Surface Renewal Approach

Surface renewal models (Chen et al., 1997b; Katul et al., 1996; Paw U et al., 1995;

Snyder et al., 1996) assume that the ramp patterns occurring in scalar time series, measured with sufficient temporal resolution, arise from an idealized air parcel motion which is illustrated in Fig. 1. An air parcel of volume V₀ is assumed to originate above the forest and to instantaneously penetrate the canopy.

During a certain residence time the parcel stays in contact with leaves and other canopy elements, exchanging heat and mass. Small-scale diffusive exchange together with chemical production (or loss) for reactive species results in a gradual enrichment (e.g., for temperature with a warmer canopy) or gradual depletion (e.g., for deposited quantities like ozone) of a scalar X within the parcel, until the parcel leaves the canopy and is replaced (renewed) by another one from aloft. The measured time series of X shows a correspondingly slow, nearly constant, temporal increase or decrease dX/dt throughout the residence time, which is concluded by a step-like change back to the initial level when the parcel is exchanged by a coherent structure.

Assuming that (i) the canopy–atmosphere exchange of X occurs exclusively through

these instantaneous replacements and (ii) the loss/gain of X at the parcel top during the entire residence time is negligible, one can derive the surface flux density F_X. It is the mean storage change in the corresponding air volume associated with all coherent structures throughout the averaging period T_a.

− dX/dt is the temporal derivative of the measured scalar and [ ]_+/- is a conditional temporal average of dX/dt depending on the direction of the flux (see Section 2.2). V₀ and A are the air parcel volume and ground area.

V₀ /A can be approximated by the height of the correspondent air parcel. In the original work of Paw U et al. (1995) (see also Katul et al.

(1996)), the canopy height h (equivalent to their measuring height z_m) was considered as the air parcel top, assuming the whole canopy depth to be involved in the renewal process (Fig. 1 (b)).

For a (approximately) horizontally uniform canopy, the average change of storage is described by the depth-averaged mean of dX/dt below the measuring height in Eq. (1). The depth average in Eq. (1) is indicated by

z. With assumptions (i) and (ii) mentioned above, the weighting coefficient α accounts for the dX/dt profile due to the vertical distribution of sources and sinks within the considered air volume V₀ and relates therefore the average temporal change in X of the whole volume to the temporal change at the measuring height z_m. Paw U et al. (1995) proposed α = 0.5 which approximates the real vertical gradient of dX/dt in the air volume by a linear increase of dX/dt with height from zero at the ground to a maximum dX/dt measured at z_m = h. Despite the differences in the vertical source/sink distribution of heat and water vapor in plant canopies, with this value of α good agreement with the corresponding eddy covariance fluxes was achieved for sensible and latent heat exchange at forest canopies (Katul et al., 1996;

Paw U et al., 1995). Well above shorter canopies like grass or wheat, α was found to be close to unity for the sensible heat flux, most likely because the main part of the air parcel influenced by the mean temperature change remains above the canopy and is therefore relatively independent of the uneven source distribution within the canopy below (Anandakumar, 1999; Snyder et al., 1996;

Spano et al., 1997a).

Fig. 1. Conceptual scheme of the air parcel motion on which the surface renewal approach is based (Paw U et al., 1995). During the residence time τs, the volume V0 is in contact with the canopy surface, and the scalar property X is exchanged. The mean duration D of the turbulent structures is directly related to the peak of the corresponding wavelet variance spectrum (scale a0; for details see text). The exchange results in a gradual increase (here for an emitted scalar quantity) in the time series X(t) measured at zm until the parcel is replaced by another one. Small-scale turbulence and the source/sink distribution within V0 control the vertical profile of the temporal change dX(z)/dt (the profile suggested by Paw U et al. (1995) is indicated by the black curve in the middle volume).

Triggered by the fact that ramp patterns were frequently observed in scalar time series measured well above the forest top (Fig. 3; see also, e.g., Gao et al. (1989)), a slightly different conceptual picture is proposed in Fig.

