On the temperature gradient in the upper part of cold ice sheets

Abstract

The influence of the surface slope on the temperature profile in the upper part of a cold ice sheet can be described by a function which is independent of the geothermal heat and the heat of friction. This function is calculated for the two-dimensional and the axisymmetric cases. In the two-dimensional case its simplest form is proportional to the horizontal velocity and to the height above bedrock reduced by a constant; another form of this function is approximately proportional to the square of velocity and height.

L'influence de l'inclinaison de la surface sur le profil de température dans la partie supérieure d'une calotte glaciaire froide peut être décrite par une fonction qui est indépendante des chaleurs géothermique et de friction. Cette fonction est calculée pour les cas bidimensionnel et axisymmetrique. Dans le cas bidimensionnel la forme la plus simple de cette fonction est proportionnelle à la vitesse horizontale et à l'altitude sur le lit rocheux diminuée d'une constante; une autre forme est donnée qui est approximativement proportionnelle au carré de ces deux facteurs.

Zusammenfassung

Der Einfluss der Oberflächenneigung auf das Temperaturprofil im oberen Teil eines kalten Eisschildes lässt sich als Funktion besehreiben, die unabhängig ist von der Erd- und der Reibungswärme. Diese Funktion ist für den zweidimensionalen und für den rotationssymetrischen Fall angegeben. Im zweidimensionalen Fail ist ihre einfachste Form proportional der Horizontalgeschwindigkeit und der um eine Konstante reduzierten Höhe üher dem Felsboden; eine andere Form ist angenähert proportional dem Quadrat dieser beiden Grössen.

I. Introduction

The negative temperature gradient in the upper part of large ice sheets has two sources: The movement of the ice in the x-direction and climatic changes. Both effects result in a temperature decrease with depth of the order of 1 deg. It is possible to draw conclusions about climatic changes if the influence of the movement is determined precisely.

Theoretical and field work on this subject has been published by Robin (1955, 1970), Bogoslovski (1958), Radok (1959), Weertman (1968), Budd (1969), Budd and others ([1970]), Radok and others {1970), Budd and Radok (1971). It seems, however, that the following useful solution for the upper part of ice sheets has been overlooked so far.

II. General solution for the two-dimensional case

The homogeneous form of the heat transport equation under steady-state conditions is

(1)

where κ is the diffusivity of ice, y the height above bedrock, x the distance from the ice divide, v_u and v/_z the vertical and horizontal velocities respectively.

In the upper part of the ice sheet, v_z is considered to be independent of y (B. Philberth, 1956; Haefeli, 1961; Weertman, 1968; Dansgaard and Johnsen, 1969[b]; Philberth, I972[b]), and

can be written

Using the continuity equation we can write

. Hence

, where y_r , the “reduced height” is equal to

. The term h₀(x) can be visualized as the height below which the ice has zero velocity and above which it is governed by the block flow law. Usually h₀(x) is taken to be zero (Robin, 1955; Weertman, 1968; Budd, 1969; Radok and others, 1970). In this paper we shall use the relation:

(2)

where h = h(x) is the total thickness of the ice sheet, v_zm is the mean value of the horizontal velocity over the total ice sheet and v_x is its value in the upper part of the ice sheet. Corresponding to y_r we shall use the “reduced ice-thickness”

. The ratio h_r/h is equal to ν_xm/v_z . For the station Jarl Joset, central Greenland, this ratio is 0.9 (Philberth, 1972[b]).

Using

and neglecting the heat diffusivity in the x-direction, Equations (1) can now be written

(3)

The simplest non-trivial solution of this equation is

(4)

where C ₁, and C ₀ and C ₂ used below, are constants.

If

is small either with respect to

or to 2 (for station Jarl Joset the first term is 0.15; the second term is > 10, down to a depth of 1 km), the following approximate solution can also be used

(5)

Solutions with higher powers of u_z need not be considered for practical use.

The combination of Equations (4) and (5) leads to the general solution

(6)

III. General solution for divergent movement

A generalization of Equations (6) for cases of divergent movement can be obtained by introducing a factor ε characterizing the divergence of the streamlines

(7)

In the two dimensional case ε = 1, If the accumulation a and the total depth of the ice h are independent of x, ε is 2 in the axisymmetric case and between 1 and 2 in intermediate cases. If a = a(x) and h = h(x), however, in the axisymmetric case ε is equal to

where z is the horizontal direction orthogonal to the streamline. If these last expressions are not constant a mean value along the streamline has to be used.

