Spurious Dianeutral Advection and Methods for Its Minimization

Yandong Lang; Geoffrey J. Stanley; Trevor J. McDougall

doi:10.1175/JPO-D-22-0174.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Background
- a. Neutral surfaces and helicity
- b. Dianeutral processes
3. The method to construct ω surfaces
- a. Theory of ω+ surfaces
- b. Discretization
4. Minimizing the slope error, s
5. Minimizing the fictitious dianeutral diffusivity
6. Minimizing the spurious dianeutral advection
7. Minimizing the spurious dianeutral advection and fictitious dianeutral diffusivity, together
8. Surfaces assessment
9. The change from the ω+ surface to the ωu⋅s surface
10. Summary and discussion
Acknowledgments.
Data availability statement.
APPENDIX A
- The Difference of the Epineutral Mixing, Thermobaric, and Cabbeling Terms between an ANS and the NTP in the WMT Equation
APPENDIX B
- Derivation of the Discretization of the Dot Product
REFERENCES

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Fig. 1.

Schematic of the C-grid. Tracers such as temperature Θ are located on water columns at the centers (black points) of the tracer cells (red dotted line). The velocity and slope error components (u_m_,_n and s_m_,_n) are located at the interface between tracer cells (red points). Water columns are labeled by a one-dimensional sequence of integers. The distance between water columns m and n is Δ_m_,_n, while the distance of the interface between tracer cells m and n is $Δ_{m, n}^{⊥}$ .

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Fig. 2.

Schematic of the slope error calculation. Casts m and n represent adjacent casts. Points A and B are on the ANS and points A′ and B′ are neutrally related. Point M is at the interface of casts m and n and has a height of z_m_,_n, the average height of A and B. To find the position of A′ and B′, Eq. (37) is solved. The slope difference between AB and A′B′ is the slope error of AB. Points C, C′, D, and D′ are used for describing the slope error when we add a height perturbation ϕ_m and ϕ_n to points A and B. After perturbing the points A and B to points C and D, the updated neutral slope is C′D′, given by Eq. (43).

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Fig. 3.

Map of e^sp = u ⋅ s (m s⁻¹), with arrows indicating lateral velocity for the ω_u_·_s surface in the North Atlantic Ocean.

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Fig. 4.

Global maps of e^sp = u ⋅ s (m s⁻¹) for the 10 surfaces detailed in section 8. The root-mean-square of e^sp is indicated above each panel.

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Fig. 5.

As in Fig. 4, but showing the slope error, |s|, on a logarithmic scale. We calculate |s| as (s ⋅ s)^1/2 following (65). The RMS listed above each caption is calculated not as the RMS of the mapped |s| on the tracer cells, but rather as the RMS of the separate zonal and meridional components of s, each weighted by the area of the U and V cells, respectively.

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Fig. 6.

As in Fig. 4, but showing the fictitious dianeutral diffusivity D^f (m² s⁻¹) on a logarithmic scale. As in Fig. 5, the RMS listed is calculated using the square of the zonal slope error and the square of the meridional slope error separately and area-weighted by the U and V cells.

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Fig. 7.

As in Fig. 4, but showing e^sp + D^f/(1000 m). Taking the vertical scale of Θ and S variations as Θ_z/Θ_zz ≈ S_z/S_zz ≈ (1000 m)⁻¹, this approximately measures the flux of Θ or S across the surface caused by the combination of e^sp and D^f.

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Fig. 8.

Bivariate histograms of e^sp and the squared slope error |s|², both on a logarithmic scale, for (a) the σ_0.75 surface, (b) the ω₊ surface, (c) the ω_u_·_s surface, (d) the $ω_{u \cdot s + s^{2}}$ surface, (e) the γ^SCV surface, and (f) the γ^T surface. The bold plus symbols show the RMS of the data. The blue region denotes where e^sp ≤ 3 × 10⁻⁹ m s⁻¹ and |s|² ≤ 10⁻⁸ m² s⁻¹, which are acceptably small limits below which an inverse study is not strongly adversely affected.

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Fig. 9.

(a)–(j) The depth of the surfaces. Panel (b) is the depth of the ω₊ surface. Panels (a), (c), (d), (g), (i), and (j) compare the depth of the σ_0.75 surface, the ω_s surface, the ω_u_·_s surface, the $ω_{s^{2}}$ surface, the γ^SCV surface, and the γ^T surface, resepectively, with the depth of the ω₊ surface. Panels (e), (f), and (h) compare the depth of the $ω_{u \cdot s Θ_{z}}$ surface, the $ω_{u \cdot s S_{z}}$ surface, and the $ω_{u \cdot s + s^{2}}$ surface, respectively, with the depth of the ω_u_·_s surface.

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Fig. 10.

The cosine of the angle between the lateral velocity u and the slope error s on (a) the ω₊ surface, (b) the ω_u_·_s surface, and (c) the σ_0.75 surface.

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Fig. 11.

(a) The neutrality error ε_ω and $ε_{ω_{u \cdot s}}$ on an ω₊ surface and an ω_u_·_s surface. The neutrality error ε is given by the vector sum of ρ_Θ∇_aΘ and ρ_S∇_aS. The ω_u_·_s surface changes ρ_Θ∇_ωΘ and ρ_S∇_ωS slightly but this results in a large change of $ε_{ω_{u \cdot s}}$ . Note that $ε_{ω_{u \cdot s}}, ρ_{Θ} \nabla_{ω_{u \cdot s}} Θ$ , and $ρ_{S} \nabla_{ω_{u \cdot s}} S$ represent the vectors for the ω_u_·_s surface. Note also that $| ε_{ω_{u \cdot s}} | > | ε_{ω} |$ as drawn here, consistent with ω_u_·_s surfaces having larger ε neutrality errors than ω surfaces. However, the ω_u_·_s surface has smaller e^sp than the ω surface because the neutrality error ε (or slope error s) is nearly perpendicular to the velocity u. (b) The neutrality error for the potential density surface, ε_σ = ρ_Θ∇_σΘ + ρ_S∇_σS. The exactly (anti)parallel vectors ρ_Θ∇_σΘ and ρ_S∇_σS are nearly perpendicular to the lateral velocity u, hence so too is ε_σ.

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Fig. A1.

Comparing the errors of (a) the epineutral mixing term, (c) the thermobaric term, and (e) cabbeling term caused by the difference between evaluating them along the NTP and $ω_{u \cdot s + s^{2}}$ surface with the (b),(d),(f) corresponding terms evaluated along the $ω_{u \cdot s + s^{2}}$ surface.

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Article Type:: Research Article

Spurious Dianeutral Advection and Methods for Its Minimization

Yandong Lang

Yandong Lang ^aSchool of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

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Geoffrey J. Stanley

Geoffrey J. Stanley ^aSchool of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia
^bSchool of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

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Trevor J. McDougall

Trevor J. McDougall ^aSchool of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

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Online Publication:: 01 Jun 2023

Print Publication:: 01 Jun 2023

DOI:: https://doi.org/10.1175/JPO-D-22-0174.1

Page(s):: 1401–1427

Received:: 29 Aug 2022

Final Form:: 23 Feb 2023

Accepted:: 27 Feb 2023

Published Online:: 01 Jun 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

An existing approximately neutral surface, the ω surface, minimizes the neutrality error and hence also exhibits very small fictitious dianeutral diffusivity D^f that arises when lateral diffusion is applied along the surface, in nonneutral directions. However, there is also a spurious dianeutral advection that arises when lateral advection is applied nonneutrally along the surface; equivalently, lateral advection applied along the neutral tangent planes creates a vertical velocity e^sp through the ω surface. Mathematically, e^sp = u ⋅ s, where u is the lateral velocity and s is the slope error of the surface. We find that e^sp produces a leading-order term in the evolution equations of temperature and salinity, being similar in magnitude to the influence of cabbeling and thermobaricity. We introduce a new method to form an approximately neutral surface, called an ω_u_·_s surface, that minimizes e^sp by adjusting its depth so that the slope error is nearly perpendicular to the lateral velocity. The e^sp on a surface cannot be reduced to zero when closed streamlines contain nonzero neutral helicity. While e^sp on the ω_u_·_s surface is over 100 times smaller than that on the ω surface, the fictitious dianeutral diffusivity on the ω_u_·_s surface is larger, nearly equal to the canonical 10⁻⁵ m² s⁻¹ background diffusivity. Thus, we also develop a method to minimize a combination of e^sp and D^f, yielding the $ω_{u \cdot s + s^{2}}$ surface, which is recommended for inverse models since it has low D^f and it significantly decreases e^sp through the surface, which otherwise would be a leading term that cannot be ignored in the conservation equations.

Corresponding author: Yandong Lang, aaronlangyandong@gmail.com

Abstract

Corresponding author: Yandong Lang, aaronlangyandong@gmail.com

Keywords: Ocean; Differential equations; Inverse methods; Optimization; Numerical analysis/modeling; Parameterization; Software

1. Introduction

Mixing processes in the interior ocean may be thought of as composed of three different processes, each operating on quite distinct physical scales [see de Lavergne et al. (2022) for a recent review of this paradigm]. At the smallest scale (millimeter scale) molecular diffusion is effective, and its main role in the ocean is to destroy tracer variance that is created by the stirring of tracer gradients by the larger-scale motions. The next largest scale is the fine-structure scale where the various physical processes have vertical scales from centimeters to a few meters. While molecular diffusion is not effective on these scales, it is the vertical overturning motion associated with ocean fine-structure that provides the sharp gradients on which the molecular diffusion can operate. The fine-structure motions are usually referred to as “small-scale turbulent mixing,” with breaking internal waves being an example process. Small-scale turbulent mixing is usually described as being irreversible, and we will continue this tradition, even though we recognize that the irreversible mixing is actually done by molecular diffusion, which operates on the sharp spatial gradients that are caused by the turbulent fine-structure overturning motions (de Lavergne et al. 2022).

The third type of turbulence in the ocean operates at much larger scales, and these phenomena are visible from space. These are the mesoscale motions, often called mesoscale eddies, that have horizontal scales ranging from 20 to 200 km. These mesoscale motions have long been thought of as flowing along density surfaces, or more specifically along the neutral tangent plane (NTP). The NTP is defined as the directions in which a fluid parcel can be (infinitesimally) displaced without generating a buoyant restoring force (McDougall 1987a). However, there is surprisingly little observational, modeling, or theoretical evidence that addresses this issue of the preferential directions of mesoscale motion. The main argument lies in the observed smallness of the dianeutral (through the NTP) diffusivity D caused by small-scale mixing, with canonical values of D ≈ 10⁻⁴ m² s⁻¹ near-bottom topography and D ≈ 10⁻⁵ m² s⁻¹ in the open ocean (Waterhouse et al. 2014). If the lateral turbulent mixing of the mesoscale eddies did not occur along the NTP, fluid would be brought back toward the NTP in buoyant plumes, which would generate larger D than is observed (McDougall and Jackett 2005; McDougall et al. 2014). Based on observations, the small-scale turbulent diffusivity is not substantially enhanced where the mesoscale turbulent activity is strong, and the observed small-scale turbulent activity can be mostly explained by our existing theories of internal tides radiating from the sea floor and the interactions between internal waves. In this way, fine-structure observations provide no hint of a separate role of mesoscale motions in order to explain the fine-structure mixing activity, and this is why the community takes the mesoscale turbulence to act along neutral tangent planes. Recent research has focused on another physical process at work between the scales of isotropic turbulent mixing and mesoscale lateral mixing, namely, submesoscale eddies. These can cause elevated levels of dianeutral mixing particularly near the surface and bottom boundaries, but their role in sustaining dianeutral mixing in the ocean interior remains particularly uncertain [see Gula et al. (2022) for a recent review]. Still, to the extent that observations have intersected with submesoscale events in the ocean interior, the above argument for mesoscale eddies would again apply, suggesting that interior submesoscale currents are closely aligned with the NTP. Our operating paradigm, therefore, is that in the ocean interior away from diabatic sources, lateral stirring preferentially acts along the NTP.

A neutral surface—a surface that is everywhere tangent to the NTP—would be a valuable practical and theoretical surface for oceanographic analysis and modeling. However, because of the nonlinear dependence of the specific volume of seawater on salinity, temperature, and pressure, neutral helicity is nonzero and consequently, neutral surfaces never exist in the real ocean (McDougall and Jackett 1988).

Without the existence of perfectly neutral surfaces, we must use approximately neutral surfaces (ANSs) that are well-defined but only approximately tangent to the NTP. Examples of ANSs include isosurfaces (level sets) of three-dimensional density variables: potential density σ (Wüst 1933); specific volume anomaly δ (Montgomery 1937); neutral density γⁿ (Jackett and McDougall 1997); pressure-independent neutral density γ^SCV (Lang et al. 2020); orthobaric density σ_υ (de Szoeke et al. 2000); and thermodynamic neutral density γ^T (Tailleux 2016). There are also bespoke two-dimensional ANSs: ω surfaces (Klocker et al. 2009; Stanley et al. 2021) and topobaric surfaces τ (Stanley 2019a,b). Ideally, an ANS is

as closely oriented to the NTP as possible, and is either
pycnotropic (a function only of in situ density and pressure) or
quasi-material (a function only of Absolute Salinity and Conservative Temperature).

Being pycnotropic confers the existence of an Ertel potential vorticity (with no baroclinic production term) and an exact geostrophic streamfunction (de Szoeke 2000; Stanley 2019b). A quasi-material density variable has the advantageous property that its sources are faithfully linked to physical sources of salt or heat. Potential density and thermodynamic neutral density are quasi-material. The pycnotropic form has evolved from specific volume anomaly to orthobaric density, both of which are globally pycnotropic, to topobaric surfaces that are locally pycnotropic, enabling the functional relationship between in situ density and pressure to be geographically dependent and thus more neutral. Finally, γⁿ, γ^SCV, and ω surfaces forgo the pycnotropic and quasi-material ideals, striving purely to be as neutral as possible.

Property i, or the degree of misalignment between an ANS and the NTP, has major consequences for ocean modeling. Mesoscale turbulence stirs fluid along the NTP and leads to an effective epineutral (along the NTP) diffusivity K that is typically 10²–10³ m² s⁻¹ (Groeskamp et al. 2020). If this lateral diffusivity K is applied in the ANS instead of the NTP, then a component of this strong diffusivity acts across the NTP, creating a fictitious dianeutral diffusivity of D^f = K|s|², where s is the slope difference between the NTP and the ANS. Since K is about 10⁶–10⁸ times larger than D, the slope error s should ideally be smaller than 10⁻⁴ to avoid D^f overwhelming the physically supported D. This stringent requirement is met by carefully constructed ANSs such as ω surfaces and topobaric surfaces, but classic density variables such as potential density fail this test (Stanley 2019a). Measuring property i has thus far focused on D^f, or closely related quantities.

One quantity closely related to D^f is the neutrality error ε, which is proportional to the slope error s times the local stratification. If we measure property i by ε, the most neutral ANS to date is the ω surface (Klocker et al. 2009; Stanley et al. 2021), which is built specifically to minimize ε. One goal of this paper is to create a variant of an ω surface that instead minimizes D^f itself.

