a. Experiments with C87

The special case of an ocean model with buoyancy dependent on a single variable, such as a single active tracer, is a useful place to begin testing the C87 diffusion scheme. In the continuum, the isoneutral portion of the diffusion tensor will not act on this tracer since the neutral directions are parallel to the isotracer surfaces. This property is true for both linear and nonlinear equations of state (section 3 and appendix B provide further discussion). To test the ability of the C87 scheme to respect this property, we ran the sector model using horizontal diffusion with salinity fixed throughout the domain at 35 psu, and allowed temperature to be active with the nonlinear equation of state. We then switched from horizontal to C87 diffusion using 10⁷ cm² s⁻¹ for the isoneutral diffusion coefficient, removed all surface forcing, and turned off tracer advection.

Figure 2a shows the solution 600 years after the switch. As expected, vertical gradients of isotherms are weakened due to the nonzero dianeutral diffusivity. However, there is a near complete loss of numerical integrity. In the north, temperature has undergone a vigorous amount of vertical mixing due to convection acting on the unstable solution. In the south, where convection is absent, the presence of unstable grid waves is clear. These waves grew unbounded in subsequent years of integration, which caused the solution to eventually blow up. Figure 2b shows the temperature at year 300 for the case in which salinity is also allowed to change. This solution for two active tracers is consistent with the single active tracer experiment. For both cases, setting the dianeutral diffusivity to zero increased the growth rate of the instability by roughly one order of magnitude.

Dispersion errors associated with grid Peclet number violations encountered with centered advection (Bryan et al. 1975; Weaver and Sarachik 1990) have traditionally been considered one of the main reasons for employing horizontal background diffusion with C87 (see comment at the end of C87). Since tracer advection is removed from these experiments, Peclet grid noise has nothing to do with this unstable behavior.

Another example of the problems with the diffusion scheme can be seen when performing the switching experiment while maintaining an Eulerian advective transport. For centered differenced advection, the solution blows up (not shown) even sooner than without advection. This behavior might be expected based on grid Peclet arguments. A reasonable question to ask is whether a completely monotonic advection scheme, such as flux corrected transport (FCT) (see Gerdes et al. 1991), is sufficient to suppress the C87 instability. Although FCT acts only on the advective fluxes, it might provide enough dissipation to stabilize the C87 scheme. Figure 2c shows that this possibility is not completely realized, even with a single active tracer for which the diffusive fluxes should vanish. After switching the diffusion from horizontal to C87, switching the advection from centered to FCT, and releasing the surface forcing the model undergoes a spindown in which the isotherms eventually flatten and spread apart due to the dianeutral diffusion and lack of surface forcing. However, by year 900, there is a nontrivial grid wave whose growth is able to overcome the stabilizing aspects of the FCT scheme. The amplitude of this grid wave continued to grow in subsequent years. Turning off all diffusion and using FCT alone resulted in smooth and flat isotherms, with no sign of grid noise.

The implementation of GM90 eddy-induced advective transport in z-coordinate models has been associated with the ability to remove the background horizontal diffusion otherwise necessary with the C87 scheme. It is noted that GM90 acts to reduce the isoneutral slopes, thus producing more horizontally aligned diffusive fluxes. Such a reduction in slopes may provide for increased numerical stability according to the linear stability analysis discussed in appendix A. However, that analysis is in error and, so, is not relevant for isoneutral diffusion. A different reasoning for why GM90 stabilizes certain experiments is provided in Griffies (1998). Unfortunately, as seen in Fig. 2d, the switching experiment with GM90 eddy-induced advection, C87 diffusion, two active tracers, and 0.5 cm² s⁻¹ dianeutral diffusivity is not stable. Indeed, when compared to Fig. 2b, the solution looks even worse, and it blows up sooner than without GM90 advection. Such behavior is consistent with the problems encountered when implementing GM90 eddy-induced advection as described by Weaver and Eby (1997). As discussed by Griffies (1998), the problems with this GM90 experiment are related to the method of implementing both the GM90 closure as well as isoneutral diffusion.

b. Comments on the switching experiments

We performed numerous other switching experiments in this and other model configurations with various permutations of subgrid-scale parameterizations. Consistently, the only way to eliminate the unstable grid waves was to add at least 10% background horizontal diffusion to the C87 diffusion scheme. As discussed in the introduction, such background diffusion is not a viable choice for realistic climate modeling. It should be noted that when running the C87 scheme with centered advection from the start of a spinup from uniform tracer fields, the sector model remained stable, with only modest grid noise. Therefore, the switching experiments provide substantially stronger tests of the numerical integrity than the spinup experiments. The reason is that the diffusive fluxes of the active tracers have the potential to be much larger in the period after switching between diffusion processes than during a spin up from rest. Within the spirit of evaluating the performance of a scheme under various conditions, some of which could be realized in more realistic models run under time-varying forcing and with bottom topography, we conclude that the C87 scheme is unsound. The remainder of this paper is devoted to providing the physical and numerical understanding necessary to interpret these problems with C87 and thereafter to derive a new scheme that aims to rectify them.

a. The diffusion tensor and the diffusion operator

A diffusion tensor K is a symmetric, positive semidefinite second-order tensor. The components F^m(T) of the diffusive flux are related to the diffusion tensor through F^m(T) = −K^mn∂_nT, where T is any tracer and K^mn are components to the diffusion tensor K. The diffusion tensor therefore acts as a matrix operator that orients the tracer gradient in the process of defining the tracer diffusive flux. The tracer will evolve due to divergences of the flux, resulting in the diffusion equation ∂_tT = −∂_mF^m(T) ≡ R(T), where R(T) is termed the diffusion operator. The simplest type of diffusion is isotropic diffusion, for which K^mn = Aδ^mn, A > 0 is a positive diffusion coefficient, and δ^mn is the Kronecker delta, which equals unity when m = n and vanishes otherwise. For horizontal/vertical diffusion, the tracer flux in the vertical is distinguished from that in the horizontal which means the diffusion tensor is anisotropic and has components K^mn = A_H(δ^mn − ẑ^mẑⁿ) + A_Vẑ^mẑⁿ, where A_H, A_V are nonnegative horizontal and vertical diffusion coefficients, respectively, and ẑⁿ is the nth component to the unit vector (0, 0, 1) in the vertical direction.

