Dureza y peso

For measuring the ground heat flux, heat flux plates are employed in most studies. Here, a plate made of a material with known thermal conductivity is inserted horizontally in the ground. The temperatures at the top and the bottom of the plate are measured, so that the ground heat flux can be evaluated from the temperature gradient. The method is satisfactory if the thermal conductivity of the heat flux plate is equal or at least close to the thermal conductivity of the soil. If this is not the case, the assumption of 1D-heat flow inherent in the idea of the heat flux plates breaks down. This problem is particularly severe in permafrost-dominated soil, where the temperature gradients can be extreme. An additional problem is the accumulation of ice in the voids around the heat flux plates, which can alter the heat flux. Furthermore, it is highly difficult to achieve an adequate quality assessment for ground heat fluxes obtained with heat flux plates. For this thesis, we therefore do not make use of heat flux plates, but employ two alternative methods.

2.3.1

Calorimetric method

If one is interested in average ground heat fluxes over longer periods of a few weeks or months, it is feasible to make use of the difference over time in the total internal energy E(t) of the soil. For this, Eq. 2.15 is integrated over the soil column between the surface z = 0 and z = zb, below which the internal

E(t2) − E(t1) =ρwLsl 0 Z zb [θw(T (t2, z)) − θw(T (t1, z))] dz (2.45) + 0 Z zb T (t2) Z T (t1) ch(T, z) dT dz

For the applicability of Eq. 2.45, it is important that only changes in the water content due to freezing or thawing of are considered, not due to infiltration. Therefore, the evaluation of the first term must be restricted to temperatures below 0◦_{C. Complications arise in case of infiltration in frozen soil, when the}

formula is no longer applicable in a strict sense, as it is not possible to differen- tiate between infiltrated water and water from thawed ice. However, infiltration is normally restricted to the uppermost soil layer, so that the error is bearable. The average ground heat flux through the surface can then be calculated as

jg,av= E(t2) − E(t1)

t2− t1 , (2.46)

if one assumes the flux through the lower boundary at z = zbto be zero. This is

fulfilled to a very good approximation, if zb is chosen sufficiently deep. In this

case, the ground heat flux is close to the geothermal heat flux, which is on the order of 10−2 to 10−1Wm−2 and thus negligible compared to normal average values of the ground heat flux. For this thesis, only rather shallow temperature and soil moisture profiles within the active layer to a depth of zb ≈ 1.5 m are

available, so that the non-zero flux through the lower boundary constitutes a major source of error.

For the evaluation of the soil water content, a profile of Time-Domain-Reflec- tometry (TDR) measurements is employed. The technique used to evaluate the soil water and ice contents during the winter is described in Boike et al. (2003b). The specific heat capacity ch(T ) is then evaluated as

ch(T ) = θw(T ) cw+ θi(T ) ci+ θmcm+ θaca. (2.47)

If the freeze curve θw(T ) is known, the integral in Eq. 2.45 can be evaluated.

In practice, most of the energy is consumed by the phase transition of the soil water, so that the impact of changing heat capacities on the evaluation of the ground heat flux is rather limited.

2.3.2

Conductive method

The second method makes use of times series of soil temperature measurements. A profile of three temperature sensors at depths z1, z2 and z3, z1> z2> z3be-

neath each other yields the time series Tmeas(z1, t), Tmeas(z2, t) and Tmeas(z3, t).

To calculate the ground heat flux jg(z, t) = −Kh(z, t)

∂

the first step is to evaluate the thermal conductivity Khof the soil or snow. We

assume a conductive 1D-heat transport and constant thermal conductivity and heat capacity between z1 and z3 and over the considered period. Furthermore,

periods are excluded when a phase change of water occurs between z1 and z3.

We then numerically solve Eq. 2.14 with Tmeas(z1, t) and Tmeas(z3, t) as Dirichlet

boundary conditions. The initial condition is chosen as a linear interpolation between the first two data points of the boundary conditions. In this case, the exact choice of the initial condition is not critical, since the solution converges to a value independent of the initial condition after few time steps. The numerical solution of Eq. 2.14 is performed with the partial differential equation solver of MATLAB (Skeel and Berzins, 1990), yielding the modeled times series of temperatures for a given dh, Tdh(z2, t), for all values of z2 with z1> z2> z3.

With Tmeas(z2, t), we can perform a least-square fit for dh by minimizing the

RMS error between Tmeas(z2, t) and Tdh(z2, t). This method relies on rapid

temperature changes which induce a time lag of the surface temperature signal in deeper soil layers characteristic for a certain dh (see Sect. 2.2.3). In the

summer period, when a strong diurnal temperature signal exists, the method generally works at the study site for depths of z1≈ −0.01 m, z2≈ −0.15 m and

z3 ≈ −0.30 m. The same procedure is used by Putkonen (1998), and the basic

idea of obtaining soil properties from a time series of temperature measurements is extended by Nicolsky et al. (2007) and Nicolsky et al. (2009).

With the heat capacity chof the soil determined from the volumetric fractions

θw and θm in soil samples (or θi in snow samples), the thermal conductivity

Kh can be evaluated as Kh = chdh. The heat flux jg(z1, t) through the upper

boundary is then calculated by jg(z1, t) = −Kh(z1, t)

∂

∂zT (z, t)|z=z1. (2.49)

Note that the required derivative of the temperature can be easily evaluated from the numerical solution of Eq. 2.14 which delivers the full temperature field between z1 and z3.

Although convective heat transport, e.g. through infiltrating rain water, is not accounted for in the conductive method, the inherent assumption of a purely conductive heat transfer has been shown to be adequate for the study area during winter (Roth and Boike, 2001), a site approximately 10 km from the study area (Putkonen, 1998) and for other permafrost areas (e.g. Romanovsky and Osterkamp, 1997).