Shallow subsurface heat transfer occurs primarily due to heat conduction and heat advection (Domenico and Schwartz, 1990), although the latent heat released or absorbed during pore water freeze–thaw can also be important in cold regions (Kurylyk et al., 2014b). The one-dimensional, transient conduction–advection equation for subsurface heat transport is (Stallman, 1963) λ∂2T∂z2-qcwρw∂T∂z=cρ∂T∂t, where λ is the bulk thermal conductivity of the soil–water matrix (W m-1 ∘C-1), T is the temperature at any point in space or time (∘C), z is the depth below the surface (m; down is positive and the land surface occurs at z = 0), q is the vertical Darcy flux (m s-1; down is positive), cwρw is the volumetric heat capacity of pure water (4.18 × 106 J m-3  ∘C-1; Bonan, 2008), t is time (s), and cρ is the bulk volumetric heat capacity of the soil–water matrix (J m-3 ∘C-1). The first term on the left of Eq. (1) represents the divergence of the conductive flux, the second term on the left represents the divergence of the advective flux, and the term on the right represents the rate of change of thermal storage. Subsurface heat transport phenomena and the physical meaning of the terms in Eq. (1) are reviewed in more detail by Rau et al. (2014) and Kurylyk et al. (2014b).

Equation (1) is often rewritten in the form (Carslaw and Jaeger, 1959) D∂2T∂z2-U∂T∂z=∂T∂t, where D is the bulk thermal diffusivity (thermal conductivity divided by heat capacity) of the soil–water matrix (m2 s-1), and U is the velocity of a thermal plume due only to heat advection (m s-1). Even in the absence of conduction, the thermal plume will not migrate at the same rate as the Darcy velocity due to differences in the heat capacities of water and the medium (Markle and Schincariol, 2007; Luce et al., 2013). An expression for U can be obtained via a comparison of Eqs. (1) and (2) U=qcwρwcρ. Often an effective thermal diffusivity term, which accounts for the combined thermal homogenizing effects of heat diffusion and heat dispersion, is utilized in place of the bulk thermal diffusivity term D in Eq. (2). However, it is still common to ignore the subsurface thermal effects of dispersion, which are often minimal in comparison to heat conduction (Kurylyk et al., 2014b; Rau et al., 2014). Equation (2) represents vertical subsurface heat transport processes and accounts for the thermal effects of heat conduction induced by a thermal gradient and heat advection induced by groundwater flow. Analytical solutions to this equation can be developed and applied to consider inter-relationships between groundwater flow, surface-temperature changes, and subsurface thermal regimes. We consider four analytical solutions to Eq. (2) (Table 1) that vary based on the nature of the surface boundary condition. These are discussed in subsequent sections.

The diel or seasonal land surface-temperature cycle can be approximated with a harmonic function. Suzuki (1960) derived an analytical solution to Eq. (2) subject to a sinusoidal surface-temperature boundary condition: Boundarycondition:T(z=0,t)=Tm+Asin⁡2πtp-φ,Solution:T(z,t)=Tm+Aexp⁡(-dz)sin⁡2πtp-φ-Lz, where A is the amplitude of the harmonic surface-temperature cycle (∘C), Tm is the mean surface temperature (∘C), p is the period of the surface-temperature cycle (s), φ is a phase shift to align the timing of the surface-temperature signal with the sinusoid (rad), d is a thermal damping term (m-1), and L is a lag term (m-1). Equation (5) thus states that the harmonic temperature signal at the surface retains its period within the subsurface but is exponentially damped and linearly lagged with depth. Stallman (1965) demonstrated that the exact expressions for d and L are d=πDp2+0.25U2D40.5+0.5U2D20.5-U2D,L=πDp2+0.25U2D40.5-0.5U2D20.5. Equations (5) to (7) are generally collectively referred to as Stallman's equation. No initial conditions are presented for the solution of Stallman (1965) as it assumes that the boundary condition has been repeating the harmonic cycle indefinitely. This solution also depends on a lower boundary condition (T = Tm) at infinite depth. Various forms of this solution have been applied/inverted to infer rates of groundwater flow due to subsurface temperature–time series arising from daily or seasonal harmonic variations in surface temperature (e.g., Anderson, 2005; Hatch et al., 2006; Rau et al., 2014). Here, we employ the solution of Stallman (1965) in a forward manner to demonstrate why seasonal changes in air and surface temperature are not manifested in subsurface thermal regimes below certain depths, and thus why groundwater-dominated streams and rivers exhibit low thermal sensitivity to seasonal weather variability. In particular, we consider the ratio of the amplitude of the seasonal groundwater temperature cycle at any arbitrary depth to the amplitude of the surface-temperature boundary condition. This dimensionless parameter, herein referred to as the exponential damping factor Ω, can be obtained from Eqs. (4) and (5) Ω=Amplitudeatdepth=zAmplitudeatdepth=0=Aexp⁡(-dz)A=exp⁡(-dz).

