## 1. Introduction

Mixing in the ocean influences Earth’s climate through its ability to alter the ocean’s circulation and uptake and distribution of tracers such as heat, oxygen, and carbon. An increased understanding of ocean mixing, both observationally and its representation in models, is necessary to better understand and model the ocean’s influence on the climate system.

Currently climate ocean models parameterize interior ocean mixing as downgradient epineutral diffusion (along-isopycnal diffusion), with diffusion coefficient *K*, and dianeutral downgradient turbulent diffusion due to small-scale mixing, with diffusion coefficient *D* (Redi 1982; Griffies 2004). The spatial- and temporal-varying magnitudes of *K* and *D* are not easily obtained from theory. Therefore, we require empirically (observationally) based estimates to improve our understanding of *K* and *D*.

Munk (1966) provided such an estimate using an approximate balance between (vertical) ocean advection and mixing along with tracer observations to obtain a steady-state estimate of *D*. Munk’s study demonstrated that observed estimates of tracers can be used to obtain estimates of ocean circulation and mixing.

To obtain estimates of the structure and magnitude of the ocean circulation from observations, Stommel and Schott (1977) and Wunsch (1978) introduced inverse methods into the field of oceanography. Ever since, many inverse studies have provided observationally based estimates of circulation (Schott and Stommel 1978; Killworth 1986; Cunningham 2000; Sloyan and Rintoul 2000, 2001) of which some used the diapycnal fluxes to provide estimates for *D* (Ganachaud and Wunsch 2000; Lumpkin and Speer 2007). Zhang and Hogg (1992) and Zika et al. (2010a) developed an inverse method that, simultaneously, solves for the circulation and both *K* and *D*.

Ocean circulation is often represented by a volumetric streamfunction, simplifying the three-dimensional time-varying (global) ocean circulation into a two-dimensional time-averaged circulation. The streamfunction has been defined using different combinations of coordinates, both geographic and thermodynamic (Bryan et al. 1985; Döös and Webb 1994; Hirst et al. 1996; Hirst and McDougall 1998; Nycander et al. 2007). Based on an averaging technique developed by Nurser and Lee (2004), Ferrari and Ferreira (2011) have defined an advective meridional streamfunction *υ* and arbitrary tracer *C*_{1}. Zika et al. (2012) generalized this technique to compute a mean advective streamfunction *C*_{1} and *C*_{2}, for an instantaneous velocity field, **u** = (*u*, *υ*, *w*). Here *C*_{1}, *C*_{2}) coordinates. Zika et al. (2012) applied this to salinity *S* and temperature *T* coordinates to obtain the advective thermohaline streamfunction

Recently, Groeskamp et al. (2014) showed how the total circulation in (*S*, *T*) coordinates is driven by thermohaline forcing, that is, surface freshwater and heat fluxes and salt and heat fluxes by diffusive mixing. They showed that the total circulation in (*S*, *T*) coordinates is in fact a summation of the advective thermohaline streamfunction *S*, *T*) coordinates and is quantified by the diathermohaline streamfunction

Calculation of

To obtain the diathermohaline streamfunction as defined by Groeskamp et al. (2014), we require both diathermal and diahaline transport. To obtain both transports we will, like Speer and Tziperman (1992) and Hieronymus et al. (2014), use salt and heat fluxes in combination with Walin’s framework to estimate water mass transformation rates in (*S*, *T*) coordinates. Here a water mass transformation is a change of the *S* and *T* properties of a water mass, which can result in along- and cross-isopycnal transport (Speer 1993; IOC et al. 2010). Speer (1993) suggested that in a steady-state ocean, the net divergence of the water mass transformation due to surface heat and freshwater fluxes, projected in (*S*, *T*) coordinates, should be balanced by mixing, thus providing constrains on mixing estimates.

