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					             MASS-BALANCE MODELLING


                  Karthaus, September 2005

                       Wouter Greuell

Institute for Marine and Atmospheric Research Utrecht (IMAU)
               Utrecht University, the Netherlands


                            AIM:
    Calculate surface mass balance from data collected at a
                climate station (not on the glacier)
          SURFACE ENERGY BALANCE

                 dm           dTi                     2
        Q0  L f      Mic pi                  [Wm ]
                  dt          dt

Energy exchange        melting /         heating / cooling
with atmosphere        freezing          of the ice or snow

  Q0      energy flux atmosphere to glacier
  Lf      latent heat of fusion (0.334.10-6 J kg-1)
  m       amount of melt water
  Mi      mass of the ice
  cpi     specific heat capacity of ice (2009 J kg-1 K-1)
  Ti      ice temperature
    FLUXES ATMOSPHERE TO GLACIER


Q0 = S ( 1 – a ) + L - L + QH + QL + QR


    S    short-wave incoming radiative flux
    a     albedo of the surface
    L    long-wave incoming radiative flux
    L    long-wave outgoing radiative flux
    QH    turbulent flux of sensible heat
    QL     turbulent flux of latent heat
    QR    heat flux supplied by rain.
COMPONENTS OF THE ENERGY BALANCE
    7 LOCATIONS - VATNAJOKULL

                                                                                E melt
                                                                                SWnet
                      200                                                       LWnet
                                                                                turb flux
 energy flux (W m )
 -2




                      150


                      100


                      50


                       0


                      -50
                            A4      A5      I6       U8      U9       R2       R5
                            (279)   (381)   (715)   (1210)   (870)   (1100)   (1140)
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o CD D ecompress or

ed to use this picture
                         AUTOMATIC WEATHER STATION
MODEL INPUT
MEASUREMENTS AND OBSERVATIONS AT A
CLIMATE STATION NEAR THE GLACIER

In case of energy-balance model, input may consist of:

To determine ablation
- 2 m temperature
- 2 m wind speed
- 2 m humidity
- cloud amount

To determine accumulation
- precipitation
   TRANSFER FORCING FROM CLIMATE
         STATION TO GLACIER

                                T, u, q, n, p


T, u, q, n, p
                T, u, q, n, p




                                    T, u, q, n, p
TRANSFER FORCING FROM CLIMATE
      STATION TO GLACIER

             Some commonly used assumptions


Variable          assumption
temperature       constant lapse rate, i.e. dT/dz constant

wind speed        constant
humidity          constant relative humidity
cloud amount      constant
precipitation     linear in elevation (used for tuning)
   2 D PICTURE OF THE TEMPERATURE
             In case the surface is melting


        dT/dz = constant (e.g. -0.007 K/m)

Free atmosphere                                      dT/dz = ?




                                              dT/dz = 0
                              ACTUAL TEMPERATURE VARIATION
                                       Temperature on glacier (ÞC)
                       averages over 46 days of the ablation season, Pasterze, Austria
                                       Sonnblick




                       7.5
                                    stations along the glacier
                         7                          U3                                                     Constant lapse-rate
                       6.5            A1     U2                                                              can be a bad
                                                                                                             description,
2 m temperature (ÞC)




                                                                                  Sonnblick
                         6
                                                                               climate station
                                                                                                             because:
                       5.5
                                       gentle                   steep              gentle
                         5             slope                    slope              slope
                       4.5
                                                                                 stations along
                                                                                   the glacier
                         4
                                                                        U4                     U5
                       3.5
                             2000     2200       2400       2600     2800      3000     3200        3400
                                                        Elevation (m a.s.l.)
                              ACTUAL TEMPERATURE VARIATION
                                       Temperature on glacier (ÞC)
                       averages over 46 days of the ablation season, Pasterze, Austria
                                       Sonnblick




                       7.5
                                    stations along the glacier
                         7                          U3                                                     Constant lapse-rate
                       6.5            A1     U2                                                              can be a bad
                                                                                                             description,
2 m temperature (ÞC)




