Glacier meteorology�Surface energy balance

• How does ice and snow melt ?
• Where does the energy come from ?
• How to model melt ?

Regine Hock

Summer School in Glaciology

McCarthy, AK, June 2026

Melting of snow and ice

• Ice and snow melt at 0°C (but not necessarily at air temperature >= 0° C)
• Depends on energy balance which is controlled by� meteorological conditions� properties of the surface

Properties of snow and ice

• Surface temperature can not exceed 0°C� - stable stratification in melt season� - glacier wind (katabatic wind)� - max surface vapor pressure 611 Pa� - max longwave outgoing radiation = 316 Wm²

Blue ice in Antarctica

Properties of snow and ice

• Surface temperature can not exceed 0°C� - stable stratification in melt season� - glacier wind (katabatic wind)� - max surface vapor pressure 611 Pa� - max longwave outgoing radiation = 316 Wm²

• High and variable� albedo

Warming of snow/ice

• First, snow/ ice needs to be warmed to melting point
• Then melt can occur

Cold content

Snow temp

Depth

Cold content Ground heat flux

= energy needed to bring the snow / ice to 0 °C.

1 g water refreezes  160 grams of snow will be warmed by 1 K

Data by Matt Nolan

SURFACE ENERGY BALANCE

7

Energy flux

atmosphere to glacier surface

Energy available for melting

Q_{atm =} Q_M + Q_G

Change in internal energy (heating / cooling

of the ice or snow)

50 W/m² is available at the glacier surface. How much does the glacier melt (cm/day)?

L_f latent heat of fusion (334000 J kg^-1)

M Melt rate

ρ Density of water (1000 kg / m3)

Convert energy into melt:

Energy balance

Q_N Net radiation

Q_H sensible heat flux

Q_L latent heat flux

Q_G ground heat flux (heat flux in the ice/snow)

Q_R sensible heat supplied by rain

-----------------------------------------------

G Global radiation

α albedo

L↓ longwave incoming radiation

L↑ longwave outgoing radiation

Radiation

Wien’s Displacement law:

inverse relationship between the wavelength of the peak of the emission of a black body and its temperature

λ_max = 2.88*10^-3 T^-1

[m] [m K] [K]

Shortwave - longwave radiation

GLOBAL RADIATION or shortwave incoming radiation

- Top of atmosphere radiation

• Solar constant = amount of incoming solar electromagnetic radiation per unit area, measured on the outer surface of Earth's atmosphere, in a plane perpendicular to the rays.
• roughly 1367 W/m², fluctuates by about 7% during a year (1412 W/m² to 1321 W/m²)  varying distance from the sun

GLOBAL RADIATION or shortwave incoming radiation

• Wavelength: 0.15-5 μm
• 2 components: �Direct / diffuse component:
• Scattering by air molecules�scattering and absorption by liquid and solid particles
• Selective absorption by water vapor and ozone

Direct & diffuse

only diffuse

Direct solar radiation

Strong spatial variation

• Controlling factors:� - site characteristics: slope, aspect, solar geometry� - atmosphere: transmissivity (cloudiness, pollutants ...)
• I increases with� - decreasing angle of incidence� - increasing transmissivity � (affected by volcanoes, pollution, clouds)� - increasing elevation (decreasing mass, less shading, multiple reflection)

I₀= solar constant=1368 Wm^-2

R= Earth-Sun radius (m=mean)

Ψ= atmospheric transmissivity

P= atmospheric pressure (0=at sea level)

β= slope angle, φ_sun Φ_slope=solar and slope azimuth angle

cos (angle of incidence θ)

θ

zenith, Z

Spatial variation of potential solar direct radiation�(clear-sky conditions)

Topographic shading Potential direct radiation� averaged over melt season

The large spatial variability of the direct component of global radiation in complex topography is responsible for much of the spatial variability in observed melt

Hock (1999)

• Clear-sky days= D approx. 15% of global radiation, Overcast days 100%
• D controlled by� - atmospheric conditions (clouds)� - spatially: albedo, skyview factor, V
• Less variable spatially than direct radiation

Modelled direct and diffuse radiation on Storglaciaren, Jun 7 – Sep 17, 1993

20-90 Wm2

90

75

W/m2

85

85

Extrapolation of global radiation

• Ratio is small under cloudy conditions
• Ratio is large under clear-sky conditions
• Ratio is proxy for cloudiness and assumed to be the same across glacier