In the present paper, we will merge both the Munk (1966) and Walin (1982) frameworks in (*S*, *T*) coordinates, obtaining a balance between surface forcing, mixing, and circulation. This is obtained using a two-dimensional extension into (*S*, *T*) coordinates by applying Walin’s framework to a volume bounded by a pair of isotherms *and* a pair of isohalines (Fig. 1). We can then provide an estimate of the diathermohaline circulation using boundary salt and heat fluxes and diffusive mixing. Representing the diathermohaline circulation by a diathermohaline streamfunction we develop the thermohaline inverse method (THIM) that can be applied to observationally based ocean hydrography and surface heat and freshwater fluxes to simultaneously obtain estimates of both *K* and *D*. We test the THIM by calculating

## 2. Diathermohaline streamfunction

This section is a summary of a derivation by Groeskamp et al. (2014) leading to an expression for the diathermohaline streamfunction *S*_{A} and Conservative Temperature Θ coordinates. Here Conservative Temperature is proportional to potential enthalpy (by the constant heat capacity factor ^{−1}) (IOC et al. 2010; McDougall et al. 2012).

*C*=

*C*(

**x**,

*t*), where

**x**= (

*x*,

*y*,

*z*). Hence,

*C*(Griffies 2004; Groeskamp et al. 2014):Here

**u**=

**u**(

**x**,

*t*) = [

*u*(

**x**,

*t*),

*υ*(

**x**,

*t*),

*w*(

**x**,

*t*)] and the forcing terms

*f*

_{C}=

*f*

_{C}(

**x**,

*t*) and

*m*

_{C}=

*m*

_{C}(

**x**,

*t*) (both in

*C*s

^{−1}) are flux divergences of

*C*due to boundary fluxes and diffusive mixing processes, respectively. Equation (1) shows that

*C*

*C*over time period Δ

*t*leads to a net shift of the geographical position of the surface of constant

*C*. This shift can be expressed as a net velocity of the surface of constant

*C*over Δ

*t*, given byHere

*C*=

*S*

_{A}(

**x**,

*t*) and

*C*= Θ(

**x**,

*t*) in Eq. (1), we can define the diathermohaline velocity vector

*C*=

*S*

_{A}(

**x**,

*t*) and

*C*= Θ(

**x**,

*t*) in Eq. (2), we can define the diathermohaline trend

*S*

_{A}and Θ. Following Groeskamp et al. (2014), for a Boussinesq ocean in which

**∇**·

**u**= 0, the diathermohaline streamfunction

*S*

_{A}, and

*S*

_{A}on a surface of constant Θ. For a statistically steady ocean,

*S*

_{A}, Θ) coordinates, while the diathermohaline trend represents the divergent component of this circulation.

**u**(

**x**,

*t*) is known. An expression that provides

## 3. The diathermohaline volume transport

The diathermohaline velocity

To express the conservation equations we consider a volume Δ*V*, bounded by a pair of isotherms that are separated by ΔΘ (= 2*δ*Θ) and a pair of isohalines that are separated by Δ*S*_{A} (= 2*δS*_{A}). The volume’s Θ ranges between Θ ± *δ*Θ and *S*_{A} ranges between *S*_{A} ± *δS*_{A}. As a result, Δ*V* = Δ*V*(*S*_{A} ± *δS*_{A}, Θ ± *δ*Θ, *t*) may have any shape in (*x*, *y*, *z*) coordinates (Fig. 1a), but it covers a square grid in (*S*_{A}, Θ) coordinates (Fig. 1b).

*V*, at coordinates (

*S*

_{A}, Θ), we define a diathermohaline volume transport vector. The diahaline volume transport, in the positive

*S*

_{A}direction through the area of the surface of constant

*S*

_{A}, for the Θ range of Θ ±

*δ*Θ is given byThe diathermal volume transport, in the positive Θ direction through the area of the surface of constant Θ, for the

*S*

_{A}range of

*S*

_{A}±

*δS*

_{A}, is given byThis derivation results in the diathermohaline volume transport vector

*V*. It would be more accurate to use conservation of mass in a non-Boussinesq ocean, but we leave this for future work. As Δ

*V*= Δ

*V*(

*S*

_{A}±

*δS*

_{A}, Θ ±

*δ*Θ,

*t*), this results in

*S*

_{A}=

*S*

_{A}(

*t*) and Θ = Θ(

*t*), where

*S*

_{A}and Θ can only vary in time within the range

*S*

_{A}±

*δS*

_{A}and Θ ±

*δ*Θ, respectively. Using this and applying the Boussinesq approximation, the conservation of volume, salt, and heat for Δ

*V*is given byHere we have used that the derivative of

*S*

_{A}(Θ), is constant over the interval

*S*

_{A}±

*δS*

_{A}(Θ ±

*δ*Θ), such that we can define the thermohaline divergence operatorThe thermohaline divergence operator [Eq. (10)] is a short-hand notation for taking the difference of the outflow and inflow of volume and accompanied tracers, through the pair of isohalines that enclose Δ

*V*that are separated exactly by Δ

*S*

_{A}, and isotherms separated exactly by ΔΘ.