                                                                                  Sonnblick
                         6
                                                                               climate station
                                                                                                             because:
                       5.5
                                                                                                           1) Air over glacier
                                       gentle                   steep              gentle
                         5             slope                    slope              slope                     colder than over
                       4.5
                                                                                                             snow-free terrain
                                                                                 stations along
                                                                                   the glacier             2) No constant lapse
                         4
                                                                        U4                     U5            rate over glacier
                       3.5
                             2000     2200       2400       2600     2800      3000     3200        3400
                                                        Elevation (m a.s.l.)
                                                        MEASURED CLIMATE SENSITIVITY
                                              46 daily means during the ablation season, Pasterze, Austria
Temperature (ÞC) on glacier (2205 m a.s.l.)




                                              10
                                                                                                                  Constant lapse-rate
                                              9
                                                                                                                    can be a bad
                                              8                                                                     description,
                                              7                                                                     because:
                                                                                                                  3) Climate
                                              6
                                                                                                                    sensitivity over
                                              5
                                                                                                                    glacier smaller
                                              4
                                                   -2       0          2           4           6              8
                                                                                                                    than over snow-
                                                        Temperature (ÞC) at climate station (3106 m a.s.l.)         free terrain
         ALTERNATIVE DESCRIPTIONS
        TEMPERATURE ALONG GLACIER

De Ruyter de Wildt, M. S., J. Oerlemans and H. Björnsson,
   2003: A calibrated mass balance model for Vatnajökull,
   Iceland. Jökull, 52, 1-20.

Greuell, W. and R. Böhm, 1998: Two-metre temperatures along
   melting mid-latitude glaciers and implications for the
   sensitivity of the mass balance to variations in temperature.
   J. Glaciol., 44 (146), 9-20.
Oerlemans, J. and B. Grisogono, 2000: Glacier wind and
   parameterisation of the related surface heat flux. Tellus, A54,
   440-452.
    SHORT-WAVE INCOMING RADIATIVE FLUX



Calculation of:

-   Incidence angle (date, time, location, slope)
-   Transmission through clear-sky atmosphere (water vapour)
-   Multiple reflection (surface albedo)
-   Cloud transmission (cloud amount)
                          CLOUD FACTOR
               causes largest uncertainty in calculated
                   incoming short-wave radiation
                1.1


                 1
                                                         Greenland
                                                          2000 m
                0.9
                                            Antarctica
                                             1200 m
Cloud factor




                0.8
                                Pasterze
                                Austria
                0.7             2205 m               Greenland
                                                      250 m
                0.6


                0.5


                0.4


                0.3
                      0   0.2    0.4       0.6           0.8         1
                                Cloud amount
         ALBEDO PARAMETERISATION


              asnow(i) = afirn + (afrsno - afirn) exp s-i
                                                      t*

              a (i) = asnow(i) + {aice - asnow(i)} exp -d
                                                       d*

          This model has five parameters:         afrsno
                                                  afirn
                                                  aice
                                                   d*
                                                   t*
Oerlemans and Knap, 1999
DIRTY ICE - PASTERZE

       a~ 0.2
    CLEAN ICE - GREENLAND ICE SHEET
a~ 0.45



                             c i m
                          Q ui kT e™ and a
                        hot D
                       P o C decompressor
                                         s c
                  ar e needed t o see t hi pi t ure.




                                 c i m
                              Q ui kT e™ and a
                            hot D
                           P o C decompressor
                                             s c
                      ar e needed t o see t hi pi t ure.
FEEDBACK ALBEDO  SNOW AND ICE MELT


                               2) Ice appears earlier
  1) Faster metamorphosis of   3) More meltwater on top of ice
     snow                      4) More water between snow grains



                  Lower albedo

                           Net short-wave
                             radiation

                    More melt
                     GLACIER SHOULD THEORETICALLY NOT
                     BE SENSITIVE TO TEMPERATURE CHANGE

                                                                               E melt
                                                                               SWnet
                                                                                           Because
                     200                                                       LWnet
                                                                               turb flux
                                                                                           i) Net short-wave
energy flux (W m )
-2




