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Permafrost and PeriglacialProcesses,Vol 7: 101-110(1996)

Active Layer Distortion of Annual Air/Soil Thermal Orbits

Hugo Beltramil

Department of Geology,St Francis Xavier University,PO Box5000,Antigonish,Nova Scotia, Canada,B2G 2W5,hugo@juliet.stfx.ca

ABSTRACT

A straightforward procedure is proposed as a first order, initial approximation for assessingthe character of the heat transfer process in the subsurface. Considering monthly averages ofair and soil temperatures as a perpendicular superposition of simple harmonic motions,'phase-space' figures can be generated to permit a rapid qualitative diagnostic of the subsur­face thermal regime. It is found that for subsurface conductive regimes the shape of the inter­ception figures is regular. For sites where an active layer and associated processes arepresent, the interception figures are highly irregular owing to non-conducive heat transfer.Implications for the prediction of soil temperatures and determination of climatic changesfrom geothermal data are discussed in this context.

RESUME

Dne procedure directe est proposee comme une approximation initiale de premier ordrepour estimer Ie caractere du transfert de la chaleur dans la zone sous superficielle. En consid­erant les moyennes mensuelles des temperatures de l'air et du sol comme une superpositionperpendiculaire de variablesharmoniques simples, des figures de 'phase-espace' peuvent etreetablies pour permettre un diagnostic du regime thermique sous la surface. Il a ete trouveque pour des regimes des sous-surfaces transferant la chaleur par conduction la forme desfigures est reguliere. Pour des sites ou une couche active et des processus associes sontpresents, les figures d'interception sont hautement irregulieres en relation avec un transfertde chaleur par un processus autre que la conduction. Des implications pour la prediction destemperatures du sol et la determination des changements climatiques a partir des donneesgeothermiques sont discutees dans ce contexte.

! "'WO'D' ,om",,,, m,delli ••" ,,""' by", wil "mp"""'''INTRODUCTION

The need to understand the relationship betweenair and soil temperatures has been a preoccupa­tion of soil scientists for a long time (e.g. Geiger,1965; Goodrich, 1982; Hinkel and Outcalt, 1993).Such interest has been mostly motivated by

applications to agriculture (e.g. de Vries, 1975;Sharrat et at., 1992) and, in the northern lati­tudes, to the study of the evolution of permafrostand its stability in the context of developmentand extraction of natural resources (e.g. Hinkeland Outcalt, 1994; Smith, 1975; Goodrich, 1982;Lachenbruch, 1959; Lachenbruch et al., 1988) and

I Previously at: Centre for Climate and Global Change Research, McGill University, Montreal, Canada.

CCC 1045-6740/96/020101-10© 1996 by John Wiley & Sons, Ltd.

Received 5 November1995

Accepted 20 March 1996

102 H. Beltrami

current attention arising from climate model pre­dictions of high latitude warming.

Recently, because of the general concernabout the possible influences of anthropogenicactivities on the climate system, a new reason hasarisen accentuating the need for understandingthe processes governing the air-soil energyexchange mechanisms and factors which influ­ence heat transfer at the air-ground interface.The realization that ground surface temperaturevariations are recorded in the subsurface as per­turbations to the equilibrium geothermal gradi­ent, such that past climatic changes in continentalareas can be reconstructed by direct measure­ment and analysis of temperature-depth profiles,led a number of groups around the world tocarry out this type of analysis. The reportedresults are promising, although there are somerestrictions on the resolution of ground tempera­ture changes from these data (see Beltrami andMareschal, 1995). For reviews see: Pollack andChapman, 1993; Vasseur and Mareschal, 1993;Beltrami and Chapman, 1994. It remains how­ever, a critical issue that the character of the heattransfer regime in the subsurface be conductivefor the validity of the climatic inferences and forthe combined analysis of geothermal and proxy(Beltrami and Taylor, 1995; Beltrami et at., 1995).

The number of processes governing the energytransfer between ground and atmosphere islarge, and the interrelation between processes isextremely complex (e.g. Goodrich, 1982). Thusinstead of attempting to model the air-groundinterface in general at each particular location, afirst order, initial approximation technique toassess the character of the subsurface heat trans­

fer regime at this interface may be useful.There have been several attempts to differenti­

ate between conductive and non-conductive heat

transfer regimes in soils. These approachesinvolve Fourier spectral analysis or non-lineardynamics measures from time series of soil tem­peratures (e.g. Outcalt and Hinkel, 1992; Hinkeland Outcalt, 1993; Outcalt et al., 1992).

In this note I examine, semi-qualitatively, therelationship between air and soil temperature bymeans of 'phase-space' diagrams, and attempt toidentify when such a relation corresponds to aconductive regime or whether, at some latitudes,non-conductive active layer processes distort thepropagation of the signal of past ground surfacetemperature variations into the subsurface, makingreconstructions of ground temperature histories(GTHs) very difficult or problematic.