In Eq. (7), *F*_{m} is the boundary mass flux into Δ*V* due to evaporation *E*, precipitation *P*, ice melt and formation, and river runoff *R*. We assumed that the boundary salt flux is zero, that is, neglecting the formation of sea spray, the interchange of salt with sea ice and salt entering from the ocean boundaries. Although the total amount of salt in the ocean remains constant, the ocean’s salinity is modified by *F*_{m} (Huang 1993; Griffies 2004). The term *F*_{Θ} is the net convergence of heat into Δ*V* due to boundary fluxes. The values *F*_{m} and *F*_{Θ} can be obtained from surface freshwater and heat flux products. The terms *M*_{Θ} are the net convergence of salt and heat into Δ*V* due to *all* interior diffusive processes and can be obtained from an ocean hydrography in combination with a mixing parameterization.

*S*

_{A}/∂Θ = ∂Θ/∂

*S*

_{A}= 0. Taking the time average, this results in the following expression for

*V*due to water mass transformation as a result of a convergence of salt or heat into Δ

*V*. This is equivalent to an extension of Walin (1982)’s framework in (

*S*

_{A}, Θ) coordinates. The vector

**J**vector as defined by Hieronymus et al. (2014), except for the last terms on the rhs of Eqs. (11) and (12).

We will now calculate the terms on the right-hand side of Eqs. (11) and (12) in detail. In section 4, we relate

### a. Boundary salt and heat fluxes

^{−1}) is given by the integral of

*E*−

*P*−

*R*over area

*A*

*, bounded by a pair of isohalines and isotherms:Modification of the ocean’s salinity through*

_{b}*S*

_{A}±

*δS*

_{A}; hence,

*δS*

_{A}approaches 0. A simple scale analysis shows that the heat flux equivalent

*F*

_{Θ}(J s

^{−1}) is the integral of the surface heat flux

*f*

_{h}(

*x*,

*y*) due to longwave and shortwave radiation, and the latent and sensible heat flux, and geothermal heating over area

*A*

_{b}:Note that, if required, one can include effects of solar penetration below the surface (Iudicone et al. 2008).

### b. Diffusive salt and heat fluxes

*K*and 2) small-scale isotropic downgradient turbulent diffusion by means of a turbulent diffusion coefficient

*D*. The isotropic nature of

*D*is discussed in McDougall et al. (2014), but has previously been regarded to be diapycnal (Redi 1982; Griffies 2004; or vertical in the small-slope approximation). To represent epineutral diffusion in Cartesian coordinates, one makes use of a rotated tensor. The component of the diffusive tracer flux (C m

^{3}s

^{−1}), through a general surface of constant

*φ*, with surface area

*A*

_{φ}due to both eddy and turbulent diffusion, projected into Cartesian coordinates is given byThis represents diffusion of any conserved tracer

*C*through any surface

*φ*. Here

*ϵ*=

*D*(1 +

*S*

^{2})/

*K*, andUsing neutral density

*γ*

^{n}(McDougall 1987; Jackett and McDougall 1997),

**S**provides the neutral direction, and

#### 1) Diffusive heat flux

*S*

_{A}range of [

*S*

_{A}±

*δS*

_{A}], can be expressed asAs isotherms and isohalines are not necessarily orthogonal in Cartesian coordinates, there will also be a downgradient diahaline diffusive transport of heat