Isoneutral/dianeutral diffusion is completely analogous to horizontal/vertical diffusion, only now it is the dianeutral direction, defined by the dianeutral unit vector γ̂, that is distinguished from the isoneutral directions, defined by two unit vectors ê₁ and ê₂. The diffusion tensor therefore takes the form

K^mnA_Iδ^mnγ̂^mγ̂ⁿ)A_Dγ̂^mγ̂ⁿ(1)

where A_I, A_D are the isoneutral and dianeutral diffusion coefficients, respectively. Writing the diffusion tensor in this form provides for a simple geometric interpretation. Namely, the isoneutral piece A_I(δ^mn − γ̂^mγ̂ⁿ) acts as a projection operator that projects out that component (∇T)_iso of the tracer gradient within the tangent plane defined by the two neutral directions. The dianeutral piece A_Dγ̂^mγ̂ⁿ projects out that component (∇T)_diap parallel to the dianeutral unit vector γ̂. Written as a matrix, the isoneutral/dianeutral (Redi) diffusion tensor takes the form (Redi 1982)

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and the small slope approximation to this tensor (GM90) is

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In these expressions, S = (S_x, S_y, 0) = (−∂_xρ/∂_zρ, −∂_yρ/∂_zρ, 0) is the isoneutral slope vector with magnitude S, and ϵ = A_D/A_I ≈ 10⁻⁷ to 10⁻⁸ is the ratio of the dianeutral to isoneutral diffusion coefficients. It is important to note that the kinematical properties discussed in the remainder of the section apply to both the full and small angle diffusion tensors.

b. Tracer variance

Integrating the diffusion equation ∂_tT = −∇·F over a source-free domain with insulating boundaries (i.e., Neumann boundary conditions) indicates that diffusion will not change the total amount of the tracer. Integrating the tracer squared ∂_tT² = −2∇·(TF) + 2∇T·F over the same domain indicates that diffusion will not increase the tracer variance

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In this expression, dx is the volume element dx dy dz, and the inequality follows since the diffusion tensor is symmetric and positive semidefinite or, equivalently, the diffusive flux is directed down the tracer gradient (∇T·F ⩽ 0). Note that the diffusion tensor could have a zero determinant, which is the case for zero dianeutral diffusion with the Redi tensor. Downgradient diffusion, and the resulting reduction of tracer variance, is the first of two fundamental properties that we aim to realize in a numerical diffusion scheme. It will be referred to as Property I in the subsequent development.

It is useful to explicitly consider the case of diffusion with the isoneutral/dianeutral diffusion tensor. In this case, the variance equation takes the form

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The first form is analogous to that resulting from horizontal/vertical diffusion, in which the unit vectors ê₁, ê₂, γ̂ replace the unit vectors x̂, ŷ, ẑ, respectively. The second form suggests the following interpretation. The term −A_I|∇T|² ⩽ 0 represents an isotropic term that acts to dissipate all gradients, and hence all curvature, just as that occurring in isotropic diffusion. The second term A_I|γ̂·∇T|² ≥ 0 represents the effects of isoneutral diffusion acting to align the tracer isolines parallel to the neutral directions. Through this alignment process, structure is added to the tracer field, which therefore acts to increase the tracer variance; hence the positive sign for this term. When the tracer is perfectly aligned along the neutral direction, cancellation occurs between the two components of the isoneutral diffusion term. In the competition between the dissipative and alignment components, the dissipative component wins since −|∇T|² + |γ̂·∇T|² = −|∇T × γ̂|² = −|ê₁·∇T|² − |ê₂·∇T|² ⩽ 0, thus ensuring that the total tracer variance will not increase.

The previous discussion suggests the following example in order to illustrate an important point. Consider a tracer field with its power concentrated in the long wavelengths; that is, it is a smooth yet nonuniform tracer field. Allow this tracer to diffuse downgradient along static neutral directions that have a lot of spatial power in high wavenumbers: for example, the density field has a grid noise structure. Isoneutral diffusion will result in the tracer becoming aligned along the neutral directions, which means that it will have power transferred to the high wavenumbers. In the process, the total tracer variance will reduce due to the dissipative nature of diffusion. Therefore, even though tracer variance will not increase with isoneutral diffusion, it is possible to introduce grid noise into the tracer field if such noise is in the density field. This point will prove fundamental in the development of section 4.

c. Balance of the active tracer isoneutral diffusive fluxes

The isoneutral diffusion operator is a nonlinear function of active tracers through the dependence of the diffusion tensor on temperature and salinity. In this way, isoneutral diffusion differs fundamentally from more traditional forms of linear diffusion in which the diffusion tensor is independent of tracers. Since the isoneutral/dianeutral diffusion tensor is constructed locally according to the dianeutral unit vector γ̂, it is necessary to discuss some ideas about neutral directions and how to compute γ̂.

In the ocean, neutral directions are those for which an adiabatic displacement of a parcel is not affected by buoyancy forces. As a familiar example, note that after a free convective event, a vertical column is typically neutrally buoyant and the vertical is correspondingly a neutral direction. McDougall (1987a) formalized this idea to all three directions, and argued that because mixing can act unopposed by buoyancy forces, neutral directions are relevant for orienting the tracer diffusive fluxes.

