A Rigorous Derivation of the Water Mass Transformation Framework, the Relation between Mixing and Diasurface Exchange Flow, and Links to Recent Theories in Estuarine Research

Knut Klingbeil; Erika Henell

doi:10.1175/JPO-D-23-0130.1

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Fig. 1.

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 $V_{>} (S)$ containing water with salinities larger than S is located below the isohaline, and the remaining volume $V_{\leq} (S)$ containing water with salinities less or equal to S above the isohaline. The integrated diahaline volume flux Q_dia is a measure for the water mass transformation. Integrated surface boundary and open boundary volume fluxes are depicted as Q_η_,≤, Q_≤, and Q_>, defined based on how the salinity of the transported water relates to S. Integrated salt fluxes are denoted by $F$ and alter the salt contents $S_{\leq} (S)$ and $S_{>} (S)$ inside the corresponding volumes. Surface boundary fluxes Q_η_,> and $F_{η, >}$ , transporting or exchanging water with salinities larger than S, are not sketched for simplicity. Additional freshwater river discharge is depicted as Q_r. The arrows indicate the direction of positive fluxes. Walin (1977) presented the budget equations for $V_{>} (S)$ and $S_{>} (S)$ ; see (1a) and (1b). In many studies on estuaries the budgets for $V_{\leq} (S)$ and $S_{\leq} (S)$ , (5a) and (5b) are used.

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Fig. 2.

Control volumes within a water column separated by an isosurface of property s. The vertical distribution of the property is given by the function s^z(z). (a),(c),(e) Examples for strictly monotonic and nonmonotonic distributions are illustrated. For a prescribed value s the filter functions $H_{\leq}$ and $H_{>}$ , defined in (14a) and (14b), identify the parts of the water column with s^z(z) ≤ s or s^z(z) > s, respectively. (b),(d),(f) Examples for the location of the isosurface in xz space in the vicinity of the water column are sketched. The local volume budget (18) describes the relation between the cumulative thickness D_≤ and horizontal transport U_≤, as well as the diasurface velocities $u_{dia}^{z}$ for the two control volumes.

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Fig. 3.

Sketch for relation between area elements and diasurface velocities $(u_{dia}^{z}, u_{dia, z}^{z})$ . The infinitesimal area element dA is located on the isosurface with the local unit normal vector n. Its horizontal projection is given by dA_z, with z being the vertical unit vector. By construction $u_{dia}^{z} d A = u_{dia, z}^{z} d A_{z}$ holds; see (23). The same relations are valid for the total and diffusive tracer fluxes $(f_{dia}^{z}, f_{dia, z}^{z})$ and $(j_{dia}^{z}, j_{dia, z}^{z})$ , respectively.

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Fig. 4.

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 $j_{dia, z}^{S}$ , mixing per salinity class m, and diahaline velocity $u_{dia, z}^{S}$ have been diagnosed during model runtime, independently from (26), (38), and (18), respectively. In (b) and (d), the diahaline fluxes from higher to lower salinity are negative. (e),(f) The quantities are calculated from (b) and (c), respectively. The almost perfect agreement between the diahaline quantities in (b) and (c), as well as in (d)–(f) demonstrates the analytically derived relations (40), (46), and (47).

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A Rigorous Derivation of the Water Mass Transformation Framework, the Relation between Mixing and Diasurface Exchange Flow, and Links to Recent Theories in Estuarine Research

Knut Klingbeil

Knut Klingbeil ^aLeibniz Institute for Baltic Sea Research Warnemünde (IOW), Rostock, Germany

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https://orcid.org/0000-0002-6736-1260

and

Erika Henell

Erika Henell ^aLeibniz Institute for Baltic Sea Research Warnemünde (IOW), Rostock, Germany

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Online Publication:: 14 Dec 2023

Print Publication:: 01 Dec 2023

DOI:: https://doi.org/10.1175/JPO-D-23-0130.1

Page(s):: 2953–2968

Received:: 11 Jul 2023

Final Form:: 08 Sep 2023

Accepted:: 18 Sep 2023

Published Online:: 14 Dec 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

In this paper we present the analytical derivation of a local water mass transformation (WMT) framework for an individual water column. We exactly formulate the mapping of the governing equations from geopotential coordinates to an arbitrary tracer space. Unique definitions for the local effective vertical diasurface fluxes are given. In tracer space we derive new relations between the local diatracer fluxes and the mixing per tracer class. The key relation 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. Horizontal integration of the governing equations in tracer space and the relations between the diatracer quantities finally recovers the well-known integral WMT formulations.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Knut Klingbeil, knut.klingbeil@io-warnemuende.de

Abstract

Corresponding author: Knut Klingbeil, knut.klingbeil@io-warnemuende.de

Keywords: Diapycnal mixing; Thermohaline circulation; Water masses/storage

1. Introduction

In his seminal work, Walin (1977) presented integral budget equations for the volume

V_{>} (S)

and salt content

S_{>} (S)

of all water within a semienclosed basin with salinities larger than a prescribed salinity S,

\frac{\partial V_{>} (S)}{\partial t} - Q_{>} (S) - Q_{dia} (S) = 0,

(1a)

\frac{\partial S_{>} (S)}{\partial t} - F_{>} (S) - F_{dia} (S) = 0.

(1b)

The volume

V_{>} (S)

and its salt content

S_{>} (S)

are illustrated in Fig. 1. The reference to the original equations in Walin (1977), and a comparison to the original notation, is given in Table 1. The integrated volume and salt fluxes entering across the open ocean boundary with salinities larger than S are given by Q_>(S) and

F_{>} (S)

. Across the isohaline the integrated diahaline fluxes Q_dia(S) and

F_{dia} (S)

are entering, with the diahaline salt flux consisting of advective (SQ_dia) and diffusive contributions (

J_{dia}

F_{dia} (S) = S Q_{dia} (S) + J_{dia} (S) .

(2)

Walin (1977) showed that for no diffusive fluxes across the open boundary, i.e.,

F_{>} (S) = \int_{S}^{\infty} S^{'} q (S^{'}) d S^{'}

with q(S) = −∂Q_>/∂S being the volume flux per salinity class, the integrated diahaline volume flux is determined by the integrated diahaline diffusive salt flux

Q_{dia} (S) = - \frac{\partial J_{dia} (S)}{\partial S} .

(3)

In analogy, Walin (1982) presented budgets for volumes

V_{\leq} (T)

of all water with temperatures smaller or equal to T, and additionally considered integrated volume and temperature fluxes across the ocean surface, Q_η_,≤(T) and

F_{η, \leq} (T)

, respectively. For

F_{η, \leq} (T) = \int_{- \infty}^{T} T^{'} q_{η} (T^{'}) d T^{'} + J_{η, \leq} (T)

, with q_η(T) = ∂Q_η_,≤/∂T being the volume flux per temperature class, and

J_{η, \leq} (T)

being the integrated nonadvective temperature flux across the ocean surface, relation (3) was extended to

Q_{dia} (T) = - \frac{\partial J_{dia} (T)}{\partial T} + \frac{\partial J_{η, \leq} (T)}{\partial T} .

(4)

Walin’s concept of analyzing budgets of water masses defined by a distinct property, e.g., salinity or temperature, is the foundation of the water mass transformation (WMT) framework. The transformation of water masses due to interior and boundary processes leads to mass fluxes (volume fluxes under the Boussinesq approximation) between water masses, i.e., across the isosurface of the defining property, and is measured by Q_dia, see (3) and (4). A review of the integral WMT framework, its derivation from differential volumes in tracer space and references for the numerous applications can be found in Groeskamp et al. (2019).

Fig. 1.

Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1

Table 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.

With the focus on integrated diahaline fluxes in classical estuaries, MacCready et al. (2002) presented the budgets for

V_{\leq} (S)

and

S_{\leq} (S)

, neglecting surface boundary fluxes but including explicit freshwater river discharge Q_r,

\frac{\partial V_{\leq} (S)}{\partial t} - Q_{\leq} (S) + Q_{dia} (S) = Q_{r},

(5a)

\frac{\partial S_{\leq} (S)}{\partial t} - F_{\leq} (S) + F_{dia} (S) = 0.

(5b)

The budgets (5a) and (5b) offer to study the water masses inside a classical estuary. The special case of an estuarine volume completely bounded by an isohaline, i.e., without open boundary fluxes (

Q_{\leq}, F_{\leq}

), was considered by MacCready and Geyer (2001) and Hetland (2005). Contrary, if the estuarine volume is completely bounded by the open boundary, see, e.g., MacCready (2011) or Burchard et al. (2019), the diahaline fluxes (

Q_{dia}, F_{dia}

) drop out. For this case, MacCready (2011) developed the concept of total exchange flow (TEF): based on the volume flux per salinity class q(S) = ∂Q_≤/∂S = −∂Q_>/∂S, the time-averaged open boundary fluxes can be decomposed into net in- and outflow contributions, and associated salinities for the in- and outflowing water masses can be calculated, finally recovering the Knudsen relations (Knudsen 1900; Burchard et al. 2018).

