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  • View in gallery

    Schematic illustrating a two-layer temperature field θ in a rectangular flat-bottomed zonal channel. Black indicates cold waters; white shows warm waters. (a) A vertical slice at constant latitude and (b) a horizontal slice at constant depth. The dashed lines indicate where the Eulerian averaging is taken. (c),(d) The Eulerian mean temperature θ viewed from the vertical and from the horizontal slice. Gray shading indicates waters of intermediate temperatures “formed” by the Eulerian averaging. (e) The mean height of isotherms z and (f) the mean latitude of isotherms y*. In (c)–(f) the zonal axes are extended for ease of visualization

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    Schematic illustrating (a) a vertical slice through the temperature field θ in the presence of topography; (b) the mean height z of the θ isotherm defined such that the area below equals the area colder than θ, Ã; and (c) the zonally averaged height ž of the θ isotherm. Note that is deeper than ž

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    Schematic illustrating (a) a temperature field θ for a vertically uniform two-layer fluid; (b) the Eulerian mean temperature θ(y, z); (c) the mean temperature θ̃(y, z) using isothermal averaging at constant y; and (d) the mean temperature θ*(y, z) using isothermal averaging at constant z. Shading as in Fig. 1

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    Instantaneous temperature field at day 4320: (a),(b) the horizontal field at 10- and 350-m depth with contour interval 2° and 1°C, respectively, where the thin lines indicate the northern and southern relaxation zones; (c),(d) meridional sections at 180 and 470 km along the channel

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    (a) Eulerian mean temperature θ, (b) mean temperature θ using isothermal averaging at constant y, and (c) mean temperature θ* using isothermal averaging at constant z. (d) The difference θ̃θ and (e) the difference θ* − θ. (e) The temperature differences at surface; the solid line is θ̃θ, and the dashed line is θ* − θ

  • View in gallery

    Schematic illustrating (a) the meridional flow V in fluid colder than θ (drawn here as below the isotherm) across the zonal section at latitude Y; (b) the zonally integrated northward transport streamfunction ψ̃(Y, θ); (c) the upward flow W in fluid colder than θ (drawn here as below the isotherm) across the horizontal section at depth Z; and (d) the zonally integrated vertical transport streamfunction ψ*(θ, Z)

  • View in gallery

    (a) Volume balance between y and y + dy given by the convergence of integrated northward flows ψ̃(y + dy, θ) − ψ̃(y, θ), the rate of change of the area Ã(y, θ) below the isotherm θ, and the diapycnal flow ẽdy through the θ surface. (b) Balance between the diabatic heating Δ supplied into the layer per unit y between the θ and θ + Δθ isotherms and the heat gain Δθ by the diapycnal flow as it crosses the diapycnal surfaces. (c) Corresponding volume balance relating to the horizontal area A*, the upward flow Ψ*, and diapycnal flow e* between z and z + dz. (d) The relationship between heat supply per unit z and diapycnal flow across the θ surface at constant z

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    (a) Meridional transport streamfunction ψ̃(y, θ) (Sv) and (b) the schematic for (a) to show the direction of transport and different heating regimes: surface relaxation (dark shading), northern relaxation (medium shading), and southern relaxation (lightest shading). Similarly, (c) and (d) are for the vertical transport streamfunction ψ*(θ, z). Isopycnic flow is restricted to the white dashed line in schematic (d)

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    The transport streamfunction remapped into (y, z) space: (a) Ψ̃ and (b) Ψ*. The respective mean isotherms θ̃ and θ* are superposed on the plots

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    Schematic illustrating the surface bounded by the intersection between the θ = θ̃(y0, z0) surface, the vertical plane through which ψ̃ passes, and the horizontal plane up through which ψ* passes. Colder water is assumed to lie to the north (y + ve). The lighter bulged-up surface 𝒜+ is the part of the surface above the horizontal plane and south of the vertical plane. The darker depressed surface 𝒜 is the part of the surface below the horizontal plane and north of the vertical plane

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Isopycnal Averaging at Constant Height. Part I: The Formulation and a Case Study

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  • 1 Southampton Oceanography Centre, Empress Dock, Southampton, United Kingdom
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Abstract

Simple Eulerian averaging of velocities, density, and tracers at constant position is the most natural way of averaging. However, Eulerian averaging gives incorrect watermass distributions and properties as well as spurious diabatic circulations such as the Deacon cell. Instead of averaging at constant height, averaging along isopycnals removes such fictitious mixing and diabatic circulations. Such isopycnal averaging is normally performed at constant latitude, that is, averaging along isopynals as they heave up and down. As a result, height information is lost and the sea surface becomes much warmer (or lighter) than with simple Eulerian averaging. In fact, averaging can be performed along arbitrarily aligned surfaces. This study considers a particular case in which isopycnal averaging is performed at constant height. Thus, this new isopycnal averaging follows isopycnals as they meander horizontally at constant z. Height information is now retained at the cost of losing latitudinal information. The advantage of this averaging is that it avoids the problem of giving a surface that is too warm. Associated with this new isopycnal averaging, a “vertical” transport streamfunction in (ρ, z) space can be defined, in analogy to the conventional meridional overturning streamfunction in (y, ρ) space. Here in Part I, this constant-height isopycnal averaging is explained and illustrated in an idealized zonal channel model. In Part II the relationship between the two different isopycnal averagings and the Eulerian mean eddy flux divergence is explored.

Corresponding author address: A. J. George Nurser, James Rennell Division for Ocean Circulation and Climate, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: g.nurser@soc.soton.ac.uk

Abstract

Simple Eulerian averaging of velocities, density, and tracers at constant position is the most natural way of averaging. However, Eulerian averaging gives incorrect watermass distributions and properties as well as spurious diabatic circulations such as the Deacon cell. Instead of averaging at constant height, averaging along isopycnals removes such fictitious mixing and diabatic circulations. Such isopycnal averaging is normally performed at constant latitude, that is, averaging along isopynals as they heave up and down. As a result, height information is lost and the sea surface becomes much warmer (or lighter) than with simple Eulerian averaging. In fact, averaging can be performed along arbitrarily aligned surfaces. This study considers a particular case in which isopycnal averaging is performed at constant height. Thus, this new isopycnal averaging follows isopycnals as they meander horizontally at constant z. Height information is now retained at the cost of losing latitudinal information. The advantage of this averaging is that it avoids the problem of giving a surface that is too warm. Associated with this new isopycnal averaging, a “vertical” transport streamfunction in (ρ, z) space can be defined, in analogy to the conventional meridional overturning streamfunction in (y, ρ) space. Here in Part I, this constant-height isopycnal averaging is explained and illustrated in an idealized zonal channel model. In Part II the relationship between the two different isopycnal averagings and the Eulerian mean eddy flux divergence is explored.

Corresponding author address: A. J. George Nurser, James Rennell Division for Ocean Circulation and Climate, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: g.nurser@soc.soton.ac.uk

1. Introduction

Simple Eulerian averaging of velocities, density, and tracers at constant position over time and/or zonal position is the most natural way of averaging. However, Eulerian averaging of density and tracers gives incorrect watermass volumes and properties (Lozier et al. 1994), while Eulerian averaging of velocities and density gives rise to spurious diabatic circulations such as the Deacon cell. Averaging on isopycnals removes this fictitious mixing, and, when applied to transports, reduces the Deacon cell (e.g., Döös and Webb 1994; McIntosh and McDougall 1996).

