Temperature profiles in a clean ice sheet with solar radiation absorption

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Temperature profiles in a clean ice sheet with solar radiation

absorption

DIVISION OF BUILDING RESEARCH

TEMPERATURE PROFILES IN A CLEAR ICE SHEET WITH SOLAR RADIATION ABSORPTION

by

L. E. Goodrich

Internal Report No. 411 of the

Division of Building Research

Ottawa February 1974

WITH SOLAR RADIATION ABSORPTION by

L. E. Goodrich

PREFACE

In the past one of the interests of the Geotechnical Section has

been the question of mid-winter cracking of ice-covers on lakes. The

purpose of the work presented in this report was to develop formulae

which could be used to make estimates of temperature distributions in clear ice -covers which could in turn be used as a basis for studying the problem of thermal stresses in lake ice.

This work was undertaken in 1967. An internal report was not

prepared as subsequent work led the author to believe that numerical techniques which have since been developed were much more suitable for

handling problems of the type envisaged. Nevertheless, enquiries are

continually being received regarding various aspects of this problem and it is to satisfy this need that an internal report is being prepared at this time.

Ottawa

February 1974

N.B. Hutcheon Director.

WITH SOLAR RADIATION ABSORPTION by

L. E. Goodrich

The problem considered in this report is that of an ice sheet being heated internally by solar radiation while heat is lost at the surface

by long-wave and turbulent exchange with the atmo sphe r e , An analytic

solution is sought for the following problem:

R. ln AR.ln

1.

ln h(T-Ta) z

=

d ice: 1 aT

a.

at

z

=

0 •

I

I L l / I L l

,

T = 0

I

/

At the upper surface, z

=

d, the net short-wave radiation is

R = R. (l - A)

o m

where R. = incoming solar radiation

1n

A = albedo.

Incoming long -wave radiation at the surface is I.

1n

4

= E c T . a1r

where 0"

=

Stefan -Bo1tzman constant

=

1.36xlO-12 lye sec- 1 - 4• K

T . = air temperature (OK)

a1r

E = emissivity of atmosphere

=

0.44

+

O.092.jW (Brunt formula for clear-sky conditions)

where w = water vapour pressure, mm Hg ,

Outgoing long-wave radiation is, assuming emissivity of ice is unity, 4

I t = 0" T (d,t)

ou

where T (d, t)

=

ice surface temperature, OK.

Heat exchange at the surface, including both sensible and latent heat, 1S as sumed to be expre s sible as

H = h{ T (d , t) - T . ) a1r The initial condition is taken as

T (

z,

0)

=

T

z/

d

o

where T = initial surface temperature

o

= T .. a1r

The lower boundary is assumed fixed at DOC and this boundary is assumed not to move, i. e. accretion or ablation are ignored.

Linearizing the net long-wave exchange 1 -1 :=:::! cr T

3

b [(E-l) T b

+

4(T -T(d,t» ]

in out oa s oa s 0

where T oabs = To

+

273 "K, the boundary conditions at the upper surface may be written: (kaT) = 1 - S(T(d,t) - T ) az z=d 0 0 where 1 o 4

=

cr(E - l) T b oa s 3 S = 4 o T b

+

h. oa s

The differential equation governing the temperature evolution within the ice in the presence of internal heating is:

1

a.

aT at Q

+

-k

where

a.

= k = thermal diffusivity

i.C

P

i.

=

density

C = heat capacity p

Q = rate of heat gain per unit volume.

For perfectly clear ice, completely free of air bubbles or other imperfections, monochromatic radiation extinction closely follows the Bouguer -Lambert exponential law:

-S(d-z) R = R e

o

where R o is the radiation flux density at the surface z = d , R represents the flux left in the beam after travelling the distance d-z into the ice, and

/ -B(d-z)

Then Q

=

oR oz

=

_{R o} ｾ e . . In the de ve Iopm.ent that follows

we consider first the rnorioc hr-ornatic case. The results are later generalized

to take account of the spectral variation of B.