Burchard (2020) supplemented (5a) and (5b) by a budget equation for the content of squared salinity. Its sink term

M_{\leq} (S) = \int_{V_{\leq} (S)} χ d V

, being the volume-integrated dissipation of salinity variance χ (Burchard and Rennau 2008), is a measure for mixing inside

V_{\leq} (S)

. With

𝕞 (S) = \frac{\partial M_{\leq}}{\partial S} (integrated mixing per salinity class),

(6)

Burchard (2020) derived the linear relation

⟨ 𝕞 (S) ⟩ = 2 S ⟨ Q_{r} ⟩

(7)

as the universal law of estuarine mixing for estuarine volumes completely bounded by the isohaline with salinity S, and assuming sufficiently long time averaging (denoted by angle brackets). Burchard et al. [2021, their (19)] showed the general relation between the integrated diahaline diffusive salt flux and the integrated mixing per salinity class

J_{dia} (S) = - \frac{1}{2} 𝕞 (S) .

(8)

Although the integrated diahaline fluxes and mixing (Q_dia,

J_{dia}

, and

𝕞

) provide valuable information about interior processes, they do not provide information on how these quantities are distributed on the isohaline within the basin. Separating the domain into individual water columns facilitates a detailed view of the horizontal distribution. Volume integrated WMT budget equations for individual water columns have been presented in, e.g., Wang et al. (2017) and Holmes et al. (2021). However, in those works no analytical derivation and definition of the local diahaline quantities was done. Under the assumption of a stable salinity stratification, Li et al. (2022) derived budget equations for the thickness D_>(S) of all water with salinities larger than S as the distance of the isohaline from the bottom, and the corresponding salt content S_>(S). They also introduced the local diahaline volume flux per unit horizontal area u_dia,_z and the local mixing per salinity class m(S). Differences in notations are again compared in Tables 1 and 2. However, because of the conceptual limitation due to the assumed stable salinity stratification, their formulated relations are not generally valid.

Table 2.

Overview of main integrated quantities in an isohaline framework with salinities smaller or equal to the specific isohaline, and comparison to earlier publications.

In this manuscript we present the analytical derivation of a local WMT framework and generally valid mathematical definitions for the corresponding local quantities. In contrast to Groeskamp et al. (2019), we start our derivations in classical geopotential coordinates and exactly formulate the mapping to tracer space. Our new unified presentation easily links the various existing integrated formulations. One major new result of our derivations is an analytical equation that shows how the overturning circulation is linked to local small-scale tracer variance dissipation. Until now, this can only be anticipated from the integrated formulation

Q_{dia} (S) = \frac{1}{2} \frac{\partial 𝕞 (S)}{\partial S},

(9)

that we obtain by combining (3) and (8).

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

Under the Boussinesq approximation, mass conservation is turned into volume conservation which is locally described by the incompressibility constraint

\frac{\partial u}{\partial x} + \frac{\partial υ}{\partial y} + \frac{\partial w}{\partial z} = 0.

(10)

In (10) the flux divergence is written in Cartesian coordinates x, y, z, and the velocity components in the corresponding directions are given by u, υ, and w, respectively. Kinematic boundary conditions at the ocean surface and the bottom are given by

\frac{\partial η}{\partial t} + u (η) \frac{\partial η}{\partial x} + υ (η) \frac{\partial η}{\partial y} - w (η) = u_{η, z},

(11a)

u (- H) \frac{\partial H}{\partial x} + υ (- H) \frac{\partial H}{\partial y} + w (- H) = 0,

(11b)

with u_η_,_z being the volume flux across the ocean surface per unit horizontal area (positive into the water column), e.g., due to precipitation, evaporation, ice melting, or freezing.

The local budget equation for an arbitrary tracer with concentration c^z(x, y, z, t) reads

\frac{\partial c^{z}}{\partial t} + \frac{\partial}{\partial x} \underset{= f_{x}}{\underset{︸}{(u c^{z} \overset{= j_{x}}{\overset{︷}{- K_{h} \frac{\partial c^{z}}{\partial x}}})}} + \frac{\partial}{\partial y} \underset{= f_{y}}{\underset{︸}{(υ c^{z} \overset{= j_{y}}{\overset{︷}{- K_{h} \frac{\partial c^{z}}{\partial y}}})}} + \frac{\partial}{\partial z} \underset{= f_{z}}{\underset{︸}{(w c^{z} \overset{= j_{z}}{\overset{︷}{- K_{υ} \frac{\partial c^{z}}{\partial z}}})}} = π .

(12)

The notation c^z is used to indicate the functional dependence on z coordinates. The tracer flux in the x direction is given by f_x and consists of advective (uc^z) and diffusive (j_x) contributions. The fluxes into the other directions are defined accordingly. Diffusive fluxes are given in terms of horizontal and vertical diffusivities K_h and K_υ, respectively, but can also be extended to consider more advanced diffusivity tensors. The source term π on the right-hand side of (12) accounts for local tracer production, e.g., due to shortwave radiation for temperature. Flux boundary conditions at the ocean surface and bottom are given by

c^{z} (η) \frac{\partial η}{\partial t} + f_{x} (η) \frac{\partial η}{\partial x} + f_{y} (η) \frac{\partial η}{\partial y} - f_{z} (η) = f_{η, z},

(13a)

f_{x} (- H) \frac{\partial H}{\partial x} + f_{y} (- H) \frac{\partial H}{\partial y} + f_{z} (- H) = 0,

(13b)

with f_η_,_z being the surface tracer flux per unit horizontal area (positive into the water column). Bottom tracer fluxes, e.g., geothermal heating, are neglected but can easily be added to the equations.

b. Mapping to an arbitrary s space

In this work we follow Walin’s construction of control volumes bounded by a distinct water mass property. But, in contrast to Walin, we will apply the construction to a water column. Let the function s^z(z) describe the vertical distribution of an arbitrary property (e.g., temperature, salinity or density) within the water column between the bottom at z = −H(x, y) and the ocean surface at z = η(x, y, t). For any prescribed property value s we need to identify the parts of the water column, i.e., the vertical positions z, where water masses with s^z(z) ≤ s or s^z(z) > s are located. This is straight forward for strictly monotonic functions s^z(z), e.g., for stably stratified temperature or salinity distributions as sketched in Figs. 2a and 2c. Only in this monotonic case there exists a unique inverse function z^s(s), which directly provides the vertical position of the prescribed s isosurface that separates the two control “volumes” in the water column. This inverse function does not exist for nonmonotonic functions s^z(z), e.g., for tracer distributions with inversions as sketched in Fig. 2e. In this nonmonotonic case the isosurface exists at multiple positions within the water column and the control volumes consist of noncontiguous parts. Following Wolfe (2014) and Klingbeil et al. (2019), a general mathematical formulation that correctly considers also these noncontiguous parts, is possible by means of the Heaviside step function

H {x}

and the filter functions

\begin{array}{l} H_{\leq} {x} = 1 - H {x} = {\begin{array}{l} 1, & for x \leq 0, \\ 0, & for x > 0, \end{array} \end{array}

(14a)

\begin{array}{l} H_{>} {x} = H {x} = {\begin{array}{l} 0, & for x \leq 0, \\ 1, & for x > 0. \end{array} \end{array}

(14b)

With these filter functions the cumulative thickness of the control volumes, i.e., all parts of the water column where s^z(z) ≤ s or s^z(z) > s, respectively, can be calculated as

D_{\leq} (s) = \int_{- H}^{η} H_{\leq} {s^{z} (z) - s} d z,

(15a)

D_{>} (s) = \int_{- H}^{η} H_{>} {s^{z} (z) - s} d z .

(15b)

Due to the filter functions in the integrands of (15a) and (15b), the integrals only count those parts of the water column where s^z(z) ≤ s or s^z(z) > s, respectively. Figure 2 illustrates the functioning of the filter functions, as well as the resulting cumulative thicknesses D_≤ and D_> for strictly monotonic and nonmonotonic functions s^z(z). For strictly monotonic functions s^z(z), and depending on the sign of ∂s^z/∂z, the cumulative thickness represents the distance of the isosurface to the ocean surface or bottom. For nonmonotonic functions the cumulative thickness consists of the thicknesses from the individual noncontiguous parts.

Fig. 2.

Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1

In the following we will present the derivations only for the “≤” case for simplicity. The unified derivation for both cases can be found in the appendixes. The differential thickness, i.e., the thickness per s class, is given by

\frac{\partial D_{\leq} (s)}{\partial s} \overset{(15 a)}{=} \int_{- H}^{η} δ {s^{z} (z) - s} d z,

(16)

and ∂D_≤/∂s ds represents the thickness of the infinitesimal ds band around the s isosurface. The Dirac delta distribution

δ {x} = \partial H {x} / \partial x

is the derivative of the Heaviside step function. It only lets vertical positions contribute to the integral, where the specific isosurface s is present. In analogy to D_≤ and ∂D_≤/∂s, the cumulative and differential formulations for any quantity ψ^z can be obtained by

Ψ_{\leq} (s) = \int_{- H}^{η} ψ^{z} (z) H_{\leq} {s^{z} (z) - s} d z,

(17a)

\frac{\partial Ψ_{\leq} (s)}{\partial s} = \int_{- H}^{η} ψ^{z} (z) δ {s^{z} (z) - s} d z .