This averaging onto isopycnals is normally performed at constant latitude. Height information is therefore lost by this averaging. The consequences of this are particularly severe at the surface: the mean surface density becomes the lightest water ever found at a given latitude (McDougall 1998; McDougall and McIntosh 2001; Killworth 2001). This may differ substantially from the Eulerian mean surface density. If, say, the air–sea heat flux is linearly proportional to sea surface temperature then the Eulerian mean is clearly more appropriate than the “isopycnal” mean for a coupled climate model in which accurate air–sea fluxes are vital.

In fact, averaging can be performed in a multitude of ways, on arbitrarily aligned surfaces. For example, in a barotropic model, Kushner and Held (1999) averaged over time along contours of potential vorticity. In this contribution we extend their approach to baroclinic flow, averaging on isopycnals at constant height. Hence, the coordinates become (x, ρ, z). Height information is now retained at the cost of losing lateral spatial information. Such averaged fields do not see the surface as a special region, and so behave in the same way at the surface as they do in the interior. As a result, the “horizontally isopycnally” averaged surface temperature is often closer to the Eulerian mean than is the “vertically isopycnally” averaged temperature.

Both properties and velocities can be averaged in this manner, and we may define a zonally averaged “vertical overturning” streamfunction in (ρ, z) space similar to the conventional zonally averaged meridional overturning streamfunction in (y, ρ) space.

The aim of this paper (Part I) is to introduce this constant-height isopycnal averaging, and to illustrate it in an idealized zonal channel model. The differences between such isopycnally averaged fields and the more conventional fields averaged at constant latitude are explained. In Part II (Nurser and Lee 2004) we discuss and compare the vertical and horizontal isopycnal transport streamfunctions with approximations derived from Eulerian mean quantities.

2. Three ways of averaging

The averaging methods described below are valid for any tracer. For immediacy, we use potential temperature. The first task is to define the “mean temperature.”

For simplicity of presentation we consider the simple zonal mean, though our formulation also applies to the time mean. We start with a 3D field θ(x, y, z) in a zonal channel of depth H(x, y) and length L(y, z) (so we allow topography). Vertical and horizontal slices through the temperature field are shown in Figs. 1a and 1b. We consider three different ways of zonal averaging.

a. Eulerian averaging over x at constant y and z

The Eulerian averaging operator cuts through the 3D field along a line of constant y and constant z (Figs. 1a and 1b), giving
i1520-0485-34-12-2721-e21
Note that L = L(y, z) is not a constant in the presence of topography.

Waters of intermediate temperature (gray shading in Figs. 1c,d) are formed as a result of averaging, leading to spurious mixing. Hence, watermass volumes are not conserved by Eulerian averaging. However, gravitational potential energy is conserved since the averaging is at constant depth and there is no vertical displacement of temperature.

b. Isothermal averaging at constant y

Here we want to average along curves of constant θ on the vertical plane of constant y. With this averaging, we define mean isotherms in terms of the “mean heights” of isotherms at constant y. In the simplest case where topography does not vary zonally [i.e., depth HH(y), and so L = L(y)] and isotherms are vertically monotonic [i.e., the height z(x, y, θ) is a single-valued function of θ], the mean height of the θ isotherm is (de Szoeke and Bennett 1993; McIntosh and McDougall 1996)
i1520-0485-34-12-2721-e22
However, we must allow for cases where θ is not vertically monotonic (e.g., allowing salinity compensation) or topography varies zonally. So, we need to define the mean height of isotherms in a more general way. First of all, define Ã(y, θa) to be the total area of fluid with water colder than θa on the vertical xz plane at latitude y (e.g., the black area in Fig. 1a or Fig. 2a):
i1520-0485-34-12-2721-e23
This is a well-defined quantity that increases with θ even when θ is not vertically monotonic. The mean height of the θ isotherm at constant y is defined to be the height (y, θ) (Figs. 1b, 2b) such that the total area with fluid below1 (i.e., the colder side) is equal to Ã(y, θ):
i1520-0485-34-12-2721-e24
This generalized mean height of isotherms is well defined even for nonmonotonic θ and varying topography.

The generalized is equal to the usual zonally averaged height of isotherms ž in (2.2) when topography does not vary zonally and θ is monotonic in the vertical. However, if topography does vary zonally, then reaches the maximum depth at a given latitude whereas ž cannot be lower than the zonal-mean depth, as illustrated in Figs. 2b and 2c and also shown in Lee and Coward (2003). Thus, is deeper than ž in the presence of zonally varying topography.

The mean layer thickness between two adjacent isotherms is defined to be the vertical area between the two isotherms divided by the length (y, θ) = L[y, (y, θ)], the zonal length of all fluid points at (y, θ). In other words, the mean layer thickness (per unit θ) is
i1520-0485-34-12-2721-e25
This is always positive since à increases with θ. We stress that the mean quantities and must be defined in terms of area integrals on the xz plane rather than as line integrals over x.

Using the mean height of isotherms, z̃, we can define mean isotherms in (y, z) space, θ̃(y, z), as inverse functions of such that θ̃ = θa at z = (y, θa). By definition, this averaging conserves the volume of water in each temperature class at each latitude. However the disadvantage is that fluid is rearranged vertically, and so height information is lost. In particular, the warmest water that exists anywhere along the zonal section will be spread out as a thin layer at the surface, and so θ̃(y, z = 0) is the warmest water at the latitude y rather than the mean surface temperature as in the simple Eulerian averaging. The difference between the surface θ̃ and θ is thus O(Θ′), where Θ′ is a typical magnitude of the zonal variability of surface temperature. The vertical rearrangement systematically moves warm, light water upwards and dense, cold water downward. Hence this rearrangement of a zonal slice always reduces the potential energy (Killworth 2001)—indeed it gives the minimum possible energy for any zonal adiabatic rearrangement (e.g., Winters et al. 1995).

c. Isothermal averaging at constant z

Here the averaging operator slices through the 3D field along isotherms at constant z, hence remaining on a horizontal plane. A similar averaging for a 2D barotropic field was first employed by Kushner and Held (1999) to average along constant potential vorticity. We extend their approach by taking into account that contours of θ on the horizontal plane are not necessarily monotonic because of large-scale meanders and cutoff eddies. Namely, it is essential to perform averaging using area integrals on the horizontal plane of constant z.

To do this, we define an area function A*(θa, z) to be the area of fluid colder than θa on the horizontal xy plane at constant z (Fig. 1b):
i1520-0485-34-12-2721-e26
As before, this is a well-defined function that always increases with θ.
Instead of the mean height of isotherms, we now need to define the mean meridional position of isotherms. The mean meridional position y*(θ, z) (Fig. 1f) of the isotherm θ on the horizontal plane of constant z is defined such that the horizontal area with fluid on the colder side of y* (e.g., northern side for the Northern Hemisphere) is equal to A*:
i1520-0485-34-12-2721-e27
If the coldest water is on the southern end, say, in the Southern Hemisphere, then the left-hand side of (2.7) should be changed to integrate from ycold (lower bound) to y* so that the integral remains positive.

Note that Kushner and Held (1999) averaged the meridional position of contours of potential vorticity rather than utilizing area integrals. Where the zonal length of the channel varies with y (as it will do on a sphere as well as a result of coastlines and topography), y* differs from the zonal mean of y(θ) in the same way that and ž differ with zonally varying topography.

Similarly to the mean layer thickness, we can define the mean meridional width (per unit θ) between two isotherms to be the horizontal area between the two isotherms divided by the zonal extent of fluid:
i1520-0485-34-12-2721-e28
where L*(θ, z) is the zonal length of fluid points at y*(θ, z). The mean meridional width is always positive since A* increases with θ. The negative sign in front of ∂y*/ ∂θ is because we “stack up” the water from the north (coldest side at highest y) southward, as is natural in the Northern Hemisphere.