*S*

_{A}, between the Θ range of [Θ ±

*δ*Θ], given byUsing Eqs. (19) and (20) we construct

*V*given by

#### 2) Diffusive salt flux

*V*is given bywhere

*S*

_{A}, between the Θ range of [Θ ±

*δ*Θ], given byand

*S*

_{A}range of [

*S*

_{A}±

*δS*

_{A}], given by

### c. Local response

*S*

_{A}and Θ properties of Δ

*V*by an amount remaining within the defined (

*S*

_{A}, Θ) grid. As a result, the observed changes in salt and heat do not lead to a diathermohaline transport, but nonetheless reduces the amount of salt and heat available for water mass transformation. Applying Reynolds decomposition and averaging we obtainNote that both

*S*

_{A}, Θ) only within the range defined by

*S*

_{A}±

*δS*

_{A}and Θ ±

*δ*Θ, respectively. Hence,

*δS*

_{A}and

*δ*Θ approach 0.

## 4. The thermohaline inverse model

As we do not know the exact spatial and temporal distribution of *K* and *D* embedded in *M*_{Θ} in Eqs. (11) and (12), we formulate THIM, which uses an inverse technique that enables us to simultaneously estimate *K*, *D*, and

*S*

_{A}direction,and in the Θ direction,Here we used Eqs. (11) and (12) to obtain the second line of Eqs. (27) and (28), andis the diahaline volume transport due to a trend in

*S*

_{A}(

*x*,

*y*,

*z*,

*t*), andis the diathermal volume transport due to a trend in Θ(

*x*,

*y*,

*z*,

*t*). Groeskamp et al. (2014) showed that

For each Δ*V*, two unique equations can be constructed and combined in the form **x** = **b**. Here **x** is a 1 × *M* vector of unknown *K* and *D* coefficients, such that *N*_{K} and *N*_{D} are the number of unknown coefficients used to represent spatial and temporal variation of epineutral and turbulent diffusion and *V* in the *S*_{A} and Θ direction, respectively. The quantity *N* × *M* matrix of their coefficients, with **b** is a 1 × *N* vector of the known forcing terms.

**b**are based on data that includes error, leading to unknown equation error. Hence, we have

*N*unknown equation errors and

*M*unknown variables, leading to

*N*+

*M*unknowns and

*N*equations. Linear dependencies may reduce the effective number of equations

*N*, always resulting in an underdetermined set of equations with an infinite number of solutions. An estimate for

**x**can be obtained using an inverse technique that minimizes

*χ*

^{2}, which is the sum of both the solution error and the equation error (Menke 1984; Wunsch 1996; McIntosh and Rintoul 1997):Rewriting the error as

**e**=

**x**−

**b**and setting ∂

*χ*

^{2}/∂

**x**= 0, the solution can be written asHere

**x**

_{0}is a prior estimate of

**x**,

_{r}is the row (equation) weighting matrix, and

_{c}is the column (variable) weighting matrix. If

**x**and

**e**are jointly normally distributed, the minimization of

*χ*

^{2}is equivalent to finding the most probable solution of

**x**, with a standard deviation given by the square root of the diagonal of the posterior covariance matrix given by (Menke 1984)

_{r}and

_{c}be diagonal with elements 1/

*σ*

_{e}and

*σ*

_{x}, respectively, such that we can rewrite Eq. (31) asTo allow for a similar influence of each variable and equation on the solution, we require all elements of

*χ*

^{2}, and therefore

**x**

_{0}be our best estimate of

**x**, and let

*σ*

_{x}be our best estimate of the error between

**x**

_{0}and

**x**, such that

*σ*

_{e}be our best estimate of the equation error

**e**, such that

*σ*

_{e}compared to

**e**, such that

**x**=

**x**

_{0}, by effectively minimizing

*σ*

_{x}is large compared to

**x**−

**x**

_{0}, then

**x**=

**b**as accurately as possible, regardless of how far

**x**is from

**x**

_{0}. To avoid fitting

**x**toward either the equations or

**x**

_{0}, we suggest that one should find a physically realistic solution from a range of combinations for

**x**

_{0},

*σ*

_{x}, and

*σ*

_{e}, for whichNote that, if prior statistics are not well known, the sensitivity of the solution to the choice of

**x**

_{0},

*σ*

_{x}, and

*σ*

_{e}may be larger than the standard deviation for a particular solution obtained from

_{p}. We refer to the combination of Eqs. (27) and (28) and the described inverse technique as the thermohaline inverse method.