Taniguchi et al. (1999a) demonstrated how an analytical solution presented by Carslaw and Jaeger (1959) could be modified to calculate the groundwater warming arising from a sudden and permanent increase in surface temperature. This increase in surface temperature could arise due to rapid and large-scale timber harvesting or other changes in land use. Menberg et al. (2014) proposed that superposition principles could be employed to modify the solution by Taniguchi et al. (1999a) by considering a series of shifts in the surface-temperature boundary condition. Herein we employ the technique by Menberg et al. (2014) and consider up to two sequential shifts in the boundary condition. The first shift, which warms the surface temperature, occurs at t = 0, and after a period of time (t = t1), the surface temperature returns to its value prior to the initial warming (T0). Such a boundary condition could approximate the sudden temporary increase in mean annual surface temperature due to a wildfire and the subsequent return to pre-fire surface temperatures due to vegetation regrowth (Burn, 1998). Alternatively, this boundary condition could represent the effect of clearcutting followed by industrial tree planting. The subsequent surface cooling due to gradual vegetative regrowth could also be represented with a series of shorter less intense cooling phases, but for the illustrative examples in the present study we assume one warming shift followed by one cooling shift of equal magnitude: Initialconditions:T(z,t=0)=T0,Boundarycondition:T(z=0,t)=T0+ΔTfor0<t<t1T0fort≥t1,Solution:T(z,t)=T0+ΔT2erfcz-Ut2Dt+exp⁡UzDerfcz+Ut2Dtfor0≤t<t1T0+ΔT2erfcz-Ut2Dt+exp⁡UzDerfcz+Ut2Dt-ΔT2erfcz-Ut-t12Dt-t1+exp⁡UzDerfcz+Ut-t12Dt-t1fort≥t1, where T0 is the uniform initial temperature (∘C), ΔT is the magnitude of the surface-temperature shift (∘C), erfc is the complementary error function, and t1 is the duration of the period characterized by warmer surface temperatures (s).

This solution and the remaining three solutions presented later also require a lower boundary condition at infinite depth (T = T0). Equation (11) can be employed to consider the subsurface warming due to a permanent step change in surface temperature (i.e., no subsequent cooling due to vegetative regrowth) by setting t1 to infinity. In this case, only the first line on the right-hand side of Eq. (11) is retained. Even when t1 is set to infinity, Eq. (11) differs slightly from the solution presented by Taniguchi et al. (1999a) because uniform initial temperatures are assumed in the present study (Eq. 9). These initial conditions ignore the influence of the geothermal gradient and imply that the recent climate has been relatively stable. We employ these simplifying assumptions given that we are primarily interested in shallower depths (e.g., < 25 m) where the influence of the geothermal gradient is not as important. Also, the boundary conditions for this solution and the following solutions do not include seasonal temperature variations. Thus, these solutions are valid for predicting the evolution of mean annual groundwater temperature.

Carslaw and Jaeger (1959) also presented an analytical solution to Eq. (2) subject to linearly increasing surface temperature. This solution was later adapted by Taniguchi et al. (1999b) and applied to study groundwater temperature evolution due to climate change. Herein, the analytical solution is presented in a slightly simpler form as thermally uniform initial conditions are assumed (i.e., initial conditions are given by Eq. 9): Boundarycondition:T(z=0,t)=T0+βt, Solution:T(z,t)=T0+β2U(Ut-z)×erfcz-Ut2Dt+(Ut+z)exp⁡UzDerfcz+Ut2Dt, where β is the rate of the increase in surface temperature (∘C s-1).

