1. Introduction
Tidal averaging is commonly used to unravel the underlying residual dynamics in coastal systems dominated by oscillating tidal flows. Residual averages can be obtained in different coordinate systems with the corresponding spatial coordinates held fixed during the averaging process. The two most intuitive cases are Eulerian and Lagrangian averaging. The Eulerian average describes the mean state at a fixed position, or equivalently the mean property of all fluid parcels when passing this single position. In contrast, the Lagrangian average describes the mean property of an individual parcel (defined by fixed material coordinates) along its trajectory and can be referenced to the mean position of this parcel [see Generalized Lagrangian Mean theory of Andrews and McIntyre (1978)]. All other types of averaging are a mixture of these two cases. This mixture can be prone to misinterpretation, because the averages are neither associated to a single fixed position nor to a single fluid parcel.
To reduce complexity, the present study considers averages obtained at a fixed horizontal position with coordinates 
Furthermore, in the literature, the presentation of residual quantities can be confusing because sometimes the nomenclature is inaccurate (e.g., labeling of residuals as being Eulerian or Lagrangian) or inconsistent (e.g., the use of the term transport velocity), definitions are not precise, or the data processing is unclear (e.g., the coordinate system). But this clarity and accuracy is required for a correct interpretation of the results.
The aim of this paper is to reduce confusion and complexity by presenting the concept of thickness-weighted averaging (TWA) as a unified approach to calculate conservative tidally averaged quantities in any vertical coordinate system. TWA was originally developed for the parameterization of eddy-induced transports in isopycnal coordinates (e.g., de Szoeke and Bennet 1993; McDougall and McIntosh 2001; Greatbatch and McDougall 2003; Young 2012, and references therein), and later was also used to derive the governing equations describing the 3D interaction of currents and surface gravity waves (Aiki and Greatbatch 2012, 2013, 2014). In this paper the concept of TWA is extended to a generalized vertical coordinate and to depth-averaged quantities. The coordinate transformation is explained in section 2a. In section 2b the commutativity of TWA and depth-averaging is discussed as a necessary condition for conservative residuals. Section 2c extends the original TWA concept to nonmonotonic coordinate functions s(z) and shows the relation to the so-called Total Exchange Flow (TEF) analysis of MacCready (2011). Section 3 demonstrates the dependence of vertical profiles of resulting residual quantities on the averaging operator and on the specific vertical coordinate, which is held fixed during the averaging process. Results are shown for one-dimensional analytical tidal flow, for field observations from a tidal channel, and for a numerical model of a two-dimensional estuary. A discussion and final conclusions are given in section 4 and section 5, respectively. Appendix A outlines the application of the TWA operator to the governing 3D equations. The procedure of TWA for discrete data is explained in appendix B. Appendix C provides details about the analytical solution for one-dimensional tidal flow with sediment transport.
2. Thickness-weighted averaging
a. Vertical coordinate transformation
The TWA concept was originally developed for density coordinates in a stably stratified ocean. It is straightforward to extend the original concept to a generalized vertical coordinate 
b. Conservative averaging
Overview of averaging operators. The arbitrary quantity ψ is masked outside the water column. For geopotential and σ coordinates the Jacobians are given by 
Related to the properties discussed above, the TWA defined in (9) shows further useful aspects:
By definition the TWA takes into account the correlation with the Jacobian, therefore no additional covariance terms appear in the low-pass-filtered governing equations in generalized vertical coordinates [see (A9)].
For σ coordinates the Jacobian is given by
With
Commutativity of averaging operators in different vertical coordinate systems. The operators are defined in Table 1. The first row shows the general operator relations in an arbitrary vertical coordinate (i.e., the generalized vertical coordinate s), whereas the second and third rows show the special cases of σ and z coordinates, respectively. Most important findings are the commutativity of the TWA and depth-averaging operators, 
c. The generalized TWA concept
3. Examples
a. Analytical tidal estuary flow
Velocity profiles computed from the analytical solution for tidal flow with three different tidal elevation amplitudes (section 3a). (a)–(c) Instantaneous velocity profiles u during one tidal cycle. (d)–(f) Residual velocity profiles: 
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Velocity profiles computed from the analytical solution for tidal flow with three different tidal elevation amplitudes (section 3a). (a)–(c) Instantaneous velocity profiles u during one tidal cycle. (d)–(f) Residual velocity profiles:
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Velocity profiles computed from the analytical solution for tidal flow with three different tidal elevation amplitudes (section 3a). (a)–(c) Instantaneous velocity profiles u during one tidal cycle. (d)–(f) Residual velocity profiles:
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
For all values of 
Analytical SPM profiles for a settling velocity of 
As in Fig. 1, but for SPM concentration c as presented by Eq. (C4b). For 
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 1, but for SPM concentration c as presented by Eq. (C4b). For
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 1, but for SPM concentration c as presented by Eq. (C4b). For
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
The resulting tidally resolved SPM fluxes 
As in Fig. 1, but for SPM flux uc. For 
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 1, but for SPM flux uc. For
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 1, but for SPM flux uc. For
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
b. Field data from a tidal channel in the Wadden Sea
