Journal of Geophysical Research: Solid Earth

Thermal response of sediment with vertical fluid flow to periodic temperature variation at the surface

Authors


Abstract

[1] Characteristics of thermal responses of sediment with vertical fluid movement to periodic temperature variation at the surface were examined using a one-dimensional analytical solution. The amplitude of the thermal response decays exponentially, and the phase is delayed linearly with increasing depth, but they depend on the direction and velocity of vertical fluid flow, thermal diffusivity of fluid-saturated sediment, and period of surface temperature variation. To examine general characteristics of the thermal response, we defined two nondimensional parameters related to thermal diffusivity of fluid-saturated sediment, vertical fluid flow velocity, period of the surface temperature variation, and specific penetration depth at which the amplitude of the thermal response decays to e−1 of that at the surface. Analysis using these nondimensional parameters shows that there are three heat transport regimes for downward flow: (1) heat transport strongly governed by advection, (2) heat transport strongly governed by conduction, and (3) transition between these regimes. For upward flow, there are also three heat transport regimes: (1) balance of heat transports by advection and conduction, (2) heat transport strongly governed by conduction, and (3) transition between these regimes. The analytical solution is used to estimate the downward fluid velocity and thermal diffusivity of sediment from temperatures measured by long-term temperature monitoring at a site of seafloor hydrothermal circulation.

1. Introduction

[2] Interactions between surface and subsurface water influences not only hydrological, physical, chemical, and biological aspects at and under the surface but also conservation and management of water resources. Quantitative estimation of fluxes of recharging or discharging water provides fundamental information. To date, fluid fluxes were estimated in various environments, such as lakes [e.g., Shaw et al., 1990; Taniguchi and Fukuo, 1996], coastal zones [e.g., Cable et al., 1996; Moore, 1996; Spinelli et al., 2002; Taniguchi et al., 2002], submarine hydrothermal areas [e.g., Schultz et al., 1992, 1996; Cooper et al., 2000], and accretionary prisms in subduction zones [e.g., Carson et al., 1990; Tryon et al., 2001], with seepage meters, diffuse flowmeters, piezometers, and by chemical tracer methods.

[3] Measurement of subsurface temperature is one of the useful methods for estimating fluid flux through the earth material because moving fluid works as a “carrier” of heat. This method is based on solving a partial equation describing flows of heat and fluid. Under thermally stable boundary conditions, Bredehoeft and Papadopulos [1965] obtained an analytical solution describing vertical temperature distribution in the layer where fluid moves vertically at a constant velocity. On the basis of this analytical solution, they presented a type curve method for estimating vertical fluid flow velocity from a vertical temperature profile. This method was used to many studies investigating vertical groundwater movement [e.g., Sorey, 1971; Cartwright, 1979; Boyle and Saleem, 1979]. Sleep and Wolery [1978] discussed an analytical solution expressing vertical temperature distribution in sediment in which total heat transported by both advection and conduction is conserved at any depth. Using this analytical solution, Williams et al. [1979] and Becker and Von Herzen [1983] estimated both heat and fluid fluxes from vertical temperature profiles measured at the Galapagos Rift, eastern Pacific Ocean. These two analytical solutions presented by Bredehoeft and Papadopulos [1965] and Sleep and Wolery [1978] are simple mathematically but apply only to steady state problems under stable boundary conditions.

[4] There are several theoretical and practical studies that estimated fluid flow velocity from temperatures of sediment with a time-dependent thermal boundary condition. Taniguchi et al. [1999a] presented a series of type curves for estimating vertical groundwater flux in sediment in which water flows vertically at a constant rate and temperature at the surface increases linearly. They used these type curves to evaluate not only the groundwater flux but also surface warming in the Tokyo metropolitan area, Japan. Taniguchi et al. [1999b] presented another set of type curves for estimating vertical groundwater flux in sediment in which water flows vertically at a constant rate and temperature at the surface is expressed as a step function. They estimated vertical groundwater fluxes in the Collie River Basin in Western Australia where the ground surface temperature increased by the land cover change from forest vegetation to pasture or crops. Suzuki [1960] obtained an algebraic equation describing thermal response of soil in which fluid flows vertically at a constant rate and temperature at the surface changes periodically. Stallman [1965] developed Suzuki's work and obtained the analytical solution by the method of undetermined coefficients. On the basis of this analytical solution, he suggested that a fluid flow velocity of 0.3 cm d−1 can be estimated by analyzing thermal response of soil to diurnal temperature variation at the surface. Using the Stallman's analytical solution, Taniguchi [1993] presented a type curve method for estimating vertical steady fluid flow velocity from seasonal change in a vertical temperature profile in a relatively shallow depth borehole. He used this method to estimate vertical fluid flux from borehole temperatures measured at Nagoya Plane, Japan.