Equation (13) has been applied in an inverse manner to consider the complex relationships between past surface-temperature changes, groundwater flow, and measured subsurface temperature–depth profiles (e.g., Miyakoshi et al., 2003; Taniguchi et al., 1999b; Uchida and Hayashi, 2005). It has also been applied to forward model future groundwater temperature evolution due to projected climate change (Gunawardhana and Kazama, 2011). Herein, the surface boundary condition (Eq. 12) is fitted to mean annual air temperature trends produced by climate models. Because it is surface temperature, rather than air temperature, that drives shallow subsurface thermal regimes, this approach tacitly assumes that mean annual surface and air temperature trends are coupled. Thus, air temperature is being used as a proxy for surface temperature in this approach. Snowpack evolution may invalidate this assumption (Mellander et al., 2007), and thus it is best employed where snowpack effects are minimal. Snowpack evolution would typically retard the rate of groundwater warming (Kurylyk et al., 2013).

It may be inappropriate to assume a linear surface temperature rise as in Eq. (13), because many climate scenarios suggest that the rate of climate warming will increase over time. Figure 2 presents the globally averaged IPCC (2007) multi-model air temperature projections for two different emission scenarios. The global air temperature series projected for the conservative emission scenario B1 is much better represented by a linear function than the air temperature series for the aggressive A2 emission scenario, which exhibits significant concavity.

In such cases, the boundary condition would be better represented as an exponential function (Kurylyk and MacQuarrie, 2014). The solution presented here is simpler than the original form given that the initial conditions are assumed to be thermally uniform (initial conditions = Eq. 9): Boundarycondition:T(z=0,t)=T1+bexp⁡(ct), Solution:T(z,t)=T0+T1-T02erfcz2Dt-U2tD+exp⁡UzDerfcz2Dt+U2tD+b2exp⁡Uz2D+ctexp⁡-zU2/4D2+c/Derfcz2Dt-U24D+ct+exp⁡zU2/4D2+ct/Derfcz2Dt+U24D+ct, where T1 (∘C), b (∘C), and c (s-1) are parameters for the surface-temperature boundary condition which can be fitted to climate model projections. Note that T1 + b must equal T0 for the boundary and initial conditions to converge at t = 0, z = 0. The original initial condition function proposed by Kurylyk and MacQuarrie (2014) superimposed linear and exponential functions, and thus the more complex form of the solution can also be applied to forward model future climate change impacts on deeper subsurface-temperature profiles. These temperature profiles can deviate from the geothermal gradient due to groundwater flow or recent surface-temperature changes (Ferguson and Woodbury, 2005; Reiter, 2005). The alternate forms of the boundary conditions presented in Eqs. (10), (12), and (14) are illustrated in Fig. 3. Each of the listed analytical solutions to the one-dimensional, transient advection–diffusion equation is provided in Table 1 with details to highlight their differences.

The analytical solutions discussed above can be utilized to estimate the influence of surface warming at any desired depth. However, groundwater discharge to streams is sourced from different depths within the aquifer depending on the recharge location and the subsurface flow paths (Fig. 4a). Because the water table slope in unconfined aquifers is typically subdued in comparison to the land surface slope (Domenico and Schwartz, 1990), soil water that recharges the aquifer further upslope typically has a longer residence time and reaches greater depths relative to the land surface than soil water recharging the aquifer close to the discharge point. Groundwater flow in aquifers is often conceptualized as occurring in different “flow channels” or “flow tubes” (Domenico and Schwartz, 1990), and groundwater discharge is a thermal and hydraulic mixture of different groundwater flow channels coming from different depths and converging at the discharge point (Hoehn and Cirpka, 2006 and Fig. 4). Thus, when employing one-dimensional solutions to investigate the thermal evolution of groundwater discharge to streams and rivers, an effective depth zeff (m) must be considered that represents the bulk aquifer depth (i.e., accounting for all discharging groundwater flow channels) as a single point within the subsurface (Fig. 4). As a first estimate, this depth may be taken as the average unsaturated zone thickness. Figure 4b shows the conceptual model employed in this study. Above the effective depth, heat transport and water flow is assumed to be predominantly vertical as is often the case within the unsaturated zone, in overlying aquitards, or even in the upper portion of the aquifer (e.g., Kurylyk et al., 2014b). Within the aquifer (located at the effective depth), groundwater discharges horizontally towards a stream, and horizontal conductive heat transport is assumed to be negligible due to the relatively low horizontal thermal gradients in this zone. Heat advection and associated thermal dispersion near the discharge point is assumed to dominate vertical heat transfer and thus create a thermally uniform zone. Thus, the aquifer is treated as a thin, horizontally well-mixed thermal reservoir discharging to a surface water body (Fig. 4b). This approach is somewhat analogous to how contaminant hydrogeology studies have considered aquifers to be well-mixed reservoirs with respect to solute concentrations (e.g., Gelhar and Wilson, 1974). Vertical heat transfer continues below the aquifer (Fig. 4b). Limitations of this approach are discussed later.