Using the area function A* (or Ã) to define the mean meridional layer width l* (or the mean layer thickness ) is more robust than using y* (or ) since the area function increases with θ, and so its θ derivative (i.e., the mean width or the mean layer thickness) is always positive, whereas y* can increase or decrease with θ depending on the large-scale meridional gradients of θ (although will still increase with θ on the large scale).

As before, the mean y* can be used to define mean isotherms θ*(y, z) in (y, z) space such that θ* = θa at y = y*(θa, z). Fluid is rearranged horizontally, moving warm, light water toward the warmer end and dense, cold water toward the colder end: clearly this averaging only makes sense if there is a systematic large-scale north–south temperature gradient, so the cold end and warm end are well defined. Since zonal variations on the horizontal plane can be very large, whereas zonal variation on the vertical slice is constrained by gravity, the loss of meridional information is more severe than the loss of the height information with isothermal averaging at constant y. In this averaging, potential energy is conserved and water masses are of course still conserved (at constant height) and hence globally.

d. Time averaging of a 3D field

Time averaging can also be performed in these three different ways. Time averages replace zonal integrals (de Szoeke and Bennett 1993; McDougall and McIntosh 2001). The “areas” Ã and A* in Figs. 1 and 2 now become integrals over dzdt and dydt, respectively. The corresponding θ̃ and θ* are calculated as functions of z at each (x, y) and y at each (x, y) by redistributing fluid vertically or horizontally, respectively. For time averaging, the averaging time scale, analogous to L, is independent of position, and so and ž are the same (where ž is defined). Of course, since time averages rather than time integrals are required, this averaging time scale is then canceled out.

e. An illustrative example

In order to illustrate the advantages and disadvantages of these averaging methods, we consider a two-layer zonal channel where warm water lies to the south and cold water lies to the north, forming a front oscillating in the zonal direction (Fig. 3a). The three zonal averagings give three different pictures.

  • Eulerian averaging. The temperature near the center takes the averaged value between the cold and the warm temperatures. Temperature is “diffused” between the southernmost and northern most positions of the front, apparently creating water of intermediate temperature (Fig. 3b). Potential energy is conserved.
  • Isothermal averaging at constant y. The isotherm heights are averaged. The warm water is assumed to lie above the cold water and, since there is more warm (cold) water in the south (north), the result of averaging is that isotherms rise to the north, and a vertically sloping mean front appears between the southernmost and northernmost positions of the real front (Fig. 3c). Warm light water is moved upward, and cold dense water is moved downward, and so potential energy is lost.
  • Isothermal averaging at constant z. The meridional position of isotherms is averaged. Warm water is assumed to be on the south side of cold water. The result of averaging is a vertically uniform temperature front (Fig. 3d). Potential energy is again conserved.

This somewhat extreme example illustrates the advantage of this novel averaging at constant z. It (i.e., θ*) does not generate extra stratification, unlike θ̃, the averaging at constant y. This can be particularly useful in regions where the physics are sensitive to the stratification, such as convective regions. Apart from that, the largest differences between isothermal averaging at constant y and z are at the boundaries, including surface, bottom, and lateral boundaries. We now see how these averagings operate in a full eddy-resolving model of a channel.

3. Model

We use the Modular Ocean Model for Array processors (MOMA) code (Webb 1996), a modified version of the z-level primitive equation Bryan–Cox model. This employs full spherical polar coordinates. The Modified Split Quick (MSQ) algorithm (Webb et al. 1998) is employed to advect tracers. It includes fourth-order advection plus a velocity-dependent biharmonic-type diffusive term. We employ a variant of this scheme, adding a vertical flux equal to the horizontal biharmonic diffusive flux times the isopycnal slope (A. J. G. Nurser 2004, unpublished manuscript). This works in exactly the same way as the skew–symmetric representation of the GM transport (Griffies et al. 1998) to ensure that the biharmonic diffusion does not drive any diapycnal mixing. This term is not fully employed in the surface and bottom grid boxes, and where isopycnal slopes become steeper than 1/100—here some mixing does occur. We use simplified physics with constant salinity and a linear equation of state with an expansion coefficient of 2.2 × 10−4°C−1. Horizontal and vertical viscosities are 104 and 5 × 10−4 m2 s−1. Bottom friction is quadratic with drag coefficient of 0.001. Vertical diffusivity is 5 × 10−5 m2 s−1, but we use no explicit lateral diffusion.

The model is run in a zonal channel centered at 45°N. We run at high horizontal resolution: 0.09° in latitude, 0.09° cos(45°) in longitude—approximately 10 km by 10 km. Vertical resolution is uniformly 20 m. The channel is 600 km (60 grid points) long and 1000 km (100 grid points) wide, so its southern boundary lies at 40.5° and the northern boundary at 49.5°. It is 1000 m (50 grid points) deep.

The model is not forced with winds. Within the southernmost degree (south of 41.5°) and the northernmost degree (north of 48.5°) model temperatures are relaxed throughout the water column to specified values: linearly varying with depth from 25°C at the surface to 0°C at the bottom in the southern zone and from 10°C to 0°C in the northern zone. The relaxation rate (the reciprocal of the relaxation time scale) varies linearly between 2 day−1 at the boundaries, and zero at the inner edges of the relaxation zone, 41.5° and 48.5°. A surface heat flux is applied so as to relax surface (actually midsurface box, at 10-m depth) temperature towards a T* varying linearly between 24.75°C at 41.5° and 9.9°C at 48.5°. This surface relaxation is with piston velocity 3 m day−1; equivalent to a relaxation time scale for the surface grid box of 7 days, or a heat flux varying with temperature by 167 W m−2 s−1.

The model was initialized with a temperature field found from interpolating bilinearly in y and z throughout the model domain from the two vertical temperature profiles described above, south of 41.5° and north of 48.5°, toward which the temperature field is relaxed. A small temperature perturbation was applied, the model went unstable, and was then run for a total of 14 yr. Statistical equilibrium was attained after 2–3 yr, but the diagnostics below are presented as averages over the last 10 yr of the run. Isothermal averaging has been achieved simply by binning the volumes (and transports) of each model grid into temperature bins 0.6°C in extent. The large number of data points ensured smooth average fields, so no interpolation was necessary.

4. The temperature field

The vigor of the eddies in the model is evident in the plots of the temperature field after 12 yr presented in Fig. 4. Horizontal slices in the surface layer, at a depth of 10 m, and at middepth, 350 m deep (Figs. 4a,b) show the large meridional excursions of the temperature contours, of up to 500 km. These excursions are less striking at the surface because of the surface relaxation forcing. There are also two cutoff eddies, seen best at 350 m, a southern cold eddy centered at around 180 km east, 100 km north, and a northern warm eddy at 400 km east, 750 km north. These penetrate deeply into the southern and northern relaxation zones (south of the thin line at 100 km and north of the thin line at 900 km, respectively) and are therefore strongly forced since their associated temperature anomalies are about 2°C colder than the equilibrium temperature of 16.25°C in the south and 2°C warmer than the equilibrium temperature of 6.5°C in the north at this depth.

Meridional–vertical slices of the temperature field at 180 km east (Fig. 4c) and 470 km east (Fig. 4d) also show the penetration of the cold and warm temperature anomalies into the southern/northern relaxation zones at 180 km east and 470 km east, respectively. Comparison of the two meridional sections shows also that the eddy variability of isothermal heights frequently reaches 200 m.

Figures 5a–c shows the three forms of zonally and temporally averaged temperature, diagnosed from the model: θ, θ̃, and θ*. In general, the three averagings give similar temperature profiles with isotherms rising to the north. The largest differences between the three averagings are at the boundaries (Figs. 5d and 5e).

