#### Study Site

Field experiments were conducted in an old-growth *Fagus sylvatica* L.*/Quercus petraea* (Matt.) Liebl. mixed forest (site no. OB 5) in the north-west German state of Lower Saxony. The site is located at 115 m asl west of Unterlüss in the Lüneburger Heide area (52°45′ N, 10°30′ E). The stand is on highly acidic, nutrient-poor, sandy soils [pH(KCl), 2·6–2·8; Spodo-dystric Cambisols] derived from fluvioglacial sand of the penultimate ice age; the profile is covered by periglacial drift sand (Leuschner *et al*. 1993). The closed stand reaches a maximum height of 31 m (average 28 m) and has a density of 222 stems per hectare. About 90% of the stems belong to *Fagus* (90–110 years old, mean diameter 37 cm); 10% to *Quercus* (180–200 years old, mean diameter 52 cm). Shrub and field layers are absent. A 9–11 cm thick organic profile (Hemimor to Hemihumimor with L, Of and Oh horizons) above the mineral soil contains a large proportion of the fine roots, revealing the shallow rooting patterns in this forest (Büttner & Leuschner 1994). The density of fine roots decreases rapidly with depth from the nutrient-rich organic Of and Oh horizons (>2000 g dm m^{−3}) to the mineral soil at 50 cm depth (<100 g dm m^{−3}). Details on fine and coarse root distribution are given by Leuschner *et al*. (2001). The climate is cool-temperate and humid suboceanic (800 mm, 8 °C). During summer, rainless periods of up to 30 days occur periodically, causing low water contents in the sandy soil profile (Backes & Leuschner 2000; Leuschner 1993). The groundwater table is far below the rooting horizon.

#### Measurements of Root Sap Flow and Water Absorption Per Root Surface Area

All field measurements on root sap flow were conducted on roots of three 100-year-old *F. sylvatica* trees that were representative for the stand with respect to tree height and diameter. On the forest floor, segments of coarse roots were uncovered in small soil pits (<0·4 m wide, 0·1–0·6 m deep) by use of compressed air (0·2–0·4 MPa) to allow the mounting of miniature sap-flow gauges on the roots (Fig. 1). Ten roots, 3–4 mm in diameter at a distance of 1–2 m from the stem, were selected for study. The root sections used for gauge mounting had a brownish, suberized periderm and were not branched. During installation, any damage to the measured root or other exposed roots in the soil pit was carefully avoided. Duckboards were used to avoid trampling the soil. After installation the pits were covered with a wooden board and aluminium foil to minimize temperature fluctuations in the gauge surroundings. Sap flow was recorded continuously for periods of 7–60 days for all 10 roots in the summers of 1997 and 1998. Root growth during the measuring period was not taken into account because growth observations in soil chambers had shown branch root increments of ≤4 cm year^{−1}, which is negligible in a total fine root surface area per root ending of about 1000 cm^{2} (H.C., unpublished results).

After measurement, the distal part of the root with all appending branch roots, rootlets and root tips was uncovered by removing the soil with compressed air (Fig. 1). The root was harvested quantitatively by horizon (10 cm depth intervals in the organic layers and mineral soil) and transported to the laboratory in sealed plastic bags. The roots had endings distal to the gauge mounting point that were between 0·3 and 2·5 m long.

In the laboratory, the fresh root material was rinsed and separated into live and dead root mass under the stereomicroscope by colour, root elasticity and the degree of cohesion of cortex and stele (Hertel 1999; Leuschner *et al*. 2001). A dark cortex and stele; or a white but non-turgid cortex and stele; or the complete loss of stele and cortex, were used as indicators of root death. The live root mass was classed for diameter (0–1, 1–2, 2–5, >5 mm; Böhm 1979). An image-processing unit (WinRhizo, Régent, Quebec, Canada) was used to determine the total root surface area per diameter class. This procedure allowed us to express the measured flow (J) in terms of water uptake per surface area (g water m^{−2} root surface h^{−1} or mmol water m^{−2} root surface s^{−1}) of the adjacent root endings. We also calculated root sap-flow density at the measuring point (g water m^{−2} root cross-sectional area h^{−1}).

#### Miniature Sap-Flow Gauges

Measurements of root sap-flow rates were conducted with the constant power heat-balance method. The basic design described by Sakuratani (1981) and Baker & van Bavel (1987) was modified substantially to accommodate the small diameters and irregular shape of the tree roots, following Senock & Ham (1993); Senock & Leuschner (1999). The gauges consist of a 2 mm thick cork–neoprene jacket with two pairs of thermocouples and a thermopile mounted on it. A Kapton film-resistance heater (Heater Designs Inc., Bloomington, CA), supplied with a constant power of 0·04–0·07 W, was placed between the thermocouple junctions. For most roots selected, the flexible jacket enabled good contact of heater, thermocouples and thermopile to the root surface, which is crucial for accurate measurements. The gauges were wrapped with additional polyurethane foam insulation (5 mm) and fixed on the root with a metal clip. Solar-powered CR10 data loggers and AM416 multiplexers (Campbell, Cambridge, UK) were used to sample gauge signals every 15 s and to calculate 15 min averages.

