The tidally resolved mass conservation equation of solutes in a Eulerian reference frame is transformed to a new reference frame, which oscillates with the tide so as to maintain constant upstream volume. This transformation is mathematically transparent and removes almost all the tidal harmonics, which are present in the original solution. Tidal harmonics make the direct time integration in a Eulerian frame difficult. In contrast, after coordinate transformation, it is straightforward to derive an approximate time-average solute equation through the application of a time-smoothing operator. Using a simple model, it is demonstrated that the transient solute dynamics in the new, oscillating reference frame is a smooth function of time. As a result, the Reynolds rules for time averaging can readily be applied in the new frame and it is shown that the resulting solute equation, which describes the long-term solute dynamics, is indeed a very good approximation to the instantaneous, tidally resolved, mass conservation equation. The practical implementation of the time-smoothing operator is illustrated in the case of a real estuary, the Western Scheldt (Belgium-Netherlands).
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