Geophysical Research Letters

Dual-probe heat pulse method for snow density and thermal properties measurement



[1] Measuring the density (ρ), thermal conductivity (k) and thermal diffusivity (α) of snow is important for modeling the energy and water balance of the earth's surface. The objective of this letter is to determine if the dual-probe heat pulse (DPHP) method can be used to measure snow density as well as thermal properties k and α. DPHP experiments with 60-s heat pulse duration and variable strength of q′ < 15 W m−1 were carried out on ice and various kinds of snow to avoid melting. There was a linear relationship between the DPHP measured snow density and that of gravimetrically measured density. With calibration, the DPHP method may be accurate for determining snow density.

1. Introduction

[2] Snow plays an important role in the earth energy and water budgets [Yang et al., 2001; Cohen and Entekhabi, 1999]. Snow water equivalent (SWE) is commonly used to evaluate the influence of snow on climate and hydrology processes. One important parameter for estimating the SWE is snow density. Snow density can be estimated by the use of gravimetric sampling, which is a laborious task and prone to human error. Remote sensing estimation [Dong et al., 2007] is limited to approximately 25-km to 50-km spatial resolution and the presence of thick snow. The use of recently developed frequency-modulated continuous-wave (FMCW) radar to determine snow density [Marshall et al., 2004; Yankielun et al., 2004] however indicates that the optimal measurement parameters vary with snowpack conditions. Acoustic pulse methods have also been used [Gubler, 1981; Goodison et al., 1988], and the reflected wave [Kinar and Pomeroy, 2007] at the interfaces of layered snow with different texture causes the empirical relationships between acoustic signal amplitude and snow density to vary with geographic area [Pomeroy and Gray, 1995]. In addition, it is not easy to process the complicated acoustic data.

[3] The single heated-needle method has been used for measuring heat conductivity of liquid and solid such as snow [Sturm et al., 2002] for over half a century [Van der Held and Van Drunen, 1949; De Vries, 1952]. The dual-probe heat pulse (DPHP) method [Campbell et al., 1991] developed from the single heated-needle method is widely used for measuring soil thermal properties [Ham, 2001; Kettridge and Baird, 2007], soil water content [Bristow et al., 1993], and soil heat storage [Ochsner et al., 2007]. Although heat pulse methods have been used to measure thermal diffusivity and conductivity, to the best of our knowledge, no report has been published on measuring snow density from the heat pulse method. Therefore, our objective is to introduce the DPHP method to measure snow density and thermal properties accurately. The advantages of this method are high accuracy, probes are relatively inexpensive to build and they can be readily automated using data acquisition systems to obtain near-continuous data.

2. Theory

[4] Most DPHP probes have at least two needles, with one acting as a heater to release a short-duration heat pulse and the other acting as a temperature sensor [Campbell et al., 1991]. The temperature change at a radial distance of r from an infinite line heat source due to a short-duration heat pulse of duration t0 in an infinite solid with zero initial temperature can be divided into two parts [De Vries, 1952; Kluitenberg et al., 1993]:

equation image

in which –Ei(–x) is the exponential integral. Here, T is temperature (K); t is time (s); k is thermal conductivity (W m−1 K−1); and α is thermal diffusivity (m2 s−1); q′ is the quantity of heat liberated per unit length per unit time (J m−1 s−1). The value of k is related to a by

equation image

where c is the specific heat capacity (J kg−1 K−1) and ρ is the bulk density (kg m−3) of the measured samples. Given r and q′ values, the value of k and α can be obtained by nonlinear fitting of the temperature-time curves with T1(r, t) of equation (1). We used Mathematica [Wolfram, 1996] for nonlinear fitting of k and α. Since the specific heat capacity of snow is the same as that of ice [Sturm et al., 2002], we used c = 2030 J kg−1 K−1. Assuming that the melting phase change near the heater needle is negligible and there is no liquid water in the snow pack when the temperature is below 0°C, the bulk density of snow was determined from the equation (2). For the wet snow and when the ambient temperature is around 0°C, the equation (1) and the method used in this study are not suitable.