THEORETICAL CONSIDERATIONS

As a first order approximation, the air tempera­ture variation can be considered as a sinusoidaloscillation with a period of one year. Similarly, ifheat is transferred by conduction, soil tempera­ture generally follow an analogous behaviour,although the amplitudes of oscillation aredamped with respect to air temperature as heatdiffuses into the ground. It is also well knownthat soil temperature lags variations of air tem­perature. From examination of records of air andsoil temperature at any location, the decay of theamplitude and the phase lag increase with depthare apparent (Figure 1); this phenomenon hasbeen very well documented (e.g. Geiger, 1965;Smith and Riseborough, 1983 and referencestherein).

Consider the superposition of two simple har­monic motions (SHMs) in perpendicular direc­tions. This problem can be easily describedanalytically, as the reader can verify by consult­ing any elementary physics book (e.g. Joss,1934).

Assume that air and soil temperature at depthZ vary in a perfectly sinusoidal fashion describedby:

Aa = A~cost(wt) (1a)

As = A~cos(wt + 8) (1b)

Perpendicular superposition of these types ofoscillation will yield a curve described by:

A oAa AOY(1 A~).s = As -0 cos 8 + s - -0 sm 8,Aa Aa

where Aao and Aso are the mean amplitudes of airand soil temperatures, As and Aa are the 'instan­taneous' soil and air temperatures respectively,and 8 is the phase lag given by:

8 = Zg Yr!iJwhere Zg is ground depth (positive downwards),T is the period of oscillation and Kg is the thermaldiffusivity of the ground. Equation (2) is theequation of an ellipse. If the phase difference iszero then (2) simplifies to the equation of astraight line, and if the phase is 90° it reduces tothe parametric representation of the ellipse. Ingeneral, the direction of the principal axes of theellipse depends on the phase difference and a fullexplanation can be found in Joss (1934).

Evaluation of (2) is straightforward and it can

Distortion of Air/Soil Thermal Orbits 103

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16 20 24 28 32 36 40 44 48

Time (Months)

<I.l

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a.E<I.lI-

Figure 1 Monthly air and soil temperature records at Val d'Or. Only a few years of data are shown for clarity.

be used to simulate an idealized case of perpen­dicular superposition of air and soil temperaturerecords in the absence of snow cover. As an illus­tration, examples for some simulated air-soiltemperature interception figures are shown inFigure 2a, for depths of 50,100 and 300 cm. Notethe change in the orientation of the ellipses' prin­cipal axes with increasing depth.

However, in regions with ground snow coverduring the winter, the situations present differentphase lags for the period without snow (8 = Zg v'Crr/Kg1) and with snow (8 = Zs v' (rdKs1) +Zg v'(rdKg1), where Zs and Ks refer to snow depth anddiffusivity respectively; active layer processeshave been neglected. Thus it would be expectedthat, under a conductive regime, a phase-spaceplot of air versus soil temperatures would be acomposite of a summer part and a winter part oftwo different interception figures, given by:

As = A~~~ cos ( - Zg VC:T) - ZS VL:T))(4)

+ A~V(l - A~)sin (- Zs '\ rrr - Zs '\ /Tf )Aa Vq V~sTStrictly speaking, the interception figure shouldappear as a half ellipse for the summer and atime varying ellipse in winter, with the directionof the principal axes, or eigenvectors, changingwith variations of snow cover.

Figures 2b and 2c show the result of simplemodel interception figures for air and soil tem­perature, at 3 and 0.5 m, in phase space for which

the lag was calculated assuming a summer diffu­sivity of 0.7 X 10-6 m2/s, estimated from meteoro­logical data; for the winter the lag was calculatedassuming a constant snow thermal diffusivity of0.3 X 10-6 m2/s (Smith, 1975). To simulate year­to-year monthly average air temperature vari­ability, white noise was added (SD = 2 0c) to a20°C amplitude air temperature synthetic sinu­soidal variation. Snow cover was simulatedassuming ground cover to be 0.10, 0.20, 0.30, 1.0,1.5, 0.5 and 0.25 m, for the months of Octoberthrough April respectively, with the rest of theyear remaining snow free. Also, to simulate yearto year variability, noise was added to theassumed snow cover variation. The background'earth' temperature was set to 0 °C for this exam­ple. The examples in Figures 2b and 2c show 200months of simulation. The changes in the orien­tation of the principal axes of these interceptionfigures vary in time following the changes of theeffective ground depth as expected from equa­tion (3).

DATA AND ANALYSIS

Records of monthly mean air and soil tempera­tures at several meteorological stations inQuebec and Ontario were obtained from Envi­ronment Canada to illustrate this application.