                     150
                                                                                             radiation dominates the
                     100
                                                                                             surface energy balance
                     50
                                                                                           ii) Net short-wave
                      0
                                                                                             radiation is not a
                     -50
                           A4
                           (279)
                                   A5      I6       U8      U9       R2
                                                                    (1100)
                                                                              R5
                                                                             (1140)
                                                                                             function of the
                                   (381)   (715)   (1210)   (870)

                                                                                             temperature


                      HOWEVER: GLACIERS ARE VERY
                      SENSITIVE TO TEMPERATURE CHANGE!!!                                                          
DIRECT IMPACT OF TEMPERATURE INCREASE ON
                  MELT

          Higher temperature




 Turbulent fluxes
 Incoming long-
    wave radiation



                          More melt
  SENSITIVITY INCREASES DUE TO ALBEDO
                FEEDBACK

         Higher temperature


                                          2) Ice appears earlier
             1) Faster metamorphosis of   3) More meltwater on top of ice
                snow                      4) More water between snow grains



Turbulent fluxes             Lower albedo
Incoming long-
   wave radiation
                                      Net short-wave
                                        radiation

                               More melt
LONG-WAVE INCOMING IS DETERMINED BY …


  L varies with the entire vertical profiles of
        temperature and water vapour

  and with cloud-base height, cloud-base temperature
        and cloud amount

  But in this case we only know:
  T2m temperature at 2 m
  e2m water-vapour pressure at 2 m
  n      cloud amount
LONG-WAVE INCOMING, PARAMETERISATION



           
   L    cs   n   ocn  T2m
              1
                        a        a
                                           4



            clear-sky        overcast
            term (cs)       term (oc)

     emittance (): is 1.0 for a black body
                                     1/ 8
                          e 2m 
         cs  0.23  c L  
                          T2m 

    Three tunable parameters: a, oc and cL
LONG-WAVE OUTGOING RADIATION



             L = s  Ts4

 where s and Ts are the emissivity and
      temperature of the surface

      but since s is close to 1.0:

              L =  Ts4
SENSIBLE HEAT FLUX (QH)              calculated with the “bulk method”

                                 u   Ts 
                                 2
                                     T
          QH  a C pa
                          z am z   z ah z 
                         ln z  L  ln z  L 
                          0      ob    T   ob 


      a       air density
      Cpa      specific heat capacity of air
      k        von Karman constant
      u        wind speed at height z
      T        air temperature at height z
      Ts       surface temperature
      z0       momentum roughness length
      zT       roughness length for temperature
      am, ah   constants
      Lob      Monin-Obukhov length (depends on u and T-Ts)
                 ROUGHNESS LENGTHS

Momentum roughness length (z0) is a function of the surface
    geometry only.
z0 increases with the roughness of the surface. Most values
    for ice and for melting snow are in the range 1 to 10 mm.

Distinguish:
z0 = momentum roughness length (wind)
zT = roughness length for temperature (depends on z0 and
       wind speed)
zq = roughness length for water vapour (depends on z0 and
       wind speed)
                                DETERMINE MOMENTUM ROUGHNESS LENGTH

The momentum roughness length is defined as the height above the
surface, where the semi-logarithmic profile of u reaches its surface
values (0 m/s). It is determined by extrapolation of measurements.

                           14                                                                      100

                           12
Height above surface (m)




                                                                       Height above surface (m)
                                                                                                    10          stable
                                                      neutral                                                 conditions
                           10
                                                     conditions                                               (katabatic
                                                                                                     1          wind)                  neutral
                            8
                                                                                                                                      conditions
                            6
                                      stable                                                        0.1
                                    conditions
                            4       (katabatic
                                      wind)                                                        0.01
                            2

                                                                                                  0.001
                                0      2         4    6       8   10                                      0        2       4      6        8       10
                                           Wind speed (m/s)                                                            Wind speed (m/s)
              LATENT HEAT FLUX

                       2 u q  q s 
     QL  a Ls
                 z a m z   z ah z 
                ln        ln 
                 z 0 L ob   zq L ob 
                                          

a       air density
Ls       latent heat of sublimation
k        von Karman constant
u        wind speed
q        specific humidity at height z
qs       surface specific humidity
z0       roughness length for velocity
zq       roughness length for water vapour
am, ah   constants
Lob      Monin-Obukhov length (depends on u and T-Ts)
           ZERO-DEGREE ASSUMPTION



