The Physical Basis of Ice Sheet Modelling (Proceedings of the Vancouver Symposium, August
1987). IAHS Publ. no. 170.
Department of Geophysics and Astronomy
University of British Columbia
Vancouver, British Columbia V6T 1W5, CANADA
Geological Survey of Canada
Department of Energy, Mines and Resources
Ottawa, Ontario K1A 0E8, CANADA
Geophysics Program AK-50
University of Washington
Seattle, Washington 98195, USA
ABSTRACT Atmospheric pressure fluctuations cause air to flow through permeable firn near the surface of an ice sheet. Viscous friction resulting from this flow could prove to be a significant heat source and may explain anomalous temperatures measured in Agassiz Ice Cap, Arctic Canada. We examine the possible magnitude of this "wind pumping" energy source by assuming that air is an isothermal ideal gas and that air flux is governed by
Darcy's law. Our results show that pressure fluctuations in the frequency range 0.1-10 Hz are likely to have the greatest influence on the thermal regime of ice sheets and that under favorable conditions the disturbance to the 10 m temperature can exceed 3°C. A causal connection between wind pumping and the observed Agassiz Ice Cap temperature anomaly has yet to be established and indeed our results are counter-intuitive. Some entirely different explanation may prove correct but this issue cannot be settled without further field measurements.
Le pompage par le vent, une source de chaleur dans les calottes polaires potentiellement non négligeable
RESUME A la surface d'une calotte polaire, les fluctuations de la pression atmosphérique provoquent des flux d'air à travers le névé perméable. La dissipation de chaleur par viscosité qui en résulte pourrait être une source de chaleur significative, et expliquer les températures anormales mesurées sur la calotte Agassiz, dans le Canada arctique. On estime l'ordre de grandeur de cette source d'énergie le "pompage par le vent," en admettant que l'air est un gaz parfait isotherme, et que son flux obéit à la loi de Darcy. Nos résultats montrent que les fluctuations de pression de fréquence 0.1-10 Hz sont celles qui doivent avoir la plus grande influence sur le régime thermique des calottes
169

170 G.K.C. Clarke et al. polaires, et que dans des conditions favorables réchauffement à 10 m de profondeur peut dépasser 3°C.
INTRODUCTION
Firn temperatures measured at three drill sites on Agassiz Ice Cap, Arctic Canada, point to the operation of a mysterious heat source. The sites are located on or near a ridge; from site to site, their separation does not exceed 2 km horizontally or 60 m in elevation (Fig. 1). What at first seems curious is that site A84, at the highest elevation, also has the warmest 10 m temperature, exceeding that of nearby site A79 by 0.45°C and that of the more distant site A77 by
2.64°C; this temperature offset is established in the upper 10 m and maintained at all greater depths (Fig. 2),
FIG. 1 Topographic map for the region of drill sites on Agassiz Ice Cap,
Ellesmere Island, Arctic Canada. Boreholes are identified by the year they were drilled: 1977, 1979 and 1984. These sites were also the sites at which surface wind and shallow firn temperatures were measured. Note that sites
A79 and A84 are located on a ridge. (The prefix A denotes "Agassiz.")
Two lines of evidence suggest that exposure to wind is one of the main differences between the Agassiz sites, those at greater elevation being the most exposed. The measured accumulation rate is least at the uppermost site and greatest at the lowermost site (Table 1).

Wind pumping 171
This fact and additional observations by Fisher et al. (1983) indicate that wind scour decreases the net accumulation at sites on the ridge, whereas little or no scour occurs a few hundred metres away from it. A comparison of 8
18
0 measured at the three sites also reveals a consistent pattern (Table 1). At the ridge sites, the retained precipitation is isotopically "warmer" than at the lower site. We conclude that winter precipitation has been scoured from sites A79 and A84, but preserved at site A77.
-25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15
Temperature (C)
FIG. 2 Measured temperature distribution in the Agassiz Ice Cap boreholes.
The A84 profile is preliminary, but the A77 and A79 profiles have been measured several times. In particular, the A77 profile has been measured many times over several years, and even the small wiggles are reproducible. Holes
A79 and A84 are air-filled; immediately after drilling, hole A77 was filled to the 60m depth level with arctic diesel oil. The more pronounced wiggles in the
A77 temperature profile are known to be caused by weak thermal convection in the oil (Fisher, unpublished).
In summary, a conspicuous difference between the three drill sites is the degree to which they are affected by wind. For this reason we examined the possibility that wind somehow raises the temperature at the anomalously warm sites. The mechanism that we propose is
"wind pumping": frictional heating associated with the diffusion of atmospheric pressure