Albedo

= average reflectivity over the spectrum 0.35-2.8 μm

ALBEDO

• Large variability in space and time
• Key variable in glacier melt modelling
• Ice albedo less variable than snow albedo, but often modified by sediment and debris cover

Large wavelength dependency

Typical values:

= ratio between reflected and incoming solar radiation

Hourly discharge

Daily albedo variations on Storglaciaren

2 stations 1 km apart show same temporal variations

 Snow: cloud effect

 Ice: same weathering processes

Incident shortwave radiation

• Cloudiness ( increase since clouds preferentially absorb near-infrared radiation  higher fraction of visible light which has higher reflectivity); effect enhanced by multiple reflection, increase up to 15%�effect is small over ice surfaces
• Zenith angle ( increase when sun is low, due to Mie scattering properties of the grains)

Jonsell et al., 2003, J. Glac.

Surface properties

• Grain size (large grain size  decreases albedo)
• Water content (water increases grain size  decreases albedo)
• Impurity content
• Surface roughness
• Crystal orientation/structure

CONTROLS ON GLACIER ALBEDO

How to model snow albedo ?

• Radiative transfer models including effects of grain size (most important control) and atmospheric controls� large data requirements, not practicable for operational purposes
• Empirical relationships�Aging curve approach: � function of time after snowfall� (Corps of Engineers, 1956)�variants include temperature, snow depth …

b, k = coefficients

α₀=minimum snow albedo

Snow albedo parameterisations

1) U.S. Army Corps of Engineers (1956)

function snow age n

and temperature (through k)

2) Brock et al. (2000)

Snow albedo is computed as a function of accumulated maximum daily temperature since snowfall

Daily albedo

Pelliciotti et al., J. Glaciol. 2005

Slope effect albedo

Jonsell et al, 2003, J.Glaciol., 164

Parallel-leveled

instrument�

Horizontally leveled

No diurnal

variation

diurnal variation

8:00 12:00 16:00

8:00 12:00 16:00

Cryoconite holes

Longwave radiation

Longwave incoming radiation

• Wavelengths 4-120μm
• Emitted by atmosphere (water vapor, CO₂, ozone)
• Function of air temperature and humidity (cloudiness)
• High values compared to shortwave radiation
• Longwave rad. balance < 0, when fog ca 0 W/m2
• Spatial variation: Topographic effects:� - reduced by obscured sky� - enhanced by radiation from slopes and air � in between
• Climate change: � Temp increase or � more cloudiness �  more L↓

glacier

L acts day & night

L↓=ε_effσT⁴

The longwave incoming radiation is the largest contribution to melt (~ 70%)

About 70 % of the longwave incoming radiation originates from within the first 100m of the atmosphere

Variations of screen-level temperatures can be regarded as representative of this boundary layer

L↓=ε_effσT⁴

PARAMETERISATION OF LONG-WAVE INCOMING RADIATION

clear-sky

term (cs)

overcast

term (oc)

L↓=Fε_clear-skyσT⁴₌ε_effσT⁴

INPUT

• air temperature (T)
• vapour pressure (e) (air humidity)
• Cloudiness (n) �(can e.g. be parameterized as function of global radiation and top of atmosphere radiation)

ε_eff=effective emissivity

T=air temperature

n= cloud fraction (0-1)

e=vapour pressure

ε_cs=clear-sky emissivity

ε_oc=overcase emissivity

L = _clear (T,e) F σ T⁴

Longwave outgoing radiation

L↑=εσT⁴+(1-ε)L↓

Longwave reflectance of snow: < 0.05

Emissivity ε > 0.95

Often assumed ε =1

What is L↑ for a melting surface ?

• σ = 5.67*10^-8 W m^-2 K^-4
• T in Kelvin = ?
• Assume black body radiation

Under which conditions can L↑ of the glacier surface be larger ?

Longwave outgoing radiation

Longwave rad. balance < 0, when fog ca. 0 W/m2

L↑=εσT⁴+(1-ε)L↓

Longwave reflectance of snow: < 0.05

Emissivity ε > 0.95

Often assumed ε =1

L↑ = 315.6 W m^-2

• σ = 5.67x10^-8 W m^-2 K^-4
• Assuming black body radiation -> ε=1

Radiation messages

• Net radiation is negative during clear-sky nights:�low atmospheric emissivity  �low incoming longwave radiation  cooling of the air
• Net radiation is higher during cloudy nights (high atmospheric emissivity  �high incoming longwave radiation � warming of the air
• Longwave incoming radiation (ca. 300 W/m²) much larger than shortwave incoming radiation (150-250 W/m²), but less variation