[5] The thermal response of sediment with vertical water flow to surface temperature variation depends on the direction and velocity of the fluid flow, the physical properties of the sediment and water, and the period of the surface temperature variation. However, the quantitative dependences of the thermal response to these parameters have not been discussed in a systematic way until now. In this paper, we examine them on the basis of Stallman's analytical solution. Finally, we adapt the analytical solution to estimate the fluid flux and thermal diffusivity of seawater-saturated sediment from temperatures measured at the TAG hydrothermal mound, 26°N on the Mid-Atlantic Ridge.

2. Thermal Response in Sediment to Periodic Temperature Variation at the Surface

[6] We assume homogeneous sediment saturated with fluid. At the surface, temperature fluctuates as a series of trigonometric functions of time t as

equation image

where Ai and θi are the amplitude and phase, respectively, of temperature variation of period Pi. In sections 2.1 and 2.2, we first introduce an analytical solution describing thermal response of sediment with no fluid flow (pure conduction) to surface temperature variation expressed as equation (1). Then an analytical solution that describes the thermal response of sediment with vertical fluid flow of a constant velocity to the surface temperature variation is presented.

2.1. Thermal Response in Sediment Under Pure Heat Conduction

[7] We assume that there is no fluid flow in the homogeneous sediment, so heat transient is entirely conductive. The thermal response of the sediment Tcond(z, t) to the surface boundary condition expressed as equation (1) is given as [Carslaw and Jaeger, 1959]

equation image

where z is the depth below the sediment surface; κm is the thermal diffusivity of fluid-saturated sediment defined as

equation image

where Km and (ρcp)m are thermal conductivity and heat capacity, respectively, of fluid-saturated sediment. Equation (3) indicates that the amplitude of the thermal response decays exponentially and the phase is delayed linearly with increasing depth. The amplitude decay and phase delay also depend on values of Pi and κm.

2.2. Thermal Response in Sediment With Vertical Fluid Flow

[8] We assume that there is vertical fluid flow (Darcy flow) with a constant velocity vf in the homogeneous sediment (vf > 0 for downward flow and vf < 0 for upward flow). The thermal response of the sediment T(z, t) to the boundary condition expressed as equation (1) is given as [Stallman, 1965]

equation image

The parameter v is a product of the fluid flow velocity vf and the ratio of the heat capacity of the fluid (ρcp)f and fluid-saturated sediment (ρcp)m defined as

equation image

where ρ and cp indicate the density and specific heat, respectively, and the subscripts f and m denote fluid and fluid-saturated sediment, respectively; αi is a function of Pi, κm and v:

equation image

Equations (4) to (6) indicate that the thermal response is a nonlinear function of κm (controls conductive component), v (controls advective component), Pi and z. When vf = 0 (pure heat conduction), equation (4) reduces to equation (2).

3. Characteristics of Thermal Response in Sediment With Vertical Fluid Flow to Surface Temperature Variation

[9] In this section, characteristics of the thermal response of sediment with vertical fluid flow to periodic temperature variation at the surface on κm, v and Pi are discussed on the basis of equation (4). To examine the dependence on κm, we assume two types of sediments with different thermal properties (ordinary marine sediment and hydrothermal deposit) saturated with seawater. The thermal conductivity Km and thermal diffusivity κm are assumed to be 1 W m−1 K−1 and 2.5 × 10−7 m2 s−1, respectively, for the ordinary marine sediment and 6 W m−1 K−1 and 2.5 × 10−6 m2 s−1, respectively, for the hydrothermal deposit. The density and specific heat of seawater are assumed to be 1030 kg m−3 and 3985 J kg−1 K−1, respectively.