172 G.K.C. Clarke et al.
Site
A84
A79
A77
Elevation m a.s.l.
1730
1700
1670
TABLE 1 Near surface properties ofAgassiz Ice Cap
5
18
0
50 year average
°l
'oo
-27.05
-27.90
-30.40
X
cm a
- 1
ice equivalent
9.4-10.1
11.5
17.5
10 m temp- erature
°C
-21.88
-22.33
-24.52
Tsile-TAll
°c
+2.64
+2.19
0.00
Relative wind speed
May 1986
(A77 = 1)
1.38
1.26
1.00 fluctuations into permeable firn. Wind blowing over any surface consists of two components: a steady time-averaged component and a turbulent fluctuating component. The latter produces a rapidly-varying pressure field that propagates into the firn. At depth, these random pressure waves are attenuated and their energy is transferred to the firn as heat. The maximum penetration depth for pressure fluctuations is set by the pore close-off density of 820 kg m
-3
. On
Agassiz Ice Cap, this density is reached at 46 m. In reality, the effective penetration depth must be considerably less than this, because high density is expected to reduce permeability and hence vapor flux. An indication of the maximum depth of effective penetration of pressure fluctuations is given by the general observation that in closed boreholes on Agassiz Ice Cap, air-suspended thermistors do not yield stable temperature readings at depths less than 10-15m.
Very near the surface any frictional heating is rapidly lost to the cold air above, so there is no disturbance of surface temperature. Thus, we predict the existence of a thermal boundary layer extending to some depth in the range 0-15 m beneath the surface.
In May 1986, we made preliminary measurements to detect this boundary layer and to associate it with a wind source. Highly sensitive totalling anemometers were installed 1 m above the surface at sites A77, A79 and A84, and the cumulative horizontal wind speed was measured continuously for 8 days. No storms occurred during the period of measurement, and wind conditions were light to calm. The anemometers could respond to very light wind gusts so that both the steady and turbulent components of wind speed were recorded. The observed cumulative wind speeds for the three sites are compared in Table 1. As expected, the totalled wind speed was greatest at the ridge sites.
Two sets of three calibrated thermistors were buried near the A77 and A79 holes at depths of 0.25, 1.00 and 1.85 m. This was done by digging a 2 m pit and inserting the thermistors horizontally about 0.4 m into an undisturbed wall of the pit. With the leads secured to the surface, the pits were filled with snow and temperatures were measured twice daily for five days. Results are shown in Fig. 3. At each of the three depths, the A79 ridge site registered warmer fim. Approximate average temperature differences T
A79
—T
ATJ
at 0.25,
1.00 and 1.85 m are respectively 0.69, 4.46 and 2.64°C (Table 2). There is a temperature difference maximum near 1 m depth, and at a depth of 2 m, die 10 m temperature difference between the two sites has been established. These observations reveal the existence of a thermal boundary layer and are consistent with the wind pumping hypothesis. The similarity of the measurements taken at a depth of 0.25 m confirms that die two sites are subjected to essentially identical forcing of the surface temperature (at least when these observations were made).
The maximum temperature difference of 4.46°C at 1.0 m suggests the presence of a heat source at this depth. The source must be transient, radier tiian steady, because at 1.85 m the temperature difference is reduced to 2.64°C. The following theoretical analysis lends further plausibility to the wind pumping hypothesis.

Wind pumping z = 0.25 m os
CD i ~ <*
3 c\l
4-» '
CO
Q ) CN, a •
E c o m CM z = 1.00 m
^ _ 9—9— z = 1.85 m
" S e - . - o « ®-
-« « «
173
14 15 16 17 18 19
May 1986
20
FIG. 3 Near-surface firn temperatures measured in May 1986 at two sites on
Agassiz Ice Cap. Measurements taken at site A79 (the site that is on a ridge and more exposed to wind) are plotted as a solid line. Measurements taken at site A77 (the site at lower elevation) are plotted as a dashed line. We expect that surface temperature variations are similar at the two sites; this is supported by the similarity of the 0.25 m temperatures. The temperature discrepancy at depths of 1.0 m and 1.85 m must therefore be attributed to some other cause, for example, wind pumping.
TABLE 2 Summary of near surface temperature measurements, May 1986
Temperature
Depth m
0.25
1.00
1.85
T
A19
°c
-19.37
-22.04
-25.84
T
A77
°c
-20.06
-26.50
-28.48
TAIT
T
ATI
°C
+0.69
+4.46
+2.64