Assumption: surface temperature = 0 ˚C

If this leads to
Q0 > 0: Q0 is consumed in melting
Q0 ≤ 0: nothing occurs

Assumption ok when melting conditions are frequent

wrong when positive Q0 causes heating of the snow
(spring, early morning, higher elevation)
                   SUB-SURFACE PROCESSES

Alternative to zero-degree assumption: model sub-surface processes on a
    vertical grid

Relevant processes:
- penetration of short-wave radiation; absorption below the surface
- refreezing of percolating melt water in snow with T < 0˚C ( = internal
   accumulation)
- retention of percolating melt water by capillary forces
- when slope is small: accumulation of water on top of ice; leads to
   superimposed-ice formation when T < 0˚C
- conduction
- metamorphosis

Output: mass balance, but also surface temperature
                 DEGREE-DAY METHOD

N =b Tpdd          N: ablation
                  b:    degree-day factor [mm day-1 K-1]
                    Tpdd: sum of positive daily mean temperatures

Why does it work:
- net long-wave radiative flux, and sensible and latent heat flux ~ proportional
  to T
- feedback between mass balance and albedo

Advantages:
- computationally cheap and easier to model
- input: only temperature needed

Disadvantages:
- more tuning to local conditions needed: e.g. b depends on mean solar zenith
   angle
- only sensitivity to temperature can be calculated
      ACCUMULATION



Treated in a very simple way:

Precipitation = snow for T < 2˚C
Precipitation = rain for T ≥ 2˚C
     ROLE OF DATA AUTOMATIC WEATHER
      STATIONS (AWS) AND MASS BALANCE
              MEASUREMENTS
AWS data:
- develop parameterizations for incoming short- and
  long-wave radiation
- Determine relation between temperature at climate
  station and temperature over glacier
- Determine wind speed
- Determine roughness lengths
- Test energy balance model

Mass-balance data
- tune the model, mainly with precipitation amount
  and gradient
- validate the model (correct simulation of interannual
  variation?)
                             SUM UP


-   surface energy balance fundamental
-   motivation for forcing from climate station; role of AWS’es
-   transfer forcing to glacier
-   parameterisations of radiative and turbulent fluxes
-   sub-surface models and zero-degree assumption
-   degree-day models
-   intermezzo: understand apparent paradox about sensitivity of
    glaciers
         READING AND MODELLING

Review about mass balance modelling:
Greuell, W., and C. Genthon, 2004: Modelling land-ice surface
    mass balance. In Bamber, J.L. and A.J. Payne, eds. Mass
    balance of the cryosphere: observations and modelling of
    contemporary and future changes. Cambridge University
    Press.


Mass balance model that includes sub-surface module:
    http://www.phys.uu.nl/%7Egreuell/massbalmodel.html
measure short-wave radiation
 with a pyranometer (glass       SOME INSTRUMENTS
           dome)

                               measure sensible heat flux
                               with a sonic anemometer




 measure long-wave radiation
 with a pyrgeometer (silicon
           dome)
                        ENERGY BALANCE AT 5 ELEVATIONS
                      300
                                                         net shortwave
                                                         net longwave
                      250                                sensible heat
                                                         latent heat
                  2
Energy flux in W/m



                      200

                      150

                      100

                       50

                        0

                      -50
                              A1      U2      U3      U4      U5
                            2205 m 2310 m 2420 m 2945 m 3225 m
                            a=0.21 a=0.29 a=0.25 a=0.59 a=0.59
                            T=6.8ÞC T=6.4ÞC T=7.1ÞC T=3.5ÞC T=3.2ÞC
                   ATMOSPHERIC MODELS
               e.g. a General Circulation Model (GCM)
       or an operational weather forecast model (e.g. ECMWF)


Advantages:
- include all of the physics contained in a surface energy-balance model
- forcing outside the thermal influence of the glacier or ice sheet
- effect of entire atmosphere on long-wave incoming radiation
   considered
- clouds computed
- accumulation computed