3.1. Amplitude Decay of Thermal Response

[10] Figure 1 shows examples of the thermal response of the ordinary marine sediments with fluid flow of velocities of ±1 × 10−6 m s−1 (32 m yr−1) to the surface temperature variation calculated by equation (4). The period of the surface temperature variation is 720 hours, with which positive and negative effects of the fluid flow on downward propagation of the temperature variation are clearly demonstrated. The amplitude of the surface temperature variation is 1 K. For comparison, we also plot profiles of thermal response of the sediment under pure heat conduction. In the purely conductive regime, the effect of the surface temperature variation is negligible (the amplitude is less than 0.005 K) below a depth of about 2.6 m (Figure 1b). In the sediment with downward fluid flow (Figure 1a), the effect of the surface temperature variation is negligible below around 8 m, indicating that the downward fluid flow acts as a carrier of the temperature variation. In the sediment with upward fluid flow (Figure 1c), on the other hand, the effect is negligible below only 1.4 m, indicating that the upward fluid flow hinders downward propagation of surface temperature variation.

Figure 1.

Temporal variations of thermal responses of ordinary marine sediment (Km = 1 W m−1 K−1 and κm = 2.5 × 10−7 m2 s−1). Amplitude and period of surface temperature variation are assumed to be 1 K and 720 hours (30 days), respectively. (a) Thermal responses for downward fluid flow of vf = 1 × 10−6 m s−1. (b) Thermal responses for pure heat conduction case. (c) Thermal responses for upward fluid flow of vf = −1 × 10−6 m s−1.

[11] Figure 2 shows diagrams for decay of amplitude of thermal responses of the ordinary sediment (Figure 2a) and hydrothermal deposit (Figure 2b) with downward fluid flow of various velocities (vf = 1 × 10−8 to 1 × 10−5 m s−1, 0.32 to 320 m yr−1) to surface temperature variation. In Figures 2a and 2b, the horizontal axis is the period of surface temperature variation (1 to 10,000 hours), and the vertical axis is the specific penetration depth (zs) where the amplitude of the thermal response decays to e−1 of that at the surface. The specific penetration depth is calculated by the equation derived from equation (4) as follows:

equation image

For comparison, the corresponding specific penetration depths for pure heat conduction are also plotted in Figures 2a and 2b as thick gray lines. For the ordinary marine sediment (Figure 2a), the specific penetration depth for vf = 1 × 10−8 m s−1 is almost the same as that for pure heat conduction. For vf = 1 × 10−7 m s−1, the specific penetration depth for a period <100 hours is almost the same as that under pure heat conductive regime. On the other hand, the specific penetration depth for a period >100 hours is deeper than that for pure heat conduction. When the downward fluid velocity becomes more rapid, the period at which the advective component is effective becomes shorter. For vf = 1 × 10−5 m s−1, the advective component predominates over the downward propagation of the surface temperature variation for the full range of periods shown in Figure 2.

Figure 2.

Diagrams for specific penetration depths of thermal responses of ordinary sediment and hydrothermal deposit with downward fluid flow of various velocities versus period of surface temperature variation. (a) Specific penetration depths of thermal response for ordinary marine sediment. (b) Specific penetration depths of thermal response for hydrothermal deposit.

[12] The dependences of specific penetration depths for the hydrothermal deposit on downward fluid velocity and period of the surface temperature variation are similar, although the minimum period for which the advective component is measurable is longer than that for the ordinary marine sediment because conduction is much more effective (Figure 2b). For vf ≤ 1 × 10−7 m s−1, the specific penetration depths for the hydrothermal deposit are deeper than those for the ordinary marine sediment over the given range of period. For vf = 1 × 10−6 m s−1; however, the specific penetration depth for the ordinary marine sediment is deeper than that for the hydrothermal deposit for a period >2000 hours. For vf = 1 × 10−5 m s−1, the reversal of the specific penetration depths of the hydrothermal deposit and ordinary marine sediment appears at a period of 20 hours. This indicates that when the fluid velocity becomes more rapid, the advective component becomes more effective in sediment with low thermal diffusivity than in sediment with high thermal diffusivity.

[13] Figure 3 shows specific penetration depths for the same sediments with upward fluid flow of various velocities (vf = −1 × 10−8 to −1 × 10−5 m s−1) and for pure heat conduction. For both sediments, specific penetration depths for vf = −1 × 10−8 m s−1 are almost the same as those for purely heat conductive regime over the given range of periods. For the ordinary marine sediment (Figure 3a), the specific penetration depth for vf = −1 × 10−7 m s−1 is almost the same as that for the pure heat conduction case for a period <100 hours. For a period >100 hours, the specific penetration depth is shallower than that for the pure heat conduction case, indicating that the advective component limits downward propagation of surface temperature variation because the direction of fluid flow is opposite to that of thermal propagation. For the hydrothermal deposit (Figure 3b), the reduction in specific penetration depth for vf = −1 × 10−7 m s−1 appears with periods >1000 hours.