Groundwater thermal sensitivity is herein defined as the change in groundwater temperature at some depth and time divided by the driving change in surface (z = 0) temperature at the same time. For example, if the surface temperature increases by 2 ∘C and the groundwater temperature has only increased by 1.4 ∘C at that same time, then the groundwater thermal sensitivity is 0.7 (1.4 ∘C/2 ∘C). The temperature changes at the surface and in the aquifer are measured with respect to the initial temperatures at those locations. This definition for groundwater thermal sensitivity S (∘C ∘C-1) can be expressed in the following manner: S(z,t)=ΔSubsurfaceTemp.ΔSurfaceTemp.=T(z,t)-T(z,t=0)T(z=0,t)-T(z=0,t=0). The groundwater thermal sensitivity is the analog to the stream thermal sensitivity defined by Kelleher et al. (2012), although the temperature changes are measured on a longer timescale for groundwater (e.g., multi-decadal vs. seasonal). Equation (16) represents the thermal sensitivity at any arbitrary depth within the aquifer. The bulk (i.e., the entire portion of the aquifer discharging to the stream or river) groundwater thermal sensitivity in Eq. (16) can be found by replacing z with zeff.

These analytical solutions assume that subsurface thermal properties are homogeneous, but in reality the bulk thermal properties of unconsolidated soils depend on many factors, including the mineral constituents, porosity, total moisture saturation, and the pore water phase (Farouki, 1981; Kurylyk et al., 2014b). Water has a much higher thermal conductivity than air; thus, the saturated zone typically is characterized by a higher bulk thermal conductivity than the unsaturated zone (Oke, 1978). Despite the existence of subsurface thermal property heterogeneities, natural variability in soil thermal properties is orders of magnitude less than the natural variability in hydraulic properties (Domenico and Schwartz, 1990), and thus homogeneous assumptions are better justified for subsurface heat transport than for subsurface water flow. Table 2 lists the bulk thermal properties for unfrozen sand, clay, and peat at three water saturations (volume of soil water/pore volume). These values are used to represent the typical ranges of thermal conductivities experienced in common unconsolidated soils. The bulk thermal diffusivities of these soils do not vary significantly at pore water saturations above 0.5.

Stallman's equation (Eqs. 5–7) can be utilized to investigate how idealized subsurface environments respond to seasonal surface-temperature changes. Figure 5 shows temperature–depth profiles for each month and temperature–time series for different depths in a soil column driven by a harmonic boundary condition at the surface (Eq. 4). The results were obtained from Eq. (5) for sandy soil (thermal properties; Table 2) and for a downwards Darcy velocity (i.e., recharge) of 0.2 m yr-1. This recharge value was chosen as a representative basin groundwater recharge (Döll and Fiedler, 2008; Healy, 2010). Stallman's equation generally matches seasonal groundwater temperature data reasonably well in shallow subsurface environments, except in locations where snowpack can make the surface temperature non-sinusoidal and the subsurface thermal envelope (Fig. 5a) asymmetrical (Lapham, 1989). Regardless, Eq. (5) and Fig. 5 both demonstrate that the seasonal subsurface-temperature variability is exponentially attenuated with depth and is barely discernible beyond a certain depth (e.g., 10–14 m).

The exponential damping factor Ω is the ratio of the amplitude of the seasonal temperature cycle at an arbitrary depth z to the amplitude of the seasonal surface-temperature cycle (Eq. 8). It is thus a measure of how the subsurface thermal regime responds to seasonal temperature variations, and it can be considered the seasonal counterpart to the groundwater thermal sensitivities derived from the analytical solutions experiencing long-term surface-temperature variability. Figure 6 illustrates that the exponential damping factor (or seasonal thermal sensitivity) Ω for a given depth decreases for the discharge scenario (black series; Fig. 6) in comparison to the recharge scenario (dashed-blue series). In a discharge scenario, the upward advective flux is impeding the downward propagation of the surface-temperature signal, and thus the surface signal is more quickly attenuated.