A major goal of this paper is to scrutinize the meaning of property i: how should we measure the neutrality of a surface? We turn the focus of property i from how epineutral diffusivity K creates fictitious dianeutral diffusivity D^f to consider an analogous erroneous effect caused by advection. If lateral advection is applied in an ANS, as in isopycnal models, a component of this acts across the NTP, creating a fictitious dianeutral advection, defined as e^sp = u ⋅ s, where u is the lateral velocity. The converse is also problematic: physically realistic advection along the NTP has a component that acts across the ANS, creating excessive diasurface flow (through the surface in question, in this case the ANS). Note that D^f and e^sp are not real oceanic processes: they depend on the choice of the ANS, which is an anthropogenic activity, unknown to the ocean. To date, oceanographers have cared mostly about D^f and mostly neglected e^sp. However, both dianeutral advection and dianeutral diffusion are crucial to the deep ocean dynamics (Stommel and Arons 1959). Both processes are leading-order terms in the evolution equations for temperature and salinity in the deep ocean.

The diasurface advection e^sp caused by the slope error of the ANSs may be thought of as having two separate causes. First, there is an irreducible amount of slope error due to the nonzero neutral helicity in the ocean, so even the most carefully constructed ANS will have nonzero slope error. We show that it is generally impossible to build an ANS such that its slope error s is orthogonal to u. Thus, there is an irreducible amount of e^sp. In this paper, we will develop a new surface, denoted a ω_u_·_s surface, that achieves this minimum amount of e^sp. Any other ANS has larger e^sp that is a manifestation not only of neutral helicity, but is also a mark of the anthropogenic hand that constructed this less-than-ideal surface, which is the second cause of e^sp.

In addition to these two types of diasurface flow associated with the definition of ANSs, e^sp and D^f, there is also diasurface advection caused by real physical processes such as small-scale turbulent mixing, thermobaricity, and cabbeling. Iudicone et al. (2008) and Klocker and McDougall (2010a) have shown that thermobaricity and cabbeling cause significant water mass transformation and significant diasurface advection, particularly in the Southern Ocean.

The diasurface flow e^sp caused by the helical nature of neutral trajectories is inherently advective. In this regard, e^sp is somewhat related to the diasurface flow driven by thermobaricity and cabbeling, which also do not require any dianeutral (isotropic, turbulent) mixing (Klocker and McDougall 2010a). Rather, these three processes rely on the epineutral turbulent flux of Absolute Salinity and Conservative Temperature, and any nonzero amount of small-scale mixing is sufficient to destroy the tracer variance that is produced by large-scale epineutral stirring motions (McDougall 1987b, 1991; Klocker and McDougall 2010b). Being inherently advective, each of these three processes affects the evolution of Absolute Salinity and Conservative Temperature through an extra diasurface velocity, and any vertical diffusive effect is minimal (Klocker and McDougall 2010a).

A major output of this paper is the introduction of the ω_u_·_s surface that minimizes e^sp. We also create variants of this surface that minimize the fictitious advective flux of either Absolute Salinity or Conservative Temperature, and another called the $ω_{u \cdot s + s^{2}}$ surface that minimizes a weighted sum of e^sp and D^f. These surfaces provide new neutral surface reference frames for water-mass transformation (WMT) analysis, and investigation of the influence of processes caused by the nonlinear nature of the equation of state on dianeutral and diasurface advection.

Section 2 provides background material on neutral surfaces and neutral helicity and a brief derivation of the density conservation equation, showing the contribution of e^sp to the vertical velocity through the ANS. In section 3, we summarize the numerical algorithm to create the ω surface. Sections 4–7 introduce the ω_s surface, $ω_{s^{2}}$ surface, ω_u_·_s surface, and $ω_{u \cdot s + s^{2}}$ surface, which minimize the s, D^f, e^sp, and a combination of e^sp and D^f, respectively. In section 8, we assess e^sp, s, and D^f on various old and new ANSs. In section 9, we discuss why the ω_u_·_s surface can achieve such small values of e^sp despite being very close in depth to the ω surface. Summary and discussion are given in section 10. Table 1 provides a nonexhaustive glossary of commonly used symbols in this paper.

Table 1

Glossary of some commonly used symbols.

2. Background

a. Neutral surfaces and helicity

In this paper, S, p, Θ, and ρ, denote the salinity, Conservative Temperature (McDougall 2003), pressure, and in situ density respectively. Regarding the use of S for salinity, we note that to date ocean models have ignored the spatial variation in the composition of seawater, and the associated nonconservative biological source terms of Absolute Salinity. Hence, the salinity variable in ocean models should be interpreted as Preformed Salinity $S_{*}$ (McDougall et al. 2021). Similarly, models that use an equation of state accepting potential temperature (rather than Conservative Temperature) nonetheless treat potential temperature as a purely conservative variable, and so their model temperature variable should actually be interpreted as Conservative Temperature (McDougall et al. 2021). In this paper we will ignore these issues and simply use the symbols S and Θ to describe the model’s salinity and temperature variables. Henceforth, we simply call Θ the temperature. The equation of state $R$ relates these by $ρ = R (S, Θ, p)$ . Partial derivatives of $R$ follow as $ρ_{S} = \partial_{S} R (S, Θ, p)$ , $ρ_{Θ} = \partial_{Θ} R (S, Θ, p)$ , $ρ_{P} = \partial_{p} R (S, Θ, p)$ , $ρ_{S p} = \partial_{S p} R (S, Θ, p)$ , and $ρ_{Θ p} = \partial_{Θ p} R (S, Θ, p)$ .

The neutral tangent plane (NTP) is defined as the plane that contains all the directions in which the local water parcel can be moved infinitesimally without experiencing a buoyant restoring force. Consider displacing a water parcel from its position r, where it is neutrally buoyant, to a new position r + dr in an adiabatic and isohaline manner. To first order, the in situ density of the parcel at the new location r + dr is ρ(r) + ρ_P∇p ⋅ dr, while the in situ density of the environment at r + dr is ρ(r) plus

\nabla ρ \cdot d r = (ρ_{S} \nabla S + ρ_{Θ} \nabla Θ + ρ_{P} \nabla p) \cdot d r .

(1)

The fluid parcel is neutrally buoyant (having the same in situ density ρ) at its new location if

(ρ_{S} \nabla S + ρ_{Θ} \nabla Θ) \cdot d r = 0,

(2)

so the NTP is defined as the plane normal to the dianeutral vector

N = ρ_{S} \nabla S + ρ_{Θ} \nabla Θ .

(3)

Projecting (3) onto the NTP yields

ρ_{S} \nabla_{N} S + ρ_{Θ} \nabla_{N} Θ = 0,

(4)

where ∇_NC = ∇C − |N|⁻²(N ⋅ ∇C)N is the gradient of a tracer C (such as S or Θ) projected onto the NTP. Equation (4) shows that, for displacements in the NTP, the contributions of salinity and temperature variations to density are balanced. McDougall et al. (2014) showed that (4) is equivalent to the McDougall (1987a) definition of the NTP,

ρ_{S} \nabla_{n} S + ρ_{Θ} \nabla_{n} Θ = 0,

(5)

where ∇_nC is the gradient of a tracer C along the NTP but with distance measured only in the horizontal. This is a projected nonorthogonal gradient, first introduced by Starr (1945) and widely used in geophysical fluid dynamics. Mathematically,

\nabla_{n} C = \partial_{x} {C |}_{n} \hat{x} + \partial_{y} {C |}_{n} \hat{y},

(6)

where

(\hat{x}, \hat{y})

are the horizontal unit basis vectors and the partial derivatives of C are taken along the NTP. Whereas ∇_NC is 3D, ∇_nC is purely horizontal. To derive a temporal version of (5), one can consider vertical and temporal displacements, letting dr be (dz, dt) and letting the gradient operator ∇ be (∂_z, ∂_t), so that (2) becomes

ρ_{S} \partial_{t} {S |}_{n} + ρ_{Θ} \partial_{t} {Θ |}_{n} = 0,

(7)

having used the standard projected nonorthogonal coordinate transformation ∂_tS|_n = ∂_tS|_z + ∂_zS∂_tz|_n and similarly for ∂_tΘ|_n (Starr 1945), while also identifying ∂_tz|_n as the ratio of the infinitesimal displacements dz and dt when dr = (dz, dt) is a neutral displacement, satisfying (2).

A neutral surface is a 2D surface that is everywhere tangent to the NTP, so (5) holds everywhere on a neutral surface. However, the NTPs can only be meshed together to form a neutral surface if the neutral helicity

H

is zero everywhere (Jackett and McDougall 1997), although this is a necessary but not sufficient condition (Stanley 2019a). Neutral helicity is defined as

H = N \cdot \nabla \times N = (ρ_{Θ} ρ_{S p} - ρ_{S} ρ_{Θ p}) \nabla p \cdot \nabla S \times \nabla Θ = ρ_{S} T_{B} \nabla p \cdot \nabla S \times \nabla Θ,

(8)

where T_B is ρ times the thermobaric parameter of McDougall (1987b):

T_{B} = - ρ_{p Θ} + ρ_{p S} \frac{ρ_{Θ}}{ρ_{S}} = - ρ_{S} \frac{\partial (ρ_{Θ} / ρ_{S})}{\partial p} .

(9)

Due to the nonlinear nature of the equation of state, (ρ_Θρ_Sp − ρ_Sρ_Θ_p) is nonzero. Also, the vectors ∇p, ∇S, and ∇Θ are generally not coplanar in the real ocean, i.e., ∇p ⋅ ∇S × ∇Θ ≠ 0 (McDougall and Jackett 2007). Thus,

H \neq 0

and neutral surfaces are ill-defined in the real ocean.

A neutral trajectory is a 1D curve that is everywhere tangent to the NTP. The nonzero nature of $H$ means that a neutral trajectory that returns to its original geographic (latitude–longitude) position will return at a different depth from which it began (McDougall and Jackett 1988), and this is why the NTPs cannot be connected globally to form a well-defined neutral surface. This helical neutral trajectory is an example of a process that transports water across any ANS even in the absence of dianeutral (turbulent, isotropic) mixing. By moving laterally along neutral trajectories, the large-scale circulation, for example, generates diasurface and even dianeutral motion through this effect of neutral helicity. The diasurface and dianeutral motion caused by motion along a helical neutral trajectory can be achieved without requiring any more than a trivially small amount of small-scale mixing, and this mixing need not be dianeutral but can be purely epineutral (McDougall and Jackett 1988). This is quite different compared with the traditional dianeutral mixing process where the mean dianeutral advection is proportional to the intensity of small-scale mixing processes.

b. Dianeutral processes

The misalignment between the slopes of an ANS and the NTP means that lateral diffusion, if prescribed as acting along the ANS, has a component acting to mix across the NTP. This is quantified by the fictitious dianeutral diffusivity of density (McDougall and Jackett 2005; Klocker et al. 2009),

D^{f} = K s \cdot s = K {| s |}^{2},

(10)

where K is the epineutral diffusivity and

s = \nabla_{n} z - \nabla_{a} z

(11)

is the slope error between the slopes of the NTP and the ANS. Alternatively,

s = \frac{g}{ρ N^{2}} ε,

(12)

where g is the gravitational acceleration, N is the buoyancy frequency satisfying

\frac{ρ N^{2}}{g} = - (ρ_{S} S_{z} + ρ_{Θ} Θ_{z}), and

(13)

ε = ρ_{S} \nabla_{a} S + ρ_{Θ} \nabla_{a} Θ

(14)

is the neutrality error, with ∇_a the projected nonorthogonal gradient in the ANS, defined like ∇_n but acting along the ANS. To derive (12) from (11), use the neutral relation (5) and the coordinate transformations

\nabla_{n} C = \nabla_{z} C + C_{z} \nabla_{n} z,

(15a)

\nabla_{a} C = \nabla_{z} C + C_{z} \nabla_{a} z .

(15b)

for any tracer C. Subtracting Eqs. (15) and using (11) yields the useful identity,

\nabla_{n} C - \nabla_{a} C = C_{z} s .

(16)

Whereas the 3D isotropic turbulent diffusivity D is caused by real physical processes (e.g., breaking internal waves), D^f is related to our human-caused definition of the ANS, but is often comparable in magnitude or larger than D. The opening argument of this paragraph can also be reversed: real lateral diffusion in the NTP has a component acting to mix across the ANS. Thus, fictitious dianeutral diffusion can cause WMT across the ANS, such as diasurface salt or temperature transport.

To derive the dianeutral velocity (through the NTP), begin with the appropriately averaged conservation equations for salinity and temperature expressed with respect to the NTP,

{S_{t} |}_{n} + u \cdot \nabla_{n} S + e S_{z} = h^{- 1} \nabla_{n} \cdot (h K \nabla_{n} S) + {(D S_{z})}_{z},

(17a)

{Θ_{t} |}_{n} + u \cdot \nabla_{n} Θ + e Θ_{z} = h^{- 1} \nabla_{n} \cdot (h K \nabla_{n} Θ) + {(D Θ_{z})}_{z},

(17b)

where S, Θ, and u are the thickness-weighted salinity, temperature, and horizontal velocity obtained by temporally averaging between a pair of closely spaced locally referenced potential density surfaces whose time-mean depth difference is h, as per the (neutral density) temporal residual mean (McDougall and McIntosh 2001; Stewart and Thompson 2015). The averaging needs to be done between adjacent pairs of locally referenced potential density surfaces because NTPs are tangent to these surfaces and the strong lateral mixing of mesoscale eddies occurs between pairs of these surfaces. In (17), the small-scale turbulent diffusivity D acts vertically as is commonly done in ocean modeling, though in reality it acts isotropically; this is done simply because D ≪ K, so the horizontal components of the small-scale turbulent mixing fluxes are trivially small compared to those of the epineutral mixing process.

The mean density conservation equation is derived by multiplying (17a) by ρ_S and (17b) by ρ_Θ and summing with the neutral relationships (5) and (7), yielding the mean dianeutral velocity averaged on the locally referenced potential density surface,

e = D_{z} - D \frac{g}{ρ N^{2}} (ρ_{Θ} Θ_{z z} + ρ_{S} S_{z z}) - K \frac{g}{ρ N^{2}} (ρ_{Θ} \nabla_{n}^{2} Θ + ρ_{S} \nabla_{n}^{2} S)

(18a)

= D_{z} - D \frac{g}{ρ N^{2}} (ρ_{Θ} Θ_{z z} + ρ_{S} S_{z z}) + e^{T_{B}} + e^{C_{B}},

(18b)

where

e^{T_{B}} = - K \frac{g}{ρ N^{2}} T_{B} \nabla_{n} p \cdot \nabla_{n} Θ,

(19a)

e^{C_{B}} = - K \frac{g}{ρ N^{2}} C_{B} \nabla_{n} Θ \cdot \nabla_{n} Θ,

(19b)

with the cabbeling parameter

C_{B} = 2 ρ_{Θ S} \frac{ρ_{Θ}}{ρ_{S}} - ρ_{Θ Θ} - {(\frac{ρ_{Θ}}{ρ_{S}})}^{2} ρ_{S S} .