As discussed by McDougall and Jackett (1988), it is not possible to define a coordinate γ that can globally describe the envelope of neutral directions, unless one neglects the affects of cabbeling and thermobaricity through linearizing the equation of state (see appendix B for further details). However, all that is necessary for our purposes is a local description, which is available for the general case of two active tracers with a nonlinear equation of state. For this description, start by noting that through any point in the ocean with pressure p, there passes a potential density surface ρ_p(θ, s), which is referenced to the same pressure p. McDougall (1987a) showed that the neutral directions at this point lie within the tangent plane to the ρ_p surface at this point. It follows that the plane’s unit normal vector ∇ρ_p|∇ρ_p|⁻¹ at this point is perpendicular to the neutral directions within this plane, so it can be identified with the dianeutral unit vector γ̂.

It is useful to relate ∇ρ_p to the active tracer gradients when computing γ̂. For this purpose, note that the log-gradient of the locally referenced potential density can be written in terms of the gradients of the active tracers through ∇ lnρ_p(θ, s) = (∂ lnρ_p/∂θ)∇θ + (∂ lnρ_p/∂s)∇s. Note the absence of pressure gradients due to the local referencing. It follows that, since this density gradient is of interest only when evaluated at the in situ pressure where the potential density is referenced, the potential density partial derivatives are equivalent to the thermal and saline expansion coefficients −∂ lnρ_p/∂θ = α_p(θ, s) ≡ α(θ, s, p) and ∂ lnρ_p/∂s = β_p(θ, s) ≡ β(θ, s, p) (section 3.7.4 of Gill 1982; McDougall 1987a). For example, the vertical component of this gradient yields the familiar buoyancy frequency N² = −gd lnρ_p/dz = g(α∂_zθ − β∂_zs).

Since it is sufficient to evaluate all quantities in this section at the local pressure p, the p label can be unambiguously dropped from both the potential density and the expansion coefficients in order to reduce clutter. It is important to remember, however, the local referencing used for all subsequent equations. With these conventions, the log-gradient of the locally referenced potential density ρ is written:

∇ρα∇θβ∇s,(6)

which yields the expression for the dianeutral unit vector:

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A very important consequence of Eqs. (6) and (7) is that the isoneutral diffusive flux of the locally referenced potential density vanishes. It follows that the isoneutral diffusive fluxes of the active tracers are coupled since

ρ⁻¹F_IραF_IθβF_Is(8)

This relation is taken as the second of two fundamental properties that we aim to realize in a numerical isoneutral diffusion scheme. It will be referred to as Property II in the subsequent development.

a. Grid-scale computational modes

For the small angle approximated diffusion tensor, the C87 discretization of the isoneutral diffusion flux of a tracer in a two-dimensional x–z model is

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The notation is standard for the MOM 2 model (Pacanowski 1996) and is defined in Fig. 3. In order to define the diffusive fluxes consistently on the model’s grid, the x flux must be placed at the east face of T-cell (i, k) and the z flux at the bottom of this same cell. For off-diagonal terms, a double spatial average ( )^{x, z} brings the z-derivative term appearing in the x flux onto the east face of a T-cell, and the x-derivative term appearing in the z flux onto the bottom face of the T-cell. The grid stencil used for computing the x component of the flux is given in Fig. 4. Similar six point stencils are used for computing the other components of the flux.

The practical difficulty encountered when discretizing isoneutral diffusion in the z models concerns how to handle the off-diagonal terms. This issue is not unique to the B grid used in the GFDL model since the physical process (tracer diffusion) concerns only tracer points and tracer cells. The discretization given by C87 considers tracer gradients and the corresponding isoneutral slopes to be independent, each of which needs to be spatially averaged in order to bring it onto the relevant face of the T-cell. In addition, the horizontal and vertical density gradients appearing in the isoneutral slopes are handled independently of one another. Therefore, the average operator appears individually on only the numerator or denominator. There is nothing fundamental about this particular discretization. Rather, it merely represents a series of convenient choices based on details of the model grid.

To start our critique of the C87 discretization, consider the z derivative of the tracer appearing in the x component to the flux

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The same form appears for the z derivative of the density appearing in the calculation of the slope. It is apparent that the combination of a z average and a z derivative allows for the presence of 2Δz structures T_i,k−1 = T_i,k+1 and ρ_i,k−1 = ρ_i,k+1 for which the discretized z derivative on the east face will vanish. Therefore, this wave, or computational mode, will be invisible to the x component of the computed isoneutral diffusive flux. Likewise for the z flux, 2Δx computational modes exist due to the combination of an x average with an x derivative. Indeed, for the 2Δx modes in the density field, the z component to the isoneutral flux vanishes identically! This property of the C87 scheme will prove to be crucial to understanding why the scheme is unstable.

In general, when working on the B grid and acting on a single field, such combinations of an average in one direction combined with a derivative in the same direction introduce computational modes. The potential for grid splitting and possible amplification of the modes must be addressed when such modes exist. Namely, if these modes are either completely invisible to the dissipation, or worse if they are amplified, then they can be very harmful to the integrity of the numerical solution. In particular, when appearing in the active tracer fields, and therefore in the density field, certain computational modes can lead to model instabilities, as we now show.

b. Increasing tracer variance

Consider a two-dimensional configuration with density containing the 2Δx form ρ_i+1,k = ρ_i−1,k. For this field configuration or mode, the previous discussion showed that the z component of the discretized isoneutral flux is identically zero. Therefore, the projection of the diffusive flux onto the tracer gradient is entirely zonal and consists of the product

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Hence, if slopes of the isoneutrals S_ρ and tracer S_T satisfy the inequality

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then the flux on the east face will be directed up the tracer gradient.

