Project: Project: Study of radiative and thermal physical Study of radiative and thermal physical properties of the snow-ice cover (drifting and properties of the snow-ice cover (drifting and

land fast ice, glaciers) in the Spitsbergen land fast ice, glaciers) in the Spitsbergen archipelago areaarchipelago area..

Andreev O.M. (AARI)

Tasks of the project:

• Conducting a number of natural observations of Spitsbergen archipelago snow-ice cover parameters, which are necessary for making of mathematical model.

• Developing of sea ice cover model, especial for Spitsbergen archipelago conditions.

• Testing adequacy of model work according to real data.

• Comparison of developed model which analogous famous models by the example of test calculations.

• Determination of model blocks, needed in further improvement.

The special one-dimensional thermodynamic model of snow-ice cover for conditions of the Spitsbergen

Archipelago area

)t(Hz0;Iz

T

zt

Tc

)t(hz0;Iz

T

zt

Tc

202

22

2

222

101

11

1

111

( 1 )

B o u n d a r y c o n d i t i o n s :

0z1

111

1z

T0z ; ( 2 )

0z2

22

hz1

112

21z

T

z

T0z

; ( 3 )

)0,t(T)h,t(T0z 212 ; ( 4 )

The special one-dimensional thermodynamic model of snow-ice cover for conditions of the Spitsbergen

Archipelago area

L o w e r b o u n d a r y c o n d i t i o n s :

)H,t(THz 22 ; ( 5 )

W

Hz2

2222

2z

T

t

HL

, ( 6 )

U p p e r b o u n d a r y c o n d i t i o n s b y s n o w m e l t i n g i s :

0T0z 11 , ( 7 )

a n d s n o w c o v e r m e l t i n g e q u a t i o n a r e d e s c r i b i n g :

0z1

11

111

z

T

L

1

t

h. ( 8 )

The special one-dimensional thermodynamic model of snow-ice cover for conditions of the Spitsbergen

Archipelago area

W i t h t h e b e g i n n i n g m e l t i n g o f i c e ( t h e s n o w h a s c o m p l e t e l y s u b m e r g e ) s y s t e m o f

t h e e q u a t i o n s b e c o m e :

Hz0;Iz

T

zt

Tc 20

2

22

2

222

; ( 9 )

0T0z 22 ; ( 1 0 )

)H,t(THz 22 ; ( 1 1 )

0z2

22

22W

Hz2

22

2222

z

T

L

1

z

T

L

1

t

H. ( 1 2 )

Turbulent flux of sensible and latent heat.

T h e a e r o d y n a m i c p a r a m e t e r i z a t i o n o f t h e s e n s i b l e a n d l a t e n t h e a t :

qTqDaULLE

TTStUcHt p

)( 00

0

, ( 1 3 )

w h e r e S t a n d D a – i s s t a b i l i t y - d e p e n d t r a n s f e r c o e f f i c i e n t s

0,)1(

0,1

21

20

5.00

RibRiStSt

RiRic

bRiStSt

( 1 4 )

w h e r e R i i s t h e b u l k R i c h a r d s o n n u m b e r

20 )(

UT

zTTgRi

a

a ( 1 5 )

Short-wave heat flux

B y Z i l l m a n ( 1 9 7 2 ) :

1.0cos085.110)7.2(cos

cos

05

0

02

0 zez

zSF , ( 1 6 )

B y S h i n e ( 1 9 8 4 ) :

0455.0cos2.110)0.1(cos

cos

03

0

02

0 zez

zSF ( 1 7 )

w h e r e S – i s s o l a r c o n s t a n t ; e – v a p o r p r e s s u r e ( m b ) ; z 0 – s o l a r z e n i t h a n g l e .

Penetrating solar radiation.

In our investigations (A ndreev and Ivanov, 2003), the top ice layer is y , and

the transm itted coefficient for sea ice is determ ined as:

y

c

Ai

)0025.0875.0(

10

(18)

w here c – is a constant, depends from ice structure (c=0.004), y – ice top layer, m .

For snow w e use next expression (A ndreev and Ivanov, 2003):

a

byi

)ln(0 (19)

w here a and b is a function of snow density (ρ1):

)003.035.2(

)06.00.45(

1

1

b

a

(20)

Penetrating solar radiation.