Figure 3.

Diagrams for specific penetration depths of thermal responses of ordinary sediment and hydrothermal deposit with upward fluid flow of various velocities versus period of surface temperature variation. (a) Specific penetration depths of thermal response for ordinary marine sediment. (b) Specific penetration depths of thermal response for hydrothermal deposit.

[14] For the ordinary marine sediment, the specific penetration depths for vf = −1 × 10−5 m s−1 and vf = −1 × 10−6 m s−1 converge to 0.024 m and 0.244 m, respectively, with increasing period of surface temperature variation (Figure 3a). Similarly, the specific penetration depth for the hydrothermal deposit with vf = −1 × 10−5 m s−1 converges to 0.146 m with measurable period (Figure 3b). The asymptotic specific penetration depth, zas, is a function of κm and v given as

equation image

In Figure 3, the periods at which the specific penetration depth converges are about 80 hours for vf = −1 × 10−5 m s−1 and about 8300 hours for vf = −1 × 10−6 m s−1 for the ordinary marine sediment (Figure 3a), and about 300 hours at vf = −1 × 10−5 m s−1 for the hydrothermal deposit (Figure 3b). For the hydrothermal deposit with vf = −1 × 10−6 m s−1, the specific penetration depth converges to 1.462 m for a period >30,000 hours, out of range of the period in Figure 3b.

3.2. Phase Shift of Thermal Response in Sediment

[15] Comparison between equations (2) and (4) leads the following inequality with regard to the phase shift at any depth if vf ≠ 0:

equation image

This inequality shows that the phase shift of thermal response in sediment with vertical fluid flow, independently of the direction, is smaller than that of the thermal response of sediment under pure heat conductive regime. This relation means that the effect of surface temperature variation propagates in sediment with vertical fluid flow faster than sediment under pure heat conduction. Furthermore, the phase shift for downward fluid flow (vf > 0) is identical to that for upward fluid flow (vf < 0) at the same absolute rate. Figure 4 shows examples of thermal responses of the ordinary marine sediment with fluid flow of vf = ±1 × 10−6 m s−1 and vf = ±1 × 10−7 m s−1 at a depth of 0.5 m. Period and amplitude of the synthetic surface temperature variation are assumed to be 720 hours and 1 K, respectively. For comparison, the purely conductive thermal response of the sediment is also plotted. The peaks of the thermal responses in the sediment with vf = ±1 × 10−6 m s−1 appear at the same time (denoted A in Figure 4), although the amplitudes of these thermal responses differ significantly. These peaks appear 24 hours earlier than that of thermal response in sediment under pure heat conduction (denoted B in Figure 4). As ∣vf∣ increases, the appearance of peak of thermal response of the sediment becomes earlier. On the other hand, the peaks of thermal responses of the sediments with of vf = ±1 × 10−7 m s−1 almost correspond to that of thermal response in the sediment under pure heat conduction. This indicates that for the ordinary marine sediment, vertical fluid flow with ∣vf∣ ≤ 1 × 10−7 m s−1 will have little influence on the phase of thermal response to surface temperature variation.

Figure 4.

Thermal responses of ordinary sediment with vertical fluid flow of various velocities at depth of 0.5 m (Km = 1 W m−1 K−1 and κm = 2.5 × 10−7 m2 s−1). Amplitude and period of the surface temperature variation are assumed to be 1 K and 720 hours (30 days), respectively.