(20)

The dianeutral velocity e is the mean vertical velocity through the NTP and is quite different to the mean vertical velocity w (see Fig. 1 of Klocker et al. 2009).

On the right-hand side of (18), the first term represents the spatial variation of the dianeutral diffusivity. The second term represents the influence of small-scale mixing caused by bona fide interior dianeutral mixing processes such as breaking of internal waves and double diffusive convection. The last terms in (18b) are the vertical velocities caused by thermobaricity and cabbeling. Thermobaricity occurs when two water parcels come together while conserving their salinities and temperatures, so that they move not along neutral trajectories but along submesoscale coherent vortex (SCV) trajectories (McDougall 1987c). The in situ density of an SCV depends only on pressure since its salinity and temperature are conserved; an SCV adjusts its pressure to always have the same in situ density as the environment. The difference between the SCV trajectory and the neutral trajectory is caused by the compressibility of the ocean ρ_P being a function of salinity and temperature. The cabbeling process occurs when two water parcels of equal density mix intimately at the same pressure. Due to the nonlinear dependence of in situ density on salinity and temperature, the in situ density of the mixed parcel is greater than that of the original water parcels: $R (S_{A}, Θ_{A}, p) = R (S_{B}, Θ_{B}, p) < R [(S_{A} + S_{B}) / 2, (Θ_{A} + Θ_{B}) / 2 p]$ . Like the diasurface and dianeutral motion caused by the helical nature of neutral trajectories, $e^{T_{B}}$ and $e^{C_{B}}$ require only a trivially small amount of small-scale turbulent mixing. Thermobaricity can cause either upwelling or downwelling, while cabbeling is a densification process that only causes downwelling.

The left-hand sides of (17) are the material derivatives of S and Θ, which can be expressed in any coordinate system. Doing so with respect to ANSs gives

{S_{t} |}_{a} + u \cdot \nabla_{a} S + e^{a} S_{z} = h^{- 1} \nabla_{n} \cdot (h K \nabla_{n} S) + {(D S_{z})}_{z},

(21a)

{Θ_{t} |}_{a} + u \cdot \nabla_{a} Θ + e^{a} Θ_{z} = h^{- 1} \nabla_{n} \cdot (h K \nabla_{n} Θ) + {(D Θ_{z})}_{z} .

(21b)

Again, multiplying (21a) by ρ_S and (21b) by ρ_Θ and summing, also using (18) to handle the right-hand side, yields the vertical velocity through the ANS,

e^{a} = e + e^{sp} + e^{tmp}

(22)

where

e^{sp} = u \cdot s,

(23a)

e^{tmp} = \frac{g}{ρ N^{2}} ({ρ_{S} S_{t} |}_{a} + {ρ_{Θ} Θ_{t} |}_{a}) = {z_{t} |}_{n} - {z_{t} |}_{a},

(23b)

having used (12)–(14) for (23a) and their temporal equivalents for (23b). In (22), e^sp represents the vertical velocity through the ANS due to the lateral flow acting along the NTP, and so is a type of diasurface velocity. Equivalently, −e^sp is the vertical velocity through the NTP caused by a layered modeling framework treating the lateral velocity u as acting within an ANS, and so is a type of dianeutral velocity. Depending on one’s application, here the terms “diasurface” and “dianeutral” are two sides of the same coin; for definiteness, we call e^sp the “spurious dianeutral advection.” Similarly, e^tmp is the vertical velocity through the ANS caused by fluid parcels moving vertically to conserve locally referenced potential density (LRPD), i.e., to stay on the NTP as it evolves with temporal changes in the ocean’s hydrography. Klocker and McDougall (2010b) showed that e^tmp is relatively insignificant as a water-mass transformation mechanism, so we focus on e^sp.

The existence of e^sp is independent of other dianeutral processes such as cabbeling and thermobaricity. All three processes (thermobaricity, cabbeling, and the helical nature of neutral trajectories) occur due to the nonlinear nature of the equation of state, and all three have been shown to have a significant influence on the deep ocean circulation and on the conservation equations of salinity and temperature (Klocker and McDougall 2010a). Thermobaricity and cabbeling are real physical processes in the ocean and they cannot be minimized on an ANS. In contrast, by adjusting the depth of an ANS, we can adjust the slope error s (and to a much lesser degree the lateral velocity u but this varies only slightly between two surfaces that are close in depth), and thereby change the spatial pattern and magnitude of e^sp. Because of nonzero neutral helicity, we cannot make s = 0 everywhere, and we show in section 6b that it is also generally impossible to orient s to be orthogonal to u everywhere, which would make e^sp = 0 everywhere. While some e^sp is therefore unavoidable, most ANSs have excessively large |e^sp| because of large slope errors. In section 6, we will create a surface that minimizes a global measure of |e^sp|.

Oceanographers used to use σ₂ surfaces as the density coordinate of inverse models for investigating water-mass transformation. To better separate the effects of epineutral and dianeutral mixing, neutral density γⁿ surfaces are now commonly used [e.g., Fig. 5 of Stanley (2019a) shows D^f is much smaller for γⁿ than for σ₂]. We argue below that using $ω_{u \cdot s + s^{2}}$ surfaces as the vertical coordinate in inverse studies is a better option, because otherwise the tracer (such as temperature or salt) tendency caused by e^sp is a leading-order term that should not be ignored.

3. The method to construct ω surfaces

Here, we review ω surfaces (Klocker et al. 2009; Stanley et al. 2021), which we modify in subsequent sections to create new types of ANSs. Section 3a reviews the relevant theory as developed by Stanley et al. (2021) for ω₊ surfaces; the original theory of ω surfaces developed by Klocker et al. (2009) is not discussed. These two surfaces, ω and ω₊ surfaces, are very similar as physical objects. The major operational distinction is that the computational algorithm for ω₊ surfaces is faster and more robust than that for ω surfaces. After this section, we will refer to ω₊ surfaces simply as ω surfaces.

Henceforth, we apply the Boussinesq approximation, under which the Boussinesq-modified equation of state R is defined by

R (S, Θ, z) = R (S, Θ, - g ρ_{B} z) = ρ,

(24)

where g is the gravitational acceleration and ρ_B is the Boussinesq reference density (Young 2010). The below Boussinesq theory can be easily adapted for the non-Boussinesq case.

a. Theory of ω₊ surfaces

The ω₊ surface algorithm begins with an initial surface, which can be any ANS. The ω₊ surface is found by iteratively improving the neutrality of the surface, as measured locally by the neutrality error ε from (14), which would be zero for a perfectly neutral surface—see (5). The goal of ω₊ surfaces is to minimize the

l^{2}

norm of ε, defined as

{| ε |}_{2} = {(\iint_{A} ε \cdot ε d A)}^{1 / 2}

(25)

where A is the lateral extent of the surface (excluding locations where the surface has out/incropped) and dA is the area increment projected onto the sphere of radius R_Earth ≈ 6.37 m. Technically, the cost function (25) should be normalized by the total area

\iint_{A} d A

to avoid a solution where the surface out/incrops entirely, but this is operationally avoided by using only local information on the surface where it exists, so the optimization does not “know” that better solutions exist by out/incropping. To minimize (25), ω₊ surfaces iteratively apply a two-step process. First, a 2D scalar field Φ is determined by solving the constrained minimization problem

\min_{Φ} {| ε + \nabla Φ |}_{2}^{2}

(26a)

subject to Φ (x_{0}) = 0

(26b)

where x₀ is a reference latitude and longitude. The constraint (26b) is needed to make the solution Φ of (26a) unique, as ∇Φ = ∇(Φ + c) for any constant c. In (26), Φ is a locally referenced potential density perturbation, so the second step is to update the depth of the surface from z to

z^{*}

, which solves

R (S (x, z^{*} (x)), Θ (x, z^{*} (x)), z (x)) = R (S (x, z (x)), Θ (x, z (x)), z (x)) + Φ (x) .

(27)

For each water column (fixed x), (27) is a 1D nonlinear equation for the depth

z^{*}

of the updated surface at which the potential density, referenced to the depth z of the original surface, has increased by an amount Φ. This two-step procedure is iterated until convergence. It succeeds in reducing |ε|₂ because the neutrality error on the updated surface of depth

z^{*}

ε^{*} \approx ε + \nabla Φ,

(28)

with the approximation due to small vertical variations of ρ_S and ρ_Θ (Stanley et al. 2021). Thus, step 1 finds the perturbation Φ that will minimize the

l^{2}

norm of the neutrality error on the updated surface, and step 2 applies this update. The original inspiration for this algorithm arose from Klocker et al. (2009) noting that the two-dimensional curl of ε is roughly fixed by the hydrography (S and Θ), so they sought to perturb ε by an irrotational vector ∇Φ.

b. Discretization

Following Stanley et al. (2021), the constrained optimization problem (26) is discretized as follows. At each iteration, suppose there are N “wet” water columns that the surface intersects; let these be indexed 1, …, N. Suppose there are E pairs of wet water columns that are adjacent, indexed (u₁, υ₁), …, (u_E, υ_E). Let Δ_m_,_n be the distance between (adjacent) water columns m and n, and let

Δ_{m, n}^{⊥}

be the distance of the interface between two grid cells centered at water columns m and n, as shown in Fig. 1. Let

A_{m, n} = Δ_{m, n} Δ_{m, n}^{⊥}

(29)

be the area associated with the region between water columns m and n; this form ensures the adjoint relationship between the gradient and convergence operators carries over to the discrete space (appendix D of Stanley et al. 2021). The salinity and temperature as functions of height z in water column m are S_m(z) and Θ_m(z), respectively. The height of the ANS in water column m is z_m. We write S_m without an argument as shorthand for S_m(z_m), the salinity on the ANS at m; likewise for Θ_m.

Fig. 1.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

The neutrality error on the ANS from cast m to cast n is discretized from (14) as [following Eqs. (15) and (16) from Stanley et al. (2021)]

ε_{m, n} = {(ρ_{S})}_{m, n} \frac{S_{n} - S_{m}}{Δ_{m, n}} + {(ρ_{Θ})}_{m, n} \frac{Θ_{n} - Θ_{m}}{Δ_{m, n}}

(30)

where

{(ρ_{S})}_{m, n} = \partial_{S} R (\frac{S_{m} + S_{n}}{2}, \frac{Θ_{m} + Θ_{n}}{2}, \frac{z_{m} + z_{n}}{2}),

(31a)

{(ρ_{Θ})}_{m, n} = \partial_{Θ} R (\frac{S_{m} + S_{n}}{2}, \frac{Θ_{m} + Θ_{n}}{2}, \frac{z_{m} + z_{n}}{2}) .

(31b)

Then (26) becomes

\min_{(Φ_{1}, \dots, Φ_{N})} \sum_{i = 1}^{E} A_{u_{i}, υ_{i}} {(ε_{u_{i}, υ_{i}} + \frac{Φ_{υ_{i}} - Φ_{u_{i}}}{Δ_{u_{i}, υ_{i}}})}^{2}

(32a)

subject to Φ_{r} = 0

(32b)

where r ∈ {1, …, N} indexes the reference cast at x₀. Writing

A_{u_{i}, υ_{i}} = {(\sqrt{A_{u_{i}, υ_{i}}})}^{2}

to bring it inside the square, the problem (32) can be re-expressed as finding a least squares solution to the overdetermined matrix problem

G_{P}

Φ = −ε₀, i.e., solving

\min_{Φ} {| G_{P} Φ + ε_{0} |}_{2}^{2},

(33)

where Φ = (Φ₁, …, Φ_N)^T,

ε_{0} = {(ε_{u_{1}, υ_{1}} \sqrt{A_{u_{1}, υ_{1}}}, \dots, ε_{u_{E}, υ_{E}} \sqrt{A_{u_{E}, υ_{E}}}, 0)}^{T}

, and

G_{P} = [\begin{matrix} G \\ P \end{matrix}]

is an (E + 1) × N matrix, where (i)

G

is an E × N matrix in which columns υ_i and u_i of row i are defined by

{(G)}_{i, υ_{i}} = A_{u_{i}, υ_{i}}^{1 / 2} Δ_{u_{i}, υ_{i}}^{- 1} = {(Δ_{u_{i}, υ_{i}}^{⊥} / Δ_{u_{i}, υ_{i}})}^{1 / 2} = - {(G)}_{i, u_{i}}

, representing the area-weighted discretized gradient, and (ii)

P

is a 1 × N matrix in which (

P

)_1,_r = 1, representing the pinning constraint. In (33), the squared

l^{2}

norm of a vector is the sum of squares of its components, e.g.,

{| Φ |}_{2}^{2} = \sum_{m = 1}^{N} {(Φ_{m})}^{2} .

(34)

Applying the normal equations to (33), Stanley et al. (2021) used a direct Cholesky solver on the square, symmetric, sparse and full-rank matrix problem

(G_{P}^{T} G_{P}) Φ = - G_{P}^{T} ε_{0},

(35)

the exact solution of which also solves the least squares problem (33). For large grids, this Cholesky solver is considerably faster than the LSQR solver that Klocker et al. (2009) used for (33).

The second step, solving (27), is discretized as follows. For each water column m ∈ {1, …, N}, the depth of the surface z_m is updated to

z_{m}^{*}

that solves

R (S_{m} (z_{m}^{*}), Θ_{m} (z_{m}^{*}), z_{m}) = R (S_{m} (z_{m}), Θ_{m} (z_{m}), z_{m}) + Φ_{m},

(36)

where S_m(z) and Θ_m(z) are the salinity and temperature in cast m evaluated at height z.

4. Minimizing the slope error, s

a. Calculating the slope error, s

Whereas ω surfaces aim to minimize the $l^{2}$ norm of the neutrality error ε, we here develop ω_s surfaces that aim to minimize the $l^{2}$ norm of the slope error s between the NTP and the ANS. This is one step toward minimizing the fictitious dianeutral diffusivity, D^f = K|s|²; we will finish this task in section 5, developing the $ω_{s^{2}}$ surface that minimizes the $l^{2}$ norm of D^f.

To calculate the slope error s, (11) and (12) provide two expressions that are theoretically equivalent. The simpler expression is s = gρ⁻¹N⁻²ε, which only uses information on the surface and is hence a local calculation. An obvious deficiency of this expression is that it can easily overestimate |s| when the stratification is small, N² ≈ 0. Various ad hoc slope regularizations can be used to overcome this, such as adding a minimum stratification threshold; such methods make this local method more stable but less accurate. Generally, this expression s = gρ⁻¹N⁻²ε is theoretically correct only locally, and hence is not well suited for discretized data.