174 G.K.C. Clarke et al.
THEORY
We take fini as a rigid porous solid permeated by an isothermal ideal gas. Let p be the density of air and n be the porosity of firn. For the following calculations n will be taken as a constant and we neglect the possibility of mass exchange between the firn skeleton and interstitial air. The total mass of gas in the porous medium is m
= j « p d
3
.
(1) giving the local form expression for mass conservation
n^- + -^-[nvipl = 0 dt fy where v,- is the air velocity. We define the volume flux of air as q i
= nv; so that Equations (2) and (3) give
(2)
(3)
[p<7d = 0 dt dxi
(4)
Assuming that air flux is governed by Darcy's law (Bear, 1972), k dd> qi
^ ax,-
(5) where k is the permeability of firn, |X is the viscosity of air, and <j> is the potential energy per unit volume of air. We choose a coordinate system for which the z axis is directed positive
downward so that <j> = p - pgz, where p is pressure, g is the gravity acceleration and z is the downward distance from some datum level. We take the firn layer as a permeable half space bounded at z = 0 by the atmosphere and having an impermeable horizon lying at some great distance below the upper surface. The presence of this impermeable horizon prevents continuous downward migration of air through the snow pack, something that is neither observed nor expected. Thus, in the absence of pressure disturbances at the z = 0 surface boundary (and neglecting thermally-driven vapor flux), air in the snow pack will be static and have constant potential energy <j> = (j>
0
= p
0
, where p
0
is atmospheric pressure. If the surface pressure has a small fluctuating component p'(x m
,t) then <j>(x m
,0 = Po + p\x m
,t); Equations (4) and (5) give
_a_ dxi f f ( P o + P')
U.
OX; dp dt
(6)
For an ideal gas, p =M
0
p/RT, where M
0
is the molecular weight of the gas, R = 8.314 J morMeg
- 1
is the universal gas constant and T is absolute temperature. Isothermal conditions will be assumed so that dp
=
M) dp dt RT dt
For small pressure fluctuations of an ideal gas, (6) and (7) give
(7)
JL dxi kPo dp'
\l dXi n
dt
Taking the ambient pressure p
0
, the viscosity u. and permeability k as constant gives vy = JXLM. kp
0
dt
(8)
(9)

Wind pumping 175 as the equation governing the diffusion of air pressure fluctuations into permeable firn.
Diffusion of surface pressure fluctuations into a permeable half space
Equation (9) can be recognized as a diffusion equation and the coefficient K = kp
Q
/\m measures the diffusivity of atmospheric pressure fluctuations into permeable firn. By analogy with the diffusion of surface temperature fluctuations into a conducting half space, the existence of a surface boundary layer for pressure fluctuations can be foreseen. Let z
0
be the thickness of this layer. Provided z
0
•«: H, where H is the depth to the impermeable horizon, the permeable half space approximation is valid. Pressure fluctuations at the surface boundary can be written as p'(x,y,0,t). If the lateral pressure gradients dp'(x,y,0,t)/dx and dp'(x,y,0,t)/dy are small, then over some characteristic distance I the pressure fluctuations within the firn can be approximated as p'(z,i). This approximation is acceptable provided that / » z
0
. Thus, assuming p' = p'{z,i),
Equation (9) can be written as a V
=
\in dp' dz
2
kPo &
The pressure fluctuation p'(z,t) can be written in terms of its Fourier transform p (z,w) as
(10) pXz,t) = ^- \ p(z,(ù)e iwt
d(ù (11) where co is angular frequency. It follows that
ÏM> = ±]&l£Ù.#*to (12)
//„(z,0 =-5- j i % » l e
< ^ C D (13)
2TCJU dzr oo p' t
(z,t) = 4~ f io>P (.z,(ù)e i<ot
d(ù (14) where subscripts z and t indicate partial differentiation. Applying (13) and (14) to (10) gives the Fourier transformed differential equation d
2
p(z,(û) lin . - , . ,,_-
—rv •
=
-tL_ ztt
(p( z
,co) (15) dz
2
kp
0
Defining
P
2
= i sgn co |4.nl(ol/^>o (16) where sgn co denotes the algebraic sign of co, the standard solution for (15) can be written in the form
p (z,co) = A(co)exp[+pz] + £(co)exp[-Pz] (17)
Noting that (/ sgn co)
1/2
= ± (1 + /' sgn co)/V2 and that the boundary conditions on p\z,i) are that p'(0,t) is specified and p\°°,t) = 0, the Fourier transformed solution may be written as
p(z,(ù) = p(0,co)exp{ - (1 + i sgn co)cxz} (18) where j_ a = (\in\(ù\/2kpo)
2
(19)
Two expressions that follow from (18) are essential to subsequent discussion:

