Santa Barbara
1
Heat transfer
• Radiation
– Energy transfer via photons
• Sensible
– Heat exchange from a change in temperature
– 2 types:
• Conduction
– Direct exchange of kinetic energy
• Conduction
– Heat carried by bulk flow, i.e. wind
– Heat exchange from a change of temperature
• Latent
– Heat exchange from a change of phase
2
Radiation
• All bodies emit electromagnetic radiation as a
function of their temperature
• This can be modeled by the Steffan-Boltzman
equation
L = es T 4
where L is radition,
e is emissivity, 0.99
s is the Steffan Boltzman constant,
5.67x10 -8 Wm -2 K -4
and T is snow temp in Kelvin, 270 K
3
Electromagnetic spectrum
4
5
Energy balance for dry snow
G = R+ H + L+ M
• Dry snow
R Net radiation
G Heat flow into/out of pack
H Sensible heat exchange
– M=0
• Wet, ripe snow
– G=0, because T is
uniform
L Latent heat exchange
M Melt
All units in W m-2
6
G, heat flow, Fourier’s Law
G = -kÑT
k,thermalconductivity, W m K
-1
ÑT ,temperature gradient, K m
-1
-1
Thermal conductivities
0.045 fiberglass
0.05-0.25 dry snow
0.56-0.61 water
16-24 stainless steel
7
Periodic:
T  10C, T0  10
Temperature, C
0
-5
-10
surface
0.1 m
0.2 m
0.3 m
-15
-20
0
6
12
Time
18
24
8
R, Net radiation
Net radiation: solar and thermal infrared
R = Net solar + longwave¯ - longwaveR = S¯ (1- a ) + F¯ - es T 4
S¯ ,incoming solar
a ,albedo
F¯ ,incoming longwave
es T 4 ,outgoing longwave (Steffan Boltzman)
where e is emissivity, s is the Steffan Boltzman constant,
and T is snow temp in Kelvin
9
H, Sensible heat
10
H, Latent heat
é k ( q2 - q1 ) ù é k ( u2 - u1 ) ù
H = r a Lv ê
úê
ú
êë ln ( z2 - z1 ) úû êë ln ( z2 - z1 ) úû
r a air density (2438 m/8000 ft), 0.53 kg m -3
-1
Lv latent heat of vaporization, 2.5x10 J kg deg
6
-1
q is specific humidity, g m-3
u is wind velocity, m s -1
k is Von Karman's constant (0.4)
z is measurement height, m
11
Convective energy transfers
warm dry air
wind
latent heat
cold dry air
latent heat
sensible heat
cold humid air
sensible heat
warm humid air
mixing
Negative
net
turbulent
transfer
Positive
net
turbulent
transfer
latent heat
sensible heat
sensible and
latent heat 12
Phase changes of water
Condensing and freezing 1 g
water vapor
Freezing 1
g water
Cooling 1 g 1
C water to 0 C
1 cal
80 cal
720 cal
13
Positive
radiation
balance
Negative
radiation
balance
14
Warming effects
• Generally, the effects of warming on avalanche
formation are minor
• Not affected:
– Layers deeper than 20-30 cm (e.g. the failure layer)
– The stress bulb depth
• Affected:
– E modulus of upper 20 cm of slab (increased bending)
– PST and ECT results
15
The stress bulb and layers deeper than
20-30 cm are not affected
Exner, T., and B. Jamieson, 2008: The effect of snowpack warming on the stress bulb below a skier.
International Snow Science Workshop.
16
Warming effect on E modulus
• No change in weak layer (wf) or layers in the slab
deeper than 20 cm.
• E modulus (stiffness) in layers < 20 cm decreased.
• PST cut length decreased after cumulative energy
inputs of 400 kJ m-2.
Reuter, B. and Schweizer, J., 2012. The effect of surface warming on slab stiffness and the fracture behavior of
snow. Cold Reg. Sci. Technol.: in press, doi: 10.1016/j.coldregions.2012.06.001.
17
Warming case study, Hammil Bowl 3/11/13
18
• Triggered at 10,550 ft on 38° N aspect at 11:30AM on
3/11/13
• 2 skiers in old skin track, crown formed 200 vertical feet above
them
• R2D2.5, crown 80 cm at deepest
• HS 285cm
38°
19
Avalanche
triggered here
20
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Weather Summary
• March 6-9: 15.5” inches of new snow, 1.5” SWE, ~10% water at
MMSP’s Sesame site (9,014 ft).
• Temperature change at CUES (9,645 ft) from low of -8 °C on March
10 to +6 °C at time of the accident, 11:30 AM on March 11.
Air temp
10
8
6
4
ºC
2
0
-2
-4
-6
-8
-10
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
22
Air temp
10
8
6
4
ºC
2
0
-2
-4
-6
-8
-10
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
Vapor Pressure
400
350
300
Pa
250
200
150
100
50
0
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
23
The E modulus in the upper 20-30 cm
and PST cut lengths can be reduced
Reuter, B. and Schweizer, J., 2012. The effect of surface warming on slab stiffness and the fracture behavior of
snow. Cold Reg. Sci. Technol.: in press, doi: 10.1016/j.coldregions.2012.06.001.
24
1200
1000
800
W/m2
UpClrAuto
UpDiffuseAuto
600
DownClr
UpLongWave
400
200
0
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
25
Air temp
10
8
6
4
ºC
2
0
-2
-4
-6
-8
-10
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
Vapor Pressure
400
350
300
Pa
250
200
150
100
50
0
3/9/13 0:00
3/9/13 12:00
3/10/13 0:00
3/10/13 12:00
3/11/13 0:00
3/11/13 12:00
26
Energy fluxes at CUES
27
Aspect and slope effects
Direct
Shortwave
300 W/m2
300 W/m2
Longwave
300 W/m2
(Reflected:
500-950
W/m2)
700 W/m2
Negative
net
radiation
1000 W/m2
Positive net
radiation
28
Steps to adjust net solar flux for TJ
Bowl
• Calculate solar declination δ and solar longitude λ
from measurement dates & times
• Calculate solar zenith angle μ0 and solar azimuth
φ0 (i.e. local sun position on flat surface) from
lat/lon, δ, and λ
• Calculate illumination angle μ from μ0 , φ0, slope
angle, and slope azimuth
• Calculate ratio of sun on flat surface to slope
s=μ/μ0
net solar N = éësD + f ùû éë1- a ùû
where D is direct solar, f is diffuse solar,
and a is snow albedo
29
CUES
TJ Bowl
30
Cumulative heat flow at snow surface (G)
Increases:
Sat 3/9 – 1761 kJ
Sun 3/10 – 3822 kJ
Mon 3/11 – 1017
kJ
Increases:
Sat 3/9 – 475 kJ
Sun 3/10 – 751 kJ
Mon 3/11 – 245 kJ
31
Reuter, B. and Schweizer, J., 2012. The effect of surface warming on slab stiffness and the fracture behavior of
snow. Cold Reg. Sci. Technol.: in press, doi: 10.1016/j.coldregions.2012.06.001.
32