(17b)

The cumulative and differential quantities in (17a) and (17b) are defined for any value of s. Therefore, these relations can be used to map data from z coordinates to an arbitrary s space, similar to a classical coordinate transformation, but applicable also for nonmonotonic functions s^z(z). See appendix B for more details.

c. Local volume budget

Following the derivation of (C15) in appendix C, vertical integration of (10) over the parts of the water column where s^z(z) ≤ s yields the cumulative volume budget in s space

\frac{\partial D_{\leq} (s)}{\partial t} + \frac{\partial U_{\leq} (s)}{\partial x} + \frac{\partial V_{\leq} (s)}{\partial y} + u_{dia, z}^{s} (s) = u_{η, z} H_{\leq} {s_{η} - s},

(18)

with the cumulative thickness D_≤ from (15a), the cumulative transports

U_{\leq} (s) = \int_{- H}^{η} u (z) H_{\leq} {s^{z} (z) - s} d z,

(19a)

V_{\leq} (s) = \int_{- H}^{η} υ (z) H_{\leq} {s^{z} (z) - s} d z,

(19b)

and the effective vertical diasurface velocity (positive into the direction of increasing s)

u_{dia, z}^{s} (s) = \int_{- H}^{η} u_{dia, z}^{z} (z) | \frac{\partial s^{z}}{\partial z} | δ {s^{z} (z) - s} d z (effective vertical diasurface velocity) .

(20)

The cumulative transports in (19a) and (19b) are the basis for generalized streamfunctions in s space (Döös and Webb 1994; Nurser and Lee 2004; MacCready 2011; Wolfe 2014). The effective vertical diasurface velocity

u_{dia, z}^{s}

in s space, defined in (20), represents the volume flux across the given s isosurface per unit horizontal area.^¹ It is a key quantity in this paper. The term effective stresses that the integrand can contribute with compensating values from multiple positions in the water column where the specific isosurface s is present (see Fig. 2f). At each vertical position in the water column, the local diasurface volume flux per unit horizontal area,

u_{dia, z}^{z}

in z space in the integrand of (20), is given by

u_{dia, z}^{z} (z) = u_{dia}^{z} (z) {| n (z) \cdot z |}^{- 1} = u_{dia}^{z} (z) {| \frac{\partial s^{z}}{\partial z} |}^{- 1} | \nabla s^{z} |,

(21)

in terms of the diasurface velocity

u_{dia}^{z}

. The relation between

u_{dia}^{z}

and

u_{dia, z}^{z}

in (21) is illustrated in Fig. 3. The diasurface velocity

u_{dia}^{z}

is the component of the relative velocity normal to the isosurface that is currently located at the specific position z and moving with velocity u_s,

u_{dia}^{z} (z) = [u (z) - u_{s} (z)] \cdot n (z) = \frac{1}{| \nabla s^{z} |} \frac{D s^{z}}{D t} (diasurface velocity),

(22)

with n(z) = ∇s^z/|∇s^z| and z denoting the unit normal vectors at the isosurface and in the vertical, respectively. The term D/Dt is the material derivative. As sketched by the geometric relations in Fig. 3, the diasurface transport across an infinitesimal isosurface area element dA can be reformulated in terms of the local diasurface volume flux per unit horizontal area

u_{dia, z}^{z}

and the horizontal projection of the isosurface area dA_z,

u_{dia}^{z} d A \overset{(21)}{=} \overset{= u_{dia}^{z}}{\overset{︷}{u_{dia, z}^{z} \underset{= d A_{z}}{\underset{︸}{| z \cdot n | d A}}}} = u_{dia, z}^{z} d A_{z} .

(23)

Relation (23) is well known from the treatment of boundary fluxes at the ocean surface and bottom, e.g., (C4a) and (C5a), because the mathematical formulation is identical for any surface (Griffies 2004, section 6.7).

Fig. 3.

Citation: Journal of Physical Oceanography 53, 12; 10.1175/JPO-D-23-0130.1

The local diasurface volume flux per unit horizontal area,

u_{dia, z}^{z}

in (21), is formally not defined for vertical isosurface elements, i.e., for ∂s^z/∂z = 0. In our framework for a vertical water column this does not pose a problem, because the formally infinite

u_{dia, z}^{z}

is multiplied with |∂s^z/∂z| = 0 in (20) and an associated horizontal area dA_z = 0 in (23). Instead, the diasurface transport through a vertical isosurface is already accounted for as part of the cumulative horizontal transports (19a) and (19b). Moreover, due to the associated modified horizontal transport divergence in (18), this “missing” contribution to

u_{dia, z}^{s}

in (20) will simply appear in an adjacent water column. Therefore, the globally integrated diasurface volume flux

Q_{dia} (s) = \int u_{dia, z}^{s} (s) d A_{z} (integrated diasurface volume flux)

(24)

again correctly accounts for all contributions. In case of a nonmonotonic coordinate distribution, with an isosurface occurring at multiple positions within the water column, the area dA_z is still only counted once for the specific s (and not for every position z) in (24), because the effect of multiple isosurfaces is already accounted for in the multiple contributions of

u_{dia, z}^{z}

u_{dia, z}^{s}

in (20).

For strictly monotonic coordinate distributions s^z(z), and depending on the sign of ∂s^z/∂z, the cumulative transports (19a) and (19b) represent the transports below or above the isosurface, respectively (see Figs. 2b,d). In this monotonic case a unique inverse function z^s(s) exists, and the effective vertical diasurface velocity in (20) only contains the contribution from the one vertical position of the isosurface,

u_{dia, z}^{s} (s) \overset{(20), (B 3)}{=} \int_{- H}^{η} u_{dia, z}^{z} [z^{s} (s^{'})] δ {s^{'} - s} d s^{'} \overset{(B 3)}{=} u_{dia, z}^{z} [z^{s} (s)] for strictly monotonic s^{z} (z) .

(25)

Finally, the term on the right hand side of (18) with u_η_,_z considers the volume fluxes across the ocean surface into water column volumes that contain the isosurface s_η. In case of geometric coordinates s_η refers to the coordinate value at the ocean surface s^z(η). In case of s being a water mass property s_η refers to the property value that is transported with the volume flux.

d. Local tracer budget

Following the derivation of (C16) in appendix C, vertical integration of (12) over the parts of the water column where s^z(z) ≤ s yields the cumulative tracer budget in s space

\frac{\partial C_{\leq} (s)}{\partial t} + \frac{\partial F_{x, \leq} (s)}{\partial x} + \frac{\partial F_{y, \leq} (s)}{\partial y} + f_{dia, z}^{s} (s) = Π_{\leq} (s) + f_{η, z} H_{\leq} {s_{η} - s},

(26)

with the cumulative tracer content

C_{\leq} (s) = \int_{- H}^{η} c^{z} (z) H_{\leq} {s^{z} (z) - s} d z,

(27)

the cumulative tracer fluxes

F_{x, \leq} (s) = \int_{- H}^{η} f_{x} (z) H_{\leq} {s^{z} (z) - s} d z,

(28a)

F_{y, \leq} (s) = \int_{- H}^{η} f_{y} (z) H_{\leq} {s^{z} (z) - s} d z,

(28b)

the cumulative tracer production

Π_{\leq} (s) = \int_{- H}^{η} π (z) H_{\leq} {s^{z} (z) - s} d z,

(29)

and the effective vertical diasurface tracer flux (positive into the direction of increasing s)

f_{dia, z}^{s} (s) = \int_{- H}^{η} f_{dia, z}^{z} (z) | \frac{\partial s^{z}}{\partial z} | δ {s^{z} (z) - s} d z (effective vertical diasurface tracer flux) .

(30)

For the interpretation of the effective vertical diasurface tracer flux, and the quantities in the following relations, we refer to the analogous explanations for the effective diasurface velocity

u_{dia, z}^{s}

in (20). The local diasurface tracer flux per unit horizontal area in the integrand of (30) is defined as

f_{dia, z}^{z} (z) = f_{dia}^{z} (z) {| n (z) \cdot z |}^{- 1} = f_{dia}^{z} (z) {| \frac{\partial s^{z}}{\partial z} |}^{- 1} | \nabla s^{z} |,

(31)

with the diasurface tracer flux

f_{dia}^{z}

being the component of the tracer flux normal to the moving isosurface that is currently located at the specific position z,

\begin{array}{l} f_{dia}^{z} (z) = c^{z} (z) u_{dia}^{z} (z) + \overset{= j_{dia}^{z} (z)}{\overset{︷}{j (z) \cdot n (z)}} \\ \overset{(22)}{=} \frac{1}{| \nabla s^{z} |} (c^{z} \frac{D s^{z}}{D t} + j_{x} \frac{\partial s^{z}}{\partial x} + j_{y} \frac{\partial s^{z}}{\partial y} + j_{z} \frac{\partial s^{z}}{\partial z}) \\ = \frac{1}{| \nabla s^{z} |} (c^{z} \frac{\partial s^{z}}{\partial t} + f_{x} \frac{\partial s^{z}}{\partial x} + f_{y} \frac{\partial s^{z}}{\partial y} + f_{z} \frac{\partial s^{z}}{\partial z}) (diasurface tracer flux) . \end{array}

(32)

For strictly monotonic coordinate distributions s^z(z) the effective vertical diasurface tracer flux in (30) only contains the contribution from the one vertical position of the isosurface, analogously to the relation for the diasurface velocity (25), such that

f_{dia, z}^{s} (s) = c^{s} (s) u_{dia, z}^{s} (s) + j_{dia, z}^{s} (s) for strictly monotonic s^{z} (z)

(33)

holds, with the diffusive contribution

j_{dia, z}^{s}

defined equivalently to

f_{dia, z}^{s}

in (30),

j_{dia, z}^{s} (s) = \int_{- H}^{η} j_{dia, z}^{z} (z) | \frac{\partial s^{z}}{\partial z} | δ {s^{z} (z) - s} d z

(34a)

= \int_{- H}^{η} (- K_{h} \frac{\partial c^{z}}{\partial x} \frac{\partial s^{z}}{\partial x} - K_{h} \frac{\partial c^{z}}{\partial y} \frac{\partial s^{z}}{\partial y} - K_{υ} \frac{\partial c^{z}}{\partial z} \frac{\partial s^{z}}{\partial z}) \times δ {s^{z} (z) - s} d z .