176 G.K.C. Clarke et al. p z
(z,(û) = -oc(l + i sgn co)j5(0,(B)exp{ - (1 + i sgn œ)az} (20) obtained by taking the z derivative of (18), and
\p
2
(z,oo)l
2
= 2a
2 exp(- 2az)lp (0,co)l
2
(21) obtained by multiplying p z
(z,co) by its complex conjugate. Equation (21) applies if the fluctuation p'(0,t) is a transient pressure fluctuation. If p'(Q,t) is a stationary random process having a mean value of zero, Equation (21) takes the form
S ldp,3z]
(z,(0) = 2a
2 exp(- 2ocz)S w
(0,co) (22) where is the power spectrum of p' z
(z,t) and S^O.co) is the power spectrum of
Viscous dissipation of pressure fluctuations
For slow flows through permeable media
a = -qi dtydxi (23) gives the rate of viscous dissipation of energy per unit volume. For the case in which
<j> = <j>
0
+ p'(z,f), Equation (5) gives a(z,f) = —
M.
12 dz
(24)
The time-averaged strength of the viscous heating function is therefore
T T
a\z) = lim -±- Ja(z,f)dt = A lim ^ jp'
2z
(z,t)dt (25)
Noting that
T
R
&pldz
\z,x) = lim -±- \p\(z,t)p' z
{z,t - x)dt (26) r-*» Li _j defines the autocorrelation function of p' z
(z,f) and that oo
R
[3pldz
\z,x) = - î - f S ldp/dz
\z,(ù)e im
d(ù (27)
2itj^, by the Wiener-Khinchin theorem (e.g. Solodovnikov, 1960, p. 98), Equations (21), (25), (26) and (27) lead to
a\z) = — R
[dp,dz
\z,0) oo
= A J _ \ s ldp/dz]
(z,e>)d(>) (28) oo
= • — - f2a
2 exp(-2az)S tp]
(0,co)dco
Thus, the time-averaged viscous heat source strength can be expressed in terms of the power spectrum of the pressure fluctuations at the firn surface.

Wind pumping 177
Thermal effects of viscous dissipation in firn layer
The diffusion of a temperature disturbance T(z,t) through firn is governed by the equation
K^-p s
c^=-a(z,t) (29) where K is the thermal conductivity, p s
is the density, and c is the specific heat capacity of firn. Time-averaging (29) gives the ordinary differential equation
K ^ = -a\z) (30) where T(z) denotes the time-averaged temperature disturbance. The form of â\z) is given by
(28); the boundary conditions on T(z) are T(0) = 0 and dT(°°)/dz = 0. The resulting solution is oo f
(z) =
_JL^ f
f l
_
eX p(-2ocz)]S
[p]
(0,Cû)dco (31)
Let us define
So = ^-jS w
(0,(ù)d(ù (32) where S
0
can be recognized as the time-averaged "power" of the random process p'(0,t), i.e.,
So = p'(0,t)
2
. For large values of z, Equation (31) indicates that T(z) approaches the fixed temperature T
0
= kS
0 I2\XK.
Thus, at some distance from the upper boundary, the temperature disturbance is independent of spectral form and depends only on the mean-square pressure fluctuation S
0
.
We have no measurements of p'(0,t) that would allow us to calculate 5^(0,(0) for the
Agassiz drill sites. To proceed further we shall consider some special cases. If the windinduced pressure disturbance is a pure sinusoid, then
S w
(0,co) = %S
0
[B((ù - cog) + ô(œ + ©o)] (33) where oo = ±co
0
is the angular frequency of the pressure disturbance and 5 is the Dirac delta function. Substituting (33) into (31) yields the temperature disturbance:
T(z) = - ^ {1 - exp[-2a((»
0
) z
]} (34)
Note that (34) gives the e-folding distance as respectively 0.69, 4.46 and 2.64°C (Table 1).
There z
0
= l/2a((Q
0
) = (kp
0
/2\in(0
Q
)
V2
. We can regard z
0
as the thickness of the thermal boundary layer caused by wind pumping. For a frequency /
0
= 10 Hz (the angular frequency co
0
= 2TC/
0
) and using plausible values for the physical constants (Table 3), the calculated value of z
0
is 1.1 m; for/
0
= 1-0 Hz, 0.1 Hz, or 0.01 Hz, z
0
is equal to 3.5 m, 11.2 m, or 35.3 m, respectively. It has already been noted that below a depth of 46 m, firn at the Agassiz Ice Cap sites is completely impermeable, so it is apparent that fluctuations at frequencies of 0.01 Hz and lower will be strongly inhibited by the lower impermeable horizon and the assumption z
0
« H is clearly invalid.
A second spectral form of interest is that of the Gaussian-shaped spectrum having spectral peaks at angular frequencies © = ±co
0
- For this case
5 w
(0,co) = —^= {expK© COQ^O
2
] + exp[-(co + (ù
0
)
2
/2a z
]} (35) crv2it where a determines the breadth of the spectral peak. The above power spectrum has the desired property (32). Hgure 4 shows results obtained by integrating (31) with the Gaussian