Instead, we use the direct expression, s = ∇_nz − ∇_az, which requires vertically nonlocal information and is stable and accurate with discretized data. Consider two discrete water columns m and n where the ANS depth is z_m and z_n respectively, as illustrated in Fig. 2. The ANS slope ∇_az from m to n is simply (z_n − z_m)/Δ_m_,_n. The required NTP slope ∇_nz is the slope of the NTP from cast m to cast n that has average height z_m_,_n = (z_m + z_n)/2. Thus, we solve for the heights z_m_,_n − ζ_m_,_n/2 on cast m and z_m_,_n + ζ_m_,_n/2 on cast n that are neutrally related, i.e., have equal potential densities referenced to the midpoint, z_m_,_n. Mathematically, we solve

R (S_{m} (z_{m, n} - \frac{ζ_{m, n}}{2}), Θ_{m} (z_{m, n} - \frac{ζ_{m, n}}{2}), z_{m, n}) = R (S_{n} (z_{m, n} + \frac{ζ_{m, n}}{2}), Θ_{n} (z_{m, n} + \frac{ζ_{m, n}}{2}), z_{m, n}),

(37)

for ζ_m_,_n. Note that ζ_n_,_m = −ζ_m_,_n. The NTP slope from cast m to cast n is [(z_n + ζ_m_,_n/2) − (z_m − ζ_m_,_n/2)]/Δ_m_,_n = ζ_m_,_n/Δ_m_,_n. We solve (37) for ζ_m_,_n using Brent’s method (Brent 1973) on an interval obtained by expanding outward, as done by Stanley et al. (2021), from an initial guess of ζ_m_,_n = 0. Combining the above discrete expressions for ∇_nz and ∇_az, the slope error from cast m to cast n is

s_{m, n} = \frac{ζ_{m, n} - (z_{n} - z_{m})}{Δ_{m, n}} .

(38)

Our above method to calculate the NTP slope gives the same results as the vertical nonlocal method (VENM) of Groeskamp et al. (2019), but is numerically more efficient, as Groeskamp et al. (2019) nested an iterative solver for the neutral slope from a fixed point to a water column inside another iterative solver to fix the average depth of the neutrally related points. Hence, we refer to our method (for the NTP slope or the slope error) as VENM also.

Fig. 2.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Note, VENM has slope regularization naturally built in, as a slope can never exceed the depth of the ocean divided by the lateral distance between adjacent water columns (Groeskamp et al. 2019). Consider a case where N² ≈ 0 on the surface but N² is large just above and below the surface, as may occur after a burst of intense turbulent mixing. The local method s = gρ⁻¹N⁻²ε would diagnose a large |s|, whereas VENM “knows about” the density structure in the full water columns, and correctly determines s.

b. Theory of the ω_s surface

Similar to the ω surface, the ω_s surface is created by iteratively optimizing an initial surface. On each iteration, a height perturbation is added to the surface. After updating the surface height from z to

z^{*} = z + ϕ

, the slope error on the new surface will be, to first order,

s^{*} \approx s - \nabla ϕ + H ϕ,

(39)

where ∇ϕ is the slope change of the surface after its height is moved from z to z + ϕ, and

H = \frac{\partial (\nabla_{n} z)}{\partial z}

(40)

is the vertical dependence of the NTP slope, arising from lateral variations of the vertical density structure. For instance, in Fig. 2 the stratification is larger in water column m than in water column n, so the NTP slope steepens as the NTP midpoint moves up. Note, the symbol z always describes height, not depth.

The ω_s surface aims to minimize the

l^{2}

norm of the slope error, again using a two-step iterative process. The first step solves the constrained optimization problem

\min_{ϕ} \iint_{A} s^{*} \cdot s^{*} d A

(41a)

subject to C [ϕ] = c,

(41b)

where

s^{*}

is expressed using (39), and

C [ϕ] = {| A |}^{- 1} \iint_{A} ϕ d A

, where

| A | = \iint_{A} d A

. We did not use the “pinning” method as in (26b) because the unconstrained optimization problem (41a) itself has a unique solution, owing to the presence of the stratification term (40); in contrast, the ω₊ optimization problem (26) is arbitrary up to an additive constant. Upon discretization of these problems, this distinction manifests as a full-rank matrix for the ω_s problem (

M_{s}

, shown in section 4c), whereas the matrix for the ω₊ problem has an eigenvector [1, …, 1]^T whose eigenvalue is precisely zero. The unique solution to the unconstrained ω_s problem involves a large overall height perturbation; adding a pinning constraint would then drag the solution back to the initial height at the reference cast, leading to a height perturbation that has a cusp at the reference cast, being zero there and generally single-signed elsewhere. The resulting ω_s surface would be dependent upon the choice of the pinning location, and two ω_s surfaces that are pinned at different locations could intersect each other. Instead, we constrain the mean height perturbation^¹ to be some nonzero value c in (41b), with c chosen to satisfy the pinning constraint, ϕ(x₀) = 0, where x₀ is the reference cast. To achieve that, let f(c) = ϕ(x₀) where ϕ solves (41) with the specified c, then solve for c in f(c) = 0. This nonlinear root finding problem is solved, again, using Brent’s method (Brent 1973) between a = 0 and b = −1.2f(a), though if f(a)f(b) > 0 then [a, b] is geometrically expanded until a sign change is detected. We use this nonzero mean method to constrain the other surfaces created in this paper as well.

After solving for ϕ, the second step is to update the surface’s height from z to

z^{*}

, simply by adding the height perturbation ϕ:

z^{*} = z + ϕ .

(42)

Whereas ω surfaces required a 1D nonlinear solver for their second step, ω_s surfaces second step is trivial, but a 1D nonlinear solver is required to calculate the NTP slope by solving (37), as per VENM.

c. Discretization

To discretize the optimization problem (41), we first require a discretized expression for H in (40). After applying the height perturbation in (42), the average height of the surface between casts m and n is

z_{m, n}^{*} = z_{m, n} + ϕ_{m, n}

where ϕ_m_,_n = (ϕ_m + ϕ_n)/2, so the new NTP slope

ζ_{m, n}^{*} / Δ_{m, n}

satisfies

R (S_{m} (z_{m, n}^{*} - \frac{ζ_{m, n}^{*}}{2}), Θ_{m} (z_{m, n}^{*} - \frac{ζ_{m, n}^{*}}{2}), z_{m, n}^{*}) = R (S_{n} (z_{m, n}^{*} + \frac{ζ_{m, n}^{*}}{2}), Θ_{n} (z_{m, n}^{*} + \frac{ζ_{m, n}^{*}}{2}), z_{m, n}^{*}),

(43)

as per (37). Now, expand each side of (43) as a Taylor series about the NTP locations along the initial surface for each water column. Specifically, expand the left-hand side about [S_m(z_m_,_n − ζ_m_,_n/2), Θ_m(z_m_,_n − ζ_m_,_n/2), z_m_,_n] and the right-hand side about [S_n(z_m_,_n + ζ_m_,_n/2), Θ_n(z_m_,_n + ζ_m_,_n/2), z_m_,_n]. The zeroth-order terms, namely, R applied to these two locations, cancel by the NTP relation, (37). Next, apply another Taylor series for the vertical S and Θ structure, e.g.,

S_{m} (z_{m, n}^{*} - ζ_{m, n}^{*} / 2) = S_{m} (z_{m, n} - ζ_{m, n} / 2) + S_{m}^{'} (z_{m, n} - ζ_{m, n} / 2) [ϕ_{m, n} - (ζ_{m, n}^{*} - ζ_{m, n}) / 2]

to first order, where the prime indicates a derivative, e.g.,

S_{m}^{'} = d S_{m} / d z

. Collecting the

ζ_{m, n}^{*} - ζ_{m, n}

terms, using ϕ_m_,_n = (ϕ_m + ϕ_n)/2, and dividing by Δ_m_,_n yields an expression, accurate to first order in ϕ, for the change in the NTP slope from the initial to the updated surface:

\frac{ζ_{m, n}^{*} - ζ_{m, n}}{Δ_{m, n}} \approx H_{m, n} \frac{ϕ_{m} + ϕ_{n}}{2}

(44)

where

H_{m, n} = \frac{2}{Δ_{m, n}} \frac{{(ρ_{S})}_{n, m} S_{n, m}^{'} - {(ρ_{S})}_{m, n} S_{m, n}^{'} + {(ρ_{Θ})}_{n, m} Θ_{n, m}^{'} - {(ρ_{Θ})}_{m, n} Θ_{m, n}^{'} + {(ρ_{z})}_{n, m} - {(ρ_{z})}_{m, n}}{{(ρ_{S})}_{m, n} S_{m, n}^{'} + {(ρ_{S})}_{n, m} S_{n, m}^{'} + {(ρ_{Θ})}_{m, n} Θ_{m, n}^{'} + {(ρ_{Θ})}_{n, m} Θ_{n, m}^{'}},

(45)

with (ρ_S)_m_,_n = ∂_SR(S_m(z_m_,_n − ζ_m_,_n/2), Θ_m(z_m_,_n − ζ_m_,_n/2), z_m_,_n) and

S_{n, m}^{'} = S_{m}^{'} (z_{m, n} - ζ_{m, n} / 2)

, and similarly for the Θ and z terms. Recalling that ζ_m_,_n = −ζ_n_,_m, we also have (ρ_S)_n_,_m = ∂_SR(S_n(z_m_,_n + ζ_m_,_n/2), Θ_n(z_m_,_n + ζ_m_,_n/2), z_m_,_n) and

S_{n, m}^{'} = S_{n}^{'} (z_{m, n} + ζ_{m, n} / 2)

, etc. Thus, the discretized form of (39) is

s_{m, n}^{*} \approx s_{m, n} - \frac{ϕ_{n} - ϕ_{m}}{Δ_{m, n}} + H_{m, n} \frac{ϕ_{m} + ϕ_{n}}{2} .

(46)

Equation (46) leads to the discrete form of (41) as

\min_{(ϕ_{1}, \dots, ϕ_{N})} \sum_{i = 1}^{E} A_{u_{i}, υ_{i}} {(s_{u_{i}, υ_{i}} - \frac{ϕ_{υ_{i}} - ϕ_{u_{i}}}{Δ_{u_{i}, υ_{i}}} + H_{u_{i}, υ_{i}} \frac{ϕ_{u_{i}} + ϕ_{υ_{i}}}{2})}^{2}

(47a)

subject to \frac{1}{| A |} \sum_{m = 1}^{N} A_{m} ϕ_{m} = c,

(47b)

where

| A | = \sum_{m = 1}^{N} A_{m}

. As in section 3b, the problem (47) can be re-expressed as finding a least squares solution to the overdetermined matrix problem

M_{s}

ϕ = −s with constraint (47b), i.e., solving

\min_{ϕ} {| M_{s} ϕ + s |}_{2}^{2}

(48a)

subject to C ϕ = c

(48b)

where ϕ = (ϕ₁, …, ϕ_N)^T,

s = {(s_{u_{1}, υ_{1}}, \dots, s_{u_{E}, υ_{E}})}^{T}

M_{s}

= −

G

H

, where

G

is as in (33),

H

an E × N matrix with

{(H)}_{i, u_{i}} = {(H)}_{i, υ_{i}} = A_{u_{i}, υ_{i}}^{1 / 2} H_{u_{i}, υ_{i}} / 2

, and

C = {(\sum_{m = 1}^{N} A_{m})}^{- 1} (A_{1}, \dots, A_{N})

. We solve (48) for ϕ using a constrained linear least squares solver (lsqlin in MATLAB). With ϕ known, the trivial vertical update (42) is then applied in each water column,

z_{m}^{*} = z_{m} + ϕ_{m} .

(49)

This two-step procedure is repeated until convergence, yielding the ω_s surface.

5. Minimizing the fictitious dianeutral diffusivity

The ω_s surface minimizes the $l^{2}$ norm of the slope error s, which significantly decreases the overall fictitious dianeutral diffusivity, D^f = Ks ⋅ s from (10). Now, we create the $ω_{s^{2}}$ surface that minimizes the $l^{2}$ norm of D^f. The theoretical and numerical details are similar to those of the ω_s surface in section 4, but for the following modifications.

Using (39), the squared slope error on the updated surface is, to first order in ϕ,

{| s^{*} |}^{2} \approx {| s |}^{2} - 2 s \cdot \nabla ϕ + 2 s \cdot H ϕ,

(50)

which leads to the minimization problem

\min_{ϕ} \iint_{A} {[K ({| s |}^{2} - 2 s \cdot \nabla ϕ + 2 s \cdot H ϕ)]}^{2} d A

(51a)

subject to C [ϕ] = c .

(51b)

To discretize (51) for ocean datasets on a staggered grid, we must first consider how to discretize s ⋅ s, given that components of s live at separate locations, namely, the faces between tracer cells. In section 6 we will consider u ⋅ s as living on the tracer cell center, allowing cancellation between positive and negative contributions to the dot product. Here, however, s ⋅ s is positive definite and the epineutral diffusivity occurs on the faces between tracer cells, so we wish to minimize the resulting fictitious dianeutral diffusivity also on the faces between tracer cells. That is, with K_m_,_n = K_n_,_m the epineutral diffusivity between adjacent water columns m and n (taken to be depth independent for simplicity), we are concerned with

K_{m, n} {(s_{m, n}^{*})}^{2}

over all pairs of adjacent water columns. Using the discrete expression (46) for

s_{m, n}^{*}

, (50) becomes

{(s_{m, n}^{*})}^{2} = s_{m, n}^{2} - 2 s_{m, n} \frac{ϕ_{n} - ϕ_{m}}{Δ_{m, n}} + s_{m, n} H_{m, n} (ϕ_{m} + ϕ_{n}),

(52)

so (51) becomes

\min_{(ϕ_{1}, \dots, ϕ_{N})} \sum_{i = 1}^{E} A_{u_{i}, υ_{i}} {K_{u_{i}, υ_{i}} [s_{u_{i}, υ_{i}}^{2} - 2 s_{u_{i}, υ_{i}} \frac{ϕ_{υ_{i}} - ϕ_{u_{i}}}{Δ_{u_{i}, υ_{i}}} + s_{u_{i}, υ_{i}} H_{u_{i}, υ_{i}} (ϕ_{u_{i}} + ϕ_{υ_{i}})]}^{2}

(53a)

subject to \frac{1}{| A |} \sum_{m = 1}^{N} A_{m} ϕ_{m} = c .

(53b)

The solution of (53) is handled similarly to the solution of (47). Specifically, the matrix problem is

\min_{ϕ} {| M_{D} ϕ + D^{f} |}_{2}^{2}

(54a)

subject to C ϕ = c

(54b)

where

{(D^{f})}_{i} = A_{u_{i}, υ_{i}}^{1 / 2} K_{u_{i}, υ_{i}} s_{u_{i}, υ_{i}}^{2}

and

M_{D}

is an E × N matrix with

{(M_{D})}_{i, υ_{i}} = A_{u_{i}, υ_{i}}^{1 / 2} K_{u_{i}, υ_{i}} s_{u_{i}, υ_{i}} (2 Δ_{u_{i}, υ_{i}}^{- 1} + H_{u_{i}, υ_{i}})

and

{(M_{D})}_{i, u_{i}} = A_{u_{i}, υ_{i}}^{1 / 2} K_{u_{i}, υ_{i}} s_{u_{i}, υ_{i}} \times (2 Δ_{u_{i}, υ_{i}}^{- 1} + H_{u_{i}, υ_{i}})

6. Minimizing the spurious dianeutral advection

Due to neutral helicity, the ω surface and the ω_s surface have residual neutrality errors (ε) or slope errors (s), which will cause nonzero e^sp, from (23a). The ω_u_·_s surface aims to minimize the $l^{2}$ norm of the spurious dianeutral advection e^sp.

a. The importance of the ω_u_·_s surface in inverse studies

Inverse studies and analysis of ocean model output both widely use the water-mass transformation (WMT) equation, which is found by eliminating dianeutral advection between Eqs. (18) and (17b), obtaining

{Θ_{t} |}_{n} + u \cdot \nabla_{n} Θ = \frac{1}{h} \nabla_{n} \cdot (h K \nabla_{n} Θ) + K Θ_{z} \frac{g}{ρ N^{2}} (T_{B} \nabla_{n} p \cdot \nabla_{n} Θ + C_{B} \nabla_{n} Θ \cdot \nabla_{n} Θ) + D ρ_{S} \frac{g}{ρ N^{2}} (Θ_{z} S_{z z} - S_{z} Θ_{z z}) .