Examining the difference between θ and θ̃ shows that θ̃, the isothermal averaging at constant y, gives surface temperatures up to 4°C warmer than Eulerian averaging (Fig. 5f). This is because the surface temperature θ̃ is the warmest water ever found at that latitude. Consequently surface warm water extends farther north than with standard Eulerian averaging. Below about 100– 200 m, temperatures are up to 1° colder. This must happen because the total heat content of the water column is conserved by both averaging methods. Since isothermal averaging at constant y gives warmer surface temperatures, it must also give colder temperatures at depth. The reason for the larger difference near the surface than at depth is that θ̃ = θ + O(Θ′) at the surface whereas θ̃θ + O(Θ′2) at depths where isotherms do not outcrop to the surface (McDougall and McIntosh 2001). See Part II for a more detailed discussion.

Similar differences are seen in comparing θ and θ*. Now isothermal averaging at constant z puts warmer water to the south, with θ* being warmer than θ by about 1.5°C to the south and conversely being cooler by about 1°–1.5°C to the north.

5. Transports and mass balance

The three different averagings not only give different mean isopycnals but also give different views of how mass is transported and, in particular, how the mass transport is related to diabatic heating. We discuss these issues here.

a. Eulerian transport streamfunction

The Eulerian transport streamfunction is the zonally integrated northward transport of fluid across a given latitude y and below the constant height surface za,
i1520-0485-34-12-2721-e51
The Eulerian streamfunction Ψ(y, z) directly implies the flow in Eulerian space: ∂Ψ/∂z gives the northward flow and −∂Ψ/∂y the upward flow.

b. Isothermal transport at constant y

The meridional transport of fluid with temperature colder than a given θa across a given latitude y is (see Figs. 6a,b)
i1520-0485-34-12-2721-e52
Consequently, the total northward transport of fluid with temperature between θ and θ + Δθ at constant y is (∂ψ̃/∂θθ.
The mean meridional velocity between two isotherms is then the total northward transport between the isotherms divided by the vertical area between the isotherms:
i1520-0485-34-12-2721-e53
Like the mean height z̃, the mean velocity must be defined in terms of areal averaging rather than zonal line averaging in order for the definition to hold for nonmonotonic θ and varying topography. Thus, the mean velocity υ̃ is an isothermal areal averaging of velocity υ at constant y.

Since the total transport (per unit temperature) is υ̃h̃, our mean velocity υ̃ includes the contribution associated with correlations between layer thickness and velocity resulting from the “standing” eddies (the analog in zonal averaging of the “bolus velocity” obtained by time averaging). Indeed, the definition of υ̃ is the usual “layer-thickness-weighted” velocity in (x, θ) coordinates whenever such coordinates are globally defined.2

While the streamfunction's isothermal derivative, ∂ψ̃/∂θ, gives the meridional transport along the isotherm, its spatial derivative, ∂ψ̃/∂y, only gives the diapycnal flow if there is a steady state. The mass (assuming incompressibility, just volume) balance for fluid colder than θ in the vertical slice at constant y is (Fig. 7a),
i1520-0485-34-12-2721-e54
where (y, θ) is the total diapycnal transport (per unit y) across the θ surface. At each latitude, the rate of change of area colder than θ must be balanced by the convergence of transport of fluid colder than θ across that latitude and the diapycnal flux across the θ surface. Thus, it is only for a statistically steady state where ∂Ã/ ∂t vanishes, that the spatial derivative of ψ̃ gives the diapycnal flux, = −∂ψ̃/∂y.
The diapycnal transport is driven by diabatic heating. Namely, the diabatic heating supplied into the layer between two isotherms of temperatures θ and θ + Δθ allows the diapycnal flux to undergo the temperature change of Δθ. Hence if the diabatic heating Q = / Dt and
i1520-0485-34-12-2721-e55
is the total heat input into the vertical area à below the θa isotherm, then the diapycnal transport can be written as (Fig. 7b)
i1520-0485-34-12-2721-e56
Thus, from (5.4) and (5.6), we have
i1520-0485-34-12-2721-e57
Diabatic heating between isotherms drives diapycnal flow and so convergence of transport colder than the isotherms and/or changes of area below the isotherms.
Integrating northward and assuming a steady state, we have
i1520-0485-34-12-2721-e58
where ℛ̃(y, θ) is the region with fluid colder than the θ and north of latitude y. In other words, in the steady state, the northward transport at constant y of waters colder than θ must give rise to a diapycnal transport across the θ surface to the north of y. This diapycnal transport is set by ℋ̃Q(y, θ), the heat input into the layer between θ and θ + Δθ to the north of y. This is the basis of watermass transformation theory (e.g., Walin 1982; Marsh et al. 2000).
We can define the mean diabatic heating as the total heating between two isotherms divided by the vertical area between the two isotherms,
i1520-0485-34-12-2721-e59
The diapycnal flux is then related to this mean diabatic heating simply by
i1520-0485-34-12-2721-e510
Differentiating (5.7) by temperature then gives the mass balance within a layer between two isotherms:
i1520-0485-34-12-2721-e511a
or, equivalently,
i1520-0485-34-12-2721-e511b
This mass balance equation implies that after isothermal areal averaging the continuity equation takes the same form as the unaveraged continuity equation.3 The averaged continuity equation is now an equation for vertical area between isotherms rather than for layer thickness. Likewise, the velocity and diabatic forcing are areally averaged rather than zonally averaged. This is more robust since neither topography nor nonmonotonic θ will affect this result.

c. Isothermal transport at constant z

Analogously, the natural streamfunction in (θ, z) coordinates is the upward transport of fluid colder than a given temperature θa across a height surface z (Figs. 6c,d):
i1520-0485-34-12-2721-e512
Note that the sign convention is such that ψ̃ and ψ* have the same sign for the case in which colder water lies to the north and isotherms rise northward (since northward flow along isopycnals implies upward flow). If the colder water lay to the south, then ψ̃ and ψ* would have opposite signs. We discuss the relationship between ψ̃ and ψ* in more detail later on.
Similar to (5.3), the mean vertical velocity is given by the total upward transport between the isotherms divided by the horizontal area between the isotherms at constant z:
i1520-0485-34-12-2721-e513
The mass balance for fluid colder than θ in the horizontal plane at constant z is (Fig. 7c)
i1520-0485-34-12-2721-e514
where now e*(θ, z) represents the total diapycnal transport per unit z across the θ surface (from cold to warm, i.e., southward)—different from (y, θ). The diapycnal flux e*(θ, z) is now driven by heat input between the θ and θ + Δθ isotherms on the horizontal plane of constant z (Fig. 7d):
i1520-0485-34-12-2721-e515
where
i1520-0485-34-12-2721-e516
is the total heat input into the horizontal area A* colder than θ at height z. Note that the heat input into the horizontally meandering layer at constant z, which drives e*(θ, z), may be quite different to the heat input into the vertically undulating layer at constant y that drives (y, θ).
From (5.14) and (5.15)
i1520-0485-34-12-2721-e517
For a steady state, the isothermal movement term again vanishes and −∂ψ*/∂z is the total diapycnal flow (from cold to warm), driven by the total heat input into the layer between isotherms at given z. Integrating from z upwards, we have
i1520-0485-34-12-2721-e518
where ℛ*(θ, z) is the region above z with fluid colder than θ. Thus, the vertical transport colder than θ across a given height must be balanced by *Q(θ, z) the total heat input between θ and θ + Δθ above the same height level. This heat input *Q(θ, z) is different from the heat input ℋ̃Q(y, θ).
The mean diabatic heating associated with this averaging is then
i1520-0485-34-12-2721-e519
It is an average over the horizontal area between isotherms at constant z, in contrast to Q̃, which is averaged over the vertical area between isotherms, at constant y. This Q* is related to the diapycnal flux e* by
i1520-0485-34-12-2721-e520
The averaged continuity equation now takes the form
i1520-0485-34-12-2721-e521
Again, this isothermal averaging also preserves the form of the continuity equation.4

d. Streamfunctions in the model

The Eulerian streamfunction Ψ and the meridional transport streamfunction ψ̃ are very familiar, whereas the vertical transport streamfunction ψ* is unfamiliar and at first hard to interpret. Figures 8a and 8c show these streamfunctions plotted out from our zonal channel model, while Figs. 8b and 8d show schematically the positions of the southern boundary, surface, and northern boundary forcing regions. The streamfunctions are, in fact, zonal–temporal means, so are the time means of the zonal integrals (5.2) and (5.12). Since the cold water lies to the north in our model, ψ̃ and ψ* have the same sense (cf. Fig. 6 and the discussion in the preceding section). The Eulerian mean streamfunction Ψ (not shown) has no significant meridional transport since the zonal–temporal mean ageostrophic velocity is small.