150 200 250 300 350 400 450

kg/m 3

1

2

3

4

5

6

7

8

%

The values of solar radiation, penetrating on 10 cm horizon (in % from surface) for measurements in 2002-2004 (points) and calculating (line) for formula

That dependence for upper 10 cm. snow layer, is following:

0.4506.0

003.005.0i

s

s0

;

where s - is snow density; 0i- is transmission coefficient. Obtained dependence satisfactorily conforms to results

of expeditional observations (describing about 70% dispersion)

Thermal-physical properties of sea ice and snow.

The heat capacity, density and heat conductivity of sea ice is calculated by:

ps

p

dT

dLS

ps

sccSc

psdT

dSLc csiibiiii

22

2 )1(1

)(1 ;

)()1)((

)(2 pssS

ps

bibi

bi

;

T

bSi ,

where b = 0.1172 W/(m 0/00).

Thermal-physical properties of sea ice and snow.

T h e h e a t c a p a c i t y o f s n o w i s r e p r e s e n t e d a s a l i n e a r f u n c t i o n o f t e m p e r a t u r e :

)364.788.92( 111 Tc

T h e e x p r e s s i o n o f t h e r m a l c o n d u c t i v i t y o f s n o w t h a t a c c o u n t s f o r b o t h d i r e c t

t h e r m a l c o n d u c t i o n a n d w a t e r v a p o r d i f f u s i o n :

5/)233(421

61

12107.210845.2 T

The salinity of sea ice and heat flux from water, using in model.

F o r t h e s a l i n i t y o f y o u n g i c e w e u s e d t h e e m p i r i c a l f o r m u l a s , r e l a t i n g i t t o i c e

t h i c k n e s s :

HS i 6.19.7 w i t h H < 0 . 3 5 m ,

HS i 4.192.14 w i t h H < 0 . 3 5 m .

S i n c e n o d a t a o n h e a t f l u x f r o m t h e o c e a n m i x e d l a y e r , w e s e t o n F w = 0 f o r t h e

c a l c u l a t i o n s o f f a s t i c e f r o m G r o n f j o r d , a n d F w = 5 W / m 2 ( H < 0 . 7 m ) a n d F w = 2 . 5 W / m 2

( H > 0 . 7 m ) – f o r t h e c a l c u l a t i o n s o f r i s i n g a n d F w = 0 f o r m e l t i n g f a s t i c e i n K o n g s f j o r d .

The temperature and relative humidity of air measuring in Barentsburg meteostation in 2003

0 30 60 90 120

Jan Feb Mar Apr May

0

4

8

12

16

20

0 30 60 90 120

Jan Feb Mar Apr May

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

5

The wind velosity (m/s) in the area of Barentsburg (2003).

The air temperature (0C) in the area of Barentsburg (2003).

The temperature and relative humidity of air measuring in Ny-Alesund meteostation in 2004

0 30 60 90 120 150 180 210 Jan Feb Mar Apr May Jun Jul- 3 0

- 2 0

- 1 0

0

1 0

0 30 60 90 120 150 180 210 Jan Feb Mar Apr May Jun Jul2 0

4 0

6 0

8 0

1 0 0

Temperature of air (0C) in the area of Ny-Alesund (2004).

The relative humidity of air (%) in the area of Ny-Alesund (2004).

The wind velocity and cloud amount measuring in Ny-Alesund meteostation in 2004

The cloud amount (ball) in the area of Ny-Alesund (2004).

0 30 60 90 120 150 180 210 Jan Feb Mar Apr May Jun Jul0

4

8

12

16

20

0 30 60 90 120 150 180 210 Jan Feb Mar Apr May Jun Jul0

2

4

6

8

10

The wind velocity (m/s) in the area of Ny-Alesund (2004).

Calculations by the one-dimensional nonstationary thermodynamic model of sea ice with using real

meteorological data of Gronfjord (2003) and Kongsfjord (2004).

Calculated after the model and observed

thickness of snow and fast ice in Gronfjord (2003 y.)

3 0 6 0 9 0 1 2 0 1 5 0 1 8 0

- 1 . 2

- 0 . 8

- 0 . 4

0

0 . 4

ICE

SNOW

H, m

1

2

3

4

t

Calculated after the model and observed thickness

of snow and fast ice in Kongsfjord (2004 y.)

-10 20 50 80 110 01 .2 00 3 0 2 .2 00 3 0 3 .2 00 3 0 4 .2 0 0 3 0 5 .2 0 0 3

-1.2

-0.8

-0.4

0

0.4

SNOW

ICE

H, m

1

2

3

4

1 – calculated fast ice thickness;2 – calculated snow thickness;3 – measured ice thickness;4 – measured snow thickness.