In general, an upgradient component to the diffusive flux vector can be the result of projecting the downgradient isoneutral flux vector onto orthogonal Cartesian axes. As such, it is important to distinguish between particular flux components, which may be upgradient, and the diffusive flux vector, which should be downgradient with respect to the tracer gradient (Property I in section 3b). The above discrete configuration is an example of how, in the C87 numerical scheme, an upgradient component in the x direction is not compensated by a downgradient component in the z direction, since the z component identically vanishes. Therefore, the numerically realized diffusive flux vector is upgradient. Furthermore, this upgradient flux can in general be distributed over the extent of the model domain, thus ensuring the increase in tracer variance. The rate of increasing variance is directly proportional to the difference between the slopes, and to the value of the diffusion coefficient. Since the strength of the growth in variance is proportional to A_I, the larger the isoneutral diffusivity, the larger the background horizontal diffusivity needed to suppress the variance increase.

Figure 5 shows a simple realization of this discussion in which a passive tracer field is diffused in the background of a density held constant in time with a 2Δy structure (ρ_i,j+1,k = ρ_i,j−1,k), constant in longitude, and stratified in depth. The passive tracer is initialized with unity on the surface level and zero below. There is no surface forcing, advection is turned off, and there is zero dianeutral diffusivity. As anticipated, the result is a uniform increase in tracer variance.

c. Active tracer fluxes and the basic problem with C87

Consider a single active tracer for which the isoneutral diffusive flux of this tracer should vanish. For example, the x component of the flux in the small angle limit is given by

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The diffusive flux vanishes because the thermal expansion coefficient α cancels between the numerator and denominator, thus providing an equivalence between the slope S_ρ = −∂_xρ/∂_zρ of the neutral direction and the slope S_θ = −∂_xθ/∂_zθ of the isotherms. Importantly, this result is valid for both linear and nonlinear equations of state.

In C87, the density gradients are computed by referencing each density to the same depth level. For example, the x slope of the neutral direction is computed on the east face of a T-cell through

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where the superscript on ρ symbolizes the depth reference level. This expression does not provide for cancellation of the thermal expansion coefficient as occurs in the continuum, unless the equation of state is linear. The reason is that density gradients are not computed explicitly in terms of the expansion coefficients, which would allow for the proper alignment of the temperature isolines (which define the correct neutral directions for this example) and the computed neutral directions. Hence, the computed neutral directions, as determined by the slope calculation, are not aligned with the isolines of locally referenced potential density. The effect produces a spurious flux of locally referenced potential density along the misaligned neutral directions. Note that it is common to check the integrity of isoneutral diffusion schemes by inserting the symbol “ρ” into the discretized expression for the isoneutral flux. Such an assessment would lead to the incorrect conclusion that C87 will not flux locally referenced potential density. The present analysis therefore points to the problem with this approach to deducing the self-consistency of the discretization.

The misalignment of the computed neutral directions might be harmless were it not for the following property of seawater. For a stably stratified column, and especially for a stretched grid in the vertical for which level k + 1 is further below level k than level k − 1 is above it, the C87 computation of the neutral directions can produce an effective expansion coefficient in the numerator that is greater than that in the denominator. This behavior is typical since α is an increasing function of temperature (Gill 1982). Therefore, the discretized slope of the neutral direction has a tendency to be steeper than the slope of temperature

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This inequality is identical to Eq. (13) for which the C87 scheme will produce upgradient diffusion if the 2Δx computational mode ρ_i+1,k = ρ_i−1,k is present.

Figure 6 illustrates an unstable profile corresponding to the above inequality combined with a 2Δx computational mode. With stable stratification, θ₁ > θ₂, and so point A is a local minimum with respect to the horizontal direction x̂. Upgradient fluxes along x̂, with zero compensating vertical flux (recall that the vertical flux vanishes for this 2Δx mode), will cause point A to cool. The converse occurs at point B. The tendency is to increase the amplitude of the wave thus providing for its instability. The upgradient fluxes induce an unbounded increase in temperature variance so long as inequality (16) is maintained. The situation for two active tracers is similar, only now the dashed lines in Fig. 6 denote isolines of locally referenced potential density. In effect, the nonlinear equation of state, combined with the upgradient fluxes and computational modes, feeds “energy” into the numerical solution, which causes the tracer variance to grow unbounded. It is this instability that must be stabilized by the introduction of A_back ≈ A_I background horizontal diffusion.

d. A nonlinear instability due to spurious density fluxes

Upgradient diffusive fluxes are generally destabilizing in a numerical diffusion scheme. Furthermore, they do not correspond to the physical process of interest. Therefore, in developing the new isoneutral diffusion scheme, much effort is focused toward correcting this problem (see section 5). As discussed here, however, reducing tracer variance is not sufficient to stabilize the profile shown in Fig. 6, or any wave structure in which the inequality (16) is respected. Stability is realized only if there is exact alignment between the locally referenced potential density surfaces (dashed lines) and the computed neutral directions (solid line).

To see why this profile is generally unstable, consider any profile in which the inequality (16) is satisfied, for which Fig. 6 provides a particular case. Point A is a local temperature maximum with respect to the solid line. A downgradient diffusive flux of temperature aligned along the solid line causes point A to cool, and the converse occurs at point B. This downgradient temperature flux amplifies the initial wave since as the temperature tries to align with the neutral direction, the neutral direction in turn steepens. This steepening is due to the nonlinear nature of isoneutral diffusion, for which the diffusion tensor is a function of the active tracers, which are themselves being diffused. Therefore, the profile is nonlinearly unstable to downgradient diffusion aligned along the spurious neutral directions. The instability will occur regardlesss of the linearity or nonlinearity of the equation of state as long as inequality (16) is satisfied. For two active tracers, the instability generalizes with the dashed lines representing locally referenced potential density isolines.