## 5. The THIM applied to a numerical climate model

In this section, we apply the THIM to the hydrography and surface fluxes of an intermediate complexity numerical climate model’s output, where the model’s *K*, and *D* are known.

### a. The University of Victoria Climate Model

We use the final 10 yr of a 3000-yr spinup simulation of the University of Victoria Climate Model (UVIC). This model is an intermediate complexity climate model with horizontal resolution of 1.8° latitude by 3.6° longitude grid spacing, 19 vertical levels, and a 2D energy balance atmosphere (Sijp et al. 2006; the case referred to as GM). The ocean model is the Geophysical Fluid Dynamics Laboratory Modular Ocean Model, version 2.2 (MOM2), using the Boussinesq approximation (*ρ* ≈ *ρ*_{0} = 1035 kg m^{−3}) and a constant heat capacity (*c*_{p} = 4000 J K^{−1} kg^{−1}) with the rigid-lid approximations applied, and the surface freshwater fluxes are modeled by way of an equivalent salt flux (kg m^{−2} s^{−1}) (Pacanowski 1996). The model conserves heat and salt by conserving potential temperature *θ* (°C) and Practical Salinity *S*_{P}. We use monthly averaged *S*_{P} and *θ*.

^{−4}m

^{2}s

^{−1}at the surface and increasing to 1.3 × 10

^{−4}m

^{2}s

^{−1}at the bottom. The model employs the eddy-induced advection parameterization of Gent et al. (1995) with a constant diffusion coefficient of 1000 m

^{2}s

^{−1}. Tracers are diffused in the isopycnal direction with a constant coefficient of

*K*

_{uvic}= 1200 m

^{2}s

^{−1}. The model adopts epineutral diffusion everywhere and uses a slope maximum of

*S*

_{max}= 1/100; any slopes exceeding this limit are set to

*S*

_{max}.

### b. Formulating the THIM for UVIC

*x*,

*y*,

*z*) coordinate tracer location of UVIC,

*S*

_{P}and

*θ*values are given, and one can define six interfaces that encloses a volume. To calculate diffusion, each interface is assumed to be a surface of constant

*S*

_{P}and

*θ*. Adopting constant diffusion through a surface reduces the integral over the surface in Eq. (16) into a multiplication with the surface. Since the unit normal of the surfaces are exactly in the

*x*,

*y*, and

*z*direction, we find the associated interfaces to be

*A*

_{yz}=

*dydz*,

*A*

_{xz}=

*dxdz*, and

*A*

_{xy}=

*dxdy*. In the UVIC model the small-slope approximation has been applied, such that the convergence of salt and heat due to the sum of both epineutral and vertical turbulent downgradient diffusion is given byHere

*S*

_{P},

*θ*) coordinates. We used the fact that

*D*/

*K*≪ 1, and

_{small}isFor the inverse model we use one unknown for both

*K*and

*D*. To take into account the vertical structure of

*D*, we multiply (∂

*S*

_{P}/∂

*zA*

_{xy})|

_{z}and (∂

*θ*/∂

*zA*

_{xy})|

_{z}by the vertical structure of

*D*, given by

*D*

_{uvic}(

*z*) × 10

^{4}. This reduces the variables for small-scale turbulent diffusion to a single parameter (

*D*= 1 × 10

^{−4}m

^{2}s

^{−1}) and yet retains the vertical structure as given by Eq. (36). The isopycnal gradients are obtained by using the locally referenced potential density values (

*σ*

_{n}) to calculate density gradients. The gradients at

*z*=

*z*

_{k}are obtained by finding

*σ*

_{n}=

*ρ*(

*S*

_{P},

*θ*,

*p*

_{k}) for the whole ocean according to McDougall et al. (2003), applied for all depth levels ranging from

*k*= 1:

*N*, where

*N*is the total number of vertical layers.