(34b)

Finally, the last term on the right hand side of (26) with f_η_,_z considers the tracer flux across the ocean surface into water column volumes that contain the isosurface s_η. In case of salinity, that is for c = S, there is no salt flux across the atmosphere–ocean interface, but potentially across ice–ocean interfaces.

e. Local squared tracer budget

Following Burchard and Rennau (2008), and according to (C3), an equation for the squared tracer can be derived from the tracer equation (12),

\frac{\partial {(c^{z})}^{2}}{\partial t} + \frac{\partial f_{x}^{(2)}}{\partial x} + \frac{\partial f_{y}^{(2)}}{\partial y} + \frac{\partial f_{z}^{(2)}}{\partial z} = 2 c^{z} π - χ,

(35)

with the fluxes of squared tracer defined in (C1) and the tracer variance dissipation

χ = 2 [K_{h} {(\frac{\partial c^{z}}{\partial x})}^{2} + K_{h} {(\frac{\partial c^{z}}{\partial y})}^{2} + K_{υ} {(\frac{\partial c^{z}}{\partial z})}^{2}] .

(36)

Flux boundary conditions at the ocean surface and the bottom are given by (C4d) and (C5d). Note, that the boundary fluxes of squared tracer are already determined by the volume and tracer fluxes according to (C4e) and (C5e).

Following the derivation of (C14) in appendix C, vertical integration of (35) over the parts of the water column where s^z(z) ≤ s yields a budget equation for the cumulative squared tracer content in s space. Its sink term is given by the cumulative content of variance dissipation and quantifies the mixing inside D_≤:

M_{\leq} (s) = \int_{- H}^{η} χ (z) H_{\leq} {s^{z} (z) - s} d z (cumulative tracer mixing) .

(37)

From (37) the mixing per s class can be defined as

m (s) = \frac{\partial M_{\leq}}{\partial s} = \int_{- H}^{η} χ (z) δ {s^{z} (z) - s} d z (mixing per s class) .

(38)

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

If the tracer is mapped to its own coordinate system, the effective vertical diatracer tracer flux (30) can be written as

f_{dia, z}^{c} (c) = c u_{dia, z}^{c} (c) + j_{dia, z}^{c} (c),

(39)

which, in contrast to (33), does not require any monotonicity constraints on the tracer distribution. The effective vertical diatracer diffusive tracer flux

j_{dia, z}^{c}

obeys

j_{dia, z}^{c} (c) \overset{(34 b)}{=} - \int_{- H}^{η} \underset{\overset{(36)}{=} χ / 2}{\underset{︸}{[K_{h} {(\frac{\partial c^{z}}{\partial x})}^{2} + K_{h} {(\frac{\partial c^{z}}{\partial y})}^{2} + K_{υ} {(\frac{\partial c^{z}}{\partial z})}^{2}]}} δ {c^{z} (z) - c} d z \overset{(37)}{=} - \frac{1}{2} \underset{\overset{(38)}{=} m (c)}{\underset{︸}{(\frac{\partial M_{\leq}}{\partial c})}},

(40)

with the cumulative mixing M_≤ defined in (37) and m(c) being the mixing per tracer class. Relation (40) is the local equivalent of the integrated formulation (8) and the correct generalization of (9) in Li et al. (2022).

2) Ocean surface boundary fluxes

The volume and tracer fluxes across the ocean surface in (18) and (26) are formally considered in water column volumes that contain the isosurface s_η. In tracer space the individual fluxes enter different tracer classes and thus are considered in different water column volumes:

u_{η, z} H_{\leq} {s_{η} - s} \to u_{η, z} H_{\leq} {c_{η} - c},

(41a)

f_{η, z} H_{\leq} {s_{η} - s} \to c_{η} u_{η, z} H_{\leq} {c_{η} - c} + j_{η, z} H_{\leq} {c^{z} (η) - c} .

(41b)

Each advective contribution enters the tracer class that corresponds to the tracer concentration c_η transported by u_η_,_z. The nonadvective tracer flux j_η_,_z is assumed to enter the tracer class that is currently located at the ocean surface, given by c^z(η). The consideration of advective contributions in tracer classes with concentrations different from the actual sea surface tracer concentration is mathematically valid, because tracer coordinates are not required to be monotonic and layers of c_η can be constructed anywhere in the water column. For example, incoming freshwater volume fluxes always enter the zero salinity class, independent of the actual sea surface salinity.

3) Budget for internal tracer content

With (39), (41a), and (41b) a budget equation for the internal tracer content C_I_,≤ can be derived by combining (18) and (26),

\begin{array}{l} \frac{\partial}{\partial t} \overset{= C_{I, \leq} (c)}{\overset{︷}{(C_{\leq} - c D_{\leq})}} + \frac{\partial}{\partial x} \overset{F_{I x, \leq} (c)}{\overset{︷}{(F_{x, \leq} - c U_{\leq})}} + \frac{\partial}{\partial y} \overset{F_{I y, \leq} (c)}{\overset{︷}{(F_{y, \leq} - c V_{\leq})}} + j_{dia, z}^{c} (c) \\ = Π_{\leq} (c) + (c_{η} - c) u_{η, z} H_{\leq} {c_{η} - c} + j_{η, z} H_{\leq} {c^{z} (η) - c}, \end{array}

(42)

with internal tracer fluxes F_Ix_,≤ and F_Iy_,≤. As motivated by Holmes et al. (2019) and Bladwell et al. (2021) for the integrated internal heat and salt content, respectively, C_I_,≤ quantifies the tracer content relative to the bounding tracer concentration c and the so-called external tracer content cD_≤. As shown below, in this work the budget for the internal tracer content will be used to derive a relation between the diatracer velocity and the diatracer diffusive tracer fluxes. Alternative expressions for the internal tracer content and the associated internal tracer fluxes can be derived by means of (D1a) and partial integration:

C_{I, \leq} (c) = - \int_{- \infty}^{c} D_{\leq} (c^{'}) d c^{'},

(43a)

F_{I x, \leq} (c) = - \int_{- \infty}^{c} U_{\leq} (c^{'}) d c^{'} + J_{x, \leq} (c),

(43b)

F_{I y, \leq} (c) = - \int_{- \infty}^{c} V_{\leq} (c^{'}) d c^{'} + J_{y, \leq} (c) .

(43c)

4) Relation between diatracer velocity and diffusive tracer fluxes

With (43a)–(43c) and (18), the budget for the internal tracer content (42) can be reformulated^² to

\int_{- \infty}^{c} u_{dia, z}^{c} (c^{'}) d c^{'} + j_{dia, z}^{c} (c) + \frac{\partial J_{x, \leq} (c)}{\partial x} + \frac{\partial J_{y, \leq} (c)}{\partial y} = Π_{\leq} (c) + j_{η, z} H_{\leq} {c^{z} (η) - c} .

(44)

Taking the derivative with respect to c yields the differential formulation

u_{dia, z}^{c} (c) + \frac{\partial j_{dia, z}^{c}}{\partial c} + \frac{\partial}{\partial x} (\frac{\partial J_{x, \leq}}{\partial c}) + \frac{\partial}{\partial y} (\frac{\partial J_{y, \leq}}{\partial c}) = \frac{\partial Π_{\leq}}{\partial c} + j_{η, z} δ {c^{z} (η) - c} .

(45)

Neglecting horizontal diffusive tracer fluxes and tracer production simplifies the general relation to

u_{dia, z}^{c} (c) = - \frac{\partial j_{dia, z}^{c}}{\partial c} + j_{η, z} δ {c^{z} (η) - c} .

(46)

Following (46), the local diatracer velocity (positive into the direction of increasing tracer concentration) across isosurfaces in the interior of the water column is always into the direction of larger

- j_{dia, z}^{c}

, i.e., into the direction of stronger diasurface diffusive tracer fluxes. Of course, the net diffusive tracer transport is always into the direction of decreasing tracer concentration, but it can be either diverging (

\partial j_{dia, z}^{c} / \partial c > 0

, i.e., dominated by the transport from the present tracer concentration toward lower one), or converging (

\partial j_{dia, z}^{c} / \partial c < 0

, i.e., dominated by the transport from higher tracer concentration toward the present one). Divergence is associated with a decreasing volume of high tracer concentrations and an increasing volume of lower ones, i.e., a volume transport from high to low tracer concentrations (

u_{dia, z}^{c} < 0

). Vice versa, convergence is associated with water mass being transformed from low to high tracer concentrations (

u_{dia, z}^{c} > 0

). Relation (46) is the local equivalent^³ of the integrated formulation (4) and the correct generalization of (18) in Li et al. (2022).