[16] The advance of phase of thermal response is related to the downward propagation rate of surface temperature variation. From equations (1), (2), and (4), we define the effective propagation rates of the surface temperature variation as (see Appendix A):

equation image
equation image

Figure 5 shows effective propagation rates for the ordinary marine sediment (Figure 5a) and for the hydrothermal deposit (Figure 5b) with various fluid flow velocities (vf = ±1 × 10−7 to ±1 × 10−5 m s−1) against the period of surface temperature variation. Corresponding effective propagation rates for the pure heat conduction case are also plotted. For the both sediments with pure heat conduction, the effective propagation rates decrease with increasing period. The effective propagation rates for the both sediments with vf = ±1 × 10−7 m s−1 are almost same as those for the sediment under pure heat conduction over the given range of the period. On the other hand, the effective propagation rates for the ordinary marine sediment with vf = ±1 × 10−6 m s−1 and vf = ±1 × 10−5 m s−1 and for the hydrothermal deposit with vf = ±1 × 10−5 m s−1 become constant with increasing period (the effective propagation rate for the hydrothermal deposit with vf = ±1 × 10−6 m s−1 becomes constant for a period >30,000 hours). The converged effective propagation rate is equal to the absolute value of v defined by equation (5), which expresses advective component on the heat transport system. Thus ∣v∣ gives the lower limit of the effective propagation rate of surface temperature variation.

Figure 5.

Diagrams for effective propagation rates of surface temperature variation of ordinary marine sediment and hydrothermal deposit with vertical fluid flow of various velocities versus period of surface temperature variation. (a) Effective propagation rates of surface temperature variation for ordinary marine sediment. (b) Effective propagation rates of surface temperature variation for hydrothermal deposit.

[17] The thermal response of sediment with vertical fluid flow to surface temperature variation results from integration of advective and conductive components of heat transport, as described in equation (4). In sediment with downward fluid flow, the moving fluid carries downward the effect of the surface temperature variation. The effective propagation rate is higher than that in the sediment under pure heat conductive regime due to the contribution of heat transport by advective component. In sediment with upward fluid flow, however, the direction of the fluid flow is opposite to that of propagation of the surface temperature variation. The interaction between the downward propagation of surface temperature variation by conduction and the upward transport of information on temperature at a deeper depth by advection results in a positive phase shift of the thermal response.

4. Nondimensional Parameter

[18] As discussed in section 2.2, the thermal response of sediment with vertical fluid flow to surface temperature variation is a nonlinear function of κm (controls conductive component), v (controls advective component), Pi and z. To examine the general dependences of the thermal response on κm, v and Pi, we define a nondimensional parameter related to these three parameters, ξ, from equation (6) as (compare equation (6))

equation image

For simplicity, the period of surface temperature variation is expressed as P in the present section. A lower value of ξ corresponds to lower κm, longer P or higher v. In contrast, a higher ξ value corresponds to higher κm, shorter P or lower v. We also define another nondimensional parameter as

equation image

Pe is the thermal Peclet number that is a measure of relative importance of heat transport by advection and conduction [Fowler, 1990]. When Pe ≫ 1, the advection component dominates heat transport. The conductive component dominates heat transport when Pe ≪ 1. By using ξ, equation (13) is reexpressed as

equation image
equation image

Figure 6 shows Pe values for thermal responses of sediments with downward fluid flow (Figure 6a) and upward fluid flow (Figure 6b).

Figure 6.

Plots of Pe versus ξ for (a) downward fluid flow and (b) upward fluid flow.

[19] For ξ < ∼0.2, Pe for downward fluid flow (vf > 0) decreases with decreasing ξ and the value of Pe is sufficiently larger than 1 (Pe > ∼400). On the other hand, Pe for upward fluid flow (vf < 0) in this range of ξ is approximately 1. In this range of ξ, the effective propagation rate is nearly equal to the corresponding value of ∣v∣ (Figure 5). For upward fluid flow, furthermore, the specific penetration depth can be approximated by a constant, zas defined by equation (8), as shown in Figure 3.

[20] For ξ > ∼40,000, Pe for downward fluid flow is nearly identical to that of upward fluid flow. In this range of ξ, Pe is sufficiently smaller than 1 (Pe < ∼0.01), indicating that advection little contributes to the thermal response of sediment to periodic temperature variation at the surface.

[21] Using these values of ξ and Pe, the downward propagation of surface temperature variation can be divided into three heat transport regimes: (1) heat transport strongly governed by advection (ξ < ∼0.2, Pe > ∼400), (2) heat transport strongly governed by conduction (ξ > ∼40,000, Pe < ∼0.01), and (3) transition between these two regimes (∼0.2 < ξ < ∼40,000, ∼0.01 < Pe < ∼400). When Pe = 1, which means that heat transports by advection and conduction have the same contribution, ξ = 17. For sediment with upward fluid flow (vf < 0), the downward propagation of surface temperature variation can also be divided into three heat transport regimes: (1) balance of heat transports by advection and conduction (ξ < ∼0.2, Pe ≈ 1), (2) heat transport strongly governed by conduction (ξ > ∼40,000, Pe < ∼0.01), and (3) transition between these two regimes (∼0.2 < ξ < ∼40,000, ∼0.01 < Pe < ∼1).