[5] Two reasons led to the use of T1(r, t) rather than T2(r, t) for calibration and fitting in this study. First, T1(r, t) is relatively simple in expression and makes the nonlinear fitting easy and accurate. Second, the heat pulse can be approximated by an infinite line source during the time t < t0, while in the relaxation period (t > t0) the finite size of the samples in the laboratory will cause deviation from the infinite line source approximation [Liu et al., 2007]. But for field situations where the samples size can be regarded as infinite and ambient temperature changes with time, one needs to use both the heating and cooling equations and to correct for the rate of temperature drift.

3. Materials and Methods

[6] The DPHP was similar to Mori et al. [2003], a five needle design. All needles were 28 mm in length and were constructed from stainless-steel tubing (1.27-mm-o.d. and 0.84-mm-i.d., Small Parts, Inc., Miami Lakes, Florida). In each of the five needles, there was a thermistor (Model 10K3MCD1, 0.46-mm diameter, 10 kΩ at 25°C; Betatherm Corp., Shrewsbury, Massachusetts) installed in the center of the needle. The heater needle was also fitted with heater wire (Nichrome A, 79-μm diam., 205 m–1, Pelican Wire Co., Naples, Florida)) with a total resistance of 34 Ω. All needles were then filled with highly thermally conductive epoxy OMEGA 101 (Omega Engineering, Stanford, CT) which is also a good electrical insulator. The heater needle was at the centre of a square and there was a temperature needle at each corner of the square for a total of five needles. Each temperature needle was 6 mm from the heater needle. All five probes were secured into predrilled holes in a 22.0-mm-diameter and 8.0-mm-thick PVC plug. The plug was then screwed on the bottom plate of a cylindrical Plexiglas container (5-cm-long and 7.9-cm-i.d).

[7] For temperature measurements, the thermisters were connected via multiplexers (Model AM16/32, Campbell Scientific, Logan, UT) to a datalogger (Model CR10xp, Campell Scientific, Logan, UT). The datalogger was used to control the heat pulse, to monitor the current through the heater, and to measure the temperatures as a function of time. Heat pulse measurements made use of thermistors and a heater-control relay circuit. The heater-control relay circuit consisted of a 12-V DC power supply controlled by a relay with the datalogger and a 1 Ω precision current-sensing resistor (RH25, 25W, Dale, Mexico) for measuring the amount of heat released from the heater. Heaters were multiplexed with AM416 multiplexers (Campbell Scientific). Heat-pulse measurements consisted of a 180-s sequence (1-s measurement interval) including 60-s heating (2 ≤ q′ ≤ 16 W m−1). Heat inputs were inferred from the measured voltage drop across the precision resistor. The apparent distance r between the heater and thermistor of the sensor needles was determined by calibration in 5 g L–1 agar stabilized water [Campbell et al., 1991]. The detailed set-up procedure of DPHP is given by Bristow et al. [1993] and Ochsner et al. [2007].

4. Snow Samples and Measurements

[8] The experiment was conducted on one ice sample and ten snow samples collected from Saskatoon, Canada. The shape and structure of new snow crystals depends on the temperature and humidity of the atmosphere in which they have grown [Furukawa and Wettlaufer, 2007]. The snow metamorphism and evolution is predominantly governed by varying but large temperature gradients [Fierz and Baunach, 2000], resulting in dynamic crystal growth and thus complicated layer textures and crystal-shape. Table 1 lists the details of selected snow properties. All of the snows were collected between November and January except the coarse grains, which were collected in April with great ambient temperature fluctuation and thus large grain size. Four (deep hoar and coarse grain snow) of the ten snows (the repacked snows in Table 1) samples were sieved through a 2-mm screen and then packed into cylindrical Plexiglas containers. For the remaining six snows, they were packed directly. The bulk density of packed snow columns was kept constant for a given snow, but bulk density varied between snows (Table 1). Plexiglas containers packed with snow were tightly wrapped with plastic film and immersed in a temperature regulated cooling bath (−17 ± 0.02°C) for 24 h before measuring. For each kind of snow, three DPHP devices were used for measuring.