The meridional transport streamfunction ψ̃ shows about 5 Sv (Sv ≡ 106 m3 s−1) southward transport of cold (θ < 10°C) waters, returning northward as warm (12°C < θ < 24°C) waters. Note that there is flow at all temperatures that ever exist at a given latitude—the streamfunction covers the θ̃ range rather than the θ range. The transport in each isopycnal layer, ∂ψ̃/∂θ, is the result of correlations between vertical area and meridional velocity across the area. Since the mean ageostrophic flow is small, this transport must be achieved by eddies. The diapycnal flow consistent with this steady meridional circulation involves (i) warming of 4–5 Sv of water in the southern relaxation zone, where the forcing attempts to keep isopycnals down. Temperatures rise from about 4° to 22°C, implying heat gain of about 77 Sv° (0.3 PW) (ii) cooling of 4 Sv of the warmest (surface) waters from about 22° to 10°C by the surface relaxation as they flow northward, a heat loss of 47 Sv° (0.18 PW) and (iii) further cooling of the 5 Sv in the northern relaxation zone where the forcing keeps the isopycnals up, giving a heat loss of 30 Sv° (0.12 PW). The weak vertical diffusion drives additional slight warming, driving ∼−0.9 Sv across the 10°C isotherm so as to give the peak value of the streamfunction (∼−5.9 Sv) at about 750 km.

The vertical transport streamfunction ψ* at first looks strange. Since temperature at constant depth generally increases southward, the temperature scale has been reversed to increase toward the left in order to retain the intuitive link left = south/right = north of the other figures. The triangular shape of the plot results from the increase in temperature range from the bottom (all water about 0°–2°C) to the surface (between 5° and 24°C). The streamfunction shows about 4 Sv of rising warmer waters (2°–22°C), cooling near the surface (between 24° and 9°C) and returning downward as cooler waters (2°– 10°C). This vertical transport is due to the correlation between horizontal area and the vertical velocity across the area, which is achieved by eddies.

The rising motion is associated with strong warming, so must be occurring largely in the southern relaxation zone, while the descending motion is associated with cooling in the northern relaxation zone. The warming and rising is strongest at depths between 250 and 500 m and temperatures between 9° and 14°C, consistent with the strongest warming between temperatures of 9° and 14°C seen in the plot of ψ̃. Note that the range of temperature associated with warming and upwards motion decreases with depth, from 5°C at 250 m to 1.5°C at 750 m. The 4 Sv of water is warming from about 4° to about 19°C—about 60 Sv° (0.32 PW). There is also cooling from 21° to 11°C near the surface (the upper side of the triangle), giving about 40 Sv° (0.21 PW) of heat loss in total. Most of this is the cooling due to surface relaxation evident also in ψ̃, but part of it is caused by cooling in the northern relaxation zone. Finally the balance of the heat loss, 20 Sv° (0.11 PW) occurs in the northern relaxation region. All three heat gains/losses are smaller in magnitude when diagnosed from ψ* than ψ̃. This is because ψ* gives information about heating of an isopycnal at a given depth anywhere in the domain, and the forcing applied here has more cancellation at constant depth than constant latitude.

Regions of purely isopycnal flow in zθ space are very restricted because at any depth almost all isotherms run into either the southern or northern relaxation zones (Figs. 4a,b), and so they are forced. There appears to be some purely isopycnal flow along the white dashed line in Fig. 8d, but this may simply result from cancellation of southern warming by northern cooling.

This vertical transport streamfunction ψ* is unfamiliar, but we believe it to be useful in diagnosing data from models and from observations where it is important to know at what depth diapycnal flow/warming is taking place—for example, where flows are driven by the potential energy input from buoyancy input at depth. The model run being diagnosed is one such, driven by the warming at depth in the southern relaxation zone, the left side of the triangle in the schematic Fig. 8d and the plot of ψ* in Fig. 8c.

Thus, the two streamfunctions ψ̃ and ψ* can be used complementarily as they indicate the meridional transport and vertical transport directly. In the steady state, they also indicate the latitudes and depths of diapycnal flows.

6. Isopycnal streamfunctions in Eulerian space

a. The isopycnal streamfunction ψ̃(y, θ)

The meridional isopycnal streamfunction ψ̃(y, θ) may be written as a function of y and z by identifying θ with its mean height (y, θ) at each value of y, that is, by writing
y,zψ̃y,θ̃y,z
This ensures that the surface, z = 0, is identified with the warmest temperatures ever encountered at that y and the bottom the coolest; hence Ψ̃ = 0 both at z = 0 and at the bottom. If θ were used instead, with its more restricted range (cooler at the surface, warmer at the bottom), δ functions would appear in the transformed streamfunction (McDougall and McIntosh 2001). We plot this isopycnal streamfunction in Eulerian space, Ψ̃(y, z), for our model run in Fig. 9a. This remapped streamfunction Ψ̃(y, z) is more than just a cosmetic exercise. It has some very useful properties.
It can be shown [in McDougall and McIntosh (2001) and the appendix] that θ̃(y, z) satisfies exactly
i1520-0485-34-12-2721-e62
where
i1520-0485-34-12-2721-eq4
is the isothermal mean horizontal velocity and
i1520-0485-34-12-2721-e63
Thus, θ̃(y, z) is forced by the mean diabatic forcing and advected by the mean velocity (υ̃, ). Moreover, (υ̃, L̃w̃) form a nondivergent flux with the streamfunction given by Ψ̃. This implies that Ψ̃(y, z) is a “true” streamfunction in the sense that its spatial derivatives give the mean “horizontal” and “vertical” flux. This is in contrast with ψ̃(y, θ) whose spatial derivative give the diapycnal flow only when in steady state. Equation (6.2) implies that, when the flow is steady and adiabatic, the streamlines of Ψ̃(y, z) follow exactly the mean isotherms θ̃(y, z).

The vertical velocity requires some explanation. Given z = za in (y, z) space, we can define a surface 𝒮za consisting of θ surfaces such that (y, θ) = za. Since θ varies in space and time, the surface 𝒮za is made up of different θ surfaces at different times such that (y, θ) = za. However, since za is fixed, the area under these θ surfaces remains constant [i.e., Ã(y, θ) = ∫∫zazH(x,y) dx dz].

The −∂Ψ̃/∂y term on the right-hand side of (6.3) is the convergence of the flow underneath 𝒮za. This flux divergence is balanced by the flux per unit y across the 𝒮za surface. Thus, the flux (y, za)(y, za) represents the flux per unit y across 𝒮za, made up of those θ surfaces that satisfy (y, θ) = za, that is, across the line of constant z = za in (y, z) space.