In this section in situ observations of along-channel current velocity and SPM concentration in a tidal channel, as presented by Becherer et al. (2016), are analyzed to compare residual quantities in σ coordinates and geopotential coordinates. The data have been observed during 2 days in May 2011 in the southern Wadden Sea of the German Bight, southeastern North Sea. During that period, the landward water was about 4 K warmer than the seaward water, with only a small salinity difference, such that a classical estuarine density gradient was established. During the campaign, the research vessel was anchored in a tidal channel of about 13 m depth. Velocity data are from a vessel-mounted acoustic Doppler current profiler (ADCP) and SPM data are from a turbidity sensor mounted in a free-falling microstructure profiler operated from the ship. Turbidity data were calibrated against water samples to obtain SPM concentrations. The tidal range was about 2 m during the observation period and due to the swing of the vessel at the anchor with the semidiurnal tidal current, the vertical position of the bottom was variable in time as well. Further details of the campaign and the data analysis are given in Becherer et al. (2015) and Becherer et al. (2016). All data were vertically interpolated to a grid of 20 equidistant σ layers and short data gaps within each layer were filled by temporal along-layer interpolation. To test the averaging procedures for water columns with only partial data coverage, missing data in the vicinity of the bottom and free surface were not filled by extrapolation (see, e.g., Giddings et al. 2014), but by zero values. Consequently, the velocity in the lowermost σ layer above the bottom and the four uppermost σ layers below the surface was set to zero. The same holds for the SPM concentration in the uppermost σ layer below the surface. Vertical interpolation to 170 equidistant geopotential layers provided the instantaneous profiles for the Eulerian residuals. All residual quantities are obtained as running means over an averaging period of two M2 cycles. The available measurements do not offer a longer averaging window, usually applied to analyze subtidal residuals.
The along-channel velocity data are presented in Fig. 4. The tidally resolved velocity (Fig. 4a) shows a tidal wave with slack after flood occurring shortly after high water. This includes a long ebb-to-flood transition during low water and a short flood-to-ebb transition during high water. Maximum ebb velocities are higher than maximum flood velocities, partly due to the curvature of the tidal channel leading to a weak ebb dominance at the anchor position (see Becherer et al. 2015). The TWA velocity in σ coordinates 
Along-channel velocity in a tidal channel (section 3b). (a) Observed velocity u. (b),(c) Residual velocities 
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Along-channel velocity in a tidal channel (section 3b). (a) Observed velocity u. (b),(c) Residual velocities
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Along-channel velocity in a tidal channel (section 3b). (a) Observed velocity u. (b),(c) Residual velocities
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Sediment concentration (see Fig. 5) is elevated during maximum ebb and flood, with slightly higher ebb values. A local concentration maximum is visible near the surface mainly during ebb and low tide, caused by SPM transport from the adjacent tidal flats (Becherer et al. 2016). The TWA concentration profile in σ coordinates 
As in Fig. 4, but for SPM concentration c.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 4, but for SPM concentration c.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 4, but for SPM concentration c.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
SPM flux profiles are presented in Fig. 6. The TWA flux in σ coordinates 
As in Fig. 4, but for SPM flux uc.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 4, but for SPM flux uc.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
As in Fig. 4, but for SPM flux uc.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
The water column is only very weakly stratified during the measurement campaign. Similar to the Eulerian sediment concentration, also 
c. Numerical model of a two-dimensional tidal estuary
Warner et al. (2005) set up an idealized two-dimensional tidal estuary for numerical model studies. Adopting the slight modifications from Burchard et al. (2019), the present application simulates an estuary of 100 km length, with a water depth that decreases linearly from 15 m at the open ocean boundary to 5 m at the river end. A sinusoidal semidiurnal tide with an elevation amplitude of 0.6 m is prescribed at the ocean side. The vertically integrated freshwater runoff from the river is set to 0.1 m2 s−1 (equivalent to 50 m3 s−1 for an estuary of 500 m width). All water up to 30 km from the ocean boundary is initialized with a salinity of 30 g kg−1. The last 20 km closest to the river end are initially covered by freshwater. A linear down-estuary salinity gradient is prescribed over the intermediate portion of the estuary. At the open ocean boundary the initial salinity is held fixed during the course of the simulation. The simulation (for details see Burchard et al. 2019) is carried out with the coastal ocean model GETM (Burchard and Bolding 2002; Hofmeister et al. 2010; Klingbeil and Burchard 2013). After a spinup of 100 tidal cycles, an almost periodic state is reached and residual quantities are calculated over the next subsequent 10 tidal cycles. Residuals are obtained in the 40 equidistant σ layers of the model, 134 equidistant z bins (Δz = 0.1 m) and 70 equidistant salinity bins 
Residual quantities at 22.5 km up-estuary from the open-ocean boundary in a two-dimensional tidal estuary (section 3c). (a)–(c) Residual velocity, salinity, and salinity flux, respectively, in z (black) and σ coordinates (red). (d)–(f) Residual velocity, differential thickness, and salinity flux, respectively in salinity space.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Residual quantities at 22.5 km up-estuary from the open-ocean boundary in a two-dimensional tidal estuary (section 3c). (a)–(c) Residual velocity, salinity, and salinity flux, respectively, in z (black) and σ coordinates (red). (d)–(f) Residual velocity, differential thickness, and salinity flux, respectively in salinity space.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
Residual quantities at 22.5 km up-estuary from the open-ocean boundary in a two-dimensional tidal estuary (section 3c). (a)–(c) Residual velocity, salinity, and salinity flux, respectively, in z (black) and σ coordinates (red). (d)–(f) Residual velocity, differential thickness, and salinity flux, respectively in salinity space.