Swnming up, we wish to find the solution of the equation:

1 aT

a

ot =

with boundary conditions:

t

=

0; T(z,o) = T (z/d) o z

=

0; T(o, t) = 0 z =: d; k(oT

=

I -S[T(d,t)-T ] oZ)z=d o 0 Let: x

=

z/d

a.

1"

=-

t d2 8= k T Rod and b = Sd

Substituting for z, d, t and T gives the equation

with boundary conditions:

1"

=

0;

8(x,

0)

=

8 x where 8

=

o 0

x = i.

,

ＨｾＩ

= L - c (8( 1 , 'f) -8 ) ox x=1 0 Sd I where c = _k

,

L

- -

0 R 0

SOLUTION BY FOURIER METHODS

-b( I-x)

Because of the time -independent forcing function, be , we may

expect to find solutions in the form:

Thus the original problem can conveniently be separated into two problems:

(1) = -be-b( I-x) wit. h 8

1 = 0 at x = 0

( 2) = with

o at x = 0

Problem (1) is readily solved by straightforward integration to give:

-b -b(l-x) 8 1(x) ::: e Ｍｾ

+

Ax where A= -b / l+L+c(l-e ) b+c8 o 1

+

c

Inserting 8 1(x) into problem (2) with b. c. ; 1" = 0, -b(1-x) -b 8 ( )

=

8 x +( e - e ) - Ax 2 x ,0 0 b x

=

0, 8 = 0 2 08 2 x = 1, -

_ox

= -c8 (1 1") 2 '

Solution of this equation for a "z-adiatfon" type boundary condition by

a Fourier series method is discussed by Carslaw and Jaeger (1), p, 126.

According to these authors the solution may be put in the form:

2 2 2 co

"

o

n

+

c

')

-0n 1" 8 2(x , 1" )

=

Ｒｾ］Ｑ

ｾ

_o

2 2 e (Sin

o

nx) an +c +c n 1

where a

=

S

Sin (Onx) 8

2(x , 0 )d x n

0

and the 0 are the positive roots of 0 = -c Tan( 6 ).

n n n

Inserting the expression for 8

2(x , 0): 1 -b(l-x) -b

r

Sin6 x {( 8 0-A)x + e -e }dx a

=

n

J

n b 0 (l+c)Sin 6 -b eb(b+c)Sin6 +6 (8 -A) n e { n n = + -0

_o

₂ b _{b2+0 2} n n

Substituting the expre ssion for A we obtain:

o

2 n a n SinO = (8 -L) n 0

0

2

n -b (b+c)SinO + 0 e

b{

n n }

o

2(b2+0 2) n n

then cO +L 8(x, T, b, c) = (8 0-L)g(x, T, c) + (c+ol )x + f(x, T,b , c) where >. 2 2 v +c n 2 2 6 +c +c n e 2 -6 T n Sin6 x n

and f(x, T, b,c)

=

e-b_e-b(l-x) + b+c(l_e- b) x

b b(c+1) co - 2b L: n=l 6 2 2 -b , +c , , (b+c)Sin6 + 6 e ' n \ ' n n

'-2-2-) , - - - )

e 6 +c +c 6 2(b2+6 2) n n n 2 . -6 T Sm6 x n n

where the 6 are the positive roots of [)

=

-cTan6 .

n n n

Finally the temperature distribution is given by:

T(X,T,b,c)

dI c T +dI /k, R d

o ( 0 0 ' 0

k

)

g + \ c+ 1 ) x + ｾ f (x, T, b, c)

MULTISPECTRAL CASE

Thus far a radiation heating function has been as sumed that is appropriate to the monochromatic case, viz:

Q(z) = R Be -B(d-z)

o

If we wish to apply the results of these calculations to the case of thin,

clear, flawless ice-sheets 1 metre or less in thickness, it is necessary to

consider the appropriateness of using a function of the type above with R as an

average value for the visible spectrum, as is usually done when discussing heating of a snow cover or a glacier by solar radiation.

Divide the spectrum into M+ 1 coefficient may be taken as constant. spectrum is approximately:

regions, in each of which the absorption The heating function integrated over the

M+l Q(z}ｾ R L: o m=l

S

r e mm -[3 z m

where

S

= value of absorption coefficient for the mt h spectral interval

m

and r

=

fraction of incoming solar radition at the ground in the m th

m

spectral interval.