To obtain the time-averaged net convergence of salt and heat in (*S*_{P}, *θ*) coordinates we (i) sum *M*_{θ} for each volume enclosed by the six interfaces in (*S*_{P}, *θ*) coordinates according to their tracer value on the (*x*, *y*, *z*) coordinate and then (ii) take the time average (Figs. 2, 3). We have chosen grid sizes in (*S*_{P}, *θ*) coordinates (Δ*θ* = 0.75 and Δ*S*_{P} = 0.1) that distinguish the different water masses in the ocean’s interior and provide an approximately equal number of equations in both *S*_{P} and *θ* directions.

*F*

_{θ}is calculated according to Eq. (15). The surface fluxes are gridded according to the surface grid

*S*

_{P}and

*θ*values (Figs. 2, 3). The term

*S*

_{P}and

*θ*of the first month (Figs. 2, 3). Using the above, (27) and (28) applied to UVIC areand,Here

*C*with

*S*

_{P}and

*θ*. Then

Writing Eqs. (41) and (42) for each grid leads to a set of equations that can be written in the form **x** = **b**. Here **x** in *K* and *D* and

### c. The a priori constraints

To provide a physically realistic estimate of **x** using the THIM, we need to include boundary conditions and specify **x**_{0}, *σ*_{x}, and *σ*_{e}. We have omitted equations for which both ^{6} m^{3} s^{−1}), as these equations do not have a signal-to-noise ratio that adds information to the solution. We have also applied this to the heat equations. We have also imposed that transport into a Δ*V* that does not exist in the ocean is zero. That is, we have set *V*’s do not exist in the ocean. The a priori expected magnitude for *K* and *D* are *N* = 2709 and *M* = 1603.

#### 1) Row weighting

*K*=

*K*

_{uvic}and

*D*=

*D*

_{uvic}into the lhs of Eqs. (41) and (42) and taking the absolute value of the difference between both sides of the equations. Assuming that 1) the errors in the equations are proportional to, but not necessarily equal to, the size of the transports given by

**e**

_{0}and 2) that the salt and heat equations involve different physical processes and may therefore have a different proportionality to

**e**

_{0}, the expected errors of the diahaline and diathermal volume transport (

*f*

_{θ}, respectively, to vary between 0.01 and 1. This allows us to study the sensitivity of the solution to our choice of the equation error.

#### 2) Column weighting

We will allow for a standard error of 25% of the expected values for *K* and *D*. This leads to

To obtain an approximation of the structure of

### d. The solution

*ϵ*

_{rms}minimum within the range defined by Eq. (35). Here

*ϵ*

_{rms}is given byThe term

*J*is the number of unknown streamfunction variables, and

*ϵ*

_{rms}represents a weighted root-mean-square value of the difference between the diathermohaline streamfunction determined by the inverse method

*ϵ*

_{rms}≥ 1, our solution is no better than the solution given by our prior estimate

*ϵ*

_{rms}< 1, the solution is more accurate than our prior estimate with a perfect solution (i.e.,

*ϵ*

_{rms}= 0. The results of this solution are discussed in the next section.

## 6. Results and discussion

In this section, we discuss the skill of the THIM by comparing the UVIC model’s variables with the inverse estimates. For a detailed physical interpretation of the circulation cells of

### a. The forcing terms

The surface salt flux binned in (*S*_{P}, *θ*) coordinates shows a diahaline transport in the direction of higher-salinity values for salty water and in the direction of lower-salinity values for freshwater (Fig. 2). A similar feature is observed for the surface heat flux binned in (*S*_{P}, *θ*) coordinates, which show diathermal transport in the direction of higher temperatures for fluid parcels with high temperatures and in the direction of lower temperatures for fluid parcels with low temperatures (Fig. 3). Hence, both the surface salt and heat fluxes lead to divergence of volume in (*S*_{P}, *θ*) coordinates. The surface divergence is balanced by convergence of volume in (*S*_{P}, *θ*) coordinates due to both the eddy and turbulent diffusive transport terms for salt and heat (Figs. 2, 3). Note that the local term is very small compared to the surface and diffusive terms, and the trend term is statistically insignificant for this particular model within the 95% confidence level of the Student’s *t* test (Groeskamp et al. 2014). Hence, the inverse method balances surface fluxes, mixing (of which the diffusion coefficients are estimated), and advection (represented by the diathermohaline streamfunction).