Citation: Journal of Physical Oceanography 49, 7; 10.1175/JPO-D-18-0083.1
The TWA velocity in σ coordinates 
The residual salinity profiles 
The residual salinity fluxes 
Figure 7e shows the residual differential thickness 
d. Analysis of results
For all examples, the profiles of the TWA velocity in σ coordinates 
Although caused by the same mechanism during averaging, it is important to classify the described peaks in 
These artifacts in the upper and lowermost coordinate levels are also present in 
4. Discussion
It was demonstrated that TWA in σ coordinates provides a simple (no masking of data outside the water column is necessary) and conservative framework for calculating residual quantities in tidal estuaries (also see Giddings et al. 2014). The averaging can be applied to a time-varying water depth and is robust also for large tidal ranges (Kjerfve 1975). Averaging in geopotential coordinates is also conservative and the interpretation of the resulting Eulerian residuals is very intuitive (see below). TWA in isohaline coordinates conservatively averages over water masses with identical salinities and offers the analysis of residual dynamics with respect to salinity classes (MacCready 2011).
The vertical profiles of residual quantities obtained in different coordinate systems show different vertical distributions of the residual quantity. These different views, extracted from the same original dataset, provide complementary information about the residual dynamics and are all equally valid. Therefore, the classical results for estuarine circulation [as proposed by Hansen and Rattray (1965)] reproduced by TWA in σ coordinates are as correct as the sometimes inverse-like estuarine circulation obtained from Eulerian averaging.
The difficulty is in the correct interpretation of these different results. Unweighted tidal averages represent the residual that a fictitious observer would measure by vertically following a specific coordinate line. The relevance of these residuals can be debated. As a special case the Eulerian average simply represents the residual that would be obtained from data measured by a sensor that is mounted at a fixed geopotential position. The interpretation is less intuitive for other coordinate systems, because the resulting residuals are obtained following a varying position and can be referenced only to a mean position (e.g., for graphical presentation). The focus on the mean position can be prone to misinterpretation, as it distracts from the correct interpretation in the transformed space of the applied coordinate system (e.g., the correct interpretation of residuals in isohaline coordinates with respect to salinity classes). Second, the mean position can be misleading, especially in the case of nonmonotonic coordinate functions 
Another word of caution is necessary in the context of the commonly expected similarity of residual quantities for converging coordinate systems. For example, in the ultimate limit of a vanishing tidal range to depth ratio, that is, 
It becomes clear that residual quantities are only meaningful as long as they are linked to the information on how they were obtained. Since a unique nomenclature (considering, e.g., the various options of vertical coordinate systems) is rather impractical, the present paper focused on a clear and consistent operator notation with a superscript indicating the vertical coordinate, which was held fixed during the averaging process. In all coordinate systems care has to be taken to exclude coordinate levels with missing data inside the water column from the analysis of residual dynamics.
5. Conclusions
This paper links and unifies concepts from different disciplines of oceanography. Thickness-weighted averaging (TWA), originally developed for ocean eddy closures in isopycnal coordinates, is presented in a generalized vertical coordinate and applied in the context of tidal averaging. In contrast to an unweighted temporal low-pass filter, TWA is a conservative operator by definition. Furthermore, TWA commutates with depth-averaging. This commutativity can be exploited in analytical studies and also renders depth-averaged TWA quantities to be invariant to the vertical coordinate system. The choice of a vertical coordinate system depends on the specific purpose of investigation. The original instantaneous data can be remapped between different coordinate systems. Vertical profiles of residual quantities depend on the vertical coordinate that is held fixed during the averaging process. In particular, there is no unique vertical distribution of the Eulerian residual transport 
Acknowledgments
This paper is a contribution to the project M5 (Reducing spurious diapycnal mixing in ocean models) of the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean” (project 274762653) and to the Research Training Group Baltic TRANSCOAST GRK 2000, both funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). The work of Elisabeth Schulz was supported by the project MOREWACC (Morphodynamic response of the Wadden Sea to climate change) funded by the DFG as BU 1199/21-1 under the umbrella of the Priority Program SPP 1889 on Regional Sea Level Change and Society. We are grateful for the comments of two anonymous reviewers.
APPENDIX A
Governing Equations
APPENDIX B
TWA with Discrete Data
If the averaging is carried out directly in the provided vertical intervals each interval represents one bin with 
APPENDIX C
Analytical Steady-State Solution
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A relation between 
The assumption of 
