For more than a century, estuarine exchange flow has been quantified by means
of the Knudsen relations which connect bulk quantities such as inflow and
outflow volume fluxes and salinities. These relations are closely linked to
estuarine mixing. The recently developed Total Exchange Flow (TEF) analysis framework, which uses
salinity coordinates to calculate these bulk quantities, allows an exact
formulation of the Knudsen relations in realistic cases. There are however
numerical issues, since the original method does not converge to the TEF bulk
values for an increasing number of salinity classes. In the present study,
this problem is investigated and the method of dividing salinities,
described by

The Total Exchange Flow (TEF) analysis framework calculates time-averaged net volume and mass transport between enclosed volumes of the ocean and ambient water masses, sorted by salinity classes. Since oscillatory inflow and outflow components occurring at the same salinity compensate for one another, TEF characterises the net exchange flow with the ambient ocean. Salinity rather than density or temperature is used as a coordinate for calculating estuarine exchange flow, since only the salt budget is entirely controlled by the exchange flow. Therefore, salt is the only conserved quantity. In contrast, temperature and thus density are additionally affected by the freshwater runoff and the surface heat fluxes.

A first bulk approach based on inflow and outflow salinity and volume
transport had been developed and applied to the exchange flow of the Baltic
Sea by

The TEF analysis framework considers a time-averaged transport of a tracer

Recently,

Since the TEF analysis framework is continuous in salinity, a discretisation
in salinity space is required when analysing data from numerical model
simulations or field observations. In their Appendix A2,

In order to obtain robust bulk values, which are less sensitive to the number
of salinity bins,

Using the maximum of

To demonstrate the different convergence behaviours of the sign method and
the dividing salinity method, we take the analytical example from

We created a time series of

Oscillating exchange flow (see Sect.

To study the convergence of the two different methods (the sign method and
dividing salinity method), one can compare the errors in discrete form to the
analytical values. Figure

Oscillating exchange flow (see Sect.

Oscillating exchange flow (see Sect.

Encouraged by the good convergence behaviour of the dividing salinity method
demonstrated in the previous section, we introduce here a general formulation
which includes inverse estuaries and exchange flows with more than two
layers. The general idea is to identify the salinities which divide

The mixing relations of

Sketch of hypothetical TEF profiles of a four-layered
system with alternating inflows and outflows,

The output from a numerical model along a transect across an estuary is
assumed to consist of

Sketch of how

The Baltic Sea, shown in Fig.

Episodic inflow events of water consisting of a mixture of saline North Sea
water and recirculated brackish Baltic Sea water

Above the halocline, driven by wind, inflows and Earth rotation, a cyclonic
circulation is generally present in the central Baltic Sea, with net
northward flow in the east of Gotland and southward flow in the west of
Gotland

In the following, the numerical properties of the TEF analysis framework are
tested against two transects of the Baltic Sea. The first transect is located
across Darss Sill (D, red transect) in the western Baltic Sea over which
part of the exchange with the North Sea is occurring; see
Sect.

Map and bathymetry of the Baltic Sea. K: Kattegat, D: Darss Sill and the Darss Sill transect (red), G: the Gotland island, GB: Gotland Basin and the Gotland transect (green), BoB: Bothnian Bay.

In their recent review paper,

The horizontal resolution of the model is about

Application of the TEF analysis framework for

The values found in this study with the dividing salinity method confirm that the
found bulk values in

Similar to the dependency of the TEF bulk values on

Exchange flow over Darss Sill: profiles of

Exchange flow over Darss Sill: comparison of

Cross section through Gotland Basin: profiles of

In this section, the capability of the extended dividing salinity method to
be applied to exchange flows or transects with more than two layers is
demonstrated. Here, example results are shown for model data of the Gotland
Basin in the Baltic Sea. The analysed transect uses the model run from

Figure

The net southward transport of

This study investigated the numerical issues of the TEF
analysis framework, proposed by

Based on our results, we propose a best-practice procedure for calculating TEF
from a numerical model:

At the level of setting up a numerical model, the spatial (horizontal and vertical) resolution should be chosen as high as possible to reproduce return flows due to lateral eddies and smaller overturns.

Once a transect for the TEF analysis has been identified, the frequency for storing the output along that transect has to be chosen. For analytical correctness, the binning of data of volume and salt fluxes into salinity classes should be done online within the hydrodynamic model at every model time step. Time-averaged model output of these binned data can directly be used for the TEF analysis. If the model only provides output within the model layers, the binning and averaging must be done offline during postprocessing. This would induce different kinds of errors: (i) instantaneous data snapshots which skip intermediate model time steps do not conserve fluxes and do not consider intermediate salinity variations; (ii) model data obtained by thickness-weighted averaging over model time steps conserve fluxes but merge data of different salinities. Both types of errors can be reduced with a sufficiently high output frequency, such that the output data still resolve the dynamics of the flow.

If the binning is not done online, required output fields are the velocity
component normal to the transect, the salinity and the grid box area along
the transect. We suggest that these variables are stored as
thickness-weighted averaged values

The results should be analysed for a large range of salinity classes

Visualisation of the exchange flow should still be done with a smooth

Please request the authors if you are interested in the code used for this publication.

For the oscillating exchange flow given in (

The algorithm finding the extrema of

Figure

Comparison of the algorithm for

The authors declare that they have no conflict of interest.

This paper is a contribution to BMBF-GROCE FKZ 03F0778. Hans Burchard and Marvin Lorenz were supported by Research Training Group Baltic TRANSCOAST GRK 2000 funded by the German Research Foundation. Knut Klingbeil was supported by the Collaborative Research Centre TRR 181 on Energy Transfer in Atmosphere and Ocean funded by the German Research Foundation (project no. 274762653), and Parker MacCready was supported by US National Science Foundation (grant no. OCE-1736242).

The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Eric J. M. Delhez and reviewed by two anonymous referees.