Figure 6a–c also indicate that the soil thermal properties greatly influence the subsurface thermal response to seasonal temperature variability. In particular, due to the significantly lower thermal diffusivity of partially saturated peat (Table 2), the surface-temperature signal is more quickly damped in the peat soil (Fig. 6c) in comparison to the results obtained for sand (Fig. 6a) and clay (Fig. 6b). However, in each of the nine scenarios presented in Fig. 6, the Ω parameter is less than 0.2 (amplitude reduced by at least 80 %) when the depth is greater than 5 m, which indicates that groundwater discharge does not have to be sourced from a very deep aquifer to decrease the stream thermal sensitivity to seasonal air temperature changes.

Beyond the depth of seasonal temperature fluctuations (Fig. 5), groundwater temperature will still be influenced by long-term surface-temperature perturbations. For instance, Fig. 7a (solid lines) shows the groundwater warming produced with Eq. (11) at different depths and for different soils due to a sudden and permanent (t1 = ∞, Eq. 10) mean annual surface temperature increase of 2 ∘C. This is approximately the long-term mean annual surface temperature increase observed by Lewis (1998) in response to deforestation. This is at the lower end of the range (1.6 to 5.1 ∘C) in the mean annual surface temperature increases noted by Taniguchi et al. (1998) following forest removal in Western Australia. The groundwater warming, rather than the temperature, is obtained by setting the initial temperature to zero (T0; Eqs. 10 and 11).

Results are presented for sandy soil and peat soil as these two soils respectively exhibit the highest and lowest thermal diffusivities given in Table 2. Due to the nature of the surface thermal boundary condition, these groundwater warming series exhibit a convex upward curvature. The results for the two depths (5 and 20 m) indicate that the lag between the surface and subsurface warming increases with increasing depth. For the sandy soil, the temperature at a depth of 20 m increases by 1.77 ∘C after 100 years, whereas at 5 m depth, this magnitude of warming was realized after only 14 years. Thus, for initially uniform conditions, deeper aquifers will generally remain colder longer than shallow aquifers, as it takes longer for the warming signal to be advected or conducted downwards. Furthermore, Fig. 7a also indicates that soils with a higher thermal diffusivity (i.e., sand) will initially transport the surficial warming signal through the subsurface more rapidly than soils with lower thermal diffusivity (i.e., peat). However, because the subsurface is slowly equilibrating with the new constant surface temperature, the solid series representing the results for the different depths and soils begin to converge as time increases.

In the case of vegetation regrowth, the surface-temperature warming due to the land cover disturbance would be temporary. As an illustrative example, Fig. 7a (dashed lines) shows the groundwater warming produced by Eq. (11) at two depths and for two soils due to a sudden 2 ∘C increase in surface temperature that persists for only 25 years (t1; Eq. 10). If desired, the equation could be further enhanced to accommodate a gradual cooling phase, rather than the instant cooling employed in the present study, using the more general formula described by Menberg et al. (2014). In Fig. 7a, the dashed and solid lines overlap prior to the cooling phase occurring at 25 years. The dashed temperature curves after 25 years represent the thermal recovery period. The groundwater warming curve for a depth of 5 m and the more diffusive soil (sand) is sharp, whereas the groundwater warming curve for a depth of 20 m and the less diffusive soil (peat) is more diffused and lagged. For example, the maximum groundwater warming (0.88 ∘C) for the peat soil at a depth of 20 m occurs at 33 years, which is 8 years after the surface warming has ceased. Thus, thermal impacts to cold-water streams caused by deforestation may persist several years after vegetation regrowth has occurred, particularly if groundwater discharge to the stream is sourced from a deeper aquifer. However, these effects would likely not be significant as the warming signal would be strongly damped at such depths.

Figure 7b shows the aquifer thermal sensitivities in response to a sudden permanent (solid lines) or temporary (dashed lines) step increase in surface temperature, which correspond to the same warming scenarios as shown in Fig. 7a. As indicated in Eq. (17), these thermal sensitivity curves are similar to the groundwater warming curves (Eq. 11 and Fig. 7a), but are scaled by a factor of ΔT-1. Hence, the thermal sensitivity curves due to a step increase in surface temperature are normalized with respect to the boundary temperature increase and are thus independent of the ΔT value. The results presented in Fig. 7 clearly demonstrate that shallow groundwater will initially warm rapidly in response to permanent deforestation and then the rate of temperature increase will decrease with time. This arises due to the initially high thermal gradient and heat conduction arising from the abrupt surface step change in temperature. The resultant impacts of groundwater warming on streambed conductive and advective heat fluxes should be considered in models that simulate stream temperature warming due to deforestation – at least for streams where groundwater discharge has been shown to influence stream temperature. Of particular note, small headwater streams, which are often groundwater dominated, can warm more rapidly than larger streams in response to deforestation because, for natural vegetative conditions, smaller streams typically experience more shading than larger rivers (e.g., Caissie, 2006).