Note that, since the surface 𝒮za is not flat, L̃w̃ is not the integrated vertical velocity across the horizontal plane z = za. It is instead (see the appendix) the sum of the motion ∂Ã/∂t of the surface of constant θ, the flux through the meridionally varying area under the isopycnal, υ̃Ã/∂y, and the diapycnal flux, = Ã/∂θ:
i1520-0485-34-12-2721-e64
Hence, identifying with θ̃/Dt, we may write the pseudo vertical velocity
i1520-0485-34-12-2721-e65
Remarkably, takes the same form as real vertical velocity w in the unaveraged space.

The averaged equations for θ̃ and differ from the instantaneous unaveraged equations only in that θ̃, υ̃, and the forcing are averages following the isopycnals as they heave up and down. In particular, note that surface forcing will appear smeared out over the isopycnals that outcrop at the surface at different times and places. The isomorphism to the unaveraged equations is obvious if we regard υ̃, Q̃, etc. as flux and forcing densities in the isothermal averaged space and note that the volume element in such space is simply dy dz, and so (υ̃, L̃w̃) is a nondivergent flux field.

The temperature variable θ̃ retains the correct volume–temperature distribution and hence variance of the original unaveraged variable θ. In the presence of eddies, θ̃ is not spuriously mixed by the averaging. The impact of eddies on θ̃ is thus

  1. to modify the advection υ̃ and and
  2. to modify the diabatic forcing by including the entirely genuine mixing across θ surfaces.
Contrast this with the Eulerian mean temperature, θ, which does not retain the same volume–temperature distribution and variance, and so suffers apparent mixing by the averaging. It is therefore diffused as well as advected by the eddy-driven flow (see Part II).

b. The isopycnal streamfunction ψ*(θ, z)

The vertical transport streamfunction ψ* can of course also be transformed into Eulerian space—now each value of θ at given z is associated with its mean meridional position y*(θ, z), so
y,zψθy,zz
This Ψ*(y, z) is plotted for our model run in Fig. 9b. Kushner and Held (1999) remapped in a similar way the time averaged barotropic transport streamfunction for eastward flow from (x, q) (where q is potential vorticity) space to (x, y) space.
Similar manipulations to those employed for ψ̃(y, z) then show that the θ* field evolves according to
i1520-0485-34-12-2721-e67
where Q* is the outcrop-mean diabatic forcing at given z, the velocity
i1520-0485-34-12-2721-eq5
is the isothermal mean vertical velocity, and
i1520-0485-34-12-2721-e68
is the pseudohorizontal velocity across surfaces of constant y*, which is equivalent to
i1520-0485-34-12-2721-e69
The temperature variable, flow field and forcing now follow the isopycnals as they move laterally. Forcing is “smeared out” laterally rather than vertically. Again, Ψ*(y, z) follows θ* when the flow is steady and adiabatic, and of course θ* is not spuriously mixed by the averaging. The flux (L*υ*, L*w*) also form a nondivergent velocity field.

Hence, we have shown that any 3D nondivergent velocity field can be projected onto a variety of 2D nondivergent fields with respect to different averaging. In Part II, we try to explain how these 2D nondivergent flows may be related.

c. Ψ̃ and Ψ* in the model

Fields of Ψ̃ and Ψ* from the model are plotted in Figs. 9a and 9b, superposed upon θ̃ and θ* respectively. In interpreting the two fields we need to keep in mind that the “z” coordinate in the Ψ̃ plot and the “y” coordinate in the Ψ* plot simply represent different values of θ remapped according to (y, θ) and y*(θ, z)—vertical information has been lost from the Ψ̃ plot and horizontal from the Ψ* plot. This information loss is clear from inspecting the instantaneous temperature field in Fig. 4, which shows xy plots of the temperature field at 10 and 350 m in Figs. 4a,b and meridional yz sections in Figs. 4c,d. Isotherms show vertical displacements of 200–300 m, but horizontal displacements of up to 600 km, emphasizing how little horizontal information is retained in the Ψ* plot.

The model is in steady state, so flow across θ̃ and θ* respectively implies diabatic flow at that y and temperature (for Ψ̃) and at that z and temperature (for Ψ*). The Ψ̃ plot shows, similar to the ψ̃(y, θ) plot, warming restricted to the south, but cooling “near the surface”— over the range of temperatures exposed to the surface at that y—as well as to the north. The deep return flow is unforced and so follows isotherms. The plot of Ψ* shows instead warming over much of the “southern” half of the domain, with cooling over the “northern” half and throughout the upper 200–300 m. It is the large meridional excursions of the isotherms that spread the boundary relaxation through the domain. Inspection of the temperature fields at 150 and 300 m shows that almost all of the temperature contours occasionally pass into either the north or south relaxation zones and so are either cooled or warmed. Because the forcing is a relaxation, the colder the isotherm (for an incursion into the southern warming region), the stronger the warming, so even an occasional incursion of a cold isotherm may cause significant warming. Similarly for incursions of warm isotherms into the northern cooling region.

Much of the vertical spreading of the cooling over the upper 200 m in Ψ* is genuine, resulting from convection, which can penetrate down to 150 m even in the warmer waters. The cooling is weakened relative to that seen in the Ψ̃(y, z) plot by incursions of the near-surface isotherms into the southern warming zone. Cooling deeper than 150–200 m is probably mostly driven by incursions of the isotherms into the northern cooling zone.

d. Comparison of Ψ̃ and Ψ*

Streamfunctions Ψ̃ and Ψ* are both functions of y and z and so may be compared directly. Note that they have the same sign only when colder water lies to the north as shown in Figs. 6c and 6d. If cold water lies to the south, then Ψ* should instead be compared with −Ψ̃. In such a case the following analysis should have the sense of Ψ̃ and y reversed.

To compare them, say, at given (y0, z0), one must remember that by definition
y0z0ψ̃y0θ̃0
where θ̃0 = θ̃(y0, z0), and
y0z0ψθ*0z0
where θ*0 = θ*(y0, z0).

In general θ̃0θ*0, and so the two streamfunctions Ψ̃ and Ψ* refer to flow relative to different isotherms. Also, they differ because the vertical plane y = y0 across, which the northward flux ψ̃(y0, θ̃0) flux is calculated is in a different place from the horizontal plane z = z0 across which the upward flux ψ*(θ̃0, z0) is calculated.

Thus,
i1520-0485-34-12-2721-e612

The Δ flow term gives the difference between the upward flux northward of the θ̃0 isotherm, ψ*(θ̃0, z0) and the northward flux below the θ̃0 isotherm, ψ̃(y0, θ̃0) (Fig. 10). This consists of two parts.

  1. The first part is the upward flux through the z0 plane south of y0 that never reaches the y0 plane. This gives a positive contribution to ψ*(θ̃0, z0) but does not contribute to ψ̃(y0, θ̃0). This flux either passes up through the part of θ̃ surface where it lies above the z0 plane but south of y0 (indicated by 𝒜+ in Fig. 10) or goes to inflate the volume bounded by the 𝒜+ and the two planes z0 and y0:
    i1520-0485-34-12-2721-e613
    where 𝒜+ = {(x, y, z): θ = θ̃0; y < y0; z > z0}, the bulged up surface in Fig. 10.
  2. The second part is the northward flux through the y0 plane, and so part of ψ̃(y0, θ̃0), but that never crosses the z0 plane, and so does not contribute to ψ*(θ̃0, z0). Hence, this gives a negative contribution to (6.12). This flux either passes up through the θ̃0 surface where it lies below the z0 plane but north of y0 (indicated by 𝒜 in Fig. 10) or inflates the volume bounded by θ̃0 surface and the planes z0 and y0:
    i1520-0485-34-12-2721-e614
    where 𝒜 = {(x, y, z): θ = θ̃0; y > y0; z < z0}, the depressed surface in Fig. 10.