5) Relation between diatracer velocity and mixing

The combination of (40) and (46) directly formulates how the diatracer exchange flow is linked to local tracer variance dissipation:

u_{dia, z}^{c} (c) = \frac{1}{2} \frac{\partial m}{\partial c} + j_{η, z} δ {c^{z} (η) - c} .

(47)

Following (47), the local diatracer velocity across isosurfaces in the interior of the water column is always into the direction of increased mixing. Relation (47) is the local equivalent of the integrated formulation (9), deduced in the introduction for salinity without salt fluxes across the ocean surface.

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

Integration of the local volume and tracer content budgets (18) and (26) in tracer space over a horizontal area A_z, and combination with (41a) and (41b) yields prognostic equations for volumes

V_{\leq} (c)

of all water with tracer concentrations ≤ c and their tracer content

C_{\leq} (c)

\begin{array}{l} \frac{\partial}{\partial t} \overset{= V_{\leq} (c)}{\overset{︷}{\int D_{\leq} (c) d A_{z}}} + \overset{= - Q_{\leq} (c) - Q_{r, \leq} (c)}{\overset{︷}{\int [\frac{\partial U_{\leq} (c)}{\partial x} + \frac{\partial V_{\leq} (c)}{\partial y}] d A_{z}}} + \overset{= Q_{dia} (c)}{\overset{︷}{\int u_{dia, z}^{c} (c) d A_{z}}} \\ = \underset{= Q_{η, \leq} (c)}{\underset{︸}{\int u_{η, z} H_{\leq} {c_{η} - c} d A_{z}}}, \end{array}

(48a)

\begin{array}{l} \frac{\partial}{\partial t} \overset{= C_{\leq} (c)}{\overset{︷}{\int C_{\leq} (c) d A_{z}}} + \overset{= - F_{\leq} (c) - F_{r, \leq} (c)}{\overset{︷}{\int [\frac{\partial F_{x, \leq} (c)}{\partial x} + \frac{\partial F_{y, \leq} (c)}{\partial y}] d A_{z}}} + \overset{= F_{dia} (c)}{\overset{︷}{\int f_{dia, z}^{c} (c) d A_{z}}} \\ = \underset{= P_{\leq} (c)}{\underset{︸}{\int Π_{\leq} (c) d A_{z}}} + \underset{= F_{η, \leq} (c)}{\underset{︸}{\int [c_{η} u_{η, z} ℋ_{\leq} {c_{η} - c} + j_{η, z} ℋ_{\leq} {c^{z} (η) - c}] d A_{z}}} . \end{array}

(48b)

The volumes are bounded by an open boundary and the isosurface with tracer concentration c. Figure 1 illustrates volumes for the special case of a stable salinity stratification. The equations derived here are also valid for nonmonotonic tracer distributions with inversions and noncontiguous volumes. The evolution of the volumes and their tracer content depends on the integrated fluxes entering across the open boundary (

Q_{\leq}, F_{\leq}

), integrated diatracer fluxes (

Q_{dia}, F_{dia}

) and integrated fluxes entering across the ocean surface (

Q_{η}, F_{η}

). Incoming integrated fluxes due to rivers (

Q_{r, \leq}, F_{r, \leq}

) are treated separately from the integrated open boundary fluxes. The integrated tracer production is denoted by

P_{\leq}

. The integrated diatracer tracer flux is given by

F_{dia} (c) \overset{(39)}{=} c Q_{dia} (c) + \underset{= J_{dia} (c)}{\underset{︸}{\int j_{dia, z}^{c} (c) d A_{z}}},

(49)

and recovers (2).

The general formulation of integrated fluxes across the ocean surface and due to rivers in (48a) and (48b) supports loads with different tracer concentrations. For the special case of freshwater fluxes in salinity space the formulation simplifies, because the associated volume fluxes per salinity class are given by q_η(S) = ∂Q_η_,≤/∂S = Q_ηδ{S_η − S} and q_r(S) = ∂Q_r_,≤/∂S = Q_rδ{S_r − S} with S_η = S_r = 0, respectively. Then the integrated fluxes become

Q_{η, \leq} (S) = Q_{η}, J_{η, \leq} = F_{η, \leq} = 0 (integrated freshwater surface fluxes),

(50a)

Q_{r, \leq} (S) = Q_{r}, J_{r, \leq} = F_{r, \leq} = 0 (integrated freshwater river discharge) .

(50b)

With these integrated freshwater fluxes the integral budgets (48a) and (48b) recover (5a) and (5b) used by, e.g., MacCready et al. (2002). Analogously, horizontal integration of (C15) and (C16) also recover Walin’s (1a) and (1b).

2) Relation between integrated diatracer diffusive tracer flux and mixing

The integrated diatracer diffusive tracer flux

J_{dia}

obeys

J_{dia} (c) \overset{(40)}{=} - \frac{1}{2} \underset{= 𝕞 (c)}{\underset{︸}{\int m (c) d A_{z}}} = - \frac{1}{2} \frac{\partial}{\partial c} \underset{= M_{\leq} (c)}{\underset{︸}{\int M_{\leq} (c) d A_{z}}},

(51)

which recovers (8). The integrated mixing per tracer class

𝕞

can be either obtained by horizontal integration of the local mixing per tracer class m, or following (6) as the derivative of the integrated tracer mixing

M_{\leq}

3) Integrated internal tracer content budget

By combination of the integral budgets (48a) and (48b), or directly by horizontal integration of the local formulation (42), the budget for the integrated internal tracer content can be obtained as

\frac{\partial}{\partial t} \overset{= C_{I, \leq} (c)}{\overset{︷}{(C_{\leq} - c V_{\leq})}} - \overset{= F_{I, \leq} (c)}{\overset{︷}{(F_{\leq} - c Q_{\leq})}} - \overset{= F_{I r, \leq} (c)}{\overset{︷}{(F_{r, \leq} - c Q_{r, \leq})}} + J_{dia} (c) = P_{\leq} (c) + \underset{= F_{I η, \leq} (c)}{\underset{︸}{(F_{η, \leq} - c Q_{η, \leq})}},

(52)

which recovers the internal heat content budget (12) in Holmes et al. (2019) and the internal salt content budget (10) in Bladwell et al. (2021). A comparison to the notation of Bladwell et al. (2021) is given in Table 2.

With the budget for the internal salt content, the mixing per salinity class inside a classical estuarine volume can be diagnosed,

𝕞 (S) = 2 {\frac{\partial}{\partial t} [S_{\leq} (S) - S V_{\leq} (S)] - [F_{\leq} (S) - S Q_{\leq} (S)] + S (Q_{r} + Q_{η})},

(53)

where (50a), (50b), and (51) has been used. This formulation generalizes (7) in Li et al. (2022) by including integrated fluxes across the ocean surface and the open boundary. For an estuarine volume completely bounded by the isohaline the integrated open boundary fluxes Q_≤(S) and

F_{\leq} (S)

do not exist (MacCready and Geyer 2001; Burchard 2020). In this case, and after sufficiently long time averaging, (53) recovers the universal law of estuarine mixing (7).

4) Relation between integrated diatracer volume and diffusive tracer fluxes

The integrated formulation of relation (46) reads

Q_{dia} (c) = - \frac{\partial J_{dia}}{\partial c} + \frac{\partial}{\partial c} \underset{= J_{η, \leq}}{\underset{︸}{\int j_{η, z} H_{\leq} {c^{z} (η) - c} d A_{z}}},

(54)

and recovers (4). In contrast to the local relation (46), by means of the divergence theorem the integral relation (54) does not require to neglect horizontal diffusive tracer fluxes in general, but only across the open boundary. For a global volume such an open boundary does not even exist, and relation (54) holds for any horizontal diffusive tracer fluxes.

5) Relation between integrated diatracer volume flux and mixing

Combination of (51) and (54), or direct integration of the local relation (47), yields

Q_{dia} = \frac{1}{2} \frac{\partial 𝕞}{\partial c} + \frac{\partial J_{η, \leq}}{\partial c},

(55)

and generalizes (9) deduced in the introduction.

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∂c^z/∂z = ∂(K_υ ∂c^z/∂z)/∂z. Assuming the resulting horizontally homogeneous tracer distribution is strictly monotonic with ∂c^z/∂z > 0, the mixing inside an arbitrary dz interval is given by dM = χdz = 2K_υ(∂c^z/∂z)²dz = 2K_υ ∂c^z/∂zdc ≡ mdc. Combination of both equations yields w∂c^z/∂z = ∂(m/2)/∂z = ∂(m/2)/∂c ∂c^z/∂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 $u_{dia, z}^{c} \overset{(25)}{=} u_{dia, z}^{z} \overset{(21), (22)}{=} w$ holds, and the simplified notation (omitting the superscripts for the actual coordinate space) is appropriate as long as the quantities are evaluated in the correct space. However, in more general settings we strongly encourage the use of the precise notation developed during the course of the manuscript, clearly indicating the coordinate space and preventing misunderstandings about derivatives. In this context, it is also important to remember that derivatives of the local quantities with respect to a tracer are evaluated at constant horizontal position (x, y), i.e., should not be misinterpreted as three-dimensional gradients in tracer direction.