The results shown in Fig. 7 are presented for a recharge scenario (q = 0.2 m yr-1). This approach is conservative because recharge environments will typically warm more rapidly in response to rising surface temperatures than discharge environments, as conduction and advection are acting in parallel in the former case. The analytical solutions provided in this study for simulating subsurface warming due to long-term surface-temperature trends (Eqs. 11, 13, and 15) are better suited for recharge environments than discharge environments, as groundwater discharge can bring up warm groundwater from deeper within the aquifer in accordance with the geothermal gradient. This phenomenon is not accounted for in the uniform initial conditions (Eq. 9). These solutions can be modified to allow for linearly increasing temperature with depth to account for the geothermal gradient (Kurylyk and MacQuarrie, 2014; Taniguchi et al., 1999a, b), but this adds complexity to the resultant sensitivity formulae. Also as previously noted, this study is primarily concerned with shallow aquifers where heat fluxes due to surface-temperature changes can dominate the influence of the geothermal gradient.

Equations (13) and (15) can be employed to investigate the sensitivity of groundwater temperatures to long-term gradual surface-temperature changes such as those experienced during climate change. The IPCC (2007) multi-model results (Fig. 2) are globally averaged results, and these data will be used to form the surface boundary conditions for the illustrative examples.

The consideration of groundwater temperature in stream temperature modeling is especially relevant in small streams where surface heat fluxes do not dominate the total energy budget. In fact, small streams are generally very dependent on groundwater inputs and temperatures, and their low thermal capacity (shallow depth and volume) makes them very vulnerable to any surface or subsurface-energy flux modifications (e.g., Matheswaran et al., 2014). This has been shown in many timber harvesting studies, where the smallest streams have experienced the greatest increase in stream temperature following forest removal (e.g., Brown and Krygier, 1970). Thus, quantifying future changes in shallow groundwater flow and temperatures is essential for a better understanding of the future thermal regimes of groundwater-dominated rivers and associated impacts to aquatic organisms (Kanno et al., 2014).

The results presented in Fig. 8 demonstrate the limitations inherent in inferring future stream warming from stream thermal sensitivities obtained from seasonal stream and air temperature data. For instance, the seasonal groundwater thermal sensitivity (Ω) values presented in Fig. 6 indicate that groundwater temperature beyond 10 m depth generally exhibits minimal sensitivity to seasonal variations in weather. Thus, stream thermal sensitivities obtained from seasonal air and stream temperature data are typically low for groundwater-dominated streams (Kelleher et al., 2012). However, as Fig. 8c and d illustrate, groundwater warming at depths greater than 10 m may still be significant in response to long-term surface-temperature changes, such as would be experienced under climate change. Due to the inter-relationships between the thermal regimes of stream and aquifers and the differences between the thermal sensitivities of shallow aquifers to short-term (Fig. 6) and long-term (e.g., Figs. 7b and 8) surface-temperature changes, it is not generally valid to extrapolate thermal sensitivities for groundwater-dominated streams obtained from sub-annual data to project long-term stream warming.

These results also demonstrate the potential limitations of using relatively short (e.g., < 25 years) records of inter-annual air and water temperature data to obtain estimations of future stream warming (e.g., Luce et al., 2014). Results for the present study (Fig. 8c and d) indicate that even at a timescale of 25 years, the thermal sensitivities of relatively shallow (e.g., 10 m) groundwater reservoirs may be low compared to thermal sensitivities that could be attained after 100 years of surface warming. These results suggest that what is interpreted as a damped stream thermal sensitivity to inter-annual air temperature variability in the case of groundwater-dominated streams may actually be a delayed thermal sensitivity due to the lag in the warming of groundwater and associated streambed heat fluxes. We acknowledge, however, that employing thermal sensitivities derived from inter-annual temperature data to project future stream warming is preferable to considering thermal sensitivities from seasonal temperature data (Luce et al., 2014). The appropriateness of these methods depends on the depth to the aquifer, the degree of groundwater contribution to the stream/river, the subsurface thermal properties, and the timescale of interest.