5. Application

[22] By using equation (4), we can estimate vertical fluid flow velocity and thermal diffusivity of sediment from the thermal response of sediment to surface temperature variation. In this section, we estimate these parameters from long-term temperature data obtained at the TAG hydrothermal mound, Mid-Atlantic Ridge 26°N.

[23] TAG hydrothermal mound (Figure 7), discovered in 1985, is one of the largest submarine hydrothermal mounds with high-temperature hydrothermal activity over the world [e.g., Rona et al., 1986; Humphris et al., 1995; Humphris and Kleinrock, 1996; Kleinrock and Humphris, 1996]. On the NW part of the central cone, there is the central black smoker complex (CBC) that is composed of numerous black smoker vents. CBC discharges hydrothermal fluid with temperature of up to 369°C [Goto et al., 2002]. The distribution of heat flow on the mound reflects the complicated hydrological system inside the mound [Becker and Von Herzen, 1996; Becker et al., 1996].

Figure 7.

Bathymetry of the TAG hydrothermal mound (bathymetric data from Humphris and Kleinrock [1996]). Contour interval is 5 m. CBC is the central black smoker complex that discharges hydrothermal fluid up to 369°C [Goto et al., 2002]. Solid square and circle indicate deployed positions of long-term temperature monitoring system Daibutsu and its probe (probe 2), respectively. Insert is an index map of location of the TAG hydrothermal mound.

[24] In August 1994, a long-term temperature monitoring system, designated Daibutsu, was deployed on the TAG hydrothermal mound (Figure 7) [Kinoshita et al., 1996; Goto et al., 2002]. This system has eight thermal probes 80 cm in length. Each probe contains five thermistors (the accuracy and resolution are 0.01°C and 0.001°C, respectively) at 17.5 cm intervals. One of these probes (probe 2) was inserted into hydrothermal deposit 20 m SE of CBC and measured temperatures of the hydrothermal deposit and bottom water a few centimeters above the deposit. Figure 8 shows the measured temperatures. P2-4 is the bottom water temperature, and P2-2 and P2-1 are the temperatures of hydrothermal deposit at depths of 0.228 and 0.307 m, respectively. The bottom water temperature periodically fluctuates every 12.4 hours, which is consistent with the semidiurnal ocean tide M2. The subsurface temperatures change according to the bottom water temperature variation. Goto et al. [2002] indicated that amplitudes of the temperature variations of the deposit could not be explained by the model that the effect of bottom water temperature variation is propagated by heat conduction only (Figure 9a). By using the model that seawater percolates into the mound at a constant rate, they suggested that the amplitudes of the temperature variations of the deposit could be explained by which seawater percolated at a velocity of 1.3 ± 0.5 × 10−5 m s−1.

Figure 8.

Bottom water (thin dashed line) and subbottom (thin solid and thick gray lines) temperatures measured at position of probe 2. Bottom water temperature fluctuates every 12.4 hours, and subbottom temperature reflects the bottom water temperature variation.

Figure 9.

Comparison between measured and calculated temperatures at position of P2-1 (subsurface depth of 0.307 ± 0.050 cm). Each temperature is normalized by the average one. (a) Best fit temperature calculated by forward modeling using equation (2) [Goto et al., 2002]. (b) Best fit temperature calculated by inversion using equation (4).

[25] From the bottom water and subsurface temperatures, we reestimated a percolation rate of seawater into the TAG hydrothermal mound and thermal diffusivity of the hydrothermal deposit by equation (4). In this estimation, we assumed that the bottom water temperature variation is expressed by equation (1). By the Bayesian inversion of Tarantola [1987], v and κm were estimated at 1.7 ± 0.1 × 10−5 m s−1 and 1.7 ± 0.1 × 10−6 m2 s−1, respectively. The temperatures calculated using these two parameters are agreement with the measured ones (Figure 9b). At probe 2 site, thermal conductivity of the hydrothermal deposit was measured as Km = 6.12 W m−1 K−1 (K. Becker, personal communication, 1998). Assuming density and specific heat of seawater to be 1030 kg m−3 and 3985 J kg−1 K−1, respectively, we calculated the percolation rate of seawater into the mound as 1.5 ± 0.1 × 10−5 m s−1 by equation (5). This value corresponds to that of Goto et al. [2002] within the possible error range. ξ and Pe values corresponding to the predominant period of the bottom water (12.4 hours) are calculated as 3.31 and 4.05, respectively, indicating that the heat transport regime was at the transition between heat transport governed by advection and conduction.