The instability represented by Fig. 6 is equivalent to the following analytical discussion. Consider a temperature profile that in some local interior region of the ocean takes the form θ = θ_o + bz + B(t) cos(2πx/L) and let the directions along which θ is diffused downgradient be determined by ρ = ρ_o + az + A(t) cos(2πx/L). Here ρ surfaces equal neutral surfaces when the slope ratio S_ρ/S_θ = (Ab)/(Ba) is unity. As suggested by Fig. 6, let ρ be directly coupled to θ, yet let it be an imperfect approximation to locally referenced potential density. The misalignment introduces an unphysical degree of freedom. The zonal wavelength L corresponds to the grid scale over which the slopes are computed in the numerical case, and the constants a, b determine a stable linear vertical stratification. Assume the isoneutral fluxes are computed with the small slope tensor and a constant diffusivity. The downgradient diffusion of temperature along the ρ surfaces induces the evolution ∂_tB(t) = A_I(2π/L)²(Ab/a − B). A normal diffusive adjustment to the misaligned slopes causes the temperature wave amplitude to change. However, because the neutral directions are directly dependent on temperature, as the temperature wave changes, so does the ρ wave. The amplitude of both waves grows as long as S_ρ/S_θ > 1. Even if the slope misalignment is small, the diffusivity A_I ≈ 10⁷ cm² s⁻¹ can provide a nontrivial growth. This growth is largest when the wave is short, as for the case when L corresponds to the grid scale in a numerical model. Variance is reduced if the diffusion is downgradient and the boundaries are either insulated or held with a fixed tracer value (see section 5). Therefore, all waves in a model will not grow; only those for which the slope misalignment is relevant. What occurs, therefore, is a reduction in variance with some of the original spectral density of variance being transferred preferentially to the grid scale. This process must saturate since variance is bounded from below (i.e., it is ≥0). At saturation, the amount of small-scale spectral density is bounded above by the variance in the initial condition.

To illustrate the importance of balancing the isoneutral fluxes of the active tracers, consider two methods that act to squelch the instability. Prescription A employs the relation ∇ lnρ = −α∇θ + β∇s [see Eq. (6)] for the computation of the neutral direction slope in the x direction

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where α^(i,k) and β^(i,k) are evaluated using the temperature, salinity, and pressure values at grid point (i, k) in both the numerator and denominator. Referring to Fig. 6, this prescription corrects the alignment of the computed neutral directions (solid line) so that it is now parallel to the locally referenced potential density isolines (dashed lines). Prescription B employs the C87 discretization of the salinity diffusive flux, including the C87 calculation of the slopes, yet the temperature flux is diagnosed through imposing the constraint αF_I(θ) = βF(s) [see Eq. (8)]. This prescription performs an alignment complementary to that done with prescription A. For a single active tracer, both prescriptions trivially stabilize the solution since all active tracer diffusive fluxes vanish. Figures 7a and 7b show the solution for the two active tracer switching experiments, which should be compared to that obtained with C87 in Fig. 2b. Almost all of the unstable behavior has been eliminated.

Ideally, both prescriptions A and B eliminate the numerical instability by zeroing out the isoneutral diffusive flux of locally referenced potential density along the computed neutral directions; that is, they balance the active tracer fluxes and hence provide a self-consistent discretization of isoneutral diffusion. In practice, prescription A performs slightly less satisfactorily than B perhaps due to truncation errors allowing for slight imbalances. Additionally, as discussed in appendix C, in order to solve the diffusion equation in regions of steep isoneutral slopes it is necessary to solve the vertical piece of the diffusion equation implicitly. The effect is to split the vertical flux F^z(T) = K^zn∂_nT into a piece (K^zz∂_zT) solved implicitly and another (K^zx∂_xT + K^zy∂_yT) solved explicitly. The split introduces the possibility of numerical mismatch between the two parts and, so, cannot in general ensure αF^z_I(θ) = βF^z_I(s). This result explains the localization to the far north of problems seen using prescription A since it is in this region that there are stronger vertical isoneutral fluxes, thus allowing more opportunity for truncation errors to produce this mismatch. Closer analysis of the solution produced with prescription B (not shown) also indicates some residual instability in the far north, thus pointing to the generality of this residual instability encountered when splitting the vertical flux.

a. Functional formalism

This section presents the mathematical framework allowing for a systematic incorporation of the downgradient and variance nonincreasing properties of diffusion into a discretization of the isoneutral diffusion operator (Property I of section 3). This framework is based on the property that, for any linear self-adjoint operator, it is possible to associate a functional, whose functional derivative is equal to that operator [Courant and Hilbert (1953); see also Goloviznin et al. (1977), Tishkin et al. (1979), Korshiya et al. (1980) for examples similar to the following]. The functional corresponding to the diffusion operator R(T) is given by

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Since the diffusion tensor is symmetric and positive semidefinite at every point in the ocean, the functional defined over any arbitrary volume is negative semidefinite (F ⩽ 0). Correspondingly, the downgradient property (∇T·F ⩽ 0) of diffusive mixing is equivalent to a negative semidefinite functional. This is an important equivalence that will hold, within a finite volume interpretation (clarified in the subsequent development), in the discrete case as well.

The total ocean is an important special volume of interest for defining the functional. Assuming no sources and using insulating boundaries,

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where the tracer variance given by Eq. (4) was employed. Hence, for this particular volume, 4F can be interpreted as the sink of tracer variance arising from the effects of downgradient diffusion.