The present study demonstrates the importance of surface-temperature forcing on groundwater temperature, particularly for shallow aquifers. The potential influence of shallow groundwater warming on stream temperatures is not generally considered in existing empirical stream temperature models. The equations proposed in this study can be used to develop an approach to approximate the timing and magnitude of groundwater temperature warming in response to long-term surface-temperature changes. As described below, this information may be integrated within existing stream temperature models that consider streambed heat fluxes.

The upper boundary condition for the equations presented in this study is the ground surface temperature. Thus, the projected trends in catchment land surface temperature due to future climate change or land cover disturbances must be obtained prior to utilizing these equations. In the case of climate change without related snowpack changes, mean annual surface-temperature trends are often assumed to follow mean annual air temperature trends (see Mann and Schmidt, 2003). This simplification facilitates the boundary condition generation because air temperature trends can be readily obtained from the output of climate models. However, in the case of land cover changes (e.g., urbanization) or snowpack evolution, mean annual air temperature trends may be decoupled from mean annual surface-temperature trends (Mann and Schmidt, 2003; Mellander et al., 2007). In this situation, a simple surface heat-flux balance model can be applied to calculate the surface-temperature changes due to changes in the climate and/or land cover. A detailed discussion on appropriate techniques for simulating these relationships can be found in Mellander et al. (2007), Kurylyk et al. (2013), and Jungqvist et al. (2014).

Once the surface-temperature trends are obtained, they can then be fitted to the appropriate boundary condition function (Fig. 3). The associated analytical solution (Table 1) and groundwater thermal sensitivity formula can be utilized to perform simulations of future subsurface warming and/or groundwater thermal sensitivity due to the surface-temperature change. It should be noted that these solutions only calculate increases in mean annual groundwater temperature and do not account for seasonality. It is generally reasonable to assume that the amplitude and timing of the seasonal groundwater cycle will not be greatly influenced by climate change (Taylor and Stefan, 2009), provided snowpack conditions or the seasonality of soil moisture will not change significantly (Kurylyk et al., 2013).

In addition to the surface-temperature boundary condition terms, the analytical solutions must be parameterized with subsurface thermal properties, vertical groundwater flow information, and effective aquifer depth. Subsurface thermal properties can be obtained from information regarding the soil type and typical water saturation of the sediment overlying the aquifer (Table 2). Vertical groundwater flow rates can be obtained from field measurements (e.g., using heat as a hydrologic tracer, Gordon et al., 2012; Lautz, 2010; Rau et al., 2014) or from regional or local groundwater recharge and discharge maps. Potential changes in groundwater recharge (Crosbie et al., 2011; Kurylyk and MacQuarrie, 2013; Hayashi and Farrow, 2014) and groundwater discharge (Kurylyk et al., 2014a; Levison et al., 2014) due to changes in climate or land cover could also be considered. The aquifer effective depth can be crudely estimated as the average unsaturated zone or aquitard thickness overlying the aquifer (e.g., Fig. 4). Such information may be available from well data, geophysical surveys, or regional maps of the groundwater table depth (Fan et al., 2013; Snyder, 2008). Further research is required to assess approaches for more accurately determining the effective aquifer depth. A reasonable range of the input variables to these equations should be considered to generate an envelope of predicted groundwater warming (see Fig. 4 of Menberg et al., 2014). Such a range could incorporate uncertainties arising from, for example, heterogeneities in soil thermal properties and inter-annual variability in groundwater recharge (Hayashi and Farrow, 2014). Table 3 lists alternative options for parameterizing the equations presented in this study. The parameter values used in the present study are representative of conditions often observed.

To determine the influence of warming groundwater on stream temperatures, the future groundwater thermal sensitivity can be applied to estimate the resultant changes to streambed heat fluxes. There are different approaches available for estimating streambed heat fluxes from subsurface temperatures depending on whether the total streambed energy flux or the apparent sensible flux is being considered (e.g., Caissie et al., 2014; Moore et al., 2005), but in either case, the streambed fluxes depend on subsurface temperature, particularly the temperature immediately below the stream. These changes in streambed heat fluxes can then be combined with simulated changes in stream surface heat fluxes, and the resultant change in stream temperature can be obtained in a deterministic stream temperature model. Such an approach to estimate long-term evolution of stream temperatures would be more realistic than considering a stream temperature model driven by air temperature only, as both surface and streambed heat fluxes can be important in stream temperature dynamics.