Figures 3 and 4 show the IS-year records ofmonthly mean air temperature versus monthlymean soil temperature at depths of 10, 20, 150and 300 cm at Val d'Or in Quebec (48°04'N,

n047'W) and at depths of 20, 50, 100 and 300 cmfor the Elora Research Station in Ontario(43°39'N,80025'W).

The deformation of the interception ellipsesarises from the insulating effect of snow coverduring the winter months at these locations.Snow cover increases the effective depth atwhich temperature measurements are beingmade; thus it changes the lag and the shape ofthe interception ellipse. This can be observed inFigure 5 which shows a three-dimensional plot ofmonthly means of air, soil temperatures (3 m)and the number of days with snow on the groundfor Val d'Or. Comparison of Figure 5 with Fig­ure 3d illustrates that the part of the ellipse cor­responding to the winter appears stretched outbecause of the increasing snow cover (i.e.increasing effective depth) as the seasonadvances.

The ellipse axes vary from summer to winterand their orientation is determined by the phaseshift of the different depth records. We alsoobserve in the above figures that, as expected,the amplitudes of temperature oscillation of soiltemperatures are damped with respect to thedriving air temperature amplitude (Geiger,1965).

The mean soil temperature in these recordsdiffers by several degrees from the mean air tem­perature; this difference varies with latitude (seeTable 1 in Beltrami and Mareschal, 1991) and itis accounted for by the insulating effect of snowcover (e.g. Smith, 1975) and the long-term meanground temperature. Figure 6 shows the phase­space diagram for air and soil temperature (10cm) for Gainsville (Florida) and Auburn(Albama); the absence of snow at these locationspreserves the 'winter' part of the interception fig­ure, i.e. there is no deformation because thereare no effective depth changes during the year,although in this particular case other factorsintervene in the air-soil temperature relation.Examination of these figures thus providesstrong comparative evidence for the snow coverrelated distortion of the interception figures inhigher latitudes.

From Figures 3 and 4, it is apparent that theground temperature does not follow air tempera­ture in a straightforward manner. Thus, in mostof Canada, ground temperatures respond to thecombined effects of air temperature and, mainly,snow cover variations (Goodrich, 1982). Fromthe figures we can see that the snow cover duringthe time period covered by the data has not

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Figure 2 (a) Interception figures for idealized air-soil tem­perature cycles for the indicated depths. The orientation of theprincipal axes varies with depth. (b),(c) Model response of anidealized soil at (b) 300 cm and (c) 50 cm to sinusoidal air tem­perature variations for 200 months. The ground was assumedto have a snow cover during the winter months. The shape ofthe interception ellipse changes from summer to winter owingto the increased phase lag induced by snow cover (see text)

104 H. Beltrami

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Distortion of Air/Soil Thermal Orbits 105

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AIR TEMPERATURE (C)Figure 3 Interception figures (phase-space plots) for monthly averages of air and soil temperatures at Val d'Or (Quebec): (a)10 em, (b) 50 em, (c) 100 em, (d) 300 em. The records span 15 years.

changed drastically as can be inferred from theregularity of the year-to-year interception fig­ures. Indeed, if this relation holds outside thesampling interval, it might be possible to inferthe bounds of the soil temperature at a givendepth for any month of the year, i.e. the interfer­ence figure can be considered as an 'attractor'within which the temperature trajectories moveabout phase-space. It can vary within the boundsof the ellipse, but it cannot lie outside the limitsof the figure. In other words the 'cloud' of pointsrepresents the region in air-soil temperaturespace with a high probability of finding a particu­lar trajectory. Alternatively, the long-term(decadal) relationship between air and soil tem­perature can be considered to be deterministic toa large extent, at least when considering monthlyaverages. Higher resolution data contain higherfrequency variability, and this is currently beinginvestigated in this context; for the determinationof climatic inferences from deep borehole geo­thermal data it is this long-term relation betweenatmosphere and ground which is important.

However, meteorological records of the sametype at locations further north show a differentstory. Figure 7 shows the monthly air tempera­tures versus soil temperatures at 20, 50, 100 and150 cm for Kuujjuaq in Quebec (56°06'N,68°25W). It is apparent that the interception fig­ures are no longer regular and are dominated bysignificant variations in winter. Soil temperaturesat this location show a large range of valuesacross the freezing point of water. The effect ofsnow cover insulation, in this case, is partiallyresponsible for the deformations, but at these lat­itudes the snow cover does not vanish from theground until spring or early summer, so snowcover variability is probably not responsible forthese irregularities. This can be inferred fromFigure 7 by noting that the soil temperatureremains close to zero for two or three months.Taking into account that these figures were con­structed from monthly averages and thus the res­olution is rather poor, we interpret the constancyof the autumn soil temperatures and the irregu­larities in the below freezing side of the figures as

106 H. Beltrami

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'fingerprints' of phase transition processes as theupper 150 cm of soil freeze; i.e. the presence ofan active layer.

