Ninth International Conference on Permafrost ... - IARC Research

Ni n t h In t e r n at i o n a l Co n f e r e n c e o n Pe r m a f r o s tThe temperature regime is determined by the well-knownheat conduction equation:2∂T2 ∂ T= a(1)2∂t∂zwhere T is the temperature, t and z the time and depth variables,respectively, and a the thermal diffusivity, given bya =with κ the thermal heat coefficient, ρ the soil density andc its specific heat. Our startingTnpoint ( z = dis,t) the= 0resolutionof equation (1) with the single harmonic boundary0condition Tn( z = d ,t) = 0 and Tn( z = 0,t) = Tncos ( ωnt).Using the separation variables method, the solution ofequation (1) is given by the 0Texpressionn( z = 0,t) = Tncos ( ωnt)sin( γ ( )0 nd − z )Tn( z,t)= Tnexp ( −iω t )sin( nd)nγwhereDeveloping the solution, we obtainwhereandwithκcρωnγn= 1+22aTn( z,t)( i)0Tn= ×2 2( cosh βn− cos ( βn))× A cos ( ω t) + B sin( ω t )n n n n( α ) ( β ) ( α ) ( β )+ cos ( α ) cos ( β ) sin h( α ) sinh( β )A = sin sin cosh cosh +n n n n nn n n( α ) ( β ) ( α ) ( β )− cos ( α ) sin( β ) sin h( α ) cosh( β )B = sin cos cosh sinh −n n n n nn n nωnωnα (n= d − z2) and βn= d(3)22a2aadding now all the harmonics corresponding to thetemperature data recorded at the depth z=0 (and consequence 0T ( z = 0,t) =∑Tn cosωntof its Fourier analysis), T z ,t T cosωt , andnalso considering the boundary condition T z = d ,t = .Hence, we can predict T ( zand = d ,t reconstruct ) = 0 the expectabletemperature record at any depth, using the linear superpositionprinciple. Simultaneously, this solution allows us to estimate thebehaviour of the diffusivity at different depths (see below).DiscussionAlthough snow in the surface is thought to be a low-passfilter for the temperature signal (Goodrich, 1982), we findthat the dependence of the diffusivity with depth is complexand determinant for interpreting the experimental data. Thus,looking at the results of the FFT power spectrum analysis0( = ) =∑0n nnnn( ) 0(2)98of the temperature signal at different depths (Fig. 2a–2b), itis clearly seen the strong dependence of the thermal wavewith frequency. Figure 2b shows the resultsνfor frequenciesn≈ 0.11 Hzaround νn≈ 0.11 Hz , where an almost linear decay of thecorresponding amplitude T nis observed, according with its-1linear dependence νwith n≈ 0depth . 11 Hz reflected in ν-1the n≈ term 0.41 hνα n. On then≈ 0.41 hother hand, to explain the tendencies shown for the amplitudes-1-1at frequencies νand wen≈ 0.41 h ν-1n≈ 0.55 hnecessarily νn≈assume 0.55 h an increase of the term β nwith depth,what implies a decrease of the -1νdiffusivity with it. This nonuniformdiffusivity is a consequence of the variation ofn≈ 0.55 hsome of the magnitudes, and depends on the density, theconductivity, or the specific heat. Because it does not seemto justify a significant variation in either density or specificheat in the depths we are considering, the decrease seemsto be related to thermal conductivity. Of course, a goodknowledge of behaviour of the diffusivity with depth allowsa better forecasting of the thermal evolution of the soil.Using the presented method to analyse temperatures, datarecorded in periods of several years will allow us to make aprecise determination of the soil properties under interest,the evolution of the active layer, and their implication onclimate. Thus, we will develop a tool implemented withthe commercial program IDL with which we will analyseautomatically the behaviour of the active layer. Even more,we will extend the presented model, taking into accountthermal sources along the depth of interest, which could bevery interesting for the study of the evolution of the activelayer in soils.AcknowledgmentsThis research was founded by the Spanish Ministerio deEducación y Ciencia project, reference code: POL2006-01918/CGL.ReferencesBlanco, J.J. et al. 2007. Active layer apparent thermaldiffusivity and its dependence on atmospherictemperature (Livingston Island, Maritime Antarctic).U.S. Geological Survey and the National Academies,USGS OF-2007-1047, Extended Abstract.Goodrich, L.E. 1982. The influence of the snow cover onthe ground thermal regime. Canadian GeotechnicalJournal 19: 421-432.Hauck, C. et al. 2007. Geophysical identification ofpermafrost in Livingston Island, Maritime Antarctica.Journal of Geophysical Research (in press).Ramos, M. & Vieira, G. 2003. Active Layer and PermafrostMonitoring in Livingston Island, Antarctica, firstresults from 2000 and 2001. In: M. Phillips, S.M.Springman, & L.U. Arenson (eds), Permafrost 2:929-933. ICOP 2003. A.B. Balkema. Rotterdam.