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).

Fig. 4.

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”.

Due to $- \int_{- \infty}^{c} u_{η, z} H_{\leq} {c_{η} - c^{'}} d c^{'} = (c_{η} - c) u_{η, z} H_{\leq} {c_{η} - c}$ the advective flux across the ocean surface in (42) cancels with the one from (18).

Here, $j_{η, z} δ {c^{z} (η) - c} = \partial (j_{η, z} H_{\leq} {c^{z} (η) - c}) / \partial c$ holds.

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

For brevity we will use

D_{\leq, >} (s) = \int_{- H}^{η} H_{\leq, >} {s^{z} (z) - s} d z

(A1)

as a combined notation for (15a) and (15b) throughout the appendixes. In both cases the differential thickness is given by

\pm \frac{\partial D_{\leq, >} {(s)}^{(A 1)}}{\partial s} = \int_{- H}^{η} δ {s^{z} (z) - s} d z .

(A2)

In (A2) and in the following the upper sign (here “+”) always refers to the “≤” case and the lower sign (here “−”) to the “>” case. In analogy to D_≤,> and ±∂D_≤,>/∂s, for any quantity ψ^z the cumulative and differential formulations in s space can be obtained by

Ψ_{\leq, >} (s) = \int_{- H}^{η} ψ^{z} (z) H_{\leq, >} {s^{z} (z) - s} d z,

(A3)

\pm \frac{\partial Ψ_{\leq, >} (s)}{\partial s} = \int_{- H}^{η} ψ^{z} (z) δ {s^{z} (z) - s} d z .

(A4)

APPENDIX B

Mapping as a Generalized Coordinate Transformation

In this paper the vertical coordinate z is replaced by an arbitrary coordinate s. For geometric coordinates, e.g., geopotential (s = z) or sigma coordinates with s = σ = (z − η)/(η + H), the coordinate distribution s^z(z) is strictly monotonic between the bottom at z = −H(x, y) and the ocean surface at z = η(x, y, t). The same holds for stably stratified temperature or salinity distributions (see Figs. 2a,c). Under this monotonicity constraint there exists a well-defined coordinate transformation to a unique inverse function z^s(s) such that an arbitrary quantity ψ can be uniquely defined in s coordinates by

ψ^{s} (s) = ψ^{z} [z^{s} (s)] for strictly monotonic s^{z} (z),

(B1)

with superscripts indicating the coordinate space the different functions are defined for. In contrast, for nonmonotonic distributions s^z(z), e.g., for tracer coordinates where single tracer values may occur at multiple positions in the water column (e.g., Fig. 2e), there exists no function z^s(s) and the usual transformation procedure (B1) cannot be applied. In this case a general mapping to s space can formally be defined by thickness-weighted quantities in infinitesimally small bins according to (B2a),

ψ^{s} (s) = \frac{\int_{- H}^{η} ψ^{z} (z) δ {s^{z} (z) - s} d z}{\int_{- H}^{η} δ {s^{z} (z) - s} d z},

(B2a)

= \frac{\pm \frac{\partial Ψ_{\leq, >} (s)}{\partial s}}{\pm \frac{\partial D_{\leq, >} (s)}{\partial s}} .

(B2b)

The equivalent formulation (B2b) is of more practical use and is based on the cumulative thickness D_≤,> and the cumulative quantity Ψ_≤,> defined in (A1) and (A3). By construction (B2a) and (B2b) map data from z coordinates to s levels, and do not require any monotonicity constraints on the coordinate distribution s^z(z). For strictly monotonic functions the integrals in (B2a) can be transformed by means of

\begin{array}{l} \int_{- H}^{η} ψ^{z} (z) δ {s^{z} (z) - s} d z = \int_{- \infty}^{+ \infty} ψ^{z} [z^{s} (s^{'})] | \frac{\partial z^{s}}{\partial s} | δ {s^{'} - s} d s^{'} = ψ^{z} [z^{s} (s)] | \frac{\partial z^{s}}{\partial s} | \\ for strictly monotonic s^{z} (z), \end{array}

(B3)

where in the last step the sampling property of the δ distribution was used. Analogously, the differential thickness ±∂D_≤,>/∂s simplifies to the magnitude of the Jacobian of the coordinate transformation,

\pm \frac{\partial D_{\leq, >}}{\partial s} = | \frac{\partial z^{s}}{\partial s} | = {| \frac{\partial s^{z}}{\partial z} |}^{- 1} for strictly monotonic s^{z} (z),

(B4)

and (B2a) recovers (B1). The formulations (B2a) and (B2b) are motivated by (13) and (14) in Klingbeil et al. (2019) from the context of generalized thickness-weighted averaging.

APPENDIX C

Derivation of the p-Generalized Tracer Budget in s Space

Because the integrations for volume, tracer and tracer-square content are similar, the following derivations consider the tracer concentration c to the power of an arbitrary integer p ≥ 0. The flux of c^p in the x direction (the fluxes into the other directions are defined accordingly) is given by

\begin{array}{l} f_{x}^{(p)} = u {(c^{z})}^{p} \overset{= j_{x}^{(p)}}{\overset{︷}{- K_{h} \frac{\partial {(c^{z})}^{p}}{\partial x}}}, \\ = p {(c^{z})}^{p - 1} f_{x} - (p - 1) {(c^{z})}^{p} u, \end{array}

(C1)

with

f_{x}^{(0)} = u

and

f_{x}^{(1)} = f_{x}

being the velocity and tracer flux, respectively. According to (C1), for p ≥ 2,

f_{x}^{(p)}

is determined by u and f_x. With the incompressibility constraint (10) it can be shown that

\begin{array}{l} \frac{\partial f_{x}^{(p)}}{\partial x} + \frac{\partial f_{y}^{(p)}}{\partial y} + \frac{\partial f_{z}^{(p)}}{\partial z} = p {(c^{z})}^{p - 1} (\frac{\partial f_{x}}{\partial x} + \frac{\partial f_{y}}{\partial y} + \frac{\partial f_{z}}{\partial z}) \\ - \frac{p (p - 1)}{2} {(c^{z})}^{p - 2} \times \underset{= χ}{\underset{︸}{2 [K_{h} {(\frac{\partial c^{z}}{\partial x})}^{2} + K_{h} {(\frac{\partial c^{z}}{\partial y})}^{2} + K_{υ} {(\frac{\partial c^{z}}{\partial z})}^{2}]}} . \end{array}

(C2)

Multiplication of the tracer equation (12) by

p {(c^{z})}^{p - 1}

and reformulating the fluxes with (C2) yields the prognostic equation for

{(c^{z})}^{p}

\frac{\partial {(c^{z})}^{p}}{\partial t} + \frac{\partial f_{x}^{(p)}}{\partial x} + \frac{\partial f_{y}^{(p)}}{\partial y} + \frac{\partial f_{z}^{(p)}}{\partial z} = p {(c^{z})}^{p - 1} π - \frac{p (p - 1)}{2} {(c^{z})}^{p - 2} χ .

(C3)

Boundary fluxes across the ocean surface and the bottom per unit horizontal area (positive into the water column) are defined by

f_{η, z}^{(p)} = f_{η}^{(p)} {| n_{η} \cdot z |}^{- 1}

(C4a)

= f_{η}^{(p)} | \nabla (η - z) |

(C4b)

= {{[c^{z} (η)]}^{p} [u (η) - u_{η}] + j^{(p)} (η)} \cdot \nabla (η - z)

(C4c)

= {[c^{z} (η)]}^{p} \frac{\partial η}{\partial t} + f_{x}^{(p)} (η) \frac{\partial η}{\partial x} + f_{y}^{(p)} (η) \frac{\partial η}{\partial y} - f_{z}^{(p)} (η)

(C4d)

\overset{(C 1)}{=} p {[c^{z} (η)]}^{p - 1} f_{η, z} - (p - 1) {[c^{z} (η)]}^{p} u_{η, z}, and

(C4e)

f_{- H, z}^{(p)} = f_{- H}^{(p)} {| n_{- H} \cdot z |}^{- 1}

(C5a)

= f_{- H}^{(p)} | \nabla (z + H) |

(C5b)

= {{[c^{z} (- H)]}^{p} u (- H) + j^{(p)} (- H)} \cdot \nabla (z + H)

(C5c)

= f_{x}^{(p)} (- H) \frac{\partial H}{\partial x} + f_{y}^{(p)} (- H) \frac{\partial H}{\partial y} + f_{z}^{(p)} (- H)

(C5d)

\overset{(C 1)}{=} p {[c^{z} (- H)]}^{p - 1} f_{- H, z} - (p - 1) {[c^{z} (- H)]}^{p} u_{- H, z},

(C5e)

with the unit normal vectors n_η = ∇(η − z)/|∇(η − z)| and n₋_H = ∇(z + H)/|∇(z + H)|, and the vertical unit vector z. For p = 0 the relations (C4d) and (C5d) recover the well-known kinematic boundary conditions (11a) and (11b), and for p = 1 the tracer flux boundary conditions (13a) and (13b). According to (C4e) and (C5e), for p ≥ 2 the boundary fluxes are determined by the ones for volume and tracer.