TURBULENT HEAT FLUXES

Turbulent heat fluxes

Turbulent heat fluxes =

Ability to transfer x gradient of relevant property

Eddy diffusivity

Turbulent heat fluxes

Sensible heat flux

• Function of temperature gradient
• Function of wind speed

Latent heat flux ( the energy released or absorbed during a change of state)

• Function of vapour pressure gradient
• Function of wind speed

Fluxes also affected by

• Surface roughness
• Atmospheric stability

Turbulent heat fluxes

Bulk aerodynamic method

• Surface temperature and vapor pressure are known
• Measurements at only one level needed

Wind speed

Temperature difference

Specific humidity difference

Exchange

coefficient

Sensible heat flux

Latent heat flux

Exchange coefficient is function of

- surface roughness

- stability function � (empirical expressions to define � stability functions)

Turbulent heat fluxes

Temperature difference

Specific humidity difference

Sensible heat flux

Latent heat flux

wind speed

P=air pressure, Z0=roughness length, Ψ=stability function, L=Monin-Obukhov length

Fluxes affected by

• Surface roughness
• Atmospheric stability

Sublimation

• To melt 1 kg snow/ice requires �334 000 J kg^-1�Latent heat of fusion
• To sublimate 1 kg of snow requires � 2 848 000 J kg^-1 �Latent heat of sublimation (8x L_f !!!)

Positive vapor gradient

 Condensation —> more energy available for melt

To melt 1 kg snow/ice requires 334 000 J kg^-1 � = Latent heat of fusion

Latent heat flux

Negative vapor gradient

 Sublimation� To sublimate 1 kg snow/ice requires 2 848 000 J kg^-1 � = Latent heat of sublimation

Sensible heat flux

Latent heat flux

Net radiation

L_s = 8*L_f  In case of sublimation 8 x less ablation for same latent heat energy input

L_s = 8*L_f  In case of sublimation 8 x less ablation for same latent heat energy input

Typical for during dry periods in the outer tropics

Sensible heat flux by rain

Tr = rain temperature

T_s = surface temperature

R = rain fall rate

ρ = density of water

C_p=Specific heat of water

radiation components

(S↓, S↑, L ↓, L↑)

temperature

humidity

wind speed & direction

1. Temp � sensible heat flux� latent heat flux� rain heat flux� (longwave incoming radiation)
2. Humidity  latent heat flux�(longwave incoming radiation)
3. Wind speed � sensible heat flux� latent heat flux
4. Global radiation�

Westfork Glacier, Alaska Range

What data do you need to compute an energy balance?

Temporal variation of energy balance components

Energy partitioning (%)

Net rad Sensible Latent Ground heat Melt

Melt energy – weather patterns

Q_M = Q_N+Q_H+Q_L+…�

Net radiation

Sensible heat

Latent heat

day-to-day variability in melt is often determined by the turbulent fluxes

Summary Energy balance

• Radiation can be measured directly; longwave often not measured, needs parameterization f(T, e, n); extrapolation of global and longwave radiation relatively straightforward, but modellng albedo is major uncertainty;
• Turbulent fluxes generally need parameterization f(T, e, u) at both point scale and larger scale; problems determining the exchange coefficients and stability functions, issues with validity of underlying theory;  large uncertainty (but often only smaller contributor to total melt energy);
• Often net radiation dominant source of energy, longwave radiation largest source (often the double the amount of shortwave incoming)
• Variations from day to day often determined by variations in turbulent heat fluxes
• Spatial variations in energy balance determined by shortwave incoming radiation
• Energy by rain generally very small
• direction of vapor pressure gradient important  sublimation reduces energy available for melt significantly
• Ice heat flux: cooling by conduction, but warming of snow mainly by refreezing of melt/rain water; flux important for cold/polythermal glaciers

How to model melt ?