178 G.K.C. Clarke et al. spectrum of Equation (35). Physical constants have been taken from Table 3 and the assumed magnitude of the mean-square pressure fluctuation is S
0
= 5000 Pa
2
, corresponding to a rootmean-square (rms) pressure fluctuation p rms
= SQ
12
= 70.7 Pa. This is a large pressure fluctuation; it has been chosen to produce a temperature effect of comparable magnitude to that found at site A84 on Agassiz Ice Cap. The spectral peak has been placed at/
0
= 0.1 Hz and the effect of varying peak width a = 2its has been tested. The variable s is a measure (in Hz) of the half bandwidth of the spectral peak. By choosing large values of s, a white spectrum can be approximated.
TABLE 3 Assumed values for physical constants
Porosity offirn, n 0.4
Permeability offirn, k 10~
8
Thermal conductivity offirn, K 0.5 W mT l deg-
1
Viscosity of air, u. lôxlO
- 6
Ambient pressure, p n
10
5
(a) (b)
0.1 0.2 0.3 0.4
Frequency (Hz)
10 15
Depth (m)
FIG. 4 (a) Normalized Gaussian power spectra for f
0
= 0.1 Hz and various choices of bandwidth parameter s = a/2%, (b) Temperature disturbances resulting from pressure fluctuations having a Gaussian power spectral density.
The physical constants used in these calculations have been extracted from
Table 3. The mean-squared pressure fluctuation has been taken as S
0
= 5000
Pa
2
, corresponding to a rms pressure fluctuation p rms
= 70.6 Pa.

Wind pumping 179
Dimensionless formulation
To explore more completely the thermal effects of varying the peak location and bandwidth, it is efficient to restate the previous results in dimensionless form. Take S
0
as a characteristic value of
S&\0,(U),
T
0
= kS
0 I2\JJC
as a characteristic temperature, z
0
= (kp
0
/2n/zco
0
)
1/2
as a characteristic distance, and a as a characteristic value of angular frequency. Define the dimensionless jxwer spectrum as S*(co) =
S^{Q,(Ù)IS
Q
,
the dimensionless temperature disturbance as
T*(z) = T(z)/T
0
, the dimensionless depth as z* = z/z
0
, and dimensionless angular frequencies co* = co/a and co*, = co
0
la. From these definitions, Equation (35) gives the dimensionless power spectrum as
S\(ù) = ^!"{exp[-(co* - a>o)
2
/2] + expRco* + co*,)
2
]/2]} and (31) and (36) give the dimensionless temperature disturbance as
(36)
TV) = 1 -
V2i
* . , * . 2 * n
{expKco - co
0
) '2] + exp[-(co + co
0 r/2]}exp[-lco /co
0 l z ]d® (37)
Figure 5 shows the temperature disturbance that results from several different choices of dimensionless frequency co
0
.
(a) (b)
C O " ci co*=0.01
\ o . i u>£=0.01 , /
/ /
/ 0.1 ©
5 9 o «~ a.
X \
0.5\
«
2
Is-
<D
I -
/ / °'
5
I / A x
I /
/ / //
^ \ c
.2
"55
E "
5
» »
C d -
/ / /
/ / /
1.5
A \ \ / \ \
2 V / \ \ \ \
/ /
E «
Q ° ~
/ / y
/ /
3 \ \ \ \ \ \ O d _
0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 5
Dimensionless Frequency Dimensionless Depth
FIG. 5 (a) Gaussian power spectrum in dimensionless variables. co
0 = ro o 1° is a dimensionless parameter that is related to both the position and breadth of the spectral peak, (b) Time-averaged temperature disturbance vs. depth in dimensionless variables.
DISCUSSION
The above theoretical analysis demonstrates the essential features of "wind pumping" as a heat source in ice sheets. Unfortunately, we do not have the pressure fluctuation measurements that would allow us to decide its role at the Agassiz sites. Lacking these measurements we have examined the literature on atmospheric turbulence and its spectral properties to determine whether the required pressure fluctuations are physically reasonable. The expression