2 for the forest–atmosphere exchange of scalars. It is a combination of the concepts for tall canopies by Paw U et al. (1995) and lower

vegetation by Snyder et al. (1996) and Spano et al. (1997a). Due to the large canopy space of the rain forest, the average storage change of the volume V₀ must be clearly influenced by the vertical source/sink distributions of X within the forest. The fact, however, that pronounced ramp patterns occur at 1.33 h without clear amplitude reduction compared to

1.05 h (see Fig. 3) indicates that the top of the relevant air volume may be above the canopy height. Between the renewal events, the top of the air volume is characterized by negligible vertical exchange. But the temporal change of X (ramp slope) above the canopy must be a result of upward transport (if X is an emitted scalar), since no direct surface exchange is possible there. Therefore it seems very likely,

as indicated in Fig. 2, that a considerable part of the air parcel that is enriched or depleted by surface exchange is sitting above the canopy.

Fig. 2 also shows the supposed vertical profile of dX(z)/dt. It has a maximum in a broad band around the canopy top and declines towards the ground owing to decreasing surface exchange.

Fig. 2. Conceptual scheme as in Fig. 1 (b) with a modified vertical extent of the air parcel. In between consecutive renewal events, considerable small-scale exchange is assumed to occur at canopy height. This exchange is responsible for the pronounced ramp pattern registered above the canopy (the black curve in the middle volume indicates the corresponding profile of dX(z)/dt ).

Towards larger heights above the canopy, dX(z)/dt is also expected to decline because of decreasing internal transport and the consequently reduced ramp amplitude. The top of the air volume is characterized by a negligible ramp amplitude. Because the volume height is most likely variable and not detectable, it is only possible to use a local spatial scale like z_m and the corresponding dX(z_m)/dt in the surface renewal approach.

Here z_m marks not the top of the whole renewed air parcel but the top of the volume which is considered in the analysis. The weighting factor α is therefore not just accounting for the vertical distribution of dX(z)/dt (for z ≤ z_m) but also for the reduction or enhancement of the average storage change caused by the internal small-scale transport through the top of the considered volume at the measuring height z_m. For forest canopies such a local approach has only been applied by Chen et al. (1997b), throughout a Douglas-fir canopy for sensible heat, resulting in a height constant α of also about 0.5.

The application of Eq. (1) requires the temporal change dX^S(z_m)/dt associated with the ramp slope in between consecutive renewal events. These events (expressed by the step-like change at the end of each single ramp) are separated by a characteristic time τs (Fig. 1 (a)).

Although fluctuations of X arising from volume-internal small-scale transport make no contribution to the depth-averaged temporal change in Eq. (1), the measured temporal change dX(z_m)/dt is influenced by them. It is therefore necessary to separate the

contributions to the time series X associated with the ramp slopes from those associated with small-scale turbulence and instrumental noise. Due to the fact that these contributions occur mainly on different time scales (Paw U et al., 1995), filtering schemes can be employed for this separation. For a filtered scalar time series X^f(t), the time derivative is dX^f(z_m)/dt ≈ dX^S(z_m)/dt. In former studies several methods were used to achieve the separation. A Butterworth band-pass filter was used by Paw U et al. (1995) and in combination with an orthonormal wavelet thresholding scheme by Katul et al. (1996).

Another method to extract the mean ramp characteristics from small-scale turbulence employs structure functions of the scalar time series (e.g., Snyder et al., 1996; Spano et al., 1997a; 2000b). All studies in which structure functions were applied for surface renewal analysis are on the basis of the results of van Atta (1977), except for Chen at al. (1997a;

1997b), who used a modified approach. In the present study a combined detection and filtering scheme on the basis of continuous wavelet transform was applied to extract the ramp pattern in the time series.

2.2 Structure Detection and Filtering