In the case where the distortion of the phase­space figures is due to snow cover variabilityonly, it would be expected that the subsurfacetemperatures would be 'well behaved' once theair temperature signal passes through the snowcover. If interception figures of these records areregular, one could assume that the heat transferinto the subsurface can be characterized or mod­elled as being mainly conductive at the resolu­tion of the data considered here (i.e. monthlythermal diffusivity). To verify this idea, we haveplotted the record of monthly soil temperature at20 cm versus the soil temperatures at 10, 50, 100and 300 cm for Val d'Or. This is shown in Figure8. It is clear from these figures that the intercep­tion ellipses are regular; thus, at this site, thenear surface heat transfer can be thought asbeing active layer free (soil temperatures remain

below 0 0c) or dominated by heat conduction, inagreement with the model assumptions.

To further illustrate the non-conductive char­acter of heat transfer in higher latitudes, Figure 9shows the 20 cm monthly mean soil temperatureversus the temperatures at 50, 100 and 150 cm forKuujjuaq. From the irregularities of the intercep­tion ellipses, for 100 and 150 cm, it can beinferred that the subsurface heat transfer regimeat this site is dominated by non-conductive pro­cesses.

DISCUSSION AND CONCLUSIONS

A straightforward method has been proposed asa preliminary step to assess the character of thethermal regime of the subsurface. It has beenfound by examining air-soil temperature phase­space diagrams that, when active layer processes

Distortion of Air/SoilThermal Orbits 107

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Figure 5 Relation between air and soil temperature (300 cm) and snow cover at Val d'Or. The plot shown here correspondsto the phase-space diagram in Figure 3d.

are present, heat transfer to the ground from airtemperature variations is not by conduction, butincludes the effects of phase changes and theassociated latent heat releases and fluid migra­tion.

The non-conductive character of the heattransfer regime in the subsurface appears asdeformations to the interception ellipse such thatthe analysis presented here can be used as a pre­liminary diagnostic tool for assessing the rela­tionship between air and soil temperatures andalso to assess the stability of permafrost at agiven site.

In the context of geothermal data analysis, it issuggested that this simple analysis be carried out,whenever possible, to assess whether the air tem­perature variations are recorded underground orwhether other thermal conditions affect or domi­nate the thermal regime in the subsurface, beforeclimatic inferences are extracted from geother­mal data; seeking a generalized model for thebehaviour of air and soil temperatures at eachsite appears problematic.

Although climatic change determination fromgeothermal data has proven to be feasible andhas a place among other palaeoclimatic indica-

tors (e.g. Hughes and Diaz, 1994), applications ofthis method of climatic inference might berestricted to latitudes where there exists noactive layer or where the active layer is very shal­low.

Work is in progress to apply this phase-spacemethodology in order to map out the areas ofconductive and non-conductive regimes of thesubsurface across Canada in an attempt to iden­tify areas where temperature-depth profilesmight contain useful information in terms of cli­matic change reconstruction, and in order toassess the spatial distribution of active layerregions in this area.

The approach presented here does notattempt to strictly model the air-soil interface.Detailed measurements of many variables areneeded to carry out such work. Furthermorethere is no guarantee that such a model wouldbe applicable to other locations. Some detailedobservations involving multivariable monitoringare being carried out and the results will be veryuseful in future detailed studies of the air-soiltemperature relationship but this type of moni­toring is unlikely to be carried out across spa­tially extensive regions in the near future.

Distortion of Air/Soil Thermal Orbits 109

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110 H. Beltrami

ACKNOWLEDGMENTS

H.B. acknowledges postdoctoral support fromFCAR (Quebec) through the McGill Centre forClimate and Global Change Research. Discus­sions with W. Pollard, D. S. Chapman, J. C.Mareschal, and S. N. Putnam are appreciated.The author is indebted to O. Jensen (McGill) forproviding infrastructure support while this workwas carried out. Comments by S. Outcalt and K.Hinkel are gratefully acknowledged. I amindebted to an anonymous reviewer for provid­ing excellent comments which helped to improvethis communication.

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Beltrami, H., Chapman, D. S., Archambault, S. andBergeron, Y. (1995). Reconstruction of high resolu­tion ground temperature histories combining den­drochronological and geothermal data. Earth andPlanetary Science Letters, 136, 437-445.

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SIpith, M. W. (1975). Microclimatic influences onground temperatures and permafrost distribution,Mackenzie Delta, Northwest Territories. CanadianJournal of Earth Sciences, 12, 1421-1438.

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