Table 1. Various Parameters and Snow Type Used in the Experiments
 SHEBA Codeaρ (kg m−3)q′(J m−1 s−1)t0 (s)α × 10−7 (m2 s−1)k (W m−1 K−1)
Sector plateb11133.081603.7490.086
Fine grain32914.772603.0980.183
Fine grain33105.830602.8760.181
Depth hoar103955.551603.4550.277
Depth hoar104345.282602.9620.261
Coarse grain-4655.640603.3900.320
Coarse grain-4715.806603.9850.381

[9] Since we assume that there is no liquid water in the snow below 0°C, we considered two factors in choosing the right combinations of q′ and t0. One was the resolution of the thermistor, which was around 0.01 K. The temperature rise at the sensor needle should be much larger than the detection limit of the thermistor. The other was the maximum temperature around the heater needle. If the maximum temperature is above zero or if liquid and solid phases coexist such as for wet snow, a portion of the energy released by the heater needle will be used for melting snow and heating liquid water (which has a higher heat capacity than ice), resulting in a higher apparent value of ρc and thus the overestimation of snow density. After several simulations based on equation (1) such as graphed in Figure 1a and some pre-experiment tests, we found the temperature difference between the heater needle and the sensor needle increased with decreasing heat pulse duration. Substitution of the thermal properties for dendrite as are listed in Table 1 into equation (1) led to the simulation results of Figure 1a. By using a 60-s heat pulse with strength q′ ≤ 15 W m−1 (not the commonly used 8-s heat pulse with strength q′ ≥ 60 W m−1), we were able to keep the temperature rise at the sensor needle above 0.5 K and keep the temperature rise of the heater needle no higher than 11 K. We therefore can minimize the effect of measuring error at the sensor needle and avoid snow melt around the heater. The thermal properties we used for dendrite are listed in Table 1.

Figure 1.

(a) Theoretical temperature rise at the sensor probe and heater probe following application of 8 seconds and 60 seconds heat pulse to the heater (data are for the dendrite). (b) Typical measured temperature at the sensor probe and heater probe following application of 60 seconds heat pulse to the heater. (c) Measured and fitted temperature T1 (r, t) at the sensor probe following application of 60 seconds heat pulse.

[10] Experimental temperature curves of dendrite and ice as measured at the sensor and heater needles are presented in Figure 1b. For dendrite, the maximum temperature rise (ΔTmax) at the heater needle was ΔTmax ≈ 10.5 K for a 60-s heat pulse of strength q′ ≈ 3 W m−1, while ΔTmax was 1.3 K for the sensor at r ≈ 6 mm away from the heater. The ΔTmax for ice after releasing a 60-s heat pulse of strength q′ ≈ 14.5 W m−1 was 2.38 K and 0.81 K at the heater and sensor needles, respectively. For our experiments, since all other snow samples were denser than dendrite, therefore, the values of ΔTmax for these snow samples was less than that of dendrite, assuming the same values of t0 and q′ were used. Due to the fact that snow dendrite and highly condensed snow samples have different bulk density, it is impossible to maintain a constant q′ value for these eleven experiments when the ΔTmax of the sensor needle lies between 0.6 K < ΔTmax < 1.5 K. As is illustrated in Table 1 and Figure 1b, to obtain nearly the same temperature variation ΔTmax, the highly condensed sample will need more heat input and thus a large q as compared with dendrite and new snow. To warrant that there was no melting phase change around the heater, q was carefully selected (Table 1) to control the temperature of the heater needle below −5°C in all of our experiments. This was done before the measurements by substituting the known snow thermal properties [Sturm et al., 2002] into equation (1) to calculate the temperature rise caused by a given heat pulse of strength q′ and duration of 60 s. For field experiments, the procedure is exactly the same except that t0 can be much larger than 60 s, since the DPHP can be treated as an infinite line source in an infinite snow sample.

5. Results and Discussion

[11] The DPHP measured temperature-time curves for snow and ice are presented in Figure 1c. As can be seen, experimental data can be fitted by T1(r, t) of equation (1) very well without noticeable differences between experiments and predictions. Hence, the error associated with the nonlinear fitting can be neglected. The averaged k values are plotted in Figure 2. The good agreement between our DPHP data and that of single heated needle data [Sturm et al., 2002] suggests that the DPHP method was reliable for measuring the k of snow.

Figure 2.

Average thermal conductivity k as a function of snow density ρ. Measured k from DPHP (open circles) and from the works of Strum et al. [2002] (solid circles). The inset shows enlarged area within dashed lines.