To motivate the form for the functional derivative relating F to R(T), consider an infinitesimal perturbation, or variation, of the tracer field T → T + δT. This tracer variation induces a variation in the functional, which is given by [see Courant and Hilbert (1953) for more discussion]

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where 2L = ∇T·F(T) = −∂_mTK^mn∂_nT is a negative semidefinite quadratic form, and T_m = ∂_mT. Dropping the total divergence term ∂_m(δTδL/δT_m) requires the application of either one of the natural boundary conditions: 1) δT(x, t) = 0 on the boundaries of the domain or 2) N̂^mδL/δT_m = 0, where N̂ is the boundary’s normal. Here δT(x, t) = 0 corresponds to taking a Dirichlet condition for the tracer (tracer specified on the boundaries), whereas N̂^mδL/δT_m = 0 corresponds to a Neumann or no-flux boundary condition, where δL/δT_m are the components to a generalized flux. These results are valid for any functional F, which can be written as ∫ dx L. Specializing now to the case of the diffusion functional with a diffusion tensor independent of the tracer—that is, linear diffusion of passive tracers—yields δL/δT = 0 and δL/δT_m = −K^mn∂_nT = F^m(T). Therefore, using δT(x)/δT(y) = δ(x − y), the desired relation between the functional and the diffusion operator is given by the compact expression

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This continuum relation has a natural finite volume generalization

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where V_i,j,k is a finite cell volume associated with tracer T_i,j,k. It is over this finite volume that the discretized operator possesses the downgradient properties. We elaborate on this important point in the subsequent development and in appendix D.

As previously mentioned, the functional formalism is strictly useful for the case of linear diffusion of the passive tracers. Since the active tracers are diffused with the same operator as the passive tracers, this restriction is of no substantial limitation. Yet, in order to prevent the nonlinear computational instability described in section 4, it is crucial to add to the functional formalism the constraint that the slopes be computed so that the active tracers closely approximate the self-consistency or balance condition on the isoneutral diffusive fluxes αF_I(θ) = βF_I(s). Otherwise, even though the scheme will not increase tracer variance, it will be subject to grid noise.

In summary, for any symmetric and positive semidefinite diffusion tensor, the functional formalism allows for a straightforward incorporation of the variance reducing properties implied by such a mixing tensor to be readily built into a discretization scheme. The procedure is to first discretize the functional F and then to take the discrete version of the functional derivative given by Eq. (22). The power of the formalism is that for any consistent discretization of F, the corresponding discretization of the diffusion operator inherits the desired downgradient properties over the corresponding finite volume. Additionally, this result means that the discrete diffusion operator R(T) will not increase the tracer variance, the eigenvalues of R(T) will all be positive, and the scheme is ensured to be at least conditionally stable in a linear sense. It follows that because the C87 diffusion operator can increase tracer variance, it does not correspond to a semidefinite functional.

The expressions for the case of isoneutral/dianeutral diffusion will prove necessary for the subsequent discussion. Writing the components of the diffusion tensor given in Eq. (1) as K^mn = (A_I − A_D)(δ^mn − γ̂^mγ̂ⁿ) + A_Dδ^mn brings the functional for isoneutral/dianeutral diffusion into the convenient form

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which allows for the identification of the diffusive flux

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In the small slope approximation, the functional is givenby

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and the corresponding small angle flux components are

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where F_h = (F^x, F^y, 0) is the horizontal diffusive flux vector, S is the isoneutral slope vector, ϵ = A_D/A_I, and ∇_h = (∂_x, ∂_y, 0) is the horizontal gradient operator.

b. Discretizing the functional

The functional given by either Eq. (23) for the full tensor, or (25) for the small angle approximation, consists of quadratic terms that take the form (∂_mT∂_nρ − ∂_nT∂_mρ)², with m ≠ n. Their discretization defines grid stencils in the corresponding two-dimensional (m, n) plane. This observation motivates a discretization of the functional where its different pieces are discretized separately within their respective two-dimensional plane. The exception to this two-dimensionality arises from the term |∇ρ|⁻², which occurs due to the two factors of the dianeutral unit vector γ̂ = ∇ρ|∇ρ|⁻¹ appearing in the full tensor. This gradient contains all three differential operators, which means that in discretizing the full tensor we must consider extending the stencil to points off a given plane. How we handle this detail will be discussed in appendix E.

In the longitudinal–depth plane, consider the term (∂_xT∂_zρ − ∂_zT∂_xρ)². A simple discretization is to use nearest neighbor tracer grid points for constructing the discrete differential operators. Since the model grid is staggered, such a choice employs second-order accurate difference operators in the functional, and will likewise be the case for the corresponding diffusion operator. Higher order in accuracy schemes can be derived by extending the stencil outward to include more points. Using the nearest neighbors, a typical component of the discretizaton of (∂_xT∂_zρ − ∂_zT∂_xρ)² consists of triads of tracer and density values. For example, one such triad contains contributions from the grid points (i − 1, k), (i, k), and (i, k − 1), where the corresponding term in the functional is (δ_xT_i−1,kδ_zρ_i,k−1 − δ_zT_i,k−1δ_xρ_i−1,k)². As seen in Fig. 8, this triad is one of 12 that contain contributions from the central tracer point T_i,k.

The triads partition the area of the longitudinal–depth plane into a series of quarter-cells (see Fig. 8). There is a single unique quarter-cell for each of the triads. For example, the triad (i − 1, k), (i, k), and (i, k − 1) is associated with the quarter-cell 4, and triad (i, k), (i, k + 1), and (i − 1, k + 1) with quarter-cell 11. The areas (actually, the volume when considering the third dimension) of each quarter-cell define the volume element associated with the triad to be used in discretizing the functional. Therefore, the discretization of the functional corresponding to that contribution from the x–z plane is given by