180 G.K.C. Clarke et al.
T
0
= kS
0 I2\IK gives an estimate of the pressure fluctuations required to increase the deep temperature by an amount T
Q
. For example, the 2.64°C difference in 10 m temperature between sites A77 and A84 could be accounted for if at site A77 S
0
~ 0 and at site A84
S
0
= p'(0,t) z
= 4224 Pa
2
. (Table 3 has been used to assign values to the various physical constants.) The corresponding rms pressure fluctuation is p rms
= 65 Pa. According to Businger
(1982, p. 30) a typical magnitude for pressure fluctuations is p' ~ 10 Pa and "in very strongwinds the dynamic pressure fluctuations may be an order of magnitude larger." Velocity, rather than pressure, fluctuations are usually measured, but to a fair approximation these can be related to each other by the expression p rms
~ pv
0 v rms
where v
0
is the steady component of wind speed and v rms
is the rms magnitude of the fluctuating component. Typically v rms l y
Q ~ 0-1
s o m a t
Prms ~ O.lpv
2
). Thus for p rms
= 65 Pa and p = 1.39 kg m
-3
as the density of air the corresponding wind speed is v
0
= 21.6 m s
-1
(77 km h
-1
). This is unquestionably a brisk wind, but it must be appreciated that the uncertainties behind the estimate are considerable. Firn permeability is known to vary over a wide range. If the actual firn permeability was an order of magnitude greater than the assumed value, the above estimates would be revised to S
0
= 422 Pa
2
, p rms
= 20.6 Pa and v rms
= 44 km h
-1
.
The spectral form, in addition to the total power, is unknown. Chiba & Kobayashi
(1984) measured the power spectrum of the fluctuating vertical component of wind velocity at
Mizuho Station in Antarctica and their Fig. 5 shows a spectral peak at 0.1 Hz. The pressure spectrum was not measured but a peak near the same frequency would be expected. Thus, for this example at least, the power of the pressure fluctuations is concentrated in the spectral range that we consider most conducive to producing a thermal effect from wind pumping.
We confess that the predicted magnitude of the wind pumping effect seems surprisingly high and may point to shortcomings in the assumptions of our model. For example, convective heat transport in the firn layer could prove significant and would reduce the thermal anomaly.
ACKNOWLEDGEMENTS G.K.C.C. thanks the Natural Sciences and Engineering Research
Council of Canada for financial support. Data from Agassiz Ice Cap were obtained with the help and financial assistance of the Polar Continental Shelf Project, Energy Mines and
Resources, Canada.
REFERENCES
Bear, J. (1972) Dynamics of Fluids in Porous Media. American Elsevier, New York, USA.
Businger, J.A. (1982) Equations and concepts. In: Atmospheric Turbulence and Air Pollution
Modelling (ed. by F.T.M. Nieuwstadt & H. van Dop), 1-36. Reidel, Dordrecht, Netherlands.
Chiba, O. & Kobayashi, S. (1984) Turbulent characteristics of the vertical wind velocity near the critical Richardson number. In: Proc. 6th Symp. on Polar Meteorology and Gla-
ciology (ed. by K. Kusonoki), 62-69. Tokyo Press, Tokyo, Japan.
Fisher, D.A. (unpublished) Mer de Glace thermal anomalies. Internal report for Polar Continental Shelf Project, Department of Energy, Mines and Resources, Canada, 1980.
Fisher, D.A., Koerner, R.M., Paterson, W.S.B., Dansgaard, W., Gundstrap, N. & Reeh, N.
(1983) Effect of wind scouring on climatic records from ice-core oxygen-isotope profiles. Nature 301, 205-209.
Solodovnikov, V.V. (1960) Introduction to the Statistical Dynamics of Automatic Control Sys-
tems. Dover, New York, USA.