1. Introduction
Idealized illustration of control volumes bounded by an isohaline with salinity S and a boundary transect toward the open ocean. For simplicity a stable salinity stratification is assumed, such that the volume
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1
Overview of main quantities in an isohaline framework with salinities larger than the specific isohaline, and comparison to earlier publications. Wang et al. (2017) and Walin (1977) presented the equivalent equations horizontally integrated over a single water column and an ocean basin with vertical transects, respectively. In the integrated equations relation (23) holds and there is no need to distinguish between diahaline and effective vertical diahaline fluxes.
Overview of main integrated quantities in an isohaline framework with salinities smaller or equal to the specific isohaline, and comparison to earlier publications.
The paper starts with the governing equations in Cartesian coordinates (section 2a) and the mathematical formulation for the mapping from vertical coordinates to a general coordinate space of an arbitrary property s in section 2b. The mapping of the local budgets for volume, tracer content and squared tracer content, as well as the definitions for the local diasurface quantities, are presented in the following sections 2c–2e. The relations between these diasurface quantities in tracer coordinates are derived in section 2f. In section 2g we show how the local budgets and relations recover the well-known integral formulations presented in the introduction. The new local relations are demonstrated in section 3. A summary and conclusions are given in section 4. In the appendixes, details can be found about the unified notation (appendix A), the relation between the general mapping and coordinate transformations (appendix B), the mapping of the budget equations (appendix C), integral relations in tracer space (appendix D), and the used Leibniz rule (appendix E).
2. Derivation of a local WMT formulation
a. Governing equations in Cartesian coordinates
b. Mapping to an arbitrary s space
Control volumes within a water column separated by an isosurface of property s. The vertical distribution of the property is given by the function sz(z). (a),(c),(e) Examples for strictly monotonic and nonmonotonic distributions are illustrated. For a prescribed value s the filter functions
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1
c. Local volume budget
Sketch for relation between area elements and diasurface velocities
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1
d. Local tracer budget
e. Local squared tracer budget
f. Mapping to tracer coordinates
In the previous sections we derived the governing equations in a general coordinate space by mapping to an arbitrary property s. In the following we will demonstrate, that for the special choice of tracer coordinates, i.e., s = c with c being the tracer in the tracer equation (12), useful relations between the diasurface (now diatracer) quantities can be derived. These relations in tracer space are the local equivalents of the integrated formulations presented in the introduction.
1) Relation between diatracer diffusive tracer fluxes and mixing
2) Ocean surface boundary fluxes
3) Budget for internal tracer content
4) Relation between diatracer velocity and diffusive tracer fluxes
5) Relation between diatracer velocity and mixing
g. Integrated budgets and relations
Finally, we will show that integration of the local budgets and relations, derived in the previous sections, recovers the integral formulations presented in the introduction.
1) Integrated volume and tracer content budgets
2) Relation between integrated diatracer diffusive tracer flux and mixing
3) Integrated internal tracer content budget
4) Relation between integrated diatracer volume and diffusive tracer fluxes
3. Demonstration
a. Idealized steady state and a word of caution
In a steady-state balance between vertical advection and vertical diffusion the tracer equation (12) reduces to w∂cz/∂z = ∂(Kυ ∂cz/∂z)/∂z. Assuming the resulting horizontally homogeneous tracer distribution is strictly monotonic with ∂cz/∂z > 0, the mixing inside an arbitrary dz interval is given by dM = χdz = 2Kυ(∂cz/∂z)2dz = 2Kυ ∂cz/∂zdc ≡ mdc. Combination of both equations yields w∂cz/∂z = ∂(m/2)/∂z = ∂(m/2)/∂c ∂cz/∂z and thus w = ∂(m/2)/∂c, which resembles the newly derived relation between the diatracer velocity and mixing (47). Indeed, for this special setting
b. Model results for an idealized tidal estuary
To demonstrate the derived relations in salinity space, the diagnostic calculation of diahaline mixing, volume and salt fluxes has been implemented into the coastal ocean model GETM (Burchard and Bolding 2002; Klingbeil et al. 2018). A simulation has been carried out for an idealized tidal estuary, which already represents realistic features of water mass transformation and overturning circulation, while still being a simple benchmark test case. Detailed results from realistic ocean modeling applications are presented in Henell et al. (2023) and Reese et al. (2023).