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where V(n) is the volume of the nth quarter cell and A(n) is the corresponding nonnegative diffusion coefficient. The superscript (n) on the tracer and density refers to the particular finite-difference discretization of the gradient for the nth quarter-cell. For example, in quarter-cell 1 shown in Fig. 8, ∂_xT⁽¹⁾ symbolizes the discrete derivative δ_xT_i−1,k−1, and ∂_zT⁽¹⁾ = δ_zT_i,k−1. Equation (28) means that the discretization of the functional F^(x−z) = Σ_i,k F^(x−z)_i,k is built by summing over the tracer points T_i,k on the lattice and, for each tracer point, summing over the 12 triads/quarter-cells that contain some contribution from the tracer point T_i,k. For purposes of discretizing the diffusion operator R(T)_i,k at a particular tracer point, it is only necessary to consider the terms appearing inside the Σ¹²_n=1 sum over the triads since the derivative ∂F^(x−z)/∂T_i,k will eliminate the Σ_i,k sum over all tracer points. Note that each of the 12 contributions to the functional F^(x−z)_i,k vanishes individually when the tracer T is replaced by locally referenced potential density ρ. This is an important property that must be respected when the detailed discretization of the density derivatives is specified (next subsection).

These details about quarter-cells, triads, and their respective volumes are important since they make precise the notion about downgradient diffusion on the lattice. Namely, these 12 quarter-cells define the finite size volume, mentioned in the discussion of Eq. (22), for which the new scheme provides downgradient fluxes of the tracer. Since the finite volume encompasses many grid cells, the locally defined diffusive fluxes discretized on the faces of the T-cell (next subsection) will not generally satisfy the downgradient property individually. In other words, the new scheme is not“positive definite” for each cell, rather it is positive-definite only over the semilocal finite volume defined by the 12 quarter-cells. This is an important qualification that must be kept in mind when interpreting results from this scheme. Note, however, that the new scheme will not increase tracer variance since variance reduction depends only on the negative semidefiniteness of the globally defined functional. The finite volume interpretation of downgradient diffusion in the new scheme is made more mathematically precise in appendix D.

c. Flux discretization with density triads

After discretizing the functional, it is necessary to take its derivative with respect to the tracer in order to obtain the discrete diffusion operator [Eq. (22)]

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where V_Ti,k is the volume of the T-cell whose center is the tracer point T_i,k. The effect from this derivative is to break or separate the tracer triads. However, it preserves the integrity of the density triads. The preservation of the density triads suggests a heuristic approach to directly discretizing the individual diffusive fluxes on the cell faces. Namely, construct the pieces of this flux using density triads as fundamental building blocks. The result of this derivation, presented in the following, is identical to that obtained when the functional derivative is directly computed and the resulting terms are combined into the divergence of a tracer flux across the T-cells [details placed in an appendix to Pacanowski (1996)]. One thing to note immediately is that the use of unbroken density triads, when weighted by their respective tracer gradients (described below), removes the computational modes. The reason is that there is no splitting of the grid, which is a characteristic of the computational modes. As discussed in section 4, density computational modes are fundamentally related to the problems with the C87 scheme. Therefore, by eliminating these modes, we already see the crucial role the density triads play in the new scheme.

For the purpose of describing the discretization, it is sufficient to discretize the diffusion flux from the small angle tensor since the basic ideas are the same for both the small angle and full slope diffusion tensors. It is also sufficient to continue considering the two-dimensional longitudinal–depth geometry. We start by discretizing the x component F^x(T) = −A_I(∂_xT + S_x∂_zT) on the east face of the T-cell (i, k) (see Fig. 3). Recall that in the C87 discretization discussed in section 4, the central practical difficulty is how to handle the off-diagonal term A_IS_x∂_zT = −A_I(∂_xρ/∂_zρ)∂_zT. When employing density triads as fundamental units, notice that there are four triads surrounding the east face of each T-cell. Each defines a discretization of the isoneutral slope S_x, so each has an associated diffusion coefficient A_I. The diffusion coefficients are selected according to one of the chosen slope constraints discussed in appendix C in order to satisfy the requirements of linear stability for steep isoneutral sloped regions. In addition, for each triad it is necessary to choose a reference point for calculating the locally referenced potential densities. This point can be chosen anywhere as long as there is a unique point for each of the triads. Since the equation of state is already computed for each tracer point, it is convenient to choose the corner of the triad as the reference point. By constructing A_IS_x∂_zT as an average over these four triads, each multiplied by their respective vertical tracer gradient, the discretized diffusive flux component F^x is correctly placed at the east face of the T-cell. Additionally, in order to account for nonuniform grids, it is important to weight each term in the average by its associated vertical grid spacing. Figure 9 summarizes this discussion by showing the stencil for the new scheme. Importantly, it requires the same six densities and tracers as for the C87 scheme (see the C87 stencil in Fig. 4), so the grid “footprint” is identical to that of C87.

Putting the pieces together yields the discretized x component of the small angle tensor isoneutral diffusion flux:

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(refer to Fig. 3 for definitions of the grid spacing factors dzt and dzw and difference operators). The sum over ip and kr represents the sum over the four triads shown in Fig. 9. The isoneutral slope is computed through the relation

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The corresponding diffusion coefficient A^(i1,k1)_{(i2,k2|i3,k3)} is set according to the relevant slope criteria discussed in appendix C. The superscripts (i1, k1) refer to the corner point of the triad, which is used for determining the reference points in calculating the densities. It is important to note the use of the same reference point for density gradients in both the numerator and denominator. Accordingly, the density gradients are computed as

δ_mρ^(i1,k1)_i2,k2ρ_θi1,k1δ_mθ_i2,k2ρ_si1,k1δ_ms_i2,k2(32)

where the labels on the density partial derivatives indicate the grid point for which the temperature, salinity, and pressure are used in evaluating these terms. In the model, the coefficients ρ_θ = (∂ρ/∂θ) and ρ_s = (∂ρ/∂s) are computed by analytically differentiating the cubic polynomial used for approximating the UNESCO equation of state (Bryan and Cox 1972), thus producing quadratic expressions which are diagnosed.