Values of r m were calculated from spectral energy distribution values for solar radiation at the ground (clear sky) due to R. Emden and

quoted by H. Reuter (2). Sm values were obtained from data compiled by

the author from various sources (3). These data are listed in Table 1.

The results of the integration are shown in Figure 1. As may

be seen, the curve deviates significantly from linearity in the first 30 or 40 cm, which means that the use of an absorption coefficient appropriate to the visible region only would re suit in a calculated heating curve which is grossly in error over a large fraction of the thickness of the ice-sheet. It is essential, therefore, to take into account the spectral variation of absorption coefficient with wavelength when considering radiation heating

of thin

«-1.

m) clear, flawless ice-sheets*. This is probably best done by

performing a simplified spectral integration, although there may be some

advantage to using a depth ､･ｰ･ｮ､･ｮｾ･ｱｵｩｶ｡ｬ･ｮｴ extinction coefficient, which

could be derived from the curve of

ｑｾｺＩ

o

*Such a conclusion probably does not apply for snow or ice with flaws because, due to internal reflections, the absorption coefficient is about 2 orders of magnitude greater, so that at certain depths the only radiation

remaining is in the visible region. The near -infrared part of the incoming

solar radiation can probably be included with the thermal radiation and be

treated to a good approximation as acting at the surface only. In the case

of glacier ice however, the problem remains complicated inasmuch as with direct radiation the monochromatic absorption coefficient is itself depth dependent owing to inte rnal refle ction s .

To calculate the t erripe r atu r e distribution in the ice, in the po lyc hr omatrc case define

M+l T = L: ITl=1 T ITl M+l

and R = R L: 1 r where r is the fraction of solar radiation in the

o 0 ITl= ITl ITl

th 1 . 1

ITl spectra inte r val , M+l

aT M+l

a2T

M+l

-b b

-;- _{ITl d ITl ITlX}

Then:

L

aT =

L

- - + R

-

L

r b e e

ax2 0 k ITl ITl

ITl=1 ITl=1 ITl= 1

with the boundary conditions:

T = 0; x = 0; x = 1; L: T = T x ITl ITl 0 L: T (0, T) ₌ 0 ITl ITl , aT d d L: _{\ ax}ITl ₎ = + I ok

- s-

(L: T (I,T)-T ) ITl x=1 k ITl 0 This is (3 )

equi valent to the SUITl of the

2 aT l o T1 d = + -aT ax2 k equations: with b. c. : T = 0; = T x o x = 0; T 1 = 0 x

=

1;

plus the M equations of the fo rrn :

( 4) aT ITl

=

+ -d R rOITlITlb e k -b ITl e b x ITl

with b. c.: T = 0; T ₌ 0 m x

=

0; _{T =} 0 m aT sd m T (l,T) x= 1 · - - =_{, ax}

-k m

Problem (3) has been solved previously and the solution of (4) may be

obtained from this by setting T = 0, I = 0, i ,e., the solution of (4) is

o 0

d

T

=

k R r f(x, T,b ,c)

m O m m

Thus the solution for the general polychromatic case is:

dI c T

+

dI /k R d M

+

1

(

0 \ { 0 0 ') 0

T(x, T, b, c) = To - - k ) g(x, T, c)

+

l+c

J

x

+

- k L: r f(x, T, b ,c)

m=l m m

For the special case where a linear temperature gradient has been established as a result of thermal radiation cooling over a sufficiently long period (as is appropriate in the morning just before sunrise) we have

I

=

kT /d

o 0

and the equation for the temperature at a later time under the influence of solar radiation heating is

T(x, T, c) R d o

=

T x

+

-o k M+l L: m=l rmf(X,T,bm ,c) where T is to be determined by o k T = I = a(E-l)(T +273)4.

d

0 0 0 RESULTS

The above expression was used to determine the temperature distribution for ice-sheets of thickness d = 100, 50, 25 ern at times

For late February at 500 N

a typical value of daily insolation on a clear day is "-' 300 ly. For a daylength of 10 hour s, this corresponds to R in = 8.3 x 10- 3 ly/ sec, and taking albedor - 10 per cent independent of

wavelength, then ｒｯｾ 7.5 x 10- 3 ly/ sec.