Following (A3) the p-generalized cumulative tracer content is defined as

C_{\leq, >}^{(p)} (s) = \int_{- H}^{η} {[c^{z} (z)]}^{p} H_{\leq, >} {s^{z} (z) - s} d z,

(C6)

with

C_{\leq, >}^{(0)} = D_{\leq, >}

and

C_{\leq, >}^{(1)} = C_{\leq, >}

. In analogy the p-generalized cumulative tracer fluxes are defined as

F_{x, \leq, >}^{(p)} (s) = \int_{- H}^{η} f_{x}^{(p)} (z) H_{\leq, >} {s^{z} (z) - s} d z,

(C7a)

F_{y, \leq, >}^{(p)} (s) = \int_{- H}^{η} f_{y}^{(p)} (z) H_{\leq, >} {s^{z} (z) - s} d z,

(C7b)

with

F_{x, \leq, >}^{(0)} = U_{\leq, >}, F_{y, \leq, >}^{(0)} = V_{\leq, >}

F_{x, \leq, >}^{(1)} = F_{x, \leq, >}

and

F_{y, \leq, >}^{(1)} = F_{y, \leq, >}

. For (C6)–(C7b) Leibniz rule (E1) yields

\begin{array}{l} \frac{\partial C_{\leq, >}^{(p)} (s)}{\partial t} = \mp \int_{- H}^{η} {(c^{z})}^{p} \frac{\partial s^{z}}{\partial t} δ {s^{z} (z) - s} d z + \int_{- H}^{η} \frac{\partial {(c^{z})}^{p}}{\partial t} H_{\leq, >} {s^{z} (z) - s} d z \\ + {[c^{z} (η)]}^{p} \frac{\partial η}{\partial t} H_{\leq, >} {s_{η} - s}, \end{array}

(C8a)

\begin{array}{l} \frac{\partial F_{x, \leq, >}^{(p)} (s)}{\partial x} = \mp \int_{- H}^{η} f_{x}^{(p)} \frac{\partial s^{z}}{\partial x} δ {s^{z} (z) - s} d z + \int_{- H}^{η} \frac{\partial f_{x}^{(p)}}{\partial x} H_{\leq, >} {s^{z} (z) - s} d z \\ + f_{x}^{(p)} (η) \frac{\partial η}{\partial x} H_{\leq, >} {s_{η} - s} + f_{x}^{(p)} (- H) \frac{\partial H}{\partial x} H_{\leq, >} {s_{- H} - s}, \end{array}

(C8b)

\begin{array}{l} \frac{\partial F_{y, \leq, >}^{(p)} (s)}{\partial y} = \mp \int_{- H}^{η} f_{y}^{(p)} \frac{\partial s^{z}}{\partial y} δ {s^{z} (z) - s} d z + \int_{- H}^{η} \frac{\partial f_{y}^{(p)}}{\partial y} H_{\leq, >} {s^{z} (z) - s} d z \\ + f_{y}^{(p)} (η) \frac{\partial η}{\partial y} H_{\leq, >} {s_{η} - s} + f_{y}^{(p)} (- H) \frac{\partial H}{\partial y} H_{\leq, >} {s_{- H} - s} \cdot \end{array}

(C8c)

Application of (E2) for the vertical flux gives

\begin{array}{l} \int_{- H}^{η} \frac{\partial}{\partial z} [H_{\leq, >} {s^{z} (z) - s} f_{z}^{(p)}] d z \\ = \mp \int_{- H}^{η} δ {s^{z} (z) - s} f_{z}^{(p)} \frac{\partial s^{z}}{\partial z} d z + \int_{- H}^{η} H_{\leq, >} {s^{z} (z) - s} \frac{\partial f_{z}^{(p)}}{\partial z} d z \\ \equiv f_{z}^{(p)} (η) H_{\leq, >} {s_{η} - s} - f_{z}^{(p)} (- H) H_{\leq, >} {s_{- H} - s} \cdot \end{array}

(C9)

With (C4d), (C5d), and (C9), summation of (C8a)–(C8c) yields

\begin{array}{l} \frac{\partial C_{\leq, >}^{(p)} (s)}{\partial t} + \frac{\partial F_{x, \leq, >}^{(p)} (s)}{\partial x} + \frac{\partial F_{y, \leq, >}^{(p)} (s)}{\partial y} \\ = \mp \int_{- H}^{η} [{(c^{z})}^{p} \frac{\partial s^{z}}{\partial t} + f_{x}^{(p)} \frac{\partial s^{z}}{\partial x} + f_{y}^{(p)} \frac{\partial s^{z}}{\partial y} + f_{z}^{(p)} \frac{\partial s^{z}}{\partial z}] δ {s^{z} (z) - s} d z \\ + \int_{- H}^{η} [\frac{\partial {(c^{z})}^{p}}{\partial t} + \frac{\partial f_{x}^{(p)}}{\partial x} + \frac{\partial f_{y}^{(p)}}{\partial y} + \frac{\partial f_{z}^{(p)}}{\partial z}] H_{\leq, >} {s^{z} (z) - s} d z \\ + f_{η, z}^{(p)} H_{\leq, >} {s_{η} - s} + f_{- H, z}^{(p)} H_{\leq, >} {s_{- H} - s} \cdot \end{array}

(C10)

The integrand of the first integral on the right hand side of (C10) can be further reformulated in terms of the local diasurface flux

f_{dia}^{(p), z}

, defined as the component of the relative flux normal to the isosurface that is currently located at the specific position z and moving with velocity u_s:

f_{dia}^{(p), z} = {(c^{z})}^{p} (u - u_{s}) \cdot n + j^{(p)} \cdot n = \frac{1}{| \nabla s^{z} |} [{(c^{z})}^{p} \frac{D s^{z}}{D t} + j_{x}^{(p)} \frac{\partial s^{z}}{\partial x} + j_{y}^{(p)} \frac{\partial s^{z}}{\partial y} + j_{z}^{(p)} \frac{\partial s^{z}}{\partial z}]

(C11a)

= \frac{1}{| \nabla s^{z} |} [{(c^{z})}^{p} \frac{\partial s^{z}}{\partial t} + f_{x}^{(p)} \frac{\partial s^{z}}{\partial x} + f_{y}^{(p)} \frac{\partial s^{z}}{\partial y} + f_{z}^{(p)} \frac{\partial s^{z}}{\partial z}]

(C11b)

\overset{(C 1)}{=} p {(c^{z})}^{p - 1} f_{dia}^{z} - (p - 1) {(c^{z})}^{p} u_{dia}^{z} .

(C11c)

The local diasurface velocity and the local diasurface tracer flux are given by

f_{dia}^{(0), z} = u_{dia}^{z}

in (22) and

f_{dia}^{(1), z} = f_{dia}^{z}

in (32), respectively. With the unit normal vector n = ∇s^z/|∇s^z|, the diasurface fluxes are positive into the direction of increasing s. In analogy to the formulations of the fluxes across the ocean surface and bottom (C4a) and (C5a), the associated local diasurface tracer flux per unit horizontal area formally can be defined as

f_{dia, z}^{(p), z} = f_{dia}^{(p), z} {| n \cdot z |}^{- 1} = f_{dia}^{(p), z} {| \frac{\partial s^{z}}{\partial z} |}^{- 1} | \nabla s^{z} | .

(C12)

Using (C11b) and (C12), the first integral on the right hand side of (C10) can be defined as the effective vertical diasurface flux

f_{dia, z}^{(p), s} (s) = \int_{- H}^{η} f_{dia, z}^{(p), z} (z) | \frac{\partial s^{z}}{\partial z} | δ {s^{z} (z) - s} d z,

(C13)

The integrand of the second integral on the right hand side of (C10) can be replaced by inserting the prognostic equation for (c^z)^p, (C3), such that the following budget equation for the p-generalized cumulative tracer content can be derived:

\begin{array}{l} \frac{\partial C_{\leq, >}^{(p)} (s)}{\partial t} + \frac{\partial F_{x, \leq, >}^{(p)} (s)}{\partial x} + \frac{\partial F_{y, \leq, >}^{(p)} (s)}{\partial y} \pm f_{dia, (z)}^{(p), s} (s) \\ = \int_{- H}^{η} [p {(c^{z})}^{p - 1} π - \frac{p (p - 1)}{2} {(c^{z})}^{p - 2} χ] H_{\leq, >} {s^{z} (z) - s} d z \\ + f_{η, z}^{(p)} H_{\leq, >} {s_{η} - s} + f_{- H, z}^{(p)} H_{\leq, >} {s_{- H} - s} \cdot \end{array}

(C14)

Neglecting bottom fluxes, for p = 0 the volume budget

\frac{\partial D_{\leq, >} (s)}{\partial t} + \frac{\partial U_{\leq, >} (s)}{\partial x} + \frac{\partial V_{\leq, >} (s)}{\partial y} \pm u_{dia, z}^{s} (s) = u_{η, z} H_{\leq, >} {s_{η} - s},