Hence the Δ flow term in (6.12) becomes
i1520-0485-34-12-2721-e615
and so
i1520-0485-34-12-2721-e616
In the statistically steady state, as in our model run, (6.16) simply states that Ψ*(y0, z0) differs from Ψ̃(y0, z0) as a result of (i) θ* differing from θ̃ and (ii) diapycnal flow across the isotherm in the quadrants y < y0; z > z0 and y > y0; z < z0.

e. Comparison of Ψ̃ and Ψ* in the model run

We can use (6.16) to explain why Ψ̃ and Ψ* in Fig. 9 look different. Consider y0 = 150 km, z0 = −350 m. Here Ψ* ∼ −1.8 Sv is much less negative than Ψ̃ ∼ −5.1 Sv. Why? First, θ̃*0 ∼ 16°C is warmer than θ̃0 ∼ 13.5°C; hence ψ*(θ̃0, z0) = −3 Sv, still ∼2 Sv less negative than Ψ̃(y0, z0) = ψ̃(y0, θ̃0). This means that the last two terms in (6.16) contribute about 2 Sv.

The model is in statistically steady state, and so the 2 Sv of transport across the 13.5°C isotherm must result from diabatic forcing due to relaxation either at the sea surface or near north/south boundaries. For example, it could be caused either by warming on the 13.5°C isotherm above 350 m depth, south of 150 km, or cooling on the isotherm below 350 m, north of 150 km. Inspection of the temperature field at 350 m, Fig. 4b, shows that the 13.5°C isotherm does indeed outcrop within a cold-core eddy inside the southern relaxation zone. Inside this outcrop the isotherm must lie above 350-m depth, subject to strong warming (the equilibrium temperature above 350 m is greater than 16.25°C here)—this could explain why ψ*(θ̃0, z0) is less negative than ψ̃(y0, θ̃0).

This outcrop within the relaxation zone is possible because the equilibrium depth of the 13.5°C isotherm at the southern boundary is only 450 m, so the eddy displacement only needs to be 100 m upward. There is, however, no outcrop within the northern relaxation zone: the equilibrium depth of the 13.5°C isotherm is the surface here. Eddies would thus have to displace the isotherm at least 350 m downward from its equilibrium height—this will not happen. However the northern cooling below 350 m does explain the less negative Ψ* farther “north”; for example, again at 350-m depth the θ = 8.5°C isotherm outcrops into the northern relaxation zone.

7. Discussion and conclusions

We have illustrated here how quantities may be isopycnally averaged in two different ways. One way is the conventional “vertical” averaging—that is, averaging along the isopycnals as they oscillate vertically at constant y. Another, following Kushner and Held (1999), is “horizontal” averaging—averaging along the isopycnals as they oscillate horizontally at constant z. Both methods remove the spurious mixing resulting from simple Eulerian averaging. The vertical isopycnic averaging generates a mean density field (here taken simply as a temperature field θ̃) by redistributing water vertically, whereas the horizontal isopycnic averaging redistributes water horizontally to create the mean temperature θ*. Warmer, lighter waters are moved upward and southward, respectively. The vertical stratification in θ̃ is given by the mean layer thickness defined as the vertical area between isopycnals divided by a length L̃, while the horizontal stratification of θ* is given by the mean layer width defined as the horizontal area between isopycnals divided by the length L*. Both isopycnal averagings are defined in terms of area integrals (at constant y or at constant z) rather than zonal line averaging in order to take into account of nonmonotonic θ and varying topography.

Since areal averaging does not require a monotonic field, it can be used to generate watermass-conserving “averages” and transport streamfunctions for tracers, such as salinity, that may not be monotonic.

The problem with the conventional vertical averaging is that the vertical redistribution increases the stratification and leads to excessive surface temperatures. Potential energy is therefore lost. Horizontal averaging does not suffer from this problem. On the other hand, horizontal information is instead lost, and this loss may be severe because of the large meridional displacements of the temperature contours.

Associated with this horizontal averaging is the isopycnal vertical transport streamfunction ψ*(θ, z), defined as the total upward flux across the horizontal z plane of all fluid colder than a given temperature. This is analogous to the isopycnal meridional streamfunction ψ̃(y, θ), the total northward flux across the vertical y plane of all fluid colder than θ. The streamfunction ψ* looks somewhat bizarre at first sight, but it gives information about the depths at which diapycnal flow is taking place. This depth information, lost in the meridional streamfunction ψ̃(y, θ), is particularly useful in estimating how much potential energy (through warming and hence expansion at depth) is being supplied to the circulation by diapycnal processes.

This vertical transport streamfunction can be remapped back into Eulerian space to give a Ψ*(y, z) by identifying temperature at a given depth with the meridional position of the isotherm in the “horizontally averaged” stratification θ*. This may be compared with the conventional meridional isopycnal transport streamfunction remapped into Eulerian space, Ψ̃(y, z), by identifying temperature with the vertical position of the isotherm in the “vertically averaged” stratification θ̃. Both of these remapped streamfunctions have the important property that their vertical and horizontal gradients give the mean horizontal and vertical velocities, respectively. In contrast, the spatial gradients of the original isopycnal streamfunctions, ∂ψ*/∂z and ∂ψ̃/∂y, only give the diapycnal flow in the steady state.

Very appealingly, each of Ψ* and Ψ̃ advect their associated temperature fields, θ* and θ̃, according to equations that have exactly the same form as the unaveraged θ equation except that velocities (and θ) are averages along isotherms. For steady adiabatic flow, Ψ* and Ψ̃ follow the θ* and θ̃ isotherms exactly—there are no other eddy forcings as would arise under Eulerian averaging. The sign of the two streamfunctions Ψ̃ and Ψ* (as defined here) is only the same if cold water is assumed to lie to the north.

For a thermally forced, statistically steady, zonal channel run with energetic large eddies, we found considerable differences between the two remapped streamfunctions. The vertical transport streamfunction Ψ* was more “smeared out” than the meridional transport streamfunction Ψ̃. The large meridional isothermal displacements cause a wide range of isothermal layers to wander into forcing regions near the boundaries, so spreading the forcing throughout much of the domain after the remapping.

The two different averagings may be used complementarily in diagnosing observational data or model output. By using both ψ̃ and ψ* (and their remapped Ψ̃ and Ψ*) we can indicate meridional and vertical transport simultaneously and, if the flow is steady, infer diapycnal flows at a given latitude or depth. Using just one can give the incorrect impression. For example, in our schematic (Fig. 3), if the fluid were circulating with purely horizontal motion, it would appear to have an overturning cell in the isopycnal meridional transport streamfunction ψ̃ even though there is no vertical motion at all (ψ* = 0)!

For clarity the discussion and examples here have involved zonal averaging in a channel. However, as discussed in section 2, θ̃ and θ* can easily be defined as 3D time-averaged fields. McDougall and McIntosh (2001) have further shown how, for standard vertical isopycnal averaging, streamfunctions may be defined for zonal and meridional flow under isopycnals, and then mapped back into real space using θ̃. A 3D version of Eq. (6.2), the advection equation for θ̃ then holds. A 3D version of the advection equation for our θ* may be developed similarly.