The estuary has a length of 100 km and a width of 500 m. The water depth decreases linearly from 15 m at the open ocean boundary to 5 m at the river end (see Fig. 4a). The estuary is forced by a harmonic semidiurnal tide with an elevation amplitude of 0.6 m and a constant salinity of 30 g kg−1 at the ocean boundary. At the river end a freshwater discharge of 50 m3 s−1 is prescribed. Horizontal diffusive fluxes are switched off. The estuarine circulation is discussed in detail in Burchard et al. (2019) and Klingbeil et al. (2019), such that we focus on the new diahaline analyses here. Figures 4b and 4c show the diagnosed diahaline diffusive fluxes and mixing between incoming high-saline ocean water and estuarine water of lower salinities. The resulting diahaline exchange flow in Figs. 4d–f indicates the transformation of the incoming high-saline water masses into lower-salinity water all the way up to the reversal point 60 km up-estuary, and the transformation of the outflowing brackish water masses into higher-salinity water again while flowing back toward the open ocean. This overturning circulation in salinity space complements the classical picture of estuarine circulation (e.g., Hansen and Rattray 1965).
Model results for an idealized tidal estuary in a periodic state. (a) Tidally averaged salinity distribution (blue contours) and salinity variance dissipation χ (colored) are shown. (b)–(d) The tidal mean diahaline diffusive salt flux
Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1
4. Summary and conclusions
In this paper we analytically derived the local water mass transformation (WMT) framework for an individual water column. The derivation is based on the mapping of the governing equations from geopotential coordinates to an arbitrary tracer space. Exact local definitions for the effective vertical diasurface velocity, tracer flux and its diffusive contribution are given in (20), (30), and (34a), respectively. Mixing is defined in terms of the dissipation of tracer variance according to (37) and (38). In tracer space new relations between the local diatracer quantities and the mixing per tracer class have been derived as (46), (40), and (47). The key relation (47) between the effective vertical diatracer velocity and the mixing per tracer class directly formulates how the overturning circulation is linked to local tracer variance dissipation. It was demonstrated for different settings. The relations could also be used to study the diapycnal mixing and circulation in bottom boundary layers in the abyssal ocean described by Ferrari et al. (2016), and consistently link it to the large-scale overturning circulation. The local quantities and relations in tracer space represent an intermediate formulation between the direct ones in geopotential coordinates, for example presented by Garrett et al. (1995) and Ferrari et al. (2016), and the ones from the integral WMT framework presented in the introduction. The effective vertical diasurface quantities offer a convenient calculation of integrated diasurface fluxes by simple integration over the horizontal area instead of the complexly shaped area of the isosurface. By horizontally integrating the budget equations in tracer space and the relations between the diatracer quantities we finally recovered the well-known integral WMT formulations. Similar to the extension of the integral formulation by Hieronymus et al. (2014), an extension of our local formulation to multidimensional tracer space is possible. A summary of local and integral quantities and equations presented in this paper, as well as a comparison with published formulations, is given in Tables 1 and 2. The local diatracer fluxes and the local diatracer mixing provide new and valuable information on the spatial distribution of these quantities, and will enable new insights for coastal and large-scale oceanography.
In this paper all diasurface fluxes measured as “per unit horizontal area” are denoted by subscript “dia,z”.
Here,
Acknowledgments.
This paper is a contribution to the project M5 of the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean” (project 274762653), funded by the German Research Foundation (DFG). We are grateful to Hans Burchard (IOW, Germany) for inspiring discussions and valuable comments. We also acknowledge the comments from two anonymous reviewers.
Data availability statement.
Model data shown in Fig. 4 have been obtained from a simulation with the coastal ocean model GETM (www.getm.eu). The model setup, simulation data, and scripts are archived at https://doi.org/10.5281/zenodo.8136528 (Klingbeil 2023a). The used model source code is archived at https://doi.org/10.5281/zenodo.8137091 (Klingbeil 2023b).
APPENDIX A
Unified Notation
APPENDIX B
Mapping as a Generalized Coordinate Transformation
APPENDIX C
Derivation of the p-Generalized Tracer Budget in s Space
APPENDIX D
Further Relations in Tracer Space
APPENDIX E
Leibniz Rule
In the rest of the manuscript we replace sz(η) and sz(−H) in the arguments of the Heaviside functions in (E1) and (E2) by sη and s−H.
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