IV. Application of the two-dimensional solution

Differentiating Equations (6) with respect to x, using T = T_s (surface temperature) and y = h(x), hence y_r = h_r , yields

(8)

where we introduced the substitution

, with a = a(x) denoting the ice value of the accumulation rate. In Equations (8) the term

is neglected for the reasons explained in connection with Equations (5). Equations (8) allows us to determine the constants C ₁, and C ₂.

Under normal conditions dT _s/dx, the horizontal gradient of the surface temperature, and a(x) increase with increasing x. If they increase at an equal rate, C- becomes zero. Using Equations (8) with C ₂ = 0, Equations (6) becomes

(9)

Near the surface, y_rh_r approximates y/h.

The solutions (4) and (9) do not take into account the heat of friction and the geothermal heat. On the other hand, Robin (1955), Dansgaard and Johnsen (1969[a]) and Philberth and Federer {1971) calculated vertical temperature profiles taking T_s as independent of x but considering the geothermal and frictional heats. Let T_g be such a profile, which is normalized by the addition of a constant so that it becomes zero at the surface.

The real temperature profile taking into account all three influences, is obtained simply by adding T_g to Equations (6), (7) and (9) respectively. This can be explained in the following way: T_g is the solution of a differential equation, which differs from Equations (1) only by a function of x and y on the right-hand side, expressing the heat of friction. In the case of a linear differential equation, the sum of the solutions of the homogeneous plus the inhomogeneous forms is also a solution of the inhomogeneous form. At the surface T _g is zero (T = T _g), but for y _r = 0 the function T and its vertical gradient are negligibly small with respect to T _g and its vertical gradient.

Of practical significance is the fact that in the upper part (region with negative temperature gradient) not only T but also T _g, can be calculated in a simple way. In the upper part T _g can be taken as independent of x (e.g. according to Robin, 1955, Equations (8) or Philberth and Federer, 1971, table I). This can be verified as follows: The upper part of the ice sheet is influenced by a heat of friction which is much smaller than the heat of friction produced below point x, because it originated at a considerably smaller x and v_s. Therefore its influence is normally much smaller than the (x-independent) geothermal heat and can be neglected or approximated by a function which does not depend on x. The horizontal gradient

depends on dh/dx and da/dx; but Weertman's (1968) Equation (6b) for dT _B/dx = 0 (his dT _u/dx) shows

to be very small in the range where T — T, is small.

V. An example: station jarl joset (Lat. 33° 30' W., Long. 71° 20' N., a 865 m A.S.L.

General values:

K = 40 m² a⁻¹; temperature lapse rate λ = 9.5 deg km⁻¹.

Local values :

h = 2.500 km; h_t = 2.250 km; x = 125 km (Philberth, 1972[b]).

Values between ice divide and Jarl Joset:

a = 0.30 m a⁻¹ ice value (independent of x; Federer and others, 1970), ε - 1 (two dimensional case; Philberth, in press), height of surface = (10) (Mäker, 1964; Lliboutry, 1968; Philberth and Federer, 1970).

Derived relations :

Multiplication of Equations (10) by λ yields

according to Equations (6) we have:

The comparison of Tfor y _r = h _r with T _g yields:

For x = 125 km (Jarl Joset) the result is:

(11)

Comparison with measured values:

At the depth of 615 m (y_r = 1 635 m) -29-30° C and at 1 005 m (y _r, = 1 245 m) -30.00° C have been measured by the thermal probe method (Philberth, 1962, 1970); that is a difference of 0.70 deg. For these two depths the Equations (11) yields a difference of 0.52 deg and the function T _g, yields a difference of -0.32 deg. Hence the total amount of the theoretical difference for steady-state conditions is 0.20 deg.

The measured value (0.70 deg) and theoretical value (0.20 deg) differ by 0.50 deg, which can be explained by palaeoclimatic changes. If a temperature jump θ (end of ice age) is assumed to be at 10 000 years B.P. (Dansgaard and others, 1969), the 0.50 deg difference corresponds to θ 5 deg, if the jump is assumed to be at 12000 years B.P., it corresponds to θ = 6 deg (Philberth, 1972[a]).