(55)

Compared with the regular conservation Eq. (17b):

Equation (55) is affected not only by epineutral diffusion, h⁻¹∇_n ⋅ (hK∇_nΘ), but also by the nonlinear dianeutral advection processes of cabbeling and thermobaricity [the second line of Eq. (55) is $- (e^{C_{B}} + e^{T_{B}}) Θ_{z}$ , recalling (19)].
Equation (55) is not influenced by the vertical variation of D.
Equation (55) is affected by dianeutral turbulent mixing only to the extent that the vertical S–Θ diagram is not locally straight, which is seen by writing $(Θ_{z} S_{z z} - S_{z} Θ_{z z}) = Θ_{z}^{3} d^{2} S / d Θ^{2}$ . Hence Eq. (55) is more accurately evaluated than (17b) because the term involving D is insensitive to the vertical heaving and squeezing of the water column.

These features are why the WMT equation in the form (55) is the choice of inverse modelers (e.g., Zika et al. 2010; Hautala 2018; Kouketsu 2018, 2021; Finucane and Hautala 2022).

Recall from (17) that the lateral velocity u on the left-hand side is the thickness-weighted average, which an inverse model decomposes as the sum of an Eulerian mean velocity and an eddy-induced (quasi-Stokes) velocity (McDougall and McIntosh 2001); the latter is parameterized as proportional to an epineutral diffusivity (perhaps but not necessarily the same diffusivity as for tracers, K) and moved to the right-hand side of (55), e.g., as done by Zika et al. (2010).

When doing inverse studies, (55) is typically applied on an ANS. Substituting (22) into (21b) and again using (18), we get the WMT equation for temperature along an ANS:

{Θ_{t} |}_{a} + u \cdot \nabla_{a} Θ + (e^{sp} + e^{tmp}) Θ_{z} = \frac{1}{h} \nabla_{a} \cdot (h K \nabla_{a} Θ) + K \frac{g}{ρ N^{2}} Θ_{z} (T_{B} \nabla_{a} p \cdot \nabla_{a} Θ + C_{B} \nabla_{a} Θ \cdot \nabla_{a} Θ) + D \frac{g}{ρ N^{2}} ρ_{S} (Θ_{z} S_{z z} - S_{z} Θ_{z z}) + \frac{1}{h} \nabla_{a} \cdot (h K Θ_{z} s) + \frac{1}{h} {[h K (\nabla_{a} Θ + Θ_{z} s)]}_{z} \cdot s + K \frac{g}{ρ N^{2}} Θ_{z} {T_{B} [(p_{z} \nabla_{a} Θ + Θ_{z} \nabla_{a} p) \cdot s + p_{z} Θ_{z} {| s |}^{2}] + C_{B} (2 Θ_{z} \nabla_{a} Θ \cdot s + Θ_{z}^{2} {| s |}^{2})}

(56)

To derive (56), we used (16) which also yields the relation ∇_n ⋅ a = ∇_a ⋅ a + a_z ⋅ s for any 2D vector field a. A similar procedure yields the WMT equation for salinity. McDougall (1991) derived a similar WMT equation along a potential density surface. In contrast, (56) is valid for a general ANS, albeit with the shortcoming that the layer thickness h remains defined in terms of the NTP, not the ANS.

On the right-hand side of (56), the last three lines are corrections that arise when the formulas on the right-hand side of (55) are written with respect to an ANS, not an NTP. For a potential density surface whose reference pressure is distant from the in situ pressure, these terms can be significant. However, for carefully defined ANSs such as ω surfaces, these error terms are small because the lateral gradients of temperature and pressure closely approximate those on the NTPs, since (∇_nΘ − ∇_aΘ) = Θ_zs and (∇_np − ∇_ap) = p_zs from (16), and |s| is small on good ANSs (shown later in Fig. 5). This argument is made more rigorous and these errors are quantified in appendix A.

On the left-hand side of (56), the e^tmp term is very small (Klocker and McDougall 2010b), hence we focus on e^sp over e^tmp. The major difference between the left-hand sides of (55) and (56) is in the lateral advection term, u ⋅ ∇_nΘ = u ⋅ ∇_aΘ + e^spΘ_z. Above, we argued that ∇_nΘ and ∇_aΘ are similar on a good ANS. However, small differences in the direction of ∇_nΘ and ∇_aΘ relative to u can create significant differences between the lateral advections u ⋅ ∇_nΘ and u ⋅ ∇_aΘ. Indeed, ∇_nΘ tends to be very closely aligned with u but this tight relationship is not maintained on an ω surface, for which the angle of ∇_aΘ deviates from the angle of u substantially (shown later in Fig. 10). Thus, e^spΘ_z can be a dominant term in (56), and performing an inverse study in an ANS while ignoring e^spΘ_z—effectively forgetting that an ANS is not perfectly neutral—can make the results of the inverse study less accurate. So, a surface with minimized $\iint_{A} {(e^{sp} Θ_{z})}^{2} d A$ or $\iint_{A} {(e^{sp} S_{z})}^{2} d A$ can significantly increase the accuracy of an inverse study that is performed without explicitly calculating the error terms in (56). Since we cannot minimize both $\iint_{A} {(e^{sp} S_{z})}^{2} d A$ and $\iint_{A} {(e^{sp} Θ_{z})}^{2} d A$ , a good first step is to design a surface, which we call the ω_u_·_s surface, that minimizes $\iint_{A} {(e^{sp})}^{2} d A$ , which will yield small though not minimized $\iint_{A} {(e^{sp} S_{z})}^{2} d A$ and $\iint_{A} {(e^{sp} Θ_{z})}^{2} d A$ .

The creation of the ω_u_·_s surface requires knowing the lateral velocity, but this information may not be available to an inverse modeller. So, an inverse model can first be done on an ω₊ surface to estimate the mean geostrophic velocity u_g and the quasi-Stokes (eddy-induced) velocity u⁺. Next, to construct the ω_u_·_s surface, we can use u_g from the ω₊ surface, ignoring the small effect of vertical shear, since the depth difference between those two surfaces is small (shown in section 9). (For even greater accuracy, we could update the velocity by running an inverse model after each iteration of the ω_u_·_s-surface algorithm.) Thus, we can perform an inverse model on an ω_u_·_s surface using u_g on the left-hand side of (56). Alternatively, we could construct the ω_u_·_s surface using u_g + u⁺ (which, neglecting the mean ageostrophic velocity, is the TRM velocity) and perform an inverse model using this velocity on the left-hand side of (56). However, this complicates the inversion because u_g + u⁺ is generally not representable by a streamfunction, whereas using u_g is represented by the mean geostrophic streamfunction which serves to connect different boxes of the inverse model.

b. Does a surface with u ⋅ s = 0 everywhere exist?

Due to the helical nature of neutral trajectories, the neutrality error ε and the slope error s on a spatially extensive surface cannot be zero everywhere. However, does a surface with u ⋅ s = 0 everywhere exist? Can we adjust a surface so that its slope error is everywhere orthogonal to the lateral velocity on the surface?

Consider a special case where the velocity is barotropic (depth independent), and there exist closed streamlines. The neutral trajectory (in three dimensions) that follows such a closed streamline laterally will, due to the helical nature of the neutral trajectories, have some vertical pitch. When we traverse this same loop in (x, y) but on an approximately neutral surface, this trajectory must have some nonzero slope error in the direction of the loop. Since the component of u in the direction of the loop is positive everywhere on the loop, we conclude that u ⋅ s is nonzero at least somewhere on this loop, and hence nonzero somewhere on the surface. Thus, in general there does not exist a surface with u ⋅ s = 0 everywhere.

c. Mathematics of the ω_u_⋅_s surface

Similar to the ω_s surface, the ω_u_·_s surface is achieved by optimizing an initial ANS by iteratively adding a height perturbation ϕ. The e^sp on the updated surface is determined by both the velocity and the slope error on the updated surface. The velocity on the updated surface,

u^{*}

, is estimated in terms of the velocity u and shear u_z on the current surface as, to first order,

u^{*} \approx u + u_{z} ϕ .

(57)

The slope error on the updated surface,

s^{*}

, is given by (39). So e^sp on the updated surface is

{(e^{sp})}^{*} = u^{*} \cdot s^{*} \approx (u + u_{z} ϕ) \cdot (s - \nabla ϕ + H ϕ) = e^{sp} - u \cdot \nabla ϕ + (u_{z} \cdot s + u \cdot H) ϕ + O (ϕ^{2})

(58)

where

O (ϕ^{2})

represents terms that are second order in ϕ, which we ignore. Setting

{(e^{sp})}^{*} = 0

as we desire (aware that we will not achieve this exactly) renders (58) as a first-order linear inhomogeneous partial differential equation (PDE) for ϕ, which can be written in a compact form,

u \cdot \nabla ϕ = f (ϕ),

(59)

where f(ϕ) = e^sp + (u ⋅ s + u ⋅ H)ϕ. Assuming |u| ≠ 0, dividing the PDE (59) by |u| shows that the gradient of ϕ along an integral curve (streamline) of u is f(ϕ)/|u|.

For Eq. (59) to possess a unique solution requires that a value of ϕ be specified at one point on each integral curve of u. Then, the unique solution ϕ can be determined by the method of characteristics. If we instead impose a single constraint as in section 4b, effectively specifying ϕ at one point, globally, then ϕ can be modified by an arbitrary additive constant on every streamline other than the one through this one constrained point: the solution is not unique, and hence the PDE is ill posed. Note that a well-posed PDE requires the solution to exist, be unique, and be stable (the solution varies continuously with variations of the initial conditions). Finally, note that if f(ϕ) were independent of ϕ, i.e., f(ϕ) = e^sp, then a necessary condition for existence of solutions of Eq. (59) is that

\oint_{P} e^{sp} {| u |}^{- 1} d l = 0

(60)

for all closed streamlines

P

. However, under generic conditions, f(ϕ) is dependent upon ϕ and a solution exists even if (60) does not hold, as ϕ adjusts to satisfy

\oint_{P} [e^{sp} + (u_{z} \cdot s + u \cdot H) ϕ] {| u |}^{- 1} d l = 0 .

(61)

d. The numerical method to minimize e^sp

To prepare for the numerical ω_u_·_s surface algorithm, we first convert the PDE (59) into an optimization problem, namely, to minimize the area-integral of

{[{(e^{sp})}^{*}]}^{2}

. Under generic conditions, the PDE (59) is solvable and there exists ϕ that makes the cost function

\iint {[{(e^{sp})}^{*}]}^{2} d A

equal zero. Under special conditions in which (59) is not solvable, this optimization problem nonetheless is solvable, albeit with a nonzero cost function. Using the expression (58) for

{(e^{sp})}^{*}

, the ω_u_·_s surface thus requires solving the optimization problem

\min_{ϕ} {\iint_{A} [e^{sp} - u \cdot \nabla ϕ + (u_{z} \cdot s + u \cdot H) ϕ]}^{2} d A

(62a)

subject to C [ϕ] = c .

(62b)

In preparation for the discretized problem, we here impose a single constraint on the area-weighted mean of ϕ; the solution to this problem is not unique, as discussed above. As in section 4, c is chosen so ϕ(x₀) = 0 at reference cast x₀, then the height of the surface is updated as

z^{*} = z + f

. This procedure is iterated until ϕ converges to zero, and thus the second-order terms of ϕ have been ignored in (62).

To discretize (62), we first define the discretized dot product of two vector fields a and b at water column m as

{(a \cdot b)}_{m} = \frac{1}{2 A_{m}} \sum_{n \in N (m)} a_{m, n} b_{m, n} A_{m, n},

(63)

where

N (m) = {u_{i} : υ_{i} = m, i \in {1, \dots, N}} \cup {υ_{i} : u_{i} = m, i \in {1, \dots, N}}

(64)

is the set of water columns that are adjacent to water column m. Equation (63) is derived in appendix B. Using (63), the discretized form of e^sp = u ⋅ s is

e_{m}^{sp} = \frac{1}{2 A_{m}} \sum_{n \in N (m)} u_{m, n} s_{m, n} A_{m, n} .

(65)

Likewise, we could discretize the PDE (59) as

\frac{1}{2 A_{m}} \sum_{n \in N (m)} (u_{m, n} s_{m, n} - u_{m, n} \frac{ϕ_{n} - ϕ_{m}}{Δ_{m, n}} + U_{m, n} \frac{ϕ_{m} + ϕ_{n}}{2}) A_{m, n} = 0

(66)

for each m ∈ {1, …, N}, where

U_{m, n} = {(u_{z})}_{m, n} s_{m, n} + u_{m, n} H_{m, n} .

(67)

Equation (66) constitutes N equations in N unknowns, (ϕ₁, …, ϕ_N), for which there is an exact solution under generic conditions. There are special conditions under which (66) is underdetermined and its solutions are nonunique, such as when u and u_z are exactly aligned with the numerical grid, so that streamlines are exactly captured on the numerical grid and information cannot propagate across these characteristics. Also, under such special conditions, the equivalent of (60) must hold for solutions to exist at all. However, under generic conditions, the advective operator’s discretization introduces some numerical diffusion (cf. section 20.1 of Press et al. 2007), so that information propagates across characteristics and (66) possesses a unique solution. However, we also wish to keep the surface’s height constant at the reference cast, which we achieve as in section 4 by controlling the mean height perturbation. This leads to N + 1 equations in N unknowns, which cannot be solved exactly and thus we seek a least squares solution. Specifically, we focus on the constrained minimization problem (62), the discretized form of which, ignoring a factor of 1/4, is

\min_{(ϕ_{1}, \dots, ϕ_{N})} {\sum_{m = 1}^{N} [\frac{1}{\sqrt{A_{m}}} \sum_{n \in N (m)} (u_{m, n} s_{m, n} - u_{m, n} \frac{ϕ_{n} - ϕ_{m}}{Δ_{m, n}} + U_{m, n} \frac{ϕ_{m} + ϕ_{n}}{2}) A_{m, n}]}^{2}

(68a)

subject to \frac{1}{| A |} \sum_{m = 1}^{N} A_{m} ϕ_{m} = c .

(68b)

The constrained minimization problem (68) can be re-expressed in matrix form as

\min_{ϕ} {| M_{u \cdot s} ϕ + e^{sp} |}_{2}^{2}

(69a)

subject to C ϕ = c .