In fact, it is possible to perform similar remappings using two conserved tracers. In this case (y, z) space would be spanned by, say, temperature θ and surfaces of any other tracer as long as temperature and the tracer are not functionally dependent. For instance, meteorologists use potential vorticity and potential temperature as quasi-meridional and quasi-vertical coordinates in modified Lagrangian mean analysis (McIntyre 1980; Butchart and Remsberg 1986; Nakamura 1995). McIntyre (1980) believed (without deriving it) that an averaged θ equation similar to (6.2) holds. Nakamura (1995) showed the mean potential vorticity equation in (potential vorticity, potential density) coordinates. He termed what we call y* the “equivalent latitude.” It would be desirable to derive systematically how to perform averagings on any suitable pair of tracer, say μ and ν. Analogues of both y* and z̃, as well as of θ̃ and θ* could be defined. It should be possible to define transport streamfunctions analogous to ψ̃ and ψ* and remap them back into Eulerian space like Ψ̃ and Ψ*. Moreover, prognostic equations for the averaged μ and ν, similar to (6.2) and (6.7), could also be derived. We hope to extend our results in this direction in the future.

In Part II (Nurser and Lee 2004) we discuss how both vertical and horizontal isopycnal transport streamfunctions can be approximated using Eulerian mean quantities and, in particular, how they are related to the transformed Eulerian mean (TEM) streamfunction.

Acknowledgments

We acknowledge useful conversations with Peter Killworth, and helpful comments from Trevor McDougall and an anonymous reviewer.

REFERENCES

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APPENDIX Derivation of the Prognostic Equation for θ̃

Here, we derive the equation by which θ̃ evolves. The lateral derivative of the streamfunction at constant θ is related to the vertical isopycnal motion and the diapycnal flow by (5.4)
i1520-0485-34-12-2721-eqa1
From the definition ψ̃(y, θ) = Ψ̃[y, (y, θ)], we have
i1520-0485-34-12-2721-eqa2
and so
i1520-0485-34-12-2721-ea1
Furthermore, dividing by ∂Ã/∂θ and using
i1520-0485-34-12-2721-eqa3
gives
i1520-0485-34-12-2721-ea2
Since
i1520-0485-34-12-2721-eqa4
and we can define as
i1520-0485-34-12-2721-eqa5
we have
i1520-0485-34-12-2721-ea3
where /Dt is the material derivative following isothermal mean velocity (υ̃, ). The identification of as a vertical velocity is made clear by writing
i1520-0485-34-12-2721-ea4
where the third equality used (A.1). Thus is the sum of the isopycnal motion ∂/∂t, the sliding upwards motion along the isopycnal, υ̃/∂y, and the diapycnal velocity (∂/∂θ)Q̃.

Fig. 1.
Fig. 1.

Schematic illustrating a two-layer temperature field θ in a rectangular flat-bottomed zonal channel. Black indicates cold waters; white shows warm waters. (a) A vertical slice at constant latitude and (b) a horizontal slice at constant depth. The dashed lines indicate where the Eulerian averaging is taken. (c),(d) The Eulerian mean temperature θ viewed from the vertical and from the horizontal slice. Gray shading indicates waters of intermediate temperatures “formed” by the Eulerian averaging. (e) The mean height of isotherms z and (f) the mean latitude of isotherms y*. In (c)–(f) the zonal axes are extended for ease of visualization

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 2.
Fig. 2.

Schematic illustrating (a) a vertical slice through the temperature field θ in the presence of topography; (b) the mean height z of the θ isotherm defined such that the area below equals the area colder than θ, Ã; and (c) the zonally averaged height ž of the θ isotherm. Note that is deeper than ž

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 3.
Fig. 3.

Schematic illustrating (a) a temperature field θ for a vertically uniform two-layer fluid; (b) the Eulerian mean temperature θ(y, z); (c) the mean temperature θ̃(y, z) using isothermal averaging at constant y; and (d) the mean temperature θ*(y, z) using isothermal averaging at constant z. Shading as in Fig. 1

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 4.
Fig. 4.

Instantaneous temperature field at day 4320: (a),(b) the horizontal field at 10- and 350-m depth with contour interval 2° and 1°C, respectively, where the thin lines indicate the northern and southern relaxation zones; (c),(d) meridional sections at 180 and 470 km along the channel

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 5.
Fig. 5.

(a) Eulerian mean temperature θ, (b) mean temperature θ using isothermal averaging at constant y, and (c) mean temperature θ* using isothermal averaging at constant z. (d) The difference θ̃θ and (e) the difference θ* − θ. (e) The temperature differences at surface; the solid line is θ̃θ, and the dashed line is θ* − θ

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 6.
Fig. 6.

Schematic illustrating (a) the meridional flow V in fluid colder than θ (drawn here as below the isotherm) across the zonal section at latitude Y; (b) the zonally integrated northward transport streamfunction ψ̃(Y, θ); (c) the upward flow W in fluid colder than θ (drawn here as below the isotherm) across the horizontal section at depth Z; and (d) the zonally integrated vertical transport streamfunction ψ*(θ, Z)

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 7.
Fig. 7.

(a) Volume balance between y and y + dy given by the convergence of integrated northward flows ψ̃(y + dy, θ) − ψ̃(y, θ), the rate of change of the area Ã(y, θ) below the isotherm θ, and the diapycnal flow ẽdy through the θ surface. (b) Balance between the diabatic heating Δ supplied into the layer per unit y between the θ and θ + Δθ isotherms and the heat gain Δθ by the diapycnal flow as it crosses the diapycnal surfaces. (c) Corresponding volume balance relating to the horizontal area A*, the upward flow Ψ*, and diapycnal flow e* between z and z + dz. (d) The relationship between heat supply per unit z and diapycnal flow across the θ surface at constant z

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 8.
Fig. 8.

(a) Meridional transport streamfunction ψ̃(y, θ) (Sv) and (b) the schematic for (a) to show the direction of transport and different heating regimes: surface relaxation (dark shading), northern relaxation (medium shading), and southern relaxation (lightest shading). Similarly, (c) and (d) are for the vertical transport streamfunction ψ*(θ, z). Isopycnic flow is restricted to the white dashed line in schematic (d)

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 9.
Fig. 9.

The transport streamfunction remapped into (y, z) space: (a) Ψ̃ and (b) Ψ*. The respective mean isotherms θ̃ and θ* are superposed on the plots

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

Fig. 10.
Fig. 10.

Schematic illustrating the surface bounded by the intersection between the θ = θ̃(y0, z0) surface, the vertical plane through which ψ̃ passes, and the horizontal plane up through which ψ* passes. Colder water is assumed to lie to the north (y + ve). The lighter bulged-up surface 𝒜+ is the part of the surface above the horizontal plane and south of the vertical plane. The darker depressed surface 𝒜 is the part of the surface below the horizontal plane and north of the vertical plane

Citation: Journal of Physical Oceanography 34, 12; 10.1175/JPO2649.1

In the case, say in the Arctic Ocean, in which the coldest water may lie at the surface rather than at the bottom, the lower and upper bounds on the left-hand side of (2.4) need to be reversed and −H replaced by 0 so that the integral remains positive.

When (x, θ) are globally defined coordinates, the definition of υ̃ reduces to the more familiar thickness-weighted zonal averages:
i1520-0485-34-12-2721-eq1
Note that ∂z/∂θ (the local thickness) is simply the Jacobian ∂(x, z)/ ∂(x, θ). Thus, strictly speaking, the phrase thickness-weighted average is misleading since averaging in (x, θ) coordinates without the Jacobian is meaningless: areal averaging is the only sensible way to perform averaging.
Locally, the continuity equation in (x, y, θ) coordinates is
i1520-0485-34-12-2721-eq2
Locally, the continuity equation in (x, θ, z) coordinates takes the form
i1520-0485-34-12-2721-eq3
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