(C15)

and for p = 1 the tracer budget

\frac{\partial C_{\leq, >} (s)}{\partial t} + \frac{\partial F_{x, \leq, >} (s)}{\partial x} + \frac{\partial F_{y, \leq, >} (s)}{\partial y} \pm f_{dia, z}^{s} (s) = Π_{\leq, >} (s) + f_{η, z} H_{\leq, >} {s_{η} - s},

(C16)

with the cumulative tracer production

Π_{\leq, >} (s) = \int_{- H}^{η} π (z) H_{\leq, >} {s^{z} (z) - s} d z

(C17)

are obtained, respectively. Taking the derivative of (C14) with respect to s and using (B2b) yields the differential formulation

\begin{array}{l} \frac{\partial}{\partial t} (\pm \frac{\partial D_{\leq, >}}{\partial s} c^{(p), s}) + \frac{\partial}{\partial x} (\pm \frac{\partial D_{\leq, >}}{\partial s} f_{x}^{(p), s}) + \frac{\partial}{\partial y} (\pm \frac{\partial D_{\leq, >}}{\partial s} f_{y}^{(p), s}) + \frac{\partial f_{dia, z}^{(p), s}}{\partial s} \\ = \int_{- H}^{η} [p {(c^{z})}^{p - 1} π - \frac{p (p - 1)}{2} {(c^{z})}^{p - 2} χ] δ {s^{z} (z) - s} d z \\ + f_{η, z}^{(p)} δ {s_{η} - s} + f_{- H, z}^{(p)} δ {s_{- H} - s}, \end{array}

(C18)

with the thickness-weighted quantities in s space,

c^{(p), s} = (\partial C_{\leq, >}^{(p)} / \partial s) / (\partial D_{\leq, >} / \partial s)

f_{x}^{(p), s} = (\partial F_{x, \leq, >}^{(p)} / \partial s) / (\partial D_{\leq, >} / \partial s)

and

f_{y}^{(p), s}

accordingly, from (B2b). For strictly monotonic coordinates, i.e., with (B4), the budget (C18) with p = 0 and p = 1 recovers the budget equations in generalized vertical coordinates which are the basis for the layer-integrated equations in ocean models, see, e.g., Griffies (2004) and Klingbeil et al. (2018).

APPENDIX D

Further Relations in Tracer Space

For tracer coordinates, i.e., for s = c, and by means of the Newton–Leibniz theorem the following useful transformation relation holds for an arbitrary quantity ψ:

\begin{matrix} \int_{- H}^{η} {(c^{z})}^{p} ψ^{z} H_{\leq, >} {c^{z} (z) - c} d z = {\int_{- H}^{η} {(c^{z})}^{p} ψ^{z} H_{\leq, >} {c^{z} (z) - (\mp \infty)} d z}^{= 0} \\ + \int_{\mp \infty}^{c} [\frac{\partial}{\partial c^{'}} \int_{- H}^{η} {(c^{z})}^{p} ψ^{z} H_{\leq, >} {c^{z} (z) - c^{'}} d z] d c^{'} \\ = \pm \int_{\mp \infty}^{c} [\int_{- H}^{η} {(c^{z})}^{p} ψ^{z} δ {c^{z} (z) - c^{'}} d z] d c^{'} \\ = \pm \int_{\mp \infty}^{c} {(c^{'})}^{p} [\int_{- H}^{η} ψ^{z} δ {c^{z} (z) - c^{'}} d z] d c^{'} \\ = \pm \int_{\mp \infty}^{c} {(c^{'})}^{p} [\pm \frac{\partial}{\partial c^{'}} \int_{- H}^{η} ψ^{z} H_{\leq, >} {c^{z} (z) - c^{'}} d z] d c^{'} \\ \overset{(A 3)}{=} \pm \int_{\mp \infty}^{c} {(c^{'})}^{p} (\pm \frac{\partial Ψ_{\leq, >}}{\partial c^{'}}) d c^{'} \\ = \int_{- \infty}^{+ \infty} {(c^{'})}^{p} (\pm \frac{\partial Ψ_{\leq, >}}{\partial c^{'}}) H_{\leq, >} {c^{z} (z) - c^{'}} d c^{'} \\ \overset{(B 2 b)}{=} \int_{- \infty}^{+ \infty} {(c^{'})}^{p} [\pm \frac{\partial D_{\leq, >}}{\partial c'} ψ^{c} (c^{'})] H_{\leq, >} {c^{z} (z) - c^{'}} d c^{'} . \end{matrix}

(D1a)

By means of (D1a), the integration on the right hand side of (C14) can be carried out in tracer space,

\begin{array}{l} \frac{\partial C_{\leq, >}^{(p)} (c)}{\partial t} + \frac{\partial F_{x, \leq, >}^{(p)} (c)}{\partial x} + \frac{\partial F_{y, \leq, >}^{(p)} (c)}{\partial y} \pm f_{dia, z}^{(p), c} (c) \\ = \pm \int_{\mp \infty}^{c} [p {(c^{'})}^{p - 1} (\pm \frac{\partial Π_{\leq, >}}{\partial c^{'}}) - \frac{p (p - 1)}{2} {(c^{'})}^{p - 2} (\pm \frac{\partial M_{\leq, >}}{\partial c^{'}})] d c^{'} \\ + f_{η, z}^{(p)} H_{\leq, >} {c_{η} - c} + f_{- H, z}^{(p)} H_{\leq, >} {c_{- H} - c}, \end{array}

(D2)

with the cumulative tracer production Π_≤,> defined in (C17), and the cumulative tracer mixing

M_{\leq, >} (s) = \int_{- H}^{η} χ (z) H_{\leq, >} {s^{z} (z) - s} d z

. Subsequent derivation with respect to c yields

\begin{array}{l} \frac{\partial}{\partial t} (\pm \frac{\partial D_{\leq, >}}{\partial c} c^{p}) + \frac{\partial}{\partial x} [\pm \frac{\partial D_{\leq, >}}{\partial c} f_{x}^{(p), c}] + \frac{\partial}{\partial y} [\pm \frac{\partial D_{\leq, >}}{\partial c} f_{y}^{(p), c}] + \frac{\partial f_{dia, z}^{(p), c}}{\partial c} \\ = p c^{p - 1} (\pm \frac{\partial Π_{\leq, >}}{\partial c}) - \frac{p (p - 1)}{2} c^{p - 2} (\pm \frac{\partial M_{\leq, >}}{\partial c}) \\ + f_{η, z}^{(p)} δ {c_{η} - c} + f_{- H, z}^{(p)} δ {c_{- H} - c}, \end{array}

(D3)

which simplifies the first term and the right hand side from the general formulation (C18).

APPENDIX E

Leibniz Rule

For ξ ∈{x, y, t} the Leibniz rule reads

\begin{array}{l} \frac{\partial}{\partial ξ} \int_{- H (ξ)}^{η (ξ)} ψ^{z} (ξ, z) H_{\leq, >} {s^{z} (ξ, z) - s} d z \\ = \mp \int_{- H}^{η} ψ^{z} \frac{\partial s^{z}}{\partial ξ} δ {s^{z} (ξ, z) - s} d z + \int_{- H}^{η} \frac{\partial ψ^{z}}{\partial ξ} H_{\leq, >} {s^{z} (z) - s} d z \\ + ψ^{z} (η) \frac{\partial η}{\partial ξ} H_{\leq, >} {s^{z} (η) - s} + ψ^{z} (- H) \frac{\partial H}{\partial ξ} H_{\leq, >} {s^{z} (- H) - s} . \end{array}

(E1)

But note that the bottom topography H will be considered as constant in time t in this manuscript.

For the vertical direction it can be shown that

\begin{array}{l} \int_{- H}^{η} \frac{\partial}{\partial z} [ψ^{z} H_{\leq, >} {s^{z} (z) - s}] d z \\ = \mp \int_{- H}^{η} ψ^{z} \frac{\partial s^{z}}{\partial z} δ {s^{z} (z) - s} d z + \int_{- H}^{η} \frac{\partial ψ^{z}}{\partial z} H_{\leq, >} {s^{z} (z) - s} d z, \\ \equiv ψ^{z} (η) H_{\leq, >} {s^{z} (η) - s} - ψ^{z} (- H) H_{\leq, >} {s^{z} (- H) - s}, \end{array}

(E2)

where the third line follows directly from the first one, using the Newton–Leibniz theorem.

In the rest of the manuscript we replace s^z(η) and s^z(−H) in the arguments of the Heaviside functions in (E1) and (E2) by s_η and s_−H.

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A Rigorous Derivation of the Water Mass Transformation Framework, the Relation between Mixing and Diasurface Exchange Flow, and Links to Recent Theories in Estuarine Research

Abstract

Abstract

1. Introduction

2. Derivation of a local WMT formulation

a. Governing equations in Cartesian coordinates

b. Mapping to an arbitrary s space

c. Local volume budget

d. Local tracer budget

e. Local squared tracer budget

f. Mapping to tracer coordinates

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

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

5) Relation between integrated diatracer volume flux and mixing

3. Demonstration

a. Idealized steady state and a word of caution

b. Model results for an idealized tidal estuary

4. Summary and conclusions

Acknowledgments.

Data availability statement.

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

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