<p><b>We present mathematical analysis of temperature oscillations in depth-dependent media by investigating the thermodynamics of sea ice and of soils. Time-series temperature measurements from thermistor strings are common in both sea ice and soils and are used to study their properties, evolution, seepage flux and a host of interactions with their environment. We use numerical tools and perturbation theory to study the propagation of high frequency, small amplitude temperature oscillations through the in-homogeneous media using one dimensional models. Analytical tools for studying such thermal waves are derived.</b></p> <p>In sea ice the absorption of solar radiation and oscillating air temperatures result in two distinct thermal wave propagation behaviours. At depths, stationary waves associated with in place solar heating are observed, whereas near the surface, travelling thermal waves are present due to the quick decay in the absorbed solar radiation and the oscillatory air temperatures. These are observed in thermistor string data taken in McMurdo Sound, Antarctica between 1996-2003. Using a variety of mathematical tools, the leading order behaviour of the diurnal temperature oscillation is approximated in terms of elementary functions and is compared with results from numerical simulations.</p> <p>The thermodynamics of soils differ from sea ice in that all the solar radiation is absorbed at the upper boundary and water movement within the soil carries heat. Macroscale in-homogeneity in the advection-diffusion equation is considered and the thermal wave propagation characteristics are studied using a WKB approximation. The leading order behaviour is shown to reduce exactly to the Stallman equations, being the solution to the thermal wave propagation in a homogeneous soil with constant, uniform water flow. We use the leading order WKB expansion to estimate errors in the homogeneous soil assumption commonly made to estimate the seepage velocity and soil diffusivity. It is shown that the diffusivity estimations are relatively stable and provide reasonably accurate results, but the seepage velocity estimations incur significant errors that should be considered. A frequency dependence in the error leads us to suggest multi-frequency analysis for detection and further studies of the effects of in-homogeneous soil thermodynamics.</p>

<p><b>We present mathematical analysis of temperature oscillations in depth-dependent media by investigating the thermodynamics of sea ice and of soils. Time-series temperature measurements from thermistor strings are common in both sea ice and soils and are used to study their properties, evolution, seepage flux and a host of interactions with their environment. We use numerical tools and perturbation theory to study the propagation of high frequency, small amplitude temperature oscillations through the in-homogeneous media using one dimensional models. Analytical tools for studying such thermal waves are derived.</b></p> <p>In sea ice the absorption of solar radiation and oscillating air temperatures result in two distinct thermal wave propagation behaviours. At depths, stationary waves associated with in place solar heating are observed, whereas near the surface, travelling thermal waves are present due to the quick decay in the absorbed solar radiation and the oscillatory air temperatures. These are observed in thermistor string data taken in McMurdo Sound, Antarctica between 1996-2003. Using a variety of mathematical tools, the leading order behaviour of the diurnal temperature oscillation is approximated in terms of elementary functions and is compared with results from numerical simulations.</p> <p>The thermodynamics of soils differ from sea ice in that all the solar radiation is absorbed at the upper boundary and water movement within the soil carries heat. Macroscale in-homogeneity in the advection-diffusion equation is considered and the thermal wave propagation characteristics are studied using a WKB approximation. The leading order behaviour is shown to reduce exactly to the Stallman equations, being the solution to the thermal wave propagation in a homogeneous soil with constant, uniform water flow. We use the leading order WKB expansion to estimate errors in the homogeneous soil assumption commonly made to estimate the seepage velocity and soil diffusivity. It is shown that the diffusivity estimations are relatively stable and provide reasonably accurate results, but the seepage velocity estimations incur significant errors that should be considered. A frequency dependence in the error leads us to suggest multi-frequency analysis for detection and further studies of the effects of in-homogeneous soil thermodynamics.</p>