ANOTHER MODELS, USING IN OUR ANOTHER MODELS, USING IN OUR RESEACHRESEACH

The thermodynamic model of sea ice of Yu. P. Doronin (1963).

The conservation of heat in the ice and snow layers are given

in one-dimensional forms as:

2

2

z

TK

t

Ti

where Ki – is snow or ice temperature conductivity coefficient.

For sea ice the temperature conductivity coefficient is:

22

22 1Tc

SLc

Ki

i

where - = 500; L - effective heat of sea water freezing.

ANOTHER MODELS, USING IN OUR ANOTHER MODELS, USING IN OUR RESEACHRESEACH

The the rmo dy na mic mo de l o f s e a ice o f M a y k ut a nd U nte rs hte ine r (1 9 7 1 ).

The co nservatio n o f heat in the ice and sno w layers are given in o ne -

d imensio nal fo rms as:

z

I

z

T

t

Tc

02

2

)(

w here fo r sea ice heat co nd uctio n is ca lcula ted b y fo rmula (3 2 ) , and vo lumetric heat

cap acity is:

2)(

T

Scc i

i

w here = 1 .7 1 5 * 1 0 7 J K m -3 p p t-1 ; (c ) i – is vo lumetric heat cap acity o f p ure ice .

ANOTHER MODELS, USING IN OUR ANOTHER MODELS, USING IN OUR RESEACHRESEACH

T h e t h e r m o d y n a m i c m o d e l o f s e a i c e o f E b e r t a n d C u r r y ( 1 9 9 3 ) .

T h e m o d e l o f E b e r t a n d C u r r y i s i m p r o v i n g m o d e l M U , a n d t h e m a i n d i f f e r e n c e s

i n t h e n e x t m o m e n t s i s c o n c l u d e d .

T h e e f f e c t i v e v a l u e o f t r a n s m i s s i o n c o e f f i c i e n t i s o b t a i n e d a s :

NNi 35.0)1(18.00

w h e r e N – i s f r a c t i o n a l c l o u d a m o u n t .

I n t h e m o d e l o f E b e r t & C u r r y a l s o u s i n g a s p e c i a l p a r a m e t e r i z a t i o n o f s n o w a n d

i c e h e a t - p h y s i c s c h a r a c t e r i s t i c s a n d a l b e d o .

ANOTHER MODELS, USING IN OUR ANOTHER MODELS, USING IN OUR RESEACHRESEACH

The thermodynamic model of sea ice of Zeebe et al (1996).

The model of Zeebe et. al. is improving model MU also, and the main differences

in the next moments is concluded.

The incoming longwave radiation flux is given by:

424 )275.01()15.273(1077.7exp261.01 afa TcTR

where Ta – is surface air temperature; cf – is cloud cover fraction.

The ice density and the thermal conductivity were determined by:

21

11FF

FV

i

ia

bbiba VVV 1

1F

SVb

where Va – is the gas volume (1.5 %); Vb – is brine volume.

The fast ice thickness in Kongsfjord (area of Ny-Alesund, 2004y.), calculated from a number of models.

1 – fast ice thickness calculated from our model;2 – fast ice thickness calculated from MU model;3 – fast ice thickness calculated from Ebert and Curry model;4 – fast ice thickness calculated from Doronin model;5 – fast ice thickness calculated from Zeebe et al model;6 – measured fast ice thickness.

0 30 60 90 120 150 180

-1

-0.8

-0.6

-0.4

-0.2

0

H, m

1

2

3

4

5

6

The model blocks, The model blocks, which need to be which need to be

subsequently improve:subsequently improve: • Block of determination of heat flux from underice

water.• Block of determination of snow and sea ice

cover albedo.• Block of determination of extinction and

transmission coefficients of solar radiation for sea ice.

• Block of determination of vertical distribution of sea ice salinity.

CONCLUSIONS:

• Unique investigations of snow and ice cover characteristics from Spitsbergen archipelago were carrying out.

• Mathematical, especially for Spitsbergen archipelago conditions, one-dimensional thermodynamic model of sea ice was elaborated.

• The developed one-dimensional non-stationary thermodynamic model of sea ice fully adequate reproduces evolution of fast ice thickness in Kongsfjorden gulf (area of Ny-Alesund, West Spitsbergen) and Gronfjord gulf (area of Barentsburg).

• Taking into account water heat flux for model calculations of evolution of fast ice thickness in Kongsfjorden gulf is necessary.

• The developed one-dimensional non-stationary thermodynamic model of sea ice more really shows changing of fast ice thickness in Kongsfjorden gulf, then another well known famous thermodynamic models of sea ice.