Sources of errors or variations in the inverse estimates, apart from weighting coefficients, are numerical diffusion and limits on the temporal and spatial resolution. The latter leads to averaging and rounding errors of the ocean’s hydrography and surface fluxes and results in unresolved fluxes at the sea surface and unresolved flux divergence in the ocean interior. For example, unresolved fluxes with periods less than a month may occur because we have used monthly averaged values. Such fluxes are expected to have the largest influence on circulations that occur near the surface, as heat and salt fluxes are expected to vary at the surface on time scales shorter than a month. The numerical diffusion and unresolved fluxes lead to *K* and *D*), the numerical noise will not allow us to calculate a streamfunction directly from Eq. (4). Hence, to obtain

### b. The solution range

We first discuss the range of solutions and then discuss the results for the optimal solution, indicated by a black dot (Fig. 5). When *f*_{θ} (and therefore **x** = **x**_{0}, and because **e**. This is expected because we have limited knowledge of

As *K* = *K*_{uvic} and *D* = *D*_{uvic} (Figs. 2, 3), one will never obtain *K* = *K*_{uvic} and *D* = *D*_{uvic}. As a result, an improved estimate of **x** = **x**_{0}, and thus moving away from *K* = *K*_{uvic} and *D* = *D*_{uvic}, but toward *K*. This is not due to an incorrect formulation of the inverse method, it is a problem imbedded in the model monthly means. The inverse solution behaves as expected, and for the whole range of solutions, both *K* and *D* are very reasonable approximations of *K*_{uvic} and *D*_{uvic} (Fig. 5c). This shows that the THIM is skilled in providing estimates of *K*, and *D*.

The optimal solution selected according to section 5d gives *f*_{θ} = 0.24, that is, errors with a magnitude of 2% and 24%, for the conservation of salt and heat, respectively. This suggests that most of the error in the equations is created by the models heat flux terms.

### c. The inverse estimate

For the optimal solution, *ϵ*_{rms} = 0.56. The standard deviation of _{p} does not exceed 0.5 Sv (and is generally much smaller) and is very small compared to

The difference *S*_{P}, *θ*) coordinates (Fig. 7). To explain the differences, we use that the dynamics of the area in (*S*_{P}, *θ*) for which *θ* > 20°C are dominated by processes that occur near the surface, while the area for *θ* < 10° and *S*_{P} = 35 ± 0.5 g kg^{−1} are also strongly influenced by processes that occur in the ocean interior (Zika et al. 2012; Döös et al. 2012; Groeskamp et al. 2014; Hieronymus et al. 2014). The magnitudes of the turbulent diffusion terms, which can be scaled by changing *D*, are large throughout the whole *S*_{P} and *θ* domain (*D* terms in Figs. 2 and 3). The magnitude of the epineutral eddy diffusion terms, which can be scaled by changing *K*, is large only in the ocean interior (*K* terms in Figs. 2 and 3). As a result, the large surface fluxes of salt and especially heat of the area in (*S*_{P}, *θ*), for which *θ* > 20°, can only be balanced by increasing the turbulent diffusion, increasing *D*. This will also result in an increased effect of turbulent diffusion in the oceans interior. The only way to compensate for this is by reducing the magnitude of the epineutral eddy diffusion, reducing *K*. This idea is supported by the estimates of the diffusion coefficients. The estimate of *K* for the optimal solution is *K* = 929 ± 7 m^{2} s^{−1}, which is close to *K*_{uvic} = 1200 m^{2} s^{−1}, but an underestimation. The estimate of *D* is *D* = 1.27 ± 0.01 × 10^{−4} m^{2} s^{−1}, which is close to *D*_{uvic} = 1 × 10^{−4} m^{2} s^{−1}, but an overestimation. We note that numerical diffusion may also increase the total diffusion in the model, possibly contributing to the fact that *D* > *D*_{uvic}. As a result of an increased *D* and reduced *K*, we then find general differences given by

The fact that the standard deviation for **x** obtained from _{p} is much smaller than the variation of the solution as a result of changing