Disadvantages:
- grid size
- computer time
            REGRESSION MODELS



               Mn = c0 + c1 Twcs + c2 Pwcs

Mn: mean specific mass balance
c i:  coefficients determined by regression analysis
Twcs: annual mean temperature at climate station with
      weights varying per month
Pwcs: idem, for precipitation
      SHORT-WAVE INCOMING RADIATION

   S = I0 cos(s) Tg+a Fms Fho Frs Tc




  example for site on
glacier tongue Pasterze,
         Austria
   averages over 46
     summer days
                                                                                                   10 0


                      FIGURES BUOYANCY




                                                                       Sensible heat flux (W/m )
                                                                       2
                                                                                                    80


                        AND ROUGHNESS                                                               60        laminar flow
                                                                                                    40


                                                                                                    20
                                                                                                                         temperature = 5 ÞC
                                                                                                     0                 roughness length =1 mm
                            60                                                                     -2 0
                                                                                                          0     2         4       6        8    10
Sensible heat flux (W/m )
2




                            50
                                                                                                                 Wind speed at 2 m (m/s)
                            40


                            30                                                                     10 0




                                                                       Sensible heat flux (W/m )
                                                                       2
                            20                                                                      80
                                           wind speed = 4 m/s                                                    temperature = 5 ÞC
                            10           roughness length =1 mm                                                  wind speed = 4 m/s
                                                                                                    60

                            0
                                 0   2      4       6      8      10                                40

                                     Temperature at 2 m (ÞC)
                                                                                                    20


                                                                                                     0
                                                                                                     0.01        0.1          1       10        100
                                                                                                                 Roughness length (mm)
                SEASONAL SENSITIVITY CHARACTERISTICS

Vatnajökull, Iceland                                                   Devon Ice Cap, Canada

        12                                                                     12

        11                                                                     11
        10                                                                     10

         9                                                                      9

         8                                                                      8
Month




                                                                       Month
         7                                                                      7
         6                                                                      6

         5                                                                      5

         4                                                                      4
         3                                                                      3

         2                                                                      2

         1                                                                      1
        -0.2   -0.15       -0.1    -0.05   0         0.05        0.1           -0.2   -0.15       -0.1    -0.05   0         0.05        0.1
                                   -1                                                                     -1
                       C     (mwe K )          C     /10 (mwe)                                C     (mwe K )          C     /10 (mwe)
                       T,k                     P,k                                            T,k                     P,k
LIMITATIONS OF DEGREE-DAY METHOD
 Calculation of degree-day factors for various points on the
 Greenland ice sheet with a sophisticated atmospheric and
 snow model (thesis Filip Lefebre)




          snow                                  ice
                                SEPARATION OF SHORT- AND LONG-WAVE
                                            RADIATION

                                      Black body radiation
                         1


                        0.8                                                Q = T4
Normalized irradiance




                        0.6
                                                                       Q  flux (irradiance)
                        0.4                                              Stefan Boltzmann
                                                                          constant (5.67.10-8
                        0.2                                               W m-2 K-4)
                                    T = 5780 K       T = 290 K
                                        Sun            Earth            temperature
                         0

                              0.1          1           10        100
                                          Wavelength (µm)
             TURBULENT FLUXES



Vertical transport of properties of the air by eddies
Turbulence is generated by wind shear (du/dz)
Turbulent fluxes increase with wind speed


Heat:           sensible heat flux
Water vapour: latent heat flux
                                            DAILY COURSE
                            site on glacier tongue (ice) in summer

                     1000


                     800
                                                                    short wave in
Energy flux (W/m )




                     600
2




                     400

                                                       long wave in
                     200
                                sensible heat
                       0
                                                      latent heat

                                                                       short wave out
                     -200
                                  long wave out

                     -400
                            0          4          8    12            16         20      24
                                                      Time
                                     NET FLUXES
                                Daily course at single site
                     700

                     600
Energy flux (W/m )



                     500
2




                                           net short wave
                     400

                     300

                     200

                     100                    sensible heat
                                             latent heat
                       0
                                                     net long wave
                     -100
                            0   4      8       12          16    20   24
                                            Time

				
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