The turbulent heat transfer coefficient h might typically have value '"'" 24 ly/day· °C (4) or

-4 /

h ｾ 2.8 x 10 ly sec.

At temperatures around -20C degrees the saturation vapour pressure is "-' 1 mm Hg so that for relative humidity 100 per cent we have E

=

.53 for

the emissivity of the atmosphere, while for dry air E

=

.44. A value of

E = O. 5 has been used for the pre sent calculations.

Thermal parameters used were:

3 .

K = 5.3 x 10- cal/cm· °C· sec

C_p = 0.5 ca

L'

grn ' °C

-2 2

a.

= 1. 4 x 1 0 em / sec

Values of absorption coefficient and spectral distribution of incoming solar radiation listed in Table I were used for the spectral integration.

The results are presented in Figures 2,3, and 4 for h

=

0

(corresponding to no wind), h

=

2.8 x 10-4ly/sec, and h

=

5.6 x

ＱＰＭＴｾＺ｣

It will be noted that in almost all cases the curves show greater

heating at the surface than at depth. This is contrary to re suIt s pre sented

by Reuter (2) who found greater warming a few centimetres below the surface. Reuter also took account of the spectral variation of absorption coefficient using values similar to those used here, however he assumed the simple

bounday condition

ＨｾｾＩ

=

constant, rather than the more realistic

surface

"radiation" type boundary condition.

For the larger value of turbulent heat transfer coefficient the

heating at depth tends to a value near that for the surface. It is concluded

that, at least for the conditions considered, the so-called IIgreenhouse effect"

ｾＺｃａｬｬ calculations were performed by digital computer as preliminary work showed the necessity of including up to 30 terms in the series in order to obtain sufficient accuracy.

is not significant. Calculations such as those of Reuter (2) that as sume fixed heat flux at the surface seriously overe stimate the magnitude of the "greenhouse effect".

The analytic formulae pre sented here are severely limited. Solar

radiation and air temperature have necessarily been assumed constant and

this cannot be considered realistic. The more general problem could be st

be examined by numerical techniques. REFERENCES

1. Carslaw, H. S. and J. C. Jaeger. Heat Conduction in Solids. Oxford,

Clarendon Press, 1959.

2. Reuter, H.

Medien.

Zur:Theorie doer Warmehaurhalter strahlungsdurchHissiger Tellus1., 3, August 1949, p.6-14.

3. Goodrich, L. E. Review of Radiation Absorption Coefficients for Clear

Ice in the Spectral Region 0.3 to 3 Microns. National Research

Council of Canada, Division of Building Research, December 1970. (NRCC 11761).

4. Williams, G. Heat Transfer Coefficients for Natural Water Surfaces.

International Assoc. of Scientific Hydrology. Publication No. 62,

ABSORPTION COEFFICIENT FOR CLEAR FLAWLESS ICE ..1

Wavelength Interval r

S

(em )

m m .45jJ. .08022 -3 0.30 - .65 x 10 -3 .45 - .50 .07392 .65 x 10 -2 .50 - .55 .07139 .104 x 10 .60 .06862 -2 .55 - • 165 x 10 .60 - .65 .06408 .25 x 10-2 -2 . 65 - .70 .05790 .45 x 10 .05235 -1 • 70 - .75 • 13 x 10 .80 .04755 -1 .75 - .20 x 10 .85 .04175 -1 .80 - .40 x 10 .03582 -1 .85 - .90 .60 x 10 .90 - 1.00 .06193 .17 1.00 - 1.15 .07076 .33 1.15 -1.35 .06811 1.3 1. 35 - 1. 85 .10343 17 • 1. 85 - 2.50 • 03746 59. 2. 50 .. 3.50 .06471 2.4 x 103

6 8 10 x 10- 3 2 4 6 8 lOx 10- 2 2 4

o

-20 100 ｾ 40

u

.