*F*

_{Salt}(

*ρ*

_{r})] and heat [

*F*

_{Heat}(

*ρ*

_{r})] transports for reference potential density

*ρ*

_{r}. Following (Zika et al. 2012) this is given byWe have applied an integration along lines of constant reference potential density

*ρ*

_{r}. The salt flux can be expressed as an equivalent freshwater flux by dividing it by a reference salinity [

*F*

_{FW}(

*ρ*

_{r}) =

*F*

_{Salt}(

*ρ*

_{r})/

*S*

_{ref},

*S*

_{ref}= 35]. At densities that cross the tropical cell 21 kg m

^{−3}<

*σ*

_{0}< 24 kg m

^{−3}, we have an overestimation of both the diapycnal freshwater and heat transport due to an increased magnitude of the tropical cell of

^{−3}<

*σ*

_{0}< 27 kg m

^{−3}(Fig. 8). However, the similarity of the shapes of the diapycnal transports show the THIM is skilled in capturing

### d. Putting the THIM in perspective

Most inverse (box) models are designed with a focus on estimating the absolute velocity vector only. Currently inverse methods are one of few methods by which one is able to provide an estimate of mixing from observations. Inverse box methods that also estimate *D* often contain unknowns at the boundaries that require both dynamical constrains and conservation statements, increasing the complexity of the system and sensitivity to prior estimates [Sloyan and Rintoul (2000, 2001), among many others]. The THIM uses boxes bounded using two pairs of tracer surfaces, analyzed in tracer coordinates, rather than Cartesian coordinates. This reduces the complexity of the system to a set of simple tracer conservation equations. The global application of the THIM leads to strong constrains on the solution, as confirmed by the small error calculated using _{P} and small variation of the solution for a wide range of choices of the row weighting. There are inverse models that provide global estimates of *D* (Ganachaud and Wunsch 2000) or local estimates of both *K* and *D* (Zika et al. 2010b). However, to our knowledge the THIM is the only inverse estimates that can provide globally constrained estimates of both *K* and *D* from observations.

Groeskamp et al. (2014) showed that *S*_{A}, Θ) coordinates due to cyclic changes of the ocean’s volume distribution in (*S*_{A}, Θ) coordinates, without motion in geographical space and can be calculated from an ocean hydrography. The advective thermohaline streamfunction *S*_{A}, Θ) coordinates, and requires global observations of **u**. However, using the THIM and the calculation of

This study has shown that premultiplying the turbulent diffusive terms with a structure function is an appropriate method to reduce the number of unknown diffusion coefficients, while allowing for its spatial variation. When applying the THIM to observations, such structure functions can be applied to reduce the number of unknowns required to capture the spatial and temporal variation of *K* and *D*. When the THIM is applied to observations, choosing *S*_{A} and Θ as tracer coordinates utilizes the extensive observational coverage of these tracers, reducing uncertainties in the solution. We therefore believe that the THIM has the potential to obtain well constrained global estimates of spatially- and temporally-varying values of *K* and *D*.

## 7. Conclusions

We have presented the thermohaline inverse method (THIM), which estimates the diathermohaline streamfunction *K*, and isotropic turbulent *D* diffusion coefficients. The THIM uses a balance between advection and water mass transformation due to thermohaline forcing in (*S*_{A}, Θ) coordinates. The thermohaline forcing, that is, the surface freshwater and heat fluxes and diffusive salt and heat fluxes, can be obtained from ocean hydrography and surface flux products.

We have tested the THIM using a model’s hydrography and surface fluxes and compared the inverse estimate of *K*, *D*, and *K*, *D*, and

## Acknowledgments

SG was supported by the joint CSIRO–University of Tasmania program in quantitative marine science (QMS) and the CSIRO Wealth from Ocean flagship and through the Office of the Chief Executive (OCE) Science Team Postgraduate Scholarship Program. BMS was supported by the Australian Climate Change Science Program, jointly funded by the Department of the Environment and CSIRO. JDZ is supported by the U.K. National Environment Research Council. We thank Louise Bell for preparing some of the figures.

We thank Daniele Iudicone, Gurvan Madec, Nathan Bindoff, and Paul Barker for valuable discussions. We are grateful for the helpful comments of Johan Nilsson and two anonymous reviewers.

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