1. Physically based energy-balance models: each of the relevant energy fluxes at the glacier surface is computed from physically based calculations using direct measurements of the necessary meteorological variables
2. Temperature-index or degree-day models: melt is calculated from an empirical formula as a function of air temperature alone

Melt modelling

Mass balance models

Model type Spatial discretization

Temp-index

regression

Temp-index or simplified energy balance

Energy

balance

0-Dim elevation bands fully distributed

Increasing model sophistication

• Assume a relationship between air temperature and melt: M=f(T), M=f(T⁺)

Temperature-index melt models

Data by R. Braithwaite

• Assume a relationship between air temperature and melt: M=f(T), M=f(T⁺)

Temperature-index melt models

Positive degree-day sum PDD = ΣT⁺

Physical basis of temp-index models

Case studies (Sicart et al., 2008, JGR): �Correlation between energy fluxes and air temp

• L has low variability compared to other fluxes
• L is poorly correlated to air temperature when cloud variations dominate its variability (usual in mountains)

Air temperature directly affects several components of the surface energy balance

L↓ =εσT⁴

f(T)

Longwave incoming rad

Sensible heat flux

f(e(T))

Latent heat flux

• L↓ is the largest contribution to melt (~ 70%) (Ohmura, 2001, Physical basis of temperature-index models)

Degree-day model

M = f_ice/snow * T⁺

[mm/day/K]

Typical values

Ice ~5-10 mm/d/K

Snow ~3-5 mm/d/K

PDD = 20 K

DDF = 80/20 = 4 mm/d/K

Degree-day (Definition Cogley et al., 2011, glossary)

The name of a derived unit, the K d, equal in magnitude to a 1 K departure of temperature, above or below a reference temperature, sustained for a period of 1 day.

Different choices of the reference temperature --> e.g. heating degree-day, freezing degree-day).

​

In glaciology: �positive degree-day (relative to the reference temperature 0 ºC).

Degree-day factor

In a positive degree-day model, the coefficient of proportionality between ablation a and the positive degree-day sum .

Degree-day factors

[mm/day/K]

Snow Ice

M = f_ice/snow * T⁺

Typical values

Ice ~5-10 mm/d/K

Snow ~3-5 mm/d/K

Degree-day factors

M = f_ice/snow * T⁺

[mm/day/K]

Spatial and diurnal variation

Derived from energy balance modeling

Hock, 1999, J.Glaciol.

Performance of degree-day model

Melt = f_ice/snow * T⁺

Model captures seasonal variations but not diurnal melt fluctuations

Modified temperature-index model

• Classical degree-day factor� M = f_ice/snow * T⁺

​

• Including pot. direct radiation� M = (f_m + a_ice/snow*I) * T⁺

Including potential direct solar radiation

Model introduces

• a spatial variation in melt factors
• a diurnal variation in melt factors

Hock 1999, J. Glaciol

Model comparison

Gornergletscher outburst floods

Photo: Shin Sugiyama

Huss et al., 2007, J. Glaciol.

Extended temperature-index models including other data than temperature

M = f_ice/snow * T⁺

M = a * R + f_melt * T

Net radiation

Shortwave bal

Degree-day model

Energy balance model

Extended temperature-index model including radiation

→ Gradual transition from degree-day models to energy balance-type expressions by increasing the number of climate input variables

Simplified energy balance model

M = (1-α)G+c₀+c₁T

Shortwave radiation

balance

Longwave balance and

turbulent fluxes

Oerlemans (2001)

→ Gradual transition from degree-day models to energy balance-type expressions by increasing the number of climate input variables

M = a * R + f_melt* T

Distributed temp-index model by Pelliciotti et al, 2005, J.Glaciol.

Temperature

Model only requires air temperature

Global radiation and albedo parameterized

Albedo

Incoming shortwave radiation

Pellicciotti et al, 2005, J. Glaciology

Brock et al. (2000)

Computed as function of accumulated maximum daily temperature since snowfall

Brock et al. (2000)

Computed as function of daily temperature range

Temperature-index versus energy balance

Temperature index Energy balance

Shortcomings Advantages

Both approaches are needed, use depends on application and data availability

Energy balance on Storglaciären

Net radiation

Sensible heat

Latent heat

Wrong conclusion if temperature taken as sole index,

​

Energy balance needed to solve problem

SUMMARY

• Both temp-index and energy balance models are useful tools, choice depends on data availability
• Awareness of limitations
• Need for more approaches of intermediate complexity and moderate data input
• Both temp-index and energy balance models need calibration (parameter tuning)

Global glacier modeling

Most global glacier models

• use temperature-index method to simulate melt
• elevation-dependent
• temperature and precipitation as input
• simple approaches to simulate geometric changes�(OGGM, PyGEM: 1-D flow model)

Literature

Energy balance:

Hock, R., 2005. Glacier melt: A review on processes and their modelling. Progress in Physical Geography 29(3), 362-391.

​

Temperature-index methods:

Hock, R., 2003. Temperature index melt modelling in mountain regions. Journal of Hydrology 282(1-4), 104-115. doi:10.1016/S0022-1694(03)00257-9.

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