It is important to highlight the direct computation of the expansion coefficients ρ_θ and ρ_s for the purposes of computing the density gradients and, hence, for computing the isoneutral slopes. As discussed in section 4, a proper alignment of the computed neutral directions will eliminate spurious and unstable fluxing of locally referenced potential density. Otherwise, the new scheme, even though it cannot increase tracer variance, would be exposed to a form of the nonlinear instability described in section 4d and hence subject to grid noise. A viable alternative to computing the density gradients as given in Eq. (32) is to directly diagnose one of the active tracer fluxes from the other (i.e., prescription B discussed in section 4d). Such a scheme was tested, but it did not alter the solution substantially. The choice of computing the slopes in terms of the expansion coefficients represents a decision to emulate as closely as possible the kinematics discussed in section 3, and in particular Eq. (7) for the dianeutral unit vector. The direct calculation of ρ_θ and ρ_s is therefore a fundamental and novel aspect of the new scheme.

The diagonal piece of the flux consists of the horizontal gradient of the tracer weighted by the four diffusion coefficients, which correspond to the four triads used for constructing the off-diagonal term. Should the slopes for the four triads all lie within the stability range and therefore not require scaling, then the sum collapses and the diagonal piece becomes A_Iδ_xT_i,k, where A_I is the unscaled diffusion coefficient.

The resulting diffusive flux can be thought of as an average of four “subfluxes” associated with each of the four triads. Each of the subfluxes correctly vanishes individually when locally referenced potential density is substituted for the tracer. In this manner, the new scheme provides for a completely symmetric or democratic sampling of the diffusive fluxes associated with each of the four vertical tracer gradients surrounding the east face of the T-cell. Should any one of these subfluxes lie adjacent to a solid boundary, its contribution to the average over the four triads is eliminated. Additionally, the vertical tracer gradient and the corresponding density triad are fundamentally coupled. Therefore, it is not possible to identify a discretization of the K^xz off-diagonal diffusion tensor component independently of the particular tracer, whereas it was possible to do so with the C87 scheme. The resulting discretization of the diffusion operator recovers the traditional five-point Laplacian in the limit of flat isoneutral directions. This operator has well-known stability properties; that is, it fluxes tracer downgradient and it has no computational modes.

The z component of the small angle tensor flux is given by

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The construction of the z component is based on the same arguments as the x component, through the use of the density triads. The new scheme does allow for the diagonal terms in the diffusion tensor to be identified independently from the tracers. Most importantly,

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which means the implicit in time algorithm for solving the vertical diffusion equation is identical to C87.

a. Effects due to the triad discretization

Consider the effects in a two-dimensional model with a 2Δx profile in salinity yet vertical stratification in temperature and allow the equation of state to be linear.³ The 2Δx salinity structure induces a 2Δx density structure. With this structure, C87 produces a zero vertical flux [Eq. (12)] and an upgradient horizontal flux, which means that the variance created in the x direction is not removed in the z direction. The effect is to move water parcels along neutral directions in θ–s space until there is a 1–1 relationship between θ and ρ. The initial spread in salinity is replaced by a spread in temperature, with the densest water colder than any water initially in the domain, and the lightest water warmer than any initially in the domain. The final range is roughly αΔθ ≈ βΔs. In cold water, α is small so that small horizontal variations in salinity will drive large horizontal variations in temperature, thus creating very cold water. By contrast, the new scheme has no computational mode and so diffuses temperature and salinity in both the vertical and horizontal. The result is the more reasonable temperature–salinity distribution shown in Fig. 10b. All the temperatures now lie within the initial range. There are a few points, however, where the salinity is slightly outside the initial range. These unphysical results may be attributed to small amounts of upgradient diffusion occurring for the individual cell faces (see discussion of the finite volume interpretation of downgradient fluxes in section 5 and appendix D).

Now reconsider the passive tracer experiment shown in Fig. 5 using the new scheme. The tracer variance for this experiment, shown in Fig. 11a, reduces immediately. Figure 11b shows a meridional slice at the surface in the middle of the basin for both the C87 and new results. This figure shows unphysical values (those values outside the range [0, 1]) for the tracer associated with the increasing variance in the C87 scheme. The new scheme produces physically realistic values.

b. Diffusion and GM90 advection

The unsuccessful relaxation experiment with GM90 advection and C87 diffusion shown in Fig. 2d, which used 0.5 cm² s⁻¹ dianeutral diffusivity, was cleanly simulated with the new diffusion scheme (not shown). A more difficult problem to solve with the C87 scheme is the case with identically zero dianeutral diffusivity. Such an experiment is possible with the new diffusion scheme (as seen in Fig. 12). The solution relaxes toward a horizontally uniform state, as expected since the GM90 scheme acts to reduce the available potential energy of the system. We found a similar smooth relaxation when allowing the Eulerian velocity to act on the tracers through the addition of a centered difference advective flux, in addition to the GM90 advection and isoneutral diffusion (not shown).

c. Computational timing requirements

The computational requirements of the new diffusion scheme are more than the C87 scheme. With the idealized sector model used in this study, the new diffusion scheme using the small angle tensor took roughly 10%–20% more computational time than the MOM 2 implementation of the C87 scheme. These numbers represent the time for the model as a whole. This result is consistent with comparisons made in a realistic coarse-resolution global ocean model (not shown). The discretization provided in appendix E for the full isoneutral diffusion tensor took roughly six times longer than the new small angle tensor. If further justification is provided for employing the full tensor (see appendix C for more discussion), then some sort of approximation to the scheme provided in appendix E should be considered in order to reduce the large time requirements for using this tensor. In general, for the small tensor, the 10%–20% increased time requirements over the C87 scheme appear modest considering the substantially improved numerical representation of the isoneutral diffusion process.