(69b)

where ϕ = (ϕ₁, …, ϕ_N)^T,

e^{sp} = {(e_{1}^{sp} / \sqrt{A_{1}}, \dots, e_{N}^{sp} / \sqrt{A_{N}})}^{T}

, and

M

_u_⋅_s= −

G

_u_⋅_s +

U

_u_⋅_s where

G

_u_⋅_s and

U

_u_⋅_s are N × N matrices. Specifically, in row m ∈ {1, …, N}, matrix

G

_u_⋅_s has

A_{m}^{- 1 / 2} \sum_{n \in N (m)} (- u_{m, n} / Δ_{m, n}) A_{m, n}

in column m and

A_{m}^{- 1 / 2} (u_{m, n} / Δ_{m, n}) A_{m, n}

in column n and matrix

U

_u_⋅_s has

A_{m}^{- 1 / 2} \sum_{n \in N (m)} U_{m, n} A_{m, n} / 2

in column m and

A_{m}^{- 1 / 2} U_{m, n} A_{m, n} / 2

in column n. Recall from (29) that

A_{m, n} / Δ_{m, n} = Δ_{m, n}^{⊥}

, so

G

_u_⋅_s has terms involving the transport fluxes,

u_{m, n} Δ_{m, n}^{⊥}

. We calculate the velocity u_m_,_n and velocity shear (u_m_,_n)_z on the surface by first interpolating the velocity data between casts m and n as a function of the height data z with the piecewise cubic Hermite interpolant (PCHIP) method (Fritsch and Carlson 1980), obtaining a function u = f(z). Then, we obtain u_m_,_n = f(z_m_,_n) and (u_m_,_n)_z = f′(z_m_,_n), where the latter involves first analytically differentiating f. Note that the method to determine the constant c in the nonzero mean of (68b) is the same as that in section 4b. We can solve (69) using lsqlin in MATLAB, but will actually solve a slightly different, regularized version of the problem to handle an undesirable property of the unconstrained linear system

M_{u \cdot s} ϕ = - e^{sp}

, discussed next.

e. Properties of the linear system $M_{u \cdot s} ϕ = - e^{sp}$

1) Ill-conditioned system

Numerically, we solve u ⋅ ϕ = f(ϕ) not by the method of characteristics, but by setting up a matrix problem for ϕ, which discretizes the problem with a finite volume method. The ill-posed PDE leads to a discretized matrix problem involving $M$ _u_⋅_s that is ill conditioned, meaning the condition number of $M$ _u_⋅_s is very large—as large as 3.66 × 10¹⁵, which shows our system is ill conditioned.

In our problem, one reason that the system is ill conditioned is because of places where |u| is very small, relative to the typical |u|. Neglecting (u_z ⋅ s + H) in (59)—as generally these U_m_,_n terms are about 1% the magnitude of the u_m_,_n terms—then ϕ is the integral of e^sp/|u| along the characteristic (plus the initial condition for ϕ specified at one point on the characteristic), so ϕ is sensitive to the values of e^sp at locations where |u| ≈ 0.

2) The checkerboard numerical mode

The solution x of an ill-conditioned system $A$ x = b, where $A$ is an N × N matrix and x and b are N-dimensional vectors, is very sensitive to the input b. The solution of such systems (in our case, ϕ) normally is not smooth; in our case, the solution ϕ will exhibit a “checkerboard” structure, with values alternating between positive and negative on the grid scale. We can see the checkerboard shape on the e^sp map of the ω_u_·_s surface, shown in Fig. 3. (Details on the calculation of Fig. 3 are given in section 8.) We will ameliorate this checkerboard mode by regularizing the problem, as has been done in Fig. 3 and is discussed next, but first let us discuss the phenomenon itself.

Fig. 3.

Map of e^sp = u ⋅ s (m s⁻¹), with arrows indicating lateral velocity for the ω_u_·_s surface in the North Atlantic Ocean.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

The root of the checkerboard numerical mode lies in the unconstrained problem (69a), so we restrict attention to that. The solution to (69a) is the solution to $M_{u \cdot s}^{T} M_{u \cdot s} ϕ = - M_{u \cdot s}^{T} e^{sp}$ , which in this case is also the exact solution to $M_{u \cdot s}^{T} M_{u \cdot s} ϕ = - e^{sp}$ , since $M$ _u_⋅_s is full rank—though just barely, being ill conditioned. It is useful to use $A \equiv M_{u \cdot s}^{T} M_{u \cdot s}$ for this discussion, however, as $A$ is real and symmetric, hence its eigenvectors (v₁, …, v_N) form a complete basis. Now, the checkerboard mode arises because $A$ is ill conditioned and the eigenvector of the smallest eigenvalue of $A$ has a checkerboard shape. Our matrix problem is $A$ ϕ = b where $b \equiv M_{u \cdot s}^{T} e^{sp}$ . The right-hand side can be expressed as a linear combination of these eigenvectors, $b = \sum_{i = 1}^{N} b_{i} v_{i}$ . Each eigenvector satisfies $A$ v_i = λ_iv_i, so the solution is $ϕ = \sum_{i = 1}^{N} b_{i} λ_{i}^{- 1} v_{i}$ . As $A$ is ill conditioned, the smallest eigenvalue λ₁ is very small relative to the other eigenvalues. Hence, if the input datum b is perturbed slightly so that b₁ changes slightly, then the solution ϕ will change dramatically according to the structure of v₁. In our case, v₁ exhibits a “checkerboard” structure, which thus appears in the height perturbation ϕ.

The reason this checkerboard structure arises is because the matrix equation (68a) couples ϕ_m on the tracer cell to ϕ_n at the four surrounding points on our rectilinear grid. In particular, consider that u is dominated by its rotational, nondivergent component as per geostrophic flow, and that the shear and stratification terms are relatively small, i.e., the U_m_,_n terms are small relative to the u_m_,_n terms in (69a). Taking this to the limit, where U_m_,_n = 0 and $\sum_{n \in N (m)} u_{m, n} A_{m, n} / Δ_{m, n} = 0$ , the diagonal of $M$ _u_⋅_s becomes precisely zero. Thus, the equation for a central “black” square does not involve the value of ϕ on that black square, but only involves ϕ on the four surrounding “red” squares, and vice versa. This decouples the system into a set of equations for the red squares, and another for the black squares, producing a checkerboard numerical mode. Physically, this manifests as two positive values of u_m_,_ns_m_,_n being nearly balanced by two negative values of u_m_,_ns_m_,_n surrounding a given water column m.

f. Tikhonov regularization

Since our matrix

M_{u \cdot s}^{T} M_{u \cdot s}

is ill conditioned and exhibits a checkerboard numerical mode, we use Tikhonov regularization (see section 19.5 of Press et al. 2007) to increase the condition number and promote a smooth solution. Specifically, we aim to find ϕ that satisfies

\min_{ϕ} {| M_{u \cdot s} ϕ + e^{sp} |}_{2}^{2} + {| λ G ϕ |}_{2}^{2},

(70a)

subject to C ϕ = c .

(70b)

where λ is a constant and

G

is the gradient matrix from (33). The cost function in (70a) is the

l^{2}

norm of

{(e^{sp})}^{*}

augmented by a penalty for height perturbations that are highly oscillatory, such as those exhibiting the checkerboard numerical mode. Writing the

l^{2}

norms in (70a) as the vector’s transpose times the vector itself, we get

{| M_{u \cdot s} ϕ + e^{sp} |}_{2}^{2} + {| λ G ϕ |}_{2}^{2} = ϕ^{T} (M_{u \cdot s}^{T} M_{u \cdot s} + λ^{2} G^{T} G) ϕ + 2 (e^{sp}^{T} M_{u \cdot s}) ϕ + e^{sp}^{T} e^{sp} .

(71)

The last term, being constant, does not affect the minimization and hence is ignored, so (71) expresses (70) as a problem of quadratic programming with equilibrium constraints, which we solve using MATLAB’s quadprog. The matrix

G

^T represents a negative divergence operator, so

G

G

in (71) represents a negative Laplacian operator [see appendix D of Stanley et al. (2021)] that limits high-frequency waves in the solutions. This Tikhonov regularization reduces the condition number of our matrix by increasing the smallest eigenvalue, which makes the problem stable (the solution is not highly sensitive to perturbations of

e^{sp}

) and decreases the checkerboard pattern that otherwise is obvious in the solution. As λ is increased, the checkerboard shape on the e^sp map decreases but |u ⋅ s|₂ increases. We choose λ² = 8 × 10⁻⁹ to balance the checkerboard shape and the magnitude of |u ⋅ s|₂.

g. $ω_{u \cdot s S_{z}}$ surface and $ω_{u \cdot s Θ_{z}}$ surface

From (56) and the similar form for S terms, we can see that we actually need to minimize

\iint_{A} {(e^{sp} S_{z})}^{2} d A

\iint_{A} {(e^{sp} Θ_{z})}^{2} d A

, as discussed in section 6a. So besides the ω_u_·_s surface, we create two more surfaces to minimize

\iint_{A} {(e^{sp} S_{z})}^{2} d A

\iint_{A} {(e^{sp} Θ_{z})}^{2} d A

, called the

ω_{u \cdot s S_{z}}

surface and the

ω_{u \cdot s Θ_{z}}

surface, respectively. For the surface that minimizes

\iint_{A} {(e^{sp} S_{z})}^{2} d A

, applying Taylor series to the first order, then S_z on the updated surface with height perturbation ϕ is

{(S_{z})}^{*} \approx S_{z} + S_{z z} ϕ .

(72)

Then

{(e^{sp} S_{z})}^{*}

on the updated surface is, to first order in ϕ,

{(e^{sp} S_{z})}^{*} \approx {(e^{sp})}^{*} S_{z} + S_{z z} e^{sp} ϕ .

(73)

where

{(e^{sp})}^{*}

is in (58). The numerical form of the (unregularized) constrained minimization problem of

\iint_{A} {(e^{sp} S_{z})}^{2} d A

\min_{(ϕ_{1}, \dots, ϕ_{N})} {\sum_{m = 1}^{N} (\frac{{(S_{z})}_{m}}{\sqrt{A_{m}}} \sum_{n \in N (m)} {u_{m, n} s_{m, n} - u_{m, n} \frac{ϕ_{n} - ϕ_{m}}{Δ_{m, n}} + [{(u_{z})}_{m, n} s_{m, n} + u_{m, n} H_{m, n}] \frac{ϕ_{m} + ϕ_{n}}{2}} A_{m, n} + \frac{{(S_{z z})}_{m} ϕ_{m}}{\sqrt{A_{m}}} \sum_{n \in N (m)} u_{m, n} s_{m, n} A_{m, n})}^{2}

(74a)

subject to \frac{1}{| A |} \sum_{m = 1}^{N} A_{m} ϕ_{m} = c .

(74b)

The matrix for (74) is

M_{u \cdot s S_{z}} = M_{S_{z}} + M_{S_{z z}}

, where matrix

M_{S_{z}}

can be achieved by weighting row m (with m ∈{1, …, N}) of matrix

M

_u_⋅_s with (S_z)_m and matrix

M_{S_{z z}}

has

{(S_{z z})}_{m} / \sqrt{A_{m}} \sum_{n \in N (m)} u_{m, n} s_{m, n} A_{m, n}

in the element (m, m). Likewise, the vector is

e^{s p}

with element m weighted by (S_z)_m. Tikhonov regularization is then added as in section 6f. A similar procedure holds for the

\iint_{A} {(e^{sp} Θ_{z})}^{2} d A

problem. Note that S_z, S_zz, Θ_z, and Θ_zz are evaluated using PCHIPs, as for u_z in section 6d. As PCHIPs are C1 interpolants, S_zz and Θ_zz may be discontinuous where the surface depth crosses model depth levels; however, this effect is not severe as on the right-hand side of (73) the S_zz term is small compared to the S_z term—not only at convergence when the S_zz term goes to zero as ϕ → 0 and

{(e^{sp})}^{*} \to e^{sp}

, but also in the first few iterations. We will discuss the different usefulness of the surfaces created in this paper in section 8.

7. Minimizing the spurious dianeutral advection and fictitious dianeutral diffusivity, together

We have constructed the ω_u_·_s surface to minimize the spurious dianeutral advection e^sp and the $ω_{s^{2}}$ surface to minimize the fictitious dianeutral diffusivity D^f. It is worthwhile to create a surface that minimizes a linear combination of these quantities, called the $ω_{u \cdot s + s^{2}}$ surface, which decreases D^f relative to the ω_u_·_s surface, though it pays the price of a larger e^sp. We create this surface by combining the matrices of the ω_u_·_s surface and the $ω_{s^{2}}$ surface.

On an ANS, misalignment from the NTP causes a spurious dianeutral temperature advection of e^spΘ_z, and a fictitious dianeutral temperature diffusive flux of D^fΘ_zz, and similarly for salinity. To minimize both the undesirable temperature fluxes individually, we want to minimize

{| e^{sp} Θ_{z} |}_{2}^{2} + {| D^{f} Θ_{z z} |}_{2}^{2}

, so we add the cost functions of the ω_u_·_s surface and the

ω_{s^{2}}

surface, weighting the former by Θ_z and the latter by Θ_zz. However, the spurious dianeutral advection and fictitious dianeutral diffusive fluxes for S and Θ are different, i.e., e^spS_z ≠ e^spΘ_z and D^fS_zz ≠ D^fΘ_zz. As such, we cannot achieve the desired minimization for both temperature and salinity fluxes. We can only minimize a chosen combination of these fluxes, such as

\iint_{A} [{(a e^{sp} S_{z})}^{2} + {(a D^{f} S_{z z})}^{2} + {(b e^{sp} Θ_{z})}^{2} + {(b D^{f} Θ_{z z})}^{2}] d A

(75)

with a = b = 1 to treat temperature and salt fluxes evenly, or with a = ρ_S and b = ρ_Θ to minimize the four contributions to the spurious dianeutral advection and diffusion of LRPD. In theory, this is easily achieved by weighting and adding cost functions together (as is done next). In practice, however, this involves terms of first-order importance that contain second derivatives S_zz and Θ_zz, which as discussed in section 6g are discontinuous across model depth levels when interpolating with PCHIPs; this means the depth of the surface would have discontinuities that would be notable when comparing against the depth of, say, an ω₊ surface. To overcome this, a C2 (continuous second derivatives) interpolant would be required, such as cubic splines. However, cubic splines cause unrealistic overshoots when applied to oceanographic data (Barker and McDougall 2020). Without a suitable C2 interpolant, we continue with PCHIPs and instead simply treat Θ_z/Θ_zz = S_z/S_zz = 1000 m ≡ Z as a constant vertical distance, typical of the e-folding scale for Θ and S. Then, we are finding ϕ to solve

\min_{ϕ} Z {\iint_{A} [e^{sp} - u \cdot \nabla ϕ + (u_{z} \cdot s + u \cdot H) ϕ]}^{2} d A + \iint_{A} {[K ({| s |}^{2} - 2 s \cdot \nabla ϕ + 2 s \cdot H ϕ)]}^{2} d A .

(76a)

subject to C [ϕ] = c .