[12] Figure 3a presents the snow density from the heat-pulse method and gravimetric method. Good linear relationships exist between the two methods. However, regardless of snow texture, the heat-pulse method systematically overestimated ρ. On average, equation (1) overestimated by 0.019, 0.055, and 0.122 Kg m−3 for the ice, coarse grain, and dendrite, respectively. Some of the scatter in the data for dendrite, depth hoar was caused by the irregular shape of snow crystals and the spatial variation within the Plexiglas container.

Figure 3.

(a) Linear regression of DPHP measured snow density ρ versus density from the gravimetrically determined ρ. (b) DPHP predicted snow density ρpredict versus gravimetrically determined ρ, the linear regression relationship of Figure 3a was used. The solid line represents 1:1 relationship.

[13] Equation (2) suggests that the overestimation of ρ can be a result of either overestimated k or underestimated α. Four mechanisms might have contributed to the overestimation of ρ in Figure 3a. The first is the existence of air, which turned the snow pack used for experiment into a two-component mixture [Brailsford and Major, 1964; Jackson and Black, 1983], i.e., the snow was saturated by air other than the vacuum. Although the air has a relative small thermal conductivity of kair ≈ 0.02 W m−1 K−1, its effect on heat conduction of snow cannot be ignored when the porosity of snow is much larger than 50%, such as dendrite and new snow. As a result, the values of k measured by DPHP were overestimated by adding the contribution from air. The second reason was the so-called thermal binding effect [Jackson and Black, 1983], i.e., air distributed in the pores of snows acts as a binder which improves thermal properties by enhancing thermal contact between the snow particles and thus increases k significantly. The third reason is the heterogeneous characteristic of snow crystal at the scale of experiment [L'Heureux, 2004]. The porosity fluctuation of snow was characterized by a smaller effective α and thus resulted in the overestimation of ρ (equation (2)). The last reason is related to the possible melting phase change around the heater needle. This effect would raise the apparent value of the volumetric heat capacity ρc and ρ. Since the coexistence of water and snow is very common in field experiments, especially when the ambient temperature is high, the overestimations could be significant. However, in this study, we controlled temperature well under zero and significant melting is not expected. We excluded the possibility of thermal contact resistance [Carslaw and Jaeger, 1959] because the effect is nearly undetectable in our pre-experiment measurements.

[14] Although, there was overestimation in ρ, we found that a good linear relationship between the data from the heat-pulse method and the gravimetric method for predicting the true value of ρ. The calculated snow density using the linear regression relationship agrees well with the gravimetrically measured values, and the data are nearly perfectly distributed near the 1:1 line (Figure 3b). Therefore, we can use DPHP methods to measure snow density accurately with proper calibration.

[15] Through optimum combination of t0 and q', DPHP method has minimum disruption of snow if installed before the snow season compared to other snow measurement methods. It is relatively easy to build and is cost-effective. Further, this probe is multipurpose, providing simultaneous and continuous measurement of snow density, thermal conductivity and thermal diffusivity.

[16] In this letter, all the measurements were taken in the laboratory. More research is needed to verify the method using field studies. In addition, for fresh snow with low density, melting is more significant when the ambient temperature is high. Fine tuning t0 and q is needed to avoid the problem.

6. Conclusions

[17] We have demonstrated that the DPHP method can be used for measuring snow density and thermal properties. The DPHP measured results from loss dendrite to more condensed coarse grain snow showed that there is a linear relationship between the DPHP measured snow density and that of gravimetrically measured density. The linear relationship can be used for predicting the true density of snow. The predicted values of density agree well with the gravimetrically measured values. Because of the simplicity, accuracy and cost-effectiveness as compared with other methods, by choosing good combination of t0 and q' to avoid the melting phase change, this method has the potential of becoming a reliable technique for measuring snow density and thermal properties in the field when the ambient temperature is below 0°C.

[18] For the installation of the heat pulse method, the two-needle design in this paper is not much different from the single-needle design of Sturm et al. [2002]. One could simply insert the needles in the snow just like inserting TDR probes into soils for measuring soil water content.


[19] This work was partially supported by the National Science and Engineering Research Council of Canada (NSERC). The financial support of China National Natural Science Foundation (grant G40671085 (G.L.)) is gratefully acknowledged.