6. Conclusion

[26] Characteristics of thermal responses of sediment with vertical fluid flow to periodic temperature variation at the surface were examined by the one-dimensional analytical solution of Stallman [1965]. The amplitude of the thermal response decays exponentially and the phase is delayed linearly with increasing depth, but these depend on direction and velocity of vertical fluid flow, thermal diffusivity of fluid-saturated sediment, and period of surface temperature variation. Downward fluid flow propagates the effect of surface temperature variation efficiently. On the other hand, upward fluid flow enhances to decay amplitude of the thermal response. In both fluid flow directions, the effect of surface temperature variation propagates faster than that in conductive heat transport. However, the thermal response of sediment with fluid flow velocity of ∣vf∣ ≤ 1 × 10−8 m s−1 is almost the same as that of sediment under pure heat conduction.

[27] To examine the general characteristics of the subsurface thermal response, we defined two dimensionless parameters, ξ (related to thermal diffusivity of sediment, fluid flow velocity and period of surface temperature variation) and Pe (thermal Peclet number) as equations (12) and (13), respectively. Examinations using these dimensionless parameters show that there are three heat transport regimes for downward fluid flow as (1) heat transport strongly governed by advection (ξ < ∼0.2 and Pe > ∼400), (2) heat transport strongly governed by conduction (ξ > ∼40,000 and Pe < ∼0.01), and (3) transition between these two regimes (∼0.2 < ξ < ∼40,000 and ∼0.01< Pe < ∼400). When ξ = 17, Pe = 1, indicating that heat transport by advection and conduction have the same contribution to the downward propagation of surface temperature variation. For upward flow, there is also three heat transport regimes as (1) balance of heat transports by advection and conduction (ξ < ∼0.2 and Pe ≈ 1), (2) heat transport strongly governed by conduction (ξ > ∼40,000 and Pe < ∼0.01), and (3) transition between these two regimes (∼0.2 < ξ < ∼40,000 and ∼0.01 < Pe < ∼1).

[28] We used the analytical solution to estimate downward fluid velocity and thermal diffusivity of hydrothermal deposit from long-term temperature data obtained at the TAG hydrothermal mound, Mid-Atlantic Ridge 26°N. The estimated fluid flow velocity and thermal diffusivity are 1.5 × 10−5 m s−1 and 1.7 × 10−6 m2 s−1, respectively. This method could be used to estimate the velocity of vertical groundwater flow in recharge and discharge areas on land. The analytical solution could also be used to evaluate a past ground surface temperature change in recharge and discharge areas [e.g., Taniguchi et al., 1999a, 1999b].

Appendix A:: Effective Propagation Rate of Surface Temperature Variation

[29] We assume homogeneous sediment with vertical fluid flow at a constant velocity vf. At the surface, temperature fluctuates as a series of trigonometric functions of time t as

equation image

where Ai and θi are the amplitude and phase, respectively, of temperature variation of period Pi. Thermal response of the sediment at the depth z to the boundary condition as equation (A1) is given as [Stallman, 1965]

equation image
equation image
equation image

where ρ and cp indicate density and specific heat, respectively, and the subscripts f and m denote fluid and sediment, respectively. Equation (A2) is reexpressed as

equation image

Thus the time τi when the effect of the surface temperature variation transports to the depth z is expressed as

equation image

The effective propagation rate Vi is derived from

equation image

The effective propagation rate for the sediment under pure heat conduction is led by the same procedure as

equation image

Acknowledgments

[30] We thank Nobuhiko Fujii for discussion about equations used in this study. We are thankful to Keir Becker for permission to use thermal conductivity data at the TAG hydrothermal mound. We are grateful to Andrew Fisher for critical reading and improvement of the manuscript. We are also grateful to Michael Manga and one anonymous reviewer for review and comments that improved the manuscript. This study was partially supported by a Grant-in-Aid for the 21st Century COE Program (Kyoto University, G3).

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