N

bor

/

J: ..Q. = ｾ r

p

ex

p (-

p

Z) R m m m I - m Q.. 0 UJ c

.Q

= 69.7 AT Z = 0 R

I

/

0 80 FIGURE BR 5156-1

o

-4 -8 -1 2 -24 -20 -16 TEMPERATURE,

-c

-28 -32 -36

o

-40 -0 .8 <,

x

.

J: f-c, UJ .6 Cl VI VI UJ ...J Z 0

_-

.4 VI Z

ｾ

d

=

100 eM UJ h

=

0 ｾ -Cl .2 FIGURE 20 BR 5156-2

o

-4 -8 -1 2 -16 -20 -24 -28 -32 -36

o

I I I I I I I I I I I I I I I I I I I I ... -40

"

.8 <,

x

.

:::c I -0... w _.6 0 Vl Vl W ...J Z 0 .4

-Vl Z w

I

d

=

100 CM :E _h

₌

_{2 .8 x 1 0- 4}

-0 .2 TEMPERATURE, °C FIGURE 2b BR 5156-3

o

-4 -8 -12 -16 -20 -24 -28 -32 -36

o

I I i i i I I i i i i i i I I I i i i I ... -40 .8 "'tl <, X

.

:::c I -Q.. .6 w 0 Vl Vl W ...I Z .4

l

d

=

100 CM 0 _h

₌

_{5.6 x 10- 4}

-Vl Z w ｾ

-0 _.2 TEMPERATURE, °C FIGURE 2c BR 5156-4

-c .8 <,

x

ｾ J: t-a.. w _.6 Cl In In w ....I Z 0 _.4

-In Z w ｾ

-Cl .2

o

-40 -36 -32 -28 -24 -20 -16 . -12 -8 -4

o

T.E MPE RA TU RE, °C

FIGURE 30 BR5156-5

a

-4 -8 -1 2 -16 -20 -24 -28 -32 -36 0' I I ! I I J I I A I , I I I I -4 I I I I I >I ""0 .8 <, X ｾ ::I: ｾ a.. w .6 Cl VI VI W

...

.)

Z d = 50 CM 0 h = 2 .8 x 1 0- 4

-VI Z w :E

-Cl .2 TEMPERATURE, °C FIGURE 3b BR 5156-6

o

-4 -8 -12 -16 -20 -24 -28 -32 -36

o

I I I I I I I I I I I I I I I , I I I I ' " -40 "1j .8 <,

x

, I f-c, w .6 0 Vl Vl w .4

L

- l

z

d

=

50 CM 0 h

=

5 .6 x 10- 4

-Vl Z w :E

-0 .2 TEMPERATURE, °C FIGURE 3c BRｾＱＵＶＭＷ

N I ｾ I -0 I eo • U 0 (I) I W C CD c.::

..,.

It) ::> ii 0 I - et: W (I)

«

_c.:: c.:: ::> w 0- _C> ::E u... W N I - ..-I ｾ ..-I ::E u -0 10 NO II _II "'0 .s: eo ..-I N <Xl o L-_...L_ _..._...L_ _....&..._...l_ _...L.._ _l.-_....L.._ _.L..._... N O ' o

pix

'Hld30 SS31NOISN3WIO

N 1 "'<t I -0 I OJ I U 0 en w ...a Ih "'<t ttl 0 c.:: _in ... ::> It: 1 t - W lD

-c

c.:: c.:: _::> w 0 e, ｾ IL. N W

...

t -1 10 _"'<t "'<t I 0 0 ｾ X UOJ 10 N N -0 II II ... I "'tl .L o

pix

'H!d3G SS31NOISN3WIG o N 0 1

o N I 'o:t I -0 I co I U 0 Q I U (l) UJ to Cl' 'o:t iO 0 ::> It:

....

UJ CD

«

Cl' Cl' _::> UJ 0 0.. ｾ U. UJ

....

N ｾ ｾ x U-o -0 Ll") NLl") 11 II "U.s::. CO

pix

'Hld3G SS31NOISN3WIG