Water-flow measurements with miniature sap-flow gauges are based on the steady-state energy balance of a heated portion of a root, defined as:

- (eqn 1)

where each term is a heat flux (W) calculated from equations describing heat flow within the system (Sakuratani 1981). *Q* is the energy supplied, *Q*_{r} the radial conduction of energy into the insulation measured by the thermopile, *Q*_{v} the apical and basal heat energy transferred by conduction along the root axis, *Q*_{f} the heat energy transported by the mass flow of water, and *S* the rate of change in heat storage in the root segment.

The energy transported by the sap flow, *Q*_{f}, is defined as:

- (eqn 2)

where *J* is the sap-flow rate (g s^{−1}), *c* the heat capacity of the xylem water (4·187 J g^{−1} K^{−1}), and *T*_{so} − *T*_{si} the temperature difference of the water flowing into and out of the heated root segment. *T*_{so} − *T*_{si} is recorded by the two pairs of thermocouples, assuming that the xylem temperature is represented by the root peridermal surface temperature. This assumption seems to be justified given that the ratio of the gauge heater width (10 mm) to the root diameter (3–4 mm) is large, and so large interior cross-sectional temperature gradients are unlikely (Sakuratani 1981; Senock & Ham 1995). The thermocouple junctions were placed at distances of 3 and 6 mm from the heater.

The portion of heat transported along the root axis (*Q*_{v}) is the product of the root cross-sectional area (*A*), the thermal heat conductivity (*K*_{r}), and the axial temperature gradient (Δ*T*) measured with the two pairs of thermocouples (*Q*_{v} = *A K*_{r}Δ*T*/Δ*x*). *K*_{r} has been calculated for woody and herbaceous tissues by Sakuratani (1984) and is 0·42 W m^{−2} for woody species.

The radial heat flux (*Q*_{r}) is given by:

- (eqn 3)

where *E* is the thermopile output (V), and *K*_{g} the gauge conductance (W µV^{−1}). *K*_{g} is determined at zero flow by combining equations 1 and 3, and setting *Q*_{f} to zero:

- (eqn 4)

Zero flow can be obtained by excision of the root segment at the end of the measurement (Baker & van Bavel 1987). In longer measurement periods (in this study, 3–6 weeks), changes in temperature and humidity inside the pit can lead to changes in *K*_{g} so that it has to be reset more frequently. We recalculated *K*_{g} daily by assuming that sap flow in *F. sylvatica* trees approaches zero typically at about 03:00 h, when leaf water potentials are at their predawn maximum. On dry and windy summer nights, when sap flow was substantially larger than zero, we used the *K*_{g} values of earlier nights with sap flow minima close to zero.

Empirical data have shown that the rate of change in heat storage in the root segment (*S* in equation 1) typically contributes <3% to the energy balance of small-diameter stems, and can usually be ignored in measurements of small stems or roots (Senock & Ham 1993).

- (eqn 5)

#### Sources of Measurement Errors in Miniature Sap-Flow Gauges

When applying the miniature sap-flow technique to small-diameter roots with low flows, errors may originate from the specific position of the thermocouples relative to the heater. Principally, the temperature difference *T*_{so} − *T*_{si} above and below the heated zone can be measured in sap-flow systems in two different ways, with the thermocouple junctions arranged either equidistantly or asymmetrically from the heater coil. In the latter system, one junction is placed directly at the end of the heater, while the second is positioned at a safe distance upstream from the heater so that it will record the baseline temperature of the unheated xylem. Equation 5 was originally based on this thermocouple arrangement (Sakuratani 1981), where zero or low night-flow rates maximize *T*_{so} − *T*_{si}. Increasing sap-flow rates lead to lower temperature differences because part of the heat is carried away with the moving xylem sap and the heated segment cools down.

The application of equation 5 to a symmetrical arrangement of the thermocouple junctions, as in the systems used by Sakuratani (1981) and in this study, has two consequences. First, *T*_{so} − *T*_{si} is zero at zero flow because the heat is distributed homogeneously around the heater. This case, however, is not defined by equation 5. Second, no uniform negative correlation between flow rate and *T*_{so} − *T*_{si} exists over the whole range of flows. Instead, *T*_{so} − *T*_{si} initially increases with increasing flow rates to a maximum when the flow is sufficient to move the zone of highest temperature in the xylem toward the *T*_{so} junction. Higher flow rates are characterized by a negative correlation between *J* and *T*_{so} − *T*_{si}, as predicted by the basic theory of sap-flow measurement.