(76b)

The discretized form of (76) is obtained by summing the cost functions in (68) and (53). Including Tikhonov regularization, this leads to the constrained minimization problem

\min_{ϕ} {| [\begin{matrix} Z M_{u \cdot s} \\ M_{D} \end{matrix}] ϕ + [\begin{matrix} Z e^{sp} \\ D^{f} \end{matrix}] |}_{2}^{2} + {| λ G ϕ |}_{2}^{2}

(77a)

subject to C ϕ = c,

(77b)

which is solved similarly to (70). For (77), we use λ² = 5 × 10⁻⁹, slightly smaller than the λ² = 8 × 10⁻⁹ used for the ω_u_·_s surface problem, as the effect of

M_{D}^{T} M_{D}

[which appears in the quadratic programming problem derived from (77)] introduces additional Laplacian-like smoothing structure (as

M

_D is akin to the discretized gradient

G

provided the grid resolution is sufficient that s does not oscillate on the grid scale).

8. Surfaces assessment

The Ocean Comprehensible Atlas (OCCA; Forget 2010) data averaged over 2004–06 is used for the numerical tests in this paper. The following 10 approximately neutral surfaces are compared together for assessment:

The potential density surface (σ_0.75 surface with reference depth z_ref = 750 m) that intersects a reference cast at (0°, 180°) in the equatorial Pacific at a depth of 1500 m. This surface has an isovalue of σ_C = 1031.11 kg m⁻³.
The ω₊ surface (Stanley et al. 2021) initialized from the σ_0.75 surface from (i) and pinned at the same reference cast as the σ_0.75 surface.
The ω_s surface,
the ω_u_·_s surface,
the $ω_{u \cdot s Θ_{z}}$ surface,
the $ω_{u \cdot s S_{z}}$ surface,
the $ω_{s^{2}}$ surface, and
the $ω_{u \cdot s + s^{2}}$ surface are all initialized from the ω₊ surface from ii. Note that all the surfaces iii–viii use the constraint on the mean height perturbation that leads to pinning at the reference cast (0°, 180°).
The pressure-invariant neutral density surface (Lang et al. 2020) γ^SCV = 27.78.
The thermodynamic neutral density surface (Tailleux 2021b,a) γ^T = 27.78, which is an analytic representation of the original, databased version of γ^T developed by Tailleux (2016). Note also that both of the γ^SCV surface and the γ^T surface are at 1500-m depth at the reference cast (0°, 180°).

We will compute e^sp, s, and D^f on these 10 surfaces. For simplicity, we take K = 1000 m² s⁻¹ globally uniform. We recognize that parameterizations of K are evolving as research progresses; we could calculate K based on Groeskamp et al. (2020), but we have instead adopted K = 1000 m² s⁻¹ so that our results can be more easily compared with future results involving spatially varying K. To map |s| and D^f, we use (63), as we used for e^sp, to get $\sqrt{s \cdot s}$ or its square on the tracer cell. However, RMS values for |s| and D^f are calculated from their values on the interfaces between tracer cells, using s_m_,_n from (38).

a. The spurious dianeutral advection e^sp

Figure 4 shows e^sp, from (65), of the 10 ANSs. The area-weighted RMSs of e^sp are listed in Table 2. The RMS of e^sp on the ω_u_·_s surface is the smallest of all, over 100 times smaller than that on the ω₊ surface. Compared with the e^sp distribution of the ω₊ surface, e^sp of the ω_u_·_s surface in the Southern Ocean has been significantly reduced. We also show the RMS of e^sp in the Southern Ocean (data from 55.5° to 65.5°S) in the second column of Table 2. The North Atlantic Ocean still has some significant e^sp because neutral helicity tends to be high in this region and also because there are many closed streamlines in that region (Fig. 3). For a closed streamline, some upwelling and downwelling (positive or negative e^sp) is due to the ocean hydrography and cannot be further minimized; the best way to deal with it in terms of the RMS is to make e^sp evenly distributed along closed streamlines. The upwelling and downwelling depends on both the velocity around the closed streamlines and the neutral helicity. The ω_u_·_s surface algorithm has successfully made e^sp on a closed streamline be closer to its average around the streamline.

Fig. 4.

Global maps of e^sp = u ⋅ s (m s⁻¹) for the 10 surfaces detailed in section 8. The root-mean-square of e^sp is indicated above each panel.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Table 2

The area-weighted root-mean-square of the quantities on the 10 surfaces. For the columns labeled “southern,” data is taken only in the latitude range of [55.5°S, 65.5°S]. The minimum value in each column is listed in bold text.

The ω_s surface has a smaller RMS e^sp than that on the ω₊ surface, due to the smaller slope error on the ω_s surface, as expected. The e^sp on the $ω_{u \cdot s Θ_{z}}$ surface and the $ω_{u \cdot s S_{z}}$ surface are similar in RMS to that of the ω_s surface, over 20 times larger that of the ω_u_·_s surface. Compared with the e^sp on the ω_s surface, the e^sp on the $ω_{u \cdot s Θ_{z}}$ surface and the $ω_{u \cdot s S_{z}}$ surface is more concentrated in certain areas rather than evenly distributed in the Southern Ocean and the North Atlantic Ocean. The e^sp on the $ω_{s^{2}}$ surface is very close to that on the ω_s surface. Compared with the ω_u_·_s surface, the $ω_{u \cdot s + s^{2}}$ surface sacrifices e^sp on the surface to achieve better D^f (shown later), so the e^sp on it is larger than that on the ω_u_·_s surface. Because σ_0.75 is a function of (S, Θ) only, the neutrality error on the potential density surface tends to be nearly perpendicular to the lateral velocity (shown in section 9), so e^sp on the σ_0.75 surface is large but less than one would expect based purely on the magnitude of the slope error: the RMS e^sp on the σ_0.75 surface is 2.8 times larger than that on the ω₊ surface, while for |s| this number is 12.3 (cf. Fig. 5). The γ^SCV surface has the largest RMS e^sp on the global ocean, while the γ^T surface does marginally better, as expected since γ^T is fit to γⁿ which is quite similar to γ^SCV, but γ^T is a function of (S, Θ) only, like σ_0.75. The e^sp result for the γⁿ surface is similar to that of the γ^SCV surface (not shown) because the lateral advection makes a similar contribution to the material derivative of γⁿ and γ^SCV (Lang et al., 2020). However, the γ^SCV surface has much better e^sp performance on the Southern Ocean than the γ^T surface and potential density surface (Table 2). Over the Southern Ocean, the RMS of e^sp on the ω₊ surface is about 4 × 10⁻⁷ m s⁻¹, while for the σ_0.75 surface this number rises to 8 × 10⁻⁷ m s⁻¹; thus, e^sp on these classic surfaces is roughly comparable to and generally larger than the dianeutral velocities driven by cabbeling and thermobaricity $e^{C_{B}}$ and $e^{T_{B}}$ from (19) which Klocker and McDougall (2010a) showed to be of order 10⁻⁷ m s⁻¹ in the Southern Ocean.

Fig. 5.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

In a modeling context, we care about the spurious dianeutral advection of both temperature, e^spΘ_z, and of salt, e^spS_z. The $ω_{u \cdot s Θ_{z}}$ surface and the $ω_{u \cdot s S_{z}}$ surface achieves smallest RMS of e^spΘ_z and e^spS_z, respectively, about 87% and 48% of the corresponding values for the ω_u_·_s surface. However, compared to the ω_u_·_s surface, the $ω_{u \cdot s Θ_{z}}$ surface has an RMS of e^spS_z about 15 times larger, while the $ω_{u \cdot s S_{z}}$ surface has an RMS of e^spΘ_z about 12 times larger. Thus, while the $ω_{u \cdot s S_{z}}$ - and $ω_{u \cdot s Θ_{z}}$ surfaces succeed at minimizing the spurious dianeutral flux of their tracer of focus, this comes at some cost to the other active tracer; the ω_u_·_s surface provides a good compromise at minimizing spurious dianeutral fluxes of both S and Θ. More details can be found in Table 2.

b. Neutrality and fictitious dianeutral diffusivity

As stated in the previous section, the ω_u_·_s surface needs to adjust the direction of the slope error to minimize e^sp, but doing this will sacrifice some neutrality. Figure 5 shows that the bigger slope error of the ω_u_·_s surface compared to that of the ω₊ surface is mainly in the Southern Ocean and the North Atlantic Ocean. The ω_s surface has the smallest slope error (Table 2) because its algorithm minimizes the slope error. The patterns of |s| on the ω_s surface and the ω₊ surface are very similar. The slope error of the $ω_{s^{2}}$ surface is very close to that on the ω_s surface. Compared with the ω_u_·_s surface, the $ω_{u \cdot s + s^{2}}$ surface has smaller slope error. Note that the γ^T surface has a larger slope error than the potential density with a carefully chosen reference pressure.

The fictitious dianeutral diffusivity D^f from (10) is used to quantify the diffusion of all our ANSs. A canonical dianeutral diffusivity is 10⁻⁵ m² s⁻¹ in the deep ocean, below the pycnocline and sufficiently high above the seafloor (MacKinnon et al. 2013). Keeping the fictitious dianeutral diffusivity D^f below this value is important for the integrity of inverse studies that ignore this term. Figure 6 shows that all 10 surfaces meet this criteria; even the potential density surface has an RMS D^f of around 2 × 10⁻⁶ m² s⁻¹, which is due to our good choice of reference depth of 750 m, halfway between the depth of the surface in the main ocean basins (around 1500 m) and the depth of its outcropping (0 m). The RMS D^f on the ω_u_·_s surface actually larger, about 8 × 10⁻⁶ m² s⁻¹, but still less than 10⁻⁵ m² s⁻¹. The $ω_{u \cdot s + s^{2}}$ surface has an RMS D^f over 10 times smaller than that on the ω_u_·_s surface, showing that the $ω_{u \cdot s + s^{2}}$ surface successfully controls the diffusion on the ω_u_·_s surface. The $ω_{s^{2}}$ surface achieves the smallest D^f of all, as expected.

Fig. 6.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

c. The comparison between advection and diffusion

From (17b), the contribution to temperature tendency from spurious dianeutral advection is e^spΘ_z and from fictitious dianeutral diffusion is D^fΘ_zz. So e^spΘ_z + D^fΘ_zz is the net spurious temperature tendency, and similarly e^spS_z + D^fS_zz is the net spurious salt tendency. As in section 7, we treat Θ_z/Θ_zz ≈ S_z/S_zz ≈ 1000 m (a characteristic value for the ocean pycnocline) and so map |e^sp| + D^f/10³ m in Fig. 7 to roughly show the total spurious temperature or salt tendency on an ANS. With this approximation, |e^sp| + D^f/1000 m on the $ω_{u \cdot s + s^{2}}$ surface is the smallest among all the 10 surfaces as expected, which is about 35 times smaller than that on the ω₊ surface. The spurious temperature (or salt) tendency e^sp + D^f/1000 m on the ω_s surface and the $ω_{s^{2}}$ surface are quite similar, and smaller than that on the ω₊ surface. The e^sp + D^f/1000 m on the $ω_{u \cdot s Θ_{z}}$ surface and the $ω_{u \cdot s S_{z}}$ surface are of a similar magnitude to the ω_s surface. The RMSs of e^sp + D^f/1000 m of the 10 surfaces are listed in Table 2.

Fig. 7.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Figure 8 shows bivariate histograms for e^sp and |s|² (slope error square). Here, we use e^sp = 3 × 10⁻⁹ m s⁻¹ and |s|² = 10⁻⁸ to be acceptable upper bounds when an ANS is used in an inverse model. We base these limits upon an advective–diffusive balance as per Munk (1966), in which D/e = Θ_z/Θ_zz ≈ 1000 m, so a canonical vertical diffusivity D of 10⁻⁵ m² s⁻¹ is balanced by a dianeutral velocity e (across the NTP) of 10⁻⁸ m s⁻¹. The e^sp through an ANS should be smaller than, say, one-third of e in order that it could be ignored when doing an inverse study. As for |s|², the magnitude of K|s|² along the ANS should be, say, a third of D if it can be ignored when performing an inverse model (and we here take a more conservative value of K = 300 m² s⁻¹). As can be seen, the ω_u_·_s surface and the $ω_{u \cdot s + s^{2}}$ surface are almost entirely located in the acceptable square in Fig. 8. The $ω_{u \cdot s + s^{2}}$ surface is more acceptable than the ω_u_·_s surface. As for the ω₊ surface, |s|² on it is small but e^sp on it is too large to be ignored in an inverse model. As for the σ_0.75 surface, the γ^SCV surface, and the γ^T surface, almost all the data are outside of the acceptable box, which means using them for an inverse study will make the results unreliable because the spurious dianeutral advection on them is comparable in magnitude or larger than the background dianeutral advection.

Fig. 8.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

9. The change from the ω₊ surface to the ω_u⋅s surface

Figure 9 shows the depth of the ω₊ surface and depth differences of the other surfaces. Since the $ω_{u \cdot s Θ_{z}}$ surface, the $ω_{u \cdot s S_{z}}$ surface, and the $ω_{u \cdot s + s^{2}}$ surface are based on the ω_u_·_s surface with specific goals, the depth of these three surfaces are compared with that of the ω_u_·_s surface, whereas other surfaces are compared against the ω₊ surface. Figure 9a shows the depth difference between the σ_0.75 surface and the ω₊ surface has an RMS of 30.26 m, and there are large regions where the depth difference between them exceeds 60 m. Figure 9c shows that the ω_s surface is very close to the ω₊ surface in depth, with an RMS difference of 1.82 m.

Fig. 9.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Figure 9d shows the depth difference between the ω₊ surface and the ω_u_·_s surface. The changes in the Southern Ocean and the North Atlantic Ocean are more obvious than in the other areas, which is consistent with neutral helicity being larger in these two regions than in other regions. The RMS of the depth difference between the ω₊ surface and the ω_u_·_s surface is 3.77 m, which means the total depth difference between these two surfaces is not very large.

Figure 9h shows that the depth difference between the $ω_{u \cdot s + s^{2}}$ surface and the ω_u_·_s surface is mainly in the North Atlantic Ocean. The reason is that the largest fictitious dianeutral diffusivity D^f of the ω_u_·_s surface is in the North Atlantic Ocean.

The ω_u_·_s surface may be thought of as adjusting the slope errors (compared with those on the ω₊ surface) to make the slope error closer to being perpendicular to the lateral velocity. The angle ψ between the velocity and the slope error is given by

\cos ψ = \frac{u \cdot s}{| u | | s |} .

(78)

Figure 10 shows that cosψ on the ω₊ surface is spatially incoherent, and typically closer to 1 than 0, which means the velocity and slope error are typically more aligned than perpendicular. The reason is that the ω₊ surface minimizes the

l^{2}

norm of ε, without caring about the angle between the velocity and ε on the surface. On the ω_u_·_s surface, however, the velocity is almost perpendicular to the slope error (cosψ ≈ 0). Considering that u is nearly constant on the scale of a few meters of vertical heave, the ω_u_·_s surface algorithm can reduce e^sp by a combination of reducing |s| and increasing the angle between u and s toward 90°. Evidently, the most effective way to minimize e^sp is to pursue the latter strategy, even at the expense of increasing |s|. Indeed, the slope error on the ω_u_·_s surface is larger than that on the ω₊ surface (Fig. 5). In places such as the Southern Ocean where |u| and |s| are both large (the latter due to large neutral helicity here), the ω_u_·_s surface orients its slope error to be particularly close to orthogonal to u.

Fig. 10.

The cosine of the angle between the lateral velocity u and the slope error s on (a) the ω₊ surface, (b) the ω_u_·_s surface, and (c) the σ_0.75 surface.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Figure 10c shows cosψ on the σ_0.75 surface revealing that u and s are more perpendicular on the σ_0.75 surface, especially in the Southern Ocean, relative to the ω₊ surface. McDougall (1987a) showed that the gradient of Θ in a potential density surface, ∇_σΘ, is parallel to the corresponding gradient in an NTP, ∇_nΘ. This is the same as saying that both a potential density surface and an NTP contain the curve pointing in the direction ∇Θ × ∇S, along which both Θ and S are constant. Given that the material change of salinity and temperature following the fluid flow is small (because mixing processes are in some sense small compared with advection), the 3D velocity is closely aligned with ∇Θ × ∇S. This means that the 2D velocity on both the NTP and the potential density surface is nearly perpendicular to ∇_σΘ and ∇_σS. Note that ∇_σΘ and ∇_σS are parallel since 0 = ∇_σσ = σ_S∇_σS + σΘ∇_σΘ. Thus, the neutrality error in a potential density surface ε_σ = ρ_S∇_σS + ρ_Θ∇_σΘ is nearly perpendicular to the lateral velocity, as illustrated in Fig. 11b. Note that nonzero neutrality error on the σ_0.75 surface is due to ρ_S ≠ σ_S and ρ_Θ ≠ σ_Θ (0 = σ_S∇_σS + σ_Θ∇_σΘ). Figure 10c confirms that on the σ_0.75 surface, s [which is parallel to ε by (12)] and u are nearly perpendicular in the Southern Ocean, where advection is particularly strong. However, due to the large magnitude of the slope error on the σ_0.75 surface, it still results in large e^sp, seen in Fig. 4.

Fig. 11.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

Between Figs. 9d and 10b, we can see that the ω_u_·_s surface changes velocity and the slope error from often parallel (on the ω₊ surface) to mostly perpendicular, and this is done with a very small depth change. How can such a small e^sp be achieved with such a small depth change? Since the depth difference between these surfaces is small and since the vertical shear of u is small in the interior ocean, the difference of u between these surfaces is also small. So the ω_u_·_s surface achieves small e^sp mainly by changing the direction of the slope error. Figure 11a shows a sketch of the change of the neutrality error from the ω₊ surface to the ω_u_·_s surface. We use the neutrality error ε to do the explanation because ε is proportional to the slope error theoretically and it can be expressed in the form of the vector sum of ρ_S∇_aS and ρ_Θ∇_aΘ. The magnitude of ρ_S∇_aS and ρ_Θ∇_aΘ are both much larger than |ε|, so a small change in either ρ_S∇_aS or ρ_Θ∇_aΘ can result in a big change in the direction of ε, seen in Fig. 11a.

10. Summary and discussion

To date, oceanographers have estimated the quality of an approximately neutral surface by focusing on the errors caused by lateral diffusion being applied in the wrong directions. However, there is also spurious advection caused by lateral advection being applied in the wrong directions, and this turns out to be a larger concern. While Klocker and McDougall (2010a) quantified the diasurface flow through an ω surface (which aims to minimize the neutrality error), they emphasized that the area-integrated transport was small, even though this diasurface flow is often a leading-order process, locally. In this paper, we reveal that the evolution equations, as expressed in density coordinates, for the salinity and temperature contain large terms due to the diasurface flow e^sp caused by the misalignment of the density surface with the neutral tangent plane, and that ignoring these terms can cause significant errors in inverse and diagnostic studies. Likewise, numerical ocean models that run using a stack of density layers have a spurious dianeutral flow −e^sp across the neutral tangent plane caused by the lateral advection being applied in the misaligned density layers.

Nonzero neutral helicity means some e^sp is unavoidable; we create a surface that minimizes e^sp. Specifically, we use numerical optimization to minimize the cost function |e^sp|₂, proportional to the area integral of (e^sp)² over the surface, with a constraint to control the mean depth of the surface or the depth at a reference latitude and longitude. The optimized surface is called an ω_u_·_s surface. The algorithm starts from an initial surface, which can be any approximately neutral surface, and |e^sp|₂ is minimized iteratively. Compared with the ω₊ surface, our ω_u_·_s surface can reduce the |e^sp|₂ by a factor of over 100 with only a slight overall depth change. The positive and negative e^sp on the ω_u_·_s surface can be significantly minimized in most of the ocean. For regions with closed streamlines, e^sp is due to the complex ocean hydrography and consequent neutral helicity, and cannot be reduced to zero. The ω_u_·_s surface is less neutral, in terms of the neutrality error ε or the slope error s or the fictitious dianeutral diffusivity D^f, than the ω₊ surface, because the ω_u_·_s surface aims to minimize e^sp while the ω₊ surface aims to minimize ε. The fictitious dianeutral diffusivity of the ω_u_·_s surface is still lower than the background diffusivity, but only slightly, and in fact is larger than D^f for a potential density surface with a well-chosen reference pressure.

We then developed a surface, called the $ω_{u \cdot s + s^{2}}$ surface, for which the spurious advection and diffusion are both low. To create this surface, we first developed a surface that minimizes |s|₂, proportional to the area-integral of the square of the slope error on the surface, and another that minimizes |D^f|₂ = |Ks²|₂, proportional to the area-integral of the fourth power of the slope error. The cost function for the latter surface, which has the smallest fictitious dianeutral diffusion of all surfaces, was then combined with the cost function of the ω_u_·_s surface, yielding a method to construct the $ω_{u \cdot s + s^{2}}$ surface. The spurious diffusion and advection across the $ω_{u \cdot s + s^{2}}$ surface can be controlled, according to the user’s needs, by altering the relative weighting of the two cost functions for e^sp and D^f. The $ω_{u \cdot s + s^{2}}$ surface achieves an RMS e^sp of 1.3 × 10⁻⁹m s⁻¹ and an RMS D^f of 6.2 × 10⁻⁷ m² s⁻¹, both well below the canonical background values of 10⁻⁸ m² s⁻¹ and 10⁻⁵ m² s⁻¹, and thus is recommended for use in inverse studies. If the velocity is not known to the inverse modeler, it can be estimated by first running an inverse model on an ω₊ surface (or another ANS), and then using that velocity to construct an $ω_{u \cdot s + s^{2}}$ surface and performing the inverse study on this $ω_{u \cdot s + s^{2}}$ surface.

On the journey to create the $ω_{u \cdot s + s^{2}}$ surface, we also created the $ω_{u \cdot s Θ_{z}}$ surface and the $ω_{u \cdot s S_{z}}$ surface, which minimize the spurious diasurface temperature and salt advection, e^spΘ_z and e^spS_z, respectively. However, this minimization comes at the cost of increasing the spurious diasurface advection of the other active tracer (salinity or temperature, respectively); generally, the ω_u_·_s surface provides a better overall outcome.

All the new surfaces presented here require the slope error s to be accurately calculated, which we achieve using a method equivalent to but numerically faster than the vertically nonlocal method (VENM) of Groeskamp et al. (2019). Note that our numerical results are specific to a model (OCCA), so the transferability of our findings to the real ocean remains to be investigated. Also note that the essence of our work is not dependent on the particular definition of the neutrality and slope errors given here [in (11) and (14)] and that are standard in the neutral theory literature (McDougall 1987a; McDougall and Jackett 1988); our new classes of ω surfaces could easily be adapted, for example, to minimize spurious advective fluxes across the planes orthogonal to the P vector of Nycander (2011) or of Tailleux and Wolf (2022). Finally, the ANSs created here are two-dimensional surfaces, but by building a stack of ANSs from the sea surface to the seafloor and labeling each surface with a density value (see appendix A of Stanley et al. 2021), these ANSs may be extended to define three-dimensional density variables (providing depth as a function of latitude, longitude, and density, which may then be inverted to provide density as a function of latitude, longitude, and depth).

Constraining the perturbation’s mean, rather than its value a reference cast, is reminiscent of the original ω-surface formulation by Klocker et al. (2009), which constrained the mean perturbation to be zero.

Acknowledgments.

The authors gratefully acknowledge Australian Research Council support through Grant FL150100090. GJS also acknowledges support from the Banting Postdoctoral Fellowship through funding reference 180031.

Data availability statement.

We used Ocean Comprehensible Atlas (OCCA; Forget 2010) data averaged over 2004–06.

APPENDIX A

The Difference of the Epineutral Mixing, Thermobaric, and Cabbeling Terms between an ANS and the NTP in the WMT Equation

The WMT equation (55) involves lateral gradients and divergences—such as h⁻¹∇_n ⋅ (hK∇_nΘ) for the epineutral term, ∇_np ⋅ ∇_nΘ for the thermobaric term, and ∇_nΘ ⋅ ∇_nΘ for the cabbeling term—that are within the NTP. Calculating these along an ANS will cause error. Here, we show that these errors on our $ω_{u \cdot s + s^{2}}$ surface are small and thus calculating the right-hand side of (55) along this surface, rather than along the local NTPs, is suitable in an inverse study.

Applying (16), the error when calculating h⁻¹∇_n ⋅ (hK∇_nΘ) along an ANS is

h^{- 1} \nabla_{n} \cdot (h K \nabla_{n} Θ) - h^{- 1} \nabla_{a} \cdot (h K \nabla_{a} Θ) = h^{- 1} \nabla_{a} \cdot (h K s Θ_{z}) + h^{- 1} s \cdot {[h K (\nabla_{a} Θ + s Θ_{z})]}_{z} .

(A1)

Similarly, the error terms in the thermobaric and cabbeling terms are

\nabla_{n} p \cdot \nabla_{n} Θ - \nabla_{a} p \cdot \nabla_{a} Θ = {| s |}^{2} p_{z} Θ_{z} + (p_{z} \nabla_{a} Θ + Θ_{z} \nabla_{a} p) \cdot s,

(A2)

and

\nabla_{n} Θ \cdot \nabla_{n} Θ - \nabla_{a} Θ \cdot \nabla_{a} Θ = {| s |}^{2} Θ_{z}^{2} + 2 Θ_{z} \nabla_{a} Θ \cdot s .

(A3)

Substituting (A1)–(A3) to the right-hand side of (55) and expressing its left-hand side along ANSs gives (56).

Figure A1 compares the error terms of the epineutral mixing, thermobaric, and cabbeling, i.e., the left-hand sides of (A1)–(A3), with these terms evaluated along the $ω_{u \cdot s + s^{2}}$ surface, i.e., just the second term on the left-hand side of (A1)–(A3). For simplicity, we take h as a constant in these calculations. This affects the quantitative maps, but not the overall conclusion, which is that the errors caused by the misalignment of the $ω_{u \cdot s + s^{2}}$ surface from the NTP are small relative to the corresponding terms evaluated along the $ω_{u \cdot s + s^{2}}$ surface.

Fig. A1.

Citation: Journal of Physical Oceanography 53, 6; 10.1175/JPO-D-22-0174.1

APPENDIX B

Derivation of the Discretization of the Dot Product

To derive (63), note that at a given point x, there exists a scalar field χ defined in a neighborhood of x such that a = ∇χ at x. Then a ⋅ b = ∇χ ⋅ b = ∇ ⋅ (χb) − χ∇ ⋅ b at x. The divergences in the last expression lend themselves to discretization on a finite volume grid. So, for a given water column m, a = ∇χ discretizes to

a_{m, n} = \frac{χ_{n} - χ_{m}}{Δ_{m, n}} .

(B1)

Using a centered second-order discretization for the divergence, a ⋅ b = ∇ ⋅ (χb) − χ∇ ⋅ b at water column m is

{(a \cdot b)}_{m} = \frac{1}{A_{m}} \sum_{n \in N (m)} \frac{χ_{m} + χ_{n}}{2} b_{m, n} Δ_{m, n}^{⊥} - \frac{χ_{m}}{A_{m}} \sum_{n \in N (m)} b_{m, n} Δ_{m, n}^{⊥} .

(B2)

Without loss of generality, let χ_m = 0 to annihilate the second sum; then (B2) leads to (63) using (29).

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Background
- a. Neutral surfaces and helicity
- b. Dianeutral processes
3. The method to construct ω surfaces
- a. Theory of ω+ surfaces
- b. Discretization
4. Minimizing the slope error, s
5. Minimizing the fictitious dianeutral diffusivity
6. Minimizing the spurious dianeutral advection
7. Minimizing the spurious dianeutral advection and fictitious dianeutral diffusivity, together
8. Surfaces assessment
9. The change from the ω+ surface to the ωu⋅s surface
10. Summary and discussion
Acknowledgments.
Data availability statement.
APPENDIX A
- The Difference of the Epineutral Mixing, Thermobaric, and Cabbeling Terms between an ANS and the NTP in the WMT Equation
APPENDIX B
- Derivation of the Discretization of the Dot Product
REFERENCES

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Spurious Dianeutral Advection and Methods for Its Minimization

Abstract

Abstract

1. Introduction

2. Background

a. Neutral surfaces and helicity

b. Dianeutral processes

3. The method to construct ω surfaces

a. Theory of ω+ surfaces

b. Discretization

4. Minimizing the slope error, s

a. Calculating the slope error, s

b. Theory of the ωs surface

c. Discretization

5. Minimizing the fictitious dianeutral diffusivity

6. Minimizing the spurious dianeutral advection

a. The importance of the ωu·s surface in inverse studies

b. Does a surface with u ⋅ s = 0 everywhere exist?

c. Mathematics of the ωu⋅s surface

d. The numerical method to minimize esp

e. Properties of the linear system Mu⋅sϕ=−esp

1) Ill-conditioned system

2) The checkerboard numerical mode

f. Tikhonov regularization

g. ωu·sSz surface and ωu·sΘz surface

7. Minimizing the spurious dianeutral advection and fictitious dianeutral diffusivity, together

8. Surfaces assessment

a. The spurious dianeutral advection esp

b. Neutrality and fictitious dianeutral diffusivity

c. The comparison between advection and diffusion

9. The change from the ω+ surface to the ωu⋅s surface

10. Summary and discussion

Acknowledgments.

Data availability statement.

APPENDIX A

The Difference of the Epineutral Mixing, Thermobaric, and Cabbeling Terms between an ANS and the NTP in the WMT Equation

APPENDIX B

Derivation of the Discretization of the Dot Product

REFERENCES

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Headings

References

a. Theory of ω₊ surfaces

b. Theory of the ω_s surface

a. The importance of the ω_u_·_s surface in inverse studies

c. Mathematics of the ω_u_⋅_s surface

d. The numerical method to minimize e^sp

e. Properties of the linear system $M_{u \cdot s} ϕ = - e^{sp}$

g. $ω_{u \cdot s S_{z}}$ surface and $ω_{u \cdot s Θ_{z}}$ surface

a. The spurious dianeutral advection e^sp

9. The change from the ω₊ surface to the ω_u⋅s surface